Numerical Methods for High Speed Vehicle Dynamic Simulation Edward J. Haug Dan Negrut* Radu Serban** Dario Solis Automotive Research Center and NSF Center for Virtual Proving Ground Simulation NADS and Simulation Center The University of Iowa Iowa City, Iowa 52242 April 30, 1999 -------------------------- *Presently Senior Development Engineer, Mechanical Dynamics, Inc., 2301 Commonwealth Blvd., Ann Arbor, MI 48105 **Presently Postgraduate Researcher, Computational Science and Engineering Program, Department of Mechanical and Environmental Engineering, University of California at Santa Barbara, Santa Barbara, CA 93106
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Numerical Methods for High Speed Vehicle Dynamic Simulation
Edward J. Haug
Dan Negrut*
Radu Serban**
Dario Solis
Automotive Research Center
and
NSF Center for Virtual Proving Ground Simulation
NADS and Simulation Center
The University of Iowa
Iowa City, Iowa 52242
April 30, 1999
--------------------------
*Presently Senior Development Engineer, Mechanical Dynamics, Inc., 2301 Commonwealth
Blvd., Ann Arbor, MI 48105
**Presently Postgraduate Researcher, Computational Science and Engineering Program,
Department of Mechanical and Environmental Engineering, University of California at Santa
Barbara, Santa Barbara, CA 93106
2
Abstract
Recent developments in numerical methods for high speed vehicle dynamic simulation in
the multi-university Automotive Research Center sponsored by the U.S. Army Tank-Automotive
Research, Development, and Engineering Center are summarized and illustrated. The prior
state-of-the-art is reviewed and computational developments focusing on both computer-aided
engineering applications with stiff equations and driver-in-the-loop real-time vehicle system
simulation are presented. To illustrate gains achieved in computational efficiency and reliability,
realistic handling models of the Army High Mobility Multi-Purpose Wheeled Vehicle are
simulated and compared to results obtained using previous methods. A demonstration of two
orders of magnitude speed-up for stiff vehicle models characteristic of those employed by
vehicle developers in computer-aided engineering is presented. Achievement of real-time
simulation on computing platforms integral to emerging driving simulators that are establishing
new capabilities for virtual proving ground simulation is presented.
1. Prior State-of-the-Art
Prior to the mid 1990s, a decade of development in the technology of multibody dynamic
simulation had advanced the state-of-the-art to the point that significant capabilities were
available, but significant limitations remained, as follows:
(1) Real-time simulation for emerging driver-in-the-loop simulators was feasible with
models accounting for up to 15 Hz frequency response (wheel hop frequency), using either
optimized symbolic (Wehage, Belczynski, 1992; Schiehlen, 1994) or complex recursive(Bae,
Haug, 1987a,b) formulations.
Simplified models with special formulations were required to achieve real-time
simulation, which were generally incompatible with computer-aided engineering (CAE) tools
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such as DADS and ADAMS that are used in engineering design. Modeling incompatibilities that
resulted required that two distinct, generally incompatible, sets of models be used for driver-in-
the-loop real-time simulation to determine duty cycles and CAE analysis in design for subsystem
integrity and system performance.
(2) Reliable and accurate numerical integration of the differential-algebraic equations
(DAE) of multibody dynamics was possible only with explicit predictor-corrector methods,
which had significant difficulty with stiff models of vehicle systems that routinely arise in
engineering design.
The engineering community had no robust implicit numerical integration algorithms to
reliably carry out dynamic analysis of stiff differential-algebraic systems (Hairer, Wanner, 1996)
arising in practice. The implementation of implicit integrators (Hairer, Wanner, 1996) requires
calculation of a derivative of the equations of motion, for numerical solution of the discretized
equations. This implies the need for the derivative of all mass and force quantities, as well as
three derivatives of constraint expressions. Except for the highly inefficient alternative of
numerical differentiation, no practical means of obtaining these derivatives was available.
(3) High frequency vehicle subsystem response characteristics that are important in driver-
vehicle interaction could only be approximately modeled in achieving real-time performance.
High fidelity modeling of vehicle subsystems such as tires, headway control devices,
hybrid-electric powertrains, and a host of the emerging actively controlled devices appearing in
modern vehicles was not possible in real-time. As a result, often unsatisfactory simplified
models were used in real-time dynamic simulation.
(4) Computational environments for multibody dynamic simulation were either
nonexistent, or integrated into proprietary software such as ADAMS and DADS.
