# Sliding Balance Control of a Point-Foot Biped Robot Based on a Dual-Objective Convergent Equation
## 4 Results
### Balance in Sliding on Uneven Terrain  
To verify the sliding capability on uneven terrain, a terrain including both uphill and downhill is modeled in the simulation scene. The robot is initially set to slide across the horizontal plane at an initial speed of 5 m/s and then slide into a trapezoidal terrain comprising an uphill, flat-top and downhill. The lengths of the uphill, flat-top and downhill sections are 2 m, the friction coefficient is 0.15, and the slope is 10 degrees. The robot is released from a balanced posture with the torso upright and the knees bent at 90 degrees. The parameters of the control system are given in Table 2 and the friction coefficient of the control system is estimated according to Equation (15) during the sliding. Figure 5 shows snapshots of the sliding simulation when the slope changes and the sliding stops. The sliding slope changes at 0.23, 0.75, 1.73 and 2.97s, after which the robot adjusts its attitude and recovers to the torso recovery state. The simulated motion is shown in the supplemental video. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Description** & **Symbol** & **Value** & **Unit** \\ \hline torso mass & \(m_{0}\) & 15.6 & Kg \\ thigh mass & \(m_{1}\) & 6.8 & Kg \\ calf mass & \(m_{2}\) & 4.5 & Kg \\ torso inertia & \(l_{0}\) & 0.49 & Kg \(\cdot\) m\({}^{2}\) \\ thigh inertia & \(l_{1}\) & 0.11 & Kg \(\cdot\) m\({}^{2}\) \\ calf inertia & \(l_{2}\) & 0.09 & Kg \(\cdot\) m\({}^{2}\) \\ torso length & \(l_{0}\) & 0.40 & m \\ thigh length & \(l_{1}\) & 0.33 & m \\ calf length & \(l_{2}\) & 0.33 & m \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Mechanical parameters used in the simulation. Figure 6 shows the results of the friction coefficient estimation. The estimated friction coefficient changes as the robot slides onto a new slope. On the uphill and downhill, the estimated friction coefficient finally stabilizes at 0.338 and \(-\)0.025, which correspond to ground angles of 10.1 and \(-\)10.0 degrees estimated using Equation (14). Therefore, the friction coefficient estimated by the contact force effectively compensates for the ground slope angle. Figure 7 shows the tracking result for the equilibrium variable. Abrupt changes in the estimated friction coefficient led to mutations in the equilibrium variable. There are a total of four mutations in the simulation, and the time from mutation to a reduction in the tracking error to less than \(\rho_{U}-\Delta\rho_{U}\) (the maximum allowable error in Equation (22) for fully activating the equilibrium buffer) is 0.14, 0.12, 0.12 and 0.18 s. This result indicates that the dynamics-based control system can quickly converge from an unbalanced state to a stable torso recovery state. \begin{table}
\begin{tabular}{c c c} \hline
**Description** & **Symbol** & **Value** \\ \hline dual-objective proportion & \(\varepsilon\) & 0.025 \\ torso control parameter & \(K_{\theta}\) & [0.50, 0.25, 0.04, 0.02, 10\(\pi\)/180, 500\(\pi\)/180, 0.0125, 0.0156] \\ equilibrium variable PD parameter & \(K_{lt_{sl}}\) & [2.5, 0.02, 3.0, 0.5, 50.0, 1.0] \\ knee PD parameter & \(K_{q_{2}}\) & [\(\pi\), 10\(\pi\)/180, 1.5\(\pi\), 0.5, 20\(\pi\), 200\(\pi\)/180] \\ equilibrium variable relaxations weight in QP & \(\rho_{lt_{sl}}\) & 1000 \\ knee relaxations weight in QP & \(\rho_{q_{2}}\) & 1 \\ max Vertical contact force proportion & \(\alpha_{max}\) & 2.5 \\ min Vertical contact force proportion & \(\alpha_{min}\) & 0.5 \\ \hline \end{tabular}
\end{table} 
Table 2: Control parameters in sliding. Figure 5: Snapshots of sliding on uneven terrain: (**a**) sliding from the bottom to the uphill, (**b**) sliding from the uphill to the flat-top, (**c**) sliding from the flat-top to the downhill, (**d**) sliding from the downhill to the bottom, (**e**) sliding having stopped. Figure 6: Trajectory of the estimated friction coefficient when the robot slides on uneven terrain. Figure 8 shows the trajectory of the postural stability criterion. The tracking error of \(U_{sl}\) will increase when the slope angle changes.