Modeling and Control of 2D Grasping under Rolling Contact Constraints between Arbitrary Shapes: A Riemannian-Geometry Approach

Abstract

Modeling, control, and stabilization of dynamics of two-dimensional object grasping by using a pair of multijoint robot fingers are investigated under rolling contact constraints and arbitrariness of the geometry of the object and fingertips. First, modeling of rolling motion between 2D rigid bodies with arbitrary shape is treated under the assumption that the two contour curves coincide at the contact point and share the same tangent. The rolling constraints induce the Euler equation of motion that is parameterized by a pair of arclength parameters and constrained onto the kernel space as an orthogonal complement to the image space spanned from all the constraint gradients. Furthermore, it is shown that all the Pfaffian forms of the rolling constraints are integrable in the sense of Frobenius and therefore the rolling contacts are regarded as a holonomic constraint. The Euler-Lagrange equation of motion of the overall fingers/object system is rederived together with a couple of first-order differential equations that express evolution of contact points in terms of quantities of the second fundamental form. A control signal called “blind grasping” is defined and shown to be effective in maintenance or stabilization of grasping without using the details of object shape and parameters or external sensing. An extension of the Dirichlet-Lagrange stability theorem to a system of DOF-redundancy under constraints is discussed by introducing a Morse-Bott function and deriving its Hessian, in a special case that the object to be grasped is a parallelepiped.

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 [2–4] that attempt to model dynamics of physical interactions between the fingerends and an object under the existence of rollings. However, the papers [2–4] 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.

Figure 1 

A pair of two-dimensional robot fingers with a curved fingertip makes rolling contact with a rigid object with a curved contour.

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)=𝑃_1𝐴 and 𝛾₀₁(𝑠₀)=𝑃_1𝐵 so that 𝑃_1𝐴=𝑃_1𝐵=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 𝑃_1𝐴(𝑠) and 𝑃_1𝐵(𝑠) expressed in the frame space 𝑂-𝑥𝑦 coincide and share the same tangent vector, as shown in Figures 2 and 3.

Figure 2 

Definitions of local coordinates $O_m$-$XY$ for the object and $O_1$-$X_1{Y}_1$ for the fingertip.

Figure 3 

Rolling contact constraint is expressed by equalizing the velocity of contact point $P_{1A}(=P_1)$ in the direction of the unit tangent vector ${\mathit{b}}_0$ with that of $P_{1B}(=P_1)$ in the same direction of ${\mathit{b}}_1$. The velocity of $P_{1A}$ in the direction of the normal ${\mathit{n}}_1$ should be equal to that of $P_{1B}$ in the same direction. It is therefore assumed that ${\mathit{b}}_0={\mathit{b}}_1$.

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 𝑂_𝑚-𝑋𝑌 fixed at the object. If we denote 𝛾₁(𝑠) and 𝛾₀₁(𝑠) by

𝛾₁(𝑠)=𝑋¹𝑌(𝑠)₁(𝑠),𝛾⁰¹(𝑠)=𝑋⁰¹𝑌(𝑠)₀₁(𝑠)(1) and define 𝐛₁(𝑠)=𝛾^1(𝑠)=dd𝑠𝑋¹𝑌(𝑠)₁𝐛(𝑠),(2)₀₁(𝑠)=𝛾^01(𝑠)=dd𝑠𝑋⁰¹𝑌(𝑠)₀₁(𝑠),(3) then 𝐛₁(𝑠) must be the unit tangent vector at the point 𝑃_1𝐴(𝑠) and 𝐛₀₁(𝑠) must be also the unit tangent vector at 𝑃_1𝐵(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

𝐛^1(𝑠)=𝜅₁(𝑠)𝐧₁(𝑠),𝐛^01(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

𝐧^1(𝑠)=−𝜅₁(𝑠)𝐛₁(𝑠),𝐧^01(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 𝑂_𝑚 expressed in the frame coordinates as follows:

Π₀=𝐫_𝑋,𝐫_𝑌,(6) where 𝐫_𝑋 denotes the unit vector of 𝑋-axis and 𝐫_𝑌 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, 𝑂_𝑚-𝑋𝑌 rotates by 𝜃 from the frame coordinates 𝑂-𝑥𝑦, then

Π₀=𝐫_𝑋,𝐫_𝑌=−cosisn𝜃𝜃csoins𝜃𝜃.(7) Similarly, define

Π₁=𝐫_𝑋1,𝐫_𝑌1,(8) where 𝐫_𝑋1 denotes the unit vector of 𝑋₁-axis of the left-hand fingertip expressed in the frame coordinates 𝑂-𝑥𝑦 and 𝐫_𝑌1 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 Π̇₀=(dΠ₀/d𝑡) and Π̇₁ are obtained as follows:

ΠΠ̇̇₀=Π₀Ω₀,Ω₀=𝜃̇−010₁1,=Π₁Ω₁,Ω₁=𝑝̇₁−0110,(9) where 𝑝₁=𝑞₁₁+𝑞₁₂+𝑞₁₃=𝑞^T₁𝐞₁, 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, 𝜃_𝑖, 𝑝_𝑖, and 𝑞_𝑖𝑗 in the counterclockwise direction. Hence, Ω₀ in (9) expresses the rotation of positive angle 𝜋/2 with associating the angular velocity 𝜃̇. In other words, Ω₀𝐛₀₁=𝜃̇𝐧₀₁ or 𝐛^T₀₁Ω₀=−𝜃̇𝐧^T₀₁ and Ω₁𝐛₁=−𝑝̇₁𝐧₁ or 𝐛^T₁Ω₁=𝑝̇₁𝐧^T₁.

