Field

One path to the tool

Every mechanism here so far has been a loop, and a loop is why a configuration on this site is a solve. An arm has no loop: its pose is a product of six transforms with nothing to converge and nothing to refuse. The difficulty does not go away — it moves to the other end, where one tool pose has eight answers and a straight line can cost more than the machine has.

elbow arm at a posture. elbow arm, drawn from 6 joint values through a product of 6 exponentials — no equation is solved anywhere in this picture, because an open chain has none to solve. The thin lines are the joint axes at this configuration, which are also the columns of the arm's Jacobian: the tool's velocity is a sum of turns about exactly those lines. Here the smallest singular value of that Jacobian is 0.3674 and the largest is 2.407, so the arm is comfortably away from a configuration where a direction of motion is lost. Drag θ₂ shoulder.

The chain that does not close

An open kinematic chain has no closure equation, so its forward problem is a multiplication rather than a solve — and its inverse problem, which for a loop was a division, has eight answers.

SCARA, as four numbers a joint. A Denavit–Hartenberg table describes each joint by the common perpendicular between its axis and the next one: how long it is, how much the axes twist across it, where along the first axis it meets, and at what angle. 3 of these 3 rows have no answer for the last two, because the axes they describe are parallel and two parallel lines have infinitely many common perpendiculars, all the same length, at every point along them. The arm is perfectly ordinary; it is the description that has run out.

Four numbers or a screw

Denavit–Hartenberg parameters against joint screws, with the ill-conditioning of the classical table measured rather than described — and shown to have a direction.

Reachable, and dexterous. A 3-link planar arm with links 1.6, 1.2, 0.7. The outer region is everywhere the tool can be put: an annulus from 0.00 to 3.50. The inner region is everywhere it can be put at every tool angle — from 1.10 to 2.10, which is 26% of the area. Both are measured by counting cells on a 300 × 300 grid and both agree with the area computed from the radii to 0.09%, which is what makes the picture a measurement.

Where the hand can go

Reachable and dexterous workspace measured by counting cells and computed from radii, with the two agreeing to a hundredth of a per cent and the hole in the middle got right the hard way.

Two routes to a Jacobian

The geometric Jacobian as transported joint screws, checked against a numerical differentiation of the tool pose, and the difference between a singularity of the mechanism and a singularity of the coordinates describing it.

8 postures, one tool pose. The tool is at one place, held one way. Eight different sets of joint values put it there, and this is all of them at once: two ways for the base joint to face the target, two for the elbow, and two for the wrist — two by two by two. Every posture here came out of a closed form and was checked by running the arm forward again; the worst disagreement is below 10⁻¹⁴ of a metre. Which of them a machine can actually use is a separate question, and it is answered by the joint limits rather than by the geometry.

Eight ways to hold the same tool

The inverse kinematics of a decoupled six-joint arm in closed form, its eight branches named, and the same count recovered by damped least squares from six hundred seeds.

The wrist is three joints and one point

Pieper's decoupling condition measured rather than assumed, the spherical wrist as a spherical mechanism, and what a construction does when the geometry it assumes has been taken away.

Three singularities, and where each one is. The elbow singularity is at the edge of the workspace, where the arm is straight and cannot reach further — nothing is lost that could have been used. The shoulder singularity is where the two ways of facing the target merge, on a cylinder of radius 0.18 m about the base axis which is also the boundary of a hole the arm cannot reach into at all. The wrist singularity is the one that stops real machines: axes four and six in line, one rotation gone, in the middle of an ordinary working volume with nothing about the tool's position to suggest it. The rank falls by exactly one in all three — and what is lost differs. The elbow and the shoulder are constrained by a pure force, along the arm and across it; the wrist is constrained by a screw of pitch −0.629, which is a force and a couple together. Those are reciprocal screws rather than singular vectors, because a singular vector's direction depends on whether the arm was written in metres or millimetres and a screw does not.

Where the arm loses a direction

The elbow, shoulder and wrist singularities of a six-joint arm, told apart by the reciprocal screw each one leaves behind rather than by a singular vector, which turns out to depend on whether the arm is measured in metres or millimetres.

How far a joint-space move bows off the line. Every joint runs from its start value to its end value at a constant rate — the simplest possible move, and the one that can never be refused, because every point along it is a set of joint values and every set of joint values is a pose. The tool does not travel in a straight line while it happens. It bows away by 402 mm over a move of 1223 mm, which is 32.9% of the distance travelled and enough to hit something that the straight line would have missed.

A straight line at constant speed

Joint-space against task-space interpolation, the bow measured in millimetres, and the cost of a straight line near a singularity fitted rather than quoted.

Three shortest paths. The tool's route between the same two poses, under three interpolations. The straight one is position interpolated along a line with the rotation carried separately, which is what most controllers do. The screw path is the single turn about a single axis that Chasles's theorem says takes any pose to any other — the only one of the three that mentions no coordinate system — and it is 18.0% longer. The joint-space path is what the arm does when nobody asks for anything in particular, and it is 26.8% longer again. The screw axis itself is drawn: pitch -0.096 m per radian.

The distance between two poses

Why the set of rigid poses has no natural metric, measured as three interpolations that disagree by 27% and a nearness that flips at a length scale of a gripper's width.

One tool pose, a curve of postures. 41 postures of S-R-S arm, every one of them holding the tool at the same pose. The tool moved 1.0e-13 of a metre over the whole sweep, which is the arithmetic's noise floor rather than a tolerance. The joint that moves most is the elbow, and it runs on a circle of radius 0.192 m about the line from shoulder to wrist — a curve computed here by walking the Jacobian's null space, and known independently as the intersection of two spheres. The two agree to 10⁻¹³.

