Trapezoidal Velocity Profile

A speed plan of constant acceleration, constant speed, then constant deceleration, tracing a trapezoid on a velocity-time graph.

The trapezoidal velocity profile is the most basic speed plan for point-to-point motion in motors and robot arms: accelerate at a constant rate a from 0 up to a maximum speed v_max, cruise at that speed, then decelerate at the same rate back to 0 — hence the trapezoid shape on a velocity-versus-time graph. The acceleration phase lasts t_a = v_max / a and covers a distance of v_max²/(2a); if the total distance L is less than v_max²/a, there isn't room to reach maximum speed and the profile degenerates into a triangle, with a peak speed of √(aL). It's simple to compute, and industrial-arm commands like MoveJ and MoveL, along with many motor drivers, use it. Its drawback is that acceleration jumps abruptly at the segment boundaries, making jerk (the rate of change of acceleration) theoretically infinite and prone to causing vibration and shock; when smoother motion is needed, an S-curve profile or a minimum-jerk trajectory is used instead. For synchronized multi-joint motion, the time needed by the slowest joint is usually computed first, and the other joints are then slowed proportionally so all arrive together.

ExampleMoving a linear axis L = 1 m with a speed limit of v_max = 0.5 m/s and acceleration a = 1 m/s²: it accelerates for 0.5 s (covering 0.125 m), cruises for 1.5 s (covering 0.75 m), and decelerates for 0.5 s (covering 0.125 m), for a total of 2.5 s.

Also called

Trapezoidal Acceleration Profile, T-Curve Velocity Profile

Related

S-Curve Velocity Profile · Minimum-Jerk Trajectory · Jerk · MoveJ / MoveL · Time Parameterization · Trajectory Interpolation

Sources

MATLAB Robotics System Toolbox: trapveltraj（Generate trajectories with trapezoidal velocity profiles）