Machine Kinematics
solved, then drawn
Concept

Conditioning — where it appears

How much a computed answer moves for a small change in what it was computed from, which can be enormous where the arithmetic is a cancellation. It is why a mechanism near a singularity produces enormous joint rates for ordinary tool motions, and why a difference of two nearly equal products has to be watched.

Named by 17 essays across 8 fields — each of them below, with the objects they name alongside it.

SCARA, as four numbers a joint. A Denavit–Hartenberg table describes each joint by the common perpendicular between its axis and the next one: how long it is, how much the axes twist across it, where along the first axis it meets, and at what angle. 3 of these 3 rows have no answer for the last two, because the axes they describe are parallel and two parallel lines have infinitely many common perpendiculars, all the same length, at every point along them. The arm is perfectly ordinary; it is the description that has run out.

Four numbers or a screw

An arm can be written down as four numbers a joint or as a line in space with a pitch on it. Both are minimal, both describe the same machine to the last bit, and one of them jumps by three hundred and fifty thousand when an axis is tilted by a millionth of a radian.

serial · Serial Six legs, six numbers. A Gough–Stewart platform at (0.00, 0.00, 2.40) with a rotation vector of (0.00, 0.00, 0.15). The six leg lengths run from 3.077 to 3.263, and every one of them was computed from the pose by a single subtraction and a square root, with no leg consulting another. Going the other way — six lengths in, one pose out — takes an iterative solve and does not have one answer. The six leg lines span a screw system of order 6; the smallest singular value of the six is 0.0316, which is how far this pose is from a configuration where they become dependent and the platform stops being controllable.

Six legs and a square root

A Gough–Stewart platform's inverse problem is one subtraction and one square root per leg, computed six times without any leg consulting another. Its forward problem has forty solutions. That gap is the whole design, and it is why flight simulators are built this way and robot arms are not.

parallel · Parallel How far inside the hull the origin actually is. The same seven arrangements with their margins drawn rather than tabulated, because the shape of this chart is the argument: the quantity is not a probability and not a percentage, it is a distance — how far the origin sits from the nearest face of the hull of the contact rows, with every row a unit vector so the number is comparable across arrangements. The two that hold come in at 0.211 and 0.091; the five that do not come in at exactly nought, and they are drawn at nought rather than left off. A margin that falls smoothly to nothing is what makes this a measurement: an arrangement approaching one that lets go says so before it does.

The test is a program, not a rank

Three independent routes to one yes-or-no: enumerate the escape cone's extreme rays by cross products, take the convex hull of the contact rows and ask where the origin is, or hand the whole thing to a simplex. They agree on every arrangement — and the first version of the third one reported a disc as held, which is the one part in the field that no number of contacts holds.

holding · Restraint Free to spin, and it cannot turn at all. An ellipse of semi-axes 1.4 and 0.9 inside four flat walls that touch it at the ends of its own axes. Every normal points at the centre, so every row's moment is nought and the four rows span two dimensions rather than three: the cone of permitted twists is the whole spin axis, a line through the origin, and the first-order answer is that the part is free to turn either way. Drag the angle and watch what happens. The ellipse's reach in the direction of the top and bottom walls is √(a²sin²θ + b²cos²θ), which is smallest at θ = 0 and grows from there, so any rotation whatever drives it into both of them — by 0.016 at this angle. A nullity is a candidate and not a motion, and this is the shape of case the fields before this one could not produce: not a mechanism at a singularity, but an ordinary part in an ordinary pocket. positioned by solving, not by drawing.

Free to turn and unable to

An ellipse in a pocket the size of its own bounding box has four contacts whose rows span two dimensions, so the cone of permitted twists is a whole line and the first-order answer is that it spins both ways. It cannot turn by any amount whatever: the penetration grows as the square of the angle, with a fitted exponent of 1.9944, and a circle in the same pocket turns for ever.

holding · Restraint Where the platform stops being controllable, at 0°. Every point is a position of the platform's centre at a fixed orientation of 0°, shaded by how far it is from a direct singularity — pale is near. Unshaded means unreachable, which is the workspace boundary and the ordinary kind of singularity. The line is det A = 0, traced through the field rather than tested for: it runs through the middle of the reachable region in 102 segments, and on it the platform can move with all three actuators locked. 3312 of 6561 sampled positions are reachable.

The workspace is not a shape you choose

A serial arm's reach is roughly a sphere and can be quoted as a number. A parallel mechanism's is the intersection of three reachability conditions, changes with every degree of orientation, and has a surface of uncontrollable poses cutting through the middle of it. There is no formula. There is a map, and it has to be computed.

parallel · Parallel Three singularities, and where each one is. The elbow singularity is at the edge of the workspace, where the arm is straight and cannot reach further — nothing is lost that could have been used. The shoulder singularity is where the two ways of facing the target merge, on a cylinder of radius 0.18 m about the base axis which is also the boundary of a hole the arm cannot reach into at all. The wrist singularity is the one that stops real machines: axes four and six in line, one rotation gone, in the middle of an ordinary working volume with nothing about the tool's position to suggest it. The rank falls by exactly one in all three — and what is lost differs. The elbow and the shoulder are constrained by a pure force, along the arm and across it; the wrist is constrained by a screw of pitch −0.629, which is a force and a couple together. Those are reciprocal screws rather than singular vectors, because a singular vector's direction depends on whether the arm was written in metres or millimetres and a screw does not.

