## 4 Aslb--A Bipedal Robot for Dynamics Locomotion
### ASLB Dynamics  
Based on the generalized coordinates, multi-body dynamics of ASLB are formulated as in (Equation (21))  
\[\mathbf{M}(\mathbf{q})\tilde{\mathbf{q}}+\mathbf{C}(\mathbf{q},\dot{\mathbf{q}})+\mathbf{G}(\mathbf{q})=\mathbf{S} \mathbf{\tau}+\mathbf{J}_{\mathcal{C},i}^{T}\mathbf{F}_{\mathcal{C},i} \tag{21}\]  
Figure 4: Geometric entities related to calculation of \(\theta_{1,R}\). where \(\mathbf{M}(\mathbf{q})\in\mathbb{R}^{12\times 12}\), \(\mathbf{C}(\mathbf{q},\dot{\mathbf{q}})\in\mathbb{R}^{12}\), and \(\mathbf{G}(\mathbf{q})\in\mathbb{R}^{12}\) are the generalized mass, Coriolis and centrifugal, and gravitational terms, respectively. \(\mathbf{S}\in\mathbb{R}^{12\times 6}\), \(\mathbf{\tau}\in\mathbb{R}^{6}\), \(\mathbf{F}_{\mathbf{C},i}\in\mathbb{R}^{3}\) and \(\mathbf{J}_{\mathbf{C},i}\in\mathbb{R}^{3\times 12}\) are the selection matrix for the actuated joints of respective legs, actuated joint torques, geometric Jacobian of the tip point of the \(i^{th}\) leg, and external force on the tip of \(i^{th}\) leg. Let \(\mathbf{x}~{}=~{}\left[\mathbf{q},~{}\dot{\mathbf{q}}\right]^{T}\in\mathbb{R}^{24}\), \(\mathbf{u}_{\tau}~{}=~{}\mathbf{\tau}\), and \(\mathbf{u}_{c}~{}=~{}\mathbf{F}_{\mathbf{C},i}~{}\forall i\) be the states, inputs to motors and external forces acting on tip point of the legs, respectively; then, the non-linear dynamics of the robot can be written as in Equation (22). \[\dot{\mathbf{x}} = \mathbf{f}(\mathbf{x},\mathbf{u}_{\tau},\mathbf{u}) \tag{22}\] \[\mathbf{f}(\mathbf{x},\mathbf{u}_{\tau},\mathbf{u}) = \left[ \begin{array}{c}\dot{\mathbf{q}}\\ \mathbf{M}(\mathbf{q})^{-1}\mathbf{\Theta}\end{array} \right]\] \[\mathbf{\theta} = -\mathbf{C}(\mathbf{q},\dot{\mathbf{q}})-\mathbf{G}(\mathbf{q})+\mathbf{S}\mathbf{\tau}+\sum_{ i}\mathbf{J}_{\mathbf{C},i}^{T}\mathbf{F}_{\mathbf{C},i}\]  
The application of the graph-theoretic modeling for the ASLB by separating the legs, which are defined as **Agent 2** and **Agent 3**, from the floating base introduces simplifications to the overall complexity of the model and optimization. One of the simplifications appears in modeling as leg dynamics does not necessarily need to be modeled with respect to \(\mathcal{F}_{I}\) using \(\mathbf{q}_{B}\) but rather is better defined with respect to the \(\mathcal{F}_{B}\). This local representation of leg dynamics leads to following outcomes for the dynamics of the agents:  
* The leg is only used to find adjacency and contact forces on the floating base and ground,
* The state space representation of the leg dynamics can be kept at velocity level. Note that the contact forces and geometric properties of contact point must be converted to \(\mathcal{F}_{B}\). For ASLB, the floating base of the robot is denoted as **Agent 1**, the right leg is denoted as **Agent 2**, and the left leg is denoted as **Agent 3**. **Agent 1** is defined with two nodes, **node 1** and **node 2**; **Agent 2** is defined with **node 3**; and **Agent 3** is defined with **node 4**. The locations of these nodes are illustrated in Figure 5, and it should be noted that **node 1** and **node 2** coincide. Without any interconnection constraint, agents are independent of each other; however, there are rigid joints connecting them. In this case there are two bi-directional edges and these are \((\textbf{node 3},\textbf{node 1})\in\mathcal{E}\) and \((\textbf{node 2},\textbf{node 4})\in\mathcal{E}\). Connections between nodes are assumed to be rigid, and then the adjacency matrix is composed of edge weights \(a_{mn}=1\). This yields an adjacency matrix as in (23). \[\mathbb{A}~{}=~{}\left[ \begin{array}{cccc}0&0&1&0\\ 0&0&0&1\\ 1&0&0&0\\ 0&1&0&0\end{array} \right] \tag{23}\]  
Finally, the relationships between cooperative agents are given in (24) by the Laplacian matrix. \(\mathbf{I}_{p}\) represents identity matrix with size \(p\).