Chapter 2. Prehension

Chapter 6 - During Contact 239
counterbalanced by a wrench in the other system. These intersecting
wrenches are called internal forces. For example, wl from the left
contact and w2 from the right contact in Figure 6.12 are internal
forces, and so are w7 and wg. But W is just a collection of unit
vectors (with magnitude of 1). The magnitude of a wrench is defined
as an intensitv. Increasing the intensities of wl and w2 will not
impart motion to the object, only hold it tighter. Algebraically, this
pair of equal and opposite wrenches correspond to a homogeneous
solution to the problem of finding contact wrench intensities that
satisfy
WF=w (4)
where F is defined to be an n-vector of contact wrench intensities for
the n contact wrenches and w is the external force (from Figure 6.10,
the external force is the weight of the object). In terms of grasping, a
solution to Equation 4 describes the forces and moments (with
magnitudes F and directions W, as transmitted through the modelled
contacts) that are necessary to counteract the weight of the object.
Included in the solution are any internal forces that are applying forces
without imparting motions.
Researchers in robotics have proposed various methods for
finding a solution for Equation 4. Salisbury (1985) defined the grip
transform G, which is built from W by turning it into a square matrix
by adding to it the magnitudes of the internal forces. Yoshikawa and
Nagai (1990) define the internal forces between three contact points as
the unit vectors directed between them, and develop grasp modes for
grasping arbitrarily shaped objects. Li and Sastry (1990) use an
optimization technique so that they determine a configuration based on
a grasp quality measure.
In terms of the model of opposition space presented in Chapter 2,
virtual fingers are applying forces against the object along the contact
normals. The Force Orientation constraint identifies the internal forces
of the virtual fingers, while the magnitudes are constrained by the
Force Magntitude constraint. These internal forces are applied along
the opposition vector of the object, using WF=w. A homogeneous
solution to WF=w in effect identifies an opposition vector between the
two wrench systems with a magnitude equal to the width of the object.
In pad opposition, the pads of the fingers must be in opposition in
order to supply internal forces along the opposition vector; for palm
opposition, the finger surfaces oppose the palm; and in side
opposition, the pad of the thumb opposes the radial hand surfaces.