A linear space of admittance control laws that guarantees force-assembly with friction

Force-assembly has been defined as an assembly process for which the use of a single, properly designed, admittance control law will guarantee the proper assembly of a given pair of mating parts. In previous work in workpart-into-fixture insertion, the conditions on a manipulators accommodation control law that ensure proper insertion despite infinitesimal positional error and finite (but bounded) friction have been identified. Through the use of an optimization routine, a control law that satisfies these force-assembly conditions at or below a friction maximum value can be obtained. This single control law, however, is not unique-there exists many other control laws that will satisfy the conditions of force-assembly at the same value of friction. This paper addresses the identification and construction of a linear space of accommodation control law parameters that ensure force-assembly with friction. First, linear sufficient conditions that ensure force-assembly with friction are identified. These linear sufficient conditions are then modified to separate the N²+N dimensional space of accommodation control law parameters into N+1 different N-dimensional subspaces. A means of efficiently generating basis nominal velocity vectors and basis accommodation matrices is presented. A nominal velocity selected using any positive linear combination of the nominal velocity basis vectors and an accommodation matrix selected using any positive linear combination of the accommodation basis matrices will guarantee force-assembly (for any value of friction less than that used in generating the basis matrices). A planar example of the construction of each accommodation control law subspace is presented and illustrated in the geometry of the fixturing task