Math & Kinematics

Dual Numbers

Dual Numbers are an extension of the real numbers of the form a + bε, where ε is a nilpotent element satisfying ε² = 0. Evaluating a function on a dual number yields its derivative in the dual component, which is the mechanism behind forward-mode automatic differentiation. In kinematics, dual numbers combine with quaternions to form dual quaternions, and with screw theory they compactly encode coupled rotation and translation.

Why it matters for physical AI

Automatic differentiation built on dual-number arithmetic supplies the exact gradients used in differentiable simulation, calibration, and trajectory optimization, while dual-quaternion kinematics gives singularity-free pose representations for control.