# Sliding Balance Control of a Point-Foot Biped Robot Based on a Dual-Objective Convergent Equation
## 1 Introduction  
Biped robots, with a legged structure similar to that of biped creatures, are naturally capable of adapting to a variety of environments [1, 2]. Biped robots perform better than wheeled and tracked robots on complex, unstructured and discontinuous ground [3, 4, 5]. To take full advantage of the characteristics of the bipedal structure and to improve the environmental adaptability of biped robots, control algorithms such as those for walking, running, jumping and balance recovery have been designed and adapted to terrains such as grass [6] and stepping stones [7]. Furthermore, biped robots are expected to be widely used in human habitats owing to their human-like shape and in the wild and high-risk environments owing to their wide terrain adaptability. In these environments, it is possible for robots to encounter terrains with low-friction coefficients, such as smooth ceramic tiles, glass floors and hard ice. For such terrains, many robots use an extra device, such as a pulley, skateboard or wheeled foot, to control sliding [8, 9, 10], which not only limits the robot's ability to adapt to a wider range of terrain but also increases the robot's mass and inertia at the end of the leg and deprives the robot of the fast motion characteristic of the point-foot structure. On these types of terrain, the high dynamic characteristics of the point-foot structure make it difficult for the robot to maintain balance by stepping, but the method of adjusting posture is still effective. Therefore, to expand the adaptive ability ofthe point-foot biped robot, this article investigates a control method of sliding and standing posture balance on ground with a low-friction coefficient. In recent years, many control methods have been applied to the balance control of biped robots. The key to balance control is to establish the control objective indicating the balance state of the robot. In bipedal control, the center of pressure (CoP) represents the action point of the contact force on the sole of the robot, which is an efficient control objective reflecting the plantar balance [11, 12, 13, 14, 15, 16]. [PERSON] et al. [14] introduced generalized momentum to design the CoP for standing balance. Similarly, [PERSON] and [PERSON] [15] defined balance control objectives more intuitively in terms of linear and angular momenta to directly determine the CoP with the ground reaction force and realize balance control on non-level and non-stationary ground. Momentum plays an important role in postural balance, and the centroidal moment point (CMP) has thus been proposed and used to describe the variation in angular momentum [17, 18, 19, 20]. [PERSON] et al. [17] referred to the CMP as the ZRAM (zero rate of change of angular momentum) point and presented three strategies (i.e., enlarging the support polygon, moving the center of mass (CoM), and changing the direction of the ground reaction force ) to recapture balance. [PERSON] et al. [19] introduced an excellent CoM-CMP regulator for reference CMP design with linear and angular momentum priority and used the regulator to construct a balance controller for the standing and walking on a humanoid robot. A comparison of these two methods (of [PERSON] et al. and [PERSON] et al.) shows that the CMP can be set outside the supporting polygon to expand the stable region of the robot. [PERSON] et al. [20] applied a capture point to evaluate the balance recovery capability and presented a push recovery controller based on a feedback controller by combining the CMP method and ankle strategy to compensate for the error at the CP. The above methods have been shown to work well in postural control and recovery. However, the CoP and CMP have mainly been aimed at the area of contact between the ground and the sole of a foot, and they are difficult to apply to the point-foot of a robot, which is in point contact with the ground. Therefore, a method of keeping the point-foot posture balance by constantly adjusting the foothold has been proposed [21, 22].