The freedom that does nothing

Kinematic redundancy as the null space of the Jacobian, the self-motion it generates walked numerically and constructed in closed form, and the mobility-one mechanism the pinned arm turns out to be.

Arms with their tools pinned down. Pin an arm's tool to the ground and the open chain is a closed loop, which the mobility count of a planar loop already handles. Kutzbach gives 6(n − 1) − 5n = n − 6 for a loop of n revolutes, and the measurement is n minus the rank of its screw system — the columns of the arm's own Jacobian, read as constraints rather than as velocities. The two agree on every row but one, and the one is the arm at a wrist singularity: the formula says the pinned arm is a structure and the mechanism has a freedom. That is the finding this site opened with, arrived at from the far end of its subject.

Pin the tool and it is a loop

A serial arm closed into a loop by pinning its tool, counted by Kutzbach and measured by the rank of its screw system, with the disagreements falling exactly on the arm's singularities.

Where an error at the shoulder ends up

A serial arm's error budget joint by joint, with each contribution shown to be a lever arm by three routes that agree, and worst case set against root sum square at a ratio of 1.81.

What a calibration can and cannot see. The singular values of elbow arm's identification Jacobian — the matrix of how the tool pose moves when each model parameter is nudged, over 11 postures. There are 36 parameters and only 30 of them can be found: the last 6 directions come out at 2.3e-8, which is the difference noise, against 6.3e-4 for the weakest real one — a gap of 3e+4. And 30 is exactly 4R + 2P + 6 for this arm's 6 turning and 0 sliding joints, which is a count from the literature meeting a rank measured from the arm's own arithmetic.

What a calibration cannot see

The identification Jacobian of a serial arm, its rank measured from singular values rather than by elimination, and the classical identifiable-parameter count recovered on four arms.

The free configurations, with joint limits. Every point is a pair of joint angles for a two-link arm; the pale region is the configurations at which neither link touches an obstacle, and the dark one is where something is in the way. The free space is in 2 pieces. The arm's joints cannot turn all the way round, so the edges of the square are edges — and now the barrier separates. The two crosses put the tool at exactly the same point, and the arm cannot get from one to the other at all.

The space of configurations

The configuration space of a planar arm among obstacles, its components counted on the torus and on the square, and the finding that joint limits rather than obstacles are what separate one posture from another.

The configuration space of a crank rocker. Every point of the square is a pair of angles — the crank's and the rocker's — and the curve is where the coupler is exactly the right length to join them. That curve is the mechanism: one equation in two angles leaves one freedom, which is the mobility. It has 2 components, and that is the two assembly branches. A built linkage cannot cross between them, because there is no path in the set to cross by — and each goes all the way round, which is what makes the crank a crank. The square's left and right edges are the same line, and so are its top and bottom.

Branches were components all along

The site's own vocabulary for multi-valued mechanism solutions, restated as connected components of one set, with the four-bar's branches counted by contour and by sweep.

An arm is a tree

Open chains as trees: the census of them, why a serial arm is one shape out of many, and what the number of independent loops separates.

A SCARA arm, and where it can put its tool. A SCARA arm at its home position, with the axes dashed and a cloud of the tool positions reached over random joint values. The displacement set of an open chain is the product of its joints' groups, one factor per joint, and the question this field asks of it is whether the product is itself a group. Here the logarithms of the reached displacements occupy 4 dimensions, so the motion lies inside Schoenflies motion and composing two of its displacements gives another one. The cloud is a fact about the reach and not about the group: a chain of finite links covers a bounded piece of its group and never the whole of it, which is a separate question and a different field's.

The arm that is a group

Schoenflies motion as the specification of a SCARA arm, with the group's dimension deciding the joint count and the group's closure deciding what the machine guarantees.

An arm's parameters and its poses

Applying the identification apparatus to a serial arm: what it recovers, why the difficulty is conditioning rather than rank, and what a pose plan looks like in three dimensions.

The arm that hits itself

Self-collision of a planar three-link arm over its joint space, and what one obstacle costs a workspace when the arm has a spare freedom and when it does not.

A straight move ends where its posture does. The six-joint arm with its stops on, moving its tool in a straight line at a fixed orientation in the posture it starts in (back · elbow up · wrist flipped). Faint: the arm at the start. Solid: where the move ends, 66.0 cm along, stopped by the base joint at its stop. Dashed: the same line continued, which this posture cannot follow. Of the arm's usable postures at the start, the best would have gone 125.5 cm.

A straight move ends at a stop

Three hundred straight moves of a six-joint arm's tool at fixed orientation, each in the posture it starts in: one move from start to stop, drawn; how far the moves get in the starting posture and in the best usable one; what ended each move; one move's six joints between their stops; each move's travel against its direction; and the survey with the corrected count of usable postures.

A seventh joint steers the base roll off its stop. The S-R-S arm with stops on every joint, moving its tool in a straight line at fixed orientation from the same start (faint). With its upper-arm roll locked, a six-joint arm, the move ends 27.0 cm along at the base roll at its stop (dashed arm). With all seven joints and the self-motion steered at every step to move the joint nearest its stop away from it, it ends 61.5 cm along at the base roll at its stop (solid arm). Shown 60.0 cm along, with the six-joint arm stopped where its move ended.

A seventh joint cannot move the elbow

Two hundred straight moves of a seven-joint shoulder-elbow-wrist arm's tool at fixed orientation, from the same starts with its upper-arm roll locked and free: one move with six joints and seven, drawn; how far the moves get with six joints, seven by the least motion, and seven steered off the nearest stop; which stops the self-motion gets round; the elbow's angle along one move, the same both ways; each move's gain, in order; and how the moves end.

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