Rapid Smooth Entry Trajectory Planning for High Lift/Drag Hypersonic Glide Vehicles

This paper presents how to apply second-order cone programming, a subclass of convex optimization, to rapidly solve a highly nonlinear optimal control problem arisen from smooth entry trajectory planning of hypersonic glide vehicles with high lift/drag ratios. The common phugoid oscillations are eliminated by designing a smooth flight path angle profile. The nonconvexity terms of the optimal control problem, which include the nonlinear dynamics and nonconvex control constraints, are convexified via techniques of successive linearization, successive approximation, and relaxation. Lossless relaxation is also proved using optimal control theory. After discretization, the original nonconvex optimal control problem is converted into a sequence of second-order cone programming problems each of which can be solved in polynomial time using existing primal–dual interior-point algorithms whenever a feasible solution exists. Numerical examples are provided to show that rather smooth entry trajectory can be obtained in about 1 s on a desktop computer.

[1]  Jing Zhang,et al.  Automated Custom Code Generation for Embedded, Real-time Second Order Cone Programming , 2014 .

[2]  Amitabh Saraf,et al.  Design and Evaluation of an Acceleration Guidance Algorithm for Entry , 2003 .

[3]  Behcet Acikmese,et al.  A model predictive control technique with guaranteed resolvability and required thruster silent times for small-body proximity operations , 2006 .

[4]  David B. Doman,et al.  Optimal Trajectory Reconfiguration and Retargeting for Reusable Launch Vehicles , 2005 .

[5]  Ping Lu,et al.  Autonomous Trajectory Planning for Rendezvous and Proximity Operations by Conic Optimization , 2012 .

[6]  Stephen P. Boyd,et al.  ECOS: An SOCP solver for embedded systems , 2013, 2013 European Control Conference (ECC).

[7]  Erling D. Andersen,et al.  On implementing a primal-dual interior-point method for conic quadratic optimization , 2003, Math. Program..

[8]  Stephen P. Boyd,et al.  Code generation for embedded second-order cone programming , 2013, 2013 European Control Conference (ECC).

[9]  Michael J. Todd,et al.  Self-Scaled Barriers and Interior-Point Methods for Convex Programming , 1997, Math. Oper. Res..

[10]  I. Michael Ross,et al.  Rapid Verification Method for the Trajectory Optimization of Reentry Vehicles , 2003 .

[11]  Behçet Açikmese,et al.  Lossless Convexification of Nonconvex Control Bound and Pointing Constraints of the Soft Landing Optimal Control Problem , 2013, IEEE Transactions on Control Systems Technology.

[12]  A. Roenneke,et al.  Re-entry control to a drag-vs-energy profile , 1994 .

[13]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[14]  N. Vinh.  Optimal trajectories in atmospheric flight , 1981 .

[15]  Richard G. Cobb,et al.  Multiple Method 2-D Trajectory Optimization Satisfying Waypoints and No-Fly Zone Constraints , 2008 .

[16]  Ping Lu,et al.  Entry Guidance: A Unified Method , 2014 .

[17]  J. Lofberg,et al.  YALMIP : a toolbox for modeling and optimization in MATLAB , 2004, 2004 IEEE International Conference on Robotics and Automation (IEEE Cat. No.04CH37508).

[18]  Andrew E. Johnson,et al.  Flight testing of trajectories computed by G-FOLD: Fuel optimal large divert guidance algorithm for planetary landing , 2013 .

[19]  Robert E. Jones,et al.  Test Results for Entry Guidance Methods for Space Vehicles , 2004 .

[20]  Suresh P. Sethi,et al.  A Survey of the Maximum Principles for Optimal Control Problems with State Constraints , 1995, SIAM Rev..

[21]  Andrew E. Johnson,et al.  ADAPT demonstrations of onboard large-divert Guidance with a VTVL rocket , 2014, 2014 IEEE Aerospace Conference.

[22]  David B. Doman,et al.  Fault Tolerant Optimal Trajectory Generation for Reusable Launch Vehicles (Preprint) , 2006 .

[23]  Kevin P. Bollino.  High-fidelity real-time trajectory optimization for reusable launch vehicles , 2006 .

[24]  Behcet Acikmese,et al.  Convex programming approach to powered descent guidance for mars landing , 2007 .

[25]  Kenneth D. Mease,et al.  Reduced-Order Entry Trajectory Planning for Acceleration Guidance , 2002 .

[26]  P. Lu,et al.  Entry Trajectory Optimization by Second-Order Cone Programming , 2016 .

[27]  A. Rao,et al.  Performance Optimization of a Maneuvering Re-Entry Vehicle Using a Legendre Pseudospectral Method , 2002 .

[28]  Behçet Açikmese,et al.  Maximum Divert for Planetary Landing Using Convex Optimization , 2014, J. Optim. Theory Appl..

[29]  Ping Lu,et al.  Onboard Generation of Three-Dimensional Constrained Entry Trajectories , 2002 .

[30]  J. C. Harpold,et al.  Shuttle entry guidance , 1978 .

[31]  Xinfu Liu,et al.  Autonomous Trajectory Planning by Convex Optimization , 2013 .

[32]  Richard G. Cobb,et al.  Three-Dimensional Trajectory Optimization Satisfying Waypoint and No-Fly Zone Constraints , 2009 .

[33]  Kenneth D. Mease,et al.  Feasible Trajectory Generation for Atmospheric Entry Guidance , 2007 .

[34]  Ping Lu.  Entry Guidance and Trajectory Control for Reusable Launch Vehicle , 1997 .

[35]  Behçet Açikmese,et al.  Lossless convexification of a class of optimal control problems with non-convex control constraints , 2011, Autom..

[36]  Behcet Acikmese,et al.  Minimum-Landing-Error Powered-Descent Guidance for Mars Landing Using Convex Optimization , 2010 .

[37]  Behcet Acikmese,et al.  Minimum Time Rendezvous of Multiple Spacecraft Using Differential Drag , 2014 .

[38]  Ping Lu,et al.  Solving Nonconvex Optimal Control Problems by Convex Optimization , 2014 .

[39]  Nguyen X. Vinh,et al.  Phugoid oscillations in optimal reentry trajectories , 1981 .

[40]  W. Chen,et al.  Reentry Vehicle Constrained Trajectory Optimization , 2011 .

[41]  Ping Lu,et al.  Gliding Guidance of High L/D Hypersonic Vehicles , 2013 .

[42]  Curtis Zimmerman,et al.  An Automated Method to Compute Orbital Re-Entry Trajectories with Heating Constraints , 2013 .

[43]  Ping Lu,et al.  Robust Trajectory Optimization for Highly Constrained Rendezvous and Proximity Operations , 2013 .

[44]  Jos F. Sturm,et al.  Implementation of interior point methods for mixed semidefinite and second order cone optimization problems , 2002, Optim. Methods Softw..

[45]  Donald Goldfarb,et al.  Second-order cone programming , 2003, Math. Program..

[46]  Ping Lu,et al.  Entry Guidance for the X-33 Vehicle , 1998 .