Machine Kinematics
solved, then drawn
Concept

Degrees of freedom — where it appears

The number of independent coordinates a body or mechanism still has once its constraints have been applied. It is what a mobility count is counting, and for a mechanism with a velocity constraint it is not the same number as the dimension of the set of configurations that can be reached.

Named by 33 essays across 10 fields — each of them below, with the objects they name alongside it.

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.

A constraint that only pushes

Every constraint on this site so far has been an equation: a pin holds two points together, a bar holds two apart, a mesh holds a ratio. A part resting against another part says only *do not come closer* — so what it may do is a cone rather than a subspace, and whether it can move at all stops being a rank.

holding · Restraint 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 Kutzbach's count against the measured mobility. Five closed loops of revolute joints. The count is 6(L − 1) − 5j, a statement about how many links and joints there are; the measurement is the number of joints minus the rank of the loop's screw system, which knows only where the axes point. They disagree for four of the five, and the one they agree on is the generic seven-joint loop — so the formula is not broken, it is blind to the special geometry that makes the other four work. The universal joint is counted at -2 degrees of freedom and is in every car built.

Six freedoms, not three

Every mechanism on this site so far has been flat, and flatness is not a simplification made for teaching — it is a special case that hides the most interesting thing constraint counting does, which is get the answer wrong about mechanisms that are in daily use.

spatial · Spatial elbow arm at a posture. elbow arm, drawn from 6 joint values through a product of 6 exponentials — no equation is solved anywhere in this picture, because an open chain has none to solve. The thin lines are the joint axes at this configuration, which are also the columns of the arm's Jacobian: the tool's velocity is a sum of turns about exactly those lines. Here the smallest singular value of that Jacobian is 0.3674 and the largest is 2.407, so the arm is comfortably away from a configuration where a direction of motion is lost. Drag θ₂ shoulder.

The chain that does not close

Every mechanism on this site so far has been a loop, and a loop is why a configuration here is a solve. An arm has no loop. Its pose is a product of six transforms, evaluated, with nothing to converge and nothing to refuse — and the difficulty does not disappear, it moves to the other end of the problem.

serial · Serial A four-bar at 60°, solved. Ground 4, crank 1, coupler 3.5, rocker 3. Every joint position here is the output of a Newton–Raphson solve on the loop-closure equations, converged to 0.0e+0 — not a placement that looked right. Grashof's condition classifies these lengths as a crank rocker, and sweeping the crank through 360° confirms it: 120 of 120 positions assemble. The transmission angle at this instant is 66.9°.

Four bars and four pins

The smallest interesting machine there is. Four lengths decide everything about it — which link can turn all the way round, how hard it pushes, where it stops and whether it can be assembled at all — and every one of those is a number that falls out of a solve rather than a judgement about a drawing.

linkages · Fourbar One, two, sixteen, two hundred and thirty. Every planar chain of mobility one, up to ten links, counted by enumeration rather than quoted. The pins column is forced: a chain of 10 links has one degree of freedom only if it has exactly (3n − 4)/2 pins, which is why no odd link count appears. Pass the count is how many graphs satisfy Grübler's rule, are connected, are simple and give every link at least two pins. Are chains is how many of those survive the fourth condition, that no proper subchain is already a structure — and the gap between the two columns is the whole of this field's first argument: at ten links 1,878 graphs pass a rule that 230 of them deserve. Mechanisms is larger again, because a chain is not a mechanism until a link is held still, and how many different mechanisms that gives is a question about the chain's own symmetry.

The mechanism is the graph

Twenty-one fields of this site have been handed a mechanism and asked what it does. Take the mechanism away and keep only which link is pinned to which, and there is still a finite list of answers: one chain of four links, two of six, sixteen of eight, two hundred and thirty of ten — and 1,878 graphs at ten links that pass every count and are not among them.

topology · Topology What each machine is sold with. Every mechanism in this field arrives with a number. The middle column is that number; the right-hand column is what the mechanism was measured to have, by the library named under the row. The coloured bar and the word under each machine are the verdict: exact means the mechanism has the number at every position, mean that it is the average of something that varies within every cycle, bounded that it is false by a stated and negligible amount, point that it is the value at one position, and quoted that the mechanism has no such quantity at all. Of the 14 rows, 6 are quoted and 3 are exact.

