Abstract
Modeling, control, and stabilization of dynamics of two-dimensional object grasping by using a pair of multijoint robot fingers are investigated under rolling contact constraints and arbitrariness of the geometry of the object and fingertips. First, modeling of rolling motion between 2D 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 constraints induce the Euler equation of motion that is parameterized by a pair of arclength parameters and constrained onto the kernel space as an orthogonal complement to the image space spanned from all the constraint gradients. Furthermore, it is shown that all the Pfaffian forms of the rolling constraints are integrable in the sense of Frobenius and therefore the rolling contacts are regarded as a holonomic constraint. The Euler-Lagrange equation of motion of the overall fingers/object system is rederived together with a couple of first-order differential equations that express evolution of contact points in terms of quantities of the second fundamental form. A control signal called “blind grasping” is defined and shown to be effective in maintenance or stabilization of grasping without using the details of object shape and parameters or external sensing. An extension of the Dirichlet-Lagrange stability theorem to a system of DOF-redundancy under constraints is discussed by introducing a Morse-Bott function and deriving its Hessian, in a special case that the object to be grasped is a parallelepiped.
Highlights
This paper aims at tackling the control problem for dextrous multifingered hands from computational perspectives, based on the assumption that a complete model of grasping must be developed even under the existence of rolling contacts and the arbitrariness of geometry of objects
In the series of our papers [5, 6], a set of Lagrange equations of motion of the overall fingers/object system under rolling constraints is derived under the assumption that rolling is interpreted as a constraint of the equal velocity of the contact point running on the fingerend sphere relative to running on the object surface, as originally formulated in the text book [7]
A mathematical modeling of 2-dimensional grasping of an object by a pair of robot fingers under rolling contact constraints is presented under the situation that the geometric shape of both the fingertips and 2D object is arbitrary
Summary
This paper aims at tackling the control problem for dextrous multifingered hands from computational perspectives (see Figure 1), based on the assumption that a complete model of grasping must be developed even under the existence of rolling contacts and the arbitrariness of geometry of objects. This paper further presents a complete model of grasping in the case that the geometry of fingerends is arbitrary and discusses a control scheme called “blind control” that need neither the use the geometric information of the fingerends and object nor the sense locations of the contact points and object mass center. The argument presented in [9] can be interpreted as a kinematic extension of the well-known theorem on curves that if two smooth curves have the same curvatures along their arclengths the two curves can be exactly superposed by using a homogeneous transformation By virtue of this mathematical standpoint of rolling contact, we can show immediately that the Euler-Lagrange equation of motion of the overall fingers/object system is characterized by a pair of arclength parameters and quantities of only the first fundamental form of contour curves. We point out a few of important and interesting problems that remain unsolved or not yet tackled
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