Round a cusp into another assembly

One command, six answers found that the site’s planar platform, with its three motors locked, can be in as many as six different poses, and it drew the practical conclusion that the pose a machine is in depends on its history. Then it added a sentence that made the history simple: the machine cannot pass from one assembly mode to another without going through a configuration where two modes coincide, which is a direct singularity. On that reading the mode is a constant of the motion, broken only by an event a controller can see coming.

For this platform the sentence is false, and it has been known to be false for mechanisms of this kind since the late 1990s. This essay finds the place where it fails on the standard platform, drives the machine through it, and measures how far from singular the platform stays while its assembly changes.

Where the assembly count changes #

The platform is the one the parallel field has used throughout: three legs of two links each, 1.4 long, from motors on a circle of radius 2 to attachment points on a platform of radius 0.6. Hold the first motor at 216° and each setting of the other two is a point in a plane of motor angles. At each point the platform has some number of assemblies, found as before by reducing the forward problem to one equation in the platform’s angle and scanning it.

Over a 41 by 41 grid of a 40° square of that plane, the counts are 2 at 1,261 points, 4 at 347 and 6 at 73. They are not scattered. They form regions, and the boundaries of the regions are exactly what the earlier essay identified from the other side: curves along which two assemblies merge and disappear, which is to say curves of direct singularities.

Most of each boundary is a smooth curve, and crossing it changes the count by two. But in two places inside this square, two boundary curves meet at a sharp point rather than crossing or joining smoothly. Those points are cusps, and the whole phenomenon lives at them.

A cusp is three assemblies becoming one #

The assemblies at a setting of the motors are the zeros of a single closure function of the platform’s angle. One command, six answers drew that function and counted its zeros. In that language the three objects on the slice are three orders of zero.

A simple zero is an ordinary assembly. A double zero, where the function touches the axis without crossing, is two assemblies merged: a direct singularity, and a point on one of the boundary curves. A triple zero, where the function, its slope and its curvature all vanish together, is three assemblies merging at once.

In a two-dimensional slice of motor angles, a triple zero happens at isolated points, because three conditions are being imposed on the two free motor angles and the platform angle. So a cusp can be located exactly, not found by looking: three equations, three unknowns, Newton’s method from a nearby start. The cusp enclosed by the loop comes out at (101.3541°, 185.3187°), with the platform at −50.6°, solved to a residual of 2.2 × 10⁻¹². The third derivative there is −9.059, which is not zero, so the point is a genuine cusp and not some higher degeneracy.

Across the whole slice at a 3° sampling there are nine such cusps. Two of them are in the square drawn.

The loop #

The loop is a circle in the plane of motor angles, radius 8°, centred at (98.15°, 185.32°). Its centre sits 3.2° to one side of the cusp, so the cusp is inside with 4.8° to spare, and the nearest of the other eight cusps is 11.1° from the centre, well outside. It was chosen from a search over circles about every cusp in the slice for the one that changes assembly while staying furthest from a singularity, with no leg ever passing straight.

The loop starts at its lowest point, (98.15°, 177.32°). There the platform has four assemblies, and they are genuinely different machines.

With the three motors at 216°, 98.15° and 177.32°, the platform can be turned to −79.4°, −44.2°, −27.3° or −8.2°, with every leg exactly its own length in each. The closure function has four simple zeros there, and the scan that found them is exhaustive to a tenth of a degree.

The motors are then driven once round the loop, 720 steps, and each of the four assemblies is followed by continuation from where it began. Each step solves the platform’s pose from the previous one, which is what a controller tracking a real machine does.

What each assembly does #

The four do four different things, and all four are informative.

Two of them die almost at once. The assemblies that start at −44.2° and −27.3° run towards each other as the motors begin to move, and 2.5° round the loop they merge and cease to exist. That is an ordinary crossing of a boundary curve, from a four-assembly region into a two-assembly one, and it is what the earlier essays led a reader to expect: assemblies are created and destroyed in pairs, at direct singularities.