The number on the box

Fourteen machines, fourteen quoted numbers, and five different things a quoted number can be. Six of the fourteen name a quantity the mechanism does not have at all; three are exactly right, and all three of them are counts. The field is built on the difference.

applied · Misconception A bevel differential, turning. cage in, hold left, with every member's speed taken from the train's null space and every angular position that speed integrated. The teeth are marked at the pitch points rather than cut as involutes — the flank is the teeth field's subject — but the count is the tooth count and the positions are the solved ones, so what turns and how fast is real. Drag it and watch which way each member goes: left 0.000 · right 2.000 · cage 1.000.

Two inputs and one output

Every mechanism in this collection so far has had one input, and its output has been a function of that input. A differential does not. Its cage turns at the mean of two wheels, so knowing one of them tells you nothing at all about where the third shaft is going — and that is not a complication of the mechanism, it is a different kind of object.

transmission · Transmission Three bars and four bars. On the left, two bars to a common point: three links, three joints, and Grübler gives 3(3−1) − 2(3) = 0. The Jacobian agrees — two free coordinates, rank 2, nothing left over — and the shape cannot change without a bar changing length. On the right, one more bar and one more joint gives mobility 1, and the whole of this site follows from that difference. The triangle is why bridges are triangulated and the quadrilateral is why machines are not.

What decides whether it moves

Before a mechanism does anything it has to be able to. Two bars pinned to a point cannot move; three can. The count that separates them is one subtraction, it is the first thing anybody computes about a machine, and it can be wrong in a way that no amount of care with the arithmetic will catch.

constraint · Mobility 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 Mobility, counted and measured. Grübler's criterion counts links and joints and knows nothing about the dimensions; the rank of the constraint Jacobian measures the dimensions and knows nothing about the topology. They agree for four of these five. The parallelogram with a redundant third bar is the exception: the formula declares it a structure with zero degrees of freedom, and it moves. The formula is the one that is wrong, because it cannot see that the third bar's constraint equations are already implied by the other two.

Counting and measuring mobility

Grübler's criterion counts links and joints and never asks how long anything is. The rank of the constraint Jacobian measures the lengths and never asks what a joint is. Two calculations with no inputs in common, producing one number — which is the only arrangement under which agreement is evidence.

constraint · Mobility Same links, same pins, different chains. Watt chain on the left and Stephenson chain on the right. They have the same number of links, the same number of pins and the same assortment — 4×2 + 2×3 — so no count of anything can tell them apart. What differs is where the pins go: on the left the two ternary links share a pin, on the right they do not, and that single fact makes two mechanisms with different coupler curves, different numbers of inversions and different position problems. It is the smallest case in the subject of the thing this field exists to say: the arithmetic is a filter and the graph is the answer.

Same links, same pins, different machines

Watt's six-bar and Stephenson's have six links, seven pins, four binary links and two ternary ones. Every count anybody can make on them agrees. They are different chains, they give two mechanisms and three, and the difference is whether the two ternary links share a pin.

topology · Topology 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 6 contacts, and the box lifts straight off. 3-2-1, six contacts. Each pad is a contact and each arrow the direction the box is free to move there — the inward normal, and the whole of what the contact contributes. The rank is 6, which is full: these 6 contacts are 6 independent constraints and a bilateral version of them would fix the box completely. The margin is nought, so the box is free to leave, and no amount of tightening the tolerances on where the pads are would change that. Rank and hold are different questions and this is the pair of pictures that separates them. positioned by solving, not by drawing.

Four in the plane and seven in space

Six independent constraints fix a body in space and six contacts fix nothing, because d vectors can span d dimensions and can never positively span them. The minimum is one more than the dimension — and it is a floor rather than an answer: six contacts on a box held it in none of four thousand random arrangements and seven held it in twenty-one.

holding · Restraint 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 this site can name.

applied · Mobility 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 Three verdicts, and only one instrument can give all three. Every 10-link graph that satisfies Grübler's count, split by what is actually true of it. 230 are mechanisms with 10 links. 1,165 carry a subchain whose own count is exactly nought — and neither of the two standing routes can see them: the count returns one and the rank returns one, and both are right, because a rigid subchain removes exactly the freedoms it is supposed to. What is false is the description. 483 carry a subchain whose count is below nought, and those the rank does catch: the surplus pins repeat a constraint already imposed, the Jacobian loses rank, and the measured mobility comes out above the count. The third instrument — a count run over every subset of the links — is the only one that answers the question at all.

