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research-article

Inverse Dynamics for Discrete Geometric Mechanics of Multibody Systems with Application to Direct Optimal Control

[+] Author and Article Information
Staffan Björkenstam

Geometry and Motion Planning, Fraunhofer-Chalmers Centre, SE-412 88 Göteborg, Sweden
staffan@fcc.chalmers.se

Sigrid Leyendecker

Chair of Applied Dynamics, University of Erlangen-Nuremberg, D-91058 Erlangen, Germany
sigrid.leyendecker@fau.de

Joachim Linn

Mathematical Methods in Dynamics and Durability, Fraunhofer Institute for Industrial Mathematics, D-67663 Kaiserslautern, Germany
joachim.linn@itwm.fraunhofer.de

Johan S. Carlson

Geometry and Motion Planning, Fraunhofer-Chalmers Centre, SE-412 88 Göteborg, Sweden
johan.carlson@fcc.chalmers.se

Bengt Lennartson

Signals and Systems, Chalmers University of Technology, SE-412 96 Göteborg, Sweden
bengt.lennartson@chalmers.se

1Corresponding author.

ASME doi:10.1115/1.4040780 History: Received January 09, 2018; Revised June 28, 2018

Abstract

In this paper, we present efficient algorithms for computation of the residual of the constrained discrete Euler-Lagrange equations of motion for tree structured, rigid multibody systems. In particular, we present new recursive formulas for computing partial derivatives of the kinetic energy. This enables us to solve the inverse dynamics problem of the discrete system with linear computational complexity. The resulting algorithms are easy to implement, and can naturally be applied to a very broad class of multibody systems by imposing constraints on the coordinates by means of Lagrange multipliers. A comparison is made with an existing software package, which shows a drastic improvement in computational efficiency. Our interest in inverse dynamics is primarily to apply direct transcription optimal control methods to multibody systems. As an example application, we present a digital human motion planning problem, which we solve using the proposed method. Furthermore, we present detailed descriptions of several common joints, in particular singularity free models of the spherical joint and the rigid body joint, using the Lie groups of unit quaternions and unit dual quaternions, respectively.

Copyright (c) 2018 by ASME
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