Does a twin with no gravity certify PD gains that sag, or cannot hold a pose at all?

research/gravity-twin

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.

5 tests

What it found

A pendulum under PD control u=k_p(θ*−θ)−k_dω; the twin omits gravity and promises zero error at every gain. The real hold sags to the root of k_p(θ*−θ)=g sinθ nearest the target (RK4 agrees to six digits; 0.182 rad at k_p=4, θ*=1), falling only as ≈1/k_p. Holding upright is stable iff k_p>g: below it the arm settles 0.2–1.9 rad away, and just above it the basin is tiny (0.116 rad at 1.02g, 0.519 at 1.05g, none below 3 rad from 1.1g; numerical scan). A gravity feedforward with calibration error leaves a sag linear in (g(g−g_ff)minus;g_ff) (±0.0185 rad at ±10%). Companion to the lag and flex twins. Stylised: one rigid joint, noiseless, no friction.

A figure from its own output

Figure produced by research/gravity-twin: sag_vs_kp — gravity-twin/experiments/sag_vs_kp.svg, written by the project's own experiment code.

Paper, code and results

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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.