path vs trajectory

In everyday talk "path" and "trajectory" sound like the same thing, but in robotics they are two distinct steps, and keeping them apart matters. A path is purely the shape of the route — the geometric line of poses the robot will pass through, from start to goal — with nothing said about timing. It is like a trail drawn on a map: it shows where to go but not how fast or when. A trajectory takes that same route and pins a clock to it: it says exactly where the robot should be at every instant, and therefore implies a speed and an acceleration at each moment. A path answers "which way?"; a trajectory answers "which way, and how fast at every step?"

Planning usually happens in that order for a good reason. First a planner finds a path — a clear, collision-free route through space — worrying only about geometry. Then a separate step, called time parameterization or trajectory generation, dresses that path with timing that the robot's motors can actually deliver: speeding up smoothly, slowing into corners, never demanding more force or velocity than the hardware allows. The same path can be turned into a slow, gentle trajectory or a fast, aggressive one. Splitting the work this way is powerful: the hard geometric question of avoiding obstacles is solved once, and the question of moving smoothly and within the robot's limits is handled afterward, on top of it.