Machine Kinematics
solved, then drawn
Concept

Instant centre — where it appears

The point a moving body is turning about at an instant, which lies on every rolling wheel's axle line and at the intersection of a linkage's extended cranks. Two independent constructions place it — the meeting of the wheel axles and the meeting of a linkage's extended cranks — and this site requires them to agree.

Named by 16 essays across 8 fields — each of them below, with the objects they name alongside it.

A ratio that is not a number. The output-to-input angular velocity of a four-bar, computed from the velocity solution at every position. It runs from -0.398 to 0.333 — it changes sign, because the rocker turns back — with a mean of 0.000 that no instant of the cycle actually exhibits. Quoting a single figure for a linkage's ratio is quoting the average of that curve. A gear pair is the case where the same phrase is honest: its ratio is 0.5000 and stays there, which is not a coincidence but the property the involute was invented to guarantee.

The ratio that is not a number

A four-bar's output-to-input speed ratio runs from −0.29 to 0.51 through one turn and changes sign on the way. Quoting a single figure for it quotes the average of that curve, which the mechanism never exhibits. A gear pair is the case where the same phrase is honest, and it is honest by construction.

wrong · Misconception Five points on one coupler. The same four-bar, with a tracing point rigidly attached to the coupler at five different places. Each curve is a sextic — degree six — and moving the attachment point a little changes it a great deal. That sensitivity is the reason coupler-curve synthesis was done with atlases of printed curves for most of the twentieth century: there is no simple inverse, so the practical method was to look one up.

What a coupler point draws

A point rigidly attached to the coupler of a four-bar traces a curve of degree six. Move the attachment a little and the curve changes a great deal. For most of the twentieth century the practical way to find the linkage that draws a wanted curve was to look it up in a book of printed atlases.

curves · Coupler What one contact forbids, drawn as a place. A single contact on one edge of a square, and the whole plane coloured by what it permits. A rotation about a point is a twist, and a twist is permitted when it does not drive the part into the obstacle; because a rotation about (x, y) is affine in the point, the condition is a half-plane and the boundary is a straight line — the line through the contact along its own surface. On one side of it only anticlockwise rotations are permitted, on the other only clockwise, and the two together are the whole plane bar the line itself. So one contact rules out exactly half of what the part could do and leaves the other half untouched, which is why the count of contacts a hold needs is one more than the dimension rather than equal to it: the first 1 of them cannot leave nothing over. The picture is exact — the regions are clipped polygons, not a sampled grid.

What one contact forbids

A rotation about a point is a twist, and a twist is affine in the point — so what a single contact permits is a half-plane of centres, with the boundary being the contact surface's own line. Reuleaux drew it in 1875 and it is exact rather than sampled, which is why every figure in this field is a picture of the plane rather than of a cone.

holding · Restraint Where the coupler is pivoting, at 70°. At any instant the coupler is turning about one point — not a pin, and usually not on the mechanism at all. Kennedy's theorem finds it: the crank and coupler share the pin at A, the coupler and rocker share B, so the coupler's centre relative to the frame must lie on both O₂A extended and O₄B extended, and it is where they cross. The dashed lines are that construction. The cross is a completely different route to the same point — the place where the coupler's solved velocity field is zero, computed from the Jacobian and knowing nothing about Kennedy. Across 119 positions the two agree to 2.7e-15. At this instant the centre lies outside the frame — the two construction lines are nearly parallel, the coupler is close to translating, and the pivot has run off rather than gone missing.

