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.

1. Introduction

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. So far the kinematics and geometry of contact between rigid bodies were solved by Montanna [1], and a set of velocity relations is given in detail [1]. However, there is a dearth of papers except the papers mentioned in [24] that attempt to model dynamics of physical interactions between the fingerends and an object under the existence of rollings. However, the papers [24] did not yet gain physical insights into the constraint forces arising from rolling contacts and show any explicit forms of them in the object wrench space. 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]. However, all discussions in [5, 6] have been restricted to the case of ball-plate rollings. Very recently, a complete model of 2-dimensional grasping of a rigid object with arbitrary shape is given as a set of Lagrange's equations of motion of the overall fingers/object system together with a pair of first-order differential equations that update arclength parameters [8]. It should be noted that modeling of the system assumes the full knowledge of geometry of a given object but design of control signals neither needs the information of object geometry nor uses any external sensing of contact points.

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. In this paper, rolling contact is interpreted as the following conditions.

(1)Two contact points on the contour curves must coincide at a single point.(2)The two contours must have the same tangent at the common contact point.

These two conditions as a whole are equivalent to Nomizu's relation [9] concerning tangent vectors at the contact point and normals to the common tangent. 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 then 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. It is further shown that the Euler-Lagrange equation should be accompanied with a pair of first-order differential equations that govern evolutions of arclength parameters, where quantities of the second fundamental form are involved. Furthermore, we show that rolling contact constraints expressed in terms of velocity relations are integrable in the sense of Frobenius. In other words, the rolling contact constraints can be regarded as a holonomic constraint. Based upon these holonomic constraint forms, we reformulate the Euler-Lagrange equation of motion of the system.

In the final two sections, we will discuss an extension of Dirichlet-Lagrange stability theorem from the standpoint of Riemannian Geometry by introducing a control signal called “blind grasping." The signal can be constructed real-time based upon the data of fingers' kinematics without referring to object kinematics, regarding the pair of robot fingers as the intrinsic world but objects to be grasped existing in the external world. We show that the blind control signal designed on the basis of the fingers-thumb opposability can be interpreted as a Morse-Bott function in the case that the object is a parallelepiped, since its Hessina in this case becomes positive semidefinite on the base manifold. Notwithstanding the redundancy in system's DOF, we prove the asymptotic convergence of a solution trajectory to the closed-loop dynamics to an equilibrium manifold that attains minimization of the Morse-Bott function. We finally point out a few of important and interesting problems that remain unsolved or not yet tackled.

2. Definitions of Tangent Vectors in Terms of Arclength Parameters

First, denote the contour curve of the left hand fingertip by 𝛾1(𝑠1) in terms of arclength parameter 𝑠1 as shown in Figure 2 and the object contour by 𝛾01(𝑠0) in terms of another length parameter 𝑠0. Suppose that the fingertip contour contacts with the object at 𝛾1(𝑠1)=𝑃1𝐴 and 𝛾01(𝑠0)=𝑃1𝐵 so that 𝑃1𝐴=𝑃1𝐵=𝑃1 in the frame coordinate space. In general, we assume implicitly that by rolling motion the contact point 𝑃1 moves on each contour with the same run-length without splipping. Therefore, we characterize the contour curves 𝛾1(𝑠) and 𝛾01(𝑠) by using the same length parameter 𝑠, at which 𝑃1𝐴(𝑠) and 𝑃1𝐵(𝑠) expressed in the frame space 𝑂-𝑥𝑦 coincide and share the same tangent vector, as shown in Figures 2 and 3.

In order to gain a physical insight into this rolling constraint, we need to denote firstly the fingertip contour curve 𝛾1(𝑠) in the local coordinates 𝑂1-𝑋1𝑌1 as shown in Figure 3 and the object contour by 𝛾01(𝑠) in the object coordinates 𝑂𝑚-𝑋𝑌 fixed at the object. If we denote 𝛾1(𝑠) and 𝛾01(𝑠) by

𝛾1𝑋(𝑠)=1𝑌(𝑠)1(𝑠),𝛾01𝑋(𝑠)=01𝑌(𝑠)01(𝑠)(1) and define 𝐛1(𝑠)=𝛾1d(𝑠)=𝑋d𝑠1𝑌(𝑠)1𝐛(𝑠),(2)01(𝑠)=𝛾01d(𝑠)=𝑋d𝑠01(𝑌𝑠)01(𝑠),(3) then 𝐛1(𝑠) must be the unit tangent vector at the point 𝑃1𝐴(𝑠) and 𝐛01(𝑠) must be also the unit tangent vector at 𝑃1𝐵(𝑠) (see Figure 3). It is well known in the mathematical theory of curves and surfaces that the further derivatives of 𝐛1 and 𝐛01 in 𝑠 can be expressed as

𝐛1(𝑠)=𝜅1(𝑠)𝐧1(𝑠),𝐛01(𝑠)=𝜅01(𝑠)𝐧01(𝑠),(4) where 𝐧1(𝑠) and 𝐧01(𝑠) are orthogonal to 𝐛1(𝑠) and 𝐛01(𝑠), respectively, and 𝜅1(𝑠) is the curvature of the fingertip contour curve and 𝜅01(𝑠) is that of the object contour in its left-hand side. It is also well known that

𝐧1(𝑠)=𝜅1(𝑠)𝐛1(𝑠),𝐧01(𝑠)=𝜅01(𝑠)𝐛01(𝑠),(5) and this equation together with (4) constitutes the Frenet-Serret formula (see [10]).

Next consider the rotation matrix of the object around its mass center 𝑂𝑚 expressed in the frame coordinates as follows:

Π0=𝐫𝑋,𝐫𝑌,(6) where 𝐫𝑋 denotes the unit vector of 𝑋-axis and 𝐫𝑌 that of 𝑌-axis expressed in the inertial frame coordinates 𝑂-𝑥𝑦 as shown in Figure 2. If the object rotates by angle 𝜃, that is, 𝑂𝑚-𝑋𝑌 rotates by 𝜃 from the frame coordinates 𝑂-𝑥𝑦, then

Π0=𝐫𝑋,𝐫𝑌=cos𝜃sin𝜃sin𝜃cos𝜃.(7) Similarly, define

Π1=𝐫𝑋1,𝐫𝑌1,(8) where 𝐫𝑋1 denotes the unit vector of 𝑋1-axis of the left-hand fingertip expressed in the frame coordinates 𝑂-𝑥𝑦 and 𝐫𝑌1 that of 𝑌1-axis. Both the matrices Π0 and Π1 are an orthogonal matrix regarded as an element of SO(2). The derivatives of Π0 and Π1 in 𝑡 denoted by ̇Π0=(dΠ0/d𝑡) and ̇Π1 are obtained as follows:

̇Π0=Π0Ω0,Ω0=̇𝜃,̇Π01101=Π1Ω1,Ω1=̇𝑝1,0110(9) where 𝑝1=𝑞11+𝑞12+𝑞13=𝑞T1𝐞1, 𝑞1=(𝑞11,𝑞12,𝑞13)T, and 𝐞=(1,1,1)T in the case of the finger mechanism (a planar robot with three joints) depicted in Figure 1. Throughout the paper, we take the positive sign of angles 𝜃, 𝜃𝑖, 𝑝𝑖, and 𝑞𝑖𝑗 in the counterclockwise direction. Hence, Ω0 in (9) expresses the rotation of positive angle 𝜋/2 with associating the angular velocity ̇𝜃. In other words, Ω0𝐛01=̇𝜃𝐧01 or 𝐛T01Ω0̇=𝜃𝐧T01 and Ω1𝐛1=̇𝑝1𝐧1 or 𝐛T1Ω1=̇𝑝1𝐧T1.

