Machine Kinematics
solved, then drawn
What a joint is

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.

Assumes A chain multiplies and Twelve kinds of freedom.

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

A serial chain gives its tool the product of its joints’ groups, and a product of groups is almost never a group. A parallel machine gives its platform something else entirely.

The platform is connected to the frame by several legs at once. It may make a displacement only if every leg permits it, so the set of displacements it may make is the intersection of what the legs permit:

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

And an intersection of groups is a group. Always. It needs no coincidence, no alignment condition and no luck: if two sets are each closed under composition and inverses, so is anything in both.

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, intersected. Every cell is a group, because an intersection of groups always is — the only construction in this field that produces closure for nothing.

That single asymmetry is why a parallel machine can be designed for a motion type while a serial one has to be checked for it.

What a leg permits

A leg is itself a serial chain, so what it permits its far end to do is the product of its own joints’ groups — and that product is usually not a group either. So the argument only works when each leg is built so that its product is one, which means arranging the leg’s joints to share a common group in the way the previous rungs describe.

That is exactly what the classical parallel architectures do. A leg of three parallel revolutes permits planar motion. A leg of a revolute and a slide on the same axis permits cylindrical motion. A leg ending in a spherical joint permits everything below whatever holds it. In every case the leg’s own joints are chosen so that the leg confines the platform to one of the twelve, and then the design question is which twelve to intersect.

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. 2 One leg, on its own: three parallel pins, permitting planar motion. A platform held by several legs like this may do only what all of them permit.

Two planes make a line

The smallest example is the one the site already has a measurement of.

Take two legs, each three revolutes with parallel axes, and make the two axis directions perpendicular. Each leg confines the platform to a planar group — the one whose normal is that leg’s axis direction — so the platform is in both, and the intersection of two planar groups with non-parallel normals is the one-dimensional group of translations along their common perpendicular.

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. 3 Two planar groups, drawn as the planes their orbits are, meeting in a line. Drag the arms towards each other: the intersection is a translation at every angle but zero.

That mechanism is Sarrus’s linkage, and its platform goes up and down and does nothing else. The spatial field measured that straightness by solving the mechanism sixty times and reporting a departure of 9.8×10−169.8 \times 10^{-16} of the span. Here it is a two-line argument with no mechanism in it, and a rung of its own is about what the two routes each contribute.

Reading the table

The intersection table is the design tool, and reading a few of its cells is the fastest way to see what the method buys.

Two planar groups with different normals give a translation. Sarrus, as above, and every straight-line mechanism built as a pair of planar arms.

Two spherical groups with different centres give the identity. Two ball joints at different points hold a body completely — which is obvious once said and is the reason a two-ball-joint link is a rigid strut rather than a mechanism.

A planar group and a cylindrical group whose axis is the plane’s normal give a translation along that axis plus a rotation about it — a cylindrical group again, or less, depending on where the axis sits.

Schoenflies motion intersected with Schoenflies motion about another direction gives all the translations. Two four-joint legs of SCARA type, with their rotation directions different, leave a platform that can be positioned anywhere and cannot turn at all. That is a Delta-type motion, and it is the specification behind every high-speed pick-and-place machine with three arms: pure translation, produced by intersecting two or three groups that each contain it.

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. 4 The lattice, read as an intersection diagram. Two groups intersect in a group at or below both of them, and the lowest common one is what the platform gets.

The lattice is the fastest way to read the table: the intersection of two of the twelve is at or below both in the diagram, and choosing legs is choosing two nodes whose meet is the wanted motion.

Three legs and a spare

A detail that looks like bookkeeping and is a real design freedom: more legs than the intersection needs.

If two legs already intersect in the wanted group, a third leg whose group also contains that motion adds nothing to the classification — the intersection of three is the same as the intersection of two. What it adds is everything else: load path, stiffness, symmetry, and a smaller working range on each leg.

That is why the standard architectures have three of something rather than two. A Delta machine has three arms and needs two; a planar parallel manipulator has three legs where two would fix the motion type. The extra leg is redundant in the group sense and not in any other, and this is one of the few places in kinematics where a redundancy is free rather than a source of overconstraint.

It is free because the intersection is idempotent in the right way: adding a set that already contains the answer changes nothing. Compare that with a serial chain, where adding a joint always adds a dimension unless the joint is exactly aligned with something — a fourth parallel pin on three parallel pins adds a freedom that is redundant in a way the arm has to be told about.

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

Why the intersection is the algebra’s

Computationally the intersection is a piece of linear algebra: intersect the two algebras, which is the null space of the two projections stacked, and the answer is the algebra of the intersection group.

That step deserves a sentence of justification because it is not true of arbitrary groups. For connected subgroups of the rigid displacements it is: the identity component of the intersection has as its algebra the intersection of the algebras, and every group in the classification is connected and exponential. So the whole computation is a rank and a null space, with no mechanism solved anywhere and nothing to converge.

