Machine Kinematics
solved, then drawn
Concept

Scale invariance — where it appears

The property that a mechanism's angular behaviour depends only on the ratios of its lengths, so scaling the whole machine changes nothing an angle sensor can read. It makes the parameter vector an exact null vector of its own identification Jacobian, by Euler's relation on a function homogeneous of degree zero.

Named by 28 essays across 16 fields — each of them below, with the objects they name alongside it.

Three machines a protractor cannot tell apart. The same four-bar at 0.60×, 1.00×, 1.50×, drawn one inside another at the same crank angle. Every one of them puts its output link at 102.914064°, and the three readings differ by 2.8e-14° — which is the solver's floor rather than a difference. A protractor on the output link is reading a function of the ratios of the lengths, so it is the same function for every member of this family, at every crank angle, exactly. Whatever such an instrument recovers, it is not the size of the machine.

The direction no protractor can see

A four-bar's output angle depends only on the ratios of its lengths. That is a sentence anybody would agree to, and it has a consequence with a number attached: the vector of the four lengths is annihilated by every row of the machine's own identification Jacobian, to 7.6 × 10⁻¹⁵, at every pose, for ever.

metrology · Parameter The coordinates that do not move. Above: the four link lengths as the whole machine is scaled from 0.4× to 2.5×, four straight lines through the origin. Below: Freudenstein's K₁ = g/a, K₂ = g/c and K₃ = (g² + a² + c² − b²)/2ac over the same range, three horizontal lines whose total variation is 1.8e-15. Each of them is homogeneous of degree zero in the lengths, so the scale ray is a level set of all three at once — and the map from a four-bar's shape to its three K's is invertible, so they are not merely invariant but complete. The identifiable quotient of a four-bar's parameter space is three-dimensional, and this site has had its coordinates since its first essay on synthesis.

The coordinates the site already had

Three of a four-bar's four parameters are recoverable, so there are three recoverable quantities. They are Freudenstein's K's, which this site has used to design function generators for as long as it has synthesised anything — and the same 3 × 3 linear system, read backwards, identifies a machine from three measured angle pairs.

metrology · Parameter What each instrument recovers. The same twenty poses of the same four-bar, read three ways. A protractor on the output link recovers 3 of the four lengths and leaves the fourth exactly invisible, because its readings are dimensionless in the lengths and scaling the machine does not move them. A coordinate machine on the tracing point recovers all six parameters — the four lengths and the two that say where the tracer sits — at a condition number of 162.3. Using both recovers the same six at 26.1, 6.2 times better, which is the case for putting two instruments on one machine: not more parameters, better-conditioned ones.

A ruler and a protractor

What a measurement recovers is decided by the units of its readings. An angle is dimensionless and cannot see a size; a position is not and can. And putting both instruments on one machine recovers no more parameters than the better of them alone — it recovers the same ones six times better conditioned.

metrology · Parameter A cycloidal cam at 60°. A 20-unit rise over 120°, a dwell, a return and a dwell. The dashed curve is the pitch curve — where the roller's centre travels — and the solid one is the surface that has to be cut, which is the pitch curve offset inward by the roller radius along its own normal. The pressure angle at this instant is 25.5°: the angle between the follower's direction of travel and the normal to the surface, and the number that decides whether the follower jams in its guide rather than sliding.

A lift is a size and a law is a shape

A cam's motion law is a dimensionless function of a dimensionless argument, so everything the cam field says about laws — which has the least acceleration, which has an impulsive jerk, which one costs least — transfers between cams of any size. What does not transfer is anything with the lift or the base circle in it.

cams · Cam Grashof's classification, predicted and then swept. Four sets of link lengths. For each, Grashof's condition predicts from the lengths alone whether the input can rotate a full turn, and the solver then attempts all 180 positions and reports how many assembled. The prediction and the measurement agree in every case, which is what licenses quoting the classification for a mechanism nobody has swept.

