Linearangular

Linearangular is a term used in some discussions of rigid-body kinematics to describe a framework that treats linear and angular components of motion within a unified linear-algebraic setting. It is not a widely standardized term, but it appears in contexts where the state of a rigid body is represented by a six-dimensional vector combining linear and angular parts, such as position or velocity with angular velocity or orientation increments.

Definition and informal formalism: A linearangular transform acts on a 6-vector s = [p; w], where p ∈ R^3 denotes a linear component (such as position or linear velocity) and w ∈ R^3 denotes an angular component (such as angular velocity or rotation increments). The action is s' = L s, with L a 6×6 matrix that may capture coupling between translation and rotation. In practice, the linearangular viewpoint is often aligned with linearizations of rigid-body motion and with representations used in state estimation and control.

Relation to established formalisms: The concept is closely related to the mathematical structures used for rigid-body motions, notably SE(3) and its Lie algebra se(3). Linearangular descriptions are commonly used to approximate or linearize SE(3) dynamics around a nominal pose, yielding a block-structured matrix that couples translational and rotational parts. This aligns with the twist representation of motion and with common practices in robotics and computer graphics.

Applications and limitations: Linearangular ideas appear in robotics control, motion planning, computer graphics, and aerospace simulations, particularly in linearized state estimation and sensor fusion. However, because linearangular is not a standard, widely adopted term, precise definitions can vary. For rigorous work, SE(3)/se(3) and twist representations are typically preferred.