A Mission Planning Tool Design for Re-Entry

Master of Science Thesis

Stefan van Doorn September 21, 2010 ii Preface

The Aerospace Engineering Masters at the Delft University of Technology is concluded with an individual research. The research is documented in a thesis work. This report presents the thesis work of the author. In addition to the thesis work, a presentation on the research is given which is open to the public. Finally, the student will defend the thesis work in private with a graduation committee. The research presented in this thesis work has been carried out at the chair of Astrodynamics and Space Missions at the aforementioned faculty. The time spent on the research is equal to 42 ECTS or 7 months. Formally, the thesis work is known under the code AE5-006. This report is intended for two types of readers. First, it is intended for the graduation committee who grades the student. Second, it is intended for people who are interested in mission planning for atmospheric entry. It can serve as a means to broaden ones knowledge on the subject or as a start for further research. In order to fully understand the contents of this thesis work, it would certainly be helpful if the reader has a Bachelor’s degree in Aerospace Engineering. In speciﬁc, the reader should have some background knowledge on ﬂight mechanics, control theory and numerical methods.

The background theory for angle of attack and bank angle planning is given in chapters2 and3. The design of a generic ﬂight simulator can be found in chapter5. If the reader is interested in angle-of-attack or bank angle planning, he/she should take a look at chapter 6 or7, respectively. The testing of guidance systems, employing the bank angle planner, can be found in chapter8. For the derivation of the ﬁrst and second derivative of drag with respect to energy, the reader is referred to appendixD.

The author would like to thank his thesis supervisor Dr.ir.Erwin Mooij in speciﬁc for his guidance during the research. One of his key qualities during a progress meeting is to give the student extra work. Of course, this is only because the student has gained new insights into the problem. Another quality of his is that he can speak very en- thusiastically about atmospheric entry and related areas. Finally, it is worthnoty to mention that during the progress meetings there was always time for a joke. Without the support of my girlfriend Marlot Jansen and parents Evelyne and Bart van Doorn this thesis work would not have been possible. PhD student ir.Jeroen Melman is thanked for his contribution of programming code and the discussions about step-size control for numerical integration. Mirjam Boere and I have worked together on the development of a generic ﬂight simulator. She is, therefore, thanked for her contribution. Finally, a

iii special word of appreciation goes out to the students from the 9th floor for keeping the author company during this work. Specifically, in random order, Tom de Groot, Mir- jam Boere, Vivek Vittaldev, Valentino Zuccarelli, Antonio Pagano, Nicoletta Silvestri, Dominic Dirkx, Hessel Gorter and Willem van der Weg. The company of Elgar van Veggel and Arthur Tindemans, both Aerospace Engineering students, was also highly appreciated during coffee and lunch breaks.

This thesis work is dedicated to my grandfather, who taught me my ﬁrst English words.

Stefan van Doorn September 21, 2010

The picture on the front page illustrates: a) the ballistic entry, b) the glide entry b and c) skipping entry [Loh, 1968].

iv Summary

A transition through the Earth’s atmosphere is inevitable if it is desired to bring or return something useful from space. Mostly, these are astronauts or samples from a celestial object. The transition is also known as the atmospheric entry. The entry is characterised by a vehicle that has a high energy. This energy needs is reduced by drag upon transition through the atmosphere. Three types of entry can be distinguished: ballistic, glide and skip entry. A typical glide entry flight has three phases: the hypersonic transition, the Terminal Area Energy Management (TAEM) phase and the actual landing. In the hypersonic transition phase, there are various threats that can pose a risk to the vehicle. The heating and structural loading can become severe enough to damage the vehicle and/or the crew. In the hypersonic transition phase, the attitude of the vehicle has to be controlled such that the vehicle safely ends up at the TAEM interface. The process of determining the attitude throughout entry is called mission planning. For on-board mission planning, simplifications need to be made on the vehicle, its environment and the flight dynamics to achieve an acceptable computation speed. As a consequence, the real trajectory will deviate from the planned trajectory. The trajectory tracker has the task to steer the vehicle towards the planned trajectory. The combination of a planning and tracking algorithm forms a guidance algorithm. The main question of this thesis work is formulated as:

Is it possible to design an on-board executable guidance algorithm, for the hypersonic transition phase, that safely targets the TAEM interface?

A simulator has been built to serve as a test bed for the guidance algorithm. A systems-based approach is taken in the modelling of the vehicle and environment. The advantage is that the software has a clear modular structure and it becomes easy to extend the simulator with new capabilities. The trajectory propagation operates on the integration of the equations of motion expressed in Cartesian components with respect to an inertial reference frame. This allows for a stable integration and an easy inclusion of complex environment models. The performance and validity of a guidance algorithm can only be assessed if tested with a validated simulator. Therefore, a validation has been carried out using the already validated software called Simulation Tool for Atmospheric Re-entry Trajectories. The design of a trajectory planner has been decomposed into an angle of attack and bank angle planner. The angle of attack planner operates on the assumption of an

v equilibrium-glide trajectory. First, a trajectory is designed in the height-velocity space. Second, this trajectory is converted to an angle-of-attack profile. The method can take path constraints and performance parameters into account. The performance parameters are the flight range, heat flux and integrated heat load. The bank angle planning algorithm is centered around an iterative search for a drag profile that corresponds to the required trajectory length. The drag profile is a linear spline consisting of three segments. From this drag profile, a bank angle profile is deduced. One bank reversal is planned. The point of initiation is determined by the minimisation of the cross-range error. The required trajectory length is updated and the drag profile search is repeated as part of the iteration. By incorporating the bank angle planning and tracking algorithms in one guidance algorithm, the main question is answered positively. The algorithm executes fast enough for an on-board implementation. Furthermore, the interface is successfully targeted under the assumption that the constraint on the final heading can be taken care off. The first recommendation is to include the meeting of the final heading constraint in the planning algorithm. Another recommendation concerns the breakpoints of the linear spline, which give rise to tracking problems. In addition, planning a linear spline does not allow for a flexible profile, while the mission definition might demand it. It is recommended to tackle these two problems by planning a smooth profile. The algorithm requires a trial-and-error process to obtain the angle of attack profile and tracking algorithm gains to ensure mission success. This limits the flexibility of the planner for on-board execution. Future work should integrate the angle of attack planning as well. A more advanced guidance algorithm has to be developed to solve the problem of manual gain selection.

vi Nomenclature

Symbols

b Characteristic length [m] C Direction cosine matrix [ ] − C Geopotential coefficient [ ] − C Aerodynamic coefficient [ ] − C Constant in the heat flux model [] c Characteristic length [m] D Drag force/acceleration [N] E Energy [J] e Ellipticity [ ] − F Force vector [N] G Universal Gravitational Constant [m3/(kg s2)] · g Gravitational acceleration vector [m/s2] h Height [m] I Moment of inertia [kg m2] · i Unit vector [ ] − J Harmonic coefficient [ ] − k Gain [ ] − L Lift force [N] M Mach [ ] − M Pitch moment [N m] · m Mass [kg] m Order [ ] − n g-load [ ] − n Degree [ ] − P Legendre polynomial [] Q Integrated heat load [J] q Pressure [N/m2] q˙ Heat flux [W/m2] R Radius [m] R Specific gas constant [J/(kg mol)] · r Radial distance [m]

vii S Geopotential coefficient [ ] − S Side force [N] S Surface [m2] S Trajectory length [m] s Distance mass element to vehicle [m] T Temperature [K] t Time [sec] U Gravitational potential [m2/s2] u Input [ ] − V Velocity [m/s2] w Weight [ ] − x Cartesian position coordinate [m] y Cartesian position coordinate [m] z Cartesian position coordinate [m] α Angle of attack [rad] β Side-slip angle [rad] β Inverse of scale height [m−1] γ Flight-path angle [rad] γ Ratio of specific heats [ ] − δ Latitude [rad] δ Control surface deflection [rad] δ Kronecker delta [ ] − δ Step-size scaling parameter [ ] − θ Angular displacement [rad] ρ Density [kg/m3] σ Bank angle [rad] τ Longitude [rad] χ Heading angle [rad] ω Planet rotational velocity [rad/s] ζ Damping [ ] −

Index (if other than above)

0 At seal-level A Aerodynamic A Airspeed AA Aerodynamic reference frame w.r.t. airspeed variables AG Aerodynamic reference frame w.r.t. groundspeed variables B Body reference frame b Body ﬂap c Convective c Circular

viii c Constant cb Central body cmd Commanded D Drag dyn dynamic e Elevon e Equatorial f Final G Gravity G Groundspeed I Inertial reference frame i Initial L Lift m Pitch moment m Order max Maximum min Minimum N Reference stagnation point n degree p Planet p Polar R Rotating reference frame ref Reference r Radiative TA Trajectory reference frame w.r.t. airspeed variables TG Trajectory reference frame w.r.t. groundspeed variables V Vertical reference frame W Wind reference frame

Abbreviations

DoF Degrees of freedom EAGLE Evolved Acceleration Guidance Logic for Entry EoM Equations of Motion HAC Heading Alignment Circle IC Initial Conditions MBB Messerschmidt-Boelkow-Bloehm STA Space Trajectory Analysis START Simulation Tool for Atmospheric Re-entry Trajectories TAEM Terminal Area Energy Management US United States

ix x List of Tables

3.1 Reference mission definition for Horus.[Mooij, 2010]...... 37 3.2 Range of Horus aerodynamic coefficients and flap deflections. [Mooij, 1995] 38 3.3 Reference mission definition for the Orbiter, X-33 and X-38...... 44

5.1 State and control deﬂection errors (absolute) at the end time...... 59

7.1 Number of iterations and convergence success for the six cases...... 112

8.1 Important distances for assessing tracker performance...... 134 8.2 Eﬀects on trajectory for constant σ(E)...... 141 8.3 Simulation characteristics for the α-proﬁles...... 143 8.4 Trajectory parameters for changing γe with constant σ(E)...... 145 8.5 Simulation characteristics for a changing γe...... 145

C.1 Coeﬃcients a c for the three linear segments...... 165 −

xi xii List of Figures

2.1 Deﬁnition of the bank angle σ ...... 7 2.2 Groundspeed vector in the rotating planetocentric reference frame. [Mooij, 1997]...... 9 2.3 Arbitrary mass distribution. [Montenbruck and Gill, 2005]...... 10 2.4 Illustration of height for an ellipsoid.[Seeber, 1993]...... 13

3.1 The TAEM interface for the X-33. [Chen et al., 2002]...... 25 3.2 Entry corridor showing different flight segments. [Harpold and Graves Jr, 1978]...... 29 3.3 The Saenger spaceplane with Horus on top. [Astronautix, 2010]...... 35 L 3.4 Horus D as a function of angle of attack and Mach number...... 36 3.5 Angle of attack for maximum lift to drag ratio...... 36 3.6 Deflection angles for trim at αmax and α(L/D)max...... 37 3.7 The reference angle-of-attack profile...... 38 3.8 The reference bank angle profile...... 39 3.9 Body flap deflection versus time...... 39 3.10 Elevon deflection versus time...... 40 3.11 Height versus velocity...... 40 3.12 Latitude versus longitude...... 41 3.13 An illustration of the X-33 . [Li, 1999]...... 42 L 3.14 Orbiter D as a function of angle of attack and Mach number...... 42 L 3.15 X-33 D as a function of angle of attack and Mach number...... 43 L 3.16 X-38 D as a function of angle of attack and Mach number...... 43 3.17 Reference angle-of-attack profile for the X-33 mission...... 44

5.1 Systems breakdown of the environment...... 52 5.2 Systems breakdown of the vehicle...... 52 5.3 Flow diagram of the trajectory propagation...... 53 5.4 Mach numbers from simulated and reference trajectory...... 60 5.5 Vertical lift coeﬃcients from simulated and reference trajectory...... 61

◦ 6.1 Equilibrium glide at αmax (= 43.9 forHorus)...... 65 6.2 Equilibrium heights for a varying α...... 66 6.3 G-load constraints for a varying α...... 67

xiii 6.4 q α constraints for a varying α...... 67 dyn · 6.5 Equilibrium glide at α(L/D)max...... 68 6.6 The entry corridor...... 69 6.7 The Orbiter entry corridor...... 70 6.8 X-38 entry corridor...... 70 6.9 X-33 entry corridor...... 71 6.10 Equilibrium glide trajectories...... 73 6.11 Minimum heat flux constraint altitude...... 73 6.12 Minimum g-load constraint altitude...... 74 6.13 Minimum q α constraint altitude...... 74 dyn · 6.14 Equilibrium glide including and excluding trim...... 76 6.15 Minimum g-load constraint altitude including and excluding trim...... 76 6.16 Typical lift-to-drag ratio’s for entry vehicles. [Regan and Anandakrishnan, 1993]...... 78 6.17 h V profiles for the three test cases...... 81 − 6.18 Angle of attack profiles for the three test cases...... 82 6.19 Flight-path angle profiles for the three test cases...... 82 6.20 h V diagram for three test cases with enforced α(V ) and γ(V )...... 84 − 6.21 Downrange versus time for the three test cases...... 84 6.22 Heat flux versus time for the three test cases...... 85 6.23 Heat load versus time for the three test cases...... 85 6.24 h V diagram for the two bank angle guidance approached...... 87 − 6.25 Heat load versus time for the three test cases...... 87 6.26 Heat load versus time for the three test cases...... 88

7.1 Mission geometry...... 90 7.2 Angle of attack test profile 1 as a function of energy...... 93 7.3 Height from test profile 1 as a function of energy...... 93 7.4 entry corridor in the drag acceleration vs. energy plane...... 94 7.5 q α constraint for different h(E) profiles...... 95 dyn · 7.6 q˙ constraint for different h(E) profiles...... 96 7.7 Normal load constraint for different h(E) profiles...... 96 7.8 Minimum drag constraint for different α(E) profiles...... 97 7.9 q α constraint for different α(E) profiles...... 98 dyn · 7.10 q˙ constraint for different α(E) profiles...... 98 7.11 Normal load constraint for different α(E) profiles...... 99 7.12 The linear spline segments in the D-E space...... 100 7.13 Found drag profile for a differing Di...... 101 7.14 h(E) profile for Di starting at Lmin...... 104 7.15 γ(E) profile for Di starting at Lmin...... 105 7.16 σ(E) profile for Di starting at Lmin...... 105 7.17 Geometry after placing bank reversal at a random energy...... 107 7.18 Triangle 1 after placing bank reversal at a random energy...... 108 7.19 Triangle 2 after placing bank reversal at a random energy...... 108 7.20 Quadrants of χe Amission and sign σ0...... 110 − 7.21 Horus drag profiles for case 1-4...... 112 7.22 Horus latitude vs. longitude for case 1-4 (x is the TAEM interface)..... 113 7.23 Horus bank angle profiles for case 1-4...... 113 7.24 Horus height profiles for case 1-4...... 114

xiv List of Figures

7.25 Horus height profiles for case 1-4...... 115 7.26 X-33 drag profiles for case 1 and 2...... 116 7.27 X-33 latitude vs. longitude for case 1 and 2 (TAEM interface is the star). 116 7.28 X-33 bank angle profiles for case 1 and 2...... 117

8.1 Ground-track for the Horus 4 case (Open-loop)...... 120 8.2 Height vs. energy for the Horus 4 case (Open-loop)...... 121 8.3 Drag vs. energy for the Horus 4 case (Open-loop)...... 121 8.4 Flight-path angle vs. energy for the Horus 4 case (Open-loop)...... 122 8.5 Ground-track for the X-33 1 case (Open-loop)...... 122 8.6 Height vs. energy for the X-33 1 case (Open-loop)...... 123 8.7 Drag vs. energy for the X-33 1 case (Open-loop)...... 123 8.8 Flight-path angle vs. energy for the X-33 1 case (Open-loop)...... 124 8.9 Ground-track for the Horus 4 case using updates...... 125 8.10 Ground-track for the X-33 1 case using updates...... 126 8.11 Ground-track for the X-33 2 case using updates...... 126 8.12 Height vs. energy for the X-33 2 case using updates...... 127 8.13 Drag vs. energy for the X-33 2 case using updates...... 127 8.14 Flight-path angle vs. energy for the X-33 2 case using updates...... 128 8.15 Drag vs. energy for the X-33 1 case using updates...... 129 8.16 Latitude vs. longitude for the X-33 2 case using closed-loop guidance (x is the TAEM interface)...... 130 8.17 Drag profile for the X-33 2 case using closed-loop guidance...... 130 8.18 Flight-path angle profile for the X-33 2 case using closed-loop guidance.. 131 8.19 Bank angle profile for the X-33 2 case using closed-loop guidance...... 131 8.20 Drag profile close-up in 0.36 < E < 0.52...... 132 8.21 Flight-path angle profile close-up in 0.36 < E < 0.52...... 133 8.22 Bank angle profile close-up in 0.36 < E < 0.52...... 133 8.23 Flight-path angle profile for the Horus reference mission...... 136 8.24 Drag acceleration profile for the Horus reference mission...... 136 8.25 Bank angle profiles for trim analysis...... 138 8.26 Angle-of-attack profiles 1 and 2...... 139 8.27 Angle-of-attack profiles 3 and 4...... 140 8.28 Angle-of-attack profiles 5 and 6...... 140 8.29 Bank angle profiles 1 and 2...... 142 8.30 Bank angle profiles 3 and 4...... 142 8.31 Bank angle profiles 5 and 6...... 143 8.32 Bank angle profiles 3 and 4...... 146

A.1 Density variation with altitude...... 155 A.2 Density variation w.r.t. Exponential atmosphere model...... 156 A.3 Temperature variation with altitude...... 156 A.4 Wind direction and magnitude as a function of height...... 157

C.1 The linear spline segments...... 163

xv xvi Contents

Preface iii

Summary v

Nomenclature viii

List of Tables ix

List of Figures xiii

1 Introduction1

2 Flight Mechanics5 2.1 Reference Frames...... 5 2.2 State Variables...... 8 2.2.1 Position and velocity...... 8 2.2.2 Attitude...... 8 2.3 Environment modeling...... 9 2.3.1 Gravity field...... 10 2.3.2 Atmosphere...... 12 2.3.3 Shape...... 12 2.3.4 Environment simplifications...... 13 2.4 Vehicle modeling...... 15 2.5 Forces...... 16 2.6 Translational flight dynamics...... 16 2.6.1 The equations of motion...... 17 2.6.2 Flight dynamics simplifications...... 19

3 Entry Mission Characteristics 23 3.1 Entry planning problem...... 23 3.1.1 Problem statement...... 24 3.1.2 TAEM interface...... 24 3.1.3 Constraints...... 25 3.2 Guidance system...... 27 3.2.1 Space Shuttle guidance...... 28

xvii 3.2.2 EAGLE planning method...... 29 3.2.3 EAGLE tracking algorithm...... 30 3.3 Control system...... 33 3.4 Trim...... 33 3.5 Reference missions...... 34 3.5.1 Horus mission...... 35 3.5.2 X-33, X-38 and Orbiter missions...... 38

4 Numerical Methods 45 4.1 Cubic spline interpolation...... 45 4.2 Secant method...... 47 4.3 Runge-Kutta methods with step-size control...... 47

5 Generic Flight Simulator Design 49 5.1 STA...... 49 5.2 Existing software and its shortcomings...... 50 5.3 Simulator architecture design...... 51 5.4 Class deﬁnitions...... 53 5.4.1 Bodies...... 54 5.4.2 Trajectory parameters...... 54 5.4.3 Forces and Moments...... 54 5.4.4 Vehicle...... 55 5.4.5 Environment...... 56 5.5 Validation...... 58

6 Angle Of Attack Planning 63 6.1 The 2D entry corridor...... 63 6.1.1 Corridor formation...... 63 6.1.2 Other vehicles...... 69 6.1.3 Atmosphere model...... 71 6.1.4 Trim...... 75 6.1.5 Parameter extraction...... 77 6.2 Design parameters...... 77 6.3 2D simulations...... 80 6.3.1 Proﬁle selection...... 80 6.3.2 Angle of attack proﬁle testing...... 81

7 Bank Angle Planning 89 7.1 Drag-Energy entry corridor...... 91 7.1.1 Creating the corridor...... 91 7.1.2 Height reference profile...... 94 7.1.3 Angle of attack reference profile...... 97 7.2 Drag acceleration profile planning...... 99 7.3 Trajectory parameters extraction...... 102 7.4 Bank reversal search...... 106 7.5 Trajectory length updating...... 110 7.6 Initial and final drag value...... 110

xviii Contents

8 Guidance System Testing 119 8.1 Open-loop guidance...... 120 8.2 Open-loop guidance using updates...... 124 8.3 Closed-loop guidance using updates...... 128 8.4 Influence of wind...... 134 8.5 Trimmed flight...... 134 8.6 Changing the angle of attack profile...... 139 8.6.1 Nominal bank angle profile...... 141 8.6.2 Including planning algorithm...... 141 8.7 Changing the initial flight-path angle...... 144 8.7.1 Using the nominal bank angle profile...... 144 8.7.2 Employing the planning algorithm...... 145

9 Conclusions and Recommendations 147 9.1 Conclusions...... 147 9.2 Recommendations...... 149

Bibliography 154

A Variations on the US 1976 Standard Atmosphere 155

B Derivation of Entry Corridor Equations 159 B.1 2D entry corridor equations...... 159 B.2 Drag-Energy entry corridor equations...... 161

C Linear Spline Segments Derivation 163

D Drag Derivatives to Energy 167 D.1 First derivative...... 167 D.2 Second derivative...... 168 D.3 Second derivative including Coriolis force...... 171

xix CHAPTER 1

Introduction

A transition through the Earth’s atmosphere is inevitable if it is desired to bring or return something useful from space. Mostly, these are astronauts or samples from a celestial object. The transition is also known as the atmospheric entry. The entry is characterised by a vehicle that has a high energy. This energy needs is reduced by drag upon transition through the atmosphere. Three types of entry can be distinguished: ballistic, glide and skip entry. In a ballistic entry, the vehicle does not generate lift. This entry type closely resembles the trajectory flown of a capsule with axial symmetry. However, in practice also a capsule generates some lift. During a glide entry, the flight-path angle is small and changes slowly with time. The vehicle needs to have sufficient lifting capabilities to maintain such conditions. The Space Shuttle Orbiter is a practical example of a vehicle that performs a glide entry. If the vehicle generates an excess of lift, it can execute a pull-up manoeuvre and possibly exit the atmosphere. If the exit is temporarily, the vehicle is in a skip entry trajectory. In theory, the vehicle could perform multiple skips. Apollo 10 has executed a small skip within the atmosphere. A typical glide entry flight has three phases:

1. Hypersonic transition

2. Terminal Area Energy Management (TAEM) phase

3. Approach and landing

In the hypersonic transition phase, there are various threats that can pose a risk to the vehicle and to the mission. First, the atmosphere can be a hostile environment. The friction between air particles and the vehicle’s surface could lead to structural failure and a severe heating of the inside. On top of the frictional heating, the vehicle is also heated by radiation coming from the surrounding air. Second, the aerodynamic force, arising from the interaction of the airﬂow with the vehicle’s body, could become so high that a human being cannot tolerate it anymore. Third, the pressure distribution on the vehicle’s surface could give a bending moment on its body that is too high. Besides the risk of damaging the vehicle or the crew, the vehicle should land at a desired location. The hypersonic transition phase ends at an altitude of about 25 km at Mach 2.5. Here, the TAEM phase starts. In this phase the energy of the vehicle is managed such that,

1 at the end, it is aligned with the runway. The entry ends with the landing in the third phase. The attitude of the vehicle should be controlled in such a manner that the vehicle lands safely at this landing site. The process of determining the attitude throughout entry is called mission planning. It is, more specifically, the attitude with respect to the airflow, as this determines the magnitude and direction of the resultant aerodynamic force. Now, the problem is to find the attitude history that leads to mission success. A logical first step is to analyse trajectories resulting from different attitude histories. To do so, one would have to model the vehicle, the gravity field and the atmosphere to simulate such trajectories. Uncertainties in the atmosphere and vehicle model can complicate the analysis. Furthermore, more detailed models are typically more accurate, but imply larger computation times. Nevertheless, flight simulation remains a better option than flying actual test models. The Space Shuttle missions have proven that it is possible to perform a successful re-entry with a winged vehicle containing a crew. Even though the taken approach works, it could be considered inflexible. The mission planning is performed before entering the atmosphere and no planning updates are performed. It would be better if a mission planning algorithm can be executed on-board during flight. This reduces pre-mission analysis and thus effort. In addition, it should reduce the risk of mission failure as the new vehicle state is taken into account during the planning. It does, however, mean that the algorithm has to perform its task in just a few seconds. To achieve this computation speed, simplifications need to be made on the vehicle, its environment and the flight dynamics. As a consequence, the real trajectory will deviate from the planned trajectory. This is where a guidance system comes into play. The task of the guidance system is to steer the vehicle towards the planned trajectory. In addition, it executes the mission planning algorithm during the flight to enhance mission success. A guidance system, therefore, consists of two parts: the trajectory planner and the trajectory tracker. The function of the tracker is to steer towards the planned trajectory.

The main question of this thesis work is formulated as:

Is it possible to design an on-board executable guidance algorithm, for the hypersonic transition phase, that safely targets the TAEM interface?

From this question, the following tasks have been derived:

Task 1: Simulator development

Task 2: Trajectory planner design

Task 3: Trajectory tracker design

Task 4: Guidance algorithm testing

The simulator serves as a test bed for both the planning and the tracking algorithm. The planning algorithm is composed of two parts: angle-of-attack and bank angle planning. A planned trajectory is an additional output of the planning algorithm. This serves as input to the trajectory tracker. The tracker has to steer the vehicle towards the reference trajectory.

2 The idea is to integrate the guidance algorithm in the Space Trajectory Analysis (STA) software. STA is an open-source astrodynamics software project initiated by ESA in August 2005 [ESA, 2010]. It is envisioned to be a research tool in the analysis phase of a space mission. It shall be capable of computing, analysing, optimising and visualising trajectories for diﬀerent types of space missions. The re-entry module of STA is used as a basis for the simulator development. The astrodynamics routines present in STA shall be used as much as possible. This allows an easier integration of the guidance algorithm in the STA software. The research that is performed in thesis work holds under the following limitations:

- The vehicle is a rigid body with a constant mass.

- The use of an ideal navigation system, i.e. the state follows from the integration.

- The use of an ideal control system with limits imposed on the angle of attack and bank angle rates. An ideal control system implies that the commanded state by the guidance system is obtained instantly.

- The planning algorithm executes instantly during ﬂight.

- The accuracy of the methods used to model the vehicle’s aerodynamics.

- The accuracy of the environment models.

To build a proper simulator, the vehicle, its environment and the flight dynamics need to be modelled. This is part of chapter2, where the flight mechanics is treated. Using this as a basis, the atmospheric entry mission is characterised further in chapter3. More specifically, the chapter elaborates on the mission planning problem and guidance algorithm. Besides the flight mechanics, the simulator and guidance algorithm require the use of numerical methods. These are presented in chapter4. Using STA as a starting point, a software architecture has been proposed for a flight simulator. This architecture is presented in chapter5. From this architecture, a flight simulator has been developed. The validation of the simulator is also presented in this chapter. The second task is the design of a trajectory planner. In chapters6 and7, an angle of attack and bank angle planning method is presented, respectively. Chapter8 presents the testing results of guidance systems incorporating the bank angle planner. In the end, chapter9 presents the conclusions recommendations.

3 4 CHAPTER 2

Flight Mechanics

The core of the flight simulator consists of the flight mechanics. The environment acts on the vehicle through forces, namely the gravitational and aerodynamics force. These forces shape the trajectory that is followed by the vehicle. The flight dynamics are the link between the forces acting on the vehicle and the trajectory that is flown. The motion of the vehicle is always described with respect to a frame of reference. The motion of the vehicle and the forces is best described in different reference frame. Section 2.1 presents the relevant reference frames to study entry missions. Besides expressing the motion with respect to a reference frame, it is described by a certain set of state variables. Different sets of state variables are used in this thesis. They are presented in section 2.2. Sections 2.3 and 2.4 describe the modelling of the environment and vehicle. For an un-powered entry, only the gravitational and aerodynamic forces act on the vehicle. Section 2.5 presents the calculation of these forces in their original reference frames. In the end, section 2.6 describes the translational flight dynamics of an un-powered entry vehicle. by using them, the three degrees of freedom (DoF) translational motion can be studied.

2.1 Reference Frames

The state of a vehicle is expressed with respect to a reference frame. The state consists of the translational and rotational state. The translational state, consisting of the position and velocity, is expressed with respect to one reference frame. The rotational state consists of the attitude and angular velocity. The attitude is typically expressed as the orientation between a body-fixed and an external reference frame. The angular velocity is given as the rotational rate of the body-fixed frame with respect to an external frame. It is expressed in components along the body-fixed frame. Below, nine reference frames are presented, all of them are right-handed orthogonal Cartesian. They are taken from [Mooij, 1997].

1. Inertial planetocentric reference frame, I-frame The origin of this reference frame is located at the center of mass of a planet or moon. The OXI YI -plane is in the equatorial plane, the ZI -axis points along the

5 rotation axis of the planet/moon. The direction of the XI -axis is deﬁned by the zero-longitude meridian at zero time. The YI -axis then completes the frame.

2. Rotating planetocentric reference frame, R-frame This reference frame is ﬁxed to the planet/moon and has therefore an angular velocity equal to the rotational velocity of the planet. The XR-axis intersects the equator at zero longitude, the ZR-axis points along the rotation axis and the YR-axis completes the system. For Earth, the zero longitude is the Greenwich meridian. This reference frame is conveniently used in atmospheric ﬂight, because the vehicle’s velocity is considered with respect to the rotating planet.

3. Body ﬁxed reference frame, B-frame This reference frame is ﬁxed to the vehicle, its origin is located at a reference point, usually the center of mass. In principle, one can choose any orientation. However, if there is one symmetry plane, it is often chosen to be the XBZB-plane. The XB- axis points forward and the ZB-axis points downward. The YB-axis completes the right-handed orthogonal system. Through this reference frame one can describe the rotation of the vehicle around its axes, namely roll, pitch and yaw.

4. Vertical reference frame, V-frame The vehicle carried vertical reference frame has its ZV -axis pointing towards the centre of mass of the central body, along the gravitational acceleration vector. For this report, it is assumed that the centre of mass is equal to the geometric centre of the planet/moon in question. The XV -axis is in a meridian plane and points towards the North, the YV -axis completes the right-handed orthogonal system. The XV YV -plane is called the local horizontal plane. The longitude τ and latitude δ suﬃce to calculate the orientation of the V -frame with respect to the R-frame.

5. Trajectory reference frame: Groundspeed based, TG-frame The groundspeed based trajectory reference frame is coupled to the groundspeed vector. The XTG-axis is directed along the groundspeed vector, relative to the R-frame. The ZTG-axis is in the vertical plane pointing downwards and the YTG- axis is in the horizontal plane completing the system. The ﬂight-path angle and heading angle, with index G, express the attitude of this frame with respect to the V -frame. The aerodynamic angles, as presented in the next section, express the orientation of the B-frame with respect to this frame. The index G is added to clarify that they are based on the groundspeed.

6. Aerodynamic reference frame: Airspeed based, AA-frame The AA-frame is coupled to the airspeed vector. The XAA-axis is in the direction of the velocity vector relative to the atmosphere. The ZAA-axis is collinear with the aerodynamic lift force (based on airspeed variables), but in opposite direction. As before, the YAA-axis completes the system. The angle of attack and side-slip angle express the orientation of the B-frame with respect to this frame. The index A is added to clarify that they are based on the airspeed.

7. Aerodynamic reference frame: Groundspeed based, AG-frame The AG-frame is also coupled to the groundspeed vector, it differs only from the TG-frame when the vehicle is banking. The definition of the bank angle can be seen in figure 2.1. The XAG-axis is collinear with the XTG-axis, the ZAG-axis is collinear with the aerodynamic lift force based on groundspeed variables, but in opposite

6 2.1. Reference Frames

direction. Once more, the YAG-axis completes the system. The aerodynamic lift force based on groundspeed variables is a virtual force in the presence of wind, because the true physical lift force is based on the airspeed. In the absence of wind they are equal and this frame coincides with the AA-frame. This reference frame is used to express the angle of attack and the angle-of-sideslip of the B-frame with respect to this frame. The bank angle is positive in the direction of a positive turn around the XTG-axis.

