A six-bar linkage is a planar kinematic chain made from six rigid links connected by seven revolute joints, giving a single degree of freedom. The extra two links over a four-bar let one input drive two coordinated output motions, or generate coupler curves a four-bar physically cannot trace. Designers reach for it when they need dwell, straight-line segments, or non-Grashof paths in one mechanism. You see it in film advance claws, stair-climbing robots, and the rear suspensions on full-suspension mountain bikes from Santa Cruz and Yeti.
Six-bar Linkage Interactive Calculator
Vary the number of links and revolute joints to see the Grubler mobility result and an animated six-bar teaching diagram.
Equation Used
Grubler mobility estimates the degrees of freedom of a planar linkage. For a six-bar made from revolute joints, n is the total link count including ground and j is the number of pin joints. The worked six-bar example uses n = 6 and j = 7, giving M = 1, so one input crank can drive the coordinated mechanism motion.
- Planar linkage with only revolute lower pairs.
- Link count n includes the fixed ground link.
- No redundant constraints, compliant joints, or higher-pair contacts are included.
- A positive mobility value estimates the number of independent input drivers required.
How the Six-bar Linkage Actually Works
A six-bar linkage exists because a four-bar runs out of design freedom fast. With four links and four pin joints, you have only three independent dimensions to play with after fixing the ground link, and that's not enough to hit precision points along a path while also controlling velocity at those points. Add two more links and one more joint and you unlock enough free parameters to shape a coupler curve with dwells, near-straight segments, or sharp transitions — all from a single rotating input.
Two topologies cover almost every six-bar you'll encounter. The Watt chain places the two ternary links (the ones with three pin connections) directly adjacent to each other. The Stephenson chain separates them with a binary link in between. That topological difference matters — Watt linkages tend to give you sequential motion stages, while Stephenson chains are the go-to for path generation and dwell synthesis. Apply the Grübler equation for planar mechanisms, M = 3(n−1) − 2j, and with n = 6 links and j = 7 revolute joints you get M = 3(5) − 14 = 1 degree of freedom. One motor in, one coordinated motion out.
Tolerance stack-up is where six-bars bite back. Each pin joint contributes radial slop, and with seven joints in series the position error at the output coupler point can compound to several times what you'd see on a four-bar. If you notice the output point drifting by more than a millimetre between cycles on a 200 mm linkage, the cause is almost always pin-bushing clearance above 0.05 mm at one or two of the high-load joints — typically the ones nearest the input crank. Bind-up at extreme positions is the other common failure mode, and it usually traces back to a link-length ratio that pushes the mechanism near a singularity. Run the position equations across the full input rotation before you cut metal.
Key Components
- Ground link (frame): The fixed reference link that grounds two of the seven pin joints. Frame stiffness sets the achievable accuracy of the whole mechanism — a frame that flexes 0.2 mm under peak load eats most of the precision budget on a 300 mm linkage.
- Input crank: The driven link, usually rotating continuously from a gearmotor or hand crank. Its length sets the basic scale of motion; doubling the crank length roughly doubles the coupler-curve size but quadruples the inertial loads at speed.
- Ternary links (×2): Links carrying three pin joints each. These are what distinguish a six-bar from a four-bar and they carry the highest bending loads in the chain. Plate thickness should be sized for the bending moment at the middle pin, not just the tension between adjacent pins.
- Coupler link: The floating link whose tracer point traces the useful output curve. Pin-to-pin tolerance on the coupler must hold to ±0.02 mm on a precision build — any more and the curve smears across the machine cycle.
- Output link or rocker: The link that delivers motion to the load, either as a rotation or as a translation through a tracer point. On a Stephenson III chain this is where you mount the working tool.
- Pin joints (×7): Seven revolute joints, typically needle-bearing or oilite-bushed. Combined radial clearance across all seven joints should stay under 0.3 mm total for the linkage to repeat to within ±0.5 mm at the output.
Where the Six-bar Linkage Is Used
Six-bar linkages show up wherever a four-bar can't quite do the job — typically when the designer needs a long dwell, a near-straight segment, or two coordinated outputs from one input. The cost is more parts and more tolerance management, so engineers only reach for them when the motion requirement genuinely demands the extra complexity. The applications below are real shipping products, not textbook examples.
- Mountain bikes: Santa Cruz VPP and Yeti Switch Infinity rear suspensions use six-bar arrangements to decouple pedalling forces from bump absorption. The instant centre traces a controlled path that a four-bar Horst-link cannot match.
- Cinema equipment: Bell & Howell 35 mm film projectors used six-bar claw-pulldown mechanisms to advance the film one frame and dwell during the shutter-open phase, hitting 24 fps with sub-frame registration.
