Nonlinear vehicle modelling

PROJECT LABORATORY 1
Adam Wittmann
Budapest University of Technology and Economics
Department of Control Engineering and Information Technology
Modelling and simulation of a non-
linear model of a four-wheel drive
electric model car
Supervisors: Gábor Rödönyi, PhD
Hungarian Academy of Sciences, Institute of Computer Science and Control
Systems and Control Lab
Ádám Bakos
Hungarian Academy of Sciences, Institute of Computer Science and Control
Systems and Control Lab
István Harmati, PhD
Budapest University of Technology and Economics
Department of Control Engineering and Information Technology

Adam Wittmann Project laboratory 1 2014/2015/2
2
TARTALOM
1 Introduction and task description................................................................................................... 3
2 Nonlinear model.............................................................................................................................. 5
2.1 Modelling paradigms and coordinate – systems .................................................................... 5
2.2 Longitudinal and yaw dynamics .............................................................................................. 6
2.3 Body forces.............................................................................................................................. 7
2.4 Wheel model ........................................................................................................................... 7
2.4.1 Wheel centre velocities................................................................................................... 8
2.4.2 Side slips and tyre slip angles .......................................................................................... 9
2.4.3 Friction coefficient and wheel forces............................................................................ 11
2.5 Simulink model and summary............................................................................................... 13
3 Linearized model ........................................................................................................................... 14
3.1 Assumptions.......................................................................................................................... 14
3.2 Tyre forces and wheel slips ................................................................................................... 15
3.3 Vehicle longitudinal and yaw dynamics ................................................................................ 16
3.4 State space representation ................................................................................................... 19
3.5 Linearized Simulink model and simulation............................................................................ 20
4 Summary........................................................................................................................................ 22
5 References..................................................................................................................................... 23

3
1 INTRODUCTION AND TASK DESCRIPTION
In the Systems and Control laboratory of the Computer and Automation Research Institute of the
Hungarian Academy of Sciences a model electric vehicle is under investigation in terms of control
algorithms and model verification.
The modelled vehicle has basically four brushless DC motors, each driving one wheel. The power
electronics and the motor-drive electronics are already developed so as the low-level controller to
drive the motors. The model vehicle’s front wheels are steerable by one single servo motor. This is
already implemented as well.
The long-term goal of this project is to reveal the control possibilities and the cornering abilities and
limits of this car. The investigation of the cornering abilities has two main parts:
 Maximum steering angle and acceleration at a given maximum side slip of the car depending
on the parameters
 Zero steering angle and cornering only by directing different torques to the four wheels.
Apart from the cornering the investigation and implementation of a controller is a goal. The point of
the control system is that the driver gives two inputs – steering angle (δ) and acceleration demand (ax)
and the vehicle behaves according to the inputs. A possible way to realize this is a reference model
which generates reference for the control loop (speed, side slip and yaw rate). The block-diagram of
the envisioned control system can be seen in Figure 1, where
 The input vector (contrary to the classical vehicles) is 𝑢 = [𝛿 𝑎 𝑥] 𝑇
 The state vector is 𝑥 = [𝑣 𝛽 𝑟] 𝑇
 The reference state vector is 𝑥 𝑟 = [𝑣𝑟 𝛽𝑟 𝑟𝑟] 𝑇
So developing and setting up a vehicle model similar to the reality is of key importance for further
development, as the long-term goal is to design a controller to the car and the controller parameters
are determined based on the vehicle model. If the model is not accurate enough, the designed
controller will not fit perfectly to the real car, only the inappropriate model.
Reference model
(linearized)
Vehicle model
(non-linear)
Controller
δ
τ
xr
u x
Figure 1 – The envisioned control loop

4
The goal if this report is to model and simulate a non-linear model of this model car, which will be the
right ‘Vehicle model’ on the block diagram, and a simpler, linearized reference model (left ‘Reference
model’) which gives a good approximation to the nonlinear model.
First a non-linear model was set up based on [1], and then was implemented in MATLAB Simulink®.
The tests were mostly run in steady-state. Then linearization and simplifications were applied on this
model and the goal was to create a simplified model, which can be a reference model in the future to
develop an appropriate controller to this model car.

5
2 NONLINEAR MODEL
This chapter is based on [1].
This modelling approach has the following aims:
 Clearly structured model where well separated parts are concentrated in sub-systems
 Level of complexity which describes the model greatly but enough for vehicle dynamics
According to the aims the model is divided into well described parts which are connected to each other
through the adequate variables. The main and most important part of the full model is the wheel
model. This is of the highest importance, because forces acting on the vehicle are developed here, and
the more accurate representation of tyres and wheel dynamics is essential, as in some cases just one
per-thousand difference is a coefficient can cause ten times larger error in wheel forces. The second
part of the model is the body force calculations and the yaw and longitudinal dynamics, where wheel
forces are converted into resultant, body forces and kinematic variables, such as longitudinal and
lateral speed, yaw rate are determined.
2.1 MODELLING PARADIGMS AND COORDINATE – SYSTEMS
In the next few points the focus is mainly on the modelling of the longitudinal and lateral dynamics of
the vehicle. This involves that roll and pitch dynamics and heave motion are not investigated and roll
and pitch angles are regarded as constant variables which does not change during the movement.
Figure 2 – Coordinate systems