Grashof is a shape test

The oldest classification in the subject compares sums of lengths, so it is unchanged by making the machine bigger — which means a protractor recovers it exactly without recovering a single length. What it does not recover is the margin, and the margin is what says whether the classification is safe.

linkages · Fourbar How much of Bennett's turn survives a bar being wrong. The same four bars and the same four twists, with one bar's length changed by the amount on the left and nothing else touched. The bar shows the fraction of 48 sampled positions of the first joint at which the loop closes to within 10⁻⁹. Two parts in a thousand already costs most of the travel. This is what it means for a mechanism to work only on a condition rather than approximately near one — and it is why Bennett's linkage was a curiosity for eighty years before anyone could machine to it.

Bennett's condition is a ratio

A spatial loop's parameters are lengths and angles together, so a scaling touches only half of them. Bennett's condition — a over sine alpha equals b over sine beta — is a relation between the two halves, and what it demands of a machine is a relation between its lengths and its twists rather than a property of either.

spatial · Spatial 20 teeth driving 32. Both flanks generated from the involute, not approximated, at a pressure angle of 20°. The orange line is the line of action — tangent to both base circles, and the only place contact happens. Its length between the two tip circles divided by the base pitch is the contact ratio, 1.612 here, which means that for 61% of the cycle two tooth pairs are carrying the load and for the rest just one. The velocity ratio is 0.6250, and it is constant because the common normal never moves. The dashed extension runs between the two base tangency points, which are 8.89 mm apart; the heavy part is where contact actually happens.

The module is a size, the ratio is a shape

A spur pair has exactly one length in it. Every quantity it produces is either proportional to that length or completely independent of it — centre distance, base pitch and contact length scale exactly; ratio and contact ratio do not move at all, not even by a part in ten to the fifteen.

gears · Tooth One crank, four machines, one area. Four crank-rockers sharing only a crank of length 1, with grounds, couplers and rockers all different, each tracing the point 30% of the way from the crank pin to the rocker pin. The curves have different shapes and sizes and different places in the plane, and the area each one encloses is 2.199115 = (1 − 0.3)·π·1² — on both assemblies of every one, with a worst difference of 3.2 × 10⁻¹⁴.

The area a coupler point encloses

Trace a point on the line through a crank-rocker's two moving pins and the region its curve encloses has area (1 − u)πa²: the crank pin's own circle, scaled by how far along the line the point sits. No ground, coupler or rocker length appears in it. Four machines that share only a crank enclose 2.199115 each, to 3 × 10⁻¹⁴.

curves · Coupler 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.

A band with a direction in it

One whole direction of a four-bar's tolerance box does nothing. A machine made a quarter of a per cent too big all over has an output error of exactly zero — and an aluminium four-bar heated by a hundred degrees has an output error of exactly zero, while one with a steel frame has 0.076°.

practice · Tolerance The circle of points going straight, at 66°. Every point on this circle is, at this instant, travelling in a straight line: its path has zero curvature there. The circle passes through the pole — where the point is not moving at all — and its diameter is 47.64, which is the pole's own speed divided by the plane's angular rate. Nothing here was assumed to be a circle. The locus is the zero set of a quadratic whose |w|² coefficient is φ′³, a real number with no cross term and no difference between its two square terms, and a general conic fitted to the sampled locus returns those coefficients at 8.1e-15 and 6.4e-15. At this position the coupler is close to translating, the pole has run off the canvas and the circle with it — the figure is the size of the mechanism, and δ here is 13.6 coupler lengths. positioned by solving, not by drawing.

