The smallest parallel robot

The parallel field has spent its length on a platform with three legs, and round a cusp into another assembly found that platform doing something the field had said it could not: changing assembly without passing a singularity, by going round a cusp. A finding like that needs a control, a mechanism of the same kind that provably cannot do it, so that the explanation can be tested rather than admired.

The control is the smallest parallel robot. It has one loop, two motors and a hand, and everything about it can be computed by hand as well as by machine. It cannot go round a cusp, because it has none, and the reason it has none takes one sentence.

Two motors and a hand #

The mechanism has two motors on the ground, 1.0 apart. Each drives an arm of length 1, and each arm carries a distal link of length 1.25 at its elbow. The two distal links meet at a single pin, the hand. That is five links counting the ground, five revolute joints, and by the mobility count two freedoms, which the two motors supply.

It is a parallel mechanism in the sense the first essay on parallel platforms defined: two separate chains from the ground to the hand, each with one motor, closing on the hand from two sides. And it has the defining reversal. The inverse problem, where the hand is given and the motor angles are wanted, is two separate two-link arms, each solved alone. The forward problem, where the motor angles are given and the hand is wanted, couples the two chains.

The forward problem is short enough to state completely. With the motors set, both elbows are known. The hand is 1.25 from each elbow. So the hand is on two circles of radius 1.25, and two circles meet in at most two points. At 100° and 60° the elbows are 1.678 apart, less than the 2.50 at which the circles would only touch, so they meet twice: at (0.229, 1.850) and at (0.098, 0.001).

Those are the robot’s two assembly modes, and there are never more.

Four ways to hold one hand position #

The inverse problem has its own multiplicity. Given the hand, each arm reaches it by bending its elbow one way or the other, just as a two-link serial arm does, and the two arms choose independently.

So a hand at (0.1, 1.6) is held by four working modes, with motor angles of 115.8° and 153.2°, 115.8° and 54.9°, 23.1° and 153.2°, or 23.1° and 54.9°. A working mode is fixed when the robot is built and cannot change during motion without an arm passing straight or fully folded, which is the edge of that arm’s reach. Each is a different machine for everything that follows: a different workspace, a different singular curve, and a different map of how well the hand is held.

The two multiplicities are the same pair the three-legged platform has, eight working modes and up to six assemblies there, reduced to their smallest form: four working modes and two assemblies.

The one singularity that matters #

A parallel mechanism’s velocity relation has two matrices, and the singularity essay explained why each has its own failure. On the five-bar both are visible in the drawing.

The matrix A collects the directions of the two distal links. Its determinant vanishes when the two distal links lie along one line, and then the hand can move along that line with both motors locked: a direct singularity, where the mechanism gains a freedom. The other matrix, B, vanishes when an arm lies along its own distal link, straight or folded, and the hand is at the edge of what that arm can reach: an inverse singularity, where a motor loses control of the hand.

The first of these is the same event as the two assemblies meeting. The distal links are collinear exactly when the two circles about the elbows are tangent, because the hand is then on the line joining the elbows, and tangent circles have one intersection rather than two. So the direct singularity and the merging of the two assemblies are one event, and this is the place to look for anything like the cusp.

Where the motors assemble the robot at all #

On a 121 by 121 grid of both motor angles over a full turn, the robot assembles at 12,517 of the 14,641 settings. At each of those it has exactly two assemblies; outside, none. The edge of the shaded region is where the elbows are exactly 2.50 apart, the circles are tangent, and the two assemblies merge. Traced through the grid, it is 220 segments of smooth curve.

Nowhere along that edge do two pieces of it meet in a point. On the three-legged platform’s slice, the edges between regions met at sharp points, and those points were the cusps. Here there are only two regions, “two assemblies” and “none”, and a single kind of edge between them. There is no third region for a second kind of edge to bound, and so nothing for two edges to meet at.

