A Predictor-Corrector Guidance Algorithm Design for a Low L/D Autonomous Re-entry Vehicle
by Carla Haroz B.S. Aerospace Engineering B.A. Russian Language and Literature The University of Texas at Austin, 1996
Submitted to the Department of Aeronautics and Astronautics in partial fulfillment of the requirements for the degree of Master of Science in Aeronautics and Astronautics
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY S 11~SACHUSETTSINSTITUTE r - OFTECHNOLOGY December 1998 1AY 1 71999 BAg- oopkPubkd rr oC'.crnd 1 Ctihes Certified by ...... Richard H. Battin Professor of Aeronautics and Astronautics Thesis Supervisor Accepted by ...... AcceptedVP~o.. 1...... JaimePeraire Associate Professor of Aeronautics and Astronautics Chairman, Department Graduate Committee 0. 2 A Predictor-Corrector Guidance Algorithm Design for a Low L/D Autonomous Re-entry Vehicle by Carla Haroz Submitted to the Department of Aeronautics and Astronautics on December 18, 1998, in Partial Fulfillment of the Requirements for the Degree of Master of Science in Aeronautics and Astronautics Abstract The Precision Landing Reusable Launch Vehicle (PL-RLV) is a low L/D, 2-stage craft with a mission plan that calls for low cost, speedy retrieval, and quick turn-around-times for successive flights. A guidance scheme that best adheres to these goals and captures the vehicle's capability is desired. During re-entry, the PL-RLV's second stage, the Precision Landing Vehicle 2 (PLV-2), will perform a reversal maneuver. This thesis concentrates on a possible re-entry guidance scheme for the PLV-2 during the terminal phase, the time from the completion of the reversal until the landing system parachutes are deployed. A simple bank-to-steer algorithm is suggested. The angle of attack is trimmed, and the bank angle (or bank rate) remains as the only means for control. The algorithm controls the time history of the vehicle's bank angle and tunes the bank angle history to meet land- ing and fuel requirements. This versatile guidance approach employs predictor-corrector methods. The guidance scheme presented generates a possibility of bankrate profiles within limitations that could be used for target acquisition. Selection of a robust target location and the nominal bankrate profile which will yield a minimum target miss are investigated. Testing shows the trade-offs between fuel cost and landing capability. Dis- persion testing with winds and density are also performed. The predictor-corrector combination can yield target miss distances on the order of hun- dreds of feet or less. Open-loop and closed-loop results display the guidance system's ability to capture the PLV-2's capability in the presence of dispersions while still meeting system requirements. Thesis Supervisor: Dr. Richard H. Battin Title: Professor of Aeronautics and Astronautics Technical Supervisor: Peter J. Neirinckx Title: Member Technical Staff, The Charles Stark Draper Laboratory, Inc. 4 Acknowledgements There are many people that I would like to thank that have made my experience at MIT and Draper Labs educationally broadening, challenging, and enriching. Thank you to Tim Brand, who gave me the opportunity to work at Draper, and to Peter Neir- inckx for supervising during the thesis process. Thank you also to Lee Norris, Doug Fuhry, and George Schmidt for their guidance and advice. I have been honored to be in the classrooms of two of the greatest minds in Orbital Mechan- ics, Dr. Richard Battin and Dr. Victor Szebehely. A special thanks to both of them for presenting the beauty of planetary motion to me and for bringing the history of US Space Exploration alive. A big thanks to my professors at the University of Texas who encouraged me to attend MIT and who are always there for advice and support: Dr. Hans Mark, Dr. Wallace Fowler, and Dr. Robert Bishop. Thank you to all my friends at Draper Labs - Gregg "TMG" Barton for all the encouragement and cookies!, Chris D'Souza for checking up on me to make sure I was still alive in my cubicle, Chris "Sparkster" Stoll for constantly battling the gremlins in my computer, Jenn "KB" Hamelin for the chats, Ed Bachelder for the tete-e-tete's and love of Trader Joe's Chocolate, and to all my Draper fellow friends with whom I made it through classes, work, and fun with - Christina, Atif, Nate "Shenckenstein", Geoff, Pat, Chisolm, and my favorite softball team, the Draper Monkeys!! Also, a thank you smile to my encouraging friends at the DLR German Space Center - Manfred, M. Klimke, M. Reichart, and Dr. Seiboldt; to Shaun, Carolyn, Benno, Tobias, Jonathan, Santiago, and my NASA buds, Terry, Greg, and Andy. Thank you to Nick Nuzzo, for all the love and support, thesis empathy, late night Draper din- ners, Swing dancing in the halls, and our thesis getaway island adventure in Greece. S'agapo. The biggest thank you goes to my family: Mom, Dad, Lezlie, Tammy, and Kim. To my older sisters, you can officially stop calling me your "LITTLE" sister now. Mom and Dad, you have always encouraged me to follow my dreams and shoot for the stars. I continue my journey on the road less traveled knowing that you are always there for me. I love you. 5 This thesis was prepared at the Charles Stark Draper Laboratory, Inc. Publication of this thesis does not constitute approval by the Draper Laboratory or the sponsoring agency of the findings or conclusions contained herein. It is published for the exchange and stim- ulation of ideas. Permission is hereby granted by the Charles Stark Draper Laboratory, Inc. to the Massachusetts Institute of Technology to reproduce any or all of this thesis. I= Carla " S. Haro z Carla S. Haroz a 6 Table of Contents 1 Introduction ...... 15 1.1 Problem Definition...... 16 1.2 Autonomous Bank-to-Steer Guidance Challenges ...... 18 1.3 Thesis O verview ...... 19 1.4 Chapter Breakdown ...... 19 2 Mission Overview and Requirements ...... 21 2.1 Flight Profile ...... 21 2.2 PLV-2 Aerodynamic and Mass Properties ...... 22 2.3 Coordinate Frames ...... 25 3 Simulation Code Structure...... 31 3.1 Simulation Environments ...... 31 3.2 General Design Features of the Re-entry Guidance ...... 31 3.3 PLV-2 Entry Guidance Code Definition...... 33 3.4 Functions of the Initial Subphases ...... 34 3.5 Terminal Subphase ...... 38 3.6 Landing System Phase ...... 38 3.7 Atmospheric Models ...... 40 3.8 Vehicle Uncertainty and Environment Dispersion Sources ...... 42 4 Guidance Design ...... 45 4.1 Guidance Scheme Definitions...... 45 4.2 Trajectory Control...... 49 4.3 Predictor...... 50 4.4 Corrector ...... 57 5 Nominal Bank Rate Profile Selection ...... 65 5.1 Profile Generation ...... 66 5.2 Bank Rate Bin Definition...... 67 5.3 Example Profiles ...... 69 5.4 Nominal Profile Selection ...... 72 6 Robustness Testing ...... 77 6.1 Range Capability ...... 77 6.2 Nominal Target Robustness Results ...... 82 6.3 Footprint Range ...... 85 7 Guidance Performance...... 91 7.1 Predictor and Corrector Performance ...... 91 7.2 Fuel Cost vs. Landing Performance...... 99 7.3 Corrector Performance On Nominal Profile ...... 112 7.4 Bin Number Selection ...... 113 7.5 Effects of Atmospheric Dispersions ...... 118 8 C onclusions ...... 127 Appendix A Analytical Study of the PL-RLV Re-entry Guidance ...... 129 Appendix B Acceleration Model for Bank Maneuvers ...... 157 Appendix C Nominal Profile Simulation Plots...... 165 7 References ...... 171 8 List of Figures Figure 1.1: The PL-RLV Concept Drawing ...... 17 Figure 2.1: Mach vs. Trim Angle of Attack During PL-RLV Descent ...... 24 Figure 2.2: Aerodynamic Parameter History During PL-RLV Descent...... 24 Figure 2.3: Relationship Between Inertial, LVLH, Body, and Velocity Axis Systems [8] 27 Figure 2.4: Relationship Between Bank Angle and Lift...... 28 Figure 2.5: Relationship Between Body Frame and LVLH Frame [1] ...... 29 Figure 2.6: Relationship Between North/East and Crossrange/Downrange Frames ...... 30 Figure 3.1: Re-entry Guidance Outputs...... 32 Figure 3.2: Sample Bank Angle Profile...... 39 Figure 3.3: Example Footprint ...... 41 Figure 3.4: Altitude vs. Density, Typical PLV-2 Descent...... 41 Figure 3.5: Altitude vs. Speed of Sound, Typical PLV-2 Descent ...... 42 Figure 4.1: Re-Entry Guidance Predictor Calls ...... 51 Figure 4.2: Closed-Loop Predictor/Corrector Guidance ...... 52 Figure 4.3: Predictions for Corrector Guidance in Entry Phase ...... 58 Figure 4.4: Minimum Miss Guidance for Reversal Stage ...... 62 Figure 5.1: Flow Chart for Selection of the Nominal Profile ...... 65 Figure 5.2: Terminal Phase Bins...... 67 Figure 5.3: Bankrate Profile Generation, Cycling Through First Bin ...... 68 Figure 5.4: Open-Loop Footprint of Landing Locations ...... 71 Figure 5.5: Location of Target Selected Within Footprint ...... 74 Figure 5.6: Acceptible Bank Rate Profiles (#1-#9) Generated for New Target ...... 75 Figure 5.7: Acceptible Bank Rate Profiles (#10-#14) Generated for New Target ...... 76 Figuie 6.1: Target A, B, C, and D Locations Within the Footprint ...... 78 Figure 6.2: Bank Rate Profile for Target A ...... 79 Figure 6.3: Bank Rate Profile for Target B...... 80 Figure 6.4: Bank Rate Profile for Target C...... 81 Figure 6.5: Bank Rate Profile for Target D ...... 82 Figure 6.6: Range Capability of Test Cases, Final Miss Within 10,000 ft...... 83 Figure 6.7: Range Capability of Test Cases, Final Miss Within 1,000 ft...... 84 Figure 6.8: Footprints at Time= 0 seconds in Terminal ...... 86 Figure 6.9: Footprints at Time= 5 seconds in Terminal ...... 86 Figure 6.10: Footprints at Time= 10 seconds in Terminal ...... 87 Figure 6.11: Footprints at Time= 15 seconds in Terminal ...... 87 Figure 6.12: Footprints at Time= 20 seconds in Terminal ...... 88 Figure 6.13: Footprints at Time= 30 seconds in Terminal ...... 88 Figure 6.14: Footprints at Time= 40 seconds in Terminal ...... 89 Figure 7.1: Bank Angle Profile for Case 7.1 ...... 93 Figure 7.2: Terminal Phase Bank Rate Profile for Case 7.1 ...... 93 Figure 7.3: Open-loop Horizontal Prediction Error: Case 7.1 ...... 94 Figure 7.4: Open-loop Horizontal Prediction Error: No Pre-terminal Bin Knowledge: Case 7.1...... 95 9 Figure 7.5: Horizontal Prediction Error, Closed-Loop Terminal Phase ...... 96 Figure 7.6: Open-loop Horizontal Prediction Error: Case 7.2 ...... 97 Figure 7.7: Closed-loop Horizontal Prediction Error: Case 7.2 ...... 98 Figure 7.8: Terminal Phase Bank Rate Profile for Case 7.2 ...... 98 Figure 7.9: Closed-Loop Target Miss Distances for the 14 Profiles ...... 100 Figure 7. 10: Corrections to the Bank Rate Profile - Case 1...... 101 Figure 7.11: Trajectory Path - Case 1 ...... 102 Figure 7.12: Trajectory Path - Case 1 ...... 102 Figure 7.13: Prediction Error- Case 1 ...... 103 Figure 7.14: Corrections to the Bank Rate Profile - Case 2...... 104 Figure 7.15: Trajectory Path - Case 2 ...... 104 Figure 7.16: Trajectory Path - Case 2 ...... 105 Figure 7.17: Trajectory Path - Case 2 ...... 105 Figure 7.18: Corrections to the Bank Rate Profile - Case 3 ...... 106 Figure 7.19: Trajectory Path - Case 3 ...... 107 Figure 7.20: Trajectory Path - Case 3 ...... 107 Figure 7.21: Trajectory Path - Case 3 ...... 108 Figure 7.22: Corrections to the Bank Rate Profile - Case 4...... 109 Figure 7.23: Trajectory Path - Case 4 ...... 109 Figure 7.24: Trajectory Path - Case 4 ...... 110 Figure 7.25: Trajectory Path - Case 4 ...... 110 Figure 7.26: Closed-Loop Prediction Error: Offset From Nominal Target ...... 112 Figure 7.27: Terminal Phase Bank Rate Profile: Offset From Nominal Target ...... 113 Figure 7.28: Open-loop Landing Locations for Large Bin #'s of 3, 4, 5, and 6 ...... 115 Figure 7.29: Open-loop Footprint Given Maximum Small Bins of 10, 20, 30, 35, 36, & 40116 Figure 7.30: Altitude vs. Density: Summer Case ...... 119 Figure 7.31: Altitude vs. Density: Winter Case ...... 119 Figure 7.32: Altitude vs. North Wind: Summer Case ...... 120 Figure 7.33: Altitude vs. North Wind: Winter Case ...... 120 Figure 7.34: Altitude vs. East Wind: Summer Case ...... 121 Figure 7.35: Altitude vs. East Wind: Winter Case ...... 121 Figure 7.36: Altitude vs. Vertical Wind: Summer Case ...... 122 Figure 7.37: Altitude vs. Vertical Wind: Winter Case...... 122 Figure 7.38: Dispersion Miss Distances (1000 ft): Summer ...... 125 Figure 7.39: Dispersion Miss Distances (6000 ft): Summer...... 125 Figure 7.40: Dispersion Miss Distances (1000 ft): Winter ...... 126 Figure 7.41: Dispersion Miss Distances (6000 ft): Winter ...... 126 Figure A. 1: Single-Switch Program Schematic ...... 130 Figure A.2: X, Y, Z Axes and Bank Angle q Defined ...... 131 Figure A.3: Final Positions When T_max = 40 sec ...... 135 Figure A.4: Final Positions When T_max = 30 sec ...... 136 Figure A.5: Final Positions When T_max = 20 sec ...... 136 Figure A.6: Final Positions When T_max = 10 sec ...... 137 Figure A.7: Final Positions When T_max = 5 sec ...... 137 Figure A.8: All Final Positions, Tswitch = 10 sec ...... 139 Figure A.9: All Final Positions, Tswitch = 5 sec ...... 139 10 Figure A.10: Final Positions, Initial Bank Angle = 90 deg ...... 140 Figure A. 11: Resulting Final Positions, Each Parameter Tweeked One at a Time...... 142 Figure A. 12: Accessible Target Case ...... 146 Figure A. 13: Another Solution to Case 1, Target Acquisition ...... 147 Figure A. 14: Case 1 With Gains Set to .50 ...... 148 Figure A.15: Case 1 With Gains Set to 1.0...... 148 Figure A. 16: Figure 12. Crossing the Inaccessible Zone...... 150 Figure A. 17: Non-instantaneous Bank Rate Change ...... 152 Figure A. 18: Non-zero Initial Positions ...... 1531...... Figure A. 19: Non-zero Initial Velocities ...... 153 Figure A.20: Wait Times between Bank Rate Changes Schematic...... 155 Figure B. 1: Acceleration Model Bankrate vs. Time ...... 157 Figure B.2: Profile Redesign Due to Time Constraints ...... 160 Figure C.1: Altitude Profile ...... 165 Figure C.2: Earth Relative Velocity Profile ...... 166 Figure C.3: Relative Flight Path Angle Profile...... 166 Figure C.4: Dynamic Pressure Profile ...... 167 Figure C.5: Dynamic Pressure x Alpha Profile ...... 167 Figure C.6: Heating Rate Profile ...... 168 Figure C.7: Stagnation Point Temperature Profile ...... 168 Figure C.8: Acceleration Profile ...... 1691...... Figure C.9: Mach Number Profile ...... 169 Figure C. 10: Angle of Attack Profile...... 170 11 12 List of Tables Table 2.1: Modeled Aerodynamic and Mass Properties of the PL-RLV 22 Table 2.2: PLV-2 Mass Properties 23 Table 3.1: Guidance Subphase Transition Criteria 34 Table 5.1: Bin Generation Conditions for Sample Run 69 Table 5.2: Initial Target and PLV-2 Setting 70 Table 6.1: Target Coordinates in ECEF and North/East Frame 78 Table 7.1: Case 7.1 Simulation Conditions 92 Table 7.2: Closed-Loop Results for the 14 Profiles 99 Table 7.3: Fuel Cost and Miss Distance Comparisons 111 Table 7.4: Large Bin Number Range Testing 114 Table 7.5: Small Bin Size Testing Results 117 Table 7.6: Dispersion Testing Results: Summer Case 123 Table 7.7: Dispersion Testing Results: Winter Case 124 13 14 Chapter 1 Introduction With the new era of satellite constellations, the advent of space station construc- tion, and renewed planning for Mars missions, comes an increased interest in Reusable Launch Vehicles (RLV's). Commercial RLV's must be able to handle particular challenges that expendable launch vehicles do not need to meet. RLV's are necessarily more complex than expendables and thus investigations of design tradeoffs are all the more important. Some RLV mission requirements can be critical, such as the desire to land the vehicle within a target area that is easily accessible to support and launch facilities, ground crews, and transportation. To ensure frequent deployment opportunities, RLV's must be able to operate under relatively quick turn-around-times between launches. Also, a desire for complexity reduction in one design area can often sacrifice system performance. A specific guidance design for a RLV, hereafter refered to as the PL-RLV (Preci- sion Landing Reusable Launch Vehicle), is presented in this thesis. Structural design sim- plification has yielded a low L/D (Lift-over-Drag) vehicle. However, this simplification is traded against the already sensitive guidance and landing requirements of the RLV's re- entry and landing phases. This trade-off requires an in-depth analysis into the precision and robustness of any re-entry guidance design. This thesis project designs and analyzes a particular re-entry autonomous guidance scheme for the PL-RLV during the terminal phase of descent, the time between a reversal maneuver and the landing system parachute deployment. A simple bank-to-steer algorithm meets both the vehicle constraints and landing requirements. 15 1.1 Problem Definition One of the most important aspects of a RLV is the ability to land at a predeter- mined target which is located close to the processing facility. As opposed to the mid-ocean splashdown techniques of the Apollo days, a more sophisticated landing scheme must be developed. This scheme could also be more simplistic than the Space Shuttle's guidance, which relies on three control variables for its precision, but at a significant design and structure cost [2]. Furthermore, due to its simplicity, the new landing scheme could be uti- lized in the future for an entry vehicle flying missions on distant planets. In that case, it would be imperative that the guidance scheme provide a precise landing since the survival of the manned or unmanned mission could hinge upon the precision of the ship's landing location. It is desired that the PL-RLV return as close to the launch site as possible - within a few miles. This will enable quick turn-around-times, more launches, less cost for retrieval, and more opportunities to deploy payloads into space. The PL-RLV is shown below dur- ing the ascent phase in Figure 1.1. 