## 3 Design of the Controller
### Theoretical Proof  
To prove Theorem 1, we need to solve the error between the desired control signal and the actual control signal and analyze the error between the expected trajectory and the actual trajectory. The convergence of the controller is proved as follows:  
Step 1: In solving the error between the desired control signal and the actual control signal, we need to use the state error of the system. Assuming \(q_{d}\) represents the expected motion state of the system, the state error can be expressed as follows:  
\[ \begin{array}{l}\Delta q_{i}(k+1)=q_{d}(k+1)-q_{i}(k+1)\\ \qquad\qquad=\left[q_{d}(k)+d_{k}B_{d}u_{d}(k)\right]-\left(q_{i}(k)+d_{k}B_{i }u_{i}(k)+\tilde{\beta}_{i}(k)\right)\\ \qquad\qquad=\Delta q_{i}(k)+\left(B_{d}-B_{i}\right)u_{d}(k)-B_{i}(u_{d}(k)-u _{i}(k))-\tilde{\beta}_{i}(k)\end{array}  \tag{26}\]  
According to the formula constraint expression, the state error of the system can be effectively scaled, which can be expressed as follows:  
\[\|\Delta q_{i}(k+1)\|\leq\|\Delta q_{i}\|+C_{B}b_{u_{d}}\|\Delta q_{i}\|+b_{B} \|\Delta u_{i}\|+\tilde{b}_{\beta} \tag{27}\]  
If we set \(h_{2}=(1+C_{B}b_{u_{d}})\), the state error (27) can be simplified again, which can be expressed as follows:  
\[\|\Delta q_{i}(k+1)\|\leq h_{2}\|\Delta q_{i}\|+b_{B}\|\Delta u_{i}\|+\tilde{b }_{\beta} \tag{28}\]  
Solving the state error by iteration method from the state error at the initial moment to the current moment, we obtain the state error expression:  
\[ \begin{array}{l}\|\Delta q_{i}\|\leq\left(\|h_{2}b_{B}\Delta u_{1}\|+\left\| h_{2}^{2}b_{B}\Delta u_{2}\right\|+\ldots\ldots+\left\|h_{2}^{i-1}b_{B} \Delta u_{i-1}\right\|\right)\\ \qquad\qquad+h_{2}(\tilde{b}_{\beta}+\tilde{b}_{\beta}+\ldots\ldots+\tilde{b }_{\beta})+h_{2}^{k}b_{q_{0}}\\ \qquad\qquad\leq\sum\limits_{j=0}^{k-1}\left(h_{2}^{k-1-j}[b_{B}\|\Delta u_{ i}\|+\tilde{b}_{\beta}]+h_{2}^{k}b_{q_{0}}\right.\end{array}  \tag{29}\]  
where \(b_{B}\) represents the maximum error of the control object, \(\tilde{b}_{\beta}\) is the maximum disturbance of the state variable, and \(b_{q_{0}}\) indicates the error of the system at the initial time. Step 2: Assuming that \(u_{d}\) as the desired control signal can control the mobile platform to track the desired trajectory. The error between the actual control signal and the desired control signal can be expressed as follows:  
\[ \begin{array}{l}\Delta u_{i+1}(k)=u_{d}(k)-u_{i+1}(k)\\ \qquad\qquad=u_{d}(k)-u_{i}(k)-L_{1}e_{i}(k+1)-L_{2}e_{i+1}(k)-L_{3}e_{i+1}(k-1 )\\ \qquad\qquad=u_{d}(k)-u_{i}(k)-L_{1}[y_{d}(k+1)-y_{i}(k+1)]\\ \qquad\qquad-L_{2}[y_{d}(k)-y_{i+1}(k)]-L_{3}[y_{d}(k-1)-y_{i+1}(k-1)]\\ \qquad\qquad=\|\Delta u_{i}\|-L_{1}(q_{d}(k+1)-q_{i}(k+1)-\alpha_{i}(k+1))\\ \qquad\qquad-L_{2}(q_{d}(k)-q_{i+1}(k)-\alpha_{i+1}(k+1))\\ \qquad\qquad-L_{3}(q_{d}(k-1)-q_{i+1}(k-1)-\alpha_{i+1}(k-1))\\ \qquad\qquad=\|\Delta u_{i}\|-L_{1}[\|\Delta q_{i}\|+(B_{d}-B_{i})u_{d}+B_{i }\|\Delta u_{i}\|-\alpha_{i}(k)]\\ \qquad\qquad-L_{2}(\|\Delta q_{i+1}\|-\alpha_{i+1}(k+1))-L_{3}(\|\Delta q_{i-1} \|-\alpha_{i-1}(k-1))\\ \qquad\qquad=\|\Delta u_{i}\|-L_{1}\|\Delta q_{i}\|-L_{1}(B_{d}-B_{i})u_{d}-L_{ 1}B_{i}\|\Delta u_{i}\|+L_{1}\alpha_{i}(k)\\ \qquad\qquad-L_{2}\|\Delta q_{i+1}\|+L_{2}\alpha_{i+1}(k+1)-L_{3}\|\Delta q_{i- 1}\|+L_{3}\alpha_{i-1}(k-1)\end{array}  \tag{30}\]Considering physical constraints, Formula (30) can be simplified to the following:  
\[ \begin{array}{c}\|\Delta u_{i+1}(k)\|\leq\|I-L_{1}B_{i}\|\|\Delta u_{i}\|+b_{ L1}\|\Delta q_{i}\|+b_{L1}c_{B}b_{u}\|\Delta q_{i}\|+b_{L1}b_{u}\\ \qquad\qquad+b_{L2}\|\Delta q_{i+1}\|+b_{L2}b_{u}+b_{L3}\|\Delta q_{i-1}\|+b_{ L3}b_{u}\\ \qquad\qquad=\|I-L_{1}B_{i}\|\|\Delta u_{i}\|+(b_{L1}+b_{L1}c_{B}b_{u})\|\Delta q _{i}\|+b_{L2}\|\Delta q_{i+1}\|\\ \qquad\qquad+b_{L3}\|\Delta q_{i-1}\|+(b_{L1}b_{a}+b_{L2}b_{\gamma}+b_{L3}b_{ a})\end{array}