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Modelling and Simulation in Engineering
Volume 2011 (2011), Article ID 684034, 8 pages
http://dx.doi.org/10.1155/2011/684034
Research Article

A Mathematical and Numerically Integrable Modeling of 3D Object Grasping under Rolling Contacts between Smooth Surfaces

RIKEN-TRI Collaboration Center for Human-Interactive Robot Research, Nagoya, Aichi 463-0003, Japan

Received 31 March 2011; Accepted 25 July 2011

Academic Editor: Antonio Munjiza

Copyright © 2011 Suguru Arimoto and Morio Yoshida. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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-𝑥𝑦𝑧 is fixed in the Euclidean space 𝐄3, and local coordinates systems denoted by 𝑂𝑖-𝑋𝑖𝑌𝑖𝑍𝑖 for 𝑖=1,2 are introduced at each robot fingerend. The local coordinates system of the object is denoted by 𝑂𝑚-𝑋𝑌𝑍 as shown in Figure 1, where 𝑂𝑚 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.

684034.fig.001
Figure 1: A pair of robot fingers grasping a 3D rigid object with smooth surfaces.
684034.fig.002
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 𝑂𝑚-𝑋𝑌𝑍, its expression in the frame coordinates is given by 𝛾0𝑠0=Π0𝛾0𝑠0,(1) where Π0 is a 3×3 rotational matrix composed of three unit orthogonal vectors 𝐫𝑋,𝐫𝑌, and 𝐫𝑍 that are expressed in the inertial frame coordinates 𝑂-𝑥𝑦𝑧 as shown in Figure 1, that is, Π0=𝐫𝑋,𝐫𝑌,𝐫𝑍.(2)

Since 𝛾0(𝑠0) is parametrized by length parameter, 𝛾0(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 𝑂𝑚-𝑋𝑌𝑍 and 𝑂1-𝑋1𝑌1𝑍1, respectively (see Figure 2). Then, it is possible to certify that 𝐧0=𝐧1,(3) at 𝑠0=𝑠1=𝑠, where 𝐧0=Π0𝐧0,𝐧1=Π1𝐧1,(4) and Π1=(𝐫𝑋1,𝐫𝑌1,𝐫𝑍1), 𝐫𝑋1 denotes the unit vector of 𝑋1-axis of 𝑂1-𝑋1𝑌1𝑍1 expressed in the inertial frame coordinates, and 𝐫𝑌1 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 𝑡 by ̇Π𝑖 and similarly the derivatives of 𝐧𝑖 and 𝐛𝑖 in 𝑡 by ̇𝐧𝑖, and ̇𝐛𝑖. If we assume that the instantaneous axis of angular velocity of the object through the mass center 𝑂𝑚 is denoted by 𝜔=(𝜔𝑥,𝜔𝑦,𝜔𝑧)T in the frame coordinates, then the angular velocity vector 𝝎 attached to the local coordinates 𝑂𝑚-𝑋𝑌𝑍 can be defined in such a way that 𝝎=Π0𝜔𝑋,𝜔𝑌,𝜔𝑍T=Π0𝝎,(5) where we define 𝝎=(𝜔𝑋,𝜔𝑌,𝜔𝑍)T. It is well known in the text books [810] that ̇Π0=Π0Ω0,(6) where Ω0=0𝜔𝑍𝜔𝑌𝜔𝑍0𝜔𝑋𝜔𝑌𝜔𝑋0.(7)

It is easy to check that, in the illustrative case of a spherical left hand fingertip shown in Figure 1, we have ̇Π1=Π1Ω1,Ω1=0̇𝑝10̇𝑝100000,(8) where ̇𝑝1=̇𝑞11+̇𝑞12 because both the rotational axes of joints 𝐽1 and 𝐽2 have the same direction in 𝑧-axis of the frame coordinates 𝑂-𝑥𝑦𝑧.

