## 3 Experiments and Results
### Angular Momentum Compensation in the Yaw Direction  
We conducted an experiment to evaluate how effectively the humanoid could compensate for the angular momentum generated by the leg motion during the flight phase. In this experiment, the humanoid was suspended in midair to perform running motion during the flight phase without any constraint in the yaw direction. The lower-body joints were controlled according to the joint angle references based on the human motion data [25]. We verified the humanoid turning angles in the yaw direction depends on whether control is provided, which depended on the angular momentum in the yaw direction. We measured the turning angle of the waist in the yaw direction using a motion capture system that included infrared cameras used to determine three-dimensional marker positions at 120 Hz. Spherical retro-reflective markers were attached to the surface of the waist parts of the humanoid near the COM of the whole body. The motion capture system calculated the segment attitude using the position data for some of the markers. When angular momentum compensation was not employed, each joint of the upper body maintained a neutral position. When angular momentum compensation was employed, the angular momentum generated by the leg motion was calculated, and the upper body moved to compensate for the angular momentum actively. The ratios of the arms contribution and trunk contribution were determined to be 0.8 and 0.2, respectively, based on the human running data [26] in which the angular momentum generated by the arms was 1.2 kgm\({}^{2}\)/s and that generated by the trunk was 0.3 kgm\({}^{2}\)/s. The lower-body motion of each joint was determined to follow an approximate sine wave as Equation (10), based on the human running data [15].  
\[\theta_{HIP}=\theta_{HIP\_INITIAL}-A_{HIP}\sin(\omega t) \tag{10}\]  
where, \(\theta_{HIP\_INITIAL}\) is the initial joint angle, \(A_{HIP}\) is the amplitude of the joint motion, \(\omega\) is the natural frequency of the joint motion, and \(t\) is the elapsed time from the start of the experiment. The experimental parameters and their values of joint movements are listed in Table 9.  
Figure 6 illustrates the rotation angle of the humanoid waist in the yaw direction. Figure 7 presents photographs of the experiment. When the angular momentum compensation control was not employed, the humanoid rotated approximately 15 deg in the yaw direction. On the other hand, when the control was used, the rotation decreased to approximately 8 deg. The results confirm the effectiveness of the proposed angular momentum compensation control for stabilization in the yaw direction. It was assumed that the friction force in the yaw direction of the hanger influenced the angular momentum because it was small but not zero.  
\begin{table}
\begin{tabular}{l l l} \hline \hline  & **Parameters** & **Value** \\ \hline Hip pitch amplitude & \(A_{HIP}\) & deg & 15 \\ Hip pitch initial angle & \(\theta_{HIP\_INITIAL}\) & deg & 10 \\ Knee pitch amplitude & deg & 20 \\ Knee pitch initial angle & deg & 150 \\ Ankle pitch amplitude & deg & 0 \\ Ankle pitch initial angle & deg & 90 \\ Movement period & s & 1.0 \\ Arm gain of angular momentum control & K & 0.8 \\ Torso gain of angular momentum control & & 0.2 \\ \hline \hline \end{tabular}
\end{table}
Table 9: Experimental conditions.