Figure 2.1: Deﬁnition of the bank angle σ

8. Trajectory reference frame: Airspeed based, TA-frame The airspeed based trajectory reference frame is similar to the TG-frame if there is no wind. The XTA-axis is directed along the velocity vector relative to the atmosphere. The ZTA-axis is in the vertical plane pointing downwards and the YTA-axis completes the system. The aerodynamic angles express the orientation of the B-frame with respect to this frame. If the vehicle is not banking, the AA-frame and the TA-frame coincide. The ﬂight-path angle and heading angle, with index A, express the attitude of this frame with respect to the V -frame. 9. Wind reference frame, W-frame The XW -axis is collinear with the wind-velocity vector, positive in northern direction for a southern wind. The ZW -axis is in the vertical plane pointing downwards, so for a wind in the local horizontal plane it is positive pointing downwards. The YW -axis completes the right-handed system. There are two angles, the ﬂight path angle for the wind vector γW and heading for the wind vector χW , to express the orientation of this frame with respect to the V -frame. The index W is added to make a distinction with the TG-frame.

One can transform a vector from one reference frame to another by using direction cosine matrices. A direction cosine matrix consists of one or the multiplication of several unit rotation matrices. One unit rotation matrix describes a rotation around an X -, Y - or Z -axis. The notation used in this thesis work is as follows. If one would like to transform a vector F expressed in frame A to frame B, one writes:

FB = CB,AFA (2.1)

A subscripts to F indicates the reference frame to which it is expressed. CB,A implies a transformation from A to B. Direction cosine matrices describing the transformation between the reference frames introduced above, are given in [Mooij, 1997].

7 2.2 State Variables

This section is divided in two subsections. Subsection 2.2.1 presents the state variables for position and velocity. Subsection 2.2.2 describes the state variables for the attitude.

2.2.1 Position and velocity In this thesis work, there are two sets of state variables used to express the position and velocity of an entry vehicle. The ﬁrst is Cartesian components, the second is spherical components. In spherical components, the translational state is usually expressed with respect to the R-frame. The advantage of this expression, is that the state is directly physically interpretable. A disadvantage is that singularities can occur if this set is used for the simulation core. An expression in Cartesian components do not suﬀer from this disadvantage. Both sets are used in this thesis work and are therefore given below. The routines for transforming between both state variable sets are given in [Mooij, 1997].

Cartesian components The Cartesian components can be used in every reference frame described in Section 2.1. The position is indicated by x ,y and z; the velocity is given by x˙, y˙ and z˙. An index is added to show to which reference frame the Cartesian components are valid.

Spherical components As said before, this set is usually used when the translational state needs to be expressed w.r.t. the R-frame. This set is given by: r : Radial distance τ : Longitude δ : Latitude VG : Groundspeed γG : Flight-path angle χG : Heading angle

The longitude is positive towards the east starting from the Greenwich meridian (0◦ τ 360◦), the latitude is positive in the direction of North starting from the equator ≤ ≤ (-90◦ δ 90◦). The radial distance r is measured between the center of mass of the vehicle ≤ ≤ and the center of mass of the body. The ﬂight-path angle γ is the angle between the local horizontal plane and the velocity vector VG , it is positive when the velocity vector is above the horizontal plane (-90◦ γ 90◦). For a glide entry, this angle is negative ≤ ≤ throughout the entry. The heading χG is the direction of the velocity vector with respect ◦ ◦ to the local North, it is measured positive in clockwise direction (-180 χG 180 ). The ≤ ≤ index G is added to clarify that the variables are based on the groundspeed and not the airspeed. Figure 2.2 presents an illustration of the spherical components in the R-frame.

2.2.2 Attitude The attitude of a vehicle can be expressed by for example, Euler angles, quaternions and a Direction Cosine Matrix (DCM). A set of Euler angles corresponds to three rotations around reference frame axes. Each rotation should be around a diﬀerent axis than the

8 2.3. Environment modeling

Figure 2.2: Groundspeed vector in the rotating planetocentric reference frame. [Mooij, 1997] one before. This gives 12 sets of Euler angles, with which one can reach any attitude with the reference frame. The aerodynamic angles, corresponding to a 2-3-1 rotation Euler angles set, express the attitude of the vehicle with respect to the TA- or TG- frame. Another set of Euler angles (yaw, pitch, roll) can be used to physically interpret the attitude of the vehicle with respect to the V - or I -frame. Both sets of angles have the disadvantage that singularities can occur at the second rotation. To circumvent this problem, one can use quaternions to express the attitude. In this thesis work, only the aerodynamic angles are used:

- Angle of attack α (-180◦ α 180◦) ≤ ≤

- Side-slip angle β (-90◦ β 90◦) ≤ ≤

- Bank angle σ (-180◦ σ 180◦) ≤ ≤ The angle of attack is measured positive for a ’nose-up’ attitude, the side-slip angle is positive for a ’nose-left’ attitude and the bank angle is positive when banking to the right. The angle of attack and side-slip angle are used to calculate the aerodynamics coeﬃcients. All three angles are required to transform the aerodynamic force from the AA-frame or AG-frame to the I -frame. On a side note, the rotational state consists of both the attitude and angular velocity. In this thesis work, only the attitude is actively used, the angular velocity is not. In [Mooij, 1997], state variable sets for the angular velocity are presented.

2.3 Environment modeling

The motion of a vehicle is inﬂuenced by forces originating from its environment. These can be forces on which the vehicle cannot exert any control, for example a third body gravitational attraction. On the other hand, the space shuttle can control the magnitude and direction of the aerodynamic force that originates from an atmosphere. It does this by deﬂecting its aerodynamic control surfaces. In order to accurately simulate an entry

9 trajectory, a properly detailed representation of the environment is needed. Subsection 2.3.1, presents an approach to model the gravity field of a celestial body. Hereafter, subsection 2.3.2 describes how the atmosphere can be modelled. Subsection 2.3.3 discusses the shape of a celestial body. Finally, subsection 2.3.4 presents simplifications that can be made to the environment. In this thesis work, no further attention will be paid to a magnetic field and radiation properties. Any forces arising from these sources are considered to be too small [Mooij, 2010]. Also, third body effects at not part of this thesis work.

2.3.1 Gravity ﬁeld

According to Newton’s law of gravitation, two particles attract each other with a force proportional to their masses and inversely proportional to the square of the distance between them [Wakker, 2007]. Hence, a celestial body exerts a gravity force on a vehicle, and vice versa. One could also imagine a gravity field around a celestial body that has a certain strength at a point in space. The strength at a point depends on the mass distribution of the body. Perhaps the simplest mass distribution that one could think of for a body is a sphere. Although this may seem a good model when looking at the Sun or the Moon, in reality it is unlikely that a celestial body is a perfect sphere. Therefore, a model is required that is able to describe the gravity field of a body with arbitrary mass distribution. Theoretical mechanics prescribes that the gradient of a potential gives the local field strength [Wakker, 2007]. The gravitational acceleration is calculated as:

¨r = U (2.2) ∇ Here, r is the radial distance vector to the centre of the celestial body. Figure 2.3 shows an body with an arbitrary mass distribution. The gravity potential U for this arbitrary mass distribution is given by equation 2.3. In essence, this equation represents a summing up of the contribution of each individual mass element.

Z dm U = G (2.3) r s | − | Here, dm = ρ(s)d3s. The distance from a mass element dm to the vehicle is indicated by s. The Universal Gravitational Constant G is equal to 6.673 10−20 km3/(kg s2). As · · before, m is the mass and ρ is the density.

Figure 2.3: Arbitrary mass distribution. [Montenbruck and Gill, 2005]

10 2.3. Environment modeling

One could expand the inverse of the distance between the vehicle and the individual mass element in a series of Legendre polynomials (a special case of spherical harmonics), prescribing that r > s. Subsequently, the longitude τ and the planetocentric latitude δ of the point r can be introduced. The gravitational potential for an arbitrary mass distribution of a planet with mass mp and equatorial radius Re can now be described as:

∞ n n Gmp X X Re U = Pnm(sin δ)(Cnm cos(mτ) + Snm sin(mτ)) (2.4) r rn n=0 m=0 Where,

Z n 2 δ0m (n m)! s 0 0 3 Cn,m = − − n Pnm(cosδ ) cos(mτ )ρ(s)d s (2.5) mp (n + m)! Re

Z n 2 δ0m (n m)! s 0 0 3 Sn,m = − − n Pnm(sinδ ) cos(mτ )ρ(s)d s (2.6) mp (n + m)! Re

m 2 m/2 d Pn,m(u) = (1 u ) Pn(u) (2.7) − dum

n 1 d 2 n Pn(u) = (u 1) (2.8) 2nn! dun −

In equations 2.5 to 2.8, Pnm is the associated Legendre polynomial of degree n and order m and Pn is the Legendre polynomial of degree n. Cnm and Snm are the geopotential coeﬃcients, they describe the dependence on the planet’s internal mass distribution. δ0m is the Kronecker delta, which equals 1 if m = 0 and 0 otherwise. The prime indicates a reference to s and not to r. Finally, u is an arbitrary function. The acceleration in the R-frame gR can be calculated by equations 2.9 to 2.11[Mel- man, 2010].

  ! 1 ∂U zR ∂U 1 ∂U gxR =  q  xR 2 2 yR (2.9) r ∂r − 2 2 2 ∂δ − x + y ∂λ r xR + yR R R

  ! 1 ∂U zR ∂U 1 ∂U gyR =  q  yR 2 2 xR (2.10) r ∂r − 2 2 2 ∂δ − x + y ∂λ r xR + yR R R

q 2 2 1 ∂U xR + yR ∂U gz = z + (2.11) R r ∂r R r2 ∂δ In practice, the geopotential coeﬃcients cannot be determined as the internal mass distribution of a planet is not exactly known. The coeﬃcients can be found by, for example satellite tracking and/or satellite altimeter data [Montenbruck and Gill, 2005]. For Earth the EGM96 model has been used [NASA/NIMA, 2010].

11 2.3.2 Atmosphere An atmosphere can be a very hostile environment for a vehicle. Friction with atmospheric gas can lead to overheating of the vehicle. The local pressure can become so high that structural damage occurs. On the contrary, the vehicle can use the atmosphere to its own advantage. The aerodynamic force generated by interaction with the atmosphere can be used to steer the vehicle to a landing site. The entry window can be signiﬁcantly enlarged compared to a non-atmospheric powered descent. To predict and simulate an atmospheric trajectory accurately, a good representation of the atmosphere is needed. For the scope of this thesis, an atmosphere model should deliver the following information at any point in the atmosphere:

- Density ρ

- Temperature T

- Speciﬁc gas constant R

- Ratio of speciﬁc heats γ

The density is used to calculate the dynamic pressure and, subsequently, the aerodynamic forces. It is, furthermore, used to calculate the stagnation point heat flux. The speed of sound can be computed from the temperature, specific gas constant and ratio of specific heats. Then, using the velocity w.r.t. the air, the Mach number can be determined. The Mach number plays a role in the calculation of the aerodynamic coefficients. It is further chosen to use a standard atmosphere instead of a reference atmosphere. Compared to a standard atmosphere, the reference atmosphere includes a dependency on time next to height. This gives a more accurate representation of the atmosphere, as it shows a variation in time due to solar/stellar activity and a rotating planet. In the future, the atmosphere class could be extended with the reference atmosphere model. The specific gas constant and the ratio of specific heats are assumed constant for an atmosphere. Pressure and temperature are taken from a table as a function of height. Information between the nodes is obtained from cubic spline interpolation, the method can be found in section 4.1. Cubic spline interpolation has proven superior over linear interpolation [Volckaert, 2007]. The US Standard Atmosphere 1976 is used for the Earth. A wind model has been taken from GRAM-99. GRAM-99 is a reference atmosphere model that is described in [Justus and Johnson, 1999]. In there, the atmosphere model is given for January 1st, 1999. The magnitude and direction profiles with height have been taken from there. These profiles can be seen in appendixA.

2.3.3 Shape In case of asteroids and sometimes moons, the shape of a celestial body can differ significantly from a perfect sphere. For example, a rotating star shows a flattening due to the centripetal force. In the current version of the flight simulator as described in chapter5, the main body can be modeled as a sphere or an ellipsoid. For trajectory simulations, this implies that the relative position between vehicle and a point on the surface (or in the atmosphere) has changed. For an entry vehicle, this has three direct effects. The line of sight distance between vehicle and TAEM point has changed. Hence, the trajectory length has changed. Also, a different location needs to be targeted. Furthermore, as can be seen in figure 2.4, height is defined perpendicular to the surface. This has implications for the atmospheric properties that vary with height.

12 2.3. Environment modeling

Figure 2.4: Illustration of height for an ellipsoid.[Seeber, 1993]

An iterative process can be used to determine the height of a vehicle with respect to an ellipsoid. This method is given in [Seeber, 1993]. This is, however, costly in computation time. It is therefore chosen to use approximative equations [Mooij, 1998]. These are given in the remainder of this section. The ellipticity is given by:

Re Rp e = − (2.12) Re

Here, Re and Rp are the equatorial and polar radii. Using the planetocentric latitude δ, the height h is approximated by:

2 h = r Re(1 e sin δ) (2.13) − − The previous equation derives from a Taylor series expansion. Only the ﬁrst term of this expansion has been retained.

2.3.4 Environment simpliﬁcations

Generally, a more detailed and accurate environment representation implies a longer simulation time. A mission planning algorithm that can be executed on-board would have to finish in a few seconds. Therefore, simplifications need to be made on the environment in order to reduce computation time. Logically, the real trajectory will deviate from the planned trajectory due to these simplifications. It is the task of the guidance system to correct for these errors. This section presents environment simplifications that are used in this thesis work. Summed up they are:

- A gravity ﬁeld with J2, J3 and J4

- A central gravity ﬁeld

- A spherical shaped celestial body

- Exponential atmosphere model

13 This spherical harmonics model can be simpliﬁed when assuming m = 0. Then, the potential is no longer a function of the longitude and all the Sn0 terms vanish too. Deﬁning Jn = Cn0, equation 2.4 can be written as: −

" ∞ n # Gmp X Re U = 1 JnPnm(sin δ) (2.14) r − rn n=2

In the previous equation, one can see that if no coefficients are taken into account, the part between brackets will be equal to 1. This corresponds to a central force field where all the mass is concentrated in the center of the body. Equation 2.14 can be simplified further by taken only J2, J3 and J4 into account. Then, the gravitational acceleration in spherical coordinates is [Regan and Anandakrishnan, 1993]:

∂ ∂ U = grir + g i = (U)ir + (U)i (2.15) ∇ δ δ ∂r r∂δ δ

Here, ir and iδ are the unit vectors in the direction of r and δ. The result is given by [Mooij, 1997]:

2 Gmp 3 Re 2 gr = [1 J2 (3 sin δ 1) r2 − 2 r − 3 Re 2 2J3 sin δ(5 sin δ 3) − r − 4 5 Re 4 2 J4 (35 sin δ 30 sin δ + 3)] (2.16) −8 r −

2 Gmp Re g = 3 sin δ cos δ[J2 δ − r2 r 1 Re 2 + J3 csc δ(5 sin δ 1) 2 r − 2 5 Re 2 + J4 (7 sin δ 3)] (2.17) 6 r −

The gravitational force FG in the V -frame is computed by equation 2.18. This force can be transformed to the I -frame using the direction cosine matrix CI,V as presented in [Mooij, 1997].

  gδ   FG,V = m  0  (2.18) gr

For the aforementioned central gravity ﬁeld, the gravitational acceleration g acting on a vehicle is computed as:

Gmp g = (2.19) r2

14 2.4. Vehicle modeling

The vector g points towards the center of the spherical body. In addition, the simpli- ﬁcation is made that the body is geometrically spherical, h = r Re. This simpliﬁes − the targeting of the TAEM interface. Finally, the atmosphere behaves according to an exponential atmosphere model. This model describes that the density varies according to equation 2.20.

−βh ρ = ρ0e (2.20)

Here, β is the inverse of the scale height and ρ0 is the density at zero height. The exponential atmosphere is isothermal, hence the temperature is constant. The exponential atmosphere derives from the two assumptions. First, there is an equilibrium of the gravitational and pressure forces. Seconds, the atmosphere behaves according to the ideal gas law. In appendixA, a comparison between an exponential atmosphere model for the Earth and the US 1976 Standard Atmosphere model is presented.

2.4 Vehicle modeling

The vehicle configuration has a major influence on the trajectory that is flown. For example, the Space Shuttle’s aerodynamic properties allow for a horizontal landing on a runway. In contrast, due to the low lifting capabilities of Apollo, the capsule had to land by deploying parachutes. Besides the aerodynamics properties, the mass and thermodynamic properties have an influence as well. The mass properties, together with the forces and moments, determine the translational and rotational accelerations that the vehicle experiences. The thermodynamic properties can be such that one trajectory is preferred over another, because the first does not give an overheating. One can think of more vehicle properties that influence the trajectory, for example a parachute or propulsion system. In this thesis work, only the aerodynamic, thermodynamic and mass properties are modeled. These properties are the most important ones for a vehicle in an un-powered hypersonic descent. To study 3 DoF trajectories, only the total mass is modeled as part of the mass properties. The thermodynamic properties are modeled by a characteristic radius and limits on the total stagnation point heat flux and total heat load. The characteristic radius, typically, comes from the nose or leading edges. The aerodynamic properties are modeled by aerodynamic coefficients and a reference area. Together with the dynamic pressure, one can calculate the corresponding aerodynamic force. This is shown in the next section. To compute an aerodynamic moment, an additional reference length is required. For the same vehicle shape, the coefficients are a function of many variables, a.o. the angle of attack, side-slip angle, Mach number and Reynolds number. In principle, one could represent each coefficient by a Taylor series including a desired number of derivatives. For this thesis work, it is chosen to make the coefficient depend on two variables maximum: the angle of attack or the side-slip angle, and the Mach number. In addition, no derivatives are taken into account. The total value for an aerodynamic coefficient is the sum of the contribution due to the body and control surfaces. For software purposes, one could think of modeling the coefficients by a table or an analytical function.

15 2.5 Forces

For a vehicle in an un-powered descent, there are two forces that can shape the trajectory. Namely, the aerodynamic and gravitational force. The total aerodynamic force FA,AA can be decomposed into three components: drag, side and lift force. The ﬁrst index A implies that the force is the aerodynamic force. The second index shows that the force is expressed with respect to the AA-frame. FA,AA can be calculated by:

   1 2  D CD ρV S − − 2 A ref F    1 2  A,AA =  S  =  CS 2 ρVASref  (2.21) − − 1 2 L CL ρV S − − 2 A ref where,

D : Drag force [N] S : Side force [N] L : Lift force [N] CD : Drag coefficient [ ] − CS : Side force coefficient [ ] − CL : Lift coefficient [ ] − ρ : Density [kg/m3] VA : Airspeed [m/s] 2 Sref : Reference area [m ]

The gravitational force FG,R is calculated in the R-frame. It may be expressed as:

  gxR F = m   G,R  gyR  (2.22) gzR

The gravitational acceleration components gxR , gyR and gzR are given by equations 2.9 to 2.11.

2.6 Translational ﬂight dynamics

The translational ﬂight dynamics are expressed by the translational equations of motion (EoM). The EoM describe the dynamics of a point mass with respect to a certain reference frame. In this section, the EoM are presented in two formats:

1. Cartesian coordinates in the I -frame

2. Spherical coordinates in the R-frame

The main advantage of using the EoM w.r.t the R-frame in spherical coordinates is that they are physically interpretable. This is what makes them an excellent candidate for trajectory planning and guidance systems. The disadvantages are: the occurrence of singularities and the inclusion of wind makes the EoM complicated (as shown in [Mooij, 1997]). An expression w.r.t. the I -frame in Cartesian coordinates does not suﬀer from these disadvantages. Including wind in Cartesian components in the I -frame

16 2.6. Translational flight dynamics is straightforward by using an additional reference frame and transformation matrices. Furthermore, a simulator based on Cartesian components in the I -frame has a lower computation time [Mooij, 2010]. Due to the previously discussed advantages, the EoM in Cartesian coordinates w.r.t. the I -frame are used as a core for the simulator. Subsection 2.6.1 presents the EoM in both reference frames. In subsection 2.6.2, simplifications are made to the flight dynamics.

2.6.1 The equations of motion The governing equation of motion for a rigid body with ﬁxed mass-distribution is [Mooij, 1997]:

d2r dV F = m I = m I (2.23) I dt2 dt Where,

FI : Summation of the external forces expressed in the I -frame [N] rI : Position w.r.t. the I -frame [m] VI : Velocity w.r.t. the I -frame [m/s]

This dynamic equation describes the motion of a body under the inﬂuence of external forces, the corresponding kinematic equation is given by:

dr I = V (2.24) dt I The previous equation can be written in Cartesian components with:

T rI = (xI , yI , zI ) T VI = (x ˙I , y˙I , z˙I )

Adding the aerodynamic and gravity force, the EoM becomes:

  x¨ dV I 1 I =  y¨  = F + F (2.25) dt  I  m A,I G,I z¨I   x˙ dr I I =  y˙  (2.26) dt  I  z˙I

In equation 2.25, FA,I and FG,I are, respectively, the aerodynamic and gravitational forces expressed in the inertial reference frame. They are give by:

FA,I = CI,AAFA,AA (2.27)

FG,I = CI,RFG,R (2.28)

17 FA,AA and FG,R can be found in equations 2.21 and 2.22 respectively. The direction cosine matrix CI,AA is composed of the following direction cosine matrix multiplications:

CI,AA = CI,RCR,VCV,TACTA,AA (2.29)

In case there is no wind, VA = VG and:

FA,I = CI,AGFA,AG (2.30) where,

CI,AG = CI,RCR,VCV,TGCTG,AG (2.31)

The direction cosine matrices presented above, can be found in [Mooij, 1997]. The EoM in spherical components w.r.t. the R-frame can be seen in equations 2.32 to 2.37 below. A derivation can be found in [Mooij, 1991] and [van Doorn, 2009].

FV 2 V˙G = + ω r cos δ (sin γG cos δ cos γG sin δ cos χG) (2.32) m cb −

2 Fγ V V γ˙ = + 2ω V cos δ sin χ + G cos γ + G G m cb G G r G 2 +ωcbr cos δ (cos δ cos γG + sin γG sin δ cos χG) (2.33)

Fχ VG cos γGχ˙ G = + 2ω VG (sin δ cos γG cos δ sin γG cos χG) + m cb − V 2 + G cos2 γ tan δ sin χ + ω2 r cos δ sin δ sin χ (2.34) r G G cb G

r˙ = VGsinγG (2.35)

V sin χ cos γ τ˙ = G G G (2.36) r cos δ

V cos χ cos γ δ˙ = G G G (2.37) r

18 2.6. Translational ﬂight dynamics

Where,

FV = D mgr sin γG + mg cos γG cos χG (2.38) − − δ Fγ = +S sin σG + L cos σG mgr cos γG mg sin γG cos χG (2.39) − − δ Fχ = S cos σG + L sin σG mg sin χG (2.40) − − δ

The gravitational acceleration is expressed in two components, gr and gδ. gr is pointing from the vehicle to the center of the body and gδ points to the North perpendicular to the surface. It needs to be mentioned that these EoM already contain two simpliﬁcations:

- There is no wind

- The gravity ﬁeld is rotationally symmetrical

◦ ◦ As can be seen from equation 2.34, a singularity occurs if γG is 90 or 90 . − 2.6.2 Flight dynamics simpliﬁcations In total, three sets of simpliﬁcations of the EoM are made in this thesis work. They are presented in this subsection. The state variables presented in this section are groundspeed based.

Reduced-order system The mission planning tool described in [Saraf et al., 2003] uses a reduced-order system for generating a reference trajectory. This system is obtained by reducing the EoM w.r.t. the R-frame in spherical components. The necessary assumptions to be made are:

- No side force: S = 0.

2 - Neglect terms involving ωcbr.

- Central gravity ﬁeld only: gδ = 0.

- Elimination of vertical dynamics by enforcing an r and γ proﬁle. Hence, the EoM for r and γ are not integrated.

In addition, the independent variable is switched from time to energy per unit mass E, using dE = DV . For example: dt −

dτ dt VG sin χG cos γG 1 =τ ˙ = − (2.41) dE dE r cos δ DV Switching to energy has as an advantage that the EoM for the velocity can be eliminated as well. The resulting reduced-order system is given by:

dτ sin χ cos γ = (2.42) dE − r cos δD

19 dδ cos χ cos γ = (2.43) dE − rD

dχ L sin σ cos γ tan δ sin χ = + Cχ (2.44) dE −DV 2 cos γ − rD

Where, using dE = DV and equation 2.34: dt −

sin δ cos δ tan γ cos χ Cχ = 2ω − (2.45) − cb DV In all the equations, D is the drag acceleration. Logically, E is the energy per unit mass. In equation 2.44, L is the lift acceleration. When evaluating equations 2.42 to 2.44 r and γ are taken from the enforced proﬁles. The main advantage of this system is that only three EoM need to be integrated. On the other hand, r(E) and γ(E) proﬁles need to be available as input.

2D EoM In [Ambrosius and Wittenberg, 2006] a 2D approach to analysing re-entry trajectories is taken. Based on that, a 2D set of EoM can be given. This set consists of four state variables: r - Radius from planet center to vehicle θ - Angular displacement (combination of latitude and longitude) V - Velocity γ - Flight-path angle

The corresponding EoM are, based on [Ambrosius and Wittenberg, 2006]:

dr = V sin γ (2.46) dt

dθ V cos γ = (2.47) dt r

dV m = D mg sin γ (2.48) dt − −

dγ mV = L mg cos γ + mV 2 cos γ/r (2.49) dt − The angular displacement θ is not required to evaluate the diﬀerential equations for r, V and γ. This state variable only serves as a measure of the distance traveled. This system presented above can be obtained from equations 2.32 to 2.37 by assuming:

- No side force: S = 0.

20 2.6. Translational ﬂight dynamics

- No planet rotation: ωcb = 0. - Central gravity ﬁeld only.

- Motion in a vertical plane: cos σ = 1

In this 2D case, the heading is not deﬁned.

Equilibrium glide ﬂight By making two further assumptions on the ﬂight path with respect to the 2D set, a special type of trajectory can be found. The assumptions are:

- Slowly varying flight-path angle: dγ 0 dt ≈ - Small flight-path angle: cos γ 1 ≈ The trajectory that satisfies these assumptions, is said to be an equilibrium glide trajectory. The EoM for the flight-path angle can be reduced to:

V 2 L + m mg = 0 (2.50) r − In essence, this equation implies that the weight of the vehicle is balanced by the aerodynamic lift and centrifugal force. If one thinks of the vertical lift, the assumption of motion in a vertical plane does not have to be made. The vertical lift is the total lift force multiplied by cos σ. Now, the weight of the vehicle is balanced by the centrifugal and vertical component of the lift force.

21 22 CHAPTER 3

Entry Mission Characteristics

The entry mission is characterised by the high energy of the vehicle at entry. If the potential energy is evaluated with respect to the landing strip, then all the energy at entry needs to be reduced to zero for the landing. This energy is reduced by the drag during the flight. The energy goes into the heating of the atmosphere and the vehicle. The interaction between the atmosphere and the vehicle can give several problems. The aerodynamic forces and pressure acting on the vehicle can become too high. In addition, the heating can degrade the structure. Besides trying to avoid these problems, the vehicle should target and land at the landing strip. These challenges lead to the formulation of a problem definition for entry planning. Section 3.1 presents this problem definition. The guidance algorithm has to provide the steering commands that safely guide the vehicle to the landing site. The guidance system of the Space Shuttle has proven to work. It is, therefore, considered to be a good starting point. A method called Evolved Acceleration Guidance Logic for Entry (EAGLE) [Saraf et al., 2003] elaborates on the Space Shuttle guidance. Both guidance algorithms are presented in section 3.2. The task of the control system is to ensure that the commanded attitude from the guidance system is obtained. Section 3.3 introduces this topic. Finally, section 3.5 presents the reference missions that were used for this thesis work.

3.1 Entry planning problem

The purpose of an entry planner is to generate reference steering commands such that the vehicle can safely target the Terminal Area Energy Management (TAEM) interface. The aerodynamic angles are the steering commands. The side-slip angle is always commanded to zero. Hence, the angle of attack and bank angle are the steering commands. Subsection 3.1.1 introduces the problem statement for the hypersonic entry phase of a glide entry. The TAEM interface is further deﬁned in subsection 3.1.2. To ensure that the vehicle and its payload lands safely, constraints are imposed on certain parameters. These are introduced in subsection 3.1.3

23 3.1.1 Problem statement The diﬃculty in planning the reference steering commands and trajectory within seconds can be decomposed in four factors:

- A TAEM interface has to be targeted.

- Constraints need to be satisﬁed.

- The vehicle dynamics are highly non-linear.

- The environment and vehicle needs to be modeled suﬃciently accurate.

The first two factors limit the vehicle’s freedom in choosing an angle of attack and bank angle. The angle of attack determines the lift-to-drag ratio throughout entry. This in turn determines the maximum range that can be achieved [Vinh, 1981]. The bank angle magnitude controls the vertical lift-to-drag ratio, its sign determines the direction of the flight. The attitude history of the vehicle cannot be freely chosen, as a too high heating could occur or the TAEM interface is missed by tens of kilometers. The third factor makes it difficult to predict where the vehicle goes for a certain angle of attack and bank angle input. For both the vehicle dynamics, environment en vehicle model simplifications need to be made in order to achieve the low computation time. It is the challenge to determine which simplifications can or cannot be made. Based on [Shen, 2002], the problem statement is given as:

Given the initial conditions upon entering the atmosphere and the final conditions at the TAEM interface, find the reference bank angle and angle-of-attack profile such that the TAEM interface is reached within certain limits, while none of the path and control constraints are violated.

Subsection 3.1.2 shall elaborate on the TAEM interface conditions. Subsection 3.1.3 shall present the path and control constraints.

3.1.2 TAEM interface During the TAEM phase, the vehicle’s energy is managed such that, at the end, it is aligned with the runway and has a correct height and velocity. As in the hypersonic transition phase, the flight is un-powered. The flight through the TAEM phase starts supersonic but ends subsonic. Hence, the aerodynamics show a significant change during the flight in this phase. One of the characteristics of the TAEM phase is the Heading Alignment Cylinder (HAC). This is an imaginary cylinder that touches the centerline of the runway. The vehicle flies along the surface of the HAC such that, at the end, it is aligned with the runway. If the vehicle arrives at the TAEM interface with a too high or low energy value, it may under- or overshoot the runway. In the TAEM phase, the vehicle can make so-called S-turns or deflect the speedbrakes to dissipate more energy [de Ridder, 2009]. Nevertheless, the vehicle should arrive at the interface satisfying certain requirements. These requirements serve as the TAEM interface conditions. On a side note, in [de Ridder, 2009] a study is performed on optimal trajectories and energy management capabilities of a winged entry vehicle in the TAEM phase. An illustration of the TAEM phase for the X-33 vehicle can be seen in figure 3.1. This vehicle is introduced in section 3.5.

24 3.1. Entry planning problem

Figure 3.1: The TAEM interface for the X-33. [Chen et al., 2002]

For the X-33, the TAEM interface point is given as 56 km from the HAC point at an altitude of about 29 km. Here, the X-33 ﬂies at Mach 2. The HAC point lies at the outer surface of the HAC, which has a diameter of about 9.3 km. The error tolerances at the TAEM phase are (from [Chen et al., 2002] and [Saraf et al., 2003]):

- 11 km error on ﬁnal range to HAC point ± - 1.8 km altitude error ± - 10◦ heading error ± - < 25◦ as ﬂight-path angle

TAEM interface conditions for Horus (introduced in section 3.5) are given as [MBB, 1988]:

- 60 km distance to HAC point

- 25 3 km altitude ± - 2.5 0.3 Mach number ± - 7◦ 5◦ flight-path angle − ± - heading towards HAC 30◦ ± 3.1.3 Constraints Constraints arising from limitations on the vehicle and human body are called path constraints. Besides these path constraints, there are control constraints. For example, maximum aerodynamic control surface deflections or the maximum force from a reaction control thruster. For a three DoF simulation, these control system constraints can be translated to constraints on α˙ , σ˙ , α¨ and σ¨. There are five path constraints imposed:

- Dynamic pressure qdyn

- q α dyn · - Stagnation point heat ﬂux q˙

25 - Integrated heat load Q

- g-load n

The maximum constraint values depend on the vehicle’s design and the presence of a crew. Below, equations are given that can be used to evaluate the constraints during ﬂight. When the constraints are actively taken care of during the planning phase, there should be more conﬁdence in satisfying the constraints during simulations.

