(Steering Sys of Car)Sdarticle

Journal of the Franklin Institute 338 (2001) 21–34

A bond graph model incorporating sensors,actuators, and vehicle dynamics for developing

controllers for vehicle safety

Donald Margolis*, Taehyun Shim

Department of Mechanical and Aeronautical Engineering, University of California, One Shields Ave. Davis,

CA 95616, USA

Received 23 February 2000; received in revised form 21 September 2000

Abstract

The process of developing controllers for vehicle dynamics requires reasonable models thatexpose the important dynamic effects without being excessively complicated. The controllers

require sensors, signal processing, and actuation, and these parts must be incorporated anddemonstrated using vehicle models. Bond graphs are a concise pictorial representation of alltypes of interacting energy domains, and are an excellent tool for representing vehicle

dynamics with associated control hardware. This paper develops a four-wheel, nonlinearvehicle dynamic model with electrically controlled brakes and steering, as well as control ateach suspension corner. Controllers are not developed here but are demonstrated through

simulation. # 2001 The Franklin Institute. Published by Elsevier Science Ltd. All rightsreserved.

Keywords: Bond graph; Vehicle dynamics; Vehicle control; System modeling

1. Introduction

There are many commercially available vehicle stability control systems. ABSbrakes, traction control, and yaw rate control are examples. All can be had onvarious vehicles of the world. The concepts behind these systems were proposed byvehicle researchers, prototyped by development engineers, and commercialized byproduct engineers. Refs. [1–7] present some of the development of such systems. At

* Corresponding author. Tel.: (530)752-1446; fax: (530)752-4158.

E-mail address: [email protected] (D. Margolis).

0016-0032/01/$20.00 # 2001 The Franklin Institute. Published by Elsevier Science Ltd. All rights reserved.

PII: S 0 0 1 6 - 0 0 3 2 ( 0 0 ) 0 0 0 6 8 - 5

each stage of the development process, models are needed to assist in thinking aboutand communicating ideas. At the concept level, these models can be quite simple. Atthe prototype testing level, the models might be required to be very detailed andcomplex. In Ref. [8], a one-wheel model of a car was presented that allowed bothvertical and horizontal vehicle motions. Steering, brake, and suspension actuatorscould be included and control concepts could be proposed and developed.Obviously, a one-wheel model is about as simple a conceptual model as possible.

Bond graphs were used to develop the one-wheel model [9]. Bond graphs are aconcise pictorial representation of all types of dynamic interactions and areparticularly well suited when actuators and control are included in an otherwisemechanical model of a vehicle. In this paper the one-wheel model of [8] is extended toinclude four wheels, but still subject to simplifying assumptions that allows themodel to expose the essence of the vehicle dynamics. Bond graphs are used for thisdevelopment. The model includes a suspension at each corner of the vehicle alongwith an actuator between each vehicle corner and the unsprung mass at that corner.The model includes individual electric brake actuators. A steering actuator isincluded that has a differential in the steering column so that an electric motor cansteer additionally to the command from the driver without forcing the driver’s handsto move. The intended use of the model is for control concept development forimproved safety in vehicles.

Models that include sensors, actuators, and controls in overall physical systemmodels are commonly referred to as Mechatronic Systems. When such models areconstructed, each component is modeled separately. For the case here, the vehicle ismodeled separately from the brake actuators, which are modeled separately from thesteering actuator, and so forth. Ultimately, these submodels must be assembled intoan overall system model suitable for analysis or simulation. For a convenientlycomputable model to result, the outputs from each submodel must be received asinputs by any attached submodel. When the submodels are represented with bondgraphs, all input/output relationships are immediately known through application ofcausality. The demonstration of this assemblage of submodels into a computableoverall model for a complex system is a major contribution of this paper.

2. Model development

Fig. 1 shows a vehicle schematic for the system. The body is modeled as rigid withbody fixed coordinates, xyz, attached at the cg and aligned in principal directions.The body has mass, m, and moments inertia Jr (roll) about the x-axis, Jp (pitch)about the y-axis, and Jy (yaw) about the z-axis. The cg is located, a, from the frontaxle, b, from the rear axle, and, h, from the ground. The half-width of the vehicle isw=2. The front wheels can be steered through the angle, d, although the steeringactuator is not shown in this figure.

