# Design, Analysis, and Experiment of a Wheel-Legged Mobile Robot
## Appendix A  
Projecting Formula (26) onto the \(X\)-axis and the \(Y\)-axis gives:  
\[\left\{ \begin{aligned} & L\cdot cos(\gamma_{2})+L_{3}\cdot cos( \gamma_{1})=L_{1}+L_{2}\cdot cos(\gamma_{4})\\ & L\cdot sin(\gamma_{2})+L_{3}\cdot sin(\gamma_{1})=L_{2}\cdot sin( \gamma_{4})\end{aligned} \right., \tag{10}\]  
where \(\gamma_{i}\) (\(i=1,2,4\)) are the angles between the positive direction of the \(X\)-axis and \(A_{1}A_{2}\), \(A_{1}A_{6}\), and \(A_{4}A_{6}\). According to Formula (10), we have:  
\[\gamma_{1}=\arctan\left[\frac{L_{2}\cdot sin(\gamma_{4})-L\cdot sin(\gamma_{2} )}{L_{1}+L_{2}\cdot cos(\gamma_{4})-L\cdot cos(\gamma_{2})}\right], \tag{11}\]  
\[L_{3}=\sqrt{\left[L_{1}+L_{2}\cdot cos(\gamma_{4})-L\cdot cos(\gamma_{2}) \right]^{2}+\left[L_{2}\cdot sin(\gamma_{4})-L\cdot sin(\gamma_{2})\right]^{2}}, \tag{12}\]  
After geometric analysis, the following formulas are derived:  
\[\gamma_{1}=\varphi_{1}+\varphi_{2}^{1}-\pi,\,\gamma_{2}=\varphi_{2}^{1},\, \gamma_{4}=\pi-\varphi_{4}^{2}, \tag{13}\]  
\[L_{1}\!\!=2L\cdot\!\sin\!\left(\frac{\varphi_{3}}{2}\right)\!,\,L_{2}\!\!=2L \cdot\!\sin\!\left(\frac{\varphi_{5}}{2}\right)\!,\,\varphi_{2}^{2}=\varphi_{4 }^{1}=\frac{\pi-\varphi_{3}}{2},\,\varphi_{4}^{3}=\frac{\pi-\varphi_{5}}{2}, \tag{14}\]  
The solutions of Formulas (11) and (12) are:  
\[\varphi_{1}=\arctan\left[\frac{\cos\left(\frac{\varphi_{3}}{2}+\varphi_{4}+ \varphi_{5}\right)-\cos\left(\frac{\varphi_{3}}{2}+\varphi_{4}\right)+\cos \left(\varphi_{2}+\frac{\varphi_{3}}{2}\right)}{2\sin\left(\frac{\varphi_{3}}{ 2}\right)+\sin\left(\frac{\varphi_{2}}{2}+\varphi_{4}+\varphi_{5}\right)-\sin \left(\frac{\varphi_{3}}{2}+\varphi_{4}\right)-\sin\left(\varphi_{2}+\frac{ \varphi_{3}}{2}\right)}\right]-\varphi_{2}-\frac{\varphi_{3}}{2}+\frac{3\pi}{2}, \tag{15}\]  
\[L_{3}\!\!=5L^{2}\!+\!2L^{2}\!\cos(\varphi_{2}+\varphi_{3}+\varphi_{4}+\varphi_{5 })-2L^{2}[\cos(\varphi_{2}+\varphi_{3}+\varphi_{4})+\cos(\varphi_{3}+\varphi_{ 4}+\varphi_{5})]+ \tag{16}\]  
\[2L^{2}[\cos(\varphi_{2}+\varphi_{3})+\cos(\varphi_{3}+\varphi_{4})+\cos( \varphi_{4}+\varphi_{5})]-2L^{2}[\cos(\varphi_{2})+\cos(\varphi_{3})+\cos( \varphi_{4})+\cos(\varphi_{5})] \tag{17}\]  
The displacement in the plane of point \(A_{6}\) on the body are expressed as:  
\[OA_{6}=OA_{1}+A_{1}A_{2}+A_{2}A_{3}+A_{3}A_{4}+A_{4}A_{5}+A_{5}A_{6}, \tag{18}\]Point \(A_{6}\) on the robot body is expressed by the following formulas after being projected on the coordinate axis:  
\[\left\{ \begin{aligned} & X_{A6}=L\text{cos}(\theta_{1})+L\text{cos}( \theta_{2})+L\text{cos}(\theta_{3})+L\text{cos}(\theta_{4})+L\text{cos}(\theta_ {5})+X_{A1}\\ & Y_{A6}=L\text{sin}(\theta_{1})+L\text{sin}(\theta_{2})+L\text{ sin}(\theta_{3})+L\text{sin}(\theta_{4})+L\text{sin}(\theta_{5})+Y_{A1}\end{aligned} ,\right. \tag{10}\]  
where \(\theta_{i}\) is the angle between rod \(A_{i}\) (\(i\) = 1-5) and the \(X\)-axis. The rotation of the wheel-leg can be divided into three repetitive cycles. When the wheel leg rotates clockwise, the point \(C_{5}\) on the \(X\)-axis and \(Y\)-axis displacement are expressed as:  
\[x_{\text{C5}}=\left\{ \begin{aligned} & ks+R_{\text{max}}\cos\delta_{1}-R_{\text{ max}}\cos\left(\theta_{7}-\frac{2}{3}k\pi+\delta_{1}\right)& kT\leq t\leq kT+t_{1}\\ & ks+s_{1}+R_{\text{max}}\sin(D_{1})-R_{\text{max}}\sin(\delta_{1} -\beta)+r(D_{2})-r\sin(D_{2})& kT+t_{1}\leq t\leq(k+1)T^{\prime} \end{aligned} \right. \tag{11}\]  
\[y_{\text{C5}}=\left\{ \begin{aligned} & R_{\text{max}}\sin \left(\theta_{7}-\frac{2}{3}k\pi+\delta_{1}\right)& kT\leq t\leq kT+t_{1}\\ & R_{\text{max}}\cos(D_{1})+r-r\cos(D_{2})& kT+t_{1}\leq t\leq(k+1)T^{\prime}\end{aligned} \right.