A Planar Pair is a lower kinematic pair where two flat surfaces stay in contact and slide freely against each other, giving 3 degrees of freedom — two translations in the plane and one rotation about the normal axis. Franz Reuleaux formalised this classification in his 1875 work Theoretische Kinematik, which still anchors how we describe machine joints today. The pair constrains the two bodies so they cannot separate or tilt, only slide and spin in-plane. You see it everywhere flatness matters — surface grinders, flatbed plotters, air-bearing metrology stages.
Planar Pair Interactive Calculator
Vary the constraint count to see the remaining joint degrees of freedom and how the sliding block motion changes.
Equation Used
The calculator applies the planar-pair degree-of-freedom count: a free rigid body has 6 possible motions, and the joint constraints remove c of them. For the worked planar pair, c = 3, so x translation, y translation, and yaw remain free.
- Rigid body in 3D space starts with 6 possible degrees of freedom.
- A true planar pair removes z translation, roll, and pitch.
- Yaw, x translation, and y translation remain free when c = 3.
How the Planar Pair Actually Works
The Planar Pair, also called the Plane-on-plane pair in machine-design textbooks, works by holding two flat faces in continuous surface contact while permitting motion only within that shared plane. Of the 6 possible degrees of freedom in space, 3 are removed — translation perpendicular to the plane, and rotation about the two in-plane axes (pitch and roll). What remains is x-translation, y-translation, and yaw. That is why a block sitting on a surface plate can be slid anywhere on the granite and spun around a vertical axis, but cannot be tilted or lifted without breaking contact.
The geometry sounds trivial but the tolerances are not. For the pair to behave as a true lower kinematic pair, both surfaces must be flat to within a fraction of the working clearance — typically 2-5 µm over 300 mm for a precision machine slide, and sub-micron for a metrology-grade granite surface plate. If flatness drifts beyond that, the contact reduces from full surface to three-point or edge contact and the joint stops behaving as a planar pair. You start seeing rocking, micro-tilts, and Abbe error in measurements. Lubrication matters too — a dry plane-on-plane pair under load will gall, stick-slip, or cold-weld; that is why surface grinder tables run on a thin oil film and why granite measuring stages use air bearings to float the moving block 3-8 µm off the reference.
The most common failure modes we see in customer builds: edge loading from non-planar mating faces, dirt particles trapped between the surfaces causing point contact and scoring, and thermal warping in long aluminium beds where the plane bows by 20-50 µm across a metre as the shop heats up. Any of those break the kinematic assumption and your 3-DOF joint quietly becomes something else.
Key Components
- Lower Plane (Reference Surface): The fixed flat datum — typically a ground steel way, lapped granite, or precision-scraped cast iron. Flatness specification drives the whole system; for a Mitutoyo Grade 0 surface plate that is 2.5 µm over 600 mm. Anything worse and the pair stops behaving kinematically.
- Upper Plane (Moving Body): The sliding element resting on the reference. Its underside must match the lower plane's flatness within roughly 1× the lubricant film thickness — call it 5 µm for oil, 1 µm for air bearings. Mass and contact area set the bearing pressure, which usually sits between 0.05 and 0.5 MPa for steel-on-cast-iron ways.
- Lubricant Film or Air Gap: The interface that prevents metal-to-metal contact. Hydrodynamic oil films run 5-20 µm thick on machine-tool ways; air bearings run 3-8 µm with regulated supply pressure around 0.4-0.6 MPa. Lose the film and you lose the pair — galling kills the surfaces in seconds.
- Retention or Preload: Gravity alone retains most planar pairs, but vertical-axis or inverted applications need vacuum preload, magnetic preload, or a gib strap. Preload force must exceed any out-of-plane disturbance load by at least 2× to keep contact unbroken.
- Wipers and Way Covers: Mandatory on any production machine. A 50 µm chip trapped between the two planes destroys flatness across centimetres of travel in one pass. Telescoping covers on a Haas VF-2 or felt wipers on a hand-scraped Bridgeport exist for exactly this reason.
Who Uses the Planar Pair
The Planar Pair shows up wherever a surface needs to translate freely in two dimensions and rotate about the normal — essentially anywhere the design problem reduces to "keep this thing flat against that thing while it moves around." It is one of the most frequently used lower kinematic pairs in industry, both as a primary joint and as a constraint reference for higher-order assemblies.
- Metrology: Mitutoyo and Starrett granite surface plates — the moving height gauge or sine bar forms a Plane-on-plane pair with the granite, providing the 3-DOF reference for dimensional inspection.
- Machine Tools: Surface grinder tables on a Chevalier FSG-1224 reciprocate on hand-scraped flat ways — a textbook planar pair carrying 100-300 kg workpieces at 25 m/min.
- Semiconductor: ASML wafer-stage air-bearing platens float on lapped ceramic with sub-micron flatness, giving a near-frictionless planar pair for sub-nanometre lithography positioning.
- Printing: HP DesignJet flatbed plotters and Roland VersaUV LEF UV printers run the print carriage assembly across a flat vacuum platen — the sheet-to-platen interface is itself a planar pair held by negative pressure.
