# Trajectory-Tracking Control of Unmanned Vehicles Based on Adaptive Variable Parameter MPC
## 2 Unmanned Vehicle Trajectory-Tracking Model  
### Unmanned Vehicle Dynamics Model  
The foundation for achieving precise tracking control is building an accurate unmanned vehicle dynamics model, which is established, as seen in Figures 1 and 2, while taking the road lateral inclination angle into consideration [18; 19]. For the unmanned vehicle model, the following presumptions are made in order to increase control efficiency and model accuracy: a small angle assumption is made for the front wheel turning angle \(\delta_{f}\), i.e., \(\cos(\delta_{f})\approx 1\). Since we focus on the traverse and roll motions of the vehicle, for the pitching motion of the vehicle, the pitch angle is assumed to be \(\theta\approx 0\).  
The unmanned vehicle dynamics model can be expressed as  
\[\left\{\begin{array}{l}m\dot{v}_{y}=-m\dot{\varphi}v_{x}+mh\ddot{\phi}+(F_{yf} +F_{yr})-mg\phi\\ I_{x}\dot{\phi}=ma_{y}h+mgh\phi-M_{x}\\ I_{z}\ddot{\varphi}=l_{f}F_{yf}-l_{r}F_{yr}\end{array}\right. \tag{1}\]  
where \(h\) is the height at the center of mass of the unmanned vehicle, \(\phi\) and \(\dot{\phi}\) are the lateral inclination angle and the rate of change of the lateral inclination angle of the body  
Figure 1: Unmanned vehicle dynamics modeling considering road sidelobes.  
Figure 2: Unmanned vehicle trajectory-tracking model considering road curvature.  
of the unmanned vehicle, and the lateral inclination angle of the body of the unmanned vehicle is mainly caused by the deformation of the suspension system. \(M_{x}\) is the lateral inclination resistance moment of the suspension system of the body of the unmanned vehicle, \(M_{x}=K_{\phi}\phi_{r}+D_{\phi}\phi\); \(\phi_{r}\) is the lateral inclination angle of the body of the vehicle, \(\phi_{t}\) is the pavement camber \(\phi_{r}=\phi-\phi_{t}\), assuming that \(\dot{\phi}_{t}\approx 0\), of which \(K_{\phi}\) is the coefficient of lateral inclination stiffness of the unmanned vehicle and \(D_{\phi}\) is the coefficient of lateral inclination damping. \(l_{f},l_{r}\) is the distance between the center of mass and the front and rear axles, respectively; \(\dot{\varphi}\) is the yaw rate of the vehicle, \(a_{y}=\dot{v}_{y}+\dot{\varphi}v_{x}-h\ddot{\phi}\) is the transverse acceleration of the center of mass of the unmanned vehicle; there are \(ma_{y}=F_{y}\cos\phi\), \(F_{y}\) for the combined force of the tire force in the \(Y\)-axis direction. \(m\) is the overall mass of the vehicle, and \(I_{x}\), \(I_{x}\) are the moments of inertia of the unmanned vehicle around the \(Z\)-axis and \(X\)-axis. \(F_{yf}\) and \(F_{yr}\) are the lateral forces on the front and rear wheels, respectively, where the lateral tire forces can be approximated as  
\[F_{yf}=C_{f}\alpha_{f},F_{yr}=C_{r}\alpha_{r} \tag{2}\]  
\(C_{f}\), \(C_{r}\) are the front and rear wheel tire cornering stiffnesses, and \(\alpha_{f}\), \(\alpha_{r}\) are the front and rear wheel lateral deflection angles. Therefore, the unmanned vehicle dynamics model in Equation (1) can be further expressed as  
\[\left\{\begin{array}{l}\dot{v}_{y}\approx F_{y}(\frac{1}{m}+\frac{h^{2}}{I_{ x}})-v_{x}\dot{\varphi}+\Big{[}\frac{h}{I_{x}}(mgh-K_{\phi})-g\Big{]}\phi\\ -\frac{h}{I_{x}}D_{\phi}\dot{\varphi}+(\frac{mgh^{2}}{I_{x}}-g)\phi_{t}\\ \ddot{\varphi}\approx\Big{[}I_{f}F_{yf}-l_{r}F_{yf}\Big{]}\,/\,I_{z}\\ \ddot{\phi}\approx\Big{[}F_{y}h+mgh(\phi+\phi_{t})-K_{\phi}(\phi-\phi_{t})-D_{ \phi}\dot{\varphi}\Big{]}\,/\,I_{x}\end{array}\right. \tag{3}\]