Cart-Table Model

Models a biped as a cart rolling on a tabletop, used to connect center-of-mass motion to the zero moment point.

The cart-table model was introduced by Shuuji Kajita and colleagues in a 2003 ICRA paper for generating walking patterns via ZMP preview control. It concentrates the robot's mass into a cart running along a massless, horizontal tabletop supported by a single thin leg, with the bottom of that leg standing in for the sole of the foot. The zero moment point — where the horizontal moment from the ground-reaction force vanishes — satisfies p = x − (z_c/g)·ẍ: x is the cart's (i.e. the center of mass's) horizontal position, z_c is the center-of-mass height, g is gravitational acceleration, and ẍ is horizontal acceleration. Accelerate the cart too hard and the zero moment point runs off the bottom of the table leg, tipping the table over. It is mathematically the same set of equations as the linear inverted pendulum, but written the other way around — 'given a reference ZMP, find the center-of-mass trajectory' — which makes it convenient for preview control that looks ahead at future reference values.

ExampleIn their paper, Kajita and colleagues generated walking trajectories using the cart-table model combined with preview control, then used a full multi-body model to compensate for the ZMP error introduced by the simplification, enabling a simulated biped robot to climb a spiral staircase.

Also called

Cart-on-a-Table Model, Table-Cart Model

Related

Zero Moment Point · Linear Inverted Pendulum Model (LIPM) · ZMP Preview Control · Bipedal Locomotion · Support Polygon · Center of Mass (CoM)

Sources

Kajita et al.: Biped walking pattern generation by using preview control of zero-moment point (ICRA 2003)
Stéphane Caron: Linear inverted pendulum model