- Walking robots: The Theo Jansen Strandbeest leg is a 12-bar built from two coupled six-bar sub-chains, generating a foot path with a flat ground-contact phase and a high swing-clearance arc.
- Aerospace: Boeing 737 and 777 main landing-gear retract mechanisms use six-bar geometries to fold the strut into a wheel well that's smaller than the strut's rotation envelope would otherwise allow.
- Industrial packaging: Bosch and IMA cartoner flap-folding stations use Stephenson III six-bars to drive a folding blade through a curved path that wraps the carton flap over without scuffing the printed face.
- Automotive: Convertible roof-stowage mechanisms on the BMW Z4 and Mercedes SLK use multiple coupled six-bars to fold a three-piece hardtop into a trunk volume roughly 40% smaller than the deployed roof.
The Formula Behind the Six-bar Linkage
The Grübler-Kutzbach equation gives you the mobility of any planar linkage — the number of independent inputs needed to fully constrain the motion. For a six-bar, you want this to come out to exactly 1. Drop below 1 and the mechanism is locked or overconstrained; go above 1 and you'd need two motors to control the output, defeating the point. At the lower bound, removing one joint pushes mobility to 2 and the linkage flops uncontrolled. At the upper bound, adding redundant constraints drives mobility to 0 and the linkage binds solid. The sweet spot is M = 1 with all link lengths sized so the input crank can rotate fully without crossing a singularity.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| M | Mobility (degrees of freedom) | dimensionless | dimensionless |
| n | Total number of links including ground | count | count |
| j1 | Number of single-DOF joints (revolute or prismatic) | count | count |
| j2 | Number of two-DOF joints (cam or gear contacts) | count | count |
Worked Example: Six-bar Linkage in a Stephenson III flap-folder for an espresso pod sealer
A coffee-equipment OEM in Trieste is designing a Stephenson III six-bar to drive the foil-flap folding blade on an aluminium espresso pod sealing line running 80 pods per minute. The input crank rotates at 80 RPM nominal, with the line specified to run between 40 RPM (slow start-up) and 120 RPM (peak throughput). The designer needs to confirm mobility before cutting the prototype links from 6 mm aluminium plate.
Given
- n = 6 links
- j1 = 7 revolute joints
- j2 = 0 higher-pair joints
- Nnom = 80 RPM
Solution
Step 1 — apply the Grübler equation with the link and joint counts to confirm the linkage has exactly one degree of freedom:
Mobility checks out at M = 1. One servo input gives one fully-constrained output path. Now the kinematic question — how does the linkage behave across the operating range?
Step 2 — at nominal 80 RPM, the input crank cycle time is:
The folding blade has 0.75 seconds to complete its full path. With a Stephenson III tuned for a 120° dwell at the fold-over position, the blade pauses for roughly 0.25 s while the foil sets — exactly the window the heat-seal jaws need to bond the flap.
Step 3 — at the low end of the operating range, 40 RPM:
At 1.5 seconds per cycle the dwell stretches to 0.50 s. The seal still works, but operators sometimes catch the blade visually mid-fold and assume the line has stalled — this is normal start-up behaviour, not a fault.
Step 4 — at the high end, 120 RPM:
At 0.50 s per cycle the dwell collapses to 0.167 s — below the heat-seal time the foil chemistry needs. In practice the line jams or seals fail above roughly 100 RPM unless you upgrade to a higher-wattage seal head. The mechanism's mobility is unchanged, but the system-level performance is bounded by the seal physics, not the linkage.
Result
The Grübler check returns M = 1, so the six-bar is correctly constrained — one servo, one output, no redundant inputs needed. At 80 RPM nominal, the 0.25 s dwell window matches the foil seal-time perfectly and the line runs clean. Across the operating range, 40 RPM gives a comfortable 0.50 s dwell while 120 RPM compresses dwell to 0.167 s and starts producing weak seals — the mechanical sweet spot is 70-90 RPM. If you build the linkage and see the blade refusing to complete a full cycle, suspect three things in order: a link-length ratio that crosses a Grashof singularity (recheck the geometry against your crank-rocker condition), a ternary link bending under load because the plate is thinner than 5 mm at the centre pin, or a seventh joint accidentally constrained as a higher pair (a press-fit pin that's actually rigid will drop mobility to zero and lock the chain solid).
