# Design and Control of an Omnidirectional Mobile Wall-Climbing Robot
## 3 Dynamic Model Analysis
### The Critical Slip State  
To facilitate the description, we propose a critical slip state for the wall-climbing robot in this section. The state is defined as when \"the friction between one or more wheels and the wall reaches the maximum static friction, none wheel slips with the wall\". In other words, when the robot reaches the critical slip state, at least one wheel has reached the maximum static friction. Therefore, we have  
Figure 4: Schematic diagram of the forces on the wall of the robot system. \[ \begin{cases}|F_{w1}|=f_{1}(\theta)\\ |F_{w2}|\leq f_{2}(\theta)\\ |F_{w3}|\leq f_{3}(\theta)\end{cases} \quad \begin{cases}|F_{w1}|\leq f_{1}(\theta) \\ |F_{w2}|=f_{2}(\theta)\\ |F_{w3}|\leq f_{3}(\theta)\end{cases} \quad \begin{cases}|F_{w1}|\leq f_{1}( \theta)\\ |F_{w2}|\leq f_{2}(\theta)\\ |F_{w3}|=f_{3}(\theta)\end{cases} \]  
The omnidirectional platform fulfills holonomic constraints, i.e., the translation and rotation will be decoupled for motion analysis. In order to analyze the translational acceleration, assuming \(\theta\) is constant, let the angle between the direction of the combined force and the \(X\)-axis of the world coordinate system be \(\gamma\) and the magnitude of the combined driving force provided by the three omnidirectional wheels in the world coordinate system is \(F_{\text{s}}\). Then, the dynamic analysis of the robot system can be expressed as follows:  
\[ \begin{bmatrix}F_{w1}\\ F_{w2}\\ F_{w3}\end{bmatrix} =F_{\text{s}} \begin{bmatrix}-\dfrac{2}{3}\sin(\theta+\pi)& \dfrac{2}{3}\cos(\theta+\pi)\\ -\dfrac{2}{3}\sin(\theta-\dfrac{\pi}{3})&\dfrac{2}{3}\sin(\theta-\dfrac{\pi}{3 })\\ -\dfrac{2}{3}\sin(\theta+\dfrac{\pi}{3})&\dfrac{2}{3}\sin(\theta-\dfrac{\pi}{3 })\end{bmatrix}\begin{bmatrix}\cos\gamma\\ \sin\gamma\end{bmatrix}  \tag{7}\]  
According to (6), we have  
\[ \begin{bmatrix}f_{1}\\ f_{2}\\ f_{3}\end{bmatrix} =-\dfrac{2\mu_{m}M_{\text{G}}}{3r} \begin{bmatrix}\sin(\theta+ \pi)\\ \sin(\theta-\dfrac{\pi}{3})\\ \sin(\theta+\dfrac{\pi}{3})\end{bmatrix} + \begin{bmatrix}\dfrac{\mu_{m}F_{A}} {3}\\ \dfrac{\mu_{m}F_{A}}{3}\\ \dfrac{\mu_{m}F_{A}}{3}\end{bmatrix}  \tag{8}\]  
where \(f_{1}\), \(f_{2}\), and \(f_{3}\) are the maximum static friction of \(W_{1}\), \(W_{2}\), and \(W_{3}\), respectively. \(\mu_{m}\) is the maximum static friction coefficient between the wheels and the wall surface. As mentioned above, at least one of the driving forces \(F_{w1}\), \(F_{w2}\), and \(F_{w3}\) reach the maximum static friction when the critical slip state is reached. Equations (7) and (8) yield  
\[ \begin{cases}\dfrac{F_{w2}}{F_{w1}}=\dfrac{-2/3\sin(\theta-\pi/3)\cos\gamma+2 /3\cos(\theta-\pi/3)\sin\gamma}{-2/3\sin(\theta+\pi)\cos\gamma+2/3\cos(\theta+ \pi)\sin\gamma}\\ \dfrac{F_{w3}}{F_{w2}}=\dfrac{-2/3\sin(\theta+\pi/3)\cos\gamma+2/3\cos(\theta +\pi/3)\sin\gamma}{-2/3\sin(\theta-\pi/3)\cos\gamma+2/3\cos(\theta-\pi/3)\sin \gamma}\\ \dfrac{F_{w1}}{F_{w3}}=\dfrac{-2/3\sin(\theta+\pi)\cos\gamma+2/3\cos(\theta+ \pi)\sin\gamma}{-2/3\sin(\theta+\pi/3)\cos\gamma+2/3\cos(\theta+\pi/3)\sin \gamma}\\ f_{1}=-\dfrac{2\mu_{m}M_{\text{G}}}{3r}\sin(\theta+\pi)+\dfrac{\mu_{m}F_{A}} {3}\\ f_{2}=-\dfrac{2\mu_{m}M_{\text{G}}}{3r}\sin(\theta-\dfrac{\pi}{3})+\dfrac{\mu_{ m}F_{A}}{3}\\ f_{3}=-\dfrac{2\mu_{m}M_{\text{G}}}{3r}\sin(\theta+\dfrac{\pi}{3})+\dfrac{\mu_{m}F_{A}} {3}\\ \end{cases}  \tag{9}\]  
If \(W_{1}\) reaches the maximum static friction first, then we have  
\[ \begin{cases}F_{w1}=sgn(F_{w1})f_{1}\\ F_{w2}=sgn(F_{w1})\dfrac{F_{w2}}{F_{w1}}f_{1}\\ F_{w3}=sgn(F_{w1})\dfrac{F_{w3}}{F_{w1}}f_{1}\\ F_{\text{s}}=\dfrac{sgn(F_{w1})f_{1}}{-2/3\sin(\theta+\pi)\cos\gamma+2/3\cos( \theta+\pi)\sin\gamma}\end{cases}  \tag{10}\]The same is true when \(W_{2}\) or \(W_{3}\) reaches the maximum static friction force first. Combining (9) and (10), \(F_{s}\), \(a_{x}\), and \(a_{y}\) can be expressed as below. \[ \begin{cases}a_{x}=\dfrac{F_{s}\cos\gamma}{m}\\ a_{y}=\dfrac{F_{s}\sin\gamma}{m}-g\end{cases}