Motorcycle Dynamics

Vehicle System Dynamics 0042-3114/02/3706-423$16.002002, Vol. 37, No. 6, pp. 423±447 # Swets & Zeitlinger

A Motorcycle Multi-Body Model for Real Time

Simulations Based on the Natural

Coordinates Approach

VITTORE COSSALTER1 and ROBERTO LOT2

SUMMARY

This paper presents an eleven degrees of freedom, non-linear, multi-body dynamics model of a motorcycle.

Front and rear chassis, steering system, suspensions and tires are the main features of the model.

An original tire model was developed, which takes into account the geometric shape of tires and the

elastic deformation of tire carcasses. This model also describes the dynamic behavior of tires in a way

similar to relaxation models.

Equations of motion stem from the natural coordinates approach. First, each rigid body is described with

a set of fully cartesian coordinates. Then, links between the bodies are obtained by means of algebraic

equations. This makes it possible to obtain simple equations of motion, even though the coordinates are

redundant.

The model was implemented in a Fortran code, named FastBike. In order to test the code, both simulated

and real slalom and lane change maneuvers were carried out. A very good agreement between the numerical

simulations and experimental test was found. The comparison of FastBike's performance with those of some

commercial software shows that ®rst is much faster than others. In particular, real time simulations can be

carried out using FastBike and it can be employed on a motorcycle simulator.

1. INTRODUCTION

The use of computer simulations in motorcycle engineering makes it possible both to

reduce designing time and costs and to avoid the risks and dangers associated with

experiments and tests. The multi-body model for computer simulations can be built

either by developing a mathematical model of the vehicle or by using commercial

software for vehicle system dynamics. Even though the ®rst method is more dif®cult

and time consuming than the second, maximum ¯exibility in the description of the

features of the model can be obtained only by using a mathematical model. In

particular, it makes it possible to properly describe the tire behavior at large camber

1Department of Mechanical Engineering, University of Padova, Italy.2Corresponding author: Roberto Lot, Department of Mechanical Engineering, University of Padova, Via

Venezia 1, 35131 Padova, Italy. Tel.:�39 049 8276806; Fax:�39 049 8276785; E-mail: [email protected];

website: www.dinamoto.mecc.unipd.it

angles, whereas multi-body codes such as ADAMS, DADS or Visual Nastran lack

such a feature. Moreover, mathematical modeling has a high computation ef®ciency,

while multi-body software require a lot of time to carry out simulations.

For the reasons above, the focus of this study was to develop mathematical models

of a tire and motorcycle. The tire model properly describes the shape of the carcass

and the position of the contact point. Moreover, it takes into account the sliding of the

contact patch and the deformation of the tire carcass. The motorcycle model was

developed based on the natural coordinates approach [1], which makes it possible to

obtain simple equations of motion and hence high computation ef®ciency.

2. MOTORCYCLE AND RIDER DESCRIPTION

The motorcycle is modeled as a system of six bodies: the front and rear wheels, the

rear assembly (including frame, engine and fuel tank), the front assembly (including

steering column, handle-bar and front fork), the rear swinging arm and the unsprung

front mass (including fork and brake pliers). The driver is considered to be rigidly

attached to the rear assembly; front and rear assembly are linked by means of the

steering mechanism. The front suspension is a telescopic type and the rear suspension

is a swinging arm type.

This vehicle model has eleven degrees of freedom, which can be associated to the

coordinates of the rear assembly center of mass, the yaw angle, the roll angle, the

pitch angle, the steering angle, the travel of front and rear suspension and the spin

rotation of both wheels (see Fig. 1).

Fig. 1. Eleven degrees of freedom motorcycle model.

424 V. COSSALTER AND R. LOT

The following forces act on the motorcycle elements: suspensions forces due to

springs and shock-absorbers, tire forces and torques, aerodynamic forces, rider steering

torque, steer damper torque, rear and front brake torques and ®nally propulsive torque,

which is transmitted from the sprocket to the rear wheel by means of the chain.

The rider's actions on the motorcycle determine both the direction of the vehicle

and the forward speed. In this model, the rider is considered to be a rigid body

attached to the rear assembly, so that the rider's movement away from the saddle and

the corresponding control action are neglected. In this way the motorcycle's direction

is controlled only by the torque exerted on the handlebars (steering torque). The

forward speed is controlled by applying the brakes (rear and front brake torques) and

by acting on the accelerator lever (propulsive force).

3. TIRE MODEL

In motorcycles the roll angle can reach 50±55�, hence it has a signi®cant in¯uence

both on tire forces and torques and on the contact patch. In this model, the actual

shape of the tire is described in detail and the deformation of the tire carcass is taken

into account. The road±tire contact is assumed to be dot-shaped and the position of the

contact point depends on the roll angle. Tire forces and torques are applied in the

contact point. The tire forces include the vertical load N, the lateral force F and the

longitudinal force S; the tire torques include the rolling friction torque My and the yaw

torque Mz.

The tire reference frame Tw is de®ned by using 4� 4 transformation matrix

notation [2], as shown in Figure 2: its origin is located in wheel center G, plane XwZw

is the symmetry plane of the wheel, the Xw axis is horizontal and points forwards, the

Yw axis is parallel to the wheel spin axis and points rightwards and the Zw axis

completes the reference frame. The frame T 0 has its origin located in contact point C,

the road plane X0Y 0 is horizontal, the X0 axis is parallel to Xw, points forwards and has

unit vector s, the Y 0 axis points rightward and has unit vector n, the Z 0 axis is vertical

and points downwards.

