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

3

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

were implemented and tested in this environment.

Figure 1. Vehicle Multibody Dynamics Computational Environment

In order to assess the benefits of developments summarized in this paper, a realistic

handling model of the Army’s high mobility multipurpose wheeled vehicle (HMMWV) shown in

Fig. 2 is used as a test problem. The fourteen-body model of this vehicle adopted for test and

evaluation is shown in Fig. 3. It consists of four independent suspension subsystems, each with a

wheel spindle body, connected by spherical joints to a pair of control arms, which are in turn

connected to the chassis via rotational joints. Each wheel assembly is connected to the chassis or

to a steering rack using a tie rod, modeled as a distance constraint. Excluding the steering degree

of freedom, which is specified as an input, this model has 10 degrees of freedom. The kinematics

model using Cartesian coordinates employs a total of 98 generalized coordinates, subject to 88

nonlinear kinematic constraint equations. Force elements acting in the model, not shown in Fig.

3, include suspension spring and shock absorber elements and tire force elements.

Model Synthesis

Derivative Computation

Topology-BasedLinear Solvers

Globally Independent Generalized Coordinates

Dual-Rate Integration forSubsystem Simulation

Real-Time,High Fidelity,

Vehicle Simulation

Stable and EfficientImplicit Integrators

for Stiff Vehicle Models

6

Figure 2. U.S. Army High Mobility Multipurpose Wheeled Vehicle (HMMWV)

Figure 3. Fourteen-Body Model of HMMWV

In this paper, q = q1, q2, ..., q n[ ]T denotes the vector of generalized coordinates that define

the state of a multibody system (Haug, 1989). The generalized coordinates are Cartesian

position coordinates and orientation Euler parameters of body centroidal reference frames.

7

Joints connecting the bodies of a mechanical system restrict their relative motion and impose

constraints on the generalized coordinates. In most cases, constraints are expressed as algebraic

expressions involving generalized coordinates; i.e., expressions of the form

F ( ) ( ), ( ),..., ( )q q q q∫ =F F F1 2 mT 0 (1)

Differentiating Eq. (1) with respect to time yields the kinematic velocity equation,

F q q q 0b g&= (2)

where subscript denotes partial differentiation; i.e., Φ q =LNMM

OQPP

∂Φ∂

i

jq, and an over dot denotes

differentiation with respect to time. Differentiating Eq. (2) with respect to time yields the

kinematic acceleration equation,

F Fq q qq q q q q qb g d i b g&& & & ,&= − ≡ t (3)

Equations (1) through (3) characterize the admissible motion of the mechanical system.

The mechanical system configuration changes in time under the effect of applied forces. The

Lagrange multiplier form of the constrained equations of motion for the mechanical system is

(Haug, 1989)

M q Q q qq( )&& ( ) ( &q q+ =F T , , t)λ A (4)

where M q( ) is the system mass matrix, λ is the vector of Lagrange multipliers that account for

workless constraint forces, and Q q q,A ,&tb g is the vector of generalized applied forces.

In order to determine a partitioning of the generalized coordinate vector q into dependent

and independent coordinate vectors u and v, respectively, a set of consistent generalized

coordinates q0 ; i.e., satisfying Eq. (1), is first determined. In this configuration, the constraint

8

Jacobian matrix is evaluated and numerically factored, using the Gauss-Jordan algorithm with

column pivoting (Atkinson, 1989),

Φ Φ Φq u vq q q0 0 0b g b g b gGaussæ Æææ (5)

The column pivoting induces a reordering of the generalized coordinates, its objective being to

produce a nonsingular sub-Jacobian with respect to u; i. e.,

det( ( ))Φ u q0 π 0 (6)

Partitioning the generalized coordinates to satisfy the condition of Eq. (6) is possible as long as

the constraint equations of Eq. (1) are independent (Haug, 1989).

Having partitioned the generalized coordinates, Eqs. (1) through (4) can be rewritten in

the associated partitioned form (Haug, 1989),

M u v v M u v u u v Q u v u vvv vuvT v, , , , , ,b g b g b g b g&& && &&+ + =Φ λ (7)

M u v v M u v u u v Q u v u vuv uuuT u, , , , , ,b g b g b g b g&& && &&+ + =Φ λ (8)

Φ (u,v) = 0 (9)

Φ Φu vu v u u v v 0( , )& ( , )&+ = (10)

Φ Φ τu vu v u u v v u v u v( , )&& ( , )&& , ,&,&+ = b g (11)

The condition of Eq. (6) and the implicit function theorem (Corwin, Szczarba, 1982)

guarantee that Eq. (9) can be solved locally for u as a function of v,

u = g(v) (12)

where the function g( )v has as many continuous derivatives as does the constraint function

Φ ( )q . Thus, at an admissible configuration q0 , there exist neighborhoods U1 of v0 and U2 of

9

u0 and a function g : U1 → U2 such that for any v ∈ U1 , Eq. (9) is identically satisfied when u is

given by Eq. (12). The analytical form of the function g v( ) is not known, but g v( )* can be

evaluated by fixing v v= * in Eq. (9) and iteratively solving for u g v* *= ( ) .

