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 ``compassgait'' 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
 obtain insights into the complex set of actions that
comprise biped locomotion, for passive gaits these ``actions''
being the manifestations of the natural dynamics of the system
 device simple control schemes such that an actively
controlled biped robot mimics the motion of its passive
counterpart
 understand the full nonlinear dynamics of simple
mechanical systems, especially the ones governed by
hybrid algebrodifferential equations
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 statespace 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 compasslike 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, wellmaintained, and free. It is available from
http://www.scilab.org/. 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
passivitymimicking control laws.
 A. Goswami, B. Espiau, and A. Keramane
 Journal of Autonomous Robots, Vol. 4, No. 3, 1997.
 Compasslike 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.