Two orientations no position can rescue

A yaw that is singular everywhere turned the standard hexapod 30° about the vertical and found it singular wherever it was put: one screw motion that none of its six locked legs could resist, at every position, because six moments became equal and the equality did not involve position. It ended by asking the same question of a planar platform. Whether any orientation of it is singular at every position, it said, is decided by whether its three leg lines can be made to meet for every position at once, and the answer can be read from its attachment points.

That last clause is true of one kind of planar platform and false of another, and the difference is the most useful thing the question turns up.

Legs that run through their own attachment points #

The planar platform drawn throughout these essays is a 3-RRR. Each of its legs is a two-link arm: a motor turns the first link about a base pivot, and the second link joins the elbow to a point on the platform. The only force a leg can put on the platform runs along its second link, so the three lines that decide whether the platform is held run from three elbows to three platform points, and a direct singularity is where those three lines meet at a point or lie parallel. The elbows move with the motors, and one set of motor angles can leave the platform several places to be, each with its own elbows and its own three lines.

A 3-RPR has straight legs that extend. Each is pinned to the base at one end and to the platform at the other, and a motor changes its length. Its line runs from its base pivot to its platform point, and nothing else. Where the three lines are is decided by the six attachment points and the platform’s pose alone.

The measure is the one used for the 3-RRR: the smallest singular value of the three rows that describe the legs’ lines as forces on the platform, the direction of each leg together with its moment about the platform’s reference point. It is zero exactly when some motion of the platform stretches no leg, which is when the lines meet or are parallel.

Similar triangles, and a centre of similarity #

Take two triangles, one a scaled copy of the other with no turn between them. The three lines through their corresponding corners meet at one point, the point about which the scaling carries one triangle onto the other: the centre of similarity. With a scale factor of κ, each platform point sits at $c + \kappa(b_i - c)$, where $b_i$ is its base pivot and c is that point, and any κ other than one gives a c.

Now let the base triangle and the platform triangle of a 3-RPR be similar, the platform a scaled copy of the base, with some turn between them as built. There is exactly one orientation of the platform that undoes that turn, and at it the platform triangle is a scaled copy of the base with a positive scale factor. Half a turn from there it is a scaled copy with the same size and a negative scale factor, the copy that has been turned through the centre. At either orientation the three leg lines pass through corresponding corners, so they meet at the centre of similarity. That holds wherever the platform is put, because moving the platform moves the centre with it rather than breaking the arrangement.

Three force lines through one point cannot resist a turn about that point, which is the planar form of a screw, with its axis standing through the meeting point and a pitch of zero. So at those two orientations the platform is singular at every position. It is the planar version of the hexapod’s singular yaw, and like that yaw it is a property of an orientation rather than of a pose.

Measured over a square of positions #

The platform in the figures has its base pivots on a circle of radius 2 at 90°, 210° and 330°, and its platform points on a circle of radius 0.6 at the same angles, so the platform triangle is the base triangle scaled by 0.3 with no turn. The attachment points predict singular orientations at 0°, with a scale factor of 0.3, and at 180°, with −0.3.

The platform’s centre is placed at every point of a 25-by-25 square 1.8 across, and at each orientation, a degree apart, the smallest singular value is computed at every position and the largest of those kept. That largest value is how well the platform can be held at its best position at that orientation. At 0° it is 1.6 × 10⁻⁸ and at 180° it is 8.6 × 10⁻⁹, which is rounding. At 90°, for comparison, the values across the square run from 0.43 to 0.995.

The second route does not compute a singular value. It asks where the three leg lines meet, as the point nearest all three, and how far that point is from each line. At 0° with the platform centred at (0.4, −0.3), the lines meet at (0.5714, −0.4286), within 1.1 × 10⁻¹⁶ of every line. The centre of similarity, computed from the base pivots and the scale factor alone, is (0.5714, −0.4286). At 180° from the same position they meet at (0.3077, −0.2308), which is again the predicted centre. At every position tried, at both orientations, the lines meet within 3.3 × 10⁻¹⁶ and exactly where the attachment points say.

The two routes share the platform and nothing else. One is a statement about a three-by-three matrix at each pose; the other is a statement about where three lines cross, predicted from six points and a number. They agree about where the singularity is and why it is there.

The orientations belong to the attachment points #

The derivation says the two orientations are set by the turn between the two triangles as built, and a second platform tests it. Keep the base and the platform’s size, and place the platform’s points at 130°, 250° and 10° instead, which is the same triangle turned 40° in the platform’s own frame.