Using the partitioning of generalized coordinates induced by Eq. (5), the system of DAE

in Eqs. (7), (8), and (11) is reduced to a state-space ordinary differential equation (ODE),

through a succession of steps that use information provided by Eqs. (9) and (10). Since the

coefficient matrix of &u in Eq. (10) is nonsingular, &u can be determined as a function of v and

&v , where Eq. (12) is used to eliminate explicit dependence on u. Next, Eq. (11) uniquely

determines &&u as a function of v, &v , and &&v , where results from Eqs. (10) and (12) are substituted.

Since the coefficient matrix of λ in Eq. (8) is nonsingular, λ can be determined uniquely as a

function of v, &v , and &&v , using previously derived results. Finally, each of the preceding results

is substituted into Eq. (7) to obtain an underlying state-space ODE in only the independent

generalized coordinates v (Haug, 1989),

$ ( )&& $ ( &M v Q q qq = , , t) (13)

where

$$M M M M M

Q Q M Q M

vv vu uv uu

v vu u uu

= - - -

= - - -

- - -

- - -

Φ Φ Φ Φ Φ Φ

Φ τ Φ Φ Φ τ

u v v u u v

u v u u

1 1 1

1 1 1

T T

T T

c hc h

(14)

3. Implicit Runge-Kutta Numerical Integration Methods

Implicit Runge-Kutta numerical integration methods (Hairer, Nørsett, Wanner, 1993)

have been well developed for the solution of ODE of the form

&y = f t,yb g (15)

A broad range of Runge-Kutta integrators for this problem can be written in the form

10

k f t c h y h a k i si n i n ij jj

i

= + +FHG

IKJ =

=Â, , ,...,

1

1 (16)

y y h b kn n i ii

s

+=

= + Â11

(17)

where tn is the current time step; yn is the approximate solution at tn ; aij , bi , and ci are constants;

s is the number of stages in integrating from tn to tn+ 1; and ki are stage variables that

approximate &y at the stage points t c hi i+ . In all cases treated here, a ij = 0 for i j< ; i.e., only

diagonally implicit Runge-Kutta (DIRK) methods (Hairer, Nørsett, Wanner, 1993) with a ii ≠ 0

are considered. If each of the diagonal elements are equal; i.e., a ii = γ, the algorithm is called a

singly diagonal Runge-Kutta (SDIRK) method. According to Eq. (16), in successive steps, ki

appears on both sides of the equation. Since f assumes a nonlinear form, an iterative method to

solve for the stage variable ki is required.

In order to make Runge-Kutta methods suitable for integration of the second order DAE

of multibody dynamics, the second argument on the right of Eq. (16) is interpreted as an

approximate solution at time t t c hi n i= + (Haug, Negrut, Engstler, 1999); i.e.,

z y h a zi n ijj

i

j= +=∑

1

& (18)

where&z j is used instead of k j . Substituting Eq. (18) into Eq. (16) and regarding the left side as

&zi , the equation can be solved for &zi . Once all stages in Eq. (16) are solved for the associated &zi ,

the results are substituted into Eq. (17) to obtain

y y h b zn n ii

s

i+=

= + Â11

& (19)

11

Applying Eq. (18) to integrate acceleration yields

& & && &~ &&z y h a z z ha zi n ijj

i

j i ii i= + ∫ +=Â

1

(20)

where

&~ & &&z y h a zi n ij jj

i

= +=

-

1

1

Â

Extending the integration formula of Eq. (19) to second order,

& & &&y y h b zn n ii

s

i+=

= + ∑11

(21)

In order to enable solution of second order differential equations using Eqs. (19) and (21),

with &&zi as the solution variable in the discretized equations of motion, it is helpful to substitute

from Eq. (20) into Eq. (19), to obtain

y y h b y h a z

y h b y h b a z

y hy h b z

n n ii

s

n ij jj

i

n ii

s

n i iji

s

j

s

j

n n jj

s

j

+= =

= ==

=

= + +FHG

IKJ

= + FHG

IKJ + F

HGIKJ

= + +

 Â

 ÂÂ

Â

11 1

1

2

11

2

1

& &&

& &&

& &&

(22)

where it is recalled that aij = 0 for j i> , the condition bii

s

=Â =

1

1 must hold (Hairer, Nørsett,

Wanner, 1993), and the coefficients b j are defined as

b b aj i iji

s=

=Â

1

(23)