A curvature is a size with a minus sign

Everything in the curvature field is a similarity invariant in shape and a reciprocal length in value. Scale a moving plane and its inflection circle scales, its cubic of stationary curvature scales, and every curvature it computes is divided by the factor — so a bigger machine traces gentler paths than its drawing suggests.

curvature · Curvature Every length wrong, every reading right. A four-bar was built to the dimensions in the upper bar of each pair and its output angle read at 30 positions. A calibration started from the nominal dimensions returns the lower bar. It reproduces every one of those readings to 1.81e-16 radians and not one of its four numbers is the machine's: they are the machine's multiplied by 0.992289, every one of them, to 3.0e-16. The shape is recovered exactly — the distance in Freudenstein's three invariants is 6.3e-16 — and the size is a free parameter the damping happened to leave near where it started. A machinist handed these numbers would build a machine that works and is not this one.

Every length wrong, every reading right

A four-bar was built out of true and measured at thirty positions. A calibration started from the nominal dimensions reproduces every reading to 1.8 × 10⁻¹⁶ radians and returns four lengths, not one of which is the machine's. They are the machine's, multiplied by 0.99229 — every one of them, to fifteen figures.

metrology · Parameter The transmission angle through one turn. μ is the angle at B between coupler and rocker, computed from each solved position rather than from a formula. It runs from 54.3° to 100.3° for these lengths. The shaded band is the usual design rule — keep μ between 40° and 140° — and this linkage stays inside it throughout. The rule is about geometry alone: nothing here knows about friction, and a mechanism with a comfortable μ can still be a poor machine.

The transmission angle has no size

The geometric half of force transmission is an angle in a triangle whose three sides scale together, so it is the same at every size — 54.31° at its worst on this machine, whatever units the drawing is in. Which means a measurement that cannot recover a single length recovers the whole of what this field computes.

transmission · Transmission A joint with no way out in the plane it is drawn in. A dovetail. The tail is wider at its far end than at the mouth it went in through, so every direction out of the mouth is blocked by a slanted face and every direction further in is blocked by the floor. In the plane of this drawing the joint cannot be taken apart at all, and the direction it does come apart in is the one the drawing does not show. The moving part touches the rest at 3 faces, each contributing one inequality on the direction it may be translated in — the direction must not have a negative component along that face's inward normal — and the set of directions that satisfy all of them is a cone in two dimensions rather than three, because a translation has no moment term. That is why a removal cone can be drawn as an angle where a mobility cone cannot. Here it is empty: every direction is refused by one face or another, so the part cannot be taken out by any translation and cannot have been put in by one either. That is a statement about the assembly and not about the part, and the direction the joint does come apart in is perpendicular to this drawing. positioned by solving, not by drawing.

A cone has no size

What a set of contacts permits is a cone of twists, and a cone is closed under positive scaling by definition — so nothing about it changes when the part it holds is made bigger. Except that a twist is a screw, a screw has a pitch, and a pitch is a length.

holding · Restraint A rolling wheel, where it was driven to. The mechanism at a configuration nothing wrote down: it was reached by integrating permitted velocities from the start of the trail, and there is no equation here whose root it is. The barred line at each wheel is the direction that wheel forbids — the subject of the whole field, and the one thing a photograph of a car cannot show. The constraint residual along the drawn history is 0.0e+0.

A wheel that cannot report its radius

Rolling relates a wheel's turning to a vehicle's travelling, and the relation has a length in it. So a rolling constraint is the one place on this site where an angle measurement does carry a size — and the size it carries is the one thing a vehicle's own odometry can never separate from its wheelbase.

rolling · Rolling Three prescribed positions of a rigid body. The whole of the design problem, before any mechanism exists. A body has to occupy these three positions — each one a place and an angle, three numbers — and what carries it between them is not yet decided. A forward analysis starts from link lengths and finds the motion. This starts from the motion, and the lengths are what has to be found. The marked points are the poles: any planar displacement is a rotation about one point, so each pair of poses has one, and the arcs show the turn each represents through the body's own origin. A pole is a property of the displacement and not a mechanism — nothing has been chosen yet. 1 of the 3 poles lies outside this frame and is not drawn; near-parallel displacements push their pole a long way off.