A suspension unit is shown at each corner of the vehicle body. They consist of aspring, damper, and force actuator. The constitutive behavior of these elements canbe specified and are not necessarily linear. The force actuator is not reticulated

D. Margolis, T. Shim / Journal of the Franklin Institute 338 (2001) 21–3422

further in this presentation. It would be an electro-mechanical, -pneumatic, or,-hydraulic device. It might be fully active or semi-active. Here it is simply acontrolled force and will be demonstrated through example below.

The wheels are each modeled by their rotary inertia, Jw, mass, mu, tire stiffness, kt,and radius, Rw. Also shown in Fig. 1 are the forces at the contact patch of the rightfront, rf, wheel. There are similar forces at each wheel.

Fig. 2 shows just the vehicle body. The velocity and angular velocity componentswith respect to the body-fixed coordinates are indicated at the cg. The vehicle hasforward velocity, U, lateral velocity, V, and vertical velocity,W. There is roll angularvelocity, or, pitch angular velocity, op, and yaw angular velocity, oy. The body fixedvelocity components are shown at the right front wheel, having been transferredfrom the cg. It has been assumed that the forward velocity, U, is much larger thanthe other velocity components. It has also been assumed that the roll and pitchangular velocities do not contribute substantially to the lateral velocity of the tire.The body-fixed velocity components at the other wheels are shown with arrows only.

Fig. 3 shows the top view of one wheel and the attached electric brake system. Theelectric motor drives a ball-screw which pushes the brake pad into the disk. There isalso the possibility of an applied torque, ta, from the engine. Also shown in the figureis the bond graph fragment representing the wheel and brake system. Thelongitudinal force, FL, is from the modulated R-element. This element has an inputof the difference between the longitudinal velocity of the wheel center and the rollingvelocity of the wheel, and outputs the longitudinal force. The constitutive law for thiselement could come from any of the excellent empirical models of pneumatic tires(Ref. [10], for instance), or it could be simply a linear relationship typically used forlinear analysis.

Fig. 4 shows the steering system comprised of a steering wheel input from thedriver and an additional input from an electric motor through a differential. This is

Fig. 1. Overall schematic.

D. Margolis, T. Shim / Journal of the Franklin Institute 338 (2001) 21–34 23

not a conventional power steering system. If a controller is to be allowed someamount of control over the steered wheels, then a system such as this would have tobe considered. The differential allows the electric motor to add to the command fromthe driver without forcing the wheel from the driver’s hands. Systems such as thishave been investigated and Refs. [11,12] develops such a system.

A bond graph fragment of the steering system is shown in Fig. 4. It includes theelectric motor and differential. The cornering force, FC, acts through the contactpatch, which is behind the vertical rotational axis of the tire by some distance, Ca,

Fig. 2. Schematic and bond graph of rigid body.

Fig. 3. Schematic and bond graph of electric brakes for one wheel.

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known as the castor. This force effects the steering system as indicated in the bondgraph.

Each corner of the vehicle has a suspension unit as shown in Fig. 5. The unsprungmass, mu, represents the tire and wheel and other mass that moves vertically with thewheel. The suspension components consist of a spring, ks, damper, bs, and a forcegenerator, FCij

. The spring and damper could be nonlinear by inserting theappropriate constitutive law while solving the equations of motion for the entire

Fig. 4. Schematic and bond graph of power steering system.

Fig. 5. Schematic and bond graph of one suspension unit.

D. Margolis, T. Shim / Journal of the Franklin Institute 338 (2001) 21–34 25

system. Realistic kinematics could also be included. However, for the intended use ofthis model, the simple linear spring and damper is fine. A bond graph fragment of thesuspension unit is also shown in Fig. 5.

For the control force, FCij, an actuator is not modeled here. The actual force

generator could be hydraulic, pneumatic, or electro-mechanical. Rather thanselecting any particular actuator, the resulting control force is specified.

3. Assembling the overall model

We are now in a position to assemble all the subsystems consisting of the body,wheels, brakes, and suspension into an overall model. Basically, it is only necessaryto attach the bond graph fragments together appropriately and use causality todetermine the complexity of equation formulation.

