# Structural Design and Kinematics Simulation of Hydraulic Biped Robot
## 3 Kinematics Analysis
### Speed Jacobian Matrix Based on POE Method  
The speed Jacobian matrix (hereinafter referred to as Jacobian) refers to the mapping matrix from the speed of the driving joint to the speed of the end effector. As the core of the robot speed analysis, the establishment of the Jacobian matrix is the basis for kinematics analysis, acceleration analysis, accuracy analysis, static/dynamic analysis, and synthesis. The traditional way to describe a robot's  
Figure 9: Coronal geometric relationship diagram. Jacobian matrix is to solve its forward kinematics using different equations, which involves a more complicated solving process and results in general. The screw theory can be employed to clearly describe the Jacobian matrix of the serial robot, which can highlight the geometric characteristics of the robot, and avoid the deficiencies of using local parameters in the differential method [28]. The screw theory and POE method can be used to easily solve the robot's Jacobian matrix. The POE formula of the robot kinematics equation is as follows:  
\[ \begin{array}{c}{}_{T}^{S}T(\theta)=\mathrm{e}^{[V_{1}]\theta_{1}}\mathrm{e} ^{[V_{2}]\theta_{2}}\cdots\mathrm{e}^{[V_{6}]\theta_{6}S}{}_{T}^{S}T(0)=\left[ \begin{array}{cc}R(\theta)&p(\theta)\\ 0&1\end{array}\right]\end{array}  \tag{30}\]  
The instantaneous space velocity of the foot can be obtained. \[ \begin{array}{c}{}_{T}^{S}V^{s}\end{array} = \begin{array}{c}{}_{T}^{S} \hat{T}(\theta)_{T}^{S}T^{-1}(\theta)\end{array}  \tag{31}\]  
According to the characteristics of the POE formula of the kinematics equation, we get:  
\[ \begin{array}{c}{}_{T}^{S}V^{s}\end{array} = \begin{array}{c}{}_{i=1}^{n} \Bigg{(}\frac{\partial\left({}_{T}^{S}T\right)}{\partial\theta_{i}}\Bigg{)}_{T }^{S}T^{-1}(\theta)=\sum\limits_{i=1}^{n}\Bigg{(}\frac{\partial\left({}_{T}^{ S}T\right)}{\partial\theta_{i}}\frac{S}{T}T^{-1}(\theta)\Bigg{)}\dot{ \theta}_{i}(i=1\cdots 6)\end{array}  \tag{32}\]  
The twist coordinate can be represented as:  
\[\begin{array}{c}{}_{T}^{S}V^{s}= \begin{array}{c}{}_{S}^{J}s(\theta)\dot{ \theta}\end{array}  \tag{33}\]  
where, the matrix \( \begin{array}{c}{}_{T}^{S}J^{s}(\theta)\in R^{6\times n}\end{array} \) is the spatial Jacobian matrix of the foot. \[ \begin{array}{c}{}_{T}^{S}J^{s}(\theta)=\Bigg{[}\Bigg{(}\frac{\partial\left( {}_{T}^{S}T\right)}{\partial\theta_{1}}\frac{S}{T}T^{-1}(\theta)\Bigg{)}^{ \vee}\Bigg{(}\frac{\partial\left({}_{T}^{S}T\right)}{\partial\theta_{2}}\frac{ S}{T}T^{-1}(\theta)\Bigg{)}^{\vee}\cdots\Bigg{(}\frac{\partial\left({}_{T}^{S}T \right)}{\partial\theta_{6}}\frac{S}{T}T^{-1}(\theta)\Bigg{)}^{\vee}\Bigg{]} \end{array}  \tag{34}\]  
The operator \([\;\cdot\;]\) in square bracket indicates an antisymmetric matrix, and \(\vee\) is its inverse operator. It maps the joint velocity vector to the corresponding twist coordinate of the foot. Let \([V_{i}]\in\)\(\mathcal{se}(3)\) be the unit twist, so any column of the spatial Jacobian matrix is as follows. \[ \begin{array}{ccl}\frac{\partial\left({}_{T}^{S}T\right)}{\partial\theta_{i }}{}_{T}^{S}T^{-1}(\theta)&=&\mathrm{e}^{[V_{1}]\theta_{1}}\mathrm{e}^{[V_{2} ]\theta_{2}}\cdots\mathrm{e}^{[V_{i-1}]\theta_{i-1}}\frac{\partial}{\partial \theta_{1}}\Big{(}\mathrm{e}^{[V_{i}]\theta_{i}}\Big{)}\mathrm{e}^{[V_{i+1}] \theta_{i+1}}\cdots\mathrm{e}^{[V_{6}]\theta_{6}S}{}_{T}^{S}T(0)_{T}^{S}T^{-1} (\theta)\\ &=&\mathrm{e}^{[V_{1}]\theta_{1}}\mathrm{e}^{[V_{2}]\theta_{2}}\cdots\mathrm{ e}^{[V_{i-1}]\theta_{i-1}}\big{|}_{\frac{\partial}{\partial\theta_{1}}}\Big{(} \mathrm{e}^{[V_{i}]\theta_{i}}\Big{)}\mathrm{e}^{[V_{i+1}]\theta_{i+1}}\cdots \mathrm{e}^{[V_{6}]\theta_{6}S}{}_{T}^{S}T(0)_{T}^{S}T^{-1}(\theta)\\ &=&\mathrm{e}^{[V_{1}]\theta_{1}}\mathrm{e}^{[V_{2}]\theta_{2}}\cdots\mathrm{ e}^{[V_{i-1}]\theta_{i-1}}\big{|}_{\frac{\partial}{\partial\theta_{1}}}\Big{(} \mathrm{e}^{[V_{i}]\theta_{i}}\Big{)}\mathrm{e}^{[V_{i-1}]\theta_{i-1}}\cdots \mathrm{e}^{[V_{1}]\theta_{1}}\end{array}  \tag{35}\]  
Convert it to twist coordinates:  
\[