## 3 Model Reduction Method and Theoretical Solution for the Contact Force \(f\)  
As shown in Figure 1, under the contact force between the foot and the ground, the \(\theta_{1}\) can be considered as a variable describing the contact motion under the contact force between the \(\theta_{2}\), the remaining redundant variable, and \(\theta_{2}\) can be represented by the \(\theta_{1}\) as \(\Phi(\theta_{1})\). Then:  
\[\left\{ \begin{array}{l}\dot{\theta}=\left[\begin{array}{cc}\dot{\theta}_ {1}&\dot{\theta}_{2}\\ \end{array}\right]^{T}\dot{\theta}_{2}=\Phi(\theta_{1})\left[\begin{array}{ cc}1&\frac{\partial\Phi(\theta_{1})}{\partial\theta_{1}}\\ \end{array}\right]^{T}\dot{\theta}_{1}=\mathrm{L}(\theta_{1})\dot{\theta}_{1} \\ \ddot{\theta}=\mathrm{L}(\theta_{1})\dot{\theta}_{1}+\mathrm{L}(\theta_{1}) \ddot{\theta}_{1}\end{array} \right., \tag{4}\]  
Bringing Equation (4) into Equation (3), multiply \(L^{T}(\theta_{1})\) on both sides of the equation to obtain:  
\[\left\{ \begin{array}{l}M_{L}(\theta_{1})\ddot{\theta}_{1}+C_{L}\Big{(}\theta _{1},\dot{\theta}_{1}\Big{)}\dot{\theta}_{1}+G_{L}(\theta_{1})=L^{T}(\theta_{ 1})\tau\\ \\ M_{L}(\theta_{1})=L^{T}(\theta_{1})M_{1}(\theta_{1})=L^{T}(\theta_{1})M( \theta_{1})L(\theta_{1})\\ C_{L}\Big{(}\theta_{1},\dot{\theta}_{1}\Big{)}=L^{T}(\theta_{1})C_{1}\Big{(} \theta_{1},\dot{\theta}_{1}\Big{)}=L^{T}(\theta_{1})\Big{(}M(\theta)\dot{L}( \theta_{1})+C(\theta,\dot{\theta})L(\theta_{1})\Big{)}\\ G_{L}(\theta_{1})=L^{T}(\theta_{1})\Big{(}G(\theta)+J_{\psi}^{T}(\theta_{1}) \Big{)}\end{array} \right., \tag{5}\]  
The reduced-order model (Equation (5)) satisfies the following properties [29]. If one defines:  
\[M_{L}(\theta_{1})=L^{T}(\theta_{1})M(\theta_{1})L(\theta_{1})>0, \tag{6}\]  
**Property 1**. : _Then \(\dot{M}_{L}(\theta_{1})-C_{L}\Big{(}\theta_{1},\dot{\theta}_{1}\Big{)}\) has oblique symmetry._  
**Property 2**. : _Then_  
\[J_{\psi}(\theta_{1})L(\theta_{1})=L^{T}(\theta_{1})J_{\psi}^{T}(\theta_{1})=0. \tag{7}\]Let \(\theta_{d}(t)\) be the ideal angle command, \(\tau_{f}^{d}\) be the ideal contact force, and let it satisfy Equation (1). Then, \(\Psi(\theta_{d})=0,\tau_{f}^{d}=J_{\psi}^{T}(\theta_{d})f_{d}\), and the control targets are actual \(\theta(t)\) tracking \(\theta_{d}(t)\) and actual \(\tau_{f}\) tracking \(\tau_{f}^{d}\). Given the constraints imposed by limited research resources, we employed a methodology integrating theoretical deduction and numerical simulation to substantiate the efficacy of the proposed control scheme. To facilitate the simulation validation of the controller, it was imperative to obtain an analytical expression for the contact force \(f\). The theoretical expression (Equation (8)) for the contact force \(f\) was derived using Equation (5):  
\[J_{\psi}^{T}(\theta_{1})f=\tau-M_{1}(\theta_{1})\ddot{\theta}_{1}-C_{1}\Big{(} \theta_{1},\dot{\theta}_{1}\Big{)}\dot{\theta}_{1}-G_{1}(\theta_{1}), \tag{8}\]  
According to Equation (8), the value of the contact force \(f\) is as follows:  
1. When \(J_{\psi}=0\), \(f\) is obtained from the expression of \(J_{\psi}^{T}(\theta_{1})\);
2. When \[J_{\psi}(\theta_{1})=J_{\psi}(a)\
eq 0,\;f_{a}=1/J_{\psi}(a)\Big{(}\tau(a)-M_{ 1}(a)\ddot{\theta}_{1}-C_{1}(a)\dot{\theta}_{1}-G_{1}(a)\Big{)};\] (9)
3. When \[J_{\psi}(\theta_{1})=J_{\psi}(b)\
eq 0,\;f_{b}=1/J_{\psi}(b)\Big{(}\tau(b)-M_{ 1}(b)\ddot{\theta}_{1}-C_{1}(b)\dot{\theta}_{1}-G_{1}(b)\Big{)}.\] (10)  
According to Property 2: \(J_{\psi}(\theta_{1})L(\theta_{1})=L^{T}(\theta_{1})J_{\psi}^{T}(\theta_{1})=0\), and the two equations generated by Equation (5) are linearly related, and the correlation coefficients are \(L^{T}(\theta_{1})\), then \(f_{a}=f_{b}\). Then, Equation (1) can be expanded as follows:  
\[\left\{ \begin{array}{l}\Psi(\theta)=l_{1}cos\theta_{1}+l_{2}cos(\theta_{1}+ \theta_{2})=0\\ l_{\psi}(\theta)=-l_{1}sin\theta_{1}-2l_{2}sin(\theta_{1}+\theta_{2})\end{array}  \right., \tag{11}\]  
When \(\theta_{1}+\theta_{2}=\pi\), \(\theta_{1}=0\), \(\theta_{2}=\pi\), then \(J_{\psi}(\theta)=0\Longrightarrow f=0\).