6
Other presuppositions are that the vehicle’s chassis behaves as a rigid body, the suspension (as the
heave motion) is neglected so wheels are rigidly connected to the body. During steering wheel tilts are
neglected. The vehicle is driven by four, independently controllable BLDC motors and the front wheels
are steerable and the steering angle of the two front wheels are regarded to be equal.
During the modelling basically two types of coordinate systems will be used (Figure 2): a coordinate
system fixed to the vehicle’s centre of gravity (x,y), where the x-axis is parallel to the vehicle’s
longitudinal axis, and the y-axis is 90°left from the x. The other coordinate systems are the four wheel
coordinate systems (xw and yw), each fixed to a wheel and the orientation of their x-axis is in the
direction of the wheel pane, while y-axis points 90° left to the x-axis.
2.2 LONGITUDINAL AND YAW DYNAMICS
The inspected movement of the whole vehicle (longitudinal and yaw dynamics) can be defined by three
main equilibrium equations as the vehicle is regarded to move in one plane. These three equations
describe the longitudinal movement in the two directions of the plane and the rotation movement
perpendicular to the plane. The three equations give derivatives of those three parameters which are
able to fully describe this flat movement of the vehicle. These are the absolute forward vehicle velocity
(v), the vehicle slip angle (β) and the yaw rate (r) around the z-axis of the vehicle. The equations are
the following:
𝑣̇ =
cos 𝛽
𝑚
𝑓𝑥 +
sin 𝛽
𝑚
𝑓𝑦 Equation 2.1
𝛽̇ = −
sin 𝛽
𝑣𝑚
𝑓𝑥 +
cos 𝛽
𝑣𝑚
𝑓𝑦 − 𝑟 Equation 2.2
𝑟̇ =
𝑚 𝑧
𝐽𝑧 Equation 2.3
where
 𝑣 [
𝑚
𝑠
] is the longitudinal velocity of the vehicle
 𝛽 [𝑟𝑎𝑑] is the side slip of the vehicle
 𝑟 [
𝑟𝑎𝑑
𝑠
] is the yaw rate of the vehicle
 𝑚[𝑘𝑔] is the mass of the vehicle
 𝐽𝑧[𝑘𝑔𝑚] is the inertia of the vehicle around the z-axis
 𝑓𝑥[𝑁] is the longitudinal force acting on the vehicle body
 𝑓𝑦[𝑁] is the lateral force acting on the vehicle body
 𝑚 𝑧[𝑁𝑚] is the moment acting on the vehicle around its z-axis

7
2.3 BODY FORCES
In the equations above on the left side stand the variables which describe the dynamic movement of
the vehicle. While on the right side there are the forces acting on the vehicle body. These forces add
up of the forces acting on the wheels resulted by wheel and tyre dynamics. However air drag is usually
considered, now we put it out from the calculation as the effect of it is negligible as the modelled and
examined vehicle is a small model car. Working in the coordinate system of the centre of gravity of the
vehicle, the forces acting on the body are the following:
𝑓𝑥 = 𝐹𝑥,𝑅𝐿 + 𝐹𝑥,𝑅𝑅 + (𝐹𝑥,𝐹𝐿 + 𝐹𝑥,𝐹𝑅)∙ cos 𝛿
− (𝐹𝑦,𝐹𝐿 + 𝐹𝑦,𝐹𝑅) ∙ sin 𝛿 Equation 2.4
𝑓𝑦 = 𝐹𝑦,𝑅𝐿 + 𝐹𝑦,𝑅𝑅 + (𝐹𝑦,𝐹𝐿 + 𝐹𝑦,𝐹𝑅)∙ cos 𝛿
− (𝐹𝑥,𝐹𝐿 + 𝐹𝑥,𝐹𝑅) ∙ sin 𝛿 Equation 2.5
𝑚 𝑧 = −(𝐹𝑦,𝑅𝐿 + 𝐹𝑦,𝑅𝑅)𝑙 𝑟
+ 𝑙 𝑓[(𝐹𝑦,𝐹𝐿 + 𝐹𝑦,𝐹𝑅)cos 𝛿
+ (𝐹𝑥,𝐹𝐿 + 𝐹𝑥,𝐹𝑅)sin 𝛿]
+
𝑙 𝑤
2
[(𝐹𝑦,𝐹𝐿 − 𝐹𝑦,𝐹𝑅)sin 𝛿
+ (𝐹𝑥,𝐹𝑅 − 𝐹𝑥,𝐹𝐿)cos 𝛿 + 𝐹𝑥,𝑅𝑅 − 𝐹𝑥,𝑅𝐿]
Equation 2.6
where
 𝛿[𝑟𝑎𝑑] is the steering angle of the two front wheels
 𝐹𝑥,𝑖𝑗[𝑁] are the longitudinal forces acting on the wheels in the coordinate system of the
wheels, which are in the direction of the wheel plane
 𝐹𝑦,𝑖𝑗[𝑁] are the lateral forces acting on the wheels in the coordinate system of the wheels,
which are perpendicular to the wheel plane
 𝑙 𝑟[𝑚] 𝑎𝑛𝑑 𝑙 𝑓[𝑚] are the distances of the centre of gravity from the rear and front axle
 𝑙 𝑤[𝑚] is the wheelbase.
2.4 WHEEL MODEL
The role of this part is to determine the forces acting on the wheels. To derive the wheel forces it is
indispensable to know the wheel slips and the tyre slip angles, for which wheel velocities and friction
coefficients are used. Also the wheel caster can be taken into consideration but for the sake of
complexity now it is ignored. Caster is the tilt of the steering axis of the wheel and as it causes only a

