A multiple-shooting differential dynamic programming algorithm. Part 2: Applications

Abstract A Multiple-Shooting Differential Dynamic Programming Algorithm is applied to a variety of constrained nonlinear optimal control problems, including classic benchmark problems, as well as a robotic arm problem and sensitive spacecraft trajectory optimization problems. The multiple-shooting extension presented in Part 1 is validated, as well as the Powell, Hestenes, and Rockafellar approach used in the treatment of equality and inequality path and terminal constraints. The results for example applications demonstrate the applicability of the algorithm to a variety of trajectory optimization problems. The advantages of the multiple-shooting approach over the single-shooting algorithm is evident in problems with high sensitivity. In particular, convergence of the multiple-shooting algorithm is demonstrated for complex spacecraft trajectory problems that are intractable using the single-shooting formulation.

[1]  H. Bock,et al.  A Multiple Shooting Algorithm for Direct Solution of Optimal Control Problems , 1984 .

[2]  Joaquim R. R. A. Martins,et al.  The complex-step derivative approximation , 2003, TOMS.

[3]  R. T. Bui,et al.  A general structure of specific optimal control , 1993 .

[4]  R. Russell,et al.  On the Computation and Accuracy of Trajectory State Transition Matrices , 2016 .

[5]  G. Lantoine,et al.  A Hybrid Differential Dynamic Programming Algorithm for Robust Low-Thrust Optimization , 2008 .

[6]  Emiliano Cristiani,et al.  Initialization of the Shooting Method via the Hamilton-Jacobi-Bellman Approach , 2009, 0910.0521.

[7]  Nicholas I. M. Gould,et al.  Convergence Properties of an Augmented Lagrangian Algorithm for Optimization with a Combination of General Equality and Linear Constraints , 1996, SIAM J. Optim..

[8]  R. Rockafellar.  The multiplier method of Hestenes and Powell applied to convex programming , 1973 .

[9]  M. Hestenes.  Multiplier and gradient methods , 1969 .

[10]  R. Broucke,et al.  Periodic orbits in the restricted three body problem with earth-moon masses , 1968 .

[11]  Ryan P. Russell,et al.  A Hybrid Differential Dynamic Programming Algorithm for Constrained Optimal Control Problems. Part 1: Theory , 2012, Journal of Optimization Theory and Applications.

[12]  R. Tyrrell Rockafellar,et al.  A dual approach to solving nonlinear programming problems by unconstrained optimization , 1973, Math. Program..

[13]  Thierry Dargent,et al.  Using Multicomplex Variables for Automatic Computation of High-Order Derivatives , 2010, TOMS.

[14]  M. J. D. Powell,et al.  On search directions for minimization algorithms , 1973, Math. Program..

[15]  Etienne Pellegrini,et al.  A multiple-shooting differential dynamic programming algorithm. Part 1: Theory , 2020 .

[16]  M. J. D. Powell,et al.  A method for nonlinear constraints in minimization problems , 1969 .

[17]  Ryan P. Russell,et al.  A database of planar axisymmetric periodic orbits for the Solar system , 2018, Celestial Mechanics and Dynamical Astronomy.

[18]  Ryan P. Russell,et al.  A Hybrid Differential Dynamic Programming Algorithm for Constrained Optimal Control Problems. Part 2: Application , 2012, Journal of Optimization Theory and Applications.

[19]  Todd Munson,et al.  Benchmarking optimization software with COPS. , 2001 .

[20]  M. J. D. Powell,et al.  Algorithms for nonlinear constraints that use lagrangian functions , 1978, Math. Program..

[21]  George M. Siouris,et al.  Applied Optimal Control: Optimization, Estimation, and Control , 1979, IEEE Transactions on Systems, Man, and Cybernetics.