# Notes for Vehicle Dynamics & Control
###### tags: `vehicle_dynamics` `note` `electrical_system` `NTURT` `vehicle_dynamics`
##### Author: @QuantumSpawner
## Table of Contents
[TOC]
## Youtube Playlist
Youtube playlist: [Vehicle Dynamics & Control](https://www.youtube.com/watch?v=Cg0L_HZYxP4&list=PLW3FM5Kyc2_4PGkumkAHNXzWtgHhaYe1d).
## Reference Frames / Coordinate Systems
### Inertial Frame
Denoted by capitalized $X$, $Y$, $Z$ that is fixed to the earth with the $Z$-axis pointing to the anti-gravitational direction.
### Vehivle Frame
Denoted by $x_v$, $y_v$, $z_v$ that is fixed to the vehicle reference point with the $x_v$-axis pointing to the front and the $z_v$-axis pointing to the top of the vehicle.
### Horizontial Frame
Denoted by $x$, $y$, $z$ that is also fixed to the vehicle reference point with the $x$ and $y$-axis being the projection of the $x_v$ and $y_v$-axis onto the horizontial plane.
### Path Frame
Denoted by $d$, $e$, $n$ with the $d$-axis being tangential to the path, $e$-axis on the road plane perpenticular to the $d$-axis pointing to the left and $n$-axis perpenticular to them such that the right-handed Cartesian coordinate system is constructed.
## Terminologies
### Wheelbase $l$
The distance between the axles of the front wheels and the rear wheels.
### Track $b$
The distance between the left and right wheel. (separately for the front and the rear axles)
### Horizontial Plane
The plane orthogonal to the gravity at the vehicle reference pount.
### Road Plane
The plane tangent to the earth surface (road) at the vehicle reference point.
### Slope angle $\lambda$
The angle between the horizontial and the road plane in the direction of the $x$-axis.
### Bank angle $\beta$
The angle between the horizontial and the raod plane in the direction of the $y$-axis.
### Vehicle Velocity $\bf v$$(t)$ and acceleration $\bf a$$(t)$
The first and the second derivatives of the inertial frame coordinates $\bf r$$(t)=(X,Y,Z)$.
### Projecting the vehicle velocity and acceleration onto the horizontial frame as
- Longitudunal velocity $v_{lon}(t)$ and acceleration $a_{lon}(t)$ in the direction of the $x$-axis.
- Lateral velocity $v_{lat}(t)$ and acceleration $a_{lat}(t)$ in the direction of the $y$-axis.
- Vertical velocity $v_{vert}(t)$ and acceleration $a_{vert}(t)$ in the direction of the $z$-axis.

### Yaw angle and rate $\psi$, $\dot{\psi}$
The angle between the $X$-axis in the inertial frame and the $x$-axis in the horizontial frame (the rotation angle with respect to the $z$-axis) with the yaw rate as its angular velocity.
### Pitch angle and rate $\theta$, $\dot{\theta}$
The angle between the $x$-axis in the horizontial frame and the $x_v$-axis in the vehicle frame (the rotation angle with respect to the $y$-axis) with the pitch rate as its angular velocity.
### Roll angle and rate $\varphi$, $\dot{\varphi}$
The angle between the $y$-axis in the horizontial frame and the $y_v$-axis in the vehicle frame (the rotational with respect to the $x$-axis) with the roll rate as its angular velocity.
## Review: Kinematics of a Rigid Body
### Kinematics of a point $P$ in space
The position $\bf r_p$$(t)$$\in\Bbb R^3$ is given by three coordinates (often an arbitrary reference frame):
- Position $\bf r_p$$(t)=(x(t), y(t), z(t))$
- Velocity $\bf v_p$$(t)=(\dot{x}(t), \dot{y}(t), \dot{z}(t))$
- Acceleration $\bf a_p$$(t)=(\ddot{x}(t), \ddot{y}(t), \ddot{z}(t))$

### Descripting the Motion of a Rigid Body
> A **rigid body** refers to a collection of infinitely many infinitesimally small mass points which are ridgidly connected, i.e. that the relative position remains unchanged over time.


The motion of a rigid body is described by the motion of a reference point $C$ of the body plus the relative motion of all other points $P$ with respect to $C$.
> - $C$: reference point that is fixed to the rigid body
> - $P$: an arbiturary point of the rigid body
> - $\bf \omega$: the angular velocity of the rigid body (in fact, the angular velocity is independent of the choice of the reference point $C$)
- position $\bf r_p$$=\bf r_c$$+\bf r_{c/p}$
- velocity $\bf v_p$$=\bf v_c$$+\bf \omega$$\times \bf r_{c/p}$
- acceleration $\bf a_p$$=\bf a_c$$+\bf \dot{\omega}$$\times \bf r_{c/p}$$+\bf \omega$$\times (\bf \omega$$\times \bf r_{c/p}$$)$

> Remark: Thus a ridgit body has six degree of freedom in space:
>- three positions $\bf r_c$ and three angles $\bf \theta$
>- three velocities $\bf v_c$ and three angular velocities $\bf \omega$
>- three accelerations $\bf a_c$ and three angular accelerations $\bf \dot{\omega}$

### Instananeous center of rotation
The reference point $C$ can be arbirtarily selected as any point of the rigid body, or even as a point outside of the body which is assumed to be fixed in the body's reference frame.
> Fact: At each time instance $t\in \Bbb R$, there exists a particular reference point $O$ (the "instantaneous center of rotation") for which $\bf v_O$$(t)=0$. In another word, each point $P$ of the rigid body performs a pure rotation about $O$: $\bf v_p$$=\bf v_O$$+\bf \omega$$\times \bf r_{o/p}$=$\bf \omega$$\times \bf r_{o/p}$

#### Example 1: Turning wheels
- wheel is completely lifted off the ground
- wheel does not move in x or y direction
- $\bf \omega$ points into the $xy$-plane

The center point of the wheel is the only point with a velocity of zero, thus is the instantaneous center of rotation $O$.
#### Example 2: Rolling wheels
- wheel is rolling without slipping
- ground is fixed to the $xy$-plane
- $\bf \omega$ points into the $xy$-plane

The contact point of the wheel and the ground is the only point with a velocity of zero, thus is the instantaneous center of rotation $O$.
> Remark: The set of all points of the rigid body which at some point becomes the instantaneous center of rotation is called **polhode** (denoted orange in the diagram below).
The set of all points in the $xy$ coordinate system which at some point becomes the instantaneous center of rotation is called **herpolhode** (denoted purple in the diagram below).
> 

## Review: Dynamics of a Rigid Body
### Translatory motion
#### Dynamics of a point $P$ in space
Let $\bf r_P$$(t)\in\Bbb R^3$ be its position in an inertial reference frame, and $\bf v_P$$(t)$ be its velocity, and $\bf a_P$$(t)$ be its acceleration.
- The **(linear) momentum** of $P$ $\bf p_P$$(t)\equiv m(t)\bf v_P$$(t)$
- By Newton's Second Law, $\frac{\mathcal d\bf p_P}{\mathcal dt}=\sum_{i}{\bf F_i}$, or if $m(t)=$const., $m\bf a_P$$=\sum_{i}{\bf F_i}$ where $\bf F_i$$(t)$ repensents all forces acting on the point $P$.

#### Dynamics of a rigid body $B$ in space
Let $\bf r_B$$(t)\in\Bbb R^3$ be its position of its center of gravity (mass) $C$ in an inertial reference frame, and $\bf v_B$$(t)$ be its velocity, and $\bf a_B$$(t)$ be its acceleration.
> The **center of gravity** of a rigid body behaves like a point mass with mass $m$ and as if all forces acting anywhere on the rigid body were acting directly on this point mass.

- The **(linear) momentum** of $P$ $\bf p_B$$(t)\equiv m(t)\bf v_C$$(t)$
- By Newton's Second Law, $\frac{d\bf p_B}{dt}=\sum_{i}{\bf F_i}$, or if $m(t)=$const., $m\bf a_B$$=\sum_{i}{\bf F_i}$ where $\bf F_i$$(t)$ repensents all forces acting on the rigid body $B$.

### Rotatory motion of a rigid body $B$ in space
For the rotatory motion, the geometric shape and the spatial distribution of its mass is important.
- Let the reference point $C$ be the body's center of gravity. (Alternatively, $C$ may be chosen as a fixed point in the body frame that is also a fixed point in the inertial frame.)
- Let $\rho(x, y, z)$ be the body's density function, so that $m=\int_{B}{dm}=\int_{B}{\rho(x, y, z)dxdydz}$.
- The **inertia tensor** $\Theta\equiv\begin{bmatrix}I_{xx}&I_{xy}&I_{xz}\\I_{yx}&I_{yy}&I_{yz}\\I_{zx}&I_{zy}&I_{zz}\end{bmatrix}$
Where the **moments of inertial angular mass** are
$I_{xx}\equiv\int_{B}{(y^2+z^2)dm}$, $I_{yy}\equiv\int_{B}{(x^2+z^2)dm}$, $I_{zz}\equiv\int_{B}{(x^2+y^2)dm}$
The **moments of deviation** are
$I_{xy}=I_{yx}\equiv-\int_{B}{xydm}$, $I_{xz}=I_{zx}\equiv-\int_{B}{xzdm}$, $I_{yz}=I_{zy}\equiv-\int_{B}{yzdm}$
- Let $\bf \omega$ be the angular velocity of the reigid body about the body frame as $\bf \omega$$\equiv(\omega_x, \omega_y, \omega_z)$.
- The **angular momentum** of the rigid body with respect to $C$ is given by $\bf L^{(C)}$$=\Theta\bf \omega$.
- By the angular momentum principle, $\frac{d}{dt}\bf L^{(C)}$$=\sum_{i}{\bf M_i^{(C)}}$, or if $\Theta(t)=$ const., $\Theta\bf \omega$$=\sum_{i}{\bf M_i^{(C)}}$ where$\bf M_i^{(C)}$$(t)$ are the moments of all forces acting on $B$ with respect to $C$.