As it is well known, horizontal tire forces depend on tread deformation and slide,

i.e., they depend on sideslip angle l, longitudinal slip k, camber angle j and vertical

load N as follows

S � Sslip k; l;j;N� �F � Fslip k; l;j;N� � �1�

In several tire models [3±5] the sideslip angle and longitudinal slip are de®ned

according to wheel kinematics, without taking into account the deformation of the tire

carcass. On the contrary, in this model slip quantities are de®ned considering the

actual contact point, which moves with respect to the rim because of the deformation

A MOTORCYCLE MULTI-BODY MODEL 425

of tire carcass. Deformability of the tire carcass is taken into account as shown in

Figure 3. The contact point lies on the vertical plane which passes through the wheel

spin axis. The tire de¯ection with respect to the rim consists of radial displacement �r,

lateral displacement �l and rotation x around the wheel spin axis. Moreover, it is

assumed that tire deformations do not alter the mass properties of the wheel.

Fig. 2. Tire kinematics and tire forces.

Fig. 3. Tire deformability.


The position of the contact point is expressed by means of its coordinates yc; zc

with respect to frame Tw as follows

C � Twf0; yc; zc; 1gT �2�Thus, the instantaneous sideslip angle is de®ned as:

l � ÿarctanVY

VX

� ÿarctan_C � n_C � s �3�

where VX is the forward speed, VY the lateral speed, s and n the unit vectors of axis X0

and Y 0 respectively.

The instantaneous longitudinal slip is de®ned as:

k � ÿ1ÿ VR

VX

� ÿ1ÿ zc� _y� _x�_C � s �4�

where VR is the rolling speed which depends both on spin velocity _y and rotational

deformation rate _x.

On the other hand, tire forces depend on carcass deformation and camber angle, as

shown in experimental tests [6, 7]

S � Selastic x;j� �F � Felastic �r; �l;j� �N � Nelastic �r; �l;j� �

�5�

In absence of tire forces, no tire de¯ection is present and the contact point

coincides with the point of tangency between the tire surface and road plane C0. Thus,

the position of the contact point only depends on the tire shape and the coordinates of

C0 with respect to frame Tw can be de®ned as a function of the roll angle, as follows

C0 � Tw 0; yt�j�; zt�j�; 1f gT �6�where functions yt�j� and zt�j�make a parametric representation of the lateral pro®le

of the carcass. In order to guarantee the condition of tangency between tire and road

plane, functions must satisfy the following relation

tan�j� � ÿ dzt

dj

�dyt

dj

Lateral and radial deformation can be calculated by subtracting expression (6)

from expression (2), obtaining

�l � yc ÿ yt j� ��r � zc ÿ zt j� �

�7�


This model is able to properly describe tire behavior both in steady state and

transient conditions. Indeed, by coupling Equation (1), which describe the behavior

of the contact patch during sliding, with Equation (5), which describe elasticity

properties of the tire carcass

Sslip k; l;j;N� � ÿ Selastic x;j� � � 0

Fslip k; l;j;N� � ÿ Felastic �r; �l;j� � � 0�8�

one obtains a description of tire behavior which is equivalent to relaxation tire models

[8±11]. To proof this, let us de®ne a linear relation between longitudinal force and

longitudinal slip

S � Ksk �9�and a linear relation between longitudinal force and rotational deformation

S � Kxx �10�where Ks and Kx are respectively the longitudinal slip stiffness and rotational stiffness

of tire. By substituting Equation (4) in Equation (9) and by rearranging terms, one

obtains:

S � KS ÿ1ÿ zc_y

VX

!ÿ KS

zc_x

VX

� KSk0 ÿ KSzc

_xVX

�11�

where k0 is the steady state value of longitudinal slip, which corresponds to the steady

state value of longitudinal force S0: The time derivation of expression (10) yields:

_x �_S

Kx�12�

By replacing Equations (12) in Equation (11) and by rearranging the terms, one

obtains:

KSzc=Kx

VX

_S� S � S0 �13�

which is a ®rst order relaxation equation, where relaxation length is s � KSzc=Kx. The

equivalence between this tire model and the relaxation model can be found for lateral

force as well.

This approach presents several advantages with respect to relaxation models. First,

it explains the physical behavior of the tire in more detail, by highlighting both the

deformability of the carcass and the sliding of the tread. Furthermore, with this tire

model only static and steady state experimental tests are required in order to

characterize tire behavior in both static and dynamic conditions.


In order to complete the model it is necessary to de®ne tire torques with respect to

the contact point. The rolling resistance torque is assumed to be proportional to the

wheel load

My � N d �14�where d is the rolling friction parameter.

Yaw torque Mz is generated by lateral force F, tire trail t and twisting torque MTz as

follows [12±14]:

Mz � ÿt l� �F �MTz j� � �15�The ®rst term depends on the sideslip angle and tends to align, the second term

depends on the roll angle and tends to self-steer.

Finally, it is not necessary to take into account overturning moment Mx , because

tire forces are applied in the actual contact point [3, 13, 14].

4. MULTI-BODY MODEL

The mathematical model of the motorcycle was developed based on the natural

coordinates approach [1]. Natural coordinates consist of cartesian coordinates of

points or direction cosines of vectors belonging to the bodies of the system. With this

approach, kinematic relationships and equations of motion are very simple. However,

the number of variables required for describing a system is larger than the number of

degrees of freedom and so additional constraint equations must be introduced.

