Machine Kinematics
solved, then drawn
Concept

Platform — where it appears

The moving body of a parallel mechanism, carried by several legs at once. Its pose is what the legs' lengths jointly decide, and finding it from them is the hard direction rather than the easy one.

Named by 11 essays across 4 fields — each of them below, with the objects they name alongside it.

Three legs, one platform. A 3-RRR planar parallel mechanism at (0.20, -0.15) turned 11.5°, elbows up/up/up. The three actuator angles were computed one leg at a time and independently, which is what makes this direction cheap. The dashed lines extend each leg's second link: those are the three forces the legs can transmit to the platform, and the mechanism is controllable exactly while they stay independent. Here they miss one another by 0.651, and the smallest singular value of the three is 0.9550. At this position the platform can be turned through 206° in all before a leg runs out of reach.

The easy problem and the hard one change places

For a robot arm, working out where the hand is takes a walk down a chain and working out what the joints must be to put it somewhere is the hard part. Connect the platform by three legs instead of one and both statements reverse — and the reversal is not a matter of degree.

parallel · Parallel Losing a freedom and gaining one. Left, the nearest inverse-kinematic singularity: a leg is straight to 1.1e-5 and cannot reach further, so the platform has lost a freedom. That is the workspace boundary and it is a serial arm's singularity. Right, the nearest direct-kinematic singularity: the three leg lines pass through one point to 0.0001, so the three forces the legs transmit are no longer independent and the platform can turn about that point with every actuator locked. It has gained a freedom, and it is 1.97 from the other configuration and nowhere near the edge of the reach.

Locked, and still moving

A serial arm goes singular at the edge of its reach, where it loses a freedom, and the failure is visible as an arm gone straight. A parallel mechanism has a second kind with no serial counterpart — it gains a freedom, in the middle of the workspace, at poses nothing about the legs' reach marks out.

parallel · Parallel 6 ways to assemble the same three actuator angles. The actuators are at 216°, 48°, 144° in every panel, so the three elbows are at the same three points throughout and only the platform differs. Each pose satisfies all three legs to 6.7e-16. Found by reducing the problem to one equation in the platform angle and scanning it at 0.100° — exhaustive to that resolution and no further, which is the honest thing to say about a root count. Over a survey of 6750 actuator triples this mechanism ranges from none to 6.

One command, six answers

Lock the three motors of a planar platform and the platform can be in as many as six different poses, every one of them satisfying every leg exactly. Which one it is in was decided by how it was assembled and where it has been since — and the number of answers is not a property of the mechanism but of where the motors happen to be.

parallel · Parallel Six legs, six numbers. A Gough–Stewart platform at (0.00, 0.00, 2.40) with a rotation vector of (0.00, 0.00, 0.15). The six leg lengths run from 3.077 to 3.263, and every one of them was computed from the pose by a single subtraction and a square root, with no leg consulting another. Going the other way — six lengths in, one pose out — takes an iterative solve and does not have one answer. The six leg lines span a screw system of order 6; the smallest singular value of the six is 0.0316, which is how far this pose is from a configuration where they become dependent and the platform stops being controllable.

Six legs and a square root

A Gough–Stewart platform's inverse problem is one subtraction and one square root per leg, computed six times without any leg consulting another. Its forward problem has forty solutions. That gap is the whole design, and it is why flight simulators are built this way and robot arms are not.

parallel · Parallel Three bars, and no rotation left. Each of a delta robot's legs ends in a parallelogram, which keeps the bar on the platform parallel to the bar on the arm. That leg therefore permits the platform no turn about either direction perpendicular to its bar: it imposes two couple constraints, drawn here as rings about the directions they act on. Three legs impose 6; together they span only 3, so 3 are redundant. What is reciprocal to them is three couples — three pure translations, and the platform cannot turn at all.

Why the platform stays flat

A delta robot has three legs and three freedoms, and there is no obvious reason those freedoms should be the three translations rather than some mixture. The reason is a parallelogram in each leg, and the argument from there to "the platform cannot turn at all" is a constraint computation that takes six wrenches and a rank.

parallel · Parallel Where the platform stops being controllable, at 0°. Every point is a position of the platform's centre at a fixed orientation of 0°, shaded by how far it is from a direct singularity — pale is near. Unshaded means unreachable, which is the workspace boundary and the ordinary kind of singularity. The line is det A = 0, traced through the field rather than tested for: it runs through the middle of the reachable region in 102 segments, and on it the platform can move with all three actuators locked. 3312 of 6561 sampled positions are reachable.

