# Modeling and Control of 2D Grasping under Rolling Contact

Authors: Suguru Arimoto; Morio Yoshida
Journal: Journal of Robotics
 (2010)

---

## 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.


---

## Body

## 1. Introduction

This paper aims at tackling the control problem for dextrous multifingered hands from computational perspectives (see Figure1), 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 byγ1(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.Figure 2
Definitions of local coordinatesOm-XY for the object and O1-X1Y1 for the fingertip.Figure 3
Rolling contact constraint is expressed by equalizing the velocity of contact pointP1A(=P1) in the direction of the unit tangent vector b0 with that of P1B(=P1) in the same direction of b1. The velocity of P1A in the direction of the normal n1 should be equal to that of P1B in the same direction. It is therefore assumed that b0=b1.In order to gain a physical insight into this rolling constraint, we need to denote firstly the fingertip contour curveγ1(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(1)γ1(s)=(X1(s)Y1(s)),γ01(s)=(X01(s)Y01(s))
and define
(2)b1(s)=γ1′(s)=dds(X1(s)Y1(s)),(3)b01(s)=γ01′(s)=dds(X01(s)Y01(s)),
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(4)b1′(s)=κ1(s)n1(s),b01′(s)=κ01(s)n01(s),
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(5)n1′(s)=-κ1(s)b1(s),n01′(s)=-κ01(s)b01(s),
and this equation together with (4) constitutes the Frenet-Serret formula (see [10]).Next consider the rotation matrix of the object around its mass centerOm expressed in the frame coordinates as follows:(6)Π0=(rX,rY),
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(7)Π0=(rX,rY)=(cosθsinθ-sinθcosθ).
Similarly, define(8)Π1=(rX1,rY1),
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 Π̇0=(dΠ0/dt) and Π̇1 are obtained as follows:(9)Π̇0=Π0Ω0,Ω0=θ̇(01-10),Π̇1=Π1Ω1,Ω1=ṗ1(01-10),
where p1=q11+q12+q13=q1Te1, 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 θ̇. In other words, Ω0b01=θ̇n01 or b01TΩ0=-θ̇n01T and Ω1b1=-ṗ1n1 or b1TΩ1=ṗ1n1T.Next, denote the position vector of pointO1 as a center of the left-hand fingertip by r1(=OO1⃗) and that of the object mass center by rm(=OOm⃗). Then, the first assumption that the two contact points P1A and P1B coincide at the common contact point P1 implies the equality(10)(r1A=)r1+Π1γ1=rm+Π0γ01(=r1B).
For the sake of convenience, we denote the unit tangent vector at P1 expressed in the frame space O-xy by(11)b̅1=Π1b1=Π1∂γ1∂s,b̅01=Π0b01=Π0∂γ01∂s.
Then, the second assumption that at the contact point the two curves share the same tangent implies(12)b̅1=Π1b1=Π0b01=b̅01.
Throughout the paper we use symbol (̇) for expressing the derivative in t in such a way that ṙ1=dr1/dt and Π̇1=dΠ1/dt. Then, differentiation of (10) in t yields(13)ṙ1+Π̇1γ1+Π1γ1′dsdt=ṙm+Π̇0γ01+Π0γ01′dsdt.
Since Π1γ1′=b̅1 and Π0γ01′=b̅01, (13) is reduced to, according to (12),(14)(ṙ1-ṙm)+Π̇1γ1=Π̇0γ01,
which from (9) can be written in the form(15)(ṙ1-ṙm)+Π1Ω1γ1-Π0Ω0γ01=0.
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 b̅01 (or equivalently b̅1), which results in(16)Rb1=(ṙ1-ṙm)Tb̅01+b1TΩ1γ1-b01TΩ0γ01=0.
On account of the skew symmetry of Ω1 and Ω0 as shown in (9) and discussed below (9), (16) is reduced to(17)Rb1=(ṙ1-ṙm)Tb̅01+ṗ1n1Tγ1+θ̇n01Tγ01=0.