4
No modular software environments were available for creating, testing, and
implementing new algorithms and methods to overcome the difficulties outlined above. The
enormous number of formulation and computational alternatives that must be systematically
investigated to create fundamentally new methods to overcome the foregoing limitations can
only be effectively assessed in an integrated computational environment.
2. Scope of High Speed Dynamic Simulation Developments Addressed
High speed dynamic simulation research supported by the Army and the National Science
Foundation during the period 1994 to1998 focused on overcoming limitations in the four areas outlined
above. Upgrades being made in Tank and Automotive Research and Development Center soldier- and
hardware-in-the-loop simulators and US Department of Transportation development of the National
Advanced Driving Simulator (NADS) motivated research aimed at achieving real-time simulation, in
order to realize the potential for fundamental new virtual proving ground capabilities. The goal was to
support both human factors analysis and vehicle duty cycle determination and their integration with CAE
tools and technologies. The principal developments that resulted are summarized in more detail in the
body of the paper. A computational environment created to support the research reported in this paper is
briefly outlined in this section.
Starting from a vehicle model generated in formulations based on Cartesian coordinates
used in DADS, a general purpose computational environment has been created to support a
broad range of mutlibody mechanical analyses. Applications supported include explicit and
implicit numerical integration of the DAE of motion for vehicle system design, as well as real-
time simulation of vehicle systems for use in emerging vehicle driving simulators. The core of
this computational environment is shown in Fig. 1, comprising six basic tools developed in an
environment supporting both design level fidelity simulation and real-time simulation for
5
hardware- and operator-in-the-loop driving simulation. All developments presented in this paper
Acceleration computation (total)Function evaluations in acceleration computationLinear solver in acceleration computationDependent coordinate recoveryFunction evaluations in dependent coordinate recoveryLinear solver in dependent coordinate recoveryOther computations (ODE integration, data transfer between subroutines, etc.)
1
10
100
1 2 3 4 5 6 7 8 9 10
step size (ms)
CPU
tim
e (s
)
GICGCP1GCP2
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8. Conclusions and Future Work
The following high-speed numerical simulation capabilities have been developed and
demonstrated, for use in CAE and in real-time driver-in-the-loop simulation:
(1) High fidelity dynamic analysis of vehicle models with stiff components, using
recently developed implicit integration algorithms, yields speed-ups of at least two orders of
magnitude over commercially available tools.
A new class of implicit SDIRK methods, developed for solution of stiff ODE, has been
adapted for solution of the DAE of multibody dynamics. Both fast low-order and more refined
higher order methods have been demonstrated. The new class of Runge-Kutta-based implicit
integrators is shown to overcome limitations in stability of lower order Newmark-based
algorithms. The more refined algorithms are targeted for highly accurate simulation of very stiff
vehicle system models, which are often encountered in vehicle system engineering. Topology-
based linear algebra tools discussed in Section 4 provide a highly efficient framework for the
computationally intense Rosenbrock and SDIRK methods. The benefits of the new implicit
numerical integrators is demonstrated for simulation of a bushed HMMWV14-body model.
This capability is currently being extended and implemented in the commercial DADS
software system for vehicle system engineering, under a grant from the U.S. National Science
Foundation. The formulation, extended to incorporate discontinuous effects due to intermittent
motion and variable model structure, will be made available to the commercial sector by LMS-
CADSI in the near future.
(2) Real-time simulation of vehicle models of comparable fidelity to those used by
industry in vehicle design for handling has been demonstrated, using the globally independent
coordinate formulation, topological linear solvers, and dual-rate numerical integration algorithms.
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In order to demonstrate the real-time capabilities of the GIC formulation, in conjunction
with topology-based linear solvers and dual-rate integrators, a 10-body model of the HMMWV is
implemented and simulated in real time on multiprocessor computers that are commonly used in
driving simulators. Simulations used to test and demonstrate the capability developed include
steering, tire, and powertrain models that are at a level of fidelity suitable for vehicle system design.
The real-time simulation environment developed is now being implemented jointly by the
research team and Army personnel for interactive simulation with Army motion-based
simulators, such as the Ride Motion Simulator shown in Fig. 6, and the National Advanced
Driving Simulator shown in Fig. 7. With this capability, these enormously capable, state-of-the-
art vehicle driving simulators will function as virtual proving grounds for engineering
development of both on-and off-road vehicles and equipment.
