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2003-01-1286

Real-Time Multi-Body Vehicle Dynamics Using A ModularModeling Methodology

Richard RomanoRealtime Technologies, Inc

Copyright 2003 Society of Automotive Engineers, Inc.

ABSTRACT

Simulations of ground vehicles are extensively used bymilitary and commercial vehicle developers to aid in thedesign process. In the past, ground vehicle simulationshave focused on non-real-time models. However withthe advancement of computers and modeling

methodologies, real-time multi-body models havebecome one of the standard tools used by vehicledevelopers. Multi-body models are composed of joint,body, and force elements which map well into a modularmodeling approach. Based on recursive techniques aset of reusable components were developed for use in agraphical simulation and modeling environment. Thecomponents were then connected to form a real-timemulti-body model of a Ford Taurus. Finally, the Taurusmodel was integrated with simulator cueing subsystemsto build a complete driving simulator. The performanceof the Taurus model was compared with test data. Itwas found that the vehicle model was both accurate and

ran much faster than real-time. Due to the modelformulation, the current set of modular components arelimited to modeling open treed systems with either afixed or mobile base body.

INTRODUCTION

Over the past twenty years offline vehicle dynamicssimulations have been used extensively in theautomotive industry. As computer performanceincreased vehicle dynamics applications were introducedinto hardware in the loop simulation and operator in theloop simulation laboratories. To meet the needs of

vehicle designers the complexity and fidelity of thevehicle dynamics used in operator in the loop simulationhas been increasing and typically multi-body dynamicsare used in this environment. Multi-body dynamics isbased on classical mechanics. The multi-body methodutilizes a finite set of elements including rigid bodies,joints, springs, dampers, and actuators. In the first stepsof standardization of a datamodel of multi-body systemsfor use with computer codes, the following assumptionshave been agreed upon [1]:

1. A multi-body system consists of rigid bodies andideal joints. A body may degenerate to aparticle or to a body without inertia.

2. The topology of the multi-body system isarbitrary. Chains, trees and closed loops areadmitted.

3. Bodies, joints and actuators are summarized inlibraries of standard elements.

Kecskemethy [2] developed a formalization orepresenting multi-body elements using a moduladataflow representation. Figure 1 shows a Rigid Linkand Joint. In the figure, q is the position vector of the

frame. The derivatives of q contain the velocities and

accelerations of the frame. Q represents the

generalized forces in each coordinate frame. Thegeneralized forces are made up of both inertial forces

and external forces and are generated by mass, springdamper, or active elements as given in Figure 2.

Figure 1: Rigid Link and Joint (from Krebs [3])

Figure 2: Mass Element and Spring (from Krebs [3])

Each element provides a function f mapping the

position q from the current frame to the next frame

The mapping functions are typically straight forward


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Kecskemethy [2] and Craig [4] give good descriptions ofmapping functions for revolute and prismatic joints. In

the case of a joint, the function f takes into account the

joint position, velocity and acceleration. As given byKrebs [3], the position q and its derivatives in Frame B

can be calculated from Frame A as:

AfAfB

AfB

AB

qJqJq

qJq

qfq

&&&&&&

&&

+=

=

= )(

and the generalized forces of Frame A can be calculatedfrom Frame B as:

B

T

fA QJQ =

where:

Af q

f

J

=

is the Jacobian matrix.

The approach used in this paper is to develop a set ofsoftware components that represent links, joints, bodies,and actuators as defined in Figure 1 and 2. By callingthe software components in a particular order a multi-body structure can be developed as a set of link, joint,mass, and force elements. By calculating inside eachjoint software component a joint torque as the dot

product between the generalized force Q acting on the

joint and the joint axis, the set of joint torques required togenerate a particular mechanism acceleration, positionand velocity can be determined. The resulting softwarestructure solves the inverse dynamics problem: whatjoint torques are required to yield a particular set ofpositions, velocities, and accelerations of a mechanism.

Introducing a set of generalized coordinates: , that

represent joint angles and position, in general the set ofjoint torques required will take the following form:

)(),()( ++= GVM &&&

where is defined as the mass matrix, V representscentrifugal and coriolis effects, and G representsgravity effects. With a set of software thatcomputationally solves the inverse dynamics problem,the difficulty is solving the forward dynamics problem.That is given a set of joint torques what is the resultingmotion of the actuator. Walker and Orin [5] developed

several methods of solving for the joint accelerations &&

given a set of joint torques. Once the joint accelerationsare known the joint velocities and positions can becalculated using numerical integration techniques. Themethod used in this paper is similar to Method 3 in [5].

For a particular time step the method can be describedas follows:

1. Using the joint positions and velocities from the

previous step calculate a torque bias vector bthat is the torque at each joint required for zero

joint acceleration: )(),( += GVb & .

2. With the joint velocities and gravity set to zero

(i.e. with the torque bias vector set to zero)calculate the joint torque vector for a unit jointacceleration for each joint separately (i.e. alother joint accelerations set to zero). The jointorques calculated are essentially a column othe mass matrix. Once all the columns of themass matrix have been computed, and giventhe actual torques acting at each joint, theacceleration of the joints can be calculated as

))((1 bM = &&

The joint accelerations can then be integrated toyield the joint positions and velocities for thenext step.