What a count cannot see

At ten links, 1,878 graphs satisfy Grübler's rule and 230 are mechanisms. The other 1,648 contain a subchain that is already a structure — and on 1,165 of them the count says one degree of freedom, the rank of the constraint Jacobian says one degree of freedom, and both are right about a mechanism that does not have ten links.

topology · Topology 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 How often a set of screws is a group. Take a subspace of the twists at random and ask whether it is closed under the Lie bracket — whether doing two of its motions in one order and undoing them in the other leaves you inside it. Every one-dimensional subspace is, trivially and importantly: a single screw always generates a one-parameter subgroup, which is the same statement as every screw is a joint somebody could build. Above one dimension, not one of eighty thousand is a group, and every one of them generates the whole of the rigid displacements at the first bracket. So a mechanism whose motion lies inside a proper subgroup is not merely unusual; it is a coincidence of measure zero — and it is the coincidence every planar mechanism, every spherical one and every Sarrus linkage on this site is built on.

Almost nothing is a group

Eighty thousand subspaces of the twists were drawn at random and closed under the Lie bracket. Above one dimension, not one of them was already closed, and every single one generated the whole of the rigid displacements at the first bracket. Two pins with parallel axes close at three; move one axis a hair and they close at six.

pairs · Pair 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 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.

A chain multiplies

An open chain's displacements are the product of its joints' groups, one factor per joint, in order. Sometimes the product is a group — three parallel pins and a slide along them give Schoenflies motion, which is a SCARA arm and is why it has four joints. Usually it is not, and then the chain's poses are a four-parameter set that needs six numbers to describe.

pairs · Pair Two numbers, and where they differ. Every mechanism in this field, with what its constraints leave and what its brackets fill. On every mechanism without a rolling contact the two columns are the same number, which is why nobody had to say which one mobility meant. Here only the rail agrees with itself — and the rail is the one mechanism in the table that cannot go anywhere new.

The count that counts the wrong thing

Mobility has meant one number for six fields, because until a wheel appeared no mechanism could tell two questions apart. A rolling wheel has two velocity freedoms and a three-dimensional reachable set, and the formula that gives 2 is not wrong — it is answering the question about instants when the question anybody asks is about intervals.

constraint · Mobility What each instrument returns, on each kind of graph. The 8-link census, three rows, and the same three questions asked of every graph in it. Grübler returns 1 in every row — it has to, because that is what the census selected on. The rank returns 1 in the first two rows and 2 in the third. Only the third column changes across all three rows, and it is the one this site did not have before this field: a mobility computed for every subset of the links rather than for the whole. Read down the middle two columns and the site's standing pair of routes is unanimous about 62 graphs, of which only 16 are what it says they are.

The count was right and the name was wrong

The constraint field has checked Grübler's count against a Jacobian rank since the foundation, and the two disagree only where the geometry is special. Here is an assembly where they agree, where both are correct, and where the mechanism does not have the number of links it is described as having.

constraint · Mobility 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 One freedom, whatever the count says. Every one of these machines has exactly one degree of freedom, measured as the number of unknowns minus the rank of the constraint Jacobian. Unbraced, the count agrees. Braced, the count says the largest machine has -153 — that it cannot move, by a wide margin — and the rank says it still turns exactly as it did. The gap is one equation per brace and every one of those equations is implied by the others. This is the constraint field's oldest example, at a scale nobody would try by hand: a count that is wrong by a hundred and fifty-three about a mechanism that works.

One freedom and four hundred links

Braced, the machine compiled from a quintic has 1,249 equations in 1,096 unknowns and a Grübler count of minus a hundred and fifty-three. It turns. The rank of its constraint Jacobian is 1,095, so its mobility is one — and every one of the hundred and fifty-four surplus equations was added deliberately.

constraint · Mobility 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 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 on this site — 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 Nine bars joining two sets of three joints, at three placements of one motion. Joints B₁, B₂ and B₃ lie on the horizontal line at -2, 1, 3, joints W₁, W₂ and W₃ on the vertical line through the same point at -1.5, 1, 2.5, and every joint of one set is barred to every joint of the other. Counted, nine bars on six joints leave no freedom. Drawn here at three placements, the joints have slid along their lines — B₂ at 0.632 and W₂ at 1.265; B₂ at 1.000 and W₂ at 1.000; B₂ at 1.265 and W₂ at 0.632 — and every one of the nine bars has the same length in all three, to 4.4e-16.