16 ~·~i~__"MIRMn ;6001 Figure 1.1: The PL-RLV Concept Drawing The central theme and challenge of this project lies in designing a terminal guid- ance algorithm that captures most of the vehicle's physical capability, is controllable, and is robust to environmental variations. An early examination of precision landing schemes and an investigation into their cost/risk trade-offs are justified since their results impact many aspects of a program, including: mission architecture and objectives, landing site selection, navigation infrastructure, trajectory design, vehicle loads and structure, and environmental knowledge requirements. Preliminary studies indicated that the trade-off between design complexity and guidance performance of the PL-RLV proved favorable with a bank-to-steer guidance scheme. See Appendix A for one of these early studies. 17 1.2 Autonomous Bank-to-Steer Guidance Challenges Bank-to-steer guidance, similar to the re-entry guidance used for the Apollo mis- sion, utilizes only a single control parameter: vehicle bank angle. The bank angle is essen- tially the angle the lift vector points with respect to the vertical in the plane perpendicular to the velocity vector at a trim angle of attack and zero sideslip. Commanding and execut- ing a change in bank angle steers the vehicle to the target. Since this guidance scheme is constrained to only one control parameter, a considerable reduction in the structure and software complexity of the design results if the scheme is successful. The price for this simplicity, however, could cause a reduction in guidance capability and landing perfor- mance. The challenge lies in creating a bank-to-steer design that meets both the mission performance requirements and is robust in its response to in-flight dispersions. Atmospheric uncertainties in air density and wind information pose a significant challenge to the trajectory control of an autonomous guidance scheme with a single con- trol parameter and for a low L/D body. The navigation uncertainties on guidance perfor- mance are typically small in comparison, and therefore will not be addressed in this study. The precision landing problem is further complicated upon inclusion of an unguided land- ing phase which exposes the vehicle to wind effects after landing system deployment, which is the case for the PL-RLV, or possible hazard detection maneuvers during the ter- minal phase. To successfully direct the vehicle to the desired landing location, the re-entry guid- ance scheme must be able to construct a realistic bank versus time profile, predict the resulting landing locations, and correct the bank plan if necessary. The profile must also be 18 able to conform to the RCS jet specifications and utilize only the fuel available. Additional trajectory control parameters, such as angle-of-attack modulation or landing system deployment time, could also be considered in the development of a guidance scheme, but are not researched in this study. 1.3 Thesis Overview The concentration of this thesis is the development of a precision autonomous guidance scheme for the low L/D PL-RLV to be used during the terminal phase. Several bank-to-steer schemes were designed using a predictor/corrector coding structure. Draper Laboratory's CSIM simulation framework of the PL-RLV was used to test this scheme. It is hoped that with minor modifications, the design will serve as a model for those missions that will take place on other celestial frontiers in the future, specifically the safe transportation of human explorers to the surface of Mars. 1.4 Chapter Breakdown Chapter 1 provides introductory information on this thesis. Chapter 2 defines the PL-RLV mission and the vehicle's properties and requirements. Chapter 3 introduces the entry guidance algorithm software structure. Chapter 4 defines the re-entry guidance schemes and the predictor/corrector algorithms utilized. Chapter 5 introduces the selection of the nominal bank profile, and chapter 6 contains the nominal design robustness testing. The guidance design performance, fuel reduction versus landing performance results, 19 optimization of the terminal phase design, and the effects of atmospheric dispersions are all presented in Chapter 7. Chapter 8 is devoted to overali results and conclusions to the thesis, as well as suggestions for future work. 20 Chapter 2 Mission Overview and Requirements Knowledge of the "big-picture" which surrounds the task at hand is always a key factor in the production of a simulation design. This chapter provides a brief overview of the PL-RLV's flight profile, as well as an introduction of certain vehicle parameters which have influenced the design of the PL-RLV. Coordinate frame definitions are provided in this chapter. 2.1 Flight Profile The PL-RLV is a fully reusable launch vehicle consisting of two stages: the PLV-1 (Precision Landing Vehicle, the first stage) and the PLV-2 (the second stage). At lift-off, the PLV- 1 helps to boost the PLV-2 on its trajectory path. The specific phases of the mission's descent phase will be described in Chapter 3. At an appropriate time after lift-off, the PLV-2 separates from the PLV-1, fires its engine, and continues on its own journey. The PLV-1 guidance is not investigated in this thesis. The PLV-2's ascent continues to the desired low-earth-orbit (LEO) or middle-earth-orbit (MEO). Orbital maneuvers are accomplished by orbital maneuvering system (OMS) engines. While in orbit, the satellite housed inside the PLV-2 is deployed. Following deployment, the PLV-2 performs orbital adjust maneuvers. At the appropriate time, the 21 PLV-2 performs a pitchover maneuver and fires the OMS engines to deorbit. Upon atmo- spheric entry, the PLV-2 guidance will fly the craft on a path toward the target until the landing system deploys. 2.2 PLV-2 Aerodynamic and Mass Properties Crucial in mission design is the knowledge of the craft's aerodynamic and mass properties. The modeled aerodynamic and mass properties of the PL-RLV used in the guidance design are shown below in Table 2.1. In the guidance simulation, the vehicle is modeled as a point mass of 923.19 slugs with an effective aerodynamic area of 201 ft2. A maximum bank angle and bank rate were introduced into guidance control in order to prevent unrealistic modeling of the vehicle's motion. Table 2.2 gives a more in-depth mass property listings for the PLV-2. Point Mass 923.19 slugs Vehicle Aerodynamic 201 ft 2 Reference Area Average Lift / Drag Ratio .1 Maximum Bank Angle 175 deg= Entry Phase 180 deg = Rev/Terminal Maximum Bank Rate 10 deg/sec Velocity at 974.25 ft/sec Landing System Deploy Table 2.1: Modeled Aerodynamic and Mass Properties of the PL-RLV 22 Table 2.2: PLV-2 Mass Properties The coefficient of lift, C1, and the coefficient of drag, Cd, are both functions of the vehicle's angle of attack and Mach number. These aerodynamic parameters are indepen- dent of the trajectory. The following plots are typical examples of the PL-RLV's aerody- namic history as it flies to the ground. The large jump in the Cd and C1 plots near 75,000 ft occurs when the landing system parachute is deployed. 23 Figure 2.1: Mach vs. Trim Angle of Attack During PL-RLV Descent a, 3- 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Lift and Drac Coefficients Figure 2.2: Aerodynamic Parameter History During PL-RLV Descent 24 2.3 Coordinate Frames Several different coordinate system reference frames are used throughout this sim- ulation. These frames are used in the design and analysis of the entire system. The follow- ing coordinate frames were used in the PL-RLV simulation: Inertial Frame The Inertial reference frame is non-rotating with an earth-centered origin. The XI axis points through zero longitude at time zero (epoch), the ZI axis through the North Pole, and the YI axis points in the direction perpendicular to the XI and ZI axes in order to complete a right handed set. Integration of the vehicle's state (equations of motion) is per- formed in this frame. Earth Centered Earth Fixed (ECEF) The ECEF frame rotates about the inertial frame z-axis at a fixed rotation rate defined by the earth's constant rotation rate (without precession). The inertial and ECEF frame are co-aligned at tirne=O.The origin is the earth's center and the axes are defined as follows: the Xecefaxis points through zero geodetic longitude and latitude, the Zecefaxis through the North Pole, and the Yecefaxis completes the right-handed set. The landing site is given and some guidance calculations are performed in this frame. 25 Local Vertical Local Horizontal (LVLH) This coordinate frame is used a great deal in trajectory design. The origin is the vehicle's center of gravity, the Zivlh axis (local vertical) points towards the earth's center, the Yvlh axis is the cross product of Zlvh and the vehicle's earth-relative velocity vector, and the Xlvlh axis (local horizontal) is defined by the cross product of Y1vlhand the Zvlh axes. Body Frame The body frame is fixed to the vehicle center of gravity with the Xb axis along the longitudinal axis (nose positive) of the PLV-2, the Zb axis pointing "down" through the lateral axis of the PLV-2, and the Yb axis completes the right handed set. The rotations between this frame and the LVLH frame express the typical aerodynamic control parame- ters: yaw, pitch, and roll. Velocity Frame The velocity frame has the Xv axis defined along the vehicle's earth-relative veloc- ity vector, the Yv axis a result of the cross product between the gravity vector and X,. The Z. axis completes the right handed set. The relationships between the Inertial, LVLH, Body, and Velocity frames are shown in Figure 2.3 below. The angles X, y, a, and 0 are defined as the co-latitude, flight path angle, angle of attack, and the pitch angle of the craft. 26 I ,I Xb a C I Zlvlh I *Zv XI Figure 2.3: Relationship Between Inertial, LVLH, Body, and Velocity Axis Systems [8] The figure below shows a different view of the Body and Velocity Frames. Most importantly for this thesis is the definition of the vehicle bank angle, 4.The angle between the Zv axis and the lift vector is the bank angle. The bank angle rotates around the earth- relative velocity vector, Xv. 27 __ Vector Xb ! v Zv __ Figure 2.4: Relationship Between Bank Angle and Lift Figure 2.5 below illustrates the relationship between the Body and Local Vertical Local Horizontal Frame. The vehicle attitude is defined by 4, 0, A; the roll, pitch, and yaw angles of the craft. When a change in bank is commanded, the maneuver is executed by changing the roll and pitch angles. 28 Xb - IsI I \ Zb Yb Figure 2.5: Relationship Between Body Frame and LVLH Frame [1] Crossrange-Downrange Vehicle Trajectory Frame This coordinate frame is used to express the landing-point to target miss compo- nents. The frame itself is tied to the vehicle trajectory instead of the target. A full lift-up trajectory, a trajectory with no reversal, is used as the reference and defines the frame. The crossrange direction is perpendicular to the trajectory at the landing location, while the downrange direction lies along the trajectory groundtrack. This frame ties more closely the trajectory controls on the miss components, i.e. more in-plane lift increases the coordi- nate along one axis, while more out-of-plane lift increases the coordinate along the orthogonal axis. North/East Frame The North/East Frame is used for plotting miss errors once the PLV-2 lands. A cross product between the Zecefand the landing location vector in the ECEF frame yields 29 the definition of the East vector, E. A cross product between the landing location (ECEF) vector and the unit vector in the E direction defines the North direction, N. The relation- ship between the crossrange/downrange and north/east frames are shown in Figure 2.6. N UdrI Lift-Up_ Landing Point E TrajectoryGround Track Ucr Trajectory Ground Track Figure 2.6: Relationship Between North/East and Crossrange/Downrange Frames 30 Chapter 3 Simulation Code Structure This chapter provides a breakdown of the code and design features used for the PL-RLV's descent simulation. Characteristics of each phase encountered during the descent trajectory are defined briefly for basic understanding of the simulation. Particular attention will be given to the simulation of the teminal phase. Sources of dispersions that need to be addressed in simulation design are also presented. 3.1 Simulation Environments A C-based 3-DOF simulation environment (C-SIM) was used as the framework to develop the guidance simulation code and run the PLV-2's trajectory simulations. The C- SIM recognizes point mass assumptions and guidance commanding. The guidance com- manding in this thesis is concerned with bank angle commands and bank rate histories for control purposes. The bank angle commands themselves are limited to avoid unreasonable vehicle motion and bank rate commands for the vehicle. In this thesis, the vehicle is assumed to be aerodynamically trimmed at every point on the trajectory. 3.2 General Design Features of the Re-entry Guidance The re-entry guidance is a closed-loop predictor/corrector algorithm with the downrange and crossrange target miss distance as the constraints. The original guidance 31 algorithms for the entry and reversal phases were designed for Draper Laboratory by Doug Fuhry [4]. The vehicle bank angle, the time of a single symmetric bank reversal, and a bankrate profile are the controls. There is no explicit control of any other parameters, such as convective heating rate, aerodynamic load, or landing system parachute deployment time. Figure 3.1 illustrates the guidance output flow. The re-entry guidance produces bank angle and bank rate commands, as well as the commanded time of the reversal and esti- mated landing system parachute deployment time, to re-entry control. A bank reversal is always commanded, and it is always executed as a lift-up bank reversal (one which passes through 0 degrees bank). II F 4, Bank Angle I 4dot, Bank Rate m = Output Mass t___ Parachlte Denlov Time -par)------. I Figure 3.1: Re-entry Guidance Outputs Since the guidance has no knowledge regarding mass decrements as a result of Reaction Control System (RCS) firings, the output mass is a constant value equal to the initial value provided by onorbit guidance. Guidance targets a geographic location in 32 Earth-centered/Earth-fixed (ECEF) coordinates at the time of landing system parachute deployment. The re-entry guidance also allows uplinks of atmospheric density, speed of sound, and ECEF wind vector profiles. 3.3 PLV-2 Entry Guidance Code Definition The PLV-2's re-entry guidance is divided into 5 main phases: pre-entry, entry, reversal, terminal, and landing system [4]. Until the vehicle reaches 400,000 ft, it is con- sidered to be in the pre-entry stage. Once the craft has passed this altitude and has a spe- cific force greater than the I-load, the code turns to the entry stage. The reversal stage begins at a commanded time determined by the predictor/corrector algorithms found in the entry phase. Immediately after the PLV-2 reversal is complete, the terminal guidance phase begins. The terminal phase and predictor/corrector algorithm is the concentration of the thesis and will be discussed in greater detail in Chapter 4. When the vehicle reaches a velocity of 975.00 ft/sec, a landing system parachute is deployed. The landing system phase continues until the desired altitude for the simulation termination is achieved. A 5,000 ft altitude was chosen to be the termination value of the simulation. From this point, it is assumed that the craft will travel on a relatively straight ECEF path down to the ground. Table 3.1 summarizes the criteria for phase transitioning. 33 Guidance Subphase Transition Criteria pre-entry Default at cyclic guidance initiation entry Geodetic altitude < 400kft (I-load) AND Measured specific force > (I-load) reversal Current time within .1 sec (I-load) of commanded bank reversal start time terminal Current bank within 0.05 deg (I-load) of commanded bank angle at reversal end landing system Velocity <= landing system parachute deploy value (975.00 ft/sec, mission load), approx 75kft Table 3.1: Guidance Subphase Transition Criteria 3.4 Functions of the Initial Subphases Each subphase of the descent profile is responsible for certain guidance com- mands. These commands are responsible for guiding the craft on a trajectory that will ulti- mately yield a minimum target miss. The requirements of each phase are described below [4]. Pre-Entry Phase During this phase, the commanded bank angle is set to a mission load value. The proper bank angle rate command sign is computed as well. A positive bank rate sign is indicative of a positive initial bank angle, and a negative sign indicates a negative initial 34 bank angle. The pre-entry guidance also sets the maximum commanded bank angle rate to a mission load magnitude times the computed sign. Entry Phase During the first pass in the Entry Phase, a function (CRDRlanding) is called to compute the Crossrange/Downrange Coordinate Frame Reference axes based upon the predicted stabilization parachute deploy positions of full lift-up and full lift-down trajecto- ries. It is important to note that these defined axes are invariant during re-entry. The sign of the initial bank command is computed based upon the crossrange offset of the targeted landing site. Guidance during this phase consists of a trajectory prediction to estimate the para- chute deploy time using the predictor inputs: current guidance bank angle, the state vector (position, velocity, and current time), the reversal start time commands, and the terminal phase bankrate profile. If this estimated position is not within 500 ft (tolerance) of the tar- get, two additional trajectory predictions with parameter perturbations will be run; one with the bank angle perturbed by 2 deg (mission load) and one with the reversal start time perturbed by 5 sec (mission load). The numerical partial derivatives of the constraints with respect to the guidance control parameters using the target miss distances from the three trajectory predictions are then computed. Corrections to the control parameters are gener- ated based upon the predicted target miss and the numerical partial derivatives. 