## 1 Introduction  
Humanoid robots are expected to be useful in various environments where people live. The reason for this is that humanoid robots, which are close to humans in behavior and functionality, are easy to adapt to the living environment designed for human beings. In addition, while other robots have only a few controllable joints, humanoid robots have more than 20 joints and can negotiate various scenarios. Locomotion is necessary to work in various situations, and so far, many studies have been performed on stable walking motion generation techniques. In recent years, running, which is a movement mode including the flight phase with a faster moving speed compared to walking, has also attracted attention, and research is being advanced on running motion generation methods to improve the movement ability of a humanoid robot. [PERSON] et al. developed a running robot with a single linear leg [1]. The bipedal robot ATRIAS has a four-bar leg mechanism that includes a series of elastic springs [2; 3]. [PERSON] et al. developed a biologically inspired robot based on a dog-leg model [4]. However, these robots do not have a human-like structure. Some studies have shown that bipedal humanoid robots can run [5; 6; 7; 8; 9]. For example, the Advanced Step in Innovative MObility (ASIMO) humanoid robot, which was designed and developed by Honda, can run at a speed of 2.5 m/s [10]. Toyota's bipedal humanoid robot can run using a zero-moment-point (ZMP)-based running control system [11]. The athlete robot developed by [PERSON] et al. has a human-like musculoskeletal system built to execute dynamic motions, such as running [12]. The bipedal robot MABEL, developed by researchers at the University of Michigan, has leg elasticity that originates from a leaf spring. It is the fastest-running of all currently available bipedal robots, having achieved a speed of 3 m/s with axial constraints on the y-axis [13]. However, present humanoid robots cannot run as fast and stably as human beings, who can run at speeds ranging between 2 m/s and 13 m/s. The reason that ordinary humanoid robots cannot run as fast and stably as humans because they require both a large power output for kicking the ground and various stabilization control methods. In general, to increase the power output, actuators having large power output capacity are required. However, high power actuators are heavy and their use in the humanoid robots renders the robots heavy. Moreover, the actuators require higher power. Therefore, it is difficult to design humanoid robots that can achieve high power output and are light enough to jump. Ordinary humanoid robots can attain a power output of approximately 3.5 W/kg for the joints in the leg; however, humans generate around 16.7 W/kg in leg joints while running [14]. In addition, various stabilization control methods such as considering the center of mass position, landing point, ground reaction force, and linear and angular momentum, are needed for stable running. However, present running stabilization methods do not consider motion during the flight phase. For example, several studies on running control have used the spring-loaded inverted pendulum (SLIP) model [15; 16; 17]. This simple model does not consider the mass of the legs, which generates angular momentum by leg swinging [18]. However, a humanoid that can perform human-like fast lower-leg swinging, generates large angular momentum during the flight phase, especially in the yaw direction. Some studies considered the angular momentum of the entire body; however, they focused mainly on only stance-phase movement [19; 20]. Thus, in these studies, humanoids could run slowly without leg swing during the flight phase to decrease the angular momentum generated by leg motion. Therefore, the humanoid requires a method to compensate for angular momentum during the flight phase for high speed running.