## 2 Analysis of the Gait Pattern
### Robot Dynamics Description
\right] \tag{15}\]  
\[T=\left[0,\tau_{\rm hip}\,0,0\right]^{T} \tag{16}\]  
where the representations of \(M\) and \(C\) are shown as Equations (17)-(32):  
\[M_{11}=2l_{sh}^{2}m_{l}+l_{sh}^{2}m_{h}+l_{th}^{2}m_{h}+4l_{th}^{2}m_{l}\sin( q_{2}/2)^{2}+2l_{sh}l_{th}m_{h}+4l_{sh}l_{th}m_{l}\sin(q_{2}/2)^{2} \tag{17}\]  
\[M_{12}=M_{21}=-l_{th}m_{l}(l_{sh}\cos(q_{2})-l_{th}+l_{th}\cos(q_{2})) \tag{18}\]  
\[M_{13}=M_{31}=l_{th}m_{l}\cos(q_{1}+q_{2})-l_{th}m_{l}\cos(q_{1})-l_{sh}m_{h} \cos(q_{1}) \tag{19}\]  
\[-l_{th}m_{h}\cos(q_{1})-2l_{sh}m_{l}\cos(q_{1})\]  
\[M_{14}=M_{41}=l_{th}m_{i}\sin(q_{1}+q_{2})-l_{th}m_{i}\sin(q_{1})-l_{sh}m_{h} \sin(q_{1})\]  
\[-l_{th}m_{h}\sin(q_{1})-2l_{sh}m_{l}\sin(q_{1}) \tag{20}\]  
\[M_{23}=M_{32}=l_{th}m_{l}\cos(q_{1}+q_{2}) \tag{21}\]  
\[M_{24}=M_{42}=l_{th}m_{t}\sin(q_{1}+q_{2}) \tag{22}\]  
\[M_{22}=l_{th}^{2}m_{l} \tag{23}\]  
\[M_{11}=2l_{sh}^{2}m_{l}+l_{sh}^{2}m_{h}+l_{th}^{2}m_{h}+4l_{th}^{2}m_{l}\sin( q_{2}/2)^{2}+2l_{sh}l_{th}m_{h}+4l_{sh}l_{th}m_{l}\sin(q_{2}/2)^{2} \tag{17}\]0  
\[M_{11}=2l_{sh}^{2}m_{l}+l_{sh}^{2}m_{h}+l_{th}^{2}m_{h}+4l_{th}^{2}m_{l}\sin( q_{2}/2)^{2}+2l_{sh}l_{th}m_{h}+4l_{sh}l_{th}m_{l}\sin(q_{2}/2)^{2} \tag{17}\]1  
\[M_{11}=2l_{sh}^{2}m_{l}+l_{sh}^{2}m_{h}+l_{th}^{2}m_{h}+4l_{th}^{2}m_{l}\sin( q_{2}/2)^{2}+2l_{sh}l_{th}m_{h}+4l_{sh}l_{th}m_{l}\sin(q_{2}/2)^{2} \tag{17}\]2  
\[M_{11}=2l_{sh}^{2}m_{l}+l_{sh}^{2}m_{h}+l_{th}^{2}m_{h}+4l_{th}^{2}m_{l}\sin( q_{2}/2)^{2}+2l_{sh}l_{th}m_{h}+4l_{sh}l_{th}m_{l}\sin(q_{2}/2)^{2} \tag{17}\]3\[M_{11}=2l_{sh}^{2}m_{l}+l_{sh}^{2}m_{h}+l_{th}^{2}m_{h}+4l_{th}^{2}m_{l}\sin( q_{2}/2)^{2}+2l_{sh}l_{th}m_{h}+4l_{sh}l_{th}m_{l}\sin(q_{2}/2)^{2} \tag{17}\]4 \[M_{11}=2l_{sh}^{2}m_{l}+l_{sh}^{2}m_{h}+l_{th}^{2}m_{h}+4l_{th}^{2}m_{l}\sin( q_{2}/2)^{2}+2l_{sh}l_{th}m_{h}+4l_{sh}l_{th}m_{l}\sin(q_{2}/2)^{2} \tag{17}\]5  
\[M_{11}=2l_{sh}^{2}m_{l}+l_{sh}^{2}m_{h}+l_{th}^{2}m_{h}+4l_{th}^{2}m_{l}\sin( q_{2}/2)^{2}+2l_{sh}l_{th}m_{h}+4l_{sh}l_{th}m_{l}\sin(q_{2}/2)^{2} \tag{17}\]6 \[M_{11}=2l_{sh}^{2}m_{l}+l_{sh}^{2}m_{h}+l_{th}^{2}m_{h}+4l_{th}^{2}m_{l}\sin( q_{2}/2)^{2}+2l_{sh}l_{th}m_{h}+4l_{sh}l_{th}m_{l}\sin(q_{2}/2)^{2} \tag{17}\]7  
\[M_{11}=2l_{sh}^{2}m_{l}+l_{sh}^{2}m_{h}+l_{th}^{2}m_{h}+4l_{th}^{2}m_{l}\sin( q_{2}/2)^{2}+2l_{sh}l_{th}m_{h}+4l_{sh}l_{th}m_{l}\sin(q_{2}/2)^{2} \tag{17}\]8  
\[M_{11}=2l_{sh}^{2}m_{l}+l_{sh}^{2}m_{h}+l_{th}^{2}m_{h}+4l_{th}^{2}m_{l}\sin( q_{2}/2)^{2}+2l_{sh}l_{th}m_{h}+4l_{sh}l_{th}m_{l}\sin(q_{2}/2)^{2} \tag{17}\]9  
\[M_{12}=M_{21}=-l_{th}m_{l}(l_{sh}\cos(q_{2})-l_{th}+l_{th}\cos(q_{2})) \tag{18}\]0  
2. Constrained force applied to the stance foot:  
The stance foot has two DOFs in the walking model. To adhere to the physical constraints, which include eliminating slippage between the stance foot and the ground (achieved via a parallel forward force) and ensuring that the stance foot remains above the ground surface (maintained by a perpendicular upward force), a constrained force is applied to the stance foot to maintain its position. The constrained condition at the stance foot can be shown as Equations (33) and (34):  
\[M_{12}=M_{21}=-l_{th}m_{l}(l_{sh}\cos(q_{2})-l_{th}+l_{th}\cos(q_{2})) \tag{18}\]1  
\[J_{r}= \begin{bmatrix}0&0&1&0\\ 0&0&0&1\end{bmatrix}  \tag{34}\]  
From Equations (11) and (33), we derive the resulting relation shown in Equation (35):  
\[J_{r}\ddot{q}=-J_{r}M(q)^{-1}\Big{(}C(q,\dot{q})\dot{q}+G(q,\gamma)-J_{r}^{T}u -T\Big{)}=0 \tag{35}\]  
From Equation (12), we derive the resulting relation shown in Equation (36):  
\[u=\Big{(}J_{r}M(q)^{-1}J_{r}^{T}\Big{)}^{-1}J_{r}M(q)^{-1}\big{(}C(q,\dot{q}) \dot{q}+G(q,\gamma)-T\big{)} \tag{36}\]  
Consequently, the stance foot can be kept in the current state, and then the dynamic oscillation phase state can be solved. 3. Heel-strike:  
During the termination of the swing phase, when the swing leg separates from the ground and makes contact with the ground again, the following assumptions are considered:  
1. Push-off and heel-strike are completed instantaneously;
2.