#### Special cases
- If the body frame coordinates $x, y, z$ are choosen as a principle axis system for the rigid body (symmetry axes), then the inertia tensor becomes diagonal:$$\Theta=\begin{bmatrix}I_{xx}&0&0\\0&I_{yy}&0\\0&0&I_{zz}\end{bmatrix}$$
Examples:
- For the (quasi-)planar motion of a rigid body in the $xy$-plane ($\omega_x=\omega_y=0$ and $M_x^{(C)}=M_y^{(C)}=0$), the angular momentum becomes: $L_z^{(C)}=I_{zz}\omega_z$ and the angular momentum principle yields: $I_{zz}\dot{\omega_z}=\sum_{i}{M_i^{(C)}}$. (assuming $I_{zz}=$const.)

## Ackermann Steering Geometry
### Wheel steer angle $\delta_{ij}$
Defined separately for each of the four wheels ($i=f$ for "front" and $i=r$ for "rear" and $j=l$ for "left" and $j=r$ for "right"), wheel steer angle is the angle between the orientation of the wheel and the $x$-axis, about the $z_v$-axis.
> Assumption: No rear wheel steering ($\delta_{rl}=\delta_{rr}=0$).
### Slip angle $\alpha_{ij}$
The angle between the velocity of the wheel's center point $v_{ij}$ and the wheel's orientation. (where $i\in\{f, r\}$ and $j\in\{l, r\}$)
### Kinematic four-wheel model
The combination of all admissible (front) wheel steer angles for the kiematic four-wheel model is called the **Adcermann steering geometry**.
>Assumption: The slip angles of all four tires are zero: $\alpha_{ij}=0$.

## Kinematic Bicycle Model
The kinematic bicycle model is a simplification of the kinematic four-wheel model, where the two rear wheels and the two front wheels are lumped into on (imaginary) rear wheel and front wheel, respectively.
### Equation of motion
Assuming a constant velocity $v$:
$\dot{X}(t)=v\cos(\psi+\beta)$
$\dot{Y}(t)=v\sin(\psi+\beta)$
$\dot{\psi}=\omega=\frac{v}{R}$
By some basic geometry and trigonometry:
$\tan\delta=\frac{l_f+l_r}{\bar{R}}\Rightarrow\bar{R}=\frac{l_f+l_r}{\rm tan\delta}$, $\tan\beta=\frac{l_r}{\bar{R}}=\frac{l_r}{l_f+l_r}\tan\delta$
$\cos\beta=\frac{\bar{R}}{R}\Rightarrow\frac{1}{R}=\frac{1}{\bar{R}}\cos\beta=\frac{1}{l_f+l_r}\tan\delta\rm cos\beta$
Hence, we obtain:
$\dot{X}(t)=v\cos(\psi+\beta)$
$\dot{Y}(t)=v\sin(\psi+\beta)$
$\dot{\psi}=\omega=\frac{v}{l_f+l_r}\tan\delta\cos\beta$
where $\beta=\arctan(\frac{l_r}{l_f+l_r}\tan\it\delta)$
## Tires: Terminology and Basics

A tire cinsists of a combination if rubber and fabric (**carcass**):
- The carcass is made of **cords** of nylon, polyester, etc. ($\Rightarrow$**plies**)
- Cord are often laied by two or more lays which are also called **plies**.
- The main types are **radial-ply** and **bias-ply** tires.

- Due to the tire load, $r_{eff}<r$

### Tire code
- section width: 25mm
- aspect ratio: 65%
- construction: R1
- (radial-ply)
- rim diameter: 19 inches
> Note: Aspect ratio is the ratio of the section hight and the section width.

### Wheel coordiante system
Let the wheel travel from the left tho the right and traveling towards us in the side view, then we can define the wheel coordinate $x_w$, $y_w$, $z_w$ as the figure above.
## Lateral Tire Models
> The purpose of tire modles is to determine the froces and moments that act on the vehicle via its tire-road contact.

Un the contact patches, **tire forces** and **tire moments** are acting on each tire:

- vertical force $F_z$
- longitudinal force $F_x$
- lateral force $F_y$
- aligning torque $M_z$
- overturning moment $M_x$
- rolling resistance moment $M_y$

> Assumption: For lateral tire models, the longituddinal force $F_x$ is neglacted. (i.e. no accerelation and braking)

> Difficulties:
> - Uneven pressure distribution in the tire contact path
> - Complexity of the tire-road contact
> 
> - Different road and tire conditions

### Analytic tire models
The idea is to derive the tire forces from first principles in mechanics, in particular Coulomb friction (adhesion + friction) in the contact path and elastic deformation of the tire.
The following elastic foundation model for lateral tire has been developed by Ernst Fiala in 1954:
In the simplest case, we may allow for a discontinuous displacement of the center line (i.e. discontinuous tire):
#### Case 1: Linear Tire Model
> Assumption: The slip angle $\alpha$ is samll that the maximum static force of $\mu F_z$ is not exceeded anywhere in the contact path (i.e. all static friction) with $\mu$ being the static friction coefficient.

with $(x,y)$ being the cordinate of the tire center line
$\Rightarrow \frac{\mu F_z}{c}\ge y$
$\Rightarrow$ Lateral tire force $F_y=\frac{1}{2a}\int_{0}^{2a}{c(x)y(x)dx}$ where $c(x)=c$ and $y(x)=x\tan\alpha$ for our sipmle case
$\Rightarrow F_y=ca\tan\alpha \fallingdotseq ca\alpha$ when $\alpha$ is small, where $ca\equiv c_\alpha$ is **cornering stiffness**
$\Rightarrow$ Aligning torque $M_z=\frac{1}{2a}\int_{0}^{2a}{c(x)y(x)(x-a)dx}\fallingdotseq\frac{1}{3}ca^2\alpha$ in small angle
####
2:
> Assumption: The slip angle $\alpha$ is large enough that the maximum static force $\mu F_z$ is exceeded at some place in the contact path (i.e. dynamic friction involved) with $\mu$ being the static and dynamic friction coefficient.

The center line displacement is given by:
$$y(x)=\begin{cases}y(x)=x\tan\alpha,&\text{if } 0<x<\bar{x}\\y(x)=\frac{\mu F_z}{c},&\text{if } \bar{x}\le x\le 2a\end{cases}$$
with $\bar{x}\tan\alpha =\frac{\mu F_z}{c}$
$\Rightarrow$ Lateral tire force $F_y=\frac{1}{2a}[\int_{0}^{\bar{x}}{cx\tan\alpha dx}+\int_{\bar{x}}^{2a}{c\frac{\mu F_z}{c}dx}]=\mu F_z-\frac{\mu ^2F_z^2}{4ac\tan\alpha}$ if $\alpha\le\frac{\mu F_z}{2ac}$ as a small angle
$\Rightarrow$ Aligning torque $M_z=\frac{1}{2a}[\int_{0}^{\bar{x}}{cx\tan\alpha (x-a)dx}+\int_{\bar{x}}^{2a}{c\frac{\mu F_z}{c}(x-a)dx}]=\frac{\mu ^2F_z^2}{4c\tan\alpha}-\frac{\mu ^3F_z^3}{16c^2a\rm tan^2\it\alpha}$ if $\alpha\le\frac{\mu F_z}{2ac}$
#### Empirical (經驗) tire models
The main idea is to fit the parameters of predefined functions to measure data from a tire est bench.
Empirical data shows that the lateral tire force $F_y$ roughly behaves as follows:
Saturation level $\alpha _{sat}\equiv\alpha -\frac{F_y}{c_\alpha}$ is the deviation of the actual slip angle of the lateral tire force minus the slip angle from the line
The shape of the tire force curve depends on many variables:
- size of the tire
construction type (radial-ply, ...)
- number and angle of plies
- general tire/thread desine
- aspect ratio

For any given tire (above variables fixed), the main factors are:
- tire inflation pressure
- tire load (vertical force $F_z$)

A frequent assumption is that $\frac{F_y}{F_z}$ follows the same curve, but actually, the curves differs with different slip angles:
##### Pacejka Tire Model
The Pacejka tire model is the most commonly used tire model in practice.
Is's based on the **Magic Foumla**:
$$F_y=D\sin[C\arctan(B\alpha -E(B\alpha -\arctan(B\alpha)))]\mu F_z$$
- $\alpha$: tire slip angle
- $\mu$: tire -road firction coefficient
- $F_z$: vertical tire force
- $B, C, D, E$: model prarmeters
where
$B$: stiffness factor
$C$: shape factor
$D$: peak value
$E$: curvature factor

This formula is an emperial modle, thus it can't be derived from any physics laws.
## The Dynamic Bicycle Model with Linear Tires
> Assumptions:
> - Wheels of the front and rear axle are lumped together into one front and one rear wheel.
> - Reference point $C$ is the vehicle's center of gravity.
> - The motion of the vehicle is restricted to the $X,Y$-plane.
> - For the lateral and yaw motion, the vehicle is considered as a rigid body, whose dynamics are determined by the fundamental laws of motion.
> - Only lateral tire forces which are generated by a linear tire model.
> - The steering angle is small: $\sin\delta\fallingdotseq\delta$, $\tan\delta\fallingdotseq\delta$, $\cos\delta\fallingdotseq1$.
> - For longitudinal motion, the velocity $v_{lon}$ is constant.

