AIR FORCE INSTITUTE OF TECHNOLOGY - DTIC
Common Aero VehicleAutonomous Reentry Trajectory OptimizationSatisfying Waypoint and No-FlyZone ConstraintsDISSERTATIONTimothy R. Jorris, Major, USAFAFIT/DS/ENY/07-04DEPARTMENT OF THE AIR FORCEAIR UNIVERSITYAIR FORCE INSTITUTE OF TECHNOLOGYWright-Patterson Air Force Base, OhioAPPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITED.
The views expressed in this dissertation are those of the author and do not reflect theofficial policy or position of the United States Air Force, Department of Defense, orthe United States Government.
AFIT/DS/ENY/07-04Common Aero VehicleAutonomous Reentry Trajectory OptimizationSatisfying Waypoint and No-FlyZone ConstraintsDISSERTATIONPresented to the FacultyGraduate School of Engineering and ManagementAir Force Institute of TechnologyAir UniversityAir Education and Training CommandIn Partial Fulfillment of the Requirements for theDegree of Doctor of PhilosophyTimothy R. Jorris, BS, MSMajor, USAFSeptember 2007APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITED.
AFIT/DS/ENY/07-04Common Aero VehicleAutonomous Reentry Trajectory OptimizationSatisfying Waypoint and No-FlyZone ConstraintsTimothy R. Jorris, BS, MSMajor, USAF
AFIT/DS/ENY/07-04AbstractTo support the Air Force’s Global Reach concept, a Common Aero Vehicle isbeing designed to support the Global Strike mission. “Waypoints” are specified forreconnaissance or multiple payload deployments and “no-fly zones” are specified forgeopolitical restrictions or threat avoidance. Due to time critical targets and multiple scenario analysis, an autonomous solution is preferred over a time-intensive,manually iterative one. Thus, a real-time or near real-time autonomous trajectoryoptimization technique is presented to minimize the flight time, satisfy terminal andintermediate constraints, and remain within the specified vehicle heating and controllimitations. This research uses the Hypersonic Cruise Vehicle (HCV) as a simplified two-dimensional platform to compare multiple solution techniques. The solutiontechniques include a unique geometric approach developed herein, a derived analytical dynamic optimization technique, and a rapidly emerging collocation numerical approach. This up-and-coming numerical technique is a direct solution method involvingdiscretization then dualization, with pseudospectral methods and nonlinear programming used to converge to the optimal solution. This numerical approach is applied tothe Common Aero Vehicle (CAV) as the test platform for the full three-dimensionalreentry trajectory optimization problem. The culmination of this research is the verification of the optimality of this proposed numerical technique, as shown for boththe two-dimensional and three-dimensional models. Additionally, user implementation strategies are presented to improve accuracy and enhance solution convergence.Thus, the contributions of this research are the geometric approach, the user implementation strategies, and the determination and verification of a numerical solutiontechnique for the optimal reentry trajectory problem that minimizes time to targetwhile satisfying vehicle dynamics and control limitation, and heating, waypoint, andno-fly zone constraints.iv
