Modeling and Control of a Motorbike

Marco De Vittori

Under supervision of Joao Fernando Cardoso Silva Sequeira

Dep. of Electrical and Computer Engineering, IST, Lisbon, Portugal

July 21, 2011

Abstract

The work developed in this thesis explains firstly the construction of both kinematic and dynamic models

for a motorcycle, and then the design of an automatic control that can command the bike avoiding instability

under normal driving conditions. All final and intermediate results have been tested in several diverse conditions

through a Matlab implementation, in order to validate the model and to educe information about the motorcycle

behavior.

Keywords: Motorbike, Control, Model, Simulation, Automation, Dynamics, Kinematics.

1 Introduction

Autonomous four-wheel vehicles already exist and the

research in this field has widely evolved in the last

decades. In what concerns motorbike control this is

not the case despite the increasing attention the sub-

ject is attracting. The development of an autonomous

motorbike is thus a natural topic of interest in future

transportation systems. The innumerable advantages

of an autonomous driving reside in transports optimiza-

tion, user comfort, and driving security, among others.

The unstable physical equilibrium intrinsic of the two-

wheels vehicles makes them somewhat more complex

to control, even from the mere theoretical perspective.

The first contribution of this study is the creation of

a realistic physical model which includes diverse kine-

matic and dynamic effects that are often not included in

other models. The underlying idea is that the more ef-

fects are included, the more information the automatic

control will have available, and, eventually, the easier

will be to find an autonomous controller capable of high

performance riding such as that found in motorbike rac-

ing.

The motorcycle modeling has been extensively

studied by a number of researchers over many years.

The study of bicycle modeling and control dates back

1

to the last years of 19th century, namely through the

work of F.J. Whipple (see for instance [1]). During

most of the 20th century, the development of the mo-

torbike industry kept the focus of the research in the

engines but it was not until the 80s that a renewed inter-

est in modeling emerged from an academic viewpoint,

with multiple researchers discussing different kinemat-

ics and addressing also dynamics and control issues.

Sharp et al, [3], studied the effects of an advanced

tyre modeling on the motorcycle dynamic simulation.

A linearized model for an uncontrolled bicycle is de-

scribed in [5]. General dynamics from the perspective

of control have been analyzed in [2]. In the paper, a

wide range of different bike architectures are studied

along with the consequences of the particular aspects

of each one of them. In an on-line article, [6], Mariotti

and Vitale explain the global issue of the rear wheel

damping system. Popov et al, [4], explain the aspects

related to the steering angle control in human driving.

2 Kinematics

2.1 Theoretical basis

The kinematics equations for a simple bicycle model

can be easily obtained as (see for instance [1]),

x = vcosψ ; y = vsinψ ; ψ =v tanδ

wcosϕ(1)

Figure 1 illustrates the meaning of the main compo-

nents in this model, v is the linear speed at the back

wheel, x direction is longitudinal with respect to the

bike’s frame, and y is perpendicular to the plane de-

fined by the rear wheel.

Figure 1: Main variables for the basis bike kinematic model

The bicycle model (1) is the basis model to de-

velop a motorbike kinematics but still does not include

key features that are readily observed in commercially

available motorbikes such as the front fork and back

swingarm.

Consider now the structure in Figure 2, where a

front fork structure is added to the basis bicycle model.

This front fork is characterized by the fork angle λ . As

λ is included in model (1) the angle between front and

rear wheels does not correspond to the δ angle between

the frame and the front fork anymore (as in Figure 1).

In fact, the front wheel rotation is now only a vector

portion of the front fork rotation, as shown in figure 2.

Figure 2: Additional geometric variables

In addition, the front fork structure commonly in-

2

cludes the front suspension, meaning that the kinemat-

ics must include a degree-of-freedom (dof) to account

for this possibility, i.e., the length of the front fork is a

variable, leading to a more realistic model. Let l rep-

resent the length of front fork. All of the motorbike’s

geometry will be affected by the movement of the front

fork, in particular the λ and w parameters. To complete

the kinematics modeling it remains to include a moving

swingarm to support the back wheel. Figure 2 shows

the typical swingarm arrangement with the angle be-

tween the swingarm and the horizontal x axis denoted

by β . For the Honda CBR600RR motorbike, the result-

ing linearized kinematics for the model with swingarm

can be found as,

x = vcosψ ; y = vsinψ

ψ =−7.7×108 vδ

cosϕ

[l3(β +132.1)2 −4253l2(β+

75.8)(β +132.1)+5×106l(β 2 +195.3β +8925.6)

−9.6×108(β 2 +516.7β +53326.4)]2[l(β+

132.1)−1959.5(β +150.3)]−3 [l2(β +132.1)

−2293.4l(β +12.2)+487247(β +366.4)]−3

(2)

Equation (2) provides a realistic kinematics that can

easily be simulated.