Let us now interpret the first postulate (a1) in a mathematical form described by 𝐫1𝑞1+Π1𝛾1(𝑠)=𝐫0(𝐱)+Π0𝛾0(𝑠),(9) where 𝐫1(𝑞1) denotes the position vector of 𝑂1 (the center of the left hand fingerend) expressed in terms of the frame coordinates and 𝑞1=(𝑞11,𝑞12)T, and 𝐫0(=𝐱) does the position vector of 𝑂𝑚 (the object mass center) also expressed in the frame coordinates and 𝐱=(𝑥,𝑦,𝑧)T. Then, differentiation of (9) in 𝑡 yields ̇𝐫1̇𝐫0+̇Π1𝛾1+Π1𝐛1d𝑠1=̇Πd𝑡0𝛾0+Π0𝐛0d𝑠0d𝑡.(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 ̇𝐫1̇𝐫0+Π1Ω1𝛾1Π0Ω0𝛾0=0.(11) According to the previous paper, multiplication of the rotation matrix of the moving frame coordinates defined by Π0Ψ0 from the right yields 𝑅𝑛1,𝑅𝑏1,𝑅𝑒1̇𝐫1̇𝐫0TΠ0Ψ0𝝎T𝛾0×Ψ0+𝝎T1𝛾1×Ψ1=0,(12) where we define 𝐞0=𝐧0×𝐛0, 𝐞1=𝐧1×𝐛1, and Ψ0=𝐧0,𝐛0,𝐞0,Ψ1=𝐧1,𝐛1,𝐞1(13) (see Figure 2), and we use Π0=Ψ0=Π1Ψ and 𝛾0×Ψ0=𝛾0×𝐧0,𝛾0×𝐛0,𝛾0×𝐞0,(14) and 𝛾1×Ψ1 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 d𝑄d𝑡𝑛1,𝑄𝑏1,𝑄𝑒1=𝑅𝑛1,𝑅𝑏1,𝑅𝑒1=0,(15) where 𝑄𝑛1,𝑄𝑏1,𝑄𝑒1=𝐫1𝐫0TΠ0Ψ0+𝛾T1Ψ1𝛾T0Ψ0,Ψ1=𝐧1,𝐛1,𝐞1(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=𝐾𝑀𝑔𝑦𝑖=1,2𝑃𝑖𝑞𝑖𝑖=1,2𝑓𝑖𝑄𝑛𝑖+𝜆𝑖𝑄𝑏𝑖+𝜂𝑖𝑄𝑒𝑖,(17) where 𝐾=𝑖=1,212̇𝑞T𝑖𝐺𝑖𝑞𝑖+𝑀2̇𝐱2+12𝝎T𝐻𝝎.(18) In these equations, 𝑀 denotes the mass of the object, 𝐻, the inertia matrix of the object around its mass center, 𝐺𝑖(𝑞𝑖), the inertia matrix of finger 𝑖, 𝑃𝑖, the potential energy of finger 𝑖, 𝑔, the gravity constant, and 𝑓𝑖, 𝜆𝑖, and 𝜂𝑖 express Lagrange multipliers corresponding to constraints 𝑄𝑛1,𝑄𝑏1,𝑄𝑒1=(0,0,0)(19) and ̇𝐱=d𝐫(𝐱)/d𝑡. Then, by applying the variational principle to 𝐿 described as 𝑡1𝑡0𝛿𝐿+𝑖=1,2𝑢T𝑖𝛿𝑞𝑖d𝑡=0,(20) with control input 𝑢𝑖 at finger