Minimum-Jerk Trajectory

A trajectory that minimizes the integral of squared jerk (the rate of change of acceleration), giving smooth motion close to how a human hand moves.

Jerk is the derivative of acceleration with respect to time; large jerk means abrupt, jarring motion. A minimum-jerk trajectory is the one that, given a fixed start, end, and duration, minimizes the integral of squared jerk. The best-known source is Flash and Hogan's 1985 paper in the Journal of Neuroscience: human point-to-point reaching movements in a plane are approximately straight with a bell-shaped speed profile, matching this model's predictions. In one dimension, with zero velocity and acceleration at both ends, the solution is a quintic polynomial: x(t) = x₀ + (x_f − x₀)(10τ³ − 15τ⁴ + 6τ⁵), where x₀ and x_f are the start and end points and τ = t/T is normalized time over total duration T. So it's exactly the special case of quintic polynomial interpolation where boundary velocity and acceleration are both zero. In robotics it's commonly used to generate point-to-point motion or smooth interpolation, reducing shock to motors and gearboxes. Minimizing the peak jerk instead of its integral leads to the S-curve velocity profile.

ExampleMoving an arm's cup from one spot on a table to another over 2 seconds: interpolating with the formula above, end-effector velocity rises smoothly from 0, peaks at the midpoint, and smoothly falls back to 0, with no jolt at either the start or the stop.

Also called

Minimum Jerk

Related

Jerk · Quintic Polynomial Interpolation · S-Curve Velocity Profile · Trajectory Planning · Minimum-Snap Trajectory / Differential Flatness · Time Parameterization

Sources

Flash & Hogan (1985), The coordination of arm movements: an experimentally confirmed mathematical model, J. Neurosci. 5(7):1688–1703