A ten-bar linkage is a planar mechanism built from 10 rigid links joined by 13 revolute pivots, giving it a single degree of freedom when sized correctly under the Grübler equation. The extra links over a four-bar or six-bar buy you a coupler curve with multiple inflection points and tuneable dwell zones — typical synthesis tasks hit positional accuracy of 0.2 mm across a 200 mm path. Engineers reach for ten links when no shorter chain can hit the required pose sequence, the way the Otto Bock 3R60 prosthetic knee uses a polycentric multi-bar to control swing-phase trajectory.
Ten-bar Linkage Interactive Calculator
Vary the number of links and revolute joints to see the Grübler mobility calculation and an animated planar linkage diagram.
Equation Used
The Grübler equation estimates planar mechanism mobility. For the ten-bar example, n is the total number of links including the frame, and j is the number of revolute pin joints. A result of DOF = 1 means one input crank can drive a fully constrained motion.
- Planar mechanism with only single-DOF revolute joints.
- Ground link is included in the link count n.
- No higher-pair joints or redundant constraints are included.
How the Ten-bar Linkage Actually Works
A ten-bar linkage is what you build when a four-bar or a Stephenson six-bar can't trace the path you need. With 10 links and 13 single-degree-of-freedom revolute joints, the Grübler equation gives you DOF = 3 × (10 − 1) − 2 × 13 = 27 − 26 = 1. One input crank, one constrained output. The extra links give the designer many more dimensional parameters to play with during kinematic synthesis — link lengths, pivot positions, and ground anchors all become variables you tune to force the coupler point through 6, 8, or even 12 prescribed positions instead of the 3 to 5 a four-bar can manage.
The geometry is unforgiving. If a single link length drifts by 0.5% in a precision build, the coupler curve can shift its inflection point by 2-3 mm. That's why a serious ten-bar build needs CNC-machined links with hole-to-hole tolerances of ±0.02 mm and pivot bores reamed to a sliding fit on the pin — usually H7/g6. Wear at any one of the 13 joints stacks up. If you let a single bushing develop 0.1 mm of radial slop, you'll typically see 0.3-0.5 mm of positional error at the coupler point because joint slop near the input amplifies through the chain.
Failure modes cluster around two issues. First, the linkage locks up at a singular configuration if any two adjacent links become collinear during the cycle — this is the multi-bar version of a four-bar going through dead-centre, and it's harder to predict because the singular poses depend on all 10 link lengths simultaneously. Second, branch-defect: the assembled linkage operates on a different solution branch than the one the synthesis software solved for, which produces a coupler curve that looks vaguely right but skips the dwell zone you wanted. You catch this on the first dry run before you motorize.
Key Components
- Ground link (frame): The fixed reference that anchors the input and output pivots. In a ten-bar, the ground link typically carries 4 or 5 of the 13 revolute joints, so its rigidity matters — flex of 0.1 mm under load translates almost 1:1 into coupler-point error. A welded steel weldment or a machined aluminium plate is standard.
- Input crank: The driven link, usually rotating continuously through 360° from a gearmotor or servo. Length sets the stroke envelope of the entire mechanism. A 50 mm input crank running at 60 RPM gives a tip speed of about 314 mm/s, which is the practical sweet spot for prosthetic and walker applications.
- Coupler links (intermediate floating links): The 6 to 7 floating links that carry no fixed pivot to ground. They define the shape of the coupler curve. Length tolerances of ±0.05 mm are typical; tighter than that gets expensive without much gain, looser than 0.1 mm and your synthesized path stops matching reality.
- Revolute joints: 13 single-DOF pin joints. Each one is a pin in a reamed bore, usually with a bronze or PTFE-lined bushing. Joint slop is cumulative — keep each joint under 0.05 mm radial play or the output drifts visibly.
- Output link with coupler point: The link carrying the point of interest — the foot of a walker, the toolhead of a packaging mechanism, the shin attachment of a prosthetic knee. Its mass affects the inertia loading every other joint sees during high-speed cycling.
- Stop pins or hard limits: Mechanical end-stops that prevent the linkage from being driven through a singular configuration during start-up or fault conditions. Sized to absorb the full stall torque of the input motor without yielding.
