Machine Kinematics
solved, then drawn
Concept

Finite difference — where it appears

A derivative estimated by evaluating a function either side of a point, which is the independent route this site checks its analytic ones against. Its error has two parts that move in opposite directions as the step shrinks — truncation and cancellation — so the step size has to be chosen rather than made small.

Named by 8 essays across 7 fields — each of them below, with the objects they name alongside it.

The coupler's motion at 66°. Every point drawn as a stub is a point of the coupler's own plane, and the stub is that point's velocity — solved, not sketched. They all point different ways and they are all consistent with one statement: at this instant the whole plane is turning about a single point, the pole, marked with a cross. It is off this frame at 1.5 coupler lengths from the crank pin, which happens whenever the coupler is close to translating. The arrow through it is the pole's own velocity, which is a quantity about the motion rather than about any point of it, and half of everything in this field follows from its direction and its size. positioned by solving, not by drawing.

The mechanism drops out

Every other field here is about a machine. This one is about the motion a machine makes — a plane sliding over a plane — and near any instant that motion is a handful of numbers with no linkage in them. Two mechanisms that agree on those numbers make the same motion, and one of them can always be thrown away.

curvature · Curvature 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 Four legs that do not cancel. Drive forward, turn, drive back, turn back — each leg exactly as long as the one it is undoing. The mechanism does not come home. What is left over is 1149.9 mm at an amplitude of 1.10, and it points along the direction the wheel forbids. The gap and the computed bracket are 31.51° apart here and 3.15° apart at a tenth of this amplitude — the agreement is a leading-order statement and the departure is the third-order remainder, which falls with the manoeuvre rather than staying put. Every point on the path was reached by a permitted velocity, so nothing here cheats; the sideways motion is assembled out of motions that are not sideways.

The motion left over by going nowhere

Drive forward, turn, drive back the same distance, turn back the same angle. Every leg is undone by another leg and the mechanism does not come home — it has moved sideways, in the one direction it is forbidden to move in. The leftover has a name, a formula, and a measured exponent of 1.997.

rolling · Rolling 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 elbow arm at a posture. elbow arm, drawn from 6 joint values through a product of 6 exponentials — no equation is solved anywhere in this picture, because an open chain has none to solve. The thin lines are the joint axes at this configuration, which are also the columns of the arm's Jacobian: the tool's velocity is a sum of turns about exactly those lines. Here the smallest singular value of that Jacobian is 0.3674 and the largest is 2.407, so the arm is comfortably away from a configuration where a direction of motion is lost. Drag θ₃ elbow.

Two routes to a Jacobian

An open chain's Jacobian is a list of its joint axes, drawn as lines in the same picture as the arm. A finite difference of its own forward kinematics is a completely different computation and has to agree — and when the two disagreed by 5 × 10⁻⁵, the fault was in a function six phases old that every spatial loop on this site had been using.

serial · Serial The bound that says the sweep missed nothing. Every corner of every body is an affine function of its link's two joints, so its speed is the same combination of the joint velocities the mechanism already solves for — and the largest corner speed anywhere on this drive is V = 1.255 per radian. A distance between point sets is 1-Lipschitz in those points, so from each sample the gap can fall no faster than V: the fine lines are those cones. Where two cones cross is the least the gap can be between the samples, and at 60 samples that is 0.0383 — positive, so nothing was missed. The same bound refuses the twelve-sample sweep of the stud machine, where the bound is -0.281.

A sweep that missed nothing

A swept clearance check looks at finitely many positions of a machine that has infinitely many, and cannot report what it did not look at. Here is a twelve-sample sweep declaring a machine clear by 0.007 while it is 0.010 inside a stud — and the bound that refuses to certify it.

bodies · Body A quantity that is not there, refusing to converge. A second central difference of a function with a bounded second derivative settles as the step is halved; across a step in the first derivative it doubles, every time. The rising line is a mutilated gear at the instant its teeth engage, where the driven wheel goes from stopped to full pitch-line speed: its successive estimates grow by a factor of 2.000000, which is the signature and not an accident of the step. The flat line is a six-slot Geneva at the same point in its cycle, whose pin enters along the slot and whose acceleration is a number. This is the only way to report an acceleration that does not exist: not by quoting a large one, but by showing the measurement refuse.

The gear with its teeth cut away

Leave teeth on part of a gear's circumference and take the rest off, and the output turns for part of the input's revolution and stops for the rest. It is the cheapest intermittent drive there is and it engages at full speed, so its output's velocity has a step and its acceleration is not a large number — it is not a number, and the way to report that is to watch a difference quotient refuse to converge.

timing · Intermittent 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

Named alongside it

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

JacobianCentrodeConstraint jacobianKinematic geometryMoving planeNewton–RaphsonRankSingularityVelocity fieldAccelerationClosed formConfiguration space

All concepts