joints, it is possible to obtain the following set of Euler-Lagrange equations: 𝑀̈𝐱𝑖=1,2𝑓𝑖𝐧0𝑖+𝜆𝑖𝐛0𝑖+𝜂𝑖𝐞0𝑖+𝑀𝑔𝐞𝑦𝐻̇=0,(21)𝝎+𝝎×𝐻𝝎𝑖=1,2𝛾0𝑖×𝑓𝑖𝐧0𝑖+𝜆𝑖𝐛0𝑖+𝜂𝑖𝐞0𝑖𝐺=0,(22)𝑖𝑞𝑖̈𝑞𝑖+12̇𝐺𝑖𝑞𝑖+𝑆𝑖(𝑞,̇𝑞)̇𝑞𝑖+𝜕𝑃𝑖𝑞𝑖𝜕𝑞𝑖𝐽T𝑖𝑞𝑖𝑓𝑖𝐧𝑖𝜆𝑖𝐛𝑖+𝜂𝑖𝐞𝑖𝑊T𝑖𝛾𝑖×𝑓𝑖𝐧𝑖𝜆𝑖𝐛𝑖+𝜂𝑖𝐞𝑖=𝑢𝑖,𝑖=1,2,(23) where 𝐞𝑦=(0,1,0)T, and the meaning of 𝑊𝑖 will be explained later. It should be noted that the sum of inner products of ̇𝐱 and (21), 𝝎 and (22), and ̇𝑞𝑖 and (23) for 𝑖=1,2 yields the energy relation 𝑖=1,2̇𝑞T𝑖𝑢𝑖=dd𝑡𝐾+𝑀𝑔𝑦+𝑖=1,2𝑃𝑖𝑞𝑖.(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,𝜅(25)𝑛0+𝜅𝑛1d𝑠0d𝑡=𝝎T0𝐞0+𝝎T1𝐞1,(26) as shown in detail in (𝐴1) of Appendix A, where 𝜅𝑛0 denotes the normal curvature of the object surface at the contact point 𝑃1, and 𝜅𝑛1 does that of the fingerend surface 𝑆1 at 𝑃1. Both the normal curvatures 𝜅𝑛0 and 𝜅𝑛1 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 ̇𝐛0=̇𝐛1 imply 𝝎T0𝐧0+𝝎T1𝐧1+𝜅𝑒0+𝜅𝑒1d𝑠0d𝑡=0,(27) where 𝜅𝑒0 denotes the geodesic curvature of the object surface at 𝑃1, and 𝜅𝑒1 has a similar meaning. This equation does not determine each 𝜅𝑒0 or 𝜅𝑒1 individually. Therefore, let us try to differentiate 𝐞0 and 𝐞1 in 𝑡. 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 𝝎T0𝐞0𝜅𝑒0𝝎T1𝐞1𝜅𝑒1d𝑠0=̇𝝎d𝑡T1𝐛1̇𝝎T0𝐛0+𝜅𝑛0𝝎T0𝐧0+𝜅𝑛1𝝎T1𝐧1,(28) as shown in (𝐴4) of Appendix A. Then, this equation together with (27) leads to 𝝎11T0𝐞0𝝎T1𝐞1𝜅𝑒0𝜅𝑒1d𝑠0=𝝎d𝑡T0𝐧0+𝝎T1𝐧1𝜉1,(29) where 𝜉1𝜅𝑛0𝝎T0𝐧0+𝜅𝑛1𝝎T1𝐧1d𝑠0+̇𝝎d𝑡T1𝐛1̇𝝎T0𝐛0.(30) Thus, it is possible to determine each geodesic curvature individually by inverting the coefficient 2×2 matrix of (29) in the following way: 𝜅𝑒0d𝑠0=d𝑡𝝎T1𝐞1𝝎T0𝐧0+𝝎T1𝐧1+𝜉1𝝎T1𝐞1+𝝎T0𝐞0,𝜅(31)𝑒1d𝑠1=d𝑡𝝎T0𝐞0𝝎T0𝐧0+𝝎T1𝐧1𝜉1𝝎T1𝐞1+𝝎T0𝐞0.(32)

