| L. Han, L. Rudolph, J. Blumenthal, and I. Valodzin
Department of Mathematics and Computer Science, Clark University, Worcester, MA 01610, USA |
| Systems involving loops have been especially challenging in the
study of robotics, partly because of the requirement to maintain
loop closure constraints, conventionally formulated as highly nonlinear
equations in joint parameters. In this paper, we present our
novel triangle-tree-based approach and parameters for planar closed
chains with revolute joints. For such a loop, the loop closure constraints
are exactly, not approximately, a set of linear inequalities
in our new parameters. Further, our new parameters provide explicit
parametrization of the system deformation space (configuration
space modulo the group of rigid motions of the system’s ambient
space respecting system specifications) and endow it with a nice
geometry. More precisely, the deformation space of a generic planar
loop with n revolute joints consists of 2n-2 copies of one and the
same convex polytope (which, when all of the link lengths are fixed,
is bounded and of dimension n - 3), glued together into either one
connected component or two (ignoring collision-free constraints), via
proper boundary identification. Such a completely solved, stratified
space of convex strata will have profound implications for these systems
and lead to great simplifications in many kinematics related
issues. For example, in essence, our approach makes path planning
for planar loops with revolute joints no more difficult than for open
chains. We also briefly point out the connection and extension of the
work presented here to other systems such as spatial loops with spherical
joints and systems involving multiple loops. |
