A computable model of grasping and manipulation of a 3D rigid object with arbitrary smooth surfaces by multiple robot fingers with smooth fingertip surfaces is derived under rolling contact constraints between surfaces. Geometrical conditions of pure rolling contacts are described through the moving-frame coordinates at each rolling contact point under the postulates: (1) two surfaces share a common single contact point without any mutual penetration and a common tangent plane at the contact point and (2) each path length of running of the contact point on the robot fingertip surface and the object surface is equal. It is shown that a set of Euler-Lagrange equations of motion of the fingers-object system can be derived by introducing Lagrange multipliers corresponding to geometric conditions of contacts. A set of 1st-order differential equations governing rotational motions of each fingertip and the object and updating arc-length parameters should be accompanied with the Euler-Lagrange equations. Further more, nonholonomic constraints arising from twisting between the two normal axes to each tangent plane are rewritten into a set of Frenet-Serre equations with a geometrically given normal curvature and a motion-induced geodesic curvature.

In relation to the recent development of robotics research and neurophysiology, there arises an important question on a study of the functionality of the human hand in grasping and object manipulation interacting physically with environment under arbitrary geometries of objects and fingertips. Another question also arises as to whether a complete mathematical model of grasping a 3D rigid object with an arbitrary shape can be developed and used in numerical simulation to validate control models of prehensile functions of a set of multiple fingers. In particular, is it possible to develop a mathematical model as a set of Euler-Lagrange equations that govern a whole motion of the fingers-object system under rolling contact constraints between each robot fingertip and a rigid object with an arbitrary smooth surface.

Traditionally in robotics research, a rolling contact constraint between two rigid-body surfaces is defined as the zero velocity of one translational motion of the common contact point on the fingertip surface relative to another on the object surface [

Even in case of 2D grasping by means of dual robot fingers with smooth fingertip surfaces, the integrability of Pfaffian forms of rolling contact constraints was shown very recently in our previous paper [

two contact points on each contour curve must coincide at a single common point without mutual penetration,

the two contours must have the same tangent at the common contact.

Owing to these postulates, the path length of one contact point running on each fingertip contour curve and that of another contact point running on the object contour must coincide, that is, the constraint can be expressed eventually in the level of position variable. Hence, it is shown in [

This paper aims at extending such a moving frame coordinates approach for mathematical modelling of 2D grasping to computable mathematical modelling of 3D grasping of a rigid object with arbitrary smooth surfaces under the following set of 3D rolling contact constraints:

two contact points on each curved surface must coincide at a single common point without mutual penetration,

the two curved surfaces have the same tangent at the common contact point, that is, each surface has the same unit normal with mutually opposite direction at the common contact point,

the two path lengths running on their corresponding surfaces must be coincident.

In the previous paper [

Consider firstly a physical situation that a pair of multijoint robot fingers is grasping a 3D rigid body as seen in Figure

A pair of robot fingers grasping a 3D rigid object with smooth surfaces.

Definition of the moving frame coordinates system centering at the rolling contact point.

Second, suppose that at some

Since

In what follows, we denote vectors

It is easy to check that, in the illustrative case of a spherical left hand fingertip shown in Figure

Let us now interpret the first postulate (a1) in a mathematical form described by

By virtue of the integrability of each Pfaffian form of rolling contact constraints, the Lagrangian of the system is written into

In order to always keep the tangent vector

We are now in a position to find a necessary condition for maintaining the equality of (

Finally, it is possible to see that the moving frames denoted by

In the Frenet-Serre equation of (

It is now possible to show a set of all the differential equations of motion of the fingers-object system under rolling contact constraints. In what follows, we use the suffix “

A computational model of dynamics of 3D object grasping and manipulation under rolling contact constraints by means of multiple multijoint robot fingers with smooth fingerend surfaces is derived on the basis of the postulates of pure rolling contact constraint. The postulates are summarized: (1) at the contact point, the fingerend and object surfaces share a common tangent plane with each normal with opposite direction and (2) the path length of contact points running on the fingerend is coincident with that running on the object surface. The postulates are adopted by referring to Nomizu’s work [

Note that

Similarly, it follows that

Similarly, it follows that

Time rate of

Derivation of

If

With derivation of

With derivation of