Nine bars that ought to be rigid

Join each of three joints to each of three others and the nine bars leave no freedom, by the count and by the rank, wherever the joints are. Put one set on a line and the other on a line at right angles and the framework moves, all the way round a loop, with no bar repeating any other: take away any one of the nine and the motion is unchanged, take away any two and it gains a freedom. Tilt the lines by a degree and it still has a freedom by rank and cannot move at all.

constraint · Mobility Three joints, and the three screws that describe them. A leg of three revolute joints, drawn as its three axes, and the principal screws of the three-system they span, drawn through the system's own centre. The three principal axes are mutually perpendicular — worst cosine 1.1e-16 — and they meet at one point, missing it by 2.6e-16. Their pitches are -0.3766, -0.0338, 0.8075, and every screw the leg leaves free has a pitch the three of them give by h₁l² + h₂m² + h₃n². Nothing in the three joint axes looks like a right angle and the system's own frame is one.

What a leg of three joints leaves free

Five essays of this field have computed the order of a screw system and drawn none of them. A three-joint leg spans a three-system; its three principal axes are mutually perpendicular and meet at a point, six numbers price every screw in the family, and the directions of the lines it contains form a cone.

spatial · Spatial What a leg of three joints permits and what it resists, on one frame. The leg of three revolute joints in grey, and the principal screws of two three-systems drawn through one centre: the twists the leg lets its platform make, and the wrenches it holds without any joint turning. They are computed separately — the second is the reciprocal complement of the first, found by its own eigenproblem — and they come out with the same centre, 5e-16 apart, and the same three axes, parallel to 0e+0. Only the pitches differ, and each wrench pitch is its twist's with the sign reversed, to 1e-15: -0.3766 against 0.3766, -0.0338 against 0.0338, 0.8075 against -0.8075.

The lines a leg turns about and the lines it is pushed along

A leg of three revolute joints permits a three-system of motions and resists a three-system of loads. The two share a centre and three axes and differ only in the sign of every pitch — and the revolute axes of the first and the lines of force of the second are the two rulings of one hyperboloid, every line of one meeting every line of the other.

spatial · Spatial The chains that never close, counted by the same routine. An open chain has one joint fewer than it has links, so its graph is a tree and its mobility is the joint count rather than one. Feeding the same enumerator that produced the closed-chain census — with the minimum-degree condition relaxed to one, because an arm's base and its end each carry a single joint — gives 1, 2, 3, 6, 11, 23, 47, 106 for three links up to ten. That is the number of unlabelled trees, a sequence anybody can look up, and reproducing it is the strongest check the enumerator gets: it was written for a different problem, tested against three mechanism counts, and asked here for a number from a different subject entirely.

An arm is a tree

The serial field's chains are the ones that never close, and as graphs they are trees. There are 106 distinct arrangements of ten links joined that way, and exactly one of them is the straight arm every essay in the field has drawn — the other 105 branch.

serial · Serial A framework built off square, and the play it asks for to move at all. The nine bars with their two lines 4° from perpendicular, as built and after the first joint has been pushed 0.2 along its line. The framework has a freedom by rank and no motion, so the push cannot be taken with the bars at their lengths. With every bar allowed to be wrong by 1.509e-4 — which is a radial clearance of 7.547e-5 at each end — a placement exists, and the worst bar in it is out by 1.509e-4. The same push on the perpendicular framework needs no allowance at all, because there it is a motion.

The right angle as a tolerance

Dixon's nine bars move only when their two lines are exactly perpendicular, and a framework built a degree off square has a freedom by rank and no motion at all. Give its joints clearance and it moves a bounded distance: the play each bar needs is proportional to the tilt and to the square of the travel, one constant serves every tilt, and all nine bars end up at that play exactly. Then the framework reaches its first crossing and the law is left three hundred times behind.

constraint · Mobility A SCARA arm, and where it can put its tool. A SCARA arm at its home position, with the axes dashed and a cloud of the tool positions reached over random joint values. The displacement set of an open chain is the product of its joints' groups, one factor per joint, and the question this field asks of it is whether the product is itself a group. Here the logarithms of the reached displacements occupy 4 dimensions, so the motion lies inside Schoenflies motion and composing two of its displacements gives another one. The cloud is a fact about the reach and not about the group: a chain of finite links covers a bounded piece of its group and never the whole of it, which is a separate question and a different field's.

The arm that is a group

A SCARA arm has four joints and a six-axis robot has six, and the usual explanation is that four is enough for the job. The better one is that the job is a four-dimensional group of displacements which is not the symmetry group of any surface — so it cannot be one joint, and four is what it costs. The arm's tool face is level everywhere it can reach, and the reason is not that anybody checked.

serial · Serial

Named alongside it

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

MobilityConstraintRankOverconstraintGrübler's criterionDisplacement subgroupLower pairScrewToleranceKinematic chainRedundant constraintHigher pair

All concepts