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Research Papers

# Simulation of Constrained Mechanical Systems—Part II: Explicit Numerical Integration

[+] 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 37235michael.goldfarb@vanderbilt.edu

Practically, accuracy of the initial data depends on the accuracy required on the kinematic constraints.

$T(q, q·)$ is the kinetic energy while $V(q)$ denotes the potential energy of the system.

In this paper and in Part I of this work [8] we used $tol=10-6$ in all examples. Note that computation of $(CeNν)+$ can also be performed with standard routines that automatically set $tol$.

The velocity of the center is expressed in the spherical coordinate system ($eρ, eα, eβ$); $eρ=cos(α)cos(β)i+cos(α)sin(β)j-sin(α)k$, $eα=-sin(α)cos(β)i-sin(α)sin(β)j-cos(α)k$, $eβ=-sin(β)i+cos(β)j+0k$. The position of P can be calculated as $rP=rC+∑i=13XiPei=rC+reρ$ where $XiP$ are the coordinates of the material point on the ball in contact. By multiplying this last relation (from the left) with $eiT$ one obtains $XiP=r(eiTeρ)$, and subsequently $vPC=∑i=13XiPe·i=∑i=13r(eiTeρ)e·i$.

The initial orientation of the ball is defined as: $e1(0)=[1, 0, 0]T$, $e2(0)=[0, 1, 0]T$, $e3(0)=[0, 0, 1]T$, while the corresponding consistent initial velocities are calculated as $e·i(0)=ω(0)×ei(0)$ where $ω(0)=ωρ(0)eρ(0)+((R-r)cos(α(0))/r)β·(0)eα(0)-((R-r)/r)α·(0)eβ(0)$ and $ωρ(0)=1 s-1$.

The desired motion of the robot is chosen to avoid knee collision and ground collision with the feet such that the necessary smoothness condition on constraints is provided.

1

Corresponding author.

J. Appl. Mech 79(4), 041018 (May 16, 2012) (6 pages) doi:10.1115/1.4005573 History: Received December 11, 2010; Revised October 21, 2011; Posted February 01, 2012; Published May 16, 2012

## Abstract

This paper presents an explicit to integrate differential algebraic equations (DAEs) method for simulations of constrained mechanical systems modeled with holonomic and nonholonomic constraints. The proposed DAE integrator is based on the equation of constrained motion developed in Part I of this work, which is discretized here using explicit ordinary differential equation schemes and applied to solve two nontrivial examples. The obtained results show that this integrator allows one to precisely solve constrained mechanical systems through long time periods. Unlike many other implicit DAE solvers which utilize iterative constraint correction, the presented DAE integrator is explicit, and it does not use any iteration. As a direct consequence, the present formulation is simple to implement, and is also well suited for real-time applications.

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## Figures

Figure 1

Ball of mass m=1 kg and r=1 m rolling in a bowl of radius R=8 m. (a) Stroboscopic view of the rolling motion. (b) Position of the center of the ball (ρ(t), α(t), β(t)). The simulations are performed with 10-2 s time step, through t ∈ [0, 100] s.

Figure 2

The semilog figures depict the constraint error, energy error, and an overall error measure ɛ=|β·cos(α)2+2r7(R-r)ωρsin(α)-β·(0)cos(α(0))2-2r7(R-r)ωρ(0)sin(α(0))| where ωρ=12∑i=13(ei×e·i)Teρ. Note that, β·cos(α)2+2r7(R-r)ωρ sin(α)=const. is a first integral of the considered nonholonomic system, and accordingly ɛ is identically zero along the exact solution (i.e., ɛ≡0). In the current simulation, Baumgarte’s constrained stabilization is implemented with optimal parameters [10].

Figure 3

Anthropomorphic biped, with model parameters given in Ref. [28] and ω=πs-1. (a) Stroboscopic plot of the oscillatory motion. (b) Phase plots. The real-time simulation is performed for t  ∈  [0, 100]s with 10-2 s time step, started from (slightly) inconsistent initial conditions: q(0)=[0.056, 1.2, 1.4, 2, 2, 0.35, 1.19, 1.19, 0]T, v(0)=0. (c) The kinematic constraints are precisely satisfied and steadily maintained.

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