12

Likewise, Eq. (20) may be substituted into Eq. (18) to obtain

z y h a y h a z , j 1,...,i 1,... , j

y h a y h a a z , j i

y hc y h a z

z h a z

i n ijj 1

i

n j1

j

n ijj 1

i

n2

ij jj 1

s

1

s

n i n2

i1

i

i2

ii i

= + +FHG

IKJ = =

= +FHG

IKJ +

FHG

IKJ < <

= + +

≡ +

= =

= ==

=

∑ ∑

∑ ∑∑

∑

& &&

& &&

& &&~ &&

l ll

ll

l

l ll

l

l(24)

where the upper limits of the double summation in Eq. (24) are changed from l and j to s , since

l j i£ £ and a jl = 0 for l j> ;

a a ail ij jlj

s

∫=Â

1

(25)

is used, noting that if l i> , each term in the sum is zero and ail = 0; and

~ & &&z y hc y h a zi n i n2

ij 1

s

≡ + +=∑ l l

Introduction of the notation for the right side of Eqs. (20) and (24) enables a broad range

of implicit Runge-Kutta methods, including the trapezoidal formula and Rosenbrock methods

(Hairer, Nørsett, Wanner, 1993), to be applied in the reduction of the DAE of mechanical system

dynamics to underlying state space equations. This structure is also applicable to implicit multi-

step methods of numerical integration. Thus, a computer code that implements a state space

reduction method for solution of DAE using the implicit formulas of Eqs. (20) and (24) can be

used in conjunction with numerous implicit integrators, with the appropriate definition of

~ &~z zi i and .

13

The approach adopted to integrate the DAE of multibody dynamics is to insert Eqs. (20)

and (24) for independent coordinates into the state-space ODE of Eq. (13). This approach is

commonly used with Newmark methods in structural dynamics (Hughes, 1987) and is applicable

for multibody dynamics (Haug, Iancu, Negrut, 1997). The resulting equations are equivalent to

making the same substitution into the descriptor form of Eqs.(3) and (4), using Eqs. (9) and (10)

to determine dependent coordinates and their first time derivatives. The resulting discretized

equations of motion involve both independent and dependent accelerations and Lagrange

multipliers as solution variables. They are solved numerically and the Runge-Kutta algorithm of

Eqs. (24), (20), (19), and (22) is used, just as in the conventional first order implementation of

Runge-Kutta methods.

Solution of the resulting set of nonlinear algebraic equations using Newton and Newton-

like iterative methods (Atkinson, 1989) requires computation of derivatives of the discretized

equations and use of the implicit function theorem to obtain required iteration coefficient

matrices. Methods for calculating derivatives of terms in the equations of motion are discussed

in Section 5 of this paper.

The descriptor form method introduced above has been implemented in conjunction with

two integration formulas. The first is an order 4, 5 stage, A- and L-stable, SDIRK formula

(Hairer and Wanner, 1996), the resulting algorithm denoted in what follows by InflSDIRK. The

second integrator uses the trapezoidal formula (Hairer and Wanner, 1996), denoted InflTrap.

Details of the implementations can be found in the work of Haug, Negrut, and Engstler (1999).

Algorithms InflSDIRK and InflTrap are used to simulate the time evolution of a14-body

stiff HMMWV model, denoted HMMWV-14. Stiffness in the model is induced by replacing the

revolute joints between upper control arms and chassis in the original model with spherical

14

joints. Each joint replacement results in two additional degrees of freedom. For each such

spherical joint, two translational-spring-damper-actuators (TSDA) (Haug, 1989), acting in

complementary directions, model bushings that control the extra degrees of freedom in the actual

vehicle. The number of degrees of freedom of this model is 19. The stiffness coefficient of each

TSDA is 2 0 107. ◊ N/m, while the damping coefficient is 2 0 106. ◊ Ns/m. Tires are modeled as

TSDA elements, with stiffness coefficient 296,325 N/m and damping coefficient 3,502 Ns/m.

The dominant eigenvalue for this model has a small imaginary part, while the real part is

- ◊2 6 105. .

The vehicle model is driven at 10 mph and encounters a bump. The bump’s shape is a

half cylinder of diameter 0.1 m. The steering rack is locked, to cause the vehicle to drive straight

ahead. This reduces the number of degrees of freedom to 18. The front wheels hit the bump at

T ª05. s, and the rear wheels hit the bump at T ª12. s. The length of the simulation is 5

seconds. Toward the end of the simulation (after approximately 4 seconds), due to the high level

of damping, the chassis height stabilizes at approximately z1 0 71= . m.

For simulation of the vehicle, four error tolerances are considered, ranging from 10-2 to 10-5.

Identical absolute and relative error tolerances (Haug, Negrut, Engstler, 1999) are considered for both

position and velocity. Timing results reported in Tables 1 and 2 are in CPU seconds. The results are

obtained on an SGI Onyx computer with a 200 MHz R10000 processor.