Where the coupler is turning

At every instant the coupler of a four-bar is rotating about a single point — not a pin, and usually not on the mechanism at all. Track that point in two different frames and you get two curves which, rolled on each other without slipping, reproduce the coupler's motion exactly. The bars are one way of producing it and not the motion itself.

curves · Coupler 4 contacts, and the centres they still allow. The same four, placed pinwheel. The same square, the same four edges, the same distance along each — and taken the same way round rather than alternately. Every row's moment then has the same sign, so no positive combination can cancel it, and the part turns. Each contact contributes one half-plane of permitted centres per sense, and the shaded regions are what survives all 4 of them: the darker one is where an anticlockwise rotation is still permitted and the lighter one where a clockwise one is. What is left is the escape, and it is a region rather than a direction: any point inside it will do as a centre. The enumeration finds 4 extreme rays, of which 4 are rotations and the rest are translations — the corners of the region and its unbounded directions respectively. positioned by solving, not by drawing.

The escape is a place

A part that is not held escapes, and the useful thing is not that it escapes but where. The extreme rays of the cone are the corners of a region of the plane and its unbounded directions are translations — so the answer to 'this does not hold' is a picture with a shape, and the shape says where the next contact has to go.

holding · Restraint A screw of pitch 0.25. A single screw drawn from its own decomposition. The six numbers of the twist go into a decomposition that returns a direction, a point on the axis and a pitch of 0.2500; the helix is then the path of a point at radius 0.5 about that axis, advancing 0.250 along it per radian turned. Over 1.15 turns the point advances 1.806 — the pitch times the angle, which is what pitch means. This screw is a screw.

Every motion is a screw

Chasles showed that any rigid displacement whatever is a turn about some line together with a slide along that same line. Not approximately, and not usually — always, with the line and the amount of slide computable from the motion. It is the fact that makes spatial kinematics a subject rather than a pile of special cases.

spatial · Spatial The circle of points going straight, at 66°. Every point on this circle is, at this instant, travelling in a straight line: its path has zero curvature there. The circle passes through the pole — where the point is not moving at all — and its diameter is 47.64, which is the pole's own speed divided by the plane's angular rate. Nothing here was assumed to be a circle. The locus is the zero set of a quadratic whose |w|² coefficient is φ′³, a real number with no cross term and no difference between its two square terms, and a general conic fitted to the sampled locus returns those coefficients at 8.1e-15 and 6.4e-15. At this position the coupler is close to translating, the pole has run off the canvas and the circle with it — the figure is the size of the mechanism, and δ here is 13.6 coupler lengths. positioned by solving, not by drawing.

A curvature is a size with a minus sign

Everything in the curvature field is a similarity invariant in shape and a reciprocal length in value. Scale a moving plane and its inflection circle scales, its cubic of stationary curvature scales, and every curvature it computes is divided by the factor — so a bigger machine traces gentler paths than its drawing suggests.

curvature · Curvature An open belt, whose wraps make one turn. Two pulleys of radius 40 and 24 mm with their centres 160 mm apart, with the strand running the same way round both. The two tangency points on each pulley are marked; the run between them is computed from the signed radius difference and nothing else. The wrap angles are 191.48° and 168.52°, and they add to exactly one turn — the turning number of a simple loop. Length 522.6633 mm, against the textbook formula's 522.6633. positioned by solving, not by drawing.

The road a wheel carries with it

A taut strand on a pulley is a rolling contact: the material at the tangency is at rest against the surface, and the ratio between two bodies on one span is the ratio of their arms. But this rolling constraint integrates, where a wheel's does not — and the difference is that a strand rolls along a line and a wheel rolls across a plane.

rolling · Rolling The two curves the pole rolls along, at 60°. The pole is a different point at every instant, and it traces one curve in the fixed plane and another in the moving one. Those are the centrodes, and the whole motion is the second rolling without slipping on the first — a statement with no mechanism in it, which is why two completely different linkages with the same centrodes produce the same motion. The moving centrode is drawn here in the position it occupies at this instant, and it touches the fixed one at the pole to 0.0e+0 of a unit. positioned by solving, not by drawing.

