Machine Kinematics
solved, then drawn
One path to the tool

Pin the tool and it is a loop

Hold an arm's tool still and the open chain becomes a closed one, which this site has known how to count since its first field. Kutzbach's criterion says a pinned six-joint arm is a structure. At each of its three singularities the measurement says it can still move — the site's founding finding, arrived at from the far end of its own subject.

Assumes Where the arm loses a direction and The formula is repaired by the thing it replaced.

16 min read 7 figures Two routes to the same numberThe singularities are the pointSolved, not drawn

This field opened by saying that an arm has no loop in it, and that this is why nothing here is solved. There is one operation that undoes that, and it takes a clamp.

Pin the tool to the ground. The chain closes. What was an open chain of six joints becomes a closed loop of six joints and six links, and everything this site built in its first six phases applies to it without alteration.

Arms with their tools pinned down. Pin an arm's tool to the ground and the open chain is a closed loop, which the first field of this site knows how to count. Kutzbach gives 6(n − 1) − 5n = n − 6 for a loop of n revolutes, and the measurement is n minus the rank of its screw system — the columns of the arm's own Jacobian, read as constraints rather than as velocities. The two agree on every row but one, and the one is the arm at a wrist singularity: the formula says the pinned arm is a structure and the mechanism has a freedom. That is the finding this site opened with, arrived at from the far end of its subject.
Fig. 1 Four arms with their tools pinned, counted and measured. Kutzbach’s criterion for a closed loop of n revolute joints is 6(n − 1) − 5n = n − 6, which knows nothing but how many joints there are. The measurement is n minus the rank of the loop’s screw system — which is the arm’s own Jacobian, read as a constraint system rather than as a velocity map. Two of the four disagree.

The count

A closed loop of nn revolute joints has nn links: n−1n-1 moving ones and the ground, joined in a ring. Kutzbach’s criterion gives

M=6(n−1)−5n=n−6M = 6(n-1) - 5n = n - 6

For six joints, zero. A pinned six-joint arm is a structure: it has exactly as many constraints as freedoms and nothing is left over. For seven joints, one — a mechanism with a single degree of freedom, which is the swivel the previous essay measured.

The measurement is the site’s usual one and it is the same matrix twice. The loop’s joint screws are the columns of the arm’s Jacobian; their rank is the dimension of the space of twists the loop can produce; the mobility is nn minus that rank. The two routes to the rank are a row reduction and an eigendecomposition of the normalised screw set, and they agree on every row of the table.

Three parallel bars, and a formula that says this cannot moveFive 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 redundant oneGrübler: 3(5−1) − 2(6) = 0Jacobian: 6 − rank 5 = 1
Fig. 2 The famous case, from the site’s first field: three parallel bars, a formula that says the mechanism is a structure, and a mechanism that visibly turns. What makes it move is the special geometry the count cannot see. A pinned arm at a singularity is the same sentence with the special geometry supplied by an accident of posture rather than by a designer.

Where it disagrees

At an ordinary posture the count and the measurement both give zero. Clamp the tool of a six-joint arm and it is rigid; nothing moves; the machine is a very expensive strut.

At each of the arm’s three singularities the rank falls to five, and the measurement gives one. The formula still says zero.

That is a working mechanism being declared immobile by a count — which is exactly what happened in this site’s very first field, where a parallelogram with a third parallel bar is counted at zero and turns. Six phases and a whole second half of the subject later, the same sentence is true about a robot arm with a clamp on its wrist.

The mechanism is the same in both cases: the formula counts joints and links and the geometry decides the answer. Three parallel bars impose a constraint that two of them have already imposed; an arm at a wrist singularity has two joints whose axes are the same line, so one of the six constraints the pin imposes is one the arm could not have violated anyway.