35 The bank reversal start time is forced to begin before the estimated landing system parachute deployment time less the time it takes for reversal execution. The bank angle magnitude is limited between 0 and 180 deg (mission loads). The two guidance control parameters, commanded bank angle or reversal time, are checked to make sure they are not outside the proscribed limits. If saturation of only one control parameter is present, a single iteration on the remaining free parameter is performed to reduce target miss. If sat- uration of either or both control parameters is detected, guidance computes a new, achiev- able offset target point. A guidance solution is iterated upon until one of the following three criteria is met: (1) Three iterations completed (2) Predicted target miss < 500 ft (3) The commanded bank angle correction < 1 deg AND commanded reversal time correction < 1 sec During entry, the proper bank angle rate command sign is computed and this rate is set to a mission load magnitude times the computed sign. Another parameter that guidance is responsible for calculating is the time at which the velocity reaches 975.00 ft/sec. This time is designated as the commanded landing system parachute deployment time. 36 Reversal Phase The reversal phase consists of a closed-loop predictor/corrector algorithm with the horizontal target miss distance as the only constraint and the bank angle immediately fol- lowing bank reversal completion as the only control. The first trajectory prediction esti- mates the landing system parachute deployment position using the current guidance command. The second trajectory prediction is made to estimate the stabilization parachute deployment position for a trajectory with a bank angle perturbed by 2 deg. The minimum achievable target miss distance is estimated using a linear approximation of the horizontal miss distance as a function of bank angle. If the predicted miss distance is more than 100 ft (Mission-load) greater than the minimum miss, guidance computes a correction to the bank angle in order to achieve the minimum miss. Just as in the entry phase, the bank angle magnitude is limited between 0 and 180 deg. Saturation of the commanded bank angle is checked as well. The guidance solution is iterated upon until one of the following three criteria is met: (1) Three iterations completed (2) Predicted target miss < 100 ft (3) The commanded bank angle correction < 1 deg The proper bank angle rate command sign is calculated, and the commanded bank angle rate is set to a mission load magnitude times this computed sign. 37 3.5 Terminal Subphase When the bank angle is within .05 degrees (Mission-load tolerance) of reversal completion, guidance enters the terminal phase. The time between the end of the reversal and the estimated beginning of the landing system parachute deployment is time spent in the terminal phase. A bank rate profile separated into bins of time is selected off-line. The selection of this nominal bank rate profile will be discussed in Chapter 5. A closed-loop predictor/corrector algorithm with the horizontal target miss distance as the only con- straint and the bank rates in each bin as the controls is used in the terminal phase. If the predicted miss distance is more than 100 ft (Mission-load) from the target, guidance com- putes a correction to the bank angle in order to achieve a smaller miss distance. The bank angle magnitude is limited between 0 and 180 deg. The maximum change in the bank rate from one bin of time to the next is limited by the bank accelera- tion, which is set at 2.5 deg/sec2. The guidance solution converges if the predicted miss is less than 100 ft. 3.6 Landing System Phase The terminal phase ends its predictions and corrections at a time, 1 second (Mis- sion-load tolerance) before the commanded landing system parachute deploy time. From this time on, the commanded bank rate is held at zero, and no predictions/corrections are made. When the PLV-2 reaches a velocity of 975.00 ft/sec, the landing system parachutes are deployed. Guidance is no longer active. A constant bank angle, the final bank angle 38 from the end of the terminal phase, is assumed at this point on. The simulation remains in this phase until the PLV-2 has reached the altitude of the target. A typical bank angle profile for a simulation run is shown in Figure 3.3. The differ- ent phases are clearly distinguishable. Here a +45 deg bank is commanded through pre- entry and entry. The reversal phase begins at approximately 533 seconds when a -45 deg bank is commanded. Once the reversal is complete, the terminal phase begins. In the ter- minal phase, several different bank changes can be commanded, as illustrated in this example. The last bank angle of terminal becomes the bank angle through the landing sys- tem phase. In this example, the simulation run reaches the termination conditions near 860 seconds. BankAngle Profile 2'a, a,-C co - 0 100 200 300 400 500 600 700 800 900 Time (sec) Figure 3.2: Sample Bank Angle Profile 39 3.7 Atmospheric Models The US62 atmosphere model [10] was the main model employed in this thesis study. The atmospheric density, speed of sound, and pressure are computed by the atmo- sphere model using the position vector of the PLV-2 as the model's input. The parameter values at 400 ft altitude increments through the descent phase are saved in table form for access by guidance. A quick table look-up of the appropriate atmospheric parameters for a certain altitude is used by the predictor model. To model the atmosphere even more precisely, monthly or seasonal atmospheres for the desired target location were generated from the GRAM95 atmosphere model [7]. Day-of-flight measurements could also be taken for greater accuracy in the predictions during actual flight. The required accuracy level of the atmosphere model depends on the PLV-2's ranging capability (ability to adapt to a change in target) and on the amount of capability (the precision of the range ability) to be used for a particular entry. A footprint of possible landing locations given different bankrate profiles can be generated at one point in time to determine the ranging capability. Figure 3.4 below shows an example open-loop footprint of these possible landing locations generated at one instant in time at the start of the terminal phase. Chapter 5 will describe the footprint characteristics in detail. The ranging capability is not very robust on the edge of this footprint. For those entries landing on the edge of the footprint, a very accurate atmosphere model is necessary [9]. For over time, the locations particularly on the edge become inaccessible in the pres- ence of unexpected dispersions. The importance of an accurate atmosphere model will be justified through the robustness and dispersion testing in Chapters 6 and 7. 40 Figure 3.3: Example Footprint Figures 3.4 and 3.5 below show a typical sampling of the atmospheric parameters during re-entry of the PLV-2. X10S Altitude vs. Density 4 3.5 3 2. 5 ~~~~~~~~...... F=_- § 2 ...... 5 1 0.!t I !I 0.01 0.02 0.03 0.04 0.05 0.06 0.07 3 Densitvsluajft 1 Figure 3.4: Altitude vs. Density, Typical PLV-2 Descent 41 Figure 3.5: Altitude vs. Speed of Sound, Typical PLV-2 Descent 3.8 Vehicle Uncertainty and Environment Dispersion Sources In the design of a flight simulation, sources of uncertainty and dispersions must be addressed in order to correctly model the actual conditions that the PLV-2 will be sub- jected to. The two most prominent categories of possible uncertainties and dispersions are listed below [9]: 1. Vehicle Uncertainties - Aerodynamics, (major source) - Mass, (minor source) - Navigation, (minor source) - Maneuver Rates, Control and Modeling of Actual Vehicle Attitude 42 2. Environmental Dispersions - Atmospheric Density Variations, (major source) - Atmospheric Wind Variations, (major source) - Other Atmospheric Properties, such as the temperature, (minor source) - Gravity, (minor source) The vehicle aerodynamics and atmospheric properties are the most important char- acteristics to be modeled precisely in the simulation. In reality, navigation errors can arise when the state vectors (position, velocity, and time) are sampled from the true environ- ment by navigation, and are then delivered to guidance at each cycle. This simulation is modeled as if all minor sources are zero or perfectly modeled. 43 44 Chapter 4 Guidance Design This chapter presents different guidance scheme options, and illustrates the rea- sons behind the PLV-2's particular guidance scheme selection. The bank-to-steer control of the craft is discussed. The predictor and corrector for each phase is defined as well. 4.1 Guidance Scheme Definitions The PLV-2 will have to undergo some form of control if it is to land within a desired minimal distance from the target. Historically, there have been two main forms of guidance control of an entry vehicle; reference trajectories and trajectory control profiles [2]. The latter was chosen for the PLV-2 guidance scheme. A look into the reference tra- jectory scheme, utilized by Apollo and the Shuttle, will illustrate the reasons this scheme was not selected for the PLV-2's mission. A reference trajectory is a complete course of travel that a craft should follow as it makes its way to the earth given nominal conditions. The control tries to keep the vehicle flying, within certain bounds, on this assumed optimal path that has been predetermined. By staying within a certain corridor of the controlled parameters, the vehicle is indirectly led to the target. In order to ensure the successful flight of an entry vehicle, great attention must be given to the energy state of the vehicle. In the past, the reference trajectory method has proven to provide a good means for control while maintaining energy state limitations. These limitations are definitely a primary concern for manned missions. In 45 fact, the Space Shuttle and the Apollo missions both utilized trajectory referencing to con- trol their crafts. The downfall of this method comes into play if the craft deviates a large amount from the reference trajectory itself. This occurs when the conditions during entry vary greatly from the expected. The entry guidance schemes for the Apollo and Space Shuttle are described in Todd Dierlam's thesis written at Draper Laboratory [2]. He mentions several defining qualities of each scheme. The Apollo guidance, designed for a craft with a low L/D=.3, was required to maintain a 3 sigma accuracy of 15 nautical miles in track and range from the desired landing site. Due to the concern for the human crew, the energy state associ- ated with the heat shield was of prime importance. A reference trajectory was determined prior to the flight, in order to meet the downrange, heating, and g-load requirements. In Apollo's case, one control variable, the bank angle, was used to define and maintain the reference trajectory. Vehicle lift was directed by the reaction control system with a limited amount of fuel usage. In order to meet crossrange requirements, guidance commanded reversals by varying the bank angle sign based upon the current crossrange error. One advantage of the Apollo design, a low L/D craft, was the generation of a small amount of lift compared to the amount of drag. This feature caused a reduction in the trajectory length which, in turn, causes the vehicle to be subjected to less atmospheric dispersions. Less exposure to possible dispersions results in greater landing accuracy. The Apollo guidance design was not chosen for the PL-RLV. The PL-RLV calls for a more precise landing scheme than Apollo. The Space Shuttle requires a more precise landing than Apollo. This craft, with an L/D = 4.0, has to deorbit from LEO and land as precisely as an airplane does on a runway 46 [5]. Three control variables, angle of attack, bank angle, and a speed brake, are used by the Shuttle's guidance to maintain the pre-determined reference trajectory. Crossrange control comes from bank reversals, although these reversals are not based upon the crossrange error, but on the difference between the current heading and the heading to the target. Like Apollo, the reference trajectory should result in a flight within the heating and g-load lim- itations. The profile which results in the desired downrange and which also avoids exceed- ing any structural and life-support limitations is the one chosen to be flown by the Space Shuttle's guidance. A guidance scheme with less control variables than the Shuttle's scheme is desired for the PL-RLV. The PL-RLV, a craft with a low L/D = .1, is also designed for precision landing. It is desired to land within 1 nautical mile of the target location even in the presence of wind variations. The single control is the bank angle. Landing within this proximity of the target is of utter importance, for the mission chosen for the study stresses the issue of quick, easy retrieval of its crafts, and a fast turn-around-time between launches. Also, a land-locked launch location would be feasible if the PL-RLV could land reliably within the 1 nautical mile boundary. Since the PL-RLV mission has a payload deploying craft with no humans on board, the vehicle energy state limitations are not as strenuous as they would be for a manned mission. The heat load and heat rate can be higher for unmanned missions. For these reasons, a guidance scheme using trajectory control profiles, not reference trajecto- ries, was designed for the PL-RLV re-entry vehicle mission. When using the trajectory control profile approach, a predictor-corrector algorithm is utilized. The predictor-corrector takes a reference bank and bankrate profile to recom- pute the control parameters at each guidance cycle. Since the control parameters are 47 updated constantly, guidance is able to adapt to the atmospheric variations as well as those variations in the vehicle conditions with great success. The predictor-corrector algorithm is responsible for the computation of the control corrections necessary to update the trajec- tory. These corrections are based on the predictions of the final conditions obtained when the assumed profile is flown to the ground. This is the main difference between the control profile approach and the reference trajectory approach. Trajectory control depends on the predicted final landing location, while the reference trajectory method controls the craft by keeping other parameters within a corridor of values. Every guidance scheme design cannot be perfect, however. One disadvantage to the trajectory control profile approach is the fact that the vehicle is guided based solely upon the final predicted state. This lack of concern for the intermediate states could violate guidance constraints, such as maximum g-load, even though the desired final conditions are achieved. Also, a poorly designed predictor-corrector algorithm can lead to ineffi- ciency and result in computationally intensive loads. The PL-RLV mission, however, is still conducive to a trajectory control profile guidance scheme, as will be shown in this thesis. The trajectory control profile approach was chosen for the PLV-2 due to the adaptability of the guidance. The PLV-2 may encoun- ter wind and other atmospheric variations that would be easily dealt with using this design. The PLV-2's predictor-corrector algorithm works accurately and models the environment well. With the PLV-2 design, the trajectory can be corrected to yield the best possible solu- tion for the PLV-2 path while in flight. 48 4.2 Trajectory Control Upon entry, the PLV-2 will experience both gravitational and aerodynamic forces. T, update the trajectory while still meeting constraints, it is necessary for the entry vehicle to control the two components of the aerodynamic force; lift and drag. These forces are indirectly controlled by the bank angle control of the PLV-2. On most entry vehicles the amount of lift is more directly controlled by the angle of attack in combination with the bank [2]. The angle of attack can be modulated with the use of a body flap, reaction control jets, or by center of gravity movement. In the PLV-2's case, a body flap would add weight and complexity to the vehicle design. Modifying the angle of attack with reaction control jets would also increase the fuel cost substantially. Preliminary studies state that control of the craft could be difficult when the center of grav- ity is shifted [2]. Controlling the magnitude of the lift with the angle of attack becomes less desirable in the face of these drawbacks. Another option was chosen for the PL-RLV design in which the PLV-2 is flown at a constant trim angle of attack which is determined by the center of gravity placement. The trimmed angle of attack reduces the complexity of the design. Trajectory control is also possible by varying the direction of lift generated instead of controlling the magnitude of the lift. This is accomplished through the rotation of the vehicle, and thus its lift vector, about the atmosphere-relative velocity vector. A bank maneuver can be commanded to rotate the vehicle. RCS jets are only means for this task on the PLV-2. Since the PLV-2 design maintains a trim angle of attack, a trade-off presents itself - guidance simplicity versus fuel cost. The simplicity of the guidance is a good 49 choice in the PLV-2's case as its relatively small size and minor maneuvers will not cause a large amount of fuel expenditure. Drag, the second aerodynamic force to be concerned with, can be directly altered with changes in the vehicle surface area or by modulating the coefficient of drag [2]. Unfortunately, these methods, such as making structural additions, add complexity to the vehicle. Simpler vehicle designs call for a more indirect method of altering the drag com- ponent of the aerodynamic force. Indirect drag control is accomplished for the PI-RLV by using the bank angle to vary the vertical lift on the vehicle. Bank angle control allows con- trol authority over both lift and drag during descent, without adding undue weight or com- plexity to the vehicle by requiring angle of attack control. 