Finally, it is possible to see that the moving frames denoted by Ψ0 and Ψ1 should satisfy the Frenet-Serre equations dΨd𝑡0=Ψ00𝜅𝑛00𝜅𝑛00𝜅𝑒00𝜅𝑒00d𝑠0,dd𝑡Ψd𝑡1=Ψ10𝜅𝑛10𝜅𝑛10𝜅𝑒10𝜅𝑒10d𝑠0.d𝑡(33)

4. Sufficient Conditions for Updating the Moving Frame Coordinates

In the Frenet-Serre equation of (31), the coefficient 𝜅𝑛0 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 𝑂𝑚-𝑋𝑌𝑍, and the other normal curvature 𝜅𝑛1 in (32) is also determined similarly. The geodesic curvatures 𝜅𝑒0 and 𝜅𝑒1 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 𝜅𝑒0 and 𝜅𝑒1 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 𝑡=0, then for any 𝑡>0, it follows that dd𝑡𝐛1=dd𝑡𝐛0,dd𝑡𝐞1d=d𝑡𝐞0,𝐛1=𝐛0,𝐞1=𝐞0.(34) To prove this, first note again that the postulates (a1) and (a2) imply 𝐧0=𝐧1,dd𝑡𝐧0d=d𝑡𝐧1.(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 𝑡>0 though both 𝐛1 and 𝐛0 are lying on the same tangent plane. In other words, suppose that 𝐛T1𝐛0=cos𝜃 and 𝜃(𝑡) stands for 𝜃(𝑡,𝑠1(𝑡),𝑠0(𝑡)), then, as shown in Appendix B, we have ̇𝜃=𝝎T1𝐧1+𝝎T0𝐧0+𝜅𝑒0+𝜅𝑒1d𝑠0d𝑡.(36) Further, as discussed in (𝐵3) of Appendix B, geodesic curvatures 𝜅𝑒0 and 𝜅𝑒1 should be defined as 𝜅𝑒0=𝜕𝜃𝜕𝑠0,𝜅𝑒1=𝜕𝜃𝜕𝑠1.(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 “twisting” between the axis of normal 𝐧1 and that of 𝐧0. Nevertheless, it is further important to note that if 𝜅𝑒0 and 𝜅𝑒1 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 ̇𝜃=0. Therefore, if at the initial time 𝐛1(0)=𝐛0(0), then the setting of (31) and (32) for geodesic curvatures 𝜅𝑒1 and 𝜅𝑒0 leads to 𝐛1(𝑡)=𝐛0(𝑡) for 𝑡>0 as far as 𝝎T1𝐞1+𝝎T0𝐞00. Since 𝐧1=𝐧0 as far as the contact is maintained, the equality 𝐛1(𝑡)=𝐛0(𝑡) implies 𝐞1(𝑡)=𝐞0(𝑡) for 𝑡0, then, from (𝐴1) of Appendix A, (25) and (26) follow. At the same time, from (26) and (27), it follows that d𝐛1/d𝑡=d𝐛0/d𝑡 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 “0𝑖” 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 Π0 and Π𝑖, update equations of length parameters, and Frenet-Serre equations for updating moving frame coordinates at contact points 