The sweep finds that platform singular at every position at exactly −40° and 140°, and at no other whole degree. At −40° the platform has been turned back by the 40° it was built with, and the triangles are scaled copies facing the same way; at 140° they are scaled copies turned through the centre. Its orientation 0°, which is its home, is held at 0.841 with the platform centred.

That comparison carries the design lesson in one line. The first platform, with both triangles at the same angles, is the obvious symmetric design, and it is singular at its own home orientation: centred at 0°, its smallest singular value is 6 × 10⁻³³. The second is the same pair of triangles with the platform’s points turned in their own frame, and its home is one of the best-held orientations it has. The two triangles and the size ratio are the same. What differs is only which orientation the attachment points have made unusable.

How quickly it is held again #

A singular orientation is a practical problem only if the platform recovers slowly, since a mechanism is in trouble well before its measure reaches zero.

Centred and turned 1° from 0°, the similar platform’s smallest singular value is 0.0259; at 5° it is 0.129 and at 10° it is 0.255. Close in the rise is a straight line, the mark of a simple zero: the free turn exists at exactly one orientation and is resisted in proportion to how far the platform has been turned from it. About 180° the rise is slower, 0.0140 at 1° and 0.139 at 10°, so the two singular orientations of one platform are not equally sharp. The platform built turned 40° recovers about −40° exactly as the similar platform does about 0°, to every digit printed: turning the points in the platform’s frame moves an orientation and changes nothing about it.

The slope is set by the leg lengths #

The recovery has a closed form, and it is a second route to the same numbers. With the platform centred, the whole arrangement has three-fold symmetry, so the three leg directions stay a third of a turn apart at every orientation. Their directions then hold the platform equally against a push in any direction, with a singular value of $\sqrt{3/2}$, about 1.22, and the turn is resisted only by the three equal moment arms: the distance from the centre to each leg’s line. For a base pivot at radius R and a platform point at radius r turned δ away from it, the centre, the pivot and the point make a triangle of area ½Rr sin δ, and the leg is that triangle’s side of length $\sqrt{R^2 + r^2 - 2Rr\cos\delta}$. The moment arm is the triangle’s height over that side. The smallest singular value is $\sqrt{3}$ times the moment arm, until it reaches $\sqrt{3/2}$.

With R = 2 and r = 0.6 that gives 0.02591 at 1° and 0.25544 at 10°, the measured values to every digit. About 180° the platform point sits on the far side of the centre, the sign in front of the cosine changes, and each leg is 2.6 long instead of 1.4. The platform point moves sideways by the same amount for the same turn, so the triangle has the same area; spread along a longer leg, that area leaves a shorter height. The slope at the zero is smaller by (R + r)/(R − r), which is 1.857, and 0.02591 divided by 0.01395 is 1.857.

Nothing in the formula mentions the angle the platform points were built at, which is why the turned platform recovers identically. It depends on the two radii alone, so the sharpness of a singular orientation is a design choice too: a small platform on a large base has R − r close to R + r, and its two singular orientations recover at nearly the same rate.

Nearly similar is nearly singular everywhere #

Similar triangles are a condition that holds or fails, and a real platform is built to a tolerance. The question that matters is what happens near it.

One point of the similar platform is moved sideways by a small amount, which leaves the two triangles a measurable distance from similar: the largest distance any platform point sits from where the best-fitting similarity would put it. For each distortion, the orientation at which the platform’s best position is held worst is found to a fraction of a degree, and that best holding is recorded.

The best holding at the worst orientation is proportional to the misfit. A misfit of 8.3 × 10⁻⁴ leaves it at 5.4 × 10⁻⁴, 0.649 times the misfit; a misfit of 2.7 × 10⁻², more than thirty times larger, leaves it at 1.7 × 10⁻², 0.642 times. Across that whole range the ratio changes by 1.05%. The worst orientation drifts too, from just past a half turn, at −180.06°, back to 178.01°, but the holding there follows the misfit as a straight line.

Linear is the most it could be. Moving an attachment point a small distance changes each row of the legs’ matrix by an amount proportional to the move, and a matrix’s singular values cannot shift by more than the size of the change made to it. So at any design, the holding at a nearly singular orientation is bounded by a constant times the misfit. What the measurement adds is that it is not smaller. Had the free turn survived the distortion to first order, the holding would grow as the square of the misfit, and the ratio in the figure would grow thirty-two-fold across the range instead of changing by 1%. A distortion removes the singular orientation at the first order it can.

That makes the tolerance a design number. A platform meant to be held to some level at every orientation needs its triangles further from similar than that level divided by about two-thirds, on this base and this square of positions. A nearly similar platform on a slightly irregular base, with attachment points 1.9 × 10⁻² from similar, about 3% of its platform’s size, is held at its worst orientation, 180°, to only 1.7 × 10⁻² anywhere in the square. An isosceles platform on the similar platform’s base, 5.7 × 10⁻² from similar, manages 4.0 × 10⁻². Neither has a singular orientation, and both have one that is nearly useless, set by how nearly similar they are.

