Machine Kinematics
solved, then drawn
Concept

Watt's linkage — where it appears

The straight-line linkage of 1784 whose coupler midpoint traces a figure of eight with a nearly straight middle. Its departure from straightness is nine per cent of its stroke, which was good enough for a beam engine and is not exactness.

Named by 4 essays across 3 fields — each of them below, with the objects they name alongside it.

How straight, over how much of the stroke. The deviation from a straight line, as a fraction of the traced span, against how much of each mechanism's stroke is used. The vertical axis covers fifteen decades. Watt's and Chebyshev's linkages are excellent over a short stroke and degrade as more is used; Peaucellier's sits at the bottom of the plot at every fraction, because it is not an approximation. The gap at full stroke is about fourteen orders of magnitude, and it is the difference between a mechanism that is nearly right and one that is right.

The straight-line problem

Before 1800 a long true flat surface was harder to make than almost anything else, so guiding a piston straight without a slide was worth solving. Watt's answer was an approximation. Measuring how good an approximation, over how much of the stroke, turns out to be a more interesting question than whether it is exact.

curves · Coupler Peaucellier's cell: exact straight-line motion from pin joints. The rhombus and the two long arms hold |OP| · |OQ| constant at 16 = 5² − 3², which is inversion in a circle about O. Inversion carries circles through the centre to straight lines, and the link CQ makes Q run on exactly such a circle — so P travels on a line, with no approximation anywhere. Measured over 160 solved positions the deviation is 6.5e-16 of the span, which is arithmetic noise rather than a small error.

Peaucellier and the exact answer

Eighty years after Watt settled for an approximation, a French army officer found a linkage that draws an exactly straight line from pin joints alone. It works by inversion in a circle, the product it holds constant is measurable, and on this site it comes out straight to 10⁻¹⁶ of its span.

curves · Coupler The Sarrus linkage at 0°. Two three-joint chains in perpendicular planes, joining a fixed plate to a moving one. The left chain's three axes are all parallel, so it allows the plate to move in its plane; the right chain's are parallel to a perpendicular direction and allow the plate to move in that one. What both permit is a straight line, and only a straight line. Six revolute joints in a single loop: Kutzbach says 0 degrees of freedom, the screw system has rank 5 and says 1, and the plate rises. Measured over 24 positions, its tilt never exceeds 2.5e-14 radians and it never leaves the axis by more than 6.8e-14 — exact, from pin joints, with no approximation anywhere in it.

Sarrus, and the straight line that is exact

The planar answer to the straight-line problem took two hundred years and arrived as an inversion cell with eight bars. There is a six-bar answer that is also exact, that was published eleven years before Peaucellier's, and that works for a reason with nothing to do with inversion — it leaves the plane.

spatial · Spatial Where enumerating the corners stops being affordable. The two routes to a tolerance band, costed against the number of toleranced lengths. Enumerating every extreme combination is 2ⁿ mechanisms at every crank position; differentiating the constraints is n linear solves. A four-bar is 16 corners and a Watt six-bar is 128, which is still cheap — 142 solves for one position — and the curve is the point rather than either number: at twenty parameters, which is an ordinary spatial mechanism, the corner route is a million mechanisms and the derivative route is twenty. Both are drawn because the corner route is not merely slower, it is the one that assumes nothing, and its answer is what the cheap route has to be checked against.

Seven lengths and a hundred corners

Nothing in a tolerance analysis is about four. A Watt six-bar has seven lengths, its corner enumeration is 128 mechanisms rather than 16, and the two routes still agree to a hundredth of a per cent — but the costs have separated — 142 solves against seven. At twenty parameters, which is an ordinary spatial mechanism, it is a million against twenty.

practice · Tolerance

Named alongside it

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

ConstraintPeaucellier's cellStrokeApproximationChebyshev's linkageExactnessInversionStraight line mechanismAllocationConstraint jacobianCoupler curveDead centre

All concepts