Next, denote the position vector of point 𝑂1 as a center of the left-hand fingertip by 𝐫1(=𝑂𝑂1) and that of the object mass center by 𝐫𝑚(=𝑂𝑂𝑚). Then, the first assumption that the two contact points 𝑃1𝐴 and 𝑃1𝐵 coincide at the common contact point 𝑃1 implies the equality

𝐫1𝐴=𝐫1+Π1𝛾1=𝐫𝑚+Π0𝛾01=𝐫1𝐵.(10) For the sake of convenience, we denote the unit tangent vector at 𝑃1 expressed in the frame space 𝑂-𝑥𝑦 by

𝐛1=Π1𝐛1=Π1𝜕𝛾1,𝜕𝑠𝐛01=Π0𝐛01=Π0𝜕𝛾01.𝜕𝑠(11) Then, the second assumption that at the contact point the two curves share the same tangent implies

𝐛1=Π1𝐛1=Π0𝐛01=𝐛01.(12) Throughout the paper we use symbol (̇) for expressing the derivative in 𝑡 in such a way that ̇𝑟1=d𝐫1/d𝑡 and ̇Π1=dΠ1/d𝑡. Then, differentiation of (10) in 𝑡 yields

̇𝐫1+̇Π1𝛾1+Π1𝛾1d𝑠=̇𝐫d𝑡𝑚+̇Π0𝛾01+Π0𝛾01d𝑠.d𝑡(13) Since Π1𝛾1=𝐛1 and Π0𝛾01=𝐛01, (13) is reduced to, according to (12),

̇𝐫1̇𝐫𝑚+̇Π1𝛾1=̇Π0𝛾01,(14) which from (9) can be written in the form

̇𝐫1̇𝐫𝑚+Π1Ω1𝛾1Π0Ω0𝛾01=0.(15) It should be remarked at this stage that the position constraint expressed by (10) is holonomic with two degrees-of-freedom and the velocity constraint of (15) is Pfaffian with two DOFs, too. However, physical meanings of these two constraints are not directly connected to the constraint conditions based upon the assumptions of (1) and (2) mentioned in the previous section. In order to gain a physical insight into the rolling contact conditions, let us take the inner product of (15) and 𝐛01 (or equivalently 𝐛1), which results in

𝑅𝑏1=̇𝐫1̇𝐫𝑚T𝐛01+𝐛T1Ω1𝛾1𝐛T01Ω0𝛾01=0.(16) On account of the skew symmetry of Ω1 and Ω0 as shown in (9) and discussed below (9), (16) is reduced to

𝑅𝑏1=̇𝐫1̇𝐫𝑚T𝐛01+̇𝑝1𝐧T1𝛾1+̇𝜃𝐧T01𝛾01=0.(17) This shows the rolling contact constraint of a Pfaffian form as a condition of zero relative velocity of the contact point running on the fingertip relative to that of it running on the object. Another rolling contact constraint can be obtained by taking the inner product of (15) and 𝐧01 in the following way:

𝑅𝑛1=̇𝐫1̇𝐫𝑚T𝐧01𝐧T1Ω1𝛾1𝐧T01Ω0𝛾01=0,(18) where Π1𝐧1=Π0𝐧01 is taken into account. This is reduced to

𝑅𝑛1=̇𝐫1̇𝐫𝑚T𝐧01+̇𝑝1𝐛T1𝛾1̇𝜃𝐛T01𝛾01=0.(19) Hence, if we define the column vectors

𝑏1=𝐛01𝐧T01𝛾01𝐽T1𝑞1𝐛01+𝐧T1𝛾1𝐞1,𝑛1=𝐧01𝐛T01𝛾01𝐽T1𝑞1𝐧01+𝐛T1𝛾1𝐞1,(20) and 𝑋1=(𝐱T,𝜃,𝑞T1)T, 𝐱=(𝑥,𝑦)T, and 𝐽1(𝑞1)=𝜕𝐫1/𝜕𝑞T1, then (17) and (19) can be rewritten into the forms


A similar argument developed above can be applied to the characterization of the rolling contact constraint at the right-hand contact point 𝑃2, at which the right-hand object contour and fingertip contour share the same tangent (see Figures 1 and 4). Therefore, it is possible to define smooth curves 𝛾2(𝑠2) and 𝛾02(𝑠2) expressed in their local coordinates 𝑂2-𝑋2𝑌2 and 𝑂𝑚-𝑋𝑌, respectively, with respect to the same arclength parameter 𝑠2 as indicated in Figure 4. Note that in this paper the direction of increase of 𝑠2 is taken to be counter to that of 𝑠1. Hence, as shown in Figure 2 we see evidently that 𝐛2=d𝛾2/d𝑠2=𝛾2 and 𝐛02=d𝛾02/d𝑠2=𝛾02, and similarly to (10) and (12) that

𝐫2+Π2𝛾2=𝐫𝑚+Π0𝛾02,(22)𝐛2=Π2𝐛2=Π0𝐛02=𝐛02,(23) where Π2=(𝐫𝑋2,𝐫𝑌2), ̇Π2=Π2Ω2, and

Ω2=̇𝑝20110,(24) where 𝑝2=𝑞21+𝑞22 according to the planar mechanism of the right-hand finger shown in Figure 1. Apparently from Figure 4 it follows that 𝐛T2Ω2=̇𝑝2𝐧2 and 𝐛T02Ω0=̇𝜃𝐧02. Thus, similarly to the derivation of (16) and (17), we have from differentiation of (22) in 𝑡 the following:

𝑅𝑏2=̇𝐫2̇𝐫𝑚T𝐛02+𝐛T2Ω2𝛾2𝐛T02Ω0𝛾02𝑅=0,𝑏2=̇𝐫2̇𝐫𝑚T𝐛02̇𝑝2𝐧T2𝛾2̇𝜃𝐧T02𝛾02=0.(25) On account of the equality Π2𝐧2=Π0𝐧02 as shown in Figures 1 and 4, we have, similarly to (18) and (19), the following:

𝑅𝑛2=̇𝐫2̇𝐫𝑚T𝐧02𝐧T2Ω2𝛾2𝐧T02Ω0𝛾02𝑅=0,(26)𝑛2=̇𝐫2̇𝐫𝑚T𝐧02̇𝑝2𝐛T2𝛾2+̇𝜃𝐛T02𝛾02=0.(27) Hence, if we denote 𝑋=(𝐱T,𝜃,𝑞T1,𝑞T2)T, 𝐽2(𝑞2)=𝜕𝐫2/𝜕𝑞2, 𝐞2=(1,1)T, and

𝑏2=𝐛02𝐧T02𝛾0203𝐽T2𝑞2𝐛02𝐧T2𝛾2𝐞2,𝑛2=𝐧02𝐛T02𝛾0203𝐽T2𝑞2𝐧02𝐛T2𝛾2𝐞2,(28) then, similarly to (21), (25) and (27) are expressed as follows:

𝑅𝑏2=T𝑏2d𝑋d𝑡=0,𝑅𝑛2=T𝑛2d𝑋d𝑡=0,(29) where 03=(0,0,0)T. For convenience, we append 𝑏1 and 𝑛1 with the two-dimensional zero vector in the following way:

𝑏1=𝑏102,𝑛1=𝑛102(30) and denote 𝑏1 and 𝑛1 by 𝑏1 and 𝑛1 renewedly for the sake of convenience. Then, (21) can be written in the form


3. Derivation of Euler-Lagrange Equation of Motion and Update Law of Length Parameters

As discussed in the previous papers [1115], a set of all possible postures of the fingers/object system depicted in Figure 1 can be regarded as a Riemannian manifold 𝑀8 with eight degrees of freedom that can be represented by 𝑀8=(𝐄2×𝑆1)×𝑇3×𝑇2, where (𝐱,𝜃)𝐄2×𝑆1, 𝑞1𝑇3, 𝑞2=(𝑞21,𝑞22)T𝑇2, 𝑝2=𝑞T2𝐞2, 𝐞2=(1,1)T, and the Riemannian metric 𝐺=(𝑔𝑖𝑗) is defined as

