## 4 Experimental Results
### Basic Performance Test
#### 4.1.2 Performance Test of Hybrid Controller with Initial Error  
The above tasks were simulated without the initial error; however, the initial position of the mobile platform maybe not completely coincide with the starting of the given trajectory during actual movement. To test the novel-type controller in the initial error state, we effectively compared the previous control law and controller against the circular trajectory's initial error. We assumed the initial position and posture of the mobile platform are expressed as \((9,0,\pi/2)\), which does not coincide with the initial position of the trajectory in this task. The specific controllers were as follows:  
**Situation 2:** There is an uncompensated control law with a speed constraint, which can be expressed explicitly as follows:  
\[L_{1}^{\alpha 2}(k)=0.5\cdot L_{1}(k);\quad L_{2}^{\alpha 2}(k)=0.25\cdot L_{1}(k); \quad L_{3}^{\alpha 2}(k)=0.25\cdot L_{1}(k); \tag{46}\]  
**Situation 3:** The iterative learning controller (21) directly controls the rotation speed of the two wheels to solve the trajectory tracking problem, which can be expressed as follows:  
\[ \begin{array}{l}u_{i+1}(k)=u_{i}(k)+L_{1}(k)\epsilon_{i}(k+1)+e^{-k\cdot d _{i}}\theta(k)L_{1}(k)\epsilon_{i}(0)\\ \theta(k)=\left\{\begin{array}{cc}\frac{2}{k+\alpha}(1-\frac{k}{k}),&(0\leq k \leq h)\\ 0,&(h<k\leq n)\end{array}\right.\end{array}  \tag{47}\]  
Figure 3: Typical trajectory tracking results of the mobile platform using the P\(-\)type iterative learning controller. (**a1**) Path change in single circular trajectory. (**a2**) Final path in single circular trajectory. (**a3**) Maximum tracking error change in single circular trajectory. (**b1**) Path change in mixed arc and line closed trajectory. (**b2**) Final path in mixed arc and line closed trajectory. (**b3**) Maximum tracking error change in mixed arc and line closed trajectory. (**c1**) Path change in cardioid\(-\)like trajectory. (**c2**) Final path in cardioid\(-\)like trajectory. (**c3**) Maximum tracking error change in cardioid\(-\)like trajectory. (**d1**) Path change in spiral\(-\)like trajectory. (**d2**) Final path in spiral\(-\)like trajectory. (**d3**) Maximum tracking error change in spiral\(-\)like trajectory. where \(L_{1}^{s2}(k)\) is the gain of the controller, which can be expressed as \(L_{1}^{s2}(k)=0.5\cdot L_{1}(k)\); the coefficient of \(h\) was chosen as 200; and the corresponding trajectory tracking effect was obtained, as shown in Figure 4.  
To illustrate the adaptability of the novel-type controller, we also tested the novel-type controller on other trajectories with initial errors. For example, the initial position and posture of the robot in the combination trajectory of arc and straight task is defined as \((4,0,\pi/2)\), the initial position and posture of the robot in the cardioid-like trajectory task is defined as \((2,0,14\pi/15)\), and the initial position and posture of the robot in the spiral-like trajectory task is defined as \((2.5,0,31\pi/30)\). Using the novel-type controller with a speed constraint, the following experimental results were obtained, as shown in Figure 5.  
Comparing the above experimental results, we found the following:  
1. Without the initial error compensation in Figure 4a1-a3, the control law (17) can achieve closed-loop motion, but the actual trajectory has a relatively fixed deviation from the given trajectory. Therefore, the previous controller cannot control the tracking of the various trajectories with the initial error. 2. With the initial error compensation in Figure 4a2,a3,b2,b3, the control law (21) can effectively achieve the circular trajectory tracking with initial error. Additionally, the controller with a velocity constraint must meet the actual physical limitations of the mobile platform. 3. According to various trajectory tracking results with initial error in Figure 5, the error compensation controller designed in this paper has good adaptability to various trajectories.