Machine Kinematics
solved, then drawn
Concept

Lower pair — where it appears

A joint whose contact is over a fitted surface rather than at a point or a line — a pin, a slide, a ball, a screw. There are six of them and no more, and each removes a different number of a body's six freedoms.

Named by 17 essays across 6 fields — each of them below, with the objects they name alongside it.

Six surfaces, six groups, six pairs. The six lower pairs, drawn as the surfaces they are. A lower pair is two bodies touching over a surface rather than at a point or along a line, and that is the same thing as saying the surface slides on itself — so what the joint permits is the surface's own symmetry group. Each caption is computed from the surface's normals and not from the pair's name: a plane gives three freedoms and planar motion, a sphere gives three and spherical motion, a plain cylinder gives two, a shaft with collars gives one rotation, a prism gives one translation, and a thread gives one screw whose pitch comes back as the thread's own lead. Eleven surfaces were tried and six groups came out, which is where the number in every textbook's table comes from.

A joint is a surface that slides on itself

Twenty-two fields of this site have declared their joints and then counted what those joints take away. A count cannot tell a pin from a slide — both are one. What a joint actually permits is a set of displacements closed under composition, and it can be computed from the shape of the surface: one linear condition per point, and the answer is a null space.

pairs · Pair A 3-stage stack at 35°. Every bar is the same length and every stage is at the same angle, which is a consequence of the solve rather than an assumption in the drawing: the pins at the crossings and the roller on the ground leave one freedom, and it has been used. The platform is at 1547 mm and the ram — the heavy line from the ground pin — is 428 mm long. Moving the ram one millimetre from here raises the platform 5.0 mm; at the bottom of the travel the same millimetre is worth 17.6 mm.

A roller is not a slider

A scissor lift has one degree of freedom, at every height and for any number of stages. Grübler's criterion agrees — if the rollers under it are counted as pins in slots. Count them as slider blocks, which is how every textbook draws a slider, and the same formula declares a machine holding a car in the air to be a structure with minus one.

applied · Mobility Every surface tried, and the group it permits. The census the six lower pairs come out of. Each row is a surface, sampled at 240 points; the freedoms column is six minus the rank of a matrix with one row per point, saying that the velocity a twist gives that point is tangent to the surface. Nothing is fitted and no shape is recognised — the surface's own normals write the matrix down. Three different surfaces of revolution give the same group, which is the content of the classification; two surfaces give nothing, which is what almost every surface gives. Eleven surfaces, six groups. The last column is the ratio of the smallest singular value kept to the largest discarded, so a row reading 10¹⁵ is not near being reclassified by anybody's tolerance.

Six, and no others

Eleven surfaces were handed to the same computation and six groups came out. A cone, a torus and an ellipsoid of revolution give the same joint; a scalene ellipsoid gives none; and the list does not grow when more surfaces are added, because a surface's symmetry group has to leave a two-dimensional set alone and only six groups can.

pairs · Pair One point, six groups, six shapes. A group has no picture, so here is the next best thing: fix one point of the moving body — the marked one — and draw everywhere the group can send it. A prismatic pair sends it along a line, a revolute round a circle, a helical along a helix, a cylindrical over a cylinder, a spherical over a sphere, a planar over a plane. Those six shapes are the six surfaces the previous figures drew, which is not a coincidence and is the field's first argument read backwards: a lower pair's surface is an orbit of its own group, which is exactly why the surface can slide on itself.

The count cannot tell a pin from a slide

A revolute, a prismatic and a helical pair all take five freedoms away in space and leave one. Grübler adds the same number for each, the constraint rank measures the same number for each, and the three joints have nothing whatever in common — one sends a point round a circle, one along a line, and one along a helix at a rate the joint decides.

pairs · Pair The twelve kinds of freedom, and which are joints. Every connected group of rigid displacements, up to where its axis points and where its origin sits. There are twelve, the height on the page is the dimension, and a line means the lower one is contained in the upper — computed by asking whether each generator of the smaller lies in the span of the larger, with all twelve built about a common axis. Filled discs are joints: six of the twelve are the symmetry group of a surface and can be a single pair, and six are not and have to be built out of a chain. There is nothing at dimension five, which is not obvious and is checked rather than assumed: twenty thousand random five-dimensional subspaces of the twists were closed under the bracket, and every one generated the whole of the six.

Twelve kinds of freedom

Every set of displacements that is closed under composition is one of twelve, up to where its axis points. Six of them are joints somebody sells. Four are motions a designer may perfectly well want and cannot buy at any price. And there is nothing at all of dimension five — checked here on twenty thousand random subspaces, every one of which generated the whole of the six.

pairs · Pair What each kind of joint takes away. Grübler's formula is M = 3(n − 1) − 2j₁ − j₂, and the 2 and the 1 in it are not conventions. A lower pair — a pin or a slide — holds two bodies together over a surface and leaves one relative freedom, so it costs 2. A higher pair — a cam against a follower, a wheel on a rail — touches at a point, the contact travels along both surfaces, and it costs 1. Five chains, each built and each measured from the rank of its constraint Jacobian, which has never heard of the formula. The last row is the one worth having: count that cam contact as a pin, as is very easily done, and the formula returns 0 where the mechanism has 1. The Jacobian does not move.

