## 2 Vehicle Modelling and Performance Bounds
For a generic vehicle condition with the lateral acceleration expressed as \(a_{y}=v^{2}/R\) (\(R\) is the radius of curvature), the achievable longitudinal acceleration according to the grip constraint is as follows:  
\[ \begin{array}{l}a_{x+}=a_{x+,grip}\sqrt{1-\Big{(}\frac{a_{y}}{a_{y,max}} \Big{)}^{2}}\text{ \ \ acceleration}\\ a_{x-}=a_{x-,grip}\sqrt{1-\Big{(}\frac{a_{y}}{a_{y,max}}\Big{)}^{2}}\text{ \ \ \ braking}\end{array}  \tag{5}\]Figure 1 shows an example of the obtained ellipse for \(v=50\) km/h. The constraint on engine power can be expressed as follows:  
\[a_{x+,power}=\frac{P-k_{x}v^{3}}{mv} \tag{6}\]  
where \(P\) is the available power. As shown in Figure 2, the locus is just a vertical line. Figure 1: Friction ellipse for \(v=50\) km/h, grip effect only (blue line: acceleration, red line: braking). Figure 2: Power limit for v = 50 km/h. The overall vehicle performance bounds are determined by the most restrictive conditions between the grip limit and power limit. In acceleration conditions, the maximum achievable longitudinal acceleration, \(a_{x+,max}\), is given by:  
\[a_{x+,max}=\min\Bigl{(}a_{x+},a_{x+,power}\Bigr{)} \tag{7}\]  
while in braking conditions, the maximum achievable longitudinal acceleration, \(a_{x-,max}\), is as follows:  
\[a_{x-,max}=a_{x-} \tag{8}\]  
In other words, at each point of the trajectory, \(a_{y}=v^{2}/R\) is calculated, then Equations (7) and (8) provide the maximum acceleration/deceleration, accounting for the effect of the friction ellipse and maximum power. The resulting overall performance bounds are depicted in Figure 3 for speeds from 10 to 250 km/h in 24 km/h steps. Equation (7) inevitably leads to a discontinuity of the derivative at the boundary between the grip and power constraints. There is no discontinuity for low speeds, meaning that the performance is limited by the grip. However, as the speed increases, the discontinuity shows up and the vehicle performance is limited by power. This means that the driver can negotiate the corner at full throttle, without the risk of skidding, as there is high availability of grip. As stated earlier, the availability of the grip increases with the vehicle speed because of the downforce effect. The vehicle will not skid, as long as the following equation is satisfied:  
\[\frac{mv^{2}}{R}\leq\mu mg+\mu k_{z}v^{2} \tag{9}\]  
from which it can be observed that if the radius of the curvature is large enough, there is no grip-related speed limit. The radius of the curvature for which this happens, denoted as the critical radius of the curvature, \(R_{crit}\), can be easily worked out as follows:  
\[R_{crit}=\frac{m}{\mu k_{z}} \tag{10}\]  
Figure 3: Performance bound for different vehicle speeds. Another way to interpret this is by looking at the left- and right-hand sides of Equation (9)--they are two parabolas, with \(v\) as the independent variable. If \(R<R_{crit}\), the parabolas have an intersection point, which represents the maximum speed that can be achieved as a result of the grip constraint. If there is no intersection, there is no speed limit. This is seen in Figure 4, which depicts the two parabolas for two values of \(R\), one lower and one greater than \(R_{crit}\).