Robot control
Linear inverted pendulum model
Definition
The linear inverted pendulum model approximates a walking robot by a mass moving at constant height above its support, with simplified angular-momentum dynamics. These assumptions make horizontal center-of-mass acceleration linear in the displacement from the support point.
Also known as: LIPM, Linear inverted pendulum
Updated
A simpler model for a complicated robot
A humanoid may have many joints, but a walking planner can first reason about its center of mass. With constant height h and negligible change in angular momentum, the horizontal dynamics become x_ddot = (g / h) * (x - p), where p is the zero-moment point on flat ground. MIT's derivation explains the assumptions behind this reduction.
If the mass moves ahead of a fixed support point, gravity-driven motion accelerates it farther forward. Moving the support point changes that acceleration. The model is an inverted pendulum because its mass lies above the support.
Uses in walking control
The linear equations make center-of-mass trajectory planning and model predictive control easier to compute. They also lead to the capture-point expression used in balance recovery.
What the approximation leaves out
The basic model does not represent swing-leg dynamics, every joint limit, or the full effects of changing body angular momentum. Footstep reachability still needs separate constraints. Capture-region research shows why timing and reachable step locations matter even when reduced dynamics give a mathematically attractive target. Jumping or large vertical motion requires a different or extended model.
Sources
Related terms
Center of mass
The center of mass is the mass-weighted average position of a body or a collection of bodies. For an articulated robot, its position changes as the links move.
Zero-moment point
The zero-moment point is a point on a chosen support plane where the net moment associated with the ground reaction wrench has zero components parallel to that plane. In flat-ground walking with the usual contact assumptions, it coincides with the center of pressure.
Capture point
The capture point is a model-dependent location where support can be placed to bring a moving robot toward rest without further steps. In the constant-height linear inverted pendulum model, the instantaneous capture point combines center-of-mass position and velocity.
Model predictive control
Model predictive control repeatedly optimizes future actions using a system model, applies the next part of the solution, and replans from updated state information. It can account for objectives and constraints over a finite prediction horizon.