- Robotics: SCARA end-effector compliance plates use a planar pair as a passive remote-centre-compliance element, letting the gripper self-align in x, y, and yaw during peg-in-hole insertion on Epson and Yamaha assembly cells.
- Theatre and Stage: Sliding stage wagons at venues like the National Theatre in London ride on flat steel decks — a low-precision but high-load planar pair carrying 2-5 tonnes of scenery.
The Formula Behind the Planar Pair
The defining number for any kinematic pair is its degrees of freedom — how many independent motions it allows. For a Planar Pair you compute this from the Grübler-Kutzbach count restricted to the joint itself, which tells you both the joint's DOF and the constraints it imposes on the bodies it connects. At the low end of the useful range, if you over-constrain the pair (add a gib, dovetail, or guide rail) you collapse 3 DOF down to 1 or 2 and the joint is no longer planar — it becomes a prismatic or cylindrical pair. At the nominal 3 DOF the pair behaves as designed. At the high end — when you let surface contact break — the constraint count drops and the joint floats, gaining unwanted pitch and roll DOF. The sweet spot is keeping all 3 in-plane motions free while rigidly removing the 3 out-of-plane ones.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| DOFpair | Degrees of freedom permitted by the joint | dimensionless | dimensionless |
| c | Number of constraints imposed by the pair (out-of-plane translations and rotations removed) | dimensionless | dimensionless |
| Fpreload | Normal preload force required to maintain contact | N | lbf |
| pcontact | Average bearing pressure across the contact area | MPa | psi |
Worked Example: Planar Pair in a pick-and-place vacuum platen on a PCB depaneliser
You are designing the floating vacuum platen on a Schunk-style PCB depaneliser cell. The platen is a 400 mm × 300 mm aluminium plate that floats on a granite reference via a planar pair, held down by vacuum preload so it can be re-positioned in x, y, and yaw by a 6-axis robot before the router engages. You need to verify the joint behaves as a true 3-DOF Plane-on-plane pair across the operating preload range (low: 200 N hold-down on a light board, nominal: 500 N, high: 1200 N for a thick FR-4 stack-up).
Given
- Acontact = 0.12 m² (400 × 300 mm)
- Fpreload,low = 200 N
- Fpreload,nom = 500 N
- Fpreload,high = 1200 N
- Flatness spec = 5 µm over 400 mm
Solution
Step 1 — confirm the joint DOF count from the constraint formula:
Three motions remain free: x, y, and yaw about z. The pair removes z-translation, pitch, and roll. Good — that matches the kinematic intent.
Step 2 — compute the nominal contact pressure at 500 N preload across the 0.12 m² platen:
That is well below the 0.05-0.5 MPa range typical for steel-on-cast-iron ways, which is exactly what you want for a low-friction reposition stage. The vacuum film and the granite reference do the work.
Step 3 — check the low end of the preload range (200 N, light single-layer PCB):
At 0.0017 MPa the platen is barely held down. Any out-of-plane disturbance — a router cutter grabbing the board edge, a vibration pulse from the robot — can momentarily break contact and let the platen lift 5-10 µm. The pair stops being kinematic for that instant and your part position shifts by tens of microns. For PCB depanelising that is borderline acceptable; for sub-micron lithography it would be catastrophic.
Step 4 — check the high end (1200 N, thick stacked FR-4):
At 0.010 MPa contact is rock-solid and the joint is unambiguously a planar pair — but friction torque resisting the yaw motion now climbs in proportion to preload. If your robot is sized to nudge the platen with 5 N·m of torque at the centre and µ = 0.15 between the lubricated faces, you are on the edge of stalling at 1200 N preload because the resisting torque scales with both preload and the moment arm of the contact patch.
Result
The nominal joint gives DOF = 3 with a contact pressure of 0. 0042 MPa at 500 N preload — exactly the regime where a planar pair operates cleanly with low friction and full kinematic constraint. At the 200 N low end the pair is under-preloaded and contact is fragile (pressure 0.0017 MPa, lift events likely under disturbance), while at 1200 N the joint is stiff but the robot starts struggling to drive the in-plane motions because friction torque scales with preload. The sweet spot sits between 400 and 700 N for this geometry. If you measure unexpected positioning errors above the predicted micron-level, check first for: (1) granite flatness drift exceeding the 5 µm spec — common after the surface plate gets moved or hit, (2) vacuum-line pressure oscillation causing preload to dip below 200 N momentarily, or (3) a chip or particle trapped under the platen creating a 3-point contact that introduces pitch and roll where you expected zero.
