Machine Kinematics
solved, then drawn
Concept

Wrench — where it appears

A force system written as six numbers — a force and the moment it makes about the origin — which is a screw just as a motion is. Writing forces and motions in the same six-number form is what makes reciprocity a single bilinear condition, so a mechanism's constraints are the orthogonal complement of its joint screws.

Named by 9 essays across 3 fields — each of them below, with the objects they name alongside it.

kelvin: 6 contacts, 6 of the six freedoms taken. A plan view of the arrangement, with each contact drawn where it acts and an arrow along the direction of the force it can carry. A ball's contact force passes through the ball's centre whatever surface it rests on, so the whole constraint system is these points and these directions and nothing else. Written as wrenches — a force along a line, which is a screw of zero pitch — their rank is 6. Six independent wrenches leave nothing free: the part has one place to be, and putting it down twice puts it in the same place.

Six points and no more

A ball resting on a surface is a joint: it takes one freedom away, and the force it can carry is a line through the ball's centre. Six of them, arranged well, take all six freedoms and leave a part with one place to be. Six arranged badly take five, and the sixth freedom is a screw with an axis that can be named.

applied · Mobility The contacts that fight each other — four-legs. With more contacts than freedoms, some weighted sum of the contact forces is zero: those contacts push against each other and not against the part. The weights are the left null space of the wrench matrix and they are drawn here. For four legs on a floor they come out as +−+− — the alternating sum — which is why a table rocks about a diagonal and never sideways, and why the gap under the fourth leg is δ₁ − δ₂ + δ₃ − δ₄ exactly.

The seventh contact

Add a contact to a part that is already exactly constrained and it adds no rank, so it constrains nothing — and it is the only contact in the set that can fail to touch. For four legs on a floor the combination that constrains nothing is the alternating sum of the four, which is why a table rocks about a diagonal and never sideways.

applied · Mobility What each loop's constraint system is made of. For each mechanism: the order of the screw system its joints span, the order of the reciprocal system — the wrenches it carries without moving, which is always six minus the first — and what those wrenches are. A planar four-bar carries one force and two couples; a mechanism whose motion lies in no subgroup carries screws of finite pitch instead, and 2 of these 6 do.

What a mechanism cannot do

A mechanism's freedoms are a subspace of screw space. Everything orthogonal to that subspace under the reciprocal product is a force the mechanism carries without moving — so the constraints are not a separate thing to be worked out, they are what is left, and one matrix gives both.

spatial · Spatial Seven arrangements, one routine, and the two that hold. Every row is the same three steps: write down one row per contact — the moment of its normal about the origin, then the normal itself — take the convex hull of those rows, and ask whether the origin is inside it. The parts differ, the numbers of contacts differ, and the routine does not. Two of the seven hold. The other five leave the part something, and the interesting column is what: four rays of rotation for the pinwheel, a translation straight out of the vee, and for the last two a whole line rather than any number of rays, which is what a rank below three means and is the case a reader has to be warned about. Note that the four contacts of the second row are the four of the first row, on the same four edges of the same square, at the same distance along each. positioned by solving, not by drawing.

Six hold nothing

Every exact-constraint coupling in these essays — Kelvin, Maxwell, three-two-one, and a Kelvin clamp with a seventh pad added — has rank six and holds the part not at all. The escape a Maxwell coupling leaves is a pure vertical translation with nothing else in it, which is not a defect: it is what a coupling is, and gravity is the seventh contact nobody draws.

holding · Mobility Two legs that share a line of force. Two legs of three revolute joints each under one platform, drawn about their common centre: the first leg's joints in one colour and the second's in another. The second leg was built so that each of its joints passes through a point of one line, the dark one, which also crosses all three joints of the first leg — the six marked points, each met to 1e-16. A force along that line does no work on any joint of either leg, so both legs resist it: it is a load path the platform has twice over. The dashed line is what comes with it: the axis of the one twist both legs permit, a screw of pitch -0.049 that passes 0.032 from the shared line at 57.2° to it. Dragging moves the shared line along the first leg's first joint and builds the second leg afresh through each position.

A shared load is a shared motion

Put two legs of three revolute joints under one platform and let them share a line of force, so that both carry the same load. The platform does not merely gain a redundant load path; it gains a motion in the same act, because three constraints and three make six. The motion is a screw about a different line, tied to the shared one by h cos φ = d sin φ on two hundred constructed pairs to 4 × 10⁻¹⁵. Pushed along it, the two legs part as the square of the push: the platform shakes and does not move.

spatial · Spatial A third leg that keeps the platform's shake. The two legs of the earlier construction — the second built through a line of force of the first, the dark line — share one twist, whose axis is the dashed line: a screw of pitch -0.049. A third leg is drawn in a third colour, built so that its three joints span a family of motions containing that twist: two joints chosen freely and the third solved for. The platform on all three legs keeps its one freedom. The dotted line is the axis of the one wrench the third leg shares with the first: a screw of pitch 0.201, not a line of force, and 0.244 from the pair's shared line. Dragging picks another third leg from the same family.