Parameters:
- $m$: vehicle mass
- $I_z$: moment of intia about $z$
- $l_f,l_r$: distance from $C$ to front and rear axle ($l_f+l_r=l$)
- $c_{\alpha ,f},c_{\alpha ,r}$: cornering stiffness of the combines front and rear tires
### Laws of motion

#### Lateral dynamics:
$ma_y^{(X,Y)}=\sum_{i}{F_{y,i}}=F_r+F_f\cos\delta\fallingdotseq F_r+F_f$
where $a_y^{(X,Y)}=\dot{v}_{lon}+\omega v_{lon}$ with $\omega v_{lon}$ being centripetal acceleration
$\Rightarrow m(\dot{v}_{lat}+\omega v_{lon})=F_r+F_f$ (1)
#### Yaw dynamics:
$I_z\dot{\omega}=\sum_{i}{M_i^{(C)}}=-l_rF_r+l_fF_f\cos\delta$
$\Rightarrow I_z\dot{\omega}=-l_rF_r+l_fF_f$ (2)
#### Tire forces:
Linear tire modle:
$F_f=-c_{\alpha ,f}\alpha _f$, $F_r=-c_{\alpha ,r}\alpha _r$ (caution on the minus sign)
Velocity geometry:
$\alpha _f\fallingdotseq\tan\alpha _f=\frac{v_{a,\eta}}{v_{a,\varepsilon}}$, $\alpha _f\fallingdotseq\tan\alpha _f=\frac{v_{B,y}}{v_{B,x}}$
Kinematics:
$v_{B,x}=v_{lon}$, $v_{B,y}=v_{lat}-\omega l_r$
$v_{A,x}=v_{lon}$, $v_{A,y}=v_{lat}+\omega l_f$
Coordiante Transformation $(x,y)\rightarrow (\varepsilon,\eta)$:
$v_{A,\varepsilon}\fallingdotseq v_{A,x}+v_{A,y}\delta$
$v_{A,\eta}\fallingdotseq-v_{A,x}\delta+v_{A,y}$
$\Rightarrow F_f=-c_{\alpha ,f}\alpha _f=-c_{\alpha ,f}\frac{v_{A,\eta}}{v_{A,\varepsilon}}=-c_{\alpha ,f}\frac{-v_{lon}\delta+v_{lat}+\omega l_f}{v_{lon}+(v_{lat}+\omega l_f)\delta}$
when $v_{lon}\ll (v_{lat}+\omega l_f)\delta$, $F_f\fallingdotseq c_{\alpha ,f}\delta-c_{\alpha ,f}\frac{v_{lat}+\omega l_f}{v_{lon}}$ (3)
Similarily, $F_r=-c_{\alpha ,r}\alpha _r=-c_{\alpha ,r}\frac{v_{B,y}}{v_{B,x}}\fallingdotseq-c_{\alpha ,r}\frac{v_{lat}-\omega l_r}{v_{lon}}$ (4)
### Linear bicycle model: State space representation
Substitution of the front (3) and rear (4) tire forces yields:
in (1): $m(\dot{v}_{lat}+v_{lon}\omega)=F_r+F_f=-c_{\alpha ,r}\frac{v_{lat}-\omega l_r}{v_{lon}}+c_{\alpha ,f}\delta-c_{\alpha ,f}\frac{v_{lat}+\omega l_f}{v_{lon}}$
$\Rightarrow\dot{v}_{lat}=-\frac{c_{\alpha ,r}+c_{\alpha ,f}}{mv_{lon}}v_{lat}+\frac{c_{\alpha ,r}l_r-c_{\alpha ,f}l_f}{mv_{lon}}\omega-v_{lon}\omega +\frac{c_{\alpha ,f}}{m}\delta$
in (2): $I_z\dot{\omega}=-l_rF_r+l_fF_f=-l_r(-c_{\alpha ,r}\frac{v_{lat}-\omega l_r}{v_{lon}})+l_f(c_{\alpha ,f}\delta-c_{\alpha ,f}\frac{v_{lat}+\omega l_f}{v_{lon}})$
$\Rightarrow\dot{\omega}=\frac{l_rc_{\alpha ,r}-l_fc_{\alpha ,f}}{I_zv_{lon}}v_{lat}-\frac{l_f^2c_{\alpha ,f}+l_r^2c_{\alpha ,r}}{I_zv_{lon}}\omega+\frac{c_{\alpha ,f}}{I_z}l_f\delta$
This is equivalent to the following state space it representation:
states: $x_1\equiv v_{lat}$, $x_2\equiv\psi$, $x_3\equiv\omega$
input: $\mu\equiv\delta$
$$\Rightarrow\frac{d}{dt}\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix}=\begin{bmatrix}\dot{v}_{lat}\\ \dot{\psi}\\ \dot{\omega}\end{bmatrix}=\begin{bmatrix}-\frac{c_{\alpha ,r}+c_{\alpha ,f}}{mv_{lon}}&0&\frac{c_{\alpha ,r}l_r-c_{\alpha ,f}l_f}{mv_{lon}}\\0&0&1\\ \frac{l_rc_{\alpha ,r}-l_fc_{\alpha ,f}}{I_zv_{lon}}&0&-\frac{l_f^2c_{\alpha ,f}+l_r^2c_{\alpha ,r}}{I_zv_{lon}}\end{bmatrix}\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix}+\begin{bmatrix}\frac{c_{\alpha ,f}}{m}\\0\\ \frac{c_{\alpha ,f}}{I_z}l_f\end{bmatrix}\mu$$
> Remark: With $\dot{X}=v_{lon}\cos\psi-v_{lat}\sin\psi$ and $\dot{Y}=v_{lon}\sin\psi+v_{lat}\cos\psi$, the model can be augmented by the global position to a nonlinear state space model.

## Vehicle Handling Performance
Handing (performance) refers to the lateral dynamics / cornering behavior of a vehicle.
The performance objectives are not specific, they can be summarized as follows:
- Responsiveness of the vehicle to driver inputs (steering).
- Case of control in case of external disturbance to the vehicle.
- Intuitive recognition of the handling limits by the driver.
The impact of changes in certain vehicle parameters (e.g. loads, tires, road surfaces) on the vehicle dynamics should be kept as small as possible.

Vehicle handling measures the performance of the vehicle-driver combination (**closed-loop performance**). However, drivers are difficult to model, because they are humans and there is no "unique" driver. Hence, usually only the dynamics of the vehicle itself are considered (**open-loop performance**).
Illustration of closed-loop system:
It makes little to no sense in practice to study this closed-loop system, especilly in normal driving situations. Driver models make sense in critical driving situations (for example, treating the driver reation time as a time delay in case of emergency), but they are not in the content of this course.
Hence we will examine important dynamic properties of the open-loop system, where $\delta (t)$ is fixed.
## Oversteer and Understeer
### Ackermann Steer Angle $\bar{\delta}$
Let $O$ be the instantaneous center of rotation of a (kinematic or dynamic) bicycle model. The Ackermann steer angle $\bar{\delta}$ is defined as the angle between the line from $O$ to the mid point of the rear wheel and the line from $O$ to the mid point of the front wheel.
Hence, $\bar{\delta}=\delta$ in kinematic bicycle model; $\bar{\delta}=\delta -\alpha _f+\alpha _r$ in dynamic bicycle model.
### Tangential Radius $\bar{R}$
Let $\bar{C}$ be the point of the kinematic bicycle model where the slip angle is zero (i.e. the direction of the local velocity equals to the orientation of the vehicle). The tangential radius $\bar{R}$ is defined as the length between the instantaneous center of rotation $O$ and $\bar{C}$.
Hence, $\bar{\delta}_{kin}=\arctan\frac{l}{\bar{R}}$ in kinematic bucycle model, $\bar{\delta}_{dyn}=\delta -\alpha _f+\alpha _r=\arctan\frac{l_l}{\bar{R}}+\arctan\frac{l_z}{\bar{R}}$.
### Steady-State Cornering
Given our discussion of vehicle handling performance, the most basic (and probably the most impotant) experiment to asses the open-loop system in steady-state cornering.
Steady-state cornering refers to the (somewhat idealized) situation where the vehicle is driving on a circle with fixed radius with a constant steering angle.
### Oversteer and Understeer
[The difference between understeer and oversteer](https://www.youtube.com/watch?v=EwmDdMzzDjY)
A dynamic bicycle model is said to be **neutrally steering** during steady-state cornering, if its steering angle $\delta _{dyn}$ equals to that of a kinematic bicycle model $\delta _{kin}$ when both are driving on a circle with the same tangential radius $\bar{R}$.
- Comparison of Ackermann angles for the same radius $\bar{R}$:
$\bar{\delta}_{kin}=\arctan\frac{l}{\bar{R}}\fallingdotseq\frac{l}{\bar{R}}$, $\bar{\delta}_{dyn}=\arctan\frac{l_l}{\bar{R}}+\arctan\frac{l_z}{\bar{R}}\fallingdotseq\frac{l_l}{\bar{R}}+\frac{l_z}{\bar{R}}=\frac{l}{\bar{R}}$
$\Rightarrow\bar{\delta}_{kin}=\bar{\delta}_{dyn}$ if both models drive on a circle with the same tangential radius $\bar{R}$.
- Substituding the steering angles $\bar{\delta}_{kin}=\delta _{kin}$ and $\bar{\delta}_{dyn}=\delta _{dyn}-\alpha _f+\alpha _r$.
$\Rightarrow\delta _{kin}=\delta _{dyn}-\alpha _f+\alpha _r$ or $\delta _{dyn}=\delta _{kin}+\alpha _f-\alpha _r$

