A straight line at constant speed

Move the arm from here to there. There are two obvious ways to say it and they produce two different motions, neither of which is wrong and only one of which can fail.

Joint space. Every joint runs from its start value to its end value at a constant rate. Six numbers interpolated linearly, and the tool goes wherever that puts it.

Task space. The tool travels on the straight line between the two positions, at constant speed, and the joints do whatever that requires.

Why a joint-space move can never fail #

The joint-space path has a property that sounds trivial and is the whole reason it is used: every point on it exists.

An open chain has no closure equation. Every set of joint values is a posture and every posture has a pose, so an interpolation in joint space cannot pass through anything unreachable, cannot be refused by an inverse solver, and cannot be trapped by a singularity — the arm may pass through one, and nothing about the motion notices. Give it two postures inside the joint limits and the path between them is inside the limits too, because a box is convex.

That is a genuinely strong guarantee. It is why a robot’s “move to home” is a joint move, why teach-pendant jogging is joint-wise, and why the first motion after a fault is almost always a joint-space one: the machine may not know where it is well enough to trust an inverse solve, but it always knows what its encoders read.

What it does not give is control over where the tool goes in between. The 402 mm bow above is not an error — it is the correct tool path for that joint motion — and it is the reason a joint move is never used near anything.

Why a task-space move can #

The task-space path holds the tool on the line, which the measurement confirms: the bow is below $10^{-9}$ m, because the path is defined by interpolating the position and the arm is solved to land on it.

Every step of it is an inverse solve, and an inverse solve can fail in three ways this site has already met. The pose may be outside the workspace. It may be inside the workspace and outside the joint limits, which happens to eleven per cent of reachable poses on this arm. Or the arm may be near a singularity, where the solve converges to something and the rates required to get there are the problem.

So the straight line is a stronger request than it looks. A joint move is a promise the arm can always keep; a task move is a request it may not be able to.

Is a third of the distance typical? #

One pair of endpoints proves nothing. A bow of 33% could be a lucky choice, and an essay that quoted it as though it were representative would be doing exactly what this site spends its time objecting to.

So the same measurement was run over sixty pairs of ordinary postures. The bow runs from 11.6% of the distance travelled to 82.4%, with a median of 29.9%. A third is not a lucky pair; it is the middle of the distribution, and the worst case is a move whose tool path strays by four fifths of the distance between its own endpoints.

Joint moves go through singularities constantly #

The vertical axis of that figure carries the second measurement, and it is the more surprising one.

Fifty-five of those sixty moves pass within σ_min = 0.05 of a singularity — a fifth of the value at an unremarkable posture — and one of them gets to $1.7 \times 10^{-5}$, which is essentially through it. Not one of the sixty is affected in the slightest. The joints keep moving linearly, the tool keeps moving smoothly, and nothing anywhere in the computation notices that the arm passed through a configuration where a direction of motion did not exist.

That is the guarantee restated as a measurement, and it makes the trade concrete. Those same fifty-five moves, demanded in task space, would have run into the rate demand below. The two descriptions of the same motion are not equally easy; they are not even comparably easy, and the difference appears exactly where the arm is least able.

It also disposes of an intuition worth disposing of: that singularities are rare places a machine mostly avoids. They are not rare. A random pair of postures has a joint-space path through the neighbourhood of one nine times in ten, and the only reason robots work at all is that most of what they do is either far from one or expressed in a coordinate system that does not care.

What a straight line costs near a singularity #

Now the measurement this essay is for.

Put the arm at distance $\varepsilon$ from its wrist singularity — $\varepsilon$ being the fifth joint’s angle, which is what the distance is here — and ask the tool to move at unit speed in the direction the arm is worst at. Solve $J\dot{\boldsymbol\theta} = \dot{\mathbf x}$ for the joint rates that would produce it, and record the largest.

At a quarter of a degree from the singularity, one metre per second of tool speed costs 289 radians a second of joint rate — about 2,760 rpm at a joint that a real machine turns at perhaps 100. The arm cannot do it, and nothing about the path says so: the tool is travelling in a straight line at a constant speed through the middle of an ordinary working volume.