AcknowledgmentsI want to thank Dr. Richard Cobb for pitting my wingman David Irvin and meagainst each other; without such fierce competition I don’t think either of us wouldhave graduated. I’d also like to thank Vince Chioma for helping me get jump startedin the program, and Vanessa Bond for being there to provide motivation whenevernecessary. The bottom line, however, is that this dissertation would really not bepossible without the encouragement and support of my loving wife.Timothy R. Jorrisv
Table of ContentsPageAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ivAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .vList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xiiiList of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xivList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xviiI.Introduction . . . . . . . . . . . . . . . . . . . . . . . .1.1 Motivation . . . . . . . . . . . . . . . . . . . . . .1.2 Problem Description . . . . . . . . . . . . . . . . .1.2.1 Terminology . . . . . . . . . . . . . . . . . .1.2.2 Hypersonic Cruise Vehicle (HCV) Overview.112341.2.3 Common Aero Vehicle (CAV) Overview1.2.4 Launch Scenario . . . . . . . . . . . . .1.2.5 Computational Objectives . . . . . . . .1.3 Technology Advancement . . . . . . . . . . . .1.4 Document Outline . . . . . . . . . . . . . . . .44567II. Previous Research . . . . . . . . . . . . . . . . . . . . . . .2.1 Reentry Trajectory Generation . . . . . . . . . . . . .2.1.1 Space Shuttle . . . . . . . . . . . . . . . . . . .2.1.2 X-33 and X-37 . . . . . . . . . . . . . . . . . .2.1.3 Feedback Linearization . . . . . . . . . . . . . .2.2 Waypoints . . . . . . . . . . . . . . . . . . . . . . . . .2.2.1 Fuzzy Logic . . . . . . . . . . . . . . . . . . . .2.2.2 Discontinuous Lagrange Multipliers . . . . . . .2.2.3 Linearization and Propagating Riccati Equation2.2.4 Steepest Descent . . . . . . . . . . . . . . . . .2.2.5 Sequential Quadratic Programming . . . . . . .2.3 No-Fly Zones . . . . . . . . . . . . . . . . . . . . . . .2.3.1 Voronoi Diagram . . . . . . . . . . . . . . . . .2.3.2 Barrier and Interior Point Methods . . . . . . .2.3.3 Receding Horizon and Artificial Intelligence . .99991111111212131313141416vi.
Page2.4 Pseudospectral Methods . . . . . . . . . . . . . . . . . . . . . . .2.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . .III. Problem Definition and Assumptions . . . . . . . . .3.1 Generic Problem Statement . . . . . . . . . . . .3.1.1 Dynamic Optimization . . . . . . . . . . .3.1.2 Discontinuous Lagrange Multipliers . . . .3.1.3 Path Inequality and Equality Constraints .3.2 Mission Assumptions . . . . . . . . . . . . . . . .3.3 Hypersonic Cruise Vehicle (HCV) . . . . . . . . .3.3.1 Vehicle Description . . . . . . . . . . . . .3.3.2 Mission Profile . . . . . . . . . . . . . . .3.3.3 HCV Assumptions . . . . . . . . . . . . .3.3.4 Cost Function . . . . . . . . . . . . . . . .3.3.5 2-D Equations of Motion . . . . . . . . . .3.3.6 Control Constraints . . . . . . . . . . . . .3.3.7 Terminal Constraints . . . . . . . . . . . .3.3.8 Path Inequality Constraints . . . . . . . .3.3.9 Interior-Point Constraints . . . . . . . . .3.3.10 Path Equality Constraints . . . . . . . . .3.3.11 HCV Summary . . . . . . . . . . . . . . .3.4 Common Aero Vehicle (CAV) . . . . . . . . . . .3.4.1 Vehicle Description . . . . . . . . . . . . .3.4.2 Mission Profile . . . . . . . . . . . . . . .3.4.3 CAV Assumptions . . . . . . . . . . . . .3.4.4 Cost Function . . . . . . . . . . . . . . . .3.4.5 3-D Equations of Motion . . . . . . . . . .3.4.6 Control Constraints . . . . . . . . . . . . .3.4.7 Terminal Constraints . . . . . . . . . . . .3.4.8 Path Inequality Constraints . . . . . . . .3.4.9 Interior-Point Constraints . . . . . . . . .3.4.10 Path Equality Constraints . . . . . . . . .3.4.11 CAV Summary . . . . . . . . . . . . . . .3.5 Numerical Methods Implementation Techniques .3.6 Solution Comments . . . . . . . . . . . . . . . . 53637373838383940414143
