A seventh joint cannot move the elbow

A straight move ends at a stop asked a six-joint arm to move its tool along a straight line at a fixed orientation, and found that half of such moves end with a joint at its stop. The reason was structural. A straight line of the tool is the commonest move asked of an arm, and moving every joint at constant speed does not make one. A six-joint arm holding a tool pose has only a finite set of postures, as eight ways to hold the same tool counted. Along a straight line it keeps the one it starts in, because changing posture means passing a singularity. With the posture fixed, the line fixes every joint, and whichever joint reaches its stop first ends the move.

A seventh joint breaks that. The freedom that does nothing showed that a seven-joint arm holding one tool pose has a whole curve of postures, a self-motion that moves the elbow round a circle while the tool stands still. On a straight move that freedom could be spent at every step keeping the joints away from their stops. The earlier essay ended by asking how much that buys. If the median move lengthened several times, the seventh joint would be worth more as a way round stops than as a way round singularities, which is how it is usually sold. If it barely lengthened, the stops would be a fence the self-motion could not get round.

The answer is neither, and the reason it is neither is exact.

One arm, locked and free #

A fair comparison needs one arm, not two. The arm here is a shoulder-elbow-wrist arm: three joints meeting at the shoulder (a base roll, a shoulder bend and an upper-arm roll), an elbow bend, and three joints meeting at the wrist (a wrist roll, a wrist bend and a flange roll). It has upper arm and forearm each 0.40 m long and a 0.126 m tool. It is the seven-joint arm of the freedom that does nothing, and the shape most collaborative arms are built to. Every joint gets a stop of the kind such arms carry: ±170° on the four rolls except the flange’s ±175°, and ±120° on the three bends.

Lock the upper-arm roll at nought and it is a six-joint arm, with the same links and the same stops on the other six joints. So every move can start from the same configuration, at the same tool pose, in the same shape, and be run twice. The only difference between the runs is whether the seventh joint may move.

Two hundred moves were run, from seeded random configurations inside the stops, each in a seeded random direction. The tool moves 5 mm a step at a fixed orientation, and each step is corrected back onto the line by Newton’s method, with the joints moving as little as possible. A move ends for one of the earlier essay’s three reasons. A joint passes its stop. Or a step’s correction will not converge, which is the edge of what the arm can reach in that shape. Or a step needs more than 0.3 rad of some joint for 5 mm of tool, sixty radians a metre, which marks a singularity’s neighbourhood.

With the seventh joint free there are two ways to run each step. The plainest takes the least total joint motion that keeps the tool on the line, and spends the self-motion on nothing in particular. The other spends it. At every step the arm finds the joint nearest its stop and moves along its self-motion, by 0.03 rad of joint per step, in the direction that takes that joint away from its stop. That is the earlier essay’s question asked literally. Three other rules were also tried on every move: the same rule more gently and more strongly, and the textbook rule that pulls every joint towards the middle of its range. The best of the four is kept as a measure of what steering can do.

Two fifths further, on the median #

With the upper-arm roll locked, the median move gets 23.3 cm before it ends. With the roll free and the least motion taken, the median is 27.8 cm. Even unsteered, the extra joint lets the arm drift into shapes the locked arm cannot reach, and that sometimes postpones a stop. Steered off the nearest stop, the median is 33.3 cm, two fifths further than the locked arm. The best of the four steering rules, taken move by move, makes almost no difference to that: 33.5 cm. The rule that steers the nearest joint at a moderate rate was the best rule on 180 of the 200 moves.

So steering does what it was asked to do, and the median does not lengthen several times. But the median is a poor summary here, because the moves split into kinds that behave quite differently.

With the roll locked, 100 moves end at a stop, 95 at the edge of the reach and 5 near a singularity. Steering moves 21 of the 100 stop endings elsewhere, and almost all of them to the edge: 79 end at a stop under the best rule and 120 at the edge. What steering cannot do is postpone the edge. Of the 95 moves the locked arm ends at the edge of its reach, 94 end at exactly the same place with the roll free and steered. Where the hand can go drew an arm’s reach as the set of places its tool can be put. A seventh joint adds shapes to that set and no places. Once the line leaves the reach, no shape will hold the tool on it.

