## 3 Vehicle Dynamics Modeling and Discretization  
### Dynamics Modeling  
For a high-speed AGV, the vehicle generally steers with a large radius and small front steering angle. Therefore, the track width is neglectable with respect to its cornering radius. Thus, it is reasonable to use a spatial single-track vehicle model, as shown in Figure 3. The vehicle dynamics can be expressed as [21]:  
\[m\dot{v}_{x}= a_{x}+mv_{y}\omega_{z}-m_{y}h_{sr}\omega_{z}\omega_{x} \tag{1a}\] \[m\dot{v}_{y}= a_{y}-mv_{x}\omega_{z}-mg\sin\left(\phi_{t}\right)+m_{s}h_{sr} \dot{\omega}_{x}\] (1b) \[I_{z}\dot{\omega}_{z}= \left(I_{f}F_{yf}-l_{r}F_{yr}\right)\] (1c) \[\dot{\phi}_{r}= \omega_{x}\] (1d) \[I_{x}\dot{\omega}_{x}= m_{s}h_{sr}(\dot{v}_{y}+v_{x}\omega_{z})+m_{s}gh_{sr}\sin\left( \phi\right)-M_{R} \tag{1e}\]  
where \(a_{x}=(F_{xf}+F_{xx}-F_{R})/m\), \(F_{xf}\), \(F_{xr}\) are the longitudinal forces at the front/rear tire, \(F_{R}\) represents the dissipative forces, including aerodynamic drag force, rolling resistance friction force, etc., \(a_{y}=(F_{yf}+F_{yr})/m\), \(F_{yf}\), \(F_{yr}\) are the lateral forces at the front/rear tire, \(M_{R}=K_{\phi}\phi_{r}+D_{\phi}\omega_{x}\), \(K_{\phi}\) and \(D_{\phi}\) are the suspension roll damping and stiffness, \(m\) is the total vehicle mass, \(m_{s}\) is the sprung mass, \(\psi\) and \(\omega_{z}\) are the yaw angle and yaw rate, \(h_{sr}\) is the height of the sprung mass from the roll center, \(l_{f}\), \(l_{r}\) are the distances from the CoG to the front and rear axle along the vehicle's \(x\)-axis, \(g\) is the gravitational acceleration, \(v_{x}\), \(v_{y}\) are the longitudinal and lateral velocities of the CoG and \(\phi_{r}\), \(\omega_{x}\) and \(\dot{\omega}_{x}\) are the roll dynamics (angle, rate and acceleration) of the sprung mass, respectively. \(\phi_{t}\) is the road bank and assumes \(\phi_{t}\approx 0\), \(\phi_{r}\) is the roll angle of the sprung mass to the vehicle's \(z\)-axis. \(\phi=\phi_{t}+\phi_{r}\), \(I_{z}\) is the moment of inertia about the vehicle's \(z\)-axis. \(I_{x}\) is the moment of inertia of the sprung mass with respect to the roll center. The cross products of the mass moment of inertia are neglected. Figure 2: Framework of the proposed MPC scheme. The relative motions of the AGV with respect to the road curvilinear coordinate system are:  
\[\dot{e}_{y}= v_{x}\sin\left(e_{\psi}\right)+v_{y}\cos\left(e_{\psi}\right) \approx v_{x}e_{\psi}+v_{y} \tag{2}\] \[\dot{e}_{\psi}= \omega_{z}-\kappa v_{x} \tag{3}\]  
where, \(e_{y}\) is the cross track error and \(e_{\psi}\) is the difference between the vehicle heading angle, \(\psi\), and the road centerline angle, \(\psi_{r}\). \(\kappa\) is the road centerline curvature. Equations (1)-(3) are derived under the following assumptions:  
* \(e_{\psi}\), \(\delta_{f}\) and \(\phi\) satisfy the small angle assumption;
* The vehicle pitch/actuators dynamics can be neglected, and the steering system is rigid;
* The disturbances such as wind lateral thrust are not considered. In this work, we mainly focus on the lateral and roll dynamics of AGVs; thus, \(a_{x}\) is assumed to be constant. Considering a front-wheel-driven vehicle, then \(F_{xr}=0\). The longitudinal front tire force \(F_{xf}\) can be computed as:  
\[F_{xf}=ma_{x}+F_{R} \tag{4}\]  
As the nonlinearity of lateral tire forces, \(F_{yf}\) and \(F_{yr}\), pose a significant challenge to real-time optimization, we use a linearized tire force model, as illustrated in \[m\dot{v}_{x}= a_{x}+mv_{y}\omega_{z}-m_{y}h_{sr}\omega_{z}\omega_{x} \tag{1a}\]0 Then, the lateral tire forces are approximated as [30]:  
\[m\dot{v}_{x}= a_{x}+mv_{y}\omega_{z}-m_{y}h_{sr}\omega_{z}\omega_{x} \tag{1a}\]1  
where \(\bar{C}_{\alpha*}\) is the equivalent cornering stiffness at \(\alpha_{*}=0\). This linearization will lead to diverging from the actual tire forces at high sideslip angle; however, this problem can be avoided by enforcing a sideslip angle threshold, \(\alpha_{lim}\).