8
small dislocation in the tyre-road contact centre, omitting it results in small changes of the wheel
distance from the vehicle’s centre of gravity, which leads to negligible differences in wheel forces.
2.4.1 Wheel centre velocities
Wheel centre velocities does not have a strict kinematic connection to the rotational velocity of the
wheels, but does have with the chassis and the body of the vehicle. In this case wheel centre velocities
are derived by the transformation of centre of gravity (CoG) velocities to a two-track (four-wheel)
model. The two-track model lets the yaw rate have a considerable effect on the wheel speeds. This
method supposes the knowledge of the CoG velocity (𝑣), the yaw rate (𝑟) and the vehicle side slip angle
(𝛽) and also the exact geometry of the vehicle (especially the distances of the wheels from the centre
of gravity).
So the wheel centre velocities consist of two components. One is originated from the transformation
of the CoG velocity, and the other comes from the rotation (yaw motion) around the vertical axis
(Figure 3). These wheel-speeds are two dimensional vectors, which have components in the direction
of the vehicle’s longitudinal axis (x axis) and perpendicular to that (y axis). The vertical axis (z axis) is
the third direction of the coordinate system located in the centre of gravity. These three constitute an
orthogonal coordinate system. The wheel centre velocities are the following:
Figure 3 – The kinematics of the vehicle

9
𝑣 𝑤,𝑓𝑙 = (𝑣 ∙ cos 𝛽 − 𝑟
𝑙 𝑤
2
) ∙ 𝑖 + (𝑣 ∙ sin 𝛽 + 𝑟𝑙 𝑓) ∙ 𝑗 Equation 2.7
𝑣 𝑤,𝑓𝑟 = (𝑣 ∙ cos 𝛽 + 𝑟
𝑙 𝑤
2
) ∙ 𝑖 + (𝑣 ∙ sin 𝛽 + 𝑟𝑙 𝑓) ∙ 𝑗 Equation 2.8
𝑣 𝑤,𝑟𝑙 = (𝑣 ∙ cos 𝛽 − 𝑟
𝑙 𝑤
2
) ∙ 𝑖 + (𝑣 ∙ sin 𝛽 − 𝑟𝑙 𝑟) ∙ 𝑗 Equation 2.9
𝑣 𝑤,𝑟𝑟 = (𝑣 ∙ cos 𝛽 + 𝑟
𝑙 𝑤
2
) ∙ 𝑖 + (𝑣 ∙ sin 𝛽 − 𝑟𝑙 𝑟) ∙ 𝑗 Equation 2.10
Apart from wheel centre velocities, rotational equivalent velocities (𝑣 𝑅) are used in the calculations.
The rotational equivalent velocity is in the direction of the wheel plane and its magnitude equals to
the product of the angular speed of the wheel (𝜔) and the static wheel radius (𝑟𝑠𝑡𝑎𝑡):
𝑣 𝑅 = 𝜔 ∙ 𝑟𝑠𝑡𝑎𝑡 Equation 2.11
, where the static wheel radius is the unloaded wheel radius (𝑟0) reduced by the effect of the ground
contact force(𝐹𝑧):
𝑟𝑠𝑡𝑎𝑡 = 𝑟0 −
𝐹𝑧
𝑘 𝑤 Equation 2.12
𝑘 𝑤 is the tyre spring stiffness.
2.4.2 Side slips and tyre slip angles
Side slip angles play a main role in the formation of wheel forces and also have a great effect on them.
This leads to the need of very accurate calculation and modelling of the side slips and also wheel slip
angles. To calculate them, tyre slip angles needed to be defined first.
Tyre slip angles are defined as the angle of the wheel plane and the wheel centre velocity (Figure 3).
In case of straight driving the tangent of the tyre slip angle can be determined as the ratio of the
component of the wheel centre velocity in the y direction and in the x direction. If the turn angle (δ) is
not zero, it must be taken into consideration at the front wheels as well. Describing the angles (𝛼𝑖𝑗) as
mathematical equations:

