Jansen's Linkage

How Jansen's linkage works

Start with the only thing that gets driven: a short crank, about 15 mm long, spinning steadily on the frame. Everything else in the leg is a rigid bar pinned to its neighbours. The trick of a good linkage is that once you fix the bar lengths and the two ground anchor points, the position of every joint is forced — there is exactly one configuration the leg can take for each crank angle. So as the crank sweeps through 360°, the foot at the far end is dragged through a single, repeatable closed loop.

Think of the leg as two stacked triangles hanging off two fixed points. The crankshaft axle is one fixed point; a second pivot sits a short distance away (the "holy numbers" put it 38 mm to the side and 7.8 mm up). The crank pin drives an upper bar pair that forms a triangle with the second fixed pivot, and that triangle's lower corner drives a lower triangle whose far corner is the foot. The crank pin also feeds the lower triangle directly through a long bar, which is what couples the two loops and shapes the curve.

Because the whole thing is just pin joints and rigid bars, you can solve it exactly. Each unknown joint sits at the intersection of two circles — one centred on each of the two known points it connects to — so the kinematics is a chain of circle-circle intersections:

Crank pin:   Pc = ( m·cosθ , m·sinθ )         m = 15 mm, θ = crank angle
Fixed pivot: P0 = ( -a , l )                   a = 38, l = 7.8

Each joint J = intersection of:
   circle( known point A , bar length r_A )
   circle( known point B , bar length r_B )

P1   = circ(P0, j ; Pc, b)        // upper joint   (j=50,  b=41.5)
P2   = circ(P0, k ; P1, c)        // upper apex     (k=61.9, c=39.3)
P3   = circ(Pc, e ; P2, d)        // knee           (e=55.8, d=40.1)
foot = circ(P3, h ; P2, f)        // toe            (h=65.7, f=39.4)

There is no algebra to "solve for the gait" — you simply step θ from 0 to 360° and read off where the foot lands. The art is entirely in choosing the bar lengths so that the resulting loop has the shape you want: flat along the bottom, high over the top.

One degree of freedom: the Gruebler count

The reason a single motor can run the whole machine is that the leg is a one-degree-of-freedom mechanism. You can verify that with Gruebler's (Kutzbach) equation for planar linkages:

DOF = 3·(L − 1) − 2·J₁ − J₂

where  L  = number of links INCLUDING the fixed frame
       J₁ = number of single-DOF (pin / revolute) joints
       J₂ = number of two-DOF joints  (none here, J₂ = 0)

Jansen leg:  L = 8 rigid bodies (frame + crank + 6 connecting bars)
             J₁ = 10 pin joints
DOF = 3·(8 − 1) − 2·10 − 0 = 21 − 20 = 1   ✓

One degree of freedom means: pin the crank angle and the whole leg is determined; turn the crank and the foot is forced to trace its curve. (People often say "eleven bars" because Jansen specifies eleven link lengths per leg counting both feet of each shared bar and the triangle sides; the moving-body count for the DOF equation is smaller. Both descriptions point at the same mechanism.) Crucially, because DOF = 1, the leg can never bind or reach a position with no solution as long as the bar lengths satisfy the loop-closure inequalities — and Jansen's numbers do, for every crank angle.

The holy numbers and where they came from

In the early 1990s Theo Jansen, a Dutch artist with a physics background, wanted a leg that walked smoothly from one rotating input. He wrote an evolutionary program: it generated random sets of leg-bar lengths, simulated the foot path, scored each one on how flat and long its ground stance was (and how cleanly the foot lifted and returned), bred the best, mutated them, and repeated over thousands of generations across several months of computer time. The winning genome is a set of thirteen lengths he calls the holy numbers:

a = 38.0    b = 41.5    c = 39.3    d = 40.1
e = 55.8    f = 39.4    g = 36.7    h = 65.7
i = 49.0    j = 50.0    k = 61.9
l = 7.8   (vertical offset of the 2nd fixed pivot)
m = 15.0  (crank radius)

All lengths are in millimetres for the reference leg.
Multiply every number by the same scale factor → identical gait at any size.

These are not arbitrary. Nudge one of them by a few percent and the flat bottom bows, the foot starts to scuff, or the recovery arc collapses. The numbers are a local optimum in a high-dimensional, sharply peaked landscape — which is exactly why a hand search would have struggled and a genetic algorithm succeeded. The whole gait being scale-invariant is what lets a tabletop model and a metres-long beach machine use the identical ratios.

The foot path: a worked trace

Sampling the kinematics above every 20° of crank rotation gives the actual foot coordinates (in mm, in the leg's own frame). Watch the Y value — it stays low and nearly constant through the back half of the cycle (the stance), then rises sharply over the front (the swing):

Crank θ	Foot X (mm)	Foot Y (mm)	Phase
0°	−8.2	−72.0	stance (planted)
60°	+7.6	−59.1	lift-off
120°	+5.8	−48.2	swing (top of arc)
180°	−24.4	−56.6	swing returning
240°	−46.8	−67.0	set-down
300°	−32.7	−75.4	stance (planted)
340°	−16.1	−74.6	stance (planted)

Across the full cycle the foot path spans about 56 mm horizontally and 28 mm vertically — close to the iconic 2:1 width-to-height ratio. The bottom stance (roughly θ = 270° through 350°) keeps the foot within a few millimetres of the lowest point while sweeping a long horizontal distance: that is the "footstep," with the set-down and lift-off transitions on either side (around θ = 240° and θ = 60°). The top arc lifts the foot ~28 mm to clear the ground on the return. Spend a moment in the interactive panel above watching the green curve fill in — the flatness is obvious the instant the trace closes.