𝐾̇𝑋=1𝑋,2̇𝑋Ṫ𝑋=𝑀𝐺(𝑋)2̇𝑥2+̇𝑦2+𝐼2̇𝜃2+𝑖=1,2̇𝑞T𝑖𝐺𝑖𝑞𝑖̇𝑞𝑖,(32)𝐺=diag𝑀,𝑀,𝐼,𝐺1𝑞1,𝐺2𝑞2,(33) where 𝑀 and 𝐼 denote the mass of the object and its inertia moment around 𝑂𝑚, and 𝐺𝑖(𝑞𝑖) the inertia matrix of finger 𝑖 for 𝑖=1,2. Particularly, a set of all postures during rolling motions of the system keeping both rolling contacts can be regarded as a subset of 𝑀8 that is subject to the constraints of (10) and (22). From the Riemannian geometric point of view, at any posture 𝑋 of the system with rolling contacts specified by some arclength parameters 𝑠1 and 𝑠2, the four equalities of (21) and (31) give rise to an assignment of the tangent vector ̇𝑋𝑇(𝑋;𝑠1,𝑠2)𝑀8, where we denote by 𝑇𝑋;𝑠1,𝑠2𝑀8 the tangent space of 𝑀8 at the given posture (𝑋;𝑠1,𝑠2) having rolling contacts at contact points 𝑃1(𝑠1) and 𝑃2(𝑠2). That is, the tangent vector should be orthogonal to all four vectors 𝑏𝑖 and 𝑛𝑖 for 𝑖=1,2. Hence, by introducing four Lagrange's multipliers 𝜆𝑖 and 𝑓𝑖 corresponding to 𝑏𝑖 and 𝑛𝑖 for 𝑖=1,2, it is possible to derive the Euler-Lagrange equation of motion in the following way:

̈1𝐺(𝑋)𝑋+2̇̇𝐺+𝑆𝑋+𝑖=1,2𝑓𝑖𝑛𝑖+𝜆𝑖𝑏𝑖=𝑢.(34) In detail, (34) can be written as follows:

𝑀̈𝐱𝑓1𝐧01𝑓2𝐧02𝜆1𝐛01𝜆2𝐛02𝐼̈=0,(35)𝜃𝑓1𝐛T01𝛾01+𝑓2𝐛T02𝛾02+𝜆1𝐧T01𝛾01𝜆2𝐧T02𝛾02𝐺=0,(36)𝑖𝑞𝑖̈𝑞𝑖+12̇𝐺𝑖𝑞𝑖+𝑆𝑖𝑞𝑖,̇𝑞𝑖̇𝑞𝑖+𝑓𝑖𝐽T𝑖𝑞𝑖𝐧0𝑖(1)𝑖𝐛T𝑖𝛾𝑖𝐞𝑖+𝜆𝑖𝐽T𝑖𝑞𝑖𝐛0𝑖(1)𝑖𝐧T𝑖𝛾𝑖𝐞𝑖=𝑢𝑖,𝑖=1,2,(37) in which symbol 𝑢𝑖 in (37) denotes a control signal treated as an external torque that can be exerted through finger joints of finger 𝑖. It should be remarked that all of equations (35) to (37) are characterized by length parameters 𝑠1 and 𝑠2 but only quantities of the first fundamental form of contour curves are involved in (35) to (37). It should be also noted that the posture of the object is governed by the first-order differential equation of (9). At the same time, it is important to find the update law of length parameters 𝑠1 and 𝑠2 through rolling contact motion of the system. In fact, note that, from the condition (1) of rolling contact mentioned in Section 1, 𝐛0𝑖=𝐛𝑖 at contact point 𝑃𝑖 for 𝑖=1,2. This also implies

dd𝑡𝐛0𝑖=dd𝑡𝐛𝑖,𝑖=1,2,(38) which reduces to

̇Π0𝐛0𝑖+Π0𝜅0𝑖𝑠𝑖𝐧0𝑖d𝑠𝑖=̇Πd𝑡𝑖𝐛𝑖+Π𝑖𝜅𝑖𝑠𝑖𝐧𝑖d𝑠𝑖d𝑡,𝑖=1,2,(39) where 𝜅𝑖(𝑠𝑖) denotes the curvature of the fingertip contour curve for finger 𝑖 for 𝑖=1,2, 𝜅01(𝑠1) the curvature of the left-hand object contour, and 𝜅02(𝑠2) that of the right-hand object contour. Since 𝐧0𝑖=𝐧𝑖 for both 𝑖=1,2, (39) is again reduced to

𝜅0𝑖+𝜅𝑖𝐧0𝑖d𝑠𝑖d𝑡=Π𝑖Ω𝑖𝐛𝑖Π0Ω0𝐛0𝑖,𝑖=1,2,(40) Thus, the inner product of 𝐧0𝑖 and (40) yields

𝜅0𝑖+𝜅𝑖d𝑠𝑖=d𝑡(1)𝑖̇𝜃̇𝑝𝑖,𝑖=1,2,(41) where we referred to the relation (as discussed below (9)):

Ω𝑖𝐛𝑖=(1)𝑖̇𝑝𝑖𝐧𝑖,Ω0𝐛0𝑖=(1)𝑖̇𝜃𝐧0𝑖,𝑖=1,2.(42) The set of Euler-Lagrange equations (35)–(37) of motion of the system should be integrated simultaneously, accompanied with integration of the first order nonlinear differential equation of (41).

4. Integrability of Pfaffian Forms of Rolling Contact Constraints

A rolling contact constraint between two rigid bodies is expressed traditionally by an equality of two contact-point velocities at the common contact point running on the one rigid body and on another body. In the case of rolling contact motion as shown in Figure 1, such zero relative-velocity relation is given in the Pfaffian forms of (29) and (31). We are now in a position to show that all these four Pfaffian forms are integrable in the sense of Frobenius [16, 17]. In fact, it is possible to prove the following.

Proposition 4.1. (1) Under the contact conditions expressed by (10) and (22), the following four geometrical relations are valid (see Figure 4): 𝑄𝑏𝑖=𝐫𝑖𝐫𝑚T𝐛0𝑖+𝐛T𝑖𝛾𝑖𝐛T0𝑖𝛾0𝑖𝑄=0,𝑖=1,2,(43)𝑛𝑖=𝐫𝑖𝐫𝑚T𝐧0𝑖𝐧T𝑖𝛾𝑖𝐧T0𝑖𝛾0𝑖=0,𝑖=1,2.(44)
(2) Each of the Pfaffian forms 𝑅𝑏𝑖 and 𝑅𝑛𝑖 for 𝑖=1,2 defined by (17), (19), (25), and (27) is integrable in 𝑡 in the following forms: d𝑄d𝑡𝑏𝑖=𝑅𝑏𝑖,d𝑄d𝑡𝑛𝑖=𝑅𝑛𝑖,𝑖=1,2.(45)

