## 2 Single-Leg Kinematics and Dynamics Modelling
### Single-Leg Dynamic Modeling  
For a single link of the leg, its active force is related to the first- and second-order kinematic differential quantities of the centroid of the link relative to the base coordinate. Considering the existence of the parallelogram closed-loop chain structure in the leg, the product exponential formula is first used to solve the kinematics of the centroid of each link [19; 20]:  
\[{}^{\mathbf{0}}T_{C_{i}}=\prod_{j}\exp(\mathbf{\hat{\xi}}_{j},\ \theta_{j})T_{C_{i}} \tag{2}\]  
where \(T_{C_{i}}\), \({}^{\mathbf{0}}T_{C_{i}}\) is the initial pose matrix and the pose matrix after homogeneous transformation of the centroid coordinate system \(\{C_{i}\}\) relative to the base coordinate system \(\{0\}\), respectively. For \(\{C_{i}\}\), \(T_{C_{i}}= \begin{bmatrix}\mathbf{I}_{3\times 3}&R_{C_{i}}\\ \mathbf{0}_{1\times 3}&1\end{bmatrix} \), \(R_{C_{i}}\) is the position vector of the centroid \(C_{i}\) relative to \(\{0\}\), \(\exp(\mathbf{\hat{\xi}}_{j},\ \theta_{j})\) is the product exponential formula, and \(\mathbf{\hat{\xi}}_{j}\), \(\theta_{j}\) are the Lie algebra of the screw coordinates and the joint rotation angle joint \(j\), respectively. From the transformation relationships among the homogeneous transformation matrix, the Jacobian matrix, and the motion screw, the mapping relationships between the velocity  
Figure 2: Diagram of the D–H coordinate system of the main kinematic chain. \(V_{C_{i}}\), acceleration \(\epsilon_{C_{i}}\) of the centroid of each link, and the active variables \(\theta_{A}\), \(\theta_{B}\), and \(\theta_{C}\) can be obtained as:  
\[\left\{ \begin{array}{ll}\mathbf{V}_{C_{i}}&=\left[\begin{array}{c} \mathbf{\omega}_{C_{i}}\\ \mathbf{v}_{C_{i}}\end{array}\right]=\left[\begin{array}{cc}\mathbf{\theta}\dot{\mathbf{ T}}_{C_{i}}\mathbf{{}^{0}}\mathbf{T}_{C_{i}}^{-1}\end{array}\right]^{\vee}\\ &=\left\{\mathbf{\xi}_{A},\text{Ad}[\exp(\mathbf{\xi}_{A},\theta_{A})]\mathbf{\xi}_{B}, \ldots,\prod\limits_{j}\text{Ad}[\exp(\mathbf{\xi}_{j},\theta_{j})]\mathbf{\xi}_{i} \right\}\cdot\mathbf{\Lambda}\cdot(\begin{array}{cc}\dot{\theta}_{A}&\dot{\theta }_{B}&\dot{\theta}_{C}\end{array}\right)^{\text{T}}\\ &=\mathbf{J}_{C_{i}}\dot{\theta}\\ \mathbf{\epsilon}_{C_{i}}&=\left[\begin{array}{c}\mathbf{\alpha}_{C_{i}}\\ \mathbf{\alpha}_{C_{i}}\end{array}\right]=\mathbf{J}_{C_{i}}\ddot{\mathbf{\theta}}+\ddot{ \mathbf{\theta}}^{\text{T}}\mathbf{H}_{C_{i}}\dot{\mathbf{\theta}}\end{array}  \tag{3}\]  
where \([\bullet]^{\vee}\) is the isomorphic mapping from the Lie algebra to the six-dimensional vector, and \(\left[ \begin{array}{cc}[\mathbf{\omega}]_{\times}&\mathbf{v}\\ 0&0\end{array} \right]^{\vee}=\left[ \begin{array}{c}\mathbf{\omega}\\ \mathbf{v}\end{array} \right]\), where \([\mathbf{\omega}]_{\times}\) is the skew symmetric matrix form of the three-dimensional vector \(\mathbf{\omega}\), \(\text{Ad}[\exp(\mathbf{\xi}_{i},\theta_{i})]\) is the adjoint transformation matrix of the screw, and \(\mathbf{\Lambda}\) is the transformation matrix between the first order differentials of the joint rotation angles (e.g., \(\theta_{D}\), \(\dot{\theta}_{F}\), \(\dot{\theta}_{G}\), \(\dot{\theta}_{I}\)) and \(\mathbf{\theta}\). \(\mathbf{J}_{C_{i}}\) is the Jacobian matrix of the centroid \(C_{i}\), and \(\mathbf{H}_{C_{i}}\) is the Hessian matrix of the centroid \(C_{i}\). By taking differentials on both sides of Equation (1), the mappings of the velocity \(V_{P}\) and acceleration \(\varepsilon_{P}\) of the foot-end \(P\) with respect to \(\theta_{A}\), \(\theta_{B}\), \(\theta_{C}\) can be obtained:  
\[\left\{ \begin{array}{ll}\mathbf{V}_{P}=\mathbf{J}_{P}\dot{\mathbf{\theta}}\\ \varepsilon_{P}=\mathbf{J}_{P}\ddot{\mathbf{\theta}}+\ddot{\mathbf{\theta}}^{\text{T}}\bm {H}_{P}\dot{\mathbf{\theta}}\end{array} \right. \tag{4}\]  
\(\mathbf{H}_{P}\) denotes the Hessian matrix corresponding to the foot-end point \(P\), and \(\mathbf{J}_{P}\) is the Jacobian matrix corresponding to the foot-end point \(P\).