All or nothing #

The stops are where the seventh joint should matter, and among the 100 moves a stop ends with the roll locked, the gain is anything but steady.

The gains are lopsided. A quarter of the moves gain nothing at all, and half gain under a tenth, at most ×1.07. Then they rise steeply: three quarters gain under ×2.55, and the top tenth gain six times or more, where the locked arm stopped within a few centimetres and the steered arm runs on for most of a metre. Very little lies in between. The self-motion either gets the arm round the stop that would have ended its move, after which it usually runs on to some other ending well beyond, or it cannot touch that stop at all.

Which of the two happens depends on which joint the stop belongs to.

Every move the base roll’s stop ends is lengthened, all twelve, by a median of ×2.94. Of the wrist roll’s fifteen, eleven are lengthened, by ×2.55. The wrist bend’s and the flange roll’s stops give way about two times in three but by less, a median of a quarter. The shoulder bend’s give way in only nine of twenty-one. The elbow’s never give way: not one of its twenty moves is lengthened by any of the four rules.

The elbow is fixed by the line #

The elbow’s twenty are not unlucky. They cannot be helped, and the reason takes one sentence of geometry.

In a shoulder-elbow-wrist arm, the elbow’s bend is the angle between the upper arm and the forearm, and those two links, with the line from the shoulder centre to the wrist centre, form a triangle. The links’ lengths are fixed, so the elbow’s angle is fixed by the distance from shoulder to wrist alone. The self-motion swings the elbow round a circle about the shoulder–wrist line, as the freedom that does nothing constructed, and that circle keeps the triangle’s three sides. It changes every joint but the elbow. Pin the tool and the arm is a loop of seven joints with one freedom, and that freedom moves six of the joints and leaves the elbow exactly where it was.

On a straight move at fixed orientation, the wrist centre sits a fixed offset back from the tool, so it moves along a straight line of its own. Its distance from the shoulder at every point of the move is therefore fixed by the line, and so is the elbow’s angle. A move that carries the wrist towards or away from the shoulder bends or straightens the elbow by an amount nothing else can alter. If that brings the elbow to its stop, the move ends there, whatever the other six joints do.

The measurement agrees with the geometry to the last digit. Along this move the steered arm’s base roll and wrist roll part from the locked arm’s by up to 0.14 rad. Its elbow angle matches the locked arm’s to 9 × 10⁻¹⁶ rad, which is the arithmetic’s rounding, and both reach the elbow’s stop together 18.0 cm along.

The edge of the reach falls to the same argument. The arm reaches the edge of its reach when the elbow straightens out, upper arm and forearm in line, and the wrist can get no further from the shoulder. That too is a statement about the shoulder–wrist distance, which the line fixes. So the two endings a seventh joint can never postpone are the same ending seen twice: the line asks the elbow for more bend, or more straightness, than the arm has.

A fence drawn before the move #

If the elbow’s angle along the line depends on the line alone, then so does the travel at which the elbow meets its stop, and it can be computed before the arm moves at all. The wrist centre starts a known distance from the shoulder and moves along a known line. The cosine rule turns each distance into an elbow angle. The elbow reaches its stop where the distance falls to $\sqrt{U^2 + F^2 + 2UF\cos\theta_{\text{stop}}}$, with $U$ and $F$ the upper arm and forearm, and the arm leaves its reach where the distance reaches $U + F$. Whichever comes first along the line is a travel that no self-motion, no steering rule and no planner can exceed.

Computed that way, the fence agrees with the measured moves wherever it should. Every one of the twenty moves the locked arm ends at the elbow’s stop ends exactly at the fence, to the 5 mm step. Ninety-four of the ninety-five that end at the edge of the reach end within a centimetre of it. No move, under any of the four rules, gets past it. And the best steering reaches it on 147 of the 200 moves. On those, the seventh joint has done everything the arm’s geometry allows, and the move ends because the line has asked the elbow for something it does not have.

