Machine Kinematics
solved, then drawn
Drawn wrongly

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.

Assumes A joint is a surface that slides on itself.

15 min read 8 figures Two routes to the same numberExact and approximateThe singularities are the point

Six things that get said about joints. None of them is careless, every one is exactly right about something, and every one is answered here with a number from the surface census, the closure test or the composition experiment.

The pattern is the one this site keeps finding: a quantity that is correct becomes a description that is not, because the quantity is a dimension and the object has more in it than a dimension.

Three of the six are about what a count cannot see, two are about what a first-order measurement cannot see, and one is about a joint that is not the kind of thing the whole apparatus applies to. That ordering is roughly the order of how much damage each does: the first three lose a distinction, the next two lose it while appearing to have measured something, and the last one applies a vocabulary where there is nothing for it to describe.

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. 1 The table the rest of this essay argues from. Twelve groups, their dimensions, and what one point sees of each.

One: a freedom count describes a joint

It describes its dimension, and the dimension is one number out of a classification with twelve entries.

Three of the twelve have dimension one — a rotation, a translation and a screw — and they are the revolute, prismatic and helical pairs. Every count on this site reports one for each. The rank of the constraint Jacobian reports one for each, because it is measuring the same dimension by a different route. And the three are not related: a point under them traces a circle, a line and a helix.

The consequence is not confined to a single joint. Three revolutes with parallel axes give planar motion; three whose axes meet at a point give spherical motion; three at random give a three-parameter set inside no proper group at all. Same joints, same count, same rank, and one of them is a wrist while another is a mechanism whose reachable poses need six numbers to describe. Swapping one pin for another that “has the same freedom count” is a change of group, and it changes everything downstream of the joint.

The helical pair is where the loss is largest, because its group carries a real number. Two threads of different lead are two different joints, not two settings of one, and every displacement either permits is different from every displacement the other permits. A count has nowhere to put a real number, and the mechanisms that live on the difference are invisible to it.

One more consequence, because it is the one that reaches furthest. The count is what type synthesis searches over: choosing which chain to build before choosing any dimension is a search over graphs whose edges are joints, and the topology field’s census ran that search with every joint a revolute and said so. With six kinds of joint the same census is a census of labelled graphs, which is a genuinely larger problem — and the labels are not decoration, because two labellings of one graph are two mechanisms with nothing in common.

Two: the screw system says which group the motion is in

It says which group the motion is in to first order, which is a different statement and is sometimes a different answer.

A screw system is the tangent space to a mechanism’s motion at a configuration. Every set equals its own tangent space near a point — that is what a tangent space is — so a measurement made near the identity returns the tangent space whatever the mechanism is.

Why this measurement is not a derivative. The honest failure mode of the field's instrument, drawn rather than hidden. Every set looks like its own tangent space near the identity — that is what a tangent space is — so a chain sampled over a thousandth of a radian reports the dimension of its velocities, which is the number the screw system already gives. Four pins at random has four joints, and sampled over 10⁻⁹ radians its displacements occupy four dimensions; sampled over two radians they occupy six. The step is at 10⁻⁶, which is where the departure from the tangent space falls below the rank tolerance — so the position of the step is a fact about arithmetic and the two plateaux are facts about the mechanism. A group is a statement about displacements you could compose, and no derivative can make it.
Fig. 2 Four pins with axes at random, measured at twelve sampling ranges. Sampled narrowly the displacements occupy four dimensions, which is the screw system’s answer; sampled at ordinary amplitude they occupy six.

Four random pins sampled over 10−910^{-9} radians report a four-dimensional set of displacements — the same answer a SCARA arm gives, whose motion really is inside a four-dimensional group. The two are told apart only above 10−610^{-6} radians, and the whole of that is a rung of this field.

The sophisticated version of the first-order route does much better: measuring how far the screw system turns separates a planar four-bar from Bennett’s linkage at 2×10−62 \times 10^{-6} degrees against 89. It still cannot name the group, and Sarrus’s linkage is the case where naming it is the entire content — a translation, one-dimensional, and a drift measurement has no way to say so.

The right reading is that a screw system and a displacement group are two different objects that a mechanism has, and that the first is the tangent space to the second when the second exists. For a mechanism whose motion is a group the screw system is constant and equal to the group’s algebra — which is exactly the drift measurement’s zero, arrived at from the other side. For a mechanism whose motion is not a group there is no second object at all, and the screw system is a tangent space to a manifold.

Three: six lower pairs is an engineering convention

It is a classification, and the six come out of a computation that has no convention in it.

Sample a surface, write one row per point saying that a twist’s velocity there is tangent, take the null space. Eleven surfaces — a plane, a sphere, a circular cylinder, a prism, a thread, a cone, a torus, an ellipsoid of revolution, a shaft with collars and two with no symmetry at all — give six distinct groups, with the weakest rank decision separated by a factor of 6.6×10156.6 \times 10^{15}.

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.
Fig. 3 The census. Three surfaces of revolution that look nothing alike give the same joint, and two ordinary surfaces give none at all.

