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This section introduces one more graphical method for representing cones of wrenches: force dual.

Recall that Reuleaux’s method represents a cone of twists by central projection to the oriented plane. Moment-labeling represents a cone of wrenches by central projection of the supplementary cone to the oriented plane. Why don’t we consider the seemingly more straightforward method, and project the cone of wrenches directly to the oriented plane without taking the supplementary cone? That is the force dual method.

Force-dual represents a polyhedral convex cone in planar wrench space by central projection to the oriented plane.

Rigid Body Statics 113 l O d 1 d l Figure 5.15

Constructing the dual of a line to obtain a point.

P

O P l

l

Figure 5.16

Applying the dual transform to a point to obtain a line.

We can develop the force dual method by defining a transformation from a planar wrench to the oriented plane:

  ffxy nz   → − fy/nz fx/nz  (5.42)

where the sign of the point is just the sign of the moment nz. If the moment is zero, we

have an ideal point. Thus, as with instantaneous centers, we have a transformation that is just a coordinate rotation away from the projection of figure 5.9.

This transformation has a simple geometrical rendering. Given a force acting along some line, construct the perpendicular to the line, through the origin. The point lies on the perpendicular, on the opposite side of the origin. Its distance from the origin is equal to the inverse distance of the original force to the origin. The third component, the sign, is just the sign of the moment.

Figure 5.17

Two examples of the force dual method. Two contacts map to a line segment in the oriented plane; three contacts map to a triangle in the oriented plane.

The mapping, as defined, is from directed lines to signed points—the image of a particular wrench is independent of the magnitude, depending only on the (directed) line of force. But we can extend the mapping, so that it also maps signed points back to directed lines. We will define the image of a point P to be the locus of images of all the lines through P, (figure 5.16). Suppose{l} is the set of directed lines through P. Then Pis defined to be {l_{}, with a direction determined by the sign of P. A simple geometric construction suffices} to show:

• P_{is a directed line} • P_{= P}

Hence the transformation is dual, which is how the the name force dual arose.

For example, let’s apply the force dual method to the problems of figure 5.12. To apply the force dual method:

1. Choose the origin and unit length wisely.

2. For each directed line of actionwi construct its dual in the oriented planewi.

3. Take the convex hull conv({w_i}).

The result is a convex figure in the oriented plane, which represents the positive linear span of the givenwi. Each point in the figure is the dual of a possible resultant.

A “wise” choice of the origin and unit length means that you should anticipate where the dual constructs will go, and keep them on the page. Graphical methods are awkward when all the action is at infinity. So don’t put the origin right on top of the contact normals. At the same time, it is convenient if the dual constructs are not right on top of the original

Rigid Body Statics 115

Figure 5.18

The force dual transform maps every contact normal to a point in the oriented plane. The resulting curve is called the “zigzag locus”.

figure. The placement of origins in figure 5.17 may seem counterintuitive at first, but note that the main features of the dual figures are on the page and not on top of the original figures.

The first example of figure 5.17 has two anti-parallel contact normals. By placing the origin between them, the moments of the normals have the same sign. The convex hull is a simple line segment also labeled. Compare this line segment with the shaded region in figure 5.13 (moment labeling) or 2.19(a) (Reuleaux’s method). Each vertex in one figure is dual to an edge in the other. The figures are projections of supplementary cones, but are different ways of representing the same set.

The second example shows three contact normals of a triangle, mapping to three points in the oriented plane. Their convex hull gives a triangle in the oriented plane, which overlaps the line at infinity. Again comparison with moment labeling (figure 5.13) and Reuleaux’s method (figure 2.19(b)) is instructive.

Since the force dual method is a little more involved than the moment-labeling method, you might wonder why we need it. The beauty of the dual mapping is that it represents each (directed) line of action as a (signed) point. So that a set of lines of action is represented by a region. Of course, the moment labeling method also represents a set by a region, but not in the usual sense that a region refers to a set of points. Because of this, the force dual method can represent an arbitrary set of lines of action, not just convex cones. For example, suppose we want to represent the set of frictionless forces that could be applied to the perimeter of an object (figure 5.18). This is easily described by a piecewise-linear

Six points

Four points Five points

Figure 5.19

Irreducible arrangements of oriented points yielding closure.

closed curve in force dual space, called the zigzag locus. But it is not convex, because we did not ask for the possible resultants of all such forces, and therefore did not take the convex hull in the oriented plane. As an exercise, the reader might consider the problems of representing this by moment-labeling.

The force dual method is ideal for problems in force closure and first-order form closure. We have already seen some examples, and there are more examples in the next section and in the exercises. The force dual method is also useful when we include friction, as we shall see in the next chapter.

The force dual method will also be useful for dynamic problems. In fact, as we shall see in chapter 8, the force dual transformation arises naturally from Newton’s laws.

One thing that the force dual method is not good for is visualization. It is not imme- diately evident what wrenches are included in a polyhedral convex cone represented using force dual. When necessary the dual transform can be used to obtain the equivalent mo- ment labeling representation. When a computer is doing the work, Brost (1991a) found that central projection onto the sphere is the most effective way of visualizing polyhedral convex cones in planar wrench space or differential twist space.

Arrangements yielding closure in the oriented plane

Force closure and first order form closure are represented in the oriented plane by a set of points whose convex hull exhausts the entire oriented plane. We have seen some examples of closure, but we might ask the more general question: what are the different arrangements of points in the oriented plane, such that the convex hull is the entire oriented plane? Can we systematically identify every topologically distinct arrangement that cannot be reduced

Rigid Body Statics 117

to a smaller set? Figure 5.19 shows all the irreducible arrangements I could identify. Any arrangement of points yielding closure should contain one of the arrangements shown in the figure.

5.9 Summary

The graphical techniques presented above can be summarized in a table: Project cone to Project supplementary oriented plane cone to oriented plane single wrench (acc’n center) line of action

single diff’l twist IC ?

wrench cone force dual moment labeling diff’l twist cone Reuleaux ?

The table notes that the force dual method, applied to a single wrench, produces the acceleration center which is described in section 8.9. Also, we see room in the table for one more method labeled “?”, which might be called “velocity dual” or “differential twist dual”.

5.10 Bibliographic notes

The material on systems of forces acting on rigid bodies, and the related equivalence theorems, was adapted from (Symon, 1971). Fundamental results on wrenches came from (Ohwovoriole, 1980), (Roth, 1984), and (Hunt, 1978). Polyhedral convex cones are developed in(Goldman and Tucker, 1956). The oriented plane is described by Stolfi (1988). Also see the earlier paper with Guibas and Ramshaw (1983).

Modelling kinematic constraint and systems of contact forces by linear inequalities in wrench space has developed through a number of papers addressing grasp planning, work-holding fixture design, and robotic assembly (Erdmann, 1984; Asada and By, 1985; Kerr and Roth, 1986; Rajan et al., 1987; Mishra et al., 1987; Brost, 1991b; Hirai and Asada, 1993). Erdmann (1984) employed cones in wrench space, and Nguyen (1988) and Hirai and Asada (1993) extended and further developed the use of cones. The force dual and moment labeling methods were described by Brost and myself (1989; 1991). Readers wishing to push deeper into the kinematics and statics of contact should also look into the formulation as a linear complementarity problem. (Pang and Trinkle, 1996) is a good place to start.

(a)

(c)

(b)

(d)

Figure 5.20

Convex hull problems for exercise 5.2.

Exercises

Exercise 5.1: Figure 5.6 shows all the different types of cones in E2, and the correspond-