4.3 Predictor As mentioned previously, trajectory profile control methods make use of a predic- tor-corrector. The predictor algorithm used in this thesis is a 3-DOF trajectory simulator. The predictor is responsible for numerically integrating the vehicle's translational equa- tions of motion forward in time by using the vehicle model, the atmospheric model, grav- ity, and the commanded bank profile. It is necessary to match these models extremely well with the environment in order to predict the final states precisely and yield robust corrector performance. 50 4.3.1 Predictor Flow The predictor is called upon through three phases of the descent: entry, reversal, and the terminal phase, as shown in Figure 4.1. - - FPredictor _ W Figure 4.1: Re-Entry Guidance Predictor Calls Figure 4.2 illustrates the logic flow of the predictor-corrector algorithm. The vehi- cle state and parameter variations are fed into the predictor. The predictor then integrates the equations of motion until the termination condition is met. The final state at the land- ing location is compared to the desired landing location, and a 2-D error vector is com- puted. A corrector is then inplemented that modifies the bank control profile in order to lessen these final errors, and it sends out the converged control commands for the vehicle to follow. This process continues during each guidance cycle. 51 Figure 4.2: Closed-Loop Predictor/Corrector Guidance The entire bank rate profile from entry to the stabilization phase is given to the pre- dictor at each guidance cycle. That is to say, the entry phase already knows what the bank rate profile will be when the vehicle is flying the terminal phase. Initially, this information is found "off-line" in order to find the most suitable bank rate profile for the initial deorbit state (at 400 kft) and the desired landing site. The selection of a suitable nominal profle is discussed in Chapter 5. 4.3.2 Equations of Motion The predictor itself is a 3-DOF trajectory simulator that integrates the equations of motion from the current state until the final conditions are met. The basic equations of motion to be integrated are as follows: d= v (4.1) dt d9V=dj= artarl (4.2) 52 The initial conditions of each prediction are given by the current estimates of posi- tion and velocity from navigation: o = (tn) (4.3) PO = U"",) (4.4) The total acceleration of the vehicle is then computed from the sum of the gravita- tional and aerodynamic accelerations: atotal = agrav +alift+ adrag (4.5) The acceleration due to lift is calculated using the bank angle, 4, in the following manner, Ujift = (sint)Ufperp + ( COS)norm (4.6) where the unit vector normal to the plane containing the position vector and the airmass-relative velocity is denoted uperp,and the unit vector normal to the airmass-rela- tive velocity and that is in the plane containing the position vector and the airmass-relative velocity is denoted Unormn = (CLQSref) (4.7) where CL = the coefficient of lift (a function of Mach and angle of attack) Q = the dynamic pressure Sref = PLV-2 aerodynamic reference area 53 m = PLV-2 mass (from Navigation) The dynamic pressure, Q, is defined by, Q = 2pv2, (4.8) where the relative velocity, vrel, is the magnitude of the airmass-relative velocity vector, and p is the airmass density. The airmass-relative velocity vector itself can be found by subtracting the wind velocity vector from the earth-relative velocity vector of the vehicle. Finally, the airmass-relative velocity vector is converted to the inertial frame. The acceleration due to drag, is determined from: (QSref'CD (CDSrefPara)' adrag =- -m eACD+ Sref JVrel- (4.9) where Cd = the coefficient of drag (a function of Mach and angle of attack) CdSrefPara= landing system parachute Cd times the reference area when deployed Lastly, the gravitational acceleration is simply calculated using: agrav = (n Dr (4.10) where the unit position vector in the inertial frame is defined by r,. 54 4.3.3 Integration of the Equations of Motion The translational equations of motion, Equations 4.1 and 4.2, are integrated using a 3rd order Runge-Kutta algorithm for integrating the position, and a 2nd order Runge- Kutta algorithm for integrating the velocity. A time step of .04 seconds was chosen as the time step of the trajectory's integration to the final state. For an adequate prediction of the environment this time step need not be .04 for all phases. The differential equations of motion aRO (4.11) aVo - _~tow Vo -t = f(t, Ro, Vo)= agravO (4.12) are integrated by stepping through the following process [4,6]: 1. Assume contact acceleration is zero. Calculate the gravitational acceleration. agravo = f(to, Ro, Vo) (4.13) 2. Update the position by using the old gravitational acceleration. R = Ro+ VoAt+ (agravo)At (4.14) 3. Compute gravitational acceleration based on intermediate state. agravl = f(t 1, R1, Vo) (4.15) 55 4. Update velocity using the verage gravitational acceleration. V = Vo + (agrav + agravO)At (4.16) 5. Correct the position using the old and new gravitational accelerations. Ri = - + (agravI -agravO) t 2 (4.17) 1 - ~t2 (4.17) 4.3.4 Termination Conditions for the Predictor After each integration time step, the predicted state is compared with the termina- tion condition. The termination condition corresponds with the target's altitude for that simulation. A more exact assessment of the final landing errors can be made if the simula- tion ends at the target's altitude. The error is then 2-D, and can easily be calculated. Time step discrepancies between the environment and the predictor can cause nonhomogeneity in the altitude termination states. It is necessary for the predictor to terminate accurately, for the predictor's final state values provide the state errors that are then dealt with in the corrector algorithm. An error in the predictor can propagate through the corrector, and cause the corrector to try to correct for errors that do not exist. 4.3.5 Final State Error Computation The final state errors are computed in the crossrange/downrange frame. The cross- range/downrange coordinate frame system is described in Section 2.5. The final horizontal position error 56 Rerror -=Rpred-Rtarget (4.18) is computed by calculating the crossrange and downrange directions as follows: CRmiss = Rerror iCR (4.19) DRmiss = Rerror iDR (4.20) where the predictor's position vector is the final predicted inertial position and the target vector is the also in the inertial reference frame. The horizontal error in position is then given by: ErrOrhorizontal= (CRmiss) 2 + (DRmiss) 2 (4.21) 4.4 Corrector The original entry and reversal phase corrector algorithm was designed by Doug Fuhry [4]. The corrector's task is to use the predicted horizontal landing errors to update the bankrate profile to be flown. The resulting profile should yield a minimal final position error. Three different correctors, each with their own constraints and controls, were used in the three main stages of the re-entry. While in the entry phase, the downrange and cross- range target miss distance comprised the two error signals while the two controls consisted of the time for the reversal maneuver to begin, t and the bank angle of the reversal, 0o. Once the craft goes into the reversal phase, the horizontal target miss distance becomes the only error signal, and the single control parameter is the final bank angle of the reversal. Upon entering the terminal phase, the two target miss distances in the crossrange and 57 downrange directions are the error signals, and the bank rates in each of the bins comprise the controls. The different corrector algorithms found in each phase will be described next. 4.4.1 Entry Phase Corrector In this phase of the re-entry flight, three predictions are made: a nominal predic- tion, a prediction with t tweaked only, and a prediction with 0 tweaked only. Y ) Reversal Time tweaked First Run Case (Nominal) weaked Figure 4.3: Predictions for Corrector Guidance in Entry Phase Prediction#1: t = ttrrnom, = _nom=> ACR, ADR Prediction#2: t = trrnom + tm, O = o0_nom==> ACRI,ADRI Prediction #3: tff = trr nom, po = 40 _nom+ Zo ==> ACR2, ADR2 ACRo and ADRo can be found from data obtained after running the nominal case, where 58 The nominal prediction is used to estimate the landing system parachute deploy time by implementing the current bank reversal angle and reversal start time commands. If this estimated position for the landing system parachute deployment is not within 500 ft of the target, the two cases with tweaked parameters are predicted. In Prediction #3, the guid- ance bank angle is perturbed by 2 degrees; and in Prediction #2, the time for reversal is altered by 5 seconds. These three trajectory predictions yield target miss distances that are used to com- pute the numerical partial derivatives of the constraints with respect to the control parame- ters. The bank angle and reversal time corrections are computed using a Taylor Series expansion which neglects all terms of second order and higher. The linear approximations of the changes in crossrange and downrange miss dis- tance are shown below: ACRo = (a At,,+( aR)A O (4.22) Ao=,a--rtrr + ( )A q) 59 ADRo = (aDR)At + (DR)O (4.23) The numerical partials defined by the three predictions are determined by: aCR CRmissl - CRmisso (4.24) atr,, t rr,,,- trr aCR CRmiss2 - CRmisso o = 0(2) - o(o) aDR DRmissl - DRmisso (4.26) atr trrl - trrO aDR DRmiss2 - DRmisso (4.27) o- 0(2) - O(O) where ( CRmissl-CRmisso) = CRmiss from Prediction 2 (the tweaked reversal time) - CRmiss from Prediction 1 (the nominal case) (trrl - tO) = Reversal Start Time from Prediction 2 - Reversal Start Time from Prediction 1 (0o(l) - 0o(o)) = Reversal Bank from Prediction 2 - Reversal Bank from Prediction 1 etc. With the knowledge of these partials and the nominal cross-range and down-range miss distances, the controls corrections can be calculated as shown below: aCR aCR [Ati arr4 ACRO1 (4.28) LAoi aDR DR ADROj Latrr o4o The new increments and gains are then subtracted from the old parameters to yield the new reversal time and bank angle at reversal: 60 trrnew) trrzold)- (K)(Atrr) (4.29) 4O(new) = O(old) -(K)(AOo) (4.30) The gains (K) are evenly weighted factors on the correction terms. 4.4.2 Reversal Phase Corrector During the reversal portion of the re-entry trajectory, a closed-loop predictor/cor- rector algorithm was designed with one constraint, horizontal miss distance, and one con- trol, the fixed bank angle of the reversal phase. A nominal prediction is made, as well as a prediction in which the reversal phase bank angle is perturbed by 2.0 degrees. The correc- tor then estimates the minimum target miss distance that is achievable by using a linear approximation of the horizontal miss distance as a function of bank angle. The search for the minimum miss distance is defined below. This technique was developed by Doug Fuhry [4]. Figure 4.5 illustrates the minimum miss guidance scheme. It is assumed that the predicted target miss distance as a function of the reversal bank angle, gp(), is a continuous, smooth function, and that a single minimum exists over the possible range of bank angles. 0o => nominal prediction 01 = o0+4& => perturbed prediction PO(-o) => predicted target miss using nominal bank 2l(l1) => predicted target miss using perturbed bank §S = _l -- => difference between the miss vectors 61 Figure 4.4: Minimum Miss Guidance for Reversal Stage The unit vector of the minimum miss distance is calculated and is then used to define the difference between the nominal miss and the minimum miss distance. apmin= UNIT(Ae x (o x p,)) (4.31) Pmin = (Po Upmin)pmin (4.32) APmin = min - Po (4.33) The bank correction to be added to the bank angle is defined by: A = SIGN(Apo Ami)(I n) (4.34) 62 4.4.3 Terminal Phase Corrector Once in the terminal stage of re-entry, the closed-loop predictor/corrector algo- rithm changes again to better handle the characteristics of this phase. Within this stage, there are two requirements, crossrange and downrange miss distances, and the bank rates in each bin represent the controls. The original terminal phase corrector was designed by Dr. Chris D'Souza [3]. The crossrange and downrange errors can be calculated using the Taylor Series expansion of the control variables, which can neglect the second and higher order terms as shown below: aCR aCR aCR ACR = -R (ABR,) + R (ABR2)... + a (ABR,) (4.35) aDR aDR aDR ADR = a--(ABR 1) + a-.R(ABR 2) +... +--- (ABR,) (4.36) 36 bins are chosen for testing purposes; thus, there would be 36 different controls. It can be seen right away that in any case where there are more than 2 bins, there are more unknowns than equations. The underdetermined system's desired corrections for each bin are determined from the minimum norm pseudoinverse that minimizes the root-mean- square (rms) bankrate deviation from the current bank profile: The A matrix is defined by the partials of the constraints (miss distances) with respect to the controls (bank rates). 63 aCRI aCR2 aCR] A '' aBRof (4.37) aDR aDR 2 aDR [ ,RIaBR 2 aBR The partials themselves are found from: aCRI (CRmiss)ominal - (CRmiss)S 1 nl perturbed (4.38) aBRI (BankRate)inl, nominal- (BankRate)Blnl. perturbed ABRI ABR 2 = A (AAT)-I CR (4.39) ~~BR. where ACR = (Predicted Landing Position)cR - (Target Position)cR ADR = (Predicted Landing Position)DR - (Target Position)DR ABR1 = (Bank Rate in Bin #l)nominal - (Bank Rate in Bin #l)ne w , etc To obtain the new bankrate for each individual bin, the correction term is sub- tracted from the original bankrate in that bin. For the first bin, this would look like: (BankRate)Bin I,new (BankRate)Bin I nominal- BR (4.40) After all the bins have their new, corrected bank rate, a prediction is made using these updates. 64 Chapter 5 Nominal Bank Rate Profile Selection Chapter 3 briefly discussed the characteristics of the terminal phase guidance scheme. This chapter will expand on that knowledge and present an in-depth look at the bankrate bin generation technique used in the terminal phase of the PL-RLV simulation. The selection of the optimal nominal bank rate profile will be discussed as well. Figure 5.1 below illustrates the steps in the selection process. These steps will be defined in the fol- lowing sections. Figure 5.1: Flow Chart for Selection of the Nominal Profile 65 5.1 Profile Generation In the terminal phase, we are interested in determining all the possible bank rate profiles that will land the PLV-2 near the target. The nominal bankrate profile will be selected from these profiles. The first step in choosing a profile is to choose an appropriate target. An off-line code is used to generate all possible bank profiles. These profiles are run open-loop at a one point in time to produce a footprint of landing locations. This foot- print illustrates the area that the PLV-2 can land within. A target can be selected within this footprint. The optimal target selection location will be explored in Chapter 6. Once a target is selected, the profile generation process repeats. New profiles are generated and run for the new target. A corrector is present in all phases up until terminal. Upon entering the terminal phase, an open-loop predictor produces a footprint of landing locations and when desired also saves acceptable profiles, typically chosen as those with a horizontal target miss distance of less than 3000 ft. After running the initial code and investigating all possible bank rate profiles yielded, these profiles can be chosen from to be used as the nominal profile in a closed- loop simulation of the terminal phase. Given the nominal profile, the terminal phase uses a predictor and corrector scheme to reduce the target miss distance even further. 66 5.2 Bank Rate Bin Definition As part of the guidance design, the terminal time profile is broken up into several "bins"- large and small. The off-line code is used to define and generate these bins. Large bins contain several smaller bins. These small bins are divided into equal slots of time. Within these large time bins, the bank rate will stay constant. Each large bin, overall, will consist of a bank rate that falls between the minimum and maximum limitations. The max- imum number of bins is chosen as 36, and the number of large bins is chosen as 5. These numbers allow for a broad range of bankrate profile possibilities, and are small enough not to utilize too much computation time. The small bins are used for the controls in the closed-loop terminal phase, as shown in Section 4.4.3. Figure 3.1 below illustrates how the time line between the end of reversal and the beginning of landing system parachute deploy can be separated into individual bins. The time steps are represented by "dt" and the bankrates in each small bins are "br". Terminal Bank Rate Profile - Bin Breakdown T. nro T rcy, T ar T nrae T rap Ad' '' " '' " "' A d ''' .W -Max Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bank Q Rate M. W t0-tt 4" I 4I I I Il I 1 -HHHI illHll YLV :4 'i N mV I fdt(8) \ .An br(8) Min ULl) -- Bank br(1) / Small Bins Rate Terminalm . ~ Time Terminal Start Finish . _ _- Figure 5.2: Terminal Phase Bins 67 Figure 3.2 below illustrates how all the possible bank rate profiles are incremented in order to ensure that every possible bin combination will be defined. All bins start out at the minimum bank rate, and a prediction is made with that profile. Next, the last bin increases its bankrate by the allotted increment, while the other bins remain at the mini- mum bankrate. A new prediction is made, and the cycling continues until all combinations have been created. By looping through all possible bank rate values in all the possible bins, the off-line code yields all the possible bank rate profiles for the PLV-2 to fly during the terminal phase and also produces the capability footprint. All of these profiles are flown by the predictor, but ultimately, only those profiles that yield a final landing location within 3000 feet of the target will be considered suitable candidates for the nominal bank profile. Those profiles that are actually acceptable have a range of generated fuel cost and the horizontal target miss distances to choose from. Beginning Profile Generation - Max j 0 Bank W Rate c -0 Cu U 1 co j~ _ sl -Min Bank Rate Terminnl Start Terminal Time Finish - -- -- Figure 5.3: Bankrate Profile Generation, Cycling Through First Bin 68 5.3 Example Profiles A sample simulation run is used to illustrate the nominal profile selection process. Table 5.1 lists the conditions for the bin generation of this example simulation run. These conditions are fed to guidance. The bank rate search limit is kept below the maximum bank rate to reduce the risk of bank rate saturation. The bank rate search increment is suf- ficient to generate a large number of profiles. Bank Acceleration 2.5 deg/sec 2 Number of Large Bins 5 Max Number of Bins 36 Maximum Bin Bank Rate +10 deg/sec (0.17453 rad/sec) Bank Rate Search Limit +9.1673 deg/sec (. 