# Sliding Balance Control of a Point-Foot Biped Robot Based on a Dual-Objective Convergent Equation
## 1 Introduction
Nevertheless, adjusting the foothold on ground with a low-friction coefficient is likely to cause the robot to slip and lose its balance. [PERSON] et al. [23] took the angular momentum and its derivative as the equilibrium point and proposed the loss balance degree (LBD) as a stability criterion with which to evaluate the degree to which a biped robot deviates from the equilibrium point. However, they did not apply the method to a specific robot model. Inspired by the revelation of the motion principle of the flywheel inverted-pendulum model using the CMP, in the present study, we combine the flywheel inverted-pendulum model with the Coulomb friction model to reconstruct a sliding model of the robot into a dual-objective convergent form including the posture stability criterion and the equilibrium control target, which not only is suitable for the sliding process but also can be extended to the case of standing balance. In adjusting the balance of the whole body, most of the joints of the robot are expected to be controlled while maintaining sliding balance. Control methods for sliding biped robots with fewer degrees of freedom (DoFs), such as the wheeled inverted-pendulum robot, have been developed over a period of decades. Many traditional controllers, such as the state-space controller [24], sliding-mode controller [25], and fuzzy controller [26], have been extensively used to control wheeled inverted-pendulum systems. Although these control methods perform well, they are difficult to adapt to more complex bipedal structures. [PERSON] and [PERSON] [27] adopted the linear quadratic regulator (LQR) method to linearize the control model of the robot and realized stable mobile motions for the i-Pentar robot, which comprises a body with a waist and an arm. To use the LQR method, the robot's model needs to be linearized, which reduces the stability region in the state-space. Meanwhile, it is difficult to use this method to solve control problems under the constraints of friction, contact forces and joint acceleration. [PERSON] [28] first used inverse dynamics to convert the task space problem to the joint space problem. Adopting the inverse dynamics control method, multi-task motion control can be nonlinearly mapped to joint motion con trol. [PERSON] et al. [29] treated the inverse dynamics problem as a quadratic programming (QP) problem and provided an optimization method to solve inverse dynamics problems with constraints. [PERSON] et al. [30] applied QP whole-body dynamics control to the wheeled humanoid robot with an arm and realized low-level centroid tracking control under the constraint of the joint torque. [PERSON] et al. [31] extended the wheeled-foot sliding equilibrium capability under non-minimum phase dynamics by considering LQR control as a constraint condition of the QP problem. However, whole-body control is mainly applied to the active movement of a robot with feet and rarely applied to the passive sliding of the point-foot robot. Nevertheless, the control objects of the above algorithms have end-effector devices that can control their speed or angle with the ground, and it is thus difficult to fit with the Coulomb friction characteristics of the ground in point-foot sliding. In this article, we establish the constraint conditions, relaxation factors and cost function for the point-foot robot conforming to sliding conditions using the QP-based whole-body control method. This article proposes a control framework based on whole-body dynamics control and dual-objective convergence for point-foot bipedal balance control, including sliding and standing balance recoveries, on low-friction terrain. In the control framework, the posture balance controller is a trajectory task generator that includes an equilibrium target generator, torso recovery equilibrium offset generator, and knee trajectory generator, while the whole-body controller is a trajectory-task-to-torque converter that adopts proportional-derivative (PD) control, physical constraints, and the optimization of whole-body dynamics (as shown in Figure 1).