Next, denote the position vector of point 𝑂₁ as a center of the left-hand fingertip by 𝐫₁(=𝑂𝑂₁) and that of the object mass center by 𝐫_𝑚(=𝑂𝑂_𝑚). Then, the first assumption that the two contact points 𝑃_1𝐴 and 𝑃_1𝐵 coincide at the common contact point 𝑃₁ implies the equality

𝐫_1𝐴=𝐫₁+Π₁𝛾₁=𝐫_𝑚+Π₀𝛾₀₁=𝐫_1𝐵.(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𝜕𝑠,𝐛₀₁=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 𝑟₁̇=d𝐫₁/d𝑡 and Π̇₁=dΠ₁/d𝑡. Then, differentiation of (10) in 𝑡 yields

𝐫₁̇+Π̇₁𝛾₁+Π₁𝛾^1d𝑠d𝑡=𝐫^𝑚̇+Π̇₀𝛾₀₁+Π₀𝛾^01d𝑠d𝑡.(13) Since Π₁𝛾^1=𝐛₁ and Π₀𝛾^01=𝐛₀₁, (13) is reduced to, according to (12),

𝐫₁̇−𝐫_𝑚̇+Π̇₁𝛾₁=Π̇₀𝛾₀₁,(14) which from (9) can be written in the form

𝐫₁̇−𝐫_𝑚̇+Π₁Ω₁𝛾₁−Π₀Ω₀𝛾₀₁=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

𝑅_𝑏1=𝐫₁̇−𝐫_𝑚̇^T𝐛₀₁+𝐛^T₁Ω₁𝛾₁−𝐛^T₀₁Ω₀𝛾₀₁=0.(16) On account of the skew symmetry of Ω₁ and Ω₀ as shown in (9) and discussed below (9), (16) is reduced to

𝑅_𝑏1=𝐫₁̇−𝐫_𝑚̇^T𝐛₀₁+𝑝̇₁𝐧^T₁𝛾₁+𝜃̇𝐧^T₀₁𝛾₀₁=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:

𝑅_𝑛1=𝐫₁̇−𝐫_𝑚̇^T𝐧₀₁−𝐧^T₁Ω₁𝛾₁−𝐧^T₀₁Ω₀𝛾₀₁=0,(18) where Π₁𝐧₁=−Π₀𝐧₀₁ is taken into account. This is reduced to

𝑅_𝑛1=𝐫₁̇−𝐫_𝑚̇^T𝐧₀₁+𝑝̇₁𝐛^T₁𝛾₁−𝜃̇𝐛^T₀₁𝛾₀₁=0.(19) Hence, if we define the column vectors

ℜ_𝑏1=⎡⎢⎢⎢⎣−𝐛₀₁𝐽_T1𝐧^T₀₁𝛾₀₁𝑞₁𝐛₀₁+𝐧^T₁𝛾₁𝐞₁ℜ_𝑛1⎤⎥⎥⎥⎦,=⎡⎢⎢⎢⎣−𝐧₀₁𝐽_T1−𝐛^T₀₁𝛾₀₁𝑞₁𝐧₀₁+𝐛^T₁𝛾₁𝐞₁⎤⎥⎥⎥⎦,(20) and 𝑋₁=(𝐱^T,𝜃,𝑞^T₁)^T, 𝐱=(𝑥,𝑦)^T, and 𝐽₁(𝑞₁)=𝜕𝐫₁/𝜕𝑞^T₁, then (17) and (19) can be rewritten into the forms

𝑅_𝑏1=ℜ^T𝑏1d𝑋₁d𝑡=0,𝑅^𝑛1=ℜ^T𝑛1d𝑋₁d𝑡=0.(21)

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 𝑂_𝑚-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=𝛾^2 and 𝐛₀₂=d𝛾₀₂/d𝑠₂=𝛾^02, and similarly to (10) and (12) that

Figure 4 

Definitions of tangent vectors ${\mathit{b}}_i\text{,}$${\mathit{b}}_{0i}$ and normals ${\mathit{n}}_i$ and ${\mathit{n}}_{0i}$ at contact points $P_i$ for $i=\mathrm{1,2}.$

𝐫₂+Π₂𝛾₂=𝐫_𝑚+Π₀𝛾₀₂,(22)𝐛₂=Π₂𝐛₂=Π₀𝐛₀₂=𝐛₀₂,(23) where Π₂=(𝐫_𝑋2,𝐫_𝑌2), Π̇₂=Π₂Ω₂, and

Ω₂=𝑝̇₂−0101,(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 𝐛^T₂Ω₂=−𝑝̇₂𝐧₂ and 𝐛^T₀₂Ω₀=𝜃̇𝐧₀₂. Thus, similarly to the derivation of (16) and (17), we have from differentiation of (22) in 𝑡 the following:

𝑅𝑅_𝑏2=𝐫₂̇−𝐫_𝑚̇^T𝐛₀₂+𝐛^T₂Ω₂𝛾₂−𝐛^T₀₂Ω₀𝛾₀₂=0,_𝑏2=𝐫₂̇−𝐫_𝑚̇^T𝐛₀₂−𝑝̇₂𝐧^T₂𝛾₂−𝜃̇𝐧^T₀₂𝛾₀₂=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:

𝑅𝑅_𝑛2=𝐫₂̇−𝐫_𝑚̇^T𝐧₀₂−𝐧^T₂Ω₂𝛾₂−𝐧^T₀₂Ω₀𝛾₀₂=0,(26)_𝑛2=𝐫₂̇−𝐫_𝑚̇^T𝐧₀₂−𝑝̇₂𝐛^T₂𝛾₂+𝜃̇𝐛^T₀₂𝛾₀₂=0.(27) Hence, if we denote 𝑋=(𝐱^T,𝜃,𝑞^T₁,𝑞^T₂)^T, 𝐽₂(𝑞₂)=𝜕𝐫₂/𝜕𝑞₂, 𝐞₂=(1,1)^T, and

ℜ_𝑏2=⎡⎢⎢⎢⎢⎣−𝐛₀₂−𝐧^T₀₂𝛾₀₂0₃𝐽^T₂𝑞₂𝐛₀₂−𝐧^T₂𝛾₂𝐞₂ℜ_𝑛2⎤⎥⎥⎥⎥⎦,=⎡⎢⎢⎢⎢⎣−𝐧₀₂𝐛^T₀₂𝛾₀₂0₃𝐽^T₂𝑞₂𝐧₀₂−𝐛^T₂𝛾₂𝐞₂⎤⎥⎥⎥⎥⎦,(28) then, similarly to (21), (25) and (27) are expressed as follows:

𝑅_𝑏2=ℜ^T𝑏2d𝑋d𝑡=0,𝑅^𝑛2=ℜ^T𝑛2d𝑋d𝑡=0,(29) where 0₃=(0,0,0)^T. For convenience, we append ℜ_𝑏1 and ℜ_𝑛1 with the two-dimensional zero vector in the following way:

ℜ_𝑏1=ℜ^𝑏10₂,ℜ_𝑛1=ℜ^𝑛10₂(30) and denote ℜ_𝑏1 and ℜ_𝑛1 by ℜ_𝑏1 and ℜ_𝑛1 renewedly for the sake of convenience. Then, (21) can be written in the form