PageIV. Analysis and Results . . . . . . . . . . . . . . . .4.1 2-D Baseline . . . . . . . . . . . . . . . . . .4.2 2-D Geometric . . . . . . . . . . . . . . . . .4.3 2-D Analytical, Bryson’s Method . . . . . . .4.3.1 Control . . . . . . . . . . . . . . . . .4.3.2 Costate Propagation . . . . . . . . . .4.3.3 Final Costates . . . . . . . . . . . . . .4.3.4 Jump Conditions . . . . . . . . . . . .4.4 2-D Comparison, Geometric versus Analytical4.5 2-D Numerical . . . . . . . . . . . . . . . . .4.6 2-D Analytical, Alternate Method . . . . . . .4.7 3-D Analytical . . . . . . . . . . . . . . . . .4.7.1 3-D Optimal Unconstrained Control . .4.7.2 3-D Optimal Constrained Control . . .4.8 3-D Numerical Results . . . . . . . . . . . . .4.9 3-D Comparison, Analytical versus Numerical4.10 Numerical Comparison of Phase Breakpoints4.11 Summary of Analysis . . . . . . . . . . . . . .444445495051525357606265666971767983V. Conclusions, Contributions, and Future Work5.1 Conclusions . . . . . . . . . . . . . . . .5.2 Contributions . . . . . . . . . . . . . . .5.3 Recommendations for Future Research .85858686Appendix A. Common Aero Vehicle (CAV) Aerodynamics . . . . . . . . .A.1 CAV Aerodynamic Data . . . . . . . . . . . . . . . . . . . . . . .A.2 CAV Aerodynamic Model . . . . . . . . . . . . . . . . . . . . . .898989Appendix B. Nondimensionalization . . . . . . . . . . . . . . . . . . . . .B.1 Spherical Earth 3-D Reentry Equations of Motion - Rotating EarthB.2 Spherical Earth 3-D Reentry Equations of Motion - Non-RotatingEarth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .B.3 Flat Earth 3-D Reentry Equations of Motion - Non-Rotating EarthB.4 Nondimensionalization . . . . . . . . . . . . . . . . . . . . . . . .B.4.1 Equations of Motion . . . . . . . . . . . . . . . . . . . . .B.4.2 Path Constraint . . . . . . . . . . . . . . . . . . . . . . . .B.4.3 2-D Equations of Motion . . . . . . . . . . . . . . . . . . .9292viii.949496969999
PageAppendix C. Shortest Path . . . . . . . . . . . . . . . .C.1 Shortest Path between Two Points . . . . . . .C.2 Derive Midpoint as Optimal Waypoint PositionC.2.1 Proof Method 1 - Equations of Motion .C.2.2 Proof Method 2 - Dynamic Optimization.101101103104106Appendix D. Midpoint Algorithm . . . . . . . . . . . . . . . . . . . . . . .109Appendix E. Pseudospectral Method . . . . . . . . . . . . . . . . . . . . .115Appendix F. Software Implementation StrategiesF.1 Capture Jumps in Costates . . . . . . .F.2 Control Multipliers . . . . . . . . . . . .F.3 Events and Cost . . . . . . . . . . . . .F.4 Singularities and Results . . . . . . . . .F.5 Nodes and Iterations . . . . . . . . . . .F.6 Breakwell Example . . . . . . . . . . . .F.6.1 Analytical Derivation . . . . . . .F.6.2 GPOCS Implementation . . . . .119119119120120120121121122F.6.3 Matlabr Code . . . . . . . . . . . . . . . . . . . . . . . .125Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .132Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .145ix.
