Math & Kinematics

Trajectory

A trajectory is a time-parameterized path through a robot's state or configuration space, specifying not just the geometric route but when each state is reached, and hence velocities and accelerations along the way. Trajectories are the common currency between planners and controllers: planners output them, tracking controllers follow them, and their smoothness and dynamic feasibility determine whether execution is possible. In learning contexts, a trajectory also denotes one recorded episode of states and actions.

Why it matters for physical AI

Robot learning datasets are collections of trajectories, and the distinction between a path and a timed trajectory matters in practice because learned policies must produce motion that respects real actuator velocity and acceleration limits.