Tuning a Robot Joint

A robot joint is told to move to a new angle. Too little damping and it overshoots and wobbles; too much and it crawls. At one exact setting the characteristic equation has a repeated root, and the joint arrives fastest without overshooting.

How it works

Repeated roots and critical damping

In one sentence

If the characteristic equation has a repeated root $r$ , the general solution of $ay'' + by' + cy = 0$ is $y = (c_1 + c_2t)e^{rt}$ .

General formula

{\htmlData{sym=0}{\theta}}'' + {\htmlData{sym=2}{K_d}}\,{\htmlData{sym=0}{\theta}}' + {\htmlData{sym=1}{K_p}}\,{\htmlData{sym=0}{\theta}} = 0, \qquad {\htmlData{sym=2}{K_d}} = 2\sqrt{{\htmlData{sym=1}{K_p}}} \;\Rightarrow\; {\htmlData{sym=4}{r}} = -{\htmlData{sym=3}{\omega}} \text{ (repeated)}, \quad {\htmlData{sym=0}{\theta}} = (1 + {\htmlData{sym=3}{\omega}} t)\,e^{-{\htmlData{sym=3}{\omega}} t}

With only one exponential available, the extra factor $t$ supplies the second independent solution; it lets the motion start at rest and still decay at the single rate $\omega$ .

What the symbols mean

$\theta$
Angle error
How far the joint is from its target angle.
$K_p$
Proportional gain
How hard the controller pushes per radian of error.
$K_d$
Derivative gain
How hard it resists the joint's speed.
$\omega$
Natural frequency
√K_p: how fast the joint would oscillate with no damping.
$r$
Repeated root
The single root −ω of the characteristic equation at critical damping.

A PD controller pushes the joint toward its target with torque $K_p\theta$ (proportional) and resists motion with $K_d\theta'$ (derivative).
The characteristic equation $r^2 + K_dr + K_p = 0$ has a repeated root exactly when $K_d = 2\sqrt{K_p}$ .
A repeated root gives only one exponential, so the second solution is $te^{rt}$ ; released at rest, $\theta = (1 + \omega t)e^{-\omega t}$ .
That critically damped response is the fastest return without overshoot.

Learn the math

Repeated RootsDiff Eq · Lesson 16, Second Order Differential Equations Reduction of OrderDiff Eq · Lesson 17, Second Order Differential Equations