This is the same move the field has made twice already — a joint’s group from a surface’s normals, a chain’s group from the span of some logarithms — and it is what makes the field cheap. A parallel machine’s motion type is available before any leg lengths are chosen, from six-vectors and a rank decision.

The one thing an intersection cannot give

Every cell of the table is a group at or below both of its arguments, and that is a ceiling as well as a guarantee.

An intersection never gives more than either leg permits. So a platform’s motion type is bounded by the poorest leg, and adding legs can only take motion away. If the wanted motion is Schoenflies, every leg must permit at least Schoenflies motion, which means every leg is at least four-dimensional and none of them can be a simple planar arm.

That is a genuine constraint on the architecture and it is where parallel design gets expensive. A wanted motion of dimension four needs legs of dimension four or more, which means legs with four or more joints each, which means a machine with a great many joints in it — three legs of five joints is fifteen joints for a platform with three or four freedoms. The count that would report such a machine as wildly redundant is reading the joints; the intersection is reading what they are for.

The lattice says which combinations are even worth trying, and its most useful line is a negative one: the spherical group is inside nothing but the whole displacement group. So a platform that must rotate about a point in three ways cannot get it from any intersection of smaller groups, and every three-rotation parallel machine has legs that permit essentially everything and is constrained only by the closure of its loops.

The design rule

Put together, the rule is short enough to state in one paragraph and it is the practical output of this whole field.

Take the wanted motion. Find two or more of the twelve groups that contain it. Build a leg for each, so that each leg’s joints all lie in that group and generate it. The platform gets the intersection, whatever the leg lengths are.

The last clause is the one that matters. The platform’s motion type does not depend on any dimension: not on the lengths, not on where the legs are attached, not on the platform’s size. Those decide the workspace, the singularities and the stiffness, and they do not decide what kind of motion it is. That is a robustness no serial architecture has, and it is the reason parallel machines are designed by type synthesis while serial ones are designed by inverse kinematics.

Sarrus's linkage, and the group it moves inTwo three-pin arms with perpendicular planes. The platform goes up and down and does nothing else. The dashed stubs are the joint axes, the solid marker is one point of link 2 and the faint curve is everywhere that point goes. Every frame is a **solve**: the joint angles are the unknowns, the closure of the loop is the equation, and a frame is drawn only where the residual comes below 10⁻⁹. What this field adds to the picture is one number. Take the displacements this link reaches, take their logarithms, and close them under the bracket: the answer is **1**, so the motion lies inside a translation and composing two of its displacements gives a third one it also reaches, to 1.6e-12. positioned by solving, not by drawing.6 joints · Sarrus's linkageinside T
Fig. 5 The rule delivered. Sarrus’s linkage drawn from solved configurations, its platform tracing a straight line, with the closure of its reached displacements reported at one dimension.

Where the mobility count and the intersection disagree

They agree about the dimension whenever both are computed correctly, and the interesting cases are the ones where the count is computed correctly and is still misleading.

A platform held by three planar legs with parallel normals has, by arithmetic, three freedoms. A platform held by three planar legs with two normals equal and one different has, by arithmetic, three freedoms too — the count reads links and joints and neither arrangement changes those. The intersections are the whole planar group and a one-dimensional translation, so the two machines have three freedoms and one.

The count is not being naive here; it is being applied to a mechanism whose axes satisfy a special condition, which is the standing failure mode of every mobility formula on this site and is the reason the site has always run a rank measurement beside the count. The rank gets it right, because it reads positions.

What the rank still cannot do is say which three freedoms, and for a parallel machine that is most of the specification. A platform with three freedoms may be a planar mechanism, a spherical one, or a pure-translation one, and those are three different machines sold for three different jobs. The intersection names it, from the legs’ geometry, before the platform exists.

Where it goes wrong

Three failure modes, and all three are visible in the same framework.

A leg whose product is not a group. If a leg’s joints do not share a common group, the leg does not confine the platform to one of the twelve, and there is nothing to intersect. The platform’s motion is then whatever the closure equations say, which is a solve rather than a classification. Most general parallel machines — a Gough platform among them — are exactly this: six legs, each permitting five dimensions of nothing in particular, and a platform with six freedoms and no motion type at all.

Legs whose groups intersect in more than was wanted. Three legs whose planar groups all have the same normal give the platform the whole planar group — three freedoms rather than one. That is not a mistake in the arithmetic; it is a design in which a condition was met by accident, and it is what Sarrus’s linkage becomes when its two arms are built in one plane.

Legs whose groups intersect in less. Add a leg whose group does not contain the wanted motion and the platform loses it. This is the ordinary way an over-enthusiastic bracing scheme turns a mechanism into a structure, and the intersection says so before anything is built.

