# Modeling of Walking-Gait Parameters and Walking Strategy for Quadruped Robots
## 3 Results and Discussion
### Kinematic Analysis of the Whole Robot Mechanism
 \tag{11}\]  
The expression of the foot-end coordinate system \(\{O_{5}\}\) in the body coordinate system \(\{C\}\) is shown in Equation (12):  
\[{}^{c}T_{5}={}^{c}T_{0}{}^{1}T_{1}{}^{1}T_{2}{}^{2}T_{3}{}^{3}T_{4}{}^{4}T_{5} \tag{12}\]  
Bring Equations (6)-(11) into Equation (12), and Equation (13) can be obtained:  
\[{}^{c}T_{5}= \begin{bmatrix}c_{234}s_{\theta}-s_{234}c_{\theta}&-c_{234}s_{ \theta}-s_{234}c_{\theta}&0&l_{0}/2+l_{2}s_{2}+l_{3}s_{23}+l_{4}s_{234}\\ -c_{1}c_{234}c_{\theta}&c_{1}s_{234}c_{\theta}-c_{1}c_{234}s_{\theta}&-s_{1}& b_{0}/2+l_{1}c_{1}+l_{2}c_{1}+l_{2}c_{2}+l_{3}c_{1}c_{23}+l_{4}c_{1}c_{234}\\ s_{1}c_{234}c_{\theta}+s_{1}s_{234}s_{\theta}&s_{1}c_{234}s_{\theta}&-s_{1}s_{2 34}c_{\theta}&-c_{1}&-l_{1}s_{1}-l_{2}s_{1}c_{2}-l_{3}s_{1}c_{23}-l_{4}s_{1}c_{ 234}-h\\ 0&0&1&1\end{bmatrix}  \tag{13}\]  
\[\text{According to }^{c}T_{5}= \begin{bmatrix}n_{x}&o_{x}&a_{x}&P_{x}\\ n_{y}&o_{y}&a_{y}&P_{y}\\ n_{z}&o_{z}&a_{z}&P_{z}\\ 0&0&0&1\end{bmatrix} = \begin{bmatrix}&P_{x}\\ R&P_{y}\\ &P_{z}\\ 0&0&1\end{bmatrix} ,\text{Equations (\ref{eq:14}) and (\ref{eq:15})}\]  
can be obtained:  
\[R= \begin{bmatrix}c_{234}s_{\theta}-s_{234}c_{\theta}&-c_{234}c_{\theta}-s_{2 34}s_{\theta}&0\\ -c_{1}c_{234}c_{\theta}&c_{1}s_{234}c_{\theta}-c_{1}c_{234}s_{\theta}&-s_{1} \\ s_{1}c_{234}c_{\theta}+s_{1}s_{234}s_{\theta}&s_{1}c_{234}s_{\theta}-s_{1}s_{ 234}c_{\theta}&-c_{1}\end{bmatrix}  \tag{14}\]  
\[P= \begin{bmatrix}P_{x}\\ P_{y}\\ P_{z}\end{bmatrix} = \begin{bmatrix}l_{0}/2+l_{2}s_{2}+l_{3}s_{23}+l_{4}s_{234} \\ b_{0}/2+l_{1}c_{1}+l_{2}c_{1}c_{2}+l_{3}c_{1}c_{23}+l_{4}c_{1}c_{234}\\ -l_{1}s_{1}-l_{2}s_{1}c_{2}-l_{3}s_{1}c_{23}-l_{4}s_{1}c_{234}-h\end{bmatrix}  \tag{15}\]  
Equation (14) differentiates time to obtain Equation (16):  
\[\begin{array}{l}\dot{P}_{x}=l_{2}c_{2}\dot{\theta}_{2}+l_{3}c_{23}(\dot{ \theta}_{2}+\dot{\theta}_{3})+l_{4}c_{234}(\dot{\theta}_{2}+\dot{\theta}_{3}+ \dot{\theta}_{4})\\ \dot{P}_{y}=-l_{1}c_{1}\dot{\theta}_{1}+l_{2 Organize Equation (16) into the matrix (17):  
\[V=J(\theta)\dot{\theta} \tag{17}\]  
Among them, \(V=\left[\dot{P}_{x}\dot{P}_{y}\dot{P}_{z}\right]^{T}\) represents the foot-end speed of the bionic dog. \(\dot{\theta}=\left[\dot{\theta}_{1}\dot{\theta}_{2}\dot{\theta}_{3}\dot{\theta }_{4}\right]^{T}\) represents the angular velocity of each joint. \(J(\theta)\) is the Jacobian matrix. It is shown in the form (18):  
\[ \begin{bmatrix}0&l_{2}c_{2}+l_{3}c_{23}+l_{4}c_{234}&l_{3}c_{23}+l_{4}c_{234}&l _{4}c_{234}\\ -l_{1}c_{1}-l_{2}c_{1}c_{2}-l_{3}c_{1}c_{23}-l_{4}c_{1}c_{234}&l_{2}s_{1}s_{2}+ l_{3}s_{1}s_{23}+l_{4}s_{1}s_{234}&l_{3}s_{1}s_{23}+l_{4}s_{1}s_{234}&l_{4}s_{1}s_{234}\\ l_{1}s_{1}+l_{2}s_{1}c_{2}+l_{3}s_{1}c_{23}+l_{4}s_{1}c_{234}&l_{2}c_{1}s_{2}+ l_{3}c_{1}s_{234}&l_{3}c_{1}s_{23}+l_{4}c_{1}s_{234}&l_{4}c_{1}s_{234} \end{bmatrix}  \tag{18}\]  
Using the Jacobian matrix, the relationship between the speed of the foot end in the body coordinate system and the speed of each limb joint can be expressed. Similarly, the kinematic models of the right-hind limb (\(RH\)), left-front limb (\(LF\)), and left-hind limb (\(LH\)) can be obtained similar to Equation (19):  
\[ \begin{array}{l}P_{RH}=\begin{bmatrix}-l_{0}/2+l_{2}s_{2}+l_{3}s_{23}+l_{4} s_{234}\\ -l_{0}/2+l_{1}c_{1}+l_{2}c_{1}c_{2}+l_{3}c_{1}c_{23}+l_{4}c_{1}c_{234}\\ -l_{1}s_{1}-l_{2}s_{1}c_{2}-l_{3}s_{1}c_{23}-l_{4}s_{1}c_{234}-h\end{bmatrix} \\ P_{LF}=\begin{bmatrix}l_{0}/2+l_{2}s_{2}+l_{3}s_{23}+l_{4}s_{234}\\ -l_{1}s_{1}+l_{2}c_{1}+l_{3}c_{1}c_{23}+l_{4}c_{1}c_{234}\\ -l_{1}s_{1}-l_{2}s_{1}-l_{2}s_{1}s_{23}-l_{4}s_{1}c_{234}-h\end{bmatrix}\\ P_{LH}=\begin{bmatrix}-l_{0}/2+l_{2}s_{2}+l_{3}s_{23}+l_{4}s_{234}\\ -l_{1}s_{1}-l_{2}s_{1}c_{2}-l_{3}s_{1}c_{23}-l_{4}s_{1}c_{234}-h\end{bmatrix} \end{array}