A Tusi Couple is a planar mechanism where a small circle rolls without slipping inside a fixed circle exactly twice its diameter, forcing every point on the rolling circle to trace a perfectly straight line through the centre of the larger circle. It solves the classic problem of converting rotary motion into pure reciprocating linear motion without using prismatic guides, slots, or sliders. The geometry produces zero side-load on the output, which is why it shows up in precision optics stages, scanning probes, and historical astronomical models. A 50 mm rolling circle inside a 100 mm fixed ring delivers exactly 100 mm of stroke.
Tusi Couple Interactive Calculator
Vary the fixed ring and rolling circle diameters to size the ideal straight-line stroke and see how ratio error creates side deviation.
Equation Used
The Tusi couple is exact when the fixed ring diameter is twice the rolling circle diameter. With a point on the rolling circle rim, the straight-line stroke is S = 2D_roll = D_fixed = 4r. The side-drift value samples the hypocycloid equation to show how a non-2:1 diameter ratio opens the trace into a curve.
- Rolling circle contacts the inside of the fixed ring without slip.
- Output point is on the rolling circle circumference.
- Pure straight-line motion requires the fixed diameter to be exactly twice the rolling diameter.
- Backlash, bearing skew, and tooth errors are ignored.
How the Tusi Couple Actually Works
The Tusi Couple — sometimes called the two-circle theorem or Cardan circles in older texts — works on a single geometric fact: when a circle of radius r rolls without slipping inside a circle of radius 2r, any fixed point on the smaller circle's circumference traces a straight line that passes through the centre of the larger circle. This is a degenerate hypocycloid, the limiting case where the curve collapses from a star shape into a line. The stroke length equals the diameter of the larger circle. Run the input shaft at a constant angular velocity and the output point oscillates with pure simple harmonic motion — no slider, no prismatic joint, no linear bearing.
Why this geometry works comes down to rolling constraint. As the small circle rotates by angle 2θ relative to its own centre, its centre orbits the fixed circle by angle θ in the opposite sense. The two angular motions cancel along one axis and add along the other, leaving a point on the rim moving along a single diameter. If the diameter ratio drifts off 2:1 — say the rolling gear has 49 teeth instead of 50 in a 100-tooth ring — the trace opens into a narrow ellipse or a pinched astroid. You will see this immediately as the output stylus walking sideways instead of staying on its line.
Failure modes are straightforward. Tooth backlash in a geared implementation lets the rolling circle hunt across the rolling contact, and the trace develops a fuzzy width instead of a clean line. Insufficient preload on the planet bearing lets the small circle skew, tipping the traced point off-axis. And if you use friction rolling instead of gear teeth, the slightest slip — even 0.1% — accumulates over revolutions and the output drifts off the diameter. For anything beyond a demonstration toy, you want internal gear teeth or a positive timing belt arrangement, not pure rolling friction.
Key Components
- Fixed outer ring (radius 2r): Provides the stationary internal rolling surface or internal gear teeth. Concentricity to the input shaft must hold to within roughly 0.02 mm for a 100 mm ring or the trace bows. In a geared version this is a fixed internal ring gear with module typically 0.5-1.0 for benchtop builds.
- Rolling inner circle (radius r): The planet that rolls inside the ring. Its diameter must be exactly half the ring's pitch diameter — a 50.0 mm planet inside a 100.0 mm ring, not 49.8, not 50.2. Tooth count ratio 1:2 is mandatory for a geared build.
- Carrier arm: Holds the planet's axis at radius r from the central axis and rotates at the input speed. The arm length must equal r within ±0.05 mm; any error shows up directly as a sideways wobble in the output trace.
- Output point (stylus or pin): A pin fixed on the planet's rim at radius r from the planet's centre. This is the point that traces the straight line. Its radial position relative to the planet centre must equal r exactly; if it sits at 0.95r the trace becomes a thin ellipse roughly 5 mm wide on a 100 mm stroke.
- Input shaft and bearing: Drives the carrier arm. Runout under 0.01 mm TIR keeps the trace clean. A precision angular contact bearing pair is standard on optics-grade builds.
Industries That Rely on the Tusi Couple
The Tusi Couple shows up wherever a designer needs reciprocating linear motion without a prismatic slider — usually because side loads, friction, or contamination on a slideway are unacceptable. It also appears in historical and educational contexts because the mathematics is elegant and the demonstration is visually striking. Most modern uses are precision-driven, not power-driven; the mechanism is excellent at low force and high accuracy and poor at high force and high speed.
- Astronomy (historical): Nasir al-Din al-Tusi's 13th-century planetary models in the Tusi-couple-driven Maragha observatory work — used to reproduce linear oscillation within a system of nested rotations without invoking Ptolemy's equant.
- Precision optics: Diametral scanning stages in interferometric flatness testers, where a probe must travel a clean 50 mm line across a reference flat with no side-load on the probe tip.
- Educational kinematics: MIT's Hart's Inversor and Tusi Couple demonstration models in the Reuleaux Collection at Cornell — used to teach degenerate hypocycloid geometry.
