Linear Inverted Pendulum Model (LIPM)

A simplified model of bipedal walking that holds the center-of-mass height constant, used for fast gait planning.

The linear inverted pendulum model is the most widely used simplification for bipedal walking, introduced by Shuuji Kajita and colleagues — a planar version at ICRA in 1991, and a 3D version, 3D-LIPM, at IROS in 2001. It represents the robot as a point mass concentrated at the center of mass, attached to a massless leg, and assumes the center-of-mass height stays constant; under that assumption, the otherwise nonlinear inverted-pendulum equation becomes linear: ẍ = (g/z_c)(x − p). Here x is the center of mass's horizontal position, p is the support point's position (the center of pressure, i.e., the ZMP), g is gravitational acceleration, and z_c is the center-of-mass height. The linear equation has a closed-form solution and is fast to compute, which suits real-time gait generation and ZMP preview control, and the capture point is also derived from it. The tradeoff is that it ignores leg mass and upper-body rotation.

ExampleWith center-of-mass height z_c = 0.8 m, g/z_c ≈ 12.3 s⁻². If the center of mass is 5 cm ahead of the support point, horizontal acceleration is about 0.6 m/s² and keeps increasing as the robot tips further, so the robot must take its next step in time to move the support point forward.

Also called

LIP, 3D-LIPM

Related

Inverted Pendulum Model (IPM) · Zero Moment Point · Capture Point · Cart-Table Model · ZMP Preview Control · Center of Mass (CoM)

Sources

Kajita et al., The 3D Linear Inverted Pendulum Mode: A Simple Modeling for a Biped Walking Pattern Generation (IROS 2001)
Kajita & Tani, Study of Dynamic Biped Locomotion on Rugged Terrain: Derivation and Application of the Linear Inverted Pendulum Mode (ICRA 1991)
Underactuated Robotics (MIT, Tedrake): Simple Models of Walking and Running / Humanoids