# Research on Low Energy Consumption Static Postures of Bionic Feet
## 2 The Kinematics and Static Analysis of a Hexapod Robot
### Kinematics Analysis  
The MDH (The Matrix of Denavit-Hartenberg) method was used to establish the one-legged model of the hexapod robot. The relationship between the angle of leg joint rotation and the position of the foot tip was given by kinematics analysis of the robot. The joint structure of the right middle leg of the robot is shown in Figure 1. The parameters are defined in Table 1. The MDH parameters of the leg joint are shown in Table 2. Among them, \(l_{k}\) represents the length of the rods in each joint of the robot leg, \(\alpha_{k}\) represents the corresponding torsion angle of the rods, \(d_{k}\) represents the distance of the joints, and \(\theta_{k}\) represents the rotation angle of the joints. According to the homogeneous transformation matrix formula, the pose transformation matrix of the robot foot relative to the heel joint coordinate system is as follows:  
\[{}^{c}T_{4}={}^{c}T_{1}{}^{1}T_{2}{}^{2}T_{3}{}^{3}T_{4}=\left[ \begin{array}{ cccc}c_{1}c_{2}c_{3}-c_{1}s_{2}s_{3}&-c_{1}c_{2}s_{3}-c_{1}s_{2}c_{3}&-s_{1}&l_{ 1}c_{1}+l_{2}c_{1}c_{2}+l_{3}c_{1}c_{23}+b/2\\ s_{1}c_{2}c_{3}-s_{1}s_{2}s_{3}&-s_{1}c_{2}s_{3}-s_{1}s_{2}c_{3}&c_{1}&l_{1}s_{1 }+l_{2}s_{1}c_{2}+l_{3}s_{1}c_{23}\\ -s_{2}c_{3}-c_{2}s_{3}&s_{2}s_{3}-c_{2}c_{3}&0&l_{2}s_{2}+l_{3}s_{23}\\ 0&0&0&1\end{array} \right], \tag{1}\]  
where \(s_{1}\) represents \(\sin\theta_{1}\), \(c_{1}\) represents \(\cos\theta_{1}\), \(c_{23}\) represents \(\cos(\theta_{2}+\theta_{3})\), \(s_{23}\) represents \(\sin(\theta_{2}+\theta_{3})\), and \(b\) represents the distance between the heel joints of the robot's two middle legs. The position of the foot tip in the coordinate system of the heel joint is \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Description** \\ \hline \(O_{c}-X_{c}Y_{c}Z_{c}\) & Robot body coordinate system \\ \(O_{j}-X_{j}Y_{j}Z_{j}\) & Coordinates of the robot heel joint, hip joint, knee joint, and foot tip. \\ \(l_{k}(k=1,2,3)\) & Lengths of the three joints of the robot’s legs \\ \(\theta_{k}(k=1,2,3)\) & Corners of the three joints of the robot’s legs \\ \(G\),\(G^{\prime}\) & Total body weight and load weight \\ \(P_{i}[P_{ix},P_{iy},P_{iz}]\) & Position of the robot’s foot tip in the joint coordinate system \\ \((i=1,2,\cdots,6)\) & Reaction force of the robot’s foot tip \\ \(\mathbf{F}[f_{ix},f_{iy},f_{iz}]\) & Reaction force of the robot’s foot tip \\ \((i=1,2,\cdots,6)\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Description of parameters in Figure 1. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Joint Number** & \(\boldsymbol{\theta}_{k}\) & \(\boldsymbol{\alpha}_{k}\) & \(\boldsymbol{l}_{k}\) & \(\boldsymbol{d}_{k}\) \\ \hline
1 & \(\theta_{c}\) & 0 & \(l_{c}\) & 0 \\
2 & \(\theta_{1}\) & \(\pi/2\) & \(l_{1}\) & 0 \\
3 & \(\theta_{2}\) & 0 & \(l_{2}\) & 0 \\
4 & \(\theta_{3}\) & 0 & \(l_{3}\) & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: MDH parameters of the right-middle leg. Figure 1: MDH model of a single leg of the robot. \[P_{i}=\left[ \begin{array}{cc}P_{ix}&P_{iy}&P_{iz}\end{array} \right]^{T}=\left[ \begin{array}{c}l_{1}c_{1}+l_{2}c_{1}c_{2}+l_{3}c_{1}c_{23}+b/2\\ l_{1}s_{1}+l_{2}s_{1}c_{2}+l_{3}s_{1}c_{23}\\ l_{2}s_{2}+l_{3}s_{23}\end{array} \right]. \tag{2}\]  
Its Jacobian matrix is  
\[\mathbf{J}=\left[ \begin{array}{ccc}-s_{1}(l_{3}c_{23}+l_{2}c_{2}+l_{1})&c_{1 }(-l_{3}s_{23}-l_{2}s_{2})&-l_{3}c_{1}s_{23}\\ c_{1}(l_{3}c_{23}+l_{2}c_{2}+l_{1})&s_{1}(-l_{3}s_{23}-l_{2}s_{2})&-l_{3}s_{1} s_{23}\\ 0&-l_{3}c_{23}-l_{2}c_{2}&-l_{3}c_{23}\end{array} \right]. \tag{3}\]  
By solving the inverse kinematics equation, the joint angles of the robot can be obtained:  
\[\left\{ \begin{array}{l}\theta_{1}=\arctan(\frac{P_{y}}{P_{x}})\\ \theta_{2}=-\arccos(\frac{P_{z}}{\sqrt{K^{2}+L^{2}}})-\psi\\ \theta_{3}=\arccos(\frac{B^{2}+P_{z}^{2}-l_{2}^{2}-l_{3}l^{2}}{2l_{2}l_{3}}) \end{array} \right., \tag{4}\]  
where  
\[ \begin{array}{l}K=l_{2}+l_{3}c_{3},L=l_{3}s_{3},\\ \psi=\arcsin(\frac{K}{\sqrt{K^{2}+L^{2}}}),B=\frac{P_{x}}{c_{1}}-l_{1}\end{array}