So the dynamic model is neutrally steering if $\alpha _f=\alpha _r$.
It's called **oversteering** when $\alpha _f<\alpha _r$, and **understeering** when $\alpha _f>\alpha _r$
### Illustration of Oversteer and Understeer
## Understeet Gradient
The understeer/oversteer behavior of a vehicle changes with the vehicle's lateral acceleration. If the lateral acceleration is small (low speed and/or large radius), the slip angles are generally small and hance the vehicle is almost neutrally steering. If the lateral acceleration of a vehicle increases, the slip angles of both vehical tries have to grow and the understeer/oversteer behavior of the vehicle becomes more prominent.
The understeer gradient quantifies the rate of change of a vehicle's understeer (or oversteer, if the gradient id negative) with its lateral acceleration.
### Stationarily Conditions during Steady-State Conering
> Assumptions:
> - Dynamic bicycle model, in particular the steering angle $\delta$ is small
> - Steady-state cornering maneuver: $\delta\equiv\delta _{dyn}=\delta _{kin}+\alpha _f -\alpha _r$
> - Linear tire model:
$F_f=c_{\alpha ,f}\alpha _f$, $F_r=c_{\alpha ,f}\alpha _f$
> - The cornering stiffness coefficients $c_{\alpha ,f}$ and $c_{\alpha ,r}$ depend linearly on the vertical tire load: $c_{\alpha ,f}=\tilde{c}_{\alpha ,f}F_{z,f}$, $c_{\alpha ,r}=\tilde{c}_{\alpha ,r}F_{z,r}$
$\sum_{i}{F_{i,z}}=0: F_{z,f}+F_{z,r}=mg$, $\sum_{i}{M_{i,y}^{(C)}}=0: l_rF_{z,r}-l_fF_{z,f}$
$\Rightarrow F_{z,r}=\frac{l_r}{l}mg=w_fmg$, $F_{z,r}=\frac{L_f}{l}mg=w_rmg$
$w_r,w_f$: fraction of weight on the front, rear tire

Balance of forces ($\vec{e}_r$-direction): $ma_r=-F_f-F_r$
in steady-state cornering, $a_r=-\frac{v_{lon}^2}{R}$ as centripetal acceleration
Balance of moments (z-direction): $I_z\ddot{\psi}=l_fF_f-l_rF_r=0$ for stationary rotation
Solving for $F_f$ and $F_r$: $F_f=\frac{l_r}{l}\frac{v_{lon}^2}{R}$, $F_r=\frac{l_f}{l}\frac{v_{lon}^2}{R}$
Substituding into the linear tire models: $\alpha _f=\frac{1}{c_{\alpha ,f}}F_f=\frac{w_f}{c_{\alpha ,f}}\frac{v_{lon}^2}{R}$, $\alpha _r=\frac{1}{c_{\alpha ,r}}F_f=\frac{w_r}{c_{\alpha ,r}}\frac{v_{lon}^2}{R}$
Steady-state cornering condition: $\delta=\delta _{kin}+\alpha _f-\alpha _r=\delta _{kin}+(\frac{w_f}{c_{\alpha ,f}}-\frac{w_r}{c_{\alpha ,r}})\frac{v_{lon}^2}{R}$
$\Rightarrow$Understeer gradient $K\equiv\frac{w_f}{c_{\alpha ,f}}-\frac{w_r}{c_{\alpha ,r}}$
### Characteristic Speed and Critical Speed
Define: $v_{char}$: characteristic velocity, $v_{crit}$: critical velocity
Illustration of understeer gradient (ideal case):
Illustration of understeer gradient (experimental case):
> Remark: The graph can olny be drawn in experimants (described in ISO 4138)
## Yaw Stability
- So far, we have analyzed the steady-state behavior ($\rightarrow$ stationary circle)
- Now we look at the transient response, in particular with respect to stability
- The linear bicycle model remains the basis for our analysis
### Linear Bicycle Model
Parameters:
- $m$: vehicle mass
- $I_z$: moment of intia about $z$
- $l_f,l_r$: distance from $C$ to front and rear axle ($l_f+l_r=l$)
- $c_{\alpha ,f},c_{\alpha ,r}$: cornering stiffness of the combines front and rear tires

State space representation:
$$\frac{d}{dt}\begin{bmatrix}v_{lat}\\\omega\end{bmatrix}=\begin{bmatrix}-\frac{c_{\alpha ,r}+c_{\alpha ,f}}{mv_{lon}}&&\frac{c_{\alpha ,r}l_r+c_{\alpha ,f}l_f}{mv_{lon}}-v_{lon}\\\frac{l_rc_{\alpha ,r-l_fc_{\alpha ,f}}}{I_zv_{lon}}&&-\frac{l_f^2c_{\alpha ,f}+l_r^2c_{\alpha ,f}}{I_zv_{lon}}\end{bmatrix}\begin{bmatrix}v_{lat}\\\omega\end{bmatrix}+\begin{bmatrix}\frac{c_{\alpha ,f}}{m}\\\frac{c_{\alpha ,f}}{I_z}l_f\end{bmatrix}\delta\equiv\dot{x}=Ax+B\delta$$
### Stability of the Linear Bicycle Model
The linear bicycle model is stable if and only if all eigenvalues of the system matrix $A$ are in the open left-half of the complex plane.
The eigenvalue $s_{1,2}$ of $A$ satisfies its characteristic equation: $det(sI-A)=0$
Substituding the system matrix of the linear bicycle model:
$det(\begin{bmatrix}s+\frac{c_{\alpha ,r}+c_{\alpha ,f}}{mv_{lon}}&&v_{lon}-\frac{c_{\alpha ,r}l_r+c_{\alpha ,f}l_f}{mv_{lon}}\\-\frac{l_rc_{\alpha ,r-l_fc_{\alpha ,f}}}{I_zv_{lon}}&&s+\frac{l_f^2c_{\alpha ,f}+l_r^2c_{\alpha ,f}}{I_zv_{lon}}\end{bmatrix})=0$
$\Rightarrow (s+\frac{c_{\alpha ,r}+c_{\alpha ,f}}{mv_{lon}})(s+\frac{l_f^2c_{\alpha ,f}+l_r^2c_{\alpha ,f}}{I_zv_{lon}})+(v_{lon}-\frac{c_{\alpha ,r}l_r+c_{\alpha ,f}l_f}{mv_{lon}})(\frac{l_rc_{\alpha ,r-l_fc_{\alpha ,f}}}{I_zv_{lon}})=0$
Determining the coefficient of the second-order polynominal:
$s^2+(\frac{c_{\alpha ,r}+c_{\alpha ,f}}{mv_{lon}}+\frac{l_f^2c_{\alpha ,f}+l_r^2c_{\alpha ,f}}{I_zv_{lon}})s+\frac{c_{\alpha ,r}+c_{\alpha ,f}}{mv_{lon}}\frac{l_f^2c_{\alpha ,f}+l_r^2c_{\alpha ,f}}{I_zv_{lon}}+(v_{lon}-\frac{c_{\alpha ,r}l_r+c_{\alpha ,f}l_f}{mv_{lon}})\frac{l_rc_{\alpha ,r-l_fc_{\alpha ,f}}}{I_zv_{lon}}=0\equiv s^2+bs+c=0$
Simplifying the cofficients:
$b=\frac{I_z(c_{\alpha ,r}+c_{\alpha ,f})+m(l_f^2c_{\alpha ,f}+l_r^2c_{\alpha ,r
})}{mI_zv_{lon}}>0$
$c=\frac{mv_{lon}^2(l_rc_{\alpha ,f}-l_fc_{\alpha ,f})+c_{\alpha ,f}c_{\alpha ,r}l^2}{mI_zv_{lon}^2}$
The roots of the second-order polynominal follow with the quadratic formula: $s_{1,2}=\frac{1}{2}(-b\pm\sqrt{b^2-4c})$
As $b>0$, let's distinguish three cases:
1. $c>\frac{1}{4}b^2$: $s_{1,2}=-\frac{1}{2}b\pm jw$ where $w\equiv\sqrt{b^2-4c}$
Since $b>0$, the eigenvalue $s_{1,2}$ both have a negative real part and therefore the system is stable.
2. $\frac{1}{4}b^2\ge c>0$: $s_{1,2}=\frac{1}{2}(-b\pm\sqrt{b^2-4c})$
Hence the eigenvalue $s_{1,2}$ are both real and negative, so the system is stable.
3. $c\le0$: $s_{1,2}=\frac{1}{2}(-b\pm\sqrt{b^2-4c})$
Hence the eigenvalue $s_{1,2}$ are both real, with one of them negative and the other one non-negative. So the system is unstable.

