An Extended Analytic Range Corrector Method for the Space Shuttle Entry Guidance Algorithm by Erin Elizabeth Evans B.S. Mechanical Engineering, California Institute of Technology (2015) Submitted to the Department of Aeronautics and Astronautics in partial fulfillment of the requirements for the degree of Master of Science in Aeronautical and Astronautical Engineering at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2018 Massachusetts Institute of Technology 2018. All rights reserved.
Signature redacted Author ...... Department of Aeronautics and Astronautics redacted May 24, 2018 Certified by ...... Signature I...... Professor Jonathan P. How, Ph.D. Professor, Department of Aeronautics and Astronautics Thesis Supervisor Certified by ...... Signature redacted...... Stephen Thrasher Guidance Engineer, C.S. Draper Laboratory Thesis Supervisor Accepted by ...... Signature redacted ...... MASSACHUSETTS INSTITUTE Professor Hamsa Balakrishnan OF TECHNOLOGY Associate Pr ofessor of Aeronautics and Astronautics JUN 28 2018 Chair, Graduate Program Committee LIBRARIES ARCHIVES 2 An Extended Analytic Range Corrector Method for the Space Shuttle Entry Guidance Algorithm by Erin Elizabeth Evans
Submitted to the Department of Aeronautics and Astronautics on May 24, 2018, in partial fulfillment of the requirements for the degree of Master of Science in Aeronautical and Astronautical Engineering
Abstract Space shuttle entry guidance with an extended analytic range corrector method is presented. The guidance method is a variation of Shuttle entry guidance in which the parameters that define the drag profile are modified using quadratic splines to make the drag profile smooth and easier to customize. In general, in order to account for off- nominal entry conditions and ensure the vehicle flies the correct range to the runway, the nominal reference drag profile is modified on-line utilizing analytic expressions for the derivative of range with respect to the relevant drag profile parameter. This new profile is then used to calculate a reference drag command in the subsequent guidance algorithm cycle. Typical implementations of Shuttle entry guidance modify the drag profile using only one variable to shift the profile by a constant value. This presents problems when the vehicle is highly constrained and can easily violate constraints such as heat load and heat rate constraints due to small drag profile variations. The methods by which the drag profile is updated are changed in order to provide multiple perturbation options. In providing multiple drag profile update parameters, a memoryless range error allocator is implemented with a vector of weights as a design variable. The allocator parameters are designed to take into account heat load while remaining within constraints using a high L/D vertical takeoff horizontal landing reusable launch vehicle simulation. The resulting algorithm seeks to leverage the high-TRL Shuttle entry guidance routine by making minimal modifications to the implementation, while increasing robustness to entry interface dispersions under tight heating constraints. A discussion of the design of the drag profile is included, in which the selection of profile update parameters is explored. Results from optimization of these parameters using a genetic algorithm are presented, as well as Monte Carlo results demonstrating that the allocator can reduce failure rates due to tight drag constraints from 42% to 0%, establishing the impact and success of this analytic range corrector method.
Thesis Supervisor: Professor Jonathan P. How, Ph.D.
3 Title: Professor, Department of Aeronautics and Astronautics
Thesis Supervisor: Stephen Thrasher Title: Guidance Engineer, C.S. Draper Laboratory
4 Acknowledgments
There is an insanely large number of people without whom this thesis and my success in grad school would not be possible. First, I would like to thank Charles Stark Draper Laboratory, for giving me the opportunity to do research so closely aligned with my interests, and for providing the resources for me to pursue my Master's Degree at MIT. I would like to thank Stephen Thrasher, my mentor, for providing endless insight and being such a patient sounding board for many ideas, both good and very bad. Thanks also to my faculty advisor, Professor Jonathan How. It was truly a pleasure and an honor to be your student, and I value all the advice and insight you provided me these past two years. I also want to extend thanks to Ross, who is my teammate, my biggest cheerleader, my day one, and my infinite source of energy. From studying for my qualifying exams with me, listening to proofs half asleep on the floor, and bringing me dinner when I'm working too late, all the way to waiting outside my qualifying exam room door for me and packing up your life to start our next adventure in Seattle, you've been there for me with patience and enthusiastic encouragement every single day. I couldn't have done it without you. To Nikita, Bjorn, Brandon, Noam, Brett, Chris, Kris, and all the members of the ACL, you made being part of the lab a fun and rewarding experience. Thanks for the Nerf fights and late nights, the long talks and coffee breaks. To my roommates, Alison, Justin, Kelly, and Scott, you made living in Boston a very exciting two years. To my family, for supporting me and giving me every possible opportunity. I would never have had the opportunity to attend MIT without you. And finally, to Ali, Brian, Pat, and my Lord Hobo family, thank you for being fantastic humans and fantastic friends. You were there when I needed you, and I'm so lucky I had you at my side through grad school.
5 6 Contents
1 Introduction 21 1.1 B ackground ...... 23 1.1.1 Shuttle Entry Guidance ...... 23 1.1.2 Literature Review ...... 29 1.2 M otivation ...... 30 1.3 Thesis Objective ...... 32 1.4 Thesis Overview ...... 33
2 Simulation Environment 35 2.1 U nits ...... 35 2.2 Reference Coordinate Frames ...... 36 2.2.1 North, East, Down ...... 36 2.2.2 Earth Centered, Earth Fixed ...... 36 2.2.3 Earth Centered, Inertial ...... 37 2.2.4 Body Centered, Body Fixed ...... 37 2.3 Coordinate Transformations ...... 38 2.3.1 North, East, Down Frame to Earth Centered, Earth Fixed Frame 38 2.3.2 Earth Centered, Earth Fixed Frame to Earth Centered, Inertial ..38 Fram e ...... 2.3.3 Earth Centered, Inertial Frame to Body Centered, Body Fixed Fram e ...... 39 2.4 Environment Models ...... 39 2.4.1 Earth Gravity Model ...... 39
7 2.4.2 Earth Atmosphere ...... 40 2.5 Vehicle M odel ...... 40 2.6 Simulation State Propagation ...... 41 2.6.1 Bank Angle and Angle of Attack ...... 41 2.6.2 Equations of Motion ...... 41 2.7 Aerodynamic Heating ...... 41 2.8 Simulation Initialization ...... 43 2.8.1 Initial Conditions ...... 43 2.8.2 Terminal Conditions ...... 43 2.8.3 Monte Carlo Parameters ...... 44
3 Shuttle Orbiter Guidance Algorithm 45 3.1 Reference Drag Profile ...... 45 3.2 Reference Drag Profile Parameters ...... 48 3.3 Equations of Motion ...... 48 3.4 Range Prediction ...... 49 3.5 Reference Trajectory Range Update ...... 52 3.6 Other Reference Trajectory Parameters ...... 54 3.7 Angle of Attack and Bank Angle Commands ...... 55 3.8 Lateral Logic ...... 57
4 Problem Description 59
5 Allocator Overview 63 5.1 Quadratic Splines .... . 64 5.2 Updated Range Equations 66
5.3 Parameter 1: Di ...... 67 5.4 Parameter 2: kq...... 68 5.5 Parameter 3: V ...... 70 5.6 Memoryless Design . . . . 71 5.7 Parameter Limits ... . . 73
8 5.8 Weight Selection ...... 7 74
6 Allocator Optimization Design 75 6.1 Genetic Algorithm Overview ...... 77 6.1.1 Crossover ...... 78 6.1.2 M utation ...... 79 6.2 Nominal Trajectory Parametrization ...... 79 6.3 Nominal Optimization Fitness Function ...... 82
7 Results 87
7.1 Monte Carlo Results under Turbulent Flow Transition Constraints . . 87 7.2 Heat Load Example ...... 89 7.3 Reference Trajectory Optimization ...... 92 7.3.1 Standard Constraints ...... 92 7.3.2 Turbulent Flow Transition Constraints ...... 94
8 Conclusion 97 8.1 Results Summary ...... 97 8.2 Future Work...... 98
9 10 List of Figures
1-1 NASA's X-38 vehicle ...... 23 1-2 Sierra Nevada's Dream Chaser vehicle ...... 24 1-3 Typical spaceplane constraints in drag-velocity space 26 1-4 Example angle of attack reference profile ...... 27 1-5 Azimuth error deadband ...... 29
2-1 NED and ECEF coordinate frames ...... 36 2-2 HL-20 with BCBF axes ...... 37 2-3 Euler angle definition ...... 39 2-4 HL-20 ...... 40 2-5 Trim angle of attack vs. Mach number ...... 42 2-6 Trim CD vs. Mach number ...... 42 2-7 Trim L/D vs. Mach number ...... 42
3-1 Typical reference drag profile in drag-velocity space ...... 47 3-2 Reference drag profile without constant drag phase ...... 47 3-3 Example trajectory update in temperature control phase ...... 53 3-4 Example trajectory update in equilibrium glide phase ...... 53 3-5 Example trajectory update in constant drag phase ...... 54 3-6 Example trajectory update in transition phase ...... 54 3-7 Bank profile with only bank modulation ...... 56 3-8 Bank profile with bank and angle of attack modulation ...... 57 3-9 Entry azimuth error definition ...... 57
11 4-1 Drag profile perturbed for range errors ...... 60
4-2 Example of restrictive drag constraints ...... 61
5-1 Refrence drag oscillations caused by inaccurate range estimates . 66
5-2 Reference drag profiles when D1 is perturbed ...... 67 5-3 Reference drag profiles when kq is perturbed, Example 1 ...... 69
5-4 Reference drag profiles when kq is perturbed, Example 2 ...... 69 5-5 Reference drag profiles when kq is perturbed, Example 3 . .. . 69
5-6 Reference drag profiles when V, is perturbed ...... 71
5-7 All possible reference profiles for a family of range errors ...... 72
6-1 Standard upper and lower drag constraints for profile optimization . . 76 6-2 Upper and lower drag constraints for profile optimization with turbu- lent flow transition ...... 77 6-3 Crossover example for a set of binary variables ...... 78 6-4 Mutation example for a set of binary variables ...... 79 6-5 Reference drag profile optimization parameters ...... 80 6-6 Effect on drag profile of small spline knot perturbation 82 6-7 Upper and lower drag constraint margins ...... 83 6-8 Drag constraint cost for two reference profiles ...... 85
7-1 Drag with t3or range error and original update method ...... 88 7-2 Drag with 3- range error and range allocator ...... 88
7-3 kq curve for range allocator Monte Carlo ...... 88 7-4 Monte Carlo results for original shuttle update method ...... 89 7-5 Monte Carlo results for range allocator ...... 89 7-6 Reference profile for heat load example using D ...... 91 7-7 Reference profile for heat load example using V...... 91 7-8 Reference profile results when using the allocator to minimize heat load 92 7-9 Nominal profile under standard constraints ...... 93 7-10 Optimized profile under standard constraints ...... 94
12 7-11 Nominal profile under turbulent flow transition constraints ...... 95 7-12 Optimized profile under turbulent flow transition constraints . ... . 96
13 14 List of Tables
2.1 Units for vehicle simulation ...... 35 2.2 Initial conditions for vehicle simulation ...... 43 2.3 Parameters dispersed for Monte Carlo simulations ...... 44
7.1 Percent Heat Load Decrease between Nominal and Optimized Profiles 93 7.2 Percent heat load decrease between nominal and optimized profiles . 95
15 16 Nomenclature
Subscripts
1O Initial value
Xb Body centered, body fixed coordinate frame xc Commanded value
1 e Earth centered, earth fixed coordinate frame xi Earth centered, inertial coordinate frame
1 m Force per unit mass
Xn North, east, down coordinate frame xt A value at time step t
Xv Vertical
Xref Reference profile value
Variables
Oz Angle of attack ht Altitude rate
Q Heat rate
7 Flight path angle
17 T Coordinate transformation
Bank angle
Azimuth of Earth relative velocity vector p Air density
Po Atmospheric density at base altitude poo Freestream air density patm Sea level standard atmospheric density
T Greenwich mean sidereal time
0 Geocentric latitude
Geocentric longitude
CD Coefficient of drag
CL Coefficient of lift cij Temperature control coefficient j for curve i
D Drag
Do Reference drag at start of temperature control phase
D1 Reference drag at end of temperature control phase
D2 Reference drag at end of equilibrium glide phase
D3 Reference drag at end of constant drag phase
D4 Reference drag at end of transition phase di Drag for quadratic spline point i
Dint Reference drag at intersection of equilibrium glide and transition phases
18 E Energy (potential + kinetic)
Eint Reference energy at intersection of equilibrium glide and transition phases fx xth control law / reference command coefficient g Gravitational acceleration h Altitude
h3 Altitude used to define the start of the transition phase hs Atmospheric density scale height
J Optimization cost
K Allocator parameter kq Quadratic curve allocator parameter
L Lift
N Total number of quadratic spline curves n Number of quadratic curves in temp control phase
Q Heat load q Dynamic pressure qj Quadrature rule coefficient
R Downrange
Ir Distance from the vehicle to the Earth's center rc Leading edge radius
RCD Range of constant drag phase
REG Range of equilibrium glide phase
19 RTC Range of temperature control phase
RTR Range of transition phase
Sref Aerodynamic reference area t Time
V Velocity
V Reference velocity at start of temperature control phase
V1 Reference velocity at end of temperature control phase
V2 Reference velocity at end of equilibrium glide phase
V3 Reference velocity at end of constant drag phase
V4 Reference velocity at end of transition phase
Vi Velocity for quadratic spline point i
V, Velocity used to define equilibrium glide phase
V.. Freestream velocity
V{a,b,c}j Quadrature rule velocity evaluation points
Vit Reference velocity at intersection of equilibrium glide and transition phases wj Weight for parameter j
20 Chapter 1
Introduction
Re-entering the earth's atmosphere is a complex procedure that requires the vehicle
to first reach the boundary of the atmosphere at precise conditions, then navigate a
narrow corridor between physical constraints on the vehicle such as"g-loading" and
minimum drag, all while ending its descent at a pre-specified state and location. If
the vehicle enters the atmosphere at an angle that is too steep, high structural loading
will cause the vehicle to break apart, in addition to excessive heating that will exceed
the capability of the vehicle's thermal protection system (TPS). Conversely, if the
flight path angle is too shallow, the trajectory will have a late range correction, also
resulting in critically high heating. Furthermore, if the angle is excessively shallow, the resulting drag will not be great enough to capture the vehicle, allowing it to skip back out of the atmosphere into an elliptic orbit with too little energy to escape the planet. Although gravity will eventually pull the vehicle back into the atmosphere, it could land far away from where it was planned to land, which could be dangerous both for the vehicle's occupants and for anyone on earth near the location.