Dynamic pressure constraints The dynamic pressure acts on the outer skin of the vehicle. A limit is imposed to ensure structural safety, the constraint is given as:

1 2 q = ρV < (q )max (3.1) dyn 2 dyn

Here, ρ is the density and V is the velocity with respect to the atmosphere. The qdynα˙ constraint is the product of dynamic pressure and angle-of-attack α. It corresponds to a bending moment M that acts on the vehicle. M is calculated as:

M = CmααqdynSref cref (3.2)

Here, cref is a reference length and Cm is the aerodynamic moment coefficient. In equation 3.2, the product q α can be seen. dyn · Heating constraints Heating can severely degrade the structural strength of the vehicle. The local heat flux influences the material strength, but also local coolant flow rates. A constraint is, therefore, put on the maximum heating rate. The maximum heating rate on the vehicle occurs at a stagnation point, this could be the vehicle’s nose, but also leading edges of wings. The heat flux consists of the convective heat flux q˙c and the radiative heat flux q˙r. The total heat input Q to the vehicle determines the temperature inside the vehicle, which cannot be too high for on-board systems and/or the well-being of a flight crew, as well as total amount of coolant required. Furthermore, the structural rigidity is degraded with increasing heat load. An increasing heat load gives an increasing temperature, which causes the structure to become more flexible. The constraints on the heat flux and total heat load are given by equations 3.3 and 3.4 respectively.

q˙c +q ˙r < q˙max (3.3)

Q < Qmax (3.4)

The convective stagnation point heat ﬂux can be calculated using equation 3.5 below [Chapman et al., 1958].

r c2 c1 ρ V q˙c = (3.5) √RN ρ0 Vc

26 3.2. Guidance system

Here, RN is the geometrical radius of the vehicle at the stagnation point. The density at sea-level is given by ρ0 and Vc is the circular velocity. c1 and c2 are constants that 8 3/2 depend on the atmosphere. In this thesis, c2 = 3 and c1 = 1.06584 10 W/m are used for · √c1 √1 1 the Earth’s atmosphere [Mooij and H¨anninen, 2009]. By substituting C = c2 , RN ρ0 Vc a simpler form is obtained:

c q˙c = C√ρV 2 (3.6) g-load The vehicle can be severely damaged if the g-load factor exceeds the structural strength. This load acts as an internal load on all joints. It should not exceed a value that either degrades the vehicle or its payload. In case one speaks of a manned ﬂight, the value is limited by the capabilities of the human body. The human tolerances with respect to accelerations are threefold [Serrano-Martinez and Parra, 1987]:

- Absolute value of the accelerations

- Exposure time

- Time rate of change of speciﬁc accelerations

Constraints have to be posed to limit these three factors. The mission planner and guidance system should make sure that there is no deterioration of human functions. And, more limiting, the ﬂight crew should be able to perform ﬂight operations during atmospheric entry. If no actions are required, then the constraints are such that no physical damage may occur. In this thesis work, a crew is considered to be present. Based on [Serrano-Martinez and Parra, 1987], the following general constraint is posed:

L cos α + D sin α n = < 2.5g (3.7) m Here m is the total mass. This constraint only holds for if the side-slip angle is zero. In essence, this constraint is the component of the g-load that acts on the ZB-axis.

Control constraints This thesis work is mainly limited to the translational ﬂight dynamics. Therefore, constraints are imposed on α˙ and σ˙ . In addition, there is a constraint imposed on the maximum α. This constraint acts as a safety margin on the controllability. In reality, the vehicle could still increase the angle of attack until a bit until stall occurs. However, this is only used for guidance and control capabilities. The values for the Horus, Space Shuttle Orbiter, X-33 and X-38 are presented in section 3.5.

3.2 Guidance system

The guidance system is composed of the planning and tracking algorithm. Using both, it commands the attitude of the vehicle. The side-slip angle is always commanded to zero. The planning algorithm determines the reference angle-of-attack, bank angle and trajectory parameters. The trajectory tracker compares this with the output from the navigation system and computes the commanded attitude. The guidance system of the Space Shuttle has proven to work, therefore subsection 3.2.1 discusses this system.

27 EAGLE has elaborated on the Space Shuttle guidance system. Also, it was tested as the best entry guidance method amongst state-of-the-art algorithms in a NASA study [Hanson et al., 2002]. Subsection 3.2.2 presents a discussion of the EAGLE planning method. Subsection 3.2.3 presents the tracking algorithm used in EAGLE.

3.2.1 Space Shuttle guidance

The planning function of the Space Shuttle Orbiter’s guidance system operates on the analytical definition of a desired drag acceleration profile [Harpold and Graves Jr, 1978]. The range-to-go during entry is a unique function of the drag acceleration profile followed. This range can be predicted by analytical techniques for geometric drag acceleration functions of the Earth-relative velocity if the flight path angle is near zero. For low velocities, the flight path angle cannot be kept near zero and the range can be predicted analytically if the independent variable is changed from the Earth-relative velocity to energy with respect to Earth. The drag acceleration profile is adjusted to account for constraints and designed to minimise the accumulated heat load. The range flown according to this drag acceleration profile is predicted by an analytical solution to the equations of motion and compared to an estimated range. The profile is then iteratively adjusted until these two converge to the same value. The Shut- tle guidance estimates the range as a great circle arc, at the TAEM height, between the entry and TAEM interface point. This is the reference drag acceleration profile found by the planning function. The altitude rate and the L/D in the vertical plane are also computed analytically from this reference drag profile, they are used by the trajectory tracker. The drag acceleration profile consists of four different types of segments. Each segment has its own analytical solution to the range flown in this segment. The four segments are:

1. Temperature Control (quadratic in drag)

2. Equilibrium Glide

3. Constant Drag

4. Transition (linear in drag)

Figure 3.2 shows the drag acceleration profile as a function of velocity for the Orbiter. As can be seen, the profile consists of five segments (two quadratic). The constraints are translated to limits on the drag acceleration. The entry corridor is defined as all the allowable drag acceleration values as a function of velocity. The control law computes the control commands by comparing the actual flown profile to the reference profile. It is basically a PID controller from which a commanded lift-to- drag ratio is obtained. Bank angle and angle-of-attack modulation are be used to achieve the commanded L/D profile. The Orbiter uses σ modulation as the primary control variable, α is used secondary to minimise the aerodynamic heating while achieving the required cross-range. Bank angle modulation is used to control the total entry range and cross-range. The magnitude of the bank angle determines the total entry range and the direction determines the cross-range. The direction of the bank angle is determined by a law operating on a heading angle corridor. When the heading angle surpasses a threshold a bank reversal is initiated. The angle-of-attack is modulated to achieve the reference profile on a short period basis, because the response to bank angle modulation

28 3.2. Guidance system

Figure 3.2: Entry corridor showing different flight segments. [Harpold and Graves Jr, 1978] is relatively slow. This reduces phugoid motion during a bank reversal and provides additional manoeuvrability to compensate for deviations. The concept of planning a drag acceleration profile has been applied in the Apollo and the Space Shuttle programs. Both have proven that this is a successful guidance strategy. It is also a strategy that can be generally applied with a fine tuning of the shape of the profile depending on the mission and vehicle. An advantage of drag acceleration planning and tracking is that the method is robust to errors introduced by aerodynamic force modeling [Mooij et al., 2007]. The Shuttle’s guidance system, however, generates a drag acceleration profile for the longitudinal motion only. By integrating lateral guidance, the cross range can be increased and extreme lateral points can be reached more accurately [Mooij et al., 2006]. Another disadvantage is that the reference profile is not designed on- board. Abort scenarii for the Shuttle are all designed separately and requires a significant effort [Hanson et al., 2002]. As opposed, on-board mission planning does not require this. Furthermore, on-board trajectory planning can take off-nominal conditions into account and hence, allow the generation of a more accurate trajectory. On-board mission planning would increase safety, by taking into account off-nominal conditions, and reduce cost by lowering the pre-mission design effort. According to [Mease and Kremer, 1994], the Shuttle’s trajectory tracker is designed for Shuttle mission conditions and therefore has a limited operating domain.

3.2.2 EAGLE planning method EAGLE can be seen as an evolution of the Orbiter’s entry guidance in which lateral motion is taken into account in the planning. To do this, range is no longer based on the great circle arc estimate of the trajectory length. As for the Space Shuttle, the strategy is to plan aerodynamic accelerations. The advantage is that they can be provided accurately by the inertial measurement unit.

29 The planner solves two problems within a successive approximation procedure:

1. Trajectory length problem

2. Trajectory curvature problem

The trajectory length problem is essentially the same as the Orbiter’s planner. A drag profile is constructed that matches an estimated trajectory length. However, one important difference is that the profile is made as a function of energy, and not velocity. This drag profile consists of three segments. The middle segment is a constant drag segment, the other two are linear with energy. More on the construction of the segments, and how this is related to range, will be said in section 7.2 From the drag acceleration profile, a bank angle magnitude profile is constructed. In the trajectory curvature problem, the reduced-order system as presented in subsection 2.6.2 is integrated with a bank reversal placed at a random energy. Then the bank reversal energy is searched for that minimises the cross-range error with respect to the TAEM interface. The cross-range error is the miss distance perpendicular to the great circle arc (at TAEM interface height) between entry and TAEM point. In section 7.4 the definition of the cross-range error is further explained. The bank angle magnitude profile is transformed into a bank angle profile that includes the correct sign and a bank reversal. The final state of the solution to the curvature problem does not have a cross-range error. However, there is no guarantee that the vehicle ends up at the TAEM interface. Probably, there is a downrange error. The downrange error is the miss distance along the great circle arc from entry to TAEM interface point. The vehicle only ends up at the TAEM interface if the correct trajectory length estimate is used. As a first guess for the trajectory length, the great circle arc between entry and TAEM point is used. This is the same approach as used in the Space Shuttle. The downrange error is minimised by adjusting the initial guess. More on this will be said in section 7.4. As mentioned before, this algorithm was tested as the best entry guidance method amongst state-of-the-art algorithms in a NASA study [Hanson et al., 2002]. It is, therefore, chosen to use this algorithm for answering the thesis question. The design of the algorithm is described in [Saraf et al., 2003]. However, not all the details are described there. Therefore, chapter7 is devoted to the implementation of the planning algorithm. The next subsection presents the tracking algorithm that is given in [Saraf et al., 2003]. In chapter8, the planning and tracking algorithm are tested.

3.2.3 EAGLE tracking algorithm

The tracker is based on the formulation of a new system that is described in [Saraf et al., 2003] and [Tu et al., 2000]. The output of this system are the drag acceleration and heading angle. The two inputs are:

L u = cos σ (3.8) D D

L uχ = sin σ (3.9) D

30 3.2. Guidance system

The system dynamics are given by equation 3.10 and 3.11:

00 D = a + buD (3.10)

0 uχ cos γ tan δ sin χ χ = + Cχ (3.11) −V 2 cos γ − rD 00 The ﬁrst equation is the short notation for D with uD substituted. Equation 3.11 is an EoM as given by equation 2.45 in which uχ is substituted. The input uD is derived by formulating a second-order linear error system as:

Z 00 00 0 0 2 (D D ) + 2ζω(q)(D D ) + ω (q )(D D) + k1 (D D)dE = 0 (3.12) ref − ref − dyn ref − ref − where,

ω(qdyn) = Scheduled undamped natural frequency. ζ = Damping ratio. k1 = Integral gain for the drag dynamics. The index ref indicates that the value is taken from the reference trajectory (the planner). The undamped natural frequency is given by equation 3.13.

" 2# qdyn qdyn ω(q) = 2 ω0 (3.13) (qdyn)ref − (qdyn)ref

Here, (qdyn)ref is a reference dynamic pressure and ω0 is the undamped natural frequency. By scheduling the natural frequency using equation 3.13, bank angle saturation is avoided where the dynamic pressure is low. Here, the vehicle’s control capabilities are marginal and drag errors cannot be removed. The input uD can be calculated by substituting equation 3.10 in equation 3.12. This tracking algorithm can be interpreted as a PID controller. The terms 2ζω(qdyn) 2 and ω (qdyn) are seen as the derivative and proportional gain. The advantage of using the formulation as given in equation 3.12 is that the damping ratio and undamped natural frequency are interpretable as opposed to the derivative and proportional gain. In [Ogata, 2009] one can find a clear desription of how the damping ratio and natural frequency affect performance parameters (e.g. rise time, overshoot, settling time) of the transient response. The transient response is how the system behaves by control input towards a reference input. PD control only cannot remove a steady state error. An integral term is added to equation 3.12 in order to remove this error. The proportional control term typically affects the rise time and the derivative control term can be seen as a damping term. In [Saraf et al., 2003], the damping ratio ζ and natural frequency ω0 are given for the second-order linear error system when time is the independent variable. This is probably because the natural frequency is better interpretable. Also, as shown below, it is another means to schedule the proportional and derivative gains. In [Tu et al., 2000], the conversion between the time and energy domain is given as:

E¨ ζE = ζt + (3.14) 2ωtE˙

31 ωt ωE = (3.15) E˙ In the previous two equations, the indices E and t indicate the energy and time domain respectively. The double derivative of energy w.r.t. time was not given but is derived as:

dE˙ d( DV ) = − = (DV˙ + DV˙ ) (3.16) dt dt − where the EoM for the velocity, given in equation 2.48, is substituted for V˙ and,

D˙ = D0E˙ = D0DV (3.17) − Thus, E¨ is given as:

E¨ = ( D0DV + DV˙ ) (3.18) − −

The input uχ is derived by setting up a ﬁrst-order error system:

Z 0 0 (χ χ ) + k2ω(q)(χ χ) + k3 (χ χ)dE = 0 (3.19) ref − ref − ref − k2 = Proportional gain factor for heading. k3 = Integral gain for the heading dynamics.

Similar to uD, uχ can be found by substituting equation 3.11 in the previous equation. In this thesis work it is considered to treat only the bank angle as a closed-loop guidance variable. The angle of attack is always taken from the reference proﬁle. In the Space Shuttle Orbiter’s guidance algorithm and EAGLE, the angle of attack is a control variable. However, both methods require an additional gain that would have to be tuned manually. To avoid time-consuming gain tuning, only the bank angle is taken as a control variable. The two inputs also lead to two values for the bank angle:

DuD σD = arccos( ) (3.20) | | L

Duχ σχ = arcsin( ) (3.21) L

The commanded bank angle σcmd is a weighted average of both:

σ = w(E)sign(σ )σD + (1 w(E))σχ (3.22) cmd ref − Here, w(E) is the energy dependent weight factor that is determined by trial-and-error.

32 3.3. Control system

3.3 Control system

The goal of the control system is to achieve the commanded attitude from the guidance system and to guarantee that this attitude is stable, i.e. trimmed. To do so, it requires knowledge of the vehicle’s state, which is provided by the navigation system. It is desired to achieve the commanded attitude as quick as possible to enhance trajectory tracking. The attitude of a vehicle can be changed by generating moments around the body axes. The following methods can be used in atmospheric ﬂight for this purpose [Mooij, 1998]:

- Aerodynamic-control surfaces

- Reaction-control jets

- Moving mass points

Deflecting an aerodynamic-control surface results in a change in the vehicle’s shape as seen by the airflow. This gives a different pressure distribution around the vehicle. Hence, the resultant aerodynamic force and moment have changed. Reaction-control jets use a cold or hot gas to generate a moment. Shifting masses inside a vehicle generate a moment around the center of mass. Also, the moment produced by aerodynamic forces will change because the center of mass itself will have shifted. Typical aerodynamic-control surfaces for an atmospheric entry vehicle are:

- Rudder

- Elevator

- Aileron

- Elevons (Combination of Elevator and Ailerons)

- Body ﬂap

The rudder is used for generating of a yawing moment, the elevator for the pitching moment and the aileron for the rolling moment. The guidance and control system of an atmospheric entry vehicle will be in closed-loop form. Closed-loop systems compare the output with the input in order to compensate for errors, e.g., environmental disturbances. Open-loop system do not do this. Therefore, a closed-loop system using feedback is preferred over an open-loop system. In this thesis work, an ideal control system is assumed. This implies that the commanded attitude by the guidance system is achieved instantly and exactly. The equations for the rotational ﬂight dynamics do not need to be integrated anymore. Only the 3 degrees-of-freedom translational motion is studied. As mentioned before, the commanded attitude of the control system should be stable. In this thesis work, trim of the motion around the YB-axis is considered, while an ideal control system is assumed. The trim algorithm is presented in the next section.

3.4 Trim

Trim is studied, because the required control deﬂections may cause a large increase in the lift and drag. In that case, they can have a large eﬀect on the trajectory. Furthermore, the generated trajectory might not be trimmable and this should be avoided.

33 It is assumed that the vehicle has one body ﬂap and two elevons for achieving trim around the YB-axis. On a side note, the purpose of the Space Shuttle’s body ﬂap is threefold [NASA, 1988]:

- It thermally shields the three main engines during re-entry

- It provides the Orbiter with pitch control trim during the atmospheric ﬂight

- It reduces the loads on the elevons because elevon deﬂections can be reduced while still maintaining trimmed ﬂight

They will apply to a winged entry vehicle in general. The remaining part of this section presents the equations to calculate trim. A vehicle in trimmed flight is in pitch dM equilibrium when M = 0 and dα < 0. The latter prescribes that a positive change in angle-of-attack results in a nose-down moment and analogous for a negative α a nose-up moment. In this way, a disturbance in α results in a moment which brings the vehicle back to the equilibrium state. Hence, the vehicle is statically stable in the longitudinal direction. This property is determined by the vehicle’s design. The trim algorithm should calculate the required body flap and elevon deflections for trim. The moment M is calculated as:

M = (Cm)totalqdynSref cref (3.23)

Here, (Cm)total = Cm0 + Cmb + 2Cme. Cm0 is the contribution to the moment by the body and the control surface deflections in neutral position. This coefficient is a function of the angle of attack and Mach number Cmb and Cme are the contributions due to the body flap and elevon respectively. Both are a function of the angle of attack, Mach number and the magnitude of the corresponding control surface deflection angle. It needs to be mentioned that these coefficients are normalised for the same reference area and length. For trim one has:

Cm0 + Cmb + 2Cme = 0 (3.24)

For a combination of an angle of attack and a Mach number, it is first attempted to achieve trim by the body flap only. In that way, the elevons remain free for changing the bank angle. In equation 3.24, the body flap deflection angle is the only unknown parameter. As Cm0 is a function of α and M and Cmb is a function of α, M and the deflection angle. Also, Cme is zero for a zero deflection. If trim cannot be achieved by the body flap only, the flap is set to its maximum deflection (in the direction that reduces (Cm) ). Then, the required elevon deflection angle is computed as before. | total| 3.5 Reference missions

The guidance algorithm should work for a certain mission deﬁnition. The mission deﬁ- nition is composed of:

- Vehicle (including path and control constraints)

- Entry conditions

34 3.5. Reference missions

- TAEM interface conditions

In total four diﬀerent vehicles were used in this thesis work, namely the HORUS, X-33, Space Shuttle Orbiter and X-38. Subsection 3.5.1 presents the HORUS mission, the other missions are presented in subsection 3.5.2. The simulator that has been developed needs to be validated. It is validated using the Simulation Tool for Atmospheric Re- entry Trajectories (START) [Mooij, 1991], an already validated software for re-entry simulations. A reference trajectory for the HORUS mission has been generated with START, this trajectory can be found in subsection 3.5.1. The validation is treated in section 5.5.

3.5.1 Horus mission In 1985, Messerschmidt-Boelkow-Bloehm (MBB) took a second look at the Saenger’s Silverbird Bomber. The design was modified into a two-stage-to-orbit horizontal take- off concept [Astronautix, 2010]. This concept can be seen in figure 3.3. The second stage could have two configurations: Horus (Hypersonic Orbital Reusable Upper Stage) for manned missions and the expendable stage Cargus(Cargo Upper Stage) for heavy payloads [NASA, 2010]. The Horus would have to make a successful re-entry into the atmosphere of the Earth such that it could be re-used.

Figure 3.3: The Saenger spaceplane with Horus on top. [Astronautix, 2010]

Horus has two elevons, two rudders and a body flap as aerodynamic control surfaces. The Horus mission characteristics can be seen in table 3.1 below [Mooij, 2010]. The aerodynamic database was provided by MBB and the numerical values are given in [Mooij, 1995]. Figure 3.4 shows the lift-to-drag ratio as a function of the angle of attack L and Mach number. The general trend is that, for the same angle of attack, D is lower L for increasing Mach. In addition, D max occurs at a larger angle of attack for increasing Mach. Figure 3.5 shows the angle of attack for maximum lift-to-drag ratio. Figure 3.6 shows the body flap and elevon deflections, required for trim, for αmax and α(L/D)max. It can be seen that for trim at α(L/D)max, the flap deflections must be larger. The force and moment coefficients are given by tabulated functions of α and the Mach number M. The increments in the aerodynamic coefficients due to control surface deflection presented in [Mooij, 1995] are a function of the angle-of-attack, control surface deflections and Mach number. The ranges for which the aerodynamic coefficients are given can be seen in table 3.2. In table 3.2, δe and δb are the elevon and body flap deflection angles. A trimmed trajectory for the mission definition has been generated with START using the body flap and two elevons. This is the trajectory used for the validation. In generating the reference trajectory, an ideal control system has been assumed. The angle of attack and bank angle profile that have been used can be seen in figure 3.7 and 3.8 respectively. Figures 3.9 and 3.10 present the flap deflections for the trimmed flight. Hereafter, figures

35 2.5

2 Mach = 2 Mach = 3 Mach = 5 Mach = 10 Mach = 20 1.5 [-] L D

0.5

0 10 15 20 25 30 35 40 45 Angle of attack [deg]

L Figure 3.4: Horus D as a function of angle of attack and Mach number.

17 Angle of attack [deg]

14 2 4 6 8 10 12 14 16 18 20 Mach [-]

Figure 3.5: Angle of attack for maximum lift to drag ratio.

36 3.5. Reference missions

20 Trimmed body ﬂap deﬂection αmax

Trimmed elevon deflection αmax 15 Trimmed body flap deflection α(L/D)max

Trimmed elevon deﬂection α(L/D)max 10

Deﬂection angle [deg] −5

−10

−15

−20 2 4 6 8 10 12 14 16 18 20 Mach [-]

Figure 3.6: Deﬂection angles for trim at αmax and α(L/D)max.

Table 3.1: Reference mission deﬁnition for Horus.[Mooij, 2010]

Mass 26, 029 [kg] Reference area 110 [m2] Nose radius 0.8 [m] Entry height 119.96 [km] Entry longitude 105.97 [◦] − Entry latitude 22.060 [◦] − Entry velocity 7438.1 [m/s] Entry ﬂight-path angle 1.4263 [◦] Entry heading 70.437 [◦] TAEM height 24.8597 [km] TAEM longitude 53.736 [◦] − TAEM latitude 49.537 [◦] TAEM velocity 660.05 [m/s] Maximum heat ﬂux 530 [kW/m2] Maximum g-load 2.5 [ ] − Maximum q α 191, 397 [N ◦/m2] · Maximum α 43.9 [◦] Maximum α˙ 5 [◦/s] Maximum σ˙ 10 [◦/s]

37 Table 3.2: Range of Horus aerodynamic coefficients and flap deflections. [Mooij, 1995]

α M δe δb [◦] [-] [◦][◦] 0.0 1.2 -40.0 -20.0 5.0 1.5 -30.0 -10.0 10.0 2.0 -20.0 0.0 15.0 3.0 -10.0 10.0 20.0 5.0 0.0 20.0 25.0 10.0 10.0 30.0 30.0 20.0 20.0 35.0 30.0 40.0 40.0 45.0

3.11 and 3.12 show the height-velocity and longitude-latitude proﬁles. These can give an indication of what the trajectory looks like.

30 Angle-of-attack [deg] 25

15 0 200 400 600 800 1000 1200 Time [sec]

Figure 3.7: The reference angle-of-attack proﬁle.

3.5.2 X-33, X-38 and Orbiter missions This section presents reference missions for the Space Shuttle Orbiter, X-33 and X-38. The author has created a database for the basic drag and lift coefficient of the Orbiter from [Group, 1980]. In [Romere and Whitnah, 1983] it is shown that these coefficients match the actual flight data fairly well. For the X-33 and X-38, a database with lift and drag coefficients was made available to the author. For the X-33 an analytical and table format model was available. Both have been compared with and were found to match

38 3.5. Reference missions

0 Bank angle [deg] −20

−40

−60

−80 0 200 400 600 800 1000 1200 Time [sec]

Figure 3.8: The reference bank angle proﬁle.

−5

Body ﬂap−10 deﬂection [deg]

−15

−20

−25 0 200 400 600 800 1000 1200 Time [sec]

Figure 3.9: Body ﬂap deﬂection versus time.

39 2

−2

−4

Elevon [deg] −6

−8

−10

−12 0 500 1000 1500 Time [sec]

Figure 3.10: Elevon deﬂection versus time.

120

110

100

70 Height [km] 60

20 0 1 2 3 4 5 6 7 8 Velocity [km/s]

Figure 3.11: Height versus velocity.

40 3.5. Reference missions

−5

−10 Latitude [deg]

−15

−20

−25 −110 −100 −90 −80 −70 −60 −50 Longitude [deg]

Figure 3.12: Latitude versus longitude. with [Murphy et al., 2001]. The lift-to-drag ratios, as function of angle of attack and Mach number, for the Or- biter, X-33 (table format) and X-38 can be seen in figures 3.14 to 3.16, respectively. Overall, the X-38 has the lowest lift-to-drag, while the Horus has the highest. The Or- L biter’s D curve shows the same behaviour as Horus. It is, however, strange that for L M = 2, the D curve shows a different behaviour for larger angles of attack. The X-33 shows the same behaviour as Horus up to about α = 35◦. From there, an increase in L Mach does not mean a decrease in D . Probably a different method is used for supersonic L Mach numbers than for hypersonic Mach numbers. The differences in D behaviour for the Orbiter and X-33 w.r.t. Horus are practically not reached due to the α profiles flown. These profiles show a large angle of attack for high Mach numbers with a decrease at a much lower Mach number. The X-38 aerodynamics has been constructed with two different methods with the boundary at Mach 4. However, there is still not a clear general L trend visible in the D profiles. Both the X-33 and X-38 programs were canceled, however the vehicle configurations are used as test vehicles in studies for advanced guidance algorithms. Figure 3.13 shows an image of the X-33 vehicle.Table 3.3 presents the reference mission definition for the three vehicles. The vehicle and mission parameters for the Orbiter were found in [Harpold and Graves Jr, 1978] and [Glass, 2008]. The vehicle and trajectory parameters for the X-33 and X-38 were found in [Shen, 2002]. Constraint values for the X-33 were found in [Shen, 2002] and [Saraf et al., 2003]. The constraint values for the X-38 were found in [Shen, 2002] only. A reference angle-of-attack profile for the X-33 mission has been taken from [Saraf et al., 2003]. It is used in the development of the bank angle planning algorithm. The profile can be seen in figure 3.17.

41 Figure 3.13: An illustration of the X-33 . [Li, 1999]

2.5

2 Mach = 2 Mach = 3 Mach = 5.5 Mach = 10 Mach = 20 1.5 [-] L D

0.5

0 10 15 20 25 30 35 40 45 50 Angle of attack [deg]

L Figure 3.14: Orbiter D as a function of angle of attack and Mach number.

42 3.5. Reference missions

1.5

2.01 3.49 6 10 20 1 [-] L D

0.5

0 10 15 20 25 30 35 40 45 50 Angle of attack [deg]

L Figure 3.15: X-33 D as a function of angle of attack and Mach number.

1.5

Mach = 2 Mach = 3 Mach = 5.5 Mach = 10 Mach = 20

1 [-] L D

0.5

0 10 15 20 25 30 35 40 45 50 Angle of attack [deg]

L Figure 3.16: X-38 D as a function of angle of attack and Mach number.

43 Table 3.3: Reference mission deﬁnition for the Orbiter, X-33 and X-38.

Space Shuttle X-33 X-38 Mass 104.33 37.65 11.57 [tons] Reference surface 249.9 149.4 21.7 [m2] Nose radius 0.3 0.8 0.8 [m] Entry height 121.9 120.1 121.9 [km] Entry longitude 243 225.5 140.6 [◦] Entry latitude 18.3 23.75 26.97 [◦] − − Entry velocity 7622 7627 7467.8 [m/s] Entry ﬂight-path angle 1.438 1.249 1.026 [◦] − − − Entry heading 38.3 49.59 58.95 [◦] TAEM Height 24.4 29.4 24.4 [km] TAEM Velocity 762 918.8 737.5 [m/s] TAEM Longitude 279.5 240.6 [◦] − TAEM Latitude 28.7 33.2 [◦] − Maximum heat ﬂux 70 75 100 [BTU/(ft2 sec)] · Maximum normal load 2.5 3.0 2.5 [ ] − Maximum q α - 7000 - [psf deg] · · Maximum dynamic pressure 342 300 300 [psf] Maximum α 45 50 50 [◦] Maximum α˙ 5 5 10[◦/s] Maximum σ˙ 2 5 9[◦/s]

Angle-of-attack [deg] 30

20 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Energy [-]

Figure 3.17: Reference angle-of-attack proﬁle for the X-33 mission.

44 CHAPTER 4

Numerical Methods

As will be mentioned in chapter5, the atmosphere model is represented by a table. Hence, the atmospheric properties are given at discrete values. To obtain the atmospheric at intermediate values, as is required during a trajectory simulation, cubic spline interpolation is used. This allows the generation of a smooth trajectory during simulation. The cubic spline interpolation method is presented in section 4.1. The bank angle planning algorithm, as will be presented in chapter7, requires the set up of problems for which the root needs to be found. The secant method, as described in section 4.2, is a root ﬁnding method that can solve these problems. Numerical integration is used in the bank angle planning algorithm and simulator core. Section 4.3 presents a general Runge-Kutta method with variable step-size control for this purpose.

4.1 Cubic spline interpolation

Polynomials are commonly used in interpolation methods. The idea isthat one can connect two points by a straight line (polynomial of degree 1), three points by a quadratic (polynomial of degree 2) and connect M points by a polynomial of degree M 1. The − interpolation process itself consists of two steps:

1. Fit an interpolating function through the data points provided

2. Evaluate the function at the required points

This is not the most efficient method if one needs information only for a few values. The best way then is to do local interpolation. Local interpolation, however, gives interpolated values that do not have continuous first or higher order derivatives. This is exactly what is required for the atmosphere table to generate a smooth trajectory. To obtain continuous first or higher order derivatives one must use the ”stiffer” interpolation provided by the spline function. A spline is a polynomial between each pair of two points, its coefficients are calculated non-locally. This non-locality ensures the desired global smoothness in the interpolated function up to the required derivative. For cubic splines this is up to the second derivative. The order of the interpolation is the number of points used in the scheme minus one. The remainder of this section presents the cubic spline

45 interpolation method.

A tabulated function yi = y(xi) is given for i = 0,...N 1 points. Linear interpolation − for a particular interval xj to xj+1 gives [Press et al., 1988]:

y = Ayj + Byj+1 (4.1) where,

xj+1 x A = − (4.2) xj+1 xj −

x xj+1 B = 1 A = − (4.3) − xj+1 xj − This equation can be extended to obtain a cubic spline interpolation method. This equation is given by equation 4.4 below.

y = Ayj + Byj+1 + Cy¨j + Dy¨j+1 (4.4) Here,

1 3 2 C = (A A)(xj+1 xj) (4.5) 6 − −

1 3 2 D = (B B)(xj+1 xj) (4.6) 6 − − The second derivatives,y¨j and y¨j+1, can be obtained by [Press et al., 1988]:

xj xj−1 xj+1 xj−1 xj+1 xj yj+i yj yj yj−1 − y¨j−1 + − y¨j + − y¨j+1 = − + − (4.7) 6 3 6 xj+1 xj xj xj−1 − − Equation 4.7 gives N 2 equations for N unknowns y¨i in the interval i = 0,...N 1. To − − solve equation 4.4, the boundary conditions for y¨0 and y¨N−1 are needed. In this thesis work, they are set to zero. In this case, one can speak of the natural cubic spline. From the foregoing, the following tridiagonal matrix is derived:

     y2−y1 y1−y0  2(h0 + h1) h1 y¨ 1 h1 − h0  ..   y¨   .   h1 2(h1 + h2) .   2   .     .  = 6    .. .. · .     . . hn−3   .      yn−1−yn−2 yn−2−yn−3 y¨n−2 hn−3 2(hn−3 + hn−2) hn−2 − hn−3 (4.8)

Here, hj = xj+1 xj. The solution to equation 4.8 is obtained with an algorithm that − solves tridiagonal systems [Press et al., 1988]. To obtain the desired value for y at x, the two values for xj and xj+1 need to be retrieved from the table. This is achieved by employing a bisection search. The bisection search is chosen for its high convergence speed [Press et al., 1988]. In essence, this algorithm successively divides the table in two parts. Each time, the part in which the value of x lies is chosen for further division. It needs to be kept in mind that the independent variable should be monotonically increasing or decreasing in the table.