16 rad/sec) Bank Rate Search Increment 2.2918 deg/sec (.04 rad/sec) Table 5.1: Bin Generation Conditions for Sample Run The entry and reversal phase for these profile generations are all closed-loop. It is not until the terminal phase that the predictor generates the open-loop profiles. Thus, an inital target value should be set. Table 5.2 below defines the initial conditions for the PLV- 2 and the target that are used during entry and reversal for this sample run 69 Target PLV-2 XECEF -1.27838717442284e7 ft 1.40469665694540e+06 ft YECEF 1.24061145352144e7 ft 1.30789611824149e+07 ft ZECEF -1.09520604018682e7 ft -1.67303063866579e+07 ft Altitude 4995.0 ft 400,000.0 ft XVelocity -2.5428430228e+04 ft/sec Yvelocity 4266.949332075 ft/sec ZVelocity 1753.2389192229 ft/sec Table 5.2: Initial Target and PLV-2 Setting At the terminal phase, the predictor flies all the profiles generated. No corrector is implemented here. Figure 5.3 below shows an example of the footprint of all possible landing locations that results when each profile is run through the predictor. A +45 to -45 degree bank reversal was commanded here. The landing location at (0,0), called the "no- action point", represents the profile with no terminal phase bankrate action. If the initial conditions, such as position and velocity, to the simulation are different the footprint will have a different origin location and shape. The size of the footprint directly correlates to the altitude at which the open-loop simulations are run. The higher the altitude, the larger the accessible area. Chapter 6 demonstrates this characteristic of the footprint. 70 4 Fia Ladn Site oll fies It 4 Final LandingSites of AllProfiles: 1--O c x 10 I 4 2 0 z -2 -4 -6 . . _ I __ - - I I ---n - 10 -8 -6 -4 -2 0 2 East(ft) x 4 - Figure 5.4: Open-Loop Footprint of Landing Locations The footprint provides many possibilities for target selection. Chapter 6 will dis- cuss the best regions within the footprint to choose the target position. Once the targeted position is chosen, the off-line code can generate possible bank rate profiles that get close to the new target. The target miss distance for each profile is computed from the rss value of the crossrange and downrange miss distance. Those profiles that result in a miss distance less than 3000 ft are acceptible for selection as possible nominal profiles. Given these accepti- ble profiles, the corrector will be able to reduce the miss distance considerably. There are other limitations in the possible bank rate combinations, however. The maximum change in the bank rate is defined by: 71 MaxABinBR = SmallBinSize x BankAccel (5.1) A profile will not be generated for those combinations that yield a change in bank rate larger than the maximum defined above. For example, if the first change in bank rate from large bin 1 to large bin 2 is greater than the maximum possible change in bank rate, that bin combination is skipped, and no profile generation will occur. ABinBR 1 > MaxABinBR (5.2) Further selection is based upon fuel. The amount of time it takes to make a maneu- ver change directly correlates to the amount of fuel that is necessary for the maneuver. More bank rate changes mean a greater amount of fuel that must be expended. The fuel cost was modeled by the time it took the bank rate to change from one bin to the next. The fuelcost is modeled as a function of the sum of the bank rate in each bin divided by the maximum bankrate limit, and is also a function of the time in each bin: FuelCost = 9L~k.1 + MaxBRin(i)!MaxBR) n(i)(5.3) i= I 5.4 Nominal Profile Selection A nominal bank rate profile can then be selected from the acceptible profiles gen- erated in the off-line code. Selection is based upon minimizing either the miss distance, 72 minimizing the fuel cost, or a compromise between the two. The degree to which a profile consumes the physical capability of the vehicle is tightly correlated to the degree of bank- ing activity in the profile. Profiles that tend to utilize the entire range of bank angles have a greater access to the achievable landing footprint. For good landing performance, the nominal profile should be chosen such that the vehicle bank rate is rarely zero. However, this maneuvering requires more fuel and thus cuts into fuel perfomance. Once the corrector is utilized, the target miss distance will be minimal, but maybe at the cost of changing the bank rates in each bin. These maneuvers use fuel. When flying a particular mission, the optimal criteria from this trade-off should be predetermined. As mentioned previously, the initial problem for the guidance design to tackle is the selection of the target point within the initial footprint. This target point should achieve the best trade-off between fuel usage and robustness to dispersions. Chapter 6 will discuss the robustness testing of target selection within the footprint. As an example, a target within the footprint was chosen as seen in Figure 5.4 below. Figure 5.5 shows the 14 possible bank rate profiles that come within 3000 ft of the chosen target in this example run. It is assumed that any one of these profiles could reach the target once the corrector is implemented. The optimal one for the mission at hand should be selected based upon fuel usage and robustness criteria. It is important to note that the fuel cost for these open-loop profiles was calculated for the full profile. The entire open-loop terminal profile might not be completed, as the stabilization time can vary once the corrector is implemented. Since some of the profiles will not be completed, many of the profiles become identical. For example, if the terminal 73 phase ends within 60 seconds, Profile #2 and Profile #3 are identical. Closed-loop testing of the profiles will provide a more accurate final target miss distance and fuel cost. The closed-loop testing of the 14 profiles can be seen in Chapter 7. x 10' FinalLanding Sites of All Profiles I I t I0 z -2 -4 __l.I -10-10 -8 -6 -4 -2 0 2 East(ft) x10o' Figure 5.5: Location of Target Selected Within Footprint 74 Profi 1 MIDeWl-2296.142 Fuel Cost - 18 304 0 O* -00 , , . .I- -0. 1 . 0 to 20 30 40 50 6o 70 80o Profll 2 M lt 2391315 Fuel Cot * 16.5256 _nc .. Profile 4 MDbt-. 290 642 Fuel Cost 17.0858 01 005...... - ' 0 -00 I ' 10 20 30 40 50 60 70 80 90 Time(sec) Profile7 MisDsl. 2475.062Fuel Cost · 17.9784 O Figure 5.6: Acceptable Bank Rate Profiles (#1-#9) Generated for New Target 75 Pmle tOM10lgd 2666362 Ful C st 16.3218 015- I 0.1 I 005 0 20 P10 30 40 IO u e 70 so 90 o10 20 30 4 70.14Ful Co 7 I O. . . . -0.02 . i I -.. AI 1I 0 10 20 30 40 50 60 70 80 90 Potle 12MuDist. 2725679 Fuel Cost 15.5579 _ _ 1 - I005 Q 0 -005 -. "0.I 10 20 30 40 50 60 70 80 I 0 Tme(.ec) Pmfile13 bIsssD- 1166.749 Fuel Cost = 13.4429 A 7 l ...... ~- -""- 2 0 2-0.02 0 I I I 0 10 20 30 40 50 60 70 80 90 Promfe14 OssOit=2936.181 Fuel Cos = 11.3279 0.02 I I I 0 a 0 . C,1-0.02 0 caD 0 10 20 30 40 50 60 70 80 90 Tone(sec) Figure 5.7: Acceptable Bank Rate Profiles (#10-#14) Generated for New Target 76 Chapter 6 Robustness Testing This chapter discusses the corrector's capability within the footprint of possible landing locations. The size of the footprint over time is also studied. Four different target areas are investigated in order to characterize performance in those regions of the foot- print. 6.1 Range Capability Once the initial footprint is determined, the target for testing purposes can be cho- sen from anywhere within that region. Testing of the footprint is performed to successfully select a target which will achieve fuel cost limitations and maximize the capability of the PLV-2. Off-line testing of the footprint will enable preflight selection of the target. Four different targets, A, B, C, and D, were selected in different regions of the footprint in order to test the vehicle capability. Figure 6.1 below shows the location of Tar- get A, B, C, and D. The target value at the origin (0,0) of the footprint plot is the resulting landing location if there were no maneuvers performed during the terminal phase. 77 Figure 6.1: Target A, B, C, and D Locations Within the Footprint The target locations in ECEF and North/East coordinates are given in Table 6.1 below. Target A i Target B Target C Target D . _ _ _ - .------. '4 _ A j A s ii 1.277055661 -1.2793245034 -1.2763761472 -1.271515701 +07 e+07 e+07 e+07 .2451867278 1.2415871446 1.2435381223 1.2441237349 +07 e+07 e+07 e+07 ZECEF (ft) 1.091585896 -1.0930181969 -1.0942369776 -1.0991867543 +07 e+07 e+07 e+07 gog~3a .266886e+04 2.5794145e+04 1.1420108e+04 -4.697762e+04 4.196583e+04 -4.68234e+02 -3.492335e+04 -7.300804e+04 .. . Table 6.1: Target Coordinates in ECEF and North/East Frame 78 Given the initial bank rate profiles to reach these targets, it is of interest to see the range of capability that each target location can achieve. This is accomplished by using the initial bank rate profile, but the target location is moved further and further away from the original target. The guidance scheme can redirect its maneuvers to try to reach these new locations. At some point, the scheme will not be able to make enough corrections to be able to hit the desired targeted location. This range of capability is directly related to the original target's location within the footprint and to the ability of the guidance algorithm.. Four cases which come from the example simulation run in Chapter 5 are explored below. 6.1.1 Case 1 - Target A Target A lies near the outer edge of the footprint. The bank rate profile to reach this target is shown below in Figure 6.2. Figure 6.2: Bank Rate Profile for Target A 79 The desired target location was increased to distances further and further from the original target. Figure 6.6 shows the range capability from Target A's location. With the original profile as an input, the guidance scheme was only able to capture other targets that were located within a small area near Target A. 6.1.2 Case 2 - Target B Target B lies north of the footprint's origin. The bank rate profile to reach this tar- get is shown below in Figure 6.3. I Time (sec) Figure 6.3: Bank Rate Profile for Target B Figure 6.6 shows the range capability from Target B's location. With the original profile as an input, the guidance scheme was able to capture other targets that were located over a wide area of the footprint. 80 6.1.3 Case 2 - Target C Target C lies near the center of the footprint. The bank rate profile to reach this tar- get is shown below in Figure 6.4. Bankrate Profile for Target C 0.04 r 0.03 0.02 0.01 0 ...... I -0.01 ...... -0.02 ...... -0.03 ...... -0.04 . . . . . I . I I 10 20 30 40 50 60 70 80 90 Time '"'~'I- c- Figure 6.4: Bank Rate Profile for Target C Figure 6.6 shows the range capability from Target C's location. With the original profile as an input, the guidance scheme captured other targets that were located within the center of the footprint. 6.1.4 Case 3 - Target D Target D is centrally located in the lower region of the footprint. The bank rate pro- file to reach this target is shown below in Figure 6.5. 81 I __ Tlmo sol Figure 6.5: Bank Rate Profile for Target D Figure 6.6 also shows the range capability from Target D's location. With the orig- inal profile as an input, the guidance scheme was only able to capture other targets within a small circular region around Target D. 6.2 Nominal Target Robustness Results The resulting range capabilities for the three cases are shown below. 82 4 in F I St of Al P i v 1n4 Final Landing Sites of All Profiles 9 z0 -10 -8 -6 -4 -2 0 2 _~~~~~~~~~~~~~~~~~~~~~~~~~East (ft) 0 x 10 Figure 6.6: Range Capability of Test Cases, Final Miss Within 10,000 ft 83 4 Site of AlIroi V IA ~~~~~~~~~~~~~~~~~~~~~~~~~....Fina Lanin t4 Final Landing Sites of All Profiles z -10 -8 -6 -4 -2 0 2 East (ft) x 104 -w ,~~~- -- - ~ - Figure 6.7: Range Capability of Test Cases, Final Miss Within 1,000 ft Target A was on the edge of the footprint, and thus had a small range of capability. Less maneuverability within the proscribed bank limits is available when a target is located near the edge of possible landing locations. Although located in the lower central region of the footprint, Target D's range capability was limited by the edge as well. Target B's location enabled the capture of a wide range of landing locations for both miss condi- tions. Target C was located in the center of the densest section of target locations. With this initial location, the guidance scheme was able to maneuver to a central range of final locations. In this example, it is obvious that Target B has the largest range of capability, 84 which is a desirable characteristic of target selection. Another characteristic, the footprint range through time, must also be considered before target selection is final. 6.3 Footprint Range In selecting the nominal target, its location within the achievable open-loop foot- print must be considered, as well as how this footprint changes through time. During the mission, the target location might need to be altered, or in-flight dispersions could happen far enough along the trajectory to make it impossible to hit the target. A target location with robust capability potential throughout time is the most desireable option. The area attainable by the PLV-2 changes as the descent time increases. If the open-loop footprint is generated near the beginning of the terminal phase, it will be much larger in area than if the footprint is generated 30 seconds into the terminal phase. Figures 6.7 - 6.13 below show the resulting footprints generated at different times during the ter- minal phase: 0, 5, 10, 15, 20, 30, and 40 seconds into terminal. The original target loca- tions (A,B,C,D) are left on the figures for comparison in the footprint size changes. 85 X 10 Final LandingSite$ of All Pofles: 10 I a 10' FinalLanding Sites of All Profiles:.0 4 2 0 -2 -4 -6 -8 i . * i *I -10 -8 -6 -4 -2 0 2 East (ft) I x 10' Figure 6.8: Footprints at Time= 0 seconds in Terminal 4 v 41 FinalLandina Sites of All Profiles:t=5 i Figure 6.9: Footprints at Time= 5 seconds in Terminal 86 4 FinalLanding Sites of All Profiles:t10 Px 10 _ _ 4 2 0 ...... -2 -4 -6 -_IR 0 * * . -10 -8 -6 -4 -2 0 2 East(ft) x 10 ------~ ~ ~ ~ I` Figure 6.10: Footprints at Time= 10 seconds in Terminal Figure 6.11: Footprints at Time= 15 seconds in Terminal 87 Figure 6.12: Footprints at Time= 20 seconds in Terminal _· x 10' Final LandingSites of All Profiles:1t=30 i 4 2 0 zZ -2 -4 -6 -R I I I -10 -8 -6 -4 -2 0 2 East() x10' Figure 6.13: Footprints at Time= 30 seconds in Terminal 88 ______ _ 10' FinalLanding Sites of AU Profile: 1t40 fl , ...... 4 ...... 2 ...... 0 .. . ~, ...... -2 -4 . . I: _~~~~~~~~~~~~~~~~· , . . . · · · _····· -6 -A -10 -8 -6 -4 -2 0 2 East(ft) x 10' Figure 6.14: Footprints at Time= 40 seconds in Terminal As seen in the figures, the sooner the open-loop footprint is generated, the larger the attainable area. This feature becomes extremely important in target selection. Disper- sions could arise along the PLV-2's descent trajectory. Early on, the corrector can easily handle these dispersions and direct the PLV-2 to the target. A dispersion arising near the end of the terminal phase, however, could result in a target miss if the target is not selected well. The craft could simply be incapable of performing the bank manuevers necessary to reach the target. Figure 6.13 shows the footprint after 40 seconds in terminal. Those "lobes" are the only regions that could be accessible at that point and time. The best choice for target selection from within the footprint can be seen from the range testing and the dispersion robustness testing. All of the characteristics of the foot- print should be taken into consideration to find a suitable target to steer towards. These 89 results display the basic, target selection criteria. Target B is the best option given its range capability and presence within the footprint over time. This target will be used for the remainder of the test cases presented unless otherwise noted. 90 Chapter 7 Guidance Performance The performance of the predictor and corrector for several cases are presented in this chapter. This chapter also explores the guidance performance in three specific areas: fuel cost versus landing performance, optimizing the bin numbers, and the effects of atmo- spheric dispersions. The corrector's performance in each of these cases will be shown. 7.1 Predictor and Corrector Performance As mentioned previously, the predictor's ability to accurately model the actual tra- jectory path is of utter importance. The more exact the predictions, the better the correc- tions will be for steering the PLV-2 towards the desired target. Cases will be presented to show the predictor and corrector fidelity throughout the entire simulation and also particu- larly in the terminal phase. 7.1.1 Case 7.1 - Target Set to Actual Open-Loop Landing Location Case 7.1 is an open-loop case with the target set to the actual landing location. The corrector is not utilized. Table 7.1 below shows the initial and final state vectors as well as the phase times during the simulation for Case 7.1. 