## 5 Simulations and Experiments  
We now present some simulations of the BHR-6 humanoid robot of Beijing Institute of Technology to illustrate the performance of the turning gait planning method. The robot parameters are shown in  
Figure 10: Rotation angle of the center of mass (COM) coordinate system. Table 1. We also present some comparisons between two different walking speeds. Through comparing different walking speeds, we can determine the function of the proposed gait planning method. The planned ZMP as-obtained by the cost function (\(f\)) is shown in Figure 11. We set the walking step to \(T_{\text{step}}\)\(=1.2\) s and sampling cycle to \(T_{\text{sample}}\)\(=0.004\) s. The walking step was \(D_{\text{step}}\)\(=0.33\) m, and the turning radius was \(r_{c}\)\(=1.06\) m; these are the planning results in the global reference frame. The ZMP was in the boundary of the supporting area and continued moving forward whether in the double supporting period or single supporting period. This planning result is similar to that of linear walking. Once the ZMP trajectory was planned, the new trajectory was generated based on the new LIP model expressed in Formula (6). The formula is easy to be discretized. We simulated two walking speed trajectories with different radii and walking speeds to test the practicability of this method. As shown in Figure 12, the trajectory was generated as the robot turned to the right. As opposed to straight-line walk, turning walk creates a trajectory offset to the direction in which the robot is moving. When the robot turns right, the trajectory of COM will shift to the right and for left turns, it will shift to the left. As shown in Figure 13, the faster the robot moves, the more the trajectory is offset. The foot width of our BHR-6 was 14 cm but when the robot walking speed was 2 km per hour, the offset achieved 4.5 cm with the radius \(r_{\text{c}}\)\(=0.67\) m. When the robot walks as slowly 1 km/h, the offset fell to 0.8 cm with the same turn radius. For the same walking speed, the offsets caused by different radii were also different. For the walking speed 2 km/h per hour, the offset was 2.5 cm with the radius \(r_{\text{c}}\)\(=1.06\) m. When the robot walks slowly, the centroid shift caused by centripetal force can be neglected and the trajectory can be planned in the same manner as for straight-line walking. We reached the results shown in Figure 14 by calculating the \(\Delta zmp\). Thus we can see the influence of robot rotation on the robot's ZMP position. Both walking speed and turning radius had significant impacts on the offset of the ZMP. A decrease in the radius and increase in walking speed caused \(\Delta zmp\) to increase. As shown in Figure 14, when the radius was 0.67 m and the walking speed was 2 km per hour, the greatest offset of ZMP was 6.2 cm; this offset markedly affects the planning of the reference ZMP trajectory. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline Degrees of freedom & 23 \\ Weight & 50.0 kg \\ Height & 1.65 m \\ Foot length & 24 cm \\ Foot width & 14 cm \\ Hip width & 15 cm \\ Contact type & Point contact \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of robot BHR-6. Figure 11: ZMP planning for the robot turning walk. Figure 13: COM trajectories in the involute reference frame (IRF) (**a**) \(v_{\rm ave}=1\) km/h, (**b**) \(v_{\rm ave}=2\) km/h. As opposed to straight-line walk, during turning walk, the hip yaw joints underwent major rotation at the angle shown in Figure 15. During a single support period, the supporting leg does not turn, and only the swinging leg turns. This means that the body does not turn. During a double support period, both hip yaw joints turn, and the body turns through an angle. Because the support area in the double support period is much larger than that in the single support period, the robot is less likely to slide on the ground when it turns fast. These phenomena correspond with the foot position planning results.