The linkage, put back from two curves

This field opened by saying the mechanism drops out, and that every planar motion is one curve rolling on another. Both are true and neither had been measured. The rolling reproduces the four-bar's own placement to a residual that quarters when the sampling halves, and the two curves lay equal arc to a part in a billion.

curvature · Curvature The body points whose three images are in a line. The three prescribed poses, faint, and a curve through them. At every point of a 110×110 grid over the moving body, the point is placed in all three poses and the signed height of its image triangle is measured; the curve is where that length is zero. Those are the points whose three images lie on a line, so the dyad they want is a slide rather than a crank. The curve is a circle. Refined onto the contour, its points lie on the circle through the three image poles to 2e-15, where the pole triangle's own circle misses them by up to 1.98. Its radius is 8.767, so inside a window three units across it reads as a gentle arc; 2 of the three image poles are in the frame and the third, P₁₃, is 6.8 units away.

Where a pin becomes a slide

Three-position synthesis gives every point of the moving body a fixed pivot, except the points whose three images fall in a line. Those want a slide, and they are not scattered: they lie on one circle, the circle through the three image poles, which a single line of algebra predicts and a contour of a measured length draws to 10⁻¹⁵.

synthesis · Synthesis Three circles at one point, at 50°. The osculating circle of the fixed centrode at the pole, the osculating circle of the moving centrode there, and the inflection circle. All three pass through the pole and all three have their centres on the common normal — which is what makes the relation between them a relation between three numbers on a line. The two centrode radii are 16.180 and 9.730, signed along that normal, and the inflection circle's diameter is 24.398 against the 24.409 the other two give. Nothing in that arithmetic knows there is a linkage. The window is 30.25 wide and the larger osculating circle is 32.36 across, so it is cut at the edge: what is drawn is the neighbourhood of the pole, where the three curvatures are the same three numbers however far the circles carrying them reach.

The two numbers are the curves' own

Every essay in this field runs on the inflection circle and the pole tangent, and every one has taken them from the motion's derivatives. They are the two centrodes' radii of curvature at the pole, combined harmonically — measurable off the curves with no mechanism in the arithmetic, and carrying a sign that changes eight times in a turn.

curvature · Curvature Holding the coupler still: the frame's inflection circle is the mirror of the coupler's. The four-bar with ground 4, crank 1, coupler 3.5 and rocker 3 at a crank angle of 225°, redrawn in the coupler's own frame, so the coupler is the horizontal bar and the ground link is what moves. The circle on one side of the pole is the ordinary inflection circle — coupler points travelling straight — carried into this frame. The circle on the other side is the inverse motion's: frame points travelling straight when the coupler is held. It is computed by differentiating the swap and, separately, by solving the coupler-held four-bar, and both routes put it at the first circle reflected through the pole, to 7e-15 on a radius of 4.45. The line through the pole is the common tangent. Dragging the crank angle moves both circles and they stay mirror images.

The frame seen from the coupler

Hold a four-bar's coupler still and let its frame move, and every construction of the curvature field has a counterpart. The frame's points that travel straight lie on the ordinary inflection circle reflected through the pole — found by differentiating the swap and, independently, by solving the coupler-held four-bar, both to 10⁻¹⁴ — and a point and the centre of curvature of its path trade places exactly. The inverse motion of one four-bar is the ordinary motion of another.

curvature · Curvature A ball bearing turned by its inner race. A bearing of pitch diameter 40 with 8 balls of diameter 8, its outer race held and its inner race turned 60°. Rolling without slipping at both contacts leaves the cage — the ring that keeps the balls apart, marked by the dark tick — turned 24.00°, a share of 0.4000 of the race, and each ball spun 120.0° backwards about its own centre, marked by its own tick. The share is (1 − d/D)/2, less than a half by d/2D; the gear-train solver, handed a planetary with a sun of 32 teeth and a ring of 48, returns 2/5 for its carrier. Dragging turns the inner race.