The equations were derived using Maple1, a software which makes it possible to

perform symbolic manipulation ef®ciently and to avoid calculation errors. Moreover,

it generates automatically the Fortran code.

4.1. Kinematic Description

Equations of motion were derived in the inertial reference frame XYZ: axes X and Y

are horizontal and lie on the road level, the Z axis is vertical and points downwards;

the unit vectors of inertial frame are, respectively, cx, cy and cz.

A body-®xed frame Ti is attached to each rigid body. The elements of the

transformation matrix are used as generalized coordinates, i.e., the con®guration of

each body is described by means of the coordinates of origin and direction cosines of

the body-®xed frame (see Fig. 4).

The rear tire reference frame Tw1 has its origin in the center of the wheel

G1 � fx1; y1; z1; 1gTand is de®ned as shown in Section 3, as well as the reference

frame T 01. Moreover, the rear wheel ®xed-frame T1 is obtained from frame Tw1

by a rotation of spin angle y1 around Yw1 axis. It is useful to de®ne the follow-

ing unit vectors: s1 � fsx1; sy1; 0; 0gTparallel to both Xw1 and X01 axes,


w1 � fwx1;wy1;wz1; 0gTparallel to axis Yw1, v1 � fvx1; yy1; vz1; 0gT

parallel to axis

Zw1 and n1 � fÿsy1; sx1; 0; 0gTparallel to axis Y 01.

The rear assembly ®xed-frame T2 has its origin in the swinging arm pin joint

P2 � x2; y2; z2; 1f gT ; plane X2Z2 is parallel to plane X1Z1, the X2 axis is perpendicular

to the steering axis, points forwards and has unit vector u2 � fux2; uy2; uz2; 0gT, the Y2

axis has unit vector w2 � w1 and ®nally the Z2 axis is parallel to the steering axis and

has unit vector v2 � fvx2; yy2; vz2; 0gT.

The front assembly ®xed-frame T3 has the origin in the point P3 � fx3; y3; z3; 1gT,

which is the intersection between the steering axis and its perpendicular plane passing

through P2 . The X3Z3 plane is parallel to the symmetry plane of the front wheel, the

X3 axis is perpendicular to the steering axis, points forwards and has unit vector

u3 � fux3; uy3; uz3; 0gT, the Y3 axis is parallel to the front wheel spin axis and has unit

vector w3 � fwx4;wy4;wz4; 0gT, ®nally the Z3 axis has a unit vector v3 � v2.

The front tire reference frame Tw4 has its origin in the center of the wheel

G4 � fx4; y4; z4; 1gTand is de®ned as shown in Section 3, as well as the reference

frame T 04 . Besides, the front wheel ®xed-frame T4 is obtained from frame Tw4 by a

rotation of spin angle y4 around Yw4 axis. The following unit vectors are de®ned:

s4 � fsx4; sy4; 0; 0gTparallel to both Xw4 and X04 axis, w4 � w3 parallel to Yw4 axis,

v4�fvx4; yy4; vz4; 0gTparallel to Zw4 axis and n4�fÿsy4; sx4; 0; 0gT

parallel to Y 04 axis.

Fig. 4. Description of multi-body system using basic points and unit vectors.


The swinging arm ®xed-frame T5 has its origin in the rear wheel center G1, the X5

axis is parallel to vector G1P2 and has unit vector u5 � fux5; uy5; uz5; 0gT, the Y5 axis

has unit vector w5 � w1 and the Z5 axis has unit vector v5 � fvx5; yy5; vz5; 0gT.

The front unsprung mass ®xed-frame T6 has the origin on the center of mass

G6 � T4fGx6;Gy6;Gz6; 1gT ; X6; Y6 and Z6 axes are parallel respectively to X3; Y3 and

Z3 and their unit vectors are u6 � u3, w6 � w4, v6 � v2 .

The con®guration of the motorcycle is described by means of a set of n � 45

coordinates, including the coordinates of points G1, P2, P3, G4, direction cosines of

unit vectors s1, v1, w1, u2, v2, u3, s4, v4, w4, u5, v5 and spin rotations of both wheels:

q � fx1; y1; z1; sx1; sy1;wx1;wy1;wz1; vx1; vy1; vz1; y1; x2; y2; z2; ux2; uy2; uz2; vx2; vy2;

vz2; x3; y3; z3; ux3; uy3; uz3; x4; y4; z4; sx4; sy4;wx4;wy4;wz4; vx4; vy4; vz4; y4; ux5;

uy5; uz5; vx5; vy5; vz5gT �16�The motorcycle has only f � 11 degrees of freedom, thus it is necessary to

formulate a set of m � nÿ f � 34 independent constraint equations:

fj � 0; j � 1 . . . m �17�By imposing the unit length condition to all unit vectors, the following 11

independent constraint equations are obtained:

f1 � s1 � s1 ÿ 1

f4 � u2 � u2 ÿ 1

f7 � s4 � s4 ÿ 1

f10 � u5 � u5 ÿ 1

f2 � w1 � w1 ÿ 1

f5 � v2 � v2 ÿ 1

f8 � w4 � w4 ÿ 1

f11 � v5 � v5 ÿ 1

f3 � v1 � v1 ÿ 1

f6 � u3 � u3 ÿ 1

f9 � v4 � v4 ÿ 1�17:1ÿ11�

By imposing the orthogonal conditions to every couple of unit vectors which belong

to the same reference frame, 15 more independent constraint equations are obtained:

f12 � s1 � w1

f15 � u2 � w1

f18 � u3 � v2

f12 � s4 � w4

f24 � u5 � w1

f13 � s1 � v1

f16 � v2 � u2

f19 � u3 � w4

f22 � s4 � v4

f25 � v5 � w1

f14 � v1 � w1

f17 � v2 � w1

f20 � v2 � w4

f23 � v4 � w4

f26 � w5 � v5

�17:12ÿ26�

The remaining 8 constraint equations are the following:

� vector G1P2 must be perpendicular to the f27 � G1P2 � w1 (17.27)

rear wheel spin axis Y1

� the magnitude of vector G1P2 must be f28 � G1P2 �G1P2 ÿ l2f (17.28)

equal to the swinging arm length lf

� vector v5 must be perpendicular to the f29 � G1P2 � v5 (17.29)

vector G1P2


� the magnitude of vector P2P3 must be f30 � P2P3 � P2P3 ÿ l223 (17.30)

equal to l23

� vector P2P3 must lie on the X2Z2 plane f31 � P2P3 � w1 (17.31)

(thus it must be perpendicular to the f32 � P2P3 � v2 (17.32)

vectors w1 and v2)

� the point R3 � G4 ÿ l1u3 must lie on the f33 � P3R3 � w4 (17.33)

steering axis Z3

(thus it must be perpendicular to vectors f34 � P3R3 � u3 (17.34)

w4 and u3)

It is worth pointing out that the natural coordinates approach made it possible to obtain

simple constraint equations, which are quadratic with respect to the coordinates.

4.2. Lagrange's Equations

Due to the presence of constraints, the Lagrange's equations become

d

dt

@K

@ _qi

ÿ @K

@qi

�Xm

j�1

lj

@fj

@qi

ÿ Qi � 0; i � 1: : n �18�

where K is the kinetic energy, li are the Lagrange multipliers and Qi the generalized

forces.

By coupling the de®nition of kinetic energy to the transformation matrix notation,

the kinetic energy of ith rigid body is

Ki � 1

2

Zm

_P2 dm � 1

2

Zm

fx; y; z; 1g _TTi

_Tifx; y; z; 1gTdm �19�

where fx; y; z; 1gTare the coordinates of point P with respect to frame Ti . Assuming

that the origin of the reference frame is the center of mass of the body and expanding

the previous equation, one obtains:

Ti � 1

2

Zm

fx; y; x; 1g

_u2i _ui � _wi _ui � _vi _ui � _Gi

_wi � _ui _w2i _wi � _vi _wi � _Gi

_vi � _ui _vi � _wi _v2i _vi � _Gi

_Gi � _ui_Gi � _wi

_Gi � _vi_G2

i

266664377775 fx; y; x; 1gT

dm

� 1

2_G2

i

Zm

dm� 1

2_u2

i

Zm

x2dm� 1

2_w2

i

Zm

y2dm� 1

2_v2

i

Zm

z2dm

� _ui � _wi

Zm

xy dm� _ui � _vi

Zm

xz dm� _wi � _vi

Zm

yz dm

� _ui � _Gi

Zm

x dm� _vi � _Gi

Zm

y dm� _wi � _Gi

Zm

z dm


By substituting the integral terms in the previous equation with moments and products

of inertia with respect to the center of mass, the kinetic energy of each rigid body can

be calculated as a function of the elements of transformation matrix Ti , as follows

Ki � 1

2mi

_G2i �

1

4Ix;i ÿ _u2

i � _w2i � _v2

i

ÿ �� 1

4Iy;i _u2

i ÿ _w2i � _v2

i

ÿ �� 1

4Iz;i _u2

i � _w2i ÿ _v2

i

ÿ �� Cxz;i _ui � _vi � Cxy;i _ui � _wi � Cyz;i _wi � _vi �20�

If the body center of mass does not coincide with the origin of the reference frame, it

is necessary to replace _Gi � f _xi; _yi; _zi; 1gTwith _Gi � _TifGxi;Gyi;Gzi; 1gT

in the

previous equation. Thus, the kinetic energy of the whole system is:

K � 1

2m1

_G21 �

1

2Iy1 _s2

1 ÿ _w21 � _v2

1 � _y1�s1 � _v1 ÿ _s1 � v1� � _y21

h i� 1

2Id1 _w2

1

� 1

2m2

_G22 �

1

4Ix2 ÿ _u2

2 � _w21 � _v2

2

ÿ �� 1

4Iy2 _u2

2 ÿ _w21 � _v2

2

ÿ �� 1

4Iz2 _u2

2 � _w22 ÿ _v2

2

ÿ �� Cxz2 _u2 � _v2 � Cxy2 _u2 � _w2 � Cyz2 _w2 � _v2

� 1

2m3

_G23 �

1

4Ix3 ÿ _u2

3 � _w24 � _v2

3

ÿ �� 1

4Iy3 _u2

3 ÿ _w24 � _v2

3

ÿ �� 1

4Iz3 _u2

3 � _w24 ÿ _v2

3


� 1

2m4

_G24 �

1

2Iy4 _s2

4 ÿ _w24 � _v2

4 � _y4 s4 � _v4 ÿ _s4 � v4� � � _y24

h i� 1

2Id4 _w2

4

� 1

2m5

_G25 �

1

4Ix5 ÿ _u2

5 � _w21 � _v2

5

ÿ �� 1

4Iy5 _u2

5 ÿ _w21 � _v2

5

ÿ �� 1

4Iz5 _u2

5 � _w21 ÿ _v2

5


� 1

2m6

_G26 �

1

4Ix6 ÿ _u2

3 � _w24 � _v2

3

ÿ �� 1

4Iy6 _u2

3 ÿ _w24 � _v2

3

ÿ �� 1

4Iz6 _u2

3 � _w24 ÿ _v2

3

ÿ �� Cxz6 _u3 � _v3 � Cxy6 _u3 � _w4 � Cyz6 _w4 � _v3 �21�

where the terms relative to wheels (i � 1 and i � 4) are slightly different from the

terms relative to other bodies because of the axial symmetric structure of the wheels

(Ix;i � Iz;i � Id;i and Cxz;i � Cyz;i � Cxy;i � 0) and because spin velocity _y1; _y4 has

been used.