IMPLEMENTATION

The links, joints, bodies, and actuators were developedand assembled as modular components in SimCreator

a

graphical simulation tool from Realtime TechnologiesInc. The link or offset component and a revolutemechanism are shown in Figure 3.

OffsetMB

Revolute

LinAccel1LinVel1

LinPos1AngAcc el1AngVel1

TM1Force1

Moment1

LinAccel3LinVel3

LinPos3AngAccel3AngVel3TM3Force3Moment3

JointAngJointAngRate

ExternalJointTorque

LinkNumber1

LocalJointAxis

LinkNumber3

LinAccel1LinVel1

LinPos1AngAcc el1

AngVel1TM1

Force1Moment1

Offset

LinAccel3LinVel3LinPos3

AngAccel3AngVel3TM3Force3Moment3

LinkNumber1 LinkNumber3

Figure 3: SimCreator Based Link and Joint


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In the SimCreator components in Figure 3, the datapassed downstream are the linear acceleration, velocityand position of the frame (LinPos1, LinVel1, LinAccel1),and the angular acceleration, velocity and transformationmatrix of the frame (AngAccel1, AngVel1, TM1). Thegeneralized forces are passed back upstream in Force1and Moment1. Finally a set of link numbers are passeddownstream. These itemized the set of joints that areupstream of the current component. In this way jointforces and accelerations can be associated with thecorrect set of joints. The velocity and position vectors(LinVel1, LinPos1, and AngVel1) are each of lengththree, containing the component values for threedimensions. The acceleration vectors, forces andmoments are each a 3xN matrix. Each index across theacceleration matrix represents the acceleration in theframe due to a unit acceleration of a particular joint.Each index across the forces and moments matricesrepresents the generalized forces due to a unitacceleration of a particular joint. These matrices areused to calculate the columns of the mass matrix. Dueto the model formulation, the current set of modularcomponents are limited to modeling open treed systems

with either a fixed or mobile base body.

VALIDATION

A double pendulum was modeled using the developedcomponents. This is shown in Figure 4.

Sum1

Gain3

Gain2

Gain1

SumGain

OffsetMB1 RevoluteJoint1 OffsetMB RevoluteJoint

Figure 4: Double Pendulum

The double pendulum was configured to have the samegeometry as Hwang [6]. Hwang performed a dynamicanalysis of a double pendulum using DADS and twoother multi-body dynamics packages. The result of theirDADS model is given in Figure 5. The results of theSimCreator model are given in Figure 6. Comparison ofthe results of several plots generated in SimCreatorshowed excellent agreement with Hwang.

VEHICLE MODEL

A vehicle dynamics model was developed similar toSayers [7]. The vehicle was modeled with a base bodywith six degrees of freedom and four prismatic jointsrepresenting each of the corners of the vehicle. Theprismatic joints were tilted to take into account anti-dive,anti-squat geometry and roll center height. The vehicle

model is shown in Figure 8. The powertrain, shown inFigure 7, and other subsystems were modeled in asimilar fashion to previous vehicle models developedusing SimCreator [8].

Figure 5: DADS Position of Second Pendulum (from Hwang [6])

Figure 6: SimCreator Position of Second Pendulum

SumFL

SumFRSumRR

SumRL

Brakes

TorqueConverter

GearBoxEngine

FrontDiffFRWheel

I

FLWheel

I

RRWheel

I

RLWheel

I

COMPONENT

Figure 4: Front Wheel Drive Powertrain

-4

-3

-2

-1

0

1

2

3

4

0 1 2 3 4 5

ydotofthesecondpendulum(

m/s)

Time (seconds)


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Figure 8: Complete Vehicle Dynamics

SimCreator components shown in Figure 8 are asfollows: PowerTrain calculates all powertrain andbrake system effects and outputs the four wheel speedsof the vehicle, SimpAero calculates the aerodynamicforces on the vehicle, SixDOFBody is the base body ofthe multi-body dynamics model, Offset calculates theposition, velocity and acceleration at an offset and

translates the generalized forces at the offset back to thecenter of gravity, CornerForces sums up thegeneralized forces of the multi-body tree structure,Corner calculates the independent suspension and thetire model at a vehicle corner, and outputs the forcesacting at the corner and the torque acting on the wheel.The Corner module contains inside it the prismatic jointas well as a tire model.

Figure 9: Lateral Response of Ford Taurus (from Salaani [10])

The vehicle dynamics model was configured torepresent a Ford Taurus using parameters collected bySalaani et al [9][10][11]. The Ford Taurus vehicleparameters collected by Salaani included mass

properties, offsets, suspension and tire characteristicsand powertrain information. In addition to parametersSalaani collected test data from an actual vehicle. Thetest data was used to validate the Ford Taurus modelFigures 9 and 10 show the roll response of the model toa lateral handling maneuver using a smooth low ratesteering input at 32 m/s. The dotted line in Figure 9represents test data. The results match up well with theroll angles predicted by the SimCreator model in Figure10. A variety of maneuvers and responses were tested

and compared with Salaani [10]. Results were found tobe reasonably accurate given that all of the vehicleparameters and all the test inputs (steering wheelaccelerator and brake force) were not available.

