Introduction to Polynomial Invariants of Screw Systems

Abstract

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