## 3 MPC Strategy  
The MPC algorithm has been widely used in motion control of mobile robots and it has three basic characteristics: model prediction, rolling optimization, and feedback correction. It has unique advantages for solving motion control problems, such as point stabilization and trajectory tracking of mobile robots. Traditional methods often neglect or simplify the robot kinematics and dynamics constraints which have a significant impact on the robot's control performance. Meanwhile, the MPC algorithm takes the constraints into consideration to improve the system robustness. The existing controllers have high requirements with respect to the accuracy of the system model and tend to obtain effective control parameters based on the model and multiple experimental tests;. However, the MPC algorithm has low requirements with respect to the accuracy of the system model and can acquire the control parameters through simple experiments. Through applying the characteristics of rolling optimization and feedback correction, the influence of time-delay in the closed-loop system can be effectively reduced or even eliminated, and the motion control is optimized to achieve improved control performance. Particularly, the MPC algorithm can handle multivariate and constrained problems effectively. The MPC problem can be described as solving the optimal solution of the cost function. At each sampling time, the outputs of the future \(N\) sampling instants are predicted according to the system model, and the cost function is constructed based on the errors between the predicted outputs and the true state outputs of the system. The optimal control inputs of the next \(N\) sampling instants are \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Para** & \(\beta_{1}\) & \(\beta_{2}\) & \(\beta_{3}\) & \(\theta_{1}\) & \(\theta_{2}\) & \(\theta_{3}\) & \(\alpha_{1}\) & \(\alpha_{2}\) & \(\alpha_{3}\) \\ \hline
**Value** & \(60^{\circ}\) & \(180^{\circ}\) & \(300^{\circ}\) & \(60^{\circ}\) & \(180^{\circ}\) & \(300^{\circ}\) & \(90^{\circ}\) & \(90^{\circ}\) & \(90^{\circ}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The parameters of system structure. Figure 6: Three-wheeled structure layout of the platform. obtained by minimizing the cost function, however, only the first control vector is used as the system input. At the next sampling time, this process is repeated and rolling optimization is performed. In this section, an MPC-based kinematic controller is designed to ensure that the mobile robot can be driven to the desired position accurately, smoothly, and stably. Define \(q=[x_{c}\;y_{c}\;\theta_{c}]^{T}\) as the state of mobile robot in coordinate \(x_{c}\theta_{c}y_{c}\) and \(S=[S_{1}\;S_{2}\;S_{3}]^{T}=R^{-1}\). Then, it can be derived that  
\[\dot{q}\left(t\right)\;=\;S\left(t\right)\;v\left(t\right) \tag{13}\]  
where  
\[S_{1}\left(t\right) = \begin{bmatrix}-0.5774\cos\theta_{c}\left(t\right)-0.3333\sin \theta_{c}\left(t\right)\\ 0.6667\sin\theta_{c}\left(t\right)\\ 0.5774\cos\theta_{c}\left(t\right)-0.3333\sin\theta_{c}\left(t\right)\end{bmatrix} ^ {T}\] \[S_{2}\left(t\right) = \begin{bmatrix}0.3333\cos\theta_{c}\left(t\right)-0.5774\sin \theta_{c}\left(t\right)\\ -0.6667\cos\theta_{c}\left(t\right)\\ 0.3333\cos\theta_{c}\left(t\right)+0.5774\sin\theta_{c}\left(t\right)\end{bmatrix} ^ {T}\] \[S_{3}\left(t\right) = \begin{bmatrix}0.002038\quad 0.002038\quad 0.002038\end{bmatrix} \]  
Based on zero-order hold (ZOH), a continuous-time system can be transformed into a discrete-time form with a sampling period T.  
\[q\left(k+1\right)=q\left(k\right)+\dot{q}\left(k\right)T \tag{14}\]  
The discrete-time system (14) can be transformed into a discrete state space system as  
\[q(k+1)=Aq(k)+Bu(k) \tag{15}\]  
where  
\[A=I\in\mathbb{R}^{n\times n},\;B=S(k)T\in\mathbb{R}^{n\times m}\]  
The parameter \(n\) is the number of state variables and \(m\) is the number of input variables. The matrix \(A\) is a constant identity matrix, and matrix B is determined by orientation of the system dynamically.