# Human-Like Walking with Heel Off and Toe Support for Biped Robot
## 1 Introduction  
Humans are the most important inspiration for research on biped robots because of their resemblance. The unique neural and morphological mechanisms enable humans to show very versatile and efficient locomotion. Consequently, realizing human-like walking on biped robots has been an attractive research field for decades. Up to now, various advanced biped robots have been developed, from childlike robots NAO [1,2] and iCub [3,4] to human-sized robots Asimo [5,6], HRP [7, 8, 9], and Atlas [10,11]. For these biped robots, the most popular control approach is based on the Zero Moment Point (ZMP) theory, i.e. the contact point of the foot with the ground where the total of horizontal inertia and gravity forces equals zero is strictly within the support polygon [12,13]. Following the ZMP criterion ensures that these biped robots walk stably and robustly on a variety of terrains. However, in order to increase the stability of walking, the stance foot of biped robots is always flat on the ground, making the gait unnatural and energy-inefficient. Actually, closer investigation on human walking reveals that, by the end of the single support phase, the heel of the stance foot rises from the ground and rotates around the toe, with the stance toe touching the ground and supporting the body [14; 15]. Even from the gait of persons with lower limb disabilities wearing prostheses can foot rotation be observed [16]. The so-called \"heel off and toe support\" yields a higher foot clearance, a higher walking speed, and less energy consumption [17; 18; 19]. Taking the human as a mechanical locomotion system, it will be under-actuated when foot rotation occurs because there is no actuation between the stance toe and the ground [20]. Although dealing with the changes in dynamic characteristics is seemingly effortless for humans, it is much more difficult for biped robots. Therefore, it is of great important to study human-like walking of biped robots with under-actuated foot rotation. In order to attain this goal, some researchers have adopted the ZMP-based method [21; 22; 23]. The desired walking pattern that includes foot rotation was generated by solving an inverse kinematics problem with a predefined ZMP and foot trajectories and achieved under servo controllers such as preview control. [PERSON] et al. [24] proposed a path-following control strategy to handle more human-like walking. The control strategy was defined in such a way that only the kinematic evolution of the state of the robot was controlled, but not its temporal evolution, allowing simultaneous control of the ZMP and the joint positions. Since the ZMP could be directly regulated, stable bipedal walking that includes prescribed foot rotation was realized. [PERSON] and [PERSON] et al. [25; 26; 27] used a human-inspired methodology to translate human locomotion to a biped robot. By analyzing human locomotion data, the human-inspired outputs of the robot were defined based on the human outputs and the representation function. Then, a controller was designed for the biped robot, enabling it to mimic human kinematics behaviors by tracking the functions of the kinematics. Although bipedal robotic walking with under-actuated foot rotation has been realized in the above research, the methods are usually very complex. Thus, a different and simpler approach is proposed in this paper. The walking control scheme adopts a hybrid controller that combines the ideas of virtual force control and foot placement control. The virtual force control, which is applied on the stance leg, assumes that there is a virtual force on the center of mass of the torso that can generate the desired torso motion including foot rotation, and the real torques are exerted on each actuated joint of the stance leg to create the same effect that the virtual force would have created. The foot placement control, which is applied on the swing leg, uses a path-tracking controller to follow the predefined curve of the swing foot to step forward.