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 n̅01 in the following way:(18)Rn1=(ṙ1-ṙm)Tn̅01-n1TΩ1γ1-n01TΩ0γ01=0,
where Π1n1=-Π0n01 is taken into account. This is reduced to(19)Rn1=(ṙ1-ṙm)Tn̅01+ṗ1b1Tγ1-θ̇b01Tγ01=0.
Hence, if we define the column vectors(20)ℜb1=[-b̅01(n01Tγ01)J1T(q1)b̅01+(n1Tγ1)e1],ℜn1=[-n̅01-(b01Tγ01)J1T(q1)n̅01+(b1Tγ1)e1],
and X1=(xT,θ,q1T)T, x=(x,y)T, and J1(q1)=∂r1/∂q1T, then (17) and (19) can be rewritten into the forms(21)Rb1=ℜb1TdX1dt=0,Rn1=ℜn1TdX1dt=0.A similar argument developed above can be applied to the characterization of the rolling contact constraint at the right-hand contact pointP2, 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) thatFigure 4
Definitions of tangent vectorsbi, b0i and normals ni and n0i at contact points Pi for i=1,2.(22)r2+Π2γ2=rm+Π0γ02,(23)b̅2=Π2b2=Π0b02=b̅02,
where Π2=(rX2,rY2), Π̇2=Π2Ω2, and(24)Ω2=ṗ2(01-10),
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 b2TΩ2=-ṗ2n2 and b02TΩ0=θ̇n02. Thus, similarly to the derivation of (16) and (17), we have from differentiation of (22) in t the following:(25)Rb2=(ṙ2-ṙm)Tb̅02+b2TΩ2γ2-b02TΩ0γ02=0,Rb2=(ṙ2-ṙm)Tb̅02-ṗ2n2Tγ2-θ̇n02Tγ02=0.
On account of the equality Π2n2=-Π0n02 as shown in Figures 1 and 4, we have, similarly to (18) and (19), the following:(26)Rn2=(ṙ2-ṙm)Tn̅02-n2TΩ2γ2-n02TΩ0γ02=0,(27)Rn2=(ṙ2-ṙm)Tn̅02-ṗ2b2Tγ2+θ̇b02Tγ02=0.
Hence, if we denote X=(xT,θ,q1T,q2T)T, J2(q2)=∂r2/∂q2, e2=(1,1)T, and(28)ℜb2=[-b̅02-n02Tγ0203J2T(q2)b̅02-(n2Tγ2)e2],ℜn2=[-n̅02b02Tγ0203J2T(q2)n̅02-(b2Tγ2)e2],
then, similarly to (21), (25) and (27) are expressed as follows:(29)Rb2=ℜb2TdXdt=0,Rn2=ℜn2TdXdt=0,
where 03=(0,0,0)T. For convenience, we append ℜb1 and ℜn1 with the two-dimensional zero vector in the following way:(30)ℜ̅b1=[ℜb102],ℜ̅n1=[ℜn102]
and denote ℜ̅b1 and ℜ̅n1 by ℜb1 and ℜn1 renewedly for the sake of convenience. Then, (21) can be written in the form(31)Rb1=ℜb1TdXdt=0,Rn1=ℜn1TdXdt=0.


## 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=q2Te2, e2=(1,1)T, and the Riemannian metric G=(gij) is defined as(32)K(X,Ẋ)=12ẊTG(X)Ẋ=M2(ẋ2+ẏ2)+I2θ̇2+∑i=1,2q̇iTGi(qi)q̇i,(33)G=diag(M,M,I,G1(q1),G2(q2)),
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 Ẋ∈T(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 ℜbi and ℜni for i=1,2. Hence, by introducing four Lagrange's multipliers λi and fi corresponding to ℜbi and ℜni for i=1,2, it is possible to derive the Euler-Lagrange equation of motion in the following way:(34)G(X)Ẍ+(12Ġ+S)Ẋ+∑i=1,2(fiℜni+λiℜbi)=u.
