Machine Kinematics
solved, then drawn
Prescribed motion

An arm is an offset that grows with the lift

Carry a cam's roller on a swinging arm instead of a slide and its pressure angle obeys the offset follower's formula exactly, with the offset replaced by the distance of the roller's line of motion from the cam's centre. That distance turns with the arm. On a cam whose strokes are mirror images the best arm is worse than a centred slide by about 5,000/L² degrees and leans the follower by its own tilt on the dwells. On a quick-rise cam the pivot's angle rebalances either side of the cam for under a degree, where a sliding offset moved to the wrong side costs eleven.

Assumes An offset trades the rise for the return.

17 min read 6 figures Two routes to the same numberSomebody had a problem

An offset trades the rise for the return moved a roller follower’s straight guide off the cam’s centre and found a trade rather than an improvement: the offset is a constant subtracted from the follower’s velocity, the velocity changes sign between the strokes, and whatever one stroke gains the other pays for. It ended on the follower most cams actually drive, a roller on an arm that swings about a pivot, and on a question it did not answer. The arm’s line of travel is an arc, the offset becomes the arm’s length and the pivot’s position, and whether the pivot can balance the two strokes as the offset did was left uncomputed.

It can, and the way it does so is not the way the offset did. The pressure angle of a swinging arm obeys the sliding follower’s formula exactly, but the quantity that was a constant offset becomes a number that changes as the arm swings. An arm is an offset that grows with the lift. Its pivot sets where the offset starts, its length sets how fast it grows, and each of those has a price the sliding follower never pays.

A roller on an arm of 50 pivoted on the left, at 35.8° of the riseThe cycloidal programme — rise 20 over 90°, return over 170° — on a prime circle of 40 with a roller of 10, followed by a roller on an arm of 50 whose pivot is on the left of the cam, at (−49.92, 42.79). The pivot angle balances the two strokes, and the cam is turned anticlockwise to 35.8°. The faint arc is the path the roller's centre swings through. The contact normal leans 23.48° from the roller's direction of motion, which is the pressure angle, and off the drawn pitch curve it is 23.48°. At this instant the roller's line of motion passes 2.98 from the cam's centre. Balanced, the arm peaks at 23.48° on both strokes; a sliding follower at its best offset of 5.88 peaks at 21.81°.φ 23.3°pivotsliding, best offset 5.88both peaks21.81°arm 50, pivot on the leftrise peak23.48°return peak23.48°at 190.9° of the returnpressure angle23.33°off the drawn curve23.33°arm's offset now12.23cycloidal, prime circle 40, arm 50, roller 10pressure angle 23.3°
Fig. 1 A roller on an arm of 50, pivoted to the left of a quick-rise cam at the angle that balances its two strokes, stopped where the rise’s pressure angle peaks. The faint arc is the path the roller’s centre swings through.

The offset formula, with the offset let loose

Keep the conventions of the offset essay: the cam turns anticlockwise about the origin, and the roller’s centre rests on the prime circle, radius R0R_0, during the lowest dwell. Carry the roller on an arm of length LL from a fixed pivot QQ, and let the follower’s lift ss be the arc its centre travels, so the arm turns through s/Ls/L as the follower rises.

Two directions at the roller’s centre CC describe everything. Call t^\hat t the direction the centre moves as the lift grows, which is along the arc, perpendicular to the arm. Call p^\hat p that direction turned a quarter turn clockwise. For a sliding follower on the offset essay’s guide these are straight up and straight to the right, and the roller’s centre is (e, R02−e2+s)(e,\ \sqrt{R_0^2 - e^2} + s).

Carrying the offset essay’s derivation through with these two directions, differentiating the pitch curve in the cam’s frame and turning its tangent back into the fixed frame, gives

tan⁡φ=s′−C⋅p^C⋅t^.\tan\varphi = \frac{s' - C\cdot\hat p}{C\cdot\hat t}.

For the slide C⋅p^C\cdot\hat p is ee and C⋅t^C\cdot\hat t is R02−e2+s\sqrt{R_0^2 - e^2} + s, and the formula is the offset essay’s, term for term. For the arm it is the same formula with both terms free to move. C⋅p^C\cdot\hat p is the perpendicular distance from the cam’s centre to the line the roller is moving along at this instant, which is exactly what an offset is. The guide fixes that line. The arm turns it.

