What a fourth leg buys

The singularity map ended with a list of what the parallel field had not computed, and one entry on it was a sentence rather than a result: adding a fourth leg to a three-freedom platform removes some singularities entirely, at the cost of the legs having to agree. It is the standard answer to the problem the map is about, a curve of uncontrollable poses running through the middle of the workspace, and the field quoted it without testing either half.

Both halves turn out to be exactly true and more specific than the sentence. The fourth leg does not make the singularities fewer; it makes them smaller by one dimension. And the agreement it costs is a number with a slope.

Why three lines fail on a curve #

The platform is the site’s planar one, with legs of two 1.4 links from motors on a circle of radius 2 to a platform of radius 0.6. With its motors locked, each leg’s outer link can push or pull on the platform only along its own line, and the singularity essay showed that the platform is held exactly when those three lines can together resist any small motion of it.

A planar platform has three freedoms, two of position and one of turning, and three lines in the plane resist all three unless they share a point. If the three lines pass through one point, a small turn about that point moves the platform without stretching any leg, and the motors cannot stop it.

Three lines passing through one point is one condition. Fix two of the lines and the third must pass through their crossing: one equation. The platform’s pose has three coordinates, so the poses that satisfy one equation form a surface in the three-dimensional workspace, and every slice at a fixed platform angle cuts that surface in a curve. That is the curve the map drew, and it is why a curve is what it drew.

A fourth leg adds a fourth line. The platform is now held unless all four lines share one point: the third through the crossing of the first two, and the fourth through it as well. That is two conditions, and two equations on three coordinates leave a curve in the workspace and isolated points in each slice.

That is the whole of the prediction, and it is made before any leg is placed.

Three lines through a point, and a fourth that is not #

The pose drawn is on the three-legged platform’s singular curve, at (−0.706, −1.221) with the platform at 0°. Its three leg lines miss a common point by only 1.4 × 10⁻⁴, and the smallest singular value of the three lines together is 1.9 × 10⁻⁴: with three legs, the platform could turn about that point with all three motors locked.

A fourth leg is attached to the base and the platform at 180°, with its elbow bent the other way from the leg at that angle. At this pose its line passes 0.144 from the three lines’ meeting point. It does not go through it, so it resists exactly the turn the other three could not, and the smallest singular value of all four lines together is 0.1010, five hundred times the three-legged value.

That is the mechanism of the whole improvement in one picture. A turn about a point is resisted by any line that does not pass through the point, and the fourth line is in the wrong place to be one of the three.

The curve becomes a point #

The prediction is tested over the slice. At a platform angle of 0°, positions are sampled on an 81 by 81 grid, and at each one the platform is solved and held with three legs and with four.

With three legs, 3,312 positions are reachable, and the singular curve runs through them in 102 traced segments. With four legs, 3,198 positions are reachable, somewhat fewer because the fourth leg must reach too, and there is no curve. What remains of the singular set in this slice is one isolated point, at (−1.449, −0.811), where all four lines meet.

Finding that point needs care, and the way it is found is part of the result. A point has no width, so a grid will never land on it, and a grid’s smallest value can never be quoted as proof that the four-legged platform has no singularity. Every local minimum of the smallest singular value on the grid is instead followed downhill, and the minima that reach zero are the singular points. Here exactly one does. At it, the three-legged platform is singular too, as it must be: four lines meeting at a point include three that do.

Two things follow for the whole workspace rather than one slice of it. The three-legged platform’s singular poses, one condition on three coordinates, form a surface through the workspace, and every slice at a fixed angle cuts it in a curve; turning the platform moves the curve, which is what the map found as it changed orientation. The four-legged platform’s singular poses, two conditions on three coordinates, form a curve through the workspace, and a slice at a fixed angle meets that curve in isolated points, usually a handful and sometimes none. A path from one pose to another that has to avoid a surface is constrained in every slice it passes through. A path that has to avoid a curve in three dimensions can almost always be bent round it.