- Medical imaging: Tusi-coupled scanning heads in some early ultrasound transducer drives, where the transducer face needs to translate without rotation.
- Animatronics and kinetic art: Reciprocating sculptural elements where a visible exposed slider would spoil the visual effect — Theo Jansen-style installations occasionally use Tusi geometry for clean linear motion of decorative arms.
- Watchmaking and complications: Linear retrograde indicators in high-end horology, where a hand must sweep along a straight scale and snap back, driven from a continuously rotating arbor.
The Formula Behind the Tusi Couple
The core formula gives you the position of the output point along its straight-line path as a function of input shaft angle. At the low end of the typical operating range — say 5-15 RPM for an optics scanner — the output moves slowly and any geometric error in the 2:1 ratio is plainly visible as a curved trace. At the nominal range, 30-60 RPM for a benchtop demonstration, the motion looks smooth and harmonic. At the high end, above 200 RPM, planet bearing dynamics and gear-mesh excitation start to add high-frequency wobble that contaminates the line. The sweet spot is wherever your bearing preload, gear quality, and dynamic balance let you run without that wobble showing up in your output.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| x(θ) | Position of output point along its straight-line path, measured from the centre of the fixed ring | mm | in |
| r | Radius of the rolling inner circle (half the radius of the fixed outer ring) | mm | in |
| θ | Rotation angle of the carrier arm about the central axis | rad | rad |
| Stroke | Total peak-to-peak travel of the output point, equal to 4r (= 2 × diameter of rolling circle) | mm | in |
Worked Example: Tusi Couple in a semiconductor wafer-inspection scanning stage
A semiconductor metrology lab in Hsinchu is building a Tusi-coupled diametral scanning stage to sweep a confocal probe across a 200 mm wafer for thickness mapping. They want a 100 mm peak-to-peak stroke at the wafer surface, driven by a brushless DC motor running the carrier arm. Target nominal carrier speed is 60 RPM. The rolling planet is a 50.0 mm pitch-diameter internal-mesh gear inside a 100.0 mm ring gear, both module 0.5, AGMA Q10.
Given
- Stroke target = 100 mm
- r (rolling circle radius) = 25 mm
- Ring radius (2r) = 50 mm
- Carrier RPM (nominal) = 60 RPM
- Carrier RPM (low end) = 15 RPM
- Carrier RPM (high end) = 180 RPM
Solution
Step 1 — confirm the stroke from the geometry. Stroke equals 4r (twice the diameter of the rolling circle, which equals the diameter of the ring):
Step 2 — at nominal 60 RPM carrier speed, compute peak output velocity. Since x(θ) = 2r·cos(θ) and θ = ω·t, the peak velocity is vpeak = 2r·ω.
vpeak,nom = 2 × 25 × 6.283 = 314 mm/s
Step 3 — at the low end of the typical scanning range, 15 RPM, the probe creeps across the wafer:
vpeak,low = 2 × 25 × 1.571 = 78.5 mm/s
This is the regime you want for high-resolution thickness maps — slow enough that a 1 kHz confocal probe gives you one sample every 78 µm of travel. Step 4 — at the high end of the range, 180 RPM, the theoretical peak velocity is:
vpeak,high = 2 × 25 × 18.85 = 942 mm/s
In theory the geometry supports it. In practice, AGMA Q10 internal mesh gears running at 180 RPM with the planet's centre orbiting at 1.5 Hz start to excite tooth-mesh harmonics around 150-200 Hz, and you will see the trace pick up a 5-10 µm sideways wobble — fatal for a confocal probe holding 1 µm thickness resolution. The sweet spot for this build is 30-90 RPM.
Result
Nominal stroke comes out at exactly 100 mm with a peak output velocity of 314 mm/s at 60 RPM. In practice the probe sweeps across the wafer in roughly half a second per stroke and the trace looks like a clean horizontal line under a dial indicator. At 15 RPM you get 78.5 mm/s — slow, smooth, and high-resolution; at 180 RPM the theoretical 942 mm/s is unreachable in a real Q10 gear stage because mesh harmonics destroy the line. If you measure a stroke of 99.4 mm instead of 100.0 mm, the most common causes are: (1) the rolling planet's pitch diameter machined undersize by 0.15 mm, which directly halves into a 0.3 mm centre offset and a 0.6 mm stroke loss, (2) carrier arm length error — if your arm holds the planet axis at 24.85 mm instead of 25.00 mm the entire trace shrinks proportionally, or (3) tangential backlash in the internal gear mesh letting the planet hunt by 50-100 µm at each direction reversal, which shows up as a fuzzy stroke endpoint rather than a hard turnaround.