A bearing is a planetary with no teeth

Roll a ball between two races and nothing but the two rolling constraints decides how fast its centre goes round. Solved, they put the cage at (1 − d/D)/2 of the inner race's speed — always less than half — and the gear field's train solver, handed a planetary with a sun of D − d teeth and a ring of D + d, returns the same fraction exactly. And a tapered roller rolls without slipping along its whole line only if its axis meets the bearing's at the apex, the wheel field's concurrency one dimension up.

rolling · Rolling A stop on the coupler, and the one direction in which it stops nothing. A four-bar — frame 4, crank 1.2, coupler 3.4, rocker 2.8, crank at 70° — whose coupler is stopped by a contact with the frame at (1.59, 2.16). The dashed line through the stop is its normal; the ringed point is the coupler's pole, where the crank's and rocker's lines meet, at (2.505, 6.883). With the normal 10° off the line to the pole the stop removes the four-bar's one freedom; through the pole it removes nothing, because the contact point is already moving along the stop's face. The graph of the stopped four-bar is the same in every case.

A contact turns and slides at once

A cam on its follower, a gear tooth or a pin in a slot takes one freedom away where a pin or a slide takes two, and it turns as well as slides. Give every joint of the four-bar, Watt's and Stephenson's chains a third label and there are 1,317 different chains. The rule that read slides off the graph extends to all of them, agreeing with the constraint rank on every one, and the two obvious shortcuts miss 36 and 769. What the graph cannot record is which way a contact's normal points — and on these chains that never matters, while a single stop added to a four-bar holds everywhere except through one point.

topology · Topology Two wheels on one frame, on a rough road. A rigid frame on two wheels of 50 mm radius, 100 mm apart — 0.5 of the road's wavelength — on the road of 10 mm amplitude and 200 mm wavelength. The dashed curve is the path every wheel's centre follows on this road, whatever the frame does; the frame only fixes where on it the second wheel is. Over each wavelength of travel each wheel counts 204.847 and 204.847 mm, the length of one period of that path, against the 200 mm travelled; the frame pitches by up to 11.53°. Dragging changes the wheelbase.

A second wheel cannot correct the first

A wheel on a rough road counts the length of its centre's path, not the distance travelled. Put two on one rigid frame and average them, and the error does not shrink at any wheelbase: both centres run on the same curve, fixed by the road and the wheel's size, and each wavelength of travel carries each of them once round it. A half-wavelength wheelbase smooths the count's ripple and rocks the frame instead. What does work is two wheels of different sizes. A bridging wheel's error falls as the inverse square of its radius, so two radii measure their own error: wheels of 0.6 and 1.2 m that overcount by 0.339% and 0.099% extrapolate to 0.018%, knowing nothing about the road.

rolling · Rolling A seat round the stop's nose holds the four-bar if it is tighter than the nose centre's path allows. The four-bar of frame 4, crank 1.2, coupler 3.4 and rocker 2.8 at a crank angle of 70°, its coupler carrying a round nose of radius 0.3 centred at (1.593, 2.160). The nose sits in a hollow seat on the frame of radius 0.8, touching it at (1.650, 2.454), on the line from the nose's centre to the coupler's pole at (2.505, 6.883), off the top of the drawing. The dashed circle is where the nose's centre may go without the nose entering the seat: a disc of radius 0.50. The curve through the nose's centre is the path the four-bar gives it, with a radius of curvature of 0.893. The seat's disc is tighter than the path, so the path leaves it either way and the four-bar cannot turn: the stop holds. The threshold is a seat of radius 1.193, the path's radius plus the nose's.

A stop aimed at its pole holds by curvature

A stop whose normal passes through its pole forbids nothing at first order, and whether it holds at all is decided by curvature. For a round nose the deciding curve is not the one it touches: the seat that just holds is the path of the nose's centre moved out by the nose, 1.193 on a four-bar where the contact point's own circle says 0.776. Past it, one contact holds both ways — loosely, as the square root of its clearance.

topology · Topology

Named alongside it

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

CentrodeContact normalEuler savaryInflection circlePath curvatureVelocity ratioConstraintCoupler curveHigher pairRolling constraintSecond orderTwist

All concepts