Machine Kinematics
solved, then drawn
Concept

Redundancy — where it appears

Having more joints than a task needs, so that one tool pose is held by a curve of postures rather than by a list of them. It buys the ability to avoid obstacles and joint limits while holding the tool still, and it costs a control problem, since the extra freedom has to be resolved by something.

Named by 13 essays across 5 fields — each of them below, with the objects they name alongside it.

Mobility, counted and measured. Grübler's criterion counts links and joints and knows nothing about the dimensions; the rank of the constraint Jacobian measures the dimensions and knows nothing about the topology. They agree for four of these five. The parallelogram with a redundant third bar is the exception: the formula declares it a structure with zero degrees of freedom, and it moves. The formula is the one that is wrong, because it cannot see that the third bar's constraint equations are already implied by the other two.

Counting and measuring mobility

Grübler's criterion counts links and joints and never asks how long anything is. The rank of the constraint Jacobian measures the lengths and never asks what a joint is. Two calculations with no inputs in common, producing one number — which is the only arrangement under which agreement is evidence.

constraint · Mobility 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.

constraint · Mobility Reachable, and dexterous. A 3-link planar arm with links 1.6, 1.2, 0.7. The outer region is everywhere the tool can be put: an annulus from 0.00 to 3.50. The inner region is everywhere it can be put at every tool angle — from 1.10 to 2.10, which is 26% of the area. Both are measured by counting cells on a 300 × 300 grid and both agree with the area computed from the radii to 0.09%, which is what makes the picture a measurement.

Where the hand can go

A robot is sold on its reach, which is one number and describes a sphere the arm touches at one posture. The set the tool can actually be put in is an annulus with a hole; the set it can be put in at every orientation is a quarter of that; and reordering the same three links leaves the first unchanged and destroys the second.

serial · Serial 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.

parallel · Parallel 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.

serial · Serial Arms with their tools pinned down. Pin an arm's tool to the ground and the open chain is a closed loop, which the mobility count of a planar loop already handles. 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.

Pin the tool and it is a loop

Hold an arm's tool still and the open chain becomes a closed one, which these essays have known how to count since their 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.

serial · Spatial Three legs and four, at 0°. The same slice of positions at a platform angle of 0°, shaded by how well the platform is held — pale is near singular. Left, three legs: 3,312 reachable samples and a singular curve through them in 102 segments. Right, the same three legs and a fourth: 3,198 reachable, because the fourth leg must reach too, and no curve. What is left of the singular set in this slice is 1 isolated point, at (-1.449, -0.811), where all four lines meet. Positions held above 0.1 go from 2,989 to 3,178, and at no sampled position is the four-legged platform held less well than the three-legged one.

What a fourth leg buys

Three leg lines fail to hold a platform when they meet at a point, which is one condition, so in every slice of the workspace the failures form a curve. Four lines fail only when all four meet at a point, which is two conditions, so the curve becomes isolated points. The fourth leg buys that and more, and it costs a machine that can no longer be assembled from any four motor angles.

parallel · Parallel A fourth leg through the meeting point, and one beside it. The platform with similar triangles at its singular orientation, at one position, with a fourth base pivot at (0, -2). Left: the fourth platform point placed where the similarity puts it, so the fourth leg's line passes 2.8e-17 from the point the other three meet at, and the smallest singular value of all four is 2.59e-9 — nothing has changed. Right: the same base pivot with the platform point moved 0.28 round the platform, so the fourth line misses the meeting point by 0.3993 and the four legs hold at 2.914e-1. The ringed point is where the three original lines meet, and it moves with the platform.

One placement of every placement

A planar platform whose two triangles are similar is singular at every position it can be put in, at two orientations. A fourth leg removes both — unless its own pair of attachment points is related by the same similarity, and then it changes nothing at all. Swept right round the platform, exactly one placement of the fourth attachment fails, the similarity names it in advance, and the holding a placement buys is proportional to how far it sits from that one point.

parallel · Parallel Two legs that share a line of force. Two legs of three revolute joints each under one platform, drawn about their common centre: the first leg's joints in one colour and the second's in another. The second leg was built so that each of its joints passes through a point of one line, the dark one, which also crosses all three joints of the first leg — the six marked points, each met to 1e-16. A force along that line does no work on any joint of either leg, so both legs resist it: it is a load path the platform has twice over. The dashed line is what comes with it: the axis of the one twist both legs permit, a screw of pitch -0.049 that passes 0.032 from the shared line at 57.2° to it. Dragging moves the shared line along the first leg's first joint and builds the second leg afresh through each position.