List of FiguresFigurePage1.Shuttle Entry - Operational Angle-of-Attack Profile. [39] . . . .92.X-33 - Ground Track for Six Analyzed Entries [81] . . . . . . .103.Fuzzy Logic - Simulation of 4 Waypoints Trajectory [3] . . . . .114.Riccati Equation - Time Optimal Trajectory of the HorizontalFlight with Constant Speed [119] . . . . . . . . . . . . . . . . .125.Threats - Ground Track for Horizontal-Plane Optimization [108]146.Voronoi Diagram [49] . . . . . . . . . . . . . . . . . . . . . . .157.Barrier - Terrain with Threats Formulated as an ExponentialFunction. On the Right is an Overhead View [100]. . . . . . . .8.Interior Point - Projection onto yz and xy Plane of the AircraftTrajectories when No-Fly Zone is Modeled as an Ellipse [70] . .9.1516Receding Horizon - Representation of a Cost-to-Go Function [53;74] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1710.A Search-Optimal Flight Trajectory in Stationary Obstacles [120]1711.Computation of the Continuous Lagrange Multipliers . . . . . .2012.Flight Path Angle Defined [41] . . . . . . . . . . . . . . . . . .3613.Baseline Trajectories: Constant Speed and Decelerating Flight4514.Turn Radius Orientation Rotated about Intermediate Waypoint(Wp ) by Angle χ. Radius (R) is constant. Endpoints (P1 andP2 ), and Intermediate Waypoint (Wp ) are fixed. . . . . . . . . .15.16.17.18.47Baseline and Geometric Trajectories: Constant Speed and Decelerating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .48Baseline, Geometric, and Analytical Dynamic Optimization Trajectories; both Constant Speed and Decelerating . . . . . . . .57Nondimensional Costates, Control, and Hamiltonian Time Histories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .59Analytical (Bryson’s method) versus Numerical Results . . . .61x
FigurePage19.Costates: Analytical (Bryson’s method) versus Numerical Results6220.Analytical (Alternate Method) versus Numerical Results . . . .6421.Numerical and Analytical Path Multiplier, µ . . . . . . . . . .6522.Control Multiplier µ for Constrained Bank Angle (σ 60 ) . .7023.Map: Seven Phase Numerical Results . . . . . . . . . . . . . .7124.States: Seven Phase Numerical Results . . . . . . . . . . . . .7225.Costates: Seven Phase Numerical Results . . . . . . . . . . . .7426.Controls: Seven Phase Numerical Results . . . . . . . . . . . .7527.Path Constraints: Seven Phase Numerical Results . . . . . . .7528.Path Constraints Expanded: Seven Phase Numerical Results .7629.Map: Seven Phase Numerical and Analytical . . . . . . . . . .7730.States: Seven Phase Numerical and Analytical . . . . . . . . .7731.Controls: Seven Phase Numerical and Analytical . . . . . . . .7832.Path Constraints: Seven Phase Numerical and Analytical . . .7833.Path Constraints Expanded: Seven Phase Numerical and Analytical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7934.Map: Number of Phases Comparison . . . . . . . . . . . . . . .7935.States: Number of Phases Comparison . . . . . . . . . . . . . .8036.Costates: Number of Phases Comparison . . . . . . . . . . . .8137.Controls: Number of Phases Comparison . . . . . . . . . . . .8238.Path Constraints: Number of Phases Comparison . . . . . . . .8239.Path Constraints Expanded: Number of Phases Comparison . .83A.1.CAV-H Aerodynamic Data and Model . . . . . . . . . . . . . .91A.2.CAV-H Nondimensional Variable c . . . . . . . . . . . . . . .91B.1.Fight Path Angle [41] . . . . . . . . . . . . . . . . . . . . . . .93B.2.Schematic of the r-V plane [41] . . . . . . . . . . . . . . . . . .93C.1.Two Point Turn from Initial Coordinate and Heading . . . . .102xi
FigureC.2.PageA constant speed trajectory from an initial point at a headingθ0 , passing though an intermediate waypoint, and completing theturn at a heading θf to intercept a final point. . . . . . . . . .104D.1.Nomenclature for Iterative Solution . . . . . . . . . . . . . . .109D.2.Initial Leg of Trajectory . . . . . . . . . . . . . . . . . . . . . .110E.1.Comparison of Legendre and Chebyshev Polynomials [31] . . .115E.2.Runge Phenomenon is Avoided Using Chebyshev Points . . . .116E.3.Chebyshev Points Projected onto the x-axis [99] . . . . . . . .117E.4.Derivative Computed Using Chebyshev Differentiation Matrices118E.5.Computation of the Continuous Lagrange Multipliers . . . . . .118F.1.Breakwell Problem: Differences in Bryson and GPOCS Results123F.2.Breakwell Problem: Matching Bryson and GPOCS Results . .124xii