-2

-1

0

1

2

3

4

-4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5

RollAngle(deg)

Lateral Acceleration (m/s/s)

Figure 10: Lateral Response of SimCreator Model

DRIVING SIMULATOR

The multi-body vehicle dynamics were inserted into adriving simulator to test its use in an operator in the loop

ForcesMergeMBForce

AckSteer

AeroOffsetSimpAero

PowerTrain

OffsetRL

OffsetRR

OffsetFL

OffsetFR

CornerForces SixDOFBody

RLCorner

RRCornerFRCorner

FLCorner

COMPONENT


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environment. A typical driving simulator has thefollowing cueing systems: audio, out the windowgraphics, driving control interface, and motion. Thesecan be encapsulated as modules and integrated with thevehicle dynamics to form the driving simulator shown inFigure 11.

Motion

Visuals

VehInput

TM2Euler

,,

Audio

Dynamics

Figure 11: Driving Simulator

PERFORMANCE

The vehicle dynamics shown in Figure 8 contains 61states including the powertrain. They were integratedusing a Runge Kutta second order method. Running thevehicle dynamics on a single 600 MHz Pentium IIIprocessor was found to take 0.2125 ms per update. Atypical update rate for the model is 2.5 ms leaving plentyof time for other calculations.

ADVANTAGES OF THE GRAPHICAL ENVIRONMENT

There are several advantages to using a graphicalsimulation tool. The simulation tool provides astructured framework to build C Code components in.The simulation tool executes the multi-body componentsin the model such that all data required for the currentcomponent has been calculated in previouscomponents. Therefore the order of calculation ofposition, velocity, and force analysis steps, which wouldnormally be laid out explicitly in recursive dynamicsformulations, are automatically determined by theframework. Graphical simulation tools provide a unifiedintegration algorithm with the ability to globally setintegration methods and integration time steps. It istypically easy to add states to the integrator from anycomponent. Any component output can be easilycollected as data during the simulation and plotted.Inputs to the model and initial conditions of the statescan be accessed from a single user interface. Thegraphical environment allows users greater insight intothe model being developed. It also allows easyconnections between components and provides for ahierarchical view of the model. In addition componentsand models are fully encapsulated and easy to share

between users. This encourages component and modereuse.

CONCLUSION

It was found that a multi-body vehicle dynamics modeand a driving simulator could be designed using modularcomponents. Links, joints, masses, and forces mappedwell into independent components. The multi-bodycomponents were found to accurately model both adouble pendulum and a Ford Taurus. The performanceof the model was found to be extremely efficient takingonly 0.2125 ms per time step to model the Ford Tauruson a 600 MHz Pentium III. This makes the model usablefor real-time simulation.

CONTACT

Richard Romano has been working in driving simulationfor twelve years focusing on motion cueing, vehicledynamics, and human factors research. Dr. Romanowas the manager of simulator research anddevelopment at the Iowa Driving Simulator and

supervised the brake system simulation group at ITTAutomotive. He is now president of RealtimeTechnologies, Inc. He may be contacted [email protected].

REFERENCES

1. Schiehlen, W., Multibody System Dynamics: Roots andPerspectives, Multibody System Dynamics 1: 149-1881997.

2. Kinematics and Dynamics of Multi-Body Systems, Eds. JAngeles and A. Kecskemthy, CISM Courses andLectures No. 360, Springer-Verlag, Wien, New York, 1995

3. Krebs, M., Vehicle Modeling for High

Dynamic Driving Simulator Applications, Proceedings othe 1st Human-Centered Transportation SimulationConference, Iowa City, IA, 2001.

4. Craig, J.J., Introduction to Robotics Mechanics andControl, Addison Wesley, New York, 1989.

5. Walker, M. W. and D. E. Orin., Efficient DynamicComputer Simulation of Robotic Mechanisms, ASMEJournal of Dynamic Systems, Measurements and ControlVol. 104, 1982, pp. 205-211.

6. Hwang, H.Y., Kim, S.S., Haug, E.J., and H.J. LaiTheoretical Cross Verifications and Comparative Studiesof Multibody Simulation Codes DADS, DISCOS, andContops, Technical Report R-66, Center for Simulationand Design Optimization, University of Iowa, 1988.

7. Sayers, M. W. and D. Han. A Generic Multibody Vehicle

Model for Simulating Handling and Braking. Journal oVehicle System Dynamics, Supplement 25, 1996, pp. 599613.

8. Romano, R., Realtime Driving Simulation Using AModular Modeling Methodology, SAE Technical PapeSeries No. 2000-01-1297, March 2000.

9. Salaani, M.K., Parameter Measurement andDevfelopment of a NADSdyna Validation Data Set for a1994 Ford Taurus, SAE Technical Paper Series No970564, February, 1997.

10. Salaani, M.K., Heydinger, G.J., and D.A. GuentherValidation Results from Using NADSdyna VehicleDynamics Simulation, SAE Technical Paper Series No970565, February, 1997.


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