What each joint takes away

Grübler's formula has a 2 in it for pins and a 1 for cam contacts, and those numbers are not conventions to be memorised. They are the number of constraints each kind of joint imposes, they are measurable as the rank of a matrix, and miscounting one of them is the commonest way the formula is got wrong.

constraint · Mobility A pin in a hole is a short link. Left: a pin of radius 0.86 in a hole of radius 1, so the clearance is 0.14. The pin's centre may sit anywhere within that of the hole's centre. Right: the same joint as it enters the kinematics — a binary link of fixed length 0.14 and free direction, with a revolute at each end. That is not an analogy. It is the same set of relative positions, so every count, every Jacobian and every solve on this site applies to it unchanged, and a four-bar with play at each pin is a mechanism with eight links and eight joints.

A clearance is a link

A pin in a hole is not a joint at a point. Its centre may sit anywhere within the difference of the two radii, so the two links it joins are connected by a body of fixed length and free direction — a binary link with a revolute at each end. That is not an analogy, and taking it literally makes a four-bar a mechanism with eight links, eight joints and five degrees of freedom.

practice · Clearance crank rocker: the closest pair at one position. The site's standard four-bar: ground 4, crank 1, coupler 3.5, rocker 3. Every joint is where the solver put it, exactly as in the linkage field; the material is the only thing added. The heavy segment joins the two closest points over every pair of parts that is tested — which excludes pairs sharing a pin, since their material surrounds that pin by construction — and its length is the gap: 0.5438 here, between coupler · frame. A negative value is a penetration depth, the distance the pair would have to be moved apart, and it is drawn in the warning colour.

Two bars that have to cross

The site's own four-bar has its coupler inside its frame by a full link width for the whole of a turn. It is not an impossible mechanism; it is a mechanism that cannot be built in one plane — and the plane it has been drawn in for twenty-three fields was a convenience nobody had to pay for.

bodies · Body One point, six groups, six shapes. A group has no picture, so here is the next best thing: fix one point of the moving body — the marked one — and draw everywhere the group can send it. A prismatic pair sends it along a line, a revolute round a circle, a helical along a helix, a cylindrical over a cylinder, a spherical over a sphere, a planar over a plane. Those six shapes are the six surfaces the previous figures drew, which is not a coincidence and is the field's first argument read backwards: a lower pair's surface is an orbit of its own group, which is exactly why the surface can slide on itself.

What a point sees

A group of displacements has no shape, so the only picture of one is what it does to something. Fix a point and the six lower pairs draw a line, a circle, a helix, a cylinder, a sphere and a plane — the six surfaces the pairs are made of. And two of the twelve sweep the same surface and are still different groups, which is the honest caption on the whole method.

pairs · Pair What it takes to build each of the twelve. The same twelve, read as a bill of materials. Six of them are one joint, because a lower pair permits the whole symmetry group of its surface and those six groups are exactly the symmetry groups surfaces have. The other five with a dimension take a chain: two slides for planar translation, three for Cartesian motion, a thread and two slides for the screw-in-a-plane group, and three parallel pins with a slide along them for Schoenflies motion — which is a SCARA arm, and is why a pick-and-place machine has four joints and not one. The group each chain produces is measured from four hundred sampled poses rather than declared, and every row agrees.

The freedom that is a set

Grübler's rule has been on this site since its first essay, and it adds up numbers. Each of those numbers is the dimension of a group of displacements, and the group has eleven siblings the number cannot distinguish. The count is not wrong; it is a projection, and this is what the projection discards.

constraint · Mobility How close a synthesis puts two pins. The distribution of the smallest pin-to-pin distance on any one link, over all 1,176 exact syntheses. Burmester's construction returns points, and points can be arbitrarily close together: the shortest here is 0.103, on a mechanism whose poses span more than two units. The shaded band is what a boss of radius 0.25 forbids — 148 of them, 12.6%. The link that is worst is most often the crank, which is not where a designer looks: the frame's two ground pivots are the pair everybody checks by eye.

A pin is not a point

A joint in the fields before this one is a name and two coordinates. A pin is a cylinder with material round it, a length through the stack of plates, and a head — and every one of those turns some construction that returns points into a construction that may return nothing buildable.

bodies · Body Which pin's play costs the most. Each pin's clearance taken one at a time, at 0.01 on links of 1 to 4, with the direction swept rather than assumed. The ranking runs A 32%, B 25%, O₂ 22%, O₄ 21% — a spread of 1.52 against the 3.43 the four lengths spread over. Clearances are more evenly weighted than length tolerances because each pin joins two links and so enters two of the four sensitivities, which is why the best bearing buys less than the best-held length does — and why it still goes somewhere the load path does not suggest.

