time parameterization / trajectory smoothing

A planner usually hands back a path: a list of points in space — go here, then here, then here — like dots on a map. But the dots say nothing about timing. They do not tell the robot how fast to travel, when to speed up, or when to ease off. Time parameterization is the step that pins a clock to that bare path: it decides, for every moment in time, exactly where the robot should be, so the silent string of points becomes a moving plan you can actually follow. The result is called a trajectory — a path with time attached.

The catch is that a robot cannot move however fast it likes. Each joint has a top speed it cannot exceed (a velocity limit) and a top rate at which it can change speed (an acceleration limit); push past either and the robot lurches, overshoots, or stalls. So time parameterization is like a careful driver planning a trip on a winding road: floor it on the straight stretches, brake well before the sharp bends, and never demand more grip than the tires can give. It stretches or compresses the timing along the path so the robot stays inside every limit while still finishing as quickly as those limits allow.

Trajectory smoothing is the close partner of this. A raw planned path is often jagged — sharp corners and sudden kinks where one straight segment meets the next. Asking a robot to follow such corners exactly would mean stopping dead and snapping to a new direction, which is jerky, slow, and hard on the hardware. Smoothing rounds those corners and blends the speed changes so motion flows in one continuous sweep, the way a skier carves a gentle curve instead of stamping out a sequence of straight lines. Together, timing and smoothing turn a geometric route into motion a real machine can perform gracefully.