\chapter*{Vehicle and Tires Models}\stepcounter{chapter}
\label{chpt: veh dynamics}
\addcontentsline{toc}{chapter}{Vehicle and Tire Models}
\textit{The synthesis of a high-performance slip control strategy requires a rigorous physical mapping of the vehicle’s motion, centered on the longitudinal degree of freedom. This section establishes the mechanical equilibrium and the kinematic relationships that govern the flow of torque from motor to tire-road interface. \\
To construct a high-fidelity control plant, the modeling must synthesize four deeply coupled physical domains: the rotational equilibrium of the wheel, the dynamic transfer of vertical loads, the kinematic definition of slip, and the inherently non-linear nature of tire friction. By deconstructing these interdependencies, the boundary between stable traction and uncontrolled spinning is defined, providing the controller with the analytical foundation necessary to operate precisely at available adhesion.}
\newpage
\section[Vehicle Model]{Vehicle Model}
The dynamic model employed in this study is based on a two-track formulation and restricted to the planar vehicle motion, in order to achieve an optimal balance between physical fidelity and computational tractability for the synthesis and validation of the anti-slip control logic.

In accordance with the primary research objective, hence the regulation of longitudinal wheel slip, the dynamics associated with body roll, pitch, and vertical heave are neglected in this approximation. This modeling assumption allows for the reduction of the system's state space to 7 fundamental Degrees of Freedom (DoF):
\begin{itemize}
    \item Longitudinal and lateral translatory motions within the $x^By^B$ ground plane;
    \item Yaw motion about the vertical $z^B$-axis;
    \item Angular rotation of the four individual wheels.
\end{itemize}
\begin{figure}[H]
    \centering
    \includegraphics[width=0.5\textwidth]{images/VehModelScheme.png}
    \caption{Force diagram in vehicle reference plane.}
    \label{fig: force diagram vehicle}
\end{figure}

Governing equations for vehicle planar dyanmics are derived from the translational force equilibrium along the longitudinal axis ($x^B$) combined with the rotational torque balance about the wheel axis, providing the analytical framework necessary to evaluate the vehicle's trajectory and stability under the influence of the four individual tire-road contact forces.

Forces, torques and velocities terms in Figure \ref{fig: force diagram vehicle} are defined below:
\begin{description}
    \item [$F_{x_{ij}}$] Longitudinal traction force acting on \textit{ij}-wheel.
    \item [$F_{y}$]\nomenclature{$F_y$}{Lateral force acting on wheel. \nomunit{[N]}} Lateral force acting on \textit{ij}-wheel.
    \item [$F_{{roll}_{ij}}$] \nomenclature{$F_{roll}$}{Rolling resistance acting on wheel. \nomunit{[N]}} Rolling resistance acting on \textit{ij}-wheel.
    \item [$F_{aero}$] \nomenclature{$F_{aero}$}{Aerodynamic drag. \nomunit{[N]}}Aerodynamic drag.
\end{description}

\subsection[Longitudinal Dynamics]{Longitudinal Dynamics}
\label{subsec: long dynamics}
According to Newton’s Second Law, the vehicle’s translation is the result of the net balance between the tractive forces generated at the four tire-road interfaces and the diverse environmental resistances opposing the motion. This global equilibrium is expressed as:
\begin{equation}
    M \dot{v}_x = \sum_{i,j} F_{x,ij} - F_{res}
\end{equation}
Where $M$ \nomenclature{$M$}{Vehicle mass\nomunit{[kg]}} represents the total vehicle mass and $\dot{v}_x$ \nomenclature{$\dot{v}_x$}{Longitudinal vehicle acceleration\nomunit{[m/s]}} is the longitudinal acceleration. While $F_{x,ij}$ is the tractive force at each driving wheel, the term $F_{res}$ \nomenclature{$F_{res}$}{Total resistive force\nomunit{[N]}} encapsulates the lumped resistance effects, primarily dominated by aerodynamic drag and rolling resistance, which, considering a flat road, can be expressed as:
\begin{align}
    F_{aero} &= \frac{1}{2} \rho C_d A v_x^2
    \\
    F_{roll} &=  f_r F_z
\end{align}
$F_{aero}$ represents the aerodynamic drag, where $\rho$ is the environmental air density, $C_d$ is the \textit{aerodynamic drag coefficient}, and $A$ is the frontal area of the vehicle.\\ $F_{roll}$ accounts for the rolling resistance, with $f_r$ being the \textit{rolling resistance coefficient}.