In fact, the inner product of (10) and 𝐛01(=𝐛1) leads to (43) for 𝑖=1 and that of (22) and 𝐛02(=𝐛2) does to (43) for 𝑖=2. The inner product of (10) and 𝐧01(=𝐧1) leads to (44) for 𝑖=1 and that of (22) and 𝐧02 does to (44) for 𝑖=2. The first equation of (45) can be derived in the following manner:

d𝑄d𝑡𝑏𝑖=̇𝐫𝑖̇𝐫𝑚T𝐛0𝑖+𝐫𝑖𝐫𝑚Tdd𝑡𝐛0𝑖+𝐛T𝑖𝛾𝑖𝐛T0𝑖𝛾0𝑖d𝑠𝑖+𝐛d𝑡𝑖T𝛾𝑖𝐛0𝑖T𝛾0𝑖d𝑠𝑖=̇𝐫d𝑡𝑖̇𝐫𝑚T𝐛0𝑖+𝐫𝑖𝐫𝑚ṪΠ0𝐛0𝑖+Π0𝐛0𝑖d𝑠𝑖d𝑡+(11)d𝑠𝑖+𝜅d𝑡𝑖𝐧T𝑖𝛾𝑖𝜅0𝑖𝐧T0𝑖𝛾0𝑖d𝑠𝑖=̇𝐫d𝑡𝑖̇𝐫𝑚T𝐛0𝑖(1)𝑖̇𝜃𝐫𝑖𝐫𝑚T𝐧0𝑖+𝐫𝑖𝐫𝑚T𝜅0𝑖𝐧0𝑖d𝑠𝑖+𝜅d𝑡𝑖𝐧T𝑖𝛾𝑖𝜅0𝑖𝐧T0𝑖𝛾0𝑖d𝑠𝑖,d𝑡𝑖=1,2,(46) where, in the derivation of the last equality, (42) is referred to. Furthermore, substituting 𝑄𝑛𝑖=0 expressed by (44) into the right-hand side of (46) yields

d𝑄d𝑡𝑏𝑖=̇𝐫𝑖̇𝐫𝑚T𝐛0𝑖+𝜅𝑖𝐧T𝑖𝛾𝑖𝜅0𝑖𝐧T0𝑖𝛾0𝑖d𝑠𝑖+d𝑡(1)𝑖̇𝜃+𝜅0𝑖d𝑠𝑖𝐧d𝑡T𝑖𝛾𝑖+𝐧T0𝑖𝛾0𝑖=̇𝐫𝑖̇𝐫𝑚T𝐛0𝑖+𝜅𝑖+𝜅0𝑖𝐧T𝑖𝛾𝑖d𝑠𝑖d𝑡(1)𝑖̇𝜃𝐧T𝑖𝛾𝑖+𝐧T0𝑖𝛾0𝑖,𝑖=1,2.(47) Thus, by applying (41) to this equation, we have

d𝑄d𝑡𝑏𝑖=̇𝐫𝑖̇𝐫𝑚T𝐛0𝑖+(1)𝑖̇𝜃̇𝑝𝑖𝐧T𝑖𝛾𝑖(1)𝑖̇𝜃𝐧T𝑖𝛾𝑖+𝐧T0𝑖𝛾0𝑖=̇𝐫𝑖̇𝐫𝑚T𝐛0𝑖(1)𝑖̇𝑝𝑖𝐧T𝑖𝛾𝑖+̇𝜃𝐧T0𝑖𝛾0𝑖=𝑅𝑏𝑖=0,𝑖=1,2.(48) The second equality of (45) can be verified in a similar way.

From this proposition, it follows that the Euler-Lagrange equation of motion of the system can be derived by applying the variational principle to the Lagrangian of the system

𝐿𝑋;𝑠1,𝑠2̇𝑋=𝐾𝑋,𝑖=1,2𝑓𝑖𝑄𝑛𝑖+𝜆𝑖𝑄𝑏𝑖,(49) where 𝐾 denotes the total kinetic energy of the system. Note that ̇𝐾(𝑋,𝑋) is independent of the shape parameters 𝑠1 and 𝑠2 but 𝑄𝑛𝑖 and 𝑄𝑏𝑖 are dependent on 𝑠𝑖 for 𝑖=1,2, respectively. The variational principle is written in this case in the following form:

𝑡1𝑡0𝛿𝐿+𝑢T1𝛿𝑞1+𝑢T2𝛿𝑞2d𝑡=0,(50) from which the set of equations (35) to (37) follow straightforwardly with control torques 𝑢𝑖 through finger joints as follows:

̈1𝐺(𝑋)𝑋+2̇̇𝑋̇𝐺(𝑋)+𝑆𝑋,𝑋+𝑖=1,2𝑓𝑖𝑛𝑖+𝜆𝑖𝑏𝑖=𝑢,(51) where 𝑢=(0,0,0,𝑢T1,𝑢T2)T. It should be noted that

𝜕𝑄𝜕𝑋𝑛𝑖=𝑛𝑖,𝜕𝑄𝜕𝑋𝑏𝑖=𝑏𝑖,𝑖=1,2.(52) Thus, (51) can be spelled out in the general form

̈1𝐺(𝑋)𝑋+2̇̇𝑋̇𝑋+𝐺(𝑋)+𝑆𝑋,𝑖=1,2𝑓𝑖𝜕𝑄𝜕𝑋𝑛𝑖+𝜆𝑖𝜕𝑄𝜕𝑋𝑏𝑖𝑢=𝐵1𝑢2,(53) where 𝐵 denotes the 8×5 constant matrix defined as 𝐵T=(03×5,𝐼5), 03×5 signifies the 3×5 zero matrix, and 𝐼5 the 5×5 identity matrix. In such a representation of the forcing term, 𝐵 is called the driving matrix in robotics.

5. Coordinate Control for Stable Grasping and Morse-Lyapunov Function

In order to design a control signal for establishing stable grasp from the practical standpoint, a family of control signals has been introduced, based upon the fingers-thumb opposability that is one of the functional characteristics of human hands as discussed in [18]. The family of controls is described by the form

𝑢𝑖=𝑐𝑖̇𝑞𝑖+(1)𝑖𝛽𝐽T𝑖𝑞𝑖𝐫1𝐫2,(54) where 𝛽 stands for a position feedback gain common for 𝑖=1,2 with physical unit [N/m]. It should be noted that the signal of the right-hand side can be constructed by the realtime measurement data on joint angles and angular velocities of both the robot fingers together with positions 𝐫1 and 𝐫2 of the centers of the fingertips. In other words, the control signal of (54) should be real time computed as a feedback signal from the measurement data of variables of the intrinsic finger world seen from the multifingered hand side. That is, any information of the geometry of the object and measurement data of location the object mass center should not be referred to in (54), because objects to be grasped, that are changeable in the situation, must be regarded as a substance in the extrinsic world.

Now, substituting 𝑢𝑖 for 𝑖=1,2 into (53) or directly into (37), we obtain

𝐺𝑖̈𝑞𝑖+12̇𝐺𝑖+𝑆𝑖̇𝑞𝑖+𝑐𝑖̇𝑞𝑖(1)𝑖𝛽𝐽T𝑖𝐫1𝐫2+𝑓𝑖𝐽T𝑖𝐧0𝑖(1)𝑖𝐛T𝑖𝛾𝑖𝐞𝑖+𝜆𝑖𝐽T𝑖𝐛0𝑖(1)𝑖𝐧T𝑖𝛾𝑖𝐞𝑖=0,𝑖=1,2.(55) We call the set of equations (35), (36), and (55) the closed-loop dynamics of the system. Clearly, the sum of inner products of ̇𝑥 and (35), ̇𝜃 and (36), and ̇𝑞𝑖 and (55) for 𝑖=1,2 yields, due to (30) and (31),

d𝐾̇𝑋+𝛽d𝑡𝑋,2𝐫1𝐫22=𝑖=1,2𝑐𝑖̇𝑞𝑖2,(56) where ̇𝐾(𝑋,𝑋) denotes the total kinetic energy of the system defined in (32). It should be noted that the total energy defined by

𝐸̇𝑋̇𝑋𝑋,=𝐾𝑋,+𝑈(𝑋),(57) where

𝛽𝑈(𝑋)=2𝐫1𝐫22,(58) is nonnegative definite in 𝑋 and ̇𝑋 and its time derivative d𝐸/d𝑡 is nonpositive definite as shown in (56). That is, if we regard the pair of position and velocity vectors ̇(𝑋,𝑋) as the state vector, then the quadratic function ̇𝐸(𝑋,𝑋) looks like a Lyapunov function as seen from the energy relation (56). Nevertheless, in the case of the overall fingers/object system shown in Figure 1, 𝑋=(𝐱T,𝜃,𝑞T1,𝑞T2)T is of 8 dimentions and hence the state is of 16-dimension. Since there are four rolling contact constraints as discussed in Sections 3 and 4, the overall system must be of four degrees-of-freedom and therefore the state ̇(𝑋,𝑋) must be of 8 dimensions. However, the total energy function ̇𝐸(𝑋,𝑋) is not positive definite with respect to ̇(𝑋,𝑋), because 𝑈(𝑋) is not positive definite in 𝑋 even if all of the four rolling contact constraints are taken into account. Thus, stabilization of grasping by means of a coordinate control of (54) cannot be treated by applying the conventional Lyapunov method to the relation of (56).

