# Modeling of Walking-Gait Parameters and Walking Strategy for Quadruped Robots
## 3 Results and Discussion
### Kinematic Analysis of the Whole Robot Mechanism  
Figure 6 shows the kinematic parameters of the bionic-robot dog. Due to the symmetry of the four limbs, the right-front limb was taken as an example for analysis. The body coordinate system {C} is established at the initial position of the center of mass of the body, the \(Yc\) axis is the forward direction, and the \(Zc\) axis is the vertical return. The coordinate system {\(O_{0}\)} is shown in the figure, and the shoulder has two joints: forward and backward swing and left and right swing. Therefore, the {\(O_{1}\)} coordinate system is fixed to the shoulder and corresponds to the left- and right-swing coordinate system. The {\(O_{2}\)} coordinate system is the coordinate system corresponding to the forward and backward swing. Since the coordinate system {\(O_{2}\)} of the hip joint swinging back and forth is coincident with the coordinate system {\(O_{1}\)} of the left- and right-swinging of the hip joint, \(l_{1}=0\) in the equation. {\(O_{3}\)} is the coordinate system corresponding to the knee joint. {\(O_{4}\)} is  
the coordinate system corresponding to the ankle joint. \(\{O_{5}\}\) is the coordinate system at the end of the foot. \(\theta_{1}\) is the angle at which the shoulder joint swings left and right. \(\theta_{2}\) is the angle at which the shoulder joint swings back and forward. \(\theta_{3}\) and \(\theta_{4}\) are the rotation angles of the knee and ankle joints, respectively. The angle of \(X_{5}\) entering the ground in the positive direction is \(\theta_{5}\). \(h\) is the thickness of the torso, and \(l_{i}\) (\(i=1\), \(2\), \(3\), \(4\)) is the length of the rod between point \(O_{i+1}\) and point \(O_{i}\). The kinematic parameters of the whole robot mechanism are now established. The transformation relationship between the coordinate system \(\{O_{0}\}\) and the body coordinate system \(\{C\}\) is shown in Equation (6):  
\[{}^{C}T_{0}= \begin{bmatrix}0&-1&0&l_{0}/2\\ 1&0&0&-b_{0}/2\\ 0&0&1&-h/2\\ 0&0&0&1\end{bmatrix}  \tag{6}\]  
The conversion relationship between the left- and right-swing coordinate system \(\{O_{1}\}\) of the shoulder joint and the coordinate system \(\{O_{0}\}\) is shown in Equation (7):  
\[{}^{0}T_{1}= \begin{bmatrix}c_{1}&-s_{1}&0&-b_{0}/2\\ 0&0&1&0\\ -s_{1}&-c_{1}&0&0\\ 0&0&0&1\end{bmatrix}  \tag{7}\]  
In Equation (7), \(C_{1}=\cos(\theta_{1})\), \(S_{1}=\sin(\theta_{1})\). The transformation relationship between the coordinate system \(\{O_{2}\}\) of the back-and-forth swing of the shoulder joint and the coordinate system \(\{O_{1}\}\) of the left- and right-swing of the shoulder joint is shown in Equation (8):  
\[{}^{1}T_{2}= \begin{bmatrix}c_{2}&-s_{2}&0&l_{1}\\ 0&0&1&0\\ -s_{2}&-c_{2}&0&0\\ 0&0&0&1\end{bmatrix}  \tag{8}\]  
Figure 6: Motion-analysis diagram of the whole robot mechanism. In Equation (8), \(C_{2}=\cos(\theta_{2})\), \(S_{2}=\sin(\theta_{2})\). The relationship of the conversion between the knee-joint coordinate system \(\{O_{3}\}\) and the shoulder-joint coordinate system \(\{O_{2}\}\) is shown in Equation (9):  
\[{}^{2}T_{3}= \begin{bmatrix}-c_{3}&-s_{3}&0&l_{2}\\ s_{3}&c_{3}&0&0\\ 0&0&1&0\\ 0&0&0&1\end{bmatrix}  \tag{9}\]  
In Equation (9), \(C_{3}=\cos(\theta_{3})\), \(S_{3}=\sin(\theta_{3})\). The transformation relationship between the ankle-joint coordinate system \(\{O_{4}\}\) and the knee-joint coordinate system \(\{O_{3}\}\) is shown in Equation (10):  
\[{}^{3}T_{4}= \begin{bmatrix}c_{4}&-s_{4}&0&l_{3}\\ s_{4}&c_{4}&0&0\\ 0&0&1&0\\ 0&0&0&1\end{bmatrix}  \tag{10}\]  
The transformation relationship between the foot-end coordinate system \(\{O_{5}\}\) and the ankle-joint coordinate system \(\{O_{4}\}\) is shown in Equation (11):  
\[{}^{4}T_{5}= \begin{bmatrix}-c_{\theta}&-s_{\theta}&0&l_{4}\\ s_{\theta}&-c_{\theta}&0&0\\ 0&0&1&0\\ 0&0&0&1\end{bmatrix}