10
𝛼 𝑓𝑙 = 𝛿 𝑤 − tan−1
(
𝑣 ∙ 𝛽 + 𝑟𝑙 𝑓
𝑣 − 𝑟
𝑙 𝑤
2
)
Equation 2.13
𝛼 𝑓𝑟 = 𝛿 𝑤 − tan−1
(
𝑣 ∙ 𝛽 + 𝑟𝑙 𝑓
𝑣 + 𝑟
𝑙 𝑤
2
)
Equation 2.14
𝛼 𝑟𝑙 = −tan−1
(
𝑣 ∙ 𝛽 − 𝑟𝑙 𝑟
𝑣 − 𝑟
𝑙 𝑤
2
)
Equation 2.15
𝛼 𝑟𝑟 = −tan−1
(
𝑣 ∙ 𝛽 − 𝑟𝑙 𝑟
𝑣 + 𝑟
𝑙 𝑤
2
)
Equation 2.16
Several approaches exist which calculate wheel slips in different ways. Now Burckhardt’s method is
used, in which the longitudinal wheel slip is calculated in the direction of the motion of the wheel in
contrast to some other solutions. The longitudinal slip is in the direction of the wheel centre velocity,
and the side slip is perpendicular to the longitudinal slip.
The rotational equivalent wheel velocity is multiplied by the cosine of the tyre slip angle, which gives
the projection of the rotational equivalent speed in the direction of the wheel centre velocity, which
is a key in the calculation of slips:
Table 1
Braking (𝑣 𝑅 cos 𝛼 ≤ 𝑣 𝑤) Driving (𝑣 𝑅 cos 𝛼 > 𝑣 𝑤)
Longitudinal slip 𝑠 𝐿 =
𝑣 𝑅 cos 𝛼 − 𝑣 𝑤
𝑣 𝑤
𝑠 𝐿 =
𝑣 𝑤 − 𝑣 𝑅 cos 𝛼
𝑣 𝑤
Side slip
𝑠 𝑆 =
𝑣 𝑅 sin 𝛼
𝑣 𝑤
𝑠 𝑆 = tan 𝛼
However, introduce another type of longitudinal slip:
𝜆 =
𝑣 𝑅
𝑣 𝑤 Equation 2.17

11
Using lambda, the equations above in Table 1 can be rewritten according to the following [2]:
𝑠 𝑆 =
𝜆 sin 𝛼
max(1, 𝜆 cos 𝛼) Equation 2.18
𝑠 𝐿 =
1 − 𝜆 cos 𝛼
Based on longitudinal and side slips, the resultant slip is the geometrical sum of this two:
𝑠 𝑅𝑒𝑠 = √𝑠 𝐿
2 + 𝑠 𝑆
2
Equation 2.20
𝑠 𝑅𝑒𝑠 =
√1 + 𝜆2 − 2𝜆 cos 𝛼
2.4.3 Friction coefficient and wheel forces
Friction coefficients, just as wheel forces are highly depending on the resultant wheel slip. However
friction (or adhesion) characteristics are mostly approximated and calculated by parametric
characteristics, which are shown in Figure 4 [2]. The definition of the friction coefficient µ [-] is the
following for each wheel:
𝜇 =
𝐹𝑓𝑟𝑖𝑐𝑡𝑖𝑜𝑛
𝐹𝑧
Equation 2.22
The calculation and approximation of the friction coefficient is determined according to Burckhardt:
𝜇(𝑠 𝑅𝑒𝑠) = 𝑐1 ∙ (1 − 𝑒−𝑐2∙𝑠 𝑅𝑒𝑠 ) − 𝑐3 ∙ 𝑠 𝑅𝑒𝑠 Equation 2.23
Figure 4 – Adhesion characteristics

12
In this equation different behaviour of the vehicle at higher velocities or in case of higher wheel loads,
however these can also influence – only decrease – the friction. There is an extended function by
Burckhardt, which incorporates these effects as well, but now it is not presented and in the model it is
not used, as our model car is supposed to have about the same full load and the same wheel load and
will not operate at very high velocities. Also the difference of the results is minimal between the two
methods and plus other variables would just complicate more the model, while the point was to keep
it as simple as it is possible regarding a sufficient accuracy. The c1, c2 and c3 parameters are defined
mostly experimentally and some of the most usual ones are collected in Table 2.
Table 2
c1 c2 c3
Asphalt, dry 1.2801 23.99 0.52
Asphalt, wet 0.857 33.822 0.347
Concrete, dry 1.1973 25.168 .05373
Cobblestones, dry 1.3713 6.4565 0.6691
Cobblestones, wet 0.4004 33.7080 0.1204
Snow 0.11946 94.129 0.0646
Ice 0.05 306.39 0
The wheel contact forces parallel (x) and perpendicular (y) to the wheel plane can be written based on
the equations (2.17)-(2.23) [2]:
𝐹𝑥 = 𝐹𝑧
𝜇(𝑠 𝑅𝑒𝑠)
𝑠 𝑅𝑒𝑠
𝜆 − cos 𝛼
𝐹𝑦 = 𝐹𝑧
𝑠 𝑅𝑒𝑠
sin 𝛼
Regarding
𝐶(𝜆, 𝛼) = 𝐹𝑧
𝑠 𝑅𝑒𝑠
Equation 2.26
as the cornering stiffness function, Fx and Fy are quite similar to each other:
𝐹𝑥 = 𝐶(𝜆, 𝛼)(𝜆 − cos 𝛼) Equation 2.27
𝐹𝑦 = 𝐶(𝜆, 𝛼) sin 𝛼
Equation 2.28

