How high a bandwidth does a rigid-body twin let you push when the real joint is flexible?

What it found

A twin with one rigid inertia says the PD loop is stable at every bandwidth. With load-side sensing the real two-mass loop is unstable above ε_c = w/w_s, the smallest root of a cubic in modal damping δ and twin damping ratio ζ: exactly independent of the mass ratio, ≈ δ/ζ for small δ (0.0143, 0.0714, 0.286 for δ = 0.01, 0.05, 0.2 at ζ = 0.7), and unstable at every gain with no modal damping. Motor-side (collocated) sensing was stable at all 7 320 grid points, so the twin is right about one sensor placement and wrong about the other. Twin and real step responses differ by under 1% at 0.9 ε_c, so ordinary validation gives no warning; a twin that overstates δ by a factor f certifies ε_c high by the same factor, and a design at fraction m of the certified bandwidth tolerates f ≈ 1/m. Stylised: one flexible mode, linear, PD only, continuous-time, noiseless.

A figure from its own output

Paper, code and results

• README and method
• White paper
• Experiments and results
• Code
• Tests

More in Digital twins & sim-to-real

• Does a twin that models each axis alone certify gains that actuator coupling destabilises?A per-axis twin of a two-axis loop certifies gains that actuator coupling destabilises: stable-gain limit 2/(1+√p) not 2 (a 10% margin fails at c=1/9), twin-deadbeat rate is √|p| not 0, and a fitted diagonal gain 1+cρ depends on log correlation while the full fit's variance grows as 1/(1−ρ²).
• A twin with an unlimited actuator says a PI step response ignores step size: what does the real loop with a command limit do?A twin with an unlimited actuator is exact for a PI step only while kp·x0 ≤ U; beyond it the integrator winds up to x0²/(2U), overshoot rises from 21% to 97% and settling from 4.9 to 944, loads d ≥ U are never rejected, and the twin-tuned faster gains settle slower (clamping recovers most).
• A fixed-step twin of a bouncing ball: how wrong are its restitution and time to rest, and does calibrating on the first bounce fix it?A fixed-step twin of a bouncing ball: the true rest time (v1/g)(1+e)/(1−e) is hypersensitive to e (2/(1−e²)), the twin's apparent restitution and settling time are off by O(dt) with scheme-dependent sign (−20% to +4% at dt=10⁻², e=0.9), and calibrating e on the first bounce is not a reliable fix.
• Does a twin fitted to the sensor-delay histogram certify gains that bursty delay makes unstable?A twin fitted to the sensor-delay histogram is blind to its persistence: i.i.d. jitter at the mean delay barely moves the mean-square gain limit (1.00 for {0,2}), but with the marginal fixed, lag-1 autocorrelation 0.9 cuts it to 0.67 (0.9 certified by the twin is unstable from ρ=0.435) and inflates variance 2.4×; almost-sure stability (≈1.84) hides it.
• Does a twin with no gravity certify PD gains that sag, or cannot hold a pose at all?A gravity-free twin certifies PD with zero error; the real pendulum sags to the root of kp(θ*−θ)=g sin θ (0.18 rad at kp=4, falling only as ~1/kp), cannot hold upright unless kp>g (basin 0.12 rad at 1.02g), and a 10%-off gravity feedforward leaves a sag linear in the error.
• A continuous-time twin certifies PD at every gain: which gains survive a zero-order-hold sampled loop?A continuous-time twin certifies PD at every gain; the real zero-order-hold loop is stable iff kp·T/2 < kd < 2/T (ωn·T < min(4ζ, 1/ζ)), overshoot is 5× the twin's at ωn·T = 1.2, raising ωn past ωn·T ≈ 1 slows settling, and deadbeat (2 samples) is the real speed limit.

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Produced with AI assistance (Claude). Results are from the code in this repository and are simulated or stylised unless the abstract says otherwise; nothing here implies affiliation with or endorsement by any lab, researcher or company cited. MIT licensed.