Abstract

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.

1. Introduction

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 [1]. Therefore, rolling contact constraints are expressed in terms of velocity relations called a Pfaffian form. Montana [2] presented a complete set of all velocity relations of a rolling contact by using the normalized gauss frame for expressing given smooth surfaces of fingertips and a 3D object. Based on Montana’s set of Pfaffian forms, Paljug et al. [3] formulated a dynamic model for the control of rolling contacts in multiarm manipulation. However, it is uncertain whether the derived model of equations of motion can be computationally integrable in time in case of rolling contacts between general smooth surfaces, since in [3] only a limited case of ball-plate contacts was numerically simulated. Another work by Cole et al. [4] tried to simulate a 3D grasping, but it is uncertain whether it could overcome the problem of arise of a nonholonomic constraint pointed out by Montana [2].

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 [5], where a complete set of computational models of Euler-Lagrange equations of motion of the whole fingers-object system and a pair of first-order differential equations expressing update laws of arc-length parameters along smooth contour curves of the object were given. Instead of the zero relative-velocity assumption of rolling contact, the following set of postulates for pure rolling contact is introduced:(1)two contact points on each contour curve must coincide at a single common point without mutual penetration,(2)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 [6] that Pfaffian constraints are integrable, and their integral forms are derived explicitly by using the moving frame coordinates. It is further shown [6] that the quantities of the second fundamental form of concerned contour curves do not appear in the Euler-Lagrange equations but play a key role in the update laws of arc-length parameters of the curves.

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:(a1)two contact points on each curved surface must coincide at a single common point without mutual penetration,(a2)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,(a3)the two path lengths running on their corresponding surfaces must be coincident.

In the previous paper [7], a set of Euler-Lagrange equations of motion of the fingers-object system have been derived by using the moving frame coordinates, but any explicit set of update laws of moving frame coordinates have been not given yet. In particular, any mathematical role of the quantities of the second fundamental form of a contact curve running on a concerned surface has not yet been studied in a mathematically explicit way. Therefore, it still remains unsolved to construct a complete set of equations of 3D grasping under rolling contact constraints in the situation of arbitrary given geometry of surfaces. This paper shows that nonholonomic constraints arising from relative twisting among the two normal axes at the contact point can be naturally resolved into determination of each of geodesic curvatures of the curves of the contact point on the fingertip surface and the object surface. Another second fundamental form of normal curvature on each surface is assumed to be extracted from a data structure of a given rigid body object, together with that of unit normal at each specified point on its surface. Thus, a set of 3D Frenet-Serret equations with normal and geodesic curvatures that update the moving frame coordinates are determined and shown to be computationally integrable together with the set of Euler-Lagrange equations of motion of the whole system.

2. Preliminary Results on Derivation of Euler-Lagrange Equations

Consider firstly a physical situation that a pair of multijoint robot fingers is grasping a 3D rigid body as seen in Figure 1. In this figure, the inertial frame denoted by 𝑂-π‘₯𝑦𝑧 is fixed in the Euclidean space 𝐄3, and local coordinates systems denoted by 𝑂𝑖-π‘‹π‘–π‘Œπ‘–π‘π‘– for 𝑖=1,2 are introduced at each robot fingerend. The local coordinates system of the object is denoted by π‘‚π‘š-π‘‹π‘Œπ‘ as shown in Figure 1, where π‘‚π‘š stands for the object mass center. Next, denote each locus of points of contact between the two surfaces by a curve 𝛾𝑖(𝑠𝑖) (3-dimensional vector in 𝐄3) with length parameter 𝑠𝑖 on its corresponding surface 𝑆𝑖(𝑖=0,1), where 𝑖=0 signifies the object, and 𝑖=1 does the left hand fingerend. It is possible to assume that, given a curve 𝛾1(𝑠1) as a locus of points of contact on 𝑆1 and another curve 𝛾0(𝑠0) as a locus of contact points on 𝑆0, the two curves coincide at contact point 𝑃1 and share the same tangent plane 𝑇1 at 𝑃1 (see Figure 2). Further, during continuation of rolling contact, the two curves 𝛾0(𝑠0) and 𝛾1(𝑠1) can be described in terms of the same length parameter 𝑠 in such a way that 𝑠0=𝑠+𝑐0 and 𝑠1=𝑠+𝑐1, where 𝑐0 and 𝑐1 are constant.


Figure 1 
A pair of robot fingers grasping a 3D rigid object with smooth surfaces.

Figure 2 
Definition of the moving frame coordinates system centering at the rolling contact point.

