## 3 Design of the Controller  
### Design of the Hybrid Learning Iteration Controller for Mobile Platform  
The hybrid controller has a two-layer hybrid structure that involves an iterative learning controller without initial error and an iterative learning controller with initial error; the basic structure of the hybrid controller is shown in Figure 2. To determine the choice of the corresponding controller, we designed the switch function based on the initial error of the mobile platform in different trajectory tracking tasks. When the initial error of the robot is equal to zero, the hybrid controller first uses the iterative controller without initial error to complete the assigned task. When the initial error exists in the system, it proceeds to the second type of iterative learning controller through the switching function. The specific design steps are as follows:  
Step 1: Design the learning iteration controller without initial error. The basic aim of the iterative learning controller is to correct the actual input signal by iteratively learning the error between the ideal output signal and the actual input signal to perform the specified trajectory tracking task. Due to the different types of error parameters obtained, we can optimize the typical P-type learning control law by increasing different types of error signals. The optimized iterative learning controller can be expressed as follows:  
\[u_{i+1}^{a}(k)=u_{i}^{a}(k)+L_{1}(k)e_{i}(k+1)+L_{2}(k)e_{i+1}(k)+L_{3}(k)e_{i +1}(k-1) \tag{17}\]  
where \(u_{i+1}^{a}(k)\) indicates the control speeds at \(k\) time in the next iteration, \(e_{i}(k+1)\) indicates the error at \((k+1)\) time in the current iteration, \(e_{i+1}(k)\) indicates the error at \(k\) time in the next iteration, \(e_{i+1}(k-1)\) indicates the error at \((k-1)\) time in the next iteration, and \(L_{1}(k),L_{2}(k),L_{3}(k)\) indicate the gain matrix of iterative learning in different error categories, and these gain matrices need to meet  
\[\|L_{1}(k)\|\leq b_{L_{1}},\|L_{2}(k)\|\leq b_{L_{2}},\|L_{3}(k)\|\leq b_{L_{3}} \tag{18}\]  
where \(b_{L_{1}}>0\), \(b_{L_{3}}>0\), \(b_{L_{2}}>0\), \(k\in N\). Theorem 3.1: _For the mobile robot described by Formula (11), in addition to satisfying the given kinematic characteristics and the specified constraint, the robot still satisfies the following conditions:_  
\[\left\{ \begin{array}{c}\|I-[L_{1}(k)+L_{2}(k)+L_{3}(k)|B(q_{i},k)\|\leq\rho <1\\ y_{i}(0)=y_{d}(0),(i=0,1,2,\ldots)\end{array} \right. \tag{19}\]  
Figure 2: Framework for the hybrid iterative learning controller (“+” represents the input, “\(-\)” represents the feedback). By using the iterations learning law (17), the output signal of the mobile system can be converted to the desired trajectory \(y_{d}(k)(0\leq k\leq n)\), namely  
\[\lim\limits_{k\rightarrow\infty}y_{s}(k)\to y_{d}(k)(0\leq k\leq n) \tag{20}\]  
Proof. : The proof of Theorem 1 is shown in Section 3.2 of the article. Step 2: Design the learning iteration controller with initial error. To achieve trajectory tracking, the initial error of the typical iterative control law (17) needs to be equal to zero. In the actual movement of the mobile platform, error may exist between the initial position of the mobile platform and the given trajectory. In this case, the iterative learning controller needs to compensate for the corresponding error in the initial moment to achieve trajectory tracking. Therefore, the iterative learning controller can be expressed as follows:  
\[u_{i+1}^{b}(k)=u_{i}^{b}(k)+L_{1}(k)e_{i}(k+1)+e^{-kdt}\theta(k)L_{1}(k)e_{i}(0) \tag{21}\]  
where \(e_{i}(0)\) represents the initial error of each iteration learning process, and \(\theta(k)\) represents the corresponding compensation function, which can be expressed as follows:  
\[\theta(k)=\left\{ \begin{array}{cc}\frac{2}{d_{k}*h}(1-\frac{k}{h}),&(0\leq k \leq h)\\ 0,&(h<k\leq n)\end{array} \right. \tag{22}\]  
**Theorem 2**.