Why the platform stays flat

The delta robot is the parallel mechanism most people have watched working: three arms swinging above a conveyor, a small triangular platform under them picking things up four times a second, and the platform staying resolutely horizontal the whole time.

The staying horizontal is the part worth explaining. Three legs, three degrees of freedom — and no obvious reason those three should be the three translations. A three-legged mechanism could perfectly well have two translations and a rotation, or some mixture that is neither. Nothing about “three legs” says which three.

The answer is a parallelogram in each leg, and getting from there to “the platform cannot turn at all” is a constraint computation.

What a parallelogram does #

Each delta leg is an upper arm driven by a motor at the base, and then a lower section made of two parallel rods of equal length running down to the platform, pinned by ball joints at all four corners.

Those two rods and the two bars they connect form a parallelogram, and a parallelogram’s defining property is that its opposite sides stay parallel. So the bar at the platform end stays parallel to the bar at the arm end, whatever the rods do.

That is the whole mechanism of it. The rods can swing — which is how the platform moves — and the bar cannot turn relative to the arm’s bar, which is how the platform stays flat.

More precisely: the leg permits the platform no rotation about either of the two directions perpendicular to its bar. It says nothing about rotation about the bar, which that leg alone leaves free.

So each leg imposes two couple constraints: a couple is a wrench of infinite pitch, with a direction and no line of action, and it is exactly the object that says “no turning about this direction”. Two per leg, about the two directions perpendicular to that leg’s bar.

Three legs, six couples, three of them wasted #

Three legs impose six couples between them. The space of couples is three-dimensional — there are only three independent directions to prevent turning about — so six couples cannot be independent.

The computation: take the six, form the subspace they span, and read its order. With the three bars 120° apart in a plane, the answer is three. Six imposed, three independent, so three are redundant, which is ν in the vocabulary the constraint field established.

Then the platform’s freedom is what is reciprocal to that three-dimensional couple system. A twist $[\omega ; v]$ is reciprocal to a couple $[0 ; m]$ when $\omega \cdot m = 0$, so being reciprocal to all three independent couples forces $\omega = 0$ exactly. What is left is $[0 ; v]$ with v free.

Three pure translations. The platform can go anywhere within reach and cannot turn about anything.

That is the derivation, and it is four lines because the parallelogram did the hard part. Everything after “each leg imposes two couples” is a rank and a complement.

The condition, and breaking it #

“Three legs impose six couples spanning three dimensions” is a claim about the three bar directions, and it is not automatic. It holds when the three directions span — that is, when they are not all parallel and not all in one plane through the origin in the wrong way.

So the derivation has a condition, and a claim with a condition should be tested by violating it.

assertDeltaNeedsItsThreeDirections builds a second mechanism with all three bars parallel. Each leg still imposes two couples, six in all — but now every leg’s two couples are about the same two directions, so the six span only two dimensions. Four are redundant instead of three, and what is reciprocal to a two-dimensional couple system is a four-dimensional twist system: three translations and one rotation, about the shared bar direction.

The platform can turn. A delta built that way would swing on its own legs, and the swinging would be uncontrolled — no motor opposes it, because no leg constrains it.

Both halves are asserted, and the second is what makes the first mean anything. Without it the check would pass on a deltaConstraint that had started returning three translations regardless of its input.

The count, and whether Kutzbach survives it #

It is worth putting the delta through the mobility arithmetic, because it is the one mechanism in this field where the naive count is spectacularly wrong and the correction is exactly the redundancy computed above.

Counted as joints and links — three arms, three parallelograms with four ball joints each, a platform — the delta has a great many joints, most of them spherical with three freedoms apiece, and Kutzbach’s arithmetic gives a number that is not three. Which number depends on how the parallelograms are counted, and every treatment of the machine has a footnote about it.

The correction is $\nu = 3$, and it is the same three this essay computed: three couple constraints imposed three times over. Add them back and the count is three, which is the number of freedoms the machine has.