Second, suppose that at some 𝑠 of the length parameter the two curves 𝛾0(𝑠0) and 𝛾1(𝑠1) coincide at 𝑃1(𝑠). Since 𝛾0(𝑠0) is described in local coordinates π‘‚π‘š-π‘‹π‘Œπ‘, its expression in the frame coordinates is given by 𝛾0𝑠0ξ€Έ=Ξ 0𝛾0𝑠0ξ€Έ,(1) where Ξ 0 is a 3Γ—3 rotational matrix composed of three unit orthogonal vectors 𝐫𝑋,π«π‘Œ, and 𝐫𝑍 that are expressed in the inertial frame coordinates 𝑂-π‘₯𝑦𝑧 as shown in Figure 1, that is, Ξ 0=𝐫𝑋,π«π‘Œ,𝐫𝑍.(2)

Since 𝛾0(𝑠0) is parametrized by length parameter, π›Ύξ…ž0(𝑠0)=d𝛾0(𝑠0)/d(𝑠0) must be expressed by the unit tangent vector 𝐛0(𝑠0) at 𝑃1(𝑠) lying on the tangent plane 𝑇1. According to (a1) and (a2), it is possible to suppose that there exist the two unit normals 𝐧0(𝑠0) and 𝐧1(𝑠1) expressed in corresponding local coordinates π‘‚π‘š-π‘‹π‘Œπ‘ and 𝑂1-𝑋1π‘Œ1𝑍1, respectively (see Figure 2). Then, it is possible to certify that 𝐧0=βˆ’π§1,(3) at 𝑠0=𝑠1=𝑠, where 𝐧0=Ξ 0𝐧0,𝐧1=Ξ 1𝐧1,(4) and Ξ 1=(𝐫𝑋1,π«π‘Œ1,𝐫𝑍1), 𝐫𝑋1 denotes the unit vector of 𝑋1-axis of 𝑂1-𝑋1π‘Œ1𝑍1 expressed in the inertial frame coordinates, and π«π‘Œ1 and 𝐫𝑍1 have a similar meaning.

In what follows, we denote vectors 𝐧𝑖 and 𝐛𝑖 for 𝑖=0,1 with upper bar when they are expressed in the inertial frame coordinates as seen in Figure 2. We also denote the derivative of Π𝑖 in time 𝑑 by ̇Π𝑖 and similarly the derivatives of 𝐧𝑖 and 𝐛𝑖 in 𝑑 by Μ‡β€Œπ§π‘–, and Μ‡β€Œπ›π‘–. If we assume that the instantaneous axis of angular velocity of the object through the mass center π‘‚π‘š is denoted by πœ”=(πœ”π‘₯,πœ”π‘¦,πœ”π‘§)T in the frame coordinates, then the angular velocity vector 𝝎 attached to the local coordinates π‘‚π‘š-π‘‹π‘Œπ‘ can be defined in such a way that 𝝎=Ξ 0ξ€·πœ”π‘‹,πœ”π‘Œ,πœ”π‘ξ€ΈT=Ξ 0𝝎,(5) where we define 𝝎=(πœ”π‘‹,πœ”π‘Œ,πœ”π‘)T. It is well known in the text books [8–10] that Μ‡Ξ 0=Ξ 0Ξ©0,(6) where Ξ©0=0βˆ’πœ”π‘πœ”π‘Œπœ”π‘0βˆ’πœ”π‘‹βˆ’πœ”π‘Œπœ”π‘‹0ξƒͺ.(7)

It is easy to check that, in the illustrative case of a spherical left hand fingertip shown in Figure 1, we have Μ‡Ξ 1=Ξ 1Ξ©1,Ξ©1=0̇𝑝10βˆ’Μ‡π‘1ξƒͺ00000,(8) where ̇𝑝1=Μ‡π‘ž11+Μ‡π‘ž12 because both the rotational axes of joints 𝐽1 and 𝐽2 have the same direction in 𝑧-axis of the frame coordinates 𝑂-π‘₯𝑦𝑧.