46 4.2. Secant method

4.2 Secant method

The Newton-Raphson method is commonly used for its high rate of convergence [Press et al., 1988]. A disadvantage is that it may not converge. The Newton-Raphson method requires the calculation of function’s derivative. However, for the problems defined in the bank planning algorithm, this is not possible. This is solved by using the secant method. The secant method applies a finite (backward) difference to approximate the function’s derivative. Equation 4.9 represents the secant method [Kaw and Kalu, 2010].

f(xi)(xi xi−1) xi+1 = xi − (4.9) − f(xi) f(xi−1) − Here, x is the independent variable and f is the function for which the root (a value of x) needs to be found. As can be seen from the equation, two initial guesses are required. If xi+1 xi is below a threshold value, the method has converged. The threshold is − problem speciﬁc. If the secant method shows a poor convergence and one knows that the solution lies within a certain range, the false-position method is a better choice. This method always tries to ﬁnd the root within this range. However, this method has a slower convergence.

4.3 Runge-Kutta methods with step-size control

Several Runge-Kutta (RK) methods exist that differ in the degree of accuracy. Typically, the accuracy is measured as an order of magnitude of the step-size. If the method has an accuracy of O(h5) the method is said to be a fifth order method (RK5). An accuracy of O(h7) implies that a seventh order method is used (RK7). If a method of order p is used, a p+1 -order method is typically used to calculate an estimate of the integration error of the p-order method. This is based on the assumption that the p+1 -order method is assumed to be the true solution for the p-order method. An RK5(6) method means that the order of the method is 5, and that an RK6 is used to calculate the integration error. This section shall introduce a set of expressions that that can be used for RK methods of different order with step-size control. This method has been derived from [Fehlberg, 1968] and [Melman, 2010]. This code was a very good starting point for general RK methods. The algorithm was, however, limited to a state vector of size six. In addition, when executed, the algorithm integrated from an initial to an end state instead of performing one step only. The general set is defined as:

f0 = f(x0, y0) k−1 X fk = f(x0 + αkh, y0 + h βkλfλ) λ=0 k−1 X p y = y0 + h ckfk + O(h ) 0 k−1 X p+1 yˆ = y0 + h cˆkfk + O(h ) (4.10) 0

47 Where α, β, c and cˆ are coeﬃcients speciﬁc to the integration, and: f : function to calculate the derivative x : independent variable y : state variable k : number of stages - 1

The number of stages that is required is speciﬁc to the order of the method. For example, an RK5(6) required eight stages and an RK7(8) method requires thirteen stages. The full expression for these methods are RK5(6)-8 and RK7(8)-13. The truncation error ∆ can be calculated by:

k−1 X ∆ = h (ˆc c )f (4.11) k − k k λ=0

In the following, the step-size control will be explained. It is taken from [Press et al., 1988]. First a scaling parameter s is introduced:

si = (tol )i + y¯i (tol )i (4.12) abs | | rel Here, i is the size of the state vector and y¯ indicates that y is a vector. In words, the scaling parameter is equal to the absolute tolerance tolabs plus state element i times the relative tolerance tolrel. Each state element has its own absolute and relative tolerance as set by the user. The scaled error is given by:

v u i−1 2 u1 X ∆ = t i (4.13) i s i=0 i

If is below 1, the step is accepted. Otherwise the step-size scaling parameter δ is calculated as:

1 1 p+1 δ = 0.9 (4.14)

Here, 0.9 is a safety factor. The new step-size hnew is calculated as hnew = hδ. Additional measures can be taken to limit the increase or decrease in step-size. The choice for the integrator’s order depends on the problem. A larger order typically implies more function evaluations as the number of stages is increases. However, larger step sizes are allowed for the same error tolerances. In this thesis work it is chosen to work with an RK56 for its eﬃcient behaviour for entry trajectory propagation [Mooij, 2010]. The coeﬃcients for this method are taken from [Fehlberg, 1968].

48 CHAPTER 5

Generic Flight Simulator Design

The Space Trajectory Analysis [ESA, 2010] tool is used as a basis for this thesis work. A short introduction of STA is given in section 5.1. The Re-Entry Module (REM) of STA is the module that is mostly used for the purposes of this thesis work. However, it has some shortcomings, as it was designed only for ballistic entry at Earth/Mars. These shortcomings and the required modifications are explained in section 5.2. A new software architecture for STA is proposed in section 5.3 as the current architecture has some shortcomings. The classes of this new architecture are presented in section 5.4. The design of the architecture and its implementation has been a joined effort with Mirjam Boere. The models used for the environment can be found in section 2.3. The translational flight dynamics are given in section 2.6.1. A general Runge-Kutta integration method with variable step size control is presented in section 4.3. Finally, section 5.5 describes the validation of the software for re-entry missions.

5.1 STA

STA is an open source tool for the simulation and optimisation of a wide range of space trajectories. For example, a ballistic re-entry can be simulated. It is an ESA initiative and is being developed by universities across Europe. In order to allow for a parallel development of the software, the tool has a modular structure. In other words, it contains several diﬀerent modules, where each module provides the functionality for one type of space trajectory. The modules are programmed using the C++ language with a Qt user interface [Qt, 2010].

Some modules are ready to use, while others need to be modified. Detailed information about these modules can be found in the relevant STA documents. For each module extension or development, the following documents need to be written: Requirement Specifications Document, Interface Control Document, Detailed Design Document, the Verification and Validation Plan and Test Report. The astrodynamics core [Ortega, 2009] contains functions and models for the core of the trajectory simulation, e.g., coordinate systems, propagators, gravity and atmospheric models. The Eigen library [Eigen, 2010] is used for mathematical functions, e.g., matrix operations. The re-entry module was used as a starting point for both thesis works.

49 5.2 Existing software and its shortcomings

The current version of the re-entry module [Volckaert, 2007] has four main functionalities, namely simulation, targeting, dispersion analysis and entry window analysis. The central body, initial time and coordinate system are inputs. The user also has the choice to select the initial point of the trajectory or the target as an input. The output can be divided into textual output and graphical output. However, it is only based on atmospheric bodies and at the time of writing only Earth and Mars are supported by STA. Entry at an atmospheric body consists of a ballistic atmospheric re-entry followed by a descent with a conventional parachute. This is, of course, very problem specific and the software is modified so that it can support different types of missions. The following key requirement was set for the software,

The software shall be able to simulate a variety of entry, descent and landing trajectories with respect to any body in the Solar system, with or without an atmosphere.

To satisfy this requirement the re-entry module has to be modiﬁed so that it can deal with all types of entry/descent trajectories. A new architecture was proposed and implemented for this purpose. The architecture design is presented in the next subsection. The software is then applied to the two thesis works:

1. Mission planning for glide entry on the Earth.

2. Optimization of lunar descent trajectories for a lunar base settlement [Boere, 2010].

These missions cannot be simulated with the current REM. Shortcomings have been identified and the following modifications are required: - Next to entry at Earth/Mars, the software must also be able to enter other atmospheric bodies and non-atmospheric bodies. A database with all Solar system bodies must be added, including all their properties. The software must be able to ’switch-off’ the atmosphere and aerodynamic expressions for non-atmospheric bodies.

- The modified software must be able to give the user the choice to define the environment, e.g., to add wind, solar radiation pressure, third body, gravity field models.)

- The REM is currently designed to enter with a ballistic trajectory followed by descent by parachute. The new software should be able to simulate also other types of trajectories, including glide trajectories and powered descent trajectories, where descent by parachute is only one of the options.

- A propulsion module should be added for powered descent. If thrusters are used, ∆V has to be added in a discrete manner.

- the aerodynamics routines should be grouped and expanded to support a lift force and control surfaces. In addition, a trim law has to be added.

- Optimization algorithms should be added to the software. The one-objective PSO and two-objective NSGA-II and DG-MOPSO algorithms should be implemented into the software. The algorithms are described in [Topputo, 2007].

50 5.3. Simulator architecture design

- A guidance class should be added to the software, where guidance algorithms for diﬀerent types of missions can be selected by the user.

- The current functions for numerical integration are not ﬂexible enough. The function interface is limited such that:

1. Integration only possible from initial to end value for the independent variable. 2. Freedom for user input in the function is limited to an array of doubles

New numerical integration methods should be added to the software. More specifically, Runge-Kutta methods of varying order.

- The secant method for ﬁnding roots, bisection method for table look-up and cubic spline interpolation for atmosphere tables should be added to the software.

5.3 Simulator architecture design

The environment acts on the vehicle through forces. In a flight simulator, this influence is modelled in the flight dynamics. The flight dynamics of the current REM version is based on the central gravity field model. This model can then be extended with several types of perturbations, e.g., solar pressure, atmospheric drag and magnetic forces. There are two main objectives against this approach:

1. A flight simulator should be able to mimic a real flight. Hence, the environment should be modelled as accurate as possible. The amount of accuracy is determined by the scope of the user’s research and the models available. From the point of accurate flight simulation, one should start from an extensive environment instead of the spherical gravity field.

2. The idea of a central gravity field with perturbations holds only for a satellite that orbits a body at low altitude. Here, the force caused by the central gravity field is the most dominant one. The definition of perturbations is very problem specific. For example, during an atmospheric entry, the aerodynamic force is the most dominant one and the J2-effect might be considered a perturbation, whereas for a satellite in LEO the atmosphere may be seen as a perturbation. Moreover, the aerodynamic force is used to control the vehicle’s trajectory and/or attitude. While the solar pressure may not be the most dominant force for a solar sail mission, it is a central force that is used to steer the vehicle. It can certainly not be regarded as a perturbation.

Instead of the approach used in the current version of the REM, it is proposed that the flight dynamics communicates with a user-defined environment. Forces are no longer assigned as perturbations beforehand, but follow from the environment and mission definition. The so-called third-body perturbation follows from the definition of more than one celestial body. The mission definition determines if forces other than following from the environment are present. For example, when and where a propulsion force is present, or when a magnetotorquer is switched on. It is further proposed to use a systems-based approach to the environment and vehicle definition. On a top level, the user shall define a mission. The user does this by defining the vehicle, its environment and input specific to the mission arc(s). A systems-based approach of the environment and vehicle can be seen in figures 5.1 and 5.2, respectively.

51 The subsystem user in figure 5.2, indicates that the user is given the opportunity to define a subsystem. Each system and subsystem shall correspond to a class in C++. These figures only show two system levels, while there can be many more levels. For example, the gravity field class can contain different gravity field models. Or, the atmosphere system could consist of a steady-state subsystem and a wind subsystem. This modular approach to the environment and vehicle definition gives a clear view of the software and makes it easy to extend them with more models.

Figure 5.1: Systems breakdown of the environment.

Figure 5.2: Systems breakdown of the vehicle.

52 5.4. Class deﬁnitions

A ﬂow diagram of how the trajectory propagation is executed can be seen in ﬁgure 5.3. Here,

A = Vehicle parameters (e.g. aerodynamic reference area). B = Parameters depending on the vehicle and environment (e.g. aerodynamic coefficient). C = Environment parameters (e.g. gravitational parameter). D = Mass properties, mass flux from the propulsion system, state velocity. E = Total inertial force and total moment around the body axes. F = Heat flux, other parameters that could be integrated.

Figure 5.3: Flow diagram of the trajectory propagation.

As can be seen from figure 5.3, the Trajectory parameters block receives input from both Vehicle and Environment. This block contains parameters that are a property of the trajectory. They are derived from both the vehicle and the environment. One could think of, for example, the density and the Mach number. In the block Forces/Moments, all the necessary frame transformations will be executed in order to give a resultant inertial force and total body moment as output. This is only the kinetics part, the block Derivatives receives the kinematics from Vehicle, which contains the state vector. The interface of the numerical integrator is designed such that it accepts a void pointer. This void pointer is subsequently transformed into a structure that contains objects of the Vehicle, Environment, Forces/Moments and Trajectory parameters classes. This flow diagram is encapsulated in the simulation manager. The simulation manager determines the step size, by taking the user-defined accuracy and GNC frequency into account. It performs, furthermore, event scheduling and takes other stop criteria into account.

5.4 Class deﬁnitions

The systems and subsystems presented in the previous section are converted to classes in C++. In addition, the classes Trajectory Parameters and Forces/Moments have been created. Besides these classes, a class Bodies is present. This class should be seen as a database consisting of important celestial bodies. This class, together with the gravity ﬁeld model, has been provided by [Melman, 2010]. Both classes have been slightly

53 modiﬁes for the purposes of the thesis works. This section shall discuss the classes on a top-level. Their main purposes and user interaction will be given. Subsections 5.4.1 to 5.4.3 present the Bodies, Trajectory Parameters and Forces/Moments respectively. In subsections 5.4.4 and 5.4.5, the class description of the subsystems of Vehicle and Environment can be found. In the current implementation, objects of the subsystems are data members of the higher level system class. For future work, a trade-oﬀ for using inheritance needs to be made.

5.4.1 Bodies

The user is able to define bodies as part of the environment definition. The bodies class should be seen as a database that contains body-specific information. Examples of these body specific parameters are the rotational velocity, mass, radius. This database also include a boolean for the presence of the atmosphere. For bodies without atmosphere, all atmospheric relations (aerodynamic force, Mach number) are not computed. It has as an input the STA ID, which is a code given to every body that has been added to the software. Currently, all Solar System planets and the Earth’s Moon are added to the database. Future work should allow the user able to select one body as the primary body, making the other selected bodies automatically third bodies. In addition, the user should be able to define a body by giving the body specific information as input.

5.4.2 Trajectory parameters

The class trajectory parameters keeps track of all state derived parameters, these are the parameters that depend on the vehicle and the environment. Examples of these parameters are height, Mach number, load factor, horizontal velocity and circular velocity. Also the state vector expressed in diﬀerent reference frames and coordinates belong to these parameters. These are not included in the state class since they are computed with information of the environment (e.g., the angular velocity of the body is needed to convert the state in I-frame to the state in the R-frame).

5.4.3 Forces and Moments

The forces and moments class collects data from the environment and vehicle class and calculates the forces and moments that act on the vehicle. The current implementation is such that in this class:

1. All forces are calculated or set to zero.

2. These forces are transformed to the inertial reference frame (currently, the thrust force from B- to I-frame, aerodynamic force from AA- to I-frame and the gravity force from R- to I-frame).

3. The resultant inertial force is computed by adding the separate inertial forces.

The resultant inertial force is used to calculate the inertial accelerations, which are used in the equations of motions. At this moment the gravity force, aerodynamic force and thrust force are used in these equations. During future work moments and other forces need to be included.

54 5.4. Class deﬁnitions

5.4.4 Vehicle The vehicle class deﬁnes the spacecraft. It contains objects of the following classes: propulsion, guidance, state, thermal protection system, parachute system, aerodynamics and geometry. These classes are further discussed hereafter.

State The class state converts the state given by the user to inertial Cartesian coordinates in the inertial reference frame. The state includes the position, velocity, heat load and mass. The user can deﬁne the position and velocity of the vehicle in three diﬀerent ways:

- Spherical coordinates in the rotating frame.

- Cartesian coordinates in the inertial frame.

- Kepler elements in the inertial frame.

Next to the initialisation of the state, also time is initialised. Time is added to this class as, according to the theory of spacetime, it is a parameter speciﬁc to the vehicle. The user is able to set the starting date (year, month, day, hour, minute, second) of the simulation. The integration time is another variable and is set to zero at the start of the simulation. By keeping track of the simulation time, the angle between both frames can be calculated, which is used for the transformations between both frames. For future work dynamic models of actuators can be added. In addition, the attitude and angular velocity should be added for studying the rotational dynamics. The user input needs to be transformed to quaternions expressing the attitude of the body frame w.r.t. the inertial frame. Furthermore, the user should be able to deﬁne the size and content of the state vector.

Propulsion The propulsion class initialises the propulsion system from an existing propulsion system or from user input (by specifying the Isp and Tmax). The user can request the Isp,Tmax and Vexh of the vehicle. Future work includes the addition of the RCS system and the location and attitude of the thrusters.

Guidance The function of the guidance class is to return the guidance parameters that are very problem dependent. The current implementation contains open loop guidance for a powered descent and open and closed-loop guidance for a lifting entry. The guidance parameters for these mission types are respectively:

- The roll, pitch, yaw angles and throttle setting.

- The aerodynamic angles, α, β and σ. β is usually commanded to zero.

The open-loop guidance parameters are loaded from a ﬁle and are listed in a table as function of a dependent variable, e.g., height, time or energy. Given the dependent variable, the bisection method is used to ﬁnd the entry in the table that is the closest entry below the dependent variable.

55 Geometry In this class, the vehicle’s geometry should be modelled. Currently, only the nose radius and a characteristic radius can be deﬁned. The aerodynamic reference lengths are placed in the Aerodynamics class.

Parachute This class contains the calculation of the drag coefficient of the parachute. In addition, the aerodynamic reference area and the deployment Mach number are defined. The user can select a file that contains the drag coefficient as a function of the Mach number. This class communicates with the aerodynamic class by giving the drag coefficient and reference area as output.

Thermal Protection System In this class, the heating models for the vehicle can be found. Its main purpose is to calculate the convective and radiative stagnation point heat flux. The class is initialised by the coefficients of the heating models. The user can set these coefficients. The stagnation point heat fluxes are used to get an estimate of the integrated heat load. In the future, more models can be added. For example, the cold- and hot-wall temperature models could be added. Furthermore, one could think of a model of an ablative heat shield.

Aerodynamics The purpose of this class is to calculate the aerodynamic coefficients. The current version is able to compute the drag, lift and pitch moment coefficients. These coefficients can be a function of the angle-of-attack, Mach number, body flap deflection and elevator trim deflections. The aerodynamic reference areas can be found here as well. In addition, the flap deflections required for trim (around the body Y-axis) can be calculated. The coefficient values are sent to Force/Moments in order to compute the aerodynamic forces and moments in their original reference frame. The user can supply files containing the aerodynamic coefficients, reference lengths and area. Linear interpolation is used for intermediate values. In the future, one could think of:

- Adding analytical models.

- Including more coeﬃcients (e.g., side force, yaw moment, derivative coeﬃcients).

- Making the coeﬃcients a function of more variables.

- Adding more control surfaces.

- Allowing the user to deﬁne coeﬃcients and control surfaces.

5.4.5 Environment In figure 5.1, a system breakdown of the environment is given. The idea is that the environment is a collection of celestial bodies. Each celestial body sytem is composed of subsystems. Besides the Shape class, they can influence the vehicle through a force. The gravitational forces exerted by other bodies than the main body is not implemented yet. The main body is determined by the sphere’s of influence of all bodies defined by

56 5.4. Class definitions the user. However, it is envisioned to solve this by creating an main body object of gravity field and an array of gravity field objects that represents all other defined bodies. This definition should then automatically change if the main body changes during the mission. The environment class contains objects of the Atmosphere, Gravity field, Shape, Mag- netic field and Radiation pressure classes. The purpose of these classes is presented below.

Gravity ﬁeld

The gravity field class models the gravity field of a body. The main functionality of this class is to return the gravitational acceleration (Cartesian components) in the R-frame. The input is the position in the R-frame (Cartesian components). The user is able to define the accuracy of the model and is given the following choices:

- Uniform gravity ﬁeld. The gravitational parameter is provided by the Bodies database.

- Uniform gravity ﬁeld + J2,J3 and J4. The gravity acceleration is computed with analytical formulas. The coeﬃcients are provided by the Bodies database.

- Spherical harmonics model. The user is able to select the order and degree of the model (order + degree < 80). This model is initialised by a file containing a list of harmonic coefficients. It is advised that standard files for the main solar system bodies are available. The user can also create his own.

Atmosphere

This class contains the atmospheric properties of a body. Its function is to return atmospheric properties for an altitude. These atmospheric properties are:

- Density

- Pressure

- Temperature

- Speed-of-sound

The user shall provide a ﬁle containing these parameters at certain altitude intervals. Using cubic spline interpolation, the parameters are found in between these intervals. Besides this approach, the analytical exponential atmosphere model is present in this class. The class also contains an object of the Wind class as data member. This class is presented below. For future work, this model can be extended with:

- Next to altitude, dependencies on time and latitude.

- Other atmospheric properties, e.g., molecular composition.

- Other methods of modelling, e.g., an analytical US76 Standard Atmosphere.

57 Wind This class is initialised by a ﬁle containing wind in three directions as a function of altitude. These wind directions are:

- West to East

- South to North

- Upwards

The function of the class is to return the wind in the V-frame for a given altitude. Cubic spline interpolation is used to obtain intermediate values. The wind model that is implemented is taken from [Justus and Johnson, 1999].

Shape In this class, the shape of the body is deﬁned. Currently, the user can only choose between a sphere and ellipsoid. The dimensions of the ellipsoid are given by the radius of the semi-major axis (or the equatorial radius) and the ﬂattening. Now, the class contains one function that calculates the height w.r.t. the surface. For an ellipsoid, this is done with an approximative method. In the future, this class can be extended with more complex shape modelling methods [Dirkx, 2010].

5.5 Validation

Once the simulator has been built, it needs to be tested to determine if it can be used for trajectory simulation. The testing is composed of two parts: verification and validation. Verification means that each algorithm or method is tested separately. The separate components still need to be linked to each other in order to create a flight simulator. Even though each method in the software is working, it does not mean that they are linked together correctly. The testing of the sum of the components is part of the validation process. The essence of verification and validation can be caught by formulating two questions:

- Veriﬁcation: Are you building the thing right?

- Validation: Are you building the right thing?

The methods in the software have been tested for simple test cases. Where possible, the results have been compared to analytical solutions. The results are not included in this thesis work. A few examples are:

- The output of the cubic spline interpolation method has been compared to the spline function in Matlab [MathWorks, 2010].

- The secant method has been tested by solving cos x x3 = 0. − - The numerical integration methods have been tested for two simple problems:

1. An accelerating car. 2. A Keplerian orbit.

58 5.5. Validation

Ideally, one would like to compare the simulator with actual flight data in the validation process. If the flight data can be reproduced to a sufficient level of accuracy, the software has been validated. Another solution is to compare the output of the simulator against one that has already been validated. This approach is taken with the software developed for this thesis work. The validated simulator that has been used is the Simulation Tool for Atmospheric Re-entry Trajectories (START) [Mooij, 1991]. The reference mission used for the validation is given in subsection 3.5.1. The reference angle-of-attack and bank angle are used during an open-loop simulation. The trim algorithm, as described in section 3.4, then calculates the control surface deflections. The behaviour of the trajectory generated by the developed simulator is the same as the reference mission from START, see section 3.5 for the reference trajectory. The differences are small till zero. The body flap and elevon deflections calculated for trim show the same behaviour as well. The result is, however, not an exact match. From the entry point errors start to develop. These errors between the simulated and reference trajectory could propagate into larger errors. The net effect could always be zero. The errors on the state variables and control surface deflections at the end time can be seen in table 5.1. The angle-of-attack and bank angle errors are zero throughout the simulation.

Table 5.1: State and control deﬂection errors (absolute) at the end time.

Parameter Error Height [m] 162 Longitude [deg] 0.072 Latitude [deg] 0.025 Velocity [m/s] 10.3 Flight-path angle [deg] 0.203 Heading angle [deg] 0.084 Body flap deflection [deg] 0.0 Elevon deflection [deg] 0.56

The errors between the simulated and reference mission can be caused by two diﬀerences between the built simulator and START. All other environment and vehicle parameters are kept the same as in START. These diﬀerences between the simulator and START software are:

1. The US76 Standard Atmosphere model used is diﬀerent. START uses an analytical model, while the simulator uses a table as input and employs a cubic spline interpolation for intermediate values. 2. START does not use extrapolation for the control surface deﬂections when the Mach number is above 20.

The errors at the start are caused by the use of a different model for the US 1976 Stan- dard Atmosphere. This would also reflect itself in different values for the aerodynamic coefficients for the body flap, as a significant part of the trajectory takes place above Mach 20. The body flap deflection error is zero in the start, because a threshold of N ◦ dynamic pressure is used. Below 100 m2 , the flap deflection is set to 15 . The Mach number and vertical lift coefficient for the simulated and reference trajectory can be seen in figures 5.4 and 5.5. From these figures, it can be seen that there is a clear offset between in the first 180 seconds. The offset in the vertical lift coefficient follows the Mach number as it

59 is directly dependent on it (the angle-of-attack error is zero). The aerodynamics are such that the value for the lift coefficient decreases for increasing Mach number. Thus, a positive difference in Mach number reflects itself in a negative difference in the vertical lift coefficient. In addition, a decreasing Mach error should show a decrease in the absolute error in the vertical lift coefficient. This is exactly what happens. Since the velocity error is zero (less than 0.02 m/s) in the beginning, it is concluded that the speed of sound is not. Since the height error stays below 2 m in the first 180 seconds, the error must be caused by the method of calculating the speed of sound. The errors caused by employing a different atmosphere model then propagate into larger errors, as different environment conditions hold for a different state. The difference in the vertical lift coefficient is more pronounced until about 250 seconds. This is caused by a higher body flap deflection angle, which in turn is caused by the extrapolation of the aerodynamics above Mach 20. The developed simulator is considered to be validated for trimmed re-entry simulations, as the induced errors are still small and the differences can be explained. The validation of the simulator for a powered descent can be found in [Boere, 2010].

Simulated 20 Reference ] − 15 Mach [

0 200 400 600 800 1000 1200 Time [sec]

Figure 5.4: Mach numbers from simulated and reference trajectory.

60 5.5. Validation

0.8

0.7

Simulated 0.6 Reference ] − [ 0.5 vertical

CL 0.4

0.3

0.2

0 200 400 600 800 1000 1200 Time [sec]

Figure 5.5: Vertical lift coeﬃcients from simulated and reference trajectory.

61 62 CHAPTER 6

Angle Of Attack Planning

For a 2D motion entry, the angle of attack is the only parameter that controls the aerodynamic force. Hence, it is the only control parameter that can be used to influence the vehicle’s motion. Assuming a 2D motion is, therefore, thought to be the starting point for the planning of an angle of attack profile. In planning the angle of attack profile, the path constraints should be taken into account. Therefore, an entry corridor is formed. This formation of this corridor is presented in section 6.1. In principle, an infinite number of profiles can be flown in the corridor. Therefore, section 6.2 presents design parameters that guide the choice for flying a certain profile. In the end, section 6.3 presents results from 2D simulations carried out to test the angle-of-attack profiles.

6.1 The 2D entry corridor

It was not found in literature how other guidance algorithms plan the angle of attack. The mission planning methods described in [Shen, 2002] and [Saraf et al., 2003] assume that the profile is given. Moreover, they do not suggest a source where the profile might come from. The method that will be described in this section uses a 2D entry corridor in the height-velocity space. It follows from the assumption of an equilibrium-glide trajectory that is described in subsection 2.6.2. The upper boundary of the corridor is formed by the equilibrium flight trajectory for the maximum CL. This section first introduces the formation of an entry corridor in subsection 6.1.1.

6.1.1 Corridor formation Each path constraint equation has a certain value for a velocity, height and possibly the angle of attack. For a certain velocity, one can ﬁnd a height such that the constraint equation evaluates to just the maximum constraint value. When the actual height is lower, the constraint is violated. Hence, the path constraints impose a minimum height for a certain velocity (and angle of attack depending on the constraint). The active path constraint is the one that gives the highest minimum height for a certain velocity (and angle of attack if required). A maximum height is imposed by the equilibrium-glide trajectory. For a velocity and angle of attack, the vehicle is not able to maintain its ﬂight-path angle when above the

63 equilibrium-glide height. As is shown below, Horus obtains the highest equilibrium-glide height when ﬂying at αmax. A corridor can be formed from the minimum and maximum height for a velocity and angle of attack. In the following, the exponential atmosphere model will be used to illustrate the method. Using this model, analytical expressions can be obtained to compute the entry corridor. The equilibrium-glide height is given by:

" # 1 mg 1 1 h = ln 1 2 (6.1) −β 2 CLSρ0 V − Vc

2 Here, Vc = gr is the local circular velocity. An extra convenience of assuming an exponential atmosphere model is that the lift coeﬃcient becomes independent of height as the temperature is constant. That is, for a given velocity and angle of attack. This does not hold for a full 3 degrees-of-freedom simulation of the translational motion. The minimum heights corresponding to the path constraints, as given in section 3.1.3, are:

Heat ﬂux

1 q˙2 h ln max 2 (2·c ) (6.2) ≥ −β ρ0C V 2

g-load

  1 nmax mg h ln  q ·  (6.3) ≥ −β 1 2 2 2 2 V S CL + CD

qdynα˙

! 1 (qdyn α)max h ln · 1 2 (6.4) ≥ −β 2 V αρ0

Dynamic pressure

! 1 qdyn h ln max 1 2 (6.5) ≥ −β 2 V ρ0

64 6.1. The 2D entry corridor

The derivation of equations 6.1 to 6.5 can be found in appendixB. The velocity range from entry down to TAEM velocity is used to compute a corresponding height for the equilibrium-glide trajectories and constraints. Thus the velocity is the independent variable. As can be seen from the equations above, the minimum height corresponding to the g-load and q α constraints depends on the angle of attack. This implies that the dyn · entry corridor varies per angle of attack. As can be seen from equation 6.1, for a certain velocity, the equilibrium flight height depends on the lift coefficient CL. The upper boundary of the entry corridor is formed by the equilibrium flight for the maximum lift coefficient, because a larger CL gives a higher equilibrium-glide height. For a Horus-type vehicle, the maximum lift coefficient is obtained at the maximum angle of attack. For now, it is assumed that one deals with a vehicle that shows this aerodynamic behaviour. Hence, the upper boundary is formed by the maximum angle of attack. The equilibrium-glide trajectory and constraints heights for the maximum angle of attack can be seen in figure 6.1. As can be seen from this figure, the equilibrium-glide flight lies entirely above the constraint heights. Also, the heat flux constraint is dominant in the initial part. The qdynα˙ constraint is not active. One could imagine that the lift coefficient in the equilibrium-glide flight is the vertical lift coefficient in a full 3 degrees- of-freedom simulation. The vertical lift coefficient, is computed as (CL)vertical = CL cos σ. By applying a bank angle, the equilibrium-glide height would move downwards, while the path constraint heights remain the same. When increasing the bank angle, a path constraint will be violated at some point. Therefore, the larger the distance between equilibrium-glide and path constraint heights, the larger the bank angle can be and a higher degree of manoeuvrability results. For Horus, the qdynα˙ includes the dynamic pressure constraint.

80 Equilibrium height Heat ﬂux 70 g-load q α dyn ·

50 Height [km]

20 1 2 3 4 5 6 7 V [km/s]

Figure 6.1: Equilibrium glide at αmax (= 43.9◦forHorus).

To see the effect of the angle of attack on the minimum and maximum height, constant angle of attack profiles are used. In figure 6.2, one can see the equilibrium heights for

65 a several constant angle of attack proﬁles. Figures 6.3 and 6.4 show, respectively, the corresponding g-load and qdynα˙ constraints. The general trend is, a higher α increases the equilibrium-glide, g-load and qdynα˙ heights. The heat ﬂux and the dynamic pressure constraints are independent of the angle of attack.

80 Heat ﬂux α = 10◦ α = 20◦ 70 α = 30◦ α = 40◦

50 Height [km]

20 1 2 3 4 5 6 7 V [km/s]

Figure 6.2: Equilibrium heights for a varying α.

Now, the problem is to find an equilibrium-glide trajectory that lies in between the maximum angle of attack trajectory and some lower trajectory that does not violate any of the constraints. In other words, it should lie in the entry corridor that is formed between the minimum and maximum height. In the initial part of the entry, the corridor between the heat flux and αmax is not very wide. In other words, the allowable angle of attack is fixed within a relatively small range. However, it can still be chosen freely within this range. At the end of the hypersonic entry, the lower boundary of the corridor is determined by the g-load or qdynα˙ constraint (depending on α and maximum constraint values). Compared to the initial part, the entry corridor is relatively wide here with a larger range in α that gives feasible trajectories. At the end, there is an interface with the TAEM phase. There are some requirements imposed on this interface, these are given in subssection 3.1.2. In [de Ridder, 2009], it was found that the third method gives a maximum range trajectory. It is, furthermore, mentioned that the second method is often used, as it is thought to give the maximum range resulting from analytical solutions, see [Vinh, 1981] and [Loh, 1968]. Three methods can be thought of that are based on the TAEM interface in order to fix the final angle of attack:

1. Determine α from equilibrium-glide with the TAEM interface height and velocity.

2. Use α that gives L/Dmax.

3. Fly at minimum drag.

66 6.1. The 2D entry corridor

80 Heat ﬂux α = 10◦ α = 20◦ 70 α = 30◦ α = 40◦

50 Height [km]

20 1 2 3 4 5 6 7 V [km/s]

Figure 6.3: G-load constraints for a varying α.