Fitting the exponent rather than quoting it is the point. A least-squares line through seven points will return some slope whatever the data does, so the fit alone proves nothing; what makes it evidence is that σ_min’s own exponent was measured independently and came out at the reciprocal. Two fits, one relationship, and they agree to two parts in a thousand.

It is also the one number in this field that transfers between arms. A joint rate is in radians per second, a σ_min is in mixed units, a condition number depends on whether the arm was written in metres or millimetres — but an exponent is a ratio of ratios and is the same in every unit system there is. This site has used that escape before: a Panhard rod’s error is second order and a Watt linkage’s fifth, fitted at 2.001 and 5.011, and those two are comparable across two mechanisms of different sizes precisely because they are exponents.

What controllers actually do, and what each one gives away #

There is no way to have the straight line and the bounded rates. Every method in service is a choice about which to give up, and it is worth naming them because the kinematics can price each one.

Damp the inverse. Add a small term to the solve so that the joint rates stay finite near a singularity. The rates are then bounded and the tool does not go where it was told — the damping works precisely by not delivering the demanded motion. The error is largest in the direction the arm is worst at, which is the direction the task most needed.

Project the demand. Ask only for the part of the motion the arm can produce, dropping the component along the lost direction. The tool follows a path that is not the one requested, and the discrepancy is again in the constrained direction, but the machine is at least doing the best it can rather than something arbitrary.

Refuse and stop. Detect that σ_min has fallen below a threshold and halt. Honest, safe, and infuriating to the person who wrote the program, because the arm stops in the middle of a move that looked reasonable.

Re-plan through a different branch. Reach the same pose with a different one of the eight postures, which may be nowhere near a singularity. This is the only one that solves the problem rather than pricing it, and it works only when the pose has another usable posture — which on this arm is often, but at 11% of poses is not.

The kinematics can say what each of those costs. It cannot say which to choose, because that is a question about the job.

The bow is not a defect either #

It is worth resisting a natural reading of the first figure. The 402 mm bow is not the joint-space path being bad; it is the tool path a linear joint motion produces, and any other tool path would require the joints to do something other than run linearly.

Which of the two is preferable depends entirely on what is around the arm. In free space the bow is irrelevant and the joint move is better — smoother in joint rates, immune to singularities, impossible to refuse. Inside a fixture the bow is everything, and it is the reason a robot’s approach and retract moves are always task-space and the long transfer between them is usually not.

That is a genuine engineering statement and it drops out of the geometry: the two coordinate systems are related by a map whose non-linearity is the whole subject of this field, and “straight” is a property of the coordinates it is measured in.

How the task path is computed, and why it may return nothing #

The task-space path in these figures is not an analytic object. It is thirty-six inverse solves, one per sample, each seeded from the answer before it — continuation, which is what a real machine does and what this site does everywhere it sweeps a mechanism.

Seeding from the previous solution matters for a reason the four-bar’s sweep established long ago: it keeps the answer on one branch. Seeded independently, each step would return whichever of the eight postures the seed happened to be nearest, and the path would jump between branches mid-move — drawing an arm that turns itself inside out between one frame and the next, which is a motion no machine can make and which the figure would present as smooth.

And the solve is allowed to fail. taskPath returns a null where the arm could not arrive, rather than the nearest posture it managed, because returning a near miss is the failure this site keeps finding elsewhere: a computation that produces a plausible number for a question with no answer. The assertion behind the figures requires every step to have been solved, so a path with a hole in it stops the build rather than being drawn with a kink.

That is why the straight-line path here can be asserted straight to $10^{-9}$ rather than described as straight. It is straight because thirty-six separate inverse solves each landed on the line, and if one of them had not, there would be no figure.

The rotation has to be interpolated too #

A pose is a position and an orientation, and everything above has been about the position. The orientation cannot be linearly interpolated — the average of two rotation matrices is not a rotation matrix — so something has to be chosen, and the choice is not free of consequence.

What the task path here does is carry the orientation along the shortest arc: take the rotation that turns the start orientation into the end one, take its axis and angle, and turn through a fraction of that angle at each step. It is the only interpolation of a rotation that mentions no coordinate system, and it is what quaternion slerp computes by another route.

