Modeling Deformations of General Parametric Shells Grasped by a Robot Hand

The robot hand applying force on a deformable object will result in a changing wrench space due to the varying shape and normal of the contact area. Design and analysis of a manipulation strategy thus depend on reliable modeling of the object's deformations as actions are performed. In this paper, shell-like objects are modeled. The classical shell theory [P. L. Gould, Analysis of Plates and Shells. Englewood Cliffs, NJ: Prentice-Hall, 1999; V. V. Novozhilov, The Theory of Thin Shells . Gronigen, The Netherlands: Noordhoff, 1959; A. S. Saada, Elasticity: Theory and Applications. Melbourne, FL: Krieger, 1993; S. P. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells, 2nd ed. New York: McGraw-Hill, 1959] 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. Generalization of the theory to an arbitrary parametric shell is therefore not immediate. This paper first extends the linear and nonlinear shell theories to describe extensional, shearing, and bending strains in terms of geometric invariants, including the principal curvatures and vectors, and their related directional and covariant derivatives. To our knowledge, this is the first nonparametric formulation of thin-shell strains. A computational procedure for the strain energy is then offered for general parametric shells. In practice, a shell deformation is conveniently represented by a subdivision surface [F. Cirak, M. Ortiz, and P. Schröder, “Subdivision surfaces: A new paradigm for thin-shell finite-element analysis,” Int. J. Numer. Methods Eng., vol. 47, pp. 2039-2072, 2000]. We compare the results via potential-energy minimization over a couple of benchmark problems with their analytical solutions and numerical ones generated by two commercial software packages: ABAQUS and ANSYS. Our method achieves a convergence rate that is one order of magnitude higher. Experimental validation involves regular and free-form shell-like objects of various materials that were grasped by a robot hand, with the results compared against scanned 3-D data with accuracy of 0.127 mm. Grasped objects often undergo sizable shape changes, for which a much higher modeling accuracy can be achieved using the nonlinear elasticity theory than its linear counterpart.