The generalized forces expression can be obtained from the virtual work dW of the

forces acting on the vehicle

dW �Xm

i�1

Qidqi �22�


In order to determine virtual works, it is necessary to calculate the virtual rotation dYi

of each rigid body with respect to its reference frame Ti . By extending the concept of

angular velocity matrix [2] to virtual rotation matrix dY � TTdT and by extracting

the components of virtual rotation from dY, the following virtual rotation operator

can be de®ned:

dY Ti� � � vi � dwi; ui � dvi;wi � dui; 0f gT �23�Virtual work contains the following terms:

dW � dWg � dWS � dWA � dWt � dWB � dWt;F � dWt;T � dWP �24�� The virtual work due to the gravity force:

dWg �X6

i�1

mig � dGi �24:1�

where g � f0; 0; g; 1gTis the gravity acceleration.

� The virtual work due to front suspension force FSf , which acts between the front

assembly and front wheel, and virtual work due to rear suspension force FSr, which

acts between the rear assembly and swinging arm:

dWs � FSf v2 � �dP3 ÿ dR3� � tsFSrcy � dY T2� � ÿ dY T5� �� 24:2�where ts � @yr=@zr is the velocity coef®cient between spring de¯ection zr and arm

rotation yr .

� The virtual work due to drag, side and lift aerodynamics forces FA �Tw1fFD;FS;FL; 0gT

, which are applied on point CA � T2fXCA; 0; ZCA; 1gT:

dWA � FA � dCA �24:3�� The virtual work due to rider steering torque t and steer damper torque tD , which

are applied between the rear and front assembly:

dWt � t� tD� �cz � dY T3� � ÿ dY T2� �� 24:4�� The virtual work due to rear brake torque MBr, which acts between the rear wheel

and swinging arm, and the virtual work due to front brake torque MBf , which acts

between the front wheel and front unsprung mass:

dWB � MBrcy � dY T1� � ÿ dY T5� �� MBf cy � dY T4� � ÿ dY T6� �� 24:5�� The virtual work due to rear tire force FT1 � T01fS1;F1;ÿN1; 0gT

, which is applied

on rear contact point C1 � Tw1f0; yc1; zc1; 1gT, and the virtual work due to front tire

force FT4 � T04fS4;F4;ÿN4; 0gT, which is applied on front contact point C4 �

Tw4f0; yc4; zc4; 1gT:

dWt;F � FT1 � dG1�G1C1�FT1 � T1dY T1� ��FT4 � dG4�G4C4 � FT4 � T4dY T4� ��24:6�


� The virtual work due to rear tire torque MT1 � T01f0;My1;Mz1; 0gTand front tire

torque MT4 � T04f0;My4;Mz4; 0gT:

dWt;M �MT1 � T1dY T1� � �MT4 � T4dY T4� � �24:7�� The virtual work due to the propulsive torque, which is transmitted from the drive

sprocket to the wheel by means of the chain. As shown in Figure 5, the drive

sprocket center is R � T2 RX ; 0;RZ ; 1f gT, whereas the chain angles are:

yc1 � arctanG1R � s1

G1R � v1

� �ÿ arcsin

rc ÿ rp

G1Rj j� �

yc2 � arctanG1R � u2

G1R � v2

� �� arcsin

rc ÿ rp

G1Rj j� �

The chain tension FC � T1 Tc sin�yc1�; 0; Tc cos�yc1�; 0f gTacts between point

P7 � T1 rc cos�yc1�; 0; rc sin�yc1�; 1f gTand point P8 � T2 RX � rp cos�yc2�; 0;RZ �

�rp sin�yc2�; 1gT

, thus the virtual work is

dWp � Fc � dP7 ÿ dP8� � ÿ Tcrcdy1 �24:8�Explicit Lagrange's equations are not shown because of their large number, while

their compact form is the following:

F q; _q; �q; k; t� � �M�q� _M _q� FTkÿQ � 0 �25�where M is the mass matrix, F is the Jacobian matrix of constraint equations (17), k is

the vector of Lagrange multipliers and Q is the vector of generalized forces. Due to

the natural coordinates approach, the mass matrix is very sparse and has only 9% non-

zero elements; moreover the evaluation of Equation (25) require less than 2,000

multiplications and less than 1,000 additions.

Fig. 5. Geometry of the chain transmission.


4.3. Tire Equations

As seen in Section 3, tire deformation is described by means of three coordinates,

hence for both the rear and front tires the following six coordinates should be de®ned:

q0 � yc1; zc1; x1; yc4; zc4; x4f gT �26�The tire behavior must be described by means of as many equations as coordinates.

