Optimal control problems on parallelizable Riemannian manifolds: theory and applications

Ram V. Iyer; Raymond Holsapple; David Doman

ESAIM: Control, Optimisation and Calculus of Variations (2005)

  • Volume: 12, Issue: 1, page 1-11
  • ISSN: 1292-8119

Abstract

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The motivation for this work is the real-time solution of a standard optimal control problem arising in robotics and aerospace applications. For example, the trajectory planning problem for air vehicles is naturally cast as an optimal control problem on the tangent bundle of the Lie Group SE(3), which is also a parallelizable Riemannian manifold. For an optimal control problem on the tangent bundle of such a manifold, we use frame co-ordinates and obtain first-order necessary conditions employing calculus of variations. The use of frame co-ordinates means that intrinsic quantities like the Levi-Civita connection and Riemannian curvature tensor appear in the equations for the co-states. The resulting equations are singularity-free and considerably simpler (from a numerical perspective) than those obtained using a local co-ordinates representation, and are thus better from a computational point of view. The first order necessary conditions result in a two point boundary value problem which we successfully solve by means of a Modified Simple Shooting Method.

How to cite

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Iyer, Ram V., Holsapple, Raymond, and Doman, David. "Optimal control problems on parallelizable Riemannian manifolds: theory and applications." ESAIM: Control, Optimisation and Calculus of Variations 12.1 (2005): 1-11. <http://eudml.org/doc/90788>.

@article{Iyer2005,
abstract = { The motivation for this work is the real-time solution of a standard optimal control problem arising in robotics and aerospace applications. For example, the trajectory planning problem for air vehicles is naturally cast as an optimal control problem on the tangent bundle of the Lie Group SE(3), which is also a parallelizable Riemannian manifold. For an optimal control problem on the tangent bundle of such a manifold, we use frame co-ordinates and obtain first-order necessary conditions employing calculus of variations. The use of frame co-ordinates means that intrinsic quantities like the Levi-Civita connection and Riemannian curvature tensor appear in the equations for the co-states. The resulting equations are singularity-free and considerably simpler (from a numerical perspective) than those obtained using a local co-ordinates representation, and are thus better from a computational point of view. The first order necessary conditions result in a two point boundary value problem which we successfully solve by means of a Modified Simple Shooting Method. },
author = {Iyer, Ram V., Holsapple, Raymond, Doman, David},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Regular optimal control; simple mechanical systems; calculus of variations; numerical solution; modified simple shooting method.; modified simple shooting method},
language = {eng},
month = {12},
number = {1},
pages = {1-11},
publisher = {EDP Sciences},
title = {Optimal control problems on parallelizable Riemannian manifolds: theory and applications},
url = {http://eudml.org/doc/90788},
volume = {12},
year = {2005},
}

TY - JOUR
AU - Iyer, Ram V.
AU - Holsapple, Raymond
AU - Doman, David
TI - Optimal control problems on parallelizable Riemannian manifolds: theory and applications
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2005/12//
PB - EDP Sciences
VL - 12
IS - 1
SP - 1
EP - 11
AB - The motivation for this work is the real-time solution of a standard optimal control problem arising in robotics and aerospace applications. For example, the trajectory planning problem for air vehicles is naturally cast as an optimal control problem on the tangent bundle of the Lie Group SE(3), which is also a parallelizable Riemannian manifold. For an optimal control problem on the tangent bundle of such a manifold, we use frame co-ordinates and obtain first-order necessary conditions employing calculus of variations. The use of frame co-ordinates means that intrinsic quantities like the Levi-Civita connection and Riemannian curvature tensor appear in the equations for the co-states. The resulting equations are singularity-free and considerably simpler (from a numerical perspective) than those obtained using a local co-ordinates representation, and are thus better from a computational point of view. The first order necessary conditions result in a two point boundary value problem which we successfully solve by means of a Modified Simple Shooting Method.
LA - eng
KW - Regular optimal control; simple mechanical systems; calculus of variations; numerical solution; modified simple shooting method.; modified simple shooting method
UR - http://eudml.org/doc/90788
ER -

References

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  1. J.T. Betts, Survey of numerical methods for trajectory optimization. Journal of Guidance, Control and Dynamics21 (1998) 193–207.  
  2. W.M. Boothby, An introduction to Differential Geometry and Riemannian Manifolds. Academic Press (1975).  
  3. P. Crouch, M. Camarinha and F. Silva Leite, Hamiltonian approach for a second order variational problem on a Riemannian manifold, in Proc. of CONTROLO'98, 3rd Portuguese Conference on Automatic Control (September 1998) 321–326.  
  4. P. Crouch, F. Silva Leite and M. Camarinha, Hamiltonian structure of generalized cubic polynomials, in Proc. of the IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control (2000) 13–18.  
  5. P. Crouch, F. Silva Liete and M. Camarinha, A second order Riemannian varational problem from a Hamiltonian perspective. Private Communication (2001).  
  6. T. Frankel, The Geometry of Physics: An Introduction. Cambridge University Press (1998).  
  7. R. Holsapple, R. Venkataraman and D. Doman, A modified simple shooting method for solving two point boundary value problems, in Proc. of the IEEE Aerospace Conference, Big Sky, MT (March 2003).  
  8. R. Holsapple, R. Venkataraman and D. Doman, A new, fast numerical method for solving two-point boundary value problems. J. Guidance Control Dyn.27 (2004) 301–303.  
  9. V. Jurdejevic, Geometric Control Theory. Cambridge Studies in Advanced Mathematics (1997).  
  10. P.S. Krishnaprasad, Optimal control and Poisson reduction. TR 93–87, Institute for Systems Research, University of Maryland, (1993).  
  11. A. Lewis, The geometry of the maximum principle for affine connection control systems. Preprint, available online at http://penelope.mast.queensu.ca/~andrew/cgibin/pslist.cgi?papers.db, 2000.  
  12. D.G. Luenberger, Optimization by Vector Space Methods. John Wiley and Sons (1969).  
  13. M.B. Milam, K. Mushambi and R.M. Murray, A new computational approach to real-time trajectory generation for constrained mechanical systems, in Proc. of 39th IEEE Conference on Decision and Control1 (2000) 845–851.  
  14. R.M. Murray, Z. Li and S.S. Sastry, A Mathematical Introduction to Robotic Manipulation. CRC Press (1994).  
  15. L. Noakes, G. Heinzinger and B. Paden, Cubic splines on curved spaces. IMA J. Math. Control Inform.6 (1989) 465–473.  
  16. H.J. Pesch, Real-time computation of feedback controls for constrained optimal control problems. Part 1: Neighbouring extremals. Optim. Control Appl. Methods10 (1989) 129–145.  
  17. J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, pp. 272–286; 502–535. Springer-Verlag, New York, second edition (1993).  
  18. H. Sussmann, An introduction to the coordinate-free maximum principle, in Geometry of Feedback and Optimal Control, B. Jakubczyk and W. Respondek Eds. Marcel Dekker, New York (1997) 463–557.  

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