\subsection{Vertical Load}
A fundamental aspect of longitudinal dynamics is the redistribution of vertical loads during acceleration and braking transients. Since the tractive force that a tire can generate is strictly proportional to its vertical load $F_{z,ij}$, the controller must account for the \textit{pitch} of the vehicle body.
\\
Assuming a rigid chassis and neglecting aerodynamic downforce for this derivation, the vertical loads on the front and rear axles are determined by the moment equilibrium around the contact patches:
\begin{equation}
    \begin{cases} 
    F_{z,F} = F_{z,F,static} - \Delta F_z \\
    F_{z,R} = F_{z,R,static} + \Delta F_z 
    \end{cases}
\end{equation}
Where the generalized term for dynamic longitudinal load transfer, denoted as $\Delta F_z$ is defined as:
\begin{equation}
\Delta F_z = \frac{M \dot{v}_x h_g + F_{aero} h_a}{L}
\end{equation}
This definition accounts for the \textit{inertial load transfer} ($\frac{M g b}{L}$) and the \textit{aerodynamic pitch moment} ($\frac{F_{aero} h_a}{L}$). \\
The explicit definition for the vertical load is finally given by: 
\begin{equation}
\begin{cases}
    F_{z,f} &= \frac{M g b - M \dot{v}_x h_g - F_{aero}h_a}{L}
    \\
    F_{z,r} &= \frac{M g a + M \dot{v}_x h_g + F_{aero}h_a}{L}
\end{cases}
\label{eq: vertical load }
\end{equation}
where $a$ and $b$ are the horizontal distances from the CoG to the front and rear axles, respectively, and $L= a+b$ is the wheelbase.

While the \textit{driven wheels} contribute only to the rolling resistance and lateral guidance, the \textit{driving} ones must manage the entirety of the driving torque output. This concentration of force on a single axle makes the system highly sensitive to  longitudinal dynamic load transfer. \\ As the vehicle accelerates, the inertial reaction at the center of gravity induces a pitch moment that increases the vertical load on the rear axle. This phenomenon provides a physical advantage for an \acrshort{rwd} layout: the normal force, and thus the maximum friction potential, are enhanced precisely at the wheels where the driving torque is applied.

The equations derived in this section provide the macroscopic framework of the vehicle's longitudinal motion dynamics. However, the system remains under-determined until the tractive force ($F_x$) is explicitly modeled. As this force is a non-linear function of the relative motion between the tire and the road, the following section is dedicated to the microscopic tire dynamics and the characterization of the friction-slip relationship. 
%------------------------------------------------------------------%
\section[Tyre-Road Interaction Model]{Tyre-Road Interaction Model}
\label{sec: tyre dynamics}
This section transitions from the rigid-body dynamics of the chassis to the physics of traction force generation. By characterizing the wheel's rotational equilibrium and defining the kinematic variables that govern the tire's capacity to transmit shear forces, the non-linear plant that the traction controller must regulate is defined. Central to this discussion is the relationship between longitudinal slip and friction coefficient, which defines the physical boundary of the vehicle's performance envelope.

\subsection{Longitudinal Slip}
\label{subsec: slip ratio}
While a comprehensive vehicle model requires the consideration of both longitudinal slip ($\sigma_L$) and lateral slip ($\sigma_S$) to describe combined-slip maneuvers, this thesis is specifically constrained to the longitudinal degree of freedom ($\sigma \equiv \sigma_L$), in the direction of the wheel velocity vector.