Two settings inside the region are special in a different way. At (60°, 120°) and at (−60°, −120°) the two elbows are at the same point, (0, 0.866) for the first. The two circles are then the same circle, and the hand can be anywhere on it: the forward problem has infinitely many answers. Those are the obvious places to look for the kind of behaviour the three-legged platform showed, and the next figure looks there.

The sign names the assembly #

The two assemblies are mirror images of each other across the line joining the elbows, because that is what the two intersections of two circles are. Reflection reverses the orientation of the pair of distal link directions, so it reverses the sign of det A. At 100° and 60° the two values are 0.995 and −0.995: the same size and the opposite sign, exactly.

That turns a geometric fact into a test. If the sign of det A names the assembly, then any continuous motion that changes the assembly must change the sign, and a continuous sign change passes through zero, which is the direct singularity. The site checks the premise directly. At 1,026 random settings of the motors, the assembly each solution belongs to and the sign of its det A were compared, and they disagreed 0 times.

The loop drawn is the most searching one available: a 25° circle of motor angles around (60°, 120°), enclosing the setting where the elbows coincide and the forward problem degenerates. Both assemblies are followed round it. The hand traces one closed circuit in each, det A never changes sign on either, its smallest magnitude anywhere is 0.240, and each assembly returns to itself.

The same check was repeated on random loops. Of the closed motor loops whose tracks completed, 305 in all, none ended in the other assembly, and 20 of them enclosed a coincidence setting. A loop round a point where infinitely many assemblies exist carries the hand once round a circle and still does not exchange the two.

The only way from one assembly to the other #

So the route between the two assemblies has to go through the edge. Hold the left motor at 100° and turn the right one down from 60°. The elbows move apart, the two intersection points of the circles move towards each other, and the two values of det A shrink together, always equal and opposite. At −21.35° the elbows are 2.50 apart, the circles touch, the distal links are in one line, and both assemblies are the same configuration with det A zero. At the last setting before that edge that assembles, they are still 0.033 and −0.033.

Beyond −21.35° there is no machine. The only continuous way from one assembly to the other is to reach that edge, pass through the single shared configuration, and come back into the region on the other assembly: exactly the direct singularity the parallel field has warned against.

Hold one motor and it is a four-bar #

That edge has a second name, and it comes from the first mechanism these essays studied.

Hold the left motor at 100° and its elbow is a fixed point, at (−0.674, 0.985). What is left free is the right motor’s pivot, the right arm, and the two distal links hinged together at the hand and pinned at their far ends to the right elbow and the fixed left elbow. That is a closed chain of four bars: a four-bar whose ground runs from the right motor to the fixed elbow, 1.532 long, whose crank is the right arm, 1, and whose coupler and rocker are the two distal links, 1.25 each.

Grashof’s condition sorts it at once. The shortest link plus the longest is 1 + 1.532 = 2.532, and the other two sum to 2.5. The condition fails, by 0.032, so this four-bar is a triple rocker: its crank, the right arm, cannot turn all the way round. It stops at dead centres, where the coupler and rocker lie in one line, and those are the positions where the two distal links are collinear and the elbows 2.50 apart. The crossing figure’s edge at −21.35° is that four-bar’s dead centre, and the five-bar’s two assemblies at this held angle are the four-bar’s two assembly branches, which on a triple rocker meet at exactly those dead centres.

So the smallest parallel robot, with one motor held, is a four-bar, and its direct singularity is a four-bar’s dead centre. The five-bar adds a second motor that moves the ground pivot of that four-bar around a circle.

The same reading says which held angles give a different kind of four-bar. With the left motor at an angle θ, the distance from the right motor’s pivot to the left elbow is 2 sin(θ/2), because both motors are 1.0 apart and the arm is 1. That distance is the four-bar’s ground, the arm and the two distal links do not change, and Grashof’s condition holds exactly when the ground is at most 1.5, which is when sin(θ/2) is at most 0.75, θ at most 97.18°. Hold the left motor anywhere below that angle and the four-bar that remains is a Grashof chain whose right arm turns fully in each assembly: along that whole line of motor settings the two assemblies never meet, because the elbows never get 2.50 apart. Hold it above 97.18° and the right arm is stopped by dead centres, as it is at 100°.

