Machine Kinematics
solved, then drawn
Several legs, one platform

Three legs and one plane

The parallel field argued that a platform stays flat from the structure of its legs, one leg at a time. The same fact is one line of linear algebra: each leg confines the platform to a group, the platform gets the intersection, and an intersection of groups is a group whatever the leg lengths are. The motion type is decided before a single dimension is chosen.

Assumes Why the platform stays flat and Legs intersect.

16 min read 8 figures Two routes to the same numberSomebody had a problem

Why the platform stays flat is the parallel field’s structural argument: follow each leg, see what it allows the platform to do, and find that the allowances agree on a plane. It is a good argument and it is made one leg at a time.

The same conclusion is a line of linear algebra, and the difference between the two routes is worth the essay.

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 on this site: 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.
Fig. 1 Every pair of the twelve groups, intersected. Every cell is a group, because an intersection of groups always is.

The rule

A platform connected to the frame by several legs may make a displacement only if every leg permits it. So its permitted set is the intersection of the legs’:

platform  =  L1∩L2∩⋯∩Lk.\text{platform} \;=\; L_1 \cap L_2 \cap \cdots \cap L_k .

If each LiL_i is a group, the intersection is a group. That needs no coincidence and no alignment: anything closed under composition and inverses, intersected with anything else that is, is closed under both.

And the intersection is computable from the legs’ geometry alone. Each leg’s group has an algebra — a subspace of the twists — and the intersection of the subspaces is the algebra of the intersection group, for connected subgroups. Two rank decisions and a null space.

No length enters. Not the leg links, not the platform’s size, not where the legs attach. Those decide the workspace, the singularities and the stiffness; they do not decide what kind of motion the platform has.

The smallest case, and the site’s own measurement

Two legs, each three revolutes with parallel axes, with the two axis directions perpendicular.

Each leg confines the platform to a planar group. The intersection of two planar groups with non-parallel normals is the one-dimensional group of translations along their common perpendicular. The platform translates.

That mechanism is Sarrus’s linkage, and the spatial field measured its straightness by solving it sixty times: a departure of 9.8×10−169.8 \times 10^{-16} of its span. The group argument gives the same answer from two normals and a cross product.

Sarrus, as two planes meeting in a lineEach 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.T — a translationone arm's planethe other arm's planetwo planar groups at 90°meet in 1 dimension
Fig. 2 The two arms’ planes, meeting in a line. Drag them towards each other and the intersection is one-dimensional at every angle but zero.

Both routes are needed and neither is a check on itself. The group argument says the straight line is exact and says why; the measurement says the mechanism as built delivers it, which nothing about a group guarantees — a mechanism has to assemble, its solve has to converge, and its travel has to exist.

Why the two routes are not the same argument

The structural argument and the group argument reach the same conclusion and they are not interchangeable, and it is worth being precise about the difference.

The structural argument follows a mechanism. It takes a particular platform with particular legs, works out what each leg allows the platform to do at the configuration in hand, and concludes. It is concrete, it is checkable by eye, and it is a statement about that mechanism.

The group argument follows the axes. It reads only which direction each leg’s joints are parallel to, produces a group per leg, and intersects. It never mentions a configuration, so its conclusion is about the whole motion at once rather than about the position drawn.

That second property is the one worth having. A structural argument made at one configuration has to be repeated at another, or supplemented by a continuity claim; a group argument covers the travel by construction, because a group is the same seen from any of its elements. The site has been quietly relying on continuity in exactly this way for several fields, and this is where the reliance is discharged.

Reading a design off the table

The table is the field’s design tool and a few cells are worth reading.

Two spherical groups with different centres give the identity. Two ball joints at different points hold a body completely, which is why a link with a ball at each end is a strut rather than a mechanism — and why a Gough platform’s legs each have to have a slide in them.

Two Schoenflies groups about different directions give all the translations. Two four-joint legs of SCARA type, rotation directions different, and the platform can be put anywhere and cannot turn. That is a Delta-type motion, and it is the specification behind every high-speed pick-and-place machine with three arms.

