# Influence of the Road Model on the Optimal Maneuver of a Racing Motorcycle
## 1 Introduction  
The influence of road geometry on vehicle motion is something we experience every day. When driving motorized or human-powered vehicles, we observe differences in acceleration capabilities when going uphill or downhill. We sometimes experience a sense of weightlessness when driving over the tops of hills, and we notice that corners are banked, even on public roads. Therefore, to accurately reproduce real vehicle motion on a racetrack, it is necessary to consider the road model, especially in minimum lap time simulations, where we expect to use results as reference data to improve the real vehicle and driver performance. Historically, the minimum lap time simulations (MLTSs) were solved assuming a predefined path, e.g., [1; 2; 3; 4]. Later on, the MTLS problem was formulated as an optimal control problem and the assumption of the free-trajectory has been introduced. In optimal maneuver problems, we can distinguish two vehicle models: transient (e.g., [5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19]) and quasi-steady-state [20; 21; 22; 23; 24]. Transient vehicle models provide information about the transient behavior of the vehicle. However, they are more computationally expensive than thefree-trajectory quasi-steady-state (QSS) approach. First presented in [20], the QSS approach is characterized by a concise system dynamic. The vehicle performance is constrained by g-g diagrams describing the vehicle-driver performance. They are computed before solving the optimal control problem (OCP), which results in a lower computational complexity of the OCP itself. According to [20], the free-trajectory QSS approach is about ten times faster than when the dynamic vehicle model is used. The idea of g-g diagrams has been discussed, for example, in [25]. An experimentally-based method to simulate the transient behavior in QSS simulations has been presented in [24]. An alternative approach to solving the minimum lap time problems is a model predictive control (MPC) framework, in which the control is based on a finite preview horizon [26; 27; 28; 29]. In [29], cascade optimization was used to mitigate the human driver's ability to learn the track and to minimize the global maneuvering time. There are also many publications related to road geometry and the trajectories of autonomous vehicles. These works include problems such as trajectory planning, tracking control, and obstacle avoidance, for example in [30; 31; 32; 33; 34]. The problem presented in this work concerns the specific and limited problem of developing an optimal trajectory for race motorcycles. A detailed historical overview of MLTS problems, along with the evolution of vehicle models and a brief description of methods used to solve OCPs, has been provided in the review article [35]. Another comprehensive study on MLTSs is presented in [36]. The three-dimensional road model was introduced in [10]. The road reconstruction procedure from real track measurements was discussed in detail in [13]. The road was modeled as a ribbon and its geometry was determined using an additional OCP, which allowed filtering out of the noise present in the measurements. Then, in [14], the optimal maneuver of the F1 car on the Barcelona-Catalunya circuit was determined and compared with the results obtained when assuming a \"flat\" road model. The track model was built based on elevation data from the global digital elevation model. The track reconstruction procedure was also described in [37], where the performances of direct and indirect methods for solving OCPs were directly compared. A more sophisticated road model has been presented in [21], when lateral-position-dependent camber, typical for some racetracks with large lateral curvature, has been introduced. The Darlington Raceway racetrack was reconstructed using high-resolution LIDAR measurements. The free-trajectory QSS simulation of NASCAR car was presented and compared with a \"flat\" road model.