Which pin to buy

The four lengths of a four-bar contribute 38, 26, 25 and 11 per cent of its output error — a spread of 3.4. Its four pins contribute 32, 25, 22 and 21 — a spread of 1.5. Clearances are more evenly shared than length tolerances, because every pin joins two links and so appears in two of the four sensitivities, and that changes what a better bearing is worth.

practice · Clearance Two permitted motions, and a composite that lifts off. A disc resting on a straight edge — a roller follower on a flat-faced cam, and the simplest higher pair there is. Two bodies touching at a point rather than over a surface, two freedoms: slide along the edge, and turn, because a disc is its own symmetry group about its centre. Both are permitted and both keep the contact exactly. Their composite does not. The faint discs are the two permitted displacements taken separately; the solid one is one followed by the other, and its centre sits 1.049 radii off the dashed line where a tangent disc's centre has to be. The excursion is exactly |t₂ sin φ₁| — the second displacement's slide times the sine of the first one's turn — derived from the two displacements rather than from the composition and agreeing to 10⁻¹⁶. A lower pair's freedoms compose and a higher pair's do not, which is why a joint's freedom count is the dimension of a group in one case and the dimension of nothing in the other.

A higher pair has no group

A disc resting on a straight edge may slide along it and may turn about its own centre. Both keep the contact exactly. Do one and then the other and the contact lifts off the edge by |t sin φ| — up to 1.2 radii over an ordinary range — so the two freedoms are real and the pair of them is not closed. The count is still right and there is nothing for it to be the dimension of.

pairs · Pair What one point can and cannot see of a group. For each of the twelve: the group's dimension, the dimension of one point's orbit under it, and the difference — the stabiliser, the motions that leave that particular point exactly where it is. The orbit is the only picture a group has, and this table is the honest caption on it. Planar motion and spherical motion are both three-dimensional and both sweep a point over a two-dimensional surface, so each leaves one motion doing nothing at all: a turn about the plane's normal in one case, a turn about the radius in the other. A point does not see the whole group, and no drawing of one trajectory can be a complete picture of what a joint permits.

Six things a joint is not

A freedom count read as a description, a screw system read as a group, a pair list read as a convention, a trajectory read as a determination, a nominal alignment read as a delivered one, and a higher pair read as a larger joint. Six claims, each of them what a careful person would say, each answered with a number.

wrong · Misconception slider crank: the closest pair at one position. Crank 1, connecting rod 3, the block sliding on the frame's own line. Every joint is where the solver put it, exactly as in the linkage field; the material is the only thing added. The heavy segment joins the two closest points over every pair of parts that is tested — which excludes pairs sharing a pin, since their material surrounds that pin by construction — and its length is the gap: -0.1400 here, between rod · guide, upper. A negative value is a penetration depth, the distance the pair would have to be moved apart, and it is drawn in the warning colour.

The block in the guide has a length

A prismatic pair is a point constrained to a line, and a point on a line of length G has a stroke of G. A block of length ℓ has a stroke of G − ℓ, because both its ends have to stay on the rails — so a guide is as long as the stroke plus the block, and a third of a short one is not stroke at all.

pairs · Pair Oldham's coupling: two slides between two offset shafts. An input hub on one axis and an output hub on a parallel axis 0.6 away, joined by a disc that slides in a slot across the input hub and carries a tongue across the output hub at 90° to the slot. At an input angle of 35° the disc has slid 0.491 along the slot and the output hub -0.344 along the tongue, and the output hub has turned through exactly the input's angle, because neither slide can change an orientation. The disc's centre, the dot, runs round the dashed circle of diameter 0.600 — the offset divided by sin 90° — and goes round it 2 times for each turn of the shafts.

A coupling that only translates

A coupling between two parallel, offset shafts turns its output at exactly the input's speed when, and only when, the relative motion of its two hubs contains no rotation — when it lies in the translation group. Oldham's two slides give that group by construction and so do two equal parallel cranks; a four-bar that is not a parallelogram gives the whole planar group and its output wanders by more than a radian. And Oldham's right angle is not what makes the ratio one: it is what makes the slides slide least.

pairs · Pair Every surface tried, and the group it permits. The census the six lower pairs come out of. Each row is a surface, sampled at 240 points; the freedoms column is six minus the rank of a matrix with one row per point, saying that the velocity a twist gives that point is tangent to the surface. Nothing is fitted and no shape is recognised — the surface's own normals write the matrix down. Three different surfaces of revolution give the same group, which is the content of the classification; two surfaces give nothing, which is what almost every surface gives. Eleven surfaces, six groups. The last column is the ratio of the smallest singular value kept to the largest discarded, so a row reading 10¹⁵ is not near being reclassified by anybody's tolerance.

The pair a catalogue sells

A plain bearing is a cylindrical pair and a catalogue calls it a bearing. Add two thrust faces and it is a revolute pair, which is a different joint and changes every mobility count downstream. The kinematic identity of a bought part is decided by which surfaces touch, and the catalogue's word for it is not the same information.

practice · Clearance

Named alongside it

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

Displacement subgroupConstraintMobilityDegrees of freedomKinematic pairHigher pairRankPitchClearanceGrübler's criterionOrbitScrew

All concepts