Discrete Variational Optimal Control
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Discrete Variational Optimal Control Fernando Jiménez · Marin Kobilarov · David Martín de Diego
Received: 5 March 2012 / Accepted: 12 October 2012 / Published online: 13 December 2012 © Springer Science+Business Media New York 2012
Abstract This paper develops numerical methods for optimal control of mechanical systems in the Lagrangian setting. It extends the theory of discrete mechanics to enable the solutions of optimal control problems through the discretization of variational principles. The key point is to solve the optimal control problem as a variational integrator of a specially constructed higher dimensional system. The developed framework applies to systems on tangent bundles, Lie groups, and underactuated and nonholonomic systems with symmetries, and can approximate either smooth or discontinuous control inputs. The resulting methods inherit the preservation properties of variational integrators and result in numerically robust and easily implementable algorithms. Several theoretical examples and a practical one, the control of an underwater vehicle, illustrate the application of the proposed approach. Keywords Variational integrators · Optimal control · Lie group · Discontinuous control inputs · Nonholonomic systems · Reduced control system
Communicated by Melvin Leok. F. Jiménez · D. Martín de Diego () Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM, Campus de Cantoblanco, UAM, C/Nicolás Cabrera, 15 28049 Madrid, Spain e-mail: [email protected] F. Jiménez e-mail: [email protected] M. Kobilarov Johns Hopkins University, 117 Hackerman Hall, 3400 N. Charles Street, Baltimore, MD 21218, USA e-mail: [email protected]
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J Nonlinear Sci (2013) 23:393–426
Mathematics Subject Classification 70Q05 · 49J15 · 37M15 · 70H03 · 37J60
1 Introduction The goal of this paper is to develop, from a geometric point of view, numerical methods for optimal control of Lagrangian mechanical systems. Our approach employs the theory of discrete mechanics and variational integrators (Marsden and West 2001) to derive both an integrator for the dynamics and an optimal control algorithm in a unified manner. This is accomplished through the discretization of the Lagrange– d’Alembert variational principle on manifolds. An integrator for the mechanics is derived using a standard Lagrangian function and virtual work done by control forces, while control optimality conditions are derived using a special Lagrangian defined on a higher dimensional space which encodes the dynamics and a desired cost function. The resulting integration and optimization schemes are symplectic and respect the state-space structure and momentum evolution. These qualities are associated with favorable numerical properties which motivate the development of practical algorithms that can be applied to robotic or aerospace vehicles. The proposed framework is general and applies to unconstrained systems, as well as systems with symmetries, underactuation, and nonholonomic constraints. In particular, our construction is appropriate for con
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