Centroidal Dynamics

• Also called: centroidal model, centroidal momentum dynamics, centroidal momentum
• Category: Control › Locomotion control
• Tags: legged-locomotion, humanoids, dynamics, balance, model-predictive-control

A reduced model of a legged or humanoid robot that describes how contact forces change the motion of its center of mass and its total angular momentum, without tracking each limb.

Definition

Centroidal dynamics describes the motion of a multi-body robot, such as a humanoid or quadruped, in terms of its overall linear and angular momentum about the center of mass. However many joints the robot has, Newton's and Euler's laws say that only external forces, mainly gravity and the contact forces at the feet or hands, can change these quantities. The rate of change of linear momentum equals the sum of external forces, and the rate of change of angular momentum about the center of mass equals the sum of external torques about it.

This gives a model of only six equations, independent of the number of joints, that captures what matters most for balance and locomotion: where the center of mass goes and how the body rotates as a result of the forces the robot applies to its surroundings. Internal joint motions affect the system only through the centroidal momentum, which is linked to joint velocities by the centroidal momentum matrix, a concept developed by Orin and Goswami.

Planners use centroidal dynamics at different levels of detail. The single rigid body model treats the whole robot as one body with fixed inertia, which works well for quadrupeds with light legs. The full centroidal model keeps the joint configuration to compute a changing inertia, which matters for humanoids with heavy limbs. Simpler still, the linear inverted pendulum ignores angular momentum and fixes the center of mass height.

In practice, a centroidal planner chooses center-of-mass trajectories and contact forces that respect friction cones and contact timing, and a whole-body controller then computes joint torques that realize them.

Why it matters

Centroidal dynamics lets controllers reason about balance, walking and jumping without solving the full dynamics of every joint, which makes real-time planning possible for humanoids and quadrupeds. It is the standard middle layer in model-based legged locomotion, between simple pendulum models and full rigid-body dynamics.

Key applications

• Model predictive control for quadruped locomotion
• Humanoid walking, running and jumping
• Planning multi-contact motions such as climbing and loco-manipulation
• Balance recovery after pushes

In practice

In a typical model-based locomotion stack, a model predictive controller runs on a single-rigid-body or centroidal model, optimizing center-of-mass motion and ground reaction forces over a short horizon subject to friction cone and contact schedule constraints. Its output goes to a whole-body controller running at a higher rate, which solves a quadratic program for joint torques that track the planned momentum while respecting joint limits. Libraries such as Pinocchio compute the centroidal momentum matrix, and optimal control frameworks such as OCS2 include centroidal models for legged robots. Teams validate the reduced model by checking how closely the robot's measured base motion follows the planned center-of-mass trajectory.

Common misconceptions

• Centroidal dynamics ignores the limbs entirely — the full centroidal model accounts for limb motion through a configuration-dependent inertia; only the single rigid body approximation ignores it
• Joint torques can directly change the robot's total momentum — only external forces such as contact forces and gravity can; the joints act by changing those contact forces
• It is the same as the inverted pendulum model — pendulum models are further simplifications that drop angular momentum and usually fix the center of mass height

Who should know this

• Legged locomotion and humanoid control engineers
• Researchers working on model predictive control for walking robots
• Motion planning researchers working on multi-contact tasks
• Graduate students in robot dynamics

Related technologies

• Pinocchio
• OCS2
• Model Predictive Control
• Whole-Body Control
• Linear Inverted Pendulum
• Single Rigid Body Model

Related terms

• Whole-Body Control
• Locomotion
• Humanoid Robot
• Friction Cone

Related projects

• Drake: MIT robotics toolbox for model-based dynamics, optimization, and control system design
• MuJoCo MPC: DeepMind's interactive real-time model predictive control framework for MuJoCo
• DART: C++ physics engine for articulated rigid body dynamics and robotics simulation
• python_simple_mppi: MPPI control derived and implemented in numpy, with four worked examples
• frax: JAX library for fast, differentiable robot kinematics, dynamics and inverse kinematics
• mjbatch: Runs thousands of MuJoCo simulations in parallel on the CPU