And every singularity of the arm is such a case. The elbow, the shoulder and the wrist all give rank five and all give a pinned mobility of one. The set of postures at which the count is wrong is exactly the singular set — not approximately, not usually: exactly, because both statements are the same rank condition written down twice.

elbow arm at a postureelbow arm, drawn from 6 joint values through a product of 6 exponentials — no equation is solved anywhere in this picture, because an open chain has none to solve. The thin lines are the joint axes at this configuration, which are also the columns of the arm's Jacobian: the tool's velocity is a sum of turns about exactly those lines. Here the smallest singular value of that Jacobian is 0.0000 and the largest is 2.603, so the arm is approaching a configuration where a direction of motion is lost. Drag θ₅ pitch.θ₁ baseθ₂ shoulderθ₃ elbowθ₄ roll θ₅ θ₆toolσ_min 0.0000 · condition Infinitythe pose is a product of exponentials, not a solve
Fig. 3 The wrist singularity, which pinned is a mechanism. Two of the six dashed lines are the same line, so the six joint screws span five dimensions instead of six — and the loop has one freedom left. Bolt this arm’s tool to a table and it will still move, slowly and in one particular way, however tightly the pin is done up.
What the last millimetre costs. At each distance ε from the wrist singularity, the tool is asked to move at unit speed in the direction the arm is worst at, and the largest joint rate that requires is recorded. The points lie on a line of slope -1.010 in the logs, so the demand goes as ε to that power — one over the distance, measured rather than quoted. The check on the fit is the other exponent: σ_min itself goes as ε to the power 1.008, and the two must be equal and opposite. At ε = 0.005 rad, a quarter of a degree, the arm needs 289 radians per second of joint rate for one metre per second of tool speed.
Fig. 4 What the pinned arm has left, measured rather than counted: the tool velocities the joints can still produce. A structure by the count, and an ellipsoid with a nonzero volume in the directions the loop does not constrain.

What it can still do, when it is a structure that moves

The freedom left at a singularity has a shape, and the shape is the constraint’s reciprocal.

Away from a singularity, an open chain’s six joint screws span the whole space of twists, so the set of wrenches reciprocal to them is empty — there is nothing the arm cannot act against, which is another way of saying it can move any way it likes. At a singularity the span drops to five and one reciprocal screw appears: the line along which the tool cannot be moved.

Pinned, that reads the other way round. The pin imposes six constraints and only five of them bite; the sixth is reciprocal to the freedom that survives. So the pinned arm’s residual motion is the twist the open arm could not produce — the same object, read from the other side of the clamp.

The previous essay measured that object at each of the three singularities: a pure force along the arm at the elbow, a pure force across it at the shoulder, and at the wrist a screw of pitch −0.629, which is a force and a couple in a fixed ratio.

The seven-joint arm is a mechanism with a mobility

The row that agrees is worth as much as the rows that do not.

A pinned seven-joint arm has M=7−6=1M = 7 - 6 = 1 by the formula and 7−6=17 - 6 = 1 by the rank. It is a one-degree-of-freedom spatial mechanism, and its one degree of freedom is the elbow swivel that the redundancy essay walked round on a circle of radius 0.192 m.

Two fields of this site arrive at that circle from different directions and neither of them needed the other. The redundancy essay found it as the null space of a 6×76 \times 7 Jacobian; the constraint field would have found it by counting a loop’s joints and links and getting one. A redundant arm’s self-motion is a closed-loop mechanism, and the reason a seven-joint arm can swivel its elbow is the reason a four-bar can turn its crank.

That also settles what the swivel’s dimension is without any linear algebra. One, because a seven-joint loop counts one, and the count is right here — the arm at an ordinary posture has no special geometry for it to be wrong about.

Three bars and four bars. On the left, two bars to a common point: three links, three joints, and Grübler gives 3(3−1) − 2(3) = 0. The Jacobian agrees — two free coordinates, rank 2, nothing left over — and the shape cannot change without a bar changing length. On the right, one more bar and one more joint gives mobility 1, and the whole of this site follows from that difference. The triangle is why bridges are triangulated and the quadrilateral is why machines are not.
Fig. 5 The question the first field opens with, which a clamp turns an arm into an instance of. Counting says structure; measuring says structure at almost every posture and mechanism at three of them; and the disagreement is not about arithmetic but about whether a formula can see where the axes are.