𝑀̈𝐱𝑖Π0Ψ0𝑖𝝀𝑖𝑀𝑔𝐞𝑦=0,(𝐸𝑥)𝐻̇𝝎+𝝎×𝐻𝝎𝑖𝛾0𝑖×Ψ0𝑖𝝀𝑖=0,(𝐸𝜔)𝐺𝑖̈𝑞𝑖+12̇𝐺𝑖+𝑆𝑖̇𝑞𝑖+𝜕𝑃𝑖𝜕𝑞𝑖+𝐽T𝑖Π𝑖+𝑊T𝑖𝛾𝑖×Ψ𝑖𝝀𝑖=𝑢𝑖,(𝐸𝑖)dΠd𝑡0=Π0Ω0,dΠd𝑡𝑖=Π𝑖Ω𝑖,(𝐸𝑟)d𝑠d𝑡𝑖=𝝎T𝐞0𝑖+𝝎T𝑖𝐞𝑖𝜅𝑛0𝑖+𝜅𝑛𝑖,(𝐸𝑠)dΨd𝑡0𝑖=Ψ0𝑖𝐾0𝑖d𝑠𝑖,dd𝑡Ψd𝑡𝑖=Ψ𝑖𝐾𝑖d𝑠𝑖,d𝑡(𝐸𝑓𝑠) where 𝐾0𝑖=0𝜅𝑛0𝑖0𝜅𝑛0𝑖0𝜅𝑒0𝑖0𝜅𝑒0𝑖0,𝐾𝑖=0𝜅𝑛𝑖0𝜅𝑛𝑖0𝜅𝑒𝑖0𝜅𝑒𝑖0,Ψ0𝑖=𝐧0𝑖,𝐛0𝑖,𝐞0𝑖,Ψ𝑖=𝐧𝑖,𝐛𝑖,𝐞𝑖,𝝀𝑖=𝑓𝑖,𝜆𝑖,𝜂𝑖T,𝐽𝑖𝑞𝑖=𝜕𝐫𝑖𝑞𝑖𝜕𝑞T𝑖,(38) and 𝛾0𝑖 denotes the contact point 𝑃𝑖 on the object surface expressed by the object local coordinates 𝑂𝑚-𝑋𝑌𝑍, and 𝛾𝑖 does that of 𝑃𝑖 on the fingerend surface of finger 𝑖 expressed by the fingerend local coordinates 𝑂𝑖-𝑋𝑖𝑌𝑖𝑍𝑖. In 𝐸𝑖, 𝑊𝑖 is an 𝑚×3 matrix depending on 𝑞𝑖, where 𝑚 denotes the degrees of freedom. In the case of a pair of robot fingers depicted in Figure 1, it is obvious to see that 𝑊1=000011,𝑊2=1000sin𝑞21sin𝑞210cos𝑞21cos𝑞21.(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 𝜅𝑒0𝑖 and 𝜅𝑒𝑖 are determined in the same manner as shown in (31) and (32). Further, computation of ̇𝝎𝑖 and ̇𝝎 appearing in (31) and (32) through 𝜉𝑖 defined by (30) can be executed simultaneously via numerical integration of 𝐸𝜔 and 𝐸𝑖. In practice, it is possible to compute ̇𝝎𝑖 by ̇𝝎=𝐻1𝝎×𝐻𝝎+𝑖𝛾0𝑖×Ψ0𝑖𝝀𝑖.(40) Analogously, it is possible to compute ̇𝝎𝑖 since 𝝎𝑖 must be expressed by a function form of 𝑉(𝑞𝑖)̇𝑞𝑖, and ̈𝑞𝑖 can be calculated by multiplying (𝐸𝑖) by 𝐺𝑖1(𝑞𝑖) 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=𝐧0 and (d/d𝑡)𝐧1=(d/d𝑡)𝐧0, dd𝑡𝐧0=̇Π0𝐧0+Π0̇𝐧0=Π0Ω0𝐧0+Π0𝜕𝐧0𝜕𝑠0d𝑠0d𝑡=Π0𝝎0×𝐛0×𝐞0𝜅𝑛0Π0𝐛0d𝑠0=𝝎d𝑡T0𝐞0𝜅𝑛0d𝑠0d𝑡𝐛0𝝎T0𝐛0𝐞0,dd𝑡𝐧1=𝝎T1𝐞1𝜅𝑛1d𝑠1d𝑡𝐛1𝝎T1𝐛1𝐞1.(A.1) If 𝐛0=𝐛1 and 𝐞0=𝐞1, then it follows that 𝝎T0𝐛0=𝝎T1𝐛1,()𝜅𝑛0+𝜅𝑛1d𝑠0d𝑡=𝝎T0𝐞0+𝝎T1𝐞1.()(𝐴2)Similarly, it follows that dd𝑡𝐛0=̇Π0𝐛0+Π0̇𝐛0=Π0𝝎T0𝐧0𝐞0𝝎T0𝐞0𝐧0+Π0𝜅𝑛0𝐧0+𝜅𝑒0𝐞0d𝑠0=𝝎d𝑡T0𝐧0+𝜅𝑒0d𝑠0d𝑡𝐞0𝝎T0𝐞0+𝜅𝑛0d𝑠0d𝑡𝐧0,dd𝑡𝐛1=𝝎T1𝐧1+𝜅𝑒1d𝑠1d𝑡𝐞1𝝎T1𝐞1+𝜅𝑛1d𝑠1d𝑡𝐧1.(A.2) These two equations imply 𝐸𝑖 and 𝝎T0𝐧0+𝝎T1𝐧1+𝜅𝑒0+𝜅𝑒1d𝑠0d𝑡=0() if 𝐛0=𝐛1 and (d/d𝑡)𝐛0=(d/d𝑡)𝐛1.(𝐴3)Similarly, it follows that dd𝑡𝐞0=𝝎T0𝐛0𝐧0𝝎T0𝐧0+𝜅𝑒0d𝑠0d𝑡𝐛0,dd𝑡𝐞1=𝝎T1𝐛1𝐧1𝝎T0𝐧1+𝜅𝑒1d𝑠1d𝑡𝐛1.(A.3) These two equations imply 𝐸𝜔 and 𝐸𝑠 if 𝐞0=𝐞1, 𝐛0=𝐛1, and (d/d𝑡)𝐞0=(d/d𝑡)𝐞1.(𝐴4)Time rate of 𝐸𝜔 reduces to ̇𝝎T0𝐛0+𝝎T0̇𝐛0=̇𝝎T1𝐛1+𝝎T1̇𝐛1.(A.4) Since ̇𝐛𝑖=(𝜅𝑛𝑖𝐧𝑖+𝜅𝑒𝑖𝐞𝑖)(d𝑠0/d𝑡) for 𝑖=0,1, the above equality reduces to 𝝎T0𝐞0𝜅𝑒0𝝎T1𝐞1𝜅𝑒1d𝑠0=d𝑡𝜅𝑛0𝝎T0𝐧0+𝜅𝑛1𝝎T1𝐧1d𝑠0+̇𝝎d𝑡T1𝐛1̇𝝎T0𝐛0𝜉1̇𝝎1,̇𝝎0.(A.5)

