Machine Kinematics
solved, then drawn
Series

Pair — the series

21 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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 collection 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.

    part 1 · pairs
  2. 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.

    part 2 · pairs
  3. 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.

    part 2 · pairs
  4. 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.

    part 3 · pairs
  5. 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.

    part 3 · pairs
  6. 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.

    part 4 · pairs
  7. 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.

    part 4 · pairs
  8. Four joints that give a group, and four that do not. Two chains of four revolute-and-slide joints, each drawn at its home position with its joint axes dashed, and each with a cloud of the tool positions it reaches. The counts are identical: four joints, four freedoms, the same Jacobian rank everywhere off a singularity. On the left the three pins are parallel and the slide is along them, and the displacement set is the Schoenflies group — every translation and one rotation direction, four dimensions, closed. On the right the axes are at random and the set is four-dimensional too, and it is inside no group smaller than all the rigid displacements. The instrument is in the caption of each panel: take the logarithms of the displacements the chain reaches and count the dimensions they occupy. Four means a group. Six means there is nothing to be inside.

    Four joints that give a group, and four that do not

    Two chains of four joints. Same joint types, same count, same mobility, same Jacobian rank, same everything measured here across twenty-two fields. One of them reaches a four-dimensional set of displacements that closes under composition; the other reaches a four-dimensional set whose logarithms fill all six dimensions. The difference is six hundredths of a radian in where two axes point.

    part 5 · pairs
  9. What every pair of groups meets in. The intersection of two subgroups is always a subgroup — that needs no computation — and which one is the useful part. This is the design rule behind every parallel machine in these essays: choose legs whose groups meet in the motion the platform is wanted to have, and it has that motion whatever the leg lengths are, with no synthesis and no tolerance. The row and column are built about different axes, at right angles, because an intersection is a statement about particular subgroups rather than about their kinds: two planar groups with the same normal meet in the whole of themselves, and two with different normals meet in a line.

    Legs intersect

    A serial chain multiplies its joints' groups and the product is almost never a group. A parallel machine's platform gets the intersection of what its legs permit — and an intersection of groups is a group, always, with no coincidence required. That is the only construction in this field that produces closure for free, and it is why a platform can be designed for a motion type instead of discovered to have one.

    part 5 · pairs
  10. 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.

    part 6 · pairs
  11. Four instruments, and only the last one names the group. Every instrument these essays have 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 in the earlier essays, stated as an integer.

    Compose two positions and see where you land

    Take two configurations a mechanism actually reaches, compose the displacements that got it there, and ask what kind of thing the result is. A planar four-bar lands inside planar motion, to 4 × 10⁻¹⁶. Sarrus lands on its own line. Bennett's linkage lands three tenths of a radian outside the four dimensions its own displacements occupy — a one-freedom motion that generates all six.

    part 6 · pairs
  12. 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.

    The instrument that is not a derivative

    Sample a chain of four random pins over a millionth of a radian and its displacements occupy four dimensions, exactly as a SCARA arm's do. Sample the same chain over two radians and they occupy six. The step is at 10⁻⁶, and where it sits is a fact about arithmetic while the two plateaux are facts about the mechanism.

    part 7 · pairs
  13. 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.

    part 7 · pairs
  14. 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.

    part 8 · pairs
  15. 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.

    part 9 · pairs
  16. A double Cardan joint in the W arrangement. Two universal joints on an intermediate shaft, the input arriving at 15° to it and the output leaving at 15°, bent the same way so that the input and output lines meet, with the intermediate shaft's two yokes in one plane. Each pin is found as the line square to its own shaft and to the pin it crosses, so the output's angle is solved rather than taken from a formula. The dashed marks are one on each hub. The shaded square is the plane that swaps the input's approach with the output's departure. At an input angle of 40° the output is at 40.00°, exactly in step.

    Constant velocity is a mirror

    Between parallel shafts a coupling keeps its output in step exactly when its hubs only translate relative to each other, which is membership of a group. Between shafts that meet, no group will do: a single universal joint at 30° and a correctly phased double Cardan joint both close to the spherical group, and one wanders by 8.2° a turn while the other keeps step to 10⁻¹⁵. What separates them is a mirror that swaps the two hubs. It either fixes every configuration of the joint or fixes none, so one symmetric position certifies a whole turn — which is what the phasing mark on a propshaft checks.

    part 10 · pairs
  17. Near its mirror, a double Cardan joint's lag is an ellipse in its two errors. A W double Cardan joint whose working angles average 15°, with a phasing error δ between its middle yokes across and a difference ε between its two working angles up, both in degrees. The curves are where the predicted lag spread over a turn, tan a·√((sin a·δ)² + ε²), is 0.05°, 0.1°, 0.15°, 0.2°. The dots are 35 joints solved from their pins, labelled with their measured spread in degrees; they meet the ellipse to 0.06% at worst. The ellipse is 3.86 times as wide as it is tall: at 15° a degree of phasing error does what 0.259° of mismatch between the angles does.

