ZMP Preview Control

Generating stable biped gaits by looking ahead at a planned future ZMP trajectory and moving the center of mass to match it in advance.

ZMP preview control was proposed by Shuuji Kajita and colleagues at Japan's National Institute of Advanced Industrial Science and Technology (AIST) at ICRA 2003, and is a classic method for generating humanoid walking patterns. The zero moment point (ZMP) is the point where the horizontal component of the ground reaction moment is zero; keeping it inside the support polygon (the foot's contact area) keeps the foot from tipping. The method first lays out a reference ZMP trajectory based on the footholds, turning the problem into: how should the center of mass (CoM) move so the actual ZMP tracks that reference? The paper uses a ‘cart-table model,’ where ZMP and CoM satisfy p = x − (z_c/g)·ẍ — p is the ZMP position, x is the CoM's horizontal position, z_c is a constant CoM height, and g is gravitational acceleration. Because CoM acceleration can only be produced by a deviation between the ZMP and the CoM, the CoM must start moving ahead of time — looking only at the current reference is too late, hence ‘preview.’ The controller takes CoM jerk as its input, minimizes ZMP tracking error, and feeds forward a weighted sum of future reference ZMP values within a preview window; the gains can be computed offline, so the online computation stays light. Later ZMP-based MPC methods add constraints on top of this foundation.

ExampleA biped walks four steps: a staircase-shaped ZMP reference is laid out based on the footholds, jumping to the center of the new support foot at each step. The preview controller sees the next step coming and smoothly shifts the CoM toward the next support foot ahead of time, keeping the actual ZMP inside the foot's support area throughout.

Also called

Preview Control of ZMP, Kajita Preview Control

Related

Zero Moment Point · Cart-Table Model · Linear Inverted Pendulum Model (LIPM) · Gait Planning · Model Predictive Control · Support Polygon

Sources

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