Abstract
One of the most effective numerical techniques for the solution of trajectory optimization and optimal control problems is the direct transcription method. This approach combines a nonlinear programming algorithm with a discretization of the trajectory dynamics. The resulting mathematical programming problem is characterized by matrices which are large and sparse. Constraints on the path of the trajectory are then treated as algebraic inequalities to be satisfied by the nonlinear program. This paper describes a nonlinear programming algorithm which exploits the matrix sparsity produced by the transcription formulation. Numerical experience is reported for trajectories with both state and control variable equality and inequality path constraints.
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References
Ashcraft, C.C., and Grimes, R.G. (1988). ‘The Influence of Relaxed Supernode Partitions on the Multifrontal Method,’ Boeing Computer Services Technical Report ETA-TR-60.
Ascher, U., Mattheij, R., and Russell, R.D. (1988). Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, Prentice Hall, Englewood Cliffs, N.J.
Betts, J.T. (1990). ‘Sparse Jacobian Updates in the Collocation Method for Optimal Control Problems,’ Journal of Guidance, Control, and Dynamics, Vol. 13, No. 3, May-June.
Betts, J.T., and Huffman, W.P. (1991a). ‘Trajectory Optimization on a Parallel Processor,’ Journal of Guidance, Control, and Dynamics, Vol. 14, No. 2, March-April.
Betts, J.T., and Huffman, W.P. (1991b). ‘Path Constrained Trajectory Optimization Using Sparse Sequential Quadratic Programming,’ AIAA-91–2739-CP, pp 1236–1259, Proceedings of the AIAA Guidance, Navigation, and Control Conference, New Orleans, LA.
Betts, J.T., and Huffman, W.P. (1992). ‘Application of Sparse Nonlinear Programming to Trajectory Optimization,’ Journal of Guidance, Control, and Dynamics, Vol 15, No. 1, January-February.
Bryson, A.E., Desai, M.N., and Hoffman, W.C., (1969)]. ‘Energy-State Approximation in Performance Optimization of Supersonic Aircraft,’ Journal of Aircraft, Vol. 6, No. 6, Nov-Dec.
Bulirsch, R., Montrone, F., and Pesch, H.J. (1991). ‘Abort Landing in the Presence of Windshear as a Minimax Optimal Control Problem, Part 2: Multiple Shooting and Homotopy,’ Journal of Optimization Theory and Applications, Vol 70, No. 2, pp 223–254, August.
Dickmanns, E.D., (1980). ‘Efficient Convergence and Mesh Refinement Strategies for Solving General Ordinary Two-Point Boundary Value Problems by Collocated Hermite Approximation,’ 2nd IFAC Workshop on Optimisation, Oberpfaffenhofen, Sept. 15–17.
Enright, P.J. (1991). ‘Optimal Finite-Thrust Spacecraft Trajectories Using Direct Transcription and Nonlinear Programming,’ Ph.D. Thesis, University of Illinois.
Gill, P.E., Murray, W., and Wright, M.H., (1981). Practical Optimization, Academic Press.
Gill, P.E., Murray, W., Saunders, M.A., and Wright, M.H., (1986a). ‘User’s Guide for NPSOL (Version 4.0): a Fortran package for nonlinear programming,’ Report SOL 86–2, Department of Operations Research, Stanford University.
Gill, P.E., Murray, W., Saunders, M.A., and Wright, M.H., (1986b). ‘Some Theoretical Properties of an Augmented Lagrangian Merit Function’, Report SOL 86–6, Department of Operations Research, Stanford University.
Gill, P.E., Murray, W., Saunders, M.A., and Wright, M.H., (1987). ‘A Schur-Complement Method for Sparse Quadratic Programming,’ Report SOL 87–12, Department of Operations Research, Stanford University.
Hargraves, C.R., and S.W. Paris, (1987). ‘Direct Trajectory Optimization Using Nonlinear Programming and Collocation,’ J. of Guidance, Control, and Dynamics, Vol. 10, No.4, July-Aug, p338.
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© 1993 Birkhäuser Verlag
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Betts, J.T. (1993). Trajectory Optimization Using Sparse Sequential Quadratic Programming. In: Bulirsch, R., Miele, A., Stoer, J., Well, K. (eds) Optimal Control. ISNM International Series of Numerical Mathematics, vol 111. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-7539-4_9
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DOI: https://doi.org/10.1007/978-3-0348-7539-4_9
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-0348-7541-7
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