What a synthesis assumes it knows

Every construction in this field is handed a demand in absolute coordinates — three positions of a coupler plane, at stated places — and returns a linkage in the same coordinates. Scale the demand and the answer scales, which means the construction's whole content is about shape and its answer carries a size it was given.

synthesis · Synthesis A link, as a distance and as a body. The same two links every field before this one has drawn as lines, drawn as the material they are made of. A bar is a rectangle with its ends rounded off to the bosses that surround its pins, and it is convex; a bell crank is two arms meeting at a shared pin, and it is not — its inner corner turns the wrong way by 0.52, which is the single fact that puts it outside every separating-axis test here. The pins are marked because they are what has not changed: the constraint equations are the same, the solve is the same, and the positions are the same. What is new is everything between the pins.

A body is all size

Twenty-five of the fields before this one compute quantities that are mostly shapes, recoverable from an angle sensor and transferable between machines of any size. This one computes clearances, footprints and swept areas, and not one of them is a shape — which makes it the only field whose whole output needs a ruler.

bodies · Body An open belt, whose wraps make one turn. Two pulleys of radius 40 and 24 mm with their centres 160 mm apart, with the strand running the same way round both. The two tangency points on each pulley are marked; the run between them is computed from the signed radius difference and nothing else. The wrap angles are 191.48° and 168.52°, and they add to exactly one turn — the turning number of a simple loop. Length 522.6633 mm, against the textbook formula's 522.6633. positioned by solving, not by drawing.

A drum is a size, a wrap is a shape

A strand's whole behaviour is decided by where it leaves each body and how far round it goes, and both are angles. So a belt drive's velocity ratio, its wrap angles and its tackle's mechanical advantage transfer between drives of any size — and the one thing that does not is how much strand there is.

strands · Strand 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.

A compiled machine and its own scale

A linkage compiled from a polynomial has bars whose lengths are the polynomial's coefficients and joints whose angles are its phases. Scale it and every coefficient scales — so the machine computes the same polynomial multiplied by a constant, which is a different polynomial with the same roots.

computing · Compute The gap is a straight line in the width. Three machines, four widths each, every width a fraction of that machine's own limit. Each set of points is collinear to the last bit of a double — the slopes wander by less than 10⁻¹³ across the range — because a feature-to-feature distance is linear in the corners of the two bodies and a bar's corners are linear in its width. The slope reads the contact: Chebyshev's linkage at -2.692, a crank rocker with a post at -1.350, a crank passing a stud at -1.350. A bar's boss grows 1.35 times as fast as its side, so −1.35 is a boss against something that is not growing and −2.70 is boss against boss. Extending each line to zero gives the widest link the machine will take, and the bisection that finds it the hard way agrees — 0.30113 against 0.30112, inside the bisection's own residual.

The gap is a straight line in the metal

Thickening every link by the same amount subtracts the same amount from every clearance, exactly, and moves the angle at which the worst one occurs by nothing at all. So a whole swept check can be done once on bars of any width and every other width read off by subtraction — until the closest pair changes hands, and never past zero.

bodies · Body The two curves the pole rolls along, at 60°. The pole is a different point at every instant, and it traces one curve in the fixed plane and another in the moving one. Those are the centrodes, and the whole motion is the second rolling without slipping on the first — a statement with no mechanism in it, which is why two completely different linkages with the same centrodes produce the same motion. The moving centrode is drawn here in the position it occupies at this instant, and it touches the fixed one at the pole to 0.0e+0 of a unit. positioned by solving, not by drawing.

