POST-OPTIMALITY EVALUATION AND ANALYSIS OF A FORMATION FLYING PROBLEM VIA A GAUSS PSEUDOSPECTRAL METHOD

The post-optimality analysis of a tetrahedral formation flying optimal control problem is considered. In particular, this four-spacecraft orbit insertion problem is transcribed to a nonlinear programming problem (NLP) using a direct method called the Gauss pseudospectral method. The Karush-Kuhn-Tucker (KKT) conditions for this NLP are then derived and are compared to the conditions obtained via a Gauss pseudospectral discretization of the Hamiltonian boundary-value problem (HBVP) that arises from applying the calculus of variations. It is found that the optimal control obtained by solving the NLP is in excellent agreement with the control obtained from the Gauss pseudospectral discretization of the HBVP. The results obtained in this paper demonstrate the accuracy of the Gauss pseudospectral method and illustrate the usefulness of the Gauss pseu-dospectral method as a means of gaining insight into the structure of optimally controlled systems.

[1]  J. Betts,et al.  A sparse nonlinear optimization algorithm , 1994 .

[2]  I. Michael Ross,et al.  Direct trajectory optimization by a Chebyshev pseudospectral method , 2000, Proceedings of the 2000 American Control Conference. ACC (IEEE Cat. No.00CH36334).

[3]  A. Rao,et al.  AAS 05-103 OPTIMAL CONFIGURATION OF SPACECRAFT FORMATIONS VIA A GAUSS PSEUDOSPECTRAL METHOD , 2005 .

[4]  P. Williams.  Jacobi pseudospectral method for solving optimal control problems , 2004 .

[5]  I. Michael Ross,et al.  DESIGNING OPTIMAL SPACECRAFT FORMATIONS , 2002 .

[6]  Michael A. Saunders,et al.  User’s Guide For Snopt Version 6, A Fortran Package for Large-Scale Nonlinear Programming∗ , 2002 .

[7]  I. Michael Ross,et al.  Costate Estimation by a Legendre Pseudospectral Method , 1998 .

[8]  I. Michael Ross.  Hybrid Optimal Control Framework for Mission Planning , 2005, Journal of Guidance, Control, and Dynamics.

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

[10]  R. Battin.  An introduction to the mathematics and methods of astrodynamics , 1987 .

[11]  Anil V. Rao,et al.  Practical Methods for Optimal Control Using Nonlinear Programming , 1987 .

[12]  Emanuel Todorov,et al.  Optimal Control Theory , 2006 .

[13]  J. Betts.  Survey of Numerical Methods for Trajectory Optimization , 1998 .

[14]  David G. Hull,et al.  Optimal Control Theory for Applications , 2003 .

[15]  Michael A. Saunders,et al.  SNOPT: An SQP Algorithm for Large-Scale Constrained Optimization , 2002, SIAM J. Optim..

[16]  Gamal N. Elnagar,et al.  The pseudospectral Legendre method for discretizing optimal control problems , 1995, IEEE Trans. Autom. Control..

[17]  David Benson,et al.  A Gauss pseudospectral transcription for optimal control , 2005 .

[18]  Stuart A. Stanton,et al.  Optimal Orbital Transfer Using a Legendre Pseudospectral Method , 2003 .

[19]  C. Hargraves,et al.  DIRECT TRAJECTORY OPTIMIZATION USING NONLINEAR PROGRAMMING AND COLLOCATION , 1987 .

[20]  T. A. Zang,et al.  Spectral methods for fluid dynamics , 1987 .