The pressure angle is folded into the range from −90° to 90°. A pivot on the other side of the cam swings the arm the other way as the follower lifts, which reverses t^\hat t, and without the fold a quantity that is the same angle would come out on the far side of the circle.

The same angle off the drawn pitch curve, on both sides of the cam

The second route is the one the offset essay used and it uses no pressure-angle formula at all. It places the roller’s centre where the arm puts it, turns that point into the cam’s frame, differences two neighbouring points of the recorded pitch curve, turns the difference back, and reads the curve’s normal against the arm’s own direction of motion. Its only input is where the roller’s centre was put; it knows nothing of s′s' or of the offset term.

Six arms were compared, three pivoted to the left of the cam and three to the right, with lengths from 30 to 150 and a range of pivot angles, every one and a half degrees of the turn. On the quick-rise programme the two routes differ by at most 1.1×10−61.1 \times 10^{-6} degrees, and on the standing programme by 4.7×10−74.7 \times 10^{-7}. With the sign of the offset term slipped, and everything else left as it is, the difference is 91.1°. The derivation is one line longer than the offset essay’s, and the check is what makes that line safe to build on.

An offset with a slope

The offset term is the one to watch, because on a slide it was the whole of the trade.

An arm's offset through a turn, at three arm lengths. The cycloidal programme — rise 20 over 120°, return over 120° — on a prime circle of 40, followed by rollers on arms of 30, 50, 150 pivoted on the left, each at the pivot angle that balances its two strokes. The curves are the distance of the roller's line of motion from the cam's centre, the quantity that plays the part of a sliding follower's offset. On the arm of 30 it is −12.51 on the low dwell and 20.09 at full lift, 1.630 for every unit of lift; on the arm of 50 it is −7.45 on the low dwell and 12.39 at full lift, 0.992 for every unit of lift; on the arm of 150 it is −2.47 on the low dwell and 4.19 at full lift, 0.333 for every unit of lift. The sliding follower's best offset, 0.00, is the same at every angle.
Fig. 2 The distance of the roller’s line of motion from the cam’s centre through a whole turn of the standing programme, for arms of 30, 50 and 150 each at its balancing pivot angle, beside the sliding follower’s best offset, which is nought.

On the standing programme — a cycloidal rise and return of 20 over 120°, with dwells of 30° and 90° — the arm of 50, balanced, has an offset of −7.45 on the low dwell and 12.39 at full lift. It is flat on the dwells, climbs through the rise, and falls back through the return along the same values, because the offset depends only on how far the arm has swung. The arm of 30 runs from −12.51 to 20.09 and the arm of 150 from −2.47 to 4.19.

The growth per unit of lift is close to R/LR/L, with RR the roller’s distance from the cam’s centre, which is about 50 here: 1.63 for the arm of 30, 0.99 for the arm of 50, 0.33 for the arm of 150. That is the geometry of a turning line. As the arm swings by a small angle, the line of motion turns by the same angle about a point roughly RR from the cam’s centre, and its distance from the centre changes by RR times the angle, which is R s/LR\,s/L.

So an arm has two numbers where a slide had one. The pivot’s angle sets the offset’s level, the value it passes through at any given lift, and moving the pivot round shifts the whole ramp up or down as moving a guide shifts ee. The arm’s length sets the ramp’s slope, which no slide has. A long arm is nearly a slide; a short one is an offset that changes by more than the lift over a single stroke.

On mirror-image strokes the slope can only cost

On the standing programme the offset essay found that no offset helps: the rise and the return are mirror images, so every degree taken off one is put on the other, and the best offset is zero, where both strokes peak at 21.22°.

An arm cannot do even that well. The rise sees the ramp climbing and the return sees it falling back, so whatever level the pivot chooses, one stroke meets a negative offset low down and the other a positive offset high up, and both are hurt somewhere.

The pressure angle through a turn, for an arm of 50 and for a sliding follower. The cycloidal programme — rise 20 over 120°, return over 120° — on a prime circle of 40. One curve is a roller on an arm of 50 pivoted on the left at its balancing pivot angle, −10.73°; the other is a sliding roller at its best offset, 0.00. The arm peaks at 23.20° on the rise and 23.20° on the return, against the slide's 21.22° and 21.22°. On the low dwell the arm's follower leans 10.73° and on the high dwell −11.94°, where the slide leans 0.00° and 0.00°. The dashed lines are the usual 30° either way.
Fig. 3 The pressure angle through a turn of the standing programme for a roller on an arm of 50 at its balancing pivot angle, and for a sliding roller at its best offset, with the dwell leans marked.