A shared load is a shared motion

Put two legs of three revolute joints under one platform and let them share a line of force, so that both carry the same load. The platform does not merely gain a redundant load path; it gains a motion in the same act, because three constraints and three make six. The motion is a screw about a different line, tied to the shared one by h cos φ = d sin φ on two hundred constructed pairs to 4 × 10⁻¹⁵. Pushed along it, the two legs part as the square of the push: the platform shakes and does not move.

spatial · Spatial A seam that holds two mechanisms. Ten rows of a strip of kagome lattice sixty cells wide, centred on the line where two differently distorted lattices meet. Below it every up-triangle's joints are moved along its sides by 0.3, 0.3, −0.3, polarization −1 in the component that counts along the seam; above it by −0.3, −0.3, 0.3, polarization +1. No bar is missing anywhere along the seam. Each joint is drawn with an area proportional to its share of the strip's two mechanisms at one wavevector. They sit on the seam and die away from it into both lattices, and the strip's outer edges, out of the picture, have none. The wavevector along the seam is 3/12 of a turn per cell.

A seam with no bar missing

Join two distorted kagome lattices of opposite polarization along a line. Every joint on the line keeps all four of its bars, so Maxwell's count says the line holds nothing. It holds two per wavevector: two mechanisms one way up, two states of self-stress the other way up, exactly the difference of the two polarizations. The self-stresses are tensions the lattice carries with no load on it, along a line where no bar is spare. In a strip sixty cells wide the prediction holds at every wavevector a quarter of the zone or more from its centre.

networks · Network A third leg that keeps the platform's shake. The two legs of the earlier construction — the second built through a line of force of the first, the dark line — share one twist, whose axis is the dashed line: a screw of pitch -0.049. A third leg is drawn in a third colour, built so that its three joints span a family of motions containing that twist: two joints chosen freely and the third solved for. The platform on all three legs keeps its one freedom. The dotted line is the axis of the one wrench the third leg shares with the first: a screw of pitch 0.201, not a line of force, and 0.244 from the pair's shared line. Dragging picks another third leg from the same family.

A third leg that keeps the shake

Two three-joint legs that share a line of force share one twist as well, and a third leg in general position takes it away. A third leg whose own freedom contains that twist does not, and building one is three conditions on twelve numbers: two joints anywhere and the third solved for. Such a leg must share a load with each of the first two — but not the line they share. What it shares with each is a screw with a pitch of its own, reciprocal to the twist through the sum of their two pitches, and only a third leg built through the pair's line shares the line itself. With any of them the platform still shakes rather than moves.

spatial · Spatial The joint space an arm's own material forbids. A three-link planar arm, at every pair of relative joint angles on a 101 × 101 grid, with the pairs at which its first and third links are inside each other shaded. It is 21.4% of the arm's own joint space, and it is a two-dimensional picture of a three-dimensional space because the first joint does not enter it: turning the whole arm about its base carries every link with it, so a collision at one base angle is a collision at all of them — checked at forty pairs of relative angles with 0 disagreements. Only the first and third links can meet; consecutive links share a pin and are excluded, exactly as in the closed-loop machines.

The arm that hits itself

A three-link arm's joint space is a cube of angles and it may not use a fifth of it, because its own material is in the way. The forbidden set does not depend on where the arm is pointing — which is why it can be drawn as a picture rather than described as a volume.

serial · Serial A seventh joint steers the base roll off its stop. The S-R-S arm with stops on every joint, moving its tool in a straight line at fixed orientation from the same start (faint). With its upper-arm roll locked, a six-joint arm, the move ends 27.0 cm along at the base roll at its stop (dashed arm). With all seven joints and the self-motion steered at every step to move the joint nearest its stop away from it, it ends 61.5 cm along at the base roll at its stop (solid arm). Shown 60.0 cm along, with the six-joint arm stopped where its move ended.

A seventh joint cannot move the elbow

A seven-joint arm can change its shape while its tool stands still, and on a straight move that freedom can be spent keeping its joints off their stops. Spent that way, it lengthens the median move from 23.3 cm to 33.3 cm, against the same arm with its extra joint locked. But the gain is all or nothing. Half the moves a stop used to end gain under a tenth, and a quarter gain more than two and a half times. One stop it never gets round: the elbow's, whose angle is set by the distance from shoulder to wrist and is the same, to 10⁻¹⁵ rad, whatever the other joints do. Nor does it ever reach past the edge of the arm's reach.

serial · Serial

Named alongside it

The objects these essays reach for when they reach for this one.

MobilityJacobianRankReciprocal screwSingularityConstraintInverse kinematicsJoint limitsNull spacePlatformRedundant constraintScrew system

All concepts