13
The wheel forces calculated above are responsible to accelerate or decelerate the vehicle. The
equation which describes this effect is the following (separately for each wheel):
𝑣̇ 𝑅 = −𝐹𝑥
𝑟𝑠𝑡𝑎𝑡
2
𝐽 𝑤
+ 𝜏 𝑑𝑟
𝐽 𝑤
Equation 2.29
, where
 𝐹𝑥[𝑁] is the wheel force in the direction of the wheel plane
 𝐽 𝑤[𝑘𝑔𝑚2
] is the inertia of the wheel
2.5 SIMULINK MODEL AND SUMMARY
In the last four subchapter a nonlinear model, according to Burckhardt, is derived. Three equations
describe the plane motion of the vehicle, one for the speed, one for the side slip and one for the yaw
rate (Equations 2.1 – 2.3, chapter 2.2). These are dependent on the body forces, which are expressed
by forces acting on the wheels (chapter 2.3). Wheel forces however have a quite non-linear
characteristic for which a wheel model was set up according to Burckhardt’s vision (chapter 2.4). This
required the deduction of wheel speeds and slips, on which the friction coefficients depend highly.
This model was created to have an approximation for the dynamic and kinematic behaviour of the
vehicle. This resulted sometimes in highly complex nonlinear functions, which require modelling
programs, such as MATLAB® Simulink®. Now the created Simulink model is introduced briefly. This
model has five inputs:
 δ [rad], the steering angle
 τij [Nm], the applied driving torques for each wheel, respectively.
The outputs of the model are the quantities on the left side of the equations (2.1) (2.2) and (2.3) in
Chapter 2.2:
 𝑣̇ [
𝑚
𝑠2] and its integrate, 𝑣 [
𝑚
𝑠
], the acceleration and the speed of the vehicle
 𝛽̇ [
𝑟𝑎𝑑
𝑠
] and its integrate, 𝛽[𝑟𝑎𝑑], the vehicle side slip and its velocity
 𝑟̇ [
𝑟𝑎𝑑
𝑠2 ] and its integrate, 𝑟 [
𝑟𝑎𝑑
𝑠
], the yaw rate of the vehicle, and its derivative
The Simulink model was built according to the chapters 2.2 – 2.4 are describing the vehicle.

14
3 LINEARIZED MODEL
Two essential aspects stand in the centre of this project laboratory. The first is to create an appropriate
model which gives a good representation of the reality and the real model car. This was introduced in
Chapter 2. The second point is to create a linearized model, which can be a reference and the base of
control design for a future control system (see the control loop in Figure 1). The point of this reference
model is to give a model which has definitely no oversteering characteristics.
3.1 ASSUMPTIONS
As this linearized model is derived from the nonlinear model presented in Chapter 2, naturally there
need to be some assumptions which let us make the simplifications and linearization. Most of them
are in connection the complex part of the wheel forces, but other parts are concerned.
In this model there are a lot of trigonometric functions which should be realized. These are
approximated by the linear part of their Taylor-series:
sin 𝛼 = 𝛼 𝑎𝑛𝑑 sin 𝛽 = 𝛽 Equation 3.1
cos 𝛼 = 1 𝑎𝑛𝑑 cos 𝛽 = 1 Equation 3.2
tan−1
𝑥 = 𝑥 Equation 3.3
In steady-state with zero longitudinal acceleration longitudinal (parallel to x-axis) wheel forces are
neglected so Fx = 0 and τ=0. Thus the longitudinal acceleration and angular acceleration of a wheel is
considered zero and depends on only the applied torque on the wheel. The connection between the
applied torque and the longitudinal force on the wheel is
𝐹𝑥,𝑖𝑗 =
𝜏𝑖,𝑗
Equation 3.4
So either Fx or τ can be considered as inputs. Fx will be.
Further assumptions are the following:
 𝛼 ∙ 𝛽 = 0, as both are small angles, and their product is negligible,
 𝛽 ∙ 𝛿 = 0, as both are small angles, and their product is negligible,
 𝛼 ∙ 𝛿 = 0, as both are small angles, and their product is negligible,
 𝐹𝑥 𝛽 = 0
 𝐹𝑥 𝛿 = 0

15
3.2 TYRE FORCES AND WHEEL SLIPS
The most complex part of the model is the calculation of the wheel forces (see equations 2.24 and
2.25). In these equations
µ(𝑠)
𝑠
causes most of the nonlinearity so the first point is to substitute this
expression. In Figure 5 µ(s) and µ(s)/s functions are plotted by red, on which Fy and Fx is highly
dependent according to equations (2.24) and (2.25).
In the two part of Figure 5 the suggested approximations are indicated with blue lines. On the right
side, where µ(s)/s is plotted, the derivative of the function is indicated by blue line, and its steepness
(c) is indicated by green, which is the usual approximation of this function for small slips. On the left
side, where µ(s) is plotted, the same approximation is plotted by green (c*s). This gives a significant
change and simplification in Fx and Fy.
Further simplification is that the maximum function in the denominator is considered as 1 because
𝜆 cos 𝛼 is nearly 1. Thus the linearized and simplified functions are the following for the wheels:
𝐹𝑥 = 𝐹𝑧
𝑠 𝑅𝑒𝑠
𝜆 − cos 𝛼
max(1, 𝜆 cos 𝛼)
≈ 𝑭 𝒛 𝒄(𝝀 − 𝟏) Equation 3.5
𝐹𝑦 = 𝐹𝑧
𝑠 𝑅𝑒𝑠
sin 𝛼
max(1, 𝜆 cos 𝛼)
≈ 𝑭 𝒛 𝒄 ∙ 𝜶 Equation 3.6
These steps gave us a significantly simpler wheel model, which makes easier the further calculations
as well.
Figure 5 - µ(s)at left and µ(s)/s at right
µ(s)/s ≈ c
µ(s) ≈ c*s
c