List of TablesTablePage1.HCV Mission Description . . . . . . . . . . . . . . . . . . . . .292.CAV Mission Description . . . . . . . . . . . . . . . . . . . . .353.Numerical Results for Each Trajectory 9,000 nmi in 2 hrs at100,000 ft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .604.Nondimensional Times for Turns and Target Arrival . . . . . .60A.1.CAV-H Aero Data Base . . . . . . . . . . . . . . . . . . . . . .89xiii
List of SymbolsSymbolPaget0initial time . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4iwaypoint number . . . . . . . . . . . . . . . . . . . . . . . . . .4tipassage time of waypoint i . . . . . . . . . . . . . . . . . . . .4jno-fly zone number . . . . . . . . . . . . . . . . . . . . . . . . .4tjcontact time of no-fly zone j . . . . . . . . . . . . . . . . . . .4tffinal time, terminal time, or time-on-target . . . . . . . . . . .4x(t)state vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . .21u(t)control vector . . . . . . . . . . . . . . . . . . . . . . . . . . . .21ttime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .21nnumber of states . . . . . . . . . . . . . . . . . . . . . . . . . .21mnumber of controls . . . . . . . . . . . . . . . . . . . . . . . . .21ẋ(t)state vector derivative . . . . . . . . . . . . . . . . . . . . . . .21fstate vector derivative function . . . . . . . . . . . . . . . . . .21Jcost function . . . . . . . . . . . . . . . . . . . . . . . . . . . .22φcost at a discrete time . . . . . . . . . . . . . . . . . . . . . . .22tda discrete time . . . . . . . . . . . . . . . . . . . . . . . . . . .22xffinal state vector . . . . . . . . . . . . . . . . . . . . . . . . . .22Lintegrand cost . . . . . . . . . . . . . . . . . . . . . . . . . . .22ψterminal constraint . . . . . . . . . . . . . . . . . . . . . . . . .22Cpath constraint . . . . . . . . . . . . . . . . . . . . . . . . . . .22Sstate inequality constraint function . . . . . . . . . . . . . . . .22νterminal constraint Lagrange multiplier . . . . . . . . . . . . .23λcostate Lagrange multiplier . . . . . . . . . . . . . . . . . . . .23µpath constraint Lagrange multiplier . . . . . . . . . . . . . . .23J adjoined cost function . . . . . . . . . . . . . . . . . . . . . . .23xiv
SymbolPageHHamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . .23Φfunction of terminal cost and terminal constraint . . . . . . . .23Ninterior-point constraint . . . . . . . . . . . . . . . . . . . . . .25πnpath constraint multiplier . . . . . . . . . . . . . . . . . . . . .25S (q)q th time derivative . . . . . . . . . . . . . . . . . . . . . . . . .25Vvelocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .30θheading angle . . . . . . . . . . . . . . . . . . . . . . . . . . . .30unormalized bank angle . . . . . . . . . . . . . . . . . . . . . . .30σmaxmaximum bank angle . . . . . . . . . . . . . . . . . . . . . . .30aacceleration constant. . . . . . . . . . . . . . . . . . . . . . .31[xf , yf ] specified final coordinate . . . . . . . . . . . . . . . . . . . . .31[xcj , ycj ] center of no-fly zone j . . . . . . . . . . . . . . . . . . . . . . .32Rjradius of no-fly zone j . . . . . . . . . . . . . . . . . . . . . . .32 xjx-displacement from no-fly zone j . . . . . . . . . . . . . . . .32 yjy-displacement from no-fly zone j. . . . . . . . . . . . . . . .32jenduser specified finite number of no-fly zones . . . . . . . . . . . .32[xi , yi ]waypoint coordinates . . . . . . . . . . . . . . . . . . . . . . .32ienduser specified finite number of waypoints . . . . . . . . . . . . .32Mno-fly zone interior-point constraint . . . . . . . . . . . . . . .33haltitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .36γflight path angle . . . . . . . . . . . . . . . . . . . . . . . . . .36CLcoefficient of lift . . . . . . . . . . . . . . . . . . . . . . . . . .36CDcoefficient of drag . . . . . . . . . . . . . . . . . . . . . . . . .36βatmospheric constant . . . . . . . . . . . . . . . . . . . . . . .37Bvehicle/mission specific constant . . . . . . . . . . . . . . . . .
cal dynamic optimization technique, and a rapidly emerging collocation numerical ap-proach. This up-and-coming numerical technique is a direct solution method involving discretization then dualization, with pseudospectral methods and nonlinear program-ming used to converge to the optimal solution. This numerical approach is applied to
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