Jansen linkage leg

A walking leg after Theo Jansen's linkage, the mechanism of his Strandbeesten ("Strandbeest" is his name for his own works, so this Jig is called after the linkage; its id keeps the old name). A crank, four bars and two rigid triangles on a fixed frame turn a rotation into a long, nearly flat stroke along the ground and a high return. Ask for the hip height (the crank axle above the ground), the plywood you have and the brass rod for the pins, and the form gives you every part to laser cut, the rod to saw and the washers to buy. Drag the crank angle and the leg walks along the path of its own sole, drawn in terracotta where the foot is down and blue where it swings. One leg, or a pair half a turn apart that always has a foot down.

What it shows

• A drive parameter walks the linkage, by formulas only. Every joint is a closed-form circle-circle intersection of the crank angle (variable.set), written as angles: the crank pin A turns about the axle O; the hip triangle's direction from the fixed pivot B is the direction of A from B plus the angle the triangle B-A-C has at B (law of cosines); the knee E the same with the other sign; the foot triangle's direction from E follows from the triangle D-E-F. Each part is drawn once at the origin (sketch.path, sketch.circle, feature.extrude) and placed by a feature.transform: turned by its direction and moved to its first joint. No assembly, no solver: the form's Crank angle slider runs it.
• The foot path is a formula curve. The position of the foot's sole as a function of the crank angle, the whole chain of intersections in one expression of t, is drawn by sketch.curve in four pieces and swept as a ribbon (feature.sweep): the stance from where the foot drops into the lowest 15 % of its step to where it leaves it, and the swing. Each piece starts at the path's lowest or highest point, where its tangent is level, so the ribbon's section stands square to it. Draw the foot path switches it off.
• Searches in formulas. Where the path is lowest, highest and longest and where it crosses the stance level has no closed form, so the document finds them itself: parabolic steps (three samples, the vertex of their parabola, a finer step) for the extremes, secant steps for the crossings. The lowest point sets the scale: the form asks the hip height, the axle's height above the ground, and the crank is (hip height − half a bar width) / 6.1223, since the sole is half a bar width below the centre of the foot's rounded tip.
• The tightest poses in closed form. Every joint angle depends on the crank only through the distance from A to B, so the linkage is most folded when the crank points at the pivot and most stretched when it points away. The joint angles over the whole turn and the least gap between parts on one layer are worked out at those two poses, in closed form, and the gap sets the range of the bar width.
• The form changes the structure. Legs builds one leg or two back to back, the second half a turn on, with a pivot rod and an axle through both (when on 22 features and 13 assertions). Open triangles cuts windows into the two walnut triangles (feature.boolean subtract by when). The rod choice brings its DIN 125 washer from a table (table.set, choicesFrom), and with it the layer pitch and every pin length.
• Cut sheets from named sets. set.define by material and face direction: one sheet of every birch part, one of the walnut triangles, true size, for two legs twice over. The set Leg parts is every part of the leg.

A drawing of a Jansen leg is one size at one crank angle. This is every size from a desk toy to a garden walker, any plywood and rod, one leg or two, and it walks.

The real thing

The linkage. Jansen's link lengths ("holy numbers"), in his units, with the crank m:

Read in two sources that agree: the table in Wikipedia's Jansen's linkage (which does not say which bar is which) and Table 1 of J. Wang, Durability-Aware Multi-Objective Optimization of the Jansen Linkage, arXiv:2606.22129 (2026). Its equations 1 to 3 give the topology used here. The bar letters are the paper's; its joints are named J1 to J5, the pivot G and the foot F, which here are A, C, D, E, F, the pivot B and the foot G. The crank axle O is at the origin, the pivot B at (-a, -l), the crank pin A = O + m (cos θ, sin θ); then C from A (bar j) and B (b), D from C (e) and B (d), E from A (bar k) and B (bar c), F from D (bar f) and E (g), and the foot G from F (h) and E (i). The hip triangle B-C-D (sides b, d, e) and the foot triangle E-F-G (g, h, i) are rigid plates. The crank turns counter-clockwise seen from the front, the foot then moves along +x on the ground, and the walker walks towards -x (to the left in the front view).

