Machine Kinematics
solved, then drawn
Concept

Connected component — where it appears

A piece of a set that a continuous path cannot leave, and therefore a set of configurations a mechanism can reach from within it and no others. It is the precise form of the statement that a built mechanism cannot change branch: two configurations in different components are unreachable from each other, however the machine is driven.

Named by 6 essays across 5 fields — each of them below, with the objects they name alongside it.

Free in every direction, and it cannot get out. A disc of radius 1 among 3 point obstacles on a circle of radius 1.100. The shaded discs are the obstacles grown by the part's own radius, which is what the part's centre may not enter — the configuration space, and for a round part it is the plane itself. The part is caged when those grown discs overlap enough to close a ring around it, which happens below R = 1/sin(π/3) = 1.154701, and here it does. At every configuration inside the cage the part is free. The three normals all point at its centre, the rank of its rows is two, the escape cone is a whole line, and none of that has anything to do with whether it can leave. A hold is a statement about velocities at one configuration; a cage is a statement about where a finite motion can go, and the second does not follow from the first in either direction. positioned by solving, not by drawing.

Free at every instant and going nowhere

Three points on a circle of 1.1 radii around a unit disc leave it free in every direction at every configuration — rank two, margin nought, the whole plane of centres shaded — and it cannot get out. The threshold is 1/sin(π/n), which is 1.154701 for three, and a flood fill of the free space agrees with the formula at every radius sampled.

holding · Restraint 20,000 four-bars, and not one with three circuits. 20,000 four-bars with the ground at one and the other three lengths drawn from 0.05 to 3, each counted exactly. crank-rocker: 3,031, of which 3,031 have two circuits, 0 have one and 0 cannot be assembled; double crank: 3,086, of which 3,086 have two circuits, 0 have one and 0 cannot be assembled; Grashof double rocker: 1,497, of which 1,497 have two circuits, 0 have one and 0 cannot be assembled; triple rocker (non-Grashof): 12,386, of which 0 have two circuits, 9,528 have one and 2,858 cannot be assembled. The most circuits any linkage has is 2.

Never three circuits

A four-bar has one circuit or two, and twenty thousand random four-bars counted exactly contain no exception. The reason is a count of four points where the two assemblies merge, which makes the configuration curve a curve of genus one, and Harnack's theorem allows a real curve of genus one two pieces and no more. A six-bar's curve has genus five or seven, and its circuits go to four and six.

algebra · Algebra Free space, in pieces. The driving angle round the circle, with the arcs at which the machine is both assembled and clear drawn heavy. One stud in the way takes a bite out of the turn and leaves 1 arc: the crank can still reach every remaining angle by going the other way. Two studs leave 2, covering 74% of the turn — and every configuration in both arcs is a perfectly good solution of the same constraint equations, on the same assembly branch, at the same mobility. Nothing a solver computes distinguishes an angle in one arc from an angle in the other; what separates them is that the machine cannot be driven from one to the other.

Free space comes in pieces

Every arc on this site has ended at a configuration the mechanism cannot reach. Put two studs in a four-bar's way and its drive falls into two arcs whose ends are configurations it reaches perfectly well and cannot occupy — and no quantity the solver computes tells one arc from the other.

bodies · Body One piece, and still not reachable. The state of a linear ratchet — a cable tie — is how far in it is pulled, and its free space is the whole interval: every state is connected to every other, with no barrier anywhere. What it does not have is a way back. From the marked state the reachable set is everything forward and only as far back as the tooth the pawl has already dropped into, so 52.1% of ordered pairs are reachable and 4.2% are reachable both ways. Connectivity is symmetric; reachability is an order, and a component count answers the first question and cannot be asked the second.

One piece, and still not reachable

The site decides whether two configurations can be joined by asking whether they are in the same connected component, and that relation is symmetric because a path run backwards is a path. A one-way mechanism breaks the symmetry and leaves the connectivity alone: its free space is a single interval with no barrier anywhere in it, and about half of the ordered pairs of states cannot be joined by any admissible motion.

timing · Intermittent 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

A two-joint arm's configurations are a torus, and drawing one on a page turns it into a square whose opposite edges are secretly the same line. Count the free space on the square and get three pieces; count it on the torus and get one. Eighty single-obstacle arrangements were tried and not one of them cut the torus in two — what does that is a pair of hard stops.

serial · Serial 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

Seven words have been used for one thing. An assembly branch, a circuit, an assembly mode, a working mode, a posture and a branch defect are all statements about the connected components of a mechanism's configuration space — and once that is said, a four-bar's two circles, a platform's six modes and a synthesis defect stop being three subjects.

serial · Algebra

Named alongside it

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

Configuration spaceAssembly branchFree spaceCircuitGrashof's conditionLimit positionBranch defectBranch pointCagingContact stateConvex hullEscape cone

All concepts