Compared to InflTrap, the algorithm InflSDIRK is more efficient because of its better

integration formula. The step-size control mechanism for the SDIRK formula is well designed

(Hairer, Wanner, 1996), and the simulation takes larger step-sizes. This results in fewer costly

matrix factorizations. Furthermore, the order of the formula is 4, and its stability properties are

very good (L-stable). These attributes strongly recommend this algorithm, especially when

15

Table 1. Timing Results InflTrap Table 2. Timing results InflSDIRK

TOL 10-2 10-3 10-4 10-5 TOL 10-2 10-3 10-4 10-5

1 s 42 61 158 463 1 s 33 52 90 1702 s 79 155 420 1198 2 s 69 124 218 4333 s 92 189 524 1521 3 s 81 150 248 4934 s 100 206 552 1568 4 s 84 155 256 500

very stiff models are integrated with medium to high accuracy requirements. For low accuracy,

the difference between the algorithms is not significant.

In order to see the impact of the proposed algorithms on simulation timing results, they

are compared with a state-of-the-art explicit algorithm, in terms of efficiency. The explicit

algorithm used is denoted ExplDEABM. It is based on the code DEABM from the suite DEPAC

of explicit integrators of Shampine and Gordon (1975). ExplDEABM is an algorithm based on a

state-space reduction method, which uses the code DDEABM to integrate independent

accelerations to obtain independent velocities and positions. Dependent positions and velocities

are recovered using the position and velocity kinematic constraint equations of Eqs. (1) and (2).

Efficient means for acceleration computation are used in ExplDEABM, (Serban, Negrut, Potra,

Haug, 1997).

Table 3 presents simulation timing results in CPU seconds for ExplDEABM, which can

be compared with the results in Tables 1 and 2. The first row contains the absolute and relative

error tolerances imposed on both positions and velocities. The first column contains simulation

lengths in seconds. The simulations are carried out using the same SGI Onyx computer.

Table 3. Timing Results for ExplDEABM

TOL 10-2 10-3 10-4 10-5

1 s 3618 3641 3667 36632 s 7276 7348 7287 72763 s 10865 11122 10949 109654 s 14480 14771 14630 14592

16

The explicit algorithm requires up to 100 times longer to complete simulations,

independent of simulation duration and error tolerances. Results obtained with ExplDEABM also

conform to theoretical predictions; i.e., for each imposed error tolerance, the algorithm required

the same CPU time to complete a simulation run. This is typical of situations in which explicit

algorithms are limited to small step-sizes by stability considerations when integrating stiff

systems (Hairer, Wanner, 1996). Furthermore, for an imposed error tolerance, the CPU time

increases linearly with the simulation length. Even after the vehicle clears the obstacle and there

is no significant source of external excitation, the explicit algorithm is constrained to take very

small steps, because otherwise it would become unstable.

4. Topologically-Based Sparse Matrix Linear Equation Solvers

The equations of motion of Eq. (4) and kinematic acceleration equations of Eq. (3), for a

multibody system, may be written in matrix form as

M0

q Qq

q

FF

TLNMM

OQPPLNM

OQP = L

NMOQP

&&l t (26)

The coefficient matrix of the linear system in Eq. (26) is of dimension ( ) ( )n m n m+ × + , and is

called the augmented matrix.

The strategy proposed for solving the linear system in Eq. (26) proceeds by formally

expressing accelerations &&q in terms of Lagrange multipliers. Assuming for the moment that the

mass matrix is nonsingular, accelerations are substituted back into the acceleration kinematic

constraint equation to obtain

Φ Φ Φq q qM M Q− −= −1 1Td iλ τ (27)

17

The matrix B Mq q≡ −F F1 T is referred to as the reduced matrix. Diagonal blocks in this

matrix have the form

B M Mq q q qjjj

ij j

ij

i i i= +− −( ) ( ) ( ) ( )F F F F

1 1 211

21

2T T (28)

if joint j connects bodies i1 and i2 . Off-diagonal blocks assume the form

B Mq qjkj

ik

i ij k= ≠−( ) ( ) ,F F1 T (29)

where joints j and k are on the same body i . Otherwise, they are zero. In Eqs. (28) and (29),

F qi

j is the derivative of the constraint function corresponding to joint j with respect to the

generalized coordinates qi of body i , and Mi− 1 is the inverse of the mass matrix of body i .

The matrix B is sparse; i.e., it has numerous zero blocks. Furthermore, the pattern of

nonzero entries in the matrix can be arranged to be as nearly as possible on the diagonal, by

adjusting the numbering system for bodies and joints in the model. Effective optimization

methods exist for body and joint numbering to minimize the bandwidth of B (Serban, Negrut,

Potra, Haug, 1997), enabling efficient factorization of the matrix for solution of the Lagrange

multiplier in Eq. (28).