80 Heat ﬂux α = 10◦ α = 20◦ 70 α = 30◦ α = 40◦

50 Height [km]

20 1 2 3 4 5 6 7 V [km/s]

Figure 6.4: q α constraints for a varying α. dyn ·

67 For Horus, the first two methods give a feasible trajectory. The first method might suffers from a disadvantage. It gives no room for banking as the vertical lift coefficient should be exactly the same as the total lift coefficient. Otherwise, the desired final conditions at the TAEM interface are not be obtained. So, the first approach degrades the vehicle’s manoeuvrability. With regard to the seconds and third method, in [de Ridder, 2009] it is found that for ◦ Horus, the difference between α(L/D)max and αDmin is lower than 2 for several TAEM phase initial conditions. In addition, the increase in range is only 3%. In addition, it is also possible to quickly and smoothly switch from an α(L/D)max trajectory to an αDmin trajectory [de Ridder, 2010]. The third method requires a search for the minimum angle of attack possible while satisfying path constraints and is a bit more complicated than the second method. Considering the above, it is chosen to aim for an α(L/D)max as end condition for the hypersonic entry phase. Figure 6.5 shows the equilibrium-glide trajectory and corresponding constraint heigths if Horus would fly at maximum L/D. In this figure, it can be seen that in the initial part, the heat flux constraint is above the equilibrium-glide trajectory. Hence, if Horus flies with α(L/D)max in this part, the heat flux constraint is violated. The other path constraints are not active in the 2D motion where no banking is assumed.

80 Equilibrium height Heat ﬂux 70 g-load q α dyn ·

50 Height [km]

20 1 2 3 4 5 6 7 V [km/s]

Figure 6.5: Equilibrium glide at α(L/D)max.

In conclusion, the lower boundary of the entry corridor is formed by the highest altitude given by the active constraint and equilibrium flight altitude for α(L/D)max. The upper boundary is formed by the equilibrium flight for αmax. The entry corridor for the Horus reference mission can be seen in figure 6.6. This figure shows that until about 4.5 km/s, the lower boundary of the corridor is constrained by the heat flux. Hereafter, α can vary freely between αmax and α(L/D)max. The g-load and qdynα˙ constraints are not taken into account here, because they are not active as can be seen from figures 6.1 to 6.5.

68 6.1. The 2D entry corridor

αmax α(L/D)max 70 Heat ﬂux

50 Height [km]

20 1 2 3 4 5 6 7 V [km/s]

Figure 6.6: The entry corridor.

6.1.2 Other vehicles

In this section, the entry corridors for the Space Shuttle Orbiter, X-38 and X-33 reference missions are presented to come up with a general method for α-profile planning. The refernce mission parameters can be found in section 3.5. The same method as in the previous subsection can be used to create the entry corridors. The result can be seen in figures 6.7 to 6.9 for the Orbiter, X-38 and X-33 respectively. From figure 6.7, it can be concluded that the Shuttle’s 2D entry corridor shows the same behaviour with respect to the constraints as the corridor created for Horus. The dynamic pressure and g-load do not play a role, for each angle of attack in between αmax and α(L/D)max. The entry corridor for the X-38 is different than those for Horus and the Orbiter. Although the aerodynamics of the X-38 are poorer in terms of the CL coefficient, the heat flux constraint is not active in this entry corridor. That is, under the assumptions made for which equilibrium-glide holds. The better thermal protection system of the X-33 allows a higher stagnation point heat flux when compared to Horus and the Orbiter. Therefore, this constraint is not active. Another important difference that can be seen is an equilibrium flight at α(L/D)max that is, for a large part, lower than the dynamic pressure constraint. Hence, flying an equilibrium-glide at α(L/D)max gives a constraint violation. This is not the case when flying at αmax, but it occurs for angles of attack close to α(L/D)max. For a general method, it cannot be assumed that one can fly freely in α α αmax. The X-33 entry corridor is again different, here not one (L/D)max ≤ ≤ constraint is active for α α αmax. The X-33 aerodynamic properties are such (L/D)max ≤ ≤ that αmax does not always give the maximum lift coefficient. Since the maximum lift coefficient determines the upper boundary, it was chosen to use the angle of attack profile that gives the maximum lift coefficient, α(CL)max. This is the profile shown in figure 6.9. The narrowing of the corridor near the end is caused by this as well, because Below Mach 6, α(CL)max is closer to α(L/D)max. From the X-33 mission, it can be concluded that the

αmax α(L/D)max Heat ﬂux 70

Height [km] 50

1 2 3 4 5 6 7 V [km/s]

Figure 6.7: The Orbiter entry corridor.

70 αmax α(L/D)max 65 qdyn Heat ﬂux 60

45 Height [km]

1 2 3 4 5 6 7 V [km/s]

Figure 6.8: X-38 entry corridor.

70 6.1. The 2D entry corridor

80 αmax α(L/D)max 75 Heat ﬂux

Height [km] 55

1 2 3 4 5 6 7 V [km/s]

Figure 6.9: X-33 entry corridor.

method should include a search for α(CL)max instead of only using αmax.

6.1.3 Atmosphere model The method described in section 6.1 relies on the use of the exponential atmosphere model. Other atmosphere models like the U.S. Standard Atmosphere 1976 model or the NASA/MSFC Global Reference Atmospheric Model-1999 are more representative of the actual atmosphere. The exponential atmosphere is a highly simpliﬁed model that leads to errors in:

1. The aerodynamic force

2. The constraint evaluations

The aerodynamic force is different, because the force coefficient is different due to a non-constant temperature (and thus speed of sound) and differing density. The constraints evaluate to a different number as either the density appears in the equation and/or the aerodynamic coefficients. This section will extend the method described in section 6.1 for an exponential atmosphere model to an arbitrary atmosphere model. A US 1976 Standard atmosphere model given in table format will be used as example. The effect of using the exponential atmosphere is that the aerodynamic coefficients are independent of the height in the method presented before, since the velocity is the independent variable. As a result, the analytical equations 6.1 to 6.5 can be obtained, as the height appears only once. By discarding this model, the height appears in more than one place for the equilibrium-glide, g-load and qdynα constraint evaluation. Namely, in ρ(h), CL(α(M),M(V, h)), CD(α(M),M(V, h)), α(M), g(h) and r(h). The angle of attack is only a function of Mach for the α(L/D)max profile. The height is, however, still the only unknown variable when the constraints are set to their maximum allowable values.

71 Thus, an iterative method can used to solve this problem. The general approach is as follows:

1. For a velocity, compute an initial-guess height with the exponential atmosphere model.

2. Calculate a second initial-guess height, preferably close to the ﬁrst.

3. Employ a secant method to ﬁnd the height.

4. Repeat steps 1-3 to ﬁll the whole velocity range.

For the equilibrium-glide height, the following equation 6.6 is solved using the secant method. For the g-load and qdynα constraint heights, the secant method is used to solve equations 6.7 and 6.8, respectively.

2 1 2 V ρV CLS + m mg = 0 (6.6) 2 r −

1 2 q 2 2 2 ρV S CL + CD nmax = 0 (6.7) − mg

1 2 (q α)max ρV α = 0 (6.8) dyn − 2 The height corresponding to the heat flux and dynamic pressure constraints is found using cubic spline interpolation, as the height appears only once in those constraint equations. The influence of the US 1976 atmosphere on the equilibrium-glide height can be seen in figure 6.10. The equilibrium-glide height for the US 1976 model is from entry until 50 km above the exponential atmosphere model. This holds for both αmax and α . The influence on the heat flux, g-load and q α can be seen in figures 6.11 (L/D)max dyn · to 6.13 respectively. The heat flux height for the US 1976 model is above the exponential atmosphere model when the density is higher. Vice versa, it is below the exponential atmosphere model when the density is lower. Where the density differences are small, the differences between the models are small. That is why in figure 6.12, the g-load constraints for α(L/D)max are small. In general, it is mainly the density variation that causes the increase or decrease in height for the same velocity. This can be seen by comparing the the offset is heights with the offset in density between the atmosphere models, see appendixA. If the density in the US 1976 model is higher, then the height for the equilibrium flight and constraints has increased. If the density is lower, then the height has decreased. The effect of the temperature is more difficult to see as the density appears more explicitly in the equations. To see the effect of the temperature variation, one should compare the equilibrium-glide trajectories at a point where the densities are the same. Figure A.2 shows that this occurs at 34.2 and 49.5 km with a temperature difference of respectively 12 and 25 K. The largest difference in Mach number is 0.78 around Mach 9.5 (for α(L/D)max) at 49.5 km. For a Horus like vehicle, this gives a small difference in the aerodynamic coefficients. This does not mean that this effect may be ignored in, for example, simulations. The temperature differences are larger at places where the density difference is also larger.

72 6.1. The 2D entry corridor

80 αmax Exponential α Exponential 75 (L/D)max αmax US1976 70 α(L/D)max US1976

Height [km] 50

1 2 3 4 5 6 7 V [km/s]

Figure 6.10: Equilibrium glide trajectories.

Exponential 70 US1976

50 Height [km]

20 1 2 3 4 5 6 7 V [km/s]

Figure 6.11: Minimum heat ﬂux constraint altitude.

73 65

αmax Exponential 60 α(L/D)max Exponential

αmax US1976 55 α(L/D)max US1976

Height [km] 40

1 2 3 4 5 6 7 V [km/s]

Figure 6.12: Minimum g-load constraint altitude.

αmax Exponential

60 α(L/D)max Exponential

αmax US1976 55 α(L/D)max US1976

Height [km] 40

1 2 3 4 5 6 7 V [km/s]

Figure 6.13: Minimum q α constraint altitude. dyn ·

74 6.1. The 2D entry corridor

It is just more difficult to see the influence of the temperature. Besides, it is a dynamic effect as deviations early in the flight propagate to larger deviations. For Horus, the effect of a different atmosphere model on the entry corridor does not seem very big. Where the heat flux height has increased, so has the equilibrium- glide height with about the same height. However, the corridor shows a narrowing in the initial part of the entry. The minimum height from the heat flux constraint has increased more than the maximum height from equilibrium-glide at αmax). The narrowing gives a smaller allowable range for the angle of attack. The narrowing is caused by the larger density difference at the height of the heat flux. The manoeuvrability (using the bank angle) has not changed significantly as the g-load and qdynα constraint heights did not change a lot. On a side note, for the exponential atmosphere model only, one could directly map all the constraints in an α V space. The equilibrium-glide equation is rewritten for − the density. Then, this equation can be substituted for the density in each constraint equation. Given a velocity, the angle of attack can be extracted and a corridor can be created in the α V space. −

6.1.4 Trim

In the method presented before, an equilibrium-glide and constraint height is calculated from a given velocity and angle of attack (αmax and α(L/D)max). For the exponential atmosphere model, the Mach number is directly dependent on the velocity (isothermal atmosphere). Thus, given a velocity and angle of attack, the control surface deflections can be calculated for trim using the trim algorithm presented in section 3.4. The effect of trim reflects itself on the value of the lift coefficient. This, in turn, results in a different value for the equilibrium-glide height and g-load constraint height. Trim can also be included if the speed of sound is a function of height, as is the case for the US 1976 atmosphere model. The method is slightly more extensive than the In figure 6.14 and 6.15, the effect of trim on the equilibrium-glide and g-load constraint heights can be seen. From these figures, it can be seen that if trim is included, the height has increased. This is caused by a positive body flap deflection (for the largest part of the entry), which gives a positive increase in CL. The effect is less noticeable, though present, for αmax w.r.t. α(L/D)max. This is caused by:

1. The body flap deflections for trim at αmax are smaller than at α(L/D)max, see figure 3.6.

2. The absolute value of CL for αmax is larger than for α(L/D)max. Hence, for the same CL increment due to trim, CL for αmax shows a smaller percentage wise increase.

These two eﬀects amplify each other and as a result, the inﬂuence of trim is barely noticeable. It can be concluded for the Horus reference mission that, when including trim:

- The constraints evaluate to a smaller value for the same angle of attack.

- To reach the same equilibrium height, a smaller angle of attack suﬃces.

80 αmax α(L/D)max 75 αmax Trimmed α Trimmed 70 (L/D)max

Height [km] 50

1 2 3 4 5 6 7 V [km/s]

Figure 6.14: Equilibrium glide including and excluding trim.

αmax 60 α(L/D)max

αmax Trimmed

55 α(L/D)max Trimmed

Height [km] 40

1 2 3 4 5 6 7 V [km/s]

Figure 6.15: Minimum g-load constraint altitude including and excluding trim.

76 6.2. Design parameters

6.1.5 Parameter extraction

Just as an h V profile can be obtained from an α-profile, an α-profile can be obtained − from an h V profile. First, CL is computed from the equilibrium-glide equation. Second, − the Mach number is determined from velocity and height. Finally, the angle of attack is calculated by linearly interpolating the aerodynamic coefficient table. Besides the inverse angle of attack, an equilibrium flight-path angle can be extracted as well. From [Ambrosius and Wittenberg, 2006]:

dh dh dρ dV sin γ = = (6.9) ds dρ dV ds The ﬁrst ratio on the right hand side of the previous equation can be obtained by numerical diﬀerentiation for an atmosphere model in table format. For an exponential atmosphere model, dρ is equal to βh. The second ratio can be obtained from the dh − equilibrium-glide equation 2.50. The last ratio is derived in [Ambrosius and Wittenberg, 2006] as:

2 Dg 1 V dV Vc = − (6.10) ds LV In section 6.3, α(V ) and γ(V ) proﬁles are extracted from h V proﬁles using the methods − presented in this section.

6.2 Design parameters

By knowing that an angle of attack profile can be derived from an h V profile, the − problem now is to find an appropriate h V profile. As can be seen from the entry − corridors for Horus and the X-38, there are a lot of h V profiles that can be flown. So, − it becomes a matter of designing one that fits a certain purpose. The question is: what is the effect of flying one h V profile over another? One could think of three design − parameters for an α-profile:

1. Range modiﬁcation

2. Manoeuvrability

3. Constraint satisfaction

This section shall treat the theoretical reasoning behind the inﬂuence of the angle of attack proﬁle on the three design parameters. In section 6.3, this is tested using 2D simulations.

Range modification According to [Vinh, 1981], the maximum range is achieved when flying at the maximum L L D . The D for typical entry vehicles can be seen in figure 6.16 below. A figure presented in [Vinh, 1981] shows similar behaviour with angle of attack on the x-axis. The aerodynamic of Horus show the same behaviour, as can be seen in figure 3.4. L The figure shows that D behaves like an inverted parabola. Moving from α that gives L L the highest D to αmax generally implies a decreasing D and decreasing range. Conversely,

77 Figure 6.16: Typical lift-to-drag ratio’s for entry vehicles. [Regan and Anandakrishnan, 1993]

moving from αmax to α(L/D)max implies an increasing range.

Manoeuvrability The 2D approach as described before, takes a zero bank angle into account. However, in practice this is not common. The 2D approach is essentially nothing else than looking at the vertical motion only. One could still use this approach while banking, if the lift force in equation 2.50 is interpreted as the vertical lift force. Banking directs the lift vector out of the vertical plane, hence the vertical component of lift decreases. This causes the equilibrium flight trajectory to be lower, as CL in equation 6.1 has been “decreased”. The advantage of having a large bank angle is that a significant lateral distance can be achieved, hence a high degree of manoeuvrability. However, the vertical component of lift can be reduced in such a way that the trajectory starts to violate path constraints. Increasing α counteracts this effect. And therefore, a higher α is associated with a higher degree of manoeuvrability.

Constraint satisfaction The constraint satisfaction is split into three parts:

- Heat-ﬂux satisfaction

- Other constraints in the h V space − - Integrated heat load

It can be seen from both the Horus, Orbiter and X-38 entry corridor that the heat flux constraint plays a large role in the initial part of the entry. The h V profile should − be in between this constraint and αmax. A higher α implies a lower heat flux, as the equilibrium-glide height (and density) is lower for the same velocity. However, there are L side-effects like a lower D .

78 6.2. Design parameters

The mission configuration for the X-38 shows that the dynamic pressure constraint lies higher in the h V space than the equilibrium-glide path for α . It is important − (L/D)max to keep in mind that the same might happen for the other vehicles if different constraint values are used. Logically, the same holds for the g-load constraint. As the method described in this section could be used for preliminary studies of vehicle design, it should be flexible enough to cope with these cases. In any case, increasing α decreases constraint values. This holds true even for the q α constraint, it is compensated by the lower dyn · density. The integrated heat load is driven by two parameters: the average vehicle heat flux and the flight time. Looking at the entry corridor, the two parameters can theoretically be influenced. First, the average vehicle heat flux is, like the stagnation point heat flux, a function of the variables density and velocity. At a given velocity, the density is lower at increasing height. This implies that a higher α gives a lower heat flux. Flight time can be reduced by increasing energy dissipation, which is given by: dE = DV . For a dt − certain velocity and height, the drag is higher for a higher drag coefficient. A higher drag coefficient is obtained by a higher α. One could also deduce a lower flight time coming from a higher α if one looks at the lift-to-drag ratio. A higher lift-to-drag ratio entry can achieve a larger range, therefore the average energy dissipation needs to be lower. This reasoning has been based on the reduction of vehicle energy from entry to TAEM value. But, the same reasoning can be applied if one looks at the reduction of the vehicle’s velocity during entry. The EoM for velocity is guiding the reasoning then. In [Ambrosius and Wittenberg, 2006] an equation for the flight time is derived that holds for an equilibrium-glide trajectory. Assuming a constant lift-to-drag ratio:

 V Vf  V L 1 + i 1 c Vc − Vc F light time = ln  V V  (6.11) 2g D 1 i · 1 + f − Vc Vc

Here, Vi and Vf are the initial and final velocity respectively. Flying a constant lift-to- drag ratio during entry is unrealistic, one could however think of a Riemann summation of L short velocity intervals in which D is kept constant in each interval. The flight time should then be longer for a higher lift-to-drag ratio trajectory. From the foregoing reasoning it is concluded that a higher α should give a lower integrated heat load. It should be kept in mind, however, that the bank angle has a major influence as well. In conclusion, in the α range between αmax and α(L/D)max for profile design one should keep in mind that:

- Increasing α, decreases: heat ﬂux, integrated heat load, range

- Increasing α, increases: manoeuvrability w.r.t. constraints

The opposite holds true for a decreasing α. A trade-off is often necessary to come up with a profile that fits the mission requirements best. It should be mentioned that the three design parameters could be more sensitive in different parts of an entry. For example, one could imagine that the range modification is not that sensitive to α in the initial part of the entry. The lift and drag force are not that high there, because of the low density. On the other hand, differences induced early in trajectory propagate into larger deviations towards the end. Simulations need to be performed to determine this.

79 6.3 2D simulations

To test the theoretical reasoning presented in section 6.2 above, numerical simulations need to be carried out. For this purpose, different angle of attack profiles need to be tested. In subsection 6.3.1, a method is presented to come up with different angle of attack profiles. A profile can be defined based on the theoretical reasoning presented in the previous section. Subsection 6.3.2 presents simulations carried out with three test profiles. These profiles are designed using the method in subsection 6.3.1.

6.3.1 Profile selection To test the effect of different angle of attack profiles on the mission, one would like to test any allowable profile in the entry corridor. This is hard to implement when seen from the practical side. In this section, a method is presented that can be used to shape the equilibrium flight path. For this purpose, the entry is split into three parts:

1. Heating phase

2. Ranging phase

3. TAEM interface matching

The heating phase is the first part of the entry, where the heat flux is the most dominant constraint. This phase ends if another constraint or the equilibrium-glide trajectory for α(L/D)max is more dominant (the height is larger). This point is called the intersection velocity. In the first phase, it should be possible to choose an equilibrium height in between αmax and the heat-flux height. This is expressed with a percentage of the height range, such that 0% corresponds to the heat flux height and 100% corresponds to the αmax height. The second phase is the ranging phase. This phase is based on influencing the range by the lift-to-drag ratio. Again using a percentage, a trajectory is chosen. In this case, the percentage is expressed in the α range and not the height range. When the percentage is put on the height range, the extracted α-profile could show upward behaviour. This is due to the non-constant offset between the equilibrium height for αmax and α(L/D)max. If the trajectory computed in this phase already exceeds the trajectory in the first part, it is chosen to fly this trajectory there for more smoothness. In the final phase, the vehicle should aim at the TAEM interface angle of attack, i.e. α(L/D)max. At some point before the TAEM velocity the trajectory should switch to the equilibrium-glide for α(L/D)max. The lowest velocity at which one should switch to α(L/D)max is found using α˙ max. The necessary velocity interval is determined by:

dα dα dt = (6.12) dV dt dV

For which it is assumed that dV = D/m. This holds under the assumptions that were dt − made earlier and is essentially saying that D/m g sin γ D/m. Using this method, − − ≈ − it was found that dV is about 35 m/s for the Horus reference mission. So, the ﬁnal switching velocity is the TAEM velocity plus dV . In the total velocity range, 35 m/s is practically a discontinuous change. It also falls within the segmentation of the velocity range. Hence, the angle of attack is forced to be α(L/D)max at the TAEM interface velocity. The switching is not actively taken care oﬀ.

80 6.3. 2D simulations

Finally, there are two more parameters that allow more freedom in the shaping. They are both percentages on a velocity range that goes from the intersection velocity to TAEM velocity plus dV . The first parameter modifies the switching velocity between the heating and ranging phase, which is normally set to the intersection velocity. By modifying this parameter, one can determine when the transition takes place between the heating and ranging phase. The second parameter is the switch between the ranging phase and the TAEM interface. It determines at which velocity, the vehicle flies with α(L/D)max. This has the effect that the range is enlarged and the vehicle is already on the correct path for the TAEM interface.

6.3.2 Angle of attack profile testing In this subsection, simulations have been performed for three angle of attack profiles. These test profiles can be qualitatively described as:

Test proﬁle 1: High angle of attack proﬁle

Test proﬁle 2: Medium angle of attack proﬁle

Test proﬁle 3: Medium to low angle of attack proﬁle with large α(L/D)max part

The h V , α(V ) and γ(V ) profiles that are used in the simulations are given in figures 6.17 − to 6.19 respectively. The α(V ) and γ(V ) profiles follow directly from the h V profiles − and vice versa. It can be seen from figures 6.17 and 6.18 that a lower h V combination − gives a lower angle of attack. This is consistent with the reasoning that a larger angle of attack gives a higher lift coeffient, and thus a larger equilibrium flight height. Figures 6.17 and 6.19 show that a higher h V combination gives a lower flight-path angle profile. −

80 Corridor Test profile 1 Test profile 2 70 Test profile 3

50 Height [km]

20 1 2 3 4 5 6 7 Velocity [km/s]

Figure 6.17: h V proﬁles for the three test cases. −

Test profile 1 40 Test profile 2 Test profile 3

Angle-of-attack [deg] 25

15 1 2 3 4 5 6 7 Velocity [km/s]

Figure 6.18: Angle of attack proﬁles for the three test cases.

−1 Test profile 1 Test profile 2 Test profile 3 −2

−3

−4

Flight-path angle−5 [deg]

−6

−7

1 2 3 4 5 6 7 Velocity [km/s]

Figure 6.19: Flight-path angle proﬁles for the three test cases.

82 6.3. 2D simulations

For the simulations, the 2D set of EoM as given in subsection 2.6.2 has been used. When using the initial conditions from the Horus reference mission, one immediate problem arises. All trajectories start to express phugoid like behaviour. The trajectories are all entries with multiple skips inside the atmosphere. The mismatch between entry initial conditions and conditions from the h V profiles causes this. More specifically, at dγ − the first point where dt = 0 (here the skip motion is initiated), the flight-path angle is smaller (more negative) than the equilibrium flight-path angle. At that point, the velocity and height differences between simulation and equilibrium-glide is orders of magnitude smaller than the difference in flight-path angle. The equilibrium flight-path angle is equal to 0.1◦ while the actual flight path angle is 2.3◦. − − dγ The skip is in principle started from a positive dt . The dynamic equation might, therefore, suggest why a skip is actually induced. The three components of this equation are:

3 1 2 z }| { dγ z}|{ mV 2 cos γ mV = L mgz }|cos γ{ + (6.13) dt − r When plotting these three components, the second and third term do not show a big change in the absolute value and behaviour with time. The first term shows an increase dγ before dt = 0 and increases more dramatically after that with a maximum at the point dγ where the altitude first starts to increase. Actually, dt shows the same behaviour as this first term. The skip is therefore induced by a larger lift force than anticipated. The main drivers of the lift force are the height and velocity as the other parameters are kept constant. As mentioned before, these differ not that much for the 2D simulation and the equilibrium-glide flight. The skip is a dynamic effect, so again, the EoM might give the solution. When looking at the derivatives for height and velocity with time, it can be seen that the flight-path dr angle appears in both. It is thought that the mismatch in flight-path angle causes dt and dV dt to be different than from the equilibrium-glide flight and induces the skip motion. This can be tested by enforcing the flight-path angle from the equilibrium flight in order dγ to produce a glide trajectory from a numerical integration of the 2D system. Hence, dt is not evaluated. The result is an equilibrium-glide trajectory, the h V diagram for the − three test profiles can be seen in figure 6.20. The result on the downrange, heat flux, heat load and flight time can be seen in figures 6.21 to 6.23. The lower the angle of attack profile, the larger the flight time to come down to the final velocity. Figure 6.21 shows that the lower the angle of attack, the larger the range travelled. In figure 6.22, it can be seen that the higher angle of attack profile has the lowest heat flux. Also, the peak occurs earlier in the trajectory. Finally, figure 6.23 indicates that the largest heat load is obtained by the lowest angle of attack profile. This is due to both the larger flight time and larger average heat flux. These 2D simulations have confirmed the theoretical reasoning from before. However a flight-path angle profile was enforced to force an equilibrium-glide flight. The question that remains is to what extent is this possible. When thinking of the lift as the vertical lift, L cos σ, a quasi equilibrium-glide equation can be formed:

mV 2 L cos σ mg + (6.14) − r

83 120

110 Test profile 1 Test profile 2 100 Test profile 3

80 ] km 70

Height [ 60

20 0 1 2 3 4 5 6 7 8 Velocity [km/s]

Figure 6.20: h V diagram for three test cases with enforced α(V ) and γ(V ). −

4 x 10 3.5

Test profile 1 3 Test profile 2 Test profile 3

2.5 ]

km 2

1.5 Downrange [

0.5

0 0 10 20 30 40 50 60 70 80 90 Time [min]

Figure 6.21: Downrange versus time for the three test cases.

84 6.3. 2D simulations

400

350 Test profile 1 Test profile 2 Test profile 3 300 ]

2 250 kW/m 200

Heat ﬂux150 [

100

0 0 10 20 30 40 50 60 70 80 90 Time [min]

Figure 6.22: Heat ﬂux versus time for the three test cases.

800

700 Test profile 1 Test profile 2 Test profile 3 600 ]

2 500 MJ/m 400

Heat load300 [

200

100

0 0 10 20 30 40 50 60 70 80 90 Time [min]

Figure 6.23: Heat load versus time for the three test cases.

85 Given a vehicle state in a 2D simulation, a bank angle can be computed that is required for equilibrium-glide ﬂight:

 V 2  m g r σ = arccos  −  (6.15) L

Another solution could be to steer towards the flight path angle profile that corresponds to the angle of attack profile. The differential equation for the flight-path angle is approximated by:

dγ γ γref = − (6.16) dt ∆t

Here, γref is the reference ﬂight-path angle. This is inserted in the EoM for the ﬂight- path angle, as given by equation 2.49, where the vertical lift L cos σ is used instead of the total lift. The bank angle is then computed by equation 6.17 below.

 γ−γref V 2  m ∆t + g r σ = arccos  −  (6.17) L

These two bank angle guidance approaches have been tested using the 2D EoM set. The trajectory in the h V space can be seen in figure 6.24. The corresponding flight-path − angle and bank angle profiles can be seen in figures 6.25 and 6.26, respectively. As can be seen from the h V diagram, both approaches were not successful in following the − equilibrium-glide flight. Moreover, the heat flux constraint is violated. The flight path angle profiles show that for a large part of the entry, the quasi equilibrium-glide approach was able to reach an equilibrium-glide flight. This is where the flight-path angle stays constant. Only in the very beginning and at the very end, it is not possible to keep the flight-path angle constant. This can be seen from the bank angle profile as well, although it is hard to see in the beginning. This also explains why, at the end, the h V diagram shows a steeper descent. The point at the algorithm − becomes effective, is when the equilibrium-glide flight height is reached. Above this height equations 6.15 cannot be solved, because the part in the arccos is larger than one. In other words, a larger lift is required. The algorithm for steering towards γref does not work in the very beginning. The bank angle profile stays zero until the reference flight-path angle is reached. Before, the part in the arccos is larger than one. The small peaks correspond to breakpoints in the angle of attack profile. During the simulations, σ was set to zero if the parts in arccos were larger than one. Hence, a larger lift was not possible. Also, increasing the angle of attack to αmax does not solve the problem. The problem needs to be solved by looking 3 DoF simulations and a more sophisticated bank angle planning algorithm. The design of a more sophisticated bank angle planning algorithm is the subject of the next chapter. One advantage of the full 3 DoF set of the EoM is that, for the reference missions, the Coriolis force helps to rotate the flight-path angle before the equilibrium-glide height has been obtained.

86 6.3. 2D simulations

120

Enforced γ(V ) 110 Steered to equilibrium glide Steered to enforced γ(V ) 100 Heat ﬂux

70 Height [km]

1 2 3 4 5 6 7 Velocity [km/s]

Figure 6.24: h V diagram for the two bank angle guidance approached. −

−1

−2

−3 Enforced γ(V ) Steered to equilibrium glide Steered to enforced γ(V ) −4

−5 Flight-path angle [deg] −6

−7

−8

1 2 3 4 5 6 7 Velocity [km/s]

Figure 6.25: Heat load versus time for the three test cases.

Steered to equilibrium glide 70 Steered to enforced γ(V )

40 Bank angle [deg] 30

0 1 2 3 4 5 6 7 Velocity [km/s]

Figure 6.26: Heat load versus time for the three test cases.

88 CHAPTER 7

Bank Angle Planning

The bank angle is one of the three aerodynamic guidance variables. The bank angle can be used to avoid a skipping flight and steering towards the target. It can have a substan- tial influence on the trajectory if the vehicle has sufficient lifting capabilities. Because of its large influence on trajectory shaping, the profile throughout entry is planned. The planning approach presented in this section is based on [Mease et al., 2002] as this algorithm showed the best performance in [Hanson et al., 2002]. The algorithm generates a bank angle profile, including magnitude and sign, from an entry point up to the TAEM phase while satisfying path and control constraints. As mentioned in subsection 3.2.2, the method is based on planning a drag acceleration profile as a function energy. The advantage of planning a drag acceleration profile is threefold:

1. The distance travelled during entry is a function of the drag proﬁle as a function of energy.

2. A bank angle proﬁle can be obtained from a drag proﬁle.

3. The drag acceleration can be provided accurately by the inertial measurement unit during ﬂight.

The algorithm presented in [Mease et al., 2002] is an iterative process that is composed of four parts:

Part 1: Create a drag acceleration proﬁle based on an estimated trajectory length S.

Part 2: Extract trajectory parameters, a.o. the bank angle proﬁle.

Part 3: Find the bank reversal point (indicated by an energy level) that minimises the cross-range error.

Part 4: Update the trajectory length using the result of the previous step. If the update is below a threshold, stop. Otherwise repeat from step 1.

The initial estimate of the trajectory length is the great circle arc between the entry and TAEM interface point. The Shuttle uses this as the only estimate of the trajectory

89 North Pole

Bmission amission

cmission

Cmission TAEM

bmission Amission

Entry

Figure 7.1: Mission geometry. length. Figure 7.1 shows the geometry of the mission projected onto a sphere. One can imagine this sphere to be the Earth, however the radius of the sphere is not limited to the Earth’s (equatorial) radius. The great circle arc is represented by bmission in figure 7.1, bmission is also called the downrange. amission and cmission are, respectively, the co-latitudes of the TAEM interface and entry point. The difference in longitude between the TAEM interface and entry point is given by Bmission. Amission can be interpreted as the heading of the TAEM interface w.r.t. the entry point. Or, as the desired initial heading of the vehicle. It is desired since such an initial heading angle minimises the cross range that needs to be flown. For the Horus reference mission, bmission and Amission can be calculated using spherical trigonometry. More specifically, applying the Law of Cosines for Sides [Wertz et al., 2001]:

cos bmission = cos cmission cos amission + sin cmission sin amission cos Bmission (7.1)

cos bmission cos cmission cos Amission = cos amission (7.2) − sin bmission sin cmission The quadrants have been checked using the acos2 function given in [Wertz et al., 2001]. Its purpose is similar to the atan2 function, namely to solve quadrant ambiguities in inverse trigonometry functions. Using the previous two equations, it was calculated that for the Horus reference mission:

◦ bmission = 57.8 ◦ Amission = 68.6

For the first part of the algorithm, one needs to find a method to create a drag acceleration profile. In addition, this profile should fit a trajectory length. In [Mease et al., 2002], a three segment linear spline profile is constructed. Such a profile has the advantage that the trajectory length can be computed analytically from it. To find the equation that describes the spline, an entry corridor is formed in the drag-energy space. In section 7.1,

90 7.1. Drag-Energy entry corridor the formation of this corridor is presented. Hereafter, section 7.2 discusses the method on how to find the equation that describes the spline. Section 7.3 elaborates on part two of the algorithm. More specifically, it introduces the extraction of trajectory parameters once a drag profile is constructed. In section 7.4 the search for a bank reversal point is explained. This corresponds to part 3 of the algorithm. The final part of the algorithm’s method is presented in section 7.5. In the end, section 7.6 presents a discussion on the selection of the initial and final drag value.