Six-bar Linkage vs Alternatives
Six-bar linkages sit between four-bars and full cam systems on the complexity-versus-capability curve. The decision usually comes down to whether you can satisfy the motion requirement with a four-bar (always try first), and if not, whether a six-bar's reachable curve family covers what you need before you commit to a cam.
| Property | Six-bar linkage | Four-bar linkage | Cam-follower mechanism |
|---|---|---|---|
| Degrees of freedom controllable from one input | 1 input, 2 coordinated outputs possible | 1 input, 1 output | 1 input, 1 output (arbitrary profile) |
| Coupler-curve flexibility | Wide — includes dwells and near-straight segments | Limited to four-bar curve atlas | Unlimited — any continuous profile |
| Part count | 6 links + 7 pins | 4 links + 4 pins | Cam + follower + return spring |
| Typical position repeatability at output | ±0.3-0.5 mm on a 300 mm linkage | ±0.1-0.2 mm on a 300 mm linkage | ±0.05 mm with preloaded follower |
| Operating speed ceiling | 300-600 RPM before joint wear dominates | 600-1500 RPM | 200-400 RPM (follower bounce) |
| Cost to manufacture (small batch) | Medium — 2-3× a four-bar | Low baseline | High — cam grinding setup |
| Best application fit | Dwell, straight-line, dual-output motions | Simple oscillation or rotation | High-speed precision profiles |
Frequently Asked Questions About Six-bar Linkage
Pick Stephenson when your motion requirement is a single complex output path — dwells, straight-line segments, or sharply asymmetric coupler curves. The separated ternary links give you more independent design parameters for path synthesis, which is why almost every published dwell mechanism in Erdman and Sandor's textbook is Stephenson III.
Pick Watt when you need two sequential or coordinated outputs from one input, like a folding mechanism that first lifts then rotates a flap. The adjacent ternary links naturally split the kinematic chain into two coupled stages, and that's a topology fight you don't want to have with a Stephenson.
Grübler counts links and joints — it doesn't check geometry. Mobility of 1 only tells you the chain is correctly constrained in topology. If the link-length ratios put the mechanism near a singularity (a position where two links momentarily align), the instantaneous transmission angle drops toward zero and the input torque required to push through goes to infinity. The mechanism binds.
Run a position analysis across the full 0-360° input rotation before you build. The transmission angle at every joint should stay above 30° for reliable motion, and above 45° if you want any reasonable mechanical efficiency. If you find an angle dipping below 30°, adjust the ground-link length first — it has the largest effect on transmission angle without changing the coupler-curve shape much.
At 1.5 mm drift on a 250 mm linkage you're at roughly 0.6% — too high for joint clearance alone unless the build is sloppy. Run a quick check: lock the input crank at one position and push the coupler point by hand. If it moves more than 0.3 mm under finger pressure, the joints are the culprit. If it doesn't move under hand load but drifts under running load, the frame is flexing.
Frame flex usually shows up because the two ground pins aren't tied to a stiff backbone. A 6 mm aluminium plate ground link will flex visibly under a 50 N load at 200 mm reach. Switch to 10 mm plate or add a triangulating gusset between the two ground pins and the drift typically drops to under 0.4 mm.
Five precision points is at the edge of what closed-form Burmester synthesis handles for a four-bar. For a six-bar, you have the freedom to hit nine or more points, but the math gets ugly fast. Most working designers don't solve it by hand — they use SyMech, MotionGen, or write a short optimiser in Python with scipy that minimises the squared distance from the coupler tracer to each precision point as the input rotates.
Start with a four-bar that gets close, then add the dyad (two extra links and the seventh joint) to correct the residual error. This staged approach converges far better than trying to optimise all six link lengths simultaneously from a random starting point.
That's almost always a transmission-angle dip you didn't catch in analysis. At one specific input position the geometry briefly approaches a singularity, the load on one pin spikes, and the friction at that pin overcomes the input torque margin. The notchy feel is the input crank momentarily bogging down then breaking through.
Check the transmission angle at every joint as a function of input crank angle. The angle that dips lowest will correspond exactly to where you feel the notch. The fix is geometric, not lubrication — re-proportion the offending link by 5-10% and the dip lifts back into the safe zone.
Yes, and it's a common move — the Stephenson III with one prismatic joint is a recognised inversion. Mobility stays at 1 because Grübler treats a slider as a single-DOF joint just like a revolute. The advantage is you get a guaranteed straight-line output without relying on the coupler curve to approximate one.
The catch is slider friction and side-load. A revolute joint takes radial load on a bearing surface; a prismatic joint takes side-load as a moment across the slider's length. If your slider is shorter than 2× its travel, expect stick-slip and accelerated wear. Linear ball bushings or a Drylin-style polymer rail handle the moment cleanly if you can fit them in the envelope.
References & Further Reading
- Wikipedia contributors. Six-bar linkage. Wikipedia
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