How Much Wind Does a Wind-Only Strandbeest Need? Evidence Tiers, Two Solvers, and Negative Results

The single most important sentence: Every number in this report comes from our simulators (MuJoCo, plus an independent planar rigid-body solver K2); not a single one comes from a physical walker. The E3 tier (traceable physical measurement) is currently empty. Every claim below carries an explicit E0–E3 evidence tier.

This is Part 5 of the Strandbeest series. After covering Theo Jansen’s twelve evolutionary periods, planar leg geometry, re-evolving the linkage with a genetic algorithm, and the open design-to-print platform (meiluosi/strandbeest-evolution, live demo), this report addresses our flagship research question:

For a walker driven by wind alone — sail for energy, gravity for help, no electricity and no pre-tensioned springs — how much wind does it need to start, and below what size does it stop working?

1. Why the Answer Is Not Obvious#

Wind force on a sail scales with area ( $L^2$ ), while weight scales with volume ( $L^3$ for solid bodies, or between $L$ and $L^3$ for thin-walled tubes), and pin friction depends on normal load or clearance fits. The sign of the scaling exponent decides everything: a larger machine can be either easier or harder for the wind to start. Intuition cannot settle this; only simulation and measurement can — and any simulation rests on physical parameters nobody has measured yet.

To keep ourselves honest, every numerical claim is graded on a four-tier evidence ladder (docs/VALIDATION.md):

Tier	Meaning	Examples in this report
E0	Single-model simulation only	Flagship scale table (MuJoCo cells), multifidelity rank correlation, robust optimization rankings
E1	At least two independent solvers agree within pre-agreed tolerance	Slow flat-ground stride (0.58 % gap); order of magnitude of start wind (4.2 %–13.9 % gap)
E2	Matches published data or literature within uncertainty	Jansen foot trajectory vs. Pop et al. (2014) experimental rig
E3	Validated by traceable physical measurement	Empty (0 records).

Verification $\neq$ Validation: two solvers agreeing proves we solved our equations correctly (verification), not that our equations match physical reality (validation).

2. What Holds Up (E2 and E1)#

2.1 Kinematics Matches Published Experimental Trajectories to Within a Millimetre (E2)#

We compared our closed-form Jansen foot trajectory against the experimental CCD-tracked trajectory published by Pop, Dolga & Pop (2014) (Applied Mechanics and Materials 658: 684–689):

Metric	Literature (Pop 2014)	Our Analytic Kinematics	Difference	Relative Error
Stroke length $L_{\text{stroke}}$	57.1 mm	57.081 mm	−0.019 mm	0.033 %
Step lift $H_{\text{lift}}$	22.5 mm	22.457 mm	−0.043 mm	0.191 %
Duty factor $D$	0.43	0.433	+0.003	0.77 %
Shape distance	—	0.0295 (2.95 % of path width)	Stance phase < 0.54 mm	Below digitization uncertainty (±0.80 mm)

This lifts our kinematics layer to E2, though it says nothing about dynamic stride, torque, or start wind.

2.2 Two Independent Physics Solvers Agree on Stride (E1)#

Start-wind thresholds sit right in the low-speed stick-slip regime where MuJoCo’s softened friction regularization is least trustworthy. Behind our PhysicsBackend boundary, we built a second, completely independent planar rigid-body solver K2: zero MuJoCo imports, 4-DOF reduced coordinates with analytic dyad projection (loop-closure error is 0.00 mm by construction), Moreau-style velocity time-stepping, and unregularized Coulomb dry friction with unilateral Signorini contact.

On slow flat-ground walking ( $\omega = 2.0\text{ rad/s}$ ) for the 6-leg small Jansen walker:

Quantity	MuJoCo	K2 (Independent)	Relative Diff	Evidence Tier
Stride per revolution	0.2760 m	0.2776 m	+0.58 %	E1
Mean body height	0.1694 m	0.1700 m	+0.35 %	E1
Start wind (15 s criterion)	0.6920 m/s	0.7226 m/s	+4.4 %	E1 (magnitude)
Mean drive torque	2.16 mN·m	2.54 mN·m	+17.6 %	E0

Stride reaches E1; torque differs by 17.6 % and remains sensitive to friction regularization (E0).

3. Flagship Experiment: Start Wind vs. Scale (E0)#

We define start wind speed as the minimum steady wind required to turn the crank one full revolution from rest within 40 s of simulated time (bracketed via binary search to 5 % relative tolerance, always reported as an interval (lo, hi]).

