## 2 Modeling of Wall-Climbing Robot
### Kinetic Analysis  
This robot had four legs for climbing on the wall. The legs were driven by the gear transmission. The gear transmission affected the stability of crawling system. Hence, the kinetic analysis of the gear transmission was so important that the relevant dynamic model was proposed. Before the model building, the following hypothesis was proposed:  
1. The internal and external mesh pairs were represented by virtual spring-damping units, time-varying stiffness, and comprehensive transmission error. 2. The frictions of each meshing pair and supporting bearings could be neglected. 3. The backlashes of gear pairs were included. Figure 2e presents the dynamic mechanics of the gear transmission system in a foot of this robot. The \(\theta_{i}\) (\(l=1\), \(2\), \(3\)) is the displacement of the gears, \(l_{i}\) (\(i=1\), \(2\), \(3\)) represents the moment of inertia of the gear, and \(e_{j}\) (\(t\)), (\(j=1\), \(2\)) is the comprehensive transmission error during the process of gear driving. The comprehensive transmission error \(e_{j}\) (\(t\)) is periodic and varies with the mesh frequency \(\omega_{j}\), which can be written as  
\[e_{j}(t)=E_{j}\cos(\omega_{j}t+\varphi_{j}) \tag{11}\]where \(E_{j}\) is the comprehensive gear meshing error amplitude and \(\varphi_{j}\) is the phase angle of the \(j_{th}\) gear pair. The \(k_{j}(t)\), \((j=1,2)\) is the time-varying mesh stiffness and can be described as  
\[k_{j}(t)=k_{mj}+k_{s}\cos(\omega_{j}t+\varphi_{kj}) \tag{12}\]  
where \(k_{mj}\) is the average amplitude, \(k_{s}\) is the varying amplitude of the \(j_{th}\) time-varying mesh stiffness of gear pair, \(\varphi_{kj}\) is the phase angle of this gear pair, \(c_{j}\), \((j=1,2)\) is the damping coefficient, \(T_{1}\) is the driving moment, and \(T_{2}\) and \(T_{3}\) are the driven moments on the gear. The dynamic mechanics' function of this gear system can be written as  
\[\left\{ \begin{array}{l}I_{2}\ddot{\theta}_{2}+c_{1}R_{2}(R_{2}\dot{\theta}_ {2}-R_{1}\dot{\theta}_{1}-\dot{e}_{1}(t))+k_{1}(t)R_{2}f(R_{2}\theta_{2}-R_{1} \theta_{1}-e_{1}(t))=T_{2}\\ I_{3}\ddot{\theta}_{3}+c_{2}R_{3}(R_{3}\dot{\theta}_{3}-R_{1}\dot{\theta}_{1}- \dot{e}_{2}(t))+k_{2}(t)R_{3}f(R_{3}\theta_{3}-R_{1}\theta_{1}-e_{2}(t))=T_{3} \\ I_{1}\ddot{\theta}_{1}-c_{1}R_{1}(R_{2}\dot{\theta}_{2}-R_{1}\dot{\theta}_{1}- \dot{e}_{1}(t))-k_{1}(t)R_{1}f(R_{2}\theta_{2}-R_{1}\theta_{1}-e_{1}(t))\\ -c_{2}R_{1}(R_{3}\dot{\theta}_{3}-R_{1}\dot{\theta}_{1}-\dot{e}_{2}(t))-k_{2}( t)R_{1}f(R_{3}\theta_{3}-R_{1}\theta_{1}-e_{2}(t))=-T_{1}\end{array} \right. \tag{13}\]  
Here, \(R_{i}\) (\(i=1,2,3\)) represents the radius of the gears. This model only included the torsional direction displacement. This gear transmission system included the gear backlash, represented by Figure 2e. The displacement function \(f(X_{i})\) that included the gear backlash can be expressed as  
\[f(X_{j})=\left\{ \begin{array}{cc}X_{j}-b&X_{j}>\text{b}\\ 0&\left|X_{j}\right|\leq\text{b}\\ X_{j}+b&X_{j}<-\text{b}\end{array} \right. \tag{14}\]  
In order to eliminate the rigid body movement, the relative displacements of the gear pair were proposed as \(X_{j}\) (\(j=1,2\)). These can be obtained as  
\[\left\{ \begin{array}{l}X_{1}=R_{2}\theta_{2}-R_{1}\theta_{1}-e_{1}(t)\\ X_{2}=R_{3}\theta_{3}-R_{1}\theta_{1}-e_{2}(t)\end{array} \right. \tag{15}\]  
Hence, the function can be written  
\[\left\{ \begin{array}{l}\ddot{X}_{1}+c_{1}R_{2}^{2}/I_{2}\dot{X}_{1}+c_{1}R_{ 1}^{2}/I_{1}\dot{X}_{1}+c_{2}R_{1}^{2}/I_{1}\dot{X}_{2}+k_{1}(t)R_{2}^{2}/I_{ 2}f(X_{1})+k_{1}(t)R_{1}^{2}/I_{1}f(X_{1})\\ +k_{2}(t)R_{1}^{2}/I_{1}f(X_{2})=-\ddot{e}_{1}(t)+R_{2}T_{2}/I_{2}+R_{1}T_{1}/I _{1}\\ \ddot{X}_{2}+c_{2}R_{2}^{2}/I_{3}\dot{X}_{2}+c_{2}R_{1}^{2}/I_{1}\dot{X}_{2}+ c_{1}R_{1}^{2}/I_{1}\dot{X}_{1}+k_{2}(t)R_{2}^{2}/I_{3}f(X_{2})+k_{2}(t)R_{1}^{2}/I_{ 1}f(X_{2})\\ +k_{1}(t)R_{1}^{2}/I_{1}f(X_{1})=-\ddot{e}_{2}(t)+R_{1}T_{1}/I_{1}+R_{3}T_{3}/ I_{3}\end{array} \right. \tag{16}\]  
The dimensionless process was used to solve this function to obtain the vibration placement.