Machine Kinematics
solved, then drawn
Concept

Higher pair — where it appears

A joint that is a contact along a line or at a point rather than a fitted surface, removing one of a planar body's three freedoms. Cams, gear teeth and rolling contacts are all higher pairs, and the single constraint each of them applies is the reason a contact can be lost while a pin joint cannot.

Named by 16 essays across 7 fields — each of them below, with the objects they name alongside it.

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

A joint is a surface that slides on itself

Twenty-two fields of this 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.

pairs · Pair A 12-tooth ratchet, holding. A ratchet wheel with a 8° tooth face and its pawl. The long line is the face extended; the short one through the contact is the normal the tooth pushes along, and both belong to the wheel and turn with it. The pawl holds, and the reason is which side of those two lines its pivot is on: 1.080 wheel radii from the face's line and 0.495 from the normal's, and on opposite sides of them. Nothing about how hard anything pushes enters the question: a normal is a direction and a moment arm is a length with a sign. Dragging turns the wheel the way it may go, and the pawl's tip is solved onto the wheel's surface at every frame: it is lifted up a tooth's back and dropped into the next notch, once per click.

A joint that works one way

A pawl either holds a ratchet or is levered out of it, and which one happens is decided by two lines through the contact. One of them is the rule a workshop quotes. The other is the boundary that rule leaves out, and a check written to confirm the quoted rule turned out to be incapable of failing.

timing · Intermittent 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 What one contact forbids, drawn as a place. A single contact on one edge of a square, and the whole plane coloured by what it permits. A rotation about a point is a twist, and a twist is permitted when it does not drive the part into the obstacle; because a rotation about (x, y) is affine in the point, the condition is a half-plane and the boundary is a straight line — the line through the contact along its own surface. On one side of it only anticlockwise rotations are permitted, on the other only clockwise, and the two together are the whole plane bar the line itself. So one contact rules out exactly half of what the part could do and leaves the other half untouched, which is why the count of contacts a hold needs is one more than the dimension rather than equal to it: the first 1 of them cannot leave nothing over. The picture is exact — the regions are clipped polygons, not a sampled grid.

What one contact forbids

A rotation about a point is a twist, and a twist is affine in the point — so what a single contact permits is a half-plane of centres, with the boundary being the contact surface's own line. Reuleaux drew it in 1875 and it is exact rather than sampled, which is why every figure in this field is a picture of the plane rather than of a cone.

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

What each joint takes away

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

constraint · Mobility One angle, read twice. Draw against the angle the locking face is tilted by. It is not a separate design quantity from recoil: the pallet's moment per unit of wheel torque is minus the rate at which the wheel is driven backwards, which is virtual work and holds to nine figures. So an escapement cannot be made to hold its own lock without also being made to push its train back — and at δ = 0 the arc has neither, which is a lock that any disturbance opens. The flat face starts above zero because the tooth rests below the corner, where a straight face is no longer tangent to anything.

One test, three mechanisms

Whether a pawl holds, whether an escapement's lock draws itself deeper, and how much a four-bar's coupler can do for its rocker are the same question asked three times: on which side of a pivot does a contact normal pass? All three are one cross product, none evaluates a force, and the three answers are used for completely different things.

timing · Intermittent 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 in this collection 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 A geared five-bar at 3 : 2, its coupler pin traced over 2 input turns. Two cranks of length 1 on pivots 3 apart, meshed through gears whose pitch circles, dashed, have radii 1.800 and 1.200, so the second crank turns 1.5000 times for every turn of the first and the other way. Coupler bars of 3 and 2.5 join them at the pin C, and the thin curve is where C goes over 2 turns of the input. The two crank pins are always between 1 and 5 apart and the bars can span 0.5 to 5.5, so the chain never meets a dead position. After 2 turns the machine is exactly where it began and the curve is closed.

One freedom, and a motion that never repeats

Mesh a gear on each crank of a five-bar and the count and the rank agree that one freedom is left, at every gear ratio. Whether the machine ever comes back to where it started is a different question, and neither instrument can see it: at 3 to 2 it is home after two turns, at 37 to 23 after twenty-three, and at the golden ratio never, with its nearest returns at the Fibonacci numbers.

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

Six things a joint is not

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

wrong · Misconception A stop on the coupler, and the one direction in which it stops nothing. A four-bar — frame 4, crank 1.2, coupler 3.4, rocker 2.8, crank at 70° — whose coupler is stopped by a contact with the frame at (1.59, 2.16). The dashed line through the stop is its normal; the ringed point is the coupler's pole, where the crank's and rocker's lines meet, at (2.505, 6.883). With the normal 10° off the line to the pole the stop removes the four-bar's one freedom; through the pole it removes nothing, because the contact point is already moving along the stop's face. The graph of the stopped four-bar is the same in every case.