Each intersection has two branches. This Jig uses the one Jansen drew, with the hip triangle standing above B and the foot below, which gives the flat-bottomed path his search was for. An own computation (Python, steps of 0.1 degree, not part of the Jig): with the other intersection for C and D the same equations give a share of 20.3 % in the lowest 15 % of the step and a stance 43.6 units long, close to Wang's Table 4 (0.202 and 43.3); on Jansen's branch they give 61.6 % and the flat bottom. The paper does not say which branch it took.

At any size the shape is the same. In crank radii: the centre of the foot's tip comes as low as 6.1223 below the axle, the foot travels 4.527 along the ground and lifts 1.497; it spends 61.6 % of the turn in the lowest 15 % of its step (Wang's definition of the stance). Since that is more than half a turn, two legs half a turn apart always have a foot down. The step length is the foot's travel relative to the leg, not how far a walker gets per turn: that depends on how many legs are down and how the body rides on them, which this Jig does not work out.

The knee nearly folds: the bars k and c close to 8.65° at E when the crank points at the pivot (and the three closing joints, C between j and b, E between k and c, F between f and g, stay between 8.65° and 142.4° over the turn; other pairs of links, such as e and f at D, do pass through a straight line, which a pin joint does not mind). That is why a Jansen leg is fussy about bar k: Wang reports that 2 % on k already breaks the assembly, and the mutation test below with k at 62.5 units bears that out.

Layers. No two parts that cross may share a layer. From the frame outwards:

Bars j and k pass right over the axle once a turn, so the crank, glued to the axle, is the axle's last layer and the drive goes behind the front frame. Three layers would do, but bar k then passes within 5.9 units of the pivot B and would run into the washer that holds bar c on B's end; k gets a layer of its own on top. The layering was chosen by an own search over every assignment (Python, a turn in steps of half a degree), then proven on the model as the Checks say. Where a pin crosses a layer without a part of its own, a birch spacer ring of the bar's width fills it, cut with the bars. Between two layers on every pin sits a DIN 125 washer, with 0.2 mm of end play.

The frame. Two frame plates, the room for the drive apart, are joined by three spacer tubes (bore the hole, outside diameter the washer's): on the pivot rod B, glued into both plates, and on two bolts at M (two crank radii above the axle) and N (as high, four thirds of a crank radius further towards +x). M and N lie outside the space any moving part sweeps over the turn: bolt the module to your chassis there, with bolts of the rod's size and a DIN 125 washer under the head and under the nut (the outputs give the grip). The axle runs in both plates, so it cannot tilt; the crank is glued to its front end and a washer glued on behind the back plate keeps it from sliding out. Drive it by a pulley or gear on the axle between the plates (the outputs give the largest radius that clears the spacer tubes by 1 mm), from a motor or from a hand-cranked shaft on your chassis; to try the leg by hand, turn the crank itself. With two legs the second leg stands behind the back plate, mirrored, its crank half a turn on the same axle.

Pins. Brass rod, sawn to the lengths in the outputs and deburred (the model chamfers the glued end by a tenth of the rod). Each pin is glued (cyanoacrylate or epoxy) into its lowest part: A into the crank, C and D into the hip triangle, E and F into the foot triangle, B into both frame plates, the axle into the crank. The other parts turn on it, and a washer glued onto the rod's front end, not to the part, keeps them on.

Washers. DIN 125 form A, brass: bore, outside diameter and thickness 3.2 / 7 / 0.5 mm for 3 mm rod, 4.3 / 9 / 0.8 for 4 mm, 5.3 / 10 / 1.0 for 5 mm, 6.4 / 12 / 1.6 for 6 mm (read from the DIN 125 table on roymech.org).

Laser cutting. Birch plywood for the frame plates, crank, bars and rings; walnut plywood for the triangles; both of the thickness you measured. Every hole is drawn the kerf smaller than the rod plus the clearance, so the beam cuts it to size; the outlines are not offset, so the parts come out half a kerf smaller all round, which a linkage does not mind. The rules held over the whole range: a hole at least as wide as the sheet is thick; a web round every hole at least the rod diameter and the sheet thickness (the brief's edge distance, and the laser's smallest feature), after the kerf; the triangles' windows rounded by the sheet thickness and at least two sheet thicknesses across. Lay every bar with the face grain along its length: the sheet export lays the parts as they stand at the crank angle set, so turn the bars in your laser software (or use cut.export, below, which lays each part along its length). The laser-burnt edges are shown dark. Sand the faces lightly, wax them, and run the leg in by hand.

