# Design and Control of an Omnidirectional Mobile Wall-Climbing Robot
## 3 Dynamic Model Analysis
### The Critical Slip State
 \tag{11}\]  
The translational acceleration \(a\) and corresponding orientation \(\alpha\) where no slip occurs are determined by \(a_{x}\) and \(a_{y}\). The relationship can be expressed as follows. \[ \begin{cases}a_{x}=a\cos\alpha\\ a_{y}=a\sin\alpha\end{cases}  \tag{12}\]  
In the practical case, the physical model gives all the known physical parameters appearing in (7)-(12). Moreover, the relationship between \(\theta\), \(\alpha\), and \(a_{max}\) under a non-slip state can be found. Assuming \(m=2.615\) kg, \(g=9.8\) m\(\cdot\)s\({}^{-2}\), \(M_{G}=1.314\) N\(\cdot\)m, \(r=0.2221\) m, \(\mu_{m}=0.6\), and \(F_{A}=150\) N, the relationship between \(\theta\), \(\alpha\), and \(a_{max}\) is shown in Figure 5.  
In Figure 5, the surface corresponds to the critical slip state, the space enclosed by the surface and the \(\theta-\alpha\) plane is the achievable non-slip motion. In order to prevent the robot from slipping, without considering the influence of \(\theta\) and \(\alpha\) on the maximum acceleration, the minimum value of \(a\) on the surface in Figure 5 is usually taken as the constraint for the maximum value of the acceleration. Here, we have \(a_{min}=7.31\) m\(\cdot\)s\({}^{-2}\). Under the specific orientation and acceleration direction angle, the maximum acceleration of the robot is constrained to the maximum value of \(a_{max}=30.45\) m\(\cdot\)s\({}^{-2}\) on this surface. It can be seen that the maximum acceleration can be increased by 316.55% by adjusting the maximum acceleration constraint in real time when considering the effect of the critical slip state. The average acceleration of the points on the surface in Figure 5 is \(a_{agg}=19.35\) m\(\cdot\)s\({}^{-2}\). It shows that the maximum acceleration is increased by 164.71% on average when considering the influence of the critical slip state. Figure 5: The relation among the acceleration, orientation, and the direction of the acceleration. Similarly, for the rotational acceleration, we have  
\[ \begin{bmatrix}F_{w1}\\ F_{w2}\\ F_{w3}\end{bmatrix} = \begin{bmatrix}-\dfrac{2}{3}\sin(\theta+\pi)&\dfrac{2}{3}\cos( \theta+\pi)&\dfrac{1}{3r}\\ -\dfrac{2}{3}\sin(\theta-\dfrac{\pi}{3})&\dfrac{2}{3}\sin(\theta-\dfrac{\pi}{3} )&\dfrac{1}{3r}\\ -\dfrac{2}{3}\sin(\theta+\dfrac{\pi}{3})&\dfrac{2}{3}\sin(\theta-\dfrac{\pi}{3} )&\dfrac{1}{3r}\\ \end{bmatrix}\begin{bmatrix}0\\ mg\\ L_{\bar{z}}\bar{\theta}\end{bmatrix}  \tag{13}\]  
According to (13), when \(\theta\) is fixed, the difference in the driving force of each wheel is fixed. Therefore, we have  
\[ \begin{cases}F_{w1}-F_{w2}=-\dfrac{2\sqrt{3}mg}{3}\sin(\theta+\dfrac{\pi}{3}) \\ F_{w2}-F_{w3}=-\dfrac{2\sqrt{3}mg}{3}\sin(\theta+\pi)\\ F_{w3}-F_{w1}=-\dfrac{2\sqrt{3}mg}{3}\sin(\theta-\dfrac{\pi}{3})\\ f_{1}=-\dfrac{2\mu_{m}M_{G}}{3r}\sin(\theta+\pi)+\dfrac{\mu_{m}F_{A}}{3}\\ f_{2}=-\dfrac{2\mu_{m}M_{G}}{3r}\sin(\theta-\dfrac{\pi}{3})+\dfrac{\mu_{m}F_{A}}{3} \\ f_{3}=-\dfrac{2\mu_{m}M_{G}}{3r}\sin(\theta+\dfrac{\pi}{3})+\dfrac{\mu_{m}F_{A} }{3}\\ \end{cases}  \tag{14}\]  
If \(W_{1}\) reaches the maximum static friction first, we have  
\[ \begin{cases}F_{w1}=f_{1}\\ F_{w2}=f_{1}-(F_{w1}-F_{w2})\\ F_{w3}=(F_{w3}-F_{w1})+f_{1}\\ \beta=\dfrac{3r(f_{1}-\dfrac{2}{3}mg\cos(\theta+\pi))}{l_{z}}\end{cases}  \tag{15}\]  
The same is true when \(W_{2}\) or \(W_{3}\) reaches the maximum static friction force first. The curve can be obtained as in Figure 6, which is the critical slip state parameter, and the region enclosed by this curve and \(\theta\)-axis is the motion state that can be achieved in the non-slip state. When the influence of the critical slip state is not considered, the minimum value of \(\beta\) on the graph curve is usually taken as the constraint on the maximum value of the angular acceleration of the robot to prevent the robot from slipping. Here, we have \(\beta_{min}=276.94\) rad\(\cdot\)s\({}^{-2}\).