A damping model can be easily obtained assuming

that the suspension elements are described by two com-

mon second order models for the front suspension and

for the back suspension respectively:

Ff = lm f + lc f + lk f

Fr =d2

dt2 (578.5sinβ )mr +ddt(578.5sinβ )cr

+578.5sinβkr

(3)

where l is front fork length, β is the swingarm angle,

m f and mb the suspended masses at front and back,

cr,c f the friction coefficients, kr,k f the elastic coeffi-

cients, and Ff ,Fb the forces applied on the forward and

back suspensions by the contact with the road.

An important effect in a motorbike model is that of

the change in the wheel radius as the bike leans in a

corner, caused by the cross section shape of the tyres.

As the motorbike leans (ϕ angle) the distance between

the wheel center and floor decreases. To include the

variation in the model one must define the bike speed

as Rrθr and specify Rr as a function depending on ϕ ,

Rr = rt cosϕ + rw cosarcsin(rt

rwsinϕ) (4)

where Rr is real wheel radius, rw = 227.7 mm and

rt = 87.2 mm for the Honda CBR600RR motorbike.

2.2 A double swingarm model

An interesting case of motorbike geometry is the dou-

ble swingarm system. It has seldom been used in pro-

duction bikes, the Yamaha GTS 1000 (Figure 3) and

Bimota Tesi, and even racing applications, which in a

sense tend to use more extreme designs, has a single

known case, the Elf 500cc, raced during the 80’s decade

The reason for developing such an innovative geome-

try has mostly been argued by manufacturers to be the

intent of separate the steering dynamics and damping

ones and eliminating the λ angle and trail.

3

Figure 3: Yamaha GTS 1000

However, one can look for a completely different

argument, based solely on the kinematics design, as it

will appear among the kinematics simulation results.

Figure 4: Double swingarm geometry

The new model geometry is illustrated in Figure 4.

Following this diagram, kinematics equations can be

calculated as a modification of the conventional model

ones,

x = vcosψ ; y = vsinψ

ψ =θgrRrδ

wcosϕ; w = scosβ + s2 cosβ2 +231

(5)

2.3 Kinematics simulations

The models obtained in the previous sections were sim-

ulated in Matlab, where the complete model is used,

with front and rear suspension and pneumatic shape ef-

fect.

Some interesting intermediate results can be ob-

served after the kinematics simulation; for example the

fact that when the driver brakes while turning, the equi-

librium leaning angle becomes smaller. In other words,

when braking during a curve, rotating speed increases,

which means that the driver can turn with a smaller ra-

dius curve.

In the experiments, simulating braking means that

l decreases 60mm approximately, whereas β augments

1.6 after some little oscillations. Another significant

issue is the fact that, although θ engine speed is con-

stant, forward velocity decreases when Rr rear wheel

radius is also decreasing due to increasing ϕ leaning

angle, owing to the tyres shape effect.

Simulations of the double swingarm model have

been run under the same conditions as in the previous

experiments. The main characteristics of the simulation

results remain very similar to the conventional motor-

bike ones. However, the effects of hard braking on driv-

ing variables are not as subject to oscillation as they are

in the front fork motorcycle model.

This is due to the fact that in brake conditions the

front swingarm angle decreases whereas the rear one

increases. As in a conventional motorbike, inclina-

tion angle during the path will depend on the wheel-

base value. The big difference is that wheelbase will

now tends to increase due to the decreasing of front

swingarm angle and in the meanwhile will tend to de-

crease due to the augment of rear swingarm angle, as it

can be seen in Figure 4. This fact leads to a non-direct

relation between the loads distribution and the wheel-

base value, which means also a different behavior of

motorbike leaning. As a consequence, when braking

hard, ψ is less disturbed, leading to a smaller wobbling

4

phenomena (Figures 5 and 6).