The alternative — interpolating three Euler angles linearly, which a surprising amount of software does — produces a different path through orientation space, one that depends on which three angles were chosen and which speeds up and slows down through the move even when the endpoints are evenly spaced. It also inherits the parameterisation’s pole: a move whose orientations pass near it demands enormous angle rates from an arm that is not being asked for anything unusual, which is the coordinate singularity doing damage rather than merely existing.

So there are two independent choices in “move the tool from here to there in a straight line” — how the position is interpolated and how the orientation is — and the next essay is about the fact that there is no way to make them one choice.

What was left out #

Three things a reader who works with robots will notice are missing, and they are all outside this site’s boundary rather than overlooked.

Time. Everything here is about the path — the set of poses the tool passes through — and not the trajectory, which is the path with a schedule on it. Acceleration limits, jerk limits, and the trapezoidal or S-curve profiles that respect them are real and they are dynamics: they involve masses.

Blending. Real machines do not stop at every waypoint; they round the corners, and how much they round depends on a tolerance the programmer sets. The geometry of that is a curve-fitting problem in joint space or task space and it is entirely kinematic — it is a genuine gap in this field rather than a boundary, and it is named as one.

Obstacles. A path that avoids something is the subject of the last essay in this field, and only half of it: whether a route exists is a question about the configuration space, which is kinematics, and finding one is a search, which belongs to algorithms-data-structures.com.

The rate diverges and the travel barely does #

The exponent of $-1$ says the joint rates are unbounded near a singularity, and it is worth asking what the whole traverse costs, because the two answers are different and only one of them is the binding constraint.

Take a straight-line path that passes at closest distance $\varepsilon$ from a singularity. Near the closest approach the smallest singular value behaves like $\sigma \approx \sqrt{\varepsilon^2 + (ks)^2}$ for $s$ the distance along the path, so the joint rate goes as $1/\sigma$ and the total joint travel over the traverse is

$$\int \frac{\mathrm{d}s}{\sqrt{\varepsilon^2 + k^2 s^2}} \;\sim\; \frac{1}{k}\log\frac{1}{\varepsilon}.$$

Logarithmic. So halving the closest approach adds a constant to the total joint motion, while it doubles the peak rate. At $\varepsilon = 10^{-4}$ the peak rate is ten thousand times the nominal and the total travel is about nine times it.

That settles which quantity a machine actually fails on. Motors have a speed limit, and the peak rate exceeds it long before anything else is troubled: 289 radians per second for a metre per second of tool speed is a demand no servo will meet, and it arrives at a quarter of a degree from the singularity. The total joint travel over the same traverse is a few times the ordinary amount, which no machine notices.

So the failure is a rate failure and not a travel one, and every remedy in the list is a rate remedy — damp the inverse, project the demand, refuse and stop. Each of them caps the peak and each accepts a path error in exchange; none of them is trying to reduce how far the joints go, because how far they go was never the problem.

It also explains why passing through a singularity is possible at all, which the $1/\varepsilon$ figure makes sound impossible. A path that goes straight through has $\varepsilon = 0$ and an infinite peak rate at one instant — and a logarithmically divergent travel, which is to say the joints still move a finite amount if the traverse is taken slowly enough near the crossing. That is exactly what a controller doing a deliberate reconfiguration does: accept an unbounded rate demand for an instant by slowing the tool to nothing, and let the joints take the finite motion the geometry requires.

The general shape #

Two coordinate systems, a non-linear map between them, and the observation that a straight line in one is not a straight line in the other. That is the whole of this essay, and it is the same observation that a cam’s motion law makes about displacement and acceleration, and that a quick-return mechanism makes about crank angle and slider position.

What is different here is that the non-linearity has somewhere to become infinite. A cam’s follower moves through its whole stroke for every turn of the shaft and there is no configuration at which the relation breaks down. An arm has three, and near them the map from joint rates to tool velocity is not merely non-linear but singular — which is why this field spends four essays on them and the cam field spends none.