7.1 Drag-Energy entry corridor

As mentioned before, the method is based on the planning of a drag acceleration profile. Setting up a drag-energy corridor is consistent with this approach. In subsection 7.1.1, the method for creating the corridor is presented. This method accepts an h(E) and α(E) profile as input. The influence of changing these input profiles on the corridor is explored in subsections 7.1.2 and 7.1.3, respectively.

7.1.1 Creating the corridor The upper boundary of the entry corridor is formed by the path constraints for the heat ﬂux, g-load, dynamic pressure and q α. The equations for the path constraint, as dyn · given in subsection 3.1.3, can be written as a limit on the drag acceleration D. The limits can be found in equations 7.3 to 7.6. The derivation of these equations can be found in section B.2 of appendixB. Throughout this chapter D refers to the drag acceleration. Although this might be confusing with the drag force, the two are essentially the same, as the mass stays constant throughout entry.

2 (q ˙c) C S D < max D (7.3) 2mC2V 2(c2−1)

(qdyn α)maxCDS D < · (7.4) mα

nmaxg D < (7.5) CL cos α + sin α CD

qmaxC S D < D (7.6) m In addition to the path constraints, which lead to a restriction on the maximum drag acceleration, one could think of a constraint on the minimum drag acceleration level. The constraint that is introduced below, is actually a constraint on the minimum lift and is given by the zero-bank equilibrium glide condition. If the lift is lower than this value, the dγ vehicle cannot maintain its flight-path angle, as dt cannot be sufficiently influenced by the vehicle. In other words, this limited manoeuvrability means that changing the angle of attack and/or the bank angle, does not lead to a sufficiently desirable reaction on the vehicle’s trajectory. One could argue that this constraint is not a direct threat to the vehicle and/or its payload. Violating this constraint does not cause a structural failure,

91 however, because of the low manoeuvrability it becomes diﬃcult to steer the vehicle to a safe landing site. Hence, mission failure is more likely to occur, because the vehicle should target a landing site. The constraint is given by:

V 2 C D > (g ) D (7.7) − r CL V 2 Here, (g ) is the minumum lift Lmin. Looking at the previous equation, for most − r missions the constraint is conservative as the Coriolis force is not taken into account. Hence, the actual minimum drag is a little lower. The constraints become dependent on only one variable if an α(E) and h(E) reference profile is available. Or, if and an α(E) and V (E) profile is available. This variable is the vehicle’s energy per unit mass. In [Saraf et al., 2003], α(E) and h(E) reference profiles are given as input. The velocity is then obtained from a height and energy value through:

1 µ µ E = V 2 (7.8) 2 − r − Rcb

Here, Rcb is the radius of the planet, which is assumed to be constant. The previous equation shows that the potential energy is taken w.r.t. the planet’s surface. The gravitational acceleration is assumed to be coming from a central gravity field and therefore g = g(h) = g(h(E)) = g(E). For the aerodynamic coefficients, it is assumed that (using the drag coefficient as an example):

CD = CD(α, M) = CD(α, M(V, h)) = CD(α(E),M(V (E), h(E))) = CD(E) (7.9) The energy at entry can be calculated from the initial conditions. Similarly, the energy at the TAEM interface can be obtained from the TAEM interface height and velocity. An entry corridor can be formed by evaluating the constraint equations 7.3 to 7.7 for the energy range from entry until TAEM interface. Figure 7.4 shows an entry corridor generated for the Horus reference mission. The α(E) and h(E) profiles are taken from test profile 1 from the angle of attack planning, the profiles can be seen in figures 7.2 and 7.3. The energy has been normalised through a division by the entry energy. The corridor is formed by the minimum drag and active maximum drag constraint. The active maximum drag constraint is the one that has the lowest drag for an energy. As can be seen from figure 7.4, the heat flux is the most dominant constraint until the energy has been halved. In addition, for this α(E) and h(E) combination, the g-load constraint does not play a role. Also, the corridor is open, i.e. for each energy there is a range in drag acceleration for which no constraint is violated. Lastly, when increasing (q ˙c)max, (qdynα˙ )max and nmax the upper boundary of the corridor moves upwards. Conversely, the upper boundary moves downward when they are decreased. Its role in the planning is twofold. First, the numeric values and shape of the entry corridor play a role in determining the breakpoints of the three segment linear spline. Second, the planned drag acceleration profile cannot exceed the upper boundary of the corridor. It is, therefore, forced to lay inside or at the boundary of the corridor. Hence, it is important to see the influence of the h(E) and α(E) profiles on the shape and numeric values of the entry corridor. The constraints have a fixed value prescribed by the vehicle design and presence of a crew. Given an α(E) and h(E), the drag acceleration corresponding to the constraints can be computed. To test the influence of one on the constraint corridor, the other is kept constant.

92 7.1. Drag-Energy entry corridor

30 Angle of attack [deg] 25

15 0 5 10 15 20 25 30 Energy [MJ/kg]

Figure 7.2: Angle of attack test proﬁle 1 as a function of energy.

50 Height [km]

20 0 5 10 15 20 25 30 Energy [MJ/kg]

Figure 7.3: Height from test proﬁle 1 as a function of energy.

93 20

14 ] 2

m/s 12 Lmin (q α) dyn · max 10 q˙max nmax 8 Drag acceleration [ 6

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Normalised energy [-]

Figure 7.4: entry corridor in the drag acceleration vs. energy plane.

7.1.2 Height reference profile As mentioned in subsection 3.2.1, the Orbiter’s trajectory planning uses a constant radius profile. The equilibrium glide height profile was used to generate figure 7.4, as this profile should better reflect the flown height than a constant height profile. The height profile shows its influence on the corridor, formed by equations 7.3 to 7.7, in three places:

1. The velocity, as it is computed from energy and height.

2. The speed of sound, because it is determined by height.

3. The gravitational acceleration, as it is determined by height.

One could think of other height proﬁles than the two that were just mentioned. In [Saraf et al., 2003], it is not explicitly mentioned what h(E) is given as input. Therefore, three options are considered in this subsection:

- Calculate an equilibrium glide height proﬁle consistent with the angle of attack proﬁle.

- Use a constant radius proﬁle.

- Use a linearly decreasing proﬁle (from entry to TAEM as a function of energy).

The advantage of the equilibrium glide profile is that it reflects the height envelope throughout entry better than a constant radius profile. The linearly decreasing profile should also provide more information on the height profile than the constant radius profile. When compared to the equilibrium glide profile, it can be said that it considers

94 7.1. Drag-Energy entry corridor the height above 85 km as well. Although probably not correct, the linear profile is added due to its simplicity. Figures 7.5 to 7.7 show the path constraints for the three separate cases. The constant radius profile has been taken at the TAEM interface height. The angle of attack profile is the same one as used in figure 7.4.

14 ]

2 Equilibrium glide Constant radius m/s 13 Linear

12 Drag acceleration [ 11

9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Normalised energy [-]

Figure 7.5: q α constraint for different h(E) profiles. dyn · The influence of the h(E) profiles on the minimum drag constraint is negligible. The other constraints show a more significant variation in the corridor region. The g-load is not an active constraint throughout the entry corridor. It should be said that for a different α(E) profile, this constraint can be active. The variation on the upper boundary of the entry corridor, caused by the h(E) profiles, is not very big. It is up to about 0.8 m/s2 for the heat flux constraint at a normalised energy of 0.5. In part 2 of the planning method, the radius (corresponds to height) profile is updated by the planned drag profile. More on this will be said in section 7.3. The three h(E) profiles given as input have been found to converge to the same radius profile. This updated profile is then used in the new iteration step. Hence, it does not matter for these three cases which h(E) profile is taken.

95 20 Equilibrium glide Constant radius 18 Linear

14 ] 2

m/s 12

8 Drag acceleration [ 6

0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Normalised energy [-]

Figure 7.6: q˙ constraint for diﬀerent h(E) proﬁles.

] 16 2 Equilibrium glide m/s 15 Constant radius Linear

Drag acceleration11 normal load [

9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Normalised energy [-]

Figure 7.7: Normal load constraint for diﬀerent h(E) proﬁles.

96 7.1. Drag-Energy entry corridor

7.1.3 Angle of attack reference profile The angle of attack shows its influence implicitly in the constraint equations through the drag and lift coefficient. In the q α constraint it appears explicitly. In order to get an dyn · idea of the influence of the angle of attack on the constraints profiles, the three profiles from subsection 6.3 are used. See figure 6.18 for these profiles. The radius profile has been set to a linearly decreasing profile. In that way, the only parameter that is varied is the angle of attack. Figure 7.8 to 7.11 show the influence of the three α(E) profiles on the constraint values.

8 Test profile 1 Test profile 2 7 Test profile 3

6 ] 2 5 m/s

3 Drag acceleration [

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Normalised energy [-]

Figure 7.8: Minimum drag constraint for diﬀerent α(E) proﬁles.

As can be seen from figure 7.8 to 7.11, the angle of attack profile has a major influence on the constraint profiles. The variation due to angle of attack is much more pronounced than the variation due to height. Figure 7.9 shows that decreasing the angle of attack, actually lowers this constraint in de D E space. It might have been expected that the − constraint would evaluate to a larger number if α is decreased as it appears explicitly in the numerator of equation 7.4. However, the decrease in α is compensated for and is surpassed by the increase in the drag coefficient. The general trend is that lowering the angle of attack, causes a lowering and narrowing of the entry corridor.

97 16 Test profile 1 Test profile 2 15 Test profile 3

14 ]

2 13 m/s 12

Drag acceleration10 [

7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Normalised energy [-]

Figure 7.9: q α constraint for diﬀerent α(E) proﬁles. dyn ·

20 Test profile 1 Test profile 2 18 Test profile 3

14 ] 2

m/s 12

8 Drag acceleration [ 6

0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Normalised energy [-]

Figure 7.10: q˙ constraint for diﬀerent α(E) proﬁles.

98 7.2. Drag acceleration proﬁle planning

17 Test profile 1 Test profile 2 16 Test profile 3 ] 2 15 m/s

11 Drag acceleration normal load [

9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Normalised energy [-]

Figure 7.11: Normal load constraint for diﬀerent α(E) proﬁles.

7.2 Drag acceleration proﬁle planning

The drag acceleration profile should lie inside the entry corridor. As mentioned before, the drag acceleration profile determines the range that is travelled. The idea is to find a drag acceleration profile in the entry corridor that fits a desired trajectory length. By planning a profile that is a linear spline, the trajectory length is an analytical function of the drag acceleration profile. By taking three segments for the spline and making the second segment constant, a drag acceleration profile can be found that suits the desired trajectory length. In [Saraf et al., 2003] it is not further mentioned how this is done. The approach taken in this thesis work is shown in the current section. First, the equations need to be found that describe the three segment linear spline. Figure 7.12 presents an illustration of the spline. The equations for the spline can be derived from this figure.

99 Figure 7.12: The linear spline segments in the D-E space.

In ﬁgure 7.12:

D1(E) - Drag segment 1 D2(E) - Drag segment 2 D3(E) - Drag segment 3 Di - Initial drag Df - Final drag Dc - Constant drag Ei - Initial energy Ef - Final energy E1 - Breakpoint 1 E2 - Breakpoint 2

The trajectory length can be calculated by:

Z tf Z Ef dE S = V dt = (7.10) 0 − Ei D(E)

Integrating the previous equation, by substituting the formulae for the drag segments gives:

E1 Ei Dc E1 E2 Ef E2 Df S = − ln + − + − ln (7.11) Di Dc Di Dc Dc D Dc − − f The derivation of this equation can be found in appendixC. As can be seen from equation 7.11, the trajectory length is determined by five parameters, E1, E2, Di, Df and Dc. According to [Mease et al., 2002], the initial and final drag values are fixed to desired values. It is not mentioned what desired implies. In [Leavitt and Mease, 2007], it is mentioned that the energy breakpoints are chosen such that the profile matches the shape of the entry corridor. It is not mentioned if this is selected manually/automatically or how parameters should be biased to match this shape. The advantage of choosing E1, E2, Di and Df , independently of the trajectory length, is that in equation 7.11 only one unknown remains. This unknown is Dc as S is considered to be the estimated trajectory

100 7.2. Drag acceleration proﬁle planning

length. Since Dc appears in multiple places in equation 7.11, it is found using the secant method. More speciﬁcally, the root is found for:

E1 Ei Dc E1 E2 Ef E2 Df f(Dc) = S − ln + − + − ln = 0 (7.12) − Di Dc Di Dc Dc D Dc − − f The approach taken in this thesis work is to set E1 to the point at which the heat flux and qdynα˙ constraints intersect. The energy at which the maximum of the Lmin constraint occurs determines the value of E2. There can be more maxima for different α(E) profiles and the highest is selected. The advantage of this approach is that both E1 and E2 are chosen automatically, and not manually by a user. The desired initial and final drag values is interpreted in two ways:

- Use the α-proﬁle, initial and TAEM conditions.

- Fix initial drag and ﬁnal drag to values such that the D(E) proﬁle lies in the entry corridor.

A figure in [Mease et al., 2002] suggests that the initial drag is set to the minimum drag constraint value at entry. However, in [Leavitt and Mease, 2007] it is suggested to use the initial drag value as specified by the entry conditions. The drag is then completely specified using the entry height and velocity. On the other hand, a figure in that paper also hints at setting the initial drag in the middle of the corridor at the entry energy. These three options are presented below, where the final drag value has been set using the final conditions. The α(E) and h(E) profiles have been taken from case 1 in subsection 6.3.

25 Corridor Di at Lmin

Di at half corridor

Di using initial conditions 20 ] 2 m/s 15

10 Drag acceleration [

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Normalised energy [-]

Figure 7.13: Found drag proﬁle for a diﬀering Di.

Figure 7.13 clearly shows that the choice for the initial drag value has a signiﬁcant impact on the drag proﬁle. This is due to the high energy at entry compared to the

101 TAEM interface energy as high energy together with low drag gives a large distance travelled. Therefore, the drag needs to be higher throughout the entry corridor in order to obtain the same total distance travelled. It is chosen to ignore the case where Di is at the half of the entry corridor. Not, because it violates the minimum drag constraint, but finding the profile is found to be quite sensitive to the initial drag value and putting Di at half the entry corridor is less representative of the real situation than when put at Lmin. When the drag value as given by the initial conditions is chosen, all of the maximum drag constraints are grossly violated. This should not happen and it is chosen to set the drag profile equal to the active constraint where it is violated. It should still be determined if Di should be put to Lmin or the actual drag acceleration. In section 7.6, this issue is further treated. A drag acceleration profile created in this way is probably the easiest profile that one could think of. It is also the most inflexible one. It does not allow for any biasing towards or away from a constraint. It could, for example, be desirable to bias towards the maximum drag boundary in order to reduce the integrated heat load. Also, the risk exists that the iteration does not converge to a Dc that gives a feasible profile. Sharp edges can lead to tracking problems as is shown in chapter8 and mentioned in [Leavitt and Mease, 2007]. Finally, it is not possible to reach all the points in the entry corridor as the corners of the entry corridor cannot be reached. Consequently, it is not possible to reach all points in the landing footprint. This includes the optimal points such as maximum range or maximum cross-range. Finding a smooth profile quickly that allows for biasing is the key to solving these problems. This, however, is left for future work. On the other hand, the elegance of this representation is that an analytical solution for the trajectory length as a function of the profile exists that allows for a quick generation of the drag profile.

- Create an entry corridor in the drag acceleration versus energy plane using the path constraints.

- Assume a drag proﬁle consisting of 3 linear segments (the second is constant in drag).

- Fix the breakpoints of this segment by matching the entry corridor shape.

- Find the proﬁle that integrates to the trajectory length by using a secant method.

7.3 Trajectory parameters extraction

Once a drag proﬁle has been found, three trajectory parameter proﬁles can be found:

1. Radius

2. Flight-path angle

3. Bank angle magnitude (excluding Coriolis)

The radius can be extracted from the drag acceleration proﬁle itself. For an energy and corresponding drag acceleration value, the only unknown parameter in the drag force equation is the radius. As the radius appears in more than one place (non-analytical and

102 7.3. Trajectory parameters extraction implicit), the secant method is again used to extract its value. The following function has been set up for which the root needs to be found at each energy:

1 2 f(E) = D (E) ρ(h)V (E, h)CD(E, h)S = 0 (7.13) profile − 2 The first derivative of drag with respect to energy can be used to extract the flight-path angle profile. It is given in equation 7.14 below, where ()0 means a differentiation w.r.t 0 energy. Using the radius profile and energy, ρ, V , g and CD are found. D and D are 0 ∂ρ given by the drag acceleration profile as a function of energy only. CD and ∂r are found using a numerical differentiation. The only remaining unknown is the flight-path angle γ.

0 0 2 CD 1 ∂ρ 2g D = D 2 + + sin γ + 2 (7.14) V CD −ρ ∂r V 1 ∂ρ Note that for an exponential atmosphere, the inverse of ρ ∂r is the density scale height. −−βh Since, using the exponential atmosphere model ρ = ρ0e :

∂ρ ∂ρ −βh = = βρ0e = βρ (7.15) ∂r ∂h − − then,

1 ∂ρ 1 = ( βρ) = β (7.16) −ρ ∂r −ρ − The second derivative of drag w.r.t. energy gives the bank angle magnitude proﬁle. Equation 7.18 presents D00. The derivation of the ﬁrst and second derivative can be found in appendixD.

00 02 ! 0 00 CD CD 0 CD 2 4D D = D 2 + D + 2 4 + CD − CD CD V − V 1 1 ∂ρ 2g V 2 1 ∂ρ 2g + + g + + Cγ (7.17) DV 2 −ρ ∂r V 2 − r −ρ ∂r V 2 − 1 1 ∂ρ 2g L + cos σ −V 2 −ρ ∂r V 2 D where, Cγ is the Coriolis force term in the EoM w.r.t. energy for the ﬂight-path angle. It is given by:

cos δ sin χ Cγ = 2ω (7.18) − cb DV The short notation for D00 is D00 = a + b(L/D) cos σ, where

00 02 ! 0 CD CD 0 CD 2 4D a = D 2 + D + 2 4 + CD − CD CD V − V 1 1 ∂ρ 2g V 2 1 ∂ρ 2g + + g + + Cγ (7.19) DV 2 −ρ ∂r V 2 − r −ρ ∂r V 2

103 1 1 ∂ρ 2g b = + (7.20) −V 2 −ρ ∂r V 2

00 00 For the linear spline segments, D is equal to zero. Assuming CD = 0 as is done in [Mease et al., 2002], the only unknown in equation 7.18 is the bank angle σ. In the next section it becomes clear why δ and χ are known as well. The height, flight-path angle and bank angle magnitude profiles corresponding to the solid line profile in figure 7.13 can be seen in figure 7.14 to 7.16 below. The solid line represents the case where Di was set to Lmin. In generating the bank angle profile, it had to be assumed that Cγ = 0, since its evaluation requires information on the latitude and heading envelope. Even though this neglects a dynamic effect, it can give an indication of what the bank angle profile looks like. This assumption is no longer made in the remainder of this thesis.

Height [km] 50

20 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Normalised energy [-]

Figure 7.14: h(E) proﬁle for Di starting at Lmin.

104 7.3. Trajectory parameters extraction

−2

−4

−6

Flight-path angle [deg] −8

−10

−12 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Normalised energy [-]

Figure 7.15: γ(E) proﬁle for Di starting at Lmin.

35 Bank angle [deg]

15 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Normalised energy [-]

Figure 7.16: σ(E) proﬁle for Di starting at Lmin.

105 7.4 Bank reversal search

The bank angle proﬁle (magnitude and sign) is found by minimising a target error and hereby integrating a reduced-order system. This system is given in subsection 2.6.2 by equations 2.42 to 2.44. It is repeated below for convenience.

dτ sin χ cos γ = (2.42) dE − r cos δD

dδ cos χ cos γ = (2.43) dE − rD

dχ L sin σ cos γ tan δ sin χ = + Cχ (2.44) dE −DV 2 cos γ − rD

The parameters r,γ and D are found using the methods described in the previous two subsections. The velocity follows from r and E. The lift acceleration is computed by multiplying the drag acceleration by the lift-to-drag ratio, which in turn is computed from . Actually, instead of computing the bank angle from equation 7.18 it is chosen to L directly substitute D sin σ in equation 2.44. This results in a quicker evaluation equation 2.44. It is calculated by:

s L L 2 L 2 sin σ = cos σ (7.21) D D − D

In [Mease et al., 2002] the following method is described. By assuming a non-rotating planet, the planet is rotated such that the entry and TAEM interface point lie on the equator. The EoM are now evaluated with respect to this new reference frame. In this new reference frame, the latitude is an indication of the cross-range. As the planned trajectory should end at the TAEM interface, it should end at the equator with zero latitude. Therefore, the latitute can be interpreted as the cross-range error. For convenience, the initial longitude can be set to zero such that the final longitude is interpreted as the downrange. The downrange is the angular distance travelled along the surface between the entry and TAEM interface point. The difference between the angular distance between entry and TAEM and the integrated longitude is interpreted as the downrange error. When a bank reversal is placed at a random energy, the final latitude corresponds to the target error that needs to be minimised. The minimisation of this error is performed by employing a secant method that finds the energy at which the final latitude is within a user defined tolerance. Due to the rotation of the reference frame, the initial conditions have changed. Both the longitude and latitude now start at zero. The initial heading angle needs to be modified as North has changed, because the planet has rotated. One disadvantage of this method is that the rotation of the planet has been neglected. This shows itself in the L EoM for the heading angle via Cχ and the computation of D sin σ through Cγ. Because the method should end at the TAEM interface within strict tolerances, it is chosen to adapt the above approach as follows.

106 7.4. Bank reversal search

The reduced-order system is evaluated using the initial conditions as given by the dχ mission deﬁnition. The Coriolis force when evaluating dE is maintained. Also, γ is substituted in the reduced-order system where required. This is opposed to the method in [Mease et al., 2002], where it was assumed that cos γ 1. ≈ When placing the bank reversal at a random energy, the geometry presented in ﬁgure 7.17 can result. North Pole

1 X TAEM

Cross-range 2 error

Downrange travelled Entry

Figure 7.17: Geometry after placing bank reversal at a random energy.

In figure 7.17, point X indicates the final point after the numerical integration. The travelled downrange and cross-range error are indicated in the figure. They are sides of triangle 2. The downrange travelled is the projected distance travelled along the great circle arc connecting the entry and TAEM interface point. The two triangles can be seen separately in figures 7.18 and 7.19.

107 North Pole

C1 X

b1 A1

Entry

Figure 7.18: Triangle 1 after placing bank reversal at a random energy.

a2 B2

C2 c2

b2 A2

Entry

Figure 7.19: Triangle 2 after placing bank reversal at a random energy.

In ﬁgures 7.18 and 7.19: a1 = Co-latitude of point X. b1 = Common side triangles 1 and 2. c1 = Co-latitude of entry point. A1 = Heading of X w.r.t. entry point. B1 = Diﬀerence in longitude between X and entry point. a2 = Cross-range error. b2 = Downrange projected on the great circle arc connecting the entry and TAEM interface point. c2 = Common side triangles 1 and 2. A2 = Heading of TAEM interface w.r.t. X. ◦ C2 = 90 .

108 7.4. Bank reversal search

All angles can be solved for by using spherical trigonometry. The following calculation steps are taken in the order presented below:

1. c1 = cmission, a1 is the co-latitude of the final integration point, B1 is the difference between the longitude of the final integration and entry point.

cos b1 cos c1 2. cos b1 = cos c1 cos a1 + sin c1 sin a1 cos B1 and cos A1 = cos a1 − sin b1 sin c1 3. c2 = b1,A2 = Amission A1 . | − | 4. Using spherical trigonometry for right spherical triangles [Wertz et al., 2001]: sin a2 = sin c2 sin A2 and tan b2 = tan c2 cos A2.

The cross-range error is equal to a2 and the downrange travelled is given by b2.A minimum cross-range error is obtained for the optimum bank reversal energy. This optimum is found by setting up a root ﬁnding problem that can be solved using a secant method. Here, the independent variable is bank reversal energy. The function for which the root needs to be found is the numerical integration that ends with the calculation of a2. This parameter needs to be reduced to zero. In practice a tolerance is placed on the error. The bank reversal is modelled by using the maximum bank rate from σrev0 to σrev0, ± ∓ where σrev0 is the magnitude of the bank angle when the reversal is started. One could think of three methods to model the bank reversal:

Method 1: Integrate w.r.t energy until the bank reversal energy, then switch to integration w.r.t time, switch back to integration w.r.t. energy when the bank reversal has ended.

dt −1 Method 2: Integrate w.r.t energy including the diﬀerential equation for time: dE = DV . Check the time when the reversal is initiated and calculate the time when the t = 2 σrev0 t reversal has ended by: rev σ˙ max . Here, rev is the required time to carry out the reversal. Method 3: Integrate w.r.t. energy and guess the change in energy during the reversal σ by: ∆E = 2 rev0 DrevstartV revstart. Where Drevstart and V revstart are the drag − σ˙ max acceleration and velocity at the start of the reversal. The former equation can be derived by combining the two equations given in the foregoing method. It is assumed that the product of drag and velocity stays more or less constant (it was found to diﬀer by 0.1% only).

The third method has the advantage of simplicity and is implemented in this thesis work. It should be noted that the first and second method are more accurate. The second method has the disadvantage that an extra differential equation needs to be integrated, which is costly in terms of computation time. The first method makes the algorithm more complicated with respect to the third method. The choice for the initial bank angle sign is based on the difference between the initial heading of the vehicle and Amission. The sign is chosen such that the vehicle is steered towards the TAEM interface. Figure 7.20 shows the quadrants of χe Amission and the − initial bank angle sign by 1. In the figure, the turning motion that the velocity vector ± should make in order to steer towards the TAEM interface is also indicated. The reduced-order system can now be integrated over the energy interval from the initial position to the TAEM interface. As said before, the secant method is used to find the reversal point that satisfies the cross-range error.

109 TAEM

Q4 Q1

+1 -1

Q3 Q2

Figure 7.20: Quadrants of χ A and sign σ . e − mission 0 7.5 Trajectory length updating

The initial estimate of the trajectory length is a great circle arc extending from entry to the TAEM interface point at TAEM interface height. For a trajectory with signiﬁcant lateral motion, this underestimates the trajectory length. The numerical integration carried out during the previous step is, therefore, likely to give a trajectory that undershoots the TAEM interface. This is caused by a high drag proﬁle that matches the initial trajectory length guess. Therefore, the drag during entry needs to be reduced. This can be achieved in step 1 if the estimate of the trajectory length is increased. The followed approach is to update the trajectory length according to equation 7.22 below.

Si+1 = Si + (R Ri) (7.22) d − re+rT AEM Here, Ri is a great circle arc computed by Ri = b2 2 . Here, b2 is the downrange resulting from the bank angle search. re and rT AEM are the radius of the entry point and TAEM interface respectively. Rd is the initial estimate of the trajectory length. If R Ri is below a threshold, the iterative process is stopped. This is partly where the | d − | key to success of this method lies in. This approach matches the downrange resulting from the bank reversal search with the actual downrange between entry and TAEM by the mission deﬁnition. Furthermore, the maximum cross-range error has been matched as well.

7.6 Initial and ﬁnal drag value

As seen in figure 7.13, the initial drag has a big influence when the linear spline segments for the drag profile are constructed. In this section, it is explored which option should be chosen for the initial drag: the actual drag specified by the initial conditions, or the drag given by the minimum lift constraint. It is chosen to calculate the final drag value from the TAEM interface conditions (and α(E) profile). Choosing the final drag value like this gives a better chance in matching the TAEM interface conditions. If the initial drag value is not matched with the actual drag as determined by the inital conditions, there are still two methods that might get the vehicle on a correct trajectory that leads to the TAEM interface:

110 7.6. Initial and ﬁnal drag value

- Closed-loop guidance algorithm

- In ﬂight re-planning

Closed-loop guidance takes the current state into account and steers towards the planned profile. In flight re-planning can significantly reduce the errors between the actual and reference trajectory. The choice for the initial drag value will be explored by looking at the Horus and X-33 vehicle and their reference missions. The mission definition for Horus and X-33 can be found in section 3.5. For Horus four cases are tested:

1. Di=De and drag proﬁle is unconstrained.

2. Di=De and drag proﬁle is maximum-drag constrained.

3. Di=De and drag proﬁle is corridor constrained.

4. Di=Lmin and drag proﬁle is unconstrained.

For the X-33 two cases are explored:

1. Di=De and drag proﬁle is unconstrained.

2. Di=Lmin and drag proﬁle is unconstrained.

The numerical integration of the reduced-order system has been carried out using the Runge-Kutte 4 method. The errors on the downrange and cross-range were set to 1.5 and 1 km respectively. In addition, the stepsize for the integrator was set to 1/5000th of the energy range between entry and TAEM conditions. The initial estimated trajectory length for the Horus and X-33 reference mission are 6506 and 8254 km respectively. The X-33 reference mission has been taken from [Saraf et al., 2003] and [Shen, 2002]. While both mention the same value for the initial heading angle (40.41◦), the definition of the heading is defined differently. In [Shen, 2002] the heading is defined from North in clockwise direction. The heading in [Saraf et al., 2003] is from East in counter clockwise direction, hence this would be an initial heading of 49.59◦ when consistent with the notation in this thesis work. The value of χe Amission is: − ◦ ◦ - 1.94 with χe = 49.59 (X-33).

◦ ◦ - 7.244 with χe = 40.41 (X-33). − ◦ ◦ - 1.87 with χe = 49.59 (Horus).

◦ When χe Amission = 7.244 it was not possible to make the algorithm converge to the − − TAEM interface. Considering this together with the value of χe Amission for the Horus − reference mission, it is believed that χe = 49.59 is the correct values for the X-33 reference mission. Below, table 7.1 presents the result of the algorithm for the different cases. For Horus, figure 7.21 to 7.23 present the final drag profiles, latitude vs. longitude and bank angle profile for the four cases. The height as a function of energy can be seen in figure 7.24. For the X-33, this can be seen in figure 7.26 to 7.28. In figure 7.21 and 7.26, the entry corridor is shown by the black lines. The Horus 2 case is tracking the maximum drag profile except for the very beginning and end of the entry. The bank angle profiles show the general behaviour that the higher the drag profile, the higher the bank angle and the later the reversal. The first relationship

111 Table 7.1: Number of iterations and convergence success for the six cases.

Case # iterations Converged Final trajectory length [km] Horus 1 3 Yes 6597 Horus 2 9 Yes 4799 Horus 3 24 Yes 16121 Horus 4 3 Yes 6677 X-33 1 3 Yes 8387 X-33 2 3 No 8344

20 Corridor Horus 1 Horus 2

] Horus 3 2 Horus 4

m/s 15

10 Drag acceleration [

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Normalised energy [-]

Figure 7.21: Horus drag proﬁles for case 1-4.

112 7.6. Initial and ﬁnal drag value

Horus 1 Horus 2 5 Horus 3 Horus 4

−5

−10 Latitude [deg]

−15

−20

−25 −110 −100 −90 −80 −70 −60 −50 Longitude [deg]

Figure 7.22: Horus latitude vs. longitude for case 1-4 (x is the TAEM interface).

100

60 Horus 1 Horus 2 40 Horus 3 Horus 4 20

−20 Bank angle [deg]

−40

−60

−80

−100 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Normalised energy [-]

Figure 7.23: Horus bank angle proﬁles for case 1-4.

