# Design and Control of an Omnidirectional Mobile Wall-Climbing Robot
## 5 Simulation Experiments
### Critical Slip State Experiments  
In the critical slip state experiment, three states of non-slip, critical slip, and slip were analyzed. The control signal is a step signal fed directly to the wheel. In the critical slip state experiments, we have no constraints on the performance of the motor in the simulation environment. For \(\theta=90^{\circ}\) and \(\alpha=0^{\circ}\), the theoretical acceleration values were compared with the simulated experimental values in the three states. Figure 8 shows the measured acceleration and theoretical acceleration curves of the robot in the simulation experiment when the driving force of \(W_{1}\), \(W_{2}\), and \(W_{3}\) are 23.76 N, 2.91 N, and \(-26.67\) N, respectively. The robot is in a non-slip state in this case. It can be seen that the theoretical acceleration curve in the figure is consistent with the acceleration measured in the simulation experiment. The pulse at 15 ms in the graph is generated by the shock caused by the system starting from a standstill. Figure 9 shows the measured acceleration and theoretical acceleration curves of the robot in the simulation experiment when the driving force of \(W_{1}\), \(W_{2}\), and \(W_{3}\) are 28.02 N, 0.80 N, and \(-28.82\) N, respectively. In this case, the robot is in the critical slip state. It can be seen that the acceleration theoretical value curve in the figure is consistent with the acceleration measured in the simulation experiment. The pulse at 17 ms in the figure is generated by the shock caused by the system starting from rest. The oscillation after  
Figure 8: Acceleration–time curve of the non-slip state. 500 ms is caused by an unstable form of friction. The wheel speed is already high at this time, and the omnidirectional wheel rollers make contact with the wall one after another, resulting in the unstable form of friction. Figure 10 shows the measured acceleration and theoretical acceleration curves of the robot in the simulation experiment when the driving forces of \(W_{1}\), \(W_{2}\), and \(W_{3}\) are 31.68 N, \(-1.05\) N, and \(-30.63\) N, respectively. The robot is slipping in this case. It can be seen that the acceleration theoretical value curve in the figure differs significantly from the acceleration measured in the simulation experiment. The difference is due to the oscillation of acceleration as the omnidirectional wheel rotates, generating sliding friction and slipping. The rollers of the omnidirectional wheel are in alternating contact with the wall, which causes the oscillation. The oscillation here is more evident than the non-slip state and the critical slip state, which is not suitable for the stable control of the robot, so the undesired slip of the omnidirectional wheel with the wall should be avoided. Figure 10: Acceleration–time curve of the slip state. Figure 9: Acceleration–time curve of the critical slip state. The control frequency of the position loop is 20 Hz, and the control frequency of the velocity loop is 200 Hz. The parameters of the prediction and control horizons are set for control frequency and model accuracy reasons. According to (22), the computation of the difference model increases significantly when the prediction and control horizons increase. Moreover, to simplify the calculation, \(\theta\) is replaced by \(\theta_{r}\) in the model, so the model mismatch is also more severe when the prediction and control horizons increase. It has been tested that when the prediction and control horizons are 1, the controller can reach the maximum control frequency with no decrease in accuracy. The tracking accuracy penalty matrix of the position loop can be expressed as follows:  
\[\textbf{Q}_{p}= \begin{bmatrix}100,000&0&0\\ 0&100,000&0\\ 0&0&10,000\end{bmatrix} \]  
The control smoothness penalty matrix of the position loop is  
\[\textbf{R}_{p}= \begin{bmatrix}0.01&0&0\\ 0&0.01&0\\ 0&0&0.001\end{bmatrix} \]  
The relaxation factor coefficient of the position loop is  
\[\rho_{p}=0\]  
Figure 12: The result of PID controller in the simulation experiment.