Machine Kinematics
solved, then drawn
What can move

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.

Assumes Counting and measuring mobility and A constraint that takes nothing away.

15 min read 8 figures Two routes to the same numberSolved, not drawnThe constant that is not constant

The mobility field’s founding argument is that a formula and a measurement can disagree about how many freedoms a mechanism has, and that when they do the measurement is right. Grübler’s count declares a working linkage immobile and the rank of its constraint Jacobian says otherwise; Kutzbach is wrong about four spatial loops in five; and the repair in both cases is to stop counting and start measuring.

That argument survives here. What does not survive is the assumption underneath it — that there is one quantity being computed and the only question is how to compute it correctly.

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.
Fig. 1 Seven mechanisms with wheels, and two columns. The left is the number the mobility field computes and the right is the number anybody asking what can this machine do means. On every mechanism drawn on this site before wheels appeared they were the same number, and here they differ on six rows out of seven.

The two questions

How many independent motions are available right now? Take the constraint rows at a configuration, take their rank, subtract from the number of coordinates. For a rolling wheel: three coordinates, one row, two.

How many dimensions can the mechanism be driven through? Take the permitted directions, close them under the Lie bracket, and see what they span. For a rolling wheel: three.

Both are correct. A wheel really can only do two things at any instant — roll and steer — and a wheel really can be brought to any position at any heading. Neither number is an error in the other and neither is a repair of the other.

The whole reason the distinction never came up is that on a mechanism whose constraints are on positions the two coincide, and every mechanism on this site until this field had constraints on positions. A four-bar’s loop closure is an equation in its angles: its permitted velocities are the tangent space to a curve, and the curve is one-dimensional, and both answers are 1. A Gough platform’s are six and six. There was one number, so mobility meant it.

Where the coincidence comes from

Frobenius’ theorem is exactly the statement that makes the coincidence a theorem rather than a habit. A distribution of permitted velocities is the tangent field of a family of surfaces if and only if it is closed under the bracket — and when it is, the mechanism lives on one of those surfaces and the surface has the dimension the rank says.

So: constraints on positions ⟹ brackets stay inside ⟹ the two counts agree. A closed-loop mechanism’s constraints are the derivative of its closure equations, so its brackets stay inside automatically, and the agreement is not luck.

Break that and the two counts come apart, and there is only one way to break it: a constraint that is not the derivative of anything. One character in a constraint row is the difference, and a rolling wheel is on the wrong side of it.

Which of these constraints is secretly about positions. How much of the bracket of two permitted directions lies outside the permitted directions, as a fraction of its own length. Frobenius' theorem says a distribution is the tangent field of a family of surfaces exactly when this is zero, so the test needs no integration and no recognition. The scale is logarithmic because the answers are seventeen orders apart: the rail returns nothing at all and everything else returns essentially the whole bracket. There is no mechanism in the middle.
Fig. 2 The test, on five mechanisms. A mechanism returning zero has its two counts equal, by theorem. The four returning one have them differing, and there is nothing in between — which is why the two numbers can be reported separately without any need to say how confident either is.

What the arithmetic has to become

The mobility field’s arithmetic is a subtraction: coordinates minus the rank of the constraints. That arithmetic is unchanged and its answer is unchanged; what changes is the label on the answer.

The new arithmetic is the growth vector, and it contains the old one. Its first entry is the velocity freedoms — the subtraction, exactly as before — and its last is the reachable dimension. A mechanism whose growth vector has one entry is a mechanism whose two numbers agree, and every mechanism the site drew before this field has a growth vector of length one.

So nothing has been overturned. A quantity that had one name has been found to be two quantities that coincide on a large class of mechanisms, and the class is characterised: it is exactly the mechanisms whose constraints integrate.

mechanism coordinates independent rows velocity freedoms reachable
a four-bar 4 3 1 1
a Gough platform 12 6 6 6
a rolling wheel 3 1 2 3
a car 4 2 2 4
a car and trailer 5 3 2 5
a ball that may not be twisted 5 3 2 5

The last three rows are the ones that would have been reported, before this field, as two freedoms — and the sentence this mechanism has two degrees of freedom would have been true and would have been read as saying that its configurations form a two-dimensional set, which is false by two, three and three respectively.

Where the mobility formulas stand

Grübler and Kutzbach count joints and links and subtract. Neither of them has any way of representing a rolling constraint, and applying them to a wheeled vehicle produces a number that is not either of the two quantities above.