The arm of 50, at its balancing pivot angle of −10.73°, peaks at 23.20° on both strokes: 1.98° worse than a centred slide. The peaks have also moved. The centred slide’s rise peaks at 55.1° of cam angle, near the middle of the stroke. The arm’s rise peaks at 43.8°, when the follower has risen only 4.91, where the ramp is still negative and adds to the velocity; its return peaks at 201.8°, with the follower still 12.69 up, where the ramp is positive and adds to the return’s velocity instead. The slope has pulled the two peaks apart, and each one now happens where the offset works against its stroke.

The price falls quickly with length, and it falls in a definite way.

Every balanced arm on the standing programme, beside the sliding follower. The programme — rise 20 over 120°, return over 120° — on a prime circle of 40. The first row is a sliding roller at its best offset: both peaks 21.22°. Then rollers on arms of 30, 50, 80, 150, 400, pivoted on the left and on the right, each at its balancing pivot angle. Left, 30: pivot angle −18.22°, both peaks 26.62°, 5.399° against the slide, leaning 18.22° on the low dwell and −19.69° on the high; left, 50: pivot angle −10.73°, both peaks 23.20°, 1.981° against the slide, leaning 10.73° on the low dwell and −11.94° on the high; left, 80: pivot angle −6.66°, both peaks 22.00°, 0.778° against the slide, leaning 6.66° on the low dwell and −7.50° on the high; left, 150: pivot angle −3.54°, both peaks 21.44°, 0.222° against the slide, leaning 3.54° on the low dwell and −4.01° on the high; left, 400: pivot angle −1.32°, both peaks 21.25°, 0.031° against the slide, leaning 1.32° on the low dwell and −1.50° on the high; right, 30: pivot angle 198.22°, both peaks 26.62°, 5.399° against the slide, leaning −18.22° on the low dwell and 19.69° on the high; right, 50: pivot angle 190.73°, both peaks 23.20°, 1.981° against the slide, leaning −10.73° on the low dwell and 11.94° on the high; right, 80: pivot angle 186.66°, both peaks 22.00°, 0.778° against the slide, leaning −6.66° on the low dwell and 7.50° on the high; right, 150: pivot angle 183.54°, both peaks 21.44°, 0.222° against the slide, leaning −3.54° on the low dwell and 4.01° on the high; right, 400: pivot angle 181.32°, both peaks 21.25°, 0.031° against the slide, leaning −1.32° on the low dwell and 1.50° on the high. The last column is the pitch curve's tightest convex radius, measured off the drawn curve, with a roller of 10 to fit inside it; the slide's is 38.86.
Fig. 4 Every arm length on the standing programme, pivoted on the left and on the right, each at its balancing pivot angle: both peaks, the premium over the centred slide, the lean on each dwell, and the pitch curve’s tightest radius.

Arms of 30, 50, 80, 150 and 400 cost 5.40°, 1.98°, 0.78°, 0.22° and 0.031°. Multiplied by the square of the arm’s length those are 4,859, 4,953, 4,976, 4,985 and 4,988, so on this cam the arm’s premium is about 5000/L25000/L^2 degrees, closer to that the longer the arm. A premium that falls as the square of the length is what an effect of the slope should look like: the slope is proportional to 1/L1/L, a balanced pair of strokes is hurt only by the part of it that the pivot cannot centre, and that shows up in the peak at second order.

Pivoting the arm on the right of the cam gives the same numbers to the thousandth of a degree, with the pivot angle reflected. On mirror-image strokes the cam cannot tell the two sides apart, and neither side escapes the slope.

A follower that leans by its own tilt

The table has a column that no slide at zero offset needs, and it is the dwell lean.

On a dwell the follower does not move, so the pressure angle is the angle between the contact normal, which is along the radius to the cam’s centre, and the roller’s direction of motion. For a centred slide those are the same line. For an arm they differ by exactly the arm’s tilt from square to that radius, and at the low dwell that tilt is the pivot angle. The balanced arm of 50 leans 10.73° on the low dwell because its pivot angle is −10.73°, and it leans −11.94° on the high dwell, where the arm has swung further.

