Machine Kinematics
solved, then drawn
What a joint is

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.

Assumes Twelve kinds of freedom and Six, and no others.

14 min read 8 figures Solved, not drawnThe singularities are the point

A group of rigid displacements is a set of displacements. It lives in a six-dimensional space, it has no shape, and nobody can draw it.

What can be drawn is what it does to something. Fix one point of the moving body and let every element of the group act on it; what the point sweeps out is the orbit, and it is the only picture a group has.

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.
Fig. 1 One point of the moving body under each of the six lower pairs. A line, a circle, a helix, a cylinder, a sphere and a plane.

The orbit is the surface

The six orbits are the six surfaces the first rung computed the groups from, and that is not a coincidence dressed up as one.

A lower pair’s group is the symmetry group of its surface. So it carries every point of the surface to another point of the surface, which means the orbit of a surface point is contained in the surface. Run the argument the other way and the surface is swept out by the orbits: a shaft’s cylinder is the orbit of any point on it under the rotation, a thread’s helicoid is the orbit under the screw, a plane is the orbit under planar motion.

The classification of joints and the classification of orbits are the same classification asked from two ends. Which surfaces slide on themselves, and which shapes a group can sweep a point over: the same six answers, because a surface that slides on itself is a surface made of orbits.

That is worth having because it makes the six visible without any algebra at all. A reader who has never seen a twist can look at six pictures and see that there are six kinds of joint and why.

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.
Fig. 2 The same six as surfaces rather than as orbits. Every line in every picture here is a curve some point traces in the picture above.

Three shapes for one freedom

The three one-freedom pairs are the clearest case, and the one that most directly answers what the count throws away.

Where R can send one point. The orbit of a single point of the moving body under a rotation, which is a circle. The revolute pair; the surface is any surface of revolution. The orbit is the only honest picture of a group: the group itself is a set of displacements and has no shape, and what a reader can see is what it does to something.
Fig. 3 A revolute pair: a circle. The radius is the point’s distance from the axis and belongs to the point; the fact that it is a circle belongs to the joint.
Where H can send one point. The orbit of a single point of the moving body under a screw, which is a helix. The helical pair; the surface is a thread, and its pitch is the surface's own lead. The orbit is the only honest picture of a group: the group itself is a set of displacements and has no shape, and what a reader can see is what it does to something.
Fig. 4 A helical pair: a helix, whose pitch belongs to the joint rather than to the point. Every point of the moving body traces a helix of the same pitch and a different radius.

A prismatic pair gives a straight line, a revolute a circle, a helical a helix. All three have dimension one and all three give the same mobility count, and the three curves have nothing in common.

The helix is the case worth dwelling on because it makes visible something a table row cannot. The pitch is a rate — length per turn — and it appears in the orbit as the slope of the helix. Change it and every orbit changes; run it to nought and every helix becomes a circle; run it to infinity and every helix becomes a line. The three one-freedom pairs are one family with a real parameter, and the orbit picture is where that is obvious.

Two-dimensional orbits, and why a joint has one

The orbits divide by dimension, and the division is the classification again.

A one-dimensional orbit is a curve, and it comes from a one-dimensional group: the three one-freedom pairs, and nothing else. A curve is not a surface, so a pair whose group is one-dimensional needs a surface of higher symmetry than its own group — which is exactly the shaft with collars, where a two-dimensional cylinder has its slide taken away by two annuli, and exactly the cone and the torus and the spheroid, all of which are two-dimensional surfaces swept by one-dimensional orbits.

A two-dimensional orbit is a surface, and it is the case that makes a lower pair possible at all: the surface is the orbit, and the group carries the whole of it onto itself with nothing left over. A circular cylinder under the cylindrical group, a sphere under the spherical group, a plane under planar motion. Three of the six.

A three-dimensional orbit is a region, and no surface is invariant under it. Every group with one is disqualified from being a joint on the spot: all the translations, Schoenflies motion, the pitched planar group, and the whole displacement group. Four of the six non-pairs, disqualified by a dimension count that takes one line.