The linkage, put back from two curves

This field opened by saying the mechanism drops out, and that every planar motion is one curve rolling on another. Both are true and neither had been measured. The rolling reproduces the four-bar's own placement to a residual that quarters when the sampling halves, and the two curves lay equal arc to a part in a billion.

curvature · Curvature 4 turns, and none of them in a plane. A strand wrapped 4 times round a drum of radius 50 whose barrel has room for 8, with each turn lying beside the last — so the pitch is the strand's own diameter, 10 here, and the path is a helix rather than a circle. The helix angle is 1.823°, the length per turn is 314.318 against the planar model's 314.159, and the whole run is 1257.27 long where a plane would have said 1256.64. The plan view is a planar wrap exactly: projected onto the plane perpendicular to the axis, the helix is a circle of the drum's own radius to 2.8e-14 traversed 4.000000 times.

The wrap that walks along the axis

Every figure in this field is drawn in a plane, and a strand that goes round twice cannot be: the second turn has to lie beside the first. The plan view of the helix that results is a planar wrap exactly — so every wrap angle survives and the length does not, by five hundred parts per million on a real rope.

strands · Strand Watt chain: 7 pins, and nothing else. A kinematic chain drawn as what it is — a graph. Each disc is a link and carries its number; each line is a pin joining two links. There are no lengths here, no angles and no positions, and every quantity this field computes survives moving any disc anywhere: the picture is a way of reading the graph and not a picture of a machine. The fill says how many pins a link carries — 4 binary, 2 ternary — which is the coarsest thing that can tell two chains apart and the first column of every census table. The count reads two numbers off this picture and nothing else: 6 links and 7 pins give 3 × 5 − 2 × 7 = 1.

A graph has no numbers at all

Every other field on this site has parameters a measurement could try to recover. This one has none. A chain is a graph, a graph is a set of links and a set of joints, and there is nothing about it that a scaling touches, a tolerance perturbs or an instrument determines.

topology · Topology Two cones, 20 teeth and 40, at 90°. The axial section, which is where every bevel quantity is read. Two cones share an apex and roll on one another along the element drawn heavy; their half-angles are 26.57° and 63.43°, adding to the shaft angle, and the ratio of their sines is the tooth-count ratio exactly. The dashed arc is the sphere of radius 22.36 on which a bevel tooth's profile actually lies. The two short lines perpendicular to the common element are the back cones; each is heading for its own axis at a distance r/cos δ from the pitch circle — 11.18 and 44.72 — and that distance is the pitch radius of the spur gear the tooth is really cut to. Both back cones lie on one line, because there is only one perpendicular to the pitch element at that point, and the two heavy stubs straddling it are the two teeth — each one addendum out from the pitch circle and 1.25 in.

A tooth that lives on a sphere

Every tooth in this field so far has been a curve in a plane, forced by the law of gearing and exact. A bevel tooth's profile lies on a sphere, no piece of a sphere flattens without stretching, and so the shape a bevel gear is actually cut to is an approximation — the only one in the field.

gears · Tooth Nine parameters, two of them invisible. A Watt six-bar has seven lengths, a fraction that says where a point rides on its rocker, and a third ground pivot with two coordinates. Reading its output link with a protractor over 28 poses gives a matrix of rank 7: two directions are invisible, at 8.24e-10 and 5.06e-10 against a largest of 7.49e+0. One is scaling the whole machine, with a zero against the fraction, because a fraction is not a length. The other is scaling the second loop alone about O₄ — that loop is a four-bar in its own right and its own size does not reach the output angle. The two wrong guesses a reader would try, scaling those five parameters about O₂ or scaling the first loop alone, are refused at 1.5e-1 and 2.0e-1.

Nine parameters, two of them invisible

A Watt six-bar has seven lengths, a fraction and a ground pivot's two coordinates. Read by a protractor on its output link, its identification Jacobian has rank seven — and the second missing direction is not a scaling of the machine at all. It is a scaling of the second loop alone, about the pivot the two loops share.

metrology · Parameter The field of action, straight and slanted. The rectangle a contact lives in: its width is the usable line of action, its height is the face width, and a new contact line enters every base pitch. On the left the teeth are straight, so the lines are vertical and each one arrives and leaves all at once — the total length in contact jumps by a whole face width, 40 mm, every base pitch. On the right the same pair with a 20° helix: the lines lean by the base helix angle, so a tooth enters at one end of the face before it has left at the other, and the total changes by 3.44 mm instead of 40. Nothing about the profile is different between the two panels.