Where the arm loses a direction

An arm has three singularities and they are three different events. Two of them are inside an ordinary working volume, all three drop the rank by exactly one, and what each one takes away is a screw — a pure force along the arm at the elbow, a pure force across it at the shoulder, and at the wrist a screw of pitch −0.629 that is a force and a couple together.

serial · Serial How far a joint-space move bows off the line. Every joint runs from its start value to its end value at a constant rate — the simplest possible move, and the one that can never be refused, because every point along it is a set of joint values and every set of joint values is a pose. The tool does not travel in a straight line while it happens. It bows away by 402 mm over a move of 1223 mm, which is 32.9% of the distance travelled and enough to hit something that the straight line would have missed.

A straight line at constant speed

Run every joint from its start value to its end value and the tool bows 402 mm off the line between them. Insist on the line instead and the arm will follow it — until the path passes near a singularity, where the joint rates a metre a second demands grow as one over the distance, measured at an exponent of −1.010.

serial · Serial One planet shaft, two centre distances. A compound epicyclic gets its enormous reduction from two meshes whose tooth counts are nearly in the same proportion. The two planet gears are on one shaft, so their axes are at one radius — and at a common module the two rings ask for radii that differ by 0.50 of a tooth. The exactness the reduction is famous for is bought with a pair of meshes running away from the centre distance they were cut at, and the difference is made up by profile shift — the same correction the teeth field applies for a different reason. It is not a rounding: it is the mechanism's own condition, and it is the reason a catalogue reduction of this kind comes in a short list of tooth counts rather than in any combination.

A hundred to one from a difference of one

A harmonic drive reduces by a hundred to one in a single stage with two gears in it, and the hundred is the flexspline's tooth count divided by the two teeth the circular spline has more than it. The same null space that answers a planetary answers it. What each of the three single-stage reductions pays for that arithmetic is different, and the compound epicyclic's price is a pair of meshes whose centre distances differ by half a tooth.

transmission · Transmission Three shortest paths. The tool's route between the same two poses, under three interpolations. The straight one is position interpolated along a line with the rotation carried separately, which is what most controllers do. The screw path is the single turn about a single axis that Chasles's theorem says takes any pose to any other — the only one of the three that mentions no coordinate system — and it is 18.0% longer. The joint-space path is what the arm does when nobody asks for anything in particular, and it is 26.8% longer again. The screw axis itself is drawn: pitch -0.096 m per radian.

The distance between two poses

Ask an arm to take the shortest route between two poses and three reasonable definitions give three different paths, of lengths 1.223, 1.443 and 1.550 metres. The disagreement is not numerical. There is no distance between two rigid poses until somebody chooses a length to measure a radian in, and on this arm the choice changes which of two poses is nearer at 74.5 millimetres.

serial · Serial How much each set of wheels forbids. Every wheel contributes the same row, and whether it is a constraint or a drive is one factor of sin γ in it — γ being the angle the rollers make with the wheel's own axle. At γ = 0 the row says the body may not move across the wheel and the wheel's speed drops out of the statement; at γ = 45° the row says nothing about the body at all and fixes the wheel's speed instead. The whole difference between a machine that shuffles and one that slides sideways is in that factor.

The wheel that forbids nothing

Every wheel contributes the same row to the same matrix, and whether that row is a constraint on the vehicle or a statement about the wheel's own speed is decided by one factor of sin γ. At γ = 0 the vehicle may not move across the wheel; at 45° the row says nothing about the vehicle at all, and sideways costs exactly what forwards costs — to the last digit, and at no other angle.

rolling · Rolling Everything one planetary can do, and the gap in the middle. A single epicyclic has three shafts, so there are six ways of choosing which is held, which is driven and which comes out. Each gives a band of reductions as the tooth counts run over every design that can be cut, assembled with three planets and kept clear of undercutting. The bands above 1 are drawn; between them is a gap running from 1.6304 to 2.5862 that no single planetary reaches in any configuration — and a reduction of exactly 2, which is the most ordinary thing anybody asks a gearbox for, is inside it. The gap's width as a factor is exactly the smallest achievable ring-over-sun ratio, 1.5862, which is a statement about how small a planet may be and how large a sun may be.

Three mechanisms, one subtraction

A micrometer's differential screw, a chain hoist's differential pulley and a robot joint's compound epicyclic look nothing like each other and are the same device. Each takes two nearly equal quantities and returns their difference, each buys its enormous ratio with that difference, and each carries the same conditioning number — |a/(a−b)| — measured here by perturbing the mechanisms rather than by quoting the formula.

transmission · Transmission Three legs and four, at 0°. The same slice of positions at a platform angle of 0°, shaded by how well the platform is held — pale is near singular. Left, three legs: 3,312 reachable samples and a singular curve through them in 102 segments. Right, the same three legs and a fourth: 3,198 reachable, because the fourth leg must reach too, and no curve. What is left of the singular set in this slice is 1 isolated point, at (-1.449, -0.811), where all four lines meet. Positions held above 0.1 go from 2,989 to 3,178, and at no sampled position is the four-legged platform held less well than the three-legged one.