### Stability vs. Understeer/Oversteer
Necessary and sufficient condition for stability: $c>0$
Explicit expression for the coefficient c: $c=\frac{mv_{lon}^2(l_rc_{\alpha ,f}-l_fc_{\alpha ,f})+c_{\alpha ,f}c_{\alpha ,r}l^2}{mI_zv_{lon}^2}>0$
Observations:
- For any given set of parameters $m,I_z, l_r,l_f,l,c_{\alpha ,r},c_{\alpha ,f}>0$: If $v_{lon}$ is sufficiently small, the vehicle is stable ($c>0$).
- If $l_rc_{\alpha ,f}\ge l_fc_{\alpha ,f}$, the vehicle is stable($c>0$) for all longitudinal velocity $v_{lon}$. (case 1)
- If $l_rc_{\alpha ,f}< l_fc_{\alpha ,f}$, the vehical becomes unstable ($c\le 0$) if the longitudinal velocity increases above a critical threshold.(case 2)

Compare these observations to the understeer gradient: $K\equiv\frac{w_f}{c_{\alpha ,f}}-\frac{w_r}{c_{\alpha ,r}}$
- The vehicle is understeering if $K>0$, neutrally steeringif $K=0$, and oversteering if $K<0$.
- $w_f=\frac{l}{l_f}$, $w_r=\frac{l}{l_r}$ represent the weight factor for the front/rear tire.
- Rewrite the understeer gradient: $K=\frac{w_f}{c_{\alpha ,f}}-\frac{w_r}{c_{\alpha ,r}}=\frac{l_r}{lc_{\alpha ,f}}-\frac{l_f}{lc_{\alpha ,r}}=\frac{l_rc_{\alpha ,r}-l_fc_{\alpha ,f}}{lc_{\alpha ,f}c_{\alpha ,r}}$
- The numerator reveals a close relationship between the understeer gradient $K$ and the two stability cases:
-- case 1: $l_rc_{\alpha, r}\ge l_fc_{\alpha ,f}\iff K\ge 0$ (understeer or neutrally steer)
-- case 2: $l_rc_{\alpha, r}< l_fc_{\alpha ,f}\iff K< 0$ (oversteer)

Conclusion:
In order to be stable, a vehicle must be understeering or neutrally steering. An oversteering vehicle is only stable up to a certain velocity and unstable above that velocity.
## Powertrain 1: Engine and Brake
### Overview of a standard powerstain
Drivetrain: The mechanism that transmits the power of the engine th the wheels. (including clutch, gearbox(transmission), driveshaft(propeller), differential and axle shafts)
### Engine
The main purpose of the engine is to deliver propulsive power to the vehicle:
$$\text{Power }P\equiv \bf F\cdot v=M\cdot\omega$$
Teh main automotivve engine types are:
- Combustion engine ($\Rightarrow$gasoline, diesel, liquefired pet
- Electric motors ($\Rightarrow$battery, fuel cell, overhead wire)
- Hybrid powertrains ($\Rightarrow$at least two different energy sources)
In particular, a combination of a combustion engine with an electric motor, e.g.
- The electric motor can accelerate the vehicle at low speeds
- It can be used as generate to recuperate kinetic energy when braking.
- Combustion engine powers an electric geneators so that is constantly running close to its optimal operating point.

Performance criteria for automotive engines:
- Peak power ($hp$ ar $kW$) and peak torque ($Nm$)
- Motor characteristic, i.e. power/torque over engine speed ($rpm$)
> Remark: These curves represent full-load power of the engine (**full throttle**)
- Power efficiecy (%) or specific fuel consumption($\frac{kg}{kWh}$)
- Emission of $\rm CO_2$ and air pollutants ($\rm NO_x,HC,CO,...$)
- Weight, size and cost of the engine and the corresponding powertrain
- range, weight and speed of fueling/charging the energy storage
### Brakes
Brake systems in vehicles serve three purposes, namely to
- decelerate the vehicle, possibly to a stopping position
- prevent acceleration on a downward slope
- keep the vehicle in a resting position (e.g. while parking)

To this end, the brakes deliver a braking torque to some or all of the wheels of th vehicle (angainst their current direction of rotation).
The main brake types (by physical principle):
- engine brakes
- friction brakes
- elecromagnetic brakes

#### Engine brakes
The engine brake refers to the braking torque that is produced by gasoline or diesel engines when the accelerator pedal is released. It is caused by friction in the drivetrain and the engine (to a minor extent), and air compression losses in the engine (majority of the braking effent).
#### Friction brakes
Friction brakes dissipate the kinetic energy of the vehicle by transforming it into heat.
There are two main types of friction brakes in autonobiles:
Further eleements of friction brake system include:
- brake froce distribution between the four wheels
- brake booster / electric brake booster
- safety mechansims: dual hydraulic circuit, emergency brake, dual brake booster (for heavy-duty vehicles)

#### Electromagnetic brakes
Electromagnetic brakes split into regenerative and non-regenerative brakes.
Noe-regenerative brake: An eddy current brake dissipates the kinetic energy as heat. However, unlike friction brakes, there is no physical contact between moving parts. Hence abrasive wear is significantly reduced.
Regenerative brake: The idea of regenerative brakes is to recover the kinetic energy of the vehicle as electric energy. This leads to an increase in energy efficiency and a reduction of the wear out of the friction brake.
The main difficulty with the process of regenerative braking is the very high power that is delivered over fairly short time periods. This requires special forms of energy storage, for example:
- 42V batteries
- supercapacitors
- flywheels
- compressed air storages

## Powertrain 2: Drivetrain
### Clutch (manual transmission)
A clutch is a mechansim device for engaging / disengaging power transmission between the engine and the wheels. Its main purpose is to bridge the **power gap** of a combustion engine.
### Torque Converter (automatic transmission)
A torque conveter is a fluid coupling between the engine and the rest of the vehicle.
### Transmission
The transmission is needed because combustion engines operate (most efficiently) at different speeds $\omega _{engine}$ and torque $M_{engine}$ from what is required at the wheels $\omega _{wheel}$, $M_{wheel}$ of the vehicle.
The drivetrain provides a "transformation of the power" between the engine and the wheels. The total transformation is a combination of the effects of (the clutch / torque converter,) the transmission and the differential.
Hence we consider the power transmission between the output of the clutch or torque converter $\omega _{shaft}$, $M_{shaft}$ and the wheels $\omega _{wheel}$, $M_{wheel}$:
- Assuming 100% efficiency of the trasmission: $\omega _{shaft}M_{shaft}=\omega _{wheel}M_{wheel}$
- The **gear ratio** of the transmission is defined as: $R\equiv\frac{\omega _{wheel}}{\omega _{shaft}}=\frac{M_{shaft}}{M_{wheel}}$
- If the efficiency of the transmission $\eta <1$: $\frac{M_{shaft}}{M_{wheel}}=\eta\frac{\omega _{wheel}}{\omega _{shaft}}=\eta\frac{1}{R}$
Real transmission in passenger vehicles usually have an efficiency $\eta >90\%$
- The ratios for each gear are usually chosen to a **geometric progression**:
where $\frac{R_2}{R_1}=\frac{R_3}{R_2}=\frac{R_4}{R_3}=...$

### Differential
A differential serves three purposes: distribution of engine power to the wheels, while allowing the wheels to turn at a different speeds, and adding to the overall gear ratio of the tranmission.
- The left and the right wheel can rotate freely in opposite directions (idea case).
- The engine torque is distributed envely between the left and the right wheel.

While this property is usually desirable under normal driving conditions (e.g. when driving a curve), it can become problematic if one of the wheels loes traction (e.g. when driving on uneven terrain).
For this reason, especially off-road vehicles feature a **differential lock**, which blocks the opposite rotation made of the two wheels.
## Longitudinal Vehicle Dynamics
Balance of forces:
$$ma_{lon}=-mg\sin\lambda -F_{aero}-\sum_{i\in I}{F_{res,i}}+\sum_{i\in I}{F_{x,i}}$$
> For simplicity, all steering angles of the car are assumed to be zero.

Longitudinal force components:
- $mg\sin\lambda$: slope force
- $F_{aero}$: aerodynamic drag force
- $F_{res,i}$: rolling resistance force(s)
- $F_{x,i}$: tire force(s)

where $i\in I\equiv\{fl,fr,rl,rr\}$
### Aerodynamic Drag Force
Under the assumption that the air flow direction is aligned with the orientation of the vehicle, the aerodynamiv drag force can be approximated as:
$$F_{areo}=\frac{1}{2}\rho C_dA_f(v_{lon}+v_{wind})^2$$
where
- $\rho$: mass density of air
- $C_d$: aeodynamic drag coefficient
- $A_f$: front area of the vehicle
- $v_{lon}$: longitudinal velocity of the vehicle
- $v_{wond}$: wind speed
#### Aerodynamic drag coefficient
The aerodynamic drag coefficient is determined by the geometric shape of the object.
For example:
- plate: $C_{d,plate}\fallingdotseq 1.15$
- cuboid: $C_{d,cuboid}\fallingdotseq 0.82$

Comparison of drag coefficients of cars:
The aerodynamice drag coefficient of vehicles is generally determined by using a wind tunnle.
#### Rolling resistance force
The main couse of rolling resistance is the tire's deformation due to the vehicle load, which is not all elastic. Hence the internal damping of the material causes a loss of energy that appears as a resistance force apposite to the tire's rolling direction.
Experiments show that $F_{res,i}$ is roughly proportional to the vehicle ture force $F_{z,i}$:
$$F_{res,i}=f_RF_{z,i}$$
where $f_R$ is the rolling resistance coefficient
Typical values for passenger vehicles are $f_R\in[0.01,0.03]$
Rolling resistance also depends on the surface of the road. For example:
- Dry asphalt surface: $f_R\fallingdotseq 0.015$
- Gravel road / dry dirt road: $f_R\fallingdotseq 0.05$
- wet soaked grass: $f_R\fallingdotseq 0.035$
- On a solid surface with standing water with depth >0.5mm, $f_R$ frows propotionally with $v_{lon}^{1.5}$
#### Tire forces
The tire forces are the forces that act on the tire due to its interaction with the ground surface. They are used to contract the longitudinal dynamics of the vehicle, i.e. acceleration and braking.
The longitudinal tire force depends on
- The **slip ratio** between the tire and the ground
- The vertical load on the tire
- The friction coefficient between the tire and the road

The longitudinal tire slip is the difference between the tire's theoratical (kinematic) speed $\omega _ir_{eff}$ and its actual speed $v_i$. The main reason of longitudinal slip is not the relative movement of the road and the conttact patch however, is the elasticity of the tire.
The slip ratio is defined as:
- $\sigma _{x,i}=\frac{\omega _ir_{eff}-v_i}{\omega _ir_{eff}}\ge 0$ under accelerating torque
- $\sigma _{x,i}=\frac{\omega _ir_{eff}-v_i}{v_i}\le 0$ under braking torque

Dependence of the longitudinal tire force $F_{x,i}$ on the slip ratio $\sigma _{x,i}$:
Dependence on other factors (vertical load $\bar{F}_{z,i}$, tire parameters, ...) are analogos to the lateral tire forces.
## Acceleration and Braking Performance
### Limits on the Accelerating / Barking forces
The propulsive force $F_{prop}$ acting on the vehicle chassis equal to the sum of all longitudinal tire forces: $F_{prop}=\sum_{i\in I}{F_{x,i}}$
If the vehicle is accelerating, then $F_{prop}\ge 0$, if it's braking, then $F_{prop}\le 0$.
For simplicity, all steering angles of the vehicle are assumed to be zero.
#### Traction limit during acceleration and braking
> Assumption: The lateral weight distribution is even and tje vehicle may be considered as a bicycle model

$F_{x,f}=F_{x,fl}+F_{x,fr}$, $F_{x,r}=F_{x,rl}+F_{x,rr}$
$F_{z,f}=F_{z,fl}+F_{z,fr}$, $F_{z,r}=F_{z,rl}+F_{z,rr}$
$F_{res,f}=F_{res,fl}+F_{res,fr}$, $F_{res,r}=F_{res,rl}+F_{res,rr}$
Dimensions:
- $h$: distance between $CoG$ and ground
- $l$: wheelbase
- $l_f$: distance between $CoG$ to front axle, $l_r$: distance between $CoG$ to rear axle

Balance of forces:
$ma_{lon}=-mg\sin\lambda -F_{aero}-F_{res,f}-F_{res,r}+F_{x,f}+F_{x,r}$ (1)
Balance of moments about the rear and front wheel tire-road contract point:
Rear: $-ma_{lon}h-mg\sin\lambda h+mg\cos\lambda l_r-F_{z,f}l-M_{aero}=0$ (2)
Front: $-ma_{lon}h-mg\sin\lambda h-mg\cos\lambda l_f+F_{z,r}l-M_{aero}=0$ (3)
(where $ma_{lon}$ here is considered as inertia force relative to an acceleration frame)
##### Case 1: Rear-drive car ($F_{prop}=F_{x,r}, F_{x,f}=0$) when accelerating
- $\frac{F_{prop}}{F_{z,r}}\le\mu$ (where $\mu$ is the tire-road friction coefficient)
$\Rightarrow$Traction limit $\bar{F}_{prop}\equiv\mu F_{z,r}$, with (3):
$\bar{F}_{prop}=\mu ma_{lon}\frac{h}{l}+\mu mg(\sin\lambda\frac{h}{l}+\cos\lambda\frac{l_f}{l})+\mu\frac{M_{aero}}{l}$
- Substituding the traction limit $\bar{F}_{prop}=F_{x,r}$ into (1):
$m(1-\mu\frac{h}{c})a_{lon}=-mg(1-\mu\frac{h}{l})\sin\lambda+\mu mg\frac{l_f}{l}\cos\lambda-F_{areo}+\mu\frac{M_{areo}}{l}-F_{rea,f}-F_{res,r}$
- Assuming $(1-\mu\frac{h}{c})>0$ and $\lambda$ is small ($\rightarrow\sin\lambda\fallingdotseq\lambda ,\cos\lambda\fallingdotseq 1$):
- $a_{lon}$ is self-enforcing (the traction limit is greater with greater $a_{lon}$), because $(1-\mu\frac{h}{l})<1$
- A slope $\lambda>0$ increases the traction limit, but decreases $a_{lon}$
- A postive aerodynamic pitch moment $M_{areo}$ may increase $a_{lon}$ ($\rightarrow$rear spoiler)

##### Case 2: Front-drive car ($F_{prop}=F_{x,f}$, $F_{x,r}=0$) when accelerating
- $\frac{F_{prop}}{F_{z,f}}\le\mu$ (where $\mu$ is the tire-road friction coefficient)
$\Rightarrow$Traction limit $\bar{F}_{prop}\equiv\mu F_{z,f}$, with (2):
$\bar{F}_{prop}=-\mu ma_{lon}\frac{h}{l}-\mu mg(\sin\lambda\frac{h}{l}-\cos\lambda\frac{l_f}{l})-\mu\frac{M_{aero}}{l}$
- Substituding the traction limit $\bar{F}_{prop}=F_{x,f}$ into (1):
$m(1+\mu\frac{h}{c})a_{lon}=-mg(1+\mu\frac{h}{l})\sin\lambda+\mu mg\frac{l_r}{l}\cos\lambda-F_{areo}-\mu\frac{M_{areo}}{l}-F_{rea,f}-F_{res,r}$
- Assuming $\lambda$ is small ($\rightarrow\sin\lambda\fallingdotseq\lambda ,\cos\lambda\fallingdotseq 1$):
- $a_{lon}$ decreases the traction limit
- A slope $\lambda>0$ has a negative effect on the traction limit and on $a_{lon}$
- A negative areodymanic pitch moment $M_{areo}$ may increase $a_{lon}$ ($\rightarrow$front spoiler)

##### Case 3: All-wheel drive car ($F_{prop}=F_{x,f}+F_{x,r}>0$)
- Clearly, for maximum traction, is holds that $\frac{F_{x,f}}{F_{z,f}}=\mu$ and $\frac{F_{x,r}}{F_{z,r}}=\mu$
- So more torque should be allocated to the rear tires with:
- Increasing $a_{lon}$
- Increasing slope $\lambda$
- Increasing areodynamic pitch moment $M_{areo}$
- In reality, the engine torque is distributed between the front and the rear axle by a central differential in a fixed ratio that is cloes to the static weight distribution.
- Most all-wheel drive cras have a differential lock to avoid unnecessary spinning of the wheels.
##### Case 4: Breaking ($F_{prop}=F_{x,f}+F_{x,r}<0$)
- Traction limit of the front wheels:$-\frac{F_{x,f}}{F_{z,f}}\le\mu$ (where $\mu$ is the tire-road friction coefficient)
$\Rightarrow$Traction limit $-\underline{F}_{x,f}\equiv\mu F_{z,f}=-\mu ma_{lon}\frac{h}{l}-\mu mg(\sin\lambda\frac{h}{l}-\cos\frac{l_r}{l})-\mu\frac{M_{aero}}{l}$
- Traction limit of the rear wheels:$-\frac{F_{x,r}}{F_{z,r}}\le\mu$ (where $\mu$ is the tire-road friction coefficient)
$\Rightarrow$Traction limit $-\underline{F}_{x,r}\equiv\mu F_{z,r}=\mu ma_{lon}\frac{h}{l}+\mu mg(\sin\lambda\frac{h}{l}+\cos\frac{l_f}{l})+\mu\frac{M_{aero}}{l}$
- Hence the maximum braking force is achieved by increasing the ratio $\frac{F_{x,z}}{F_{x,r}}$ with:
- Increasing deceleration of the vehicle ($a_{lon}<0$ getting smaller)
- Decresing slope $\lambda$
- Decreasing aerodynamic pitch moment $M_{aero}$
- In reality, a good braking system is not just about the maximum braking force / shartest braking distance. It is also about maintaining stability / steerability during the braking maneuver.

#### Power-limit acceleration
Above a certain velocity, the acceleration of a vehicle is limited by the power of the engine.
> Assumption: The clutch / torque converter is locked and the engine is continuously operated at its power limit $\bar{P}_{eng}$.

