Screwdetermines

Screwdetermines is a term used in screw theory to describe a class of algebraic invariants derived from sets of screws, which are twists or wrenches represented in Plücker coordinates. The concept provides a compact way to capture linear dependencies and compatibility conditions among multiple screws in a rigid body system.

Construction and interpretation: Given a collection of n screws s_i, each represented by a Plücker coordinate pair (ω_i, v_i) in six dimensions, form a 6×n matrix M whose columns are the coordinates [ω_i; v_i]. The screwdetermines are defined as the maximal minors of M, i.e., the determinants det(M_J) for all 6-element subsets J of {1,...,n}. A nonzero minor indicates a set of screws that is linearly independent in the screw space, while a zero minor signals a dependence among the corresponding screws. These invariants are homogeneous and scale appropriately with the screw coordinates.

Properties: Screwdetermines are invariant under rigid motions up to sign and scale. They reflect the geometric compatibility of the screws with respect to a chosen reference frame. Because the construction uses Plücker coordinates, the invariants respect the projective nature of screw representations. However, for sets with fewer than six screws, the minors may be underdetermined, and the interpretation focuses on relations among larger subcollections.

Applications: In robotics and mechanism analysis, screwdetermines can be used to assess the independence of a set of screw motions or forces, identify overconstrained configurations, and aid in kinematic synthesis. They offer a compact algebraic tool to detect potential singularities or redundancy in a system’s motion-planning or force-closure analysis.

See also: Screw theory, Plücker coordinates, twists, wrensches, determinants.