Proportions (own judgement, not from a source). Bars at most 0.6 crank radii wide: wider, the parts crowd each other and the leg turns into paddles. Bar k at most 60 sheet thicknesses long: longer, a plywood bar flexes out of its plane.

Making it. Cut both sheets; saw and deburr the rod; glue each pin into its lowest part, square to it (a scrap with a hole the same size makes a jig), and let it cure. Glue the pivot rod into both frame plates through its spacer tube, and bolt the box together at M and N through theirs. Stack the parts layer by layer with a washer between every two, glue the retaining washers on, and turn the crank by hand: nothing may rub.

What the Jig does not do: strength. Plywood of 3 mm at the largest size the form allows for it (a hip height of 271 mm, bar k 180 mm long) makes a sculpture, not a vehicle: a walker that carries a load needs its bars and pins checked by whoever builds it.

Parameters

At the defaults (one leg, hip height 150 mm, 3 mm plywood, 3 mm rod, clearance 0.2 mm, kerf 0.12 mm, bars 10 mm, open triangles) the outputs read: crank radius 23.68 mm, step length 107.22 mm, step height 35.46 mm, stance 61.58 % of the turn, least gap between parts on one layer 10.92 mm (at the most folded pose), longest part 113.74 mm (the foot triangle); brass rod 14.7 mm for pins A and E, 11.0 mm for D and F, 7.3 mm for C, 34.0 mm for the pivot B and 30.3 mm for the axle, 123.0 mm in all; two M3 mounting bolts with a grip of 27.0 mm; three spacer tubes 20.0 mm long; a pulley or gear of up to 42.87 mm radius; 25 washers.

Checks

45 assertions, 34 of them active for one leg and 43 for two:

• The bars close the linkage, measured. The pins C, E and F sit centred in the far holes of bars j, k and f (the distance from the rod's surface to the hole is half the clearance less half the kerf, within 0.001 mm): a bar of the wrong length, a moved or a loosened hole fails it. The back leg's three the same with two legs. Every other hole of every part is held by two measures together: a hole moved by more than its clearance makes the pin overlap the part, and a hole of the wrong size changes the plywood volume, which must equal its closed form to a millionth.
• Nothing overlaps: the 25 (or 41) parts, pins with their washers included, have no volume in common.
• Parts on one layer pass each other by at least the running clearance of 1 mm, measured at the crank angle set: the hip and foot triangles, bar c and the ring at D, bar f and the ring at E, and bar k and the washers on pins D and F; and the back leg's three tight pairs with two legs.
• The three closing joints keep 8° to 172° between their two links over the whole turn: C between j and b, E between k and c, F between f and g (closed form at the most folded and the most stretched pose, which bound these angles): the least is 8.65°, the knee E folded; the most 142.4°, at F stretched.
• The mounting bolts clear the leg: each bolt at M and N, modelled with a head or nut and its washer as a cylinder of the washer's diameter and the washer's thickness plus one bolt diameter high on both faces, keeps at least the 1 mm running clearance from every moving part (measured at the crank angle set, with two legs from both). They come nearest at 53° (M, bar j) and 56° (N, the crank); measured there, the least is 4.72 mm (M) and 13.70 mm (N) for the smallest pair, and 11.00 and 24.75 mm at the defaults.
• Two derivations agree (measured against an independent closed form): the foot triangle's lowest point and the hip triangle's leftmost point are where the radical-line construction of the same joints puts the foot G and the corner D (Wang's form of the intersection, with the branches chosen by side), within 0.001 mm.
• The hip height is the axle above the ground: the foot's lowest point never goes below the hip height under the axle, at the crank angle set (and the back foot's with two legs); the drawn path of the sole reaches exactly that depth at its lowest point (0.001 mm).
• The drawn path is the foot's path: its highest point is where the search found it (0.001 mm); its length is the step the search found (within 0.005 mm plus 50 millionths of it, as closely as 3DCraft proves the box of a swept spline); the stance pieces stay in the lowest 15 % of the step and the swing pieces above it, to within half the line's width. The foot rises through that level at 7.1 and 4.0 of Jansen's units per radian (the line is 0.25 units wide each side), so this pins the stance's two ends to within 2.0° and 3.6° of crank, and the stance share to within 1.6 points; the share itself comes from the secant search.
• The pair is half a turn apart: the centre of mass of the two crank pins A lies on the axle (x and z, 0.001 mm).
• Laser: the web round a hole (measured on a spacer ring) is at least the rod diameter and the sheet thickness after the kerf; no wall of the foot triangle is thinner than the sheet (thickness).
• Counts: 25 parts for one leg, 41 for two; 2 frame plates and 9 birch parts per leg, and 2 walnut triangles per leg, on the cut sheets.
• Closed forms: the birch parts' volume equals the sum of Steiner's rule for every bar, ring and frame plate (outline area plus perimeter times the rim plus a disc, less the holes); the walnut triangles' volume equals Heron's area with Steiner's rim, less the holes and the windows; both within a millionth.
• The rod list: every rod, tube and bolt the outputs list (pins A and E, D and F, C, the pivot, the axle, the spacer tubes, and the bolts' grip plus head and nut) is measured as long as the output says (0.001 mm).

