Velocity Profiles and Time Scaling

Time scaling decides how fast to move along a path. The classic trapezoidal profile speeds up at maximum acceleration, cruises at maximum speed, then slows down. An S-curve profile also limits jerk, rounding the corners of the trapezoid, which is gentler on mechanics and payloads at the cost of a little time.

Profile Distance Max speed Max acceleration Max jerk (S-curve)

The trapezoid

Accelerate at $a$ to $v$, cruise, then decelerate. If the move is too short to reach full speed, the trapezoid becomes a triangle with peak speed $\sqrt{a D}$:

$$t_a = \frac{v}{a} \qquad T = \frac{D}{v} + \frac{v}{a} \quad \text{if } D \ge \frac{v^2}{a}$$

The S-curve replaces each jump in acceleration with a ramp at the jerk limit $j$, so acceleration itself changes smoothly.

What to notice

The trapezoid's acceleration jumps straight from zero to its maximum, so its jerk is a spike, which excites vibration in arms and makes payloads slosh. The S-curve's acceleration rises and falls smoothly, and its jerk stays bounded, but the move takes slightly longer. Shorten the distance and the cruise phase disappears. Industrial motion controllers and robot arms offer both profiles.