Simple bipedal locomotion models - the compass gait model

This work is done with Ahmed Keramane, Benoit Thuilot and Bernard Espiau from INRIA, Grenoble, France.

It has already been demonstrated that suitably designed unpowered mechanical biped robots can walk down inclined planes with steady periodic gaits. The motive power for this motion comes from the conversion of the robot's gravitational potential energy as it descends down the slope. The focus of our work is a systematic study of the passive gait of a particularly simple biped robot model called the ``compass-gait'' model. The robot is kinematically equivalent to a double pendulum, possessing two kneeless legs with point masses and a third point mass at the ``hip'' joint. Three parameters, namely the ground slope angle and the normalized mass and length of the robot describe its gait. We show that in response to a continuous change in any one of its parameters the symmetric and steady stable gait of the robot gradually evolves through a regime of bifurcations characterized by progressively complicated asymmetric gaits eventually arriving at an apparently chaotic gait where no two steps are identical. The robot can maintain this gait indefinitely. We establish the stability of this gait by employing numerical techniques.

Our main objectives behind this study were to

Our research suggested that the stable motion of a given robot model is completely determined by the ground slope angle. In other words, the stable gait adopted by a robot on a given slope is unique. This hints towards a strong underlying organizing principle. The result awaits an analytical proof.

An interesting technique we employed to prove the stability of compass gait is through the evolution of a small volume element of phase space fluid. A necessary (but not sufficient) condition for the stability of such gaits is the contraction of volume of phase fluid. For our frictionless robot the volume contraction, which we compute, is caused by the dissipative effects of the ground impact model. In the chaotic regime the fractal dimension of the robot's strange attractor (2.07) compared to its state-space dimension (4) also reveals strong contraction. Use of this technique was inspired by the famous book on Variational Methods by Cornelius Lanczos.

We also developed a novel graphical technique based on the first return map that compactly captures the entire evolution of the gait, from symmetry to chaos. Additional passive dissipative elements in the robot joint result in a significant improvement in the stability and the versatility of the gait and provides a rich repertoire for simple control laws.

Our work suggested placing passive biped robots in the more general perspective of passive machines that include not just massive links but also the other passive elements such as springs and dampers. The work can be potentially useful in the modeling and synthesis of natural human locomotion as the additional passive elements in the robot resemble the inherent damping and compliance of human joints. Our initial investigations showed that a damper placed in the ``hip'' joint of the robot significantly increases the stability and versatility of the resulting gaits.

A list of my papers on this topic:

A study of the passive gait of a compass-like biped robot: symmetry and chaos
A. Goswami, B. Thuilot, and B. Espiau
International Journal of Robotics Research Vol. 17, No. 12, 1998.
Download figures of the above paper.

Download all the code used in this paper and much. The main code is written in Scilab, which is similar to Matlab (and can be easily converted to Matlab), very solid, well-maintained, and free. It is available from There is also some Matlab plotting code, and data included here.

Download another zip file with figures, codes and stuff on compass gait.

Limit cycles in a passive compass gait biped and passivity-mimicking control laws.
A. Goswami, B. Espiau, and A. Keramane
Journal of Autonomous Robots, Vol. 4, No. 3, 1997.
Compass-like biped robot Part I: Stability and bifurcation of passive gaits
A. Goswami, B. Thuilot, and B. Espiau
INRIA Research Report No. 2996, October 1996.
Bifurcation and chaos in a simple passive bipedal gait
B. Thuilot, A. Goswami, and B. Espiau
IEEE Int. Conf. on Robotics and Automation, Albuquerque, NM, April 1997.
Limit cycles and their stability in a passive bipedal gait
A. Goswami, B. Espiau, and A. Keramane
IEEE Int. Conf. on Robotics and Automation, Minneapolis, MN, April 1996.
Compass gait revisited
B. Espiau and A. Goswami
IFAC Symp. on Robot Control (SyRoCo), Capri, Italy, September 1994.
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