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Research Papers

# Simulation of Constrained Mechanical Systems — Part I: An Equation of Motion

[+] Author and Article Information
David J. Braun1

School of Informatics, University of Edinburgh, 10 Crichton Street, Edinburgh EH8 9AB, UKdavid.braun@ed.ac.uk

Michael Goldfarb

Department of Mechanical Engineering, Vanderbilt University, VU Station B 351592, Nashville, TN 37235,michael.goldfarb@vanderbilt.edu

In an ideal computational environment, the constraints would be exactly satisfied, which would result in, $Δq=0$, $Δν=0$, and $Aqv=bq$. In that case, $q·=v$, and Eq. 17 would reduce to $q··=a+R-1Cν+(bν-Aνa)+R-1NνR-TQ0$. The last two terms in this equation originate from the general solution of the linear Eq. 10. The same general solution obtained from a linear Eq. 9 would result in $q··=a+M-1/2B+(bν-Aνa)+M-1/2(I-B+B)M-1/2Q0$, where $M=M1/2M1/2$ and $B=AνM-1/2$. This equation is the one proposed by Udwadia and Kalaba [11].

In an ideal computational environment, the kinematic constraints and the energy law would be exactly satisfied, (i.e., $Δeν=0$), and thus the constrained acceleration could be exactly computed $q··=a+R-1Cν+(bν-Aνa)$. In that case, Eq. 20 would reduce to $Q0=RT[CeNν]+Ae(q··-a-R-1Cν+(bν-Aνa))$, and as such it would not perform any correction, $Q0=0$.

The numerically exact solution is an ideal reference along which the constraints are exactly satisfied.

The constraint accuracy is reported by depicting the error on the kinematic constraints.

The energy accuracy is reported by depicting the energy error.

1

Corresponding author.

J. Appl. Mech 79(4), 041017 (May 16, 2012) (9 pages) doi:10.1115/1.4005572 History: Received November 11, 2010; Revised October 21, 2011; Posted January 31, 2012; Published May 16, 2012

## Abstract

This paper presents an equation of motion for numerical simulation of constrained mechanical systems with holonomic and nonholonomic constraints. In order to avoid the error accumulation typically experienced in such simulations, the standard equation of motion is enhanced with embedded force and impulse terms which perform continuous constraint and energy correction along the numerical solution. To avoid interference between the kinematic constraint correction and the energy correction terms, both are derived by taking the geometry of the constrained dynamics rigorously into account. In this light, enforcement of the (ideal) holonomic and nonholonomic kinematic constraints are performed using ideal forces and impulses, while the energy conservation law is considered as a nonideal nonlinear nonholonomic constraint on the simulated motion, and as such it is enforced with nonideal forces. As derived, the equation can be directly discretized and integrated with an explicit ODE solver avoiding the need for expensive implicit integration and iterative constraint stabilization. Application of the proposed equation is demonstrated on a representative example. A more elaborate discussion of practical implementation is presented in Part II of this work.

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## Figures

Figure 1

Slider-crank mechanism:  l = 1  m, m = 1  kg. The simulation is performed for t ∈ [0, 100]s started from q(0) = [2/4, 2/4, π/4, 32/4, 2/4,-π/4]T, v(0) = [0, 0, 0, 0,0, 0]T and conducted with 5 × 10 - 2 s time step. The stroboscopic view represents the motion for t ∈ [96.9, 99]s.

Figure 2

Horizontal motion of the slider, xs = x2 + (l/2)cos(θ2) vs time. The picture shows that the solution obtained with the proposed DAE integration method (with 5×10 - 2  s time step) is well matched with the numerically exact ODE solution.

Figure 3

From top to bottom the semilog pictures depict: |Φq|, |Φ·q|, |Φe|, and ɛ = dx2+dy2, where dx and dy are measured with respect to the numerically exact ODE solution. The integration here is performed with 10 - 2  s time step.

Figure 4

From top to bottom the log-log pictures depict the maximum constraint error, the maximum energy error and the maximum overall error. The accuracy/step-size characterizations are performed through a t ∈ [0, 100]s simulation. On the last plot, the CPU time corresponding to integration with and without energy correction is presented.

Figure 5

Double four-bar linkage: l = 1m, m = 1 kg for each link. Whenever the links become horizontal, y1 = y2 = y3 = 0, the linkage moves through a kinematic singularity.

Figure 6

Double four-bar linkage: l = 1m, m = 1 kg for each link. The simulations are performed through t ∈ [0, 1000]s, with time step 10 - 2 s and q(0) = [0, 1, 1, 1, 2, 1]Tv(0) = [1, 0, 1, 0, 1, 0]T. The ODE solutions are obtained using the unconstrained formulation of the problem: θ·· + (7g/6l)cos(θ) = 0, θ(0) = π/2, θ·(0) =  - 1 (θ denotes the orientation of the base link, see Fig. 5).

Figure 7

Evolution of the (least accurate) kinematic constraints, the energy conservation law and the overall accuracy ɛ3 = dx32 + dy32 where dx3, dy3 is measured between the solutions depicted in Fig. 6, and the numerically exact solutions for (x3, y3)

## Errata

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