The decision to omit lateral slip dynamics from the control synthesis is motivated by the project's focus on pure longitudinal traction optimization and anti-spin regulation for Rear-Wheel Drive (RWD) architectures. By isolating the longitudinal component, the controller can more precisely manage the transmissible torque during high-acceleration transients, where the primary objective is the prevention of wheel-spin and the maintenance of the tire within its stable longitudinal friction region.
\begin{figure}[H]
    \centering
    \includegraphics[width=0.4\textwidth]{images/SlipWheelScheme.png}
    \caption{Wheel slip model.}
    \label{fig: wheelslipmodel}
    
\end{figure}
In this context, the \textbf{longitudinal slip ratio} $\sigma$ is defined as the normalized difference between the rotational velocity of the driving wheel and the average rotational velocity of the driven wheels:
\begin{equation}\label{eq: slip theoretical definition}
    \sigma_{Rj} = \begin{cases}
        \frac{{\omega_{driving, Rj}}- \omega_{driven, avg}}{\omega_{driving, Rj}}, \quad \text{traction}
        \\
        \frac{{\omega_{driving, Rj}} - \omega_{driven, avg}}{\omega_{driven, avg}}, \quad \text{braking}
    \end{cases}
    \end{equation}
Where $\omega_{driving, Rj}$\nomenclature{$\omega$}{Wheel rotational speed \nomunit{[rad/s]}}\nomenclature{$R$}{Effective tire radius \nomunit{[m]}} is the rotational speed of the j-th rear wheel, and $\omega_{driven, avg}$ is the reference free rolling speed for slip ratio evaluation, retrieved as the average rotational speed of the j-th front wheels, given by $\frac{{\omega_{FL}}+\omega_{FR}}{2}$. 
\paragraph{Control-oriented model definition.} Slip definition is adapted to prevent mathematical instabilities that typically occur at near-zero velocities, and to account for both driving and braking regimes within a single, unified expression:
\begin{equation} \label{eq: slip mathematical definition}
\sigma_{Rj} = \frac{{\omega_{driving, Rj}} - \omega_{driven, avg}}{\max(|\omega_{driving, Rj}|, | \omega_{driven, avg} |, \epsilon)}
\end{equation} 
where $\epsilon$ represents a numerical tolerance which ensures that the denominator remains non-zero during standstill conditions ($V_{vhcl}, \omega = 0$). In this context, vehicle's speed is retrieved directly from the average driven wheels rotational speed as:
\begin{equation}
    V_{vhcl} =  \omega_{driven, avg}\cdot R_F
\end{equation}

\subsection{Wheel's Dynamics}\label{subsec: wheel dynamics}
By applying the principle of angular momentum conservation to the j-th driving wheel, the governing differential equation is established as:
\begin{equation} \label{eq: driving trq}
    I_w \dot{\omega}_{Rj} = T_{drive,Rj} - T_{ext,Rj} - F_{x,Rj} R_R 
\end{equation} 
\begin{figure}[H]
    \centering
    \includegraphics[width=0.4\textwidth]{images/WheelForceDiagram.png}
    \caption{Single wheel free body diagram.}
    \label{fig:wheel force diag}
    
\end{figure}
In this formulation, $I_w$ represents the  moment of inertia of the wheel assembly (including the integrated rotor in case of \acrshort{iwm}), $T_{drive,Rj}$ is the driving torque at the wheel, provided by the electric motor, and $T_{ext,Rj}$ represents the sum of all external resistive torques, including the hydraulic braking torque $T_{b,Rj}$.

\paragraph{Control-oriented model definition.} It is important to note that for the actual controller synthesis, the fundamental equations derived above may be further manipulated to account for specific hardware configurations and external disturbances. Furthermore, in the context of Sliding Mode Control, the resistance forces $F_{res}$ and the non-linearities in $T_{ext,ij}$ are often lumped into a single \textit{uncertainty term} \cite{SMCandObservation}. This abstraction allows the adaptive law to compensate for exogenous factors without requiring an exact analytical measurement of every external variable.