The workspace is not a shape you choose

A serial arm's reach is roughly a sphere and can be quoted as a number. A parallel mechanism's is the intersection of three reachability conditions, changes with every degree of orientation, and has a surface of uncontrollable poses cutting through the middle of it. There is no formula. There is a map, and it has to be computed.

parallel · Parallel Kutzbach, plus the constraints counted twice. Each row is 6(n − j − 1) + Σf, then the redundant constraints ν measured from the two legs' constraint systems, then the mobility measured from the rank of the whole loop's screw system. The first column is wrong for 5 of 6 of these mechanisms; adding ν repairs every one. The catch is that ν is not a property of the joint graph — Bennett's linkage and a spatial four-bar with one twist changed have the same graph and different ν — so the corrected formula needs the measurement it was supposed to replace.

The formula is repaired by the thing it replaced

Kutzbach's count is wrong about most of the mechanisms worth building, and every textbook gives the same repair — add back the constraints that were imposed twice. The repair works on every loop this site has. It is also not a formula, because the number it adds cannot be read off the joint graph.

constraint · Spatial What every pair of groups meets in. The intersection of two subgroups is always a subgroup — that needs no computation — and which one is the useful part. This is the design rule behind every parallel machine on this site: choose legs whose groups meet in the motion the platform is wanted to have, and it has that motion whatever the leg lengths are, with no synthesis and no tolerance. The row and column are built about different axes, at right angles, because an intersection is a statement about particular subgroups rather than about their kinds: two planar groups with the same normal meet in the whole of themselves, and two with different normals meet in a line.

Three legs and one plane

The parallel field argued that a platform stays flat from the structure of its legs, one leg at a time. The same fact is one line of linear algebra: each leg confines the platform to a group, the platform gets the intersection, and an intersection of groups is a group whatever the leg lengths are. The motion type is decided before a single dimension is chosen.

parallel · Parallel What every pair of groups meets in. The intersection of two subgroups is always a subgroup — that needs no computation — and which one is the useful part. This is the design rule behind every parallel machine on this site: choose legs whose groups meet in the motion the platform is wanted to have, and it has that motion whatever the leg lengths are, with no synthesis and no tolerance. The row and column are built about different axes, at right angles, because an intersection is a statement about particular subgroups rather than about their kinds: two planar groups with the same normal meet in the whole of themselves, and two with different normals meet in a line.

Legs intersect

A serial chain multiplies its joints' groups and the product is almost never a group. A parallel machine's platform gets the intersection of what its legs permit — and an intersection of groups is a group, always, with no coincidence required. That is the only construction in this field that produces closure for free, and it is why a platform can be designed for a motion type instead of discovered to have one.

pairs · Pair Three joints, and the three screws that describe them. A leg of three revolute joints, drawn as its three axes, and the principal screws of the three-system they span, drawn through the system's own centre. The three principal axes are mutually perpendicular — worst cosine 1.1e-16 — and they meet at one point, missing it by 2.6e-16. Their pitches are -0.3766, -0.0338, 0.8075, and every screw the leg leaves free has a pitch the three of them give by h₁l² + h₂m² + h₃n². Nothing in the three joint axes looks like a right angle and the system's own frame is one.

What a leg of three joints leaves free

Five essays of this field have computed the order of a screw system and drawn none of them. A three-joint leg spans a three-system; its three principal axes are mutually perpendicular and meet at a point, six numbers price every screw in the family, and the directions of the lines it contains form a cone.

spatial · Spatial What a leg of three joints permits and what it resists, on one frame. The leg of three revolute joints in grey, and the principal screws of two three-systems drawn through one centre: the twists the leg lets its platform make, and the wrenches it holds without any joint turning. They are computed separately — the second is the reciprocal complement of the first, found by its own eigenproblem — and they come out with the same centre, 5e-16 apart, and the same three axes, parallel to 0e+0. Only the pitches differ, and each wrench pitch is its twist's with the sign reversed, to 1e-15: -0.3766 against 0.3766, -0.0338 against 0.0338, 0.8075 against -0.8075.

The lines a leg turns about and the lines it is pushed along

A leg of three revolute joints permits a three-system of motions and resists a three-system of loads. The two share a centre and three axes and differ only in the sign of every pitch — and the revolute axes of the first and the lines of force of the second are the two rulings of one hyperboloid, every line of one meeting every line of the other.

spatial · Spatial

Named alongside it

The objects these essays reach for when they reach for this one.

ConstraintParallel mechanismScrewDirect singularityReciprocal screwScrew systemWorkspaceForward kinematicsInverse kinematicsMobilityPitchSingularity

All concepts