Let us now interpret the first postulate (a1) in a mathematical form described by 𝐫1ξ€·π‘ž1ξ€Έ+Ξ 1𝛾1(𝑠)=𝐫0(𝐱)+Ξ 0𝛾0(𝑠),(9) where 𝐫1(π‘ž1) denotes the position vector of 𝑂1 (the center of the left hand fingerend) expressed in terms of the frame coordinates and π‘ž1=(π‘ž11,π‘ž12)T, and 𝐫0(=𝐱) does the position vector of π‘‚π‘š (the object mass center) also expressed in the frame coordinates and 𝐱=(π‘₯,𝑦,𝑧)T. Then, differentiation of (9) in 𝑑 yields ̇𝐫1βˆ’Μ‡π«0ξ€Έ+Μ‡Ξ 1𝛾1+Ξ 1𝐛1ξ‚΅d𝑠1ξ‚Ά=Μ‡Ξ d𝑑0𝛾0+Ξ 0𝐛0ξ‚΅d𝑠0ξ‚Άd𝑑.(10) If during rolling of the contact point the tangent vector 𝐛1 of the fingerend has the same direction as that of 𝐛0 (of the object), that is, if 𝐛1=𝐛0, then on account of (a3), (10) reduces to ̇𝐫1βˆ’Μ‡π«0ξ€Έ+Ξ 1Ξ©1𝛾1βˆ’Ξ 0Ξ©0𝛾0=0.(11) According to the previous paper, multiplication of the rotation matrix of the moving frame coordinates defined by Ξ 0Ξ¨0 from the right yields 𝑅𝑛1,𝑅𝑏1,𝑅𝑒1ξ€Έβ‰œξ€·Μ‡π«1βˆ’Μ‡π«0ξ€ΈTΞ 0Ξ¨0βˆ’πŽT𝛾0Γ—Ξ¨0ξ€Έ+𝝎T1𝛾1Γ—Ξ¨βˆ—1ξ€Έ=0,(12) where we define 𝐞0=𝐧0×𝐛0, 𝐞1=𝐧1×𝐛1, and Ξ¨0=𝐧0,𝐛0,𝐞0ξ€Έ,Ξ¨βˆ—1=ξ€·βˆ’π§1,𝐛1,βˆ’πž1ξ€Έ(13) (see Figure 2), and we use Ξ 0=Ξ¨0=Ξ 1Ξ¨βˆ— and 𝛾0Γ—Ξ¨0=𝛾0×𝐧0,𝛾0×𝐛0,𝛾0Γ—πž0ξ€Έ,(14) and 𝛾1Γ—Ξ¨βˆ—1 has a similar meaning. Equation (13) means the three equalities 𝑅𝑛1=0, 𝑅𝑏1=0, and 𝑅𝑒1=0 that constitute the set of three Pfaffian forms expressing the rolling contact constraint of zero-relative velocity. In the previous paper [7], it is shown that the Pfaffian forms of (13) are integrable with the integral forms d𝑄d𝑑𝑛1,𝑄𝑏1,𝑄𝑒1ξ€Έ=𝑅𝑛1,𝑅𝑏1,𝑅𝑒1ξ€Έ=0,(15) where 𝑄𝑛1,𝑄𝑏1,𝑄𝑒1ξ€Έ=𝐫1βˆ’π«0ξ€ΈTΞ 0Ξ¨0+𝛾T1Ξ¨βˆ—1βˆ’π›ΎT0Ξ¨0,Ξ¨1=𝐧1,𝐛1,𝐞1ξ€Έ(16) provided that 𝐛0=𝐛1.

By virtue of the integrability of each Pfaffian form of rolling contact constraints, the Lagrangian of the system is written into 𝐿=πΎβˆ’π‘€π‘”π‘¦βˆ’π‘–=1,2π‘ƒπ‘–ξ€·π‘žπ‘–ξ€Έβˆ’ξ“π‘–=1,2𝑓𝑖𝑄𝑛𝑖+πœ†π‘–π‘„π‘π‘–+πœ‚π‘–π‘„π‘’π‘–ξ€Έ,(17) where 𝐾=𝑖=1,212Μ‡π‘žTπ‘–πΊπ‘–ξ€·π‘žπ‘–ξ€Έ+𝑀2‖̇𝐱‖2+12𝝎T𝐻𝝎.(18) In these equations, 𝑀 denotes the mass of the object, 𝐻, the inertia matrix of the object around its mass center, 𝐺𝑖(π‘žπ‘–), the inertia matrix of finger 𝑖, 𝑃𝑖, the potential energy of finger 𝑖, 𝑔, the gravity constant, and 𝑓𝑖, πœ†π‘–, and πœ‚π‘– express Lagrange multipliers corresponding to constraints 𝑄𝑛1,𝑄𝑏1,𝑄𝑒1ξ€Έ=(0,0,0)(19) and ̇𝐱=d𝐫(𝐱)/d𝑑. Then, by applying the variational principle to 𝐿 described as ξ€œπ‘‘1𝑑0𝛿𝐿+𝑖=1,2𝑒Tπ‘–π›Ώπ‘žπ‘–ξƒ°d𝑑=0,(20) with control input 𝑒𝑖 at finger joints, it is possible to obtain the following set of Euler-Lagrange equations: π‘€Μˆξ“π±βˆ’π‘–=1,2𝑓𝑖𝐧0𝑖+πœ†π‘–π›0𝑖+πœ‚π‘–πž0𝑖+π‘€π‘”πžπ‘¦π»Μ‡ξ“=0,(21)𝝎+πŽΓ—π»πŽβˆ’π‘–=1,2𝛾0𝑖×𝑓𝑖𝐧0𝑖+πœ†π‘–π›0𝑖+πœ‚π‘–πž0𝑖𝐺=0,(22)π‘–ξ€·π‘žπ‘–ξ€ΈΜˆπ‘žπ‘–+12Μ‡πΊπ‘–ξ€·π‘žπ‘–ξ€Έ+𝑆𝑖(π‘ž,Μ‡π‘ž)Μ‡π‘žπ‘–+πœ•π‘ƒπ‘–ξ€·π‘žπ‘–ξ€Έπœ•π‘žπ‘–βˆ’π½Tπ‘–ξ€·π‘žπ‘–ξ€Έξ‚†π‘“π‘–π§π‘–βˆ’πœ†π‘–π›π‘–+πœ‚π‘–πžπ‘–ξ‚‡βˆ’π‘ŠTπ‘–ξ€½π›Ύπ‘–Γ—ξ€·π‘“π‘–π§π‘–βˆ’πœ†π‘–π›π‘–+πœ‚π‘–πžπ‘–ξ€Έξ€Ύ=𝑒𝑖,𝑖=1,2,(23) where πžπ‘¦=(0,1,0)T, and the meaning of π‘Šπ‘– will be explained later. It should be noted that the sum of inner products of ̇𝐱 and (21), 𝝎 and (22), and Μ‡π‘žπ‘– and (23) for 𝑖=1,2 yields the energy relation 𝑖=1,2Μ‡π‘žT𝑖𝑒𝑖=dd𝑑𝐾+𝑀𝑔𝑦+𝑖=1,2π‘ƒπ‘–ξ€·π‘žπ‘–ξ€Έξƒ°.(24)