A contact turns and slides at once

A cam on its follower, a gear tooth or a pin in a slot takes one freedom away where a pin or a slide takes two, and it turns as well as slides. Give every joint of the four-bar, Watt's and Stephenson's chains a third label and there are 1,317 different chains. The rule that read slides off the graph extends to all of them, agreeing with the constraint rank on every one, and the two obvious shortcuts miss 36 and 769. What the graph cannot record is which way a contact's normal points — and on these chains that never matters, while a single stop added to a four-bar holds everywhere except through one point.

topology · Topology A seat round the stop's nose holds the four-bar if it is tighter than the nose centre's path allows. The four-bar of frame 4, crank 1.2, coupler 3.4 and rocker 2.8 at a crank angle of 70°, its coupler carrying a round nose of radius 0.3 centred at (1.593, 2.160). The nose sits in a hollow seat on the frame of radius 0.8, touching it at (1.650, 2.454), on the line from the nose's centre to the coupler's pole at (2.505, 6.883), off the top of the drawing. The dashed circle is where the nose's centre may go without the nose entering the seat: a disc of radius 0.50. The curve through the nose's centre is the path the four-bar gives it, with a radius of curvature of 0.893. The seat's disc is tighter than the path, so the path leaves it either way and the four-bar cannot turn: the stop holds. The threshold is a seat of radius 1.193, the path's radius plus the nose's.

A stop aimed at its pole holds by curvature

A stop whose normal passes through its pole forbids nothing at first order, and whether it holds at all is decided by curvature. For a round nose the deciding curve is not the one it touches: the seat that just holds is the path of the nose's centre moved out by the nose, 1.193 on a four-bar where the contact point's own circle says 0.776. Past it, one contact holds both ways — loosely, as the square root of its clearance.

topology · Topology The seat at the pole, and the friction cone at its contact. The four-bar of frame 4, crank 1.2, coupler 3.4 and rocker 2.8 at 70° of crank, its coupler's round nose of radius 0.3 in a seat of radius 0.8 on the frame, aimed through the coupler's pole. At the contact the shaded wedge is the friction cone for a coefficient of 0.1: every force the seat can press with lies within 5.7° of the normal. The normal itself runs through the pole, so it has no lever about it; the cone's edges do. Under a moment of 0.001 on the crank against a contact of stiffness 10⁶ the crank turns 1.70e-4 radians, against 1.48e-3 with no friction; released, it stays at 1.70e-4.

A seat with friction stays where it is pushed

A stop aimed through its pole holds a four-bar by curvature, because its normal has no lever about the pole. Friction has one, and the obvious guess is that it lets such a stop hold at first order. It does something narrower and stranger. It cannot hold what curvature releases, since a gap that opens carries no force for friction to be a share of. It makes a seat that holds up to twenty-six times stiffer, turning its cube-root give into a square root. And it makes any stop whose pole lies inside its friction cone stay where a load pushed it: released, the seat at the pole does not come back.

topology · Topology A V of two faces, each turned off the line to the pole. The four-bar of frame 4, crank 1.2, coupler 3.4 and rocker 2.8 at 70° of crank, its coupler's round nose of radius 0.3 resting in a V of two flat faces on the frame. Each face's normal is turned off the line from the nose to the coupler's pole so that at the contact it misses the pole by 12.79°, one each way, so that each face holds the crank one way at first order. The shaded wedges are the friction cones for a coefficient of 0.2, half-angle 11.3°: the pole lies outside each, so released from either side the crank springs back. Each face's moment per radian, through a contact of stiffness 10⁶, is 72 thousand.

Return and stiffness come from the same aim

A stop with friction looked like a trade: aim it near its pole and friction multiplies its stiffness seventeen times, aim it outside the friction cone and it springs back when released. The trade is an artefact of how the gain was counted. Friction's factor multiplies a stiffness that vanishes at the pole, and measured outright a stop aimed further away is stiffer at every friction — 271 per radian at 1.1° and 104,000 at 21.3° with none, 4,631 and 184,000 at a coefficient of 0.3. Return and stiffness are bought by the same aim. What the aim costs is the other side: a face aimed off its pole holds nothing the other way, so a stop that must return both ways is a V of two faces, and across a friction band from 0.1 to 0.2 a V aimed 12.3° each way returns from both and is 8,800 times stiffer than the seat at the pole.

topology · Topology A V carried on a slide and pressed onto the nose. The four-bar of frame 4, crank 1.2, coupler 3.4 and rocker 2.8 at 70° of crank, its coupler's round nose of radius 0.3 held in a V of two flat faces. The V rides on a slide along its own axis and a spring of force F presses it onto the nose, so it can be pushed back as the crank turns. Each face's normal is turned 45° off the V's axis, and misses the line to the coupler's pole at its contact by 47.6°. The shaded wedges are the friction cones for a coefficient of 0.2. Turning the crank past the V takes a moment of 1.39 F; released, the V pushes the crank back with 0.64 F.

A V pressed onto its nose has a window

A V of two faces holds a four-bar's coupler both ways and returns it, as long as each face is aimed off the coupler's pole by more than the friction angle. Squeeze the nose between both faces and two different things can happen. A rigid V cut a hundred-thousandth tight cannot lock, because the crank moves the nose across the V and never into it; it only stops short of home, inside a dead band that is μ/tan ψ of the play an equally loose fit would leave. A V pressed on by a spring, as a detent is, rides back along its own axis, and that is the direction a wedge locks in. It returns from an aim of arctan μ at the contact and locks at 90° less arctan μ about its axis, so its aim has a window, and the V that is most nearly reversible inside it is a right angle.

topology · Topology

Named alongside it

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

MobilityContact normalConstraintGrübler's criterionLower pairDegrees of freedomClearanceDisplacement subgroupKinematic pairRankConstraint jacobianDesign rule

All concepts