Ph.D Student | Or Yizhar |
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Subject | Frictional Equilibrium Postures for Robotic Locomotion - Computation, Geometric Characterization and Stability Analysis |

Department | Department of Mechanical Engineering |

Supervisor | Professor Elon Rimon |

Full Thesis text |

Automated planning of quasistatic legged locomotion on rough and unstructured terrains requires tools for identification and computation of feasible equilibrium postures, which are robust under inertial forces generated by motion of the internal limbs, and possess dynamic stability with respect to position-and-velocity perturbations.

The goal of this thesis is computation and graphical characterization of robust and stable equilibrium postures. Lumping the kinematic structure of the robot into a single rigid body with a variable center of mass, the problem is reduced to finding the region of center-of-mass locations achieving robust and stable equilibrium postures for a given set of frictional point contacts under gravity.

First, the thesis formulates the center-of-mass region achieving feasible equilibrium and provides its geometric characterization in two and three dimensions, assuming Coulomb's friction model. The results are then generalized to guarantee robustness with respect to a given set of bounded disturbance forces and torques.

Next, the thesis defines the notion of *frictional
stability* for equilibrium postures, which requires that the dynamic
response converges to equilibrium within a small neighborhood of the original
posture, under any small initial perturbation that may include contact
separation, rolling or sliding. The dynamics of planar mechanical systems with
frictional contacts is then formulated, and two related difficulties, namely,
dynamic ambiguity and dynamic inconsistency, are analyzed. The criterion of *strong
equilibrium*, which eliminates dynamic ambiguities, is reviewed, and the
novel criterion of *persistent equilibrium*, which eliminates dynamic
inconsistencies, is developed. These two criteria are formulated in terms of
mass distribution and center-of-mass location, and are then proven to be a key
component in achieving frictional stability.

Under initial contact separation, the
dynamic response may involve collisions at the contacts. The dynamics of a
planar rigid body undergoing sequential impacts at two contacts is then formulated
as a *hybrid dynamical system*, composed of phases of continuous dynamics
interleaved by discrete impact events. The *clattering stability*
condition, guaranteeing that one or two contacts are re-established in finite
time, is then derived by using the technique of Poincaré map. The
clattering stability condition depends on the coefficient of restitution, the
mass distribution, and center-of-mass location.

Finally, frictional stability is addressed,
using the concept of *completed dynamical system* that concatenates the
phases of constrained dynamics with the phases of hybrid dynamics. The two components
of persistent equilibrium and clattering stability are proven to be sufficient conditions
for frictional stability of planar two-contact postures.