91 Initial Inertial Position (ft) 1.404696656945e6 1.30789611824149e7 -1.67303063866579e7 Initial Inertial Velocity (ft) -2.542843022765e4 4266.94933207518261042 1753.23891922291841183 Time at Start (sec) 0.0 Final Inertial Position (ft) -1.35308617120651e7 1.15858368540514e7 -1.09531053051175e7 Final Inertial Velocity (ft) -623.7359138965 -1177.1030328138 179.515068968216 Time at Finish (sec) 854.24 Target ECEF (ft) -1.27833779269691e7 1.24056875285493e7 -1.09531053051175e7 Entry Start Time (sec) 133.04 Reversal Start Time (sec) 533.04 Terminal Start Time (sec) 543.04 Stabilization Start Time 709.28 (sec) Table 7.1: Case 7.1 Simulation Conditions Figure 7.1 presents the bank angle history for this test case. A +45 degree bank is initially commanded until the reversal time (533.04 sec) when a -45 degree bank is com- manded. The terminal phase starts at 543.04 seconds into the simulation. The bank rate profile for the terminal phase can be seen in Figure 7.2. 92 Figure 7.1: Bank Angle Profile for Case 7.1 U.ln 4 , 0.05-. 0- ci i -0.05- -0.1 750 7nn 6,w 4 650 - 3 600 20 550 10 *-_ . 500 0 time (ec) Bin Number 0.1 : 0.05 . . . . . InitialProfile ED O ~-- i Hr \-- - ...- Final Profile . -0.05 . .. ' i 1-N. I . . n -0.1 , .' .' ' ...... - 02468 1012141618202224262830323436 Bin Number Figure 7.2: Terminal Phase Bank Rate Profile for Case 7.1 93 Figure 7.3 shows the error between the actual landing location and the predicted landing location. For Case 7.1, the actual landing location is the target location. In this case, the pre-terminal phases know about the terminal bank rate profile and the predictor uses this knowledge in its calculations. During entry, the error is on the order of 1250 feet, but by the end of terminal, the error is near 70 feet. The jumps near 220 and 310 seconds are due to jumps in the atmosphere tables. The reversal can be seen near 533.04 seconds. The error increases slightly at one point in the terminal phase when the bank rates change from negative to positive values. A small timing error when the bank rates change could cause this bump, but since it is on the order of 20 feet, it can be neglected at this level. HorizontalMiss vs. Time I I I I I 1200I- 1000I- F 800 'a 600 r0 400 200 I v- 100 200 300 400 500 600 700 800 900 Time secl Figure 7.3: Open-loop Horizontal Prediction Error: Case 7.1 94 Figure 7.4 below shows the same test run as Case 7.1, but the pre-terminal phases do not know about the terminal bank rate profile. A zero bankrate profile is assumed after the reversal. The error at the beginning of entry is on the order of 22,000 feet, and at the landing system parachute deploy, the error is near 70 feet. The difference between Figure 7.3 and 7.4 shows how important the knowledge of the bin activity is for the predictor. Predicting with a better accuracy at the beginning of the trajectory will enable the guid- ance scheme to handle dispersions encountered along the way with better success. x 10 HorizontalMiss vs. Tine I I I I I I I 2 L 1.5 ...... - 'a 8 1 0..5 /' I I I - 100 200 300 400 500 600 700 800 900 - Timerf .el Figure 7.4: Open-loop Horizontal Prediction Error: No Pre-terminal Bin Knowledge: Case 7.1 Predicting the actual landing location with such a small error gives the corrector a sound base to work from. Figure 7.5 shows the same simulation run, but with the terminal phase corrector implemented. 95 HorizontalMiss vs. Time Time secl Figure 7.5: Horizontal Prediction Error, Closed-Loop Terminal Phase With the terminal phase corrector on, the target miss distance at the end of terminal is 61 feet. Only minute corrections were necessary for this run, as the target position was initially set at the real open-loop landing location. These plots show how well the predic- tor and corrector work together to minimize the landing location error. 7.1.2 Case 7.2 - Target Offset From Open-Loop Landing Location Case 7.2 is an open-loop case with the target offset from the actual landing loca- tion. The new target in ECEF coordinates is [-1.27829366885425e+07 1.24074256101389e+07 -1.09515194167939e+07] ft. The corrector is not utilized. Figure 7.6 below shows the horizontal predictor error with respect to the actual landing location and the target location. 96 PLV-2Descent Rf'W~ is · · · s W WRTActu I I · ·· ·I·- - - - WRTTarget ...... 7:.T... 2500lF ...... Z2000 g 1500 • S 0 x 1000 . 500 ... il ...... 100 200 300 400 500 600 700 800 900 Time(sec) Figure 7.6: Open-loop Horizontal Prediction Error: Case 7.2 Once the corrector is utilized, the prediction error decreases. Figure 7.7 below shows the closed-loop response. The predictor is now able to predict the actual landing location within 100 feet. With corrections, the final miss distance is 1135.5687 feet. Figure 7.8 displays the terminal phase bank rate changes that were necessary for target acquisi- tion. 97 PLV-2 Descent 6 0 'aCo e a- o 0 r 0O Time (sec) Figure 7.7: Closed-loop Horizontal Prediction Error: Case 7.2 BankRate Corrections n 0 05 a0 ,· -005 40 time (sec) BinNumber V 0.1 0.0 2 0.05,mu . . . _;__. InitialProfile Xu0- X ~r : l Final Profile c -0.05 m -0.1 . 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 Bin Number Figure 7.8: Terminal Phase Bank Rate Profile for Case 7.2 98 7.2 Fuel Cost vs. Land'ng Performance As mentioned in Chapter 5, there is a trade-off between fuel cost and landing per- formance when selecting a profile. Four different open-loop profiles will be investigated to show how the closed-loop guidance scheme performs under different conditions. The cases chosen to investigate come from the acceptable profiles presented in Chapter 5. The closed-loop performance for all the profiles is shown below in Table 7.2. Profile Number Fuel Cost (sec) Target Miss Distance (ft) 1 11.033 95.23 2 11.304 16.24 3 11.41 89.76 4 11.51 68.05 5 10.82 97.81 6 11.58 91.73 7 10.98 1432.206 8 10.64 103.98 9 10.62 47.019 10 10.97 75.97 11 10.23 54.39 12 10.756 926.43 13 9.92 71.81 14 10.31 641.06 Table 7.2: Closed-Loop Results for the 14 Profiles 99 All but four of the cases come within 100 feet of the target. All of the cases come within 1433 feet of the target. All but one profile (Profile #9), had the target miss distance decrease. The small increase in Profile #9's miss distance is due to predictor error and the different start time for the landing system phase once the corrector was implemented. Fig- ure 7.9 below shows the closed-loop miss distances in relation to the target. _ Closed-LoopMiss Distances "^A 15C 100 50 x#2 : 0 x#9 + x#10 z 11 # 6 x# x #8 -50 x x#13 _.5 -100 -150 -0 -200 -150 -100 -50 0 50 100 150 200 EastMiss (ft) Figure 7.9: Closed-Loop Target Miss Distances for the 14 Profiles 7.2.1 Case 1 - Initial Conditions: High Fuel Cost, Low Miss Distance The first case, Profile #1, has an initial high fuel cost of 18.306 seconds, and a low miss distance of 2296.142 ft. When the profile is used as the nominal bank rate profile in closed-loop simulation, the resulting bank rate changes occur. See Figure 7.10. With the 100 presence of the corrector, the final miss distance is 95.23 ft at a fuel cost of 11.033 sec- onds. Figures 7.11 and 7.12 show the PLV-2's path towards the target as well as the pre- dicted landing locations along the way. Most of the predictions are located within 100 feet of the target. __ _ BankrateCorrections 0.1 3 0.05 O < r- 1z-0.05 m -0.1 750 ___ 40 700 600 0 time(sec) Bn Number 13, :av, InitialProfile - - - Final Profile I ccra: ByM 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 Bin Number Figure 7.10: Corrections to the Bank Rate Profile - Case 1 101 Figure 7.11: Trajectory Path - Case 1 PLV-2 Descent 2.! - a, 1.5 1 0.5 3000 2 3000 Nor EastMiss (ft) Figure 7.12: Trajectory Path - Case 1 102 Figure 7.13 shows the prediction error for Case 1. PLV-2 Descent 2500SPAR ~I ~~~I I- -- WRTWRTActuaTargqt 2000 C s .221 C1000 r0 500 n I I I I I I 630 640 650 660 670 680 690 700 710 Time(sec) Figure 7.13: Prediction Error- Case 1 7.2.2 Case 2 - Initial Conditions: Average Fuel Cost, Average Miss Distance The second case, Profile #2, has an initial average fuel cost of 16.5256 seconds, and an average miss distance of 2391.315 ft. Figure 7.14 shows the resulting bank rate changes in closed-loop simulation. With the presence of the corrector, the final miss dis- tance is 11.304 ft at a fuel cost of 16.24 seconds. Figures 7.15 and 7.16 show the trajectory path and the predicted landing locations. 103 BankrateCorrections (U 4 T 21 Ir 5I 40 0.15 0.1 a 0.1_- .: Initial Profile ' 005 Final Profile 0- m-0.05 . 0 2 4 6 8 1012141618202224262830323436 Bin Number Figure 7.14: Corrections to the Bank Rate Profile - Case 2 Figure 7.15: Trajectory Path - Case 2 104 C PLV-2Descent <:r- 30 F EastMiss (ft) Figure 7.16: Trajectory Path - Case 2 The prediction error for Case 2 is shown in Figure 7.18. PLV-2 Descent I I . I . WRTAci 90 i ' I: I - - - WRTati Act .. WRT Targ . 70 ri 60 0C ...... I0N 40 ...... - 30 ...... 20 ...... II ...... I ...... _ . I I ------ 10 ...... -...... I : : : 600 650 700 750 800 850 900 Time(sec) Figure 7.17: Trajectory Path - Case 2 105 7.2.3 Case 3- Initial Conditions: Low Fuel Cost, High Miss Distance The third case, Profile #14, has an initial low fuel cost of 11.3279 seconds, and a high target miss distance of 2936.181 ft. Figure 7.18 shows the resulting bank rate changes in closed-loop simulation. With the presence of the corrector, the final miss distance is 641.06 ft at a fuel cost of 10.31 seconds. Figures 7.19 and 7.20 show the trajectory path and the predicted landing locations. BankrateCorrections 02- 0.1 0- 40 time(sec) BinNumber .- 0.2 . . . . a 0.1 . . InitialProfile Final Profile Cu2 . . . . .t e -02.1 4 6 . 0 2 4 6 8 1012141618202224262830323436 - Bin Number --- Figure 7.18: Corrections to the Bank Rate Profile - Case 3 106 Figure 7.19: Trajectory Path - Case 3 . PLV-2 Descent 2.5\2.5 .,. ~ *- .. 1ol.5- 3000 21 20 , - f · .- 2000 000. ' 2000 ooo...\ .-p... .. -.. 30 000 2000 NorthMiss (ft) 3000 - 1000 -3000 __ __ Figure 7.20: Trajectory Path - Case 3 107 Figure 7.21 displays the prediction error for Case 3. PLV-2 Descent . /, i I I I WRTActua ------WRTTargEt 600 500 ..· w c 400 . I I, I I .' I I 200 *I * r I : 100 I' I / O1 il & Il , I 600 650 700 750 800 850 900 Time(sec) Figure 7.21: Trajectory Path - Case 3 7.2.4 Case 4- Initial Conditions: Low Fuel Cost, Low Miss Distance The fourth case, Profile #9, has an initial low fuel cost of 14.2069 seconds, and a low target miss distance of 2.5433e-4 ft. Figure 7.22 shows the resulting bank rate changes in closed-loop simulation. With the presence of the corrector, the final miss distance is 47.109 ft at a fuel cost of 10.62. Figures 7.22 and 7.23 show the trajectory path and the predicted landing locations. 108 Figure 7.22: Corrections to the Bank Rate Profile - Case 4 Figure 7.23: Trajectory Path - Case 4 109 L PLV-2Descent lo8; ,, ...... x lo' ...... - V ...... 1.. ...:' · · ·· · · -1min . . · · ·· · · . :.... ' · · · ···· ······ ···· .. ·· .. :·· ··· ····- ·:· -0 min ' ' 0 - '.,,,, : : ...... " ,,. -' ' X ' ' -1.000. ... 2000 -2000 0 1000 NorthMiss (t) 3000 -2000 -3000 East Miss () Figure 7.24: Trajectory Path - Case 4 PLV-2 Descent 0 .26 ° no 0.I 0 640 650 660 670 680 690 700 710 720 Time (sec) Figure 7.25: Trajectory Path - Case 4 110 7.2.5 Fuel Cost vs. Landing Performance Discussion Four different initial profiles resulting in a range of fuel cost and target miss values were shown. The corrector was able to lessen the target miss distance in all but one case. Those cases with the most bin activity and bank rate changes suffered in fuel cost. Profile #9 is the optimal selection for the nominal profile, as it resulted in a small fuel cost, and achieved a minimal target miss value. When the open-loop profiles are generated, the one with the smallest miss distance will inevitably be one with a small closed-loop miss dis- tance as well. The closed-loop fuel cost does not vary greatly between the different pro- files, but a profile like #9 which as an initially low miss distance and fuel cost will have a small closed-loop fuel cost as well. Figures for other characteristics of the nominal profile simulation, such as the heat loading and dynamic pressure profiles, are given in Appendix C. Open-loop Closed-loop Open-loop Closed-loop Miss Distance Miss Distance Fuel Cost Fuel Cost (ft) (ft) (sec) (sec) Profile #1 2296.142 95.23 18.3604 11.033 Profile #2 2391.315 16.24 16.5256 11.304 Profile #14 2936.181 641.06 11.3279 10.31 Profile #9 2.5433e-4 47.019 14.2069 10.62 Table 7.3: Fuel Cost and Miss Distance Comparisons 111 7.3 Corrector Performance On Nominal Profile The nominal profile target was offset by .01 deg East in order to evaluate the cor- rector's performance on the nominal profile (Profile #9). The new target coordinates in the ECEF frame are: [ -1.27953660797014e+07 1.2413594041297e+07 -1.0930143171118e+07] ft. Figure 7.26 shows the predictor error with respect to the actual landing point and also the target point. The target was acquired in this case with a 50.2198 ft miss distance. The changes to the bank rate profile can be seen in Figure 7.27. PLV-2 Descent earn ant . . ... I( - WRTActu - - - WRT Target 2000 o i 1500 SD 02 M *8oo o 500 II.~~~~~~~~~~~~~~~ - .. ..-... . .I Id, - - I n 600 650 700 750 800 850 900 Time (secl Figure 7.26: Closed-Loop Prediction Error: Offset From Nominal Target 112 BankRate Corrections x,< U.Ut ^ 0.04 :e 0.02 cc 0 c -0.02 -0.04 750 ____ 40 05...... Initial Profile I ..- Final Profile -0.05 0 2 4 6 8 1012141618202224262830323436 -- BinNumber Figure 7.27: Terminal Phase Bank Rate Profile: Offset From Nominal Target 7.4 Bin Number Selection The number of large bins dictates the amount of bank rate profile possibilities that can be generated by the open-loop code. A range of large bin numbers from 3 to 6 were tested in order to find a good balance between the number of profiles generated with an emphasis on reducing the computational load. The maximum amount of small bins was left at 36. Table 7.4 lists the large bin number, the amount of open-loop profiles generated within limitations for each large bin number, and the number of profiles that fall within the acceptable range. The input target for this study is the same target for the original foot- print. The coordinates are given in Table 5.2. 113 Large Bin Number # Total Profiles # Acceptable Pro- files 3 85 1 4 381 1 5 1731 6 6 7887 32 Table 7.4: Large Bin Number Range Testing From the footprint plots below, it can be seen that a large bin number of less than 5 results in a very small range of landing locations. A large bin number greater than 5 is dense with landing possibilities, however a larger bin size would take up too much compu- tation time with little to gain in the profile selection compared to the bin size of 5. A large bin amount of 5 covers a wide range of profile possibilities, yet does not strain the compu- tational load. 114 FlmI L8K84V.QSitmol #4 Pto6.a I.SSis.3. 8Sf. * 36 - s I. o' FIhi Lndng S of Al Pfi. Lk. * 4, 8Bi . 36 lS l avow Fhl drt~ Ses ot mP.l LlnlI. 3, St K· 36KK- K K KU KKI~a *4 K KIK 2 . Kx KKIK KK8*1 KK K M~~~jK K x~r l m · ' KKKKKKl K i· K K · K .:r.K .K KI K K Kr. KI K' K 4 Ks rKK~, K KK 88 .OK K I~ KK~K5 , M!2q t ",.m K I a g 1K K· i _4 .n- 4 ~~~V -v- ..- 0KKKK' . KKKKK58 'K _ i I i -4 -4 88 I 88 Al Sfm.~~~~~~~~.r a 4 I I I I I -10 -8 -4 -4 -2 0 2 -1 O -4 -4 -4 -2 0 E£80 () Ifl E () tt K-'T 'T¥ r ...... ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~I 110'4 Fel Lnd,g S"Oo A1 ProllM LBm - S. S. 36 XtO FinalLining St of Al Pro: LBm . 6. SS . 38 I a" -2C i- -4 -6 . "B - I -4 -4 -2 I -10t- -4 -4 -2 0 -10 -8 -4-2 0 EuAt() K10 Em ) . sn' A As Figure 7.28: Open-loop Landing Locations for Large Bin #'s of 3, 4, 5, and 6 In the closed-loop code, the amount of small bins affects the fidelity of the guid- ance scheme. A correction can be made on each bin individually. A range of the maximum number of small bins was investigated as well. In one set of test, the number of large bins was left at 5, while the maximum number of small bins used in the open-loop testing var- ied from 10, 20, 30, 35, 36, and 40. As seen in the figures blow, when 36 bins are used, the footprint essentially covers the same area as when 30-40 bins are used. Thus, the capabil- ity when using 30-40 bins is the same. The more small bins, the more controls available 115 once the corrector is utilized. A maximum bin number of 36 offers a good balance between the open-loop and closed-loop capability. __ __ , lo' Fll LdinG 6Set d A Pl.U L . SBh s. 10 xitO Fxu L.ndigsto l AI P.Io e: U.L, 8-5.S 20 1 r x~~ x -- ~ ' w 'Kn · KKKS·K K S I~ N l : rK KA I*IK~ K K x x ,x · K NONrr K. .tr -2 KKUKII~KKKI II IC Z 4 K ' K~~x, - I I . -2 Z- K~" K·· .KY:K -4 I~~~~~~~~~lt 1 -4 -6 18 ---o .4 -6 -4 -2 o 2 - -J - -2 0 Eat () x10 I Lx _ __ _I__ I x tO FitiEA (10 , _x10 Fml LXIdngt Sts. tAl PM. LBm ., SB.m. 30 t' Fn Lnd Sx d ANP.r.. IL81n: .. S Sa * 3S 6 · sI, I I I 4 2 -2 -A -a I -4 -1 -8 -4 -4 -2 0 1 -1 0 - -4 -4 -2 0 East (R) Eat55(t) . It' · · · I iu A.- - 7- ._r ._41 x10 Fnal LendngSes ofAU P.ro. L.ns . Sn. 36 10 FNI L.ndng Steso A P ofils LBo · 5. SrSu . 40 | ] I I _ I ; I i I 2 4 -20 2 S I-, -4 . . . . . i 0 AC & , - - . -tO -a -a -4 -2 0 2 -10 East(S) · O' Eu ( .Xs~~~~~~~~~~~ lo'0 . , . I Figure 7.29: Open-loop Footprint Given Maximum Small Bins of 10, 20, 30, 35, 36, & 40 116 Closed-loop testing was also performed to justify the choice for maximum number of bins. The nominal profile was flown, but the time increments were changed such that the number of bins was ranged from 18, 36, 72, and finally to 108. Table 7.5 below shows the final miss distances for the same simulation run (Target B, Profile 9) given different maximum small bin numbers. Number of Horizontal Miss Distance Small Bins from Target (ft) 18 44.29 36 47.019 72 2.574757019 e4 108 1.993549189 e4 Table 7.5: Small Bin Size Testing Results The greater number of small bins, the smaller the bin times. Each bin time can be given a correction to better steer the PLV-2to the target. But, if the bin times are too small, there will not be enough time for the bank maneuvers to complete. For those bins greater than 36, the horizontal miss distance was large. For a bin number of 18, there was a small reduction in the miss distance of three feet. Since a greater number of bins will handle dis- persions better, this reduction in miss distance due to a smaller number of bins is not the best option for the bin number. The balance can be made by selecting the small bin num- ber of 36. 