Modeling and control for 2-D grasping of an object with arbitrary shape under rolling contact

Modeling, control, and stabilization of dynamics of two-dimensional object grasping by using a pair of multi-joint robot fingers are investigated under rolling contact constraints and arbitrariness of the geometry of the object and fingerends. First, a fundamental testbed problem of modeling and control of rolling motion between 2-D rigid bodies with arbitrary shape is treated under the assumption that the two contour curves coincide at the contact point and share the same tangent. The rolling constraint induces the Euler equation of motion that is parameterized by a common arclength parameter and constrained onto the kernel space as an orthogonal complement to the image space spanned from the constraint gradient. By extending the analysis to the problem of stable grasp of a 2D-object with arbitrary shape, the Euler- Lagrange equation of motion of the overall fingers/object system parametrized by arclength parameters is derived, and shown to be accompanied with a couple of first-order differential equations that express evolutions of contact points in terms of the second fundamental form. Further, it is shown that rolling contact constraints are integrable in the sense of Frobenius and hence regarded as being holonomic. A control signal called "blind grasping" is defined and shown to be effective in stabilization of grasping without using the details of object shape and parameters or external sensing.