- Given the efficiency $\eta$ of the drivetrain, the maximum power availbale at the driven wheels is $\bar{P}_{wheels}=\eta\bar{P}_{eng}$.
- With $\bar{P}_{wheels}=\bar{M}_{wheels}\omega$, where $\omega$ is the wheel speed and $\bar{M}_{wheels}$ is the total wheel torque: $\bar{M}_{wheels}=\frac{1}{\omega}\eta\bar{P}_{eng}$
- With the definition of the tire slip under acceleration $\sigma_x=\frac{\omega r_{eff}-v}{\omega r_{eff}}$:
$\Rightarrow\omega=\frac{1}{1-\sigma _x}\frac{v}{r_{eff}}\Rightarrow\bar{M}_{wheels}=\frac{r_{eff}}{v}(1-\sigma _x)\eta\bar{P}_{eng}$
- The wheel torque is related to the propulsive forve by $\bar{M}_{wheels}=F_{prop}r_{eff}$:
$$\Rightarrow F_{prop}=(1-\sigma _x)\eta\frac{\bar{P}_{eng}}{v}$$
## Coupled Tire Models
The tire models provide a mathmetical formula for the forces and moments acting on a tire in its contact patch.
So far, we haved looked at lateral tire models (coupling $F_y$ and $F_z$ with the slip angle $\alpha$) and longitudinal tire modles (coupling $F_x$ and $F_z$ with the slip ratio $\sigma _x$).
However, in practice, lateral and longitudinal tire models are not independent of each other:
- The maximum **combined force** $F\equiv\sqrt{F_x^2+F_y^2}$ is limted by the **friction circle**, which is defined by the tire-road-friction coefficient $\mu$.
- Even below the friction limits, an increase in $|\alpha |$ decreases $|F_x|$ and an increase in $|\sigma _x|$ decreases $|F_y|$.

### Derated Fiala Tire Model
The longitudunal tire force is given by:
$$F_x=\begin{cases}c_\sigma\sigma _x,&\text{if }|\sigma _x|<\frac{\mu F_z}{c_\sigma} \\ \mu F_z\rm sgn\it (\sigma _x),&\text{if }|\sigma _x|\ge\frac{\mu F_z}{c_\sigma}\end{cases}$$
The limits of the lateral tire force are derated by the factor $\varepsilon\equiv\frac{\sqrt{(\mu F_z)^2-F_x^2}}{\mu F_z}$ to keep the combined force inside the friction circle:
$$F_y=\begin{cases}c_\alpha\alpha,&\text{if }|\alpha|<\frac{\varepsilon\mu F_z}{c_\alpha}\\ \varepsilon\mu F_z\rm sgn\it(\alpha),&\text{if }|\alpha|\ge\frac{\varepsilon\mu F_z}{c_\alpha}\end{cases}$$
### Pacejka-Sharp Tire Model
The main assumption of the Pacejka-Sharp model is that the longitudinal and the lateral tire stiffness are the same, i.e. $c\equiv c_\alpha =c_\sigma$. (which is rarely the same)
Define the lateral slip ratio $\sigma _y\equiv\frac{v_x\tan\alpha}{\omega r_{eff}}$, where $v_x$ is the wheel's longitudinal velocity, $\omega$ is its rotational speed, and $r_{eff}$ is its effective radius.
The **combined tire slip** $\sigma\equiv\sqrt{\sigma _x^2+\sigma _y^2}$, and the combined tire force:
$$F=\begin{cases}c\sigma-\frac{1}{3\mu F_z}(c\sigma)^2\rm sgn\it(\sigma)+\frac{1}{27(\mu F_z)^2}(c\sigma)^3,&\text{if }\sigma <\frac{3\mu F_z}{c}\\ \mu F_z\rm sgn\it (\sigma),&\text{if }\sigma\ge\frac{3\mu F_z}{c}\end{cases}$$
The longitudinal and lateral tire forces are then given by:
$$\left \{\begin{array}{c}F_x=\frac{\sigma _x}{\sigma}F\\ F_y=\frac{\sigma _y}{\sigma}F\end{array}\right.$$
### Dugoff Tire Model
Let $c_\sigma$ be the longitudinal tire stiffness and $c_\alpha$ be the lateral tire stiffness. The longitudinal and lateral tire forces are given by:
$$\left \{\begin{array}{c}F_x=c_\sigma\frac{\sigma _x}{1+\sigma _x}f(\lambda)\\ F_y=c_\sigma\frac{\tan\alpha}{1+\sigma _x}f(\lambda)\end{array}\right.$$
where $f(\lambda)\le1$ can be seen as a **scaling factor**. The factor $\frac{1}{1+\sigma _x}\fallingdotseq 1$in both expressioins if the slip ratio $\sigma _x\ll 1$.
> Remark: If we neglect $f(\lambda)$ and $\frac{1}{1+\sigma _x}$, then what we get is a linear tire model.

First, let $\tilde{F}_x$ and $\tilde{F}_y$ where $F_x=\tilde{F}_x$ and $F_y=\tilde{F}_y$.
Define the parameter $\lambda\ge 0$ as $\lambda\equiv\frac{\mu F_z}{2\sqrt{\tilde{F}_x^2+\tilde{F}_y^2}}$. Where $F_z$ is the vertical tire force and $\mu$ is the tire-road fraction coefficient.
The scaling factor $f(\lambda)\in [0,1]$ is defined as:
$$f(\lambda)\equiv\begin{cases}(2-\lambda )\lambda ,&\text{if }0\le\lambda\le 1\\ 1,&\text{if }\lambda >1\end{cases}$$
- If $\sqrt{\tilde{F}_x^2+\tilde{F}_y^2}\rightarrow 0\Rightarrow\lambda\rightarrow\infty\Rightarrow f(\lambda)=1$
- If $\sqrt{\tilde{F}_x^2+\tilde{F}_y^2}\rightarrow\frac{1}{2}\mu F_z+X\rightarrow\lambda <1\Rightarrow f(\lambda)<1$
## Automotive Suspensions
The main functions of an automotive suspension system are the following:
- To enhance the ride quality by isolating the vehicle cahssis from the vertical accelerations caused by the roughness of the road.
- To keep a steady contact between the tire and the road with a proper vertical tire force, in order to maintain the maneuverability of the car.
- To optimize the handling performance of the vehicle in different dymanic situations (understeer and oversteer, limits of handling, etc.)
- To minimize the wear of the tires and parts of the suspension and the chassis die to dynamic loads.

Additional factors that affect the choice/design of suspensions are coast, weight, dimensions, robustness, and ease of assembly.
### Solid Axle Suspension
- The axle contains a rigid beam that links both wheels together.
- The axle may be powered ("live axle") or unpowered ("dead axle").
- Advantages: simple design, inexpensive, very robust
- The are used for some cars and commonly in trucks, trailers and off-road vehicles


### Independent Suspension
- Independent suspension allow each wheel of the axle to move up and down independently.
- Advantages: more flexible design options, better ride comfort and tire-road traction, lower space requirements
- Used for almost all production cars

### Solid Axle Suspension
#### Leaf-spring suspension
- The leaf also provide damping due to friction between the leafs.
- The leaf spring suspension was common in automobiles untile the 1970s.
- The main drawbacks of leaf spring suspensions is that there is little control over important design parameters:
- damping ratio
- geometry of deformation ("elastokinematics")

A common construction from of a live axle with leaf-spring suspension is the **Hotchkiss drive**.
It was the common layout for front-engine rear wheel drive cars until the 1970s. It is still used for many light and heavy trucks and some SUVs today.
#### Multi-link solid axle suspensions
So-called "links" or "arms" are solid bars that are used to control the kinematics of a suspension. They can also be used to enchance the dynamics of a leaf-spring suspension, such as
- a **trailing arm** (**anti-tramp bar**)
- a **triangular linkage**

The links / arms restrict the degrees of freedom of the relative motion between the solid axle and the chassis (or make them very stiff). However, they must be designed such that at least two degrees of freedom are still permitted:
- translatory motion in the vertical direction
- rotatory motion around the longitudinal axis

An important example is the **four-link suspension**:
- The driveshaft and differential are rigidly connected with the solid axle.
- Four links are used to attach the axle to the vehicle frame: tow upper links and rwo lower links.
- The lower links (**trailing arms**) are used to restrict the relative longitudinal motion between axle and vehicle frame.
- The upper arms are used to restrict the relative lateral motion between axle and vehicle frame.
- The well-defined kinematics (by means of the four links) allow for the leaf spring to be replaced by a coil spring and a linear damper.
### Independent Suspensions
#### Double wishbone / A-arm suspension
The double wishbone / A-arm suspension offers many independent design parameters, e.g. for wheel alignment during turns. It is frequently used in production vehicles and race cars.
#### McPherson strut
The McPherson strut is less coastly, less complex and lighter than the double wishbone suspension. It is very common in front-drive production vehicles.
## Anti-dive and Anti-squat Suspension Geometry
Due to inertial forces and the vertical elasticity of the suspensions, a vehicle performs a pitching motion during acceleration and braking:
> Acceleration: postive pitch angle $\Delta\theta >0$ due to a compression of the rear suspensions (**squat**).

> Braking: negative pitch angle $-\Delta\theta <0$ due to a compression of the front suspensions (**dive**).

Anti-dive and anti-squat geometry is about how to reduce the dive and squat motion.
Anti-dive and anit-squat has (almost) no effect on the weight distribution, but just on how much of the vertical forces passes through the springs versus the links / control arms of the suspensions.
### Virtual Pivot Points
Consider the kinematics of a suspension from a sidewards perspective. There exists a virtual point that is the instaneous center of rotation of the relative motion of the wheel with respect to the chassis. It shall be called the **virtual pivot point**.
#### Double wishbone suspension
#### McPherson strut suspension
> Remarks:
>- The virtual pivot point exists for all (independent and solid axle) suspensions.
>- Every kinematic construction is only a (more or less accurate) approximation of the true virtual pivot point.
>- For some suspensions, the location of the virtual pivot points is more accurate perdictable ("kinematically well defined" as two examples above) than for others.

