Math & Kinematics

B-spline

A B-spline is a piecewise polynomial curve defined as a weighted combination of control points using basis functions of a chosen degree over a knot vector, providing local control: moving one control point reshapes only a neighborhood of the curve. B-splines of degree k are k-1 times continuously differentiable at simple knots, which makes them a standard representation for smooth robot joint trajectories and Cartesian paths in trajectory optimization.

Why it matters for physical AI

Smooth, locally editable trajectory parameterizations keep accelerations and jerks within actuator limits, and optimizing over a handful of control points is far cheaper than optimizing every waypoint of a dense trajectory.