3. Necessary Conditions for Updating Moving Frame Coordinates

In order to always keep the tangent vector 𝐛1 of the fingerend at the contact point 𝑃1 to coincide with the tangent vector 𝐛0 of the object surface at the same common contact point, we first show that the following two equations should be satisfied necessarily: 𝝎T0𝐛0=𝝎T1𝐛1,ξ€·πœ…(25)𝑛0+πœ…π‘›1ξ€Έd𝑠0d𝑑=𝝎T0𝐞0+𝝎T1𝐞1,(26) as shown in detail in (𝐴1) of Appendix A, where πœ…π‘›0 denotes the normal curvature of the object surface at the contact point 𝑃1, and πœ…π‘›1 does that of the fingerend surface 𝑆1 at 𝑃1. Both the normal curvatures πœ…π‘›0 and πœ…π‘›1 should be determined in accordance with the geometric structure of their corresponding surfaces, once the direction of each locus of contact points, that is, 𝐛1 or 𝐛0, is given. Similarly, as shown in (𝐴2) of Appendix A, the conditions 𝐛0=𝐛1 and Μ‡β€Œπ›0=̇𝐛1 imply ξ€·πŽT0𝐧0+𝝎T1𝐧1ξ€Έ+ξ€·πœ…π‘’0+πœ…π‘’1ξ€Έd𝑠0d𝑑=0,(27) where πœ…π‘’0 denotes the geodesic curvature of the object surface at 𝑃1, and πœ…π‘’1 has a similar meaning. This equation does not determine each πœ…π‘’0 or πœ…π‘’1 individually. Therefore, let us try to differentiate 𝐞0 and 𝐞1 in 𝑑. However, as shown in (𝐴3) of Appendix A, we rederive only (25) and (26).

We are now in a position to find a necessary condition for maintaining the equality of (25) for the time being. To this end, it is important to see that the time rate of the equality (25) reduces to πŽξ€½ξ€·T0𝐞0ξ€Έπœ…π‘’0βˆ’ξ€·πŽT1𝐞1ξ€Έπœ…π‘’1ξ€Ύd𝑠0=Μ‡πŽd𝑑T1𝐛1βˆ’Μ‡πŽT0𝐛0+ξ€·βˆ’πœ…π‘›0𝝎T0𝐧0+πœ…π‘›1𝝎T1𝐧1ξ€Έ,(28) as shown in (𝐴4) of Appendix A. Then, this equation together with (27) leads to ξ‚΅πŽ11T0𝐞0βˆ’πŽT1𝐞1ξ‚Άξ‚€πœ…π‘’0πœ…π‘’1d𝑠0=ξ‚΅βˆ’ξ€·πŽd𝑑T0𝐧0+𝝎T1𝐧1ξ€Έπœ‰1ξ‚Ά,(29) where πœ‰1β‰œξ€·βˆ’πœ…π‘›0𝝎T0𝐧0+πœ…π‘›1𝝎T1𝐧1ξ€Έd𝑠0+Μ‡πŽd𝑑T1𝐛1βˆ’Μ‡πŽT0𝐛0.(30) Thus, it is possible to determine each geodesic curvature individually by inverting the coefficient 2Γ—2 matrix of (29) in the following way: πœ…π‘’0d𝑠0=dπ‘‘βˆ’πŽT1𝐞1ξ€·πŽT0𝐧0+𝝎T1𝐧1ξ€Έ+πœ‰1𝝎T1𝐞1+𝝎T0𝐞0,πœ…(31)𝑒1d𝑠1=dπ‘‘βˆ’πŽT0𝐞0ξ€·πŽT0𝐧0+𝝎T1𝐧1ξ€Έβˆ’πœ‰1𝝎T1𝐞1+𝝎T0𝐞0.(32)

