Machine Kinematics
solved, then drawn
The ledger

Every mechanism, counted and measured

Each closed loop this site solves, with its mobility counted from the joint graph and measured from the rank of its joint screws. Both columns are computed while this page is built. Where they disagree, the formula is wrong — and it is wrong about most of the mechanisms anybody has deliberately built.

Kutzbach's criterion counts a spatial mechanism's freedoms from its joint graph alone: 6(n − j − 1) + Σf. It needs no lengths, no angles and no solve, which is why it is taught first. It is wrong about 8 of the 10 loops below.

The measurement in the third numeric column is the number of joints minus the rank of the matrix whose columns are the joint screws — the same matrix the solver uses to position the mechanism, so it costs nothing extra and knows only where the axes point. Between the two sits ν, the constraints imposed more than once, measured from the two legs' reciprocal systems rather than assumed. Adding it back repairs the count on every row here: the corrected column and the measured column agree on all 10.

Every number on this page was computed while it was built. Nothing is transcribed, so a row that contradicts an essay means the essay is out of date rather than that the table is.

The ledger

7 of 10 move · 6 of those are overconstrained

Mechanism links joints Kutzbach ν corrected measured carries drift
Planar four-bar Four parallel axes. The mechanism this whole site is about, counted as the spatial loop it is. 4 4 -2 3 1 1 one force and two couples 0
Universal joint Four axes through one point. In every car with a driveshaft. 4 4 -2 3 1 1 three forces 0
Spherical four-bar The general case of the universal joint, with four arcs instead of two right angles. 4 4 -2 3 1 1 three forces 0
Sarrus linkage Two three-joint chains in perpendicular planes. An exact straight line from pin joints alone. 6 6 0 1 1 1 one couple 0
Bennett's linkage The only mobile spatial four-bar, and it moves at one ratio of lengths to twists. 4 4 -2 3 1 1 three screws 49.4°
Bricard's six-bar Line-symmetric: a half-turn about one line carries the linkage onto itself. 6 6 0 1 1 1 one screw 56.7°
Bennett's, detuned The same four bars with one length 5% out. Same graph, same count, and it is a structure. 4 4 -2 2 0 0 two screws —
Spatial four-bar, generic twists Bennett's bars with one twist changed. The control the paradoxical case needs. 4 4 -2 2 0 0 two screws —
Six-joint loop, generic axes Nothing relating the six axes. Counted at zero, and the count is right. 6 6 0 0 0 0 nothing —
Seven-joint loop, generic axes The case Kutzbach is right about, and the reason the formula is not simply broken. 7 7 1 0 1 1 nothing 0

Reading the last two columns

What it carries is the wrench system reciprocal to the loop's joint screws: the forces and couples the mechanism transmits without moving. A planar linkage carries a force out of its plane and two bending couples; a spherical one carries three forces through its centre; a mechanism whose motion lies in no subgroup of the rigid displacements carries screws, which have an axis and a pitch and are neither.

Drift is how far that system turns as the mechanism moves through its own range, as a principal angle between subspaces. It is the number that separates the two ways of being overconstrained: the mechanisms confined to a subgroup read 0 to 0 degrees, which is an arccosine's own precision near one rather than a movement, and the paradoxical ones read 49.4 and 56.7. A dash means the loop does not move, so there is nothing to compare; a zero on a loop whose screw system already fills all six dimensions means only that a full space cannot turn.

Two ways to be counted wrong

Not every disagreement in the Kutzbach column is the same failure. A count of −2 on a mechanism with mobility 1 is a formula missing the redundancy that makes the mechanism work, and every such row here is a mechanism somebody built on purpose. A count of −2 on a loop with mobility 0 is a wrong number that reaches the right conclusion, since both say the thing does not move. The rows in bold are wrong; the ones that matter are the wrong ones whose measured column is not zero.

The same ledger, drawn

Kutzbach, plus the constraints counted twice. Each row is 6(n − j − 1) + Σf, then the redundant constraints ν measured from the two legs' constraint systems, then the mobility measured from the rank of the whole loop's screw system. The first column is wrong for 6 of 7 of these mechanisms; adding ν repairs every one. The catch is that ν is not a property of the joint graph — Bennett's linkage and a spatial four-bar with one twist changed have the same graph and different ν — so the corrected formula needs the measurement it was supposed to replace.
Fig. 1 The count, the redundancy that repairs it and the measurement, for seven of the rows above. The figure is built from the same computation the table is, so the two cannot drift apart.
Counting the mechanism that was built. Grübler's and Kutzbach's counts applied to a four-bar with points for pins and to the same four-bar with holes for pins. Each clearance joint is one extra link and one extra joint, so the planar count rises from 1 to 5 — four extra freedoms, each of them a hundredth of a millimetre wide, and the formula has no units in which to say so. In space the same substitution takes the count from -2, which is the number that says a planar four-bar cannot exist, to 18 for short pins and 10 for long ones. That is not a repair to the formula. It is the formula being right about a mechanism nobody was asking it about.
Fig. 2 The same ledger asked about a mechanism that was built rather than specified. Give the four pins holes instead of points and the planar count rises from 1 to 5 and the spatial one from −2 to 18 — all of them correct, and none of them able to say that four of the freedoms are a hundredth of a millimetre wide.

The other ways in

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