The bond graph of the body is shown in Fig. 6. Along the top are the inertialelements that totally characterize the kinetic energy of the body. The modulatedgyrator (MGY) elements are necessary when using body-fixed coordinates. It shouldbe noted that the general representation of a rigid body with body-fixed coordinatesrequires modulated gyrator coupling among all velocity components and among allangular velocity components [9]. Here it has been assumed that the forward velocity,U, is large compared to all other velocity components and that angular velocitycomponents are all small and result in small body angles. This allows the modulatedgyrator couplings to be reduced to those shown in Fig. 6. The bond graph reflects allthe modeling assumptions discussed previously concerning the contribution of thedifferent velocity and angular velocity components to the slip, slip angle, and vertical

Fig. 6. Bond graph of body showing interaction with the corners of the vehicle.

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velocity at each wheel. The causality is such that the inertial elements prescribeoutput ‘‘flows’’, and determine the longitudinal, lateral, and vertical velocities ateach wheel. These velocity components are distributed along the bottom of Fig. 6.

Fig. 7 shows a bond graph of the rest of the system. At the top of the figure, all thevelocity components from Fig. 6 are exposed. These ‘‘flows’’ are causal inputs to thelongitudinal and lateral slip R-elements, as well as the inputs to the top of eachsuspension element. The longitudinal, lateral, and vertical force components are alllabeled on the bond graph. These forces are the causal inputs to the body model ofFig. 6. The electric brake models are clearly exposed, as are the control forces for thesuspension.

The resulting overall model possesses all integral causality, and there are noalgebraic problems. This means that equation formulation is straightforward andwill result in first-order explicit differential equations. It also means that theequations could be derived automatically by a computer. Each inertial element willcontribute one state each yielding six state equations. The suspension, as currentlymodeled, will contribute 3 states at each corner, or a total of 12 state equations. Thesteering actuator will contribute only one state, and the electric brakes, as modeled

Fig. 7. Bond graph showing interaction of velocities at vehicle corner with longitudinal, transverse, and

vertical force generators.

D. Margolis, T. Shim / Journal of the Franklin Institute 338 (2001) 21–34 27

here, will not contribute any additional states. The rotary inertia of each wheelcontributes an additional 4 states to the system.

The entire overall nonlinear system can be represented by 23 first-order stateequations. These will be of the form

_x ¼ f ðx; uÞ; ð1Þwhere x is the state vector and u is the vector of inputs. The inputs include thevelocity at the bottom of each tire, the commanded steering from the driver, thecurrent in the steering control motor, the current into each brake actuator, andthe control forces at each suspension unit.

4. Demonstration of the overall model

The equations were derived directly from the bond graph and simulated usingACSL [13]. ACSL is one of several excellent commercial software packages used tosolve explicit first-order differential equations. The complete set of equations is notshown here, but the modulated-R elements representing the tire force generation arediscussed, as well as the control used for this demonstration.

The longitudinal forces, FLij, are calculated from the modulated-R element

indicated in Fig. 3 and again in Fig. 7. This element has the slip velocity as input andoutputs the force. The longitudinal slip, sij , is given by

sij ¼Rwowij

ÿ vlij

vlij�� �� ð2Þ

and the force is calculated from

FLij¼ SlsijNoij

Nij

Noij

ð3Þ

where Sl is the slope of the slip curve. The lateral force at each wheel, FCij, is the

output from the modulate-R elements indicated in Fig. 7. These elements use thelateral slip angle, aij, to compute the output force. The input to these -R elements isthe lateral slip velocity such that

aij ¼vtij

vlijð4Þ

and

FCij¼ CaNoijaij

Nij

Noij

ð5Þ

For both longitudinal and lateral forces, the normal force, Nij, comes from the tirestiffness as indicated in Fig. 5.

For the demonstration here, the suspension control is providing ‘‘skyhook’’damping at each corner. This means that the control force is always opposing theabsolute vertical velocity at each corner. This type of control has been shown toprovide excellent ride while also controlling wheel hop.

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The steering actuator and brake actuators are both being used to assist steering thevehicle to maintain a desired yaw rate. The desired yaw rate is that of a neutralhandling vehicle. For one case, only the steering actuator is used to adjust the yawrate, and in the other case only the brakes are used to adjust the yaw rate. In bothcases there is road input at the wheels creating normal force variations. The roadinput is a random velocity input at each wheel scaled so as to produce reasonablevertical cg acceleration. The roadway would be characteristic of a rough road. Thecontroller is a simple PI control with gains selected to yield reasonable response. Ofcourse, a much more sophisticated and robust control could be designed, but that isfor what the model is to be used.