For planar mechanisms, the mass matrix M is positive definite. For spatial systems, it is

singular and positive semidefinite. It has been shown (Serban, Negrut, Potra, Haug, 1997) that

associating the Lagrange multiplier corresponding to the Euler Parameter normalization

constraint for a body with its acceleration components, a modified positive definite matrix M is

obtained. Therefore, since the constraint Jacobian is assumed to have full row rank, the reduced

18

matrix B is positive definite. Since Lagrange multipliers are available from the solution of Eq.

(27), accelerations are easily computed using the equations of motion,

Mq Q q&&= − F Tl (30)

The mass matrix M has the block diagonal structure

M

M 0 00 M 0

0 0 M

=

L

N

MMMM

O

Q

PPPP

1

2

LK

L L L LL nb

(31)

For both planar and spatial systems, the block diagonal terms have closed form inverses

(Serban, Negrut, Potra, Haug, 1997), so M is easily inverted. The proposed algorithm for

solving Eq. (27) for Lagrange multipliers and accelerations is thus as follows:

(1) Assemble the reduced matrix B , based on Eqs. (28) and (29).

(2) Solve the linear system of Eq. (27) for l.

(3) Recover the generalized accelerations, based on Eqs. (30) and (31).

Steps 1 and 2 are discussed in detail by Serban, Negrut, Potra, and Haug (1997). They are

responsible for the better efficiency of the method, when compared to classical direct methods.

During these two steps, advantage is taken of the topology-induced sparsity. Step 3 requires the

solution of nb systems of linear equations of dimension 3 3¥ for planar systems and 7 7× for

spatial systems, to recover generalized accelerations. This process can be carried out in parallel.

At the core of the proposed algorithm for solution of the augmented linear system of Eq.

(26) is an efficient factorization of the reduced matrix B (Serban, Negrut, Potra, Haug, 1997).

The factorization takes advantage of the sparsity induced by the topological graph of the

19

mechanical system. The following four algorithns, based on combinations of solution sequences

and linear solvers, are compared:

(1) The Lapack dense linear solver dgetrf-dgetrs, based on LU decomposition for general

matrices, is applied to the augmented linear system of Eq. (26) to evaluate both accelerations and

Lagrange multipliers.

(2) The Harwell sparse solver ma47, based on LU decomposition of symmetric matrices, is used

to solve the augmented linear system of Eq. (26).

(3) The Harwell Cholesky decomposition routine ma35, for banded symmetric, positive definite

matrices, is used to factor the reduced matrix B and solve Eq. (27) for the Lagrange multipliers

corresponding to joint constraints. Accelerations and Lagrange multipliers for Euler parameter

constraints are then recovered using Eq. (30).

(4) The Lapack Cholesky decomposition routine (dpbtrf) for banded symmetric, positive definite

matrices, replaces the Harwell routine of algorithm 3.

Numerical results presented in Table 4 are based on the proposed algorithms and refer to

the HMMWV-14 model presented in Sections 2 and 3. For this example, the proposed solution

technique is compared with the usual direct approach of solving for the generalized accelerations

and Lagrange multipliers by factoring the augmented matrix. The most efficient solution

technique is the last. The speed-up obtained from the usual strategy for solving the augmented

system, using a sparse LU decomposition (algorithm 2), is a factor of 4.2. The small difference

between algorithms 3 and 4 is due to the different internal storage algorithms in the subroutines

dpbtrf from Lapack and ma35 from Harwell.

20

Table 4. Timing Results for HMMWV-14 Numerical Experiments

Algorithm 1 Algorithm 2 Algorithm 3 Algorithm 4

22.88 ms 12.62 ms 3.17 ms 3.07 ms

5. Analytical Differentiation Identities

As shown in Section 3, implicit integration of the equations of multibody dynamics

requires the availability of a derivative of the differential equations of motion and third order

derivatives of the algebraic constraint equations. Regardless of which implicit integrator is used,

derivatives of all terms in Eqs. (1) through (4) with respect to both generalized coordinates and

velocities are required. The derivatives to be evaluated are as follows:

As shown by Serban and Haug (1998a), in the setting of Cartesian coordinates with Euler

parameters for orientation, a relatively small number of basic identities enables computation of

all required derivatives. Explicit forms for derivatives of basic quantities are obtained.

Therefore, this approach is expected to be more efficient than using automatic differentiation

methods (Bischof, et.al., 1996). When evaluating higher order derivatives, automatic

differentiation codes differentiate a lower order derivative and thus cannot take advantage of

simplifying identities that may exist, such as Euler parameter normalization constraints.

In the proposed approach, derivatives of four basic expressions that are used as building

blocks in generating required kinematic and kinetic derivatives are obtained. The identities

derived are subsequently applied to obtain higher order derivatives of the four basic constraints

on vectors in and between pairs of bodies (DOT-1 constraint, DOT-2 constraint, distance

( &&) , ( ) , , ( &) , ( &&) , (( &) &) , (( &) &)& &Mq Q , Q q q q q q qq qT

q q q q q q q q q q q q qΦ Φ Φ Φ Φλ

21

constraint, and spherical constraint) (Haug, 1989), which are then combined to generate

derivatives of constraints corresponding to any kinematic joint. A similar treatment is applied to

obtain derivatives of common force elements.

