Modeling Deformations of General Parametric Shells Grasped by a Robot Hand
Yan-Bin Jia and Jiang Tian
Abstract
A shell is a body enclosed between two closely spaced and curved
surfaces. Classical theory of
shells~\cite{TimoshenkoWK59,Novozhilov59,Saada93,Gould99} assumes a
parametrization along the two lines of curvature on the middle surface of a
shell. Such a parametrization, while always existing locally, is very difficult,
if not impossible, to derive for most surfaces. Also, since it is not
necessarily by arc length, generalization to an arbitrary parametric shell is
not immediate from differential geometry. This paper first extends the linear
and nonlinear shell theories to describe extensional, shearing, and bending
strains on a shell, given a displacement field, in terms of geometric invariants
including the principal curvatures and vectors, and the related directional and
covariant derivatives. To our knowledge, this is the first non-parametric
formulation of thin shell strains. A computational procedure is then offered for
general parametric shells.
The deformation of a shell is conveniently represented by a subdivision
surface~\cite{CirakOS00}. Through minimization of the potential energy, we have
simulated deformations of algebraic surfaces under point and area loads, and
compared the results over a couple of benchmark problems with their analytical
solutions. Experimental validation involves regular and freeform shell-like
objects (of various materials) grasped by a robot hand, where computed
deformations are compared with scanned 3-D data (accuracy 0.127mm). On modeling
large deformations, a much higher accuracy can be achieved using the nonlinear
elasticity theory than its linear counterpart. The presented work points to the
nonlinear theory in research on robot grasping of deformable objects, which
often undergo sizable shape changes.