## 2 Vehicle Modelling and Performance Bounds  
This paper focuses on mono-dimensional tracks, i.e., predefined trajectories. Let \(\lambda\) be a generic parameter. Given a curve \(\gamma(\lambda)\), which may or may not be closed, the vehicle negotiating \(\gamma(\lambda)\) will be represented as a mass point that moves along the curve, similar to [20]. The curve can also be expressed as a function of the arc length (curvilinear abscissa), \(s\), as \(\gamma(s(\lambda))\). A condition that needs to be fulfilled is that the car is always on the track, i.e.,  
\[x(\lambda)\equiv\gamma(s(\lambda)) \tag{1}\]  
where \(x(\lambda)\) is the position vector with respect to a fixed arbitrary reference frame. Once the curve is assigned, together with the initial conditions, one can determine the optimal law of the driver'sbehaviour on the accelerator and brake pedals, as functions of the position on the track, so as to minimise the time to cover \(\gamma(s(\lambda))\):  
\[\min_{a_{x}(s)}\int\frac{1}{v(a_{x}(s))}\text{d}s\text{ subject to }\|a_{x}(s)\|\leq h \big{(}a_{y}(s),v(a_{x}(s))\big{)} \tag{2}\]  
where \(h\) represents the physical constraints or performance bounds, \(a_{y}\) is the lateral acceleration, and \(v=\dot{s}\) is the vehicle speed. \(h\) depends on both the grip and engine power. The mathematical modelling of the physical constraints was designed so as to capture the most relevant aspects of road dynamics. Three main factors were considered, namely:  
* Adherence of tyres to the asphalt (grip);
* Engine power;
* Aerodynamic loads, such as drag force and lift/downforce. When trying to minimise lap time for generic driving conditions, the upper limit of the vehicle performance may depend on either the grip or engine power. For example, in case of deceleration on a straight line or in case of cornering, the bottleneck is grip. However, in case of acceleration on a straight line, depending on the conditions, the maximum performance may be limited by either the engine power or grip. The grip constraint can be studied through the well-known friction ellipse [21]. Given its symmetry in terms of lateral acceleration behaviour, only half of the ellipse will be considered. Mathematically, the locus is determined once the values of the semi-axes are given. Because of the effect of aerodynamics, the size of the ellipse is speed dependent. The longitudinal and vertical aerodynamic forces, \(F_{x}\) and \(F_{z}\), respectively, can be expressed as follows:  
\[ \begin{array}{l}F_{x}=\frac{1}{2}\rho c_{x}Sv^{2}=k_{x}v^{2}\\ F_{z}=\frac{1}{2}\rho c_{z}Sv^{2}=k_{z}v^{2}\end{array}  \tag{3}\]  
where \(\rho\) is the air density, \(S\) is the frontal area, and \(c_{x}\) and \(c_{z}\) are the drag and downforce coefficients, respectively. The tyre model herein adopted is based on the assumption that the maximum available tyre-road frictional force is proportional to the vertical load on the tyre, through the friction coefficient \(\mu\). Therefore, the ellipse semi-axes can be obtained directly from the dynamic equilibrium equations:  
\[ \begin{array}{l}ma_{x+,grip}=\frac{1}{2}\mu mg+\Big{(}\frac{1}{2}\mu k_{z}-k_ {x}\Big{)}v^{2}\text{ \ \ acceleration}\\ ma_{x-,grip}=-\mu mg-(\mu k_{z}+k_{x})v^{2}\text{ \ \ \ braking}\\ ma_{y,max}=\mu\Big{(}mg+k_{z}v^{2}\Big{)}\end{array}  \tag{4}\]  
where \(m\) is the vehicle mass, \(g\) is the gravitational acceleration, \(a_{x+,grip}\) is the maximum achievable longitudinal acceleration when the vehicle accelerates, \(a_{x-,grip}\) is the maximum achievable longitudinal acceleration when the vehicle brakes, and \(a_{y,max}\) is the maximum achievable lateral acceleration. The last three quantities determine the ellipse semi-axes. In the first equation, the coefficient \(\frac{1}{2}\) reflects the assumption that the vehicle is either front-wheel-drive (FWD) or rear-wheel-drive (RWD), as well as the simplified hypotheses that load transfers are neglected and that the downforce effect is evenly distributed between the front and rear axles of the vehicle.