16
Another part of the wheel model are the wheel velocities and the calculated wheel slips (equations
(2.13)-(2.16)). Regarding the simplification points in chapter 3.1 trigonometric functions are linearized.
Also another assumption is regarded here, which was not listed in the first part of this chapter. In the
denominator of the mentioned equations the yaw rate has a product, which is considered much
smaller than the speed:
𝑟
𝑙 𝑤
2
≪ 𝑣 Equation 3.7
Thus the left part of this equation is neglected. Then the linearized wheel slip angles:
𝛼 𝑓𝑙 = 𝛿 𝑤 − 𝛽 −
𝑟𝑙 𝑓
𝑣 Equation 3.8
𝛼 𝑓𝑟 = 𝛿 𝑤 − 𝛽 −
𝑟𝑙 𝑓
𝑣 Equation 3.9
𝛼 𝑟𝑙 = −𝛽 +
𝑟𝑙 𝑟
𝑣 Equation 3.10
𝛼 𝑟𝑟 = −𝛽 +
𝑟𝑙 𝑟
𝑣 Equation 3.11
These expressions are used during the calculation of the lateral tyre forces, Fy,ij.
3.3 VEHICLE LONGITUDINAL AND YAW DYNAMICS
The longitudinal and yaw dynamics will be simplified thanks to two effects. The first is the simpler
models and equations of the wheel model introduced above. The second is some other approximation
in the dynamic equations. They have the same formula as in the nonlinear case:
𝑣̇ 𝑚 = 𝑓𝑥 + 𝛽𝑓𝑦 Equation 3.12
𝛽̇ 𝑣𝑚 = −𝛽𝑓𝑥 + 𝑓𝑦 − 𝑟𝑣𝑚
Equation 3.13
𝐽𝑧 𝑟̇ = 𝑚 𝑧 Equation 3.14
And the body forces and moments, and the linearization:
𝑓𝑥 = 𝐹𝑥,𝑅𝐿 + 𝐹𝑥,𝑅𝑅 + (𝐹𝑥,𝐹𝐿 + 𝐹𝑥,𝐹𝑅)∙ cos 𝛿
− (𝐹𝑦,𝐹𝐿 + 𝐹𝑦,𝐹𝑅) ∙ sin 𝛿 Equation 3.15

17
𝑓𝑥 = 𝐹𝑥,𝑅𝐿 + 𝐹𝑥,𝑅𝑅 + 𝐹𝑥,𝐹𝐿 + 𝐹𝑥,𝐹𝑅
− (𝐹𝑧,𝑓 𝑐 ∙ 𝛼 𝑓𝑙 + 𝐹𝑧,𝑓 𝑐 ∙ 𝛼 𝑓𝑟) ∙ 𝛿 Equation 3.16
𝑓𝑦 = 𝐹𝑦,𝑅𝐿 + 𝐹𝑦,𝑅𝑅 + (𝐹𝑦,𝐹𝐿 + 𝐹𝑦,𝐹𝑅) ∙ cos 𝛿
− (𝐹𝑥,𝐹𝐿 + 𝐹𝑥,𝐹𝑅) ∙ sin 𝛿 Equation 3.17
𝑓𝑦 = 𝐹𝑦,𝑅𝐿 + 𝐹𝑦,𝑅𝑅 + 𝐹𝑦,𝐹𝐿 + 𝐹𝑦,𝐹𝑅 − (𝐹𝑥,𝐹𝐿 + 𝐹𝑥,𝐹𝑅) ∙ 𝛿
Equation 3.18
+ 𝑙 𝑟[(𝐹𝑦,𝐹𝐿 + 𝐹𝑦,𝐹𝑅)cos 𝛿
+ (𝐹𝑥,𝐹𝐿 + 𝐹𝑥,𝐹𝑅) sin 𝛿]
+
𝑑 𝑟
2
[(𝐹𝑦,𝐹𝐿 − 𝐹𝑦,𝐹𝑅)sin 𝛿
+ (𝐹𝑥,𝐹𝑅 − 𝐹𝑥,𝐹𝐿) cos 𝛿 + 𝐹𝑥,𝑅𝑅 − 𝐹𝑥,𝑅𝐿]
Equation 3.19
+ 𝑙 𝑟[(𝐹𝑦,𝐹𝐿 + 𝐹𝑦,𝐹𝑅) + (𝐹𝑥,𝐹𝐿 + 𝐹𝑥,𝐹𝑅)𝛿]
+
𝑙 𝑤
2
[(𝐹𝑧,𝑓 𝑐 ∙ 𝛼 𝑓𝑙 + 𝐹𝑧,𝑓 𝑐 ∙ 𝛼 𝑓𝑟)𝛿 + 𝐹𝑥,𝐹𝑅
− 𝐹𝑥,𝐹𝐿 + 𝐹𝑥,𝑅𝑅 − 𝐹𝑥,𝑅𝐿]
Equation 3.20
Pointing out the simplifications introduced in chapter 3.1, and that in fx the sum of the longitudinal
wheel forces appear too, the equation (3.12) of the longitudinal motion can be rewritten in the
following form:
𝑣̇ 𝑚 = 𝑓𝑥 = ∑ 𝐹𝑥,𝑖𝑗
𝑖,𝑗
Equation 3.21
Then considering equation (3.17) for the lateral forces and (3.13) for vehicle slip, still keeping in mind
that 𝐹𝑥 ∙ 𝛿 = 0 and 𝐹𝑥 ∙ 𝛽 = 0
𝑓𝑦 = 𝑐[𝐹𝑧,𝑓(𝛼 𝑓𝑙 + 𝛼 𝑓𝑟) + 𝐹𝑧,𝑟(𝛼 𝑟𝑙 + 𝛼 𝑟𝑟)]
Equation 3.22
𝛽̇ 𝑣𝑚 = −𝛽 ∑ 𝐹𝑥,𝑖𝑗
𝑖,𝑗
+ 𝑐[𝐹𝑧,𝑓(𝛼 𝑓𝑙 + 𝛼 𝑓𝑟) + 𝐹𝑧,𝑟(𝛼 𝑟𝑙 + 𝛼 𝑟𝑟)]
− 𝑟𝑣𝑚
Equation 3.23

