Real-Time Multi-Body Vehicle Dynamics Using A Modular Modeling Methodology

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1 Real-Time Multi-ody Vehicle Dynamics Using Modular Modeling Methodology Copyright 2003 Society o utomotive Engineers, nc. Richard Romano Realtime Technologies, nc. STRCT Simulations o ground vehicles are extensively used by military and commercial vehicle developers to aid in the design process. n the past, ground vehicle simulations have ocused on non-real-time models. However with the advancement o computers and modeling methodologies, real-time multi-body models have become one o the standard tools used by vehicle developers. Multi-body models are composed o joint, body, and orce elements which map well into a modular modeling approach. ased on recursive techniques a set o reusable components were developed or use in a graphical simulation and modeling environment. The components were then connected to orm a real-time multi-body model o a Ford Taurus. Finally, the Taurus model was integrated with simulator cueing subsystems to build a complete driving simulator. The perormance o the Taurus model was compared with test data. t was ound that the vehicle model was both accurate and ran much aster than real-time. Due to the model ormulation, the current set o modular components are limited to modeling open treed systems with either a ixed or mobile base body. 1. multi-body system consists o rigid bodies and ideal joints. body may degenerate to a particle or to a body without inertia. 2. The topology o the multi-body system is arbitrary. Chains, trees and closed loops are admitted. 3. odies, joints and actuators are summarized in libraries o standard elements. Kecskemethy [2] developed a ormalization o representing multi-body elements using a modular datalow representation. Figure 1 shows a Rigid Link and Joint. n the igure, q is the position vector o the rame. The derivatives o q contain the velocities and accelerations o the rame. Q represents the generalized orces in each coordinate rame. The generalized orces are made up o both inertial orces and external orces and are generated by mass, spring, damper, or active elements as given in Figure 2. NTRODUCTON Over the past twenty years oline vehicle dynamics simulations have been used extensively in the automotive industry. s computer perormance increased vehicle dynamics applications were introduced into hardware in the loop simulation and operator in the loop simulation laboratories. To meet the needs o vehicle designers the complexity and idelity o the vehicle dynamics used in operator in the loop simulation has been increasing and typically multi-body dynamics are used in this environment. Multi-body dynamics is based on classical mechanics. The multi-body method utilizes a inite set o elements including rigid bodies, joints, springs, dampers, and actuators. n the irst steps o standardization o a datamodel o multi-body systems or use with computer codes, the ollowing assumptions have been agreed upon [1]: Figure 1: Rigid Link and Joint (rom Krebs [3]) Figure 2: Mass Element and Spring (rom Krebs [3]) Each element provides a unction mapping the position q rom the current rame to the next rame. The mapping unctions are typically straight orward.

2 Kecskemethy [2] and Craig [4] give good descriptions o mapping unctions or revolute and prismatic joints. n the case o a joint, the unction takes into account the joint position, velocity and acceleration. s given by Krebs [3], the position q and its derivatives in Frame can be calculated rom Frame as: q q& q&& = ( q ) = J = J q& q&& + J& q& and the generalized orces o Frame can be calculated rom Frame as: where: Q is the Jacobian matrix. J = J T = q Q The approach used in this paper is to develop a set o sotware components that represent links, joints, bodies, and actuators as deined in Figure 1 and 2. y calling the sotware components in a particular order a multibody structure can be developed as a set o link, joint, mass, and orce elements. y calculating inside each joint sotware component a joint torque τ as the dot product between the generalized orce Q acting on the joint and the joint axis, the set o joint torques required to generate a particular mechanism acceleration, position and velocity can be determined. The resulting sotware structure solves the inverse dynamics problem: what joint torques are required to yield a particular set o positions, velocities, and accelerations o a mechanism. ntroducing a set o generalized coordinates: Θ, that represent joint angles and position, in general the set o joint torques τ required will take the ollowing orm: τ = M ( Θ) Θ && + V ( Θ, Θ& ) + G( Θ) where M is deined as the mass matrix, V represents centriugal and coriolis eects, and G represents gravity eects. With a set o sotware that computationally solves the inverse dynamics problem, the diiculty is solving the orward dynamics problem. That is given a set o joint torques what is the resulting motion o the actuator. Walker and Orin [5] developed several methods o solving or the joint accelerations Θ & given a set o joint torques. Once the joint accelerations are known the joint velocities and positions can be calculated using numerical integration techniques. The method used in this paper is similar to Method 3 in [5]. For a particular time step the method can be described as ollows: 1. Using the joint positions and velocities rom the previous step calculate a torque bias vector b that is the torque at each joint required or zero joint acceleration: b = V ( Θ, Θ& ) + G( Θ). 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 or a unit joint acceleration or each joint separately (i.e. all other joint accelerations set to zero). The joint torques calculated are essentially a column o the mass matrix. Once all the columns o the mass matrix have been computed, and given the actual torques acting at each joint, the acceleration o the joints can be calculated as: 1 Θ & = M ( Θ)( τ b) The joint accelerations can then be integrated to yield the joint positions and velocities or the next step. MPLEMENTTON The links, joints, bodies, and actuators were developed and assembled as modular components in SimCreator a graphical simulation tool rom Realtime Technologies, nc. The link or oset component and a revolute mechanism are shown in Figure 3. Linccel1 LinVel1 LinPos1 ngccel1 ngvel1 TM1 Force1 Moment1 LinkNumber1 Linccel1 LinVel1 LinPos1 ngccel1 ngvel1 TM1 Force1 Moment1 LinkNumber1 Oset OsetM LocalJointxis Revolute Figure 3: SimCreator ased Link and Joint Linccel3 LinVel3 LinPos3 ngccel3 ngvel3 TM3 Force3 Moment3 LinkNumber3 Linccel3 LinVel3 LinPos3 ngccel3 ngvel3 TM3 Force3 Moment3 LinkNumber3 Jointng JointngRate ExternalJointTorque