Revolute legs do not inherit it #

The 3-RRR platform drawn throughout these essays is built on exactly the similar triangles above: base pivots on a circle of radius 2 at 90°, 210° and 330°, platform points on a circle of radius 0.6 at the same angles. If the answer could be read off attachment points alone, that platform would be singular everywhere at 0° and 180° too.

It is not. At 0° the 3-RRR reaches 3,312 of the 6,561 positions sampled in the figure, and its direct singularities are three arcs through the reachable region rather than the whole of it, which is what the map of where it fails found and drew. Sampled the way the 3-RPR was, on a 61-by-61 grid, it reaches 1,867 of 3,721 positions, its smallest singular value runs from 1.1 × 10⁻⁴ up to 0.900, and the singular arcs cross 67 cells. At 180° it can reach only 51 of those positions, and every one of them is held, at between 0.275 and 0.386.

The reason is the elbows. The 3-RRR’s leg lines run from elbows to platform points, and an elbow is wherever the motor has put the first link, not at the base pivot. Scaling the base triangle onto the platform triangle puts the three platform points on lines through the base pivots and the centre of similarity; it says nothing about where three elbows are. For the 3-RRR to be singular at every position at one orientation, its three second links would have to meet for every placement of the platform, which is a condition on its link lengths and working modes as well as its attachment points, and similar triangles do not supply it. It is the same reason the five-bar’s direct singularity sits where its distal links line up: a place the motors decide, not the pivots.

So the yaw essay’s closing sentence is exactly right for a platform whose legs run through their own attachment points, and it is wrong for the one the field actually uses. An orientation singular at every position can be read off the attachment points when the legs’ lines pass through the attachment points, as they do on a 3-RPR and on the hexapod, whose six legs are also straight lines between attachment points. On a platform with jointed legs the attachment points are not enough, because the lines that hold the platform do not pass through them.

What a designer does about it #

Do not build a 3-RPR with similar triangles, and least of all with them facing the same way. The symmetric design that looks most natural is singular at its home orientation and half a turn from it, at every position, and no controller can do anything there.

If the triangles must be similar, turn the platform’s points in its own frame until the two singular orientations are outside the range the task needs. The 40° turn tried here moves them to −40° and 140°, and leaves the home orientation well held.

Treat near-similarity as a tolerance. Because the best holding at the worst orientation is proportional to the misfit, a platform whose attachment points are a few per cent from similar has an orientation at which it is held only a few per cent as well as it could be, anywhere. The fix is the same as for exact similarity: make the triangles different on purpose, by more than the holding the task needs.

None of this carries to a 3-RRR. A jointed-leg platform has its own architecture singularities, and they are not found by comparing triangles.

One base, one square, unlimited legs #

The proportional constant belongs to this base and this square. The 0.649 was measured with a base of radius 2, a platform of radius 0.6 and positions over a square 1.8 wide, distorting one point in one direction. Another base, another square or another direction of distortion gives another constant, as the two other designs show.

Leg strokes are not limited. Every position in the square is treated as reachable, since a 3-RPR leg is modelled as a line of any length. A real platform has a least and greatest leg length, which would remove some positions from every figure and would not change the singular orientations.

Only the conditions for a 3-RRR’s second links to meet everywhere are named, not solved. That they are not supplied by similar triangles is measured; what does supply them is not worked out here.

What comes next: the 3-RRR’s own singular orientation #

The 3-RRR’s own orientation singular everywhere. A jointed-leg platform can in principle be designed so that its three second links meet for every position at some orientation, and the condition involves the link lengths, the working modes and the attachment points together. Whether the family of 3-RRRs with these attachment points contains such a design, and how close the platform drawn here comes to one, is the question this essay deliberately left open.

The hexapod with similar hexagons. The hexapod’s singular yaw found every yaw singular when its base and platform hexagons were similar. The proportional law found here suggests that a hexapod whose hexagons are nearly similar is held at its worst yaw in proportion to the misfit, with a constant of its own, and the same sweep would say. Because the hexapod drawn here is not a general one, the constant would need measuring on a general hexapod as well.

A fourth leg on the similar 3-RPR. A fourth leg reduced the 3-RRR’s singular curves to points. On a 3-RPR with similar triangles, a fourth leg through the same centre of similarity would change nothing, and one through any other point should remove the singular orientation altogether. Which placements of a fourth attachment point do each is a question with a clean geometric answer.