In detail, (34) can be written as follows:(35)Mẍ-f1n̅01-f2n̅02-λ1b̅01-λ2b̅02=0,(36)Iθ̈-f1(b01Tγ01)+f2(b02Tγ02)+λ1(n01Tγ01)-λ2(n02Tγ02)=0,(37)Gi(qi)q̈i+{12Ġi(qi)+Si(qi,q̇i)}q̇i+fi{JiT(qi)n̅0i-(-1)i(biTγi)ei}+λi{JiT(qi)b̅0i-(-1)i(niTγi)ei}=ui,i=1,2,
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, b̅0i=b̅i at contact point Pi for i=1,2. This also implies(38)ddtb̅0i=ddtb̅i,i=1,2,
which reduces to(39)Π̇0b0i+Π0κ0i(si)n0idsidt=Π̇ibi+Πiκi(si)nidsidt,i=1,2,
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 n̅0i=-n̅i for both i=1,2, (39) is again reduced to(40)(κ0i+κi)n̅0idsidt=ΠiΩibi-Π0Ω0b0i,i=1,2,
Thus, the inner product of n̅0i and (40) yields(41)(κ0i+κi)dsidt=(-1)i(θ̇-ṗi),i=1,2,
where we referred to the relation (as discussed below (9)):(42)Ωibi=(-1)iṗini,Ω0b0i=-(-1)iθ̇n0i,i=1,2.
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 Figure1, 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):
(43)Qbi=(ri-rm)Tb̅0i+biTγi-b0iTγ0i=0,i=1,2,(44)Qni=(ri-rm)Tn̅0i-niTγi-n0iTγ0i=0,i=1,2.
(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:
(45)ddtQbi=Rbi,ddtQni=Rni,i=1,2.In fact, the inner product of (10) and b̅01(=b̅1) leads to (43) for i=1 and that of (22) and b̅02(=b̅2) does to (43) for i=2. The inner product of (10) and n̅01(=-n̅1) leads to (44) for i=1 and that of (22) and n̅02 does to (44) for i=2. The first equation of (45) can be derived in the following manner:(46)ddtQbi=(ṙi-ṙm)Tb̅0i+(ri-rm)Tddtb̅0i+(biTγi′-b0iTγ0i′)dsidt+{(bi′)Tγi-(b0i′)Tγ0i}dsidt=(ṙi-ṙm)Tb̅0i+(ri-rm)T{Π̇0b0i+Π0b0i′dsidt}+(1-1)dsidt+(κiniTγi-κ0in0iTγ0i)dsidt=(ṙi-ṙm)Tb̅0i-(-1)iθ̇(ri-rm)Tn̅0i+(ri-rm)Tκ0in̅0idsidt+(κiniTγi-κ0in0iTγ0i)dsidt,i=1,2,
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(47)ddtQbi=(ṙi-ṙm)Tb̅0i+(κiniTγi-κ0in0iTγ0i)dsidt+{-(-1)iθ̇+κ0idsidt}(niTγi+n0iTγ0i)=(ṙi-ṙm)Tb̅0i+(κi+κ0i)niTγidsidt-(-1)iθ̇(niTγi+n0iTγ0i),i=1,2.
Thus, by applying (41) to this equation, we have(48)ddtQbi=(ṙi-ṙm)Tb̅0i+(-1)i(θ̇-ṗi)niTγi-(-1)iθ̇(niTγi+n0iTγ0i)=(ṙi-ṙm)Tb̅0i-(-1)i{ṗiniTγi+θ̇n0iTγ0i}=Rbi=0,i=1,2.
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(49)L(X;s1,s2)=K(X,Ẋ)-∑i=1,2(fiQni+λiQbi),
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 Qni and Qbi are dependent on si for i=1,2, respectively. The variational principle is written in this case in the following form:(50)∫t0t1{δL+u1Tδq1+u2Tδq2}dt=0,
from which the set of equations (35) to (37) follow straightforwardly with control torques ui through finger joints as follows:(51)G(X)Ẍ+(12Ġ(X)+S(X,Ẋ))Ẋ+∑i=1,2(fiℜni+λiℜbi)=u,
where u=(0,0,0,u1T,u2T)T. It should be noted that(52)∂∂XQni=ℜni,∂∂XQbi=ℜbi,i=1,2.