The other 53 moves stop short of the fence, every one of them at the stop of a joint the self-motion does move: the shoulder bend 17 times, the flange roll 13, the wrist bend 11, the base roll 7 and the wrist roll 5. Those are the moves where a better rule might do better. The median fence is 38.3 cm against the best steering’s median of 33.5 cm, so the room left for a cleverer controller is real but bounded, about five centimetres on the median move.

Why the other stops give way unevenly #

The other joints do move with the self-motion, and how much each moves explains the rest of the bars. The base roll turns the whole arm about the vertical. When the elbow swings round its circle the upper arm’s direction changes, and with it the base roll, often by a great deal. So a base-roll stop can nearly always be stepped away from, and the gain is large because once clear the arm has room.

The wrist roll is similar, because the self-motion turns the forearm about the shoulder–wrist line and the wrist must turn back to hold the tool’s orientation. The shoulder bend is the one that resists. It measures how far the upper arm leans from the vertical. Swinging the elbow round its circle moves that lean only within a band set by how the shoulder–wrist line itself is tilted, and in more than half of its moves the stop sits outside the band the circle can reach. That reading fits the counts; it was not tested separately.

So a seventh joint is a way round some stops, not all, and how often it works depends on the joint. The wrist is three joints and one point built the six-joint arm’s inverse around the same split, a position problem for the first joints and an orientation problem for the wrist. The seventh joint enters the position problem as a free angle round the circle. It helps every joint whose angle that circle changes, and leaves the one joint the circle does not touch.

What a programmer can take from it #

Steer the self-motion, but do not expect the median to multiply. On this arm, steering the joint nearest its stop away from it lengthens the median straight move by two fifths and shortens almost none: one move in a hundred, under the best rule. Choosing between steering rules matters little; the moderate one was best nine times in ten.

Check the elbow before the move, not during it. The fence above is a few lines of arithmetic on the wrist centre’s line and the shoulder, and it predicted every elbow ending and all but one edge ending exactly. If the line asks the elbow past its stop or the arm past its reach, no redundancy resolution will save the move, and the line must be changed. If it does not, the move’s length is up to the steering.

A seventh joint does not extend the reach. Ninety-four of ninety-five moves that leave the reach end exactly where they did without it. Where a seventh joint helps is inside the reach, against the rolls and the wrist’s stops.

A singularity was never the problem. As in a straight move ends at a stop, very few moves end near a singularity, five of two hundred with the roll locked. The singularities where the arm loses a direction catalogued are what a seventh joint is usually sold against, and on straight moves they were never what stopped the arm.

The limits of two hundred moves #

Four rules, not the best possible. Each rule steers greedily, one step at a time, by the joint nearest its stop now. A rule that looked ahead along the line might step round a stop the greedy rules walk into. The fence bounds how much it could gain, since nothing passes it, but on the 53 moves that stop short of it the true limit lies somewhere between the best rule and the fence, and it was not found.

One arm and one set of stops. The elbow’s invariance holds for any arm whose shoulder and wrist are each three joints through a point, which most seven-joint arms are. The counts for the other joints depend on these stops and this arm’s proportions.

Collisions are not modelled. The space of configurations drew an arm’s configurations as a space with forbidden regions. Here the only forbidden regions are the stops. A real cell has fixtures and a body the self-motion must avoid, which usually takes away more of it than it gives back.

Still open: the moves that stop short of the fence #

On 53 of the 200 moves the best steering rule stops at a stop the self-motion could in principle have avoided, short of the travel the line allows. The greedy rules decide each step by the joint nearest its stop now. A stop that is far away now but lies straight ahead, the shoulder bend’s most of all, is invisible to them until it is too late to swing the elbow round to avoid it.

The distinct argument would find, for those 53 moves, the farthest any continuous self-motion could carry the arm. It would be drawn as a plane of travel along the line against the upper-arm roll’s angle, with each point usable when the arm holds the tool there inside every stop, and the limit the farthest point joined to the start through usable points. It would then ask whether a controller that plans the self-motion over the whole line closes the gap to the fence. If most of the 53 reach the fence under a plan, the shoulder bend’s stop is a failure of greed and the fence is the whole truth. If they do not, some stops close off the line in ways no self-motion can get round, and the fence is only the outer of two.