## 4 Aslb--A Bipedal Robot for Dynamics Locomotion
### Aslb Kinematics  
Kinematic model of the floating platform starts with a set of unactuated joints that gives six DoFs to the mobile platform. These joints are collected in a column matrix \(\mathbf{q}_{B}\in\mathbb{R}^{3}\times SO(3)\). Starting from the inertial frame, joints are located in a sequence such that first translational joints \(\mathbf{q}_{t,B}~{}=~{}[q_{1},q_{2},q_{3}]^{T}\in\mathbb{R}^{3}\) are located in respective \(x\), \(y\) and \(z\) directions. Assuming the euler sequence of \(YZX\), rotational joints are \(\mathbf{q}_{r,B}~{}=~{}[q_{4},q_{5},q_{6}]^{T}\in\mathbb{R}^{3}\). Body frame \(\mathcal{F}_{B}\) is kinematically represented with respect to \(\mathcal{F}_{I}\) by \(\mathbf{q}_{B}~{}=~{}[\mathbf{q}_{t,B}^{T},\mathbf{q}_{r,B}]^{T}\) such that \(\mathbf{r}_{B}(\mathbf{q}_{t,B})\) and \(\mathbf{C}_{IB}(\mathbf{q}_{r,B})\) are translation and rotation matrices from \(\mathcal{F}_{I}\) to \(\mathcal{F}_{B}\), respectively. There are two legs attached to the body, and respective joints are represented by \(\mathbf{\theta}_{i}\in\mathbb{R}^{n_{i}}\), where \(n_{i}=5\) and \(i=R,L\). In total, the combined DoF for legs are given as \(n_{a}~{}=~{}10\)  
\begin{table}
\begin{tabular}{l l} \hline \hline \(\mathbf{q}_{B}\) & set of unactuated joints \\ \(\mathbf{q}_{t,B}\) & translational joints \\ \(\mathbf{q}_{r,B}\) & rotational joints \\ \(\mathbf{r}_{B}()\) & translation matrix from one frame to another \\ \(\mathbf{C}_{IB}()\) & rotation matrix from frame \(I\) to frame \(B\) \\ \(\mathbf{\theta}_{i}\) & vector of joints for each leg \\ \(\mathbf{\theta}_{a,i}\) & vector of active joints for leg, \(i\) \\ \(\mathbf{\theta}_{p,i}\) & vector of passive joints for leg, \(i\) \\ \(\mathbf{r}_{r,i}\) & position of point \(v\) \\ \(\mathbf{R}_{*}\) & elementary rotation matrix for \(*\) axis \\ \(\mathbf{v}_{r,i}\) & velocity relationship \\ \(\mathbf{J}_{r,i}\) & Jacobian relating velocity to active and passive joints \\ \(a_{*}\) & lengths between joints \\ \(\mathbf{M}()\) & generalized mass matrix \\ \(\mathbf{C}(...)\) & Coriolis and centrifugal terms \\ \(\mathbf{G}()\) & gravitational terms \\ \(\mathbf{S}\) & control selection matrix for actuated joints of respective legs \\ \(\mathbf{\tau}\) & actuated joint torques \\ \(\mathbf{F}_{C,i}\) & external force on tip of \(i^{\text{th}}\) leg \\ \(J_{C,i}\) & geometric Jacobian of tip point for \(i^{\text{th}}\) leg \\ \(\omega_{0}\) & angular speed for the linear inverted pendulum model (LIPM) \\ \({}^{p}\mathcal{R}_{C,K}\) & contact state \(x_{c}\) at instant \(K\) in phase \(p\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Summary of key variables used for the description of the ASLB kinematics and dynamics.  
Figure 1: Three-dimensional model (**left**) and manufactured prototype of ASLB (**right**).  
The composition of \(\theta_{i}\) for each leg is given as \(\theta_{i}\ =\ [\theta_{0,i}\ \theta_{1,i}\ \theta_{2,i}\ \theta_{3,i}\ \theta_{4,i}]^{T}\), where there are three active and two passive joints, respectively \(\theta_{a,i}\ =\ [\theta_{0,i}\ \theta_{1,i}\ \theta_{3,i}]^{T}\in\mathbb{R}^{3}\) and \(\theta_{p,i}(\theta_{a,j})\ =\ [\theta_{2,i}(\theta_{a,i})\ \theta_{4,i}( \theta_{a,i})]^{T}\). Passive joints can be written as a function of \(\theta_{a,j}\) based on a velocity constraint as described in Section 4.2.1. As a result, total joint space of ASLB is given by \(\mathbf{q}\ =\ [\mathbf{q}_{B},\ \mathbf{\theta}_{a,R},\ \mathbf{\theta}_{a,L}]^{T}\in\mathbb{R}^{12}\). The kinematic structure of the right leg is illustrated in Figure 2.