113 120

110

100 Horus 1 Horus 2 90 Horus 3 Horus 4 80 ] km 70

Height [ 60

20 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Normalised energy [-]

Figure 7.24: Horus height profiles for case 1-4. makes sense, because to be at a higher drag for the same energy, the height needs to be lower. By banking more, the height is decreased. The smaller height gives a larger lift force, this combined with a larger bank angle gives the vehicle a higher degree of manoeuvrability. The higher degree of manoeuvrability allows the vehicle to execute the bank reversal later, while still returning to the TAEM interface. The two breakpoints in the drag profile give a change in the bank angle profile. A larger change in the slope of the drag profile, gives a more noticeable change in the bank angle profile. The change is not discontinuous but linear. The non-smooth behaviour at the end of the trajectory is 0 0 caused by CD, as can be seen in figure 7.25. CD is computed by numerical differentiating the CD(E) profile that was retrieved during the trajectory parameters extraction. On a side note, The velocity differences are small, they are in the order of 10 m/s at each energy. This is expected, because the kinetic energy is much larger than the potential energy. For Horus, table 7.1 shows that case 2 and 3 lead to converged trajectory lengths that can be considered wrong. It is thought that this is caused by the mismatch in the converged spline profile and the flown constrained drag profile. Hence, the trajectory length does not correspond to the drag profile planned. It is not surprising then that the iterative process converges to a wrong trajectory length, but does allow the vehicle to end up at the TAEM interface. Case 1 grossly violates the path constraints, while case 4 does not violate any. This suggests that one should choose Di = Lmin. The guidance algorithm has to account for this mismatch by employing feedback control and/or plan updates. To solve the constraint violation, the entry point could be moved such that the required trajectory length increases. The constraint values could also be changed, but this implies a change in vehicle design. The found drag profile might then match the actual initial drag and fall in the entry corridor. Matching the initial drag value should enhance the trajectory

114 7.6. Initial and ﬁnal drag value

−7 x 10

6 Horus 1 Horus 2 5 Horus 3 Horus 4 4

−1

−2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Normalised energy [-]

Figure 7.25: Horus height profiles for case 1-4. tracking and increase mission success. The high drag profile of case 1 signals that, to match the trajectory length, the flown drag profile needs to be close to the upper boundary of the entry corridor. To make a successful entry with Horus and this bank angle planner, the entry point probably needs to be moved. This will be explored in the next chapter, where the planning algorithm is incorporated in a guidance system. The X-33 mission conditions are exactly such that planning from the actual initial drag value, gives a spline profile that does not exceed an active path constraint. Moreover, the algorithm was not able to converge when planning from Di = Lmin. Figure 7.27 shows that it is not able to reach the TAEM interface. This is caused by the drag profile being lower than the Lmin constraint for most of the energy values, as can be seen from figure 7.26. The derived bank angle profile is in general low and even close to zero for a large part, as can be seen in figure 7.28. Logically, if the bank angle is close to zero, the manoeuvrability is marginal. The result of this is that the algorithm could not find a reversal energy such that the cross-range error is within the user defined limit. The above reasoning suggests that Di should be set to De. Again, the breakpoints of the drag profile show themselves in the bank angle profile as almost discontinuities. The search for a bank angle profile suggests a different initial drag value for the Horus and X-33 mission definition. It could be true that a different mission definition requires a different initial drag value. However, it would be more elegant if one approach leads to the solution. To find an algorithm that is generic, there can be only one answer. Or, a more complicated planner must be designed that determines the optimal value for Di, which is costly in terms of computation time. In the next chapter, the algorithm is tested as part of a guidance system. It is there further explored which initial drag value should be chosen.

115 25

20 Corridor X-33 1

] X-33 2 2

m/s 15

10 Drag acceleration [

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Normalised energy [-]

Figure 7.26: X-33 drag proﬁles for case 1 and 2.

X-33 1 30 X-33 2

0 Latitude [deg]

−10

−20

−30 220 230 240 250 260 270 280 Longitude [deg]

Figure 7.27: X-33 latitude vs. longitude for case 1 and 2 (TAEM interface is the star).

116 7.6. Initial and ﬁnal drag value

40 X-33 1 X-33 2 20

−20 Bank angle [deg] −40

−60

−80

−100 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Normalised energy [-]

Figure 7.28: X-33 bank angle proﬁles for case 1 and 2.

117 118 CHAPTER 8

Guidance System Testing

The bank angle planning method requires further testing by performing trajectory simulations for the reference missions of the X-33 and Horus vehicle. The planning algorithm is seen to be a part of the guidance system. In total, three types of guidance systems will be tested:

1. Open-loop guidance.

2. Open-loop guidance using plan updates (semi closed-loop guidance).

3. Closed-loop guidance control using plan updates.

It is expected that the open-loop guidance system will not lead to mission success, i.e., path constraints are violated and/or the target is not reached. This hypothesis arises from the mismatch in the initial flight-path angles as given by the mission definition and the planning algorithm. This holds for both the Horus 4 and X-33 1 cases. The Horus 4 case has a mismatch in the initial height as well. Therefore, it is thought that plan updates and/or closed-loop guidance are necessary. Plan updates give the opportunity to start with a clean slate. It is thought that the planning algorithm is best executed when the vehicle is within the entry corridor instead of below the minimum drag constraint. That is, because a drag value below the minimum drag constraint easily leads to an inverse sine or cosine of an absolute value that is larger than 1 in equation 7.21. Hence, in practice, the bank angle wants to steer towards L a larger D than possible. In theory, if plan updates are carried out at a high enough frequency, mission success could be guaranteed as deviations from the reference trajectory are accounted for. Closed-loop guidance tries to ensure mission success in a different manner. The deviation from the reference trajectory are accounted for by using, for example, PID control. Deviations from the reference trajectory can be caused by:

- Mismatch between actual and planned initial conditions.

- Non-ideal navigation system.

- Inaccurate environment model (e.g. gravity ﬁeld, atmosphere, planet’s shape).

119 - Inaccurate vehicle model (e.g. aerodynamics, mass distribution).

- Unexpected events (e.g. collision, vehicle structural failure).

Section 8.1 presents the results of trajectory simulation using the open-loop guidance system. Hereafter, section 8.2 gives the result of using an open-loop guidance system using plan updates. The theory behind the trajectory tracker, employed in the closed- loop guidance, is presented in section 3.2.3. The test results for this guidance system can be found in section 8.3. Hereafter, the performance of guidance system is tested using a wind model. The results can be found in section 8.4. A trimmed-ﬂight analysis is given in section 8.5. In the end, in sections 8.6 and 8.7, a preliminary sensitivity study on changing α(E) and γe can be found.

8.1 Open-loop guidance

This section presents the trajectory simulation using open loop guidance for the Horus 4 and X-33 1 case. Figures 8.1 to 8.4 present the simulated and planned trajectory for the Horus 4 case. The simulated and planned trajectory for the X-33 1 case can be seen in ﬁgures 8.5 to 8.8.

5 Simulated Planned 0

−5

−10 Latitude [deg]

−15

−20

−25 250 260 270 280 290 300 310 Longitude [deg]

Figure 8.1: Ground-track for the Horus 4 case (Open-loop).

Figures 8.1 to 8.4 show that the vehicle is not able to reach the landing site and that the heat flux constraint was violated along the way. A skipping flight has been induced that shows itself in the figures for the drag acceleration, height and flight-path angle. The ground track shows that an undershoot of the TAEM interface has occurred. The E distance travelled is given by: S = R f dE . When integrating this equation for − Ei D(E) the planned and simulated trajectory, S = 6680 and S = 5850 respectively. Hence, the undershoot of the TAEM interface is caused by the higher drag throughout entry. The

120 8.1. Open-loop guidance

120

110

100 Simulated 90 Planned

70 Height [km] 60

20 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Energy [-]

Figure 8.2: Height vs. energy for the Horus 4 case (Open-loop).

12 Entry corridor Simulated Planned 10

6 Drag acceleration [m/s]

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Energy [-]

Figure 8.3: Drag vs. energy for the Horus 4 case (Open-loop).

121 1

−1 Simulated Planned

−2

−3 Flight-path angle [deg]

−4

−5

−6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Energy [-]

Figure 8.4: Flight-path angle vs. energy for the Horus 4 case (Open-loop).

Simulated 20 Planned

0 Latitude [deg]

−10

−20

−30 220 230 240 250 260 270 280 Longitude [deg]

Figure 8.5: Ground-track for the X-33 1 case (Open-loop).

122 8.1. Open-loop guidance

120

110

100 Simulated Planned 90

70 Height [km] 60

20 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Energy [-]

Figure 8.6: Height vs. energy for the X-33 1 case (Open-loop).

20 Entry corridor Simulated ]

2 Planned

m/s 15

10 Drag acceleration [

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Energy [-]

Figure 8.7: Drag vs. energy for the X-33 1 case (Open-loop).

123 1

Simulated 0 Planned

−1

−2

−3 Flight-path angle [deg]

−4

−5

−6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Energy [-]

Figure 8.8: Flight-path angle vs. energy for the X-33 1 case (Open-loop). bank angle profile is in principle designed for the reference trajectory. The differences in the initial height and flight-path angle for the planned and simulated trajectory are too large to give a successful mission. The planned flight-path angle profile shows a discontinuous change at a normalised energy of about 0.45. At that point, there is a breakpoint in the drag profile. The figures for the X-33 1 case show that also here mission was not successful. A late skip can be seen and the normal load and dynamic pressure constraints are violated. Even though the drag (and height) starts at the same value, the vehicle undershoots the TAEM interface and path constraints are violated. At entry, the only difference is the initial flight-path angle γe. However, changing γe for the simulated trajectory to its planned value did not give a successful trajectory. When moving γe towards the planned value in steps was not successful either. At some point, the trajectory changed into a skip trajectory. One solution could be to steer the vehicle towards the reference drag profile. However, in the initial part of the trajectory where the drag is low, the tracking algorithm presented in subsection 3.2.3 is not able to give a corrective steering command. Executing plan updates could steer the vehicle towards the reference trajectory. Therefore, this approach is taken in the next section. On a side note, the planned flight-path angle profile presented in figure 8.8 shows some discontinuities. The discontinuities at about 0.41 and 0.8 are caused by the breakpoints in the reference angle of attack profile. The other two are caused by the breakpoints in the planned drag profile.

8.2 Open-loop guidance using updates

When employing updates of the bank angle planning algorithm every 50 seconds, the mission is still not a success for both the Horus 4 and X-33 1 case. Figures 8.9 and 8.10

124 8.2. Open-loop guidance using updates present the trajectories in the entry corridor for these two cases. It needs to be mentioned that the ground track has improved and the X-33 1 case has a closest approach of 155 km at the TAEM interface energy. Were it not for the fact that the maximum drag has been exceeded, this miss distance of 155 km would still lead to mission failure. On the premise that the bank angle profile followed in the initial part of the entry might have an influence, it was attempted to test case X-33 2. Figures 8.11 to 8.14 show that a significant improvement has been made with respect to the previous results. The X-33, now has a missed distance of 31.8 km to the TAEM interface at the interface energy. The ground track does not show a large difference at this scale. The same could be said for the height flown as depicted in figure 8.12. The error in the flight path angle at the last update is also much less than at entry. The updates show themselves in the planned drag profile until a normalised energy of about 0.64, that is where the last update is done. The last update takes place just before the bank reversal. After the bank reversal no more updates a executed. The flown drag profile, figure 8.13, shows a much smaller oscillation around the planned profile. The oscillation is caused by the mismatch in the flight-path angle at the last update.

5 Simulated Planned 0

−5

−10 Latitude [deg]

−15

−20

−25 250 260 270 280 290 300 310 Longitude [deg]

Figure 8.9: Ground-track for the Horus 4 case using updates.

It is worth mentioning that even though the algorithm did not converge the first time, it did converge at the updates. The explanation for this is as follows. In figure 7.26, the drag profile is presented that is found when executing the bank angle planning algorithm from the entry point. In this figure, it can be seen that the drag profile is for the largest part below the minimum drag constraint. As a result, the extracted bank angle profile is in general very low. The vehicle is, therefore, not able to steer towards the TAEM interface. Or, in other words, it cannot correct for the initial heading offset from the desired initial heading. As mentioned in the previous chapter, the desired initial heading is the heading of the TAEM interface w.r.t. the entry point. Due to the low drag acceleration at entry, the vehicle travels a relatively large distance in the first part

125 40

10 Simulated Planned

0 Latitude [deg]

−10

−20

−30 220 230 240 250 260 270 280 290 Longitude [deg]

Figure 8.10: Ground-track for the X-33 1 case using updates.

10 Simulated Planned 0 Latitude [deg]

−10

−20

−30 220 230 240 250 260 270 280 Longitude [deg]

Figure 8.11: Ground-track for the X-33 2 case using updates.

126 8.2. Open-loop guidance using updates

120

110

100 Simulated 90 Planned

70 Height [km] 60

20 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Energy [-]

Figure 8.12: Height vs. energy for the X-33 2 case using updates.

20 Entry corridor Simulated ]

2 Planned

m/s 15

10 Drag acceleration [

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Energy [-]

Figure 8.13: Drag vs. energy for the X-33 2 case using updates.

127

1 Simulated Planned 0

−1

−2

Flight-path angle−3 [deg]

−4

−5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Energy [-]

Figure 8.14: Flight-path angle vs. energy for the X-33 2 case using updates.

of the entry, when compared to starting at Di = Lmin. At a plan update, the balance between the estimated trajectory length S and constant drag acceleration segment Dc is such that the planned profile is above the minimum drag constraint. The extracted bank angle profile is in magnitude large enough to steer towards the TAEM interface. So why is the X-33 1 case much worse than the X-33 2 case? The low initial drag causes the planned bank angle to be zero or very low in the beginning for the X-33 1 case. The transition to the reference flight-path angle is not as smooth and the trajectory almost shows a skip. The drag stays low and needs to be compensated for by a higher drag later in the entry. This planned drag profile has two disadvantages, it exceeds the path constraints and the slope of the first segment is very steep. The drag acceleration profile for the X-33 1 case including updates can be seen in figures 8.15. To reduce the miss distance of the X-33 1 case further, a closed-loop guidance algorithm based on feedback control is added. This is the subject of the next section.

8.3 Closed-loop guidance using updates

After implementing the closed-loop guidance algorithm, the result becomes much better for the X-33 2 case. By manually tuning the gains, the following ﬁnal conditions are obtained:

- Closest approach: 340 m distance.

- Height: 29.2 km.

- Velocity: 920 m/s.

- Flight-path angle: 9.5◦. −

128 8.3. Closed-loop guidance using updates

20 Entry corridor Simulated ]

2 Planned

m/s 15

10 Drag acceleration [

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Energy [-]

Figure 8.15: Drag vs. energy for the X-33 1 case using updates.

- Heading: 128.5◦.

When looking at the TAEM interface conditions that were presented in subsection 3.1.2, all constraints are satisfied besides the heading angle. The heading angle should have ◦ ◦ been 68.6 10 (Amission + allowable heading error). In [Saraf et al., 2003], this problem ± is solved by moving the TAEM interface point along the circle. The movement does not significantly alter the trajectory shape [Saraf et al., 2003]. Therefore, it is assumed that this is not a difficult problem to solve. The execution time of the planner is 1.1 second maximum, depending on the iterations made. Figure 8.16 present the latitude versus the longitude near the TAEM interface. This figure clearly shows the small distance between the simulated trajectory and the TAEM interface point. The stop criterion for the simulation is the final energy of the TAEM interface conditions. The gain values that were used are: √ 2 ζ = 2 2π ω0 = 8 rad/s −9 k1 = 1 10 · √ 2 p 2 At ζ = , the resonant frequency ωr vanishes as ωr = ω0 1 2ζ [Ogata, 2009]. The 2 − integral gain and error are only switched on close to the TAEM interface, in order to avoid excessive oscillations and saturation caused by the integral term. Figures 8.17 to 8.19 show the drag acceleration, flight-path angle and bank angle profile of the planned and simulated trajectory. The simulated and planned drag profile show a close correspondence. The non-smooth behaviour of the bank angle profile until a normalised energy of about 0.62 is caused by the plan updates and closed-loop control that steers towards the reference profile. To determine why the simulated trajectory does not follow the planned one, a closer look

129

28.75 Simulated Planned

28.7

Latitude [deg] 28.65

28.6

279.42 279.44 279.46 279.48 279.5 279.52 279.54 Longitude [deg]

Figure 8.16: Latitude vs. longitude for the X-33 2 case using closed-loop guidance (x is the TAEM interface).

20 Entry corridor Simulated ]

2 Planned

m/s 15

10 Drag acceleration [

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Energy [-]

Figure 8.17: Drag proﬁle for the X-33 2 case using closed-loop guidance.

130 8.3. Closed-loop guidance using updates

Simulated −1 Planned

−2

−3 Flight-path angle [deg] −4

−5

−6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Energy [-]

Figure 8.18: Flight-path angle proﬁle for the X-33 2 case using closed-loop guidance.

100

60 Simulated 40 Planned

−20 Bank angle [deg]

−40

−60

−80

−100 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Energy [-]

Figure 8.19: Bank angle proﬁle for the X-33 2 case using closed-loop guidance.

131 has to be taken at normalised energy interval from 0.36 to 0.52. The close-up of the drag profile can be seen in figure 8.20. The flight-path angle and bank angle profiles in this region can be seen in figure 8.21 and 8.22 respectively.

8.4

Simulated 8.35 Planned

8.3 Drag acceleration [m/s]

8.25

8.2 0.36 0.38 0.4 0.42 0.44 0.46 0.48 0.5 0.52 Energy [-]

Figure 8.20: Drag proﬁle close-up in 0.36 < E < 0.52.

The bank reversal takes place open-loop from an energy of about 0.513 until 0.481. Right after the initiation of the bank reversal, the drag acceleration decreases. This is expected, because the vertical lift increases. Also, the flight-path angle increases w.r.t. the planned profile. Even though the bank angle is in open-loop, this does not mean that the flight-path angle should follow the open-loop flight-path angle profile as well. Since, the flight-path angle profile is derived from the planned drag profile and not a result of the bank-reversal search simulation. After the decrease in drag, the drag becomes larger than the planned profile. This is caused by the breakpoint in the drag profile. This effect is perhaps best explained by looking at the flight-path angle profiles. The breakpoint causes a discontinuous change in the flight-path angle profile as D0 has changed. Due to this discontinuous change, the vehicle finds itself at a lower flight-path angle when compared to the planned value. As a result of this, it has the tendency to decrease its altitude faster. The density increases and the drag does so too. The increase in bank angle after returning to closed-loop guidance might seem strange at first. The drag profile clearly shows a larger drag than the reference value. One might expect that the drag is going to be decreased by lowering the bank angle, instead it is increased. The increase as commanded by the PD-control actually gives a smooth transition to the constant drag by increasing the bank angle. Again, the flight-path angle profile shows why this happens. At the moment that the closed-loop guidance is switched on, the flight-path angle is larger than its reference value (due to the reversal). If this lower flight-path angle was maintained, the vehicle would cross the reference drag profile sooner. In fact, when looking at the graph of the drag profile, the slope decreases at an energy of 0.481. By increasing the bank angle, this transition becomes smoother and a lower overshoot results.

132 8.3. Closed-loop guidance using updates

−0.3

Simulated −0.4 Planned

−0.5

−0.6 Flight-path angle [deg]

−0.7

−0.8 0.36 0.38 0.4 0.42 0.44 0.46 0.48 0.5 0.52 Energy [-]

Figure 8.21: Flight-path angle proﬁle close-up in 0.36 < E < 0.52.

Simulated 60 Planned

0 Bank angle [deg]

−20

−40

−60 0.36 0.38 0.4 0.42 0.44 0.46 0.48 0.5 0.52 Energy [-]

Figure 8.22: Bank angle proﬁle close-up in 0.36 < E < 0.52.

133 The peak starting from a normalised energy of 0.414 is caused by a breakpoint in the angle of attack profile. Here, the angle of attack starts decreasing from 46◦ to 20◦ in the reference profile. The second jump in the reference flight-path angle is a direct effect of L this, because if α , D and γ . The closed-loop guidance reacts to this by increasing ↓ ↑ ↑ L the bank angle and thereby lowering the vertical D . Something similar happens close to the TAEM interface, where the second breakpoint of the drag profile occurs. The same reasoning, except for the bank reversal, as above applies here. Here, the vehicle is at a lower drag after the breakpoint and the bank angle is increased to compensate for this. Although the commanded bank angle is always passed through a rate filter that takes σ˙ into account, it has to be realised that huge demands are going to be placed on the control system. A more sophisticated tracking and/or planning algorithm could solve this problem. One could, for example, include the angle of attack as the second closed-loop control variable. In addition, the breakpoints in the planning algorithm should be removed. They give rise to tracking errors. Including the bank reversal in the reference drag profile should give less tracking errors as well. Or, one can think about increasing the angle of attack during the reversal. This lowers the lift-to-drag ratio. The implementation of these measures are, however, left for future work. In the remaining sections, a preliminary study is executed in which the performance of the planning and tracking algorithms is tested. In section 8.4, a wind model taken from GRAM99 is used. Hereafter, section 8.5 presents the results of including trim in the analysis. Finally, section 8.6 and 8.7 treat the influence of changing α(E) and γe, respectively.

8.4 Inﬂuence of wind

The performance of the tracker can partly be tested by including a wind model during the simulations. The wind model given in [Justus and Johnson, 1999] was used. Figure A.4 in appendixA shows the wind directions and their magnitude as a function of height for this model. Table 8.1 shows ﬁve important distance parameters that are important to assess the performance of the tracker.

Table 8.1: Important distances for assessing tracker performance.

No wind Wind Closest approach 0.34 1.32 km Down-range error tolerance 1 1 km Cross-range error tolerance 1.5 1.5 km Down-range error from planning 0.33 0.4 km Cross-range error from planning 0.02 1.18 km

From the information in table 8.1, it is deduced that the larger closest approach distance, in the case including wind, is caused by the converged values of the planner. As both fall within the error tolerances, it is concluded that the tracker has performed well for this wind model. More wind models should be tested to come with a more deﬁnitive answer.

8.5 Trimmed ﬂight

Until now, only the 3 DoF translational motion has been treated in mission planning. However, if a 3 DoF ﬂight is successfully ﬂown, it does not mean that it actually can

134 8.5. Trimmed flight in a 6 DoF flight. Here, the rotational motion of the vehicle is analysed as well. More specifically, the required control surface deflections and/or impulses given by the reaction control thrusters are computed. They are required in order to obtain the angle of attack, side-slip and bank angle as commanded by the guidance system. This implies that the commanded attitude cannot be obtained in infinite time and the vehicle is not able to follow the reference commands perfectly. Hence, deviations from the reference trajectory will occur. Future work should assess the performance of the algorithm in 6 DoF simulations. One step in between a 3 DoF and 6 DoF simulation is to include trim in the analysis. The basic trim analysis is to compute the control surface deflections required for trim around the body Y-axis. Depending on the direction of the deflections, the lift and drag coefficient either increase or decrease with respect to the zero-deflection state. Trim around the body Y-axis means that the moment around that axis is zero. The advantage of a trimmed flight is that the moment around the axis can easily be changed in either positive or negative direction. Hence, this guarantees a quick response if an attitude change is demanded. In addition, the vehicle would not be constantly rotating because of a residual moment. This is better for the well-being of a crew. The aerodynamic database for the X-33 control surfaces was not available, therefore the Horus vehicle has been used for the trimmed-flight analysis. The algorithm does not work for the Horus vehicle and reference mission. A skip trajectory is induced, path constraints are violated and the TAEM interface is not successfully reached. There can be two causes:

- The guidance system is not ﬂexible enough.

- The mission deﬁnition is such that it cannot be ﬂown.

The second cause is not the case as the opposite is shown by the reference mission presented in subsection 3.5.1. Hence, the planning and/or the tracking algorithm is not flexible enough. The solution is to find a mission definition that can be flown. There are four main differences between the Horus and X-33 missions definitions:

1. The trajectory length is much larger for the X-33 reference mission.

2. The lift-to-drag ratio is much larger for the Horus vehicle.

3. The allowed heat ﬂux is much smaller for Horus than for the X-33.

4. The initial ﬂight-path angle is 0.2◦ smaller for Horus than for the X-33 mission.

The first two differences are related. A larger lift-to-drag ratio implies a larger range flown and the vehicle would have to bank more. This effect is enhanced by the smaller trajectory length for the Horus mission. The second and third differences are also related. Figure 8.23 presents the planned and simulated flight-path angle for the Horus mission. The smaller (more negative) initial flight-path angle results in a steeper descent and a larger heat flux as denser atmosphere is reached with a higher velocity. This can be counteracted by the larger lift-to-drag ratio, but the density, and thus the aerodynamic force, is small in the initial part of the entry. The smaller allowable heat flux for Horus w.r.t. the X-33 enhances this effect. The offset between the initial and planned flight-path angle results in an oscillating motion around the planned flight-path angle. The skipping motion can be seen as well.

135

−1

Simulated Planned −2

Flight-path angle−3 [deg]

−4

−5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Energy [-]

Figure 8.23: Flight-path angle proﬁle for the Horus reference mission.

12 Entry corridor Simulated ]

2 Planned 10 m/s

6 Drag acceleration [

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Energy [-]

Figure 8.24: Drag acceleration proﬁle for the Horus reference mission.

136 8.5. Trimmed ﬂight

From the last plan update, this motion is still present. The remaining part of the trajectory shows a larger drag with respect to the planned profile, resulting in an undershoot of the TAEM interface. The skip trajectory can be avoided by increasing the bank angle. The planned profile, when starting planning from Lmin has a too small bank angle in the beginning. Hence, the vertical lift is too high and a skip is induced. In this case, the skipping motion gives a violation in the heat flux and the minimum drag constraint. For both reasons, the skip should be avoided. This is left for future work. To fly a successful mission with Horus, the entry point was placed backwards. This was still not a success. Now, the Horus vehicle is placed in the X-33 mission. After searching an angle of attack profile with trial-and-error, the path constraints are no longer violated. In addition, the gains were tuned and the natural frequency was changed 2π to 16 rad/s. There is, however, a large miss distance of 11 km while the planner has converged successfully. This is caused by the bank reversal and the attempt of the tracking algorithm to correct for the corresponding decrease in drag with respect to the reference profile. The effect is more noticeable than for the X-33 vehicle. The larger lift-to-drag ratio of Horus causes the larger difference in drag, which has to be corrected for. The influence of trim on the guidance system can be determined irrespective of this final miss distance. Therefore, the 10 km is accepted and the mission is used as a baseline. The remainder of this section shall compare two guidance systems to the nominal case:

- Excluding trim in the planning, including trim in the simulation.

- Including trim in the planning and the simulation.

More specifically, it is tested if the planner can still execute fast enough for an on-board implementation if trim is included in the planning. In addition, it is tested if the tracking algorithm can cope with the different aerodynamic coefficients if trim is not included in the planning. Furthermore, a qualitative description is given on the trajectory behaviour when trim is fully included w.r.t. the non-trimmed flight. Figure 8.25 presents the bank angle profiles. When trim is excluded the planning algorithm executes between 1.4 and 1.8 seconds, depending on the number of iterations used. The integration for the bank angle search is composed of 600 steps. The planning algorithm’s speeds is greatly influenced when trim is included. Now, the algorithm executes between 20 and 26 seconds. All of this is caused by the extra overhead of calculating the control surface deflections for a trimmed state and the extra terms in the lift and drag coefficient. It is concluded that the algorithm cannot be used on-board any more. There are three possible solutions to decrease the execution time:

1. Decrease the number of data points (keep the tolerances and precision due to the number of data points in mind).

2. Use analytical formulae for the aerodynamic coeﬃcients (computation of trimmed state might, however, increase computation time).

3. Optimise the software code for computational eﬃciency.

In ﬁgure 8.25, it can be seen that the tracker is correcting a lot in the early part of the entry when trim is not included in the planning. This can be seen from the peaks occurring between 0.6 < E < 0.95. The peaks right after the bank reversal are caused

137 100

60 Nominal Proﬁle Trim in simulation 40 Trim in simulation and planner

−20 Bank angle [deg]

−40

−60

−80

−100 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Energy [-]

Figure 8.25: Bank angle profiles for trim analysis. by the tracking algorithm, which compensates for the decrease in drag caused by the reversal. The peaks starting at E = 0.44 and E = 0.26 are caused by the breakpoints in the planned drag acceleration profile. Finally, the set of peaks close to the end are induced by switching on the integral gain and error. When excluding trim in the planning, the largest differences occur between the actual and reference drag. The miss distance is, as a result, increased to 20 km. When trim is included in the planning, the miss distance is 0.8 km more w.r.t. the non-trimmed flight. This trajectory has a smaller difference between the actual and reference drag throughout entry. The difference of 20 km is considered to be too large. This can be caused by the simplicity of the algorithm or the lack of gain tuning that has been applied. The lift-to-drag ratios throughout entry are the same for the cases with trim included. They are throughout entry about 0.02 below the nominal case. Hence, the trimmed state has a lower lift-to-drag ratio than the nominal state. The converged values for Dc are:

- Nominal: 9.9 m/s2

- Trim in simulation: 10.5 m/s2

- Trim in simulation and planner: 9.7 m/s2

Dc is lower when trim is included in the planning, because the final drag value is higher and the distance that needs to be travelled is the same. As a result, the overall magnitude of the bank angle profile is lower as well. When drag is excluded in the planning, the L tracker tries to steer towards a larger D by lowering the bank angle. The drag profile ends up below the nominal profile. Less energy has been dissipated w.r.t. the nominal profile and, therefore, the converged Dc is larger. With respect to the bank reversals, it can be said that the reversals for the nominal and planner included cases are at about the same energy. When trim is excluded from

138 8.6. Changing the angle of attack proﬁle the planning, the reversal takes place later. This is caused by the larger absolute value of the bank angle in second halve of the entry. The larger bank angle gives more cross-range which allows a later bank reversal as the cross-range error can more easily be corrected.

8.6 Changing the angle of attack proﬁle

The bank angle and angle of attack are the two most important angles in mission planning for re-entry. It is, therefore, interesting to determine what the inﬂuence of varying the angle of attack is on:

- The planner’s ability to converge.

- The ability to track the reference proﬁle.

- The path constraint values.

For this purpose, six variations on the the nominal angle of attack profile are tested. The nominal angle of attack profile was presented in figure 3.17. These six profiles can be seen in figures 8.26 to 8.28, where the nominal profile can also be seen. Changing the angle of attack profile has a direct influence on the lift-to-drag ratio. For the X-33 L (and Horus) hypersonic aerodynamics, an increase in α implies a decrease in D and vice versa. This holds for α α αmax. (L/D)max ≤ ≤

45 Nominal Profile Profile 1 Profile 2 40

35 Angle-of-attack [deg] 30

20 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Energy [-]

Figure 8.26: Angle-of-attack proﬁles 1 and 2.

First, trajectories are simulated using the bank angle profile (in open-loop) that has been calculated for the nominal angle of attack profile. This should answer the question what the influence on the performance parameters is. The results are presented in subsection 8.6.1. Then, new bank-angle profiles for the X-33 reference mission are computed for the six profiles to test the planning and tracking performance. In subsection 8.6.2, the results can be found.

139

45 Nominal Profile Profile 3 Profile 4 40

35 Angle-of-attack [deg] 30

20 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Energy [-]

Figure 8.27: Angle-of-attack proﬁles 3 and 4.

45 Nominal Profile Profile 5 Profile 6 40

35 Angle-of-attack [deg] 30

20 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Energy [-]

Figure 8.28: Angle-of-attack proﬁles 5 and 6.

140 8.6. Changing the angle of attack proﬁle

8.6.1 Nominal bank angle profile In this section, the effect of varying the angle of attack using the same bank angle profile is presented. More specifically, the effects are measured on the distance travelled, time- of-flight, total heat load and maximum heat flux. These results can be seen in table 8.2. The impact on the normal load and q α is such that a lower α gives a smalle dyn · constraint value.

Table 8.2: Eﬀects on trajectory for constant σ(E).

Nominal 1 2 3 4 5 6 Distance [km] 8410 8580 8330 8470 8360 8710 9080 Flight time [min] 24.7 25.4 24.4 24.9 24.6 25.4 26.3 MJ Q[ m2 ] 384 404 375 388 380 398 417 kW q˙ [ m2 ] 443 443 443 443 441 446 452

The distance travelled has been calculated using an Euler integration. Also, the results have been rounded to the nearest ten kilometres. From the results, it is concluded that a decrease in α increases both range and flight time. The opposite is true for an increase in α. Furthermore, manipulation of α in the early part of the entry has a more pronounced effect on the range that is achieved. The maximum heat flux occurs at 67% of the entry energy. That is why profiles 1 and 2 do not affect this value. Also, profiles 3 and 4 barely show a variation in the heat flux as the difference in angle of attack is still small there. Decreasing α before the maximum heat flux has two effects. First, the maximum heat flux obtained is larger. Second, the point of maximum heat flux occurs earlier (in terms of energy value). For the lower α profiles, the energy dissipation is less. Due to the low density, the drag is low and de velocity barely shows a variation w.r.t. the nominal profiles. The flight-path angle is still about the same and causes the vehicle to hit the denser layers of the atmosphere with a larger velocity. This results in the larger heat flux. To conclude, when α is decreased, the heat load is increased as well. This is a combination of the larger heat flux and flight time. The opposite holds true for an increase in α. In chapter6, the angle of attack planning was considered. The results found in this subsection correspond to the results found in that chapter.

