Thirteen Numbers That Make a Leg Walk

One leg, one crank#

The Strandbeest leg is the Jansen linkage: a crank plus seven moving rigid bodies joined by ten revolute joints, one degree of freedom (per the arXiv paper cited below). Turn the crank once and the foot traces a closed loop. The 13 lengths (unit unstated):

with $m$ the crank. I checked the numbers against Wikipedia’s Jansen’s linkage article, which does not say which link joins which joints. The topology comes from an arXiv paper (2606.22129, on durability-aware multi-objective optimisation of the Jansen linkage) that I did not read in full; independently, a brute-force search for a flat-bottomed foot path also produced this assignment among its candidates.

Every joint is a circle intersection#

Put the fixed pivot $G$ at the origin, the crank pivot at $(a, l)$, and the crank tip $C$ at angle $\theta$. Every other joint is at known distances from two known points:

$F$ is the foot. Two circles meet in two points; the choice fixes the assembly branch, and I pick signs so the foot is the lowest point of the leg. In code each joint is one line of data:

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{ id: "K", centers: ["G", "C"], radii: ["b", "j"], side: 1 }

The solver walks the list in order, so the GA only changes lengths and never the topology.

Why a D#

The foot path has a nearly straight stretch on the ground and an arc through the air on the return. In my implementation the ground stroke is about 43% of a revolution and the lift is about a third of the path width (22.5 vs 67.9). A flat stroke means the body does not bob while a foot is down; a test pins this property down.

Evidence level: E2. These three numbers were later checked against a published test-rig dataset. Pop, Dolga and Pop published the foot trajectory of their own small leg test rig in 2014 in Applied Mechanics and Materials (CCD camera tracking); I digitised it and compared it item by item with the analytical kinematics in this repository:

Over the ground stroke the point-to-curve distance is 0.00–0.54 mm, below the ±0.80 mm digitisation uncertainty of the paper itself (line thickness and a 300 dpi raster grid). So “the path is flat and the lift is about 22.5” is not only self-consistent inside my own model: it agrees with a published measurement, which is what E2 means. Source: Applied Mechanics and Materials 658: 684–689. The normalised mean shape distance to the digitised literature trajectory is 0.0295 (2.95 % of the path width).

Several legs, staggered#

One leg spends half a turn in the air. The Strandbeest puts many legs on one crankshaft with staggered phases; in code, $n$ legs are $n$ time shifts of the same path:

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const feet = multiLegFeet(spec, evenPhases(8), 120);

The picture is schematic: legs are drawn as hip-to-foot lines, planted feet (orange) stay fixed relative to the ground while the body glides over them. I checked that with 12 legs (6 per side) at least one foot is always down and support is stable over 90% of the cycle (that is a hard constraint in the wind objective). With one leg it fails: it is in the air for half the turn.

Summary#

• 13 lengths plus a topology make a complete leg, solved with circle intersections.
• The D-shaped path is a result of the proportions, not a coincidence; it agrees with a published 2014 test-rig trajectory (E2).
• To drag the sliders or evolve it yourself: code in strandbeest-evolution, or open the live demo in your browser (nothing to install).

Next: evolving it from random lengths.