# Simultaneous Prevention of Rotational and Translational Slip for a Humanoid Robot
## 5 Simulations and Experiments
### Simulations  
The physical parameters of the virtual biped robot model are the same as the actual robot. The robot walked on a floor in the simulations. The physical parameters of the ground is set according to the actual ground in the experiments. The coefficient of static friction is estimated to be 0.5. Then, the threshold of the coefficient of friction, \(\mu_{0}\), is set to be 0.3.  
The comparison of without and with proposed slip prevention control is shown in Figure 6. The robot is commanded to walk along the straight dotted lines. Figure 6a shows that, without the slip prevention control proposed in this paper, the rotation slip occurs since the swing leg of the fast walking robot brings too large a yaw moment. Hence, the robot cannot walk along the dotted line. When the proposed slip prevention control is applied, this slip is prevented successfully, and the robot can walk along the dotted line, as Figure 6b shows.  
Figure 7 shows the yaw moment with and without rotational prevention control during biped walking in simulation. As the blue line depicts, the robot would suffer from large rotational slip without the proposed method since the yaw moment has large fluctuations because of the huge moment of the swing leg. When the proposed human-inspired walking pattern generation is utilized, the arm swinging and waist rotation effectively reduce the yaw moment as the red line depicts.  
Figure 6: Humanoid robot walking simulations.  
Thus, the proposed rotational slip prevention control, which generates the walking pattern based on the yaw moment compensation mechanism in humans, is effective.  
Figure 8a shows the ratio of the horizontal reaction force to the vertical reaction force, which is \(F_{x}/F_{z}\). As Equation (3) indicates, if the ratio of horizontal force to vertical force is smaller than the coefficient of the ground friction, the possibility of translational slip is smaller. When the proposed CoM acceleration compensation control works as the red line shows, the reaction force ratio is smaller than that without control which is plotted in blue line. That means the proposed CoM acceleration compensation control prevents the translational slip to a certain degree.  
Meanwhile, Figure 8b compares the ZMP in the forward direction with and without translational slip prevention control during biped walking. The original ZMP as the blue line depicts, is produced by the walking planning method in [8]. Thus, this ZMP is modified according to the CoM acceleration compensation and is depicted in the red line. Both ZMP are within the maximum stability margin. However, the red line is more close to the boundary of the stability margin, which means that the proposed method sacrifices part of the ZMP based on stability for slip prevention. This is realistic and quite helpful for humanoid walking in real environments.  
Figure 8: Reaction force ratio and Zero Moment Point (ZMP) in walking simulations. (**a**) Blue line: without proposed control. Red line: with proposed control. (**b**) Blue solid line: without proposed control. Red solid line: with proposed control. Short dotted line: maximum ZMP boundary. Long dotted line: minimum ZMP boundary.  
Figure 7: Yaw moment in walking simulations. Blue line: without rotational slip prevention. Red line: with rotational slip prevention.