That makes the lean predictable before any sweep is run, and it makes the lean a direct cost of balancing. The shorter the arm, the more pivot angle it needs to centre its ramp, and the more it leans: 18.22° for the arm of 30, 3.54° for the arm of 150. A follower clamping a workpiece through a long dwell on the arm of 30 pushes its pivot sideways by nearly a third of the contact force the whole time it waits, where a centred slide pushes straight along its guide.

The quick-rise cam: the pivot does what the offset did

The quick-rise programme is the one the offset essay’s trade was made for: a cycloidal rise of 20 over only 90°, a dwell of 40°, a return over a leisurely 170°, and a dwell of 60°. Centred, a slide peaks at 27.37° on the rise and 15.33° on the return, and an offset of 5.88 balances them at 21.81°.

Two strokes that cross at a pivot angle of −3.20°. The cycloidal programme — rise 20 over 90°, return over 170° — on a prime circle of 40, followed by a roller on an arm of 50 pivoted on the left, with the arm's angle on the low dwell swept 20° either side of the balance. One curve is the rise's peak pressure angle and the other the return's. They cross at −3.200°, where both are 23.48° and the pivot is at (−49.92, 42.79). The dashed line is the sliding follower at its best offset, 21.81°.
Fig. 5 The rise’s and the return’s peak pressure angles for a roller on an arm of 50 pivoted to the left of the quick-rise cam, as the arm’s angle on the low dwell is swept twenty degrees either side of the balance.

The arm’s pivot angle does the same job. Swept from about −23° to 17°, it moves the rise’s peak down and the return’s up, exactly as moving the guide did, and the two cross at a pivot angle of −3.20°, where both are 23.48°. The pivot is then at (−49.92, 42.79), a little below the level of the roller and almost directly to its left.

The arm is 1.67° worse than the best slide at that length, a smaller premium than on the standing programme, and the offset at the balance runs from −2.23 on the low dwell to 17.44 at full lift. The rise peaks low on that ramp and the return high on it, which is the right way round for this cam: the fast rise needs a positive offset early, and the idle return has margin to spend when the ramp reaches it.

The wrong side costs under a degree

The offset essay’s warning about direction was sharp. The offset helps the stroke whose velocity has its sign, so the same offset on the other side of the guide, or the same cam run backwards, charges the fast stroke instead: 32.85° on the quick-rise cam against 21.81°, eleven degrees for a sign.

Every balanced arm on the quick-rise programme, beside the sliding follower. The programme — rise 20 over 90°, return over 170° — on a prime circle of 40. The first row is a sliding roller at its best offset: both peaks 21.81°. Then rollers on arms of 30, 50, 80, 150, 400, pivoted on the left and on the right, each at its balancing pivot angle. Left, 30: pivot angle −12.31°, both peaks 27.00°, 5.190° against the slide, leaning 12.31° on the low dwell and −23.65° on the high; left, 50: pivot angle −3.20°, both peaks 23.48°, 1.665° against the slide, leaning 3.20° on the low dwell and −16.98° on the high; left, 80: pivot angle 1.50°, both peaks 22.31°, 0.501° against the slide, leaning −1.50° on the low dwell and −12.94° on the high; left, 150: pivot angle 4.88°, both peaks 21.86°, 0.048° against the slide, leaning −4.88° on the low dwell and −9.62° on the high; left, 400: pivot angle 7.15°, both peaks 21.77°, −0.039° against the slide, leaning −7.15° on the low dwell and −7.15° on the high; right, 30: pivot angle 203.88°, both peaks 27.70°, 5.889° against the slide, leaning −23.88° on the low dwell and 15.89° on the high; right, 50: pivot angle 197.93°, both peaks 24.28°, 2.464° against the slide, leaning −17.93° on the low dwell and 7.13° on the high; right, 80: pivot angle 194.53°, both peaks 22.94°, 1.126° against the slide, leaning −14.53° on the low dwell and 2.24° on the high; right, 150: pivot angle 191.78°, both peaks 22.23°, 0.416° against the slide, leaning −11.78° on the low dwell and −1.49° on the high; right, 400: pivot angle 189.73°, both peaks 21.92°, 0.104° against the slide, leaning −9.73° on the low dwell and −4.10° on the high. The last column is the pitch curve's tightest convex radius, measured off the drawn curve, with a roller of 10 to fit inside it; the slide's is 30.51.
Fig. 6 Every arm length on the quick-rise programme, pivoted on the left and on the right, each at its balancing pivot angle, beside the sliding follower at its best offset.