Hypersonic entry guidance algorithms operate during this crucial period between the deorbit burn just before the vehicle reaches the Entry Interface (EI) and a tran- sition point where the vehicle is handed off to an algorithm designed to align the vehicle with a runway or intended splashdown location, such as a Terminal Area En- ergy Management (TAEM) algorithm. During this period, the algorithm provides reference commands to the vehicle's control system, which are typically based on a
21 predetermined reference trajectory. For the Shuttle Orbiter, this hypersonic entry guidance period begins approximately at an altitude of 400,000 ft and a range 4100 NM away from the runway, and it continues until the vehicle altitude is about 80,000 ft, and 50 NM from the runway [18]. Reusable launch vehicles (RLV's) are one class of spacecraft that utilize hyper- sonic entry guidance algorithms repeatedly while undergoing the thermal stress and dynamic loads caused by atmospheric re-entry on a recurring basis due to the reusabil- ity of the vehicle. As a result, the performance of entry guidance algorithms when applied to RLVs is of particular importance, since the vehicle must be fit for reuse after each flight, and the impact of the algorithm's performance is magnified through multiple uses. Within the class of RLVs, there are two prevailing types of vehicles: high lift to drag ratio (L/D) spaceplanes and low L/D capsules [121. High L/D vehicles are typically winged, and have 2 degrees of freedom (DOF) that can be commanded during flight, bank angle and angle of attack, while low L/D capsules are rotationally symmetric and only have 1 DOF, bank angle. High L/D vehicles typically experience lower vertical acceleration, commonly referred to as "g-loading" during flight, which makes it preferable for missions with sensitive cargo or human passengers. Retrieval and recovery after landing is also substantially simpler, since the vehicle lands at a precise destination on land and does not require at-sea capsule recovery operations. Although the Space Shuttle Orbiter was the first major high L/D launch vehicle, there have been, and continue to be, many others under development. These include experimental vehicles such as Lockheed Martin's X-33 161, Orbital Science's X-34 [8], NASA's X-38 [141, and Boeing's X-37 [7]. NASA's shift towards using commercially developed vehicles for manned spaceflight and International Space Station(ISS) oper- ations, as well as the growth of private space industry, has also resulted in commercial vehicles in this category, such as Orbital Science's Prometheus proposed for NASA's Commercial Crew Development (CCDev) program [261 and Sierra Nevada Corpo- ration's Dream Chaser, which was selected for NASA's Crew Resupply 2 Mission (CRS2) [13].
22 Figure 1-1: NASA's X-38 vehicle [141
1.1 Background
1.1.1 Shuttle Entry Guidance
The Shuttle entry guidance algorithm, developed in 1978 by Jon Harpold and Claude Graves Jr., and outlined in detail in [18], has two primary objectives: "to guide the Orbiter along a path that minimizes the demands on the Orbiter system's design throughout the Orbiter missions and to deliver the Orbiter to a satisfactory energy state and vehicle attitude at the initiation of the terminal area guidance system". The algorithm was required to satisfy missions with a deorbit range that varies by up to 500 NM with orbital inclinations from 28.50 to 1040, as well as Orbiter weights between 150,000 lb and 227,000 lb. Furthermore, since the vehicle was manned, it had to be sufficiently flexible to generate abort trajectories throughout the mission.
Constraints
In order to quantify and predict vehicle failure, a number of constraints that limit the drag acceleration (and therefore altitude of the vehicle) at a given velocity are implemented. The majority of these constraints are most easily plotted with velocity
23 Figure 1-2: Sierra Nevada's Dream Chaser vehicle 113] on the x-axis and drag on the y-axis, or drag-velocity space, in order to visualize the entry corridor through which the velocity-scheduled reference drag profile must be designed. In general, dynamic pressure, q, is reduced whenever possible in order to reduce aerodynamic control surface hinge moments,
q = -pV2. 2
In order to reduce the internal structural weight of the Shuttle Orbiter, aerodynamic loads must also be decreased during reentry. This can essentially be translated into limiting the vertical acceleration, perpendicular to the Orbiter's longitudinal axis. In addition, aerodynamic heat rate must also be limited during re-entry in order to limit the mass of the insulation required for the TPS. These three constraints, heat rate, vertical acceleration, and dynamic pressure, form the upper limit on drag acceleration. The lower constraint on drag acceleration is the minimum bank equilibrium glide condition, which occurs when the flight path angle, -y, is constant for a given bank angle, 0, and crossrange control is absent,
L V2 L C g - - (1.2) cos r/
Violating this constraint will cause the vehicle to exhibit phugoid motion, and even- tually lose drag control authority. Given the angle of attack, this condition can be
24 translated into a lower drag limit,
D> ( - )(-- (1.3) rD
This constraint can generally be thought of as the shallowest glide slope the vehicle can fly given its attitude constraints. In addition to these constraints, consideration must also be given to heat load, which is found by integrating heat rate over the flight, and is a function of air density and velocity, as seen in Equation 1.4. As noted in [18], "a general trend of decreasing heat load, as the altitude is reduced and hence drag acceleration is increased, is well established". Because this constraint requires the integration of the vehicle state over time,
f Q jipV 3 dt, (1.4) t O
it cannot be plotted in drag-velocity space. A typical set of constraints are plotted in Figure 1-3, where multiple heat rate constraints for various points on the vehicle are all present. It should be noted that the velocity axis is decreasing to the right in all relevant plots in the work presented.
Reference Profile
The nominal trajectory is defined using reference profiles for angle of attack (a) and drag acceleration (D). The remaining reference values, altitude rate (h) and vertical L/D, which is the component of L/D in the plane formed by the position and Earth- relative velocity vectors ((L/D),), can be computed analytically from the angle of attack and drag acceleration reference profiles. The deviation of the vehicle from these reference values is then used in the guidance control law,
(L/D)v,c = (L/D)v,reg + fi(D - Drej) + f2 (h - href) + f4 J(D - Dref)dt. (1.5)
25 -Dynamic Pressure -Vert. Acceleration -- Eq. Glide -*-Heat Rate
Velocity
Figure 1-3: Typical constraints in drag-velocity space. The velocity axis is decreasing to the right in all relevant plots in the work presented.
This vertical L/D command is achieved using angle-of-attack modulation and bank angle modulation. The angle of attack reference profile is typically designed first and focuses on primarily satisfying heat rate and crossrange considerations. An example angle of attack reference profile can be found in Figure 1-4. In this nominal profile, the angle of attack remains at ~ 40 degrees throughout the majority of the trajectory, until it decreases to ~ 15 degrees starting at 10,000 fps, in order to transition to the subsequent Terminal Area Energy Management algorithm. This angle of attack decrease is required to move from the back side of the L/D curve required for high deceleration flight during the hypersonic phase to the front side of the L/D curve required for the airplane-like flight used while preparing to land. The reference drag profile is then typically designed to satisfy the remaining con- straints. According to 18], examination of the entry corridor shows that "an entry profile consisting of pseudoequilibrium glide, quadratic, linear, and constant drag segments as a function of Earth-relative speed can be linked together to provide the capability to achieve essentially any desired profile within the corridor". As a result, reference drag profiles in Shuttle guidance are separated into four phases. The first is
26 45-
40 EARLY OFT a PROFILE
as -OPERATIONAL
50- ANGLE Of ATTACK, CIG 23
20-
Is
0 25 20 15 10 5 0 RHATIV SPEED, TIOUSANDS OF FrS
Figure 1-4: Example angle of attack (a) reference profile [18]. the temperature control phase, delineated by a series of continuous quadratic curves, the second is the pseudoequilibrium glide phase, which is also quadratic in veloc- ity. The subsequent two phases are the constant drag phase, and then the transition phase, which is linear in energy.
Range Prediction
The predicted range for a given reference profile can be approximated using fVdV R Vd- (1.6) D(V)'
This approximation assumes that the flight path angle, y, is sufficiently small and D is sufficiently large to assume cos -y ~ 1 and D + g sin y ~ D. This approximation is valid for periods of high velocity, since the velocity is sufficiently high, and the descent rate is sufficiently low, for the flight path angle to be nearly 0 [18]. Once the small flight path angle assumption is no longer valid during the latter part of entry, the independent variable is changed from velocity, V, to energy, E,
V 2 E = gh + , (1.7) 2
27 d E R, D(E)' 18
This independent variable switch is motivated by the accuracy of the approximation D + g sin-y ~D, which is not accurate for low velocities. Both Equations 3.15 and 1.8 for range can be used to obtain analytic equations for range during each phase of the trajectory.
Reference Guidance Commands
After computing the reference vertical L/D command using Equation 1.5, this com- mand must be converted into bank angle (o-) and angle of attack (a) commands. Both the bank angle magnitude and angle of attack are modulated to track commanded vertical L/D. Bank angle is generally calculated according to Equation 1.9, while angle of attack is calculated according to Equation 1.10,
cos-1 (LID)c (1.9) L/D
Aac = CD(Dref - D)/fio. (1.10)
While the magnitude of the bank angle command is determined using Equation 1.9, the direction is controlled by a series of bank reversals with locations that are determined by an azimuth error deadband, as seen in Figure 1-5.