𝑅_𝑏1=ℜ^T𝑏1d𝑋d𝑡=0,𝑅^𝑛1=ℜ^T𝑛1d𝑋d𝑡=0.(31)

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

As discussed in the previous papers [11–15], 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, 𝑞₁∈𝑇³, 𝑞₂=(𝑞₂₁,𝑞₂₂)^T∈𝑇², 𝑝₂=𝑞^T₂𝐞₂, 𝐞₂=(1,1)^T, and the Riemannian metric 𝐺=(𝑔_𝑖𝑗) is defined as

𝐾𝑋,𝑋̇=12𝑋̇^T=𝑀𝐺(𝑋)𝑋̇2𝑥̇²+𝑦̇²+𝐼2𝜃̇²𝐺=diag𝑀,𝑀,𝐼,𝐺₁+_𝑖=1,2𝑞^T_𝑖̇𝐺_𝑖𝑞_𝑖𝑞_𝑖̇,(32)𝑞₁,𝐺₂𝑞₂,(33) where 𝑀 and 𝐼 denote the mass of the object and its inertia moment around 𝑂_𝑚, and 𝐺_𝑖(𝑞_𝑖) 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 𝑋̇∈𝑇_{(𝑋;𝑠1,𝑠2)}𝑀₈, where we denote by 𝑇_{𝑋;𝑠1,𝑠2}𝑀₈ 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 ℜ_𝑏𝑖 and ℜ_𝑛𝑖 for 𝑖=1,2. Hence, by introducing four Lagrange's multipliers 𝜆_𝑖 and 𝑓_𝑖 corresponding to ℜ_𝑏𝑖 and ℜ_𝑛𝑖 for 𝑖=1,2, it is possible to derive the Euler-Lagrange equation of motion in the following way:

𝐺(𝑋)𝑋̈+12𝐺̇+𝑆𝑋̇+_𝑖=1,2𝑓_𝑖ℜ_𝑛𝑖+𝜆_𝑖ℜ_𝑏𝑖=𝑢.(34) In detail, (34) can be written as follows:

𝐼𝑀𝜃̈𝐱−̈−𝑓𝑓₁₁𝐧₀₁−𝑓₂𝐧₀₂−𝜆₁𝐛₀₁−𝜆₂𝐛₀₂=0,(35)−𝜆₂𝐛^T₀₁𝛾₀₁+𝑓₂𝐛^T₀₂𝛾₀₂+𝜆₁𝐧^T₀₁𝛾₀₁𝐺𝐧^T₀₂𝛾₀₂=0,(36)_𝑖𝑞_𝑖𝑞_𝑖̈+12𝐺̇^𝑖𝑞_𝑖+𝑆_𝑖𝑞_𝑖,𝑞_𝑖̇𝑞_𝑖̇++𝜆𝑓_𝑖𝑖𝐽^T_𝑖𝑞_𝑖𝐧_0𝑖−(−1)^𝑖𝐛^T_𝑖𝛾_𝑖𝐞_𝑖𝐽^T𝑖𝑞_𝑖𝐛_0𝑖−(−1)^𝑖𝐧^T_𝑖𝛾_𝑖𝐞_𝑖=𝑢^𝑖,𝑖=1,2,(37) in which symbol 𝑢_𝑖 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, 𝐛_0𝑖=𝐛_𝑖 at contact point 𝑃_𝑖 for 𝑖=1,2. This also implies

dd𝑡𝐛_0𝑖=dd𝑡𝐛_𝑖,𝑖=1,2,(38) which reduces to

Π̇₀𝐛_0𝑖+Π₀𝜅_0𝑖𝑠_𝑖𝐧_0𝑖d𝑠_𝑖d𝑡=Π̇^𝑖𝐛_𝑖+Π_𝑖𝜅_𝑖𝑠_𝑖𝐧_𝑖d𝑠_𝑖d𝑡,𝑖=1,2,(39) where 𝜅_𝑖(𝑠_𝑖) 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 𝐧_0𝑖=−𝐧_𝑖 for both 𝑖=1,2, (39) is again reduced to

𝜅_0𝑖+𝜅_𝑖𝐧_0𝑖d𝑠_𝑖d𝑡=Π^𝑖Ω_𝑖𝐛_𝑖−Π₀Ω₀𝐛_0𝑖,𝑖=1,2,(40) Thus, the inner product of 𝐧_0𝑖 and (40) yields

𝜅_0𝑖+𝜅_𝑖dd𝑠𝑡^𝑖=(−1)^𝑖𝜃̇−𝑝̇_𝑖,𝑖=1,2,(41) where we referred to the relation (as discussed below (9)):

Ω_𝑖𝐛_𝑖=(−1)^𝑖𝑝̇_𝑖𝐧_𝑖,Ω₀𝐛_0𝑖=−(−1)^𝑖𝜃̇𝐧_0𝑖,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): 𝑄𝑄_𝑏𝑖=𝐫_𝑖−𝐫_𝑚^T𝐛_0𝑖+𝐛^T_𝑖𝛾_𝑖−𝐛^T_0𝑖𝛾_0𝑖=0,𝑖=1,2,(43)_𝑛𝑖=𝐫_𝑖−𝐫_𝑚^T𝐧_0𝑖−𝐧^T_𝑖𝛾_𝑖−𝐧^T_0𝑖𝛾_0𝑖=0,𝑖=1,2.(44)
(2) Each of the Pfaffian forms 𝑅_𝑏𝑖 and 𝑅_𝑛𝑖 for 𝑖=1,2 defined by (17), (19), (25), and (27) is integrable in 𝑡 in the following forms: dd𝑡𝑄^𝑏𝑖=𝑅_𝑏𝑖,dd𝑡𝑄^𝑛𝑖=𝑅_𝑛𝑖,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:

