Hart's Inversor is a six-bar planar linkage that converts circular motion at one tracing point into exact linear motion at another tracing point — no approximation, no slot, no slide. The defining component is the antiparallelogram (or crossed four-bar) frame built from four bars in a specific 2:1 length ratio, which forces three collinear points to remain collinear through the full range of motion. Harry Hart published it in 1875 as a simpler alternative to the eight-bar Peaucellier-Lipkin cell. It still appears today in straight-line tracing rigs, classroom kinematics demos, and precision-positioning research at places like the Cornell KMODDL collection.
Hart's Inversor Interactive Calculator
Vary the long and short antiparallelogram bar lengths to see the inversion constant, 2:1 ratio error, tolerance margin, and animated inverse tracing geometry.
Equation Used
The Hart inversor antiparallelogram defines an inversion constant K from the long and short bar lengths: K = L^2 - l^2. The inverse points satisfy OP x OQ = K. This calculator also checks the short bar against the nominal 2:1 Hart ratio, where l should equal L/2.
- Planar rigid-bar Hart inversor geometry.
- Nominal Hart antiparallelogram uses a 2:1 long-to-short bar ratio.
- Bar lengths are center-to-center pin distances.
- Positive tolerance margin means the short bar is within the selected length tolerance.
Inside the Hart's Inversors
Hart's Inversor takes the same mathematical idea as the Peaucellier-Lipkin cell — circular inversion through a fixed point — but builds it from six bars instead of eight. The trick is the antiparallelogram, also called a crossed four-bar. You take four bars: two long ones of equal length, two short ones of equal length, and connect them so the long bars cross over each other. As the linkage flexes, any straight line drawn across three specific points on the four bars stays straight, and those three points satisfy the inversion property OP × OQ = constant. Add a fifth bar that constrains one of those points to move on a circle through the fixed pivot O, and the third point is geometrically forced to trace a perfect straight line.
The geometry is unforgiving. The four antiparallelogram bars must obey the length ratio exactly — if the long bars are 100 mm centre-to-centre, the short bars must be a specific matching value derived from the inversion constant, typically held to ±0.05 mm in a precision build. Get the lengths wrong by even 0.5% and the output point traces a shallow arc instead of a line. The same is true for the pivot holes. A 6.0 mm bore that drifts to 6.1 mm gives you radial slop that shows up as a wavy output trace, not a clean straight line.
Common failure modes are predictable. Worn pivot bushings let the bars cock out of plane and the trace develops a vertical jitter. A bent bar — even by 0.2 mm over 100 mm length — biases the trace into a banana shape. And if the driving crank that constrains the input point isn't centred on the geometric pivot O, the inversion property breaks and you get a spiral instead of a line. Build it right and it runs essentially forever. Build it sloppy and it never works at all.
Key Components
- Antiparallelogram (crossed four-bar): Four rigid bars — two long (L), two short (l) — pinned so the long bars cross. This is the heart of the mechanism. The crossed geometry forces three collinear points (O, P, Q) on the bars to satisfy OP × OQ = L² − l², the inversion constant. Bar-length tolerance must be ±0.05 mm or better in a 100 mm build.
- Fixed pivot O: The geometric centre of inversion, anchored to the frame. Point P moves on a circle through O (driven by the constraining link), which forces point Q to trace a straight line perpendicular to the line through O and the circle's centre. If O drifts laterally by 0.1 mm during operation, the output trace bows visibly.
- Constraining link: A fifth bar pinned between the frame and point P, length equal to the radius of the input circle through O. This is what reduces the problem from generic inversion to straight-line generation. Its pivot must lie exactly on the circle's centre — not approximately, exactly.
- Tracing point Q: The output point on the antiparallelogram that draws the straight line. Stylus or tool mounts here. Travel range is typically 30–50% of the long-bar length L; outside that range the linkage approaches singularity and the trace degrades.
- Pivot pins and bushings: Five pin joints transmit all the geometry. Bronze or needle bushings with ≤0.02 mm radial play are standard for precision work. Plain steel pins running in clearance holes will give you a 0.3 mm wavy trace — fine for a teaching model, useless for a measuring rig.
Where the Hart's Inversors Is Used
You won't find Hart's Inversor on a factory floor today — modern linear guides and ballscrews have replaced it for most production tasks. But it still shows up wherever a designer needs guaranteed exact straight-line motion from a purely revolute-jointed mechanism, with no sliding contact, no rails, and no lubrication-sensitive surfaces. That makes it useful in vacuum chambers, cryogenic rigs, classroom kinematics demonstrations, and historical-instrument restoration. The reason it persists is simple: it has fewer parts than Peaucellier, fits in a flatter envelope, and uses only revolute joints which are the cleanest joint type to seal, lubricate, and machine.
- Education / kinematics teaching: Cornell University's KMODDL (Kinematic Models for Design Digital Library) collection includes a working Hart's Inversor model used to demonstrate exact straight-line generation alongside the Peaucellier-Lipkin cell.