Equation (8) can be re-written as follows

p1 � Sslip;1 k1; l1;j1;N1� � ÿ Selastic;1 x1;j1� � � 0

p2 � Sslip;4 k4; l4;j4;N4� � ÿ Selastic;4 x4;j4� � � 0

p3 � Fslip;1 k1; l1;j1;N1� � ÿ Felastic;1 �r;1; �l;1;j1

ÿ � � 0

p4 � Fslip;4 k4; l4;j4;N4� � ÿ Felastic;4 �r;4; �l;4;j4

ÿ � � 0

�27:1ÿ4�

Equations (3), (4) and (7) make it possible to express slip quantities and tire

deformations as a function of generalized coordinates, whereas camber angles can be

calculated as follows:

j1 � arcsin wz1� �j4 � arcsin wz4� � �28�

The remaining equations are obtained by imposing the contact between the tire and

road plane Z � 0, as follows:

p5 � C1� �z� z1 � wz1yc1 � vz1zc1 � 0

p6 � C4� �z� z4 � wz4yc4 � vz4zc4 � 0�27:5ÿ6�

It is worth pointing out that Equations (27.1±4) are differential equations because

slip quantities (3) and (4) contain time derivation of coordinates x and x0. On the

contrary, Equations (27.5±6) are algebraic.

4.4. State Space Formulation

Lagrange's Equation (25), constraint Equation (17) and tire Equation (27) form a set

of 85 second order differential-algebraic simultaneous equations (DAEs) of index 3

[15], with the following unknowns: 51 generalized coordinates and 34 Lagrange

multipliers.

In order to obtain a 1 index DAEs problem, algebraic constraint Equation (3)

should be replaced by differential equations using the Baumgarte stabilization method

[16], as follows:

/0 � �/� 2&o _/� o2/ �29�where constant � and o are properly chosen.


The DAEs problems of index 1 can be numerically solved using the DASSL solver

[17], however the transformation of DAEs into a set of ordinary differential equations

(ODEs) makes it possible to increase integration speed. For this purpose, the

Lagrange multipliers are replaced with the following differential expression:

k � c� t0 _c �30�where constant t0 is properly chosen. Moreover, tire Equation (27) should be replaced

by the following set of ODEs

p0 � p1; p2; p3; p4; p5 � t0 _p5; p6 � t0 _p6f gT �31�In addition, the 2nd order Lagrange's Equation (25) should be reduced to a 1st order

ODEs. The system is then described by means of the following 2n� m� 6 � 130

state variables

y � x; v; c; x0f gT �32�and the following state space equations

G y; _y; t� � �F

vÿ _q/0

p0

8>><>>:9>>=>>; � 0 �33�

Although the number of equations is rather high with respect to the number of degrees

of freedom, each equation is simple and the evaluation of expression (33) require less

than 3,000 multiplications and less than 2,000 additions. These equations have been

implemented in a Fortran code, using the implicit solver DASSL for numerical

integration.

5. COMPARISON BETWEEN COMPUTER SIMULATIONS

AND EXPERIMENTAL MEASUREMENTS

In order to validate the multi-body model, some experimental tests were carried out on

an Aprilia RSV 1000 motorcycle; they were then compared to the simulation results.

The geometrical and inertial characteristics of the motorcycle and the non-linear

elastic and damping characteristics of the suspensions were measured at the

Department of Mechanical Engineering (DIM) at the University of Padua [18, 19].

Tire parameters were also measured with department's equipment [20], whereas the

driver inertia properties were estimated as shown in reference [21]. The charac-

teristics of the motorcycle are given in Appendix and in Figures 10, 11 and 12.

The motorcycle was equipped with a measurement system: roll and yaw rate,

steering angle, spin velocity of both wheels and steering torque were measured and


stored on a data recorder [19]. Data post-processing made it possible to calculate

vehicle forward speed and roll angle as well.

In order to reproduce the experimental maneuvers by means of numerical

simulations, steering torque t was calculated according to measured steering torque

tm and measured roll angle jm, as follows:

t � tm � kj jm ÿ j� � �34�where j is the simulated roll angle and kj the control gain. The chain propulsive force

and the front brake torque were calculated based on measured speed um, as follows:

S � mr _um � ku um ÿ u� �� Tc � r1

rc

S MFf � 0; S � 0 �acceleration�Tc � 0 MFf � ÿr4S; S < 0 �braking�

8<: �35�

where S is the longitudinal thrust, mr the generalized mass, u the simulated speed and ku

the control gain. Rear brake was not used in either the real or simulated maneuvers.

Figure 6 shows the comparison of the experimental measurements with the

numerical simulation for a lane change maneuver. The lane change width was 3.6 m

and the lane change length was 40 m. It was not possible to measure the trajectory of

the motorcycle, so the experiments were compared with simulations by analyzing

steering torque (Fig. 6a), vehicle speed (Fig. 6b), roll angle (Fig. 6c) and steering

angle (Fig. 6d). The ®gure shows that at the beginning of the maneuver the rider is

driving straight and increasing speed. When he starts to apply positive steering torque

(point A), the vehicle begins to capsize on the left-hand side. Afterwards, when the

steering torque is zero (point B) the magnitude of roll angle is still increasing; when

the steering torque reaches its minimum (point C), the roll angle is increasing and the

vehicle begins to capsize on the right-hand side. Then, the rider straightens the vehicle

(from point D) and ®nally decreases the speed.

The agreement between experimental and simulated data is very good: the overall

error (RMS) of steering torque is less than 3% of its peak value, the overall error of

vehicle speed is less than 0.5% of its peak value, the overall error of roll angle is about

the 9% of its peak value and the overall error of steering angle is about 26% of its peak

value. The steering angle has the maximum error, because of some steering oscillations

that are present in the simulation but that were not found in the experimental test.