Their signs say which kind of event it is. With the elbows placed where the motors put them, the assembly at −44.2° starts with det A at +0.116 and the one at −27.3° at −0.165, and over those 2.5° the two values fall towards each other, to +0.081 and −0.100, before the pair merges. Two assemblies of opposite sign meeting at a zero of det A is exactly what a fold of the singular curve is, and it is how the three-legged platform loses a pair of assemblies wherever the loop crosses a fold rather than circling a cusp.

One comes back as itself. The assembly that starts at −8.2° goes round the loop and returns to −8.2°.

One comes back as another. The assembly that starts at −79.4° goes all the way round, never merges with anything, and at the end of the loop, with the motors back at exactly 216°, 98.15° and 177.32°, the platform is at −27.3°. That is not a pose near where it started. It is one of the four assemblies the loop started among, a different machine by every measure, turned 52° from the pose it left.

Nothing about the motors distinguishes the start from the finish. The encoders read the same three numbers. The platform is in a different assembly.

How near it came to a singularity #

The claim is not only that the assembly changed but that it changed without passing a direct singularity, so the distance from singularity has to be measured all the way round, not assumed.

The measure is the smallest singular value of the matrix whose rows are the three leg lines, the same quantity the singularity map shades by. It is zero at a direct singularity and about 0.9 at the platform’s home pose. On the track that changes assembly its lowest value anywhere on the loop is 0.0716. For comparison, the two tracks that do meet a singularity fall from 0.053 and 0.065 to zero within the first 2.5° of the loop.

Two sign records go with it. The determinant of that same matrix keeps one sign along the whole track, which is what never crossing a direct singularity means. And every leg’s entry in the other matrix of the velocity relation, the one that vanishes when a leg goes straight, keeps its sign too, never falling below 0.537 in size. So no leg passes straight either, and the working mode, the choice of elbow on each leg, is unchanged. The machine changed its assembly and nothing else, and it did so at a stated distance from every kind of singularity the platform has.

Drawn frame by frame, the motion is undramatic. The motors move a few degrees each way. The platform slides and turns steadily, its smallest singular value between 0.106 and 0.494 in the frames drawn, with no leg approaching straight and no moment of visible difficulty. A person watching the machine would see nothing special happen, and at the end the machine would be in a pose its motor readings share with three others.

Why a cusp allows it #

The mechanism is easiest to see on the closure function itself.

At a point on a boundary curve two zeros of that function merge. Along one boundary curve it is one particular pair that merges, say the first and second zeros counted by platform angle; along another it is a different pair, the second and third. At a cusp the two curves meet and all three zeros merge together.

Now go round the cusp. Starting in the region with more assemblies, cross the first boundary: the first and second zeros annihilate, and the third survives. Continue round and cross back over the second boundary: a new pair is born, and it is born as the first and second zeros of the new function, next to the survivor. By the time the loop is closed the function has as many zeros as it started with, but the zero that survived both crossings has been relabelled. It entered as the third assembly and leaves as the first.

That is why the tracked assembly never meets a singularity: it is always the survivor. The pairs that are destroyed and created are other assemblies, and the loop passes through their singularities without the tracked one ever being involved. The two assemblies that die 2.5° into the loop are one such pair.

The same loop, moved clear of the cusp #

An explanation that credits the cusp has to predict what happens without one, and the prediction is testable with the same computation: move the loop so that it encloses no cusp, and every assembly that survives the loop must come back as itself.

Within the part of the slice from 60° to 140° in the second motor and 145° to 225° in the third, the cusp search finds two cusps: the one enclosed above, at (101.35°, 185.32°), and a second at (108.24°, 180.74°), 11.1° from the loop’s centre. The loop is moved 20° up the third motor’s axis, to a centre of (98.15°, 205.32°), keeping its radius of 8°. Its nearest cusp is then 20.25° from its centre, so it encloses neither.

At its starting point the platform has two assemblies. Both are followed round the loop, both complete it, and both come back as themselves. Neither comes near a singularity on the way: the smallest singular value on the two tracks never falls below 0.286 and 0.315. Nothing merges and nothing is relabelled, because there is no place inside the loop where three assemblies meet.

