## 3 Analytical Model Conceptual Development and Discussion
### Highway Vehicle Dynamics Studied by Spherical Coordinates  
The situation in spherical coordinates is presented in Figure 6. Then, one would have two coordinates normal to the motion's trajectory, \(r\), \(\psi\), and one tangential to it: \(\phi\). As can be noticed in the figure, the \(\phi\) coordinate's unit vector goes into (i.e., enters) the page's plane. In this case, \(r\), \(\psi\), and \(\phi\) set up an orthonormal coordinate system. Figure 5: Dynamic situation description in terms of cylindrical-polar coordinates: “normal”, “tangential”, and \(z\) coordinates; that is, \((r;\phi;z)\). Plane of motion considered unrotated (i.e., parallel) with respect to the horizontal plane. In this figure, \(\tilde{N}\), \(\tilde{F}\), and \(\tilde{W}\) are the forces acting on the vehicle: normal, friction, and weight, respectively. The angle \(\theta\) is the highway’s considered super-elevation. \(\hat{r}\), \(\hat{\phi}\), and \(\hat{z}\) are the 3D cylindrical-polar coordinate system unit vectors. In this system, the position can be established as follows in terms of Cartesian unit vectors:  
\[\overrightarrow{r}=r\sin\psi\cos\phi\hat{i}+r\sin\psi\sin\phi\hat{j}+r\cos\psi \hat{k} \tag{17}\]  
The calculation of the second derivative of \(\overrightarrow{r}\), i.e., acceleration, produces  
\[\overrightarrow{a}=\bigg{(}cos\psi\Big{(}rcos\psi\hat{\psi}-2 rsin \psi\psi\hat{\psi}+2 cos\phi r\psi\Big{)}-sin\psi\ \Big{(}sin\phi\big{(}r\hat{\psi}+2\hat{\
u}\big{)}-cos\phi\hat{\
u}+rcos\phi\] \[\big{(}\hat{\psi}^{2}+\hat{\varphi}^{2}\big{)}\Big{)}\bigg{)}\hat {i}+\bigg{(}sin\psi\Big{(}sin\phi\big{(}\hat{r}-r\dot{\psi}^{2}\big{)}+rcos\phi \ddot{\phi}+2 cos\phi r\dot{\phi}\Big{)}+rsin\phi\Big{(}cos\psi\ddot{\psi}-sin \psi\dot{\psi}^{2}\Big{)}\] \[+2 cos\psi\psi\Big{(}sin\phi\dot{r}+rcos\phi\phi\Big{)}\bigg{)} \hat{j}+\bigg{(}cos\psi\Big{(}\dot{r}-r\dot{\psi}^{2}\big{)}-sin\psi\Big{(}r \ddot{\psi}+2\dot{r}\dot{\psi}\Big{)}\bigg{)}\hat{k} \tag{18}\]  
In this particular case, the acceleration calculation in Equation (18) has to take into account that \(\dot{r}=0,\ddot{r}=0,\dot{\psi}=0,\ddot{\psi}=0\), and \(\ddot{\phi}=0\). In terms of the system's unit vectors, it yields  
\[\overrightarrow{a}=-r\sin\psi\cos\phi\dot{\psi}^{2}\dot{r}-r\sin\psi\sin\phi \dot{\psi}^{2}\dot{\psi}+0\dot{\phi} \tag{19}\]  
The conditions above of the model implemented are due to the dynamic situation since there is no velocity or acceleration along radial (\(r\)) and zenithal (\(\psi\)) coordinates. Meanwhile, there is no acceleration for the azimuthal coordinate (\(\phi\)). Equation (5), applied to the situation in Figure 6, and taking into account Equation (19) (with \(v=r\dot{\phi}\)), yields, in spherical component vector representation; that is, \(\hat{r},\dot{\psi},\dot{\phi}\),  
\[(0;-N;0)+(-mgcos\psi;mgsin\psi;0)+(-\mu N;0;0)\] \[=m\bigg{(}-sin^{2}\psi\frac{v^{2}}{r};-sin\psi cos\psi\frac{v^{2} }{r};0\bigg{)} \tag{20}\]  
In this case, the \(r\) coordinate can be considered (a subject on how the actual plane of motion is taken into account) both an instantaneous radius of curvature of the vehicle's trajectory and the radial coordinate of the system used to analyze the problem. For example,  
Figure 6: Dynamic situation description in terms of spherical coordinates: “radial”, “azimuthal”, and “zenithal” coordinates; that is, \((r;\phi;\psi)\). Plane of motion considered unrotated with respect to the horizontal plane. In this figure, \(\vec{N}\), \(\vec{F}\), and \(\vec{W}\) are the forces acting on the vehicle: normal, friction, and weight, respectively. The angle \(\theta\) is the highway’s considered super-elevation; \(R\) is the radius of curvature (curvature parameter considered). \(\hat{r}\), \(\dot{\phi}\), and \(\ddot{\psi}\) are the 3D spherical coordinate system unit vectors.