Figure 7 shows the comparison of a real slalom maneuver with a simulated one,

by representing steering torque (Fig. 7a), vehicle speed (Fig. 7b), roll angle (Fig. 7c)

and steering angle (Fig. 7d). The pylon distance is 14 m and the vehicle speed is about

13.5 m/s. During the slalom maneuver, both roll and steering angles are delayed in

phase from steering torque of about 90�. Once again, the agreement between

experimental and simulated data is very good: the overall error of steering torque is


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less than 10% of its peak value, the overall error of vehicle speed is less than 3% of its

peak value, the overall error of roll angle is about the 15% of its peak value and the

overall error of steering angle is about the 13% of its peak value.

6. COMPARISON OF THE PERFORMANCES OF THE MULTI-BODY MODEL

WITH PERFORMANCE OF MULTI-BODY COMMERCIAL SOFTWARE

In this section simulations carried out using FastBike are compared with simulations

carried out using Dads1 and Visual Nastran1.

The features of Visual Nastran and Dads motorcycle models are about the same as

FastBike. It is worth pointing out that these multi-body software do not have a suitable

tire model, so it was necessary to implement the tire model presented in [14] and [13].

In this model the tire is rigid and has a toroidal shape.

The Figure 8 shows simulations of a lane change maneuver carried out using

different codes. The agreement between the data is excellent, both for the steering

torque (Fig. 8a) and roll angle (Fig. 8b).

Even if commercial software for multi-body analysis greatly reduces the time

needed for modeling systems, the time required for simulation is greater. Figure 9

compares the CPU time needed to carry out 1 s of simulation on a AMD K7 800 MHz

processor. The only code that allows real time simulation is Fast bike, which is about

10 times faster than Dads and about 100 times faster than Visual Nastran.

7. CONCLUSIONS

An original mathematical model of a tire and motorcycle was presented.

Fig. 9. CPU time on a processor AMD 800 MHz.


The tire model was developed in order to describe tire behavior at a large camber

angle. The shape of the tire and position of the contact point were described in detail.

The model is based on the physical description of tire forces genesis: the sliding of the

contact patch generates tire forces, which produce a deformation of the carcass of the

tire. By taking into account simultaneously both phenomena, an accurate description

of tire properties is obtained. It was demonstrated that this model is equivalent to

relaxation tire models.

The motorcycle multi-body model has eleven degrees of freedom and includes the

main features of a motorcycle, taking into account the non-linear properties of tires

and suspensions. The very good agreement between the numerical simulations and

experimental tests demonstrates the feasibility and correctness of the model.

The equations of motion were developed based on the natural coordinates

approach. This method made it possible to obtain simple equations of motion and

hence high computation ef®ciency was obtained. The comparison of the per-

formances of the FastBike code with the performance of DADS and Visual Nastran

showed that the ®rst is much faster than the others. In particular, real time simulations

can be carried out using FastBike and it can also be used on a motorcycle simulator.

For the same reason, it can be useful for solving optimization problems.

ACKNOWLEDGEMENTS

The authors would like to thank A. Doria for his suggestion regarding the organization

of the paper and D. Bortoluzzi and N. Ruffo for their contribution during the

experimental tests.

This research was partially supported by funds from the Italian Ministry for

Universities and for Scienti®c and Technological Research (MURST 40% funds).

REFERENCES

1. Jalon, J.C. de and Bayo, E.: Kinematic and Dynamic Simulation of Multibody Systems. Springer, 1994.

2. Sush, H. and Radcliffe, C.W.: Kinematics and Mechanism Design. Wiley, New York, 1978, Chapter 3.

3. Sakay, H.: Study on Cornering Properties of Tire and Vehicle. Tire Science and Technology 18 (1990),

pp. 136±139.

4. Pacejka, H.B. and Bakker, E.: The Magic Formula Tyre Model. Vehicle System Dynamics 21 (1991),

pp. 1±18.

5. Pacejka, H.B. and Sharp, R.S.: Shear Force Development by Pneumatic Tyres in Steady State

Conditions: A Review of Modelling Aspects. Vehicle System Dynamics 20 (1991), pp. 121±176.

6. Wang, Y.Q., Gnadler, R. and Schieschke, R.: Vertical Load-De¯ection Behaviour of a Pneumatic Tyre

Subjected To Slip And Camber Angles. Vehicle System Dynamics 25 (1996), pp. 137±146.

7. Berritta, R., Cossalter, V. and Doria, A.: Identi®cation of The Lateral and Cornering Stiffness af

Scooter Tyres Using Impedance Measurements. Proc. 2nd International Conference on Identi®cation

in Engineering Systems, Swansea, UK, 1999, pp. 669±678.


8. De Vries, E.J.H. and Pacejka, H.B.: Motorcycle Tyre Measurements and Models. Proc. 15th IAVSD

Symposium: The Dynamics of Vehicles on Road and Tracks. Budapest, Hungary, 1997, pp. 280±298.

9. Maurice, J.P. and Pacejka, H.B.: Relaxation Length Behavior of Tyres. Vehicle System Dynamics 27

(1997), pp. 339±342.

10. Zegelaar, P.W.A. and Pacejka, H.B.: Dynamic Tyre Responses to Brake Torque Variations. Vehicle

System Dynamics 27 (1997), pp. 65±79.

