Chapter 2. Prehension

240 THE PHASES OF PREHENSION
6.3.2 Fingertip force and joint torques
We have just considered the issue of transmitting forces through
the fingertips in order to counterbalance the weight of the object. The
vector F identifies the required intensities of the wrenches at the
fingertips. The second issue to consider is how those forces are
generated in an intrinsic body space, such as muscle lengths. In
robotics, a similar but somewhat simpler problem is to determine the
forces and torques at the joints that wil generate these desired fingertip
contact forces. Multi-jointed fingers in a hand apply forces at the
fingertips that are a function of the joint angles and the forces and
moments at the joints. What is needed is a relationship between the
contact forces and the joint torques.
Starting with one finger with three joints, a coordinate frame is
placed at the fingertip, as seen in Figure 6.13. The infinitesimal joint
angles of the jth finger, &j=[&jl,&j2,&j31' *, can be transformed to a
fingertip position (x,y,z) and onentation ( Q~, qZ) as follows:
where J is the Jacobian matrix that maps from joint space (the joint
angles) to Cartesian space (end point position and orientation). Let tj
be the twist system at the jth finger. Then, for the jth finger, Equation
5 can be written as:
t. J = J.4. J J (6)
For a five fingered hand, equation (6) expands into the following: