# Influence of the Road Model on the Optimal Maneuver of a Racing Motorcycle
## 5 Optimal Maneuver Simulation Results
### Comparison of Simulation Results on Adopted Road Models  
The calculated optimal lap times are 89.658 s in the 3D-local case, 90.844 s in the 3D-global case, and 91.211 s on the 2D road model. The best recorded lap time was 90.485 s (AQ), which is 0.829 s slower than the lap time in the 3D-local simulation. Figure 13 shows the measured and computed vehicle speed \(V\). The 3D-local case is closest to the experiment. The highest difference occurs in turn 10a and is equal to 3.0 m/s. The deviations in other corners do not exceed 1.6 m/s. For the other two simulations, there are significant differences in numerous corners. In the case of the 3D-global model, the largest deviations are observed in corners 5, 6, chicane 10a-10b, and turn 12. They are equal to 3.2 m/s, \(-\)3.6 m/s, \(-\)3.0 m/s, and 3.6 m/s, respectively. For the two-dimensional road model, the largest deviations are observed in corners 1, 2, and 6, where they differ from the experiment by \(-\)1.9 m/s, 3.6 m/s, and \(-\)2.8 m/s, respectively. The three-dimensional road models allowed for a better representation of the vehicle speed on sections with significant road inclination \(\mu\) (between turns 2 and 3, and turns 4 and 5). The maximum speed is achieved between turns 9 and 10 and is equal to 74.2 m/s (AQ), 73.2 m/s (3D-local), and 72.9 m/s in the 3D-global and 2D cases. The minimum speed in the slowest corner number 7 is 20.5 m/s (AQ), 20.2 m/s (3D-local), 19.2 m/s (3D-global), and 20.5 m/s (2D). The measured maximum speed is 2 m/s higher than the average maximum speed of the other laps and was reached in the aerodynamic shadow behind another rider. The thin, black line in Figure 13 represents a speed trace from the lap with a maximum speed closest to the calculated average value. Due to the aerodynamic shadow, the real vehicle gains about 0.210 s. The vehicle in the 3D-local case gains the most advantage over the AQ during braking before turns 3a, 6, 10a, 12, and on the straight  
Figure 13: Velocity profiles (**top**) and time difference (**bottom**) as a function of elapsed distance: (**a**) comparison between experimental data and 3D-local case, (**b**) comparison between GPS speed and vehicle speed computed in 3D-global and 2D cases. section between turns 5 and 6, which is analyzed in detail in Section 5.3. In the 3D-global case, the vehicle has an advantage (over AQ) of 0.606 s at \(s_{dist}=1850\) m, but then loses 1.505 s until \(s_{dist}=3607\) m. A significantly higher speed in the last corner reduces the time loss at the end of the lap to 0.359 s. The vehicle from the 2D simulation is slower than the AQ by no more than 0.350 s until turn 6. In corner 6 and 7, the vehicle loses approximately 0.400 s. After that, the time difference remains approximately constant and is equal to 0.726 s at the end of the lap. Figure 14 shows the vehicle's longitudinal acceleration \(\dot{a}_{x}\) and lateral acceleration \(\dot{a}_{y}\). The good agreement with the experimental data, in terms of the rate of change of the accelerations in the transient phases, is a consequence of the hyperbolic constraints introduced on the controls. The three-dimensional road model allowed a good prediction of the longitudinal acceleration \(\dot{a}_{x}\) during the thrust phase. This can be attributed to the inclusion of 3D effects (road inclination \(\mu\) in this case) and the speed-dependent driving force in the vehicle model. In the 2D road model case, higher values of longitudinal acceleration \(\dot{a}_{x}\) are observed in the initial acceleration phase. For both simulations on the three-dimensional road, greater deceleration occurs in sections where the road flattens (1808-1882 m, 3309-3424 m, 3780-3864 m). The \"simulated optimal rider\" takes advantage of the higher vertical tire load resulting from changes in the road normal curvature \(\Omega_{y}\). This will be explained in Section 5.2.