screwtheoretic

Screw theory, also known as the theory of screws, is a mathematical framework for describing rigid body motions and forces in three-dimensional space. It uses the concept of a screw, a line in space endowed with a pitch, to unify translations and rotations into a single geometric primitive. The line is most commonly represented by Plücker coordinates, consisting of a direction vector and a moment about the origin. A screw can encode instantaneous motion, namely a twist, or a static force system, namely a wrench.

Twists describe instantaneous rigid-body motions: an angular velocity vector ω together with a linear velocity v of a reference point. When ω is zero, the motion is pure translation; when v is related to ω in a particular way, the twist lies along a screw axis. Wrenches describe instantaneous force-couple systems: a force vector F and a torque τ, capturing how forces exert moments about the origin.

The mathematics of screw theory is closely tied to Lie groups, specifically SE(3), and their Lie algebra se(3). The product of exponentials formula expresses a finite rigid-body motion as an ordered exponential of a twist, enabling compact kinematic and dynamic analyses. Dual representations, including dual quaternions, are often used in computation.

Applications are extensive in robotics, where screw theory underpins forward kinematics, kinematic chain analysis, and constraint handling; in statics for mechanism design; in computer graphics and vision for motion estimation; and in biomechanics for modeling joint motion. Advantages include a distance- and rotation-agnostic language for combining motions and forces, though practical work requires careful numerical handling of Plücker coordinates and unit conventions.