mls94-complete[1]

F

O

Figure 5.23: Spherical finger rolling on a plane. The finger is only allowed to roll on the object and not slip or twist.

(u, v, 0). The sphere requires multiple coordinate charts to describe the entire surface, so we shall restrict ourselves to the chart

ρ cos u cos v

cf (u, v) = −ρ cos u sin v , ρ sin u

where ρ is the radius of the sphere and −π/2 < u < π/2, −π < v < π. The curvature, torsion, and metric tensors are:

The equations governing the evolution of the contact point are

6.3Grasp kinematics with rolling

We are now in a position to describe the grasp kinematics when the fingers are allowed to roll and possibly slide on the object (depending on the contact model). Figure 5.24 shows the coordinate frames which we shall use. We assume that the finger and object shapes are completely known and that local, orthogonal, surface parameterizations are available.

253

Figure 5.24: Grasp coordinate frames for moving contacts.

Abstractly, the derivation of the grasp kinematics is identical to the fixed contact case except that the grasp map and hand Jacobian depend on the instantaneous contact location, which in turn depends on ηi, the contact coordinates for each finger. Thus, the contact kinematics have the form

where η = (η1, . . . , ηk) R5k is the vector of contact coordinates and Ai(ηi) R5×6 encodes the contact kinematics for each finger, as given in equation (5.28). One additional fact is needed to extend the previous analysis to the moving contact case: BcTi Vocb i = 0. This condition is a statement that the contact point is not allowed to move in any of the constrained directions. A complete derivation is left as an exercise.

Equation (5.40) describes the grasp kinematics in terms of a set of ordinary di erential equations. To analyze the properties of the grasp, we make use of the following result. Let W SE(3) be the set of all configurations gof for which the two objects are in contact.

Proposition 5.12. Smooth dependence of η on gof

There is a smooth local bijection between η R5 and gof W SE(3) if the matrix

Kf + Rψ KoRψT

is full rank.

The proof of this proposition requires application of the implicit function theorem to the mapping η 7→gof W and can be found in [77].

254

Since ηi is a smooth function of gof = gpo−1gpfi and gpfi is a smooth function of θ, we can write this constraint as

where xo := gpo. Equation (5.41) is a direct extension of equation (5.15) to the moving contact case. All of the definitions and properties which held for fixed contact grasps also hold for moving contact grasps.

For most applications, finding η as a function of gof is not computationally feasible and we must use η directly. To make this explicit, let ηi = (αfi , αoi , ψi) represent the contact coordinates for the ith contact. In order to calculate Jh and G, we must compute the transformations

the only new quantity which really needs to be calculated is Ad−goc1i . Without loss of generality, we assume that the contact frame is aligned

with the normalized Gauss frame for the object and hence

where noi is the outward pointing surface normal. Furthermore, poci is

Finally, although in principle the contact coordinates can be obtained by integrating the contact kinematics, in practice it is much more likely that one would need to measure each ηi directly in order to maintain accuracy (see, for example, [7]). These and other implementation details do not alter the overall structure and properties of the grasp, but can have a dramatic e ect on the performance of the system.

255

7Summary

The following are the key concepts covered in this chapter:

1.A contact is described by a mapping between forces exerted by a finger at a point on the object and the resultant wrenches in some

object reference frame. The contact basis Bci : Rmi → Rp describes the set of wrenches that can be exerted by the finger, written in the contact coordinate frame. For contacts with friction, the friction

cone F Cci Rmi models the range of allowable contact forces that can be applied. The friction cone satisfies the following properties:

(a) F Cci is a closed subset of Rmi with non-empty interior.

(b) f1, f2 F Cci = αf1 + βf2 F Cci for α, β > 0.

2.A grasp is a collection of fingers which exert forces on an object.

The net object wrench is determined from the individual contact forces by the relationship Fo = Gfc, where G Rp×m is the grasp

map:

AdT−1 : Rp → Rp is a the wrench transformation between the

goci

object and contact coordinate frames. The contact forces must all lie within the friction cone F C = F Cc1 × · · · × F Cck .

3.A grasp is force-closure when finger forces lying in the friction cone span the space of object wrenches

G(F C) = Rp.

A grasp is force-closure if and only if the grasp map is surjective and there exists an internal force fN which satisfies GfN = 0 and fN int(F C).

4.The fundamental grasp constraint describes the relationship between finger velocity and object velocity:

where θ Rn is the vector of finger joint angles and xo := gpo is the configuration of the object frame relative to the palm frame. The hand Jacobian Jh Rm×n is defined as

256

the twist transformation between the base and contact frames. For contacts in which rolling does not occur, G is a constant matrix.

5.The relationships between the forces and velocities in a multifingered grasp are summarized in the following diagram:

6.A grasp is manipulable when arbitrary motions can be generated by the fingers

R(GT ) R(Jh).

A force-closure grasp is manipulable if and only if Jh is surjective.