A planar group and a cylindrical group about the plane’s normal give the rotation and the translation on that axis. Which is a cylindrical group again, and is how a platform is given a screw-free turn-and-lift.

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.
Fig. 3 The lattice, read as an intersection diagram. Two groups meet at or below both of them, so choosing legs is choosing two nodes whose meet is the wanted motion.

Three legs where two would do

A structural detail that the group view explains cleanly.

If two legs already intersect in the wanted group, a third leg whose group also contains that motion changes nothing about the motion type. The intersection of three is the intersection of two.

What it adds is load path, stiffness, symmetry and a smaller range demanded of each leg. So the standard architectures having three of something rather than two is a decision made entirely outside kinematics, and it is free in the kinematic sense — which is not true of a serial chain, where a fourth parallel pin on three parallel pins adds a freedom the controller has to be told about and which does nothing.

Redundant legs are cheap and redundant joints are not, and the difference is the difference between an intersection and a product.

A platform’s group and its workspace

Two things a reader could reasonably conflate, and separating them is what makes the “no length enters” claim survive contact with a real machine.

The platform’s motion type is the intersection of the legs’ groups. It is a set of kinds of displacement, it has a dimension, and it is decided by axis directions.

The platform’s workspace is the set of poses it can actually reach. It is bounded, it is decided by every length in the machine, and it is generally an awkward shape with holes in it where legs collide and edges where they run out.

Every pose in the workspace is an element of the group; not every element of the group is in the workspace. So the claim that no length enters is a claim about the first and says nothing about the second, and the parallel field’s own work on the map of where it fails is entirely about the second.

The two questions also fail differently. A workspace shrinks continuously as links shorten. A motion type does not shrink at all until a degeneracy, and then it changes discontinuously. Anybody reading a parallel machine’s specification is reading both, and they behave nothing alike.

What the count can and cannot see

The parallel field’s own mobility arithmetic gives the platform’s number of freedoms, and it does so correctly whenever its assumptions hold.

Three legs whose planar groups have parallel normals give a platform with three freedoms. Three legs where two normals agree and one differs give a platform with one. The arithmetic reads links and joints, which are identical in the two cases, so it returns the same number for both — and it is the ordinary special-geometry failure the site has run a rank measurement beside since the foundation.

The rank gets the dimension right. What it cannot do is say which three freedoms, and for a platform that is most of the specification: three freedoms may be planar motion, spherical motion or pure translation, and those are three different machines. The intersection names it, from the legs’ geometry, before the platform exists.

Four instruments, and only the last one names the group. Every instrument this site has 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 on this site for six phases, stated as an integer.
Fig. 4 Four instruments on six loops. The last two columns are the ones that name a group rather than counting a dimension.

The ceiling the intersection puts on a design

The rule guarantees a group and it also caps what the group can be, which is the half a designer notices when a specification will not close.

An intersection never gives more than either leg permits. So every leg must permit at least the wanted motion, and the wanted motion’s dimension is a floor on every leg’s dimension. A platform that must have Schoenflies motion needs legs that each permit at least four dimensions, which means at least four joints each — three legs of five joints is fifteen joints for a platform with four freedoms, and the count that reports such a machine as wildly redundant is reading the joints rather than what they are for.

The lattice makes the worst case immediate. The spherical group is inside nothing but the whole displacement group. So a platform that must rotate three ways about a point cannot get it from an intersection of anything smaller, and every such machine has legs permitting essentially everything, constrained only by the closure of its loops. That is a design that has left the type-synthesis world entirely, and it is why three-rotation parallel wrists are hard.

Where the type is lost

Only one thing changes the answer, and it is a degeneracy rather than a tolerance.

Two planar groups with the same normal are the same group, and their intersection is the whole of it: three dimensions instead of one. That is Sarrus’s linkage built flat, and the platform there can slide sideways and turn.

