Divergent Component of Motion

Also called: DCM, extrapolated center of mass, XCoM
Category: Control › Locomotion control
Tags: bipedal-locomotion, balance, humanoids, legged-robots, reduced-order-models

The unstable part of a walking robot's center-of-mass dynamics, combining position and velocity into one point that must be controlled to keep the robot balanced.

Definition

Divergent component of motion (DCM) refers to a quantity used in legged robot balance control that isolates the unstable part of the center-of-mass (CoM) dynamics. For the linear inverted pendulum model it is defined as ξ = x + ẋ/ω, the CoM position plus its velocity divided by the pendulum's natural frequency ω = √(g/z), where z is the CoM height.

Writing the dynamics in terms of the DCM splits a second-order system into two first-order ones. The CoM is attracted to the DCM, ẋ = −ω(x − ξ), which is stable: the CoM simply follows. The DCM is pushed away from the center of pressure, ξ̇ = ω(ξ − p), which is unstable. A controller therefore only needs to keep the DCM under control by placing the center of pressure or the next footstep appropriately, and the CoM takes care of itself.

In the planar linear inverted pendulum, the DCM coincides with the instantaneous capture point described by Pratt and colleagues, the point where the robot would need to step to come to rest, and with the extrapolated center of mass used by Hof in human balance studies. Englsberger, Ott and Albu-Schäffer used the DCM for three-dimensional bipedal walking control, introducing the virtual repellent point and the enhanced centroidal moment pivot so that the formulation also handles vertical CoM motion.

DCM-based controllers typically plan a reference DCM trajectory backward from the final footstep, then use feedback on the measured DCM error to adjust the commanded center of pressure. Because they act on one first-order quantity, they are simple to implement and tune.

Why it matters

The DCM reduces balance control to steering a single point, making walking controllers simpler and more robust. It also gives an intuitive test of balance: if the DCM leaves the region the feet can reach or support, the robot must step or it will fall.

Key applications

Real-time walking control for humanoid robots
Push recovery and footstep adjustment
Balance assessment in exoskeletons and human movement studies
Reference trajectory generation for multi-step walking

In practice

A humanoid team using DCM-based walking first plans footsteps, then computes a reference DCM trajectory that ends at rest over the final stance and passes through the support area at each step. At every control cycle the controller estimates the CoM position and velocity, computes the measured DCM, and shifts the desired center of pressure in proportion to the DCM error; a whole-body controller converts that into joint torques. Exoskeleton and rehabilitation researchers use the same quantity to judge how close a person is to losing balance.

Common misconceptions

The DCM is a physical point on the robot — it is a computed quantity combining CoM position and velocity and can lie outside the body
Controlling the CoM directly is the same as controlling the DCM — the DCM separates out the unstable part, so a controller that acts on it handles balance more directly
The DCM and capture point are unrelated ideas — in the planar linear inverted pendulum the instantaneous capture point and the DCM coincide

Who should know this

Humanoid and biped locomotion engineers
Exoskeleton and rehabilitation robotics researchers
Control researchers working on balance and push recovery

Related technologies

Linear Inverted Pendulum
Capture Point
Zero Moment Point (ZMP)
Whole-Body Control
Model Predictive Control

Related terms

Related projects

Extreme Parkour with Legged Robots: RL system for legged robot parkour trained in under 20 hours
PACE Sim2Real: Systematic sim-to-real transfer via actuator dynamics identification
Zeroth Bot: Low-cost 3D-printed humanoid robot platform for sim-to-real reinforcement learning