Thus, (51) can be spelled out in the general form(53)G(X)Ẍ+(12Ġ(X)+S(X,Ẋ))Ẋ+∑i=1,2(fi∂∂XQni+λi∂∂XQbi)=B(u1u2),
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(54)ui=-ciq̇i+(-1)iβJiT(qi)(r1-r2),
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, substitutingui for i=1,2 into (53) or directly into (37), we obtain(55)Giq̈i+{12Ġi+Si}q̇i+ciq̇i-(-1)iβJiT(r1-r2)+fi{JiTn̅0i-(-1)i(biTγi)ei}+λi{JiTb̅0i-(-1)i(niTγi)ei}=0,i=1,2.
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 q̇i and (55) for i=1,2 yields, due to (30) and (31),(56)ddt{K(X,Ẋ)+β2∥r1-r2∥2}=-∑i=1,2ci∥q̇i∥2,
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(57)E(X,Ẋ)=K(X,Ẋ)+U(X),
where(58)U(X)=β2∥r1-r2∥2,
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=(xT,θ,q1T,q2T)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 Figure5 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,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.(59)Um=U(X*)=minU(X),
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 vectorr1-r2 can be expressed in terms of length parameters si for i=1,2 as follows:(60)r1-r2=-Π1γ1+Π2γ2+Π0(γ01-γ02),
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(61)r1-r2=(s1-s2)b̅01+(-b1Tγ1+b2Tγ2)b̅01-lwn̅01+(n1Tγ1+n2Tγ2)n̅01.
Hence, by partially differentiating this equation in si, we obtain (the details will be given in the Appendix)(62)∂(r1-r2)∂si=(-1)iκi(si){(niTγi)b̅0i+(biTγi)n̅0i},i=1,2,
where κi(si) denotes the curvature of the fingertip contour for finger i(i=1,2). Thus, by regarding U(X)=U(s1,s2),(63)∂U(s1,s2)∂si=(-1)iβκi(r1-r2)T{(niTγi)b̅0i+(biTγi)n̅0i},i=1,2.
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 potentialU(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(64)biTγi=0,niTγi=ri,i=1,2.
Thus, in this case, (63) can be written in the form(65)∂U(s1,s2)∂si=(-1)iβ(r1-r2)Tb̅0i,i=1,2.
This means that ∂U/∂si=0 arises if and only if (r1-r2) is orthogonal to b̅0i 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 Figure5, 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,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 ui 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 Figure1, 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]):(66)ui=-ciq̇i+(-1)iβJiT(qi)(r1-r2)-αi{pi(t)-pi(0)}ei,i=1,2,
where αi denotes a positive constant, and(67)pi(t)=qiT(t)ei,i=1,2,
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(68)Giq̈i+{12Ġi+Si}q̇i+ciq̇i-(-1)iβJiT(r1-r2)+αi{pi-pi(0)}ei+fi{JiTn̅0i-(-1)i(biTγi)ei}+λi{JiTb̅0i-(-1)i(niTγi)ei}=0,i=1,2.
Regarding the closed-loop system of (68), (35), and (36) with four constraints (43) and (44), we have the energy relation(69)ddt[K(X,Ẋ)+β2∥r1-r2∥2+∑i=1,2αi2{pi-pi(0)}2]=-∑i=1,2ci∥q̇i∥2.
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 curvaturesκi(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.Figure 6
Minimization of the squared norm∥r1-r2∥2 over rolling motions is attained when the straight line P1P2̅ connecting the two contact points becomes parallel to the vector (r1-r2); that is, O1O2̅ becomes parallel to P1P2̅.Problem 2 (stability problem of precision prehension).
To find a necessary and sufficient conditon under which minimization of the potentialU(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 connectingO1 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 OO1̅ and O'O2̅ 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 Figure1 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(70)f1=f2=fd(>0),λ1=-λ2=λd
to satisfy ẍ=0 in (35) since all n̅0i and b̅0i(i=1,2) are invariant in this case. Further, to satisfy θ̈=0 in (36), we shall examine the conditions(71)s1=s2,b1Tγ1=b2Tγ2.
Then, substituting these two equalities into (61) yields(72)r1-r2=-lwn̅01+(n1Tγ1+n2Tγ2)n̅01=-(lw-n1Tγ1-n2Tγ2)n̅01,
which shows that (r1-r2) must be perpendicular to b̅01(=b̅02). It is also reasonable to show that, under the equality conditions of (71), ẍ=0 and θ̈=0 can be expressed as(73)-f1n̅01-f2n̅02-λ1b̅01-λ2b̅02=0,-f1(b01Tγ01)+f2(b02Tγ02)+λ1(n01Tγ01)-λ2(n02Tγ02)=0.