One more statement holds everywhere, and it is a theorem before it is a measurement. The four-legged matrix contains the three-legged matrix as three of its rows, and adding a row can never decrease a matrix’s smallest singular value. So at no position can four legs hold the platform less well than three. The comparison was made at every sampled position and the count of exceptions is zero, which is a check on the arithmetic rather than a discovery.

How much better held #

Removing a curve of exact singularities matters less than the region around it where the platform is poorly held, since a machine near a singularity is already in trouble. The measure is the share of the workspace held with the smallest singular value above a threshold.

Above a threshold of 0.1, the three-legged platform holds 90.2% of its reachable positions in this slice and the four-legged one 96.0%. Above 0.3 the difference is much larger: 67.0% against 87.4%. The count of positions held above 0.1 rises from 2,989 to 3,178.

The gain has a price in reach that the same counts state. With the fourth leg, 114 positions the three legs reached, 3.4% of them, can no longer be reached, because the fourth leg’s two links cannot span the distance from its motor to its attachment point there. The edge of the four-legged platform’s reach is therefore partly a new edge: the positions where the fourth leg is straight or folded, which is that leg’s own inverse singularity. A redundant leg adds one more way of losing a motor’s control of the platform at the boundary, in exchange for removing a way of losing it in the middle. Here the exchange is 114 positions given up at the edge against 189 more positions held above 0.1 inside, and that is a trade a designer can weigh for a task, not a free improvement.

The shape of the two curves says where the gain is. At low thresholds both platforms are held nearly everywhere, because a singular curve has no area and its bad neighbourhood is thin. At higher thresholds the three-legged platform’s held share falls away quickly, because the band of mediocre holding around its curve is wide, and the four-legged platform’s falls slowly, because it has no curve for such a band to surround.

Where the fourth leg goes #

A fourth leg is not one design. Where it is attached, on the base and on the platform, and which way its elbow bends decide which line it adds at each pose.

Six placements are compared over the positions three legs reach on a 61 by 61 grid, where three legs hold 89.9% above 0.1. At 180° on both base and platform with the elbow down, the choice drawn above, the four-legged platform still reaches 96.6% of those positions and holds 96.0%. At 270° with the elbow up it reaches 95.5% and holds 94.9%, and at 30° with the elbow up it reaches 95.5% and holds 94.7%.

Two placements show what can go wrong. Attached at 270° on the base and at the opposite side of the platform, 90°, the fourth leg cannot reach nearly half of the positions the other three can, and the four-legged platform reaches only 54.5% of them and holds 53.2%: a redundant leg that costs more workspace than it buys.

The last placement is the control. At 90° on both base and platform with its elbow up, the fourth leg exactly duplicates the first. It reaches everything, 100.0%, and holds 91.0%, a slight rise that comes only from counting one line twice. It adds no direction the platform did not already have, and its singular curve is the three-legged one, 74 segments of it on this grid, unmoved. A redundant leg removes singularities only if its line is a new line.

The price: four motors that must agree #

The second half of the quoted sentence is the cost, and it is where redundancy stops being free.

The mobility count says what the fourth leg does to the machine. The three-legged platform has eight links counting ground and platform, and nine revolute joints, so three freedoms, and with three motors locked it is a structure with none. The four-legged platform has ten links and twelve joints: still three freedoms, now driven by four motors. Lock all four and the count is −1. The machine with its motors locked is overconstrained by one, and an overconstrained structure only assembles if its dimensions agree.

The misfit is measured directly. At three poses, the three original legs put the platform where it should be, and the fourth motor is set wrong by a small angle. The fourth leg’s outer link would then have to change length to still reach its attachment point, and it cannot, so the difference is carried as something else.