Planar Pair vs Alternatives
When a designer reaches for a Planar Pair (Plane-on-plane pair), the realistic alternatives are a prismatic pair (single-axis slide) or a spherical pair (3-DOF rotational joint). Each gives a different mix of DOF, load capacity, and precision — the right choice depends on whether you need rotation, translation, or both, and how much surface area and stiffness the application can spare.
| Property | Planar Pair | Prismatic Pair (linear slide) | Spherical Pair (ball joint) |
|---|---|---|---|
| Degrees of freedom | 3 (x, y, yaw) | 1 (single linear axis) | 3 (all rotational) |
| Typical flatness/precision spec | 2-5 µm over 300 mm | 5-15 µm straightness over 1 m | Ball roundness 1-3 µm |
| Load capacity per unit footprint | High — distributed over full surface area, 0.05-0.5 MPa typical | Medium — concentrated on rails or rolling elements | Low to medium — point or small-area contact |
| Friction characteristics | Sliding, requires lubricant film or air bearing; µ ≈ 0.05-0.15 lubricated | Rolling element bearings give µ ≈ 0.001-0.005 | Sliding ball-in-socket; µ ≈ 0.1-0.2 |
| Cost (precision-grade) | Granite plate 600 mm: $1,500-4,000 | THK SR-series rail set: $400-1,500/m | Hephaist precision spherical: $200-800 |
| Lifespan under continuous duty | 10,000+ hours with proper lubrication and chip wipers | 20,000+ hours rolling element, lower for plain bushings | 5,000-15,000 hours depending on load and lube |
| Best application fit | Surface plates, vacuum platens, compliance stages, stage wagons | Linear actuators, CNC axes, drawer slides | Rod-end joints, gimbals, tow hitches |
Frequently Asked Questions About Planar Pair
Because the kinematic pair only removes pitch when both surfaces are perfectly flat across the entire travel. In practice, granite plates sag under their own weight if poorly supported, and the slider's underside has its own flatness error. The pitch DOF is removed locally at every position, but the local zero-pitch reference rotates as the slider moves across a curved or twisted plane.
Quick diagnostic: put an electronic level on the slider and traverse the full length. Anything above 2-3 arc-seconds of pitch change tells you the granite isn't supported on its three Airy points correctly, or the plate has been cantilever-loaded by a heavy fixture and warped.
Yes — same joint, two names. Reuleaux's original classification used the geometric description (Plane-on-plane pair) to emphasise the surface-on-surface contact. Modern textbooks abbreviated this to Planar Pair. Both refer to the same lower kinematic pair with 3 in-plane degrees of freedom and 3 removed out-of-plane constraints.
Depends on what you're optimising. Stacked prismatic slides give you deterministic, measurable position via encoders on each axis and rolling-element friction near zero — that's why every CNC machine uses them. But you carry two stages of straightness error stacked in series, and you cannot rotate the work in yaw without a third axis.
A planar pair gives you all 3 DOF in one joint with shared flatness reference, which is mechanically simpler and inherently includes yaw, but you give up encoder feedback per axis and friction is 10-100× higher. Use stacked prismatic for closed-loop precision motion. Use a planar pair for passive compliance, manual repositioning, or where the controlling reference is the surface itself (metrology, vacuum platens, compliance stages).
Air bearings have a non-linear stiffness curve. The 3-8 µm air gap is maintained only above a threshold supply pressure — drop below it and the gap collapses to zero in milliseconds because there's no compliance buffer. Once the surfaces touch under load, you get instant galling because the precision-lapped faces are not designed for sliding contact without the film.
Fix: install a pressure switch interlock that disables stage motion below 0.35 MPa (or whatever your bearing manufacturer specifies). Some Aerotech and PI stages do this in firmware. Without the interlock, one regulator hiccup wrecks a $20,000 stage.
Rule of thumb: preload force must exceed the worst-case out-of-plane disturbance load by at least 2×, and the resulting contact pressure should land in the 0.005-0.05 MPa range for a low-friction reposition stage or 0.05-0.5 MPa for a stiff machine-tool way.
Worked logic: if you have 50 N of cutting reaction trying to lift the platen, you want at least 100 N of preload, then divide by your contact area to check the pressure stays in range. If the pressure climbs above 0.5 MPa you'll over-stiffen the joint and friction will resist your in-plane motion. Below 0.005 MPa, vibration will lift the joint intermittently.
That's a classic sign of a dimple or burr on the grinder's flat way that has scored the table's underside. The planar pair is now riding on a high spot, which means at that one position the table tilts a few microns and the wheel cuts shallow. Same position every cycle = same defect every cycle.
Pull the table, blue the way, and look for the witness pattern. Hand-scrape the high spot back to flat — this is the original reason machine builders learned hand scraping in the first place. Don't try to fix it with shims or oil; you have to restore the flatness of the reference surface.
It can carry moment loads up to the point where the moment causes one edge of the contact area to lift. The joint distributes pressure across the surface, and as long as the resultant of the applied load stays inside the contact footprint (the classic "middle-third rule" used in foundation design), full surface contact is maintained and the joint remains a true planar pair.
Once the moment is large enough that the resultant falls outside the footprint, the joint tips and you get edge contact — which is now a line contact, not a surface contact, and behaves as a different kinematic pair with different DOF. This is why heavily loaded planar pairs on machine tools use long footprints in the direction of expected moment loads.
References & Further Reading
- Wikipedia contributors. Kinematic pair. Wikipedia
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