# Influence of the Road Model on the Optimal Maneuver of a Racing Motorcycle
## 5 Optimal Maneuver Simulation Results
### Analysis of Tire Vertical Load  
Road geometry affects the vertical tire load according to the relationship  
\[m\widetilde{\text{g}}=N_{r}+N_{f}, \tag{36}\]  
Figure 16: Comparison of (**a**) vehicle lateral position, (**b**) radius of curvature modulus. Figure 17: Comparison of the optimal trajectories; enlarged view of (**a**) turn 11, (**b**) chicane 10a–10b. which belongs to the equations of the vehicle model. The variable \(\widetilde{\wp}\) is called apparent gravity and is given by the Equation (30). Based on the vehicle model, the following equations can be derived for the vertical load on the front tire \(N_{f}\) and the vertical load on the rear tire \(N_{r}\):  
\[N_{r}=\frac{\left(m\widetilde{a}_{x}h+F_{d}h_{p}\right)\cos\widetilde{\varphi}+ (w-b)m\widetilde{\wp}}{w}, \tag{37}\]  
\[N_{f}=m\widetilde{\wp}-N_{r}, \tag{38}\]  
where \(\widetilde{\varphi}\) is a vehicle roll angle relative to the road surface given by  
\[\widetilde{\varphi}=\arctan\left(\frac{\widetilde{a}_{y}}{\widetilde{\wp}} \right). \tag{39}\]  
Variations in road geometry can significantly affect normal tire load, increasing the likelihood of front or rear wheel lift during hard acceleration or braking, as well as altering allowable tire longitudinal and lateral forces. The calculated vertical load on the front wheel, shown in the upper graph of Figure 18, makes it possible to predict (in 3D-local case) that the front wheel lift can occur in two track segments during the ride over the top of the hills, after turns 5 and 11 (between the vertical dotted lines marked in Figure 18). The simulation results are confirmed by the measurements recorded by the data acquisition system. The front wheel lift is indicated by local dips in the front wheel speed and the near zero value of the front suspension deflection (middle graph on Figure 18). At the top of the hill after turn 5, the front suspension was in the position of maximum extension for more than 40 m, while the linear velocity of the front wheel speed decreased from 37.5 m/s to 32.8 m/s. In addition, the actual rider had to stop the pitching motion toward the rear wheel by partially closing the throttle at \(s_{dist}=1488\) m (bottom plot on Figure 18). Figure 18: From top to bottom: vertical force acting on the front wheel, front suspension deflection and (in blue) wheel speed, throttle position. In the second identified track segment, the same features mentioned above are visible in the experimental data. There are numerous dips in the front wheel speed, and the suspension deflection is close to zero. The rider and the \"simulated driver\" try to counteract the front wheel lift by driving over the hill with a non-zero lateral acceleration \(\dot{a}_{y}\), as shown in Figure 14b. According to the g-g diagram shown in Figure 4a, leaning the motorcycle to a certain extent allows a higher longitudinal acceleration than in straight motion. When comparing the OCP results for the adopted road models, it can be seen again that the 3D-local case is closest to the experiment. In the 3D-global case, the vertical front wheel load in the identified road segments also takes values close to zero, but occurs in the shorter distance. A \"flat\" road model gives significantly different results. The vertical load on the front wheel in the aforementioned road segments is significantly higher. By comparing the results for 3D and 2D roads, it is possible to determine whether the low vertical load on the front tire is only due to high longitudinal acceleration, or whether it is additionally enhanced by varying road geometry. Another interesting conclusion can be drawn from Figure 19, which shows the total tire vertical load. Intuitively, the total tire vertical load on a two-dimensional road is constant and equal to \(m_{g}\). For the 3D road models, the vertical load on the tires increases when driving through a valley bottom, or decreases when driving over a hill, as in the road sections shown above.