On-board Profile Updates
In order to account for range errors that occur during flight, the nominal reference drag profile must be adjusted onboard. Shuttle entry guidance produces this func- tionality by shifting the profile for the phase that the vehicle is currently in, which allows the vehicle to both accommodate and make corrections for significant naviga- tion errors before transition to the TAEM algorithm, and drive the trajectory back to nominal at the start of each phase. The modified range update algorithm presented in this thesis deals primarily with this update method.
28 30-
AZIMUTH ERROR DEADDAND, * VEHU x*TO TARGET 0
35 30 25 20 15 10 5 0 RELATIVE SPEED, THOUSANDS OF PS
Figure 1-5: Azimuth error deadband [18].
1.1.2 Literature Review
Previous range error update efforts include [16] in which the profile is scaled uniformly by a constant, and [24] - [23], in which optimal trajectories are found that satisfy the range requirement and minimize heat load plus a regularization term to promote smooth profiles. In these algorithms, the reference drag profile is scheduled on energy, resulting in more accurate range estimates for larger flight path angles than profiles scheduled on velocity, and is optimized off-line with respect to heat load. Reference [30] similarly modifies the way the reference drag profile is defined, using cubic splines scheduled with respect to energy that are then optimized for terminal time heat load using sequential quadratic optimization. Reference [10] defines a reference profile scheduled with energy, and implements an LQR controller on range-to-go as well as altitude, flight path angle, bank angle, and angle of attack, to account for range error. Although not as robust as algorithms with the ability to re-generate trajectories onboard due to its inability to handle significant changes in the system model which might occur due to a large number of possible vehicle failures, it produces smooth reference control profiles with trajectory- independent gains. Reference [31] uses a set of trajectories computed offline, selecting a reference trajectory based on the current state, including range to the runway. This method guarantees convergence, but requires significant development time before each
29 mission since each trajectory computed off-line must be optimized. Nonlinear programming is utilized to achieve a similar goal of modifying trajecto- ries for range error in [15], in which the reference drag profile is scheduled on energy. In this method, the ranging authority within an entry corridor is maximized by turn- ing the problem into an unconstrained quadratic problem that can be solved directly on-board without iterative methods, where the updated trajectory uses deviation from the nominal trajectory as the performance criterion. Other methods for updating trajectories on-line include Predictor-Corrector guid- ance, such as that in [34] and 13], where a trajectory is found by solving a two point boundary value problem, and parameters such as the magnitude of a constant bank angle or bank direction switching times are searched over to obtain a solution. In this particular case, the cost function takes range error into account, and seeks to minimize it along with other variables. A modification of the standard predictor-corrector guid- ance implementation is presented in [21], where the trajectory is defined using a cubic spline functions of specific energy with respect to range-to-go, and reference drag ac- celeration is obtained as the derivative of specific energy with respect to range-to-go. [22] also presents a modification to standard predictor-corrector guidance in which weights are implemented on the range-related error coefficients to improve algorithm flexibility. Another similar method is outlined in 133], in which a reference drag profile is forgone for a profile whose range error is corrected on-line by calculating the flight path angle, and as a result the angle of attack, needed to obtain the required range.
1.2 Motivation
Differences in range between the true range to the runway and the range to be flown by the reference trajectory can be caused by a variety of reasons. These range errors can occur due to atmospheric and aerodynamic dispersions, wind, sensor errors, and even poor reference bank angle tracking due to dispersions in vehicle center of mass. These errors are inevitable and all guidance algorithms must include methods for
30 accounting for them. The Shuttle entry guidance algorithm, which has become the basis for entry guid- ance for a number of modern atmospheric entry vehicles, is heavily restricted in its flexibility when generating reference trajectories on-board. As a result is incapable of navigating tight, complex constraints, such as those that might arise for a highly optimized vehicle. As trajectories are regenerated to account for range errors, the reference drag profile is only shifted along one parameter. For the first three phases, this involves shifting the profile vertically, therefore the algorithm is more suited for vehicles that have sufficient margin throughout the entirety of the entry corridor. Algorithms have been developed that allow for more flexibility than the Shuttle entry guidance algorithm in on-line trajectory generation including those outlined in section 1.1.2. Overall the methods outlined in this section are more complex than the Shuttle entry guidance algorithm, both in implementation and in guarantees of convergence. The results are often not deterministic, and their performance can be hard to assess using Monte Carlo methods, which ultimately results in algorithms that are time consuming and difficult to check out for flight. Guidance algorithms for high L/D vehicles are still being developed and improved upon, particularly in the areas of flexible on-board trajectory generation and robust highly constrained trajectory generation. This is primarily in response to the en- trance of improved computational power since the first guidance algorithms were designed, as well as the tighter constraints likely to become more common as thermal protection systems are pushed to become more lightweight for overall vehicle weight reduction. Because these algorithms are being designed for reusable launch vehicles, any improvement in vehicle weight, mission-specific development time, and stress on vehicle components is magnified, making the study of optimized trajectory generation significantly more important.
31 1.3 Thesis Objective
The objectives of this thesis are to develop a range error allocation method for the hypersonic Shuttle entry guidance algorithm to increase its ability to update the reference trajectory under constraints that are extremely tight, and carry out the optimization of the reference trajectory's parameters to fully explore the allocator's utility and benefits. This range error allocation method should improve the flexibility of the reference drag profile, enabling it to be modified onboard based on which constraints are expected to be tightest and which regions have the least margin. Overall, this algorithm modification should improve the robustness of the guidance algorithm and expand the usable region in which the vehicle can operate. Specifically, the range error allocation method includes a description of a number of reference profile modification parameters as well as a profile update method that is memoryless with respect to range error, thereby decreasing the effort required to verify the allocation method for flight. The developed method also includes a strategy to keep all reference profile parameters within safe ranges, so that a feasible reference profile is always generated, regardless of the size of the range error. Finally, the development of this algorithm modification includes a method for determining analytic first order approximations necessary for updating the reference profile, as well as guidelines for determining range error allocation weights. The overall design of this method leverages the high-TRL Shuttle entry guidance routine by minimizing modifications to the implementation. This algorithm is developed and tested against the baseline Shuttle entry guidance algorithm to ensure that the new range error allocator effectively improves algorithm performance for a number of stressing cases. This includes an optimization in which the initial reference profile is optimized for constraint margin with respect to both drag constraints and heat load constraints.
32 1.4 Thesis Overview
This chapter provides an introduction to shuttle entry guidance methods and moti- vates the need for the range corrector method outlined in later chapters. Chapter 2 describes the simulation used to develop and subsequently evaluate guidance algo- rithms. Chapter 3 details the unmodified shuttle entry guidance algorithm originally described in [18]. The deficiencies of the unmodified entry guidance made apparent by a more restrictive set of vehicle constraints are outlined in the Problem Descrip- tion in Chapter 4, along with motivations for the structure of the solution to this problem. Chapter 5 lays out the design of the range update method used to satisfy more restrictive vehicle constraints, and Chapter 6 describes the optimization method used to obtain optimal parameters for this range update method. Finally, the results of this optimization as well as Monte Carlo results are presented in Chapter 7, with conclusions presented in Chapter 8.
33 34 Chapter 2
Simulation Environment
A high-fidelity simulation of a high L/D spaceplane is used to develop the algorithm presented in this thesis. This simulation operates using MATLAB® and Simulink®, and includes aerodynamics, heating, navigation, environment and actuator models, among others. This chapter gives an overview of the relevant details of this simulation. Detailed equations describing the simulation are included where relevant.
2.1 Units
This simulation as well as all equations and values used in this thesis use imperial units. The units for each dimension are listed in Table 2.1.
Table 2.1: Units for vehicle simulation.
Length foot (ft) Mass slug (slug) Force pounds-force (lbf) Angle radian (rad) Time second (s) Angular rate radians per second (rps) Pressure pounds per square foot (psf)
35 k n (North)
In (N
Meridian ( / On (East)
C: ECEF n: Local NED
Figure 2-1: North East Down and Earth Centered, Earth Fixed Coordinate Frames [4].
2.2 Reference Coordinate Frames
2.2.1 North, East, Down
The north-east-down (NED) frame (is, 3s, kn) is a Cartesian coordinate frame centered at the spacecraft's center of gravity. The i,. vector points along the northern axis, tangent to parallels on the earth, the ja vector points along the Eastern axis, tangent to meridians on the earth, and the k, vector points towards the center of the earth. This frame is shown in Figure 2-1.
2.2.2 Earth Centered, Earth Fixed
The Earth centered, Earth fixed (ECEF) coordinate frame (ie, Je, ICe) is centered at the center of the earth and rotates with it. The le vector is along the axis that runs from the origin to the surface of the earth at 0* latitude and 00 longitude. The Ike vector is along the axis that runs from the origin to the surface of the earth at the north pole. Finally, the Je vector is orthogonal to se and fke according to the right hand rule.
36 Figure 2-2: HL-20 with Body Centered, Body Fixed Coordinate Axes [271.
2.2.3 Earth Centered, Inertial
The earth centered, inertial (ECIC) coordinate frame (ii, ji, /i) is centered at the center of the earth like the ECEF coordinate frame, but differs from that frame because it is inertial, and so rotates with respect to the surface of the earth. The time convention used with this coordinate frame is the J2000 convention. The k-i vector points along the axis that runs from the origin to the north pole. The zi vector points along the axis that runs from the origin to the surface of the earth at 00 latitude and 0' longitude at time t = 0, the start of the simulation. The je vector completes the orthogonal coordinate frame according to the right hand rule.
2.2.4 Body Centered, Body Fixed
The body centered, body fixed coordinate frame (lb, jb, kb) is centered at the center of mass of the vehicle. The zb vector points towards the front of the vehicle, and the kb vector points down. The third vector, Jb completes the coordinate axes following the right hand rule.
37 2.3 Coordinate Transformations
A coordinate transformation Tb transforms a set of coordinate axes z from frame a to frame b.
Za (2.1)
kaj
Zb = TaZa. (2.2)
Because transformation matrices are orthogonal, the reverse transformation can be obtained from its inverse or its transpose,
7Tb = (7-)- 1 b) T (2.3)
2.3.1 North, East, Down Frame to Earth Centered, Earth Fixed Frame
In this transformation, 0 denotes the latitude, and ( denotes the longitude of the vehicle, - sin(0) cos(() - sin() - cos(0) cos(()
= sin(0) sin(() cos(() - cos(O)sin(() (2.4)
L cos(O) 0 - sin())
2.3.2 Earth Centered, Earth Fixed Frame to Earth Centered, Inertial Frame
In this transformation, T denotes the Greenwich mean sidereal time, which is a mea- sure of the angle that the Earth has rotated through since an initial time to,
Cos(T) sin(T) 0
= - sin (T) cos(T) 0 (2.5)
0 0 1
38 (Pb(Roll) 4"(North)
6n J (East) .... amer0n(Pitch)
On(Yaw)
kn(Down)
Figure 2-3: Definition of yaw, pitch and roll with respect to body centered, body fixed frame [29].
2.3.3 Earth Centered, Inertial Frame to Body Centered, Body Fixed Frame
In this transformation, 'Vb denotes yaw, O4 denotes pitch, and #b denotes roll. These
angles are defined according to Figure 2-3,
1 0 0 COS Ob 0 - sinOb COS Ob sin /b 0
T b 0 coSqb sin b 0 1 0 -sin V)b COS Ob 0 . (2.6)
0 -SinJ COS#O sinOb 0 COSOb 0 0 1
2.4 Environment Models
2.4.1 Earth Gravity Model
The gravity model calculates the gravitational acceleration and the body frame mo- ments acting on the vehicle due to gravity using a point mass fourth degree zonal harmonic (J4) equation based on the work in 121 and [28]. In this gravity model, third
39 -I
Figure 2-4: HL-20 127]. body gravity effects from the moon and the sun are ignored.
