# Whole-Body Motion Generation for Balancing of Biped Robot
## 1 Introduction  
Humanoids, similar to humans, can be utilized in various fields [1] and are increasingly accepted in public services [2, 3], replacing humans in existing social infrastructures. Humanoids need to perform tasks across a wide range of areas that other types of robots are not expected to handle. Since the dynamics of humanoid robots are highly complex, motion planning and control are particularly challenging [4, 5]. Since deriving the dynamic equations of humanoid robots is highly complex, and utilizing them in motion planning and control is very challenging, simplified models have often been used. One widely used simplified model is the linear inverted pendulum model (LIPM), where the robot is represented as a single mass, and a support point called the zero moment point (ZMP) is fixed to describe the robot's motion [6]. Although LIPM can easily generate the trajectory of the center of mass (COM), there exist differences in dynamics between the model and actual robots. To minimize the differences, the gravity-compensated inverted pendulum model (GCIPM) was introduced in [7], which incorporates the dynamics of the swinging leg ignored by LIPM. To reduce the model differences further, the dual linear inverted pendulum model (D-LIPM) was introduced, which splits the model into horizontal and vertical components while avoiding the complexity of nonlinear models [8]. Nonetheless, discrepancies between simplified models and actual robots, along with external disturbances, often cause instability during motion. To mitigate this, ZMP tracking controllers have been developed to minimize these inconsistencies. In [9], a linear quadratic optimal controller was used for ZMP feedback control. In [10], a preview controller was designed based on a future reference ZMP. The form of the preview controller is similar to that of model predictive control (MPC). In the field of robot control, MPC has beenwidely researched over the past decade. By reflecting the current state and replanning the motion of the robot at each step, it can make the robot more robust to disturbances [11]. In [12], the ZMP preview controller was improved by designing a linear model predictive control (LMPC), avoiding nonlinearity, taking advantage of shorter computation times, and tracking the reference ZMP even in the presence of disturbances. LMPC has extended to vertical motion and successfully replicated movements similar to human walking [13]. However, control strategies that simply track the ZMP have limitations in maintaining balance when walking under large disturbances or on uneven terrain, unless the ZMP is modified. To overcome the limitations of fixed ZMP tracking, the concept of the capture point (CP) was adopted to predict and stabilize the motion of biped robots under disturbance. CP was initially proposed in the context of representing the region on the ground where a robot has to take a step to come to a complete stop without falling [14] and is mathematically equivalent to other formulations such as the extrapolated center of mass (XCoM) [15] and the divergent component of motion (DCM) [16]. These concepts share the goal of capturing the divergent nature of bipedal dynamics and have been widely adopted in balance control and motion planning. CP not only reflects the instantaneous state of the COM but also enables a structural decomposition of the second-order LIPM dynamics into two first-order systems: one is a stable system in which the COM follows the CP, and the other is a naturally unstable system in which the CP is driven by the ZMP. This structure allows efficient stabilization by controlling only the unstable part. Since CP dynamics are inherently divergent and driven by ZMP, stabilizing CP effectively modifies the ZMP. In [17], a simple method was proposed to generate the COM trajectory that is more robust against disturbances by adjusting the ZMP based on the CP concept. In [18], CP tracking and a CP end-of-step controller were designed to stabilize the dynamics of the LIPM by modifying the reference ZMP.