A Discrete Geometric Optimal Control Framework for Systems with Symmetries

This paper studies the optimal motion control of mechanical systems through a discrete geometric approach. At the core of our formulation is a discrete Lagrange-d’Alembert- Pontryagin variational principle, from which are derived discrete equations of motion that serve as constraints in our optimization framework. We apply this discrete mechanical approach to holonomic systems with symmetries and, as a result, geometric structure and motion invariants are preserved. We illustrate our method by computing optimal trajectories for a simple model of an air vehicle flying through a digital terrain elevation map, and point out some of the numerical benefits that ensue.

[1]  R. Stephenson.  A and V , 1962, The British journal of ophthalmology.

[2]  J. Marsden,et al.  Introduction to mechanics and symmetry , 1994 .

[3]  Richard M. Murray,et al.  Geometric phases and robotic locomotion , 1995, J. Field Robotics.

[4]  J. Marsden,et al.  Discrete Euler-Poincaré and Lie-Poisson equations , 1999, math/9909099.

[5]  Vijay Kumar,et al.  Motion planning for cooperating mobile manipulators , 1999, J. Field Robotics.

[6]  James P. Ostrowski.  Computing reduced equations for robotic systems with constraints and symmetries , 1999, IEEE Trans. Robotics Autom..

[7]  Vijay Kumar,et al.  Optimal Gait Selection for Nonholonomic Locomotion Systems , 2000, Int. J. Robotics Res..

[8]  Mark B. Milam,et al.  A new computational approach to real-time trajectory generation for constrained mechanical systems , 2000, Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187).

[9]  Sonia Martínez,et al.  Optimal Gaits for Dynamic Robotic Locomotion , 2001, Int. J. Robotics Res..

[10]  J. Marsden,et al.  Discrete mechanics and variational integrators , 2001, Acta Numerica.

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

[12]  E. Celledoni,et al.  Lie group methods for rigid body dynamics and time integration on manifolds , 2003 .

[13]  R. Olfati-Saber,et al.  Collision avoidance for multiple agent systems , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[14]  Jerrold E. Marsden,et al.  Nonsmooth Lagrangian Mechanics and Variational Collision Integrators , 2003, SIAM J. Appl. Dyn. Syst..

[15]  A. D. Lewis,et al.  Geometric Control of Mechanical Systems , 2004, IEEE Transactions on Automatic Control.

[16]  Munther A. Dahleh,et al.  Maneuver-based motion planning for nonlinear systems with symmetries , 2005, IEEE Transactions on Robotics.

[17]  J.E. Marsden,et al.  Optimal Motion of an Articulated Body in a Perfect Fluid , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[18]  J. Marsden,et al.  Discrete mechanics and optimal control , 2005 .

[19]  S.D. Ross,et al.  Optimal flapping strokes for self-propulsion in a perfect fluid , 2006, 2006 American Control Conference.

[20]  Taeyoung Lee,et al.  Optimal Control of a Rigid Body using Geometrically Exact Computations on SE(3) , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[21]  B. Mettler,et al.  Nonlinear trajectory generation for autonomous vehicles via parameterized maneuver classes , 2006 .

[22]  Steven M. LaValle,et al.  Planning algorithms , 2006 .

[23]  Howie Choset,et al.  Motion Planning for Dynamic Variable Inertia Mechanical Systems with Non-holonomic Constraints , 2006 .

[24]  Jerrold E. Marsden,et al.  Geometric, variational integrators for computer animation , 2006, SCA '06.

[25]  J. Marsden,et al.  Dirac structures in Lagrangian mechanics Part II: Variational structures , 2006 .

[26]  E. Hairer,et al.  Geometric Numerical Integration , 2022, Oberwolfach Reports.

[27]  Sina Ober-Blöbaum,et al.  Optimal Reconfiguration of Formation Flying Spacecraft ---a Decentralized Approach , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[28]  Gaurav S. Sukhatme,et al.  Optimal Control Using Nonholonomic Integrators , 2007, Proceedings 2007 IEEE International Conference on Robotics and Automation.