Industries That Rely on the Ten-bar Linkage
Ten-bar linkages show up wherever the design requires a complex output trajectory that simpler chains can't deliver — multi-position dwells, precise foot placement during gait, or a coupler curve with both straight-line and circular segments in one cycle. They're more common than people realise; you just don't notice them because they hide inside prosthetics, packaging machines, and aerospace deployment systems where the path matters more than the link count. When a four-bar can't do the job and a six-bar gets close but misses one or two precision points, engineers add the extra links rather than switching to a cam or a CNC servo.
- Prosthetics: The Otto Bock 3R60 EBS polycentric prosthetic knee uses a multi-bar linkage to control swing-phase trajectory and provide stance flexion — the geometry shifts the instantaneous centre of rotation through the gait cycle to mimic anatomical knee motion.
- Packaging machinery: Bosch and IMA cartoning machines use ten-bar linkages on flap-folding stations to give a long dwell at the closed position followed by a fast retract, hitting 120 cartons per minute without a cam.
- Walking robots: Theo Jansen's larger Strandbeest derivatives have been extended to 10-link versions for research walkers at TU Delft, where the additional links allow tuning of foot-clearance height independently of stride length.
- Aerospace deployment: Solar array hinge mechanisms on ESA satellites use multi-bar linkages with up to 10 links to achieve a deploy-and-latch sequence in a single passive driven motion, locking out at the deployed pose without a separate latch actuator.
- Automotive seating: Recaro and Faurecia premium reclining seat mechanisms use 8 to 10-bar linkages to give a coupled head-rest, back-rest, and lumbar motion from a single recline lever input.
- Textile machinery: Karl Mayer warp knitting machines use multi-bar lapping motions where the guide bar follows a complex coupler curve with two distinct dwells per cycle, synthesized as a ten-bar to replace older cam drives.
The Formula Behind the Ten-bar Linkage
The Grübler-Kutzbach mobility equation tells you whether your proposed ten-bar will move at all, and if so with how many independent inputs. It's the first calculation you run before any link-length synthesis, because if the mobility comes out wrong the whole design is dead. At the low end of complexity you want DOF = 1 — a single input fully determines every other link's position, which is what a single-motor mechanism needs. DOF = 0 means the linkage is a structure and won't move. DOF = 2 or higher means you need multiple coordinated inputs, which is rarely what you want in a packaging machine or prosthetic. The sweet spot is DOF = 1 with no redundant constraints, and that pins down exactly how many revolute joints a 10-link planar chain must have.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| DOF | Degrees of freedom of the planar linkage | dimensionless | dimensionless |
| n | Total number of links including the ground link | count | count |
| j1 | Number of single-DOF (revolute or prismatic) joints | count | count |
| j2 | Number of two-DOF joints (cam-follower, pin-in-slot) | count | count |
Worked Example: Ten-bar Linkage in a polycentric prosthetic knee mechanism
A medical device startup in Utrecht is designing a polycentric prosthetic knee using a ten-bar linkage to give controlled stance flexion plus a long swing-phase dwell. They need to verify the mobility before committing to CNC tooling for the titanium links. The chain has 10 links total (1 ground, 1 input shin attachment, 8 floating links) connected by 13 revolute pin joints with bronze bushings. They want to confirm DOF = 1 across the full operating range of input crank angle from 0° (full extension) through nominal walking flexion of 60° up to maximum stance flexion of 90°.
Given
- n = 10 links
- j1 = 13 revolute joints
- j2 = 0 joints
Solution
Step 1 — at the nominal mid-stance configuration, count the moving links. The ground link doesn't count toward mobility, so we use n − 1 = 9.
Step 2 — count the constraints from the 13 revolute joints. Each revolute removes 2 planar DOF.
Step 3 — at nominal 60° flexion, compute mobility:
One degree of freedom — perfect. The shin attachment fully determines the position of every other link, which is what you want from a passive prosthetic knee driven by the wearer's hip motion.
At the low end of the operating range (0° full extension), the mobility calculation is identical because Grübler counts joints, not angles. DOF = 1 holds across the whole cycle in theory. But at this configuration, links 3 and 7 in this particular synthesis become nearly collinear within 2°, which puts the mechanism close to a singular pose — instantaneous DOF rises locally and the knee feels loose to the wearer.