When to Use a Tusi Couple and When Not To
Designers usually reach for a Tusi Couple when a slider-crank or a Scotch yoke would be the obvious choice but side-load, friction, or visible-slider problems rule them out. Here is how the three compare on the engineering dimensions that actually drive the decision.
| Property | Tusi Couple | Scotch Yoke | Slider-Crank |
|---|---|---|---|
| Output motion purity | Pure SHM, zero side-load on output point | Pure SHM but yoke slot loads sideways on slider | Near-SHM with second-harmonic distortion that grows with rod/crank ratio |
| Typical operating speed | 10-200 RPM (gear-mesh limited) | Up to 1500 RPM in compact builds | Up to 6000+ RPM (used in IC engines) |
| Load capacity | Low — limited by planet gear tooth load, typical 5-50 N output | Medium — yoke and slider take real force, 100-1000 N typical | High — can transmit kW-class power |
| Positioning accuracy | ±5-20 µm achievable with Q10 gears and preloaded bearings | ±50-100 µm typical (slot wear) | ±100 µm or worse without external guide |
| Mechanical complexity | Internal ring gear plus planet plus carrier — three precision parts | Two parts (yoke and crank pin) plus a slider | Three parts (crank, rod, piston) plus a guide |
| Cost (precision benchtop scale) | High — internal gear is the cost driver | Low — machinable in any shop | Low to medium |
| Best application fit | Precision low-force scanners, optics, demonstration | Compressors, valve drives, indexing | Engines, pumps, high-power reciprocators |
Frequently Asked Questions About Tusi Couple
You are almost certainly seeing carrier arm length error or output-pin radial error, not gear ratio error. The 2:1 ratio is enforced by tooth count, so it is either right or off by a full tooth — there is no in-between. But the carrier arm length and the radial position of the output pin on the planet are continuous machining dimensions and any error there shows up as a bowed trace.
Quick check: pull the planet, measure the carrier arm pivot-to-pivot length with calipers, and measure the output pin centre-to-planet-centre distance. Both numbers must equal r within roughly 0.05 mm for a 50 mm ring build. A 0.1 mm bow in the middle of the trace usually traces to one of those two dimensions being off by 0.08-0.12 mm.
Friction rolling fails on accumulated slip. Even a 0.1% slip per revolution — invisible by eye — adds up to a measurable trace drift after 20-30 cycles, and the output point walks off the diameter line. There is no restoring force in pure rolling friction; once you slip, the geometry is permanently shifted.
For toy demonstrations running for a minute or two it works fine. For anything that needs to hold its line over hours of operation, you need positive engagement: an internal ring gear with a planet pinion, or a timing belt wrapped around the planet and anchored to the ring. Both eliminate slip entirely.
Almost always the gear mesh and dynamic balance, not the bearings. The carrier arm is an offset rotating mass — the planet sits at radius r from the central axis — so even a 50 g planet at 25 mm radius generates a centrifugal force of about 5 N at 60 RPM and 50 N at 200 RPM. That force pulses the carrier bearings at the carrier rotation frequency.
Combine that with tooth-mesh excitation at (planet tooth count × planet RPM) and you get a forcing spectrum that excites any structural resonance in the mounting frame between roughly 50 and 500 Hz. Most builds hit a wall around 200-300 RPM where the trace gets visibly fuzzy. The fix is counterweighting the carrier and stiffening the mount, not changing the bearings.
Tusi Couple if you need continuous rotary input and oscillating output — for example a motor-driven scanning probe sweeping back and forth indefinitely. The mechanism handles unlimited revolutions naturally.
Scott-Russell if you need a single stroke driven from a lever or a finite-range input. Scott-Russell is mechanically simpler (four pivots, no gears) and cheaper to build to high accuracy because pivot bearings are easier than precision internal gears. But it cannot accept continuous rotation — the input arm sweeps a limited arc. The decision usually comes down to whether your input is a crank/motor (Tusi wins) or a lead screw/lever (Scott-Russell wins).
You have hit a structural resonance. The carrier arm carrying the planet is an unbalanced rotating mass, and at one specific carrier RPM the centrifugal forcing frequency matches a natural mode of your mounting bracket or base plate. Below and above that RPM the vibration disappears.
Two fixes. First, add a counterweight to the carrier arm 180° opposite the planet, sized to cancel the planet's mass moment about the central axis. This kills the forcing at the source. Second, if you cannot counterweight, stiffen the mounting — most resonances on benchtop builds are in the 80-200 Hz range and come from a thin aluminium base plate. A 12 mm steel base typically pushes the resonance above your operating range.
You can, but the planet gear teeth are the bottleneck. The output force at the pin is the tangential force the planet applies through its rim, which is the gear-mesh tangential load divided by the radius ratio. For a module 0.5 internal mesh in steel, allowable tangential tooth load is roughly 30-80 N depending on face width and material, and that translates to similar peak output force at the pin.
If you need more than about 100 N of reciprocating force, switch to a Scotch yoke or slider-crank — they handle kN-class loads in the same envelope. The Tusi Couple is a precision mechanism, not a power mechanism. Trying to push it past its tooth-load limit chips teeth, opens backlash, and destroys the trace accuracy you went to the Tusi Couple for in the first place.
References & Further Reading
- Wikipedia contributors. Tusi couple. Wikipedia
Building or designing a mechanism like this?
Explore the precision-engineered motion control hardware used by mechanical engineers, makers, and product designers.