At an error of 0.1° the required change of length is 2.317, 2.117 and 1.660 thousandths of a unit at the three poses. The first-order prediction, the fourth leg’s own entry in the matrix B at each pose multiplied by the angle, gives 2.316, 2.116 and 1.659, from entries of 1.327, 1.212 and 0.951. The lines are straight to the eye across half a degree. So the misfit is proportional to the motor’s error, with a constant that depends on the pose and is already computed as part of the velocity relation.

That misfit is not an error in a drawing. It is a quantity that has to go somewhere in a built machine: into clearance in the joints if there is any, into elastic strain in the links if there is not, or into a platform that will not assemble at all. That is the same bargain every overconstraint makes, the stiffness and the removed singularities paid for with a requirement that four independent motors agree to within whatever the joints can absorb. A control system for such a machine cannot position four motors independently; it has to command three and let the fourth follow, or regulate the force in the redundant loop, and both are outside what this site models.

A fourth encoder names the assembly #

The fourth leg brings one further benefit, and it answers a question the field raised and left open.

One command, six answers showed that at some motor angles the three-legged platform has six assemblies, and that no reading of three motors can tell which one the machine is in. It concluded that a parallel machine needs a redundant sensor, one reading not a function of the three motors’ angles, and a later essay showed that the assembly can change without a singularity, so that no count of singularities crossed can stand in for that sensor.

A fourth leg is such a sensor. At motor angles of 216°, 48° and 144° the three-legged platform has its six assemblies, at platform angles of −91.8°, −16.9°, 48.0°, 136.9°, 4.9° and 38.3°. Each puts the platform somewhere different, so a fourth leg attached to it would have to be at a different angle to reach: 5.3°, 15.2°, −22.9°, −29.3°, −136.4° and −150.2° respectively. The closest two of those six readings are 6.44° apart. A fourth encoder able to tell 3.2° either way therefore names the assembly outright.

So the redundant sensor the field asked for arrives with the redundant leg, and it arrives with a specification: the resolution the fourth encoder needs is half the smallest separation between the readings of the assemblies it must tell apart.

There is a structural way to say the same thing. The first essay on parallel platforms built everything on the reversal that a parallel machine’s forward problem is hard: three motor readings, several poses. With a fourth leg the forward problem has four equations in the platform’s three coordinates. It is overdetermined, and a generic set of four readings is consistent with at most one of the poses the first three allow, which is what six distinct readings for six assemblies means. Redundancy does not only remove singularities from the workspace; it removes the ambiguity from the forward problem, and both come from the same extra line. The one thing it does not remove is the need for the four motors to be commanded consistently, which is the misfit above looked at from the controller’s side.

What this essay does not establish #

The comparison is made in one slice of the workspace, at a platform angle of 0°. The argument that four lines leave isolated points in every slice holds at every angle, but the counts, the shares and the one singular point are for this slice only, and the singular set of the four-legged platform through the whole workspace, a curve, is not traced.

The misfit is kinematic. How it divides between clearance, strain and refusal to assemble depends on joint play and link stiffness, which the practice field models for clearances and which is not combined with this platform here. Neither is any control scheme for four motors with three freedoms.

One fourth-leg design is studied closely and six are compared. No search was made for the best placement, and “best” would need a statement of what the platform is for.

What comes next #

The cusps under a fourth leg. The three-legged platform’s cusps are where two singular curves of a slice meet. With the curves reduced to points there is nothing left to meet, and the singularity-free change of assembly found round a cusp should become unavailable or unnecessary. Tracking the same loop of motor angles on the four-legged platform, with the fourth motor slaved to the other three, would say which.

A hexapod’s singularities have their own structure. The spatial platform has six legs for six freedoms, and its singularities are not curves of three lines meeting but conditions on six lines in space. One of them is strange enough to need its own essay: a yaw at which the standard hexapod is singular wherever the platform is.

Redundancy as tolerance. The misfit slope is a sensitivity like any in the tolerance field, and it varies over the workspace. Mapping it would say where a four-legged platform is most sensitive to its motors disagreeing, which is also where it would need the most compliance, and the two maps could be compared directly.