117 7.5 Effects of Atmospheric Dispersions As the PLV-2 re-enters the atmosphere, it will be subjected to dispersions caused by wind, density changes, and pressure changes. An atmospheric modeling program, GRAM-95, was used to generate dispersion data. GRAM-95 data also provides an insight to monthly and seasonal variations. Case 4 (Target B, Profile 9) was used as the nominal case to test the guidance scheme robustness to different atmospheric dispersions in two different seasons, winter and summer. Figures 7.30- 7.37 below show the sampled mean, 1-sigma, and 2-sigma wind variation and density variation profiles for test cases. 118 Case x 10~~ Altitude vs. Density/Mean Denslty~~~IllSummerl x 10' Aftftudevs. Density/MeanDensfty: Summer Case i 0.5 0 6 0.7 0.8 0.9 1 I.1 1.2 1.3 1.4 1.5 Densityslugs/ft 31 Figure 7.30: Altitude vs. Density: Summer Case Figure 7.31: Altitude vs. Density: Winter Case 119 Figure 7.32: Altitude vs. North Wind: Summer Case Figure 7.33: Altitude vs. North Wind: Winter Case 120 Figure 7.34: Altitude vs. East Wind: Summer Case s I n Altitudevs. East Wind:Winter Case MS R 0 EastWind ft/secl Figure 7.35: Altitude vs. East Wind: Winter Case 121 I A1le..i . %!A.41~1 tUAin, C).. e VerticalWind ftt/secl I Figure 7.36: Altitude vs. Vertical Wind: Summer Case L __ s I 1l Altitudevs. Vertical Wind: Winter Case 0 a0 't, D Vartil Wnd Aft/s1 Figure 7.37: Altitude vs. Vertical Wind: Winter Case 122 The terminal phase dispersion testing results are given in Table 7.6 and 7.7. Only a few of the cases did not come within 1 nautical mile (-6000 feet). Most of the cases came within 100 feet of the target even in the face of dispersions. In general, the mean and 1- sigma cases had lower target miss distances than the 2-sigma cases. The East wind disper- sions caused the greatest miss distances. Dispersion CRmiss (ft) DRmiss (ft) Nominal -33.5030 -32.9906 Summer Case East Wind - mean -728.1376 -3708.5116 East Wind - lo -1010.6408 -5658.0987 East Wind - 2 3113.3935 -9759.5360 North Wind - mean 1.0227 -55.8933 North Wind - l -184.6070 47.0619 North Wind - 2a -315.0263 -76.6906 Vertical Wind - mean -33.5008 -32.9992 Vertical Wind - l -34.5593 -26.8857 Vertical Wind - 2c -27.2190 -24.3609 Density - mean -23.9225 94.0316 Density - lo 4617.7372 -3846.9591 Density - 2(c -276.1927 -2377.3139 Table 7.6: Dispersion Testing Results: Summer Case 123 Dispersion CRmiss (ft) DRmiss (ft) Nominal -33.5030 -32.9906 Winter Case East Wind - mean -3.25027859e4 -8.39587232e4 East Wind - la -2922.0533 4123.0254 East Wind - 2 -102.3566 -912.2753 North Wind - mean -26.8952 -30.8330 North Wind - l -250.1215 55.0452 North Wind - 20 123.5137 -1432.2408 Vertical Wind - mean -33.4941 -32.9920 Vertical Wind- lo -34.5368 -34.5368 Vertical Wind - 2c -24.5079 -26.4444 Density - mean -6687.2872 2018.8684 Density - 1a -75.7021 -243.6869 Density - 2 -214.2973 -3046.1677 Table 7.7: Dispersion Testing Results: Winter Case Figures 7.38 - 7.41 below show the miss distances in relation to the target and the nominal case. The Summer test cases generally had lower miss distances than those in the Winter. This would be expected as the winter time would yield greater atmospheric varia- tions. 124 SummerDispersion Test:ng: Miss Distance 0 =2 o La 0 DownrangeMiss (ft) Figure 7.38: Dispersion Miss Distances (1000 ft): Summer SummerDispersion Testing: Miss Distance a0, oi be -\ N 0 I'j O DownrangeMiss (ft) Figure 7.39: Dispersion Miss Distances (6000 ft): Summer 125 Winter Dispersion Testing:Miss Distance 1uuu. nnA ...... ' ' " " - !. -" ~,,".. . .East Wind 800 jr. · '.o NorthWind Vertical Wind 600 ../ .. .. 0 Density /O Nominal 400 ( 9 200 / : . . ; \C ii 0 I -200 U C -400 -600 -800 -A , , , ... i . r - . -1000-,100 -800 -600 -400 -200 0 200 400 600 800 1000 DownrangeMiss (t) Figure 7.40: Dispersion Miss Distances (1000 ft): Winter WinterDispersion Testing Miss Distance CE t co P2ru, DownrangeMiss (ft) Figure 7.41: Dispersion Miss Distances (6000 ft): Winter 126 Chapter 8 Conclusions A robust bank-to-steer guidance scheme was designed that could be used for a reusable launch vehicle designed for precision landing, such as the PL-RLV.The presence of the terminal phase bins allows for more diversity in target acquisition. Bank rate pro- files can be chosen preflight based upon fuel cost requirements and the minimum target miss distance desired. Several different bank rate profiles can lead to the target, so profile selection is at the discretion of the mission design. It was proven that the pre-terminal knowledge of the nominal profile can reduce the pre-terminal predictor error on orders of approximately 20,000 feet. This reduction will allow for better handling of dispersions that the craft will encounter throughout its descent. The landing footprint, all bank rate profile possibilities, and the partials of the con- straints with respect to the controls can all be calculated off-line. This off-line code allows for minimal computation load. A more accessible target can be chosen preflight from the footprint generated as well. The bin number testing provides good insight as to bin number selection for future simulations. The predictor/corrector pairing proved to work well. When given a nominal open- loop profile, the closed-loop guidance can minimize the target miss error to less than 100 feet. Even in the presence of atmospheric dispersions, the guidance scheme was robust. 127 The nominal profile reached the target within requirements when tested with wind and density variations of 1 and 2-sigma. All of these advantages to the designed guidance makes this scheme a pliable option for any bank-to-steer entry vehicle. Suggestions for Future Work In regards to the designed guidance scheme, additional testing could be performed on: 1) The use of bank angle profiles instead of bank rate profiles 2) Modulating the bin length 3) Additional dispersions In order to provide the most accurate modeling of the PLV-2's flight, a 6 DOF sim- ulation could be created, as well. An acceleration model could also be implemented to help model the bank changes more precisely. Appendix B provides a sample acceleration model which could be implemented [4]. This thesis provides a robust guidance scheme design that can be utilized. The sug- gestions for future work can be used for improvement on the design. Future work should combine advantages from all possible approaches in order to provide the most optimal guidance scheme. 128 Appendix A Analytical Study of the PL-RLV Re-entry Guidance I. Purpose The re-entry guidance requirements for the PL-RLV's second stage (PLV-2) present a complex problem due to the constraints placed on the vehicle and its performance. Upon re-entry from orbit, the PLV-2 will experience control until landing system parachute deployment. The vehicle is also required to land in a specified area which adds complex- ity to the problem. This analysis makes certain simplifications to the problem in order to gain insight used to guide the development of the re-entry guidance algorithms. The re- entry guidance task is to predict the bank rate profile effect on landing locations, correct the bank rate plan, and direct the vehicle to the desired landing location. This study is only the first of several planned analyses each exploring certain guidance algorithms. II. Set Up Given the current position and velocity from navigation as well as a target position, this simulation calculates the best possible combination of two bank rates (ThetadotA & ThetadotB) and a time at which to switch instantaneously from the first to the second bank rate (t_switch) to minimize landing error. The simulation begins at a specified altitude and controls flight down to an altitude of 70,000 ft. After this point, the landing system para- 129 chute deploys, all automated control ceases, and the vehicle will enter the coast phase. See Figure A. 1 below. Start Altitude -~~~ _ 4L A, - , -Re LAt= Il ~lIl 70,000 ft 0 .r A (a Coast Interval At = 108.5 sec t _· , v 5,000 ft iF I ·, .. · l' Z Z Z w j~ - - - - - 0 tswitch tmax ------L- -- Figure A.1: Single-Switch Program Schematic 130 III. Equations of Motion Implemented The coordinate system for this simulation is defined as follows: - I Y Y 0 = bank X Figure A.2: X, Y, Z Axes and Bank Angle O Defined Assumptions and Initial Conditions: * Falling straight down in the Z direction (Simulation does not actually include gravity, but simply time spent in a particular bank phase) * O0 = Initial Bank Angle = 0 [rad] * Instantaneous bank rate change · a = 11.28 ft./sec 2 , horizontal acceleration in bank direction, due to lift Vxo = VyO= 0 ft/sec , VzOis irrelevant since this simplified simulation is not concerned with the vertical channel = · X = Y o ft * t_max = 20 sec, time until landing system parachute deploy * t_init = 0 sec 131 The equations of horizontal motion used during bank flight are defined as follows: = -At +O o (A.1) At = t-t o (A.2) d(At) = dt (A.3) a. = a cosO (A.4) ay = a sinO (A.5) v, = a fcos(- At+ o)dt (A.6) a cosOO . a sinG V= sinAt + .o cosAt + (CIx) (A.7) = a sin( Ato)dt = a -sin(6-A+ O)dt (A.8) -a coso c a sin · v = 0 cost+ 0 sin 0At + (Cly) (A.9) -(3 H -a - cos oO a- sin0o r = 2 cos0At + 2 sin At + (Clx) · At + (C2x) (A. 10) (0) 2 (6)2 -a- cos0 o a-sin o ry= 2 sin At+ -cosAt + (Cly) ·At +(C2y) (A. 11) (G)2 (6)2 where Clx, C2x, Cly, and C2y are all constants. 132 Propagation to the ground after landing system parachute deploy (at 70,000 ft) is calcu- lated with these equations below: Final Position(X) = PositionX(70,000 ft) + VelocityX(70,000 ft) x tcoast Final Position(Y) = PositionY(70,000 ft) + VelocityY(70,000 ft) x t_coast The approximate ranges of final positions can be determined by looping through the fol- lowing steps for values of ThetadotA and ThetadotB: * Use Equations of Motion (EOM) and ThetadotA on the interval, t = t_init - tswitch, to determine conditions at t_switch. * Update bank angle, assuming instantaneous bank rate change. · Use updated bank angle and ThetadotB EOM on the interval, t = t_switch - t_max * Finally, propagate to ground during coast interval (t-108.5 sec) using the propagation equations. T_coast, the approximate time for the vehicle to coast from 70,000 ft. to 5,000 ft., was determined from the nominal profile obtained during previous PL-RLV testing. 133 IV. Aim-Point Envelopes A program was generated to calculate all possible final positions for a given range of ThetadotA, ThetadotB, and t_switch. These contours approach "bow-tie" shaped contours about the origin as time-to-go becomes small. (See Figures A. 3, 4, 5, 6, and 7 on the fol- lowing pages) Bow-tie regions define where the vehicle can land and also the landing areas that are impossible to reach given the initial conditions. It is important to note that all bow-tie contours in this study had an initial bank angle of zero, which is aligned with the initial non-zero acceleration vector. Given a different initial bank angle, the contours would rotate about the origin by the initial bank angle, thus sweeping out all areas around the ori- gin when all possible initial banks are looked at. An example of initial bank angle effect can be see in Figure A.10. Each final location is not necessarily reached by just one combination of the parameters. This simulation proved the ability to reach the same aimpoint with different sets of param- eters (ThetadotA, ThetadotB, and t_switch). All areas covered within the defined bow-tie contour are accessible. The complete density accessibility of aimpoint locations is insured by minute tweeking of parameter incrementation (e.g. t_switch = 1.456 instead of 1.5) That is to say, all final locations within the contour can be selected as aimpoints. 134 The following parameter ranges were chosen since they result in all final positions within the parameter limits: ThetadotA = -15 to +15 in steps of 1 deg/sec ThetadotB = -15 to +15 in steps of 1 deg/sec T_switch = 0 to t_max in steps of 1 sec An initial bank angle of 0 deg was used for the contours found in Figures A.3, 4, 5, 6, and 7. The initial bank angle direction lies along the horizontal X-axis for these cases. Final Positions,Lmax - 40 sec no It ...... - . 40 c 20 C ._ C .; -20 -40 miPS! I - . . . · -40 -30 -20 -10 0 10 20 30 40 50 60 Distancein the X Direction[kft] Figure A.3: Final Positions When T_max = 40 sec 135 Final Positions, tmax - 30 sec :1 I C I a I .ciCD C 0 -30 -20 -10 0 10 20 30 40 50 Distance in the X Direction [ktt] Figure A.4: Final Positions When T_max = 30 sec Final Positions, t_max = 20 sec An '4U I I I I I I I 30 ...... 20 C 0o = 10 Sa) -c 0) - a) c -10 ...... 5Cu -20 ...... -30 ...... - ...... __I·I [ I I -40 I I -40v-40 -30 -20 -10 0 10 20 30 40 Distance in the X Direction [kft] __ Figure A.5: Final Positions When T_max = 20 sec 136 Final Positions, tmax = 10 sec 40! I 30...... FE 20 10 ...... I...... 10 ., O . . . . : . . . 4 . I I 8 . . : I. [ -20 -30 ...... , ...... : . .. -40 -30 -20 -10 0 10 20 30 40 Distance in the X Direction [kft] Figure A.6: Final Positions When T_max = 10 sec Final Positions, tmax -5 sec I 30 .. .; ...... 20 ... .. I ...... ·.·..· 0 10 0 ...:...... c ...... -1, ...... -20 -30 .. . . -0 , [ I. I -40 -30 -20 -10 0 10 20 30 40 Distance in the X Direction [kft] Figure A.7: Final Positions When T_max = 5 sec 137 Contour Observations By increasing the time until drogue deploy (increase in t_max), the contour area increases and begins to close in around the target. The inaccessible zone becomes smaller, thus increasing the range of final positions. As t_max decreases, the contour area decreases and widens the opening around the target. The inaccessible zone becomes larger, thus limiting the possible final positions. Small holes resulted in the high t_max contours. These holes are thought to vanish by tweeking certain parameters (i.e. time or bank rate increments), and thus they are not true inaccessible zones. Most of the "upper" half of the contour, located in the positive X and Y quadrant, is the result of +ThetadotA's and + ThetadotB's. The "lower" half results mainly from (-Theta- dotA's) and + ThetadotB's . A +ThetadotA and B usually makes up the upper half, while a -ThetadotA and B compose the lower half of the contour. There is some overlap around the zero y-axis when the magnitude of ThetadotA is much smaller than ThetadotB (e.g. ThetadotA = 5 deg/sec and ThetadotB = -15 deg/sec) or when the switch time only allows ThetadotA to occur for a short time (e.g. t_switch = 1 sec). When both ThetadotA and ThetadotB are 0 deg/sec, the final positions lie along the zero y-axis. The original test case, tmax set to 20 sec, was investigated further to define characteris- tics of the contour. Setting t_switch = 5 sec for this test case results in final positions that cover almost all areas possible, except the small outer edge slivers of the tswitch = 10 case. Shown in Figures A.8 and A.9 below. 138 All Final Positions, Tswitch-1 0 sec .CD I I I I I~~~~ 20 15 * +r3" 10 ' 5 -5 BiG-10 -15 -20 i _-2 I · I I ~~I I --10 -5 0 5 10 15 20 25 30 Distance In the X Direction [kft] __ Figure A.8: All Final Positions, Tswitch = 10 sec All Final Positions, Tswitch-=5 sec . . - . -. ' -. .' -.. 20 ,.. 15 ·~~~~~~~~~~~~···: ~~~~~~~~~~..'*'*. -- '~~~~ '':i' r' ,4- ' ' . · · · : ' " '- .-o C ..- .- .. · ',' . . ' ,',',','... ° · '. -5o · e~~~~ · . o .- o ~~~~~~.~.1 I -10 -15 I ~I ~ ~ ~ ~ ~ ~ ~ ~~~~~~i -20 _or -10 -5 0 5 10 15 20 25 30 Distance in the X Direction [kft] Figure A.9: All Final Positions, Tswitch = 5 sec 139 As mentioned previously, a change in the initial bank angle does not alter the form of the contour, but rotates the contour. The test case in Figure A. 10 was given the same condi- tions as Figure A.5, but an initial bank angle of 90 deg was used instead of 0 deg. Other factors, such as the addition of an atmospheric model, will likely transform the contours. In the effort to reduce the original simplifications and therefore generate a more precise simulation, future work will analyze the effects of the atmosphere and other factors on the contours. Final Positions, t_max = 20 sec JA ,a I I I I I I I I I I I I I~~'' 3C 20 k 10 p E 0 ...... C c ...... - .10 ...... _0 : : ...... -20 : : . . . . . . -30 - ...... - .... : . . . . . NiA I[[ I. I . I. I. 40 -30 -20 -10 0 10 20 30 40 Distance in the X Direction [kft] Figure A.10: Final Positions, Initial Bank Angle = 90 deg 140 V. Predictor/Corrector Basics The main goal of the single-switch simulation is to calculate the most desirable combina- tion of the three parameters (ThetadotA, ThetadotB, and t_switch) that will result in a final position close to the target position. These combinations must also lie within the limita- tions of the parameters. All bank rate limits were set between ± 15 deg/sec; and the time interval from t_init to t_max (drogue deploy) was set to 20 sec. except when testing the effects of an increased or decreased t_max (i.e. tmax = 40,30,10,5 sec.) A predictor/corrector was added to calculate the final positions resulting from the given input. The corrector program in turn checks to see if the predictor's final positions are close enough to the target position. In these simulations, a + 200 ft difference in the two positions was considered acceptable. If the final position is not close enough, the correc- tor tweeks each parameter one at a time, noting the resulting final positions for each case. (See Figure A. 11 below) The sensitivities for tweeking each parameter were set at: inc_thetadotA = .01 deg/sec inc_thetadotB = .01 deg/sec inc_tswitch = .02 sec With this data produced by the different test cases, the corrector implements a least squares method in order to find the changes needed for each parameter. These changes are 141 then added to the old parameter values in an effort to steer the new final positions closer and closer to the target. _ __ _ _ Y X (4) ThetadotA tweeked adotB tweeked Figure A.11: Resulting Final Positions, Each Parameter Tweeked One at a Time Once the multiple test case data is supplied, the corrector's steps are as follows: AX = X. A A + B + x tS (A. 12) 8yA A 8 + y- At (A.13) SOB st, where: AX = Target_X - Position_X(l) AY = Target_Y - Position_Y(1) (1) = the first run case as shown in Fig.A. 11 142 (2), (3), (4) = tweeked cases as shown in Fig.A. 11 ax S8x(3) - Sx( 1 ) (A.14) F6B esb(3)- 8B(1 ) Sx = Sx(4)- Sx(1) (A.15) seA seA(4)- 8A(1) Sx _ x(2) - x( 1 ) ,etc. (A.16) Et s 8ts(2)- t,( 1) * Solve for ThetadotA, ThetadotB, and Ats , Least Squares Method * Add new increments and gains to the three parameters: New ThetadotA = Old ThetadotA + (Gain)(A ThetadotA) , etc. The gains used in these simulations were as follows: K_ThetadotA = .1 K_ ThetadotB = .1 K_Tswitch = .1 __ Gains were added due to the non-linearity of the problem, as indicated by preliminary simulations. Further work will investigate these values. 