Divergent Component of Motion (DCM)

The Divergent Component of Motion (ξ) is a state variable defined for the Linear Inverted Pendulum Model (LIPM). It is a linear combination of the Center of Mass (CoM) position (x) and velocity (ẋ): ξ = x + (ẋ / ω₀), where ω₀ = √(g / z₀) is the natural frequency of the pendulum, g is gravity, and z₀ is the constant CoM height. This formulation explicitly separates the unstable (divergent) part of the CoM dynamics from the stable (convergent) part.

The DCM represents the point on the ground toward which the CoM is currently falling. If the robot could instantaneously place a foot at the DCM and pivot around it with a massless leg, the CoM would come to rest exactly above that foot. Therefore, controlling the DCM is synonymous with controlling the robot's falling direction. A stable walking pattern is achieved by strategically placing footsteps to 'capture' the evolving DCM, preventing it from diverging beyond reach.

The DCM is fundamentally linked to the Capture Point (CP). For a robot with a point-foot model and instantaneous foot placement, the DCM and Capture Point are identical. The Capture Point is specifically defined as the point on the ground where the robot can step to bring its CoM to a complete stop. In more general planning contexts, the DCM is the variable that is steered, and the planned footstep location becomes the instantaneous Capture Point for that phase of motion.

DCM dynamics are simpler than full centroidal dynamics, making them ideal for real-time Model Predictive Control (MPC). A standard approach is the DCM Trajectory Generator:

• The future evolution of the DCM is predicted using its linear dynamics.
• An MPC optimizer solves for future footstep locations that keep the DCM within a viable region (e.g., inside the support polygon of future steps).
• This generates a dynamically feasible DCM trajectory, from which the required CoM motion and Ground Reaction Forces (GRFs) are derived. This abstraction enables planning over many steps in milliseconds.

While the classic DCM is defined for the 2D LIPM, it can be extended to 3D for full bipedal control. The 3D DCM is a vector point on the ground plane. Its dynamics are decoupled in the frontal (side-to-side) and sagittal (forward-backward) planes, allowing separate planning. This 3D formulation is crucial for generating omnidirectional walking and handling disturbances in any horizontal direction. The height z₀ remains constant in this model, a key simplification of the LIPM.

The DCM and Zero-Moment Point (ZMP) are complementary stability metrics with different uses:

• ZMP: A measure of dynamic balance. It's the point where the net horizontal moment is zero. The controller's goal is to keep the ZMP within the support polygon.
• DCM: A planning variable for future stability. It predicts where the CoM is heading. The planner's goal is to place footsteps to guide the DCM. In essence, the ZMP is an output of the current motion, while the DCM is a target for future action. Modern hierarchical controllers often use DCM-based planners to generate CoM trajectories that ensure the ZMP remains stable.

The Capture Point is the specific point on the ground where a legged robot can place its foot to bring its Center of Mass (CoM) to a complete stop in one step. It is the instantaneous goal of the DCM.

• Direct Relationship: The DCM is the future location of the Capture Point if no control action is taken. Planning a footstep at the current Capture Point instantly brings the DCM to that location, arresting its divergence.
• Control Target: In the 3D Linear Inverted Pendulum Model, the Capture Point is the horizontal projection of the DCM onto the ground plane. Foot placement controllers directly target the Capture Point for instantaneous stabilization.

The Linear Inverted Pendulum Model (LIPM) is the simplified dynamic template from which the DCM is derived. It is the foundational analytical model for bipedal walking.

• Core Assumptions: Models the robot as a point mass (the CoM) atop a massless leg. Assumes constant CoM height and linearized dynamics, which decouples horizontal motion.
• DCM Derivation: The DCM (ξ) is defined from the LIPM's equations of motion: ξ = x + (ẋ / ω), where x is the CoM position, ẋ is the CoM velocity, and ω = sqrt(g / h) is the natural frequency of the pendulum. This formulation explicitly separates the stable (CoM) and unstable (DCM) components of the system's dynamics.

The Zero-Moment Point (ZMP) is a stability criterion defining the point on the ground where the net horizontal moment of inertial and gravitational forces is zero. It is fundamentally different from, but used in conjunction with, the DCM.

• Dynamic vs. Kinematic: The ZMP is a result of the executed motion and ground reaction forces. The DCM is a state variable used for planning that motion.
• Control Relationship: In LIPM-based control, a planned ZMP trajectory (e.g., moving from foot to foot) is used to govern the evolution of the DCM. The DCM dynamics are: ξ̇ = ω (ξ - p), where p is the ZMP. The ZMP acts as a control input to steer the divergent component.

A Reduced-Order Model (ROM) is a simplified dynamical representation that captures the essential behavior of a complex system for planning and control. The DCM is a state within the most influential ROM for bipedal walking.

• Purpose: ROMs like the LIPM ignore high-dimensional complexities (e.g., limb dynamics, joint flexibility) to make real-time trajectory optimization tractable.
• Hierarchical Control: The DCM, derived from the LIP ROM, is used for high-level step timing and placement planning. These plans are then tracked by a full-body controller (like Whole-Body Control) that manages the complete robot dynamics to realize the ROM's motion.

Model Predictive Control (MPC) is an advanced control paradigm that uses a system model to predict future states and optimize a sequence of control inputs. DCM-based walking controllers are frequently implemented within an MPC framework.

• Prediction & Optimization: The LIPM, with the DCM and ZMP, provides a simple, linear model perfect for MPC. The controller solves an optimization problem at each control cycle to find the optimal future ZMP trajectory that guides the DCM to stable locations (future Capture Points).
• Real-time Replanning: This allows the robot to reactively adjust its footstep plan in real-time to account for disturbances, pushing, or uneven terrain, making DCM-MPC a standard approach for robust dynamic walking.

Centroidal dynamics describes the relationship between the net external wrench (force and torque) acting on a robot and the motion of its Center of Mass (CoM) and its angular momentum. The DCM is a specialization of this for simplified walking.

• Full Dynamics Context: The LIPM and DCM assume zero centroidal angular momentum and a constant CoM height, which are major simplifications of the full centroidal dynamics.
• Advanced Extensions: For more complex motions (running, jumping, or manipulation while walking), the basic DCM is extended into concepts like the Angular Divergent Component of Motion (ADCM) or used within more complete centroidal momentum planning frameworks that account for rotational dynamics.