Chapter 2. Prehension

246 THE PHASES OF PREHENSION
grasping plane, the potential function of the grasp is equal to the sum
of the potentials of all its springs:
where ki is the spring constant, Oi (x,y,8) is the compression at finger
Fi, and n is the number of fingers in the grasp.
Given U, we can analyze if the grasp is in stable equilibrium. In
order to do this, Nguyen first identifies the equilibrium state which,
following Fearing’s first condition (Equations 13 and 14), is where
the sum of all forces and moments in the grasping plane is zero. This
is equivalent to the first partial derivatives of the potential function
U(x,y,$) being all zero, or:
This is equivalent to Equation 4.
Secondly, Nguyen evaluates the potentional function using the Taylor
expansion about the equilibrium state as follows:
where x = (X,Y,~)~.
In order to determine if the grasp is in stable
equilibrium, we look at whether it is stable and also whether it is in
equilibrium. As we saw, in order for the grasp to be in equilibrium,