\subsection[Pacejka's Magic Formula and Tire Force Non-linearity]{Pacejka's Magic Formula and Tire Force Non-linearity}
\label{subsec: pajecka}
To understand the challenge of longitudinal traction control, the physical mechanism by which a pneumatic tire generates force must be examined. \\
Unlike the friction between two rigid bodies, tire-road friction is a result of both molecular adhesion and hysteretic deformation of the rubber. This relationship is fundamentally non-linear and is most comprehensively described by the \textbf{Pacejka Magic Formula} \cite{Pacejka}.

The dependency of the traction force generated by the wheel at a specific wheel's slip condition is described through a semi-empirical trigonometric structure:
\begin{align}
    F_{x,ij} &= F_{z,ij}\ \mu_{ij}
    \\
    \mu_{ij}(\sigma_{ij}) &= D \sin \left( C \arctan \left( B \sigma_{ij} - E (B \sigma_{ij} - \arctan(B \sigma_{ij})) \right) \right)
\end{align}
where $B$, $C$, $D$ and $E$ are Pacejka's coefficients, each corresponding to a specific physical attribute of the tire-road interaction:
\begin{itemize}
    \item Stiffness Factor ($B$): represents the initial slope of the curve, corresponding to the elastic longitudinal stiffness of the tire carcass and tread blocks before macro-sliding begins.

    \item Shape Factor ($C$): governs the width of the peak, and  determines how sharp the transition between static and dynamic friction is.
    \item Peak Factor ($D$): defines the maximum friction coefficient $\mu_{max}$, according to the road surface condition (e.g., dry asphalt vs. icy road).

    \item Curvature Factor ($E$): accounts for the drop-off in friction after the peak, characterizing the transition into the fully developed sliding regime.
\end{itemize}
These empirical coefficients, identified for a specific tire carcass, enable the Magic Formula to reconstruct the tire's characteristic curves with high fidelity across a broad operational envelope. Although the model provides a robust approximation for most driving maneuvers, its predictive accuracy may diverge under highly non-linear or transient conditions. \\
These characteristic mappings analytically describe the evolution of the longitudinal force ($F_x$), the lateral force ($F_y$), and the \textit{self-aligning torque} ($M_z$) \nomenclature{$M_z$}{Self-aligning torque \nomunit{[Nm]}} as a coupled function of the state variables: vertical load ($F_z$), longitudinal slip ($\sigma$), sideslip angle and camber angle.

\subsubsection[Sensitivity to Vertical Load]{Sensitivity to Vertical Load ($F_x \ \text{vs.} \ \sigma$)}
The longitudinal force is not solely a function of slip but is deeply coupled with the vertical load.
\begin{figure}[H]
    \centering
    \includegraphics[width=0.65\textwidth]{images/LONGITUDINAL-FORCE-AND-LONGITUDINAL-SLIP-AT-VARIOUS-VERTICAL-LOADS.png}
    \caption{Longitudinal force and longitudinal slip mapping for various vertical loads \cite{figurePacejkaFx}.}
    \label{fig: Pacejka Fx sigma Fz}
\end{figure}
As illustrated in the $F_x-\sigma$ map, an increase in vertical load, resulting from the longitudinal load transfer discussed in Section \ref{subsec: long dynamics}, shifts the entire curve upward. However, it is important to observe that the force does not increase linearly with $F_z$, but is subjected to a \textit{load sensitivity}. This degressive behavior explains how, as the rear axle gains load during acceleration, the tire stiffness and peak force capacity change, making the plant of the traction controller time-varying and non-linear.

\paragraph{Control-oriented model definition.} Despite the Pacejka model provides superior fidelity for offline vehicle simulations, its high degree of non-linearity and large parameter set make it computationally expensive for real-time \textbf{Optimal Slip Identification}. Specifically, it requires iterative numerical methods that are often incompatible with the fixed-step execution constraints of embedded vehicle control units.
Consequently, to facilitate the design of an adaptive law capable of autonomous surface identification, the focus must shift toward a model that preserves the fundamental non-linear phenomenology of the $\mu-\sigma$ curve while offering a differentiable and analytically manageable form.
For this reason, the \textbf{Burckhardt Tire Model}, which preserves the essential characteristics of the Pacejka curve with a reduced parameter set, will be utilized in Chapter \ref{chpt: comparative design of baseline} to identify the optimal slip and adapt the control law to varying road conditions.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection[Burckhardt Tire Model]{Burckhardt Tire Model}
\label{subsec: burckhardt}
The Burckhardt Tire Model offers a simplified alternative to the Magic Formula, replacing Pacejka’s extensive parameter set with a three-coefficient exponential structure. This formulation is especially effective for adaptive control and real-time estimation, as it explicitly characterizes tire's adhesion limit through a continuous and differentiable function of the longitudinal slip \cite{BurckhardtTheory}\cite{BurckTheoryJazar}.
The longitudinal friction coefficient is expressed as:
\begin{equation}\label{eq: burckhardt mu_sigma relationship}
\mu_{ij}(\sigma_{ij}) = c_1 (1 - e^{-c_2 \sigma_{ij}}) - c_3 \sigma_{ij}
\end{equation}
Where $c_1$, $c_2$ and $c_3$ are the road surface dependent coefficients defining the curve's profile. The physical significance of these parameters is defined as follows:
\begin{description}
    \item \textbf{$c_1$}: Governs the asymptotic value of the friction coefficient, effectively scaling the peak magnitude $\mu_{max}$.
    \item \textbf{$c_2$}: Dictates the exponential growth rate, representing the normalized longitudinal stiffness of the tire in the linear elastic region.
    \item \textbf{$c_3$}: Introduces the slope of the sliding region, characterizing the friction drop-off beyond the saturation point.
\end{description}

\begin{table}[H]
    \centering
    \caption{Burckhardt Tire Model Coefficients}
    \label{tab:road_coefficients}
    \begin{tabular}{lccc}
        \toprule
        & $c_1$ & $c_2$ & $c_3$ \\
        \midrule
        Asphalt, dry     & 1.2801 & 23.99  & 0.52   \\
        Asphalt, wet     & 0.857  & 33.822 & 0.347  \\
        Concrete, dry    & 1.1973 & 25.168 & 0.5373 \\
        Cobblestone, dry & 1.3713 & 6.4565 & 0.6691 \\
        Cobblestone, wet & 0.4004 & 33.708 & 0.1204 \\
        Snow             & 0.1946 & 94.129 & 0.0646 \\
        Ice              & 0.05   & 306.39 & 0      \\
        \bottomrule
    \end{tabular}
\end{table}
\begin{figure}[H]
    \centering
    \includegraphics[width=0.8\textwidth]{images/BurckhardtFrictionSlipPlot.png}
    \caption{Burckhardt tire–road friction curves for different road conditions. \cite{BurckhardtFigure}.}
    \label{fig: Pacejka Fx sigma Fz}
\end{figure}
\raggedbottom
In the initial low-slip stage, the friction coefficient increases rapidly. Physically, this represents the elastic deformation of the tire tread blocks, dominated by the term $(1 - e^{-c_2 \sigma_{ij}})$. \\
As slip increases, the tire reaches its maximum friction potential ($\mu_{max}$) at the optimal slip ratio ($\sigma_{opt}$). This is the mathematical ideal region where the gain from the exponential adhesion term is exactly balanced by the loss from the linear term ($c_3 \sigma_{ij}$). It coincides with the condition $\partial\mu/\partial\sigma = 0$, which ensures maximum longitudinal acceleration and optimal vehicle stability.
\paragraph{Control-oriented model definition.} The structural simplicity of the Burckhardt formulation is particularly beneficial for the synthesis of the traction control law. Friction coefficient equation is analytically differentiable, allowing for a direct derivation of the optimal slip condition. By applying the first-order necessary condition for a maximum at the friction peak ($\partial\mu/\partial\sigma = 0$), the optimal slip ratio can be retrieved in a closed-form expression.

\nomenclature{$i$}{Index for axle identification (\textit{R} for rear, \textit{F} for front)}
\nomenclature{$j$}{Index for wheel identification (\textit{L} for left, \textit{R} for right)}