This article presents three multibody formulations with improved efficiency in order to achieve real-time simulations for the forward dynamic of two real-life road vehicles. The bigger is a semitrailer truck with 40 degrees of freedom (DOF). Two topological and semirecursive formulations are used as well as a global formulation based on the use of Euler parameters and flexible joints. The first semirecursive formulation carries out a double velocity transformation and the integration is done by means of the explicit fourth order Runge–Kutta method. The second semirecursive formulation and the global one use a penalty scheme at position level and orthogonal projections at velocity and acceleration levels. In both cases the integrator was the implicit Hilbert–Huges–Taylor (HHT) method. The double velocity transformation method involves the coordinate partitioning of the constraint Jacobian matrix which leads to the costly solution of a redundant but consistent with the constraints linear system of equations. The choice of a unique set of independent coordinates may not be valid for a complete simulation and additional repartitioning would be required. Based on previous experience and as the examples show in this article, a careful initial choice of the independent coordinates can remain valid for complete simulations involving common maneuvers. This represents a numerical advantage for dense matrix methods and can be further exploited if sparse matrix techniques are employed. This has been the case for both of the vehicles used, reaching real-time simulations even with the semitrailer truck. The implicit semirecursive formulation involves the numerical evaluation of the stiffness and damping matrices, which hamper obtaining real-time simulations. For the semitrailer truck, this computation represents the 76% of the total simulation time. The numerical computation of these matrices is carried out by columns and its algorithm is straightforwardly parallelizable. Using a quad-core processor and with a simple and efficient OpenMP implementation, it has been possible to achieve a speedup of 3.25 reducing the simulation times under the real-time limit. The sparse matrices of Euler parameters formulation show very different sparsity degrees, difference that grows with the size of the multibody model. This poses a challenge to sparse matrix implementations in order to be able to efficiently perform matrix operations without increasing fillings or handling zero entries. This has been successfully accomplished using a new sparse matrix representation. This one is not a feature of general purpose sparse software, requiring at some stages the implementation of our own algorithms. Reductions in time of three orders of magnitude have led to real-time simulations even with the semitrailer truck.

References

References
1.
Gear
,
C. W.
,
1971
,
Numerical Initial Value Problems in Ordinary Differential Equations
,
Prentice-Hall
, Englewood Cliffs, NJ.
2.
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
.10.1115/1.2803258
3.
Baumgarte
,
J.
,
1972
, “
Stabilization of Constraints and Integrals of Motion in Dynamical Systems
,”
Comput. Methods Appl. Mech. Eng.
,
1
(
1
), pp.
1
16
.10.1016/0045-7825(72)90018-7
4.
Bayo
,
E.
, and
Ledesma
,
R.
,
1996
, “
Augmented Lagrangian and Mass-Orthogonal Projection Methods for Constrained Multibody Dynamics
,”
Nonlinear Dyn.
,
9
(
1
), pp.
113
130
.10.1007/BF01833296
5.
Yen
,
J.
,
Haug
,
E. J.
, and
Tak
,
T. O.
,
1991
, “
Numerical Methods for Constrained Equations of Motion in Mechanical System Dynamics
,”
Mech. Struct. Mach.
,
19
(
1
), pp.
41
76
.10.1080/08905459108905137
6.
Potra
,
F. A.
, and
Yen
,
J.
,.
1991
, “
Implicit Numerical Integration for Euler–Lagrange Equations Via Tangent Space Parametrization
,”
Mech. Struct. Mach.
,
19
(
1
), pp.
77
98
.10.1080/08905459108905138
7.
Haug
,
E. J.
, and
Yen
,
J.
,
1992
, “
Implicit Numerical Integration of Constrained Equations of Motion via Generalized Coordinate Partitioning
,”
ASME J. Mech. Des.
,
114
(
2
), pp.
296
304
.10.1115/1.2916946
8.
Kim
,
S. S.
, and
Vanderploeg
,
M. J.
,
1986
, “
A General and Efficient Method for Dynamic Analysis of Mechanical Systems Using Velocity Transformations
,”
ASME J. Mech. Transm. Automation Des.
,
108
(
2
), pp.
176
182
.10.1115/1.3260799
9.
