## 2 Angular Momentum Compensation  
### Requirements for Angular Momentum Compensation  
We identified the characteristics of upper-body movement of human running on the basis of previous research in human and sport sciences, and we based our determination of the requirements for a control method and an upper-body mechanism on these characteristics. Table 1 summarizes the characteristics we identified. Humans utilize the torso and arms to compensate for angular momentum in the yaw direction generated by the movement of the lower body during the flight phase, during which the human is not in contact with the ground [26]. Here, the trunk refers to the parts higher than the lumbar vertebrae, and the trunk joints refer to the virtual joint between the trunk and the lumbar vertebrae, which consolides the complex movements of the spine. Rapid leg swinging during the flight phase is responsible for the large angular momentum produced. To compensate for the large angular momentum, both the torso yaw joint and the shoulder pitch joints swing. The shoulder pitch joints swing widely and rapidly because of the inertial moment of the arms in the yaw direction. Most of that upper-body movement is in the yaw direction. In addition, the shoulder roll and yaw points can change arm postures. A human extends his/her elbow joints according to his/her running speed [27]. The elbow joints can be used to adjust the inertial moment of the arms when large angular momentum is generated by quick lower-body movements. Therefore, we determined that the requisite joint motions of the upper-body mechanism are the three torso joint motions (pitch, roll, and yaw), the three shoulder joint motions (pitch, roll and yaw), and one elbow joint motion inspired by the human configuration. Furthermore, we determined the requirements for the link lengths and mass properties of the upper-body mechanism based on human data [32; 33] (see Tables 2 and 3; note that in the tables, S.D. means the standard deviation). In Table 3, the moment of inertia is with respect to the center of mass position of each link. These parameters influence the angular momentum; however, ordinary humanoid robots have very light arms so as to decrease the required output power of the leg joint, and thereby cannot generate the large angular momentum to compensate that generated by the leg. Therefore, we assumed that employing human-like parameters would be useful to utilize the human-inspired angular momentum compensation method. The requirements of the total mass and total height are 60 kg and 1600 mm based on human data [33]. In addition, we determined the requirements of the joints. The requirements for the movable angles of the joints were determined based on those of a human [34]. The angular velocity was determined based on human running data obtained in our previous research [25]. The torque was calculated from the angular velocity and mass properties of each link. These requirements are listed in Table 4. \begin{table}
\begin{tabular}{c c} \hline \hline
**Joints** & **Characteristics** \\ \hline Trunk Roll & Bending for angular momentum compensation \\ Trunk Pitch & Bending for angular momentum compensation \\ Trunk Yaw & Swing for generating angular momentum \\ Shoulder Roll & For changing the position of the center of mass \\ Shoulder Pitch & Wide and rapid swing for generating large angular momentum \\ Shoulder Yaw & For changing the position of the center of mass \\ Elbow Pitch & Changing the position of the center of mass and moment of inertia of the arm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Characteristics of the upper body during human running.