Contact that runs along the tooth

A straight tooth engages along its whole face at once, so the amount of contact at a mesh is a square wave. Slant the tooth and a second contact ratio appears that has no tooth count in it and no pressure angle — bought with face width and helix angle alone, and able to reach a value at which the total length of contact stops varying at all.

gears · Tooth A machine measuring its own shape. A four-bar with an encoder at each end of its one freedom. Every pose gives one scalar equation — Freudenstein's, which is linear in the three invariants — so 40 poses make a three-column least squares with a condition number of 8.758 and no instrument outside the machine anywhere in it. With perfect encoders the invariants come back to 8.64e-15; with encoders good to a milliradian they come back to 5.40e-3, an amplification of about 6.26. What comes back is a shape and only a shape: the same three numbers describe this machine and one a quarter of the size, and nothing an encoder can read separates them.

A machine that measures itself

Put an encoder at each end of a one-freedom loop and every pose gives one scalar equation. The equation is Freudenstein's, it is linear in three unknowns, forty poses make a three-column least squares at a condition number of 8.76 — and no instrument outside the machine is involved anywhere.

metrology · Parameter Which numbers have a size. Twelve quantities from eight fields of mechanism kinematics, each scaled by taking every length in its mechanism up and down together and fitting the power its value follows. A shape lands on zero, a length on one, a curvature on minus one, an area on two, and every row lands on an integer to 3.3e-11. The sorting is not a units argument — it is measured from each field's own library, by asking that library the same question. What it says is that most of what kinematics computes is a shape: a mobility, a Grashof class, a transmission angle, a contact ratio and a velocity ratio are all unchanged by making the machine bigger, so all of them are recoverable from a measurement that cannot recover a single length. The row worth stopping at is the tolerance band: held to a fixed ±0.01 it lands on minus one, because ±0.01 is a length and a bigger machine held to the same absolute tolerance is a better machine. Written as a percentage the same band lands on zero. One quantity, two drawing conventions, two different kinds of number.

Which numbers have a size

Take every length in a mechanism up and down together and fit the power each computed quantity follows. A transmission angle lands on zero, a coupler point's speed on one, a path curvature on minus one, an enclosed area on two — and a tolerance band held to a fixed ±0.01 lands on minus one, which nobody would guess.

metrology · Parameter Which numbers have a size. Twelve quantities from eight fields of mechanism kinematics, each scaled by taking every length in its mechanism up and down together and fitting the power its value follows. A shape lands on zero, a length on one, a curvature on minus one, an area on two, and every row lands on an integer to 3.3e-11. The sorting is not a units argument — it is measured from each field's own library, by asking that library the same question. What it says is that most of what kinematics computes is a shape: a mobility, a Grashof class, a transmission angle, a contact ratio and a velocity ratio are all unchanged by making the machine bigger, so all of them are recoverable from a measurement that cannot recover a single length. The row worth stopping at is the tolerance band: held to a fixed ±0.01 it lands on minus one, because ±0.01 is a length and a bigger machine held to the same absolute tolerance is a better machine. Written as a percentage the same band lands on zero. One quantity, two drawing conventions, two different kinds of number.

A count is neither

Every quantity in the scaling survey lands on an integer power — zero for a shape, one for a length, two for an area. A mobility lands nowhere. It has no dimension at all, it does not move under any perturbation, and the probe that sorts the rest of the site's numbers returns nothing for it.

metrology · Parameter

Named alongside it

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

IdentifiableGrashof's conditionCalibrationUnidentifiable directionDesign ruleFreudenstein invariantsIdentification jacobianSimilarityTransmission angleApproximationBase circleInvolute

All concepts