That is the redundancy essay’s point restated on a machine in a factory. ν is not in the joint graph — a delta with three parallel bars has the same joints, the same links and the same graph, and its ν is four — so the correction needs the geometry. What makes the delta a useful example is that the geometry in question is three directions, which is about as little geometry as a mechanism can depend on, and it is still enough to change the answer.

What the model is, stated plainly #

This is a place where the site is modelling rather than solving, and the boundary should be visible.

The computation above takes as given that a parallelogram contributes two couple constraints about the directions perpendicular to its bar. That is a statement about what a four-ball-joint parallelogram does, and it is asserted here rather than derived from the four spherical joints.

It is a standard and correct statement, and it is where the delta’s behaviour comes from. But it is an input to this computation, not an output of it, and a reader should know which. What the site computes is everything downstream: the span of the six couples, the redundancy, the reciprocal system, and what happens when the bar directions change.

The alternative — building the full mechanism as three closed loops of four spherical joints each and taking ranks — is available and is not here. It would derive the two-couple statement rather than assuming it, at the cost of a good deal of machinery for a result that is not in doubt. Naming the choice is what keeps it honest.

There is one thing the assumed model quietly gets right and is worth crediting: the parallelogram also has a passive freedom of its own — the coupler bar can spin about the axis of the two rods — which does not affect the platform’s orientation and which real deltas either accept or suppress with a fourth central shaft. Modelling the leg by its constraints rather than by its joints makes that freedom simply not appear, which is correct for the question being asked and would be wrong for a question about the rods.

Reading the freedom system as three translations #

One notational point is worth clearing up, because the library’s own vocabulary reads oddly here and would mislead anyone checking the numbers.

screwKind classifies a screw of infinite pitch as a couple. Read as a force system that is exactly right: a couple has a direction and no line of action. Read as a motion, the same six numbers are a pure translation — no angular part, a direction, and no axis.

So when the delta’s freedom system comes back as “three couples”, what that says is three screws of infinite pitch, which as motions are three pure translations. The word is the force-system name for a shape of screw that has two readings, and this is the one place on the site where the twist reading is the one wanted.

That is a small thing and it is exactly the sort of small thing that produces a wrong caption. The classification is about the six numbers; what they mean depends on whether they were built as a motion or as a force, and nothing in the six numbers records which.

Between the good case and the broken one #

The condition is tested at its two extremes — three independent bar directions, and three parallel ones — and a rank is a discrete quantity, so that test reports a pass and a failure with nothing in between. The mechanism has a great deal in between, and it is where the workspace boundary comes from.

Move the three bar directions gradually towards coplanarity and the six couples still span three dimensions right up until they do not. The rank is 3, then 3, then 3, and then abruptly 2. Nothing in the count degrades on the way, so a check that reads only the rank sees a perfectly healthy mechanism at a configuration where the platform is nearly free to tilt.

What does degrade is the conditioning, and it degrades continuously. The three-dimensional couple system is spanned by vectors that are becoming nearly dependent, so the constraint against one particular tilt is carried by an ever-smaller combination — and a constraint carried by a small combination is one that a small compliance, a small clearance or a small manufacturing error can defeat. The platform does not tilt because the geometry forbids it; it tilts because the forbidding has become weak.

That is the same distinction this site draws everywhere between a rank and a number, and it is unusually consequential here because the delta is three times over-constrained. Redundancy means the constraint is carried by more rows than are needed, which is protection in the good region and is not protection near the boundary — the redundant rows go nearly dependent together, since they are all functions of the same three bar directions.

So a delta’s usable workspace is bounded by a conditioning threshold rather than by a rank change, and the threshold is a judgement rather than a geometric fact. That is why published delta workspaces are quoted with a qualifier, why two manufacturers with identical link lengths advertise different working volumes, and why the boundary is a smooth surface rather than the locus of a determinant vanishing.