Liang
,
C. G.
, and
Lance
,
G. M.
,
1987
, “
A Differentiable Null Space Method for Constrained Dynamic Analysis
,”
ASME J. Mech. Transm. Automation Des.
,
109
(
3
), pp.
405
411
.10.1115/1.3258810
10.
Radu
,
S.
, and
Haug
,
E. J.
,
2000
, “
Globally Independent Coordinates for Real-Time Vehicle Simulation
,”
ASME J. Mech. Des.
,
122
(
4
), pp.
575
582
.10.1115/1.1289389
11.
Kurdila
,
A.
,
Papastavridis
,
J. G.
, and
Kamat
,
M. P.
,
1990
, “
Role of Maggi's Equations in Computational Methods for Constrained Multibody Systems
,”
J. Guid. Control Dyn.
,
13
(
1
), pp.
113
120
.10.2514/3.20524
12.
Laulusa
,
A.
, and
Bauchau
,
O. A.
,
2008
, “
Review of Classical Approaches for Constraint Enforcement in Multibody Systems
,”
ASME J. Comput. Nonlinear Dyn.
,
3
(
1
), p.
011004
.10.1115/1.2803257
13.
Wehage
,
R. A.
, and
Haug
,
E. J.
,
1982
, “
Generalized Coordinate Partitioning for Dimension Reduction in Analysis of Constrained Dynamic Systems
,”
ASME J. Mech. Des.
,
104
(
1
), pp.
247
255
.10.1115/1.3256318
14.
Serna
,
M.
,
Aviles
,
R.
, and
Garciadejalon
,
J.
,
1982
, “
Dynamic Analysis of Plane Mechanisms With Lower Pairs in Basic Coordinates
,”
Mech. Mach. Theory
,
17
(
6
), pp.
397
403
.10.1016/0094-114X(82)90032-5
15.
Negrut
,
D.
,
Serban
,
R.
, and
Potra
,
F. A.
,
1997
, “
A Topology-Based Approach to Exploiting Sparsity in Multibody Dynamics: Joint Formulation*
,”
Mech. Struct. Mach.
,
25
(
2
), pp.
221
241
.10.1080/08905459708905288
16.
Rodríguez
,
J. I.
,
Jiménez
,
J. M.
,
Funes
,
F. J.
, and
de Jalón
,
J. G.
,
2004
, “
Recursive and Residual Algorithms for the Efficient Numerical Integration of Multi-Body Systems
,”
Multibody Syst. Dyn.
,
11
(
4
), pp.
295
320
.10.1023/B:MUBO.0000040798.77064.bc
17.
García De Jalón
,
J.
,
Callejo
,
A.
, and
Hidalgo
,
A. F.
,
2012
, “
Efficient Solution of Maggi's Equations
,”
ASME J. Comput. Nonlinear Dyn.
,
7
(
2
), p.
021003
.10.1115/1.4005238
18.
Anderson
,
E.
,
Bai
,
Z.
,
Bischof
,
C.
,
Blackford
,
S.
,
Demmel
,
J.
,
Dongarra
,
J.
,
Du Croz
,
J.
,
Greenbaum
,
A.
,
Hammarling
,
S.
,
McKenney
,
A.
, and
Sorensen
,
D.
,
1999
,
LAPACK User's Guide
,
3rd ed
,
Society for Industrial and Applied Mathematics
,
Philadelphia, PA
.
19.
Davis
,
T. A.
, and
Natarajan
,
E. P.
,
2010
, “
Algorithm 907: KLU, a Direct Sparse Solver for Circuit Simulation Problems
,”
ACM Trans. Math. Software
,
37
(
3
), Article No.
36
.10.1145/1824801.1824814
20.
González
,
M.
,,
González
,
F.
,
Dopico
,
D.
, and
Luaces
,
A.
,
2008
, “
On the Effect of Linear Algebra Implementations in Real-Time Multibody System Dynamics
,”
Comput. Mech.
,
41
(
4
), pp.
607
615
.10.1007/s00466-007-0218-2
21.
