# Sliding Balance Control of a Point-Foot Biped Robot Based on a Dual-Objective Convergent Equation
## 2 Dual-Objective Convergence Equation for the Biped Robot
### Torso Recovery Strategy  
By controlling the equilibrium variable \(U\) to track the equilibrium target \(U^{ref}\), both \(E\) and \(\dot{U}\) are stabilized. To achieve whole-body control, however, all the controlled DoFs must be used up. Hence, the generalized coordinate of the joint of knee \(q_{2}\) is expected to be another controlled variable that is used to drive the angle of the knee to a constant reference angle \(q_{2}^{ref}\). By effectively controlling \(U\) and \(q_{2}\), the robot will gradually tend to balance and eventually stay in some stationary posture, because the convergence of the triple variable (\(E\), \(\dot{U}\) and \(q_{2}\)) can be mapped to the convergence of the three-DoF posture coordinate \(q_{p}\). However, while this method can stabilize the robot, it is unable to converge the torso to a desired angle. The following torso recovery strategy is thus adopted to adjust the angle of the upper body.  
* First, reduce the difference (i.e., error) between \(U\) and \(U^{ref}\) so that \(E\) begins to converge in the desired way.
* Then, wait for the amplitude of \(E\) to gradually decrease to within a permissible range, which indicates that the robot enters a stable state.
* Finally, add an equilibrium offset \(\Delta U\) to the desired equilibrium target \(U^{ref}\) according to the feedback states of the torso. Therefore, the angle of the torso \(\theta\) gradually recovers to the reference value.  
To put the above strategy into practice, we designed a torso position feedback control method based on coefficient regulation, which is summarized as  
\[\beta_{E}=clamp\left(\frac{\rho_{E}-|E_{n}|}{\Delta\rho_{E}},0,1\right), \tag{21}\]  
\[\beta_{U}=clamp\left(\frac{\rho_{U}-\left|U_{n-1}^{ref}-U_{n-1}\right|}{ \Delta\rho_{U}},0,1\right), \tag{22}\]  
\[\Delta U_{\theta}=clamp\left(\frac{\theta^{ref}-\theta}{\Delta_{\theta}},-1,1\right), \tag{23}\]  
\[\Delta U_{\theta}=clamp\left(\frac{\dot{\theta}}{\Delta_{\theta}},-1,1 \right), \tag{24}\]  
\[\Delta U_{n}=\beta_{E}\cdot\beta_{U}\cdot\left(\beta_{\theta}\cdot\Delta U_{ \theta}+\beta_{\dot{\theta}}\cdot\Delta U_{\dot{\theta}}\right), \tag{25}\]  
\[clamp(x,a,b)=\left\{\begin{array}{ll}x,&a<x<b\\ a,&x<a\\ b,&x>b\end{array}\right.. \tag{26}\]  
Here, \(n\) is the current control period. \(\beta_{E}\) is the coefficient regulator of the posture balance criterion \(E\), and it increases linearly from 0 to 1 as \(|E_{n}|\) decreases from \(\rho_{E}\) to \(\rho_{E}-\Delta\rho_{E}\). \(\beta_{U}\) is the same regulator as \(\beta_{E}\), and the product of these two coefficients \(\beta_{E}\beta_{U}\) is the torso recovery coefficient. \(\Delta U_{\theta}\) and \(\Delta U_{\dot{\theta}}\) are the proportional and differential limiters, \(\Delta_{\theta}\) and \(\Delta_{\dot{\theta}}\) are the linear ranges of the limiters. \(\beta_{\theta}\) and \(\beta_{\theta}\) are the proportional and differential constants of the feedback controller. \(\Delta U_{n}\) is the equilibrium offset of the current control period. The parameters in the equation are represented by a constant vector \(K_{\theta}=\left[\rho_{E},\Delta\rho_{E},\rho_{U},\Delta\rho_{U},\Delta_{ \theta},\Delta_{\theta},\beta_{\theta},\beta_{\theta}\right]\). After introducing the torso control strategy, compared with Equation (2), the equilibrium target \(U^{ref}\) is modified as  
\[U_{n}^{ref}=-\varepsilon E-\Delta U_{n}. \tag{27}\]