3 n the SimCreator components in Figure 3, the data passed downstream are the linear acceleration, velocity and position o the rame (LinPos1, LinVel1, Linccel1), and the angular acceleration, velocity and transormation matrix o the rame (ngccel1, ngvel1, TM1). The generalized orces are passed back upstream in Force1 and Moment1. Finally a set o link numbers are passed downstream. These itemized the set o joints that are upstream o the current component. n this way joint orces and accelerations can be associated with the correct set o joints. The velocity and position vectors (LinVel1, LinPos1, and ngvel1) are each o length three, containing the component values or three dimensions. The acceleration vectors, orces and moments are each a 3xN matrix. Each index across the acceleration matrix represents the acceleration in the rame due to a unit acceleration o a particular joint. Each index across the orces and moments matrices represents the generalized orces due to a unit acceleration o a particular joint. These matrices are used to calculate the columns o the mass matrix. Due to the model ormulation, the current set o modular components are limited to modeling open treed systems with either a ixed or mobile base body. VLDTON model is shown in Figure 8. The powertrain, shown in Figure 7, and other subsystems were modeled in a similar ashion to previous vehicle models developed using SimCreator [8]. Figure 5: DDS Position o Second Pendulum (rom Hwang [6]) 4 double pendulum was modeled using the developed components. This is shown in Figure 4. OsetM1 RevoluteJoint1 OsetM RevoluteJoint ydot o the second pendulum (m/s) Gain Sum Gain2 Sum Time (seconds) Figure 6: SimCreator Position o Second Pendulum Gain1 Gain3 Figure 4: Double Pendulum The double pendulum was conigured to have the same geometry as Hwang [6]. Hwang perormed a dynamic analysis o a double pendulum using DDS and two other multi-body dynamics packages. The result o their DDS model is given in Figure 5. The results o the SimCreator model are given in Figure 6. Comparison o the results o several plots generated in SimCreator showed excellent agreement with Hwang. SumFR SumFL FRWheel FrontDi rakes COMPONENT SumRL SumRR RRWheel VEHCLE MODEL vehicle dynamics model was developed similar to Sayers [7]. The vehicle was modeled with a base body with six degrees o reedom and our prismatic joints representing each o the corners o the vehicle. The prismatic joints were tilted to take into account anti-dive, anti-squat geometry and roll center height. The vehicle FLWheel TorqueConverter Engine Figure 4: Front Wheel Drive Powertrain Gearox RLWheel

4 cksteer COMPONENT PowerTrain FRCorner OsetFR Simpero erooset OsetRR RRCorner CornerForces SixDOFody FLCorner OsetFL OsetRL RLCorner Figure 8: Complete Vehicle Dynamics SimCreator components shown in Figure 8 are as ollows: PowerTrain calculates all powertrain and brake system eects and outputs the our wheel speeds o the vehicle, Simpero calculates the aerodynamic orces on the vehicle, SixDOFody is the base body o the multi-body dynamics model, Oset calculates the position, velocity and acceleration at an oset and translates the generalized orces at the oset back to the center o gravity, CornerForces sums up the generalized orces o the multi-body tree structure, Corner calculates the independent suspension and the tire model at a vehicle corner, and outputs the orces acting at the corner and the torque acting on the wheel. The Corner module contains inside it the prismatic joint as well as a tire model. ForcesMergeMForce properties, osets, suspension and tire characteristics, and powertrain inormation. n addition to parameters, Salaani collected test data rom an actual vehicle. The test data was used to validate the Ford Taurus model. Figures 9 and 10 show the roll response o the model to a lateral handling maneuver using a smooth low rate steering input at 32 m/s. The dotted line in Figure 9 represents test data. The results match up well with the roll angles predicted by the SimCreator model in Figure 10. variety o maneuvers and responses were tested and compared with Salaani [10]. Results were ound to be reasonably accurate given that all o the vehicle parameters and all the test inputs (steering wheel, accelerator and brake orce) were not available Roll ngle (deg) Figure 9: Lateral Response o Ford Taurus (rom Salaani [10]) Lateral cceleration (m/s/s) Figure 10: Lateral Response o SimCreator Model The vehicle dynamics model was conigured to represent a Ford Taurus using parameters collected by Salaani et al [9][10][11]. The Ford Taurus vehicle parameters collected by Salaani included mass DRVNG SMULTOR The multi-body vehicle dynamics were inserted into a driving simulator to test its use in an operator in the loop