That is the argument the census rung makes algebraically, and here it is as a property of pictures: a joint’s orbit fits inside a surface, and most groups’ do not.

What a point cannot see

Now the limit, because the orbit method is persuasive and slightly dishonest and this field should say so in its own voice rather than be caught at it.

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.
Fig. 5 For each of the twelve: its dimension, the dimension of one point’s orbit, and the difference. The difference is the stabiliser — the motions that leave that particular point exactly where it is.

An orbit’s dimension is the group’s dimension minus the dimension of the stabiliser: the subgroup of motions that leave the chosen point exactly where it is. For the one-freedom pairs the stabiliser is trivial and the orbit has the group’s own dimension. For two of the twelve it is not.

Planar motion is three-dimensional and sweeps a point over a plane, which is two. The motion left over is the turn about the normal through that point: it moves the body and does not move the point.

Spherical motion is three-dimensional and sweeps a point over a sphere, which is two. The motion left over is the turn about the radius through that point.

So a single point under a planar pair and a single point under a spherical pair each trace a two-dimensional surface, and the surfaces are different — a plane and a sphere — which happens to save the comparison here. It does not save it in general. A trajectory is evidence about a group and never a determination of one, and the honest procedure is the one the rest of this field uses: take the displacements themselves, take their logarithms, and close them under the bracket. That reads the whole group, stabiliser included.

The site has met this before in a different costume. A calibration cannot see a parameter it does not move, and a marked point cannot see a motion that does not move it, and both are the same statement about a null space.

Two points, and the ambiguity goes

The repair is cheap and worth stating because it is what every mechanism figure in this field actually does.

Watch two points instead of one, not on the same axis, and the stabiliser has nowhere to hide: a turn about the normal through the first point moves the second. Watch three in general position and the whole displacement is determined, since three points fix a rigid body. So an orbit picture with a marked triangle rather than a marked dot is a complete picture of a group, and the reason this field does not draw one is that it is unreadable — three interpenetrating orbit surfaces on a page is a worse picture than one plus a sentence.

The mechanism figures avoid the problem for free, because a mechanism is drawn as bars and pins and a reader sees the whole body’s pose rather than one of its points.

A spherical four-bar, and the group it moves inFour pins whose axes meet at a point. The dashed stubs are the joint axes, the solid marker is one point of link 1 and the faint curve is everywhere that point goes. Every frame is a **solve**: the joint angles are the unknowns, the closure of the loop is the equation, and a frame is drawn only where the residual comes below 10⁻⁹. What this field adds to the picture is one number. Take the displacements this link reaches, take their logarithms, and close them under the bracket: the answer is **3**, so the motion lies inside rotations about a point and composing two of its displacements gives a third one it also reaches, to 3.1e-16. positioned by solving, not by drawing.4 joints · a spherical four-barinside S
Fig. 6 A spherical four-bar, drawn from solved configurations. The marked point traces an arc of a sphere; the links are what make the rest of the pose visible, and it is the pose rather than the point that the closure measurement reads.

A cylinder is two orbits at once

The two-dimensional orbits repay a closer look, because they are where the group’s structure shows up in the shape rather than in a caption.

Where C can send one point. The orbit of a single point of the moving body under a cylindrical pair, which is a cylinder. The circular cylinder, which turns and slides. The orbit is the only honest picture of a group: the group itself is a set of displacements and has no shape, and what a reader can see is what it does to something.
Fig. 7 A cylindrical pair’s orbit. The parameter lines running round it are the rotation and the ones running along it are the slide, and the surface is the two of them together.

A cylindrical group is a rotation and a translation on the same axis, and its orbit is drawn as a mesh of exactly those two families: circles from the rotation, lines from the slide. Every point of the cylinder is reachable from every other in exactly one way, which is what it means for the group’s dimension and the orbit’s to be equal.

A spherical group’s orbit is a sphere, and it is drawn as latitudes and meridians for the same reason — except that here the group has three dimensions and the surface has two, so there is more than one way to get from a point to another, and the ambiguity is the stabiliser. That difference is invisible in the drawing and is the whole content of the table above.