Figure 5: Braking effect on ψ for the conventional motorbike model

Figure 6: Braking effect on ψ for the double swingarm model

3 Dynamics

3.1 Kinetic energy

The motorbike dynamics model in this chapter is de-

veloped using the Lagrangian approach. In the first

step the free Lagrangian is computed from the total ki-

netic energy. The kinetic energy in the motorbike is

due to any moving component, namely frame, engine

and wheels. It is assumed that energy associated to el-

evation changes can be neglected. Some kinetic ener-

gies will not be included in the system because their

values can be assumed to be irrelevant in comparison

with the included ones. For example, the translation

energy of the front and rear suspensions, the rotational

energy of the swingarm and the front fork and the trans-

lation energy of the engine pieces are not included. The

kinetic energy is thus calculated through following ex-

pressions,

Kϕψ =12

Ω A Ω =12

(ϕ ψ

)m

h2 hb

hb b2

ϕ

ψ

Kxy =

12

(x y

)m

x

y

K

θ f=

14

m f r2f θ

2f ; K

θr=

14

mrr2r θ 2

r ; Kθ=

14

mmr2mθ 2

(6)

where Ω is the angular speed vector, A is the inertia ma-

trix, which depends on the distances between mass cen-

ter and the rotating axis (h = h(l,β ) and b = b(l,β ) are

coordinates of total mass center), m is the total mass,

m f is the front wheel mass, mr is the rear wheel mass,

mm is the engine’s rotating part mass, r f is the front

wheel radius, rr is the rear wheel radius, rm is the en-

gine’s rotating part radius, θ f = θ f (θ ,ϕ) is the front

wheel angular speed, θr = θr(θ) is the rear wheel an-

gular speed, θ is the engine angular speed. All masses

and inertias are simplified as point-masses (m) or thin

disks (m f , mr and mm).

To complete the equations system that represents

the dynamic model, the kinematics model of section 2

will be used, in addition to some other kinematic equa-

tions:

x = θgrRr cosψ ; y = θgrRr sinψ

θ f =θgrRr

cosδ r f; θr = grθ

(7)

Where gr is the rear wheel-engine angular speeds ratio.

The kinetic energy is therefore calculated with the

5

following equations,

K = Kϕψ +Kxy +Kθ f+K

θr+K

θ

K =m(ϕh+ ψb)2

2+

mR2r θ 2

mgr2

2+

m f R2r θ 2

mgr2

4cos2(δ cosγ)

+mrr2

r θ 2mgr2

4+

mmr2mθ 2

m

4(8)

3.2 Lagrangian

Following the Lagrange method procedure, the dynam-

ics equations are obtained by calculating the kinetic en-

ergy derivatives.

ddt

∂K∂ ϕ

− ∂K∂ϕ

= Γϕ ;ddt

∂K∂ θ

− ∂K∂θ

= Γθ (9)

This system of equations can also be written in matrix

form,

M

ϕ

θ

+V = τ (10)

The dynamics model obtained from the free, that is un-

constrained, Lagrangian it is required to account for the

constraints imposed by the kinematics of the bike and

the contact with the ground. Motorbike dynamics falls

under the general class of constrained multibody me-

chanical systems, but in a real motorbike the variety

of situations rends difficult the identification of these

constraints. Therefore a common strategy when identi-

fying constrained dynamics is to rely on a priori knowl-

edge on the physical effects affecting the vehicle.

The external forces/torques, τext , are thus simpli-

fied and they will be considered the direct consequence

of the reaction forces applied on the center of the to-

tal mass. As it has already been used in [1], contact

forces, which is calculated at each instant and depends

on the global system state, are now reduced to the sum

of gravitational and centrifugal forces that the motor-

bike receives in the contact points between the wheels

and the floor referred to the center of mass. These

forces are calculated as,

N = mg ; T = mRrgrθ ψ (11)

where N is the vertical force due to gravitation and de-

pends on the total mass, m, and the gravitational accel-

eration, g. T is the centrifugal force mainly due to ψ

rotating speed. These forces are applied ideally on the

motorbike total mass center, and have to be related to

ϕ as torques,

M

ϕ

θ

+V =

Γϕ

Γθ

+ Nhsinϕ −T hcosϕ

0

(12)

3.3 Dynamics simulations

After having implemented it, the dynamics model must

be validated before being used for control analysis. To

this purpose, several tests have been made, encom-

passing different situations. Namely, the spectrum of

simulation comprehends diverse torques applied by the

driver, starting conditions, external forces and driving

environments.

Some special tests have been made to complete the

study of the model behavior. For example some simula-

tions without gravity force or without centrifugal force

show the torque related to the gyroscopic effect. This

negative torque is due to the system’s tendency of con-

serving the inertial moment. In fact, when the wheels

and the engine are rotating in y direction while motor-

6

bike is turning around z axis, a torque in x appears as

results of the following equation:

Γgyroscopic = Ω×A Ω (13)

where Ω is the vector of the angular speeds and A is the

inertia matrix.