Adding surfaces does not add groups, and the reason is a dimension count rather than a survey: a surface is two-dimensional, so no group whose orbits are three-dimensional can leave one invariant, which eliminates all the translations, Schoenflies motion and everything above them at a stroke.

The convention story also predicts the wrong thing about the near misses. If the six were the ones somebody found convenient, two translations without a turn would be on the list — it is a motion every X–Y table has and every drawing board uses. It is not, because the surface that would give it is a plane and a plane’s own symmetry group is larger.

There is also a positive result hiding in the census that the convention story cannot produce. The helicoid is generated with a lead of 0.40.4; nothing tells the computation what the lead is; and the pitch of the twist that comes back is 0.4000000000000.400000000000, out by 5×10−165 \times 10^{-16}. A convention does not have a twelfth decimal place.

Four: watching where a point goes identifies the joint

It identifies the orbit, which is the group’s dimension minus the stabiliser’s, and two of the twelve lose a dimension that way.

Planar motion is three-dimensional and sweeps a chosen point over a plane, which is two. Spherical motion is three-dimensional and sweeps a chosen point over a sphere, which is two. In each case there is one motion that moves the body and does not move the point — a turn about the normal, a turn about the radius — and a single marked point is blind to it.

Where S can send one point. The orbit of a single point of the moving body under rotations about a point, which is a sphere. The spherical pair; the surface is a sphere. 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 spherical group’s orbit: a sphere, two-dimensional, from a three-dimensional group. What the point cannot see is the turn about its own radius.

The site has met the same shape of blindness before: a calibration cannot see a parameter that does not move the tool, and the two are the same statement about a null space. The repair is the same too — watch more than one point, or read the displacements rather than the positions, which is what every measurement in this field does.

There is a version of the claim that survives, and it is worth separating out because it is the useful half. A trajectory is evidence about a group: a path that leaves a plane rules out planar motion, and a path confined to a sphere is consistent with spherical motion and with several smaller things. What it never is, on its own, is a determination — and the four loops whose paths are drawn side by side include three that are orbits of groups and one that is not an orbit of anything, which no amount of looking at the fourth curve would establish.

Five: a nominally aligned machine behaves like an aligned one

Not in this quantity, because the quantity is not continuous.

Three revolute joints with exactly parallel axes have a displacement span of three and their motion is planar. Tilt two of the axes by 0.060.06 radians and the span is six. There is no intermediate value at any tilt: the dimension jumps at exactly zero, because being in a group is a measure-zero condition among sets of twists — eighty thousand random subspaces and not one of them closed.

How many dimensions each chain's displacements occupy. Every chain in the field, with the dimension its reached displacements' logarithms occupy. A chain of n joints always has n freedoms; what varies is whether those freedoms compose. Where the bar equals the joint count the motion is inside a group and the group is named; where it reaches six there is no proper group containing the motion, and the two chains that do are the ones whose axes were chosen at random. Nothing about the joints themselves differs — three pins are three pins, and the two rows differ only in where the axes point.
Fig. 5 The two rows in the middle are the same three joints, once with axes exactly parallel and once tilted by six hundredths of a radian. Every count, every rank and every mobility is identical.

What is not discontinuous is the defect — how far the brackets lie outside the span, as a fraction of their own length. It goes to zero with the tilt, it is what the closure test measures, and it is the number to quote for a machine made to a tolerance. Reading the dimension where the defect was wanted gives either false precision or false alarm, which is the same trap the practice field describes for every exact statement on this site.

The engineering consequence is sharper than it looks. A SCARA arm’s tool face is level at every configuration because its three axes are parallel, and an arm whose axes are out of parallel does not have slightly worse Schoenflies motion — it has none, and its tool tilts by an amount proportional to the misalignment as it sweeps. The specification that keeps the machine in its group is a parallelism, and nothing in a mobility count identifies it as the specification that matters.

Six: a higher pair is a joint with more freedoms

It has more freedoms and it is not the same kind of object, because its permitted set is not a group.

A disc on a straight edge may slide along the edge and may turn about its own centre, and both keep the contact exactly. Compose the two and the contact lifts off the edge by ∣t2sin⁡φ1∣|t_2 \sin \varphi_1|, which agrees with the composition to 4×10−164 \times 10^{-16} and reaches 1.2 radii over an ordinary range.

Two permitted motions, and a composite that lifts offA 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 1.164 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.where a tangent disc's centre must be1.164 rtwo freedoms, both permittedcomposite off the edge by 1.164 radii
Fig. 6 Two permitted displacements, and a composite that is not one. At the largest turn the slider carries, the disc ends up more than a radius clear of the surface it was resting on.

So the two freedoms are real, the count that reports them is right, and there is no group for the count to be the dimension of. Every property this field extracts from a group is unavailable: no orbit that is a surface, no guarantee that transports, no intersection rule, no name from a list of twelve.

The three-dimensional case makes the same point without composing anything. A ball on a plane has five freedoms, and there is no five-dimensional subgroup of the rigid displacements at all — checked on twenty thousand random five-dimensional subspaces, every one of which generated all six.

A seventh that is not on the list

One claim that sounds like it belongs here and does not, because it is true.