Mechanisms, and where their points go

The reason to build the orbit vocabulary at all is that it turns a set of drawn trajectories into a statement about groups.

Four loops, and where one point of each of them goes. The orbit of one point of the moving link in four overconstrained loops, drawn from solved configurations. Sarrus's platform runs along a straight line, because its displacements are a one-dimensional group of translations. A spherical four-bar's coupler point stays on a sphere. A planar four-bar's stays in a plane. Bennett's does none of those, and the reason is not that its curve is complicated: its displacements are inside no proper subgroup at all, so there is no surface for the point to be confined to. The first three paths are orbits of groups and the fourth is not an orbit of anything.
Fig. 8 One point of the moving link in four overconstrained loops. Three of the four paths are orbits of groups and the fourth is not an orbit of anything.

Read the four together and the field’s whole taxonomy is in one figure. The trajectory of a point gives the dimension of what confines it and, when the group is one of the six that a surface can have, the shape as well. What it never gives on its own is whether there is a group there at all — and the fourth panel is the case where there is not.

Sarrus’s platform runs along a straight line. That is a one-dimensional orbit of a one-dimensional group of translations — the joint every prismatic pair gives, produced here by six revolutes and no slide anywhere. The spatial field measured it straight to 9.8×10−169.8 \times 10^{-16} of its span by solving the mechanism sixty times; it is a straight line because a translation group’s orbits are lines, and two rungs from here there is a two-line argument for why the group is a translation group.

A spherical four-bar’s coupler point stays on a sphere. A two-dimensional orbit of the three-dimensional spherical group, and the point stays exactly on it — which the spatial field also checks directly, as a constraint that never drifts.

A planar four-bar’s coupler point stays in a plane, and inside that plane traces a sextic. The sextic is a fact about the four lengths; the plane is a fact about the group.

Bennett’s coupler point does none of those, and the reason is not that its curve is complicated. Its displacements are inside no proper subgroup at all, so there is no surface for the point to be confined to and no orbit for its path to be part of. The first three paths are orbits and the fourth is a curve, and telling those apart is what the rest of this field is for.

Where the orbit picture earns its keep

Two arguments in this field are made almost entirely in orbits, and they are worth flagging now so the pictures later are read as evidence rather than as decoration.

The first is why some motions cannot be a joint. Schoenflies motion sweeps a point over a three-dimensional region; no surface contains it; therefore no lower pair gives it; therefore a machine that needs it has four joints. The whole argument is a dimension of an orbit, and the classification rung is where it is spelled out.

The second is why Sarrus’s platform goes straight. Two arms, each holding the platform inside a planar group whose orbits are planes; the platform is in both, so its points are in both planes; two non-parallel planes meet in a line. That argument has no solve in it at all, and the spatial field’s measurement of the same fact took sixty converged configurations. Both are worth having: the orbit argument says why it must be a line and the measurement says that the mechanism as built delivers one.

There is a third use that is quieter and runs through every figure here. An orbit is a set the mechanism cannot leave, so an orbit picture is a refutation waiting to happen: draw the surface, draw the mechanism’s actual traced path, and any departure is visible without measuring anything. That is the same reason the site draws a coupler curve and the linkage that traced it in one figure rather than two.

The picture and the number

Two habits come out of this rung and both are used from here on.

Draw the orbit, quote the closure. Every figure in this field that shows where something goes is paired with a dimension that says which group it went in. The picture is what a reader can check against their intuition and the number is what a reader can check against another mechanism.

Never infer the group from the picture. The stabiliser is invisible in an orbit, the sampling range decides what a set of displacements even looks like, and a curve that lies in a plane may belong to a mechanism that is not planar. The picture is the caption on the measurement, not the other way round.

A probe is a joint, and it cannot see its own symmetry

The stabiliser is what an orbit cannot show, and there is a way of reading that which makes the whole limitation predictable rather than something to be remembered case by case.

Watching a point is attaching a ball joint. A marked point has no orientation, so it responds to exactly the part of a motion that moves points and is blind to exactly the part that fixes them — which is the stabiliser. That is the same information a spherical pair transmits: a spherical pair’s own permitted set is three rotations about the contact centre, and those are precisely the motions a point at that centre does not see.

