Optimal three finger grasps

The authors address the problem of optimal force distribution among three point fingers holding a planar object. A scheme that reduces the nonlinear optimization problem to an easily solved generalized eigenvalue problem is proposed. This scheme generalizes and simplifies results of Z. Ji and B. Roth (1988). The generalizations include all possible geometric arrangements and extensions to three dimensions and to the case of variable coefficients of friction. For the two-dimensional case with constant coefficients of friction, it is proved that except for some special cases, the optimal grasping forces, in the sense of minimizing the dependence on friction, are those for which the angles with the corresponding normals are all equal (in absolute value)