dd𝑡𝑄^𝑏𝑖=𝐫_𝑖̇−𝐫_𝑚̇^T𝐛_0𝑖+𝐫_𝑖−𝐫_𝑚^Tdd𝑡𝐛_0𝑖+𝐛^T_𝑖𝛾^𝑖−𝐛^T_0𝑖𝛾^0𝑖d𝑠^𝑖d𝑡+𝐛^𝑖^T𝛾_𝑖−𝐛^0𝑖^T𝛾_0𝑖d𝑠^𝑖=𝐫̇d𝑡_𝑖−𝐫_𝑚̇^T𝐛_0𝑖+𝐫_𝑖−𝐫_𝑚^TΠ̇⁰𝐛_0𝑖+Π₀𝐛^0𝑖d𝑠_𝑖+(1−1)d𝑠_𝑖d𝑡d𝑡+𝜅^𝑖𝐧^T_𝑖𝛾_𝑖−𝜅_0𝑖𝐧^T_0𝑖𝛾_0𝑖d𝑠^𝑖=𝐫̇d𝑡_𝑖−𝐫_𝑚̇^T𝐛_0𝑖−(−1)^𝑖𝜃̇𝐫_𝑖−𝐫_𝑚^T𝐧_0𝑖+𝐫_𝑖−𝐫_𝑚^T𝜅_0𝑖𝐧_0𝑖d𝑠_𝑖d𝑡+𝜅^𝑖𝐧^T_𝑖𝛾_𝑖−𝜅_0𝑖𝐧^T_0𝑖𝛾_0𝑖d𝑠^𝑖𝑖=1,2,d𝑡,(46) where, in the derivation of the last equality, (42) is referred to. Furthermore, substituting 𝑄_𝑛𝑖=0 expressed by (44) into the right-hand side of (46) yields

dd𝑡𝑄^𝑏𝑖=𝐫_𝑖̇−𝐫_𝑚̇^T𝐛_0𝑖+𝜅_𝑖𝐧^T_𝑖𝛾_𝑖−𝜅_0𝑖𝐧^T_0𝑖𝛾_0𝑖d𝑠^𝑖+−(−1)_𝑖d𝑡𝜃̇+𝜅_0𝑖d𝑠_𝑖d𝑡𝐧^T𝑖=𝐫_𝑖̇𝛾_𝑖+𝐧^T_0𝑖𝛾_0𝑖−𝐫_𝑚̇^T𝐛_0𝑖+𝜅_𝑖+𝜅_0𝑖𝐧^T_𝑖𝛾_𝑖d𝑠_𝑖−(−1)_𝑖d𝑡𝜃̇𝐧^T_𝑖𝛾_𝑖+𝐧^T_0𝑖𝛾_0𝑖,𝑖=1,2.(47) Thus, by applying (41) to this equation, we have

dd𝑡𝑄^𝑏𝑖=𝐫_𝑖̇−𝐫_𝑚̇^T𝐛_0𝑖+(−1)^𝑖𝜃̇−𝑝̇_𝑖𝐧^T_𝑖𝛾_𝑖=−𝐫(_𝑖̇−1)^𝑖𝜃̇𝐧^T_𝑖𝛾_𝑖+𝐧^T_0𝑖𝛾_0𝑖=𝑅_𝑏𝑖−𝐫_𝑚̇^T𝐛_0𝑖−(−1)^𝑖𝑝̇_𝑖𝐧^T_𝑖𝛾_𝑖+𝜃̇𝐧^T_0𝑖𝛾_0𝑖=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

𝐿𝑋;𝑠₁,𝑠₂=𝐾𝑋,𝑋̇−_𝑖=1,2𝑓_𝑖𝑄_𝑛𝑖+𝜆_𝑖𝑄_𝑏𝑖,(49) where 𝐾 denotes the total kinetic energy of the system. Note that 𝐾(𝑋,𝑋̇) is independent of the shape parameters 𝑠₁ and 𝑠₂ but 𝑄_𝑛𝑖 and 𝑄_𝑏𝑖 are dependent on 𝑠_𝑖 for 𝑖=1,2, respectively. The variational principle is written in this case in the following form:

^𝑡1_𝑡0𝛿𝐿+𝑢^T₁𝛿𝑞₁+𝑢^T₂𝛿𝑞₂d𝑡=0,(50) from which the set of equations (35) to (37) follow straightforwardly with control torques 𝑢_𝑖 through finger joints as follows:

𝐺(𝑋)𝑋̈+12𝐺̇(𝑋)+𝑆𝑋,𝑋̇𝑋̇+_𝑖=1,2𝑓_𝑖ℜ_𝑛𝑖+𝜆_𝑖ℜ_𝑏𝑖=𝑢,(51) where 𝑢=(0,0,0,𝑢^T₁,𝑢^T₂)^T. It should be noted that

𝜕𝜕𝑋𝑄^𝑛𝑖=ℜ_𝑛𝑖,𝜕𝜕𝑋𝑄^𝑏𝑖=ℜ_𝑏𝑖,𝑖=1,2.(52) Thus, (51) can be spelled out in the general form

𝐺(𝑋)𝑋̈+1+2𝐺̇(𝑋)+𝑆𝑋,_𝑖=1,2𝑋̇𝑋̇𝑓^𝑖𝜕𝜕𝑋𝑄^𝑛𝑖+𝜆_𝑖𝜕𝜕𝑋𝑄^𝑏𝑖=𝐵𝑢𝑢¹₂,(53) where 𝐵 denotes the 8×5 constant matrix defined as 𝐵^T=(0_3×5,𝐼₅), 0_3×5 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

𝑢_𝑖=−𝑐_𝑖𝑞_𝑖̇+(−1)^𝑖𝛽𝐽^T_𝑖𝑞_𝑖𝐫₁−𝐫₂,(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 𝑢_𝑖 for 𝑖=1,2 into (53) or directly into (37), we obtain

𝐺_𝑖𝑞_𝑖̈+12𝐺̇^𝑖+𝑓_𝑖+𝑆_𝑖𝑞^𝑖̇+𝑐_𝑖𝑞_𝑖̇−(−1)^𝑖𝛽𝐽^T_𝑖𝐫₁−𝐫₂+𝜆_𝑖𝐽^T_𝑖𝐧_0𝑖−(−1)^𝑖𝐛^T_𝑖𝛾_𝑖𝐞_𝑖𝐽^T𝑖𝐛_0𝑖−(−1)^𝑖𝐧^T_𝑖𝛾_𝑖𝐞_𝑖=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 𝑥̇ and (35), 𝜃̇ and (36), and 𝑞_𝑖̇ and (55) for 𝑖=1,2 yields, due to (30) and (31),

dd𝑡𝐾𝑋,𝑋̇+𝛽2‖‖𝐫¹−𝐫₂‖‖²=−_𝑖=1,2𝑐_𝑖‖‖𝑞_𝑖̇‖‖²,(56) where 𝐾(𝑋,𝑋̇) denotes the total kinetic energy of the system defined in (32). It should be noted that the total energy defined by