Duff
,
I. S.
, and
Reid
,
J. K.
,
1996
, “
The Design of MA48: A Code for the Direct Solution of Sparse Unsymmetric Linear Systems of Equations
,”
ACM Trans. Math. Software
,
22
(
2
), pp.
187
226
.10.1145/229473.229476
22.
Pacejka
,
H. B.
,
2006
,
Tyre And Vehicle Dynamics
,
Elsevier Science Serials
, Butterworth-Heinemann, Oxford, UK.
23.
Iglberger
,
K.
, and
Rüde
,
U.
,
2009
, “
Massively Parallel Rigid Body Dynamics Simulations
,”
Comput. Sci. Res. Dev.
,
23
(
3–4
), pp.
3
4
.10.1007/s00450-009-0066-8
24.
Iglberger
,
K.
, and
Rüde
,
U.
,
2011
, “
Large-Scale Rigid Body Simulations
,”
Multibody Syst. Dyn.
,
25
(
1
), pp.
81
95
.10.1007/s11044-010-9212-0
25.
Tasora
,
A.
,
Negrut
,
D.
, and
Anitescu
,
M.
,
2008
, “
Large-Scale Parallel Multi-Body Dynamics With Frictional Contact on the Graphical Processing Unit
,”
Proc. Inst. Mech. Eng. Part K
,
222
(
4
), pp.
315
326
.10.1243/13506501JET391
26.
Negrut
,
D.
,
Tasora
,
A.
,
Mazhar
,
H.
,
Heyn
,
T.
, and
Hahn
,
P.
,
2012
, “
Leveraging Parallel Computing in Multibody Dynamics
,”
Multibody Syst. Dyn.
,
27
(
1
), pp.
95
117
.10.1007/s11044-011-9262-y
27.
Anderson
,
K. S.
, and
Duan
,
S.
,
2000
, “
Highly Parallelizable Low-Order Dynamics Simulation Algorithm for Multi-Rigid-Body Systems
,”
J. Guid. Control Dyn.
,
23
(
2
), pp.
355
364
.10.2514/2.4531
28.
Kübler
,
R.
, and
Schiehlen
,
W.
,
2000
, “
Modular Simulation in Multibody System Dynamics
,”
Multibody Syst. Dyn.
,
4
(
2–3
), pp.
107
127
.10.1023/A:1009810318420
29.
Postiau
,
T.
,
Sass
,
L.
,
Fisette
,
P.
, and
Samin
,
J. C.
,
2001
, “
High-Performance Multibody Models of Road Vehicles: Fully Symbolic Implementation and Parallel Computation
,”
Veh. Syst. Dyn.
,
35
, pp.
57
84
.
30.
Arnold
,
M.
,
Burgermeister
,
B.
,
Führer
,
C.
,
Hippmann
,
G.
, and
Rill
,
G.
,
2011
, “
Numerical Methods in Vehicle System Dynamics: State of the Art and Current Developments
,”
Veh. Syst. Dyn.
,
49
(
7
), pp.
1159
1207
.10.1080/00423114.2011.582953
31.
Reinders
,
J.
,
2007
, Intel Threading Building Blocks: Outfitting C++ for Multi-core Processor Parallelism, O'Reilly Media, Sebastopol, CA.
32.
Anderson
,
K.
,
1999
, “
A Hybrid Parallelizable Low-Order Algorithm for Dynamics of Multi-Rigid-Body Systems: Part I, Chain Systems*1
,”
Math. Comput. Modell.
,
30
(
9–10
), pp.
193
215
.10.1016/S0895-7177(99)00190-9
33.
González
,
F.
,
Luaces
,
A.
,
Lugrís
,
U.
, and
González
,
M.
,
2009
, “
Non-Intrusive Parallelization of Multibody System Dynamic Simulations
,”
Comput. Mech.
,
44
(
4
), pp.
493
504
.10.1007/s00466-009-0386-3
34.
Lee
,
J. K.
,
Kang
,
J. S.
, and
Bae
,
D. S.
,
2014
, “
An Efficient Real-Time Vehicle Simulation Method Using a Chassis-Based Kinematic Formulation
,”
Proc. Inst. Mech. Eng. Part D
,
228
(
3
), pp.
272
284
.10.1177/0954407013507912
35.
Cuadrado
,
J.