A mechanism’s mobility is what the count says it is, corrected by the redundancies. That is right, it is what the constraint field has been doing since the foundation, and nothing in this field disturbs it. The count gives a dimension; the rank measures the same dimension independently; where they disagree the count is the one that is wrong, and the site has several worked cases of exactly that.

It is worth saying explicitly because an essay of six refutations invites the reading that the counting apparatus is discredited. It is not. Every mobility number this site has published stands, every rank measurement stands, and the two-route discipline that produced them was and is the right one for the question it answers.

What has changed is that there is now a third route, to a different question, and the third route can be wrong in ways the first two cannot — it depends on a sampling range, it is computed from solved configurations only, and it reports a dimension that jumps discontinuously. Those are stated in its own rung rather than buried, and the reason to state them is the same reason this essay exists.

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.
Fig. 7 The twelve as a bill of materials, which is the shortest answer to the third claim and half of the answer to the first: what a motion costs in joints is decided by whether it is the symmetry group of a surface, and six of the twelve are.

Check one level past the claim

The observation that each claim fails one step past where the checking stops is the useful half of the list, and it can be turned into an instruction rather than left as a diagnosis.

Look at where the checking stopped in each case. The freedom count was checked on one joint, where it is correct, and fails on a chain. The screw system was checked at one configuration, where it is correct, and fails over a range. The pair census was checked against the surfaces somebody uses, where it is correct, and would fail against a census if there were more groups to find. The nominal alignment was checked on a perfect machine and fails on a made one. The higher pair was checked on one displacement and fails on a composite.

Every one of those is the same move: the claim is verified on the smallest instance and asserted on the general one. Which gives the rule its form — check one level past the claim. For a statement about a joint, exhibit a chain. For a statement about an instant, sweep a range. For a statement about a derivative, compose two finite motions. For a statement about an ideal geometry, perturb it.

That is cheap, and it is cheap in a specific way worth noticing: the extra level is almost always available with the machinery already written. Composing two displacements needs no new code once displacements exist. Sweeping a range needs a loop. Perturbing an alignment needs a parameter. In every case on this list the refuting computation is smaller than the essay describing it.

It also predicts where the next one will come from, which is what a rule of this kind is for. Every quantity this field computes at a configuration is a candidate for failing over a range; every quantity computed for one pair is a candidate for failing on two. The list has six entries because six such quantities have been looked at, and the arithmetic says there are more of them than have been checked.

And it explains why the failures cluster at exactly two joints and two configurations rather than at ten. A claim that survives one extra level usually survives every level, because the structure that would break it breaks it immediately — composition fails at two elements or not at all, and a subspace that is not a subalgebra shows it at the first bracket. The dangerous step is always the first one past the checking, which is also the cheapest one to take.

What the six have in common

Every one of them is a correct statement about a dimension being read as a statement about an object.

The count is a dimension. The rank is a dimension. The screw system is a tangent space, whose dimension is the same dimension again. All three are right, all three are cheap, and all three were the only instruments this site had for twenty-two fields.

Four instruments, and only the last one names the group. Every instrument this site has for an overconstrained loop, on the same six mechanisms. Kutzbach's count gives −2 for a planar four-bar and −2 for Bennett's. The rank of the constraint Jacobian gives three and three. Both are right and neither separates them. The last two columns are this field's: the span is how many dimensions the logarithms of the displacements the moving link actually reaches occupy, and closes at is the dimension after those are closed under the bracket. A planar four-bar closes at three and the three are planar motion; Sarrus closes at one, a translation, which is the exact straight line the spatial field measured by solving the mechanism sixty times. Bennett closes at six: its displacements occupy four dimensions and no group smaller than all of them contains those four. That is what "paradoxical" has meant on this site for six phases, stated as an integer.
Fig. 8 Four instruments on six mechanisms. The first two agree about a dimension on every row, including the two rows where the mechanisms have nothing in common.

What a dimension leaves out is which of twelve things it is the dimension of, and the answer decides what a chain of joints produces, what a loop of them can do, whether a platform translates or turns, and whether a mechanism that ought not to move does. None of that is recoverable from a number, and all of it is recoverable from the surfaces.

There is a second thing they share, and it is the reason the list is worth writing rather than merely being right. Every one of the six is exactly correct in the case a person would check. A single joint, watched on its own, is described adequately by its freedom count. A mechanism sampled near one configuration is described adequately by its screw system. A machine whose axes are parallel to a thousandth behaves, over a short stroke, like one whose axes are parallel. A point’s path does identify the group for the six lower pairs, because their orbits are six different shapes.

The claims fail one step past where the checking stops — at three joints instead of one, at a finite stroke instead of an instant, at a long stroke instead of a short one, at twelve groups instead of six. That is the same shape as the chains field’s six, which are all exactly right up to eight links and wrong at ten, and it is the reason this site keeps building instruments rather than arguments: an instrument can be run one step past where anybody looked.

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. 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. 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.

About the same objects

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

The objects this essay names

Each one links to every other essay that touches it.

Displacement subgroupHigher pairKinematic pairLie bracketLower pairMobilityOrbitScrew systemStabiliserTolerance