Finally, it is possible to see that the moving frames denoted by Ξ¨0 and Ξ¨1 should satisfy the Frenet-Serre equations dΞ¨d𝑑0=Ξ¨00πœ…π‘›00βˆ’πœ…π‘›00βˆ’πœ…π‘’00πœ…π‘’00ξƒͺd𝑠0,dd𝑑Ψd𝑑1=Ξ¨10πœ…π‘›10βˆ’πœ…π‘›10βˆ’πœ…π‘’10πœ…π‘’10ξƒͺd𝑠0.d𝑑(33)

4. Sufficient Conditions for Updating the Moving Frame Coordinates

In the Frenet-Serre equation of (31), the coefficient πœ…π‘›0 called the normal curvature is determined by the geometric shape of the object surface at point 𝑃1 denoted by 𝛾0(𝑠0) in the local coordinates π‘‚π‘š-π‘‹π‘Œπ‘, and the other normal curvature πœ…π‘›1 in (32) is also determined similarly. The geodesic curvatures πœ…π‘’0 and πœ…π‘’1 are determined via the instantaneous motion of rolling contact, so that they satisfy (27) and (28). We now show under the postulates (a1) to (a3) that if πœ…π‘’0 and πœ…π‘’1 are determined by (31) and (32), respectively, and the tangent vectors 𝐛0(0) and 𝐛1(0) of the moving frame coordinates at the initial time are chosen to coincide with each other and at the same time to satisfy (25) at 𝑑=0, then for any 𝑑>0, it follows that dd𝑑𝐛1=dd𝑑𝐛0,ddπ‘‘πž1d=βˆ’dπ‘‘πž0,𝐛1=𝐛0,𝐞1=𝐞0.(34) To prove this, first note again that the postulates (a1) and (a2) imply 𝐧0=βˆ’π§1,dd𝑑𝐧0d=βˆ’d𝑑𝐧1.(35) Next, note that the sum of (31) and (32) implies (27). At this stage, suppose that 𝐛1 is not coincident with 𝐛0 at some 𝑑>0 though both 𝐛1 and 𝐛0 are lying on the same tangent plane. In other words, suppose that 𝐛T1𝐛0=cosπœƒ and πœƒ(𝑑) stands for πœƒ(𝑑,𝑠1(𝑑),𝑠0(𝑑)), then, as shown in Appendix B, we have Μ‡πœƒ=𝝎T1𝐧1+𝝎T0𝐧0+ξ€·πœ…π‘’0+πœ…π‘’1ξ€Έd𝑠0d𝑑.(36) Further, as discussed in (𝐡3) of Appendix B, geodesic curvatures πœ…π‘’0 and πœ…π‘’1 should be defined as πœ…π‘’0=πœ•πœƒπœ•π‘ 0,πœ…π‘’1=πœ•πœƒπœ•π‘ 1.(37) It is important to remark that (36) was first derived by Montana [2] as a nonholonomic constraint of rolling. Equation (36) can be interpreted by Murray et al. [1] as the nonholonomic constraint that governs the rotating motion of one tangent plane to the fingerend relative to another tangent plane of the object surface caused by relative β€œtwisting” between the axis of normal 𝐧1 and that of 𝐧0. Nevertheless, it is further important to note that if πœ…π‘’0 and πœ…π‘’1 are set as shown in (31) and (32), respectively, then the right hand side of (36) becomes zero due to (27). That is, (31) and (32) imply Μ‡πœƒ=0. Therefore, if at the initial time 𝐛1(0)=𝐛0(0), then the setting of (31) and (32) for geodesic curvatures πœ…π‘’1 and πœ…π‘’0 leads to 𝐛1(𝑑)=𝐛0(𝑑) for 𝑑>0 as far as 𝝎T1𝐞1+𝝎T0𝐞0β‰ 0. Since 𝐧1=βˆ’π§0 as far as the contact is maintained, the equality 𝐛1(𝑑)=𝐛0(𝑑) implies 𝐞1(𝑑)=βˆ’πž0(𝑑) for 𝑑β‰₯0, then, from (𝐴1) of Appendix A, (25) and (26) follow. At the same time, from (26) and (27), it follows that d𝐛1/d𝑑=d𝐛0/d𝑑 in view of the first two equations of (𝐴2) of Appendix A. Thus, it is concluded that all equalities in (34) follow.