18
𝛽̇ 𝑣𝑚 = 𝑐[𝐹𝑧,𝑓(𝛼 𝑓𝑙 + 𝛼 𝑓𝑟) + 𝐹𝑧,𝑟(𝛼 𝑟𝑙 + 𝛼 𝑟𝑟)] − 𝑟𝑣𝑚
Equation 3.24
The third equation for the yaw dynamics is:
𝑚 𝑧 = 𝑐[𝐹𝑧,𝑓 𝑙 𝑓( 𝛼 𝑓𝑙 + 𝛼 𝑓𝑟 ) − 𝐹𝑧,𝑟 𝑙 𝑟( 𝛼 𝑟𝑙 + 𝛼 𝑟𝑟 )]
+
𝑙 𝑤
2
[𝐹𝑥,𝐹𝑅 − 𝐹𝑥,𝐹𝐿 + 𝐹𝑥,𝑅𝑅 − 𝐹𝑥,𝑅𝐿]
Equation 3.25
Then inserting in α into the equations their final, linearized and simplified form are:
𝑣̇ =
1
𝑚
(𝐹𝑥,𝐹𝑅 + 𝐹𝑥,𝐹𝐿 + 𝐹𝑥,𝑅𝑅 + 𝐹𝑥,𝑅𝐿) Equation 3.26
𝛽̇ = 𝛿
2𝑐𝐹𝑧,𝑓
𝑚𝑣
− 𝛽
2𝑐
𝑚𝑣
(𝐹𝑧,𝑓 + 𝐹𝑧,𝑟)
+ 𝑟
2𝑐
𝑚𝑣2 (𝐹𝑧,𝑟 𝑙 𝑟 − 𝐹𝑧,𝑓 𝑙 𝑓) − 𝑟
Equation 3.27
𝑟̇ = 𝛿
2𝑐𝑙 𝑓
𝐽𝑧
𝐹𝑧,𝑓 + 𝛽
2𝑐
𝐽𝑧
(𝐹𝑧,𝑟 𝑙 𝑟 − 𝐹𝑧,𝑓 𝑙 𝑓)
− 𝑟
2𝑐
𝐽𝑧 𝑣
(𝐹𝑧,𝑟 𝑙 𝑟
2
+ 𝐹𝑧,𝑓 𝑙 𝑓
2
)
+
𝑙 𝑤
2𝐽𝑧
[𝐹𝑥,𝐹𝑅 − 𝐹𝑥,𝐹𝐿 + 𝐹𝑥,𝑅𝑅 − 𝐹𝑥,𝑅𝐿]
Equation 3.28
Considering these three equations, it can be clearly seen that the first equation contains only the
longitudinal wheel forces, which means that only the torque (or force) inputs influence the longitudinal
acceleration, and the cornering angle, the fifth variable has no effect on it. Since that, however in the
other two equations () and () v appears, it can be handled as an operator. So the yaw dynamics is clearly
described by the last two equations. This way, this system is decoupled to longitudinal and yaw system.

19
3.4 STATE SPACE REPRESENTATION
Thanks to the decoupling of the system and the lot of simplification and linearization rules, the system
now can be handled much easily and it is possible to shape it into the well-known state-space
representation:
𝑥̇ = 𝐴𝑥 + 𝐵𝑢
Equation 3.29
, where
 x is the vector of state variables – state vector
 u is the vector of inputs – input (or control) vector
 A is the system matrix
 B is the input matrix
Considering δ and the four Fx,ij as inputs, the first equation, the first sub-system after decoupling is the
following:
[𝑣̇] = [0
1
𝑚
1
𝑚
1
𝑚
1
𝑚
]
[
𝛿
𝐹𝑥,𝐹𝑅
𝐹𝑥,𝐹𝐿
𝐹𝑥,𝑅𝑅
𝐹𝑥,𝑅𝐿]
Equation 3.30
The second and the third equations now can be written into one system, where the input variables
are the same as in the least case, while the state variables are β and r:
[ 𝛽̇
𝑟̇
]
= [
−
𝑐
𝑚𝑣
2(𝐹𝑧,𝑓 + 𝐹𝑧,𝑟) −1 +
𝑐
𝑚𝑣2
2(𝐹𝑧,𝑟 𝑙 𝑟 − 𝐹𝑧,𝑓 𝑙 𝑓)
𝑐
𝐽𝑧
2(𝐹𝑧,𝑟 𝑙 𝑟 − 𝐹𝑧,𝑓 𝑙 𝑓) −
𝑐
𝐽𝑧
2(𝐹𝑧,𝑟 𝑙 𝑟
2
+ 𝐹𝑧,𝑓 𝑙 𝑓
2
)
] [
𝛽
𝑟
]
+ [
𝑐
𝑚𝑣
2𝐹𝑧,𝑓 0 0 0 0
𝑐𝑙 𝑓
𝐽𝑧
2𝐹𝑧,𝑓
𝑙 𝑤
2𝐽𝑧
−
𝑙 𝑤
2𝐽𝑧
𝑙 𝑤
2𝐽𝑧
−
𝑙 𝑤
2𝐽𝑧
]
[
𝛿
𝐹𝑥,𝐹𝑅
𝐹𝑥,𝐹𝐿
𝐹𝑥,𝑅𝑅
𝐹𝑥,𝑅𝐿]
Equation 3.31