𝐸𝑋,𝑋̇=𝐾𝑋,𝑋̇+𝑈(𝑋),(57) where

𝑈(𝑋)=𝛽2‖‖𝐫¹−𝐫₂‖‖²,(58) is nonnegative definite in 𝑋 and 𝑋̇ 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,𝑋̇) as the state vector, then the quadratic function 𝐸(𝑋,𝑋̇) 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=(𝐱^T,𝜃,𝑞^T₁,𝑞^T₂)^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,𝑋̇) must be of 8 dimensions. However, the total energy function 𝐸(𝑋,𝑋̇) is not positive definite with respect to (𝑋,𝑋̇), 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=(𝐱^T,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,𝑋̇) must be positive definite in (𝑋,𝑋̇) under the four holonomic rolling contact constraints, because the scalar function 𝑈(=(𝛽/2)‖𝐫₁−𝐫₂‖²) must have a minimum at some posture 𝑋=𝑋^∗ under the constraints. That is,

Figure 5 

Precision prehension of a 2D object with parallel flat surfaces by a pair of single-DOF finger.

𝑈_𝑚=𝑈𝑋^∗=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 𝐫₁−𝐫₂ can be expressed in terms of length parameters 𝑠_𝑖 for 𝑖=1,2 as follows:

𝐫₁−𝐫₂=−Π₁𝛾₁+Π₂𝛾₂+Π₀𝛾₀₁−𝛾₀₂,(60) which follows from subtraction of (22) from (10). When the object is rectangular as shown in Figure 5, all 𝐛_0𝑖 and 𝐧_0𝑖(𝑖=1,2) are invariant under the change of 𝑠_𝑖(𝑖=1,2). As seen from Figure 5, if we denote the object width by 𝑙_𝑤 then we have

𝐫₁−𝐫₂=𝑠₁−𝑠₂𝐛₀₁+−𝐛^T₁𝛾₁+𝐛^T₂𝛾₂𝐛₀₁−𝑙_𝑤𝐧₀₁+𝐧^T₁𝛾₁+𝐧^T₂𝛾₂𝐧₀₁.(61) Hence, by partially differentiating this equation in 𝑠_𝑖, we obtain (the details will be given in the Appendix)

𝜕𝐫₁−𝐫₂𝜕𝑠_𝑖=(−1)^𝑖𝜅_𝑖𝑠_𝑖𝐧^T_𝑖𝛾_𝑖𝐛_0𝑖+𝐛^T_𝑖𝛾_𝑖𝐧_0𝑖,𝑖=1,2,(62) where 𝜅_𝑖(𝑠_𝑖) denotes the curvature of the fingertip contour for finger 𝑖(𝑖=1,2). Thus, by regarding 𝑈(𝑋)=𝑈(𝑠₁,𝑠₂),

𝜕𝑈𝑠₁,𝑠₂𝜕𝑠_𝑖=(−1)^𝑖𝛽𝜅_𝑖𝐫₁−𝐫₂^T𝐧^T𝑖𝛾_𝑖𝐛_0𝑖+𝐛^T_𝑖𝛾𝑖^𝑖=𝐧1^0𝑖,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, 𝑂_𝑖 must be the center of the hemispheric fingertip of finger 𝑖 with radius 𝑟_𝑖=1/𝜅_𝑖(𝑠_𝑖), where 𝜅_𝑖(𝑠_𝑖) is constant with respect to 𝑠_𝑖 (see Figure 5). Hence, it follows that

𝐛^T_𝑖𝛾_𝑖=0,𝐧^T_𝑖𝛾_𝑖=𝑟_𝑖,𝑖=1,2.(64) Thus, in this case, (63) can be written in the form

𝜕𝑈𝑠₁,𝑠₂𝜕𝑠_𝑖=(−1)^𝑖𝛽𝐫₁−𝐫₂^T𝐛_0𝑖,𝑖=1,2.(65) This means that 𝜕𝑈/𝜕𝑠_𝑖=0 arises if and only if (𝐫₁−𝐫₂) is orthogonal to 𝐛_0𝑖 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/𝜕𝑠_𝑖𝜕𝑠_𝑗) 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,𝑔_𝑖𝑗} constrained by four holonomic constraints. Thus, the overall scalar function 𝐸(𝑋,𝑋̇) defined by (57) becomes positive definite. This means that the equality relation of (56) shows that 𝐸(𝑋,𝑋̇) can be regarded as a Lyapunov function for the closed-loop system (53) in which control inputs 𝑢_𝑖 for 𝑖=1,2 defined by (54) are substituted. Therefore, the equilibrium point at which 𝑋̇=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]):

𝑢_𝑖=−𝑐_𝑖𝑞_𝑖̇+(−1)^𝑖𝛽𝐽^T_𝑖𝑞_𝑖𝐫₁−𝐫₂−𝛼_𝑖𝑝_𝑖(𝑡)−𝑝_𝑖𝑖(=0)1,𝐞2_𝑖,,(66) where 𝛼_𝑖 denotes a positive constant, and

𝑝_𝑖(𝑡)=𝑞^T_𝑖(𝑡)𝐞_𝑖,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

𝐺_𝑖𝑞_𝑖̈+12𝐺̇^𝑖+𝛼_𝑖+𝑆_𝑖𝑞^𝑖̇+𝑐_𝑖𝑞_𝑖̇−(−1)^𝑖𝛽𝐽^T_𝑖𝐫₁−𝐫₂+𝜆_𝑖𝑝_𝑖−𝑝_𝑖(0)𝐞_𝑖+𝑓_𝑖𝐽^T_𝑖𝐧_0𝑖−(−1)^𝑖𝐛^T_𝑖𝛾_𝑖𝐞_𝑖𝐽^T𝑖𝐛_0𝑖−(−1)^𝑖𝐧^T_𝑖𝛾_𝑖𝐞_𝑖=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

dd𝑡𝐾𝑋,𝑋̇+𝛽2‖‖𝐫¹−𝐫₂‖‖²+_𝑖=1,2𝛼_𝑖2𝑝^𝑖−𝑝_𝑖(0)²=−_𝑖=1,2𝑐_𝑖‖‖𝑞_𝑖̇‖‖².(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 𝜅_𝑖(𝑠_𝑖) for 𝑖=1,2 of the fingertips are continuously differentiable in 𝑠_𝑖 and bounded between some 𝑎(>0) and 𝑏(>0) as shown in Figure 6. The problem is posed as follows.