2.4.2 Earth Atmosphere
The atmosphere model uses NASA's Global Reference Atmospheric Model for Earth from 2007. The randomized model takes in the latitude, longitude, and altitude of the vehicle as well as the date and time being simulated to provide temperature, speed of sound, pressure, and air density. The model also provides randomized wind representative of the time and location of the vehicle.
2.5 Vehicle Model
The vehicle modeled is a high L/D lifting body, similar to NASA's HL-20, as seen Figure 2-4. The mass of the modeled vehicle is about 750 slug, with a reference area of 323 ft2 . The modeled vehicle has 7 control surfaces: right elevon, left elevon, rudder, left upper bodyflap, left lower bodyflap, right upper bodyflap, and right lower bodyflap. The aerodynamic model that provides both forces and moments is a lookup table that is a function of angle of attack, sideslip, and Mach. The values provided for this aerodynamic model is determined through a combination of wind
40 tunnel data, flight test data, and computational fluid dynamics models.The trimmed angle of attack, coefficient of drag (CD), coefficient of lift (CL), and lift to drag ratio (L/D) for this vehicle are provided in Figures 2-5 - 2-7. Note that Mach decreases to the right, similar to velocity in all other figures in this thesis.
2.6 Simulation State Propagation
2.6.1 Bank Angle and Angle of Attack
The two outputs of the Shuttle entry guidance algorithm are bank angle (#) and angle of attack (av). Angle of attack is defined as the angle between the body's x axis (b) and the wind relative velocity. Bank angle is defined as the angle formed by rotating the lift vector around the velocity vector in the ECEF frame, which can also be defined as the angle between the lift vector (perpendicular to the drag vector) and or "up" in the ECEF frame.
2.6.2 Equations of Motion
The translational velocity and state, as well as the angular velocity and state, are determined by integrating the angular and translational accelerations using Simulink's integration functionality. These are obtained from the aerodynamic models developed for the vehicle which take into account gravity, wind, air density, and control surface locations among other parameters.
2.7 Aerodynamic Heating
Aerodynamic heating rates for a set of thermal control points along the vehicle such as the nose and control surfaces are determined using various simulation techniques and are stored in a lookup table that takes in Mach, drag, air density, velocity, angle
41 Trim Angle of Aftack vs. Mach Number
Figure 2-5: Trim Angle of Attack vs. Mach Number.
Trim CDCL vs. Mach Number
-C 0
Mach
igure 2-6: Trim CD, CL vs. Mach Number.
Trim L/D vs. Mach Number
0 -j
Mach
Figure 2-7: Trim L/D vs. Mach Number.
42 3.15p1/2 of attack, sideslip angle, and control surface deflections. The general model is based on Chapman's model for aerodynamic heating [51, where rc is the leading edge radius, p,, is the freestream air density, patm is the sea level standard atmospheric density, and V,, is the freestream velocity, ) Patm 7924.8
2.8 Simulation Initialization
2.8.1 Initial Conditions
The initial conditions for the vehicle are described in Table 2.2. Table 2.2: Initial conditions for vehicle simulation.
Altitude 400,000 ft Downrange 3650 NM Crossrange 0 NM Inclination 520 Angle of attack 390 Bank angle 00 Earth centered, earth fixed V 24860 ft/s
2.8.2 Terminal Conditions
For this simulation, the target landing site is Kennedy Space Center (KSC). The latitude and longitude of KSC are (28.5739' N, 80.64900 W). The simulation ter- minates when the vehicle successfully completes the entry guidance, terminal area energy management, and approach and landing routines, and lands on the runway, decelerating on rollout to 3 ft/s. The simulation will also terminate if the vehicle exceeds predefined maximum roll, pitch, or yaw rates, maximum attitude angles, or minimum altitude equivalent to crashing.
43 2.8.3 Monte Carlo Parameters
A Monte Carlo simulation is used to evaluate the entry guidance algorithm under dispersed conditions. Table 2.3 lists the relevant parameters that are dispersed for each instance of a Monte Carlo simulation. Each parameter is dispersed with a standard deviation that is similar to what would be expected for a standard high L/D atmospheric reentry flight.
Table 2.3: Parameters dispersed for Monte Carlo simulations.
1 Initial velocity 2 Initial flight path angle 3 Initial crossrange 4 Initial downrange 5 Initial altitude 6 Initial heading 7 Mass 8 Moment of inertia 9 Center of mass 10 Reentry date 11 Atmospheric model seed 12 Sensor noise 13 Sensor bias 14 Altimeter accuracy
44 Chapter 3
Shuttle Orbiter Guidance Algorithm
As outlined in Section 1.1.1, the shuttle orbiter entry guidance algorithm provides a and 0 reference commands based on a predetermined nominal trajectory. This trajectory is primarily defined via angle of attack and drag profiles. Downrange is controlled by shifting the drag profile, while crossrange is controlled via an azimuth angle error deadband. The following sections outline the guidance algorithm in greater detail.
3.1 Reference Drag Profile
The reference drag profile is separated into 4 phases: temperature control, equilibrium glide, constant drag, and transition. The temperature control reference drag phase is a series of n quadratic curves, where each curve i is defined using Equation 3.1. The majority of the work outlined here can be found in greater detail in [18 and [17],
2 Di(V) = D1(c1 + ci2 V + ci 3V ). (3.1)
The equilibrium glide reference drag is obtained from the equation for drag with the constant flight path angle, assuming that D + g sin - ~ D,
D(V) = (g - - (L) . (3.2) r D
45 However, in order to provide more flexibility in profile shaping, this reference profile is modified and subsequently simplified to include an arbitrary constant, V,
1V2 D(V) = D (3.3) 1 V- 2
The third phase is constant drag, which defined by the constant D2 ,
D(V) = D2 . (3.4)
The final phase is defined using energy as the independent variable, rather than velocity. This is because the approximations required for range estimation assume sin -y e 0 when using velocity but not when using energy. When the vehicle approaches lower velocities at the end of the entry guidance sequence, this small flight path angle assumption does not hold. Energy is defined here according to Equation 3.5, while the corresponding reference drag is defined according to Equation 3.6,
1 E = gh + -V 2, (3.5) 2
D(V) = D4 + D4 - D3 (E(V) - E 4). (3.6) E4 - E3 Example reference drag profiles are included in Figures 3-1 and 3-2. It should be noted that it is possible to obtains a combination of reference parameters such that the second and fourth phases intersect, removing the third phase, as seen in Figure 3-2.
46 Temperature Equilibrium Constant Transition Control Glide Drag V I III I 1 2 3 4
(V2,D2) (V3,D3)
Cu
(V4,D4) (V1,D1)
(Vo, Do)
VS Velocity
Figure 3-1: Typical reference drag profile in drag-velocity space.
Temperature Equilibrium Transition Control Glide
(V3,D3% (V2,D2)
(Vint,Dint CD
(V1, D) (V4,D14)
(VoDo
Vs Velocity
Figure 3-2: Reference drag profile in drag-velocity space without phase 3.
47 3.2 Reference Drag Profile Parameters
Of the profile parameters used in Figures 3-1 and 3-2, some are used to define the profile, and these will be referred to as initial parameters, while others can be deter- mined based on those initial parameters. The initial parameters used to define the profile are (V, DO), (V1 , D1 ), Vs, (V3, D 3), (V4, D4 ), and h3 , as well as n sets of points (vi, di) that define the quadratic curves in the temperature control phase. From these,
(V2 , D2) and (Vint, Dint) can be determined,
V2= V2k_ (2_72), (3.7)
D2 = D3. (3.8)
The possible intersection point of phases 2 and 4 is calculated by setting the reference drag equations for each phase equal, and solving V and D at this point. Since phase
4 is a function of energy rather than velocity, this requires the initial parameter h3 , which is an estimate of the altitude at the start of the transition phase,
2 D3-D4 D3-D 4 4D i D V D3-D4) +3V 3V ?V (D4 T V /,y__V -2VJV2 Vint = -V4 (2Di , (3.9) V2 -VI2
12 Eint= gh + -int, 2 (3.10)
Dint= D4 + (Eint - E 4 ). (3.11) E3 - E4
3.3 Equations of Motion
The equations of motion used for derivation of basic vehicle dynamics in the guidance algorithm are Equations 3.12a - 3.12e. These equations ignore Coriolis and centripetal accelerations that result from Earth's rotation because they are sufficiently small to be approximated as 0,
48 V = D +g sin -y, (3.12a)
- cos -y + L, cos 0, (3.12b) (r V 2 V cos yi - cos2 y sinO tan # + Lm sin#, (3.12c) r
R = V cos -Y, (3.12d)
h = V sin -. (3.12e)
In addition, the atmospheric density is assumed to vary with altitude according to Equation 3.13, where po is the atmospheric density at base altitude and h, is the atmospheric density scale height,
h p=poe h. (3.13)
3.4 Range Prediction
Using Equations 3.12a and 3.12d, we can obtain an analytic equation for range from the initial velocity, Vo to the final velocity, Vf,
dR dR/dt (3.14) dV dV/dt'
jvf idV V cos idV R = - v!f . = - (3.15) v V v0 D + g sin Since y ~~0, and D > g sin -y, simplify Equation 3.15 by assuming that cos -y ~ 1 and D + g sin y D, fj VdV (3.16) vo D As the velocity of the vehicle decreases, the total lift experienced by the lifting body decreases and the descent rate increases. As a result, the magnitude of the flight path angle increases. Although -y is small for the entire flight, during the last
49 phase of entry guidance when the vehicle is descending faster and velocity is lower, the drag, D, is also significantly lower. As a result, it is no longer accurate to assume D + g sin -y a D. In order to obtain an accurate estimate of range for this last phase, the independent variable is switched from velocity, V, to energy, E, since range as a function of energy only requires the assumption cos y ~ 1, and not the assumption D + g sin -y ~ D. Range for this phase is determined using the vehicle's kinetic and potential energy, E = gh + -V2 (3.17) 2
Differentiate with respect to altitude, h, to yield,
= g + V (3.18) dh dh
Use Equations 3.12a and 3.12e to determine dV/dh, and substitute,
dV D-gsiny (3.19) dh Vsiny '
dE -D =E .- (3.20) dh sin (2 We can also find dR/dh and combine this with Equation 3.20 to obtain an analytic equation for range similar to Equation 3.16 that also uses cos y 1,
dR tany, (3.21) dh
Vf cos ydE R =D (3.22)
dE (3.23) v0 D Using the equations for reference drag for each of the 4 phases, the downrange to be flown during each phase can be estimated. The temperature control phase downrange
50 is calculated as follows using Equations 3.1 and 3.16,
n IVC+= VdV RTC =-E 2 (3.24) D1(ci + ci 2 V + ci3V )
Since this integral is difficult to evaluate directly, a quadrature rule is used. In general, a quadrature rule is defined by a set of points to evaluate along each interval i, here (Vai, Vbj, Vj), as well as a set of constants to weight each evaluated point,
(qi, q2, q3 ). Given n + 1 knot locations (vi, di) to define n quadratic curves, and
quadratic coefficients for each spline, (cii, c22, ci3 ), the range can be approximated,
n Vai V- Vci RTC -Z(K+i -1KI) q1 D +q ( + q3 "(V .) (3.25) D(Va) 2 D(Vbi)1 D(Ve)
The equilibrium glide phase can be found by simply integrating the analytic reference drag equation from Equation 3.3,
V2 V 2S2D _ V2 _ T/r REC = - I V7iy 2)dV log (3.26) 'E Jv0 V2 _ V2 D1(V2-_V2 ) kSV 30~
When the initial and final velocities are substituted in (V and V2 respectively) using Equation 3.7, the analytic range for the second phase can be simplified as
V 2 _ V2o D3 RE-b 2D, (3.27) ( , .
