Chapter 2. Prehension

242 THE PHASES OF PREHENSION
equation) to the torques at the joints (right-hand side). In effect, it is a
statement about the workl4, or virtual work, of the system. The virtual
work on the left-hand side of the equation describes the work put into
the system, and the virtual work on the right side of the equation
describes the work done by the system.
Combining Equations 8 and 9, we get:
Simplifying,
or
F~ J$ = zT+
F~ J = rT
Z= J T ~
where JT is the transpose of J. The Jacobian J can be used to
determine what torques and forces z must be applied at the joints to
cause it to exert a net generalized force (or wrench) F at the end point.
The goal in opposition space planning is to find an opposition
space for the given opposition vector. The opposition vector is a
homogeneous solution to WF=w where w is the weight of the object
and the magnitude of the opposition vector is equal to the width of the
object. For pad opposition, the orientation of the finger pads is a
function of the real finger joint angles (proximal and distal
interphalangeal joints and metacarpophalangeal joint), and not the
virtual finger state variables angle @i and length hi. Therefore, the
Jacobian J is a non-linear mapping between the virtual finger forces
applied at the grasping surface patch and the virtual finger state
variables.
6.3.3 Stable grasp
In Chapter 2, many human hand prehensile functions were
discussed, and in this chapter, the use of those functional features has
been noted. Yet, underlying many prehensile tasks, a fundamental
task constraint remains to ‘not drop the object.’ In a formal sense,
what does it mean to grasp and not drop an object? Two aspects of
14Work is defined as a force acting over a distance.