Figure 6. Ride Motion Simulator
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Figure 7. National Advanced Driving Simulator
Acknowledgement: This research was supported by the US Army Tank-Automotive Research,
Development, and Engineering Center through the Automotive Research Center (DoD contract
number DAAE07-94-C-R094), a multi-university Center led by the University of Michigan.
References
Atkinson, K. E., 1989, An Introduction to Numerical Analysis, Wiley & Sons, New York.
Bae, D.-S., Haug, E. J., 1987a, “A Recursive Formulation for Constrained Mechanical System
Dynamics. Part I: Open-Loop Systems”, Mechanics of Structures and Machines, Vol. 15, pp.
359-382.
Bae, D.-S., Haug, E. J., 1987b, “A Recursive Formulation for Constrained Mechanical System
Dynamics. Part 11: Closed-Loop Systems”, Mechanics of Structures and Machines, Vol. 15, pp.
481-506.
34
Bernard, J. E., 1973, “Some Time-Saving Methods for the Digital Simulation of Highway
Vehicles,” Simulation, pp. 161-165.
Bischof, C., Roh, L., Mauer, A., 1996, “ADIC: A Tool for the Automatic Differentiation of C
Program”, Technical report, Mathematics and Computer Science Division, Argonne National
Laboratory.
Corwin, L. J., Szczarba, R. H., 1982, Multivariable Calculus, Marcel Dekker, New York.
Gillespie, T. D., 1992, Fundamentals of Vehicle Dynamics, Society of Automotive Engineers.
Hairer, E., Nørsett S. P., Wanner, G., 1993, Solving Ordinary Differential Equations I. Nonstiff
Problems, Springer-Verlag, Berlin.
Hairer, E., Wanner, G., 1996, Solving Ordinary Differential Equations II. Stiff and Differential-
Algebraic Problems, Springer-Verlag, Berlin.
Haug, E. J., 1989, Computer-Aided Kinematics and Dynamics of Mechanical Systems, Allyn
and Bacon, Boston.
Haug, E. J., Iancu, M., Negrut, D., 1997, “Implicit Integration of the Equations of Multibody
Dynamics in Descriptor Form,” Advances in Design Automation, 1997 ASME Design
Automation Conference.
Haug, E. J., Negrut, D., Engstler, C., 1999, “Implicit Runge-Kutta Integration of the Equations of
Multi-Body Dynamics in Descriptor Form,” Mechanics of Structures and Machines, to appear.
Gear, C. W., Wells, D. R., 1984, “Multirate Linear Multistep Methods,” BIT 24, pp. 484-502.
35
Liang, C. G., Lance, G. M., 1987, “A Differentiable Null Space Method for Constrained Dynamic
Analysis,” Journal of Mechanisms, Transmissions, and Automation in Design, Vol. 109, pp. 405-411.
Schiehlen, W., 1994, “Symbolic Computations in Multibody Systems”, Computer-Aided
Analysis of Rigid und Flexible Mechanical Systems, (M. F. O. S. Pereira and J. A. C. Ambrosio
eds.), Kluwer Academic Publishers, pp. 101-136.
Serban, R., Haug, E. J., 1988a, “Analytical Derivatives for Multibody System Analyses,”
Mechanics of Structures and Machines, Vol. 26, pp. 145-173.
Serban, R., Haug, E. J., 1998b, “Dual Coordinates for Real-Time Vehicles Simulation,” Journal
of Mechanical Design, submitted.
Serban, R., Negrut, D., Potra, F. A., Haug, E. J., 1997, “A Topology Based Approach for
Exploiting Sparsity in Multibody Dynamics in Cartesian Formulation,” Mechanics of Structures
and Machines, Vol. 25, pp 379-39.
Shampine, L. F., Gordon, M. K., 1975, Computer Solution of Ordinary Differential Equations.
The Initial Value Problem, Freeman and Company, San Francisco.
Solis, D., 1996, “DAE Multirate Methods for Dynamic Systems with Interacting Subsystems,”
Ph.D. Thesis, The University of Iowa.
Wehage, R. A., Belczynski, M., 1992, “High-resolution Vehicle Simulations Using Precomputed
Coefficients,” Transportation Systems, ASME, Vol. 44, pp. 311-325.
36
Wells, D. R., 1982, “Multirate Linear Multistep Methods for the Solution of Systems of Ordinary
Differential Equations,” Report UIUCDCS-R-82-1093 Department of Computer Science,