In fact, the first equality of (73) follows from (71) and n̅01=-n̅02 and b̅01=b̅02 and the second equality follows in such a way that(74)-fd{b01T(γ01-γ02)}-λd{n01T(γ01-γ02)}=-λdn01T(γ01-γ02).
Hence, if λd=0, then the second equality of (73) becomes valid. Furthermore, in the case of Problem 2, biTγi≠0 in general and hence (63) does not imply ∂U/∂si=0(i=1,2) though (r1-r2) must be perpendicular to b̅0i. 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 functionU(s1,s2) plus the extra term (the third term in the bracket [] of (69)) with respcet to q1, q2, and θ, we put(75)P(X)=β2∥r1-r2∥2+∑i=1,2αi2N̂i2,
where we define(76)N̂i=pi-pi(0),i=1,2.
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(77)U(X)=β2∥r1-r2∥2=β2{d2(s1,s2)+l2(s1,s2)}=U(s1,s2),
where, by denoting the object width by lw,(78)d(s1,s2)=s1-s2-b1Tγ1+b2Tγ2,l(s1,s2)=-lw+(n1Tγ1+n2Tγ2)
(see Figures 4 and 6). First we show that(79)dP=SNdθ+∑i=1,2ΔNieiTdqi,
where(80)SN=β{(s1-s2)l+lwd},(81)ΔNi=βNi+αiN̂i,i=1,2,(82)Ni=(-1)i(biTγi)l-(niTγi)d,i=1,2,
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(83)ddtP=ddtU(s1,s2)+∑i=1,2αiN̂idN̂idt=∑i=1,2[(-1)iβ{(niTγi)d-(-1)i(biTγi)l}κidpidt+αiNdpidt]=∑i=1,2[β{(niTγi)d-(-1)i(biTγi)l}(θ̇-ṗi)+αiN̂iṗi]=β{(n1Tγ1+n2Tγ2)d+(b1Tγ1-b2Tγ2)l}θ̇+∑i=1,2(βNi+αiN̂i)ṗi=β{(l+lw)d+(s1-s2-d)l}θ̇+∑i=1,2ΔNiṗi,
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 O1O2̅ is parallel to P1P2̅ and ΔNi=0. More explicitly, if we define(84)Δfi=fi+βl(s1,s2),Δλi=λi-(-1)iβd(s1,s2),i=1,2,
then the closed-loop dynamics of (35), (36), and (68) can be rewritten into(85)Mẍ-∑i=1,2(Δfin̅0i+Δλib̅0i)=0,(86)Iθ̈+∑i=1,2(-1)i{Δfi(b0iTγ0i)-Δλi(n0iTγ0i)}+SN=0,(87)Gq̈i+{12Ġi+Si}q̇i+ciq̇i+ΔNiei+Δfi{JiTn̅0i-(-1)i(biTγi)ei}+Δλi{JiTb̅0i-(-1)i(niTγi)ei}=0,i=1,2.
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 ofP(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(88)ddtSN=(v1+v2)dθdt-v1dp1dt-v2dp2dt,dΔNidt=-vidθdt+(αi+vii)dpidt-vijdpidt,
for i≠j, where(89)vi=-βl{1κi-lwlniTγi},vii=-βl{1κi+niTγi-(niTγi)2+(biTγi)2l},vij=-β{(n1Tγ1)(n2Tγ2)-(b1Tγ1)(b2Tγ2)},
for i=1,2 and j≠i. Note that l<0 by definition, and all v1, v2, v11, and v22 are positive provided that(90)1κi>-niTγi,i=1,2.
Thus, if we denote z=(θ,p1,p2)T, then we obtain(91)∂P∂z=(∂P∂θ,∂P∂p1,∂P∂p2)T=(SN,ΔN1,ΔN2)T,(92)∂2P∂z∂zT=[∂2P∂θ2∂2P∂θ∂p1∂2P∂p1∂p2∂2P∂p1∂θ∂2P∂p12∂2P∂p1∂p2∂2P∂p2∂θ∂2P∂p2∂p1∂2P∂p22]=(v1+v2-v1-v2-v1α1+v11-v12-v2-v21α2+v22).