In order to tackle a control problem of stable grasping from the standpoint as a combination of Riemannian geometry and Lyapunov's direct method, we first consider a simpler case that the object has a pair of flat side surfaces that are parallel as shown in Figure 5 and both the robot fingers are of a single degree-of-freedom. In this case, 𝑋=(𝐱T,𝜃,𝑞1,𝑞2)T is of 5 dimensions and hence the overall fingers/object system is of single degrees-of-freedom. Therefore, the total energy ̇𝐸(𝑋,𝑋) must be positive definite in ̇(𝑋,𝑋) under the four holonomic rolling contact constraints, because the scalar function 𝑈(=(𝛽/2)𝐫1𝐫22) must have a minimum at some posture 𝑋=𝑋 under the constraints. That is,

𝑈𝑚𝑋=𝑈=min𝑈(𝑋),(59) where “min" is taken over all possible postures that are subject to rolling motion. For the sake of convenience, we will discuss the details of such a set of all possible postures reachable from a starting posture of the system by only movements of rolling motion. For the time being, we discuss what condition specifies the minimal posture that minimizes the artificial potential 𝑈(𝑋).

First, note that the difference vector 𝐫1𝐫2 can be expressed in terms of length parameters 𝑠𝑖 for 𝑖=1,2 as follows:

𝐫1𝐫2=Π1𝛾1+Π2𝛾2+Π0𝛾01𝛾02,(60) which follows from subtraction of (22) from (10). When the object is rectangular as shown in Figure 5, all 𝐛0𝑖 and 𝐧0𝑖(𝑖=1,2) are invariant under the change of 𝑠𝑖(𝑖=1,2). As seen from Figure 5, if we denote the object width by 𝑙𝑤 then we have

𝐫1𝐫2=𝑠1𝑠2𝐛01+𝐛T1𝛾1+𝐛T2𝛾2𝐛01𝑙𝑤𝐧01+𝐧T1𝛾1+𝐧T2𝛾2𝐧01.(61) Hence, by partially differentiating this equation in 𝑠𝑖, we obtain (the details will be given in the Appendix)

𝜕𝐫1𝐫2𝜕𝑠𝑖=(1)𝑖𝜅𝑖𝑠𝑖𝐧T𝑖𝛾𝑖𝐛0𝑖+𝐛T𝑖𝛾𝑖𝐧0𝑖,𝑖=1,2,(62) where 𝜅𝑖(𝑠𝑖) denotes the curvature of the fingertip contour for finger 𝑖(𝑖=1,2). Thus, by regarding 𝑈(𝑋)=𝑈(𝑠1,𝑠2),

𝑠𝜕𝑈1,𝑠2𝜕𝑠𝑖=(1)𝑖𝛽𝜅𝑖𝐫1𝐫2T𝐧T𝑖𝛾𝑖𝐛0𝑖+𝐛T𝑖𝛾𝑖𝐧0𝑖,𝑖=1,2.(63) First, consider the simplest case when both the fingertips are spherical, which is called as the following:

Problem 1 (stability problem of ball-plate pinching). To find a necessary and sufficient condition under which minimization of the potential 𝑈(𝑋) in a set of all possible postures movable from a given starting posture by rolling motions is realized.

In this stabilization problem of ball-plate pinching, 𝑂𝑖 must be the center of the hemispheric fingertip of finger 𝑖 with radius 𝑟𝑖=1/𝜅𝑖(𝑠𝑖), where 𝜅𝑖(𝑠𝑖) is constant with respect to 𝑠𝑖 (see Figure 5). Hence, it follows that

𝐛T𝑖𝛾𝑖=0,𝐧T𝑖𝛾𝑖=𝑟𝑖,𝑖=1,2.(64) Thus, in this case, (63) can be written in the form

𝑠𝜕𝑈1,𝑠2𝜕𝑠𝑖=(1)𝑖𝛽𝐫1𝐫2T𝐛0𝑖,𝑖=1,2.(65) This means that 𝜕𝑈/𝜕𝑠𝑖=0 arises if and only if (𝐫1𝐫2) is orthogonal to 𝐛0𝑖 for 𝑖=1,2. In other words, 𝑈(𝑠1,𝑠2) becomes extremal if and only if the four points 𝑂1, 𝑃1, 𝑃2, and 𝑂2 are lying on the common straight line. At such a posture, it is easy to check that 𝑠1=𝑠2 and the Hessian matrix 𝐻=(𝜕2𝑈/𝜕𝑠𝑖𝜕𝑠𝑗) becomes positive definite. Thus, we conclude that minimization of the shape function 𝑈(𝑠1,𝑠2) is attained if and only if all 𝑂1, 𝑃1, 𝑃2, and 𝑂2 are lying on a straight line.

In the case that both the robot fingers are of a single degree-of-freedom mechanism as shown in Figure 5, the total degrees-of-freedom of the overall system becomes of one DOF, because originally the system has five DOFs but there are four holonomic constraints as discussed in Sections 3 and 4. Hence, in this case, 𝑈(𝑋) becomes positive definite with respect to position variables under the four constraints. That is, 𝑈(𝑋) must be a Morse function introduced on a single-dimensional submanifold of the base Riemannian manifold 𝑀={𝑋,𝑔𝑖𝑗} constrained by four holonomic constraints. Thus, the overall scalar function ̇𝐸(𝑋,𝑋) defined by (57) becomes positive definite. This means that the equality relation of (56) shows that ̇𝐸(𝑋,𝑋) can be regarded as a Lyapunov function for the closed-loop system (53) in which control inputs 𝑢𝑖 for 𝑖=1,2 defined by (54) are substituted. Therefore, the equilibrium point at which ̇𝑋=0 and the four points 𝑂1, 𝑃1, 𝑃2, and 𝑂2 are on a common straight line becomes asymptotically stable in the sense of Lyapunov, even if the function 𝑈(𝑋) is defined on a Riemannian submanifold.

If each robot finger has multi-DOFs as shown in Figure 1, the overall fingers/object system becomes redundant in its degrees-of-freedom. For example, the left robot finger shown in Figure 1 has three joints and the right one has two joints. Then, the total degrees-of-freedom of the system becomes of four DOFs under the four holonomic constraints. Therefore, there arises an infinite number of possible postures realizing the minimum of potential 𝑈(𝑋); that is, a set of such possible postures themselves constitute a manifold. We call it an equilibrium manifold. In the case of a pair of robot fingers with two and three joints, respectively, pinching a rectangular object, the equilibrium manifold denoted by 𝐸𝑀3 becomes of three dimensions. In order to avoid a possible occurence of abundant motion (it is called a self-motion in robotics [19]) owing to the redundancy in the system's DOF, we have adopted the following form of control signals (see [5]):

𝑢𝑖=𝑐𝑖̇𝑞𝑖+(1)𝑖𝛽𝐽T𝑖𝑞𝑖𝐫1𝐫2𝛼𝑖𝑝𝑖(𝑡)𝑝𝑖𝐞(0)𝑖,𝑖=1,2,(66) where 𝛼𝑖 denotes a positive constant, and

𝑝𝑖(𝑡)=𝑞T𝑖(𝑡)𝐞𝑖,𝑖=1,2,(67) and 𝐞1=(1,1,1)T and 𝐞2=(1,1)T in the case of a pair of robot fingers shown in Figure 1. Substituting the control inputs of (66) into (53) yields

𝐺𝑖̈𝑞𝑖+12̇𝐺𝑖+𝑆𝑖̇𝑞𝑖+𝑐𝑖̇𝑞𝑖(1)𝑖𝛽𝐽T𝑖𝐫1𝐫2+𝛼𝑖𝑝𝑖𝑝𝑖𝐞(0)𝑖+𝑓𝑖𝐽T𝑖𝐧0𝑖(1)𝑖𝐛T𝑖𝛾𝑖𝐞𝑖+𝜆𝑖𝐽T𝑖𝐛0𝑖(1)𝑖𝐧T𝑖𝛾𝑖𝐞𝑖=0,𝑖=1,2.(68) Regarding the closed-loop system of (68), (35), and (36) with four constraints (43) and (44), we have the energy relation

d𝐾̇𝑋+𝛽d𝑡𝑋,2𝐫1𝐫22+𝑖=1,2𝛼𝑖2𝑝𝑖𝑝𝑖(0)2=𝑖=1,2𝑐𝑖̇𝑞𝑖2.(69) The details of physical importance of the last term in the control signals of (66) will be disclosed in the next section when nonspherical fingertips are treated.