Real-world Strandbeests and builds

Machine / build	Legs	Drive	Notes
Animaris Rhinoceros (Jansen)	Dozens, paired	Wind → crankshaft (and PET-bottle air storage)	~3.2 tonnes, roughly 6 m long; steel skeleton with a polyester skin (unusually for Jansen, not PVC), walks on Dutch beaches
Animaris Umerus	Multiple banks	Wind-pumped compressed air "muscles"	Stores wind energy in recycled bottles to keep walking in lulls
LEGO Technic Strandbeest (fan MOCs / LEGO Ideas builds)	6–12	Single electric motor or hand crank	Off-the-shelf demonstration of the one-input principle
3D-printed desk models	4–12	Small DC gearmotor	Holy numbers scaled down ~3–5×; popular maker project
Strandbeest-inspired rovers (research)	4–8	Geared motor per crankshaft	Studied for low-control-cost legged locomotion on sand/soil

The signature engineering point of every one of these is the same: one rotating input, many legs, no per-leg control. Jansen builds his sculptures almost entirely from cheap 3/4-inch PVC electrical conduit, joined with cable ties and tape, precisely because the linkage tolerates loose joints — a millimetre of slop at a pin barely shifts a foot path set by bars tens of centimetres long.

Jansen's linkage vs other rotation-to-motion mechanisms

	Jansen linkage	Four-bar linkage	Crank-slider	Cam-follower	Wheel
Output	Closed foot-path loop (walking)	Coupler curve / rocking	Straight-line reciprocation	Arbitrary programmed lift	Continuous roll
Links	~8 bodies / 11 lengths	4	4 (one is a slider)	2–3 + cam	1
Degrees of freedom	1	1	1	1	1
Inputs needed for many feet	1 shared crankshaft	1 each	1 each	1 each (or shared camshaft)	1 axle
Flat ground segment?	Yes (by design)	Only approximate (straight-line linkages)	N/A (linear)	If profiled for it	Continuous contact
Terrain adaptation	None (fixed path)	None	None	None	Rolls over smooth ground
Best at	Walking on loose/uneven ground with one motor	Rocking, oscillation, path generation	Pumps, engines, presses	Valve timing, indexing	Hard, flat roads

Design tradeoffs and failure modes

• Fixed gait, zero adaptability. The foot path is welded into the bar lengths. The leg steps identically whether the ground is flat sand or a curb — there is no sensing and no way to change stride. This is the price of needing only one actuator. Obstacles taller than the recovery arc (~28 mm on the reference leg, scaled up on real machines) simply stop it.
• Sensitivity of the holy numbers. Because the flat bottom sits at a sharp optimum, sloppy manufacturing of the ratios bows the stance and makes the foot scuff or stub. Absolute slop at a pin matters far less than getting the bar-length ratios right; builders cut the long bars carefully and tolerate loose joints.
• Joint wear and friction. A row of legs is a row of plain pin bearings under load. On sand-blown beach machines, grit in the joints is the dominant wear and friction source; Jansen accepts it as part of the aesthetic and rebuilds. A robotic version wanting efficiency would use proper bushings or rolling-element pivots.
• Phasing and load sharing. If too few legs are on the ground at once (poor crank phasing, or too few legs per side), the body drops between steps and the gait lurches. The fix is more legs and correct 180°-and-staggered phasing so contact never drops below the number needed to support the weight.
• Turning is hard. A pure Jansen drivetrain walks straight. Steering requires changing the relative crank speed or stance length on each side — Jansen does it mechanically with a separate "steering" arrangement, and robot builders add a second drive. It is not a free feature of the linkage.
• Lower efficiency than a wheel on hard ground. On a smooth floor a wheel wins easily on energy per metre. The Jansen leg earns its keep only where wheels bog down — loose sand, soft soil — and where you value walking with a single, simple input.

Common misconceptions

• "It needs a computer to walk." No. There is no controller, no feedback, no balance loop. The geometry does all the work; spin the crank and it walks. That is the whole point and the reason it is so widely copied.
• "The foot moves in a perfect straight line." Nearly, not exactly. The bottom is approximately straight and approximately constant-velocity; "approximately" is what the genetic algorithm optimised. A truly exact straight line needs different mechanisms (Peaucellier-Lipkin, Hoeken, Chebyshev), but those don't give the high recovery arc you need for walking.
• "Any leg-shaped linkage walks." Most random eleven-bar leg linkages produce ugly loops that scuff, stub, or barely lift. The flat-bottom-plus-high-arc combination is rare; that scarcity is exactly why Jansen needed an evolutionary search and why the numbers are treated as special.
• "It's just a four-bar linkage." It contains four-bar loops, but a single four-bar can't produce the long flat stance. You need the second, coupled loop sharing the crank to constrain the foot onto the flat-bottomed curve.
• "Bigger means redesign." No — the gait is scale-invariant. Multiply every holy number by the same factor and a desk model and a metres-long beast walk identically; only the absolute stride and clearance scale.