Numerical experiments have shown significantly improved efficiency of computation

using the expressions derived, compared with results obtained employing finite differences. For

the 14-body model of the HMMWV, Table 5 presents maximum absolute differences between

analytical and numerical values of the kinematic derivatives, for different values of the

perturbation used in computing finite differences.

Table 5. Kinematic Derivative Comparison (Analytical vs. Finite Difference)

Perturbation 10-4 10-5 10-6 10-7 10-8

Φ q 3.291�10-3 3.291�10-4 3.291�10-5 3.293�10-6 3.241�10-7

Φ q q

Tλd i 4.012�10-2 4.012�10-3 4.013�10-4 4.002�10-5 5.864�10-6

Φ q qq&d i 5.301�10-2 5.599�10-3 5.599�10-4 5.655�10-5 1.148�10-5

Φ q q qq q& &d ie j 10.5970 1.0587 0.1059 1.054�10-2 2.254�10-3

Φ q q qq q& &

&d ie j 6.582�10-3 6.582�10-4 6.538�10-4 1.482�10-4 1.322�10-4

A library of subroutines that calculate the required derivatives outlined above has been

implemented in the computational environment described in Section 2. This library is used for

calculating derivatives required for implicit integration in Section 3, constructing the matrices

factored using the topological method of Section 4, and in implementing the globally

independent coordinate, real-time simulation method described in Section 7.

6. Dual-Rate Numerical Integration Methods

If a wide range of frequencies are present in the system to be simulated, different step-

sizes can be used within a multi-rate integration formula to speed up computations. Multi-rate

22

methods possess the potential for real-time vehicle simulation, due to the inherent structure of

the vehicle. Predominant frequencies in the motion of vehicles are in the range of 1 to 2 Hz for

the sprung mass (chassis) and 10 to 15 Hz for the un-sprung mass (suspension system), during

wheel hop (Gillespie, 1992). Much higher frequencies are observed for wheel spin and tire force

calculation (Bernard, 1973). High frequency content is also encountered when modeling vehicle

subsystems such as powertrain, steering, braking, and active suspension.

The equations of motion employed here for dual-rate integration of constrained

mechanical systems are

M q q q Q q,q xqb g b g b g&& &,+ =Φ λT

(32

Φ q 0b g= (33)

& ,&, ,x q q x= Γ tb g (34)

where Eq. (34) is a set of n x ODE that is used to model interacting vehicle subsystems such as controls

and tire forces that influence forces on the vehicle, often containing high frequency effects.

During the simulation of vehicle systems, problems arise when subsystems with high

frequency content require small step-size for a stable solution, making the simulation process

inefficient. If this small step-size is used for all the variables in the system, the overall

simulation performance will suffer. Within computer hardware constraints imposed by

microprocessor technology, it may not be possible to solve the governing equations of the

vehicle model in real time. As a result, multi-rate numerical integration techniques that may

overcome the limitations of existing computational engines are needed.

Solis (1996) reviewed ODE multi-rate integration methods. Among the methods

presented, the slowest first approach (Wells, 1982) was chosen to be applied to the vehicle

simulation problem. Although this method was designed for integrating ODE, its application

23

was made possible by reducing the DAE that describe the vehicle motion, to an ODE on a

manifold, expressed in terms of the constraint equations. This reduction can be carried out using

a number of DAE solvers available in the literature. For the present work, the generalized

coordinate partitioning method (Haug, 1989) is combined with the globally independent

coordinate method (Serban, Haug, 1998) to reduce the DAE to underlying ODE.

The slowest first approach consists of integrating the slow variables first, using

extrapolated values of the fast variables (if implicit formulas are used for the slow integration),

and then integrating the fast variables using interpolated values of the slow variables. Because

the slow and fast variables are coupled, extrapolation of the fast variables in the slowest first

approach is subject to numerical errors. This is not relevant if explicit integration formulas are

used to integrate the slow variables, but it becomes critical when implicit formulas are used.

In Fig. 4, the smooth line represents slow variable ( q qi i and &) response and the zig-zag

line represents a typical fast variable ( x i+rk ) response. The slow variables are integrated using a

step- size h , and the fast coordinates are integrated using a step-size r = h / m r , where m r is a

fixed integer ratio between the step-sizes for both scales. Here, qip+ 1 and &qi

p+ 1 are predicted

position and velocity values for the slow variables. Finally, qi sr+* represents the interpolated

value of the slow variable for use in integrating for the fast variables.

This technique was implemented (Solis, 1996) and applied to the HMMWV-14 model,

using the null space method (Liang, Lance, 1987), instead of the coordinate partitioning-globally

independent coordinate approach of Section 7 to reduce the DAE to ODE. Very similar results

were obtained with a ratio of 6 between slow and fast integration step lengths.

