# Influence of the Road Model on the Optimal Maneuver of a Racing Motorcycle
## 4 Results of Track Reconstruction  
The influence of the road model on the optimal maneuver is discussed using the example of the Supersport motorcycle and the Road Atlanta circuit located in the United States. The simulation results are divided into two sections. The first section includes a comparison of the road model obtained using global and local DEMs. In the second section, a comparison of the optimal maneuver computed on adopted road models is presented. The 3D simulations are compared with experimental data and the simulation on a \"flat\" two-dimensional road. The formulated OCPs were solved using the orthogonal collocation method implemented in the GPOPS-II package [41] linked to the MATLAB R2019a software. The IPOPT and the ADiGator [42] were used as the optimizer and automatic differentiation tools, respectively. Numerical differentiation was employed in the three-dimensional minimum time maneuver problem. Comparison between Results Obtained Using Local and Global DEMs  
The track edges were discretized using Google Earth Pro 7.3.6 software. A total of 1355 points were manually selected (652 on the right and 703 on the left side of the road). Due to discrepancies between satellite images taken in different years, the image closest to the orthophoto map available in [43] was selected based on analysis of the location of several checkpoints spread over different sections of the track. Three road models have been built:  
1. In the first model, later referred to as 3D-local, the elevation of the points was determined using a local DEM provided in TIFF file format by the United States Geological Survey [43]. This database has a resolution of 1 m and 0.1 m root mean square elevation error;
2. In the second model, hereafter referred to as 3D-global, the elevation data were assigned using a tool available at [44]. The assigned elevation data were derived from the NED1 global DEM, characterized by 1 arc-second horizontal resolution;
3. In the third case, the simple two-dimensional track model was built--later referred to as 2D. It was obtained by omitting variables \(z,\ \mu,\phi,\ \mu^{\prime},\phi^{\prime}\) in the state vector and \(\mu^{\prime\prime},\phi^{\prime\prime}\) in the control vector in the track identification OCP formulated in Section 2.1. The geographical coordinates of the points located on the track edges were converted to a rectangular coordinate system using the Universal Transverse Mercator projection [45]. The Road Atlanta track contains numerous hills and valleys, banked corners, and turns with large and small curvatures. The latter caused difficulties in choosing a universal value for \(w_{\theta}\), which is mainly responsible for smoothing the geodesic curvature \(\Omega_{2}\). A small value of \(w_{\theta}\) allowed accurate representation of corners with small radii of curvature, but led to an increase in simulation time in the associated minimum lap time problem. As the road geometry became more detailed, the number of mesh points required increased. On the other hand, a large value of \(w_{\theta}\) made it possible to obtain a smooth geodesic curvature \(\Omega_{2}\), but this was at the expense of accuracy in the representation of corners with small curvature radii. As a result, the error in the tight corners between the numerically identified edges and the provided data increased significantly. It was therefore decided to \"manually\" combine the results obtained for a penalty factor \(w_{\theta}=4\times 10^{4}\) and \(w_{\theta}=4\times 10^{6}\). The results for \(w_{\theta}=4\times 10^{4}\) have been used within turns from 3a to 3c, 10a, 10b, and, in turn, 12. A better representation of the track boundaries was achieved without any negative effect on the computation time in the associated lap time minimization problem. Other penalty factors were chosen as: \(w_{\phi}=w_{\mu}=4\times 10^{6}\) and \(w_{w}=1\times 10^{4}\).