Optimization for control and planning of multi-contact dynamic motion
2017
The fundamental promise of robotics centers on the
ability to productively interact with a complex and changing world.
Yet, current robots are largely limited to basic tasks in
structured environments and act slowly and cautiously, afraid of
incidental contact. In this thesis, we consider a class of control
and planning problems for robots dynamically interacting with their
environment. We address challenges that arise from non-smooth
motions induced by contact, where discontinuities result from
impact events and frictional forces. First, we examine the problem
of trajectory optimization in contact-rich environments, and
present two algorithms for synthesizing motions which make and
break contact. The novel contact-implicit trajectory optimization
algorithm lifts the problem and reasons over the set of possible
contacts forces. In doing so, we eliminate the requirement for an a
priori sequencing of the active contacts, and avoid explicit
combinatorial complexity. We also introduce a direct collocation
algorithm for optimizing high-accuracy trajectories, given an
arbitrary contact schedule. This approach eliminates drift in the
numerical integration of contact constraints, even when constraints
result in closed kinematic chains and require non-minimal
coordinates. Second, this thesis concerns questions of control
synthesis and provable stability verification of a robot making and
breaking contact. To verify stability, we introduce an algorithm
for discovering polynomial Lyapunov functions, where the system
dynamics include impacts and friction. We leverage the measure
differential inclusion representation of non-smooth contact
mechanics to efficiently optimize over Lyapunov functions in
multi-contact settings. Since avoiding hazardous falls is a primary
necessity for bipedal walking robots, we use similar tools to
characterize the capabilities of multiple simple models used for
balancing and push recovery. Using the notions of barrier functions
and occupation measures, we explicitly bound the set of
disturbances from which a robot can recover by balancing or
stepping. The primary contributions of this thesis are
computational in nature, and we heavily leverage modern approaches
to both general nonlinear programming and convex optimization.
Sums-of-squares, an approach to polynomial optimization utilizing
semidefinite programming, plays a central role in our methods for
formal stability analysis.
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