B. Preliminary Remarks on Geodesic Curvature

𝐛0𝑖=Π0𝐛0𝑖(tangentvectorsonobjectsurface𝑖),𝐧0𝑖=Π0𝐧0𝑖(normalvectorsonobjectsurface𝑖),𝐛𝑖=Π𝑖𝐛𝑖(tangentvectorsonngerendsurface𝑖),𝐧𝑖=Π𝑖𝐧𝑖(normalvectorsonngerendsurface𝑖),(B.1) both Ω𝑖 and Ω0 are skew symmetric.(𝐵1)Derivation of 𝜕𝜃/𝜕𝑡 where 𝜃(𝑡,𝑠𝑖(𝑡),𝑠𝑖0(𝑡)).

If 𝐛T1𝐛0=cos𝜃, then 𝐛T1ΠT1Π0𝐞0𝜋=cos2+𝜃=sin𝜃,𝜕𝜃𝜃𝜕𝑡cos𝜃=𝐛𝜕𝑡T1ΠT1Π0𝐞0=𝐛T1̇ΠT1Π0𝐞0+𝐛T1ΠT1̇Π0𝐞0=𝐛T1ΩT1ΠT1Π0𝐞0+𝐛T1ΠT1Π0Ω0𝐞0=Ω1𝐞1×𝐧1TΠT1Π0𝐞0+𝐛T1ΠT1Π0Ω0𝐧0×𝐛0=𝝎T1𝐧1𝐞1𝝎T1𝐞1𝐧1TΠT1Π0𝐞0+𝐛T1ΠT1Π0𝝎T0𝐛0𝐧0𝝎T0𝐧0𝐛0=𝝎T1𝐧1𝐞T1ΠT1Π0𝐞0𝐛T1Π1Π0𝐛0𝝎T0𝐧0𝝎=T1𝐧1+𝝎T0𝐧0cos𝜃,𝜕𝜃𝜕𝑡=𝝎T1𝐧1+𝝎T0𝐧0.(B.2)(𝐵2)With derivation of ̇𝜃, 𝐛T1ΠT1Π0𝐞0̇𝝎=sin𝜃,𝜃cos𝜃=T1𝐧1+𝝎T0𝐧0cos𝜃+𝐛T1ΠT1Π0𝜅𝐞0𝐛0d𝑠0+𝜅d𝑡𝑛1𝐧1+𝜅𝑒1𝐞1TΠT1Π0𝐞0d𝑠1𝝎d𝑡=T1𝐧1+𝝎T0𝐧0cos𝜃𝜅𝑒0cos𝜃d𝑠0d𝑡𝜅𝑒1cos𝜃d𝑠1,̇d𝑡𝜃=𝝎T1𝐧1+𝝎T0𝐧0+𝜅𝑒0+𝜅𝑒1d𝑠1.d𝑡(B.3)(𝐵3)With derivation of 𝜅𝑒0 and 𝜅𝑒1, 𝐛T1ΠT1Π0𝐞0𝜕=sin𝜃,𝜕𝑠0𝐛T1ΠT1Π0𝐞0=𝜕𝜃𝜕𝑠0𝐛cos𝜃,T1ΠT1Π0𝜅𝑒0𝐛0=𝜕𝜃𝜕𝑠0cos𝜃,𝜅𝑒0cos𝜃=𝜕𝜃𝜕𝑠0𝜅cos𝜃,𝑒0=𝜕𝜃𝜕𝑠0,𝜅𝑒1=𝜕𝜃𝜕𝑠1.(B.4)

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