6. Hessian Matrix of the Morse-Lyapunov Function

Now we are in a position to discuss a stability problem of precision prehension (pinching) under the general fingertip geometry (nonspherical fingertip). However, we assume that the object is rectangular and both the curvatures 𝜅𝑖(𝑠𝑖) for 𝑖=1,2 of the fingertips are continuously differentiable in 𝑠𝑖 and bounded between some 𝑎(>0) and 𝑏(>0) as shown in Figure 6. The problem is posed as follows.

Problem 2 (stability problem of precision prehension). To find a necessary and sufficient conditon under which minimization of the potential 𝑈(𝑋) (or 𝑈(𝑠1,𝑠2)) is realized in a set of all movable postures starting from an initial posture by rolling contact motions. At the same time, to prove the asymptotic convergence of the orbit of motion to some possible equilibrium state.

First, as predicted reasonably from the analysis of Problem 1, we assume that, even in the general case of Problem 1, all the three lines connecting 𝑂1 and 𝑃1, 𝑃1 and 𝑃2, and 𝑃2 and 𝑂2 are lying on the line (𝐫1𝐫2) for minimization of 𝐫1𝐫2. If it were true, then in the case of a pair of robot fingers with a single DOF (see Figure 5) the axes 𝑂𝑂1 and 𝑂𝑂2 must be parallel and hence such a condition could not be satisfied in general, because the base points 𝑂 and 𝑂 are assumed to be fixed in the frame coordinates with some specified value of the length 𝑂𝑂.

Now, we treat a pair of robot fingers with multi DOFs as shown in Figure 1 and consider again a minimization problem of 𝑈(𝑠1,𝑠2) together with a problem of finding a necessary condition for maintaining a steady state for the object (i.e., ̈𝐱=0 in (35) and ̈𝜃=0 in (36)). First, to attain a steady state of the object, we need the conditions

𝑓1=𝑓2=𝑓𝑑(>0),𝜆1=𝜆2=𝜆𝑑(70) to satisfy ̈𝐱=0 in (35) since all 𝐧0𝑖 and 𝐛0𝑖(𝑖=1,2) are invariant in this case. Further, to satisfy ̈𝜃=0 in (36), we shall examine the conditions

𝑠1=𝑠2,𝐛T1𝛾1=𝐛T2𝛾2.(71) Then, substituting these two equalities into (61) yields

𝐫1𝐫2=𝑙𝑤𝐧01+𝐧T1𝛾1+𝐧T2𝛾2𝐧01𝑙=𝑤𝐧T1𝛾1𝐧T2𝛾2𝐧01,(72) which shows that (𝐫1𝐫2) must be perpendicular to 𝐛01(=𝐛02). It is also reasonable to show that, under the equality conditions of (71), ̈𝐱=0 and ̈𝜃=0 can be expressed as

𝑓1𝐧01𝑓2𝐧02𝜆1𝐛01𝜆2𝐛02=0,𝑓1𝐛T01𝛾01+𝑓2𝐛T02𝛾02+𝜆1𝐧T01𝛾01𝜆2𝐧T02𝛾02=0.(73) In fact, the first equality of (73) follows from (71) and 𝐧01=𝐧02 and 𝐛01=𝐛02 and the second equality follows in such a way that

𝑓𝑑𝐛T01𝛾01𝛾02𝜆𝑑𝐧T01𝛾01𝛾02=𝜆𝑑𝐧T01𝛾01𝛾02.(74) Hence, if 𝜆𝑑=0, then the second equality of (73) becomes valid. Furthermore, in the case of Problem 2, 𝐛T𝑖𝛾𝑖0 in general and hence (63) does not imply 𝜕𝑈/𝜕𝑠𝑖=0(𝑖=1,2) though (𝐫1𝐫2) must be perpendicular to 𝐛0𝑖. In this case, we must bear in mind that length parameters 𝑠1 and 𝑠2 are not independent to each other.

In order to derive the Hessian of the shape function 𝑈(𝑠1,𝑠2) plus the extra term (the third term in the bracket [] of (69)) with respcet to 𝑞1, 𝑞2, and 𝜃, we put

𝛽𝑃(𝑋)=2𝐫1𝐫22+𝑖=1,2𝛼𝑖2𝑁2𝑖,(75) where we define

𝑁𝑖=𝑝𝑖𝑝𝑖(0),𝑖=1,2.(76) From (63), we see that the shape function 𝑈(𝑠1,𝑠2) that expresses the first term of the right-hand side of (75) reduces to

𝛽𝑈(𝑋)=2𝐫1𝐫22=𝛽2𝑑2𝑠1,𝑠2+𝑙2𝑠1,𝑠2𝑠=𝑈1,𝑠2,(77) where, by denoting the object width by 𝑙𝑤,

𝑑𝑠1,𝑠2=𝑠1𝑠2𝐛T1𝛾1+𝐛T2𝛾2,𝑙𝑠1,𝑠2=𝑙𝑤+𝐧T1𝛾1+𝐧T2𝛾2(78) (see Figures 4 and 6). First we show that

𝑑𝑃=𝑆𝑁𝑑𝜃+𝑖=1,2Δ𝑁𝑖𝐞T𝑖𝑑𝑞𝑖,(79) where

𝑆𝑁𝑠=𝛽1𝑠2𝑙+𝑙𝑤𝑑,(80)Δ𝑁𝑖=𝛽𝑁𝑖+𝛼𝑖𝑁𝑖𝑁,𝑖=1,2,(81)𝑖=(1)𝑖𝐛T𝑖𝛾𝑖𝐧𝑙T𝑖𝛾𝑖𝑑,𝑖=1,2,(82) in which we omit writing 𝑠1 and 𝑠2 in 𝑑(𝑠1,𝑠2) and 𝑙(𝑠1,𝑠2) for simplicity. In fact, it follows from (63), (61), and (41) that

ddd𝑡𝑃=𝑈𝑠d𝑡1,𝑠2+𝑖=1,2𝛼𝑖𝑁𝑖d𝑁𝑖=d𝑡𝑖=1,2(1)𝑖𝛽𝐧T𝑖𝛾𝑖𝑑(1)𝑖𝐛T𝑖𝛾𝑖𝑙𝜅𝑖d𝑝𝑖d𝑡+𝛼𝑖𝑁d𝑝𝑖=d𝑡𝑖=1,2𝛽𝐧T𝑖𝛾𝑖𝑑(1)𝑖𝐛T𝑖𝛾𝑖𝑙̇𝜃̇𝑝𝑖+𝛼𝑖𝑁𝑖̇𝑝𝑖𝐧=𝛽T1𝛾1+𝐧T2𝛾2𝐛𝑑+T1𝛾1𝐛T2𝛾2𝑙̇𝜃+𝑖=1,2𝛽𝑁𝑖+𝛼𝑖𝑁𝑖̇𝑝𝑖=𝛽𝑙+𝑙𝑤𝑠𝑑+1𝑠2𝑙̇𝑑𝜃+𝑖=1,2Δ𝑁𝑖̇𝑝𝑖,(83) where, in the last equality, (81) is used. By using (80), (83) reduces to (79). Equation (79) means that a local minimum of 𝑃 as a function 𝑃(𝑋,𝑠1,𝑠2) of 𝑋 and 𝑠1 and 𝑠2 is attained when 𝑆𝑁=0 and Δ𝑁𝑖=0(𝑖=1,2). This condition is satisfied when 𝑂1𝑂2 is parallel to 𝑃1𝑃2 and Δ𝑁𝑖=0. More explicitly, if we define

