Machine Kinematics
solved, then drawn
Concept

Newton–Raphson — where it appears

Solving equations by repeatedly linearising them and stepping to the linearised root. Its convergence is quadratic, so the iterate that first satisfies a stopping test has usually overshot it by several orders.

Named by 11 essays across 8 fields — each of them below, with the objects they name alongside it.

The output is a band, not an angle. The rocker's angle through one turn of the crank, for a four-bar whose four lengths are each specified to ±0.01. The line is the nominal mechanism; the band is where the output of an actual one lies, found by building all sixteen extreme combinations of the four lengths at every crank angle and solving each. The band is not a constant width: it is 0.73° at its widest, near 30°, and 0.36° at its narrowest — a factor of 2.0. Which of those a designer is told depends entirely on where the mechanism was measured.

A length is a range

Every figure on this site so far has been drawn from four numbers. No four numbers were ever cut. Give each of them a tolerance of ±0.01 and the rocker's output stops being an angle and becomes a band 0.73° wide at one part of the turn and 0.36° wide at another — and which of those a designer is told depends only on where somebody measured.

practice · Tolerance A four-bar at 60°, solved. Ground 4, crank 1, coupler 3.5, rocker 3. Every joint position here is the output of a Newton–Raphson solve on the loop-closure equations, converged to 0.0e+0 — not a placement that looked right. Grashof's condition classifies these lengths as a crank rocker, and sweeping the crank through 360° confirms it: 120 of 120 positions assemble. The transmission angle at this instant is 66.9°.

Four bars and four pins

The smallest interesting machine there is. Four lengths decide everything about it — which link can turn all the way round, how hard it pushes, where it stops and whether it can be assembled at all — and every one of those is a number that falls out of a solve rather than a judgement about a drawing.

linkages · Fourbar Three legs, one platform. A 3-RRR planar parallel mechanism at (0.20, -0.15) turned 11.5°, elbows up/up/up. The three actuator angles were computed one leg at a time and independently, which is what makes this direction cheap. The dashed lines extend each leg's second link: those are the three forces the legs can transmit to the platform, and the mechanism is controllable exactly while they stay independent. Here they miss one another by 0.651, and the smallest singular value of the three is 0.9550. At this position the platform can be turned through 206° in all before a leg runs out of reach.

The easy problem and the hard one change places

For a robot arm, working out where the hand is takes a walk down a chain and working out what the joints must be to put it somewhere is the hard part. Connect the platform by three legs instead of one and both statements reverse — and the reversal is not a matter of degree.

parallel · Parallel Every path, in the plane of one unknown. The 16 tracked paths of the 3-RPR platform, projected onto the complex plane of x. Each curve starts at a solution of the start system and ends at a solution of the target or leaves the frame on its way to infinity. This run is γ random, start constants complex, and it found 6 solutions.

Following a root from a problem already solved

Homotopy continuation solves a system nobody can solve by deforming one that anybody can, and following every root as it moves. The whole method rests on the deformation being generic, and the folklore says that is what the γ-trick is for. Running all four combinations says the folklore names one of two places the randomness can live, and either will do.

algebra · Algebra Two routes to the same derivative. How much the output angle moves when the coupler length moves, through one turn, computed twice. One route rebuilds the mechanism at b ± 10⁻⁶ and solves both from scratch; the other differentiates the constraint equations and solves one linear system against the analytic Jacobian. They lie on top of each other — the strip beneath plots the difference on a four-decade log scale, and its largest value anywhere in the turn is 2.9e-10 against a sensitivity of order 0.41. That is the only independent check there is of the Jacobian itself, whose coupler rows carried four wrong signs from the foundation phase to 2026-08-12 without ever drawing anything wrong.

Two routes to a sensitivity

How far the output moves when a link length moves can be found by rebuilding the mechanism and solving it again, or by differentiating the constraint equations and solving one linear system. The two agree to two parts in a hundred million across a whole turn — and the second route is an independent test of the constraint Jacobian itself, whose coupler rows had carried wrong signs unnoticed.

practice · Tolerance 6 ways to assemble the same three actuator angles. The actuators are at 216°, 48°, 144° in every panel, so the three elbows are at the same three points throughout and only the platform differs. Each pose satisfies all three legs to 6.7e-16. Found by reducing the problem to one equation in the platform angle and scanning it at 0.100° — exhaustive to that resolution and no further, which is the honest thing to say about a root count. Over a survey of 6750 actuator triples this mechanism ranges from none to 6.

