Machine Kinematics
solved, then drawn
Concept

Function generation — where it appears

Designing a mechanism so that its output angle is a wanted function of its input angle. A four-bar can match such a function exactly at a handful of positions and nowhere else, and the residue between them is the structural error.

Named by 8 essays across 4 fields — each of them below, with the objects they name alongside it.

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.

The problem the other way round

Every essay before this one starts from link lengths and finds the motion. That is the reader's problem, because lengths are what a drawing shows. It is not the designer's problem, which is the reverse — and the reverse is hard enough that for a century the practical method was to look the answer up in a book.

synthesis · Synthesis 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 A four-bar that computes log₁₀. The output rocker's angle, mapped back into y, against the input crank's angle mapped back into x. The pale curve is log₁₀ x and the solid one is what the linkage does. They agree exactly at the 3 precision points and nowhere else; the largest disagreement over the range is 0.2232° of rocker, which is 7.47e-4 in y. The chebyshev spacing put the precision points at 1.0670, 1.5000, 1.9330.

Three problems called synthesis

Prescribing a path, prescribing a whole pose, and prescribing a relation between two angles are three different problems with three different counts, and the word synthesis covers all of them. The third has a property the others do not — eliminate the coupler angle and the design equation becomes linear, so a four-bar that computes a logarithm falls out of a 3 × 3 solve with no iteration at all.

synthesis · Synthesis One member of the family is a pair of wheels. The same demand — one plus 0.4 sin φ — scaled by a constant, and for each the angle the driven wheel is out by after one turn of the input. It has to be zero, or the teeth do not line up with themselves and there is no wheel. Only the unscaled member closes; five per cent either way leaves the output 18° out, which is a third of a tooth on a thirty-tooth wheel and a mechanism that seizes on its second turn. Non-circular gearing is a search rather than a drawing for exactly this reason: the closure condition is one equation on a whole function, and almost no function satisfies it.

The demand that cannot be met

Ask for an output rate and the two pitch curves follow with no design step in between — so the interesting question is not how to draw them but which demands admit any pair of wheels at all. The answer is one equation on a whole function, it is about the demand's mean and nothing else, and five per cent of error leaves the output eighteen degrees out after a turn.

meshing · Meshing How wrong the trapezoid is, arm angle 15.6°. The difference between the outer wheel's angle and the angle that would put all four wheels on one circle. It is zero at straight ahead by construction — both wheels point forwards — and it reaches 2.06° at 35° of lock. "One hundred per cent Ackermann" names a condition this linkage meets at 1 angle and nowhere else, and no four-bar can do better than a handful: the condition is not a rational function of the crank angle, and the linkage is.

The steering that is never right

For four wheels to roll without scrubbing, the two front wheels must point at different angles, and the relation between them is a cotangent condition no four-bar can satisfy. The trapezoid under every car meets it at straight ahead and, if the arm angle is chosen well, at exactly one other angle — 0.34° out at worst instead of 2.06°.

applied · Fourbar The error between the precision points. chebyshev: worst 0.2232°, RMS 0.1475°; uniform: worst 0.3415°, RMS 0.2296°. The error is zero at each precision point by construction and nowhere else. chebyshev has the smallest maximum here, and which curve is best depends entirely on which measure is asked for.

Where the precision points go

A linkage that matches a function at three points is wrong between them, and where the three points are put decides how wrong. Chebyshev spacing cuts the worst error by a third against even spacing, for free — and the reason has nothing to do with mechanisms. It is a fact about a polynomial the linkage has never heard of.

synthesis · Synthesis The error between the precision points. chebyshev: worst 0.2232°, RMS 0.1475°; least-squares: worst 0.3900°, RMS 0.1203°; minimax: worst 0.2105°, RMS 0.1463°. The error is zero at each precision point by construction and nowhere else. minimax has the smallest maximum here, and which curve is best depends entirely on which measure is asked for.

The linkage that is only nearly right

Stop demanding that a linkage pass exactly through three points, and ask instead that it be close everywhere. Three linkages result, none of them passing exactly through anything, and each is the best by a different measure — the least-squares fit beats the interpolant on average error and loses to it on the worst case. An optimiser gives exactly what it was asked for and nothing else.

synthesis · Synthesis How many points a linkage will take. function generation: 5 free numbers, 1 net constraint per point, 5 precision points; motion generation: 4 free numbers, 1 net constraint per point, 5 precision points; path generation: 9 free numbers, 1 net constraint per point, 9 precision points. Every count in the literature that differs from these differs about what is being counted as free, not about the geometry.

How many points may be prescribed

Five poses, five angle pairs, nine points — three numbers that get quoted as properties of a four-bar and are properties of what somebody decided to count as free. Derive them instead, and the fifth precision point turns a linear solve into a system with 128 paths, twelve finite solutions, four real ones, and exactly one linkage anybody could build.

synthesis · Synthesis

Named alongside it

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

Precision-pointFreudenstein equationStructural errorKinematic synthesisMinimaxMotion generationPath generationPrecision positionAckermannApproximate synthesisApproximationAssembly branch

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