Section-Map Stability Criterion for Biped Robots

The importance of stability for dynamical systems is well-known. Any real system, including biped robots, need to be working under all kinds of disturbances. Whether the biped robot can effectively keep the planned motion under these disturbances is a fundamental property, and that is the explanation of stability intuitively. Stability of biped walking is the key problem in the theoretical framework of biped robots. Roughly speaking, the research of biped robots can be classified as the following three aspects: stability criterion, walking pattern planning, and walking pattern control. The purpose of stability criterion is to give the condition that the robot can realize stable walking under some control strategy. The purpose of walking pattern planning is to generate a desired gait offline or online, and it plays the role of feed-forward (Huang et al., 2001). The purpose of walking pattern control is to modify the planning walking pattern based on sensory information, and it plays the role of feedback (Huang & Nakamura, 2005). Among the above three aspects, stability criterion is the most fundamental and important, and it is the basis of walking pattern planning and real-time control. Although some researchers have proposed several walking control methods which are not based on stability criterion (Raibert, 1986; Geng et al., 2006); however if these methods can not ensure walking stability from the aspect of theory, then it will need many trials on hardware before success, and it is difficult to generate them to other platforms. Presently, there are the following three stability criteria for biped walking. The first criterion is zero moment point (ZMP) criterion. The ZMP was originally defined as the point in the ground plane about which the net moments due to ground contacts become zero in the plane of ground (Vukobratovic & Juricic, 1969). As long as the ZMP lies strictly inside the support polygon of the foot, then the support foot will not rotate about its extremities, and the desired trajectories of the robot’s joints are dynamically feasible, just like a stationary manipulator. Takanishi et al. (1985) and Hirai et al. (1998) have proposed the methods of pattern synthesis based on ZMP offline. Recently, Kajita et al. (2001), Lim et al. (2002), and Nishiwaki et al. (2002) discussed the methods of online pattern generation. The ZMP criterion is not a necessary condition for stable walking. The ZMP criterion results in a flat-footed and short-step walking style which is less dynamic than human beings. During normal walking, human do not always obeys the ZMP requirement and the foot does not always remain flat on the ground. Humans, even with prosthetic legs, use foot rotation to decrease energy loss at impact (Kuo, 2002). Based on the ZMP criterion, the robot can only realize static walking or quasi-dynamic walking, as shown in Fig. 1(a) and (b). During the dynamic walking of human beings, the under-actuated degree-of-freedom (DOF) emerges between the support foot and the ground, as shown in Fig.1 (c).