An arm pivoted on the right of the quick-rise cam has its ramp running the other way, falling as the follower lifts. That is the arm’s version of the wrong-side offset. But an arm on the right still has its own pivot angle to choose, and choosing it moves the ramp’s level back to where the fast stroke wants it. Rebalanced, the arm of 50 on the right peaks at 24.28° against 23.48° on the left: 0.80° for the wrong side, not eleven. At 30 the difference is 0.70°, at 80 it is 0.63°, at 150 it is 0.37° and at 400 it is 0.14°.

What is left is only the slope’s sign. On the right the ramp climbs on the return and falls on the rise, which is the wrong way round for a fast rise, and the pivot angle can centre the ramp but cannot turn it over. The price is small because the level was always most of the effect, and the level is something both sides can set.

The leans carry the rest of the bill. The balanced right-hand arm of 50 leans −17.93° on the low dwell, against 3.20° for the left-hand one, because it needs a much larger pivot angle to bring its ramp to the level the rise wants.

A long arm, and hundredths of a degree

One entry in the quick-rise table is worth singling out, because it is the only place an arm does better than a slide.

The left-hand arm of 400 peaks at 21.774° on both strokes. The best slide peaks at 21.813°. The long arm beats the best offset by 0.039°. Its offset runs gently from 4.98 on the low dwell to 7.45 at full lift, so the fast rise, which peaks lower, meets slightly less offset than the slow return, which peaks higher and has margin to give up. A slide can offer only one offset to both. The arm of 250 does better than the slide by 0.030° and the arm of 1,000 by 0.024°, so the gain has a best length and fades on either side of it.

That result needed care to establish. Between two designs whose peaks differ by four hundredths of a degree, a grid of cam angles a tenth of a degree apart can mis-rank them, so every peak in the tables is found on a half-degree grid and then refined by a golden-section search on the bracket round it, and the slide’s best offset was refined the same way before the two were compared.

It is also the smallest effect in this essay, and it should be read at its size. The slope of an arm is a second design variable that a slide does not have, and on a cam whose strokes are unequal it can be spent to advantage. What it buys at best, on this cam, is less than a twentieth of a degree, and every arm of 150 or shorter on the same cam costs more than it could ever buy: the arm of 150 already pays 0.048°.

What the arm does to the metal

The last column of both tables is the pitch curve’s tightest convex radius, measured by fitting circles through neighbouring points of the curve the arm’s roller actually traces. The surface that is cut is that curve moved inward along its own normals by the roller’s radius, as for any roller, so where the pitch curve is convex the cut surface’s radius is this one less the roller’s. It sets whether the cam can be cut with a given roller, and the offset essay found an offset barely moved it.

An arm moves it a little more, and in a direction that depends on the side. On the quick-rise cam the best slide’s tightest radius is 30.51; the balanced left-hand arms give 28.24 at 30 and 29.13 at 50, slightly tighter, and the right-hand arms 32.90 and 31.99, slightly looser. On the standing programme the arms give 36.84 to 38.73. With a roller of 10 every one of these cams can be cut, with at least 18 to spare. The arm is a tool for the pressure angle, and like the offset it should be re-checked against curvature rather than expected to help or hurt it.

One programme, and the arm’s own transmission

The arm’s own transmission is not computed. The pressure angle measures how the cam pushes the roller; an arm also has to deliver its motion to whatever it drives, and that has a transmission angle of its own which a slide does not.

The 5000/L25000/L^2 law is measured on one programme, one prime circle and one lift, under one motion law. The constant will scale with the lift and the prime circle, and with the law, which sets the peak velocity each stroke asks for and which the comparison of motion laws shows varies by more than half between the common ones. Nothing here derives it. The quick-rise programme’s premiums do not follow a single power of the length at all, so the law is a property of mirror-image strokes rather than of arms.

Only the pivot angle was balanced. The arm’s length was stepped through five values, not searched, and a design that chose both together would do at least as well as the best row of each table.