5. A Numerically Integrable Set of Differential Equations Under Rolling Constraints

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 β€œ0𝑖” for expressing variables on quantities of the object at contact point 𝑃𝑖 of the 𝑖-th finger and the suffix 𝑖 for those of the fingerend surface of finger 𝑖 at 𝑃𝑖. For convenience, we use 𝝎 instead of 𝝎0. In the following, we give the set of Euler-Lagrange equations, first-order equations of rotation matrices Ξ 0 and Π𝑖, update equations of length parameters, and Frenet-Serre equations for updating moving frame coordinates at contact points π‘€Μˆξ“π±βˆ’π‘–Ξ 0Ξ¨0π‘–π€π‘–βˆ’π‘€π‘”πžπ‘¦=0,(𝐸π‘₯)π»Μ‡ξ“πŽ+πŽΓ—π»πŽβˆ’π‘–π›Ύ0𝑖×Ψ0𝑖𝝀𝑖=0,(πΈπœ”)πΊπ‘–Μˆπ‘žπ‘–+12̇𝐺𝑖+π‘†π‘–ξ‚‡Μ‡π‘žπ‘–+πœ•π‘ƒπ‘–πœ•π‘žπ‘–+𝐽T𝑖Π𝑖+π‘ŠTπ‘–π›Ύπ‘–Γ—ξ€ΎΞ¨βˆ—π‘–π€π‘–=𝑒𝑖,(𝐸𝑖)dΞ d𝑑0=Ξ 0Ξ©0,dΞ d𝑑𝑖=Π𝑖Ω𝑖,(πΈπ‘Ÿ)d𝑠d𝑑𝑖=𝝎T𝐞0𝑖+𝝎Tπ‘–πžπ‘–πœ…π‘›0𝑖+πœ…π‘›π‘–,(𝐸𝑠)dΞ¨d𝑑0𝑖=Ξ¨0𝑖𝐾0𝑖d𝑠𝑖,dd𝑑Ψd𝑑𝑖=Ψ𝑖𝐾𝑖d𝑠𝑖,d𝑑(𝐸𝑓𝑠) where 𝐾0𝑖=0πœ…π‘›0𝑖0βˆ’πœ…π‘›0𝑖0βˆ’πœ…π‘’0𝑖0πœ…π‘’0𝑖0ξƒͺ,𝐾𝑖=0πœ…π‘›π‘–0βˆ’πœ…π‘›π‘–0βˆ’πœ…π‘’π‘–0πœ…π‘’π‘–0ξƒͺ,Ξ¨0𝑖=𝐧0𝑖,𝐛0𝑖,𝐞0𝑖,Ψ𝑖=𝐧𝑖,𝐛𝑖,πžπ‘–ξ€Έ,𝝀𝑖=𝑓𝑖,πœ†π‘–,πœ‚π‘–ξ€ΈT,π½π‘–ξ€·π‘žπ‘–ξ€Έ=πœ•π«π‘–ξ€·π‘žπ‘–ξ€Έπœ•π‘žT𝑖,(38) and 𝛾0𝑖 denotes the contact point 𝑃𝑖 on the object surface expressed by the object local coordinates π‘‚π‘š-π‘‹π‘Œπ‘, and 𝛾𝑖 does that of 𝑃𝑖 on the fingerend surface of finger 𝑖 expressed by the fingerend local coordinates 𝑂𝑖-π‘‹π‘–π‘Œπ‘–π‘π‘–. In 𝐸𝑖, π‘Šπ‘– is an π‘šΓ—3 matrix depending on π‘žπ‘–, where π‘š denotes the degrees of freedom. In the case of a pair of robot fingers depicted in Figure 1, it is obvious to see that π‘Š1=ξƒͺ000011,π‘Š2=1000sinπ‘ž21sinπ‘ž210cosπ‘ž21cosπ‘ž21ξƒͺ.(39) It should be remarked again that 𝐸𝑓𝑠 expresses a set of Frenet-Serre equations for determining each moving frame coordinates at contact point 𝑃𝑖, and then the geodesic curvatures πœ…π‘’0𝑖 and πœ…π‘’π‘– are determined in the same manner as shown in (31) and (32). Further, computation of Μ‡πŽπ‘– and Μ‡πŽ appearing in (31) and (32) through πœ‰π‘– defined by (30) can be executed simultaneously via numerical integration of πΈπœ” and 𝐸𝑖. In practice, it is possible to compute Μ‡πŽπ‘– by Μ‡πŽ=π»βˆ’1ξƒ―ξ“βˆ’πŽΓ—π»πŽ+𝑖𝛾0𝑖×Ψ0𝑖𝝀𝑖.(40) Analogously, it is possible to compute Μ‡πŽπ‘– since πŽπ‘– must be expressed by a function form of 𝑉(π‘žπ‘–)Μ‡π‘žπ‘–, and Μˆπ‘žπ‘– can be calculated by multiplying (𝐸𝑖) by πΊπ‘–βˆ’1(π‘žπ‘–) from the left.

