# Trajectory Tracking Control Strategy of 20-Ton Heavy-Duty AGV Considering Load Transfer
## 2 Establishment of Heavy-Duty AGV Vehicle Model
### Pose Linear Error Model of Double Ackermann Steering  
By establishing a pose linear error model, the deviation between the current state of the AGV and the desired trajectory can be described more accurately so that the prediction model can be easily established when designing the model predictive control. In order to reduce the complexity of the model and improve the solution speed, the motion model of the double Ackerman steering heavy-duty AGV is simplified into a bicycle model, as shown in Figure 3.  
_XOY_ is the geodetic coordinate system. _CM_ is the centroid of AGV, and its coordinate is (\(x\), \(y\)). \(V\) is the centroid velocity. \(\beta\) is the centroid sideslip angle of AGV's mass center. \(\theta\) is the angle between the AGV longitudinal and the horizontal axis of the coordinate system. \(l_{f}\) and \(l_{r}\) are the longitudinal distances from the center of mass of AGV to the center of the front wheel and rear wheel, respectively, and \(l_{f}=l_{r}=1.5\) m.  
It can be obtained from Figure 3:  
\[\tan\beta=\frac{l_{r}\tan\delta_{f}+l_{f}\tan\delta_{r}}{l_{f}+l_{r}} \tag{8}\]  
The load of AGV is large. When the centroid sideslip angle of the center of mass is large, it is easy to cause problems such as reduced stability of the vehicle and increased tire wear. Therefore, it is necessary to reduce the centroid sideslip angle of the center of mass to reduce the above effects. It can be obtained from (8) that when \(l_{f}=l_{r}\), in order to make the theoretical centroid sideslip angle of the center of mass zero, it is necessary to satisfy  
\[\delta_{r}=-\delta_{f} \tag{9}\]  
When \(\delta_{r}=-\delta_{f}\), \(l_{1}=l/2\) can be obtained from (5), so the parameters of the steering trapezoid in the front and rear steering transmission mechanism of AGV, as shown in Figure 4, can be determined by (6) and (7). The motion state model of AGV is shown in (10):  
Figure 4: Steering trapezoid diagram. Figure 3: Kinematics model of bicycle. \[. \begin{bmatrix}\dot{x}\\ \dot{y}\\ \dot{\theta}\end{bmatrix} = \begin{bmatrix}v\cdot\cos(\beta+\theta)\\ v\cdot\sin(\beta+\theta)\\ v\cdot\cos(\beta)\frac{\tan\delta_{f}-\tan\delta_{f}}{f_{f}+l_{r}}\end{bmatrix} . \tag{10}\]  
where \((\dot{x},\dot{y},\dot{\theta})\) is the derivative of AGV pose \((x,y,\theta)\) at different moments. \(X=[x,y,\theta]^{T}\) and \(u=[\delta_{f},\delta_{r},v]^{T}\) are selected as the state variable and input control variable of AGV, respectively. The (10) can be expressed as  
\[\dot{X}=f(X,u) \tag{11}\]  
For a given reference trajectory, it can be described by the motion trajectory of the reference vehicle. Each point on it satisfies the above motion state equation, and r represents the reference variable. The general form is  
\[\dot{X}_{r}=f(X_{r},u_{r}) \tag{12}\]  
where \(X_{r}=[x_{r},y_{r},\theta_{r}]^{T}\), \(u_{r}=[\delta_{fr},\delta_{rr},v_{r}]^{T}\). Expanding (11) at the reference point \((x_{r},y_{r},\theta_{r})\) using a Taylor series and neglecting higher-order terms, the following expression can be obtained:  
\[\dot{X}=f(X_{r},u_{r})+\frac{\partial f(X,u)}{\partial X}(X-X_{r})+\frac{ \partial f(X,u)}{\partial u}(u-u_{r}) \tag{13}\]  
The linear time-varying error model of the AGV pose obtained by subtracting (12) and (13) is as follows:  
\[\dot{X}_{e}=\dot{X}-\dot{X}_{r}= \begin{bmatrix}\dot{x}-\dot{x}_{r}\\ \dot{y}-\dot{y}_{r}\\ \dot{\theta}-\theta_{r}\end{bmatrix} =A_{t}X_{e}+B_{t}u_{e} \tag{14}\]  
\[\text{where }A_{t}= \begin{bmatrix}0&0&-v_{r}\sin(\theta_{r})\\ 0&0&v_{r}\cos(\theta_{r})\\ 0&0&0\end{bmatrix} ,X_{e}=[x-x_{r},y-y_{r},\theta-\theta_{r}]^{T},u_{e}=[\delta _{f}-\delta_{fr},\delta_{r}-\delta_{rr},v-\\ v_{r}]^{T},B_{t}=