3.1 One Untested Pin-Friction Assumption Dominates Everything#

At baseline scale ( $L = 1$ ) with ideal frictionless pins, the 6-leg walker starts at (0.479, 0.500] m/s (estimate 0.489 m/s). One-at-a-time sensitivity sweeps across 9 assumption parameters revealed:

joint_friction (Coulomb friction per printed pin) is the sole dominant parameter: raising it from $0$ to $0.0005\text{ N}\cdot\text{m}$ more than doubles the start wind to (1.02, 1.07] m/s; at $0.002\text{ N}\cdot\text{m}$ it reaches (1.27, 1.33] m/s (a 2.59× shift).
The energy account explains why: under motor drive on flat ground, 96.5 % of input work is dissipated in foot-ground slip; under sail drive at 4 m/s wind, pin friction and damping consume 82 % of input work (ground contact takes only 17 %).
The “minimum viable size” boundary is drawn by friction, not scale: under a $4.17\text{ m/s}$ (~15 km/h) reference ceiling with pin friction at $0.002\text{ N}\cdot\text{m}$ , $L = 0.02$ cannot start even in a $30\text{ m/s}$ gale ((30, inf]) and $L = 0.03$ needs $18.4\text{ m/s}$ , bounding the minimum viable scale to $L \in (0.03, 0.1]$ . With ideal pins, the smallest tested scale starts at just $0.80\text{ m/s}$ .

3.2 The Scale Curve Is Non-Monotonic#

We pre-registered our scaling predictions before running the 30-cell grid (sail torque $\propto L^3 v^2$ , gravity load torque $\propto L^{p+1}$ , predicting $v \propto L^{(p-2)/2}$ ):

Thin-walled scaling ( $p = 1$ ): pre-registered slope $-0.5$ ; measured small-scale slope $-0.513$ (global $-0.337$ ). Bigger machines need less wind ( $0.80\text{ m/s}$ at $L = 0.35$ down to $0.39\text{ m/s}$ at $L = 3$ ).
Critical scaling ( $p = 2$ ): pre-registered slope $0.000$ ; measured global slope $-0.033$ .
Solid scaling ( $p = 3$ ): the curve is a non-monotonic V, dropping from $0.529\text{ m/s}$ at $L = 0.35$ to a minimum of $0.415\text{ m/s}$ near $L \approx 0.6$ , then rising with slope $+0.595$ at large scales ( $0.865\text{ m/s}$ at $L = 3$ , matching the pre-registered $+0.5$ ). This refutes our earlier blanket claim in WIND.md §3 that “larger bodies are more favourable to the wind”.

4. Five Hard Negative Results#

Kinematic screening anti-correlates with dynamic performance among top candidates ( $\rho_s = -0.80$ ): While kinematic evaluation is 10,000× cheaper than dynamics, within the top 5 candidates selected by kinematics, the rank correlation with dynamic stride is $-0.8000$ (K2) / $-0.7000$ (MuJoCo). The #1 kinematic candidate ranks 5th in dynamics, while the #3 dynamic walker is ranked 9th by kinematics and eliminated. Rule: use kinematic screening only to discard the bottom 25 %, never to rank or pick a winner.
Robust optimization (CVaR) flips its winner across solvers: Under K2, a CVaR-optimized design beats the nominal winner by $+0.273\text{ m}$ on the worst-case wind/slope scenario; re-evaluated in MuJoCo, it loses by $-0.271\text{ m}$ . Per our validation rules, rankings that fail cross-solver stability are mechanism demonstrations, not conclusions.
Adding sail area and gear ratio to the genome bought nothing: At equal weights and compute budget, the fixed sail beat the evolved sail on all three metrics (start wind 4.4 % lower, gust CVaR 11.8 % higher, nominal distance 29.4 % higher). Furthermore, the cheap dynamometer proxy inside the loop correlated poorly with true start-wind search ( $\rho = 0.303$ ) and inverted the ranking of the top pair.
Contact stiffness and ground friction are unidentifiable together: Bayesian calibration on synthetic data recovers intervals that contain the truth, but the posteriors span most of the prior — point estimates are unusable; only the interval is honest.
A silent NaN once scrambled genetic-algorithm rankings: Linkages that assembled at 24 probe angles but failed on the 90-point fine grid returned NaN in TypeScript vs. -5.05 in Python, causing arbitrary sort order in JS. Fixing this and locking it in contracts/fitness/ required regenerating our historical GA baselines.

5. Call for the First E3 Measurement#

Our kinematics is backed by literature (E2) and our stride by two independent solvers (E1), but every statement about torque and start wind remains inside the model (E0).

If you have a 3D-printed or Lego Strandbeest, check out our bilingual guide docs/HOW-TO-MEASURE.md. Contributed data defaults to CC0 1.0 (public domain + scientific attribution), and a single traceable measurement of crank torque $T(\theta)$ or start wind speed will become this project’s very first E3 evidence record.

How Much Wind Does a Wind-Only Strandbeest Need? Evidence Tiers, Two Solvers, and Negative Results

🌿 What is a Digital Garden?

1. Why the Answer Is Not Obvious#

2. What Holds Up (E2 and E1)#

2.1 Kinematics Matches Published Experimental Trajectories to Within a Millimetre (E2)#

2.2 Two Independent Physics Solvers Agree on Stride (E1)#

3. Flagship Experiment: Start Wind vs. Scale (E0)#

3.1 One Untested Pin-Friction Assumption Dominates Everything#

3.2 The Scale Curve Is Non-Monotonic#

4. Five Hard Negative Results#

5. Call for the First E3 Measurement#

Article concept network

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Twelve Periods of the Strandbeest

Monte Carlo Tree Search

Causal Effects & Selection Bias