The change is discontinuous. There is a straight line at eighty-nine degrees between the arms, at forty-five, at five, and at a millionth of a degree; at exactly zero there is a planar mechanism. So the design condition is not an alignment to be held but an arrangement to be avoided, and a condition that holds on an open set is one a machine shop can meet.

Sarrus, as two planes meeting in a lineEach 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.T — a translationone arm's planethe other arm's planetwo planar groups at 16°meet in 1 dimension
Fig. 5 The same figure near the degeneracy. Fifteen degrees between the arms, and the intersection is still a one-dimensional translation.

That is the opposite situation from a serial chain, where being in a group requires an exact alignment and any perturbation destroys it. A parallel machine’s motion type is robust; a serial one’s is a knife edge. It is a genuine architectural advantage, it is purely kinematic, and it is usually argued for on grounds of stiffness instead.

The leg has to be a group first

The whole rule rests on a condition that is easy to state and easy to lose: each leg’s own joints must share a group.

A leg is a serial chain, so what it permits is the product of its joints’ groups, and a product of groups is a group only when all the factors lie inside one common group and fill it. Three parallel revolutes do. Three revolutes with unrelated axes do not — their product is a three-dimensional set whose logarithms span all six, and there is no group for the intersection to intersect.

So the rule inherits the serial architecture’s knife edge one level down. A leg whose three axes are nominally parallel and actually out by a few arcminutes does not confine the platform to a planar group; it confines it to something near one, and the platform’s motion is near a translation rather than being one.

That is the honest version of the robustness claim two sections up. The arrangement of the legs is robust — an open set of angles between them all give the same answer — and the construction of each leg is not. Sarrus’s linkage needs its three axes per arm parallel to the same accuracy any planar mechanism does, and what it gets for free is that the two arms may be at any angle to each other but zero.

What the group view does not cover

Most parallel machines, and the field should be clear about it.

The rule applies when each leg’s own joints share a group. A leg of five joints with unrelated axes permits a five-dimensional set that is not a group, and then there is nothing to intersect: the platform’s motion is whatever the closure equations say, computed by a solve. A Gough platform is exactly that — six legs, six freedoms, no motion type — and everything the parallel field says about it is a solve or a rank.

So the group view covers the type-synthesised parallel machines, which are the ones designed for a motion, and says nothing about the general ones, which are designed for a workspace. Both kinds exist and are built for different reasons.

Three parallel pins, and where it can put its tool. Three parallel pins 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 3 dimensions, so the motion lies inside planar 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.
Fig. 6 One leg on its own: three parallel pins, permitting planar motion. A leg like this can be intersected with another; a leg whose joints share nothing cannot.

The instrument that checks it

The group argument is a derivation and this site does not publish derivations without a measurement beside them.

The measurement is the closure test: sweep the assembled mechanism, take the displacements the platform reaches from converged configurations only, take their logarithms, and close them under the bracket. For Sarrus’s linkage it reports one dimension, of type TT, with a composition defect of 1.6×10−121.6 \times 10^{-12} — which is the solver’s own floor rather than the arithmetic’s, since every configuration is a root found to 10−1310^{-13}.

That is the derivation confirmed on the object rather than on the axes. And it is a check that could fail: a mechanism whose arms were slightly out of the plane would report a closure of six despite the design intent, and the number would say so.

For the general parallel machines the same test returns six and no type, which is the correct answer and is the shape of the boundary in the next section.

Span, and what it closes to. Two bars per loop: how many dimensions the reached displacements occupy, and how many they occupy after the brackets are added. For the four trivial loops the two bars are equal — the motion is already inside a group and bracketing adds nothing. For Bennett's linkage and the Bricard six-bar the first bar is four and the second is six, and the gap between the two bars is the whole of what paradoxical means: a one-degree-of-freedom motion that occupies four dimensions of displacement and generates all six.
Fig. 7 The test on six loops. Sarrus’s pair of bars are both one; a mechanism whose legs’ groups did not intersect cleanly would show the gap the last two rows show.