4 Control

4.1 Motivation

The control system has been designed using a con-

trol strategy that can mimic the behavior of a human

rider. Humans ride bikes, motorbikes, drive cars using

a relatively simple set of principles; namely, estimat-

ing information on (i) the current state of the vehicle,

(ii) the expected variation in the state, and (iii) the rid-

ing/driving history. These have close relations with the

well known PID structure.

Basically, there are two values that have to be con-

trolled: firstly the error between the real ψ angle and

the one that corresponds to the trajectory to be followed

by the motorbike and secondly the sum of the torque

due to the centrifugal force, the one due to the gravity

force and the gyroscopic ones.

A human rider has available four actuation points.

The first is the engine torque, which is strongly con-

nected to the forward velocity. Braking actions can be

also represented with the engine torque variable1. The

second actuation point is the steering angle in the front

fork, imposed by applying a torque through the handle-

bar. The third point is the torque applied through the

footpegs, used by riders to control the leaning angle.

The fourth point is the mass center position variation on

y axis, obtained by changing the position of the body,

namely the contact of the upper torso with the seat and

the position of the legs.

4.2 PID simulations

To implement this PID control additional Matlab code

was implemented. The results for the empirical con-

trol simulations show that with well-chosen P, D and I

values, the motorbike follows a reference trajectory.

Many driving conditions have been simu-

lated. For example, when the forward veloc-

ity is 100km/h and the curve radius is 50m, the

system manages to control the motorbike thanks

to some specific PID parameters and the equilib-

rium is reached with δ = 1.218 and ϕ = 53.37.

Figure 7: Autonomous driving during a 50m-radius curve. Leaning angle

[] (red); forward velocity [Km/h] (green); rotation speed [/s] (blue); steer-

ing angle [10x] (magenta); equivalent mass for ϕ-torque [Kg] (black)

1In fact, the current motorbike engines have excellent motor-braking properties.

7

The system response can be modified by changing

the values of the PID control constants. Many further

driving condition have been simulated with several PID

parameters in order to study the the model functioning.

The six control constants have to be carefully chosen in

order to avoid the diverse negative effects that appears

after having assigned both excessively small or exces-

sively big values.

5 Results

The first three curves of the Montmelo circuit, located

in the outskirts of Barcelona, Spain, have been chosen

to simulate the model in realistic conditions. The con-

trol model can thus be tested in fast and slow curves,

left and right rotation, under braking and accelerating

conditions. The simulation must analyze if the control

strategies are able to make the motorbike follow the cir-

cuit path by changing the values of control variables δ

and Γϕ .

The circuit has been simulated with control con-

stants that have been empirically found using a trial

and error strategy until acceptable results have been ob-

tained. This strategy reflects in some sense the driving

learning by a human rider. In fact, by changing the con-

trol parameters for each circuit section, the trajectory is

gradually improved in order to make coincide the mo-

torbike trajectory with the curves of real path

The evolution of the variables during the simula-

tion is shown in Figure 8. Γϕ torque has been limited

to a correspondent feet force of 280Kg because that is

the approximate limit of human applicable force. The

engine torque will be set in order to obtain forward ve-

locities (green line) that are normally used in each sec-

tion of the circuit. The steering angle δ (magenta line)

and rotation speed ψ (blue line), that are strongly re-

lated with each other, respond as a second degree sys-

tem reaching the final value after a delay time of ap-

proximately 1s. The inclination angle ϕ (red line) has

a smooth evolution when its value is modified. The

values for permanent δ variable respect the kinematic

equations, since the system search the equilibrium posi-

tion for a determined curve at a concrete speed. These

values are approximately 1.6 degrees in the first two

curves and 0.8 in the third faster curve.

The torque Γϕ (black line) appears when entering

in the first curve and decreases gradually after having

managed to incline the motorbike. The torque does not

disappear after having arrived to the required leaning

because the forward speed is constantly increasing. A

greater torque is also needed after the curve in order to

recover the vertical position.

Controlled model has been also simulated in other

conditions, such as without forces limitations and with

leaning angle borderline of 60.

6 Conclusions

During this study a great improvement in the motorcy-

cle modeling has been made. The major achievements

of the thesis can be assessed on the one side to the kine-

matics and dynamic equations that model the motor-

bike system, and on the other side to the results of the

controlled model simulations.