Pinning less than all of it

A clamp takes six freedoms away. Most of the constraints a real arm meets take fewer, and the arithmetic runs the same way with a different number in it.

Hold the tool point still but let the tool turn — a ball joint, or a spherical bearing on the end of a rod — and three constraints are imposed. The measurement gives rank three and a mobility of three: the arm can still turn its tool about the fixed point in three independent ways, which is a spherical mechanism and is what a joystick is.

Hold the orientation and let the point move — a tool that has to stay square to a surface but may slide on it — and again three constraints, rank three, mobility three.

what is held constraints mobility
the tool on a plane 1 5
the tool on a line 2 4
the tool point, freely turning 3 3
the point and one rotation 4 2
the whole pose 6 0

Every row is 6−rank⁡6 - \operatorname{rank} of the selected rows of one matrix, and the mobility falls one per constraint because at an ordinary posture the rows are independent. That last clause is the whole content of the table: the constraints add up only when the arm is not singular, and the singular postures are exactly where a constraint is free.

This is also where the pinning stops being a thought experiment. A robot pressing a tool against a surface is the first row; a robot with a part in a fixture that allows it to slide along a slot is the second; a robot holding a workpiece against a rotating spindle is somewhere in the middle. Each is a mechanism whose mobility is a rank calculation, and every one of them can lose or gain a freedom at a posture where the arm’s own geometry is special.

One leg, and the platform with six

There is one more reading of the pinned arm and it puts this field beside the parallel one.

A Gough–Stewart platform is a moving body connected to the ground by six legs. A pinned arm is a moving nothing — the tool is the ground — connected by one chain of six joints. Same subject, opposite extreme: six short chains of few joints each against one long chain of many.

The arithmetic is the same arithmetic. A platform’s mobility is the number of actuated freedoms minus the rank of its leg wrench system, and its direct singularity is where those six leg lines become linearly dependent and the platform gains a freedom it cannot control. A pinned arm’s mobility is its joint count minus the rank of its joint screws, and its singularity is where those six lines become dependent and the pinned arm gains a freedom.

Two mechanisms, one rank condition, and the same consequence: a machine that moves when it should not. What differs is which end is being held — and that, as this essay keeps finding, is a boundary condition rather than a change of subject.

Two ways of saying “does not move”

Two rows of the table disagree in a different manner and this site has met that too.

A pinned SCARA — four joints — is counted at 4−6=−24 - 6 = -2 and measured at 4−4=04 - 4 = 0. A pinned planar three-link arm is counted at 3−6=−33 - 6 = -3 and measured at 0. In both cases the formula and the measurement agree that the thing does not move; they disagree about by how much, and a negative mobility is not a quantity any mechanism has.

The expansion phase met this exactly and named the distinction: a disagreement of −2-2 against 00 is of a different kind from a disagreement of 00 against 11, because both numbers mean the same thing physically. What the negative count is really reporting is redundant constraint — the arm’s axes are all parallel, or all vertical, so the pin’s six constraints are not independent and some of them are asking for something the geometry has already guaranteed.

That is the same statement as “a planar four-bar is an overconstrained spatial mechanism”, which the expansion phase made about the whole first half of this site. A planar arm pinned is overconstrained by three; a SCARA pinned is overconstrained by two; and neither is a defect. They are simply mechanisms whose special geometry the count cannot see.

The corrected count, and where it comes from

The spatial-depth phase closed this gap with

M=6(n−j−1)+∑fi+νM = 6(n - j - 1) + \sum f_i + \nu

where ν\nu is the number of redundant constraints — and its finding was that the correction term is not in the joint graph. It has to be measured from the geometry, which means the formula meant to avoid the geometry needs it.

