# Predictive Control for Small Unmanned Ground Vehicles via a Multi-Dimensional Taylor Network
## 3 Materials and Methods  
### System Description of the SUGV  
A simplified SUGV model is shown in Figure 1. Two front wheels and two rear wheels are combined into a single front wheel and rear wheel, which are simplified into a two-degrees-of-freedom vehicle model. The following assumptions were made: we ignored the effect of steering and suspension; we kept the vehicle's longitudinal speed constant and only considered the lateral movement along the \(y\) axis and the yaw movement around the \(z\) axis; we ignored the effects of horizontal and vertical aerodynamics; and we considered the side characteristics of the tire in the analysis of the tire force [36; 37; 38]. The _XOY_ coordinate system in Figure 1 is a fixed geodetic coordinate system, and _xoy_ is the structure of the coordinate system, which changes with the movement of the body. The force analysis of the above model along the \(y\) axis and around the \(z\) axis is obtained. \[\left\{ \begin{array}{l}May_{y}=F_{cf}\cos\delta_{f}+F_{cr}\\ I_{z}\dot{w}=I_{f}F_{cf}\cos\delta_{f}-I_{r}F_{cr}\end{array} \right. \tag{1}\]  
where \(M\) is the vehicle's weight; \(I_{f}\) is the distance between the center of the front axle and the center of mass of the vehicle; \(I_{r}\) is the distance between the center of the rear axle and the center of mass of the vehicle; \(I_{z}\) is the moment of inertia of the vehicle around the \(z\) axis; \(\delta_{f}\) is the front wheel angle of the vehicle; \(w\) is the yaw velocity of the vehicle; and \(a_{y}\) is the accelerated velocity of the vehicle along the \(y\) axis based on the _xoy_ coordinate system, which mainly consists of the movement along the \(y\) axis and the centripetal acceleration \(V_{x}w\). We can obtain:  
\[a_{y}=\ddot{y}+V_{x}w \tag{2}\]  
where \(F_{cf}\) and \(F_{cr}\) are the lateral force of the tire applied to the front and rear wheels of the vehicle, respectively. From the defined dynamic model, we can see that the side characteristic of the tire is considered in the analysis of the tire force. When the front wheel angle \(\delta_{f}\) and the sideslip angle \(\beta\) are small, the sideslip characteristic of the tire is in a linear range. So, we can obtain  
\[F_{cf}=2C_{f}\alpha_{f} \tag{3}\]  
\[F_{rf}=2C_{r}\alpha_{r} \tag{4}\]  
where \(C_{f}\) and \(C_{r}\) are the cornering stiffnesses of the front and rear tires, respectively. Since there are two front and rear wheels, the force is twice that of a single tire. \(\alpha_{f}\) and \(\alpha_{r}\) are the tire sideslip angle and the angle between the direction of the tire velocity vector and the x axis, respectively. At a small angle, the size of the two angles can be approximately expressed as:  
\[\tan\alpha_{f}\approx\theta<V_{f},\ x>=\frac{V_{y}+l_{f}\dot{\varphi}}{V_{x}} \tag{5}\]  
\[\tan\alpha_{r}\approx\theta<V_{r},\ x>=\frac{V_{y}-l_{r}\dot{\varphi}}{V_{x}} \tag{6}\]  
Figure 1: The SUGV model. According to the coordinate system relationship, the side slip angle is \(\beta=\tan\beta=\frac{V_{p}}{V_{x}}\). From (1)-(6), the differential equation of the vehicle dynamics model is obtained. \[\left\{ \begin{array}{l}\dot{w}=\frac{l_{C}^{2}C_{f}+l_{C}^{2}C_{r}}{\frac{V_{ x}}{V_{x}}}w-\frac{l_{C}C_{f}+l_{C}C_{r}}{L_{z}}\beta+\frac{l_{C}C_{f}}{L_{z}} \delta_{f}\\ \dot{\beta}=(-\frac{l_{C}f_{C}+l_{C}C_{r}}{mv_{x}^{2}}-1)w-\frac{C_{f}+C_{r}}{ mv_{x}}\beta+\frac{C_{f}}{mv_{x}}\delta_{f}\end{array} \right. \tag{7}\]  
The control input and output variables of the controlled vehicle are the front wheel steering angle \(\delta_{f}\) and the yaw velocity \(w\), respectively. According to the vehicle dynamics model, a relation between the output and the input of the controlled system is deduced  
\[\ddot{w}(t)=-a_{1}\ddot{w}(t)-a_{2}w(t)+b_{1}\dot{\delta}_{f}(t)+b_{2}\delta_{ f}(t) \tag{8}\]  
where \(a_{1}=\frac{m(l_{C}^{2}C_{f}+l_{C}^{2}C_{r})+l_{z}(C_{f}+C_{r})}{mv_{x}L_{z}}\), \(a_{2}=\frac{-mv_{2}^{2}(l_{C}f_{C}-l_{C}C_{r})+C_{f}C_{r}}{mv_{x}^{2}L_{z}}\), \(b_{1}=\frac{l_{C}f_{C}}{L_{z}}\), \(b_{2}=\frac{C_{f}C_{r}L}{mv_{x}L_{z}}\).