A third leg that keeps the shake

Two three-joint legs that share a line of force share one twist as well, and a third leg in general position takes it away. A third leg whose own freedom contains that twist does not, and building one is three conditions on twelve numbers: two joints anywhere and the third solved for. Such a leg must share a load with each of the first two — but not the line they share. What it shares with each is a screw with a pitch of its own, reciprocal to the twist through the sum of their two pitches, and only a third leg built through the pair's line shares the line itself. With any of them the platform still shakes rather than moves.

spatial · Spatial Turning one joint about the shared line tunes the shake's first coefficient to nought. Two legs of three revolute joints each under one platform, sharing a line of force, the dark line, that crosses every joint of both — so they share one twist and the platform shakes along it. The second leg's first joint is turned about the shared line by 0°, which keeps it crossing the line and keeps the shared twist. The shake's second-order coefficient, the reciprocal product of the shared wrench with half the difference of the two legs' bracket terms, is 2.62e-2 here. It passes through nought at 36.488° and again at 109.612°; at those turns the platform still shakes, one order higher.

A shake has an order

Two legs of three joints that share a line of force let a platform shake: pushed along the one twist they share, the legs part as the square of the push. The square's coefficient is a bracket of the legs' own joints, computable without pushing at all, and it can be tuned to nought by turning one joint. The shake does not become a motion. It moves up an order, to the cube, and a second tuned joint moves it to the fourth power. Each order is one more condition, and only legs whose freedoms are groups meet them all.

spatial · Spatial A pair moves until its gap reaches what its joints' play can take up. Two legs of three revolute joints each, sharing one twist, pushed along it: the gap left between their nearest poses, along the one wrench they share, for the pair built through the shared line, the pair with one joint tuned and the pair with two. The horizontal line is the gap the six joints' play can take up, 4.52 times a radial play of 10⁻⁶. Each pair travels until its curve meets the line: built through the shared line, 1.31e-2; one joint tuned, 6.63e-2; two joints tuned, 1.39e-1.

A shake travels as a root of the play

Two three-revolute legs that share a twist let their platform shake along it: the gap between their nearest poses grows as the square of the push, or the cube once one joint is tuned, or the fourth power once two are. With play in the joints, the pair moves until that gap uses up what the play can take, and that is a root: the square, cube or fourth root of the play. At a play of a millionth the three pairs travel 0.013, 0.066 and 0.139 along their twist. A tuned joint is worth more the tighter the joints are, five times the travel at a millionth and eleven at a hundred-millionth.

spatial · Spatial Where two coefficients vanish, a curve; where three do, two points. The second leg's three joints each turned about the shared line by ψ₁, ψ₂ and ψ₃. Every point of the curve makes the shake's second- and third-order coefficients nought, σ₂ = σ₃ = 0, found by continuation from the earlier two-joint tuning (black) both ways, over an arc of 7.25 rad, until it reaches configurations where the two legs come close to sharing a second twist (crosses), where every coefficient runs off to infinity. Left: ψ₂ against ψ₁; right: ψ₃ against ψ₁. The curve is drawn in one colour where σ₄ is negative and another where it is positive; it changes sign at the two dots, (106.31°, 140.35°, 24.66°) and (136.39°, 57.24°, 18.34°), where σ₄ is nought too. Marked: 50% of the way along, at (107.0°, 141.4°, 25.2°), where σ₄ is positive.

Three tuned joints stop at the fifth order

Two legs of three revolute joints that share a line of force let their platform shake, and every joint of the second leg turned about that line clears one more order: the square of the push, then the cube, then the fourth power. A third tuned joint was expected to buy the fifth. Read off the two legs' own Taylor series instead of a push test, the set where the second- and third-order coefficients vanish is a curve in the three tuning angles, and the fourth-order coefficient crosses nought on it twice. At both points the gap falls as the fifth power of the push, confirmed independently with slopes of 4.97 and 4.89. The fifth-order coefficient is −0.0042 at one and 0.109 at the other, and with three angles spent there is nothing left to clear it: three tuned joints stop at the fifth order. And a higher order is not always more travel: in a millionth of play the second of them moves less than the fourth-order pair.

spatial · Spatial

Named alongside it

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

Screw systemReciprocal screwLegTwistPlatformRankClearanceConstraintDegrees of freedomExact-constraintSecond orderMobility

All concepts