Research Papers

Adaptive Semi-Implicit Integrator for Articulated Mechanical Systems

[+] Author and Article Information
Joe Hewlett

Department of Mechanical Engineering and
Centre for Intelligent Machines,
McGill University,
Montréal, QC H3A 2A7, Canada
e-mail: joseph.hewlett@mail.mcgill.ca

Laszlo Kovacs, Alfonso Callejo, József Kövecses, Jorge Angeles

Department of Mechanical Engineering and
Centre for Intelligent Machines,
McGill University,
Montréal, QC H3A 2A7, Canada

Paul G. Kry

School of Computer Science and
Centre for Intelligent Machines,
McGill University,
Montréal, QC H3A 2A7, Canada

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received June 30, 2016; final manuscript received December 17, 2016; published online March 9, 2017. Assoc. Editor: Paramsothy Jayakumar.

J. Comput. Nonlinear Dynam 12(5), 051003 (Mar 09, 2017) (10 pages) Paper No: CND-16-1312; doi: 10.1115/1.4035671 History: Received June 30, 2016; Revised December 17, 2016

This paper concerns the dynamic simulation of constrained mechanical systems in the context of real-time applications and stable integrators. The goal is to adaptively find a balance between the stability of an over-damped implicit scheme and the energetic consistency of the symplectic, semi-implicit Euler scheme. As a starting point, we investigate in detail the properties of a recently proposed timestepping scheme, which approximates a full nonlinear implicit solution with a single linear system, without compromising stability. This scheme introduces a geometric stiffness term that improves numerical stability up to a certain time-step size, but it does so at the cost of large mechanical dissipation in comparison to the traditional constrained dynamics formulation. Dissipation is sometimes undesirable from a mechanical point of view, especially if the dissipation is not quantified. In this paper, we propose to use an additional control parameter to regulate “how implicit” the Jacobian matrix is, and change the degree to which the geometric stiffness term contributes. For the selection of this parameter, adaptive schemes are proposed based on the monitoring of energy drift. The proposed adaptive method is verified through the simulation of open-chain systems.

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Fig. 1

Single pendulum with compliant constraints. The constraint surface, minor deviations from it, forces at successive time-steps, and the origin of the geometric stiffness are indicated.

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Fig. 3

Double pendulum. Net energy (without constraint energy) of the original and geometric stiffness methods.

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Fig. 4

Single pendulum. The choice of α is equivalent to the choice of the instant at which the Jacobian is evaluated, indicated by the dashed line. The geometric stiffness force is assumed dissipative, since it opposes the motion.

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Fig. 5

Three relationships between (E − E0)/Emax and α

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Fig. 7

Net energy for a double pendulum with c=10−5 m/N, h=10−2 s, Emax=0.01Es, and β = 0

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Fig. 6

Double pendulum with c=10−5 m/N, h=10−2 s, Emax=0.01Es, and β = 0. Value of α over a short duration (top) and a longer duration (bottom).

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Fig. 8

Double pendulum with c=10−5 m/N, h=10−2 s, Emax=0.01Es. Trajectories of the implicit biased (top left) and explicit biased (top right) formulations; net energy (bottom left) and α values (bottom right), β = 0.

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Fig. 9

Double pendulum with fixed β = 0.01 (top), β = 1 (bottom), c=10−5 m/N, h=10−2 s, Emax=0.01Es. Net energy (left) and α values (right).

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Fig. 10

Double pendulum with Emax=Es, c=10−5 m/N,h=10−2 s. Maximum value of |α| (top) and net energy (bottom).

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Fig. 11

Chain system simulated for 1.7 s with (a) the original method, (b) the geometric stiffness method, and (c) the adaptive method (hyperbolic tangent). One snapshot in every 0.1 s.

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Fig. 15

Chain system. Comparison of natural coordinate and Euler parameter trajectories in a 1.7 s simulation with; (a) the original method (α = 0); (b) the geometric stiffness method (α = 1); and (c) the adaptive method (hyperbolic tangent, α = adaptive).

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Fig. 12

Chain system. Energy drift and α value in a 1.7 s simulation with (a) the original method (α = 0), (b) the geometric stiffness method (α = 1) and (c) the adaptive method (hyperbolic tangent scheme).

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Fig. 13

Chain system. Net energy drift (without constraint violation energy) in a 1.7 s simulation.

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Fig. 14

Chain system. Effect of time-step size on (a) the maximum total energy drift and (b) the final total energy drift.



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