One command, six answers

Lock the three motors of a planar platform and the platform can be in as many as six different poses, every one of them satisfying every leg exactly. Which one it is in was decided by how it was assembled and where it has been since — and the number of answers is not a property of the mechanism but of where the motors happen to be.

parallel · Parallel The gap is a function, and it has corners. The smallest gap over every tested pair, at each of 360 solved positions of the crank. Two things are visible that a check at the ends could not report. The minimum, 0.1041, is in the middle of the travel and not at either end. And the curve has 4 corners, each one a change in which pair is closest — the colours — so the function is piecewise smooth rather than smooth, and its minimum is not where a derivative vanishes. Refining off the sample grid by golden section moves the answer by 1.4e-5, which is what a corner rather than a smooth minimum looks like.

A gap with corners in it

The clearance between two parts is a function of the crank angle, and it is not a smooth one. It has a corner wherever the closest pair of features changes hands, so its minimum is not where a derivative vanishes and is not at either end of the travel.

bodies · Body A defect that draws nothing wrong. The same four-bar swept 360 times with the pre-correction Jacobian and with the corrected one, at eight coupler-point offsets. Corrected, every position is reached at every offset. Uncorrected: 5 of the offsets lose nothing at all, and then it loses 58, 159, 267 of 360. The picture was never wrong — a refused position is simply not drawn — so the only symptom was a sweep with fewer frames in it than it asked for.

The solver was refusing a quarter of the sweep

Four numbers in this site's Jacobian had the wrong sign, from the foundation phase until now. Every picture it ever drew was correct, because a wrong derivative does not move a converged answer — it just makes Newton crawl, until the stall rule declares the position unreachable. The symptom was a sweep quietly returning fewer frames than it asked for, and no gate in the fleet has a rule against that.

wrong · Misconception 8 postures, one tool pose. The tool is at one place, held one way. Eight different sets of joint values put it there, and this is all of them at once: two ways for the base joint to face the target, two for the elbow, and two for the wrist — two by two by two. Every posture here came out of a closed form and was checked by running the arm forward again; the worst disagreement is below 10⁻¹⁴ of a metre. Which of them a machine can actually use is a separate question, and it is answered by the joint limits rather than by the geometry.

Eight ways to hold the same tool

A six-joint arm asked to put its tool at one place, held one way, has eight answers. Not approximately eight and not eight found by looking — two for the base, two for the elbow, two for the wrist, each exact to a hundredth of a femtometre, and a search from six hundred starting postures finds those eight and no ninth.

serial · Serial A lower bound that happened to be tight. The site's own Gough platform at its home pose. The search starts Newton from a spread of guesses and reports what it lands on: 16 from 400, 16 from 1200, 16 from 4000. Tracking every one of the 1458 Bézout paths says there are 28 poses in the complex numbers and 16 of them are real. The search was right. Nothing available to the search could have said so.

The search that was right

This site has reported sixteen assemblies for its Gough platform and labelled the number a lower bound found by search, everywhere it appears. Tracking every path says there are twenty-eight poses and sixteen of them are real. The lower bound was tight. Nothing available to the search could have said so, and a second method that was supposed to settle it turns out to have the same defect one level up.

algebra · Algebra The machine compiled from a rectangular hyperbola. xy − 0.5, compiled: 20 bars and 20 joints, painted by what each part is for. The two-link arm at the pivot carries the tracing point; the reflectors and means build each term's angle; the rigid offsets fix the constant φₖ and the amplitude; the translators carry those directions out along the summing chain, whose last vertex is held on a line. That last constraint is the equation. Every joint drawn is the output of a Newton–Raphson solve on 36 equations, converged to 4.2e-16, and the polynomial at the tracing point is 2.2e-16.

The machine, compiled

Twenty bars, twenty joints, and one degree of freedom. Every position is a converged solve on thirty-five equations, none of which mentions the polynomial — and the polynomial at the tracing point reads 1.3 × 10⁻¹⁴ across the whole working arc.

computing · Compute

Named alongside it

The objects these essays reach for when they reach for this one.

Assembly branchConstraintFour-barJacobianLoop closureToleranceAssembly-modeFinite differenceForward kinematicsGrashof's conditionHomotopy continuationInverse kinematics

All concepts