A rigid-body twin of a cart carrying liquid predicts no slosh: how wrong is that, what does a slosh-safe move cost, and what does a stale frequency calibration do to it?

What it found

Real cart: 80 kg body with a damped spring-mass slosh mode (20 kg liquid, 0.5 Hz at full fill); twin: one rigid mass. The twin gets the centre of mass exactly (RK4 error ~10⁻¹⁴ m) but predicts zero slosh; for a bang-bang rest-to-rest move the real residual is 4μa₀sin²(ωT/4)/ω², matched by exact propagation and RK4. The twin’s minimum-time move (3 m, 4.90 s) leaves 10.7 cm of slosh, 10× a 1 cm tolerance; the fastest move meeting it takes 7.60 s (55% slower) because zero-residual times are whole multiples of two slosh periods. Calibrating ω at a full tank is harmless down to half fill (−5% frequency, 9.5 mm) but at 25% fill the smallest safe null-tuned move takes 28 s vs 8.7 s recalibrated, worse than the tuning-free bound 24.7 s; at fixed acceleration the first unsafe null order is exactly ⌊asin(√ρ)/(π|ε|)⌋+1. Stylised: one linear slosh mode, bang-bang force, no shaper optimisation, no nonlinear slosh.

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

• Can an input shaper make a liquid-carrying cart’s move robust to a stale slosh-frequency calibration, and how fast can it be?Input-shaped moves of a liquid-carrying cart: ZV/ZVD residual sin(πε/2), sin²(πε/2) exactly multiplies the bang-bang residual, so a stale frequency calibration that breaks the null-tuned move (28 s) is harmless to a 5.9 s ZV move; closed-form tolerated error and a fill-band design that spends the tolerance on a shorter shaper.
• What is a digital twin worth for training a controller?What a digital twin is worth for training a controller: exact LQ sim-to-real regret with a stability cliff, a validation blind spot (zero score gap yet mistrained control without probing), and a twin worth exactly σ²/(vδ²) real samples at any accuracy.
• How wide should domain randomization be, and what does it buy?Domain randomization over a twin, exactly: cliff-limited width, quartic price when the twin was right, cliff pushed out ~15%, but uniform-average training is not minimax and the best width tracks the prior.
• What does it cost to keep a digital twin in sync with a drifting plant, and is probing worth it?Keeping a digital twin in sync with a drifting plant: closed-form Kalman tracking variance and control regret (matched to simulation), a frozen twin loses half its edge in ln 2/(2(1−ρ)) steps, probing never pays for one drifting parameter, and a two-parameter closed-loop blind direction gives a mean-field dither law ν* ∝ q^{1/6} that self-exciting certainty-equivalent control mostly makes unnecessary.
• How often, and how long, should a digital twin be re-identified as the plant drifts?Re-identifying a drifting twin by dedicated experiments: closed-form interval T*∝q_d^{−2/3} with downtime, noise and drift regret exactly equal at the optimum, excess ∝q_d^{1/3}, a 4× drift misjudgement costs ≤26%, but the second-order optimum destabilises 0.7–15% of cycles and a safe schedule is 37–200× worse than an always-on Kalman twin.
• What should you pay a digital-twin provider for?What to pay a digital-twin provider for: cost-paid twins differ from Brier-paid ones, certainty-equivalence loss is quartic in spread, payment caps make the elicited twin jump at a closed-form threshold, and Brier mispays probing by 15×.

All 164 Digital twins & sim-to-real projects →

Other research topics

Proper scoring & elicitation28Effort & peer prediction9Verification games & slashing19Detection & tolerances14Markets as routers13Prediction-market design9Paying contributors8Decentralised training systems23Digital twins & sim-to-real164LLM chip design loops40Robot manufacturing loops53Ambient AI assistant46Self-replicating manufacturing34Simulation-first design64Engineered clothing45Task-specific AI accelerators40Algorithmic chip-design loops42Open-source purpose-built robots44The software-engineering loop43Adversarial human imitation41Open-source residential construction41Autonomous multi-angle capture34Sensing, imaging and 3D31Energy supply and dense energy storage39Robot societies among people31Agentic co-design of low-cost robots27Laundry-hanging robots25Machine learning on qualitative data23Robots on the building site27Civilizations & group psychology1History & human sciences1

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.