Δ𝑓𝑖=𝑓𝑖𝑠+𝛽𝑙1,𝑠2,Δ𝜆𝑖=𝜆𝑖(1)𝑖𝑠𝛽𝑑1,𝑠2,𝑖=1,2,(84) then the closed-loop dynamics of (35), (36), and (68) can be rewritten into

𝑀̈𝐱𝑖=1,2Δ𝑓𝑖𝐧0𝑖+Δ𝜆𝑖𝐛0𝑖𝐼̈=0,(85)𝜃+𝑖=1,2(1)𝑖Δ𝑓𝑖𝐛T0𝑖𝛾0𝑖Δ𝜆𝑖𝐧T0𝑖𝛾0𝑖+𝑆𝑁=0,(86)𝐺̈𝑞𝑖+12̇𝐺𝑖+𝑆𝑖̇𝑞𝑖+𝑐𝑖̇𝑞𝑖+Δ𝑁𝑖𝐞𝑖+Δ𝑓𝑖𝐽T𝑖𝐧0𝑖(1)𝑖𝐛T𝑖𝛾𝑖𝐞𝑖+Δ𝜆𝑖𝐽T𝑖𝐛0𝑖(1)𝑖𝐧T𝑖𝛾𝑖𝐞𝑖=0,𝑖=1,2.(87) Apparently, steady states are attained when 𝑆𝑁=0, Δ𝑁𝑖=0, Δ𝑓𝑖=0, Δ𝜆𝑖=0 for 𝑖=1,2.

Now, let us derive the Hessian matrix of 𝑃(𝑋,𝑠1,𝑠2) by calculating d𝑆𝑁/d𝑡 and dΔ𝑁𝑖/d𝑡 for 𝑖=1,2. In a similar way to the derivation of (83), we obtain

d𝑆d𝑡𝑁=𝑣1+𝑣2d𝜃d𝑡𝑣1d𝑝1d𝑡𝑣2d𝑝2,d𝑡dΔ𝑁𝑖d𝑡=𝑣𝑖d𝜃+𝛼d𝑡𝑖+𝑣𝑖𝑖d𝑝𝑖d𝑡𝑣𝑖𝑗d𝑝𝑖,d𝑡(88) for 𝑖𝑗, where

𝑣𝑖1=𝛽𝑙𝜅𝑖𝑙𝑤𝑙𝐧T𝑖𝛾𝑖,𝑣𝑖𝑖1=𝛽𝑙𝜅𝑖+𝐧T𝑖𝛾𝑖𝐧T𝑖𝛾𝑖2+𝐛T𝑖𝛾𝑖2𝑙,𝑣𝑖𝑗𝐧=𝛽T1𝛾1𝐧T2𝛾2𝐛T1𝛾1𝐛T2𝛾2,(89) for 𝑖=1,2 and 𝑗𝑖. Note that 𝑙<0 by definition, and all 𝑣1, 𝑣2, 𝑣11, and 𝑣22 are positive provided that

1𝜅𝑖>𝐧T𝑖𝛾𝑖,𝑖=1,2.(90) Thus, if we denote 𝐳=(𝜃,𝑝1,𝑝2)T, then we obtain

𝜕𝑃=𝜕𝐳𝜕𝑃,𝜕𝜃𝜕𝑃𝜕𝑝1,𝜕𝑃𝜕𝑝2T=𝑆𝑁,Δ𝑁1,Δ𝑁2T,𝜕(91)2𝑃𝜕𝐳𝜕𝐳T=𝜕2𝑃𝜕𝜃2𝜕2𝑃𝜕𝜃𝜕𝑝1𝜕2𝑃𝜕𝑝1𝜕𝑝2𝜕2𝑃𝜕𝑝1𝜕𝜕𝜃2𝑃𝜕𝑝21𝜕2𝑃𝜕𝑝1𝜕𝑝2𝜕2𝑃𝜕𝑝2𝜕𝜕𝜃2𝑃𝜕𝑝2𝜕𝑝1𝜕2𝑃𝜕𝑝22=𝑣1+𝑣2𝑣1𝑣2𝑣1𝛼1+𝑣11𝑣12𝑣2𝑣21𝛼2+𝑣22.(92) It is possible to verify that the Hessian of 𝑃 with respect to 𝐳 in a neighborhood of the posture satisfying 𝜕𝑃/𝜕𝐳=0 is positive definite by choosing 𝛼𝑖> appropriately so as to satisfy 𝛼𝑖>𝛽𝑙/𝜅𝑖 for 𝑖=1,2 (note that 𝑙<0). Finally, the Hessian of 𝑃 with respect to the position state vector 𝑋 is given by

𝐻𝑋=𝜕2𝑃𝜕𝑋𝜕𝑋T=𝜕𝐳T𝜕𝜕𝑋2𝑃𝜕𝐳𝜕𝐳T𝜕𝐳𝜕𝑋T=𝐷T𝜕2𝑃𝜕𝐳𝜕𝐳T𝐷=𝐷T𝐻𝐳𝐷,(93) where 𝐷 is a constant 3×8-matrix of the form

𝐷=001000000001110000000011,(94) because 𝐳 can be expressed by 𝐳=𝐷𝑋. Thus, the Hessian matrix 𝐻𝑋 is degenerate, but it is possible to see that the function 𝑃(𝑋) can be regarded as a Morse-Bott function (see [20]).

7. A Proof of Dirichlet-Lagrange Stability for Precision Prehension

We define the equilibrium manifold by the set of all postures that have a form depicted in Figure 6 and maintain the contacts at some points 𝑃1 and 𝑃2. More rigorously, this equilibrium manifold denoted by 𝐸𝑀3 can be regarded as a set of all of 𝑋 that satisfies 𝑆𝑁=0, Δ𝑁1=0, and Δ𝑁2=0 under the situation that 𝑝𝑖(0) for 𝑖=1,2 are given. The submanifold 𝐸𝑀3 is of 3 dimensions, because it satisfies 𝜕𝑃/𝜕𝐳=0 and maintains both contacts (i.e., the 2-dimensional contact constraints (12) and (22)) that determine automatically the position of the object mass center. In other words, 𝐸𝑀3 corresponds to a set of motions of the three excess degrees-of-freedom of the robot fingers.

Suppose one posture 𝑋 of the system belonging to the submanifold 𝐸𝑀3 and a Riemannian ball 𝐵(𝑋,𝑟0) that is defined by

𝐵𝑋,𝑟0=𝑋𝑋𝑀,𝑔𝑖𝑗𝑋,𝑑,𝑋<𝑟0,(95) where 𝑑(𝑋,𝑋) signifies the Riemannian distance defined by

𝑑𝑋,𝑋=inf𝑋(𝜏)𝑡0𝑔𝑖𝑗̇𝑋𝑖(̇𝑋𝜏)𝑗(𝜏)d𝜏,(96) where the infimum is taken over all of the orbits in {𝑀,𝑔𝑖𝑗} connecting 𝑋 and 𝑋 from 𝜏=0 to 𝜏=𝑡. We denote the 𝑖𝑗-entry of 𝐺(𝑋) introduced in (32) and (33) by 𝑔𝑖𝑗. We will show that, for any 𝜀>0, there exists 𝛿(𝜀)>0 such that any solution trajectory 𝑋(𝑡) as a solution to the closed-loop dynamics of (85) to (87) starting from any posture 𝑋(0) with rest state inside 𝐵(𝑋,𝛿(𝜀)) remains in 𝐵(𝑋,𝜀) and converges asymptotically to some posture on 𝐸𝑀3 as 𝑡. Note that numerical values of 𝑟0, 𝜀, and 𝛿(𝜀) are given on the basis of physical dimension [kg m], that is, the dimension of the Riemannian distance defined in (96). We split the tangent space 𝑇𝑋𝑀 into the horizontal space spanned from the constraints of (28) and (30) and the kernel space as the orthogonal complement to the horizontal space. This is carried out by defining

