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.

Copyright © 2017 by ASME
Your Session has timed out. Please sign back in to continue.


Featherstone, R. , 1983, “ The Calculation of Robot Dynamics Using Articulated-Body Inertias,” Int. J. Rob. Res., 2(1), pp. 13–30. [CrossRef]
Orlandea, N. , Chace, M. A. , and Calahan, D. A. , 1977, “ A Sparsity-Oriented Approach to the Dynamic Analysis and Design of Mechanical Systems—Part 1,” J. Eng. Ind., 99(3), pp. 773–779. [CrossRef]
Orlandea, N. , Calahan, D. , and Chace, M. , 1977, “ A Sparsity-Oriented Approach to the Dynamic Analysis and Design of Mechanical Systems—Part 2,” J. Eng. Ind., 99(3), pp. 780–784. [CrossRef]
Bayo, E. , and Ledesma, R. , 1996, “ Augmented Lagrangian and Mass-Orthogonal Projection Methods for Constrained Multibody Dynamics,” Nonlinear Dyn., 9(1), pp. 113–130. [CrossRef]
Samin, J.-C. , and Fisette, P. , 2013, Symbolic Modeling of Multibody Systems, Springer, Berlin.
García de Jalón, J. , and Bayo, E. , 1994, Kinematic and Dynamic Simulation of Multibody Systems (The Real-Time Challenge), Springer-Verlag, New York.
Brogliato, B. , Ten Dam, A. , Paoli, L. , Genot, F. , and Abadie, M. , 2002, “ Numerical Simulation of Finite Dimensional Multibody Nonsmooth Mechanical Systems,” ASME Appl. Mech. Rev., 55(2), pp. 107–150. [CrossRef]
Bauchau, O. A. , and Laulusa, A. , 2008, “ Review of Contemporary Approaches for Constraint Enforcement in Multibody Systems,” ASME J. Comput. Nonlinear Dyn., 3(1), p. 011005. [CrossRef]
Marsden, J. E. , Ortiz, M. , and West, M. , 2003, “ An Overview of Variational Integrators,” Finite Element Methods: 1970s and Beyond, CIMNE, Barcelona, Spain, pp. 85–146.
Lee, J. , Liu, C. K. , Park, F. C. , and Srinivasa, S. S. , 2016, “ A Linear-Time Variational Integrator for Multibody Systems,” Cornell University Library epub, https://arxiv.org/abs/1609.02898
Baraff, D. , 1996, “ Linear-Time Dynamics Using Lagrange Multipliers,” 23rd Annual Conference on Computer Graphics and Interactive Techniques, ACM, New Orleans, LA, Aug. 4–9, pp. 137–146.
Stewart, D. E. , and Trinkle, J. C. , 1996, “ An Implicit Time-Stepping Scheme for Rigid Body Dynamics With Inelastic Collisions and Coulomb Friction,” Int. J. Numer. Methods Eng., 39(15), pp. 2673–2691. [CrossRef]
Lacoursière, C. , 2007, “ Ghosts and Machines: Regularized Variational Methods for Interactive Simulations of Multibodies With Dry Frictional Contacts,” Ph.D. thesis, Umeå University, Umeå, Sweden.
Baraff, D. , and Witkin, A. , 1998, “ Large Steps in Cloth Simulation,” 25th Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH'98, ACM, pp. 43–54.
Tournier, M. , Nesme, M. , Gilles, B. , and Faure, F. , 2015, “ Stable Constrained Dynamics,” ACM Trans. Graphics, 34(4), p. 132. [CrossRef]
Bottasso, C. L. , Dopico, D. , and Trainelli, L. , 2008, “ On the Optimal Scaling of Index Three Daes in Multibody Dynamics,” Multibody Syst. Dyn., 19(1–2), pp. 3–20. [CrossRef]
Nikravesh, P. , 1988, Computer-Aided Analysis of Mechanical Systems, Prentice-Hall, Upper Saddle River, NJ.
Haug, E. J. , 1989, Computer Aided Kinematics and Dynamics of Mechanical Systems, Vol. 1, Allyn and Bacon, Boston, MA.


Grahic Jump Location
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.

Grahic Jump Location
Fig. 3

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

Grahic Jump Location
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.

Grahic Jump Location
Fig. 5

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

Grahic Jump Location
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).

Grahic Jump Location
Fig. 7

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

Grahic Jump Location
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.

Grahic Jump Location
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).

Grahic Jump Location
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).

Grahic Jump Location
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.

Grahic Jump Location
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).

Grahic Jump Location
Fig. 13

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

Grahic Jump Location
Fig. 14

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

Grahic Jump Location
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).




Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In