Figure 6 

Minimization of the squared norm $\parallel {\mathit{r}}_1-{\mathit{r}}_2{\parallel }^2$ over rolling motions is attained when the straight line $\overline{P_1{P}_2}$ connecting the two contact points becomes parallel to the vector $({\mathit{r}}_1-{\mathit{r}}_2);$ that is, $\overline{O_1{O}_2}$ becomes parallel to $\overline{P_1{P}_2}.$

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 (𝐫₁−𝐫₂) for minimization of ‖𝐫₁−𝐫₂‖. 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 𝑂′𝑂₂ 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., 𝐱̈=0 in (35) and 𝜃̈=0 in (36)). First, to attain a steady state of the object, we need the conditions

𝑓₁=𝑓₂=𝑓_𝑑(>0),𝜆₁=−𝜆₂=𝜆_𝑑(70) to satisfy 𝐱̈=0 in (35) since all 𝐧_0𝑖 and 𝐛_0𝑖(𝑖=1,2) are invariant in this case. Further, to satisfy 𝜃̈=0 in (36), we shall examine the conditions

𝑠₁=𝑠₂,𝐛^T₁𝛾₁=𝐛^T₂𝛾₂.(71) Then, substituting these two equalities into (61) yields

𝐫₁−𝐫₂=−𝑙_𝑤𝐧₀₁+𝐧^T₁𝛾₁+𝐧^T₂𝛾₂𝐧₀₁=−𝑙_𝑤−𝐧^T₁𝛾₁−𝐧^T₂𝛾₂𝐧₀₁,(72) which shows that (𝐫₁−𝐫₂) must be perpendicular to 𝐛₀₁(=𝐛₀₂). It is also reasonable to show that, under the equality conditions of (71), 𝐱̈=0 and 𝜃̈=0 can be expressed as

−𝑓₁−𝑓₁𝐧₀₁−𝑓₂𝐧₀₂−𝜆₁𝐛₀₁−𝜆₂𝐛₀₂=0,𝐛^T₀₁𝛾₀₁+𝑓₂𝐛^T₀₂𝛾₀₂+𝜆₁𝐧^T₀₁𝛾₀₁−𝜆₂𝐧^T₀₂𝛾₀₂=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

−𝑓_𝑑=−𝐛𝜆^T_01𝑑𝛾₀₁−𝛾₀₂−𝜆_𝑑𝐧^T₀₁𝛾₀₁−𝛾₀₂𝐧^T₀₁𝛾₀₁−𝛾₀₂.(74) Hence, if 𝜆_𝑑=0, then the second equality of (73) becomes valid. Furthermore, in the case of Problem 2, 𝐛^T_𝑖𝛾_𝑖≠0 in general and hence (63) does not imply 𝜕𝑈/𝜕𝑠_𝑖=0(𝑖=1,2) though (𝐫₁−𝐫₂) must be perpendicular to 𝐛_0𝑖. 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

𝑃(𝑋)=𝛽2‖‖𝐫¹−𝐫₂‖‖²+_𝑖=1,2𝛼_𝑖2𝑁^2𝑖,(75) where we define

𝑁_𝑖=𝑝_𝑖−𝑝_𝑖(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

𝑈(𝑋)=𝛽2‖‖𝐫¹−𝐫₂‖‖²=𝛽2𝑑²𝑠₁=𝑈𝑠₁,𝑠₂+𝑙²𝑠₁,𝑠₂,𝑠₂,(77) where, by denoting the object width by 𝑙_𝑤,

𝑑𝑙𝑠𝑠₁₁,𝑠₂=𝑠₁−𝑠₂−𝐛^T₁𝛾₁+𝐛^T₂𝛾₂,,𝑠₂=−𝑙_𝑤+𝐧^T₁𝛾₁+𝐧^T₂𝛾₂(78) (see Figures 4 and 6). First we show that

𝑑𝑃=𝑆_𝑁𝑑𝜃+_𝑖=1,2Δ𝑁_𝑖𝐞^T_𝑖𝑑𝑞_𝑖,(79) where

Δ𝑆𝑁_𝑁𝑖=𝛽𝑠₁−𝑠₂𝑙+𝑙_𝑤𝑑,(80)𝑁=𝛽𝑁_𝑖+𝛼_𝑖𝑁_𝑖,𝑖=1,2,(81)_𝑖=(−1)^𝑖𝐛^T_𝑖𝛾_𝑖𝑙−𝐧^T_𝑖𝛾_𝑖𝑑,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

dd𝑡𝑃=dd𝑡𝑈𝑠¹,𝑠₂+_𝑖=1,2𝛼_𝑖𝑁_𝑖d𝑁_𝑖=d𝑡_𝑖=1,2(−1)^𝑖𝛽𝐧^T_𝑖𝛾_𝑖𝑑−(−1)^𝑖𝐛^T_𝑖𝛾_𝑖𝑙𝜅_𝑖d𝑝_𝑖+𝛼_𝑖d𝑡𝑁d𝑝^𝑖=d𝑡_𝑖=1,2𝛽𝐧^T𝑖𝛾_𝑖𝑑−(−1)^𝑖𝐛^T_𝑖𝛾_𝑖𝑙𝜃̇−𝑝̇_𝑖+𝛼_𝑖𝑁_𝑖𝑝̇_𝑖=𝛽𝐧_T1+_𝑖=1,2𝛾₁+𝐧^T₂𝛾₂𝑑+𝐛^T₁𝛾₁−𝐛^T₂𝛾₂𝑙𝜃̇𝛽𝑁^𝑖+𝛼_𝑖𝑁_𝑖𝑝̇^𝑖=𝛽𝑙+𝑙_𝑤𝑑+𝑠₁−𝑠₂−𝑑𝑙𝜃̇+_𝑖=1,2Δ𝑁_𝑖𝑝̇_𝑖,(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 𝑆_𝑁=0 and Δ𝑁_𝑖=0(𝑖=1,2). This condition is satisfied when 𝑂₁𝑂₂ is parallel to 𝑃₁𝑃₂ and Δ𝑁_𝑖=0. More explicitly, if we define