The range for the third phase is simple due to the simple reference drag,
YfVdV _f 07 RCD - 19f - 219 (3.28) v0 D3 2D3
Substituting for Vo and Vf, yields a simpler analytic equation for range during the third phase, 2 _ y2 V2 _ V 2 (3.29) 21D3 2D,
The fourth phase is calculated using Equation 3.23, since energy is the independent
51 variable,
RTR - dE (3.30) 1EO D 4 + - (E - E 4 )'
ED-E D 4 (Ef - E4) 3E (3.31) RR = 3-E log . D3 - D4 D4 + D-D (Eo - E4)
Substitute E3 = Eo, and E4 = Ef and simplify to yield
RTR = log D . (3.32) D3 - D4 D3)
In general, the range flown by the nominal profile is equal to the sum of the ranges for each phase. However, when the equilibrium glide and transition phases intersect, the sum of all 4 phases is not an accurate representation of range. However, because
the calculation of Vit and Dint and the corresponding shortened ranges REG and RTR are significantly more complicated, the range is still estimated in the original shuttle
entry guidance algorithm using the sum of all four range calculations. Because V3
> V2, the constant drag range will be a negative value. By adding the four ranges to compute the total range flown by the nominal trajectory, it is assumed that the negative constant drag phase range is approximately equivalent to the portions of the equilibrium glide and transition ranges not flown in [181.
3.5 Reference Trajectory Range Update
In order to null range errors that occur during flight, the shape of the reference profile is adjusted. In general, it is advantageous to keep the vehicle as close to the nominal profile as possible, in order to take advantage of the constraint margins under dispersed conditions that were prioritized when designing the profile. Furthermore, the nominal profile must be followed as closely as possible in order to preserve the post-communications blackout footprint that was designed into the nominal profile. In order to achieve this, only the phases nearest the phase currently being executed are modified, which keeps the remaining phases untouched at their nominal values.
52 CONSTANT DRAG LEVEL = CONSTANT Vc(I SHORT RANGE D C M ;NOMNAL m*-LONG RANGaE
R1 R2 | R3 R4 VB1 Vcg CONSTANT
Figure 3-3: Example trajectory update in temperature control phase 118].
CONSTANT DRAG LEVEL = CONSTANT SHORT +-V RANGE VC9 VCg D LONG MW RANGE--xMI
R1 = 0 R2 R3 R4= 0 Vol VC9 CONSTANT
Figure 3-4: Example trajectory update in equilibrium glide phase [181.
This behavior can be seen in Figures 3-3 - 3-6. Analytic equations for the partial derivative of range with respect to the param- eter being modified for each phase are used to update the trajectory. The partial derivatives of range for each of the four phases are found in Equations 3.33 - 3.35d.
By factoring D1 out of the reference drag equation for temperature control (Equation 3.1), the partial derivative for RTC simplifies significantly,
&RTc a n Vai Vi Vi OD 1 aD1 [ k(Vi+1 - vi) "-D(Vai) D(V+2) + q3 D(Vi)J, (3.33)
=RTCRTc (3.34) 8DI D1
53 SHORT RANGE NOMINAL D LONG MRANGE.
R4= R10 R2=01 R3 CONSTANT
Figure 3-5: Example trajectory update in constant drag phase 1181.
SHORT RANGE f D NOMINAL Mv LONG R AN GE-.
RI = 0 R2 0 R3 0 R4 -4--v
Figure 3-6: Example trajectory update in transition phase [18].
The other partial derivatives are more straightforward,
OREG V2 _ ) + (3.35a) OiD, ( log( &RCD V2 _ V2 (3.35b) 0D1 2D , 2 &RCD 2D (3.35c) D3 2D , E3_- E4 D3 10g(D4 +D 4 - D3 ) (3.35d) i9D3 D 3(D3 -D4)2
3.6 Other Reference Trajectory Parameters
The control law used to command vertical L/D to the vehicle is a function of the reference trajectory's drag (Dref), altitude rate (href), and L/D, as well as the control
54 law gains, fi - f3,
L L =)ref + fi(D - Drej) + f2(h - href)D + f3 J (D - Dref)dt.( (3.36)
href and Lrf are determined according to Equations 3.37 and 3.38. The derivations of these two equations can be found in [181,
-hs -- + 2D CD(3.37) D V CD
-- _. ( - 3D) 4D 3 D V2 D 2 L ODD CD D ODD.(38 D V V2 hs g) hs D - CD CD - V)+CD By substituting the reference drag equation and its associated derivatives for a given phase as well as the current atmospheric density scale height (h,) and current drag coefficient (CD) based on the angle of attack, the remaining reference values can be determined.
3.7 Angle of Attack and Bank Angle Commands
After determining the reference trajectory parameters, these are used to determine bank angle and angle of attack. The tracking of long period deviations from the reference trajectory is primarily accomplished by modulating bank angle magnitude, while smaller, short period deviations are tracked using angle of attack modulation. This is to take advantage of the ability to modulate both angle of attack and bank angle, since vertical L/D can be modified at a much faster right by changing the drag coefficient than by modifying the direction of the lift vector. Short period deviations are the result of transient effects such as bank reversals, phugoid motion, and density gradients. Because the reference angle of attack profile was designed with heat rate constraints in mind, it is desirable to quickly drive the angle of attack back to its nominal value. Equation 3.39a determines the magnitude of the commanded bank angle, where
55 30 - WREINENCE DRAG PROME '" "'A CTUAL DRAG PROML 40-
DRAG -BANK REVIRSAL ACCE9IRAMI4O F /S E 2 C 2 0 - A Do-ANK K IEWASALR V R
10-
35 30 2 20 15 10 3 0 RELATIVE SPEED, THOUSANOS OF FS
Figure 3-7: Bank profile with only bank modulation 1181.
the first term solves for the component of the L/D in the vertical direction, and the second term drives the angle of attack back to its nominal value. Equation 3.39b determines the commanded deviation of the angle of attack from the nominal reference profile. It is a function of the reference drag deviation as well as the coefficient of drag, CD. Figures 3-7 and 3-8 demonstrate the effect that angle of attack modulation has on the algorithm's ability to track reference drag commands,
+fI(a -ao), (3.39a) = Cos-, D)c L/ D 1
Aac = CD(Do - D)/fio. (3.39b)
56 --- RE1mNC1 DRAG PROM1 - - -ACUM. 0A1 PROELI 40
DRAG so $ANN REVERSAL ACCOURATI0N, 2 FT/SIC -
to
35 25 2sto Is to s RELATIVE SPIE1D, WIOUSANOS Of FPS
Figure 3-8: Bank profile with bank and angle of attack modulation [18].
3.8 Lateral Logic
While the magnitude of the bank angle is determined based on the reference drag profile, its direction is a function of the azimuth error in order to control crossrange. When the azimuth error violates a deadband that is scheduled on velocity, the bank angle magnitude is switched in a bank reversal. The azimuth error is defined in Figure 3-9, and an example of the azimuth error deadband can be found in Figure 1-5.
XRUNWAY
YRUNWAY ORBITER AZIMUTH RADIUS OF V LANDING SITE ZRUNWAV
AIfNsEGENT CIRCLE t 0
EARTH CENTER
Figure 3-9: Entry azimuth error definition [18].
57 58 Chapter 4
Problem Description
As the design of high L/D atmospheric reentry missions are refined and improved with each new vehicle, the margin available to the vehicle during reentry must decrease correspondingly. In an effort to reduce the empty weight of the vehicle, the load- bearing structure and the thermal protection system are quick ways to shave mass. However, by reducing the mass of these subsystems, the maximum drag that the vehicle can experience due to heat rate, vertical acceleration, and dynamic pressure is decreased. As a result, the entry corridor between upper and lower drag constraint decreases. The range update method used in the Shuttle entry guidance algorithm only allows the profile to be shifted by a single parameter. In general, the current phase is simply shifted up or down by a constant value, which is adequate when the entry corridor is sufficiently wide to accommodate such a shift. However, as the corridor is narrowed for newer vehicles with tighter constraints, even a small constant shift threatens to violate constraints. This behavior can be seen in Figure 4-1 when the turbulent flow transition heat rate constraint is violated with only a small range error update. Once the margin between reference drag and the upper drag constraint decreases sufficiently, the range of the reference profile can only be modified by utilizing pockets of margin, which potentially occur in phases other than the current phase, such as the example in Figure 4-2. In these instances, it is only possible to obtain the correct range with a significantly more flexible reference profile. This type of behavior is
59 C
'4-
Nominal -Low Range -High Range
Velocity
Figure 4-1: Drag profile perturbed for positive and negative range errors. exemplified in the case when heat load is the most restricting constraint. In order to reduce heat load for a constant range, previous trajectory design experience would indicate that drag should be increased earlier in the flight at high velocity, in order to trade for low drag at lower velocities when the atmosphere is denser and produces higher heat rates. However, if the reference trajectory must be modified for a longer range during the temperature control phase of reentry, the reference drag for both temperature control and equilibrium glide will decrease by a constant value. Because this new longer range profile is being flown for a longer period of time, the heat load will be higher. This could be mitigated by increasing the reference drag during the early part of the profile, but instead this key period has an even lower reference drag than before due to the inflexible range update algorithm. The current reference profile is restricted to being defined using the four phases comprised of quadratic and linear segments. By also restricting the update method to only shift the profile by a constant, the algorithm is artificially restricted to only being able to operate in large, simply shaped entry corridors. It is certainly unwise to put large requirements on the thermal and structural components of the vehicle simply because the guidance algorithm's reference profile is inflexible under unusual
60 C)
Velocity
Figure 4-2: Restrictive drag constraints that require a more flexible drag profile. constraints. As a result, the range update method must be made more adaptable. One advantage of the current range update method is that it is memoryless with respect to guidance algorithm inputs. The current profile is only a function of the current state and current range to the runway, and is not dependent on a history of range to the runway measurements. This is advantageous because it allows the total family of possible reference profiles to be quickly assembled and tested before flight. One only has to search over 1 parameter per phase to obtain all the possible profiles that can be generated. Although it is clear that the range update algorithm must be modified to create more flexible reference drag profiles, it is desirable to maintain the algorithm's memoryless property. The Space Shuttle orbiter entry guidance algorithm underwent a significant test campaign before its first flight, which included 5 manned drop tests from approxi- mately 24,000 ft [20]. Following these tests, the Space shuttle fleet also completed 135 missions [25]. As a result, the Shuttle entry guidance algorithm not only has a high technology readiness level (TRL), but is also well understood both in its behavior and limitations. This algorithm is much less costly to implement for future vehicles than entirely novel algorithms, and as a result is an excellent basis for future high
61 L/D RLV's. Therefore, it is important to leverage the high TRL by maintaining as much of the original algorithm structure as possible when proposing modifications. In summary, the range update method in use in the Shuttle Orbiter's entry guid- ance algorithm is inadequate for narrow entry corridors. Its lack of flexibility in modifying the shape of the profile and in updating phases other than the current phase results in an algorithm that cannot take advantage of areas of high margin and lacks robustness. However, the algorithm's high TRL and memoryless properties are advantageous for reducing the total time required to check out the algorithm for flight. As a result, any proposed improvements to the algorithm that attempt to improve the robustness of the range update algorithm should seek to preserve these important qualities.