### Anti-dive and anti-squat
> Acceleration: rear wheel drive vehicle, where vpp is the virtual pivot point of rear wheels
- Balance of moments about $A$: $\Delta F_{z,r}l-F_xh=0\Rightarrow\Delta F_{z,r}=\frac{h}{l}F_x$
- Balance of moments about the vpp: $M_{vpp}=\Delta F_{z,r}d-F_ze=\frac{h}{l}F_zd-F_xe\Rightarrow M_{vpp}=(\frac{h}{l}-\frac{e}{d})F_xd$
- 100% anti-squat is achieved if $\frac{e}{d}=\frac{h}{l}$, x% anti-squat is achieved if $\frac{e}{d}=\frac{x}{100}\frac{h}{l}$
> Braking: $\varepsilon\equiv$ fractionof front axle brake force, where vpp is the virtual pivot point of the front wheels
- Balance of moments about $B$: $-\Delta F_{z,f}+F_{barke}h=0\Rightarrow\Delta F_{z,f}\frac{h}{l}F_{brake}$
- Balance of moments about the vpp: $M_{vpp}=\Delta F_{z,f}d-\varepsilon F_{barke}e=\frac{h}{l}F_{barke}d-\varepsilon F_{brake}e\Rightarrow M_{vpp}=(\frac{h}{\varepsilon l}-\frac{e}{d})\varepsilon F_{brake}d$
- 100% anti-dive is achieved if $\frac{e}{d}=\frac{h}{\varepsilon l}$, x% anti-dive is achieved if $\frac{e}{d}=\frac{x}{100}\frac{h}{\varepsilon l}$
### Pratical remarks:
- 100% anti-dive and 100% anti-squat are rarely used in practice, because some degree of pitch rotation improves the driving feel.
- Typical valuse for anti-dive and anti-squat are 60% to 80%.
- The design of anti-dive and anti-squat cannot be considered independently of other dynamic properties / performance measures of the vehicle.
## Roll Center and Roll Dynamics
Due to interial forces and the vertical elasticity of the suspensions, a vehicle performs a roll rotation during a corning maneuver:
> A postive roll rotation $\Delta\varphi >0$ during a left turn.

The rotation axis of the chassis is called the **vehicle roll axis**:
> Remarks:
> - The location of the front / rear roll center is determained by the kinematics of the front / rear suspension.
> - Later on, we will look at how to determine the location of the roll center of a suspension.
> - The assumption of a fixed vehicle roll axis is a (good) approxmation. In reality, the roll axis chasges (slightly) with the vehicle's roll and pitch angle.
> - Rotating the chassis around the vehicle axis causes a counteracting moment by the suspension. The corresponding rotational stiffness (with the unit $\frac{Nm}{rad}$) is called **front /rear roll stiffness** for the front /rear suspension.

### Rear suspension (front suspension is analogous)
In the free body diagram, only the difference of forces are considered as compared to the static equilibrium of the car.
The changes in the vertical tire force $\Delta F_{z,rr}$ and $\Delta F_{z,rl}$ compensate for the moment of the lateral tire forces $F_{y,rr}$ and $F_{y,rl}$.
The tire forces are transmitted to the vehicle chassis in parts via the stiff parts of the suspension and in parts vir the springs ($\rightarrow$ roll rotation):
$\Delta F_{z,ri}=F_{j,ri}+F_{s,ri}$
$F_{j,ri}=\frac{h_r}{h}\Delta F_{z,ri}$
$F_{s,ri}=(1-\frac{h_r}{h})\Delta F_{z,ri}$
where $i\in\{f,r\}$ and $F_{j,ri}$ is **jacking force**
Extreme cases:
- $h_r=h$: $F_{j,ri}=\Delta F_{z,ri}, F_{s,ri}=0\Rightarrow$ No roll rotation of the chassis.
- $h_r=0$: $F_{j,ri}=F_{z,ri}\Rightarrow$ All of the vertical tire forces cause a roll angle.

### Roll dynamics and roll stiffness
where frc: front roll center; rrc: rear roll center
In the stationary case ($\ddot{\varphi}=0$ and $\dot{\varphi}=0$), the quasi-static roll angle $\Delta \varphi$ of the chassis is given by $M_\varphi = k_\varphi\Delta\varphi\Rightarrow\Delta\varphi=\frac{M_\varphi}{k_\varphi}$.
where $F_{s,fl}=F_{s,fr}=k_f\frac{b}{2}\Delta\varphi$, $F_{s,rl}=F_{s,rr}=k_f\frac{b}{2}\Delta\varphi$ and $k_f,k_r$: stiffness of the front / rear suspension
A balance of moments around the vehicle roll axis yields $M_\varphi=F_{s,fl}\frac{b}{2}+F_{s,fr}\frac{b}{2}+F_{s,rl}\frac{b}{2}+F_{s,rr}\frac{b}{2}=(2k_f+2k_r)\frac{b^2}{4}\Delta\varphi\Rightarrow k_\varphi=(k_f+k_r)\frac{b^2}{2}$
During a ststionary cornering maneuver:
$ma_{lat}=F_{y,fl}+F_{y,fr}+F_{y,rl}+F_{y,rr}$
The moment $M_\varphi$ caused by the interial force $ma_{lat}$ is given by:
$M_\varphi\fallingdotseq (h-h_c)ma_{lat}$.
The height of the vehicle roll axis at the center of gravity $h_c=\frac{l_fh_r+l_rh_f}{l}\Rightarrow M_\varphi =(h-\frac{l_fh_r+l_rh_f}{l})ma_{lat}$.
The corresponding roll angle $\Delta\varphi$ is thus given by: $\Delta\varphi =\frac{M_\varphi}{k_\varphi}=\frac{(h-\frac{l_fh_r+l_rh_f}{l})ma_{lat}}{(k_f+k_r)\frac{b^2}{2}}$.
Hence the elastic forces of the suspension are:
$F_{s,fl}=F_{s,fr}=k_f\frac{b}{2}\Delta\varphi =\frac{k_f}{k_f+k_r}(\frac{h}{b}-\frac{l_fh_r+l_rh_f}{l})ma_{lat}$
$F_{s,rl}=F_{s,rr}=k_f\frac{b}{2}\Delta\varphi =\frac{k_r}{k_f+k_r}(\frac{h}{b}-\frac{l_fh_r+l_rh_f}{l})ma_{lat}$
> Remarks:
> - The total roll stiffness $k_\varphi$ is given by the sum of the front and rear roll stiffness: $k_\varphi=k_{\varphi,f}+k_{\varphi,r}$, where $k_{\varphi,f}=k_f\frac{b^2}{2}$, $k_{\varphi,r}=k_r\frac{b^2}{2}$
> - Ther difference in the vertical tire forces consist of the sum of
> - the jacking forces $F_{j,fl}$, $F_{j,fr}$, $F_{j,rl}$, $F_{j,rr}$ (passing through the rigid parts of the suspension)
> - the elastic forces $F_{s,fl}$, $F_{s,fr}$, $F_{s,rl}$, $F_{s,rr}$ (passing through the springs of the suspensions)
> - During stationary cornering, the left vertical forces $F_{j,fl}$, $F_{j,rl}$, $F_{s,fl}$, $F_{s,rl}$ have the opposite direction (and the same magnitude) as those on the righthand side $F_{j,fr}$, $F_{j,rr}$, $F_{s,fr}$, $F_{s,rr}$, respectively.
> - The magnitude of the fornt elastic force pair $F_{s,fl}$, $F_{s,fr}$ and the rea elastic force pair $F_{s,rl}$, $F_{s,rr}$ is determined by the ratio between the roll stiffness of the front supension $k_{\varphi, f}$ and that of the rear suspension $k_{\varphi,r}$.

### Why does the difference between the left and the right tire forces matter?
The lateral tire force depends on the vertical tire force. For the dynamic bicycle model, the left and right tires at the front and those at the rear were lumped together, and so were the corresponding lateral tire forces: $F_{y,f}=F_{y,fl}+F_{y,fr}$, $F_{y,r}=F_{y,rl}+F_{y,rr}$
In gernal, the lateral tire forces do not grow proportionally with the vertical loads:
- The lateral tire force grows subproportionally with the vertical tire force.
- Hence, if both tires have the same vertical load $F_z*$, the total lateral force $2F_y*$ is greater than the one generated by two tires with ther vertical loads $F_z*\pm\Delta F_z$
A higher roll stiffness of the front / rear axle leads to a lower (total) lateral tire stiffness of the front /rear axle during a cornering maneuver.
> Remarks:
> - A higher roll stiffness of the frontacle decreases the total tire stiffness of the front axle, and hence increases the vehicle's understeer.
> - Since oversteer of a vehicle should be avoided, especially at high speeds (which leads to instability), designers usually pick the front rool stiffness large enough to avoid oversteer enven in dynamic cornering maneuvers.
> - The roll stiffness of the front / rear axle can be modified by the addition of **anti roll bars**, which alter the stiffness of the suspension only with respected to the rolling mode but not ,e.g. the pitching mode.
> - Due to the (underdesible) jacking forces, the roll center of a suspension is usually kept closely above the road surface.
> - To avoid excessive roll rotation of the chassis, the center of gravity should be as close as possible to the vehicle roll axis.

### Geometry of the roll center
Analogously to the anti-dive and anti-squat suspension geometry, there exist a **virtual pivot point** of the suspension with respect to the roll direction.
#### Double wishbone suspension:
The same principles apply for McPherson struts.