Every row of the table above is another instance. For the pinned six-joint arm at a singularity, ν=1\nu = 1; for the SCARA, ν=2\nu = 2; for the planar arm, ν=3\nu = 3. Each of those is the rank deficiency of the screw system and each is invisible to any count of joints and links.

The pleasant thing is that the arm cases make ν\nu easy to interpret. On the SCARA it is two because four parallel vertical axes cannot produce a tilt about any horizontal axis, so two of the pin’s six constraints are free. On the planar arm it is three for the same reason with one more axis’s worth of it. On the six-joint arm at the wrist it is one, and the one is the pair of coincident axes.

Three singularities, and where each one is. The elbow singularity is at the edge of the workspace, where the arm is straight and cannot reach further — nothing is lost that could have been used. The shoulder singularity is where the two ways of facing the target merge, on a cylinder of radius 0.18 m about the base axis which is also the boundary of a hole the arm cannot reach into at all. The wrist singularity is the one that stops real machines: axes four and six in line, one rotation gone, in the middle of an ordinary working volume with nothing about the tool's position to suggest it. The rank falls by exactly one in all three — and what is lost differs. The elbow and the shoulder are constrained by a pure force, along the arm and across it; the wrist is constrained by a screw of pitch −0.629, which is a force and a couple together. Those are reciprocal screws rather than singular vectors, because a singular vector's direction depends on whether the arm was written in metres or millimetres and a screw does not.
Fig. 6 The open-chain reading of the same three postures. The rank is the same number in both tables — it is one matrix — and it is doing two jobs: here it says the tool has lost a direction of motion, and there it says the pinned arm has gained one. The clamp is what turns the sentence round.
Arms with their tools pinned down. Pin an arm's tool to the ground and the open chain is a closed loop, which the first field of this site knows how to count. Kutzbach gives 6(n − 1) − 5n = n − 6 for a loop of n revolutes, and the measurement is n minus the rank of its screw system — the columns of the arm's own Jacobian, read as constraints rather than as velocities. The two agree on every row but one, and the one is the arm at a wrist singularity: the formula says the pinned arm is a structure and the mechanism has a freedom. That is the finding this site opened with, arrived at from the far end of its subject.
Fig. 7 The seven-joint case, where the pinned loop keeps a genuine freedom. The count and the rank agree here for the same reason they disagreed above — the constraints are independent, so subtracting them is the right arithmetic.

What the pinned arm is not

Three things this closure does not do, and they are worth marking because a bridge invites too much traffic.

It does not make the arm’s forward problem hard. Pinned or not, the arm’s pose at given joint values is the same product of exponentials. What the pin changes is which joint values are admissible, and that is a constraint on the configuration space rather than a solve. The pinned arm’s configuration space is a five-dimensional slice of the seven-dimensional joint space for a redundant arm, a set of isolated points for a six-joint one, and empty for most tool positions — because there is no reason a given clamp position should be reachable at all.

It does not make the loop’s mobility a statement about the arm’s usefulness. A pinned six-joint arm having mobility zero is a description of a clamped machine, not a compliment. It is the positive mobility at a singularity that is the finding, and it is a finding about a failure.

It does not import the parallel field’s second kind of singularity. A parallel platform has two — one where a leg runs out and one where the platform escapes. A pinned serial arm has a single chain, so there is only ever one rank to lose, and its two “kinds” of singularity are the same kind seen through the clamp: the open arm loses a direction and the pinned arm gains one.

Why pinning is not a trick

It would be reasonable to object that clamping a robot’s wrist is not something anybody does, and that the whole exercise is a way of making the arm fit the site’s existing machinery.

Two answers, and the second is the useful one.

It is done. A robot pressing a part into a fixture, holding a workpiece against a machine tool, or driving a screw into something rigid has its tool constrained by the world. The constraint is not always all six components, but whatever it is, the arm is in the position this essay describes and the question of whether it can still move — and how — is a real one. An arm that can move with its tool clamped is an arm that can be pushed out of position by a load it cannot resist.