- Vacuum and cryogenic instrumentation: Used in research-grade sample translators where a sliding bearing would outgas or freeze — for example linear feedthroughs in low-temperature physics rigs at university labs that need a few millimetres of dead-straight travel without rails.
- Heritage instrument restoration: Restorers of 19th-century drawing and surveying instruments rebuild Hart linkages in pantograph-style tracing rigs originally sold by makers like W. F. Stanley of London.
- Precision metrology fixtures: Bench-top straightness-reference fixtures where the guide itself must be the reference — no slide can be used because the slide's straightness is what's being checked.
- Robotics research: Compliant or 3D-printed straight-line manipulators where designers want to avoid prismatic joints. Used in published research demonstrators of single-DOF planar arms.
- Museum and demonstration models: The Science Museum in London and the Tekniska Museet in Stockholm both display working linkage models in this family for public education on 19th-century mathematical mechanisms.
The Formula Behind the Hart's Inversors
The defining equation of Hart's Inversor is the inversion relationship between the input and output tracing points. Practically you use it to size the bars: pick the long-bar length L based on your travel range, pick the short-bar length l based on the inversion constant you want, and the output trace length follows. At the low end of the typical operating range — say the tracing point at 20% of L from O — you're well inside the linkage's clean zone and the straight-line error is negligible (sub-micron in a 100 mm build). At nominal travel (around 35% of L) you hit the sweet spot of stroke vs accuracy. Push to the high end (50%+ of L) and you approach geometric singularity, where small bar-length errors get amplified into visible deviation from straight.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| OP | Distance from fixed pivot O to input tracing point P (constrained to move on a circle) | mm | in |
| OQ | Distance from fixed pivot O to output tracing point Q (the straight-line tracer) | mm | in |
| L | Length of each long bar in the antiparallelogram (centre-to-centre between pivot holes) | mm | in |
| l | Length of each short bar in the antiparallelogram (centre-to-centre between pivot holes) | mm | in |
| k | Inversion constant, k = L² − l², characterising the mechanism | mm² | in² |
Worked Example: Hart's Inversors in a precision draughting-instrument restoration
A scientific-instrument workshop in Edinburgh is restoring a Victorian Hart's straight-line linkage that originally sat inside a brass-cased planimeter from the 1880s. The long bars measure L = 120 mm and the short bars measure l = 60 mm centre-to-centre. The restorers want to know what travel they get on the output tracing point Q when the input point P moves on a 30 mm radius circle, and where the trace starts to degrade.
Given
- L = 120 mm
- l = 60 mm
- rP = 30 mm (radius of input circle through O)
- OPnominal = 60 mm (P at 0.5 × L from O at nominal position)
Solution
Step 1 — calculate the inversion constant k from the bar lengths:
Step 2 — at the nominal mid-stroke position, OP = 60 mm. Solve for OQ:
Step 3 — at the low end of the input travel, P sits closer to O. Take OPlow = 45 mm (P moved 15 mm along its circle toward O). Output is:
This is the long end of the trace. Q sweeps outward as P sweeps inward — that's the inversion at work. The motion here is in the clean zone: bar-length errors of ±0.05 mm produce sub-0.01 mm deviation from a straight line.
Step 4 — at the high end of input travel, OPhigh = 75 mm:
Total Q-travel from low to high: 240 − 144 = 96 mm of straight-line stroke. But notice — push OP much above 80 mm and the antiparallelogram approaches its singular configuration (long bars near collinear). The trace degrades from a clean line to a shallow arc with visible deviation around 0.3 mm or more, which is why Victorian instrument makers limited the working stroke to roughly 35% of L on either side of nominal.
Result
At nominal mid-stroke the output point Q sits 180 mm from pivot O, with about 96 mm of usable straight-line travel between the low and high stroke ends. That 96 mm stroke is the working range a draughtsman would actually use — wide enough to draw a useful line on a planimeter wheel, narrow enough that straightness error stays below 0.02 mm with properly-machined bushings. Push beyond it and the trace bows visibly within one cycle. If the restored linkage traces a banana-shaped curve instead of a line, the three most likely causes are: (1) bar-length error greater than 0.1 mm on either L or l, which throws off the inversion constant and bends the trace; (2) the constraining link's frame pivot mounted off the true centre of the input circle, which converts the line into a spiral; or (3) one of the antiparallelogram bars sprung or warped during reassembly, putting a permanent bias into the geometry that no amount of pivot tightening will remove.
