Abstract:
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 Plücker 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.