8.6.2 Including planning algorithm In this subsection, new trajectories are planned for the six angle of attack profiles. The corresponding simulated bank angle profiles can be seen in figures 8.29 to 8.31. Informa- tion on the bank planner algorithm and maximum path constraint values can be found in table 8.3. The bank angle profile for the nominal case is different from before. This is caused by the use of a different aerodynamic model for the X-33. In this subsection, a model given in table format was used instead of an analytical model. One important difference between using an table instead of an analytical model is the execution time. When using an analytical model, the execution time is much less. The peak right after the bank reversal for the nominal profile is caused by a combination of the bank reversal and a breakpoint in the drag profile. The first breakpoint of the drag profile falls in the bank reversal. After the reversal, the drag is higher and not lower than the nominal case due to the breakpoint. Therefore, a reduction in the bank angle is required, explaining the peak in the direction of a smaller bank angle. The upward peak

141 100

60 Nominal Profile Profile 1 40 Profile 2

−20 Bank angle [deg]

−40

−60

−80

−100 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Energy [-]

Figure 8.29: Bank angle proﬁles 1 and 2.

100

60 Nominal Profile Profile 3 40 Profile 4

−20 Bank angle [deg]

−40

−60

−80

−100 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Energy [-]

Figure 8.30: Bank angle proﬁles 3 and 4.

142 8.6. Changing the angle of attack proﬁle

100

60 Nominal Profile Profile 5 40 Profile 6

−20 Bank angle [deg]

−40

−60

−80

−100 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Energy [-]

Figure 8.31: Bank angle profiles 5 and 6. right after it is caused by the breakpoint in the angle of attack profile. Profile 1 shows a peak at E = 0.6 as a result of a breakpoint in the angle of attack profile. The recovery from the bank reversal can be seen in the peak right after it. Actually, the drag acceleration does not return to the planned profile before E = 0.26. The tracking algorithm gives an excessive reaction to the bank reversal. This should be solved by gain tuning. The same happens for case 6. The slope change due to the breakpoint in the angle of attack profile for case 6 is more pronounced than elsewhere. The tracking algorithm responds with an excessive command due to the high proportional gain. Again, gain tuning should solve this. All other peaks are the result of the bank reversal, another breakpoint in the angle of attack profile, a breakpoint in the drag acceleration profile and the switching on of integral control. This holds for profile 3 as well.

Table 8.3: Simulation characteristics for the α-proﬁles.

Nominal 1 2 3 4 5 6 TAEM miss [km] 2.7 29.7 15.2 0.65 10 1.6 16.6 m Dc [ s2 ] 8.26 8.63 8.28 8.27 8.2 8.47 8.74 MJ Q [ m2 ] 384 396 377 386 381 389 397 kW q˙ [ m2 ] 443 437 443 449 435 454 474

The bank planning algorithm converged successfully for all six profiles. However, only profiles 3 and 5 have successfully reached the TAEM interface. The other profiles have a too large miss distance and it is concluded that the tracking algorithm has failed there. The miss distance is too large when seen from the performance of the guidance system. It should not be a problem for the TAEM guidance [Mooij et al., 2007]. The large miss distance is caused by a much more pronounced difference in the planned and simulated drag profile. For case 3 and 5, the difference between the planned

143 and simulated drag profile was less than the nominal case. Therefore, the miss distance of the TAEM interface is also less. Of all the profiles, numbers 3 and 5 are the ones that resemble the nominal profile the most. This indicates that the set of gains works the best for profiles with such an envelope. None of the path constraints were violated during the simulation. The different angle of attack profiles do not result in a large variation in the maximum path constraint values. The difference in the normal load and q α is not significant. Besides profile 2, dyn · the general trend is the lower the angle of attack profile, the larger the converged value for Dc, the value for the planned constant drag segment. A lower angle of attack causes L an increase in D and gives a tendency to increase the range. Therefore, a larger drag is needed to end up at the TAEM interface. Another point of view is, it tries to match the vertical lift-to-drag ratio of the nominal profile by changing the magnitude of the bank angle. This effects shows itself in the magnitude of all the bank angle profiles, hence including profile 2. Thus, the lower the angle of attack, the larger the magnitude of the bank angle profile.

8.7 Changing the initial ﬂight-path angle

As mentioned before, the initial flight-path angle has a large influence on the trajectory. The reference initial flight-path angle is varied six times:

1. 0.2◦ − 2. 0.1◦ − 3. 0.05◦ − 4. +0.05◦

5. +0.1◦

6. +0.2◦

Like in the previous section, the influence is tested on the planner’s ability to converge, the ability to track the reference profile and the path-constraint values. The latter is tested by taking the calculated bank angle from the reference profile during open-loop simulations. The result is presented in subsection 8.7.1. Subsection 8.7.2 presents, hereafter, the results for an active planner and the tracking algorithm. In both subsections, 2π the nominal trajectory has changed w.r.t. the previous section. A value of 16 was chosen for ω0. This gave a better tracking performance.

8.7.1 Using the nominal bank angle proﬁle

When the initial flight-path angle is increased by 0.2◦, a skipping trajectory is induced. The flight-path angle becomes more negative from the start until it is positive. Now, the skip has started. In addition, both case 1 and case 5 include a small skip in the trajectory. In table 8.4, the effects of a different initial flight-path angle γe on the trajectory parameters can be found. This illustrates the sensitivity of the trajectory to the settings of the tracker. From table 8.4, it is concluded that a larger γe:

- Increases range.

144 8.7. Changing the initial ﬂight-path angle

Table 8.4: Trajectory parameters for changing γe with constant σ(E).

Nominal 1 2 3 4 5 Distance [km] 8400 6890 7480 7880 9170 10560 Flight time [min] 24.9 21.5 22.8 23.7 26.6 29.7 MJ Q[ m2 ] 385 351 365 374 398 418 kW q˙ [ m2 ] 436 545 472 438 458 488

- Increases ﬂight time.

- Increases heat load.

The opposite holds for a smaller γe. The heat flux, however, is larger for all cases. The different behaviour for the heat load can be explained by the energy dissipation. A larger γe gives a slower energy dissipation. A smaller γe results in a faster energy dissipation. Here, the flight time is a more determining factor for the heat load than the maximum heat flux. When the trajectories are plotted as a function of energy, it is as if all the trajectories are in a damped phugoid motion w.r.t. the nominal case. This is why they all experience a larger heat flux. At some point they are below the reference trajectory, while the velocity barely shows a deviation.

8.7.2 Employing the planning algorithm

◦ Again, when γe is increased by 0.2 , a skip trajectory is induced. This time cases 4 and 5 show a small skip as well. The mission has failed as well for an increase of 0.1◦. The larger initial ﬂight-path angle gives a large distance travelled in the initial part of the entry. The required trajectory is so small that the converged Dc is much above the active path constraints. Case 1 and 2 show the opposite behaviour. The distance travelled is relatively small due to smaller initial ﬂight-path angle. This has to be compensated for by a lower value of Dc. The bank angle magnitude is then small as well. Hence, lateral manoeuvrability is small and the algorithm could not converge to the TAEM interface. Only for case 3 and 4 the algorithm has converged successfully and no path constraints were violated. Table 8.5 gives the simulation characteristics for these cases.

Table 8.5: Simulation characteristics for a changing γe.

Nominal 3 4 TAEM miss [km] 0.4 27.4 12.5 m Dc [ s2 ] 8.0 7.2 12.5 MJ Q [ m2 ] 385 388 369 kW q˙ [ m2 ] 436 421 518

Both case 3 and 4 show a relatively large miss distance. This could be caused by the gain selection, which is kept the same as for the nominal case to test the performance of the tracker. A lower γe gives a lower value for Dc as more energy has been dissipated in the beginning of the entry. The opposite holds true for a larger γe. The bank angle profiles for case 3 and 4 can be seen, w.r.t. the nominal profile, in figure 8.32. Here, it also holds that a larger bank angle magnitude profile has a later reversal and a smaller bank angle magnitude profile has an earlier reversal. A larger γe has a lower energy dissipation and a larger distance travelled in the initial part of the entry. Hence,

145 100

60 Nominal Proﬁle 40 -0.05 +0.05 20

−20 Bank angle [deg] −40

−60

−80

−100 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Energy [-]

Figure 8.32: Bank angle proﬁles 3 and 4. at a plan update, the algorithm will try to increase the drag (and thus bank angle). This causes a larger maximum heat ﬂux w.r.t. the nominal value. The opposite holds true for a lower γe.

146 CHAPTER 9

Conclusions and Recommendations

The main question of this thesis work was formulated as:

Is it possible to design an on-board executable guidance algorithm, for the hypersonic transition phase, that safely targets the TAEM interface?

From this question, the following tasks were derived:

Task 1: Simulator development

Task 2: Trajectory planner design

Task 3: Trajectory tracker design

Task 4: Guidance algorithm testing

In section 9.1, the conclusions are given that were drawn during this thesis work. Hereafter, section 9.2 presents the recommendations for future work related to the main question.

9.1 Conclusions

To answer the main question, a simulator was developed that can be used to study entry missions. This simulator serves as a test bed for the planning and tracking algorithms. The performance of a guidance algorithm design can only be assessed if it is tested with a simulator that has been validated. A trajectory simulation performed with the developed simulator has been compared with a reference trajectory from START. The angle of attack and bank angle profiles from the reference mission were used in an open-loop simulation. During the simulation, the control surface deflections for trim are computed. After assessing the differences between the two trajectories, it is concluded that the simulator can be used to study un-powered entry missions with winged vehicles. There are minor differences in the trajectories, for which the cause is twofold. First, a different model is used for the US 1976 Standard Atmosphere. Second, the developed simulator

147 uses extrapolation for the aerodynamic coefficients of the control surfaces after Mach 20, while START does not. The design of a trajectory planner is decomposed into an angle of attack and bank angle planner. Because the planning methods use approximations to the flight dynamics, they can only be tested in simulations under which these approximations hold, or in combination with a trajectory tracker. By incorporating the bank angle planning and tracking algorithms in one guidance algorithm, the main question is answered positively. The algorithm executes fast enough for an on-board implementation. Furthermore, the TAEM interface is successfully targeted under the assumption that the constraint on the final heading can be taken care off. The remainder of this section shall present conclusions drawn regarding the planning algorithms and guidance system testing.

Angle of attack planning With regard to the angle of attack planning, the following conclusions have been drawn. When increasing the angle of attack proﬁle with respect to a reference proﬁle:

- The range decreases.

- The heat ﬂux decreases.

- The heat load decreases.

The opposite is true for a decrease in angle of attack.

Bank angle planning The bank angle planning method is found to be highly sensitive to the initial drag choice. A low value for the initial drag leads to large values for the constant drag segment. This has the result that the path constraints are easily violated. On the other hand, a large initial drag value could lead to a low constant drag segment. This segment could fall below the minimum drag constraint. The result is a low manoeuvrability that reflects itself in a low bank angle magnitude. The planner is, hereafter, not able to converge to the TAEM interface. It is, therefore, concluded that planning a three segment linear spline profile is highly inflexible.

Guidance system implementation The open-loop guidance including plan updates gives a significant improvement over open-loop guidance only. However, only the closed-loop guidance algorithm could successfully lead the vehicle to the TAEM interface. The tracking algorithm was found to perform sufficiently well in handling a GRAM-99 wind model as a disturbance. Including trim in the planning results in a too high execution time for an on-board implementation. However, including trim in the simulation and not in the planning did not give a successful trajectory. Gain tuning might have solved this. When the angle of attack profile was modified, the planning algorithm was able to converge for all six off-nominal profiles. However, the tracking algorithm could only bring two out of six to a mission success. It is concluded that the performance of the tracker is highly sensitive to the gains that are used. The Horus vehicle could not successfully approach the TAEM interface for both the Horus and X-33 reference mission. It is concluded that the planning algorithm is not flexible enough, as it is possible to fly the mission definition for Horus.

148 9.2. Recommendations

9.2 Recommendations

Both the guidance algorithm and analysis performed can be improved to remove the limitations and assumptions made to answer the main question. For this purpose, the recommendations are presented in this section. They are presented below in three groups:

1. Planning algorithm.

2. Tracking algorithm.

3. Removing the limitations of the study.

Planning algorithm recommendations

- Include the meeting of the ﬁnal heading constraint in the planning. A method is given in [Saraf et al., 2003].

- Combine the angle of attack planning with the bank angle planning. Now, the bank angle planning method uses an angle of attack proﬁle as input. This limits the algorithms capability to execute autonomously. Possible points of departure:

1. Start from the angle of attack planning method. More specifically, one could find a profile by an iterative process based on a required range. One could link the required range to the range travelled in a 2D simulation that follows from a certain profile. Based on the difference between the two, one could change the angle of attack profile. 2. Create an entry corridor of three variables: the angle of attack, drag acceleration and energy. Then, a method needs to be found that searches for a drag profile in this corridor.

- Remove the breakpoints in the angle of attack and planned drag profile. These have been found to give rise to tracking errors. In addition, planning a linear spline does not allow for a flexible profile, while the mission definition might demand it. The drag profile planning should, therefore, allow for more shaping in the corridor. This could also solve the sensitivity for the initial drag choice. For improving the planned drag profile one could start from [Leavitt and Mease, 2007].

- Include a planning update algorithm after the bank reversal to enhance TAEM interface targeting.

Tracking algorithm recommendations

- Incorporate a gain selection method. While the algorithm works, it has been shown that the performance is highly sensitive to the gains used in the tracking algorithm. More specifically, when trim or a different angle of attack profile is included in the planning, it is not possible to reach the TAEM interface. This is caused by the tracking algorithm, as the bank angle planner did converge successfully. Incorpo- rating a gain selection method would improve this.

- Include a more advanced trajectory tracker. A starting point could be the approximate receding-horizon control law described in [Lu, 1999].

149 - Include the angle of attack as a control variable. One could start from [Saraf et al., 2003].

Removing the limitations of the study The research that has been performed in thesis work holds under the following limitations that were presented in the introduction. Logically, the answer to the main question also only holds under these limitations. In a real mission, these limitations are not present and should therefore be removed. Below, recommendations are given for each limitation.

- Include a control system. This allows the study of the rotational motion besides the translational motion only. The control system puts a limit on the degree to which the guidance commands can be achieved. Hence, the steering towards the reference trajectory is inaccurate. This will cause larger deviations and as such is an eﬀect that needs to be studied.

- Incorporate a navigation system. Steering becomes less eﬀective if one does not know for sure from what vehicle state is steered. The impact of this should, therefore, be studied.

- Execute the planning algorithm in parallel during the simulations. In the thesis work, when the algorithm is executed, the vehicle’s state is frozen. The new planned trajectory is taken into account immediately after the planning algorithm has finished. In reality, the state of the vehicle has changed. Especially in the initial part of the trajectory, where the velocity is low and the drag is high, this effect is noticeable and could have a negative effect on the performance.

- Include aerodynamic stability and control derivatives. By including them, the vehicle’s motion is simulated more accurately. However, including them in the planning could lead to large execution times for the planning algorithm. Consequently, if not included in the planning, the vehicle will start to deviate from the reference trajectory.

- Include ﬂexible vehicle models (same reasoning as previous recommendation).

- Include more elaborate environment models in the simulation (same reasoning as previous recommendation).

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154 APPENDIX A

Variations on the US 1976 Standard Atmosphere

This appendix presents difference between the US 1976 Standard Atmosphere model and an exponential model. In addition, a wind model taken from [Justus and Johnson, 1999] is presented. The exponential model is evaluated with ρ0 = 1.225 and β = 1/7050. Figure A.1 presents the density variation with altitude. Figure A.2 shows the percentage difference in density given by the US 1976 model with respect to the exponential atmosphere model. The temperature variation with height can be seen in figure A.3.

1.4

Exponential atmosphere 1.2 US 1976 Standard Atmosphere

1 ] 3 0.8 kg/m

0.6 Density [

0.4

0.2

0 0 5 10 15 20 25 30 35 40 Height [km]

Figure A.1: Density variation with altitude.

The wind model given in [Justus and Johnson, 1999] can be seen in ﬁgure A.4.

155 60

Diﬀerence in density−20 [%]

−40

−60 0 20 40 60 80 100 120 140 Height [km]

Figure A.2: Density variation w.r.t. Exponential atmosphere model

380

360 Exponential atmosphere US 1976 Standard Atmosphere 340

320

] 300 K

280

Temperature [ 260

240

220

200

180 0 20 40 60 80 100 120 140 Height [km]

Figure A.3: Temperature variation with altitude.

156 80

West to East 40 South to North Upward 20

0 Velocity [m/s] −20

−40

−60

−80 0 20 40 60 80 100 120 140 Height [km]

Figure A.4: Wind direction and magnitude as a function of height.

157 158 APPENDIX B

Derivation of Entry Corridor Equations

In this appendix, the derivation of the equations for the formation of the 2D and drag- energy entry corridor can be found. They are presented in sections B.1 and B.2, respectively.

B.1 2D entry corridor equations

In this section, the derivation of the equations for 2D entry corridor are given. These equations are used in section 6.1.1. First, maximum height corresponding to equilibrium ﬂight is derived. Hereafter, the minimum heights corresponding to the heat ﬂux, g-load, qdynα and dynamic pressure constraints are given.

Maximum height from equilibrium glide ﬂight The equilibrium glide ﬂight is given by equation 2.50. For clarity it is repeated below.

V 2 L + m mg = 0 (2.50) r − 1 2 Now, rewriting this equation and substituting L = 2 ρV CLS gives:

2 1 2 V ρV CLS + mg 1 = 0 (B.1) 2 gr − 2 Substituting the circular velocity Vc = gr and the exponential atmosphere model ρ = −βh ρ0e :

2 1 −βh 2 V ρ0e V CLS mg 2 1 = 0 (B.2) 2 − Vc − Rewriting for e−βh leads to:

V 2 mg 1 V 2 e−βh = − c 1 2 (B.3) 2 ρ0CLSV 159 Assuming a constant Vc and g gives for the equilibrium ﬂight height:

1 1 mg 2 2 1 V − Vc h = ln 1 (B.4) −β 2 ρ0CLS Minimum height from the heat flux c The convective stagnation point heat flux is given by equation 3.6: q˙c = C√ρV 2 . Assum- ing a zero radiative heat flux, a minimum height can be found as a function of velocity. Squaring equation 3.6 gives:

2 2 2·c2 q˙c = C ρV (B.5)

Substituting the exponential atmosphere model gives:

2 −βh q˙c ρ0 exp = (B.6) C2V 2·c2 Rewriting the previous equation gives the height as a function of a velocity and convective heat ﬂux. This can be seen in equation B.7. If the maximum heat ﬂux is substituted, the height is a minimum height for a velocity.

2 1 q˙c h = ln 2 2·c (B.7) −β ρ0C V 2 Minimum height from the g-load Rewriting equation 3.7 for the g-load generated by the lift and drag gives:

1 2 q 2 2 √L2 + D2 2 ρV S CL + CD n = = (B.8) mg mg

Substituting the exponential atmosphere model for ρ and assuming that CL and CD are not a function of height gives:

  1 n mg h = ln  q·  (B.9) −β 1 2 2 2 2 ρ0V S CL + CD

Again, if the maximum g-load is substituted, the equation evaluates to a minimum height.

Dynamic pressure constraints To come up with a minimum height corresponding to the dynamic pressure constraints, the following approach is used:

1 2 - Write the dynamic pressure as 2 ρV . - Substitute the exponential atmosphere model for ρ.

- Rewrite to height as a function of velocity.

160 B.2. Drag-Energy entry corridor equations

This gives for the minimum height hmin from the dynamics pressure constraints:

! 1 (qdyn α)max h = ln · min 1 2 (B.10) −β 2 V αρ0

! 1 qdyn h = ln max min 1 2 (B.11) −β 2 V ρ0

B.2 Drag-Energy entry corridor equations

In this section, the derivation of the equations for the drag-energy entry corridor is given. These equations are used in section 7.1. The lower boundary of the corridor is formed by the zero-bank equilibrium glide ﬂight. The upper boundary is formed by the path constraints. First, the equation is derived for the minimum drag acceleration. Hereafter, the equations for the maximum drag acceleration following from the path constraints are derived. The drag acceleration D is given by:

1 2 ρV CDS D = 2 (B.12) m Minimum drag from zero-bank equilibrium glide The zero-bank equilibrium glide equation is given by equation 2.50. It is repeated as:

V 2 L + m mg = 0 (2.50) r − This equation can be rewritten for the minimu lift force as:

V 2 Lmin = mg m (B.13) − r The lift force from the previous equation is the minimum lift to maintain equilibrium glide flight. Actually, one can interpret this as the vertical lift when realising that L is in 2 fact L cos σ if the assumption of 2D motion is no longer made. Hence, if L > m V mg, the r − bank angle can be used to maintain equilibrium glide flight. Therefore, the lift calculated by equation is the minimum lift in order to maintain the flight-path angle. The minimum drag acceleration Dmin is calculated using the minimum lift and the lift-to-drag ratio as:

2 Lmin CD V CD Dmin = = (g ) (B.14) m CL − r CL

Maximum drag from (convective) heat ﬂux The convective heat ﬂux can be calculated by equation 3.6:

c q˙c = C√ρV 2 (3.6)

161 Isolating the density and using q˙c = (q ˙c)max gives:

2 (q ˙c) ρ = max (B.15) C2V 2c2 The drag acceleration can be rewritten for the density as:

mD ρ = 1 2 (B.16) 2 V CDS Now, equating the previous two equations:

2 mD (q ˙c) = max (B.17) 1 2 2 2c2 2 V CDS C V The maximum drag acceleration that corresponds to ﬂight at the constraint limit can be obtained by rewriting equation B.17:

2 (q ˙c)maxCDS Dmax = (B.18) 2mC2V 2(c2−1) Maximum drag from the g-load The g-load that is collinear with the ZB-axis can be computed by:

L cos α + D sin α 1 C cos α + C sin α n = = ρV 2S L D (B.19) mg 2 mg Isolating the dynamic pressure from drag acceleration in equation B.12:

1 mD ρV 2 = (B.20) 2 CDS

Equating the previous two equations and using n = nmax gives for the maximum drag leads to:

nmaxg Dmax = (B.21) CL cos α + sin α CD Maximum drag from the dynamic pressure constraints The maximum drag for the qdynα and qdyn constraints can be obtained similar to the maximum drag for the g-load constraint. The dynamic pressure is isolated and equated to equation B.20. The maximum drag acceleration corresponding to the qdynα and qdyn constraints have been presented before in equations 7.4 and 7.6, respectively. They are repeated below for completeness.

(qdyn α)maxCDS D < · (7.4) mα

qmaxC S D < D (7.6) m

162 APPENDIX C

Linear Spline Segments Derivation

This appendix presents the derivation of the trajectory length covered by three linear spline segments of drag as a function of energy. The second segment is a constant drag segment. The segments can be seen in ﬁgure C.1.

Figure C.1: The linear spline segments.

In the ﬁgure,

D1(E) - Drag segment 1 D2(E) - Drag segment 2 D3(E) - Drag segment 3 Di - Initial drag Df - Final drag Dc - Constant drag Ei - Initial energy Ef - Final energy E1 - Breakpoint 1 E2 - Breakpoint 2

163 The trajectory length S is given by:

Z tf S = V dt (C.1) 0

Here, tf is the ﬁnal time and V is the velocity given as a function of time. The distance travelled as a result of a D(E)-proﬁle can be found by looking at the rate of work performed. The rate of work performed is equal to power. For an atmospheric entry, the reduction in energy is given by dE = DV . Substituting this in equation C.2 gives: dt −

Z tf Z Ef dE S = V dt = (C.2) 0 − Ei D(E) Expanding the previous relation for a drag proﬁle of three linear spline drag segments:

Z Ef dE Z E1 dE Z E2 dE Z Ef dE S = = (C.3) − Ei D(E) − Ei D1(E) − E1 D2(E) − E2 D3(E) Where,

Ef - Final energy Ei - Initial energy

The general equation for a linear function is: y = a+b(x c). One can derive the function − representations for the three segments by looking at ﬁgure C.1.

Di Dc D1(E) = Dc + − (E E1) (C.4) Ei E1 − −

D2(E) = Dc (C.5)

Dc Df D3(E) = Df + − (E Ef ) (C.6) E2 E − − f The next step is to substitute equations C.4-C.6 in equation C.2 and performing the integration. First, one needs to have an expression for the inverse integration of a linear equation. The general equation for a linear function is: y = a + b(x c). The integral − now becomes:

Z x2 dx (C.7) a + b(x c) x1 − Using the substitution rule for integration with u = a + b(x c). Then, du = bdx and: −

1 Z du 1 = ln u (C.8) b u b

164 Substituting the expression for u back in the previous equation gives for the integral:

x Z 2 x2 dx 1 1 ln(a + b(x2 c)) = ln(a + b(x c)) = − (C.9) a + b(x c) b − x1 b ln(a + b(x1 c)) x1 − − For a constant segment y = a the integration becomes:

x Z 2 x2 dx 1 x2 x1 = x = − (C.10) x1 a a x1 a

For each segment, the coeﬃcients a c can be expressed as in table C.1. − Table C.1: Coeﬃcients a c for the three linear segments −

D1(E) D2(E) D3(E)

a Dc Dc Df D −D b Di−Dc 0 c f Ei−E1 E2−Ef c E1 0 Ef

The trajectory length can now be expressed as:

E1 Ei Dc E1 E2 Ef E2 Df S = − ln + − + − ln (C.11) Di Dc Di Dc Dc D Dc − − f

165 166 APPENDIX D

Drag Derivatives to Energy

In this appendix, the ﬁrst and second derivative of drag acceleration with respect to energy are derived. Section D.1 the ﬁrst derivative is derived. The derivation of the second derivative of drag acceleration to energy is given in section D.2. The drag acceleration is given by equation D.1. In this appendix D represents the drag acceleration and no the drag force.

1 D = ρV 2C S/m (D.1) 2 D D.1 First derivative

The derivative of drag with respect to energy is given by a partial diﬀerentiation:

a b c d e f z}|{ z}|{ z}|{ z}|{ z }| { z }| { dD ∂D ∂ρ ∂D ∂V ∂D ∂C = + + D (D.2) dE ∂ρ ∂E ∂V ∂E ∂CD ∂E Below, the terms a-f will be given or derived in logical order. Term a:

∂D 1 = V 2C S (D.3) ∂ρ 2 D Term b: Since E = E(r, V ), the partial diﬀerentiation of density with respect to energy is expanded as:

∂ρ ∂ρ ∂r ∂ρ ∂V = + (D.4) ∂E ∂r ∂E ∂V ∂E ∂ρ ∂r ∂ρ By knowing that ∂V = 0 and obtaining ∂E from the equations of motion, ∂E is found to be:

∂ρ ∂ρ sin γ = (D.5) ∂E − ∂r D 167 Term c:

∂D = ρV C S (D.6) ∂V D Term d:

∂V ∂V ∂t = (D.7) ∂E ∂t ∂E In equation D.7 the diﬀerential equation for velocity with respect to time given in equation 2.32 can be substituted. By assuming a non-rotating planet and a central gravity ﬁeld, equation D.7 becomes:

∂V 1 D + g sin γ = ( D g sin γ) − = (D.8) ∂E − − DV DV Term e:

∂D 1 = ρV 2S (D.9) ∂CD 2 Term f:

∂C D = C0 (D.10) ∂E D Equation D.10 introduces the notation that ()0 means a diﬀerentiation w.r.t energy. Substituting the terms a-f in equation D.2 gives:

0 0 2 CD 1 ∂ρ 2g D = D 2 + + sin γ + 2 (D.11) V CD −ρ ∂r V D.2 Second derivative

The second derivative of drag to energy starts with the diﬀerentiation of D0 to energy. This diﬀerentiation is formulated as:

A 2 0 z }| 0 { ∂ D ∂D 2 CD ∂ 2 CD 2 = 2 + + 2 + D + (D.12) ∂E ∂E V CD ∂E V CD B C z }| { z }| { ∂ sin γ 1 ∂ρ 2g ∂ 1 ∂ρ 2g + + + + sin γ (D.13) ∂E −ρ ∂r V 2 ∂E −ρ ∂r V 2

The terms A-C need to be expanded as is done in the following.

Term A:

1 2 0 z }| { z }|0 { ∂ 2 CD ∂ 2 CD 2 + D = D[ 2 + ] (D.14) ∂E V CD ∂E V CD

168 D.2. Second derivative

Expanding term 1:

∂ 2 ∂ 2 ∂V 4 D + g sin γ 4 D + g sin γ = = − = − (D.15) ∂E V 2 ∂V V 2 ∂E V 3 DV V 4 D

Here, equation D.8 has been substituted for the dynamic EoM for the velocity with respect to energy.

Expanding term 2:

0 0 0 00 02 ∂ CD ∂ CD ∂CD CD CD = 0 = 2 (D.16) ∂E CD ∂CD CD ∂E CD − CD Collecting terms 1 and 2 gives for term A:

" 00 02 # 4 D + g sin γ CD CD D −4 + 2 (D.17) V D CD − CD

Term B:

∂ sin γ ∂ sin γ ∂γ ∂γ = = cos γ (D.18) ∂E ∂γ ∂E ∂E

∂γ The equation of motion for the ﬂight path angle can be substituted for ∂E . This equation can be derived from equation 2.33. Assuming a zero side force S and again assuming a central gravity ﬁeld and a non-rotating planet gives:

∂γ ∂γ ∂t L cos σ g cos γ V cos γ 1 L cos σ g cos γ cos γ = = + − = + (D.19) ∂E ∂t ∂E V − V r DV − DV 2 D2 − rD

Where, L is the lift acceleration instead of the lift force.

Term C:

1 2 z }| { z }| { ∂ 1 ∂ρ 2g ∂ 1 ∂ρ ∂ 2g + sin γ = sin γ + sin γ (D.20) ∂E −ρ ∂r V 2 ∂E −ρ ∂r ∂E V 2

The derivation of term 1 is as following:

∂ 1 ∂ρ ∂ 1 ∂ρ 1 ∂ ∂ρ = + ∂E −ρ ∂r ∂E −ρ ∂r −ρ ∂E ∂r ∂ 1 ∂r ∂ρ 1 ∂2ρ ∂r = (D.21) ∂r −ρ ∂E ∂r − ρ ∂r2 ∂E ∂ 1 ∂ρ sin γ 1 ∂2ρ sin γ = + (D.22) ∂r ρ ∂r D ρ ∂r2 D

169 In the previous equation, the EoM for the radius r has been substituted. Starting from equation 2.35 it is found to be:

∂ r ∂ r ∂t 1 sinγ = = V sinγ − = − (D.23) ∂E ∂t ∂E DV D

Now, using equation D.8, term 2 is given as:

∂ 2g ∂ 2g ∂V 4g D + g sin γ = = − (D.24) ∂E V 2 ∂V V 2 ∂E V 3 DV

Term C can now be written as:

∂ 1 ∂ρ 2g sin2 γ ∂ 1 ∂ρ 1 ∂2ρ 4g D + g sin γ + sin γ = + sin γ (D.25) ∂E −ρ ∂r V 2 D ∂r ρ ∂r ρ ∂r2 − V 4 D

D00 can be formulated by collecting terms A-C. By neglecting the term proportional to sin2 γ and assuming that D + g sin γ D, D00 can be written as: ≈

00 02 ! 0 00 CD CD 0 CD 2 4D D = D 2 + D + 2 4 + CD − CD CD V − V 1 1 ∂ρ 2g V 2 1 1 ∂ρ 2g + + g + (L/D) cos σ (D.26) DV 2 −ρ ∂r V 2 − r − V 2 −ρ ∂r V 2

In short notation, this is written as:

L D00 = a + b cos σ (D.27) D

Where, logically, a and b are given as:

00 02 ! 0 CD CD 0 CD 2 4D a = D 2 + D + 2 4 + CD − CD CD V − V 1 1 ∂ρ 2g V 2 + + g (D.28) DV 2 −ρ ∂r V 2 − r

1 1 ∂ρ 2g b = + (D.29) −V 2 −ρ ∂r V 2

170 D.3. Second derivative including Coriolis force

D.3 Second derivative including Coriolis force

The Coriolis force is the largest of the ﬁctitious forces. One could therefore argue that it needs to be included in the expression for D00. The Coriolis force can be included dγ in equation D.19 as given in the previous section. There it is included where dE is substituted. The Coriolis force Cγ can be found by substituting it from the dynamic dγ equation dt as given in equation 2.33:

1 Cγ = 2ω cos δ sin χ − (D.30) cb DV D00 is now still give as in equations D.27 to 7.20, except that equation D.28 is now written as:

00 02 ! 0 CD CD 0 CD 2 4D a = D 2 + D + 2 4 + CD − CD CD V − V 1 1 ∂ρ 2g V 2 1 ∂ρ 2g + + g + + Cγ (D.31) DV 2 −ρ ∂r V 2 − r −ρ ∂r V 2

171