𝑄Φ=𝑛1,𝑄𝑏1,𝑄𝑛2,𝑄𝑏2T,ΦT𝑋=𝜕ΦT=𝜕𝑋𝑛1,𝑏1,𝑛2,𝑏2,ΔΛ=Δ𝑓1,Δ𝜆1,Δ𝑓2,Δ𝜆2T(97) and rewriting (85) to (87) into the following general form similarly to (53):

̈1𝐺(𝑋)𝑋+2̇̇𝑋̇𝐺(𝑋)+𝑆𝑋,𝑋+ΦT𝑋̇𝜕ΔΛ+𝐶𝑋+𝜕𝑋𝑃=0,(98) where 𝐶=diag(0,0,0,𝑐1𝐼3,𝑐2𝐼2). We bear in mind that 𝜕𝑃/𝜕𝑋=𝐷T(𝜕𝑃/𝜕𝐳) according to (91) and (94). Then, let us introduce the transformation

̇Φ𝑋=T𝑋Φ𝑋ΦT𝑋1/2̇Φ̇𝜼,Φ,Ψ𝑊=T𝑋Φ𝑋Φ1/2,,Ψ(99) where Ψ is an 8×4-matrix whose column vectors have a unit norm, and orthogonal to each other and to every column vector of ΦT𝑋. That is, 𝑊 is an orthogonal matrix and hence

𝑊T=Φ𝑋ΦT𝑋1/2Φ𝑋ΨT=𝑊1.(100) Since ̇Φ must be the null vector, substituting (99) into (98) yields

𝐺̈1𝜼+2̇𝐺+𝑆̇Ψ𝜼+Ṫ𝐶Ψ𝜼+ΨT𝐷T𝜕𝑃𝜕𝑧=0,(101) where 𝐺=ΨT𝐺Ψ, and

𝑆=ΨT1𝑆Ψ2̇ΨT1𝐺Ψ+2ΨT𝐺̇Ψ.(102) It should be noted that 𝑆 is skew symmetric, too. It is also important to note that the dynamics of (98) appear to be not fully dissipated according to the definition 𝐶 mentioned below (98). Nevertheless, from (17) and (25) it follows that

̇𝐫1̇𝐫2𝐧01+̇𝑝1𝐧T1𝛾1+̇𝑝2𝐧T2𝛾2+̇𝜃𝐧T01𝛾01+𝐧T02𝛾02=0.(103) Since the coefficient of ̇𝜃 is 𝑙𝑤, (103) reduces to

𝑙2𝑤̇𝜃2𝑐𝜃1̇𝑞12+𝑐𝜃2̇𝑞22(104) with positive constants 𝑐𝜃1 and 𝑐𝜃2. Similarly, we see from (19) and (27) that

̇𝐫1+̇𝐫2T𝐧01̇𝐫2T𝑚𝐧01+̇𝑝1𝐛T1𝛾1+̇𝑝2𝐛T2𝛾2̇𝜃𝐛T01𝛾01+𝐛T02𝛾02=0.(105) From (25) and (17) we also have

̇𝐫1+̇𝐫2T𝐧012̇𝑟T𝑚𝑏01+̇𝑝1𝐧T1𝛾1̇𝑝T2𝐧T2𝛾2̇𝜃𝐧T01𝛾01+𝐧T02𝛾02=0.(106) Since ̇𝐫𝑚=(̇𝑥,̇𝑦) and (𝐧01,𝐛01)𝑆𝑂(2), it follows from (104) and (105) that

̇𝐱2𝑐01̇𝑞12+𝑐02̇𝑞22+𝑙2𝑤̇𝜃2(107) with some constants 𝑐01>0 and 𝑐02>0. Inequalities (104) and (107) imply that the damping coefficient matrix 𝐶=ΨT𝐶Ψ in (101) is positive definite, and hence the dynamics of (101) are fully dissipated. On the other hand, the inner product of (101) and ̇𝜼 yields

ḋ̇𝜼d𝑡(𝐾(𝑋,𝜼)+𝑃)=T𝐶̇𝜼,(108) where

̇1𝐾(𝑋,𝜼)=2̇𝜼T𝐺̇1𝜼=2̇𝑋Ṫ̇𝑋,𝐺(𝑋)𝑋=𝐾𝑋,(109) where ̇𝐾(𝑋,𝑋) is defined by (32). It is also important to note that the coefficient matrix 𝐽T=ΨT𝐷T of 𝜕𝑃/𝜕𝐳 is nondegenerate (in any regular position of grasping like Figure 1) and hence it is possible to construct a scalar quantity

𝑉=Δ𝜃,Δ𝑝1,Δ𝑝2T𝐽+𝐺̇𝜼,(110) where 𝐽+=𝐽T(𝐽𝐽T)1. Then, in a similar way to the proof of stability of PD feedback for redundant systems discussed previously in [8] and [13], we find a constant 𝛼>0 and another constant 𝜎>0 by choosing 𝛽>0, 𝛼𝑖>0, 𝑐𝑖>0(𝑖=1,2) adequately such that

1𝛼̇̇1+𝛼𝐸(𝑋,𝜼)𝐸(𝑋,𝜼)+𝛼𝑉1+𝛼̇d1𝛼𝐸(𝑋,𝜼),̇̇d𝑡𝑊(𝑋,𝜼,𝛼)𝜎𝑊(𝑋,𝜼,𝛼),(111) where ̇̇𝑊(𝑋,𝜼,𝛼)=𝐸(𝑋,𝜼)+𝛼𝑉. The speed of convergence depends on 𝜎 depending on 𝛼, but eventually it can be regulated by the choice of control gains 𝛽, 𝛼𝑖, 𝑐𝑖(𝑖=1,2). Thus, the asymptotic stability of some 𝑋 on the equilibrium manifold 𝐸𝑀3 is ensured in the sense of Riemannian distance. However, the design problem of such control gains according to the physical scale of a multifingered hand must be important from the robotics research viewpoint but has not yet been fully investigated.

8. Conclusions

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. In a special class of rigid objects having a pair of parallel planar surfaces, a general class of control signals is proposed, which can be constructed only from the robot finger kinematics without sensing the object. An extension of the Dirichlet-Lagrange stability theorem to the case of precision prehension under the existence of rolling contact constraints and redundancy in system's degrees-of-freedom is discussed. A preliminary result on computer simulation is presented in [21]. Finally, it should be remarked that any extension of stabilization control for precision prehension to the case of arbitrary shape objects has not yet been tackled.


In (61), note that 𝐛01(=𝐛02) and 𝐧01(=𝐧02) are invariant under the change of 𝑠𝑖(𝑖=1,2). Hence, by partially differentiating the right-hand side of (61) in 𝑠1, we obtain

𝜕𝐫1𝐫2𝜕𝑠1=𝐛01+𝜅1𝐧T1𝛾1𝐛T1𝛾1𝐛01+𝜅1𝐛T1𝛾1+𝐧T1𝛾1𝐧01=𝜅1𝐧T1𝛾1𝐛01+𝐛T1𝛾1𝐧01,(A.1) since 𝛾1=𝜕𝛾1/𝜕𝑠1=𝐛1. It is important to remark that the content of {} in (A.1) dentoed by 𝝃1 as in (78) can be depicted as a vector 𝝃1 shown in Figure 6, which is the vector 𝑂1𝑃1 itself (=Π1𝛾1) rotated by angle 𝜋/2 in the frame coordinates. Similarly, we have