Coordinate Transformations and Logical Operations for Minimizing Conservativeness in Coupled Stability Criteria

Often it is desirable to guarantee that a manipulator will remain stable when contacting any member of some set of environments. Coupled stability criteria based on passivity may be used to provide such a guarantee, but may be arbitrarily conservative depending on the environment set. In this paper, two techniques for reducing conservativeness are introduced. The first is based on a canonical coordinate transformation which enables an environment set viewed in the frequency domain to be conformally mapped to the interior of the unit circle. A stability criterion is then derived via the small gain theorem. The second technique uses logical combinations of such criteria to reduce conservativeness further. Both techniques are illustrated with nontrivial examples.