It makes the two halves of the subject one subject. Everything on this site before this field was a loop, everything in this field is a chain, and pinning shows they are the same objects with a different boundary condition. A mechanism’s mobility, its singularities, its screw system and its redundant constraints all mean the same things on both sides; only the vocabulary was different, because one tradition writes about machines with cranks and the other about machines with tools.

The clamp is a bridge, and what crosses it in both directions is a rank.

What this ladder has been about

This essay sits on the spatial anchor rather than the arm’s own, at the top of the ladder that started with six freedoms rather than three, and it is worth saying why.

That ladder has been one argument all the way up: a count of joints and links cannot see the geometry, and the mechanisms worth building are the ones where the geometry is special. Its rungs are the universal joint, Sarrus, Bennett, the spherical four-bar, Bricard, the corrected count with its ν term, and the two kinds of overconstraint that no count separates.

Every one of those is a closed loop that somebody designed to exploit a coincidence. The pinned arm is the first case on the ladder where the special geometry is not a design at all — it is an accident of where the arm happens to be standing, it appears and disappears as the machine moves, and it is a fault rather than a feature. Bennett’s linkage moves because its lengths satisfy a condition. A pinned arm moves because its wrist happens to be flat, which nobody wanted and which the machine will pass through again in a few seconds.

Same mathematics, opposite intent. It is the strongest evidence this phase produced that the two halves of mechanism kinematics are one subject: the classical overconstrained linkages and a modern robot’s singularity are the same rank deficiency, and the only difference is whether somebody arranged it.

The table of partial pinnings is worth reading as one computation rather than as a list of cases, because that is what it is. Every row is 6−rank⁡6 - \operatorname{rank} of a selected subset of rows of the same matrix — the arm’s own Jacobian — with the selection saying which components of the tool’s motion are being held. Six rows selected is a clamp; three is a ball joint; the three rotational rows alone is a tool held square to a surface and free to slide on it. So a designer with one matrix can ask about any partial constraint whatever, including ones no fixture obviously produces: hold two components of position and one of orientation, hold the tool on a line, hold it in a plane. Each is a different subset, each is one rank, and the mechanism’s remaining mobility falls out. That is a considerably more general instrument than pin the tool and count the loop, and it costs nothing extra — the matrix was computed to answer the unpinned question, and the pinned ones are subsets of it.

Where this goes next

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. Kutzbach, plus the constraints counted twice. Each row is 6(n − j − 1) + Σf, then the redundant constraints ν measured from the two legs' constraint systems, then the mobility measured from the rank of the whole loop's screw system. The first column is wrong for 5 of 6 of these mechanisms; adding ν repairs every one. The catch is that ν is not a property of the joint graph — Bennett's linkage and a spatial four-bar with one twist changed have the same graph and different ν — so the corrected formula needs the measurement it was supposed to replace. The formula is repaired by the thing it replaced Kutzbach's count is wrong about most of the mechanisms worth building, and every textbook gives the same repair — add back the constraints that were imposed twice. The repair works on every loop this site has. It is also not a formula, because the number it adds cannot be read off the joint graph. One tool pose, a curve of postures. 41 postures of S-R-S arm, every one of them holding the tool at the same pose. The tool moved 1.0e-13 of a metre over the whole sweep, which is the arithmetic's noise floor rather than a tolerance. The joint that moves most is the elbow, and it runs on a circle of radius 0.192 m about the line from shoulder to wrist — a curve computed here by walking the Jacobian's null space, and known independently as the intersection of two spheres. The two agree to 10⁻¹³. The freedom that does nothing Give an arm a seventh joint and one tool pose stops having eight answers and starts having infinitely many. The arm changes shape while the tool stands still — measured at 3.3 × 10⁻¹² of a metre over forty-one postures — and the elbow runs on a circle that two entirely different computations agree about to a tenth of a picometre.

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.

Kutzbach's criterionLoop closureMobilityOpen chainOverconstrainedRankReciprocal screwRedundancyScrew systemSingularity