That is the control the explanation needs. Without it, a change of assembly on a small loop could have been some accident of the loop’s size or of where it happened to start. With it, the same loop does or does not change the assembly according to one thing: whether a cusp is inside.

It is tempting to go one step further and ask what a loop enclosing both cusps does. A loop of the same radius centred between them, about 4.1° from each, was driven round, and it does carry one assembly into another, but not cleanly: along it the legs pass through straight configurations hundreds of times, which is an inverse singularity and a change of working mode. That loop is measuring a different event, and whether two cusps enclosed together cancel each other’s relabelling on a loop that avoids every singularity is left open here.

Why the sign does not settle it #

There is a tempting test for which assembly a platform is in: the sign of det A. At a direct singularity det A passes through zero, so two assemblies on opposite sides of a boundary have opposite signs, and one might hope the sign names the assembly.

On this platform it cannot. The assembly that starts at −79.4° and the one at −27.3° that it becomes both have det A negative, and the track between them keeps that sign throughout, as it must if it never crosses a singularity. Two different assemblies of the same motor angles share a sign, so a sign is not a name. Handed the platform’s four assemblies at the loop’s start, a test requiring each sign to identify one assembly fails, as it must: two positive and two negative.

The contrast is the smallest parallel robot, a planar five-bar, where the same test passes on every sample, because that mechanism’s forward problem has only two answers and a cusp needs three.

What it does to the claim that the mode is constant #

The earlier essay’s practical advice survives with its reasoning changed.

A controller that tracks by continuity is still right. At every step of the loop the pose was the continuation of the previous one, and nothing jumped. The machine never enters an ambiguous state during motion, which is the property continuous tracking needs.

The mode is not a constant of the machine. It is a constant of the motion’s history, and a history can include a loop round a cusp. Two runs of the same machine that start in the same assembly and end at the same motor angles can end in different assemblies, depending on which side of a cusp their motor paths passed. That is a statement about the topology of the motor paths and nothing else.

A redundant sensor is still needed, for exactly the reason the earlier essay gave: switched on cold, the motor readings are shared by several poses. What changes is that no bookkeeping of singularities crossed can replace it, since the assembly can change without crossing one.

And one thing becomes possible. A pose in another assembly, which a planner forbidden to cross singularities would have treated as unreachable, can be reached by going round a cusp. The usable workspace of a cuspidal platform is larger than the singularity-free region around its starting assembly, and how much larger is a question about where its cusps are.

What this essay does not establish #

The loop lies in one slice of motor space, with the first motor held at 216°. The full set of cusps in the three-dimensional space of motor angles is a set of curves, and it is not mapped. Nine cusps in this slice at a 3° sampling is a count at a resolution; a denser sampling might find cusps closer together.

Whether the motion is physically possible is not checked. The links are lines here, and whether the platform and legs pass through each other somewhere round the loop is a question for the bodies field.

The result for this class of mechanism is cited, not rediscovered: Innocenti and Parenti-Castelli described singularity-free changes of assembly for a planar platform in 1998, and McAree and Daniel related them to cusps of the singular set shortly afterwards. What is measured here is that the standard platform is cuspidal, where one of its cusps is, and how far from singular a change of assembly on it can stay.

What comes next #

The mechanism that cannot do this. A platform whose forward problem has two answers has no cusps, and its assembly is named by a sign. The smallest parallel robot is that mechanism, and it is the control for everything measured here.

What a fourth leg does to the cusps. A redundant leg removes the singular curves of a slice, leaving isolated points. The cusps were meetings of those curves, so a four-legged platform should have none in the slice, and the question of which assembly it is in should become a question a fourth encoder answers directly.

How large the reachable set really is. A planner allowed to go round cusps can reach every assembly connected to the start through singularity-free loops. Counting, for each setting of the motors, how many of its assemblies are reachable from a home pose without a singularity would turn the earlier essay’s “one to six” into a number with a meaning for a machine in service.