6. Conclusions

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 [11] in which it is assumed that any relative twist motion does not arise. The proposed model is composed of a set of 2nd-order Euler-Lagrange equations derived by using the moving frame coordinates and 1st-order Frenet-Serre equations together with 1st-order differential equations governing update laws of length parameters and rotational motions of the local coordinates. The nonholonomic constraint arising from possible relative twist of the two normal axes at the contact point is resolved into determination of the geodesic curvatures of the fingerend and object surfaces. This leads to a conclusion that the whole set of simultaneous differential equations with constraints are numerically integrable (as a preliminary result of numerical simulation, see [12]).

Appendices

A. Necessary Conditions

(𝐴1)Note that 𝐧1=βˆ’π§0 and (d/d𝑑)𝐧1=βˆ’(d/d𝑑)𝐧0, dd𝑑𝐧0=Μ‡Ξ 0𝐧0+Ξ 0̇𝐧0=Ξ 0Ξ©0𝐧0+Ξ 0ξ‚΅πœ•π§0πœ•π‘ 0ξ‚Άd𝑠0d𝑑=Ξ 0ξ€½πŽ0×𝐛0Γ—πž0ξ€Έξ€Ύβˆ’πœ…π‘›0Ξ 0𝐛0d𝑠0=ξ‚΅πŽd𝑑T0𝐞0βˆ’πœ…π‘›0d𝑠0ξ‚Άd𝑑𝐛0βˆ’ξ€·πŽT0𝐛0ξ€Έπž0,dd𝑑𝐧1=ξ‚΅πŽT1𝐞1βˆ’πœ…π‘›1d𝑠1ξ‚Άd𝑑𝐛1βˆ’ξ€·πŽT1𝐛1ξ€Έπž1.(A.1) If 𝐛0=𝐛1 and 𝐞0=βˆ’πž1, then it follows that 𝝎T0𝐛0=𝝎T1𝐛1,(βˆ—)ξ€·πœ…π‘›0+πœ…π‘›1ξ€Έd𝑠0d𝑑=𝝎T0𝐞0+𝝎T1𝐞1.(βˆ—βˆ—)(𝐴2)Similarly, it follows that dd𝑑𝐛0=Μ‡Ξ 0𝐛0+Ξ 0̇𝐛0=Ξ 0πŽξ€½ξ€·T0𝐧0ξ€Έπž0βˆ’ξ€·πŽT0𝐞0𝐧0ξ€Ύ+Ξ 0ξ€·πœ…π‘›0𝐧0+πœ…π‘’0𝐞0ξ€Έd𝑠0=ξ‚΅πŽd𝑑T0𝐧0+πœ…π‘’0d𝑠0ξ‚Άdπ‘‘πž0βˆ’ξ‚΅πŽT0𝐞0+πœ…π‘›0d𝑠0ξ‚Άd𝑑𝐧0,dd𝑑𝐛1=ξ‚΅πŽT1𝐧1+πœ…π‘’1d𝑠1ξ‚Άdπ‘‘πž1βˆ’ξ‚΅πŽT1𝐞1+πœ…π‘›1d𝑠1ξ‚Άd𝑑𝐧1.(A.2) These two equations imply 𝐸𝑖 and ξ€·πŽT0𝐧0+𝝎T1𝐧1ξ€Έ+ξ€·πœ…π‘’0+πœ…π‘’1ξ€Έd𝑠0d𝑑=0(βˆ—βˆ—βˆ—) if 𝐛0=𝐛1 and (d/d𝑑)𝐛0=(d/d𝑑)𝐛1.(𝐴3)Similarly, it follows that ddπ‘‘πž0=ξ€·πŽT0𝐛0𝐧0βˆ’ξ‚΅πŽT0𝐧0+πœ…π‘’0d𝑠0ξ‚Άd𝑑𝐛0,ddπ‘‘πž1=ξ€·πŽT1𝐛1𝐧1βˆ’ξ‚΅πŽT0𝐧1+πœ…π‘’1d𝑠1ξ‚Άd𝑑𝐛1.(A.3) These two equations imply πΈπœ” and 𝐸𝑠 if 𝐞0=βˆ’πž1, 𝐛0=𝐛1, and (d/d𝑑)𝐞0=βˆ’(d/d𝑑)𝐞1.(𝐴4)Time rate of πΈπœ” reduces to Μ‡πŽT0𝐛0+𝝎T0̇𝐛0=Μ‡πŽT1𝐛1+𝝎T1̇𝐛1.(A.4) Since ̇𝐛𝑖=(πœ…π‘›π‘–π§π‘–+πœ…π‘’π‘–πžπ‘–)(d𝑠0/d𝑑) for 𝑖=0,1, the above equality reduces to πŽξ€½ξ€·T0𝐞0ξ€Έπœ…π‘’0βˆ’ξ€·πŽT1𝐞1ξ€Έπœ…π‘’1ξ€Ύd𝑠0=ξ€·dπ‘‘βˆ’πœ…π‘›0𝝎T0𝐧0+πœ…π‘›1𝝎T1𝐧1ξ€Έd𝑠0+Μ‡πŽd𝑑T1𝐛1βˆ’Μ‡πŽT0𝐛0β‰œπœ‰1ξ€·Μ‡πŽ1,Μ‡πŽ0ξ€Έ.(A.5)

