12 research outputs found
Model Reduction Near Periodic Orbits of Hybrid Dynamical Systems
We show that, near periodic orbits, a class of hybrid models can be reduced
to or approximated by smooth continuous-time dynamical systems. Specifically,
near an exponentially stable periodic orbit undergoing isolated transitions in
a hybrid dynamical system, nearby executions generically contract
superexponentially to a constant-dimensional subsystem. Under a non-degeneracy
condition on the rank deficiency of the associated Poincare map, the
contraction occurs in finite time regardless of the stability properties of the
orbit. Hybrid transitions may be removed from the resulting subsystem via a
topological quotient that admits a smooth structure to yield an equivalent
smooth dynamical system. We demonstrate reduction of a high-dimensional
underactuated mechanical model for terrestrial locomotion, assess structural
stability of deadbeat controllers for rhythmic locomotion and manipulation, and
derive a normal form for the stability basin of a hybrid oscillator. These
applications illustrate the utility of our theoretical results for synthesis
and analysis of feedback control laws for rhythmic hybrid behavior
Exponentially Stabilizing Controllers for Multi-Contact 3D Bipedal Locomotion
Models of bipedal walking are hybrid with continuous-time phases representing the Lagrangian stance dynamics and discrete-time transitions representing the impact of the swing leg with the walking surface. The design of continuous-time feedback controllers that exponentially stabilize periodic gaits for hybrid models of underactuated 3D bipedal walking is a significant challenge. We recently introduced a method based on an iterative sequence of optimization problems involving bilinear matrix inequalities (BMIs) to systematically design stabilizing continuous-time controllers for single domain hybrid models of underactuated bipedal robots with point feet. This paper addresses the exponential stabilization problem for multi-contact walking gaits with nontrivial feet. A family of parameterized continuous-time controllers is proposed for different phases of the walking cycle. The BMI algorithm is extended to the multi-domain hybrid models of anthropomorphic 3D walking locomotion to look for stabilizing controller parameters. The Poincaré map is addressed and a new set of sufficient conditions is presented that guarantees the convergence of the BMI algorithm to a stabilizing set of controller parameters at a finite number of iterations. The power of the algorithm is ultimately demonstrated through the design of stabilizing virtual constraint controllers for dynamic walking of a 3D humanoid model with 28 state variables and 275 controller parameters
Exponentially Stabilizing Controllers for Multi-Contact 3D Bipedal Locomotion
Models of bipedal walking are hybrid with continuous-time phases representing the Lagrangian stance dynamics and discrete-time transitions representing the impact of the swing leg with the walking surface. The design of continuous-time feedback controllers that exponentially stabilize periodic gaits for hybrid models of underactuated 3D bipedal walking is a significant challenge. We recently introduced a method based on an iterative sequence of optimization problems involving bilinear matrix inequalities (BMIs) to systematically design stabilizing continuous-time controllers for single domain hybrid models of underactuated bipedal robots with point feet. This paper addresses the exponential stabilization problem for multi-contact walking gaits with nontrivial feet. A family of parameterized continuous-time controllers is proposed for different phases of the walking cycle. The BMI algorithm is extended to the multi-domain hybrid models of anthropomorphic 3D walking locomotion to look for stabilizing controller parameters. The Poincaré map is addressed and a new set of sufficient conditions is presented that guarantees the convergence of the BMI algorithm to a stabilizing set of controller parameters at a finite number of iterations. The power of the algorithm is ultimately demonstrated through the design of stabilizing virtual constraint controllers for dynamic walking of a 3D humanoid model with 28 state variables and 275 controller parameters
Data-Driven Methods to Build Robust Legged Robots
For robots to ever achieve signicant autonomy, they need to be able to mitigate performance
loss due to uncertainty, typically from a novel environment or morphological
variation of their bodies. Legged robots, with their complex dynamics, are particularly
challenging to control with principled theory. Hybrid events, uncertainty, and
high dimension are all confounding factors for direct analysis of models. On the other
hand, direct data-driven methods have proven to be equally dicult to employ. The
high dimension and mechanical complexity of legged robots have proven challenging
for hardware-in-the-loop strategies to exploit without signicant eort by human
operators. We advocate that we can exploit both perspectives by capitalizing on
qualitative features of mathematical models applicable to legged robots, and use that
knowledge to strongly inform data-driven methods. We show that the existence of
these simple structures can greatly facilitate robust design of legged robots from a
data-driven perspective. We begin by demonstrating that the factorial complexity of
hybrid models can be elegantly resolved with computationally tractable algorithms,
and establish that a novel form of distributed control is predicted. We then continue
by demonstrating that a relaxed version of the famous templates and anchors hypothesis can be used to encode performance objectives in a highly redundant way, allowing
robots that have suffered damage to autonomously compensate. We conclude with
a deadbeat stabilization result that is quite general, and can be determined without
equations of motion.PHDElectrical Engineering: SystemsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/155053/1/gcouncil_1.pd