Take a differential-drive robot as a linkage: a chassis, two wheels, two revolute joints. Kutzbach in the plane gives 3(n−1)−2j3(n-1) - 2j with n=3n = 3 and j=2j = 2, which is 6−4=26 - 4 = 2 — and the 2 is a coincidence. The formula has counted the two wheel rotations as the freedoms and knows nothing about the chassis moving at all, because the wheel-to-ground contact is not one of its joint types. Change the question to include the chassis’s three coordinates and the formula has no way to be asked.

This is a different failure from the ones the mobility field is built on. There, the formula was right in form and wrong in arithmetic, because it assumed constraints were independent when the geometry made them dependent — and the Jacobian’s rank was the repair. Here the formula is not wrong; it is not about this. A rolling contact is not a lower pair, its constraint is not a position constraint, and no amount of counting joints will produce a number that answers either of the two questions.

How much each set of wheels forbids. Every wheel contributes the same row, and whether it is a constraint or a drive is one factor of sin γ in it — γ being the angle the rollers make with the wheel's own axle. At γ = 0 the row says the body may not move across the wheel and the wheel's speed drops out of the statement; at γ = 45° the row says nothing about the body at all and fixes the wheel's speed instead. The whole difference between a machine that shuffles and one that slides sideways is in that factor.
Fig. 3 The constraint rows a set of wheels contributes, and their rank. This is the arithmetic that replaces the joint count for a wheeled machine: one row per rolling wheel, rank taken at a configuration, and the answer read off. Four wheels can contribute rank 2, and the two surplus rows are made dependent by the steering geometry rather than by any formula’s assumption.

What survives from the field’s founding argument

Three things, and it is worth listing them because the previous section could be read as saying the mobility field’s method has been superseded.

Measuring beats counting, unchanged. The rank of a constraint matrix at a configuration is still the right way to get the velocity freedoms, and a formula that assumes the rows are independent is still wrong whenever the geometry makes them dependent. A steered vehicle is an instance: four wheels contribute four rows and the rank is two, and no count of wheels would have said so.

The disagreement is where the interest is, unchanged. The mobility field’s best mechanisms are the ones where the formula and the measurement differ — Bennett’s linkage, Sarrus’, the universal joint. This field’s best mechanisms are the ones where the two measurements differ, which is a disagreement one level up.

Two routes, unchanged and strengthened. Here the two routes to the reachable dimension are a rank computation on brackets and a fitted exponent on a flown manoeuvre, and they share nothing. That is a stronger pair than Grübler against a Jacobian, because Grübler and the Jacobian at least share the mechanism’s joint structure.

What has changed is one sentence: mobility is no longer a complete answer to what can this mechanism do. It was never claimed to be, and on every mechanism before this one it happened to be.

The displacement is second order in the amplitude. Seven amplitudes, each wiggle flown and its net displacement measured. On logarithmic axes the points lie on a straight line of slope 2.000 — two, to three figures, on a measurement that was never told what to expect. That is the practical content of the whole field: halving the room a mechanism has to manoeuvre in quarters what each manoeuvre wins, so the number of them goes up by four.
Fig. 4 The second route to the second number. Seven manoeuvres, each flown, and a slope of 2.000 — which says the mechanism reaches a direction its constraint rows forbid, and says how quickly. Nothing in this computation takes a rank, and nothing in the rank computation flies anything.

Two mechanisms that look alike and are not

The clearest way to see that the second column carries information the first does not is a pair with identical first columns.

A rolling wheel and a trolley on a rail each have three coordinates, one constraint row of the same trigonometric shape, and two velocity freedoms. Every count that the mobility field can take gives the same answer for both. One of them reaches a three-dimensional set and the other reaches a line, and no amount of care with the first column would ever discover it.

That is the argument for reporting both, and it is the same argument the mobility field made for measuring rather than counting: a quantity that cannot distinguish two mechanisms that behave differently is not the quantity anybody wanted.

What each of them can reach. Nine hundred control histories of four legs each, from the same starting configuration, with the resulting position plotted. The wheel's cloud is two-dimensional and fills the region; the trolley's is one-dimensional and lies exactly on its rail — the same number of coordinates, the same number of constraints, the same count of freedoms, and a reachable set of a different dimension. Nothing here is a matter of degree.
Fig. 5 The two mechanisms, driven nine hundred times each. Identical in every column the older arithmetic has, and different in the one this essay adds.

The word that has to be split

Three phrases get used interchangeably in the literature a mechanical engineer meets, and after this field they cannot be.

Degrees of freedom is used for both quantities and for a third — the number of actuators a machine has, which is a fact about how it is built rather than about what it can do. A redundant arm has seven actuators and six task dimensions; a differential drive has two actuators and two velocity freedoms and three reachable dimensions. Nothing forces the three to agree.

