Mechanics and Manipulation of a Kirchhoff Elastic Rod

SpeakerDr. Timothy Bretl
Organization University of Illinois at Urbana-Champaign
LocationRoom 1230; Engineering Building II
DateAugust 31, 2012 12:50 PM

The configuration space is a key concept in robot motion planning. There are systematic ways of deriving an appropriate configuration space for manipulation of rigid objects. Doing so is much harder for manipulation of deformable objects --- indeed, it has not even been clear that a useful notion of configuration space (i.e., finite-dimensional with well-defined coordinate charts) exists for these objects. I will show that it is possible to establish an appropriate configuration space for manipulation of canonical "deformable linear objects" like an elastic rod or a planar elastic kinematic chain. This result leads to simple algorithms for manipulation planning that are easy to implement and that work well in practice.

As one example, I will consider a flexible wire of fixed length that is held at each end by a robotic gripper. Any curve traced by this wire when in static equilibrium is a local solution to a geometric optimal control problem, with boundary conditions that vary with the position and orientation of each gripper. I will prove that the set of all local solutions to this problem over all possible boundary conditions is a smooth manifold of finite dimension that can be parameterized by a single chart. Coordinates for this chart are given by the initial value of costates that arise in necessary and sufficient conditions for optimality and that have a direct interpretation as forces and torques. These coordinates allow us to formulate the problem of manipulation planning as finding a path **of the rod** through its set of equilibrium configurations, something that has been suggested before but was previously thought impossible.

Biography

Timothy Bretl received his B.S. in Engineering and B.A. in Mathematics from Swarthmore College in 1999, and his M.S. in 2000 and Ph.D. in 2005 both in Aeronautics and Astronautics from Stanford University. Subsequently, he was a Postdoctoral Fellow in the Department of Computer Science, also at Stanford University. Since 2006, he has been with the University of Illinois at Urbana-Champaign, where he is an Assistant Professor of Aerospace Engineering and a Research Assistant Professor in the Coordinated Science Laboratory. He received the National Science Foundation Faculty Early Career Development Award in 2010.

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