143 Run through loop until within a certain acceptable radius of the target error2 = (TargetX - NewAimPointX)2 + (TargetY -NewAimPointY)2 For these simulations, error < 200 ft = stop, close proximity to target. 144 Case 1: Predictor / Corrector Reaches Target Within Parameter Limits Test case 1 shows an example of target acquisition within the parameter bounds. Each "+" shows the predictor/corrector's path to the optimal landing location, beginning with the location defined by the initial parameters and ending with the target location. The target location in this case (5000 ft , 15000 ft ) can be found within the "upper" part of the contour. The initial conditions defined a location (- 7000 ft, 13500 ft) in the "upper" part of the contour as well. (Note: the plot's axes show the location within the contour.) This case demonstrates the corrector's path to the target while trying to optimize the parameter combinations. The initial and final conditions for case 1 are listed below. Note that all final parameters are within the parameter limits. (See page 5 for parameter limits) Initial Guess Final ThetadotA = 5.0 deg/sec 6.389 deg/sec ThetadotB = 15.0 deg/sec 12.976 deg/sec t_switch = 10.0 sec 10.0 sec Note that t_switch did not change, indicating that the landing location is most sensitive to changes in ThetadotA and B than to changes in t_switch within this region of the contour. Different regions of the contour are sensitive to different parameters, as the following cases will show. 145 Case 1: Achieving The Target Within The Parameter Limits 15.2j , I I 1 4...... 14.8 .. 14.4. 14.14 . .... _...... *...... '..... 13.4 5 3 5!.6 6.5 7 7.5 Distance in the X Direction [(kttJ -- -- Figure A.12: Accessible Target Case To demonstrate that more than one solution for target acquisition is possible, an identical case to case 1 was run, but tswitch was changed from 10 to 11 seconds. In this case, the target was also achieved within the parameter limits. Initial Guess Final ThetadotA = 5.0 deg/sec 6.6273 deg/sec ThetadotB = 15.0 deg/sec 12.8949 deg/sec t_switch = 11.0 sec 11.0 sec 146 Target Acquisition with T_switch - 11 K F5 'Ia .G[ co 5 Distance in the X Direction [kft] Figure A.13: Another Solution to Case 1, Target Acquisition Changing the parameter gains can result in different corrector paths to identical target locations. The parameter gains were each set to .10 in case 1 (shown on page 13). Gains of .5 and 1.0 were also tested, and their results are shown below. In general, as the gain was increased, the final bank rates were increased slightly. The higher the gain, the faster the speed of convergence as well. Gain = .5 Gain = 1.0 Final ThetadotA = 6.3902 deg/sec 6.3915 deg/sec Final ThetadotB = 12.9773 deg/sec 12.9794 deg/sec Final t_switch = 10 sec 10 sec 147 Figure A.14: Case 1 With Gains Set to .50 Nominal C -a.with GOIln. 1.0 1. -~~ ~ ~ ~ ~~~~~~~~~~~~~~~~- . 1'6 .G 14.6 . I . . 14 13.5 _ _ _ 14 _ _ ' . 65 5.5 a e.6 7 7.5 Distanc in the X Direction [kIt] -- Figure A.15: Case 1 With Gains Set to 1.0 Since the gains of 1.0 reach the target, the possibility that the gains are unnecessary arise. Although the target is acquired in this case, the overshoot is noticeable and could prevent the corrector from converging given a different target/initial guess setup. 148 Case 2: Target Placed Within the Inaccessible Zone In Case 2, the target (8000 ft, 0 ft) was placed inside the contour's inaccessible area of Figure A.5 in order to verify that the target could not be achieved. Initial parameters and the resulting parameters for this case were: Initial Guess Final ThetadotA = 5.0 deg/sec 313.77 deg/sec ThetadotB = 15.0 deg/sec 14.310 deg/sec t - switch = 10.0 sec 1.889-- ---sec Although the corrector did converge on a final set of parameters that reached the target, final parameter values were grossly out of range, thus proving that a target cannot be reached if located in the "inaccessible zone" of the contour within the limits of the param- eters. Case 3: Target Placed Across the Inaccessible Zone In Case 3, initial conditions were given for a position at the "top" of the contour (~ 7000 ft, 13500 ft) in Figure A.5, and the target position (5000 ft, -15000 ft) was given a location at the "bottom" of the contour. Initial parameters and the resulting final parameters were as follows: 149 Initial Guess Final ThetadotA 5.0 deg/sec 154.00 deg/sec ThetadotB 15.0 deg/sec -5.56 deg/sec t_switch 10.0 sec 6.7841 sec Figure A. 16 shows the predicted aimpoint path as the corrector tries to close in onto the target. For cases such as this, the corrector forces the predicted aimpoint to enter the inaccessible zone, where the parameters, as in Case 2, are driven out of range. Upon re- entering the accessible zone near the target, the corrector is forced to use parameters out of the desired range even if they achieve the target. _ _ _ Case 3: Crossing the Inaccessible Zone C .o a 0 '1 S0 -c/ C. - -4 -2 0 2 4 b u Distance in the X Direction [kft] Figure A.16: Figure 12. Crossing the Inaccessible Zone 150 The ability of the corrector to reach the target with a different solution than the desired one (all parameters in range) proves that there are multiple solutions to target acquisition and that the corrector is inadequate in cases where the initial guess is poor. A new scheme for the corrector, in its elementary stage, is currently being analyzed in order to avoid prob- lems like Case 3 points out. This corrector adds a second stage after the corrector has con- verged that modifies the parameters towards their desired ranges while holding the aimpoint position constant. VII. Current Work (March 1997) The single-switch simulation provided preliminary insight on parameter combinations, their resulting effect on final positions, and final position contour bounds. Current simula- tions in progress update the single-switch simulation to a more realistic representation of the re-entry process. The results yielded from the single-switch simulation and the current simulations listed below will help mold re-entry guidance algorithm development for the PL-RLV. The current simulations in progress include: · Non- instantaneous change of bank rate: Intending to simulate the use of RCS thrusters, the bank rate profile now incorpo- rates bank acceleration over certain times, as seen in Figure 13. Over these times, the evaluation of a Fresnal Integral will be necessary for an analytic predictor. 151 Figure A.17: Non-instantaneous Bank Rate Change * Addition of an Atmosphere / Drag Model · Investigation of Initial Condition Effects Tests have already shown that non-zero initial positions result in translation of the final position by the values of X0 and YO. Similarly, non-zero initial velocities do not change the final position contour shape, but push the positions farther in the velocity vec- tor's direction. Figure A. 18 shows the results of adding an initial X and Y of 10,000 ft to the nominal test case found in Figure A.5. Figure A. 19 displays the results of adding an initial X and Y velocity of 100 ft/sec to the nominal case in Figure A.5. It is easy to see that the initial positions translate the contour of Figure A.5 by 10,000 ft in both the X and Y directions. The initial velocities push the original contour in the velocity vector's direc- tion as shown in Figure A.19. 152 Figure A.18: Non-zero Initial Positions _ _ InitialX and Y Velocities=100, T_switch = 5 r ac o C 0 C Distancein the X Direction[kft] Figure A.19: Non-zero Initial Velocities 153 With the addition of an atmosphere model, it is expected that the final position contours will change shape and warp slightly when the initial bank angle and velocities are not aligned. Other areas to be investigated include: * Perfected corrector scheme to handle all cases * Work towards the reduction of the number of iterations needed by the corrector * Investigation of "Wait" Time Model The "wait time" model (see Figure A.20) allows for coast times before ThetadotA implemnent n, in-between ThetadotA and ThetadotB implementation, and even after ThetadotB. This model significantly increases the dimension of the problem. The wait times and times for the bank rates do not necessarily have to equal the maximum time until drogue deploy. Test cases investigated thus far set the parameters within these limits: ThetadotA: 2.5, 5.0, 7.5, 10 deg/sec ThetadotB: +ThetadotA t= 0 sec tA must be at least 5 sec tB must be at least 5 sec 154 I i - e 7ZZZZZ~ - ~--- s V/JIlL! on - XtA tw-- tB taIa Figure A.20: Wait Times between Bank Rate Changes Schematic 155 156 Appendix B Acceleration Model for Bank Maneuvers I. Acceleration Model Defined Due to time constraints, this thesis assumed instantaneous response to changes in bank. For future investigation, an acceleration model could be employed for the reversal and even the terminal bank rate changes. A model is suggested below for the acceleration dur- ing the reversal phase [4]. This design can also be followed for the acceleration during the terminal phase bank rate changes. Figure B.1: Acceleration Model Bankrate vs. Time where he = the commanded bank angle ~0 = the bank acceleration from to to t1 157 ~i = the bank acceleration from t to t2 , which is zero ;2 = the bank acceleration from t2 to t3 , which is -~o ;3 = the bank acceleration after t3 , which is zero The time differences are defined by: At I = - to (B.1) At2 = t2 - to (B.2) At 3 = t3 - t o (B. 3) The bank angle and bank rates are defined as a function of time in each phase as: Acceleration Phase 4) ¢(t) = 0 + Io(t-t0 ) (B ~(t) = o(t- to) (B.-i) Coast Phase ¢(t) = 0 + o(At )2 + )c(t- t ) (B.6) ¢(t) = doAt = c (B.7) Deceleration Phase ¢(t) = ¢° + Ib0(Atl)(t)= 2+ t.kc(t2 - tl) + 21.tI 2(t - t2) 2 (B.8) (B.9) ¢(t) = 4oAtl + 4 2 (t- t2) 158 The acceleration model should be placed in the predictor calculations. The bank angle and bank rate calculations will be different depending on the actual time when the predictor is called. A prediction call made during the entry phase will model all of the acceleration, coasting, and deceleration phase of the maneuver. A call made within the reversal will depend on the time from reversal start to yield the bank angle and bankrate calculations. The following subsections define the bank angle and bank rate calculations when called for at different times: prior to reversal, during the acceleration phase, during the coast phase, and during the decent phase [4]. II. Prior to Reversal If the current time is before the reversal time, the switch times can be calculated by first finding the time it takes to accelerate to the commanded bank rate and the amount the bank angle changes during the acceleration phase. Atacc c (B.1 0) Aacc = 5¢o(Atacc), (B.11) 2 159 If the change in the bank angle during the acceleration phase is greater than half of the total maneu ver angle desired (IAaaccl > 0o), the acceleration model should be redefined as follows [4]: Figure B.2: Profile Redesign Due to Time Constraints Acceleration occurs over half of the maneuver and deceleration occurs over the second half. At = At2 (B. 12) At3 = 2At, (B.13) If there is enough time to perform the maneuver, the original profile design should be used. The switch times can be found from: At = Atacc (B.14) 160 At=At,- 2 - (i(o)(A,) 2 At 2 = At +-°,. (B.15) (~o)(at,) At3 = At, + At2 (B.16) III. Current Time in Acceleration Phase of Reversal When the current time is in the acceleration phase of the reversal, the change in the bank angle that remains to be completed and the time since the start of the reversal need to be calculated. The current time and bank angle are the references. A = cp- ef (B.17) At = te f -trevstart (B.18) The current bank rate is given by: }ref = ' 0At (B.19) It is necessary to calculate the bank change during the remaining acceleration and also the bank change during deceleration from the commanded bankrate. A4tacc= o(At, - At) + ~,ef(Atl - At) (B.20) Adc = o(Atl)(At 3 - At2) + I)2(t3 - t2) (B.21) 161 If (IAacc + Adecl > IA~I ) there is not enough time to complete the acceleration to the commanded rate and to decelerate to a zero rate at the commanded bank angle. In this case, the switch times are calculated as: At = At - + 2()+ A (B.22) . 2jO* At 2 = At I (B.23) lref At3 = 2At -At- + (B.24) If (IAacc + OAdec!< IAOI) then there is enough time to complete the maneuver, and the switch times are calculated as: At = Atl + IA1- IA4acc+ AodeI (B.25) At3 = At2 + _5 (B.26) 0 162 IV. Current Time in Coast Phase of Reversal During the coast phase, the total bank change that remains and the time since the start of the reversal are defined as: A = c-r, (B.27) At = t f -retrevstart (B.28) The switch times can then be found from: At 3 = At 2 -ref (B.29) at 2 = at + a + 2 f (B.30) If (At2 - At) < 0 , then the desired end conditions cannot be reached with the maneuver modeled. It is necessary to start decelerating right away. The switch times become: At 2 = At (B.31) At3 = At 2 2~ /(e2rf (B.32) 163 V. Current Time in Deceleration Phase of Reversal When the current time is in the deceleration phase, the final switch time can be cal- culated from: At = Atref - AtrevsIart (B.33) 2 At - At re2f (ref) )+ 2 (B.34) 3 42 il42 'tZ If radicand of quadratic equation is negative, the commanded bank angle will not be reached prior to reversal of the bankrate sign. The maneuver end should be set to the time at which bankrate passes through zero. (B.35) At 3 = At-3~ re2 164 Appendix C Nominal Profile Simulation Plots The following plots were generated for the nominal profile trajectory presented in this thesis. (Target B, Pro- file #9) Figure C.1: Altitude Profile 165 I _ _ 0 100 200 300 400 500 600 700 800 900 __ Time(seci _ _ __ Figure C.2: Earth Relative Velocity Profile _ _ _ PLV-2:Descent i Ii mi Figure C.3: Relative Flight Path Angle Profile 166 Figure C.4: Dynamic Pressure Profile PLV-2: Descent 90o0 I I I I 8000 x q-alpha = 8575.4957 7000 * *t121785.31 0 ft. 0 z 652 sec 50006000-. . . . . ,- 5000..:..II Di'O!r gI5. 3 4000 . ii3000 ...... 12000 ...... 1000_ /. ,1 I , , 0 100 200 300 400 500 600 700 800 900 Time {lsecI ..1- --- Figure C.5: Dynamic Pressure x Alpha Profile 167 Figure C.6: Heating Rate Profile PLV-2:Descent L3VU i i i , . X , , . I 3000 ft 2500 a gh S. 2000 . . . . J.i. . . I . . . . . I. . . d &1500 9xoI v,S:i o 1000 ...... I..... I...... ;... 500 I I I I 0 100 200 300 400 500 600 700 800 900 Time (seC) Figure C.7: Stagnation Point Temperature Profile 168 _ _ PLV-2:Descent e i; .5 Il n u Ti (.tn eia I -LN OW 1Io Tima (secl Figure C.8: Acceleration Profile Figure C.9: Mach Number Profile 169 PLV-2: Decent 14 _10 , , , , LB ...... , ...... · 2 I i !Alphat I ! ! 0 100 200 300 400 500 600 700 800 90C Tms (sec) .3 . O.252...... ;...... ; ...... ;, ...... ; ...... : .....:; ...... 0 I 0.115 . I u 0 1. . . 0.c05 ..... boo i . . l L. nI "' .w 0 100 200 300 400 500 600 700 800 900 Time(sec) -- Figure C.10: Angle of Attack Profile 170 References [1] Barchers, J. D., Entry Guidance for Abort Scenarios, S.M. Thesis, Department of Aeronautics and Astronautics, MIT, June 1997. [2] Dierlam, T. A., Entry Vehicle Performance Analysis and Atmosperic Guidance Algorithm for Precision Landing on Mars, S.M. Thesis, Department of Aeronautics and Astronautics, MIT, June 1990. [3] D'Souza, C. N., PhD., Notes and Personal Correspondence, Draper Laboratory, Jan. - Dec. 1998. [4] Fuhry, D. P., Simulation Code, Notes, and Personal Correspondence, Draper Laboratory, Jan.- Dec. 1998. [5] Harpold, J. C., C. A. Graves, Jr., Shuttle Entry Guidance, Journal of the Astronautical Sciences, Vol. 27, No. 3, July-Sept 1979, pp. 239-268. [6] Hildebrand, F. B., Advanced Calculus for Applications, Prentice-Hall Inc., Englewood Cliffs, New Jersey, 1976. [7] Justus, C. G., et al., The NASA/MSFC Global Reference Atmospheric Model -1995 Version (GRAM-95). [8] Regan, F. J., S. M. Anandakrishnan, Dynamics of Atmospheric Re-Entry, AIAA Publishing, Washington, D. C., 1993. [9] Spratlin, K. M., An Adaptive Numeric Predictor-Corrector Algorithmfor Atmospheric Entry Vehicles, S.M. Thesis, Department of Aeronautics and Astronautics, MIT, May 1987. [10] US Standard Atmosphere 1962, Prepared by the National Aeronautics and Space Administration, United States Air Force, and United States Weather Bureau. 171 THESIS PROCESSING SLIP FIXED FIELD: ill. name index biblio , COPIES: Archive Aero Dewey Eng Hum Lindgren Music Rotch Sc:ience TITLE VARIES: O NAME VARIES: fw b1 l t IMPRINT: (COPYRIGHT) COLLATION: | • ADD: DEGREE: · DEPT.: SUPERVISORS: NOTES: cat'r: date: page: DEPT:' _ . '- ·YEAR: r - 'tDEGREE: '-?t , ', , pNAME: ? ' k,