,
Dopico
,
D.
,
Gonzalez
,
M.
, and
Naya
,
M. A.
,
2004
, “
A Combined Penalty and Recursive Real-Time Formulation for Multibody Dynamics
,”
ASME J. Mech. Des.
,
126
(
4
), pp.
602
608
.10.1115/1.1758257
36.
Chapman
,
B.
,
Jost
,
G.
,
van der Pas
,
R.
, and
Kuck
,
D. J.
,
2007
,
Using OpenMP: Portable Shared Memory Parallel Programming
,
MIT
, Cambridge, MA.
37.
Dopico
,
D.
,
Lugris
,
U.
,
Gonzalez
,
M.
, and
Cuadrado
,
J.
,
2005
, “
IRK VS Structural Integrators for Real-Time Applications in MBS
,”
J. Mech. Sci. Technol.
,
19
(
S1
), pp.
388
394
.10.1007/BF02916159
38.
Hilber
,
H. M.
,
Hughes
,
T. J. R.
, and
Taylor
,
R. L.
,
1977
, “
Improved Numerical Dissipation for Time Integration Algorithms in Structural Dynamics
,”
Earthquake Eng. Struct. Dyn.
,
5
(
3
), pp.
283
292
.10.1002/eqe.4290050306
39.
Cuadrado
,
J.
,
Cardenal
,
J.
,
Morer
,
P.
, and
Bayo
,
E.
,
2000
, “
Intelligent Simulation of Multibody Dynamics: Space-State and Descriptor Methods in Sequential and Parallel Computing Environments
,”
Multibody Syst. Dyn.
,
4
(
1
), pp.
55
73
.10.1023/A:1009824327480
40.
Nikravesh
,
P. E.
,
Kwon
,
O. K.
, and
Wehage
,
R. A.
,
1985
, “
Euler Parameters in Computational Kinematics and Dynamics. Part 2
,”
ASME J. Mech. Transm. Automation Des.
,
107
(
3
), pp.
366
369
.10.1115/1.3260723
41.
Nikravesh
,
P. E.
,
Wehage
,
R. A.
, and
Kwon
,
O. K.
,
1985
, “
Euler Parameters in Computational Kinematics and Dynamics. Part 1
,”
ASME J. Mech. Transm. Automation Des.
,
107
(
3
), pp.
358
365
.10.1115/1.3260722
42.
Hussein
,
B. A.
, and
Shabana
,
A. A.
,
2011
, “
Sparse Matrix Implicit Numerical Integration of the Stiff Differential/Algebraic Equations: Implementation
,”
Nonlinear Dyn.
,
65
(
4
), pp.
369
382
.10.1007/s11071-010-9898-9
43.
Shabana
,
A. A.
, and
Hussein
,
B. A.
,
2009
, “
A Two-Loop Sparse Matrix Numerical Integration Procedure for the Solution of Differential/Algebraic Equations: Application to Multibody Systems
,”
J. Sound Vib.
,
327
(
3–5
), pp.
557
563
.10.1016/j.jsv.2009.06.020
44.
Duff
,
I. S.
,
Erisman
,
A. M.
, and
Reid
,
J. K.
,
1989
,
Direct Methods for Sparse Matrices
,
Oxford University
,
NY
.
45.
Gustavson
,
F.
,
1978
, “
Two Fast Algorithms for Sparse Matrices: Multiplication and Permuted Transposition
,”
ACM Trans. Math. Software
,
4
(
3
), pp.
250
269
.10.1145/355791.355796
46.
Kim
,
S.-S.
,
2002
, “
A Subsystem Synthesis Method for Efficient Vehicle Multibody Dynamics
,”
Multibody Syst. Dyn.
,
7
(
2
), pp.
189
207
.10.1023/A:1014457111573
47.
Newmark
,
N. M.
,
1959
, “
A Method of Computation for Structural Dynamics
,”
J. Eng. Mech. Div.
,
38
(
3
), pp.
67
94
.
48.
Davis
,
T. A.
,
2006
,
Direct Methods for Sparse Linear Systems
,
Society for Industrial and Applied Mathematics
,
Philadelphia, PA
.10.1137/1.9780898718881
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