动态运动原语（DMP）是一种轨迹控制/规划方法，用于从演示中学习机器人的复杂动作。DMP通过一个2nd order动态系统模型实现，保证了系统在目标位置的稳定性。通过加入非线性的力函数，DMP可以适应任意轨迹，并通过局部加权回归（LWR）学习指定的轨迹。DMP的优势在于其动态系统特性，允许简单地调整目标来实现空间上的轨迹缩放。
Let’s talk about the Dynamic Movement Primitive (DMP) for robots learning from demonstration. In this article, we make an assumption that you readers all have the background of control theory and robotics.
The Basics about DMP
Dynamic movement primitives (DMPs) are a method of trajectory control / planning. It was motivated by the desire to find a way to represent complex motor actions that can be flexibly adjusted without manual parameter tuning or having to worry about instability.
To begin with, we take some time to discuss 2nd order dynamic system as follows:
τ y ˙ = z \tau\dot{y}=z τy˙=z τ z ˙ = α ( β ( g − y ) − z ) \tau\dot{z}=\alpha(\beta(g-y)-z) τz˙=α(β(g−y)−z) For the sake of simplicity, we take τ = 1 \tau = 1 τ=1. Then the equations above reduces to a time-invariant linear system which has been deeply studied in linear control theory. The only equilibrium of the system above is z = 0 , y = g z=0,y=g z=0,y=g. Here we take g g g as the goal position, α , β \alpha,\beta α,β are constant parameters to be chosen such that the system is strictly damping (e.g. α > 4 β > 0 \alpha > 4\beta > 0 α>4β>0). This is why the DMP is always stable at the goal position — the dynamics of the system is always dominated by a stable linear system.
To make the DMP fit any trajectory, it is necessary to add some terms in the equations above without any effects on its stability:
τ y ˙ = z \tau\dot{y}=z τy˙=z τ z ˙ = α ( β ( g − y ) − z ) + f ( x ) \tau\dot{z}=\alpha(\beta(g-y)-z) + f(x) τz˙=α(β(g−y)−z)+f(x) where τ x ˙ = − α x x \tau \dot{x} = -\alpha_{x}x τx˙=−αxx α x > 0 \alpha_x>0 αx>
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