24

Figure 4. Dual-Rate Integration Schematic

qi+ rk*

xixi+ rk

xi+ 1

xi− rxi− 2r&qi- 3

&qi- 2

&qi- 1

&qi

&&qi

&qip+ 1

qip+ 1

q

qi- 2

qi- 1

qi

Next, advantages of using multi-rate methods are discussed in terms of function

evaluation and numerical costs. Assuming the existence of two clearly identifiable time scales,

the process of integrating the slow subsystem over one step of size h is referred to as a macro

step, and the process of integrating the fast variables with a step-size r = h / m r is referred to as a

micro step. A complete multi-rate step contains one macro step and m r micro steps.

As the coupling between slow and fast variables decreases, the error accumulation during

integration decreases. Wells (1982) shows that multi-rate methods based on the slowest first

approach are convergent under certain conditions. Bounds for the absolute stability of multi-rate

methods are also presented. These technical conditions are derived from the absolute stability of

linear multi-step integration formulas.

The numerical cost of completing a multi-rate step can be measured in terms of several

contributing costs. Let C s be the cost of evaluating the slow time scale derivatives, C f be the cost of

evaluating the fast time scale derivatives, C intrp be the cost of interpolating the slow variables to provide

values needed by the fast subsystems, and C integ be the cost of performing one integration step using an

order k explicit multistep formula. The cost of one multi-rate integration step C mr is thus

25

C m C n C C (m n n ) Cmr r f coup interp s r f s integ= × + × + + × + ×d i (35)

In Eq. (35), n f , n s , and n coup are, respectively, the number of fast variables, slow variables, and

interpolated slow variables for use in the fast subsystems. If a single-rate integration method is used, the

step-size is dictated by the fast variables. The cost of single-rate integration in the same time interval is

thus

C m C m C m (n n ) Csr r f r s r f s integ= × + × + × + × (36)

It is assumed that the integration cost C integ is approximately equal to the interpolation cost

C interp , and the number of slow variables n coup that are interpolated (or extrapolated) is equal to the total

number of slow variables n s . The first assumption is reasonable if a multistep integration formula and a

consistent interpolation formula are used. The second assumption tends to reduce the overall advantage

of multi-rate methods, since coupling between slow and fast subsystems is often through a few variables

and not the whole set of slow variables. With these assumptions, Eq. (35) becomes

C m C n C C (m n n ) Cm r f s integ s r f s integr= × + × + + × + ×d i (37

which can be rearranged and simplified as

C m C C m n n n Cmr r f s r f s s integ= × + + × + + ×d ie j (38)

The condition under which a multi-rate integration formula is advantageous is

m 1 C n C 0r s s integ− × − × >b g (39)

which, in terms of the step ratio m r , is

m 1n C

Crs integ

s

> +×

(40)

Equation (40) shows that when an explicit multistep integration formula (only one

function evaluation per step) is used in a multi-rate integration method, the multi-rate integration

26

has a computational advantage over single-rate methods if the function evaluation cost per

variable is higher than the interpolation cost per variable. In constrained mechanical systems,

this is generally the case. With the reduction in computational cost in the solution of the system

equations, a reduction in computational time is also obtained. This goal justifies the use of

multi-rate formulas.

7. Globally Independent Coordinates

Numerical methods for real-time multibody system simulation and analysis must resolve

two conflicting requirements; (1) high fidelity of the model for high accuracy of results, and (2)

high computational efficiency for low simulation time. In the past, a number of modeling

approaches have been used to achieve one or the other of these goals. Commercial analysis tools

such as DADS and ADAMS that emphasize high fidelity are based on Cartesian coordinates,

while more recent codes designed for real-time simulation employ joint relative coordinate

recursive formulations (Bae, Haug, 1987a,b).

It is desirable that the same model be used for both fast and high fidelity simulations.

The Cartesian coordinate formulation with Euler parameters for orientation (Haug, 1989) is well

suited for multibody system modeling, from the point of view of systematic analysis, because of

the simple structure of the resulting equations. The main disadvantage of this formulation is the

dimension of the problem to be solved. A large system of DAE must thus be solved, which is a

computationally demanding task. State-space methods for solving the equations of motion, such

as those presented in Section 3, are based on a parameterization of the constraint manifold via a

set of independent coordinates. The equations of motion can theoretically be reduced then to a

set of ODE in these independent coordinates. However, for general mechanisms, such a

parameterization of the constraint manifold is possible only locally. Therefore, the underlying

27

ODE can be constructed only locally. When the manifold parameterization is no longer valid,

new independent coordinates must be selected, based on matrix decomposition techniques that

are in general computationally expensive.