B. Preliminary Remarks on Geodesic Curvature

𝐛0𝑖=Ξ 0𝐛0𝑖(tangentvectorsonobjectsurface𝑖),𝐧0𝑖=Ξ 0𝐧0𝑖(normalvectorsonobjectsurface𝑖),𝐛𝑖=Π𝑖𝐛𝑖(tangentvectorsonfingerendsurface𝑖),𝐧𝑖=Π𝑖𝐧𝑖(normalvectorsonfingerendsurface𝑖),(B.1) both Ω𝑖 and Ξ©0 are skew symmetric.(𝐡1)Derivation of πœ•πœƒ/πœ•π‘‘ where πœƒ(𝑑,𝑠𝑖(𝑑),𝑠𝑖0(𝑑)).

If 𝐛T1𝐛0=cosπœƒ, then 𝐛T1Ξ T1Ξ 0𝐞0ξ‚€πœ‹=cos2ξ‚βˆ’+πœƒ=βˆ’sinπœƒ,πœ•πœƒπœƒπœ•π‘‘cosπœƒ=ξ€·π›πœ•π‘‘T1Ξ T1Ξ 0𝐞0ξ€Έ=𝐛T1Μ‡Ξ T1Ξ 0𝐞0+𝐛T1Ξ T1Μ‡Ξ 0𝐞0=𝐛T1Ξ©T1Ξ T1Ξ 0𝐞0+𝐛T1Ξ T1Ξ 0Ξ©0𝐞0=ξ€½Ξ©1ξ€·πž1×𝐧1ξ€Έξ€ΎTΞ T1Ξ 0𝐞0+𝐛T1Ξ T1Ξ 0Ξ©0𝐧0×𝐛0ξ€Έ=πŽξ€½ξ€·T1𝐧1ξ€Έπž1βˆ’ξ€·πŽT1𝐞1𝐧1ξ€ΎTΞ T1Ξ 0𝐞0+𝐛T1Ξ T1Ξ 0πŽξ€½ξ€·T0𝐛0𝐧0βˆ’ξ€·πŽT0𝐧0𝐛0ξ€Ύ=ξ€·πŽT1𝐧1ξ€ΈπžT1Ξ T1Ξ 0𝐞0βˆ’π›T1Ξ 1Ξ 0𝐛0ξ€·πŽT0𝐧0ξ€Έξ€·πŽ=βˆ’T1𝐧1+𝝎T0𝐧0ξ€Έcosπœƒ,πœ•πœƒπœ•π‘‘=𝝎T1𝐧1+𝝎T0𝐧0.(B.2)(𝐡2)With derivation of Μ‡πœƒ, 𝐛T1Ξ T1Ξ 0𝐞0βˆ’Μ‡ξ€·πŽ=βˆ’sinπœƒ,πœƒcosπœƒ=βˆ’T1𝐧1+𝝎T0𝐧0ξ€Έcosπœƒ+𝐛T1Ξ T1Ξ 0ξ€·βˆ’πœ…πž0𝐛0ξ€Έd𝑠0+ξ€·πœ…d𝑑𝑛1𝐧1+πœ…π‘’1𝐞1ξ€ΈTΞ T1Ξ 0𝐞0d𝑠1ξ€·πŽd𝑑=βˆ’T1𝐧1+𝝎T0𝐧0ξ€Έcosπœƒβˆ’πœ…π‘’0cosπœƒd𝑠0dπ‘‘βˆ’πœ…π‘’1cosπœƒd𝑠1,Μ‡dπ‘‘πœƒ=𝝎T1𝐧1+𝝎T0𝐧0+ξ€·πœ…π‘’0+πœ…π‘’1ξ€Έd𝑠1.d𝑑(B.3)(𝐡3)With derivation of πœ…π‘’0 and πœ…π‘’1, 𝐛T1Ξ T1Ξ 0𝐞0πœ•=βˆ’sinπœƒ,πœ•π‘ 0𝐛T1Ξ T1Ξ 0𝐞0=βˆ’πœ•πœƒπœ•π‘ 0𝐛cosπœƒ,T1Ξ T1Ξ 0ξ€·βˆ’πœ…π‘’0𝐛0=βˆ’πœ•πœƒπœ•π‘ 0cosπœƒ,βˆ’πœ…π‘’0cosπœƒ=βˆ’πœ•πœƒπœ•π‘ 0πœ…cosπœƒ,𝑒0=πœ•πœƒπœ•π‘ 0,πœ…π‘’1=πœ•πœƒπœ•π‘ 1.(B.4)