It also explains the shape of the machine. A delta’s arms are kept well away from the configurations where the bars approach coplanarity, which is what limits its reach outward and downward, and the limit arrives long before anything runs out of joint travel. A user meeting that boundary experiences it as the machine becoming imprecise rather than as it refusing to move, which is exactly what a conditioning limit feels like and is not at all what a singularity feels like.

The check as written is still the right one, and the reason is worth stating. A test at the extremes establishes that the derivation depends on the condition, which is what an assertion is for; measuring the conditioning across the workspace is a different job, and one this field would need a norm and a threshold to do. Naming the gap is the honest position, and it says exactly what a delta’s specification is short of.

The redundancy, and what it buys and costs #

Three redundant constraints is a lot. It is the same ν as a planar four-bar’s, on a machine with three legs and ball joints rather than four bars and pins.

What it buys is the reason the machine exists. The platform’s orientation is held by three legs all doing the same job, so it is held stiffly — the platform does not tilt under an off-centre load the way a machine relying on one constraint would. Combined with the motors being on the ground and the moving parts being three light rods, that is what makes a delta the fastest pick-and-place mechanism there is.

What it costs is the standing price of overconstraint, and here it is unusually visible. Three redundant constraints means the three parallelograms must agree about what horizontal is. If one leg’s two rods are not exactly equal in length, its bar is not exactly parallel to the arm’s bar, and its couple constraint disagrees with the other two. The platform then cannot satisfy all three and the mechanism deforms — or, in a machine with ball joints and clearance, takes up the clearance and stops being precise.

This is why a delta’s rods are made in matched pairs and why the ball joints are held by springs rather than by circlips. The kinematics says the constraint is imposed three times over. The manufacturing has to make the three impositions agree.

Why the workspace is the shape it is #

One consequence of the translation-only result is worth drawing out because it is what a user of the machine actually meets.

A delta’s platform cannot turn, so a delta has three freedoms and cannot orient what it picks up. Every real delta therefore has a fourth axis: a shaft down the middle, driven from the base, that spins the end effector about the vertical. It is a serial degree of freedom bolted onto a parallel mechanism, and it exists because the parallel part cannot supply it.

That is a design pattern the field’s opening essay predicts in general — a parallel stage for the stiff, fast, short-range motion with a small serial addition for the orientation — and the delta is its cleanest instance. The parallel part gives three fast translations; the serial shaft gives one slow rotation; and the split is not a compromise anyone chose, it is what the constraint computation says the parallel part can and cannot do.

The other thing three legs could have given #

It is worth naming what a delta is not, because the alternatives are real machines and the difference is a design decision rather than an accident.

Three legs and three freedoms could equally well have produced a mechanism with two translations and a rotation — which is the planar platform of this field’s first essay, with its legs confined to a plane. It could have produced one translation and two rotations, which is the arrangement used in several telescope and machine-tool heads where a surface has to be tilted and moved along its normal. It could have produced three rotations about a point, which is a spherical parallel mechanism and is used as a robot wrist.

All four are three-legged, all four have three freedoms, and which three is decided entirely by what each leg’s constraint system is. The delta gets translations because its legs constrain rotation; a spherical wrist gets rotations because its legs constrain translation; the planar platform gets the planar subgroup because its legs are planar.

So “three legs, three freedoms” is not a description of a machine. It is a description of a count, and the count is the least informative thing about any of these mechanisms. What distinguishes them is a three-dimensional subspace of screw space, and that is not something a joint diagram shows.

Where this sits in the field #

The delta is the third mechanism of this field and it is the one whose behaviour is entirely a constraint result.

The planar platform is about the forward and inverse problems and their asymmetry. The six-legged platform is about the same asymmetry at the scale where the forward problem stops being tractable. Neither needed the screw machinery to be stated, though both are clearer with it.

The delta cannot be explained without it. “Why does the platform stay flat” has no answer in terms of positions and lengths — the positions are exactly what changes — and it has a four-line answer in terms of what each leg cannot resist. That is the argument for the whole of the constraint field made on a machine people have watched work, and it is why this essay sits at the top of this ladder rather than at the bottom.