What a fourth leg buys

Three leg lines fail to hold a platform when they meet at a point, which is one condition, so in every slice of the workspace the failures form a curve. Four lines fail only when all four meet at a point, which is two conditions, so the curve becomes isolated points. The fourth leg buys that and more, and it costs a machine that can no longer be assembled from any four motor angles.

parallel · Parallel A reduction made out of a difference. A compound epicyclic's reduction is 1/(1 − z₁z₄/z₂z₃) — a ratio of two integers, so it is exactly what a catalogue says, at every position, forever. It is the only number in this field that survives being measured, and the reason is that it is a count. What it is not is insensitive: each row's denominator is a difference of two nearly equal products, and the last two rows differ by one tooth on one ring.

A ratio that is a count

A compound epicyclic's reduction is a ratio of two integers, because it is a ratio of tooth counts — 2176/106 for the drive here, exact at every position and on every unit ever made. It is the only quoted number in this field that survives being measured, and it is fragile in a way exactness does not protect against: one tooth on one ring moves it by forty per cent.

applied · Tooth What a calibration can and cannot see. The singular values of elbow arm's identification Jacobian — the matrix of how the tool pose moves when each model parameter is nudged, over 11 postures. There are 36 parameters and only 30 of them can be found: the last 6 directions come out at 2.3e-8, which is the difference noise, against 6.3e-4 for the weakest real one — a gap of 3e+4. And 30 is exactly 4R + 2P + 6 for this arm's 6 turning and 0 sliding joints, which is a count from the literature meeting a rank measured from the arm's own arithmetic.

What a calibration cannot see

A six-joint arm's model has thirty-six parameters and a measurement can find thirty of them. The other six are not hard to measure — they are combinations that move the tool by exactly nothing, at every posture, and no instrument ever built will separate them. The count is 4R + 2P + 6, and it comes out of a rank on four different arms.

serial · Serial A fourth leg through the meeting point, and one beside it. The platform with similar triangles at its singular orientation, at one position, with a fourth base pivot at (0, -2). Left: the fourth platform point placed where the similarity puts it, so the fourth leg's line passes 2.8e-17 from the point the other three meet at, and the smallest singular value of all four is 2.59e-9 — nothing has changed. Right: the same base pivot with the platform point moved 0.28 round the platform, so the fourth line misses the meeting point by 0.3993 and the four legs hold at 2.914e-1. The ringed point is where the three original lines meet, and it moves with the platform.

One placement of every placement

A planar platform whose two triangles are similar is singular at every position it can be put in, at two orientations. A fourth leg removes both — unless its own pair of attachment points is related by the same similarity, and then it changes nothing at all. Swept right round the platform, exactly one placement of the fourth attachment fails, the similarity names it in advance, and the holding a placement buys is proportional to how far it sits from that one point.

parallel · Parallel What a belt ratio's tolerance is made of. The three ways a variator's ratio can be wrong when the parts are wrong, at a nominal ratio of 1.848: a belt 0.3% long, a centre distance 0.2 mm out, and a sheave 0.15 mm from where it was commanded. Worst case they add to 2.98% and combined in quadrature they are 2.04% — the two conventions the practice field already argues about. The last row is a gear train's ratio under the same treatment, and it is empty: there is no length in a tooth count for a tolerance to be on.

The ratio that has a tolerance

Every dimension in this collection has been given a range at some point, and the ratios never were, because a gear ratio is a count and a count has no tolerance. A belt ratio is a quotient of two solved lengths, so every length in the mechanism is in it — and the amplification from belt length to ratio runs from 2.7 to 6.4 across the travel, which puts a whole per cent on a mechanism whose gearbox equivalent has none at all.

practice · Tolerance Three circles at one point, at 50°. The osculating circle of the fixed centrode at the pole, the osculating circle of the moving centrode there, and the inflection circle. All three pass through the pole and all three have their centres on the common normal — which is what makes the relation between them a relation between three numbers on a line. The two centrode radii are 16.180 and 9.730, signed along that normal, and the inflection circle's diameter is 24.398 against the 24.409 the other two give. Nothing in that arithmetic knows there is a linkage. The window is 30.25 wide and the larger osculating circle is 32.36 across, so it is cut at the edge: what is drawn is the neighbourhood of the pole, where the three curvatures are the same three numbers however far the circles carrying them reach.

The two numbers are the curves' own

Every essay in this field runs on the inflection circle and the pole tangent, and every one has taken them from the motion's derivatives. They are the two centrodes' radii of curvature at the pole, combined harmonically — measurable off the curves with no mechanism in the arithmetic, and carrying a sign that changes eight times in a turn.

curvature · Curvature

Named alongside it

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

JacobianRankSerial manipulatorDirect singularitySensitivityVelocity ratioInverse kinematicsNull spaceReductionScrewSingularityTwist

All concepts