As a first step into this largely unexplored area, we conduct investigation on the modeling of the deformation of grasped objects. At the current stage, we are look into computation of the deformations of thin shells under applied loads. A shell is a body enclosed between two closely spaced and curved surfaces. Classical theory of shells assumes a parametrization along the lines of principal curvature on the middle surface of a shell. Such a parametrization, while always existing locally, is not known for many surfaces, and deriving one can be very difficult if not impossible. This paper generalizes the classical strain-displacement equations and strain energy formula to a shell with an arbitrary parametric middle surface. We show that extensional and shearing strains can all be represented in terms of geometric invariants including principal curvatures, principal vectors, and the related directional and covariant derivatives. Computation of strains and strain energy is also described for a general parametrization.

We represent the displacement field on a shell as a B-spline surface. By minimization of potential energy, we have simulated deformations of algebraic surfaces under applied loads, and performed experiments on an aluminum soda can and a stretched cloth using a three-fingered Barrett Hand. The simulation and experimental results match with good accuracy. The presented work is an initial step in our research on robot grasping of deformable objects.

Deformations on a plane, a cylinder, and an ellipsoid under applied point loads as computed using linear elasticity theory. For technical details, we refer the reader to our IROS '08 paper listed below.

On modeling large deformations, the linear elasticity theory has proven inaccurate, and the nonlinear theory needs to be applied. The three figures below compares the scanned image (left), and the modeling results by nonlinear (center) and linear (right) methods, all of a contact region of a tennis ball grasped by a BarrettHand at the antipodal configuration. Nonlinear modeling generates more accurate result than linear modeling (average errors 0.61mm vs 1.25mm).



The next figure displays the deformed tennis ball with the top and bottom contact regions computed based on the nonlinear elasticity theory, and the middle (undeformed) region generated by a 3D scanner. The red lines mark the region boundaries.

• Jiang Tian and Yan-Bin Jia. Modeling deformable shell-like objects grasped by a robot hand [abstract](179K). Submitted to IEEE International Conference on Robotics and Automation, Kobe, Japan, May 12-17, 2009.

• Yan-Bin Jia and Jiang Tian. Deformations of general parametric shells: computation and robot experiment [abstract](529K). In Proceedings of the IEEE/RSJ International Conference on Intelligent Robotics and Systems, pp. 1796-1803, Nice, France, Sep 22-26, 2008.
  
  

This research is supported by the NSF grant IIS-0742334 .

Last updated on Februray 23, 2008.