62 Chapter 5
Allocator Overview
The proposed range allocator design that follows is a new method for modifying reference drag profiles that improves the guidance algorithm's flexibility and helps to ensure that drag profiles generated on-line will still satisfy highly restrictive con- straints. A similar design was presented in [11], although the design presented there was not memoryless with respect to range error, and was therefore more difficult to check out for flight. In order to improve the flexibility of the reference drag profile, additional ways to modify the profile are added to the guidance algorithm. The allocator is designed to utilize J total parameters that are capable of modifying the profile in different ways. Each of these parameters, xj is varied at each time step t to find the value xj,t that nulls the range error at that time step. A parameter K is used to link all of the modifying parameters, so that all parameters xj are updated at each time step t according to a predetermined ratio. This ratio is defined through a set of weights, wj, where the Jth weight is associated with xj. Any parameter xj,t can then be determined using Equation 5.1, where xj,o is the value of xj at t = 0, and Kt is the upated value of K at time step t,
t = Xjo + w Kt. (5.1)
In order to effectively null range error at every time step, a first order Taylor
63 Series expansion is used to update Kt similar to the approach used to update D1 in the original Shuttle entry guidance,
DR DR O = , (5.2) t j=1 t zAR Kt+1 = Kt + R (5.3)
By selecting an appropriate set of xj and wj based on the constraints and the expected downrange dispersions, the reference drag profile will have significantly im- proved flexibility under a wider range of constraints. The following sections will discuss the selection of xj for a range error allocator for Shuttle entry guidance as well as the selection of wj and restrictions on these parameters during flight.
5.1 Quadratic Splines
Quadratic splines in the temperature control phase provide more flexibility in defin- ing the drag profile in tight, restrictive entry corridors. In order to intelligently interpret and modify the reference drag profile, it is first advantageous to redefine the temperature control phase as a quadratic spline that is also smooth with respect to the temperature control phase and equilibrium glide phase juncture. In addition, smoothness requirements improve the reference command tracking of the vehicle by eliminating instant changes in reference profile slope. In general, a quadratic spline is a continuously differentiable piecewise quadratic function, where the ith quadratic function is Di (v). Any quadratic spline can be uniquely determined by a series of knots (vi, di) and the slope at one knot, usually at either end of the spline. For N + 1 total knots, there are 3N total unknowns. The continuity requirement provides 2N constraints, since each quadratic curve must pass through 2 separate knots. In addition, the smoothness requirement, which applies at every interior point, provides N - 1 constraints. By specifying the slope at one of the end points, we can obtain 2N + N - 1 + 1 = 3N constraints. The quadratic coefficients, (ai, bi, ci), for each point (vi, di), can be found by arranging these constraints into matrix form and
64 simply inverting the matrix,
di aivi + bivi + ci, (5.4)
v 1 0 0 0 ... 0 0 0 0 0 0- - - 2 -al-2 V V2 1 0 0 0 ... 0 0 0 0 0 0 b1 d2 000 v2 v2l ... 0 0 0 0 0 0 Ci d3 2 0 00 V .. a21 d3 3 V3 1 0 0 0 0 0 0 b
0 0 0 0 0 0 ... 0 0 0 v _1 v_1 I C2 -(5.5) 0 0 0 0 0 0 ... 0 0 0 v 1 2vi 1 0 -2vj -1 0 0 0 0 ... 0 0 0 0 0 0 an 0 0 0 0 2V2 1 0 -2v2 -1 0 ... 0 0 0 0 0 0 bn 1 0 -2vn -1 0. . C. L 0 0 0 0 0 0 0 0 0 ... 2Vn- 1
However, using this method to calculate spline coefficients in flight-level code would be ill-advised due to the computational complexity and potential instability of inverting a matrix. Instead, the quadratic coefficients can be computed iteratively using the following method 1321. First, let zi = D'(vi) and assume that the quadratic is of the form of Equation 5.6. As you can see, the constraints Di(vi) = di, D'(vi) = zi, and D'(vi+1) = zj+1 are satisfied,
Di(V) = Zi+1 - Zi2 (V - v,) 2 + z,(V - v,) + di. (5.6) 2(V,+1 - v,)
By enforcing the requirement that Di(vi+1) = dj+1 and solving for a relationship between zi and zi+1 , we can compute the slope at every knot,
zj+j = -zi + 2 ( (5.7) Vi+1 - i The slope at the lowest velocity knot is determined by requiring that the transition from the temperature control phase to the equilibrium glide phase be smooth. The slope of the equilibrium glide phase at V is then used for zo,
2 dDEG 2 V1 V zV (5.8) dVdVV=V1 V2 - D1V
65 Nominal Profile Reference Drag
(II
Velocity (fps) Figure 5-1: Reference drag oscillations caused by inaccurate range estimates.
5.2 Updated Range Equations
It is possible to select reference drag parameters such that the equilibrium glide and transition phases instersect. In this instance, the shuttle entry guidance makes the assumption that continuing to sum the range of each of the four phases to obtain the total range is sufficiently accurate. However,.when implementing this more com- plex range error update method, an inaccurate range can result in reference drag oscillations such as those shown in Figure 5-1. When the equilibrium glide and transition phases intersect, range equations that take into account (Vint, Dint) and Eint from Equations 3.9 - 3.11 are used,
V2 - V 2 y2t _V2 REG 2V1_ 2 (5.9a) 2D,
RCD = 0, (5.9b)
(D 4_=1 E4 - E3 RR = log D 3 - (5.9c)
66 - Nominal Shorter Range -Longer Range
C cc
Velocity
Figure 5-2: High and low range reference drag profiles when D1 is perturbed.
5.3 Parameter 1: D1
The parameter D1 was used in the initial implementation of the Shuttle guidance algorithm, and was kept as a range error allocator parameter because it modifies the profile simply and has a large impact on profile range for a given small perturbation.
Figure 5-2 demonstrates high and low range profiles that result from perturbing D1.
The partial derivative of total range with respect to D1 is the sum of the derivative over the first three phases,
(9D1 D i 'T (5.10a)
MREG V12 - V22 lo D3 1(51b aD, 2D 2 Di)+1,(.1)
8RcDV2 _ 2 aRD 2D ' (5.10c)
(9R - BRTC + REG + BRCD(51 d o9Di Di D, D, 51d
(5.10e)
67 5.4 Parameter 2: kq
The second parameter implemented to modify the reference drag profile is kq, which defines the extremum of a quadratic curve that is defined by a vector of predetermined coefficients. This quadratic curve is then added to the set of quadratic splines in the temperature control phase. This augmented drag profile modified to be the sum of two quadratics is described in Equation 5.11. Figures 5-3 - 5-5 demonstrate the effect of kq on a given reference drag profile for various settings of the quadratic coefficients
CqI - cq2, where the modifying quadratic being scaled by kq was set to 0 at the end of the temperature control phase, effectively keeping the reference drag at that location constant. A number of considerations were taken into account when selecting this parameter. The simplicity of the calculation of &R/Dkq was an important factor, particularly in comparison to other more obvious options such as using the value of a single spline knot as an allocator parameter. Additionally, the ability to tune the quadratic coefficients to respond to a variety of known problems was considered. If the quadratic curve is set to 0 for a given velocity, changing kq will not change the reference drag value at that location. This is ideal for locations where the size of the entry corridor is significantly reduced. Similarly, if the margin is known to be greater in a specific velocity region, the magnitude of the quadratic curve should be greater in that region to take advantage of the wider constraint corridor. Finally, the quadratic curve can be shaped with an aim towards preserving the initial shape of the temperature control spline. This flexibility allows the reference drag profile to follow the shape of restrictive heat rate constraints that are typically most restrictive in high velocity regions. The augmented reference drag with the additional scaling quadratic yields
2 2 D = D1(cii + ci 2 V + ci3V ) + kq(cqi + Cq2V + Cq3V ). (5.11)
68 kq Quadratic Curve Reference Drag w/ Varying kq
--- Nominal -Low Range High Range
Figure 5-3: High and low range reference drag profiles when kq is perturbed.
kq Quadratic Curve Reference Drag w/ Varying kq
-High Range Low Range- -Nominal
Figure 5-4: High and low range reference drag profiles when kq is perturbed for a second set of quadratic coefficients.
kq Quadratic Curve Reference Drag wI Varying kq Low Range -Nominal -High Range
Figure 5-5: High and low range reference drag profiles when kq is perturbed for a third set of quadratic coefficients.
69 The new range equation with kq implemented is
R~c-Zn~ Va i Vbi qVCZ, 5.2 ) 3 D' ci) '(512) RrC - (Vi+1 - Vi) iD'(ai) + D' +
where qi - q3 and Va - Vi are the same as those used in Equation 3.25. The
derivative of range with respect to the magnitude of scaled quadratic, kq, is
OR -( I - V) Li, Va(cqi + Cq2Va + Cq3Va) 2 Ok q &4 D1D'(V)DD(a)
Vb(cql + Cq2V + Cq3Vb 2) (5.13) 2 DD'(V) Vc(cqi + cq2Vc + Cq3Vc2) D1D(VC)2
5.5 Parameter 3: Vs
The parameter V, which is used to define the intersection of the equilibrium glide
phase and D = 0, was chosen in part to provide multiple methods for modifying the equilibrium glide phase. By modifying V, it is possible to push the effects of
a range error towards later in the trajectory, which is advantageous if constraints
are particularly tight through the majority of the temperature control phase of the
entry corridor. Figure 5-6 demonstrates the effects of changing V, on a reference
drag profile. The resulting equations used to update V, at each time step are found
in Equations 5.14 and 5.15. The equation for &R/DV, when the equilibrium glide
transition phases intersect using Vint can be calculated similarly,
OR _ ® +RCD =v v + v8 (5.14)
OR V D V D -- = log( 2 + 8 1 - 2 . (5.15) aV7 D, D 1 D 2 D1
70 C
1a) -Nominal -High Range -Low Range
Velocity
Figure 5-6: High and low range reference drag profiles when V is perturbed.
5.6 Memoryless Design
This set of parameters was selected to span a much larger set of possible drag profiles, with a number of likely limiting cases in mind. By choosing to use two parameters that are already used to define the profile, the structure of this algorithm continues to resemble the structure of the Shuttle heritage algorithm, preserving its simplicity. Furthermore, the selection of a quadratic curve that is added to the temperature control phase spline utilizes a significantly simpler set of equations to compute 9R/&K compared to other alternatives evaluated. Alternatives that were considered include modifying individual quadratic spline points, modifying individual cubic spline points, and modifying the magnitude of a quadratic that linearly scaled the entire spline, as seen in Equation 5.16 for an individual spline, i. Although these alternatives could have provided more parameters to modify in the allocator and more flexibility, the complexity of &R/&K in each of these outweighed the potential benefits,
Di(V) = DI(ci + ci2V + ci3V2 )kq(cql + Cq2V + cq3V 2). (5.16)
By applying these three allocator parameters to Equations 5.1-5.3, we obtain the equations used to update the reference drag profile for Shuttle entry guidance. All
allocator parameters move according to a ratio determined by w 1 - w3 . As a result,
71 0)
CM, a,
Velocity (fps)
Figure 5-7: All possible reference profiles for a given family of range errors. the reference profile at any given time step can be fully determined by K and V, and is not a function of the history of range error values. This property helps accelerate the flight code checkout process, since all possible reference profiles can be generated by simply selecting a range of expected downrange values and solving for the K value that satisfies that range. An example of such a family of profiles is shown in Figure
5-7,
OR 9R aR DR = WI + W 2 + w 3 , (5.17a) aK iD1 Dkq DVS K+ = Ki + DR (5.17b)
DIj+1 = Di,o + wiKi+1 , (5.17c)
kq,i+i = kq,o + W 2 Ki+1 , (5.17d)
Vs,i+l = Vs,O + w3Ki+. (5.17e)
72 5.7 Parameter Limits
To ensure that all constraints are satisfied and that the slope of the reference profile is not too extreme for the vehicle to follow, a set of limits have been placed on each of the allocator parameters,
Vs,min < Vs < Vs,max, (5.18)
kq,min < kq < kq,max, (5.19)
2 2 Dmin < D1(cni + ci2 V + ci 3V ) + kq(cqi + Cq2V + cq3V ) < Dmax. (5.20)
When a parameter j reaches a limit and is saturated, the allocator sets parameter j to the value at the limit it reached. This method relies on the guidance engineer selecting weights that allow the vehicle to remove range error within expected limits.