For the demonstration the vehicle is steered to the left over a 1 second time periodand then held at a constant steer angle. The parameters used in this simulation areshown in Table 1. The vehicle is ‘‘understeer’’ as can be seen in Fig. 8. The vehicleattains a steady state yaw rate of about 80% of that of a neutral steer car. The lateralacceleration a is about 0.2g but is modulated by the normal force variation caused bythe rough road (see Fig. 9).

Fig. 10 shows the yaw response of the vehicle with a steering controller. Theundersteer vehicle now attains the yaw rate of a neutral steer vehicle by controllingthe current into the steering motor as shown in Fig. 11. Thus additive steering can beused to adjust the handling characteristics of a vehicle.

We now attempt to use right/left brake control to steer the vehicle and adjust itshandling characteristics. For this case the vehicle is steered by the driver as was donepreviously, and the vehicle would understeer if it was not controlled. For thisundersteering vehicle executing a left turn, the brakes are applied at the left/rear if

Table 1

Vehicle simulation parameters

Parameter Value Parameter Value

Vehicle Suspension

Distance, cg to front axle, a 1.17m Stiffness, ks 14 900N/m

Distance, cg to rear axle, b 1.68m Passive damping, bs 475N s/m

Height of car cg above road, h 0.55m Controlled damping, bc 3328N s/m

Track, w 1.54m

Mass of car, m 1513kg Brake

Yaw moment of inertia, Jy 2345.53 kgm2 Distance brake pad and disk, rd 0.076m

Pitch moment of inertia, Jp 2443.26 kgm2 Ball screw gain, gb 0.00032m/rad

Roll moment of inertia, Jr 637.26 kgm2 Motor torque constant, Tb 0.6076Nm/A

Motor winding resistance, Rb 0.143O

Tire Steering

Wheel radius, Rw 0.32m Caster, Ca 0.018m

Unsprung mass, mu 38.42 kg Motor torque constant, Ts 0.6076Nm/A

Wheel moment of inertia, Jw 1.95 kgm2 Motor inertia, Jm 4.22e-4 kgm2

Tire stiffness, kt 150 000N/m Steering gear ratio, g 1/18.3

Cornering stiffness per axle, Ca 61 754N/rad Gear ratio, r2 1/8.6

Slope of slip curve, Sl 8.33 Gear ratio, r1 1/10

D. Margolis, T. Shim / Journal of the Franklin Institute 338 (2001) 21–34 29

Fig. 8. Yaw response of the uncontrolled vehicle.

Fig. 9. Lateral acceleration of the uncontrolled vehicle.

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Fig. 10. Yaw rate of the vehicle using the additive steering controller.

Fig. 11. Current input to the steering motor.

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Fig. 12. Yaw rate response of brake controlled vehicle.

Fig. 13. Control current at the left/rear brake motor.

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increased yaw rate is needed and at the right/front if decreased yaw rate is needed.Fig. 12 shows the yaw rate response for the brake controlled vehicle and Fig. 13shows the brake motor current at the left/rear. For this case the controller attemptsto reduce the yaw rate error and does so quickly at the beginning of the turn.However, as the vehicle speed decreases do to braking (see Fig. 14), and the normalforce at the left/rear decreases due to the roll response of the vehicle, the controller isunable to keep the vehicle responding like a neutral steer vehicle. Fig. 13 shows thecurrent input to the brake motor at the left/rear. The current rises immediately dueto the yaw rate error and then decreases as the yaw rate error is decreased. Thecurrent then increases as the yaw rate error grows and the controller is unable tomaintain control. The spikes in the current of Fig. 13 are due to restricting the wheelslip to be less than 12%. This is similar to conventional ABS brakes.

5. Conclusions

Bond graphs were used to develop a four-wheel nonlinear vehicle dynamic modeluseful for controller development. The model includes six degrees-of-freedom for thebody, steering at the front, suspension, and roadway input. A novel additive steeringcontroller is included as are electric brakes at each wheel and controlled suspension.Bond graphs and causality were used to develop the model in small pieces whichwere assembled into an overall model by connecting the pieces. The system was

Fig. 14. Forward speed for the steering and brake control.

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demonstrated for a steering control and brake control both used to adjust the yawrate of the inherently understeer vehicle to be that of a neutral steer vehicle.

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