# Influence of the Road Model on the Optimal Maneuver of a Racing Motorcycle
## 3 Three-Dimensional Free-Trajectory Quasi-Steady-State MLTS
### Vehicle Model and Related Vehicle Performance Constraint  
Assuming quasi-steady-state conditions, vehicle performance is limited by the envelope of the g-g diagram, referred to as the g-g-g diagram in the case of spatial motion on a 3D road. The g-g or g-g-g diagram represents the limit of tire adhesion when lateral and longitudinal accelerations occur simultaneously, taking into account the vehicle's ability to accelerate and decelerate. The shape of the g-g diagram envelope is speed-dependent, primarily due to aerodynamic drag and lift. Figure 3 shows a generalized rigid-body model of a vehicle. The equations describing the model are the same as in [22]. In addition, the model includes rolling resistance and considers a speed-dependent driving force, \(F_{xr}\), analogous to [24]. Yaw and side-slip accelerations are omitted, the suspension is rigid, the steering angle is zero, the tires are infinitely narrow discs, and the driver is rigidly attached to the chassis. The longitudinal friction coefficient \(\mu_{x}\) and the lateral friction coefficient \(\mu_{y}\) are the same for the front and rear tires. The vehicle parameters introduced in Figure 3 are explained in Table 1, where their values for the Supersport motorcycle used in the simulations are given. Vehicle parameters were estimated based on measurements made on a motorcycle in very similar specification to those used in the MotoAmerica championship. Aerodynamic properties and tire friction coefficients were estimated from recorded data. Figure 3: Generalized rigid-body motorcycle model. \(F_{x}\), \(F_{y}\), and \(N\) are the longitudinal, lateral, and vertical forces acting on the front (subscript \(f\)) and the rear (subscript \(r\)) tire. \(F_{d}\) is the drag force. The center of gravity and center of pressure are indicated by the acronyms CoG and CoP, respectively. The variables \(\widetilde{a}_{x}\), \(\widetilde{a}_{y}\), and \(\widetilde{\mathfrak{g}}\) shown in Figure 3, and the vehicle speed \(V\), allow the parameterization of the g-g-g diagram. The relationship between \(\widetilde{a}_{x}\), \(\widetilde{a}_{y}\), \(\widetilde{\mathfrak{g}}\), and the accelerations \(\hat{a}_{x}\) and \(\hat{a}_{y}\) expressed in the vehicle frame can be determined by analyzing the motion of the vehicle's CoG. Assuming constant CoG position relative to the road surface equal to \(\left[0,\ 0,-h\right]^{T}\), the following relationships can be obtained:  
\[\widetilde{a}_{x}=\hat{a}_{x}-\mathrm{g}(\cos\mu\mathrm{sin}\,\phi\mathrm{sin }\,\hat{\chi}-\sin\mu\mathrm{cos}\,\hat{\chi}), \tag{28}\]  
\[\widetilde{a}_{y}=\hat{a}_{y}-\mathrm{g}(\sin\mu\mathrm{sin}\,\hat{\chi}+\cos \mu\mathrm{sin}\,\phi\mathrm{cos}\,\hat{\chi}), \tag{29}\]  
\[\widetilde{\mathfrak{g}}=\mathrm{gcos}\,\mu\mathrm{cos}\,\phi+V\hat{\omega}_{ y}, \tag{30}\]  
where \(\hat{\omega}_{y}\) is the angular velocity of the vehicle frame about the \(\hat{y}\)-axis. The angular velocity \(\hat{\omega}\) of the vehicle frame can be determined from a matrix equation  
\[\hat{\omega}= \begin{bmatrix}\hat{\omega}_{x}\\ \hat{\omega}_{y}\\ \hat{\omega}_{z}\end{bmatrix} =\mathbf{R}(\mathbf{e}_{z},\hat{\chi})^{T} \begin{bmatrix}\Omega_{x}\\ \Omega_{y}\\ \Omega_{z}\end{bmatrix} \hat{s}+ \begin{bmatrix}0\\ 0\\ \hat{\chi}\end{bmatrix} =\begin{bmatrix} \begin{pmatrix}(\Omega_{x}\mathrm{cos}\, \hat{\chi}+\Omega_{y}\mathrm{sin}\,\hat{\chi})\hat{s}\\ (\Omega_{y}\mathrm{cos}\,\hat{\chi}-\Omega_{x}\mathrm{sin}\,\hat{\chi})\hat{s} \\ \Omega_{z}\hat{s}+\hat{\chi}\end{pmatrix} , \tag{31}\]  
where \(\mathbf{R}(\mathbf{e}_{z},\hat{\chi})^{T}\) is the rotation matrix used to express the angular velocity \(\Omega_{B}\) of the road frame in the vehicle frame. Equation (31) is the sum of the angular velocity \(\Omega_{B}\) of the ribbon frame and the angular velocity of the vehicle frame relative to the ribbon frame equal to \(\left[0,\ 0,\ \hat{\chi}\right]^{T}\). The effect of road geometry on the g-g diagram is illustrated in Figure 4a.