If only one parameter has a nonzero weight, resulting in only one parameter being used to update the profile, and that parameter reaches a limit, the allocator will be unable to update the profile to remove the full range error.
It is possible to design an allocator that uses alternative weight settings to remove range error when the original weights cause a parameter to reach a limit. However, although this design is more robust to large range dispersions, it is not memoryless with respect to range measurements. Using this parameter limiting method, when a parameter j reaches a limit and is saturated, the allocator sets wj = 0. The allocator then re-evaluates whether any parameter limits are violated with the modified set of weights. If a second parameter Wk also violates the parameter limits, the allocator sets
Wj = wk = 0 so that only the remaining functional parameter is being updated. The weights are reset to their initial value at every time step to ensure that all available parameters are used to update the profile at every time step.
This allocator is currently designed to work only during the temperature control phase of the guidance algorithm. As a result, the allocator is restricted to functioning within a limited velocity range, where the lower velocity limit is set at V + AV, where AV is an offset that prevents large and potentially unflyable reference profile
73 perturbations due to a small derivative, OR/axi for a given parameter xi.
5.8 Weight Selection
The tunable variables available in the range error allocator are the points that are used to define cql, cq2, and Cq3, as well as the weights wi, w 2, and w3. A number of factors must be taken into account when selecting the parameters of the range allocator. First, the location of the most restrictive constraints will determine whether some locations of the drag profile should be restricted to be unchanging during the mission. The variance of the downrange distribution under dispersed conditions both after the de-orbit burn and during flight informs how much range error the allocator can be expected to account for in flight, and can be used to determine likely constraint violation locations. Each of the three available parameters has a limited total range error it can account for during flight due both to the location of the constraints and the design of the reference profile, and this should be taken into account when selecting the weights, wI - w3 . A recommended starting point for wI-w 3 is found in Equations 5.21a - 5.21c. This initial setting would allow the parameters to move together without encountering a limit after accounting for a small range error,
wi = Di,max - Di,min, (5.21a)
w2= kq,max -kq,min, (5.21b)
W 3 = Vs,max - Vs,min. (5.21c)
The severity of the consequences of violating each individual constraint should also be considered during weight selection, which would drive the designer to ensure all pos- sible reference profiles remain further from more critical constraints. Finally, results under dispersed conditions such has a Monte Carlo simulation should also be analyzed to understand the vehicle's ability to track reference drag commands in various flight regimes.
74 Chapter 6
Allocator Optimization Design
In order to fully understand the potential of this range allocator, an optimization was carried out on the modified Shuttle entry-guidance algorithm, in which the nominal reference profile that is defined before the start of the vehicle's flight is optimized.
For this first optimization scenario, two sets of constraints in drag-velocity space are used, and the cost function is a combination of constraint margin across heat load, heat rate, vertical acceleration, and dynamic pressure.
For all optimizations carried out on the range allocator and shuttle entry guidance algorithm, a genetic algorithm was used. A genetic algorithm was selected for a number of reasons. First, genetic algorithms do not require the calculation of a gradient, which is essential for cases when the computation cost of each iteration of the cost function is high. Because it is not possible to obtain a gradient directly from the high-fidelity simulation, a gradient for this application could only be computed via a finite difference method, which would require many function calls, each with a high computation cost.
In addition, the performance of genetic algorithms does not rely on the smoothness of the cost or "fitness" function. Typically, an optimization requires bounding values for each parameter being optimized, which keeps the cost function within a smooth, well behaved area. However, because cases where the vehicle fails have nearly infinite cost, and it is impossible to predict whether a given set of parameters will cause the vehicle to fail in the simulation, the cost function for this optimization is guaranteed
75 CL
4.-
0) 0
Velocity (fps)
Figure 6-1: The upper and lower drag constraints used for nominal reference drag profile optimization. to not be smooth or even continuous. Such a poorly behaved cost function can cause issues with gradient calculation, particularly when finite difference methods are used. Because genetic algorithms do not require gradients,they are ideal for this application. Genetic algorithms are ideal for both avoiding the common issue of getting stuck in local minima and creating a more diverse set of candidates that more fully explore the solution space. It is unclear in advance what the correct optimized solution should look like, and so any initial guess is likely to lead the optimization towards an incorrect local minimum rather than towards the globally optimal solution. Finally, genetic algorithms are ideal for applications involving a large number of variables that must be optimized, particularly when those variables can be ordered so that related variables are near each other, similar to the structure of a chromosome.
76 C,, (A
I
Velocity (fps)
Figure 6-2: The upper and lower drag constraints used for nominal reference drag profile optimization with an additional turbulent flow transition heat rate constraint.
6.1 Genetic Algorithm Overview
The genetic algorithm, first laid out rigorously in [191, is an optimization method that is based on the genetic processes of biological organisms. By mimicking the behavior of natural populations that evolve over generations according to natural selection and "survival of the fittest", genetic algorithms can be used to "evolve" generations of solutions to obtain a suitably optimal solution to a real world problems. The algorithm is structured into generations of solutions, where each candidate in the new "child" generation is produced by first selecting two candidates from the previous
"parent" generation according to their relative fitness, carrying out genetic crossover, and then mutating each element of the gene with a predetermined low probability.
A genetic algorithm operates using a population of possible solutions, each with an assigned fitness score that indicates how good the solution is. This can be viewed as the opposite of a cost score typically used in optimization algorithms. Each candi- date solution is represented as a specific setting of variables joined together to form a string of parameters, which is often referred to as a chromosome. The algorithm operates by generating and then evaluating populations of possible solutions, where each subsequent generation is created via operations that are analogous to "cross
77 breeding", "natural selection", and "mutating". The random selection of a new gener- ation from the previous one prioritizes the more fit members of the population, and as a result the less fit members are less likely to get selected, and will eventually die out. In expanding the biological analogy, a set of parameters determined by a particular chromosome is commonly called a genotype, and the object constructed via the genotype is the via phenotype. According to [91, each individual gene con- verges when 95% of the population have the same value, and the entire optimization converges when all genes converge. It should also be noted that by convention genetic algorithms typically seek to maximize fitness rather than minimize cost, so all fitness values discussed are with reference to a maximizing optimization.
6.1.1 Crossover
Crossover is carried out by taking two individuals from the previous generation and cutting and swapping the chromosomes at a randomly selected position, according to Figure 6-3. According to [11, crossover is not usually applied to all pairs of individuals, and is instead applied with a likelihood between 0.6 and 1.0. If crossover is not carried out, the new generation's candidates are simply duplicates of the original two candidates.
Crossover point Crossover point
Parents io 1 0o0o0 1 1 1 0 0 0 1 1o0 1 0 0 1 O
Offspring 1 o 1 o oiooio oo 1 1 0 O1 1
Figure 6-3: Example of a crossover for a simple optimization over a set of binary variables [1].
78 6.1.2 Mutation
After crossover, mutation is applied to each child in the new generation. This is carried out by independently altering each gene with a very small probability, as in Figure 6-4.
Mutation point
Offspring 1 o 1 o o 1 0 0 1 o
Mutated Offspring 1 o 1 o 1 1 o 0 1 0
Figure 6-4: Example of a mutation for a simple optimization over a set of binary variables 11].
6.2 Nominal Trajectory Parametrization
The nominal reference drag profile has been parametrized using 6 parameters, Xi -X 6 . 3 of these define a quadratic curve that is used to scale the temperature control quadratic spline knots, while 3 are used to define the equilibrium glide, constant drag, and transition phases. These 6 parameters are described in Figure 6-5,
XI
X2
X3 X = .(6.1) X4
X5
X6
The first 3 parameters in Y, 1 - 3, are used to determine the quadratic coefficients { a, b, c}, which translate each quadratic spline knot (vi, di) according to Equation 6.2a to scaled spline knot (vi, d'). The point (X1, X2) is the location of the vertex of the
79 3) a
44,
Velocity (fps)
U- 3 2 O
Velocity (fps)
Figure 6-5: These 6 parameters are used to modify the nominal reference drag profile during optimization. parabola, and X3 determines the magnitude of the parabola's curvature,
d' = di(cvi + bvi + a), (6.2a)
C = X3 (6.2b)
b -2cxi (6.2c) b 2 a =X2 + - (6.2d) 4c
The second 3 parameters in Y, 4 - 6 are used to determine the subsequent three phases. 4 defines D2 , the magnitude of the constant drag phase, 5 defines V3 , the start velocity of the transition phase, and 6 defines V, the velocity used to specify the equilibrium glide phase,
X4= D2, (6.3)
5 = V3, (6.4)
80 X6 = Vs. (6.5)
E3 is also updated as V3 is modified. In order to approximate the vehicle's altitude and therefore energy at V3, a series of approximations are made. First, the planned reference angle of attack at V3 is used to obtain CD using the trim angle of attack and trim CD vs. Mach number curves from Figures 2-5 and 2-6. Then, because the drag D 3, velocity, V3 and CD are known, air density can be calculated,
2D3 (6.6)
V3 Sref CD
From this, the altitude h3 is interpolated using NASA's Global Reference Atmo- spheric Model for Earth, and the energy E3 is calculated.
V2 E = gh + 3 (6.7) 3 3 2 Although upon first glance it would be most intuitive to optimize the location of each spline knot, in practice, moving each knot independently results in drastically different profiles. Figure 6-6 illustrates this for the case when the lowest velocity knot is perturbed by a small amount. By perturbing only this knot, as could occur if the chromosome for this profile were mutated, the profile is drastically different in a way that does not reasonably represent the small input change. By choosing to modify the profile using a quadratic scaling curve, a large and diverse family of curves can be explored that are both feasible for the vehicle to fly and produce a well behaved fitness function with input parameters similar to each other producing similar results.
81 Velocity (fps)
Figure 6-6: Effect on drag reference profile of perturbing I spline knot a small amount.
6.3 Nominal Optimization Fitness Function
In order to represent vehicle failure, the fitness function is set to -I X 1020 when a failure is detected. Simulation or vehicle failures can occur for a number of reasons. These include failures stemming from simulated vehicle performance such as failure to successfully land and rollout down the runway, violation of a drag constraint with commanded or actual reference drag, or violation of a heat load constraint. Failures can also occur before even running the simulation due to issues with the proposed reference profile, such as the inputs producing profile parameters with imaginary components, spline knots being outside a feasible range, and the reference drag itself being outside a feasible range. Finally, if the range associated with the proposed profile is too dissimilar from the planned downrange in the simulation, the guidance code will struggle to remove the range error, and this is also counted as a failure. If the simulation has not failed, its fitness, J, is evaluated as a function of the margin between drag corridor constraints, J1, and heat load constraints, J2, where C2 is a constant that allows the user to balance the importance of heat load and drag
82 0 .u
UU CF
aij
Velocity (fps)
Figure 6-7: The upper and lower drag constraint margins, ui and 1i respectively, are used to evaluate the fitness of a reference drag profile.
constraints,
J = Ji + c2 J2. (6.8)
As seen in Equation 6.9, J is a sum over a series of n points along the reference profile, where ui and 1i are the upper and lower drag constraint margin respectively at point i,
1 (c 1 1 Ji Z= + . (6.9) i=1 Ui+1iUS+ 1/2 ii1/2 This equation has a term to reward large margins both above and below the profile, where ci is a constant that can be used to prioritize margin on one side over the other, which is useful for ensuring that there is sufficient margin for the hard upper constraints, sacrificing lower margin near the less severe equilibrium glide constraint. Each term in the summation is also scaled by 1/(ui + 1j) in order to reward larger margins in tighter portions of the entry corridor, where being centered in the middle of the entry corridor is more advantageous. Figure 6-7 illustrates ui and 1i on an example reference profile.