It is a small calculation and it does two jobs. It explains the shape of the region of motor angles where the robot assembles, whose edge touches the vertical line at 97.18° and cannot cross it. And it ties the parallel field’s most elementary mechanism back to the linkage field’s first classification: every singular configuration of a five-bar is the dead centre of some four-bar that holding one motor would leave behind.

Why two answers forbid a cusp #

The explanation for the whole difference between the two mechanisms fits in one sentence. A cusp is three assemblies merging, and the five-bar only ever has two.

On the three-legged platform the closure function had up to six zeros, and going round a cusp relabelled them: a surviving zero entered as the third and left as the first, while other pairs were destroyed and created around it. That needs at least three zeros in play. With two, the only event available is the pair merging and vanishing, and a zero that is carried round any loop cannot be relabelled, because there is no other zero it could be relabelled past without meeting it.

The sign argument is the same fact from the other side. With two assemblies that are mirror images, a sign is a complete name. With six, some of which share a sign, a sign is a partial name, and the cusp is where the partial name fails.

It also answers a question the parallel field has not asked: which parallel mechanisms can do what the three-legged platform does. Not the five-bar, and not any mechanism whose forward problem is a pair of circles. A mechanism needs a forward problem with at least three solutions before it can be cuspidal at all, and three is necessary rather than sufficient.

Where the hand is held well #

For one working mode, both elbows on the same side, the hand can reach 3,655 of the 10,201 positions sampled on a grid. The outer edge of that region is where an arm is straight or folded, the inverse singularity, the limit of reach. Across the middle of it runs the line det A = 0, 107 segments of it: the positions where the two distal links lie in one line.

That line separates the region into 1,827 positions where det A is positive and 1,828 where it is negative, which is to say into the two assemblies. On the five-bar the direct singularity is not an obstacle inside a workspace so much as the border between the two workspaces the two assemblies have. A hand on one side of the line is in one assembly and cannot be taken to the other side except across the line.

The shading says what the line costs in practice. Near it the smallest singular value falls towards zero, and the hand is poorly held in the direction along the distal links well before it reaches the line. That is the same penalty the three-legged map found around its singular curves, in a mechanism simple enough that the penalty’s cause can be read straight off the drawing.

What this essay does not establish #

One set of dimensions is measured: motors 1.0 apart, arms 1, distal links 1.25. The held-motor reading of the edge as a four-bar’s dead centre, and the angle of 97.18° below which that four-bar is a Grashof chain, are arithmetic on those dimensions and would move with them. The argument that a five-bar has no cusps rests on its forward problem having two solutions, which is true for any dimensions. The counts, the edge of assembly and the singular line are for these dimensions only.

The sampled checks are samples. That the sign of det A names the assembly follows from the reflection argument; the 1,026 random settings and the 305 completed loops confirm that the solver’s labels agree with it, and are not the proof.

Collisions are not considered. On a real five-bar the arms and distal links occupy space, and some of the loops of motor angles above would drive links through each other.

What comes next #

A fourth leg on the three-legged platform. The three-legged platform’s direct singularities are curves in every slice of its workspace, and the cusps sit where those curves meet. What a fourth leg buys adds a redundant leg and finds those curves reduced to isolated points, which removes the cusps along with them.

The five-bar’s workspace for a task. The field has computed where each working mode can reach. Which working mode and which assembly a five-bar should be built in for a stated task, a rectangle the hand must cover with the smallest singular value above a threshold, is a design question the maps above already contain the ingredients for, and nothing here has asked it yet.

Three answers and no cusp. Three solutions are necessary for a cuspidal mechanism and not sufficient. A planar platform with three assemblies somewhere and no cusps anywhere would show what else is needed, and would make the criterion a measurement instead of a citation.