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Research Papers

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,
Göteborg SE-412 88, Sweden
e-mail: staffan@fcc.chalmers.se

Sigrid Leyendecker

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

Joachim Linn

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

Johan S. Carlson

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

Bengt Lennartson

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

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received January 9, 2018; final manuscript received June 28, 2018; published online July 30, 2018. Assoc. Editor: Paramsothy Jayakumar.

J. Comput. Nonlinear Dynam 13(10), 101001 (Jul 30, 2018) (15 pages) Paper No: CND-18-1012; doi: 10.1115/1.4040780 History: Received January 09, 2018; Revised June 28, 2018

In this paper, we present efficient algorithms for computation of the residual of the constrained discrete Euler–Lagrange (DEL) 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.

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Figures

Grahic Jump Location
Fig. 1

Example of a kinematic tree with NB = 7. For link 3, we have κ(3) = {1, 2, 3}, ν(3) = {3, 4, 5, 7}, and μ(3) = {4, 5}.

Grahic Jump Location
Fig. 4

Multiple pendulum with NB links

Grahic Jump Location
Fig. 2

Recursive algorithm for computing the gradients of the kinetic energy, ∇¯qiT and ∇viT, of a kinematic tree of rigid bodies

Grahic Jump Location
Fig. 3

Algorithm for computing the generalized force acting on a kinematic tree from wrenches applied to the rigid bodies

Grahic Jump Location
Fig. 9

(a) Identity configuration of the mechanical system in the optimal control problem in Sec. 5.1.2, i.e., q = IQ and (b) sphere approximation used for collision avoidance in the optimal control problem

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Fig. 10

Reaction forces in the lower back and ankles during the lifting task

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Fig. 5

Computation times for (a) the constraint function, (b) the Jacobian of the constraints, and (c) the Hessian of the Lagrangian

Grahic Jump Location
Fig. 6

Multibody model of the humanoid. The pelvis is chosen to be the first link in the tree, with index 1, connected to the fixed base, which has index 0. See Table 1 and Appendix A.6 for more details on the bodies and joints of the model.

Grahic Jump Location
Fig. 7

(a) Contact between foot and ground plane, where f∈ℝ3 is the reaction force at the contact point p∈ℝ3 and μ∈ℝ is the coefficient of friction, and (b) octagonal friction cone approximation

Grahic Jump Location
Fig. 11

The transformation, Xρ(i),i, from the body fixed frame of body i, to the body fixed frame of its parent, is a composition of a configuration-dependent transformation XJi(qi), and a constant transformation, XTi, locating joint i in the parent frame, i.e., Xρ(i),i=XTiXJi

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Fig. 12

Recursive algorithm for calculation of the transformations in the tree

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Fig. 8

Snapshots of the solution at (a) t = 0.00 s, (b) t = 0.19 s, (c) t = 0.37 s, (d) t = 0.56 s, (e) t = 0.75 s, and (f) t = 0.94 s

Grahic Jump Location
Fig. 13

(a) The universal joint, also known as Cardan or Hooke's joint, gives rotational motion about two orthogonal axes. (b) The configuration of the ball and socket joint belongs to SO(3). Motion in SO(3) can be decomposed into swing and twist motion.

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