So the ambiguity between planar and spherical motion at one point is not an accident of those two groups. It is the statement that both have a one-dimensional stabiliser of a general point — the rotations about the point for planar motion, the rotations about the radius through it for spherical — and a probe with a three-dimensional blind spot cannot separate two groups that differ inside it.

Read that way, the repair is obvious before it is derived. Watching two points is attaching a rigid link with a ball at each end, which has a one-dimensional stabiliser of its own — the rotation about the line joining them — so it sees everything except motions about that axis. Watching a point and a direction is attaching something with no stabiliser at all, and it sees everything.

A measurement’s blindness is its own symmetry group, then, and the choice of what to watch is a choice of probe with a stabiliser attached. That is the same statement as what a calibration cannot see and it is why the two arguments have the same shape: a parameter invisible to a measurement is a parameter lying in the measurement’s stabiliser.

It also says how much watching is enough, which a case-by-case treatment cannot. To identify a group of dimension dd from orbits, the probe’s stabiliser must intersect the group trivially — and the cheapest probe with a trivial stabiliser is two points off a common axis. That is why every mechanism figure in this field draws bars and pins rather than one dot: a drawn mechanism is a probe with no symmetry, for free, and its pictures cannot suffer the ambiguity a single marked point does.

The orbit of a point that is on the axis

One more thing an orbit picture depends on, and it is the reason every figure in this field marks a point in general position rather than a convenient one.

Put the marked point on a revolute’s axis and its orbit is a single point: the stabiliser is the whole group and the picture says nothing at all. Put it on the axis of a helical pair and the orbit is a straight line, which is a prismatic pair’s picture drawn by a screw. Put it at the centre of a spherical pair and it does not move.

None of that is an edge case to be dismissed; it is the normal situation in a real mechanism, where the interesting points are on axes because that is where the next joint is. It means an orbit figure has to state which point it is drawn for, and that the shape in it belongs to a pair — the group and the point — rather than to the group alone.

The library’s orbit routines take the point as an argument for that reason, and the figures in this field all use the same one, off every axis and away from every centre, so that two orbit pictures can be compared. Where a figure needs a different point it says so.

That is the same discipline the site applies to positions — nothing is drawn that was not solved — moved one level up, to the thing being drawn rather than to where it is.

The next rung takes the vocabulary and starts composing with it. A chain’s displacements are the product of its joints’ groups, one factor per joint, and the orbit of the product is where the tool can go — so the question what can this arm reach and the question what group is this arm in turn out to be the same question asked at two dimensions apart, and only one of them has a clean answer.

Where this goes next

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. 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. Sarrus, as two planes meeting in a line. Each arm of Sarrus's linkage is three pins with parallel axes, so each arm holds the platform inside a planar group — the one whose normal is that arm's axis direction. The platform has to satisfy both, so what it may do is the intersection, and the intersection of two planar groups whose normals are not parallel is the one-dimensional group of translations along their common perpendicular. The platform goes up and down and does nothing else, and that is the exact straight line the spatial field measured to 10⁻¹⁶ of its span — arrived at here with no mechanism solved and no tolerance anywhere. Drag the arms towards each other: the answer is a translation at every angle but zero, where the two groups become one group and the intersection jumps to three dimensions. That is the linkage built flat, and it is the configuration in which it stops being a straight-line mechanism. Two planes meeting in a line Sarrus's linkage draws an exact straight line out of six pin joints, and the spatial field proved it by solving the mechanism sixty times and measuring a departure of 9.8 × 10⁻¹⁶. Here the same fact comes out of two planes and a cross product, with no mechanism solved anywhere — and the two routes are not redundant, because only one of them can tell you the linkage as built delivers it.

About the same objects

Not linked from either essay — found by the objects both name.

What links here

Essays that link to this one from their own argument.

The objects this essay names

Each one links to every other essay that touches it.

Displacement subgroupLower pairOrbitPitchScrewSelf-sliding surfaceStabiliserTwist