The lean is computed as a direction, not a load. Whether a lean of 18° on a long dwell matters depends on the force behind it and on the pivot’s bearing, and neither is modelled.

What comes next: an oscillating flat face

An oscillating flat face. A flat face removes the pressure angle and replaces it with convexity. On a swinging arm the face’s contact moves along the arm as well as across it, the convexity condition acquires the pivot’s distance, and the contact offset that the flat-face essay found equal to ds/dθds/d\theta becomes a quantity with the arm’s length in it. Whether a pivot can make a cam convex that a sliding flat face could not is the flat face’s version of this essay’s question.

A roller yoke on an arm. A cam that holds its follower both ways closed two faces round a cam and found the programme had to be its own reflection. Two rollers on one rocking arm, either side of the cam, measure across it along a line that turns with the arm, and whether the reflection rule survives, loosens or tightens is a question this essay’s formula is set up to answer.

The balancing pivot as a shape. The scaling essay sorts which numbers of a cam scale with its size. The balancing pivot angle is an angle, the arm length a length, and the premium’s constant a mixture of both; whether the balancing angle for a given ratio of arm to prime circle is a property of the programme alone would turn this essay’s tables into something a designer could reuse on a cam of any size.

Where this goes next

A roller offset by 5.88, at 42.3° of the rise. The cycloidal programme — rise 20 over 90°, return over 170° — on a prime circle of 40 with a roller of 10, drawn with the follower fixed above and the cam turned anticlockwise to 42.3°. The follower's line of travel is 5.88 to the right of the cam's centre. The contact normal leans 21.81° from that line, which is the pressure angle, and measured off the drawn pitch curve it is 21.81°. At this offset the peak on the rise is 21.81° and on the return 21.81°; centred, they are 27.37° and 15.33°. An offset trades the rise for the return Moving a roller follower's line of travel off the cam's centre lowers the pressure angle on one stroke by raising it on the other. When the rise and the return are mirror images the best offset is zero. When the cam rises in 90° and returns over 170°, an offset of 5.88 takes the worse stroke from 27.37° to 21.81° by handing the return 6.48° it did not need. A 34-unit roller on a 20-unit base: the cam cannot be made. The outer curve is the pitch curve — where the roller's centre has to go, which is just the base radius plus the follower's displacement, plotted in polar coordinates. The inner curve is what has to be cut: the pitch curve offset inward by the roller's radius along its own normal. Where the pitch curve bends more tightly than the roller is wide, that offset crosses itself, and the metal the profile needs has already been removed. The tightest radius of curvature here is 19.43 at 85°, against a roller of 34 — so the cam is undercut by 14.57 and no amount of care in manufacture recovers it. The pressure angle, meanwhile, peaks at a perfectly comfortable 21.2°: this cam fails a test it never looked like failing. The cam that cannot be cut A cam has to be big enough for two reasons, and they are not the same reason. One is that the follower will jam in its guide if the pressure angle is steep. The other is that the roller will gouge the profile if the curvature is tight — and there is a combination where the pressure angle is comfortable and the cam still cannot be manufactured at all. A flat face on a base circle of 30, where the profile bends tightest. The cycloidal programme — rise 10 over 120°, return over 120° — cut for a flat-faced follower on a base circle of 30, turned to 212.4°, where the profile's radius of curvature is smallest. The contact sits −5.37 from the follower's axis, which is ds/dθ, and the dashed circle is the one that fits the profile there: R₀ + s + s″ = 24.668, and 24.668 fitted through the drawn curve. The face's pressure angle is zero at every angle. What it needs instead is for that radius to stay positive, and it does so on every base circle above 5.332. A flat face asks for a convex cam A flat-faced follower is pushed at right angles to its face, so its pressure angle is zero at every angle and the constraint that sizes a roller's cam disappears. What replaces it is convexity: the profile's radius at the contact is R₀ + s + s″, and a base circle below 5.332 leaves a cycloidal cam with a hollow the face cannot reach — held 0.446 high on a base of 2.

What this makes readable

Essays that name this one as a prerequisite.

About the same objects

Not linked from either essay — found by the objects both name.

What links here

Essays that link to this one from their own argument.

The objects this essay names

Each one links to every other essay that touches it.

CamDwellMinimaxOffset followerOscillating followerPitch curvePressure anglePrime circleRoller follower