Motion on lie groups and its applications in control theory. Discrete time optimal control applied to pest control problems. In this paper we study a discrete variational optimal control problem for the rigid body. Discrete geometric optimal control on lie groups 2011, m. We consider the optimal control of mechanical systems on lie groups and develop numerical methods that exploit the structure of the state space and preserve the system motion invariants. Discrete mechanics and variational integrators 2001, j. An overview of lie group variational integrators and their applications to optimal control international conference on scientific computation and differential equations, saintmalo, france, july 9, p.
Geometric structures, symmetry and elements of lie groups 3 similarities. Discrete time optimal control applied to pest control problems 481 the paper is organized as follows. Discrete optimal control of interconnected mechanical. Our approach is based on a coordinatefree variational discretization of the dynamics that leads to structurepreserving discrete equations of motion. Introduction real physical systems often admit complex con. Mostow notes by gopal prasad no part of this book may be reproduced in any form by print, micro. In mathematics, a discrete subgroup of a topological group g is a subgroup h such that there is an open cover of g in which every open subset contains exactly one element of h. Optimal control, pontryagins principle, control systems on manifolds, lie groups 1 pontryagins principle let us consider the simplest version of a controltheoretical problem. A key property of a lie group is that a curved space can be studied, using. A general theory is accompanied by concrete examples.
Lie group integrators for animation and control of vehicles. Five lectures on lattices in semisimple lie groups 5 b1 a12a21 0. Controllability and optimal control for leftinvariant problems on lie groups are addressed. Discrete shapes can be described and analyzed using lie groups, which are mathematical structures having both algebraic and geometrical properties. This paper is concerned with the animation and control of vehicles with complex dynamics such as helicopters, boats, and cars. Remsing rhodes university geometric optimal control palermo, 9 july 20 1 44. The goal of this paper is to derive a structure preserving integrator for geometrically exact beam dynamics, by using a lie group variational integrator. Discrete geometric optimal control on lie groups marin kobilarov and jerrold e.
We show how these representations generate extremals for optimal control problems. In keeping with omission of the transpose on vectors, u, x, p will be used for. Sukhatme usc abstractthis paper studies the optimal motion control of mechanical systems through a discrete geometric approach. Some important control systems in mechanics, physics, geometry etc. In this paper we establish necessary conditions for optimal control using the ideas of lagrangian reduction in the sense of reduction under a symmetry group. We construct necessary conditions for optimal trajectories. The author demonstrates an overlap with mathematical physics using the maximum principle, a fundamental concept of optimality arising from geometric control, which is applied to timeevolving systems governed by physics as well as to manmade systems governed by. Lie group integrators preserve symmetry and group structure for systems with. Inverse dynamics for discrete geometric mechanics of. Thepcompact groups seem to be the best available homotopical analogues of compact lie groups 10, 11, 12, but analytical objects like lie algebras are not available for them. Outline 1 invariant control systems introduction equivalences 2 optimal control and pontryagin maximum principle invariant optimal control problems costextended systems 3 quadratic hamiltonpoisson systems liepoisson structure examples 4 outlook rory biggs rhodes geometric control on lie groups march 29, 20 3 40. Weissobservation and control for operator semigroups h.
Quantum mechanics and quantum information theory 12. A discrete optimal control problem was formulated for the. To our knowledge the only sources that discuss versions of the pmp for discrete time geometric optimal control problems are kipka and gupta 2019 and phogat, chatterjee, and banavar 2018. If both a12 and a21 are equal to 0, then the product. These structures, borrowed from mathematical physics, are both algebraic groups and smooth manifolds. Optimal control, pontryagins principle, control systems on manifolds, lie groups 1 pontryagins principle let us consider the simplest version of a control theoretical problem. Discrete geometric optimal control on lie groups ieee. The resulting scheme preserves both the discrete momentum maps and symplectic structures, and exhibits almostperfect energy conservation. While the discrete optimal control method presented here is illustrated using the lie group so3 of rotation matrices, and its corresponding lie algebra so3 of skewsymmetric matrices, we have derived the method with su. Dynamical systems control systems reachable sets and controllability af. We consider the optimal control of mechanical systems on lie groups and develop numerical methods that exploit the structure of the. A simple proof of the discrete time geometric pontryagin. Discrete geometric optimal control of multibody systems marin kobilarov dept. Lie groups relies on the same ideas which, supported by additional machinery from homotopy theory, give structure theorems for pcompact groups.
