Input Shaping (Residual Vibration Suppression)

Splitting a motion command into several time-offset sub-commands so the vibrations they excite cancel each other out.

Input shaping is a feedforward vibration-suppression method, given its modern form by MIT's Singer and Seering in 1990. Flexible systems such as slender arms, crane cables, and belt-driven shafts keep vibrating at their natural frequency after a motion finishes. The technique convolves the original command with a short train of impulses: the simplest, ZV (zero vibration) shaper, uses two impulses spaced half a vibration period apart, with amplitudes split according to the damping ratio (equal halves when undamped) — the vibration excited by the first impulse is exactly canceled by the second. The cost is that the motion takes about half a period longer, and it's sensitive to frequency-estimation error, which led to more robust but slower variants such as ZVD and EI. It needs only the natural frequency and damping ratio — no extra sensors — and can be layered on top of smooth trajectories such as S-curve velocity profiles.

ExampleThe open-source 3D-printer firmware Klipper uses input shaping to remove ‘ringing’ artifacts on a print's surface: it first measures the frame's resonant frequency, then picks a shaper such as ZV, MZV, or EI; ZV delays the motion by only half a period but is the most sensitive to frequency error.

Also called

Command Shaping, ZV Shaper, Residual Vibration Suppression

Related

Natural Frequency · Damping Ratio · Flexible Joint · Feedforward Control · S-Curve Velocity Profile · Trajectory Planning

Sources

Zaber: Input Shaping for Vibration Reduction（引 Singer & Seering 1990） (Chinese)
Klipper documentation: Resonance Compensation