# Structural Design and Kinematics Simulation of Hydraulic Biped Robot
## 3 Kinematics Analysis
### Forward Kinematics  
The coordinate system of the right leg is established in the form of the POE, as shown in Figure 7.  
Figure 6: Finite element analysis of critical components. (**a**) U-shaped bracket. (**b**) Joint axis. (**c**) Joint sleeve. \(\omega\) is the rotation axis of the rigid body, \(r\) is any point on the axis, and Matrix \([V]\) is the twist produced by pure rotation, shown as:  
\[[V]=\left[ \begin{array}{cc}[\omega]&v\\ 0&0\end{array} \right] \tag{5}\]  
Here, the antisymmetric matrix \([\omega]\in so(3)\), \(v=-\omega\times r\)  
The matrix index of the twist \([V]\) is  
\[\mathrm{e}^{[V]\theta}=\left[ \begin{array}{cc}R(\theta)&p(\theta)\\ 0&1\end{array} \right] \tag{6}\]  
The rotation matrix is obtained by Rodrigues formula, namely  
\[R(\theta)=\mathrm{e}^{[\omega]\theta}=I+[\omega]\sin\theta+[\omega]^{2}(1- \cos\theta) \tag{7}\]  
The vector is calculated as follows:  
\[p(\theta)=\big{(}1-\mathrm{e}^{[\omega]\theta}\big{)}(\omega\times v)+\omega a \gamma^{T}v\theta \tag{8}\]  
The directions of the line vectors of various joint axes are:  
\[\omega_{1}=\left[ \begin{array}{c}0\\ 0\\ 1\end{array} \right]\omega_{2}=\omega_{6}=\left[ \begin{array}{c}1\\ 0\\ 0\end{array} \right]\omega_{3}=\omega_{4}=\omega_{5}=\left[ \begin{array}{c}0 \\ 1\\ 0\end{array} \right]\] \[[\omega_{1}]=\left[ \begin{array}{ccc}0&-1&0\\ 1&0&0\\ 0&0&0\end{array} \right][\omega_{2}]=[\omega_{6}]=\left[ \begin{array}{ccc}0&0 &0\\ 0&0&-1\\ 0&1&0\end{array} \right] \tag{9}\]  
Figure 7: Coordinate system of the right leg of biped robot. Take one point on various joint axes as  
\[r_{1}=\left[ \begin{array}{c}0\\ -l_{1}\\ 0\end{array} \right]r_{2}=\left[ \begin{array}{c}0\\ -l_{1}\\ -l_{2}\end{array} \right]r_{3}=\left[ \begin{array}{c}0\\ -l_{1}\\ -l_{2}-l_{3}\end{array} \right] \tag{10}\] \[r_{4}=\left[ \begin{array}{c}0\\ -l_{1}\\ -l_{2}-l_{3}-l_{4}\end{array} \right]r_{5}=r_{6}=\left[ \begin{array}{c}0\\ -l_{1}\\ -l_{2}-l_{3}-l_{4}-l_{5}\end{array} \right]\]  
Under the shape and position when \(\theta=0\), the transformation matrix of the tool coordinate system \(\{T\}\) relative to the basic coordinate system \(\{S\}\) is:  
\[\overset{S}{{}_{T}}T(0)=\left[ \begin{array}{cccc}1&0&0&0\\ 0&1&0&-l_{1}\\ 0&0&1&-l_{2}-l_{3}-l_{4}-l_{5}-l_{6}\\ 0&0&0&1\end{array} \right] \tag{11}\]  
The twists of various joints are:  
\[\mathbf{e}^{[V_{1}]\theta_{1}}=\left[ \begin{array}{cccc}\cos\theta_{1}&- \sin\theta_{1}&0&-l_{1}\sin\theta_{1}\\ \sin\theta_{1}&\cos\theta_{1}&0&-l_{1}(1-\cos\theta_{1})\\ 0&0&1&0\\ 0&0&0&1\end{array} \right] \tag{12}\]  
\[\mathbf{e}^{[V_{2}]\theta_{2}}=\left[ \begin{array}{cccc}1&0&0&0\\ 0&\cos\theta_{2}&-\sin\theta_{2}&-l_{1}(1-\cos\theta_{2})-l_{2}\sin\theta_{2} \\ 0&\sin\theta_{2}&\cos\theta_{2}&l_{1}\sin\theta_{2}-l_{2}(1-\cos\theta_{2})\\ 0&0&0&1\end{array} \right] \tag{13}\]  
\[\mathbf{e}^{[V_{3}]\theta_{3}}=\left[ \begin{array}{cccc}\cos\theta_{3}&0& \sin\theta_{3}&(l_{2}+l_{3})\sin\theta_{3}\\ 0&1&0&0\\ -\sin\theta_{3}&0&\cos\theta_{3}&-(l_{2}+l_{3})(1-\cos\theta_{3})\\ 0&0&0&1\end{array} \right] \tag{14}\]  
\[\mathbf{e}^{[V_{4}]\theta_{4}}=\left[ \begin{array}{cccc}\cos\theta_{4}&0& \sin\theta_{4}&(l_{2}+l_{3}+l_{4})\sin\theta_{4}\\ 0&1&0&0\\ -\sin\theta_{4}&0&\cos\theta_{4}&-(l_{2}+l_{3}+l_{4})(1-\cos\theta_{4})\\ 0&0&0&1\end{array} \right] \tag{15}\]  
\[\mathbf{e}^{[V_{5}]\theta_{5}}=\left[ \begin{array}{cccc}\cos\theta_{5}&0& \sin\theta_{5}&(l_{2}+l_{3}+l_{4}+l_{5})\sin\theta_{5}\\ 0&1&0&0\\ -\sin\theta_{5}&0&\cos\theta_{5}&-(l_{2}+l_{3}+l_{4}+l_{5})(1-\cos\theta_{5}) \\ 0&0&0&1\end{array} \right] \tag{16}\]  
\[\mathbf{e}^{[V_{6}]\theta_{6}}=\left[ \begin{array}{cccc}1&0&0&0\\ 0&\cos\theta_{6}&-\sin\theta_{6}&-l_{1}(1-\cos\theta_{6})-(l_{2}+l_{3}+l_{4}+ l_{5})\sin\theta_{6}\\ 0&\sin\theta_{6}&\cos\theta_{6}&l_{1}\sin\theta_{6}-(l_{2}+l_{3}+l_{4}+l_{5})(1- \cos\theta_{6})\\ 0&0&0&1\end{array}