Mobility on this site has always meant the first quantity, and it continues to. The one change is that it is no longer the answer to how much can this mechanism do, and the essays in the rolling field say which they mean each time.

Controllability is the second quantity, borrowed from a different literature, and it is worth using because it names the right thing: the question of whether the reachable set is everything. A mechanism is controllable when its growth vector ends at the dimension of its configuration space, which is a decidable property and not a matter of degree.

The reason to be strict about this is that the three words are used loosely in exactly the place where the loose usage does damage. A car has two degrees of freedom is a true statement about actuators, a true statement about velocity freedoms, and a false statement about where a car can go — and the third reading is the one a person means when they ask.

What the mechanism ledger has to say now

This site keeps a ledger of every closed loop it solves, counted two ways, with the count and the measurement in adjacent columns and the mechanisms where they disagree marked. That ledger’s whole premise is that two columns are enough.

For a wheeled mechanism it is not, and the honest statement of what changes is small: the ledger’s measured column answers the instantaneous question, and a third column would be needed for the other. Nothing in the ledger is wrong — every mechanism in it is a closed loop with position constraints, so its third column would repeat its second — and the reason to say so is that a reader arriving from this field would otherwise reasonably ask.

The mechanisms in this field are not in that ledger for a structural reason rather than an editorial one: they are not loops. A wheel on a plane has no closure equation to take the Jacobian of, and the object whose rank is taken here is the matrix of constraint rows, which is a different matrix arrived at differently.

How many wiggles it takes. The growth vector: how many independent directions are available after one bracket, two, three. The first number is what the constraints leave and the last is the dimension of the configuration space, so the length of the row is how deep the manoeuvring has to go. A car needs one bracket more than a trolley and a car with a trailer one more again — and the ball changes by one depending only on whether it may be twisted.
Fig. 6 The arithmetic that covers both cases. Reading the first entry gives the mobility field’s number; reading the last gives the reachability; and a row of length one is a mechanism where the distinction does not arise. Every mechanism this site drew before wheels would be a row of length one.
A differential drive, where it was driven toThe mechanism at a configuration nothing wrote down: it was reached by integrating permitted velocities from the start of the trail, and there is no equation here whose root it is. The barred line at each wheel is the direction that wheel forbids — the subject of the whole field, and the one thing a photograph of a car cannot show. The constraint residual along the drawn history is 0.0e+0.3 coordinates · 1 rows · 2 controlspositioned by solving, not by drawing
Fig. 7 The mechanism the joint count fails on, drawn. A chassis, two wheels and two revolute joints is a linkage a formula can count; a chassis, two wheels and two rolling contacts with the ground is not, because a rolling contact is not one of the pairs any of the formulas has a column for.

Where a designer needs which number

The distinction is not academic and the two numbers get used by different people for different things, which is worth setting out because it is the practical reason to keep both.

The velocity freedoms decide the actuators. A machine needs at least as many independent actuators as it has velocity freedoms if it is to command all of them, and no more than that is useful without redundancy. A differential drive has two and takes two motors; a mecanum platform has three and takes four, one of which is redundant.

The reachable dimension decides whether the machine can do the job. A warehouse robot that must arrive at a station facing a particular way needs a reachable dimension of three, and it has one — but at a price its designer has to know, because the price is what decides how much clear floor the station needs in front of it.

The growth vector decides the floor space. This is the number that gets left out. A machine whose worst direction is two brackets deep needs room proportional to the square root of the accuracy it must achieve in that direction, and a machine three brackets deep needs the cube root. That is a layout constraint, it is computable before anything is built, and neither of the other two numbers contains it.

The three questions are asked by three different people and answered by three different quantities, and the reason they were ever confused is that on a machine made of links they are the same quantity.

What to say when somebody asks how many degrees of freedom a car has

The question is asked constantly and it has three defensible answers, which is a symptom rather than a paradox.

Two, if the question is what a driver controls: the accelerator and the steering wheel. This is the velocity-freedom count and it is what the mobility field computes.

Three, if the question is what a car’s pose is: xx, yy and heading. This is the dimension of the space the car lives in, and it is what a person laying out a car park cares about.

Four, if the steering angle is counted as part of the configuration, which it has to be for the mechanism’s own arithmetic to work — a car cannot change its steering angle instantly, so the angle is a state.

All three are in the ledger’s row for a car: 2, 4 and 4, with the 3 being the pose part of the 4. The reason the question feels slippery is that it is three questions, and the reason it never felt slippery on a four-bar is that there the three answers coincide.

