# Structural Design and Analysis of Multi-Directional Foot Mobile Robot
## 2 Design and Analysis of Closed-Chain Leg Structure
### Kinematics Analysis of Closed-Chain Single-Leg Mechanism
The \(X\) direction coordinate of the leg end is  
\[X=r_{1}\cos\theta_{1}+r_{4}\cos\theta_{4}+r_{7}\cos\theta_{7}+r_{9}\cos\theta _{9}. \tag{27}\]  
The \(Y\) direction coordinate of the leg end is  
\[Y=r_{1}\sin\theta_{1}+r_{4}\sin\theta_{4}+r_{7}\sin\theta_{7}+r_{9}\sin\theta _{9}. \tag{28}\]  
Deriving both ends of Equations (6) and (7) according to time, the angular velocity \(\omega\) can be obtained as  
\[-\omega_{3}r_{3}\sin\theta_{3}+\omega_{2}r_{2}\sin\theta_{2}-\omega_{1}r_{1} \sin\theta_{1}=0, \tag{29}\]  
and  
\[\omega_{3}r_{3}\cos\theta_{3}\omega_{2}r_{2}\cos\theta_{2}+\omega_{1}r_{1} \cos\theta_{1}=0. \tag{30}\]  
Combined with the above formulas, we can obtain  
\[\omega_{2}=\dfrac{\omega_{1}r_{1}\sin(\theta_{3}-\theta_{1})}{r_{2}\sin( \theta_{2}-\theta_{3})}, \tag{31}\]  
and  
\[\omega_{3}=\dfrac{\omega_{1}r_{1}\sin(\theta_{2}-\theta_{1})}{r_{2}\sin( \theta_{2}-\theta_{3})}. \tag{32}\]  
We solve both sides of Equations (17) and (18) to obtain  
\[\omega_{7}=\dfrac{r_{2}\omega_{2}\sin(\theta_{2}-\theta_{8})+r_{4}\omega_{4} \sin(\theta_{8}-\theta_{4})}{r_{2}\sin(\theta_{7}-\theta_{8})}+\dfrac{r_{6} \omega_{6}\sin(\theta_{6}-\theta_{8})}{r_{8}\sin(\theta_{7}-\theta_{8})}, \tag{33}\]  
and  
\[\omega_{8}=\dfrac{r_{2}\omega_{2}\sin(\theta_{2}-\theta_{7})+r_{4}\omega_{4} \sin(\theta_{7}-\theta_{4})}{r_{8}\sin(\theta_{7}-\theta_{8})}+\dfrac{r_{6} \omega_{6}\sin(\theta_{6}-\theta_{7})}{r_{8}\sin(\theta_{7}-\theta_{8})}, \tag{34}\]  
where \(\omega_{2}=\omega_{4},\omega_{3}=\omega_{6}\), \(\omega_{1}\) is the angular velocity of the drive motor. The second-order derivative with respect to time in Equations (6), (7), (17) and (18) can be used to calculate the accelerations of four unknown angles, which are recorded as \(\alpha_{2},\alpha_{3},\alpha_{7}\), and \(\alpha_{8}\), respectively. The unknown angular acceleration can be calculated as follows:\[ \begin{cases}\alpha_{2}=\dfrac{E\cos\theta_{3}+F\sin\theta_{3}}{r_{2}\sin(\theta_{2}- \theta_{3})}\\ \alpha_{3}=\dfrac{E\cos\theta_{2}+F\sin\theta_{2}}{r_{3}\sin(\theta_{2}- \theta_{3})}\\ \alpha_{7}=\dfrac{E\cos\theta_{8}+F\sin\theta_{8}}{r_{7}\sin(\theta_{7}- \theta_{8})}\\ \alpha_{8}=\dfrac{E\cos\theta_{7}+F\sin\theta_{7}}{r_{8}\sin(\theta_{7}- \theta_{8})},\end{cases}  \tag{35}\]  
where  
\[ \begin{cases}E=r_{0}\omega_{0}^{2}\cos\theta_{0}+r_{3}\omega_{3}^{2}\cos \theta_{3}-r_{2}\omega_{2}^{2}\cos\theta_{2}-r_{1}\omega_{1}^{2}\cos\theta_{ 1}\\ F=r_{0}\omega_{0}^{2}\sin\theta_{0}+r_{3}\omega_{3}^{2}\sin\theta_{3}-r_{2} \omega_{2}^{2}\sin\theta_{2}-r_{1}\omega_{1}^{2}\sin\theta_{1}\\ G=r_{2}\omega_{2}^{2}\cos\theta_{2}-r_{4}\omega_{4}^{2}\cos\theta_{4}+r_{6} \omega_{6}^{2}\cos\theta_{6}+r_{8}\omega_{8}^{2}\cos\theta_{8}-r_{7}\omega_{ 7}^{2}\cos\theta_{7}+\\ r_{2}\alpha_{2}\sin\theta_{2}-r_{4}\alpha_{4}\sin\theta_{4}+r_{6}\alpha_{6}\sin \theta_{6}\\ H=r_{2}\omega_{2}^{2}\sin\theta_{2}-r_{4}\omega_{4}^{2}\sin\theta_{4}+r_{6} \omega_{6}^{2}\sin\theta_{6}+r_{8}\omega_{8}^{2}\sin\theta_{8}-r_{7}\omega_{ 7}^{2}\sin\theta_{7}+\\ r_{2}\alpha_{2}\sin\theta_{2}-r_{4}\alpha_{4}\sin\theta_{4}+r_{6}\alpha_{6} \sin\theta_{6}.\end{cases}  \tag{36}\]  
Thus far, all unknown angular displacements \((\theta_{2},\theta_{3},\theta_{4},\theta_{6},\theta_{7},\theta_{8},\theta_{9})\), angular velocities \((\omega_{2},\omega_{3},\omega_{4},\omega_{6},\omega_{7},\omega_{8},\omega_{9})\), and angular accelerations \((\alpha_{2},\alpha_{3},\alpha_{4},\alpha_{6},\alpha_{7},\alpha_{8},\alpha_{9})\) involved in the two loops have been calculated. Through analysis and programming with the MATLAB (2017b) software, the leg-end motion trajectory of the single-leg closed chain mechanism is obtained.