20
3.5 LINEARIZED SIMULINK MODEL AND SIMULATION
According to the linearized model, described in chapter 3, a Simulink model was built, which
structure follows the state-space model introduced in chapter 3.4.
The model has three inputs – a steering angle and two forces, which are applied on the left and the
right wheels (there are four input forces actually, just the left and the right ones are set to be the same
in this variation, but there is a possibility to apply different forces on all wheels). The outputs are the
speed, the vehicle slip and the yaw rate (after integration).
Three groups of simulations were run. In the first at given speeds the steering angle was changed. The
outputs – side slip, yaw rate – and the front and rear wheel side slips were investigated:
Table 3 - Simulation at 5 [m/s]
v = 5 [m/s]
delta [°] beta [°] r [rad/s] alfaf [°] alfar [°]
0 0 0 0 0
3 1,171 0,1961 0,4796 -0,272
5 1,952 0,3269 0,7991 -0,4531
7 2,733 0,4577 1,119 -0,6343
10 3,904 0,6538 1,598 -0,906
15 5,857 0,9806 2,398 -1,36
20 7,809 1,307 3,197 -1,813
25 9,761 1,634 3,996 -2,266
Figure 6 – Simulink model of the linearized vehicle model

21
Table 4 - Simulation at 10 [m/s]
The results are shown in Table 3 and Table 4. It can be seen, that increasing the steering angle the side
slip, the yaw rate and the absolute value of the wheel side slips are growing. In Table 5 the steering
angle was kept constant while the speed was changed. The simulation data are also plotted in a
diagram in Figure
Table 5 - Simulations at delta = 20[°]
delta = 20 [°]
v [m/s] beta [°] r [rad/s] alfaf [°] alfar [°]
3 7,932 0,7805 3,12 -1,966
5 7,809 1,307 3,197 -1,813
8 7,504 2,119 3,387 -1,432
10 7,216 2,682 3,56 -1,068
15 6,159 4,199 4,213 0,2596
20 4,512 5,968 5,225 2,33
25 2,047 8,15 6,741 5,428
These results clearly show that the vehicle slip decreases while the speed increases, and the other
investigated quantities also grow with the speed.
v = 10 [m/s]
delta [°] beta [°] r [rad/s] alfaf [°] alfar [°]
0 0 0 0 0
3 1,082 0,4021 0,5346 -0,1605
5 1,804 0,6704 0,89 -0,2669
7 2,526 0,9385 1,246 -0,3738
10 3,608 1,341 1,78 -0,5339
15 5,412 2,01 2,673 -0,8023
20 7,216 2,682 3,56 -1,068
25 9,021 3,352 4,45 -1,334

22
4 SUMMARY
In this project laboratory a 1:10 sized electric model car was modelled. It has four, independently
controllably wheels with BLDC motors and a servo motor for steering. First a nonlinear model was set
up according to Burckhardt’s model. After that the aim was to create a linearized model which is easier
to handle and to calculate. With a series of simplifications and assumptions a linearized model was
derived, which can be represented by a well-known state-space model. This linearized model is similar
to the bicycle model, but differ from it in having four wheels and a different cornering stiffness
function. Finally these models were implemented in Simulink, and the simulations run are presented.
These simulation results show that increasing the steering angle results higher side slips and yaw rates,
but when the speed is increased, the vehicle side slip falls.
-4
-2
0
2
4
6
8
10
3 5 8 10 15 20 25
delta = 20 [°]
beta [°] r [rad/s] alfaf [°] alfar [°]
Figure 7 - Plot of simulations in Table 6

23
5 REFERENCES
[1] G. Rödönyi, „A dynamic model for a heavy truck,” Hungarian Academy of Sciences, Budapest,
2007..
[2] U. Kiencke és L. Nielsen, Automotive Control Systems, Germany: Springer-Verlag Berlin
Heidelberg, 2005..

Nonlinear vehicle modelling

Recommended

Recommended

More Related Content

Viewers also liked

Viewers also liked (12)

Similar to Nonlinear vehicle modelling

Similar to Nonlinear vehicle modelling (20)

Nonlinear vehicle modelling