What a car forbids, and what it permits. The constraint rows above and the permitted directions below, at one configuration. The two were written from opposite ends of the same geometry — the rows from what may not happen, the fields from what may — and the largest product between any row and any field is 1.4e-17. That is this field's version of the two routes the rest of the site runs on, and every figure here rests on it: the pictures are drawn by integrating the fields and captioned with what the rows forbid.
Fig. 8 A car’s rows and fields, which is where all three answers can be read off. Four columns is the third answer; two field rows is the first; and what the second answer needs is the observation that one of the four coordinates is a steering angle rather than a place.

The pair is the specification, and the gap is the interesting number

The practical upshot of two counts rather than one is that neither of them is what a machine should be described by. The description is the pair, and the quantity a designer actually reasons with is the difference between them.

Call the difference the gap: the reachable dimension less the velocity freedoms. A rolling wheel has two velocity freedoms in a three-dimensional reachable set, so its gap is one. A ball rolled without twisting has two controls in a five-dimensional set, so its gap is three. A closed-loop linkage of any kind has a gap of nought, which is what made the distinction invisible for six fields.

The gap counts the directions that exist and cannot be taken directly — the ones that have to be manoeuvred into. That makes it the right number for the question a designer of such a machine actually asks, because every unit of it is a direction whose cost obeys the inverse law this field keeps measuring: the room available raised to a power, and the driving required as its reciprocal. A gap of one is a wheel’s sideways displacement at ε2\varepsilon^2 per manoeuvre. A gap of three is a ball’s orientation, two brackets deep, and correspondingly worse.

So the two numbers have different jobs and it is worth saying which is which. The velocity count is what the actuators must supply: two motors for a differential drive, and no more, because a third would be fighting the constraints. The reachable dimension is what the planner must reason about: three coordinates for a wheel, five for a ball, and a plan is a path in that space rather than in the smaller one. A machine whose designer used one number for both jobs has either specified a motor it does not need or written a planner that cannot express where it is going.

And the gap is what the two disagree about, which is exactly the region where manoeuvring lives. A mechanism with no gap needs no manoeuvres and can be commanded directly in every direction it can reach, which is why a linkage’s controller is a different kind of object from a mobile robot’s. Nothing in a count of joints and links sees that distinction, and this field exists because a wheel is the first mechanism on this site for which it is not nought.

One caution about the gap, since it is being proposed here as a design number. It is a generic quantity, computed from brackets at a configuration, and a mechanism can have a different gap somewhere else in its configuration space — a wheel at a configuration where two of its constraint rows coincide is a different arithmetic problem from the same wheel elsewhere. So the gap is a fact about a neighbourhood rather than about a machine, and quoting it as a single number for a mechanism carries the same assumption every generic statement on this site carries: that the configuration is not special. Where that assumption fails is exactly where this field’s singular arrangements are, and it fails for the same reason and with the same warning attached.

Where this goes next

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. Three parallel bars, and a formula that says this cannot move. Five links and six pin joints, so Grübler's criterion gives 3(5−1) − 2(6) = 0 and calls it a structure. The Jacobian has rank 5 against 6 free coordinates, so it measures one degree of freedom — and the sweep assembles 59 of 60 positions, which settles the matter. The third bar removes no freedom because its constraint is already implied by the other two, and a formula that counts joints cannot notice that they happen to be parallel. Mechanisms of exactly this kind carry drafting machines, anglepoise lamps and locomotive coupling rods, where the redundant bar is there for load sharing and for keeping the linkage out of its change point. The mechanism Grübler says cannot move Three parallel bars between two frames. Five links, six pins, and the criterion every engineering course teaches gives zero degrees of freedom — a structure. It is a mechanism, it is in drafting machines and locomotive coupling rods, and the formula cannot see why. A rolling wheel, where it was driven to. The mechanism at a configuration nothing wrote down: it was reached by integrating permitted velocities from the start of the trail, and there is no equation here whose root it is. The barred line at each wheel is the direction that wheel forbids — the subject of the whole field, and the one thing a photograph of a car cannot show. The constraint residual along the drawn history is 0.0e+0. A constraint that takes nothing away A rolling wheel forbids one direction of motion and removes no coordinate from the mechanism's description. It cannot slide sideways and it can still be brought to any position at any heading — and the gap between those two sentences is the whole of this field, because in every mechanism built of pins and slides the two agree.

About the same objects

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

What links here

Essays that link to this one from their own argument.

The objects this essay names

Each one links to every other essay that touches it.

Constraint jacobianDegrees of freedomGrowth vectorKutzbach's criterionMobilityNonholonomicRankReachable setRolling constraintVelocity freedom