Mutation tests (each in a scratch copy of build.json, at the defaults unless said): bar k 1 % long fails the overlap, pin E's centring and the birch volume; C on the other branch fails pin F's centring and corner D; F on the other branch fails the foot's position, the overlap and the layer-1 clearance; bar f's far hole moved 0.1 mm fails the overlap and pin F's centring; bar j's far hole 0.1 mm larger fails pin C's centring and the birch volume; the hip triangle's hole D 0.1 mm larger fails the walnut volume; bar f moved to layer 2 fails the overlap; bars 0.75 mm wider than the range allows, at 192°, fail the layer-3 clearance; the lowest point searched from the other dip of the path (30°) and the stance ending 2° late both fail the stance check; the output for pins D and F 1 mm long fails their rod check (and the pivot's, which shares its formula); the axle's output 1 mm short fails the axle's; the D and F pins drawn 1 mm longer fail their rod check; two legs with the back crank at +170° fail both half-turn checks; bar k at 62.5 units fails the joint-angle check, the ground checks and the stance check, and the sweep breaks; the bolt at N put back half a frame offset ahead of M (where an earlier build had it), at 312°, fails its clearance and the overlap.

Range. tools/jigs.py check runs check-range over 101 points (the defaults, 64 corners, every choice and 32 random points): all pass, none unproven. The assertions measure the leg at the crank angle set; at one setting of the form they prove that angle. Beyond the sampled points, a sweep by hand with param.set every 15° and at 11, 12, 191 and 192° passed every assertion for: two legs at the smallest size (100 mm, bars 9.32 mm, solid triangles); two legs at the widest bars at 150 mm (14.01 mm) with 2.5 mm plywood, 0.5 mm clearance and 0.45 mm kerf; the largest pair, 6 mm rod and 6.4 mm plywood (579 mm, bars 54.09 mm, 60 mm of drive room); and the largest single leg (271 mm, bars 25.32 mm, foot path off). The tightest gap between parts on one layer was 4.44 mm (the smallest pair, at 192°, next to the closed form's tightest pose at 191.6°). Over the whole range the width cap of 0.6 crank radii binds before the 1 mm clearance does: at the cap the parts on one layer keep 0.28 crank radii apart.

Export

• File › Export…, Sheet of set: Cut sheet, birch plywood and Sheet of set: Cut sheet, walnut plywood, DXF or SVG. Every part at true size with its holes, each on its own layer, laid out as it stands at the crank angle set; give the dialog your bed's width, and turn the bars to lay the grain along them in your laser software. The holes are already drawn the kerf smaller: turn the laser software's kerf offset off. For two legs each sheet holds both legs' parts.
• With an agent: cut.export {"path": "parts.svg", "width": "600 mm"} with Draw the foot path off lays every plywood part along its length (grain along the bar) and says how many of each to cut; birch and walnut parts share that sheet, each on its own layer.
• STEP: switch Draw the foot path off and export the model: what is left is the named set Leg parts, every part of the leg.
• The brass rod, spacer tubes, mounting bolts and washers are in the outputs.

The README in the model's repository has the rest (Files, Rebuild, Difficulty, Platform notes, Licence): the whole README on GitHub.