Coordinates that are independent for the entire range of admissible motion, called

globally independent coordinates (GIC), are particularly attractive for multibody system

simulation. The existence of such coordinates eliminates the need for local independent

coordinate selection to parameterize the constraint manifold. For the special case of vehicle

multibody systems, Serban and Haug (1998) have shown that information on functional intent

and design criteria enables definition of a subset of generalized coordinates that remain

independent for the entire admissible range of motion. These coordinates are dual to force

elements of the vehicle system, including suspension springs, shock absorbers, and control

elements such as the steering rack.

In addition to the immediate advantage of avoiding the computationally expensive

redefinition of local parameterization in a state-space formulation, efficiency enhancements are

obtained from the fact that optimal computational sequences for dependent coordinate recovery

can be defined using the GIC formulation. Once values of independent generalized coordinates

and velocities are obtained by integration, dependent states and velocities are recovered from the

nonlinear constraint equations, usually using a Newton-Raphson or Newton-like method

(Atkinson, 1989). The dimension of this nonlinear system of equations is large, especially when

Cartesian coordinates with Euler parameters for orientation are used. Availability of GIC

associated with force and/or control elements in the multibody system results in a decoupling of

the nonlinear constraint equations into smaller sets of independent equations. Since the number

of equations being solved has a significant influence on the computational effort in solving sets

28

of nonlinear equations, recovery of dependent coordinates can be performed efficiently by

solving lower dimensional subsets of equations. Even greater efficiency is realized when these

solutions are obtained in parallel on a multiprocessor computer.

As an example of dynamic simulation using GIC, a 10-body HMMWV model is

presented. This model is obtained from the 14-body model of Fig. 3 by replacing the upper

control arms with composite revolute-spherical joints. Globally independent coordinates are

selected as lengths of suspension springs, position of the steering rack, position coordinates of

the center of gravity of the chassis, and three independent Euler parameters. A realistic slip-

based tire model is employed, requiring use of the dual-rate integration method of Section 6 in

integration for wheel rotation. In the simulation, the vehicle is initially moving at a speed of 5

m/s, and is driven for 4 s over a series of bumps.

In order to highlight the improved computational efficiency of the GIC formulation, this

method is compared to the usual generalized coordinate partitioning technique (Haug, 1989).

The topology-based linear solver presented in Section 4 is used to compute generalized

accelerations. The following three cases are compared:

GIC; The globally independent coordinate method, using the topology based linear solver

GCP1; The generalized coordinate partitioning method, using the topology based linear solver

GCP2; The generalized coordinate partitioning method, using sparse linear solver ma47 from

the Harwell library for acceleration computation

The last case serves as a reference, since it is the method currently used in the commercial

multibody dynamics software system DADS. All three cases use a constant step-size third order

Adams-Bashforth (AB3) integrator. All computations are performed on an SGI computer with

200MHz R10000 processors.

29

For integration step-sizes ranging from 10 ms to 1 ms, CPU times for a four second

simulation are plotted in Fig. 5. It can be seen that the computational effort for one integration

step with the GIC formulation is about 4.5 ms. This shows that real-time simulation can be

achieved with the GIC formulation, as long as the integration step-size is 4.5 ms or greater,

which is acceptable for this vehicle model. For the experiment considered, the generalized

coordinate partitioning algorithms use the same subset of independent coordinates throughout the

entire simulation. Therefore, the improved efficiency of the GIC formulation, as compared to

the GCP1 algorithm is due entirely to the dependent coordinate recovery algorithm. The speed-

up obtained for this portion of the numerical simulation is best seen in Table 6, which presents

the timing profile for one step of numerical integration for each of the three algorithms

considered. The proposed computational sequence for dependent coordinate recovery is faster

than the conventional method by a factor of 6.1. Overall, it is a factor of 4.9 times faster.

This example illustrates the feasibility of the GIC formulation in achieving real-time

simulation, with a relatively high fidelity vehicle handling model. The formulation holds great

potential for implementation on shared memory parallel processors, which will enable real-time

simulation of even higher fidelity design oriented models in a virtual proving ground

environment.

30

Figure 5. CPU Time vs. Step-Size for 10-Body HMMWV Simulation

Table 6. Timing Profile for One Integration Step

GIC GCP1 GCP2ms % total %

relativems % total %

relativems % total %

relativeTotal 4.538 100.00 13.988 100.00 22.163 100.00(A) 2.250 49.58 100.00 2.188 15.64 100.00 10.500 47.37 100.00(A1) 0.850 37.78 0.813 37.16 0.325 30.95(A2) 1.400 62.22 1.375 62.84 10.175 69.05(D) 1.700 37.45 100.00 10.363 74.08 100.00 10.288 46.42 100.00(D1) 1.100 64.71 2.288 22.08 2.563 24.91(D2) 0.600 35.29 8.075 77.92 7.725 75.09(O) 0.548 12.96 1.437 10.28 1.375 6.21(A)(A1)(A2)(D)(D1)(D2)(O)

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

31

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.

32

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

33

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.

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