7.The contact kinematics describe how the contact points move along the surface of the fingers and object. For an individual rolling contact, the contact kinematics are

˙

ψ = Tf Mf α˙ f + ToMoα˙ o.

where (Mi, Ki, Ti) are the geometric parameters for a given coordinate chart on the surface. The contact kinematics allow G and Jh to be computed using η = (αf , αo, ψ) rather than solving for η in terms of gpo.

8Bibliography

This chapter grew out of notes written for a workshop organized by Roger Brockett for the American Mathematical Society Annual Meeting in Liousville, Kentucky in January of 1989 [77]. Early treatments of the kinematics and statics of grasping can be found in the dissertations of Salisbury [101] and Kerr [48], on which much of Section 2 is based. See also the

257

work of Nakamura [80]. The material on force-closure and grasp planning is drawn from a number of sources, most notably Nguyen [82] and the seminal paper by Mishra, Schwartz, and Sharir [71]. Finger kinematics and issues of manipulability were described in Li et al. [62].

The kinematics of rolling contact were originally derived by Montana [72] and have been extended to the case of compliant contacts [73]. An alternate derivation of the equations of rolling contact can be found in [77].

258

9Exercises

1.Construct the grasp map for the grasps shown below and determine if the grasp is force-closure. Assume that all contacts are frictionless point contacts.

2.Construct the grasp map for the grasps shown below and write the friction cone conditions with respect to the contact basis you choose. Determine if the grasp is force-closure. Assume that all contacts are point contacts with friction.

3.Construct the grasp map for the grasps shown below and write the friction cone conditions with respect to the contact basis you choose. Determine if the grasp is force-closure. Assume that all contacts are soft-finger contacts.

259

Derive the wrench basis and friction cone for the contact models shown below. Assume the coe cient of sliding friction is µ > 0.

5.Verify the lower bounds given in Table 5.3 by constructing the following objects. Assume that vertex contacts are not allowed.

(a)Construct a 2-D object which cannot be grasped by two point- contact-with-friction fingers and µ = tan 30◦. Prove explicitly that the object cannot be grasped.

(b)Construct a 3-D object which cannot be grasped by three point-contact-with-friction fingers and µ = tan 15◦. Prove explicitly that the object cannot be grasped.

(c)True or false: For any convex 3-D object, a force-closure grasp can be constructed using at most three soft-finger contacts at smooth points (i.e., no vertex contacts). Prove or construct a counterexample.

(d)(contributed by J. Canny). Prove that for any 3-D object with smooth boundary (not just piecewise smooth), a force-closure grasp can be constructed using at most three point contacts with friction or two soft-finger contacts.

6.When controlling a multifingered robot hand, it is important to insure that desired contact forces lie strictly inside the friction cone

to avoid slip. That is, we require that for any Fo, there exists fc

Show that if G(F C) = Rp, then G(int(F C)) = G(F C) using the following steps:

260

Hence, a grasp is prehensile if and only if the grasp is force-closure.

7.Prove Theorem 5.6: Show that a planar grasp with two point contacts with friction is force-closure if and only if the line connecting the contact point lies inside both friction cones.

8.For the objects given below, find all force-closure grasps using two contacts with friction. On each the objects, draw the independent contact regions corresponding to a maximal square contained in a

force-closure region. Assume that the coe cient of friction for all contacts is µ = tan 45◦.

9.Find all the force-closure grasps for two point contacts with friction grasping a circle. Express your answer as a sketch of the forceclosure regions versus the two finger locations on the circle. Indicate on the circle a set of independent contact regions corresponding to a maximal square contained in the force-closure region.

10.Derive the contact constraints for the hands shown below. Determine if the grasps are force-closure and/or manipulable at the configuration shown.

11.Calculate the contact constraints for the grasp shown in Figure 5.17 with l3 > 0.

261

12.Consider the two-fingered grasp shown in Figure 5.17. Equate the locations of the fingertips with the contact locations on the box. Di erentiate this algebraic constraint and show that it is equivalent to the answer given in the example at the end of Section 5.

13.Give an example of two surfaces in contact which has singular relative curvature form.

14.Calculate the geometric parameters for an ellipsoid, a parabaloid, and a torus.

15.The figure below shows an elliptic fingertip in contact with a flat object. The principal axes of the ellipse have length a, b, and c, respectively.

(a)Give an orthonormal coordinate chart for the fingertip around the point of contact as shown in the figure.

(b)Assume the fingertip is in rolling contact with the object. Derive the equations of contact.

(c)Compute the velocity of the fingertip relative to the object which satisfies rolling constraint and produces a contact velocity of α˙ o = (0, v), v R.

16.Derive the equations of contact for a unit sphere in rolling contact with a sphere of radius ρ.

17.Kinematics of planar contact

The kinematics of contact for two planar objects can be obtained by

restricting equation (5.28) to the plane. Let gof = (p, R) SE(2) be the relative configuration of the objects.

(a) Let so R be the point of contact on the object and sf R be the point of contact on the finger. Assume that the surface is

parameterized by arc length, so that k∂ci k = 1 where ci : R →

∂si

R2 is a coordinate chart. Show that the contact coordinates

262