Inverted Pendulum Model (IPM)

A simplified model of a robot as a pole on the ground balancing a point mass, used to analyze and control balance.

An inverted pendulum is a pendulum with its center of mass above its pivot — an unstable equilibrium that tips over at the slightest disturbance, and must be actively controlled, moving the pivot back underneath the center of mass, to stay upright. A standing person is roughly an inverted pendulum pivoting at the feet, and a Segway balances on exactly the same principle. In bipedal and humanoid research, lumping the robot's entire mass into a single point at the center of mass and treating the legs as massless struts gives the inverted pendulum model, which describes which way the body is tipping and where the foot should land using very few equations. In 2001, Shuuji Kajita and colleagues added the further assumption that the center-of-mass height z_c stays constant, giving the linear inverted pendulum model (LIPM): in the horizontal direction, ẍ = (g/z_c)·x, where x is the center of mass's horizontal offset from the pivot and g is gravitational acceleration. Making the equation linear makes real-time gait planning tractable, and methods like ZMP preview control and the capture point are both built on top of it.

ExampleThe CartPole environment commonly used to introduce reinforcement learning is exactly a cart-pole inverted pendulum: the agent can only push the cart left or right, and the goal is to keep the pole balanced upright.

Also called

Cart-Pole

Related

Linear Inverted Pendulum Model (LIPM) · Zero Moment Point · Capture Point · Center of Mass (CoM) · Spring-Loaded Inverted Pendulum · Balance Control

Sources

Inverted pendulum - Wikipedia
The 3D linear inverted pendulum mode: a simple modeling for a biped walking pattern generation (Kajita et al., IROS 2001)