Introduction to Polynomial Invariants of Screw Systems

Screw systems describe the infinitesimal motion of multi–degree-of-freedom rigid bodies, such as end-effectors of robot manipulators. While there exists an exhaustive classification of screw systems, it is based largely on geometrical considerations rather than algebraic ones. Knowledge of the polynomial invariants of the adjoint action of the Euclidean group induced on the Grassmannians of screw systems would provide new insight to the classification, along with a reliable identification procedure. However many standard results of invariant theory break down because the Euclidean group is not reductive. We describe three possible approaches to a full listing of polynomial invariants for 2–screw systems. Two use the fact that in its adjoint action, the compact subgroup SO(3) acts as a direct sum of two copies of its standard action on R3. The Molien–Weyl Theorem then provides information on the primary and secondary invariants for this action and specific invariants are calculated by analyzing the decomposition of the alternating 2–tensors. The resulting polynomials can be filtered to find those that are SE(3) invariants and invariants for screw systems are determined by considering the impact of the Pl¨ucker relations. A related approach is to calculate directly the decomposition of the symmetric products of alternating tensors. Finally, these approaches are compared with the listing of invariants by Selig based on the existence of two invariant quadratic forms for the adjoint action.