Discrete Mechanics and Optimal Control for Constrained Multibody Dynamics

Leyendecker, Sigrid; ̈baum, Sina Ober-Blo; Marsden, Jerrold E.; Ortiz, Michael

doi:10.1115/detc2007-34574

Cited by 14 publications

(12 citation statements)

References 42 publications

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“…A variational discrete null space method is proposed in (Leyendecker et al 2008). See also (Leyendecker et al (2007)). A study of the -convergence of VI's to the corresponding continuum action functional and the convergence properties of the discrete trajectories to stationary points of the continuum problem is presented in Schmidt et al (2009).…”

Section: Variational Integrationmentioning

confidence: 98%

A Brief Introduction to Variational Integrators

Lew

2016

Structure-Preserving Integrators in Nonlinear Structural Dynamics and Flexible Multibody Dynamics

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In this chapter, a brief introduction to the formulation of variational methods for finite-dimensional Lagrangian systems is presented. To this end, the first two sections focus on describing the Lagrangian and Hamiltonian points of view of mechanics for systems evolving on manifolds. Special attention is paid to the construction of the Lagrangian function and to the role of Hamilton's variational principle in the deduction of the balance equations. The relation between the symmetries of the Lagrangian function and the existence of invariants of the dynamics along with the symplectic nature of the flow are also addressed. In the third section, the discussion turns towards the formulation of a time-discrete analogue of the theory. The cornerstone of such a construction is given by a discrete analogue of Hamilton's variational principle which provides a systematic procedure to construct discrete approximations to the exact trajectory of a mechanical system on both the configuration space and the phase space. The approximation properties and the geometric characteristics of the resulting discrete trajectories are explained. Finally, we apply the variational methodology to construct symplectic and momentum-conserving time integrators for two problems of practical interest in engineering and science.

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Section: Variational Integrationmentioning

confidence: 98%

A Brief Introduction to Variational Integrators

Lew

2016

Structure-Preserving Integrators in Nonlinear Structural Dynamics and Flexible Multibody Dynamics

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“…No other force and torque are applied to this tree-structured system; therefore, in (42), the last matrix reduces to ⎡ [1,2,3], where the absolute reparametrization q n = F(u n , q 00 ) is used instead of (11) here. The motion starts and ends at rest.…”

Section: Optimal Control Of a Rigid Body With Rotorsmentioning

confidence: 99%

“…Null space matrix: Using I 12 = I−m 1 1 ⊗m 1 1 −m 1 2 ⊗ m 1 2 , the relation between the angular velocities reads x 2 = I 11 ·x 1 +˙ 2 n 1 . With regard to (36), the null space matrix for the E pair is given by…”

Section: Planar Pairmentioning

confidence: 99%

“…Actuation of the planar pair: The three relative degrees of freedom allowed by the planar joint can be actuated by translational forces (E) 1 , (E) 2 ∈ R acting in the directions of m 1 1 , m 1 2 and a torque (E) ∈ R about n 1 . In (41), they are accounted for using…”

Section: Planar Pairmentioning

confidence: 99%

“…The new method's acronym is DMOCC. The idea of this combination has been introduced briefly in [1] and is investigated in detail for three-dimensional multibody systems consisting of rigid bodies interconnected by joints in this work.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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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An energy‐momentum conserving method for the optimal control of multibody dynamics

Siebert

Betsch

2009

Proc Appl Math and Mech

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The present work deals with inverse multibody dynamics problems, more precisely with optimal control problems for dynamical systems governed by differential‐algebraic equations. The used numerical integration method relies on the Hamilton formulation of the equations of motion. We will apply an energy and momentum conserving time discretization which typically exhibits superior numerical stability properties. The rigid body equations of motion are basically formulated using the director formulation, which requires the implementation of internal constraints. To eliminate the constraints the system will be reduced to its minimal dimension by applying the discrete nullspace method [3]. For the derivation of the necessary conditions of optimality the calculus of variation is used [4]. This approach leads to an indirect transcription method. The arising algorithm yields a sequence of discrete configurations together with a sequence of actuating forces. We test the formulation with the help of representative numerical example problems. (© 2009 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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Discrete Mechanics and Optimal Control for Constrained Multibody Dynamics

Cited by 14 publications

References 42 publications

A Brief Introduction to Variational Integrators

A Brief Introduction to Variational Integrators

Discrete mechanics and optimal control for constrained systems

An energy‐momentum conserving method for the optimal control of multibody dynamics

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