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(93)HX=∂2P∂X∂XT=(∂zT∂X)∂2P∂z∂zT(∂z∂XT)=DT(∂2P∂z∂zT)D=DTHzD,
where D is a constant 3×8-matrix of the form(94)D=(001000000001110000000011),
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 Figure6 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 postureX* of the system belonging to the submanifold EM3 and a Riemannian ball B(X*,r0) that is defined by(95)B(X*,r0)={XX∈{M,g̅ij},d(X*,X)<r0},
where d(X*,X) signifies the Riemannian distance defined by(96)d(X*,X)=infX(τ)∫0t∑g̅ijẊi(τ)Ẋj(τ)dτ,
where the infimum is taken over all of the orbits in {M,g̅ij} connecting X* and X from τ=0 to τ=t. We denote the ij-entry of G(X) introduced in (32) and (33) by g̅ij. 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(97)Φ=(Qn1,Qb1,Qn2,Qb2)T,ΦXT=∂ΦT∂X=(ℜn1,ℜb1,ℜn2,ℜb2),ΔΛ=(Δf1,Δλ1,Δf2,Δλ2)T
and rewriting (85) to (87) into the following general form similarly to (53):(98)G(X)Ẍ+(12Ġ(X)+S(X,Ẋ))Ẋ+ΦXTΔΛ+CẊ+∂∂XP=0,
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(99)Ẋ=(ΦXT(ΦXΦXT)-1/2,Ψ)(Φ̇η̇),W=(ΦXT(ΦXΦ)-1/2,Ψ),
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 ΦXT. That is, W is an orthogonal matrix and hence(100)WT=((ΦXΦXT)-1/2ΦXΨT)=W-1.
Since Φ̇ must be the null vector, substituting (99) into (98) yields(101)G̅η̈+(12G̅̇+S̅)η̇+(ΨTCΨ)η̇+ΨTDT∂P∂z=0,
where G̅=ΨTGΨ, and(102)S̅=ΨTSΨ-12Ψ̇TGΨ+12ΨTGΨ̇.
It should be noted that S̅ 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(103)(ṙ1-ṙ2)n̅01+ṗ1n1Tγ1+ṗ2n2Tγ2+θ̇(n01Tγ01+n02Tγ02)=0.
Since the coefficient of θ̇ is -lw, (103) reduces to(104)lw2θ̇2≤cθ1∥q̇1∥2+cθ2∥q̇2∥2
with positive constants cθ1 and cθ2. Similarly, we see from (19) and (27) that(105)(ṙ1+ṙ2)Tn̅01-2ṙmTn̅01+ṗ1b1Tγ1+ṗ2b2Tγ2-θ̇(b01Tγ01+b02Tγ02)=0.
From (25) and (17) we also have(106)(ṙ1+ṙ2)Tn̅01-2ṙmTb̅01+ṗ1n1Tγ1-ṗ2Tn2Tγ2-θ̇(n01Tγ01+n02Tγ02)=0.
Since ṙm=(ẋ,ẏ) and (n̅01,b̅01)∈SO(2), it follows from (104) and (105) that(107)∥ẋ∥2≤c01∥q̇1∥2+c02∥q̇2∥2+lw2θ̇2
with some constants c01>0 and c02>0. Inequalities (104) and (107) imply that the damping coefficient matrix C̅=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 η̇ yields(108)ddt(K(X,η̇)+P)=-η̇TC̅η̇,
where(109)K(X,η̇)=12η̇TG̅η̇=12ẊTG(X)Ẋ=K(X,Ẋ),
where K(X,Ẋ) 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(110)V=(Δθ,Δp1,Δp2)TJ+G̅η̇,
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(111)1-α1+αE(X,η̇)≤E(X,η̇)+αV≤1+α1-αE(X,η̇),ddtW(X,η̇,α)≤-σW(X,η̇,α),
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, 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.



---

*Source: 926579-2010-04-20.xml*