Dynamic Movement Primitives

Encoding a demonstrated trajectory as a ‘spring-damper plus a learnable force term’ dynamical system, reproducible with a different endpoint or speed.

DMPs are a trajectory-representation method proposed by Ijspeert, Nakanishi, and Schaal in 2001–2002, with a systematic summary published in 2013. The core is a second-order ‘spring-damper’ system plus a learnable force term: τ·ż = α(β(g − y) − z) + f(x), τ·ẏ = z. Here y is position, z a scaled velocity, g the goal point, τ controls speed, α and β are fixed gains, and f(x) is a weighted sum of Gaussian basis functions whose weights can be fit directly from a single demonstration by linear regression; x comes from a ‘phase’ system τ·ẋ = −αₓx that decays to 0 over time, guaranteeing the force term eventually vanishes and the system converges stably to g. So changing g gives a new endpoint, and changing τ changes speed, while a mid-motion push causes the system to return to the trajectory on its own. DMPs have long been a standard entry point for beginners in imitation learning. Their limitation is representing only one fixed trajectory, unable to describe multiple ways of doing the same task — a gap that probabilistic movement primitives (ProMP) and similar methods fill by modeling a distribution instead.

ExampleRecord one kinesthetic-teaching demonstration of an arm carrying a cup from point A to point B, and fit DMP weights from it; to move to point C instead, just change g to C's coordinates, and to slow it down, increase τ — the arm generates a new, similarly shaped trajectory without being re-taught.

Also called

DMP, Dynamical Movement Primitives, DMPs

Related

Motion Primitives · Probabilistic Movement Primitives · Imitation Learning · Kinesthetic Teaching · Gaussian Mixture Regression / Task-Parameterized GMM

Sources

Saveriano et al.: Dynamic Movement Primitives in Robotics: A Tutorial Survey (arXiv:2102.03861)