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 5°meet in 1 dimension
Fig. 6 The second failure mode, at five degrees rather than ninety. The intersection is still one-dimensional here and it becomes three at exactly zero — the mechanism does not lose its straight line gradually, it loses it at one angle.

An intersection is exact and a product is approximate

There is a final asymmetry between the two constructions that is worth naming because it decides how much each can be trusted on a real machine.

A product of subgroups is a group only when an alignment condition holds exactly, and the condition fails under any perturbation: the dimension jumps at zero tilt. So a serial machine’s membership of a group is a nominal property that a real machine has only approximately, and the right quantity to quote is the bracket defect rather than the dimension.

An intersection is a group whether or not anything is aligned. But which group depends on the legs’ geometry, and there the same sensitivity reappears one level in: two planar groups with normals at ninety degrees meet in a translation, and so do two at eighty-nine, and so do two at one — but two at exactly zero meet in a plane. So the motion type of an intersection is robust across a wide range of geometry and falls over a cliff at the degenerate configurations.

That is a much better situation than the serial one, and it is the honest version of the design rule’s last clause. The platform’s motion does not depend on the leg lengths, and it does depend on the legs not being arranged in a degenerate way — which is a condition on a whole open set rather than on one exact value, and is therefore a condition a machine shop can meet.

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. 7 Where the two constructions meet the site’s older instruments. Sarrus’s row is an intersection: one dimension, a translation, and a defect at the solver’s floor.

Error moves the axis and leaves the type

The exactness of an intersection is stated above as an advantage over a product, and it has a sharper form worth writing out, because it is the strongest claim this field makes about manufacture.

Take Sarrus’s linkage and build it badly: two planar legs whose normals are meant to be perpendicular and are three degrees out. The intersection of two planar groups with normals n1n_1 and n2n_2 is the set of motions in both — translations lying in both planes, which is the single direction n1×n2n_1 \times n_2, together with rotations about both normals, of which there are none unless the normals agree.

So the intersection is still a one-dimensional translation group, for any two normals that are not parallel. Three degrees out, thirty degrees out, it makes no difference to the kind of thing the platform does: it translates along a line, exactly, with no rotation and no second direction.

What the error changes is where the line points. The direction n1×n2n_1 \times n_2 moves continuously with the normals, so a badly built Sarrus translates along a slightly different axis than the drawing says. That is a positioning error and it is measurable and correctable; it is not a change in the mechanism’s character.

Set that beside the serial case and the asymmetry is complete. Three nearly-parallel pins do not give planar motion approximately — they give a set whose logarithms span all six dimensions, which is a different kind of object, and the mechanism has stopped being what it was designed to be. A serial chain’s group is lost to error and a parallel machine’s is not.

That is the honest reason parallel architectures are used where motion type is the requirement — a platform that must not rotate, a stage that must stay level, a mechanism that must translate along one line. The type is guaranteed by an intersection that no tolerance can break, and what tolerance costs is the axis’s direction, which is exactly the sort of error a machine is calibrated for.

It also says what a parallel machine still has to be made carefully for. The intersection’s type is robust and its placement is not, so every parallel machine’s accuracy problem is a placement problem — where the axis is, where the plane is, where the centre is — and none of them is a question about whether the mechanism does the right kind of thing.

What the count sees of this

None of it. A mobility count applied to a parallel machine adds up links and joints and returns a number, and the number is the dimension of the intersection when the arithmetic is right. It cannot name the group, so it cannot distinguish a platform that translates from one that turns, and it cannot see the design rule at all — because the rule is about which groups the legs are in, and a count has no representation of that.

The site’s own count against rank is the same story: two routes to the dimension, and this field is a third route to the object. Why the platform stays flat was argued in the parallel field from the structure of the legs, and it is an intersection argument written out longhand; three legs and one plane is the same argument with the group named.

Where T3 can send one point. The orbit of a single point of the moving body under all translations, which is everywhere within reach. The motion of a Cartesian machine; three joints, no pair. 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 What a Delta-type machine’s platform gets: all the translations, no rotation, from intersecting two Schoenflies groups about different directions. A three-dimensional group with no pair, produced by intersecting two four-dimensional ones.

The next rung takes the smallest of these intersections and follows it all the way down, because it is the one case where the site already has an independent measurement to check the group argument against.

Where this goes next

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. 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. 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. 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.

What this makes readable

Essays that name this one as a prerequisite.

About the same objects

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The 8 of 9 essays linking to this one that name the most of the same objects.

The objects this essay names

Each one links to every other essay that touches it.

ConstraintDisplacement subgroupLegMobilityOrbitParallel mechanismPlatformSchoenflies motionSubalgebra