The good case is the generic one

There is an inversion between the serial and parallel arguments that is worth stating on its own, because it is the strongest form of the robustness claim and it is easy to lose among the caveats.

For a serial chain, being inside a group requires an exact coincidence among the axes — three of them parallel, three concurrent. That is a condition of measure zero: a random chain satisfies it never, and a built one satisfies it only to the extent that manufacture holds it.

For a parallel platform whose legs are already groups, the situation is reversed. The intersection is a group for any arrangement of the legs; what would spoil the wanted answer is the degenerate case where two legs’ groups coincide — two planar groups with the same normal, two spherical groups with the same centre. And that is the condition of measure zero. A random arrangement of two planar legs has different normals, and the intersection is the one-dimensional translation the design wanted.

So the two architectures have their coincidences on opposite sides. A serial chain has to hit a coincidence and a parallel platform has to avoid one, and hitting is much harder than avoiding.

The practical form of that is what a check has to look like in each case. A serial chain’s alignment is verified by measuring an equality — how far from parallel are these axes — and the answer wanted is zero, which is a quantity a machine can only approach. A parallel platform’s arrangement is verified by measuring an inequality with margin — how far from coincident are these two normals — and the answer wanted is comfortably away from zero, which is a specification a workshop can meet with room to spare.

Anybody who has held a tolerance knows which of those two is the easier job. A margin is a design decision that costs nothing once made; an equality is a running fight with every process in the shop. That is the real content of a parallel machine gets its group for nothing, and it is not that no condition exists — it is that the condition is one nobody has to work to satisfy.

What to take from it

Three sentences.

A platform gets the intersection of its legs’ groups, and an intersection of groups is a group. No alignment condition, no coincidence, no dependence on any length.

The motion type is therefore a design input rather than an outcome. Choose the motion, find groups containing it, build a leg for each, and the platform has it whatever the dimensions turn out to be.

And the only thing that can take it away is a degenerate arrangement — two legs’ groups coinciding — which is a case to be avoided rather than a tolerance to be held.

The reason this is worth having as a separate rung of the parallel field, rather than as a remark in the pairs field, is that it changes how a platform is designed. The field’s own opening essay says the easy problem and the hard one change places when a mechanism goes parallel: the forward problem becomes the hard one, and the platform’s pose for given actuator lengths takes a solve with up to forty answers. All of that is still true.

What the intersection adds is that one whole question — what kind of motion does this platform have — comes out before any of that, from six-vectors and a rank decision, and does not need the solve at all. It is the only question about a parallel machine that gets easier rather than harder.

Where T can send one point. The orbit of a single point of the moving body under a translation, which is a straight line. The prismatic pair; the surface is a prism. The orbit is the only honest picture of a group: the group itself is a set of displacements and has no shape, and what a reader can see is what it does to something.
Fig. 8 The end of the smallest case: the platform’s group drawn as the orbit of one point, which is a straight line. It is the group a prismatic pair gives, produced by six pin joints and no sliding surface anywhere.

Where this goes next

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 on this site: 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. Three bars, and no rotation left. Each of a delta robot's legs ends in a parallelogram, which keeps the bar on the platform parallel to the bar on the arm. That leg therefore permits the platform no turn about either direction perpendicular to its bar: it imposes two couple constraints, drawn here as rings about the directions they act on. Three legs impose 6; together they span only 3, so 3 are redundant. What is reciprocal to them is three couples — three pure translations, and the platform cannot turn at all. Why the platform stays flat A delta robot has three legs and three freedoms, and there is no obvious reason those freedoms should be the three translations rather than some mixture. The reason is a parallelogram in each leg, and the argument from there to "the platform cannot turn at all" is a constraint computation that takes six wrenches and a rank. 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.

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.

ConstraintDisplacement subgroupLegMobilityParallel mechanismPlatformSchoenflies motionSubalgebraWorkspace