In order to obtain the intrinsic geometric description, the. We can perform linear combinations and lie brackets. A discrete geometric optimal control framework for. A discrete geometric optimal control framework for systems with symmetries. Index termsgeometric control, differential dynamic programming, discrete optimal control, lie groups i. The benefit of such an approach is that it makes use of the special structure of the system, especially its symmetry structure, and thus. Multisymplectic lie group variational integrator for a. Dual quaternion variational integrator for rigid body dynamic simulation 2016, j. Optimal control for holonomic and nonholonomic mechanical. This paper is intended to be a selfcontained survey of the ongoing research on lie group variational integrators, and their applications to celestial and astrodynamics simulations, and geometric and optimal control. B has lie algebra b, b is a closed connected subgroup of g, and b is its own normalizer in g. Motivated by recent developments in discrete geometric mechanics, we develop a general framework for integrating the dynamics of holonomic and nonholonomic vehicles by preserving their statespace geometry and motion invariants. Introduction to geometric control theory controllability.
Our approach is based on a coordinatefree variational discretization of the dynamics that leads to. A discrete geometric optimal control framework for systems with symmetries marin kobilarov usc mathieu desbrun caltech jerrold e. Discrete geometric optimal control on lie groups core. We will construct these geometric integrators using discrete variational calculus on lie groups, deriving a discrete version of the secondorder eulerlagrange equations. This paper is devoted to a detailed analysis of the geodesic problem on matrix lie groups, with left invariant metric, by examining representations of embeddings of geodesic flows in suitable vector spaces. Discrete geometric optimal control on lie groups abstract. Complex geometry and representations of lie groups 205 subgroup b c g corresponding to a borel subalgebra b c g is defined to be the gnormalizer of b, that is, 1. Geometric integrators are numerical methods that preserve the geometric. Solving optimal control problems by exploiting inherent dynamical systems structures. Discrete geometric optimal control on lie groups ieee xplore. Discrete geometric optimal control on lie groups ieee journals. Tata institute of fundamental research, bombay 1969. Motivated by recent developments in discrete geometric mechanics we develop a general framework for integrating the dynamics of holonomic and nonholonomic vehicles by preserving their statespace geometry and motion invariants.
This paper considers leftinvariant control affine systems evolving on matrix lie groups. An introduction to optimal control problem the use of pontryagin maximum principle. Any leftinvariant optimal control problem with quadratic cost can be lifted, via the celebrated maximum. Embedded geodesic problems and optimal control for matrix. Sachkov, discrete symmetries in the generalized dido problem. At the core of our formulation is a discrete lagranged. Solution of discretetime optimal control problems on. Geometric structurepreserving optimal control of the. Our work extends the recently developed geometric lie group in tegrators. For example, the integers, z, form a discrete subgroup of the reals, r with the standard metric topology, but the rational numbers, q, do not. The geometry of optimal control solutions on some six. It can be shown, for example, that the set of rigid body transformations behaves as a differentiable manifold 1. Both spatial and temporal discretization are implemented in a geometry preserving manner. This paper presents the approach of computational geometric optimal control for the dynamics of rigid bodies on a lie group.
Computational geometric mechanics, control, and estimation. Oneparameter groups of spiral similarities and focus for linear ode. In this paper we will discuss some new developments in the design of numerical methods for optimal control problems of lagrangian systems on lie groups. The abstract formulation of discrete optimal control problems on lie groups is. A central point in our development is to perform a uni. Their combined citations are counted only for the first article.
Synthetic description preservation of ratio of distances on a line and linear algebraic. Insection 3, we formulate the optimal dual control prob. Discrete time lagrangian mechanics on lie groups, with an application to the lagrange top. Lecture notes of an introductory course on control theory on lie groups. Discretetime maximum principle on matrix lie groups arxiv. Discrete geometric optimal control of multibody systems. Computational geometric optimal control optimal control on lie groups sec.
The former establishes a pmp for a class of smooth control systems evolving on lie groups under mild structural assumptions on the system dynamics. Discrete geometric optimal control on lie groups article in ieee transactions on robotics 274. Trajectory planning for cubesat short timescale proximity operations. At the core of our formulation is a discrete lagrangedalembert. Symmetry, integrability and the hamiltonjacobi theory. Marsden abstractwe consider the optimal control of mechanical systems on lie groups and develop numerical methods which exploit the structure of the state space and preserve the system motion invariants. This paper studies optimal control of multibody systems by constructing numerical methods operating intrinsically in the state space manifold and exploiting its lie group structure. We construct a general optimization framework for systems on lie groups and demonstrate its application to rigid body motion groups as well as to any real matrix. The techniques developed here are designed for lagrangian mechanical control systems with symmetry. Discrete mechanics and optimal control dmoc, see 9. Invertibility of control systems on lie groups siam. Index termsdiscrete mechanics, geometric optimization, lie groups, optimal control, underactuated systems. This article is concerned with the animation and control of vehicles with complex dynamics such as helicopters, boats, and cars.
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