Δ𝜆Δ_𝑖𝑓_𝑖=𝑓_𝑖+𝛽𝑙𝑠₁,𝑠₂,=𝜆_𝑖−(−1)^𝑖𝛽𝑑𝑠₁,𝑠₂,𝑖=1,2,(84) then the closed-loop dynamics of (35), (36), and (68) can be rewritten into

𝐼𝑀𝜃̈𝐱+̈−_{𝑖=1,2𝑖=1,2}Δ𝑓^𝑖𝐧_0𝑖+Δ𝜆_𝑖𝐛_0𝑖=0,(85)𝐺𝑞_𝑖̈(−1)^𝑖Δ𝑓_𝑖𝐛^T_0𝑖𝛾_0𝑖−Δ𝜆_𝑖𝐧^T_0𝑖𝛾_0𝑖+𝑆_𝑁=0,(86)+12𝐺̇^𝑖+𝑆_𝑖𝑞^𝑖̇+𝑐_𝑖𝑞_𝑖̇+Δ𝑁_𝑖𝐞_𝑖++ΔΔ𝜆𝑓_𝑖𝑖𝐽^T_𝑖𝐧_0𝑖−(−1)^𝑖𝐛^T_𝑖𝛾_𝑖𝐞_𝑖𝐽^T𝑖𝐛_0𝑖−(−1)^𝑖𝐧^T_𝑖𝛾_𝑖𝐞_𝑖=0,𝑖=1,2.(87) Apparently, steady states are attained when 𝑆_𝑁=0, Δ𝑁_𝑖=0, Δ𝑓_𝑖=0, Δ𝜆_𝑖=0 for 𝑖=1,2.

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

dd𝑡𝑆^𝑁=𝑣₁+𝑣₂d𝜃d𝑡−𝑣¹d𝑝₁d𝑡−𝑣²d𝑝₂dΔ𝑁_𝑖d𝑡,d𝑡=−𝑣^𝑖d𝜃d𝑡+𝛼^𝑖+𝑣_𝑖𝑖d𝑝^𝑖d𝑡−𝑣^𝑖𝑗d𝑝_𝑖d𝑡,(88) for 𝑖≠𝑗, where

𝑣_𝑖=−𝛽𝑙1𝜅_𝑖−𝑙^𝑤𝑙𝐧^T𝑖𝛾_𝑖𝑣_𝑖𝑖,=−𝛽𝑙1𝜅_𝑖+𝐧^T_𝑖𝛾_𝑖−𝐧^T𝑖𝛾_𝑖²+𝐛^T_𝑖𝛾_𝑖²𝑣_𝑖𝑗𝑙,=−𝛽𝐧^T₁𝛾₁𝐧^T₂𝛾₂−𝐛^T₁𝛾₁𝐛^T₂𝛾₂,(89) for 𝑖=1,2 and 𝑗≠𝑖. Note that 𝑙<0 by definition, and all 𝑣₁, 𝑣₂, 𝑣₁₁, and 𝑣₂₂ are positive provided that

1𝜅_𝑖>−𝐧^T_𝑖𝛾_𝑖,𝑖=1,2.(90) Thus, if we denote 𝐳=(𝜃,𝑝₁,𝑝₂)^T, then we obtain

𝜕𝑃𝜕𝐳=𝜕𝜕𝑃𝜃,𝜕𝜕𝑝𝑃₁,𝜕𝑃𝜕𝑝₂^T𝜕=𝑆_𝑁,Δ𝑁₁,Δ𝑁₂^T,(91)₂𝑃𝜕𝐳𝜕𝐳^T=⎡⎢⎢⎢⎢⎢⎢⎣𝜕²𝑃𝜕𝜃²𝜕²𝑃𝜕𝜃𝜕𝑝₁𝜕²𝑃𝜕𝑝₁𝜕𝑝₂𝜕²𝑃𝜕𝑝₁𝜕𝜃𝜕²𝑃𝜕𝑝²¹𝜕²𝑃𝜕𝑝₁𝜕𝑝₂𝜕²𝑃𝜕𝑝₂𝜕𝜃𝜕²𝑃𝜕𝑝₂𝜕𝑝₁𝜕²𝑃𝜕𝑝²²=⎛⎜⎜⎝𝑣₁⎤⎥⎥⎥⎥⎥⎥⎦+𝑣₂−𝑣₁−𝑣₂−𝑣₁𝛼₁+𝑣₁₁−𝑣₁₂−𝑣₂−𝑣₂₁𝛼₂+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 𝛼_𝑖> appropriately so as to satisfy 𝛼_𝑖>−𝛽𝑙/𝜅_𝑖 for 𝑖=1,2 (note that 𝑙<0). Finally, the Hessian of 𝑃 with respect to the position state vector 𝑋 is given by

𝐻_𝑋=𝜕²𝑃𝜕𝑋𝜕𝑋^T=𝜕𝐳^T𝜕𝑋𝜕²𝑃=𝐷_T𝜕𝐳𝜕𝐳^T𝜕𝜕𝑋𝐳^T𝜕²𝑃𝜕𝐳𝜕𝐳^T𝐷=𝐷^T𝐻_𝐳𝐷,(93) where 𝐷 is a constant 3×8-matrix of the form

𝐷=⎛⎜⎜⎝000000100010010010001001⎞⎟⎟⎠,(94) because 𝐳 can be expressed by 𝐳=𝐷𝑋. Thus, the Hessian matrix 𝐻_𝑋 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 𝑆_𝑁=0, Δ𝑁₁=0, and Δ𝑁₂=0 under the situation that 𝑝_𝑖(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

𝐵𝑋^∗,𝑟₀=𝑋𝑋∈𝑀,𝑔_𝑖𝑗,𝑑𝑋^∗,𝑋<𝑟₀,(95) where 𝑑(𝑋^∗,𝑋) signifies the Riemannian distance defined by

𝑑𝑋^∗,𝑋=inf_𝑋(𝜏)^𝑡0𝑔_𝑖𝑗𝑋̇_𝑖(𝜏)𝑋̇_𝑗(𝜏)d𝜏,(96) where the infimum is taken over all of the orbits in {𝑀,𝑔_𝑖𝑗} connecting 𝑋^∗ and 𝑋 from 𝜏=0 to 𝜏=𝑡. We denote the 𝑖𝑗-entry of 𝐺(𝑋) introduced in (32) and (33) by 𝑔_𝑖𝑗. 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 𝑇_𝑋𝑀 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

