Matt Strong

Matt Strong

CS PhD at Stanford University

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Biped Walking Pattern Generation by using Preview Control of Zero-Moment Point

Shuuji Kajita, Fumio Kanehiro, Kenji Kaneko, Kiyoshi Fujiwara, Kensuke Harada, Kazuhito Yokoi, Hirohisa Hirukawa

Seminal September 2003 · IEEE International Conference on Robotics and Automation (ICRA)

TL;DR The authors model a humanoid as a cart on a table and generate its center-of-mass trajectory with a preview controller that tracks the upcoming ZMP reference. A second preview pass corrects the simple model's error on the full HRP-2P multibody model, cutting the maximum ZMP error from 2.3 to 1.2 cm (x) and from 1.6 to 0.4 cm (y), and a simulated HRP-2P walks on spiral stairs.

What they do

A walking pattern generator must find a center-of-mass (CoM) motion whose ZMP follows a reference set by the footholds and step period. Earlier solvers (FFT, an O(N) trinomial method) work in batch over fixed segments, and the authors’ earlier inverted-pendulum method kept balance by moving the feet, so it could not walk on stepping stones. Here pattern generation becomes a ZMP tracking servo problem, solved by a preview controller that uses the future ZMP reference.

How it works

  • Cart-table model. With the CoM on a horizontal plane at height zc, the 3D linear inverted pendulum gives p = x − (zc/g)ẍ for x and y: a cart running on a massless table with a tiny foot.
  • ZMP as servo output. Taking CoM jerk as the input makes this a strictly proper system with state (x, ẋ, ẍ). The CoM must start moving before the ZMP reference steps to the new foot, so the controller needs future reference values.
  • Preview control. A discrete LQ servo (after Katayama et al.) combines integral action on the ZMP error, state feedback, and a weighted sum of the reference over the next NL steps. The preview gain becomes very small beyond 2 s.
  • Multibody correction. Inverse kinematics makes the pelvis (offset from the CoM) follow the trajectory. The full model’s ZMP error is buffered and fed, delayed by the preview period, to a second preview controller that computes a compensation.

Results

All in simulation with HRP-2P parameters (154 cm, 58 kg):

  • A 1.6 s preview tracks the ZMP reference over three steps with a smooth CoM. At 0.8 s the ZMP undershoots sagittally and overshoots laterally, but the system stays stable thanks to the state feedback.
  • The cart-table pattern on the multibody model has a maximum ZMP error of 2.3 cm (x) and 1.6 cm (y). With compensation (0.75 s preview) it drops to 1.2 cm and 0.4 cm.
  • In OpenHRP the robot walks on spiral stairs (0.1 m steps, 24° per step, 0.7-1.3 m radius), with CoM height from a sloped constraint plane per step that keeps the same zc.

Limitations

Only dynamic simulation; implementation on the real robot is left as the next target. Footholds, step timing and the ZMP reference must be fixed in advance, CoM planes for all steps share one zc, and too short a preview period (0.8 s) degrades tracking.

Figures

From Figure 3: The cart-table model. A cart of mass m runs on a massless table of height zc whose foot is too small t...
From Figure 3: The cart-table model. A cart of mass m runs on a massless table of height zc whose foot is too small to hold it still. Accelerating the cart keeps the table upright, and the ZMP is the point px on the foot where the moment is zero.
From Figure 7: Three steps forward generated by preview control with a 1.6 s preview period, sagittal (x, top) and la...
From Figure 7: Three steps forward generated by preview control with a 1.6 s preview period, sagittal (x, top) and lateral (y, bottom). The CoM (dashed) is smooth and the resulting ZMP follows the reference.
From Figure 11: After preview-control compensation, the ZMP computed from the HRP-2P multibody model (bold) follows t...
From Figure 11: After preview-control compensation, the ZMP computed from the HRP-2P multibody model (bold) follows the cart-table ZMP (thin) in both x and y.
From Figure 13: Snapshots of the simulated HRP-2P walking on spiral stairs.
From Figure 13: Snapshots of the simulated HRP-2P walking on spiral stairs.
Abstract

We introduce a new method of a biped walking pattern generation by using a preview control of the zero-moment point (ZMP). First, the dynamics of a biped robot is modeled as a running cart on a table which gives a convenient representation to treat ZMP. After reviewing conventional methods of ZMP based pattern generation, we formalize the problem as the design of a ZMP tracking servo controller. It is shown that we can realize such controller by adopting the preview control theory that uses the future reference. It is also shown that a preview controller can be used to compensate the ZMP error caused by the difference between a simple model and the precise multibody model. The effectiveness of the proposed method is demonstrated by a simulation of walking on spiral stairs.

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