Discrete Mechanics and Optimal Control

Junge, Oliver; Marsden, Jerrold E.; Ober‐Blöbaum, Sina

doi:10.3182/20050703-6-cz-1902.00745

Cited by 140 publications

(121 citation statements)

References 12 publications

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“…An optimal control problem based on discrete mechanics is studied in [8]. The discrete equations of motion and the boundary conditions are imposed as constraints, and the optimal control problem is solved by a general-purpose parameter optimization tool.…”

Section: So(3)mentioning

confidence: 99%

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Optimal Attitude Control of a Rigid Body Using Geometrically Exact Computations on SO(3)

2008

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An efficient and accurate computational approach is proposed for optimal attitude control of a rigid body. The problem is formulated directly as a discrete time optimization problem using a Lie group variational integrator. Discrete necessary conditions for optimality are derived, and an efficient computational approach is proposed to solve the resulting two point boundary value problem. The use of geometrically exact computations on SO(3) guarantees that this optimal control approach has excellent convergence properties even for highly nonlinear large angle attitude maneuvers. Numerical results are presented for attitude maneuvers of a 3D pendulum and a spacecraft in a circular orbit.

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Section: So(3)mentioning

confidence: 99%

“…In [8], an optimal control problem based on discrete mechanics is considered. The control inputs at each discrete step are considered as optimization parameters, and the discrete equations of motion and the boundary conditions are imposed as constraints.…”

Section: Problem Formulationmentioning

confidence: 99%

Optimal Attitude Control of a Rigid Body Using Geometrically Exact Computations on SO(3)

2008

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“…Thereby, the state and control variables are discretized directly in order to transform the optimal control problem into a finite-dimensional nonlinear constrained optimization problem that can be solved by standard nonlinear optimization techniques such as sequential quadratic programming (see [14,15]). In contrast to other methods such as, for example, shooting [16][17][18], multiple shooting [19,20] or collocation methods [21,22], relying on a direct integration of the associated ordinary differential equations parameterized by states and controls or the controls only (see also [23] and [24] for an overview of the current state of the art), a recently developed method DMOC (see [25,26]) is used here. It is based on the discretization of the variational structure of the mechanical system directly.…”

Section: Introductionmentioning

confidence: 99%

“…In the context of variational integrators, as in [27], the discretization of the Lagrange-d'Alembert principle leads to the structure-preserving time-stepping equations that serve as equality constraints for the resulting finite-dimensional nonlinear optimization problem. In [26,28,29] DMOC was first applied to low-orbital thrust transfers and the optimal control of formation flying satellites including an algorithm that exploits a hierarchical structure of that problem.…”

Section: Introductionmentioning

confidence: 99%

Discrete mechanics and optimal control for constrained systems

Leyendecker

Ober‐Blöbaum

Marsden

et al. 2010

Optim. Control Appl. Meth.

127

105

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SUMMARYThe equations of motion of a controlled mechanical system subject to holonomic constraints may be formulated in terms of the states and controls by applying a constrained version of the Lagrange-d'Alembert principle. This paper derives a structure-preserving scheme for the optimal control of such systems using, as one of the key ingredients, a discrete analogue of that principle. This property is inherited when the system is reduced to its minimal dimension by the discrete null space method. Together with initial and final conditions on the configuration and conjugate momentum, the reduced discrete equations serve as nonlinear equality constraints for the minimization of a given objective functional. The algorithm yields a sequence of discrete configurations together with a sequence of actuating forces, optimally guiding the system from the initial to the desired final state. In particular, for the optimal control of multibody systems, a force formulation consistent with the joint constraints is introduced. This enables one to prove the consistency of the evolution of momentum maps. Using a two-link pendulum, the method is compared with existing methods. Further, it is applied to a satellite reorientation maneuver and a biomotion problem.

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“…In our approach, modeled after [11], the discrete equations of motion are derived from a discrete variational principle, and this induces constraints on the configuration at each discrete time step.…”

Section: Introductionmentioning

confidence: 99%

Geometric structure-preserving optimal control of a rigid body

et al. 2009

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Abstract. In this paper we study a discrete variational optimal control problem for the rigid body. The cost to be minimized is the external torque applied to move the rigid body from an initial condition to a pre-specified terminal condition. Instead of discretizing the equations of motion, we use the discrete equations obtained from the discrete Lagrange-d'Alembert principle, a process that better approximates the equations of motion. Within the discrete-time setting, these two approaches are not equivalent in general. The kinematics are discretized using a natural Lie-algebraic formulation that guarantees that the flow remains on the Lie group SO(3) and its algebra so(3). We use Lagrange's method for constrained problems in the calculus of variations to derive the discrete-time necessary conditions. We give a numerical example for a three-dimensional rigid body maneuver.

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Discrete Mechanics and Optimal Control

Cited by 140 publications

References 12 publications

Optimal Attitude Control of a Rigid Body Using Geometrically Exact Computations on SO(3)

Optimal Attitude Control of a Rigid Body Using Geometrically Exact Computations on SO(3)

Discrete mechanics and optimal control for constrained systems

Geometric structure-preserving optimal control of a rigid body

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