Optimal Configuration of Tetrahedral Spacecraft Formations 1

The problem of determining minimum-fuel maneuver sequences for a four-spacecraft formation is considered. The objective of this paper is to find fuel-optimal spacecraft trajectories that transfer four spacecraft from an initial parking orbit to a desired terminal reference orbit while satisfying a set of constraints on the formation at the terminal time. Trajectories involving both one and two allowable maneuvers per spacecraft are considered. The resulting nonlinear optimal control problem is solved numerically using a recently developed direct transcription method called the Gauss pseudospectral method. The results presented in this paper highlight interesting features of the fuel-optimal formation and control. Furthermore, by showing that the discretized first-order optimality conditions from an indirect formulation are satisfied, a post-optimality analysis of the results demonstrates the accuracy and usefulness of the Gauss pseudospectral method.

[1]  Anil V. Rao,et al.  Direct Trajectory Optimization and Costate Estimation via an Orthogonal Collocation Method , 2006 .

[2]  Mark Campbell,et al.  Mission Objectives in Tetrahedral Formation Maneuvering , 2005 .

[3]  Ella M. Atkins,et al.  Spacecraft Formation Optimization with a Multi-Impulse Design , 2005 .

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

[5]  A. Rao,et al.  POST-OPTIMALITY EVALUATION AND ANALYSIS OF A FORMATION FLYING PROBLEM VIA A GAUSS PSEUDOSPECTRAL METHOD , 2005 .

[6]  F.Y. Hadaegh,et al.  A survey of spacecraft formation flying guidance and control. Part II: control , 2004, Proceedings of the 2004 American Control Conference.

[7]  I. Michael Ross,et al.  Pseudospectral Knotting Methods for Solving Optimal Control Problems , 2004 .

[8]  L.M. Mailhe,et al.  Initialization and resizing of formation flying using global and local optimization methods , 2004, 2004 IEEE Aerospace Conference Proceedings (IEEE Cat. No.04TH8720).

[9]  Yuichi Tsuda.  GLOBAL OPTIMIZATION OF MANEUVER SCHEDULE FOR MULTIPLE SPACECRAFTS FLYING IN FORMATION , 2004 .

[10]  Anil V. Rao,et al.  EXTENSION OF A PSEUDOSPECTRAL LEGENDRE METHOD TO NON-SEQUENTIAL MULTIPLE-PHASE OPTIMAL CONTROL PROBLEMS , 2003 .

[11]  S. Ploen,et al.  A survey of spacecraft formation flying guidance and control (part 1): guidance , 2003, Proceedings of the 2003 American Control Conference, 2003..

[12]  Steven P. Hughes,et al.  Formation Tetrahedron Design for Phase I of the Magnetospheric Multiscale Mission , 2003 .

[13]  I. Michael Ross,et al.  Legendre Pseudospectral Approximations of Optimal Control Problems , 2003 .

[14]  I. Michael Ross,et al.  A Direct Method for Solving Nonsmooth Optimal Control Problems , 2002 .

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

[16]  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).

[17]  I. Michael Ross,et al.  A Spectral Patching Method for Direct Trajectory Optimization , 2000 .

[18]  L. Trefethen.  Spectral Methods in MATLAB , 2000 .

[19]  Gamal N. Elnagar,et al.  Pseudospectral Chebyshev Optimal Control of Constrained Nonlinear Dynamical Systems , 1998, Comput. Optim. Appl..

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

[21]  Gamal N. Elnagar,et al.  Pseudospectral Legendre-based optimal computation of nonlinear constrained variational problems , 1998 .

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

[23]  Bengt Fornberg,et al.  A practical guide to pseudospectral methods: Introduction , 1996 .

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

[25]  J. E. Cuthrell,et al.  Simultaneous optimization and solution methods for batch reactor control profiles , 1989 .

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

[27]  J. E. Cuthrell,et al.  On the optimization of differential-algebraic process systems , 1987 .

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

[29]  C. R. Hargraves,et al.  DIRECT TRAJECTORY OPTIMIZATION USING NONLINEAR PROGRAMMING AND COLLOCATION , 1986 .

[30]  G. Reddien.  Collocation at Gauss Points as a Discretization in Optimal Control , 1979 .

[31]  Donald E. Kirk,et al.  Optimal control theory : an introduction , 1970 .