A grid twin over-states every route by a heading-dependent factor: does a finer grid fix it, and can it rank two routes the wrong way round?

research/grid-twin

A grid twin over-states every route by a heading-dependent factor that cell size does not shrink (8-connected: mean 8(√2−1)/π = 1.0548, worst 1.0824 at 22.5°), calibrating the mean leaves −5.2% to +2.6%, and the twin ranks two routes (lengths 1 and 1.03) the wrong way round 19% of the time; rotating the map flips a fixed pair for 31% of orientations.

11 tests

What it found

Real route: a straight line of length L; twin: shortest path on a k-connected lattice. The grid/Euclid ratio depends on heading and connectivity, not cell size: 8-connected it is cosφ+(√2−1)sinφ, mean 8(√2−1)/π=1.0548 and worst 1.0824 at 22.5° (4-connected mean 1.2732, worst 1.4142; 16-connected 1.0144, 1.0275), matched by Dijkstra; refining the cell from 8 m to 0.1 m leaves a 100 m route at 22.5° at 1.0824. Calibrating by the mean leaves −5.2% to +2.6% (8-connected). For two routes of Euclidean length 1 and 1.03 at random headings the twin ranks the longer one shorter with probability 0.194 (0 at ratio ≥ 1.0824), and for a fixed pair 22.5° apart rotating the map flips the ranking for 31% of orientations. Stylised: free space only, no obstacles, length as a proxy for time or energy.

A figure from its own output

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

Paper, code and results

More in Digital twins & sim-to-real

A global-shutter twin of a rolling-shutter camera ignores target motion: how biased is a twin-trained orientation estimate, and what does the same frame reveal?A global-shutter twin of a rolling-shutter camera ignores target motion: a moving rectangle is sheared and stretched by exact closed forms, a twin-trained orientation estimate is biased by half the edge tilt (1.79° at 1000 px/s), and one frame of a known-size target identifies pose and velocity (0.027° and 16 px/s at 0.25 px noise). Stylised.
A snapshot twin of a spinning lidar ignores that the platform moves during the sweep: how biased is the map, what does one scan reveal, and what do yaw and acceleration do?A snapshot-scan twin of a spinning lidar ignores motion during the sweep: length biased by exactly vτ/2 (0.5 m at 10 m/s), parallel walls converge by vτ/2π·(1/d₁+1/d₂), one scan gives speed to 0.13 m/s, yaw scales bearings by −κβ/(1+κ), acceleration biases odometry by a·t_f without accumulating.
A satellite-positioning twin with white range noise certifies an error radius the real receiver does not meet: how wrong is it once multipath is correlated, and does averaging fix it?A white-noise satellite-positioning twin vs a receiver with correlated multipath: exact HDOP √(4/n) and bias map 2b/n, averaging loses the 1/√T gain (95% radius covers 33% at T=10, 5% at T=100), and a residual integrity test tuned on the twin false-alarms 21% at equal multipath. Stylised.
A point-mass twin of a rolling body ignores rotational inertia: how wrong are acceleration, slip and coast distance, and what does one calibration fix?Point-mass twin of a rolling body ignores rotational inertia: acceleration overstated by exactly 1+k, slip threshold k tanα/(1+k) invisible, coast distance 40–100% off, one calibration cannot fix a mixed fleet.
A pinhole twin of a radially distorted camera mis-ranges the near field: how far off does a twin-trained braking policy stop, and does focal calibration fix it?A pinhole twin of a radially distorted camera mis-ranges the floor by exactly 1/(1+k₁h²/R²) (+8.1% at 1 m for k₁=−0.3), a twin-trained braking policy stops 9 cm short of 1 m, focal calibration trades the near-field error for a far-field one, and one grid image identifies k₁.
An instantaneous-exposure twin of a finite-exposure camera certifies detection at every speed: what is the real speed limit, and what does a blur-matched filter buy back?An instantaneous-exposure twin of a finite-exposure camera certifies detection at every speed; a moving bar is an exact trapezoid (peak C·min(1, w/d)), a twin-tuned peak detector has a hard speed limit wC/(σz), and a blur-matched box filter raises it quadratically (5–55×).

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

Other research topics

Proper scoring & elicitation28 Effort & peer prediction9 Verification games & slashing19 Detection & tolerances14 Markets as routers13 Prediction-market design9 Paying contributors8 Decentralised training systems23 Digital twins & sim-to-real164 LLM chip design loops40 Robot manufacturing loops53 Ambient AI assistant46 Self-replicating manufacturing34 Simulation-first design64 Engineered clothing45 Task-specific AI accelerators40 Algorithmic chip-design loops42 Open-source purpose-built robots44 The software-engineering loop43 Adversarial human imitation41 Open-source residential construction41 Autonomous multi-angle capture34 Sensing, imaging and 3D31 Energy supply and dense energy storage39 Robot societies among people31 Agentic co-design of low-cost robots27 Laundry-hanging robots25 Machine learning on qualitative data23 Robots on the building site27 Civilizations & group psychology1 History & 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.