5 environment. typical driving simulator has the ollowing cueing systems: audio, out the window graphics, driving control interace, and motion. These can be encapsulated as modules and integrated with the vehicle dynamics to orm the driving simulator shown in Figure 11. Vehnput Dynamics Visuals Figure 11: Driving Simulator PERFORMNCE φ,θ,ψ TM2Euler udio Motion The vehicle dynamics shown in Figure 8 contains 61 states including the powertrain. They were integrated using a Runge Kutta second order method. Running the vehicle dynamics on a single 600 MHz Pentium processor was ound to take ms per update. typical update rate or the model is 2.5 ms leaving plenty o time or other calculations. DVNTGES OF THE GRPHCL ENVRONMENT There are several advantages to using a graphical simulation tool. The simulation tool provides a structured ramework to build C Code components in. The simulation tool executes the multi-body components in the model such that all data required or the current component has been calculated in previous components. Thereore the order o calculation o position, velocity, and orce analysis steps, which would normally be laid out explicitly in recursive dynamics ormulations, are automatically determined by the ramework. Graphical simulation tools provide a uniied integration algorithm with the ability to globally set integration methods and integration time steps. t is typically easy to add states to the integrator rom any component. ny component output can be easily collected as data during the simulation and plotted. nputs to the model and initial conditions o the states can be accessed rom a single user interace. The graphical environment allows users greater insight into the model being developed. t also allows easy connections between components and provides or a hierarchical view o the model. n addition components and models are ully encapsulated and easy to share between users. This encourages component and model reuse. CONCLUSON t was ound that a multi-body vehicle dynamics model and a driving simulator could be designed using modular components. Links, joints, masses, and orces mapped well into independent components. The multi-body components were ound to accurately model both a double pendulum and a Ford Taurus. The perormance o the model was ound to be extremely eicient taking only ms per time step to model the Ford Taurus on a 600 MHz Pentium. This makes the model usable or real-time simulation. CONTCT Richard Romano has been working in driving simulation or twelve years ocusing on motion cueing, vehicle dynamics, and human actors research. Dr. Romano was the manager o simulator research and development at the owa Driving Simulator and supervised the brake system simulation group at TT utomotive. He is now president o Realtime Technologies, nc. He may be contacted at raromano@ix.netcom.com. REFERENCES 1. Schiehlen, W., Multibody System Dynamics: Roots and Perspectives, Multibody System Dynamics 1: , Kinematics and Dynamics o Multi-ody Systems, Eds. J. ngeles and. Kecskeméthy, CSM Courses and Lectures No. 360, Springer-Verlag, Wien, New York, Krebs, M., Vehicle Modeling or High- Dynamic Driving Simulator pplications, Proceedings o the 1st Human-Centered Transportation Simulation Conerence, owa City,, Craig, J.J., ntroduction to Robotics Mechanics and Control, ddison Wesley, New York, Walker, M. W. and D. E. Orin., Eicient Dynamic Computer Simulation o Robotic Mechanisms, SME Journal o Dynamic Systems, Measurements and Control, Vol. 104, 1982, pp Hwang, H.Y., Kim, S.S., Haug, E.J., and H.J. Lai, Theoretical Cross Veriications and Comparative Studies o Multibody Simulation Codes DDS, DSCOS, and Contops, Technical Report R-66, Center or Simulation and Design Optimization, University o owa, Sayers, M. W. and D. Han. Generic Multibody Vehicle Model or Simulating Handling and raking. Journal o Vehicle System Dynamics, Supplement 25, 1996, pp Romano, R., Realtime Driving Simulation Using Modular Modeling Methodology, SE Technical Paper Series No , March Salaani, M.K., Parameter Measurement and Development o a NDSdyna Validation Data Set or a 1994 Ford Taurus, SE Technical Paper Series No , February, Salaani, M.K., Heydinger, G.J., and D.. Guenther, Validation Results rom Using NDSdyna Vehicle Dynamics Simulation, SE Technical Paper Series No , February, 1997.

6 11. Salaani, M.K., and G.J. Heydinger, Powertrain and rake Modeling o the 1994 Ford Taurus or the National dvanced Driving Simulator, SE Technical Paper Series No , March, 1998.