    Two errors, one ellipse

    A double Cardan joint keeps step only while a mirror fixes its motion, and any real one is a little off. Split each error by what the mirror does to it and only the half it reverses costs anything: a phasing error spreads the lag at sin a · tan a per degree, a difference between the working angles at tan a, and raising both angles together costs nothing at all. The two costs leave their marks an eighth of a turn apart, so they add as the sides of a right triangle — a root sum of squares that no assumption of independence was needed for.

    part 11 · pairs
  18. Six balls, two grooves each, and a cage. A Rzeppa joint bent 20°, input shaft along z and output shaft turned about x. Each ball lies where the plane of its inner-race groove, through the input's axis, meets the plane of its outer-race groove, through the output's axis; at equal input and output angles every ball is in the plane bisecting the two shafts, and the cage holds them there. Left, from the side: the two shafts and the bisecting plane edge-on, the balls on it. Right, along the cage's normal: the six balls round their circle, the plane of the bend drawn across it. At an input angle of 0° a cage clearance of c lets the output lag or lead by up to 0.350 c radians, c in ball-circle radii.

    A Rzeppa cage adds its clearances

    A Rzeppa joint keeps constant velocity because its cage holds every ball in the plane that bisects the two shafts. Give the cage's windows a clearance c and the output can lag or lead the input. Six balls might be expected to limit that lag through the worst of them, or as a root sum of squares. They do neither. Opposite balls are one constraint, the cage's two tilts spend two of the three, and what is left adds the three pairs' allowances like clearances in series: 0.350 c at a 20° bend, twice what the worst ball alone would allow. The lag falls to nought each time a ball crosses the plane of the bend, grows with the bend itself, and a four-ball joint cannot hold it at all.

    part 12 · pairs
  19. A slide spends one ball of every pair. A 6-ball Rzeppa joint bent 20°, at the symmetric position, with its cage slid along its own normal by 0 of the window clearance c and then tilted to allow the largest lag. Each column is one ball's window, from −c to +c, and the dot is where the ball sits in it at that lag: −1.00, 1.00, −1.00, 1.00, −1.00, 1.00 c. The lag is 0.3500 c. Opposite balls lie on one line through the joint's centre, so a lag or a tilt moves them opposite ways and a slide moves them the same way: whatever the slide gives one ball of a pair it takes from the other, and the lag falls as 1 − s/c.

    A floating cage frees three balls and not six

    A Rzeppa cage that may slide along its own normal as well as tilt has a third freedom, and counting constraints against freedoms says a six-ball joint should then lose its hold on the lag altogether. It loses nothing: a slide moves both balls of an opposite pair the same way, so it spends one ball's clearance for every one it frees, and the six-ball lag stays at 0.350 c. A three-ball joint, whose balls have no opposites, loses its bound outright, because three ball centres always lie in some plane. Offset grooves then add a second limit, set by the worst ball alone, which takes over from the cage only past an offset of about 0.35 times the tracks' play over the windows'.

    part 13 · pairs
  20. Past the play, a caged joint shares its load and a grooved one leans on one ball. Two six-ball Rzeppa joints bent 20°, at an input angle of 15°, each twisted 0.03 of a window clearance past its free lag, with every contact a spring of the same stiffness and the cage and balls settled where the stored energy is least. Bars: each ball's share of the total contact load. Left, grooves offset 0.1, where the cage sets the play: 0.17, 0.20, 0.19, 0.16, 0.14, 0.15, 5.91 balls' worth. Right, grooves offset 0.7, where the grooves do: 0.00, 1.00, 0.00, 0.00, 0.00, 0.00, 1.00 balls' worth. Dotted: an even sixth.

    A tight groove leans on one ball

    A six-ball Rzeppa joint's free play is set by the tighter of its cage and its offset grooves, and the two limits look alike from outside. They are not alike once the play is taken up. With every contact made a spring of the same stiffness and the joint twisted past its free lag, a joint whose cage binds spreads the first load across four to six balls' worth and, a third of a window clearance on, gives every ball between 0.17 and 0.19 of it. A joint whose grooves bind puts the first load on one ball alone over most of a ball pitch, and two clearances on it still spreads it over fewer than six. At the same torque its hardest-pressed ball carries 1.5 to 3.0 times the caged joint's. Tightening the tracks to cut the play buys the play with uneven loading.

    part 14 · pairs
  21. Tighten the windows and the cage's limit drops below the grooves'. The six-ball Rzeppa joint bent 20° with its grooves offset 0.7 and a track play of one: across a ball pitch, the free lag the grooves alone allow (solid) and the cage alone allows with windows of clearance 0.5 (dashed). The joint's play is the smaller of the two. The cage is the limit wherever its curve is the lower, here at every input angle; it becomes the limit everywhere once the windows' clearance is below 0.559 of the track play. The largest play over the pitch is 0.163.

    A tight cage spreads the load until the tracks join in

    A Rzeppa joint whose offset grooves set its play loads one ball first, and at equal torque its hardest-pressed ball carries up to three times a joint whose cage sets the play. Tighten the cage's windows below 0.559 of the track play and the cage becomes the limit at every input angle: the first load spreads over 3.6 balls' worth or more everywhere, the play falls, and at low torque the worst ball carries no more than a caged joint's. The cage does not stay in charge. Once the twist reaches the gap between the two limits the tracks engage, unevenly, and the worst ball climbs back towards the untightened joint's. A tighter cage widens the gap and pushes that torque up, so the windows' clearance is a choice about the torque at which even loading should end.

    part 15 · pairs

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