Hart's Inversors vs Alternatives
Hart's Inversor competes with two other approaches whenever a designer needs straight-line motion from a rotary input: the Peaucellier-Lipkin cell (the original exact straight-line linkage) and the Watt or Chebyshev approximate straight-line linkages. The choice comes down to whether you need mathematical exactness, how many parts you can tolerate, and how much travel you need.
| Property | Hart's Inversor | Peaucellier-Lipkin Cell | Watt's Linkage (approximate) |
|---|---|---|---|
| Number of links | 6 bars, 5 revolute joints | 8 bars, 6 revolute joints | 3 bars, 4 revolute joints |
| Straight-line accuracy | Mathematically exact (limited only by bar tolerance) | Mathematically exact (limited only by bar tolerance) | Approximate — typical deviation 0.1–1% of stroke |
| Useful stroke as % of largest bar | ~35% of L either side of nominal | ~40% of long-bar length | ~25% of coupler length |
| Build complexity / cost | Medium — 4 critical bar-length tolerances | High — 8 bars, 2 length classes, very fussy | Low — 3 bars, no exactness constraint |
| Sensitivity to pivot wear | High — 0.05 mm play causes visible error | High — same sensitivity as Hart | Medium — error budget already non-zero so wear contributes proportionally |
| Best application fit | Precision rigs needing exact straight line in flat envelope | Historical / educational; physics demos | Industrial linkages where ~0.5% straightness is acceptable (steam engine valve gear, vehicle suspension) |
| Lifespan / reliability | Excellent — only revolute joints, no sliding wear | Excellent — only revolute joints | Excellent — fewest joints, longest service life |
Frequently Asked Questions About Hart's Inversors
Bar length is only one of three geometric inputs. The other two are the position of the fixed pivot O and the centre of the input circle that constrains point P. If the constraining-link pivot is off the true geometric centre of the input circle by even 0.1 mm, the inversion property still holds locally but the locus of Q is no longer a line — it becomes a high-aspect-ratio S or shallow figure-of-eight.
Diagnostic check: drive the linkage by hand through its full range and mark Q's position at five points. If the deviation from straight is symmetric about the midpoint, your bars are off. If it's asymmetric (S-shaped), it's almost always a pivot-location error.
Mathematically yes — the inversion equation has no scale dependence. Practically, no. The mechanism's accuracy depends on bar-length and pivot-position tolerances being a small fraction of the bar length itself. At a 5 mm bar length you'd need sub-micron pivot tolerances and bushings with under 1 µm radial play, which is achievable in research MEMS but not in a workshop build.
Below about 30 mm long-bar length, flexure-based compliant straight-line mechanisms (like a double-parallelogram flexure) outperform Hart on both cost and achievable straightness because they eliminate pivot clearance altogether.
If you need a flat, slim envelope and you want to minimise part count, choose Hart — six bars and five joints versus Peaucellier's eight bars and six joints. If you need maximum stroke for a given largest-bar length, Peaucellier wins by roughly 15%. If you're building a teaching model where students need to see how the geometry works, Peaucellier is more visually obvious because the rhombus is easier to recognise than the antiparallelogram.
For precision applications the two are kinematically equivalent in accuracy — the deciding factors are packaging and assembly difficulty, and Hart usually wins on both.
The constant k tells you the product OP × OQ at every position. To get a specific stroke at point Q, decide the range of OQ you want (say 100 mm to 200 mm), then the corresponding OP range is k/200 to k/100. Pick L and l so that this OP range fits comfortably within the geometric limits of the antiparallelogram — typically OP between 0.3L and 0.7L.
Rule of thumb: usable Q-stroke is roughly k × (1/OPmin − 1/OPmax). If your design wants 100 mm of Q-travel and you've sized OP to swing between 40 mm and 80 mm, you need k ≈ 8000 mm² — giving for instance L = 100 mm and l ≈ 45 mm.
You're hitting the singular configuration of the antiparallelogram. As the long bars approach collinearity (which happens at the extremes of P's circle), the mechanism's instantaneous stiffness collapses and it can flip into the parallelogram branch instead of the antiparallelogram branch. Once it flips, the inversion property is gone and the linkage jams or traces a totally different curve.
Fix: limit the input crank to a smaller swing angle — typically ±60° from the perpendicular position, never ±90°. Adding a stop pin on the input side is cheaper and more reliable than trying to design out the singularity.
For a demonstration model that traces with reasonable visual straightness over a 100 mm-class build, PLA bars work fine if you use steel bushings pressed into the pivot holes — never run a steel pin directly in a PLA hole, the hole ovals out within 50 cycles and you lose the geometry. Print bars with at least 4 mm cross-section thickness so they don't flex under their own weight, which would bend the trace.
For any precision use, aluminium or steel bars are mandatory. The mechanism's straightness is limited by bar bending stiffness as much as by joint clearance, and a 100 mm PLA bar at 4 mm thickness sags about 0.05 mm under its own weight at the midpoint — already past the precision threshold.
References & Further Reading
- Wikipedia contributors. Hart's inversor. Wikipedia
Building or designing a mechanism like this?
Explore the precision-engineered motion control hardware used by mechanical engineers, makers, and product designers.