Φ_T𝑋Φ=𝑄_𝑛1,𝑄_𝑏1,𝑄_𝑛2,𝑄_𝑏2^T,=𝜕Φ^T𝜕𝑋=ℜ^𝑛1ΔΛ=Δ𝑓₁,ℜ_𝑏1,ℜ_𝑛2,ℜ_𝑏2,,Δ𝜆₁,Δ𝑓₂,Δ𝜆₂^T(97) and rewriting (85) to (87) into the following general form similarly to (53):

𝐺(𝑋)𝑋̈+12𝐺̇(𝑋)+𝑆𝑋,𝑋̇𝑋̇+Φ^T𝑋ΔΛ+𝐶𝑋̇+𝜕𝜕𝑋𝑃=0,(98) where 𝐶=diag(0,0,0,𝑐₁𝐼₃,𝑐₂𝐼₂). We bear in mind that 𝜕𝑃/𝜕𝑋=𝐷^T(𝜕𝑃/𝜕𝐳) according to (91) and (94). Then, let us introduce the transformation

𝑋̇=𝑊Φ=^T_𝑋ΦΦ_𝑋T𝑋Φ^T_𝑋^−1/2,ΨΦ𝜼̇̇,Φ_𝑋Φ^−1/2,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 Φ^T_𝑋. That is, 𝑊 is an orthogonal matrix and hence

𝑊^T=Φ^𝑋Φ^T_𝑋^−1/2Φ_𝑋Ψ^T=𝑊⁻¹.(100) Since Φ̇ must be the null vector, substituting (99) into (98) yields

𝐺𝜼̈+12̇𝐺+𝑆𝜼̇+Ψ^T𝐶Ψ𝜼̇+Ψ^T𝐷^T𝜕𝑃𝜕𝑧=0,(101) where 𝐺=Ψ^T𝐺Ψ, and

𝑆=Ψ^T𝑆Ψ−12Ψ̇^T𝐺Ψ+12Ψ^T𝐺Ψ̇.(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

𝐫₁̇−𝐫₂̇𝐧₀₁+𝑝̇₁𝐧^T₁𝛾₁+𝑝̇₂𝐧^T₂𝛾₂+𝜃̇𝐧^T₀₁𝛾₀₁+𝐧^T₀₂𝛾₀₂=0.(103) Since the coefficient of 𝜃̇ is −𝑙_𝑤, (103) reduces to

𝑙^2𝑤𝜃̇²≤𝑐_𝜃1‖‖𝑞₁̇‖‖²+𝑐_𝜃2‖‖𝑞₂̇‖‖²(104) with positive constants 𝑐_𝜃1 and 𝑐_𝜃2. Similarly, we see from (19) and (27) that

𝐫₁̇+𝐫₂̇^T𝐧₀₁−2𝐫^T_𝑚̇𝐧₀₁+𝑝̇₁𝐛^T₁𝛾₁+𝑝̇₂𝐛^T₂𝛾₂−𝜃̇𝐛^T₀₁𝛾₀₁+𝐛^T₀₂𝛾₀₂=0.(105) From (25) and (17) we also have

𝐫₁̇+𝐫₂̇^T𝐧₀₁−2𝑟^T_𝑚̇𝑏₀₁+𝑝̇₁𝐧^T₁𝛾₁−𝑝̇^T₂𝐧^T₂𝛾₂−𝜃̇𝐧^T₀₁𝛾₀₁+𝐧^T₀₂𝛾₀₂=0.(106) Since 𝐫_𝑚̇=(𝑥̇,𝑦)̇ and (𝐧₀₁,𝐛₀₁)∈𝑆𝑂(2), it follows from (104) and (105) that

‖𝐱̇‖²≤𝑐₀₁‖‖𝑞₁̇‖‖²+𝑐₀₂‖‖𝑞₂̇‖‖²+𝑙^2𝑤𝜃̇²(107) with some constants 𝑐₀₁>0 and 𝑐₀₂>0. Inequalities (104) and (107) imply that the damping coefficient matrix 𝐶=Ψ^T𝐶Ψ in (101) is positive definite, and hence the dynamics of (101) are fully dissipated. On the other hand, the inner product of (101) and 𝜼̇ yields

dd𝑡(𝐾(𝑋,𝜼̇)+𝑃)=−𝜼̇^T𝐶𝜼̇,(108) where

𝐾(𝑋,𝜼̇)=12𝜼̇^T𝐺𝜼̇=12𝑋̇^T𝐺(𝑋)𝑋̇=𝐾𝑋,𝑋̇,(109) where 𝐾(𝑋,𝑋̇) is defined by (32). It is also important to note that the coefficient matrix 𝐽^T=Ψ^T𝐷^T 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,Δ𝑝₂^T𝐽⁺𝐺𝜼̇,(110) where 𝐽⁺=𝐽^T(𝐽𝐽^T)⁻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, 𝛼_𝑖>0, 𝑐_𝑖>0(𝑖=1,2) adequately such that

1−𝛼1+𝛼𝐸(𝑋,𝜼̇)≤≤𝐸1+(𝑋𝛼,𝜼̇)+𝛼𝑉d1−𝛼𝐸(𝑋,𝜼̇),d𝑡𝑊(𝑋,𝜼̇,𝛼)≤−𝜎𝑊(𝑋,𝜼̇,𝛼),(111) where 𝑊(𝑋,𝜼̇,𝛼)=𝐸(𝑋,𝜼̇)+𝛼𝑉. The speed of convergence depends on 𝜎 depending on 𝛼, but eventually it can be regulated by the choice of control gains 𝛽, 𝛼_𝑖, 𝑐_𝑖(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.

Appendix

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

𝜕𝐫₁−𝐫₂𝜕𝑠₁=𝐛₀₁+−𝜅₁𝐧^T₁𝛾₁−𝐛^T₁𝛾^1𝐛₀₁+−𝜅₁𝐛^T₁𝛾₁+𝐧^T₁𝛾^1𝐧₀₁=−𝜅₁𝐧^T1𝛾₁𝐛₀₁+𝐛^T₁𝛾₁𝐧₀₁,(A.1) since 𝛾^1=𝜕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 𝑂₁𝑃₁ itself (=Π₁𝛾₁) rotated by angle 𝜋/2 in the frame coordinates. Similarly, we have

𝜕𝐫₁−𝐫₂𝜕𝑠₂=𝜅₂𝐧^T2𝛾₂𝐛₀₂+𝐛^T₂𝛾₂𝐧₀₂.(A.2)