Screw Theory

Treats any rigid-body motion as a rotation about some axis combined with a translation along that same axis, like turning a screw.

Screw theory studies the geometry of rigid-body motion and the forces acting on rigid bodies. In 1763, Mozzi proved that any rigid-body motion can be viewed as a rotation about some axis combined with a translation along that same axis — like driving a screw — and that axis is called the screw axis (this result later became known as Chasles' theorem); Robert Ball systematized the theory in his 1876 book The Theory of Screws. At its core are two 6-dimensional quantities: the twist (angular velocity ω and linear velocity v combined) describes a rigid body's velocity, and the wrench (torque and force combined) describes the forces acting on it. Its benefit is that both revolute and prismatic joints can be written uniformly as a single screw axis, without needing to build a separate coordinate frame on every link the way DH parameters do. Brockett used this framework to introduce the product of exponentials formula in 1983–1984, and Lynch and Park's 2017 textbook Modern Robotics teaches kinematics, Jacobians, and dynamics entirely in this language.

ExampleTurning a screwdriver: for every full turn, the screw advances one thread pitch along its axis — that's a screw motion. A robot's revolute joint is a screw with zero pitch (pure rotation, no translation), and its prismatic joint can be viewed as a screw with infinite pitch (pure translation, no rotation).

Also called

Theory of Screws, Screw Axis

Related

Twist · Wrench · Product of Exponentials Formula · Lie Group · Exponential Map · Adjoint Representation

Sources

Wikipedia: Screw theory
Wikipedia: Product of exponentials formula
Modern Robotics: Mechanics, Planning, and Control (Lynch & Park)