The term for rewarding heat load margin, J2 , is similar to Equation 6.9. It sums over all points on the vehicle that have a defined heat load constraint, and is scaled by the magnitude of the constraint, so that margin for lower, tighter constraints is
83 prioritized,
J2 = -Z QY' n (6.10)
Figure 6-8 plots the margin and resulting cost for two example profiles. Reference 2, in red, approaches the upper drag constraint at two different points. This results in two points of higher cost relative to the rest of the profile. However, since the total margin between the upper and lower constraints is lower at the first point, the total margin cost is higher. This is to ensure that the profile is centered in the entry corridor through the most straining, tight regions. Similarly, reference 1, in blue, is approximately centered in the entry corridor, and so has a consistently lower margin cost for both the upper and lower margin.
84 Reference Drag
- -Ref 1 Lower Limit Upper Limit
Upper Margin Rd I
Velocity (fps)
Lower Margin
1 -Rd I-
Margin Cost
-- Ref 2 Lower Ref 2 Upper - Ref I Upper - - Ref I Lower
Velocity (fps)
Figure 6-8: The drag constraint margin cost, J2, for two distinct profile candidates shows the penalty for tight drag constraint margins. Reference profile 1 is equidistant from both upper and lower constraints, and consistently has a lower margin cost than reference profile 2, which aproaches the upper constraint in two separate places. The margin cost for reference 2 shows that margin cost is a function of the total distance between upper and lower margins.
85 86 Chapter 7
Results
7.1 Monte Carlo Results under Turbulent Flow Tran- sition Constraints
Two Monte Carlo simulations were carried out with 300 runs each in order to analyze the allocator's impact on the guidance algorithm's ability to satisfy restrictive con- straints including a turbulent flow transition heat rate constraint. Both Monte Carlo tests used the same nominal profile. The resulting drag flown for the nominal range as well as ranges 3 o are shown in Figure 7-1 and 7-2. From these Figures, it is clear that the original shuttle guidance method will produce a significant number of failures with respect to the upper drag constraint. The weights used for the allocator were w, = 0.25, W2 = 1.00, W3 = 0.00, and the quadratic curve used is plotted in Figure 7-3. The Monte Carlo results using this profile are plotted in Figures 7-4 and 7-5. The original update method produced 125 failures out of 300, resulting in a failure rate of 42%, while the allocator produced 0. It is clear that the drag dispersion through the tightest part of the entry corridor is smaller using the allocator. This is primarily due to the modulation of kq, which is set to 0 at the end of the temperature control phase, so that the variation in reference drag is reduced significantly there.
87 -Nominal Profile -Constraints - Nominal Range -- 3a Range +3 a Range
0) 0
Velocity
Figure 7-1: Vehicle drag under 3a range error using original shuttle update method.
- Nominal Profile - Constraints -Nominal Range - -3a Range _+3 a Range
0) 0
Velocity
Figure 7-2: Vehicle drag under 3a range error using range allocator.
-Low Range -Nominal -High Range
a,
Velocity
Figure 7-3: kq curve for range allocator update method.
88 -Nominal Profile Constraints MC Run - Pass Run - Fail VMC
0) (1 0
Velocity
Figure 7-4: Vehicle drag for 300 Monte Carlo runs using original shuttle update method. Runs that violate the upper drag constraint are plotted in red.
- Nominal Profile - Constraints -- MC Run - Pass
C, 0
Velocity
Figure 7-5: Vehicle drag for 300 Monte Carlo runs using range allocator. All runs satisfy the upper drag constraint.
7.2 Heat Load Example
In addition to using the allocator to satisfy more complex drag constraints such as a turbulent flow transition heat rate constraint, the allocator can be used to reduce heat load and satisfy tight heat load constraints. In general, heat load can be mitigated by
89 maintaining high drag in the early, high velocity part of the trajectory and decreasing drag later in the trajectory. Decreasing drag across the entire profile will increase the range flown, which increases flight time and therefore increases heat load. Conversely, increasing drag across the entire profile increases heat rate, which also tends to result in a higher heat load. To demonstrate the value of this allocator in satisfying this constraint, two sets of parameter weights are examined using deorbit conditions that stress the heat load constraint. Typically, the heat load stressing case involves an initial state that has a longer than nominal downrange as well as a steeper flight path angle, while the opposing heat rate stressing case involves an initial state that has a shorter than nominal downrange and a shallower flight path angle.
The first set of allocator weights used are w, = 1.0, w 2 = 0.0, and w 3 = 0.0, which only uses D1 to account for range error, and is equivalent to the initial Shuttle guidance implementation. When this allocator encounters a high range, heat load stressing case, the drag profile will be shifted down across both the temperature control and equilibrium glide segment, as seen in Figure 7-6. However, this will result in lower drag at high velocity, which is likely to ultimately increase heat load.
The second set of allocator weights used in this example are w, = 1.0, W2 = 0.0, and w3 = ".7(V,xVs,n) which uses both D1 and V, to account for range error. Because the equilibrium glide phase is modified independently of the temperature control phase, the reference drag at high velocity remains largely equivalent to the nominal profile. This is expected to result in a lower heat load in comparison to the first set of weights. Example reference profiles for this second set of weights can be found in Figure 7-7. Both sets of weights were implemented and used to run a vehicle simulation with initial conditions that will result in a heat load stressing case. The resulting vehicle drag in Figure 7-8 is similar to what should be expected based on the previously generated reference profiles. While the initial reference profile defined before flight is the same for both the original shuttle update case and the allocator case, the different weights result in different drag histories in flight once the vehicle encounters a large range error upon deorbit. When heat load is analyzed for these two cases, the heat
90 0) C
C.)
-Nomninal -Low Range -High Range
Velocity
Figure 7-6: High and low range reference profiles using the original shuttle update method
CD 0 a C
-Nominal I -Low Range -High Range
Velocity Figure 7-7: High and low range reference profiles only perturbing V, load was decreased in many areas of concern, in some locations by up to 1.25%, which is significant for atmospheric entry problems such as this.
91 -Constraint -Reference Profile -Heat Rate Case -D 1 Only -Vi Only
0)
Velocity
Figure 7-8: Reference drag profile results when using the allocator to minimize heat load.
7.3 Reference Trajectory Optimization
The nominal trajectory was optimized for both a standard set of drag constraints and a set including a turbulent flow transition heat rate constraint. In both cases, the drag constraints were weighted 3 x more than the heat load constraint in order to ensure sufficient drag margins for dispersed conditions, and the upper constraint was weighted 2.5x more than the lower constraint, since violating the upper constraint results in vehicle failure, while violating the lower constraint does not. Using Equa-
tions 6.9 and 6.10, ci = 2.50 and c2 = 0.33. Figures 7-9 and 7-11 show the nominal profile that was being used for general guidance development and evaluation before the optimization with respect to both sets of drag constraints. This profile represents the best effort of a guidance engineer using hand-tuning methods.
7.3.1 Standard Constraints
The optimized profile for the standard set of constraints is not substantially different from the nominal, but it does result in an increase in drag at high velocities in
92 two different locations, which is consistent with the general guideline that high drag at velocity decreases heat load at the end of the mission. The heat load results are consistent with this observation, with a decrease in heat load for every point evaluated, as high as 2.76%. There are also deviations between the reference and true drag, particularly in the transition phase; these are due to bank reversals.
- Reference Profile -- Constraints -Vehicle Drag
0) 10
Velocity
Figure 7-9: Nominal reference profile and resulting vehicle drag under standard con- straints.
Table 7.1: Percent Heat Load Decrease between Nominal and Optimized Profiles
Thermal Control Point % Decrease Bump 1.10 Center 1.05 Chine 0.84 Fin 2.76 Left Flap 1.00 Right Flap 0.83 Stagnation Point 1.84
93 - Reference Profile - Constraints -Vehicle Drag
0D
Velocity
Figure 7-10: Optimized reference profile and resulting vehicle drag under standard constraints.
7.3.2 Turbulent Flow Transition Constraints
The optimized profile for constraints that include a restrictive turbulent flow transi- tion constraint similarly have two periods of high drag relative to the entry corridor at the beginning of the trajectory in order to reduce heat load. In addition, the profile is close to the center of the entry corridor through the turbulent flow transition "spike", which is consistent with the high cost in the cost function for low margin in these tight parts of the corridor. Similarly to the results for the standard constraints, the true vehicle drag does not track the reference drag well through the transition phase, due to the bank reversals occurring in this phase.
Every thermal control point evaluated experiences a decrease in heat load between the nominal and optimized profile. However, the total decrease in heat load is less for every point except the right flap. This is likely due to the more restrictive constraints requiring that the overall drag during the temperature control phase be lower, which results in higher heat load.
94 Table 7.2: Percent Heat Load Decrease between Nominal and Optimized Profiles
Thermal Control Point % Decrease Bump 0.06 Center 1.02 Chine 0.82 Fin 2.56 Left Flap 0.95 Right Flap 0.95 Stagnation Point 1.27
K Rflfrence Prrdilne - Constraints -Vehicle Drag
0) L. 03
Velocity
Figure 7-11: Nominal reference profile and resulting vehicle drag under constraints with turbulent flow transition modeled.
95 -,Reference Profile -- Constraints . -Vehicle Drag
CM
Velocity
Figure 7-12: Optimized reference profile and resulting vehicle drag under constraints with turbulent flow transition modeled.
96 Chapter 8
Conclusion
8.1 Results Summary
The modified range update method presented here significantly improves the flex- ibility and robustness of the shuttle orbiter entry guidance algorithm. Instead of varying the reference profile with one parameter as was done in the original shuttle guidance method, three parameters are selected to modify the reference drag profile. By having the ability to perturb three parameters on-line, vehicles can satisfy much more complex constraints than was previously possible. This ability is essential as more highly-optimized RLVs are developed within the environment of the rapidly growing commercial space industry. Because a single RLV will be used repeatedly for atmospheric entry, the lighter thermal protection systems and simpler structures that result from being able to satisfy tighter constraints have a magnified effect. This range update method is also designed with implementation time and flight checkout time in mind. By requiring that the 3 parameters be perturbed according to a pre-defined ratio with weights w, all possible reference drag profiles that can be generated on-line can be predicted by varying K, or by equivalently varying range error. As a result, the profile generated during flight is only a function of the current state and current range measurement, and is not dependent on past states or past range measurements. Furthermore, the three allocator parameters are selected based on the shuttle guidance algorithm's structure in order to minimize total impact on the
97 algorithm. By "hijacking" previously defined parameters in the guidance algorithm, the high-TRL of the shuttle algorithm is leveraged for future use. The versatility and utility of the allocator are demonstrated, showing that it can be used to satisfy a variety of constraints. By implementing an allocator that primarily uses kq and Di to update the reference profile range, it is possible to reduce vehicle failures from 42% to 0% in Monte Carlo tests. This result is significant for the development of thermal protection systems in the future, since the mass of these systems can be reduced if their resultant restrictive constraints can now be satisfied. Furthermore, heat load can be reduced for the most stressing cases by modifying the allocator parameters to primarily use V, so that range updates mainly impact the second phase of the trajectory. Finally, these reference profiles can be easily optimized using a genetic algorithm to ensure that the full utility of the allocator is realized. By optimizing the allocator according to drag constraint margin and heat load margin, profiles can be found that have sufficient constraint margin to survive dispersed conditions while still decreasing vehicle heat load.
8.2 Future Work
Further work on this extended range update method would allow for its flexibility and utility to be fully understood. Since the optimization scenario involving the reference drag profile provided promising results, a genetic optimization involving the six allocator parameters under extreme entry interface conditions would likely also produce promising results. Other opportunities for improvement would involve the implementation of cubic splines, rather than quadratic splines. Although the calculation of cubic spline coefficients is more complicated than quadratic splines, it can be done iteratively without inverting a large matrix, similar to quadratic splines. In addition, each spline knot can be perturbed without disturbing the entire curve, which is useful both for simpler profile optimization, and for the ability to use each knot location as an allocator parameter.
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