8

Figure 8: Autonomous motorbike under PID-based control. Leaning angle [] (red); forward velocity [Km/h] (green); rotation speed [/s] (blue); steering

angle [10x] (magenta); equivalent mass for ϕ-torque [Kg] (black)

The kinematics model can be considered highly re-

alistic thanks to the gradually incorporated improve-

ments (the front fork inclination, the damping sys-

tem with the consequent variable front fork length and

swingarm angle and the pneumatic shape effect on the

real back wheel radius). The results reveal some inter-

esting aspects. One of this item is the fact that the pneu-

matic shape causes a variation in the forward speed

when the motorcycle inclines. Moreover, changes in

suspensions loads affect the equilibrium kinematic po-

sition. During hard braking, the load on the front wheel

augment whereas the one on the back wheel decreases

and, as a consequence, geometry variation produce a

change on the relation between leaning angle and steer-

ing angle. The simulations show indeed that leaning

angle is smaller when braking, so that, in that case,

maintaining a curved trajectory would require a minor

torque on the inclination direction as dynamic analysis

will explain.

Besides, a kinematic model for the motorbikes

based on a double swingarm damping system has been

developed. The results of the different simulations,

along with some empirical and intuitive considerations,

expose advantages and disadvantages of this particular

kind of motorcycles. A benefit is the fact that when

the loads on the wheels change (i.e. when braking or

accelerating) some variations appear in the swingarms

angles, which could compensate each other in terms of

maintaining the wheelbase length constant. In addition,

tensions received by the system from the floor are not

transmitted directly to the driver’s hand, since there is

no front fork and steering angle is independent form

the damping system. Contrary to these features, some

maneuverability is lost due to the fact that the angle of

the front wheel is not directly controlled anymore, but

a complex systems of special bars transmits the move-

9

ment of the handlebars to the wheel.

The dynamics model was obtained through the La-

grangian method based on the kinetic energy of the sys-

tem. After having calculated the free Lagrangian equa-

tion system, model has been completed with the incor-

poration of the external forces due to the contact be-

tween the motorbike and the floor. The model has been

finally validated with several simulations that tested di-

verse driving situations. The results of the simulations

show the validity of the model and demonstrate how

the equilibrium position is reached or lost by the sys-

tem by modifying the variable values. Moreover, they

explain the importance of each one of the dynamic and

kinematic variable for the system and how they affect

the behavior of the motorbike.

The system by itself is unstable, because it is not

capable of maintaining the equilibrium position for the

driving situation. Therefore a PID control has been de-

signed. The objective of the feedback control is on the

one side to eliminate the difference between the torque

due to the centrifugal effect, the gravitational force and

the gyroscopic one by applying a torque that corre-

sponds to the change of the mass center position pro-

duced by the movement of the driver’s body; on the

other side the steering angle attempts to remove the er-

ror that appears between the rotation of the bike end the

trajectory to be followed.

The final model has been finally tested on a real

driving situation. The behavior of the motorbike model

has been simulated in first part of the circuit of Mont-

melo, where a rich variety of driving conditions can be

found.

The arrival point of this thesis can be considered

the starting one for several further developments. Pos-

sible future work should extend the control implemen-

tation to all feasible situations in order to ensure stabil-

ity and performance requirements, namely in the fully

autonomous motorbike scenario.

References

[1] David J.N. Limbeer and Robin S. Sharp Bicy-

cles, Motorcycles, and Models IEEE Control

Systems Magazine, October, 2006

[2] Karl J. Astrom, Richard E. Klein, Anders

Lennartsson Bicycle Dynamics and Control

IEEE Control Systems Magazine, pp 26-47, Au-

gust 2005

[3] R.S. Sharp and S. Evangelou and D.J.N. Lime-

beer Advances in the Modelling of Motorcycle

Dynamics Multibody System Dynamics, pp 251-

283, number 12, 2004

[4] A.A. Popov and S. Rowell and J.P. Meijaard A

review on motorcycle and rider modelling for

steering control Vehicle System Dynamics, pp

775-792, volume 48, number 6, 2010

[5] A.L. Schwab and J.P. Meijard and J.M. Pa-

padopoulus Benchmark results on the linearized

equations of motion of an uncontrolled bicycle

Proc. 2nd Asian Conf. Multibody Dynamics, Au-

gust 2004

[6] Francesco Vitale and Gabriele Virzı Mar-

iotti Innovative progressive suspensions

http://www.atnet.it/lista/casuni.htm, 2007 [On-

line]10