11. Guo, K., Liu, Q. and Yangpin, H.: A Non-Steady Tire Model for Vehicle Dynamic Simulation and

Control. Proc. AVEC International Symposium on Advanced Vehicle Control: AVEC'98, Nagoya,

Japan, 1998.

12. Fujioka, T. and Goda, K.: Tire Cornering Properties at Large Camber Angles: Mechanism of the

Moment around the Vertical Axis. JSAE Review 16 (1995), pp. 257±261.

13. Berritta, R., Cossalter, V., Doria, A. and Lot, R.: Implementation of a Motorcycle Tyre Model in a

Multi-Body Code. Tire Technology International, March 1999.

14. Cossalter, V., Doria, A. and Lot, R.: Steady Turning of Two Wheel Vehicles. Vehicle System Dynamics

31 (1999), pp. 157±181.

15. Gear, C.W.: Differential-Algebraic Equation Index Transformations. SIAM Journal on Scienti®c and

Statistical Computing, 9 (1988), pp. 39±47.

16. Baumgarte, J.: Stabilization of Constraints and Integrals of Motion in Dynamical Systems. Computer

Methods in Applied Mechanics and Engineering 1 (1972), pp. 1±16.

17. Petzold, L.R.: A Description of DASSL: A Differential/Algebraic System Solver. In: R.S. Stepleman

(ed.): IMACS Transactions on Scienti®c Computation 1, 1982, pp. 430±432.

18. Da Lio, M., Doria, A. and Lot, R.: A Spatial Mechanism for the Measurement of the Inertia Tensor:

Theory and Experimental Results, ASME Journal of Dynamic Systems, Measurement and Control, 121

(March 1999), pp. 111±116.

19. Bortoluzzi, D., Doria, A., Lot, R. and Fabbri, L.: Experimental Investigation And Simulation Of

Motorcycle Turning Performance. 3rd International Motorradkonferenzen, Monaco, 2000.

20. Cossalter, V., Da Lio, M. and Berritta, R.: Studio e Realizzazione di una Macchina per la

Determinazione delle Caratteristiche di Rigidezza e Smorzamento di un Pneumatico Motociclistico. V

Convegno di Tribologia, Varenna, 8±9 Ottobre 1998 (in italian).

21. Bartlett, R.: Introduction to Sports Biomechanics. -E & FN Spon-London, 1997.

APPENDIX

MOTORCYCLE CHARACTERISTICS

Motorcycle Geometric and Mechanical Properties

m1 16.2 kg Rear wheel mass

Ia1 0.66 kgm2 Rear wheel axial inertia

Id1 0.33 kgm2 Rear wheel diametrical inertia

m2 223 kg Rear assembly mass (including rider)

(Gx2;Gy2;Gz2) (0.255, 0.000, ÿ0.0202) m Coordinates of rear assembly CoM with

respect to frame T2

Ix2; Iy2; Iz2 (24.4, 26.2, 30.3) kgm2 Rear assembly moments of inertia

Cxz2;Cyz2;Cxy2 (0.0, 0.0, 0.0) kgm2 Rear assembly products of inertia


l23 0.730 m Distance between rear arm pin and steer

axis

m3 8.75 kg Front assembly mass

(Gx3;Gy3;Gz3) (0.023, 0.000, ÿ0.098) m Coordinates of front assembly CoM with

respect to frame T3

Ix3; Iy3; Iz3 (0.29, 0.14, 0.21) kgm2 Front assembly moments of inertia

Cxz3;Cyz3;Cxy3 (0.0, 0.0, 0.0) kgm2 Front assembly products of inertia

10 Nms Damping coef®cient of steering damper

m4 12.0 kg Front wheel mass

Ia4 0.47 kgm2 Front wheel axial inertia

Id4 0.22 kgm2 Front wheel diametric inertia

l1 0.034 m Front wheel offset

ZF;0 0.517 m Center of wheel position (with respect to

frame T3) when the suspension is

completely extended

lf 0.535 m Rear arm length

m5 10.0 m Rear arm mass

Fig. 10. Suspension properties.


(Gx5;Gy5;Gz5) (0.275, 0.000, ÿ0.052) m Coordinates of rear arm CoM with

respect to frame T5

Ix5; Iy5; Iz5 (0.20, 0.80, 0.80) kgm2 Rear arm moments of inertia

y5,0 ÿ.165 rad Rear arm rotation (respect frame T2)

when the suspension is completely

extended

zr � 0:13526 � y5 ÿ 0:138 � y25 ÿ 0:036 � y3

5 Relation between spring travel zr and

arm rotation y5

m6 7.00 kg Unsprung front mass

Gx6;Gy6;Gz6 (ÿ0.029, 0.000, ÿ0.189) m Coordinates of unsprung mass CoM

with respect to frame T3

Ix6; Iy6; Iz6 (0.22, 0.18, 0 .07) kgm2 Unsprung mass moments of inertia

rp 0.041 m Sprocket radius

rc 0.104 m Wheel sprocket radius

Fig. 11. Front tire properties.


(ap; bp) (0.080, 0.030) m X±Z coordinates of sprocket center with

respect to frame T2

CDA 0.28 Ns2/m2 Drag force coef®cient (FD � CDA � u2)

Global Properties

m 276.8 kg Total mass

p 1.421 m Wheel base

e 0.43 rad Castor angle

h 0.636 m Height of the center of mass

b 0.675 m Horizontal position of the center

(with respect to the rear wheel)

Fig. 12. Rear tire properties.


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Motorcycle Dynamics