Machine Kinematics
solved, then drawn
Concept

Closed form — where it appears

A solution written as a finite expression rather than found by iteration, which for an arm's inverse problem needs three axes to meet at a point. Having one is what lets a controller solve the inverse problem thousands of times a second, and whether an arm has one is decided by its geometry when it is designed.

Named by 8 essays across 4 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 the reflector, solved. A rhombus whose far vertex is held on a line through the pivot. One side is the input, the line is the mirror, and the other side comes out reflected in it — which is where negation, doubling and addition all come from. The relation it satisfies is (μ, θ) ↦ 2μ − θ, and across a sweep of 41 positions the worst departure from it is 1.3e-13 radians. Every joint here is the output of a Newton–Raphson solve on the bar lengths; nothing is placed by the formula the picture is about.

Four bars that add two angles

A rhombus on two links from one pivot points along the bisector of their angle, exactly, because a rhombus has equal sides. Hold its far vertex on a line and it reflects instead. From those two facts come negation, doubling and addition — and every whole-number combination of two angles a compiled machine needs.

computing · Compute The collineation axis at 66°. The coupler line extended and the frame line extended meet at Q, and the line from the pole through Q is the collineation axis. Bobillier's theorem is that the axis and the pole tangent make equal angles with the two rays PA and PB, in opposite senses — so having the axis gives the pole tangent, which is otherwise the one quantity here that needs the motion differentiated. Measured over 50 pairs of conjugate points the relation holds to 2.5e-14 radians. positioned by solving, not by drawing.

A construction with no arithmetic in it

Everything in this field so far has needed the motion differentiated. Bobillier's theorem gets the pole tangent — the one quantity that otherwise needs a derivative — out of two lines that are already drawn on the mechanism, and the inflection circle follows from three points and a pair of compasses.

curvature · Curvature 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 The parking shuffle, three times over. Forward on left lock, forward on right lock, back on left lock, back on right lock — and repeat. The heading comes back to where it started and so does the position along the road, both exactly and at every leg length, because the four legs are a symmetric set. What is left is 255 mm of pure sideways translation per cycle at a leg of 0.80 m, which is exactly 4R sin(φ) tan(φ/2) with φ = s/R. Halve the leg and it quarters.

Parking is an exponent

Four legs — forward on left lock, forward on right lock, back on left lock, back on right lock — return a car to its own heading and to its own place along the road, exactly, and move it sideways by 4R sin φ tan(φ/2). Halve the room and the gain quarters, so the number of shuffles goes up by four and the distance driven doubles.

rolling · Rolling The path a towed wheel takes. The front wheel is given a path; the rear one obeys a single equation — roll along your own heading, and stay attached. The rod is drawn every twelfth sample and is never imposed: the integrator carries the axle's position and heading and nothing else, and the distance from hitch to axle comes out constant to 1.4e-13 m over the whole run. The rear track cuts every corner, which is off-tracking, and it is the reason a long vehicle needs a wide turn.

The path a towed wheel takes

A towed axle obeys one line: roll along your own heading, and stay attached. Nothing tells it to keep its distance from the hitch and it keeps it to 10⁻¹³ anyway, it settles onto a circle of exactly √(R² − L²), and the residual against that is not the integrator — it is the difference between a circle and the polygon it was sampled as, and it falls by four when the sampling doubles.

rolling · Rolling 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 θ₅ pitch.

The wrist is three joints and one point

Three axes meeting at a point is what makes a six-joint arm's inverse problem solvable in closed form, and it is why every industrial arm is built that way. Move one of those axes by ten millimetres and the construction goes on returning eight confident answers, every one of them out by three and a half.

serial · Serial Six words, and the shortest of them. Every path a car that may not reverse and may not turn tighter than R can take between two placements is one of six shapes: three arcs, or two arcs with a straight between. All six are drawn; the shortest is LSR at 2.2557 R and the longest is RSR at 14.788 R. A numerical shooting solve that shares no line of code with the closed forms returns 2.2557 R, which agrees to 4e-16.

A circle for the first millimetre

The shortest path for a car that may not reverse, from here to a point one millimetre to the side at the same heading, is 31.417 m for a five-metre turning radius. The shortest path to a point twenty metres to the side is 31.416 m. The cost of going sideways is not monotonic in how far sideways, and below a crossover at 2.956 R it is exactly 2πR + δ.

rolling · Rolling

Named alongside it

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

Rolling constraintTurning radiusArcAssembly branchDubins pathInstantaneous centreInverse kinematicsNonholonomicPieper conditionPole tangentReachable setSerial manipulator

All concepts