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.

References

References
1.
Schiehlen
,
W.
, 1997, “
Multibody System Dynamics: Roots and Perspectives
,”
Multibody Syst. Dyn.
,
1
(2)
, pp.
149
188
.
2.
Shabana
,
A. A.
, 1997, “
Flexible Multibody Dynamics: Review of Past and Recent Developments
,”
Multibody Syst. Dyn.
,
1
(2)
, pp.
189
222
.
3.
Brogliato
,
B.
,
ten Dam
,
A. A.
,
Paoli
,
L.
,
Génot
,
F.
, and
Abadie
,
M.
, 2002, “
Numerical Simulation of Finite Dimensional Multibody Nonsmooth Mechanical Systems
,”
Appl. Mech. Rev.
,
55
(
2
), pp.
107
150
.
4.
Eberhard
,
P.
, and
Schiehlen
,
W.
, 2006, “
Computational Dynamics of Multibody Systems: History, Formalisms, and Applications
,”
J. Comput. Nonlinear Dyn.
,
1
(
1
), pp.
3
12
.
5.
Laulusa
,
A.
, and
Bauchau
,
O. A.
, 2008, “
Review of Classical Approaches for Constraint Enforcement in Multibody Systems
,”
J. Comput. Nonlinear Dyn.
,
3
(1)
, p.
011004
.
6.
Bauchau
,
O. A.
, and
Laulusa
,
A.
, 2008, “
Review of Contemporary Approaches for Constraint Enforcement in Multibody Systems
,”
J. Comput. Nonlinear Dyn.
,
3
(1)
, p.
011005
.
7.
Gear
,
C. W.
, 2006, “
Towards Explicit Methods for Differential Algebraic Equations
,”
BIT
,
46
(3)
, pp.
505
514
.
8.
Braun
,
D.
, and
Goldfarb
,
M.
, 2010, “
Simulation of Constrained Mechanical Systems — Part I: An Equation of Motion
,”
J. Appl. Mech.
, (In press).
9.
ten
Dam
,
A. A.
, 1992, “
Stable Numerical Integration of Dynamical Systems Subject to Equality State-Space Constraints
,”
J. Eng. Math.
,
26
(2)
, pp.
315
337
.
10.
Ascher
,
U. M.
,
Chin
,
H.
, and
Reich
,
S.
, 1994, “
Stabilization of DAEs and Invariant Manifolds
,”
Numerische Mathematik
,
67
(2)
, pp.
131
149
.
11.
Burgermeister
,
B.
,
Arnold
,
M.
, and
Esterl
,
B.
, 2006, “
DAE Time Integration for Real-Time Applications in Multi-Body Dynamics
,”
Z. Angew. Math. Mech.
,
86
(
10
), pp.
759
771
.
12.
Petzold
,
L.
, 1992, “
Numerical Solution of Differential-Algebraic Equations in Mechanical Systems Simulation
,”
Physica D
,
60
(
1–4
), pp.
269
279
.
13.
Blajer
,
W.
, 2002, “
Elimination of Constraint Violation and Accuracy Aspects in Numerical Simulation of Multibody Systems
,”
Multibody Syst. Dyn.
,
7
(3)
, pp.
265
284
.
14.
Betsch
,
P.
, 2006, “
Energy-Consistent Numerical Integration of Mechanical Systems With Mixed Holonomic and Nonholonomic Constraints
,”
Comput. Methods Appl. Mech. Eng.
,
195
(50-51), pp.
7020
7035
.
15.
Kövecses
,
J.
,
Piedboeuf
,
J.-C.
, and
Lange
,
C.
, 2003, “
Dynamics Modeling and Simulation of Constrained Robotic Systems
,”
IEEE/ASME Trans. Mechatron.
,
8
(
2
), pp.
165
177
.
16.
Leimkuhler
,
B.
,
Petzold
,
L. R.
, and
Gear
,
C. W.
, 1991, “
Approximation Methods for the Consistent Initialization of Differential-Algebraic Equations
,”
SIAM J. Numer. Anal.
,
28
(
1
), pp.
205
226
.http://www.jstor.org/stable/2157941http://www.jstor.org/stable/2157941
17.
Nikravesh
,
P. E.
, 2008, “
Initial Condition Correction in Multibody Dynamics
,”
Multibody Syst, Dyn
,
18
(1)
, pp.
107
115
.
18.
Udwadia
,
F. E.
,
Kalaba
,
R. E.
, and
Phohomsiri
,
P.
, 2004, “
Mechanical Systems With Nonideal Constraints: Explicit Equations Without the Use of Generalized Inverses
,”
J. Appl. Mech.
,
71
(
5
), pp.
618
621
.
19.
Kim
,
S. S.
, and
Vanderploeg
,
M. J.
, 1986, “
QR Decomposition for State Space Representation of Constrained Mechanical Dynamical Systems
,”
ASME J. Mech., Transm., Autom. Des.
,
108
(2), pp.
183
188
.
20.
Neto
,
M. A.
, and
Ambrósio
,
J.
, 2003, “
Stabilization Methods for the Integration of DAE in the Presence of Redundant Constraints
,”
Multibody Syst. Dyn.
,
10
(
1
), pp.
81
105
.
21.
Fierro
,
R. D.
,
Hansen
,
P. C.
, and
Hansen
,
P. S. K.
, 1999, “
UTV Tools: Matlab Templates for Rank-Revealing UTV Decompositions
,”
Numer. Algorithms
,
20
(
2-3
), pp.
165
194
.
22.
Stewart
,
G. W.
, 1999, “
The QLP Approximation to the Singular Value Decomposition
,”
SIAM J. Comput.
,
20
(
4
), pp.
1336
1348
.
23.
Singh
,
R. P.
, and
Likins
,
P. W.
, 1985, “
Singular Value Decomposition for Constrained Dynamical Systems
,”
J. Appl. Mech.
,
52
(
4
), pp.
943
948
.
24.
Golub
,
G.
, and
Loan
,
C. V.
, 1996,
Matrix Computations
,
3rd ed
.,
The John Hopkins University Press
,
Baltimore, MD
.
25.
Abramowitz
,
M.
, and
Stegun
,
I. A.
, 1972,
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables
,
Dover Publications
,
New York
.
26.
Braun
,
D. J.
, and
Goldfarb
,
M.
, 2009, “
A Control Approach for Actuated Dynamic Walking in Biped Robots
,”
IEEE Trans. Robot.
,
25
(
6
), pp.
1292
1303
.
27.
Braun
,
D. J.
,
Mitchell
,
J. E.
, and
Goldfarb
,
M.
, 2012, “
Actuated Dynamic Walking in a Seven-Link Biped Robot
,”
IEEE/ASME Trans. Mechatron.
,
17
(1)
pp.
147
156
.
28.
Braun
,
D. J.
, and
Goldfarb
,
M.
, 2009, “
Eliminating Constraint Drift in the Numerical Simulation of Constrained Dynamical Systems
,”
Comput. Methods Appl. Mech. Eng.
,
198
(
37–40
), pp.
3151
3160
.
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