At the high end (90° maximum stance flexion), DOF = 1 mathematically, but if any link length drifts by more than 0.3 mm from synthesis values, the linkage hits a branch-defect transition and the coupler point jumps to the wrong solution branch. You'll see this as a sudden 5-8 mm shift in the instantaneous centre of rotation — the wearer feels it as a jolt or buckle.
Result
The nominal mobility is DOF = 1, confirming the ten-bar will function as a single-input mechanism driven by the wearer's hip and gravity. In practice this means one independent motion variable controls the entire knee — exactly what a passive prosthetic needs. Across the operating range, DOF stays at 1 from 0° to 90° in theory, but the linkage approaches near-singular geometry below 5° flexion (loose feel) and approaches branch-defect risk above 85° flexion if link tolerances aren't held. If a built prototype shows DOF behaviour different from predicted, suspect three things: a redundant constraint from over-tightened pin fits creating a passive overconstraint that drops apparent DOF to 0 at certain angles, a swapped pin diameter on one joint creating slop that lets the mechanism behave momentarily as DOF = 2, or a mis-machined link length pushing the assembly onto the wrong solution branch where the coupler curve looks similar but the dwell zone is missing.
When to Use a Ten-bar Linkage and When Not To
A ten-bar isn't always the right call. Most engineers default to four-bar or six-bar chains because they're easier to synthesize, easier to manufacture, and easier to debug. You move up to ten links only when the path requirements force you. Here's how the trade space looks against the realistic alternatives.
| Property | Ten-bar linkage | Four-bar linkage | Cam-follower mechanism |
|---|---|---|---|
| Maximum precision points on coupler curve | 9-12 points | 5 points | unlimited (cam profile) |
| Synthesis complexity | high — requires Burmester theory or commercial software | low — graphical or analytical | moderate — direct profile generation |
| Typical link-length tolerance for working build | ±0.02 to ±0.05 mm | ±0.1 mm | cam profile ±0.01 mm |
| Maximum continuous operating speed | ~150-300 RPM input | ~600 RPM input | ~1000 RPM input |
| Joint count and wear surfaces | 13 revolutes | 4 revolutes | 1 cam contact + follower bearings |
| Cost (small batch) | high — 10 precision links + 13 pinned joints | low | moderate — requires precision cam grinding |
| Best application fit | multi-dwell paths, prosthetics, complex deployment | general-purpose path generation | high-speed repetitive motion with custom profile |
| Reliability over 10 million cycles | good if joint slop managed; cumulative wear is the killer | excellent — fewer wear surfaces | depends on cam material and lubrication |
Frequently Asked Questions About Ten-bar Linkage
You've almost certainly assembled the linkage on the wrong solution branch. A ten-bar has multiple valid assembly configurations for the same set of link lengths — the synthesis software picks one branch when it solves the position equations, but during physical assembly you can flip one or more links into the mirror-image branch without realising it. The coupler curve looks vaguely similar but misses the dwell zones.
Diagnostic check: at the home position (input crank at 0°), measure the angle of every floating link against your synthesis output. If even one link is reflected across its pivot axis, you're on the wrong branch. Disassemble, flip the offending link, reassemble. Mark each link with a clear orientation arrow during fabrication to prevent it next time.
Use a six-bar first whenever the path requirements allow it. Six-bar can hit 6-7 precision points on a coupler curve, which covers maybe 80% of real-world synthesis problems including most packaging dwells and walker geometries. Move to ten-bar only when you need more than 7 precision points, OR when you need two independent dwell zones in a single cycle, OR when the required path has both a long straight-line segment and a circular-arc segment that a six-bar can't reconcile.
Rule of thumb: try synthesizing your problem as a six-bar first. If the residual error at any precision point exceeds your application tolerance after exhausting Burmester synthesis, then add the four extra links and resynthesize. The cost and complexity jump is significant — 13 joints versus 7 — so don't take it lightly.
You can't find them by inspection — there are too many link combinations. Run a full kinematic simulation through 360° of input rotation in 1° steps and watch the determinant of the Jacobian matrix. When the determinant approaches zero, you're at or near a singular pose. Any pose where |det(J)| drops below 5% of its average value over the cycle is a singularity risk under load.
The practical fix is either to redesign so singularities fall outside your operating range, or to install hard mechanical stops that prevent the input crank from reaching the dangerous angle. Don't rely on motor control to avoid singularities — a power-off coast can drift the linkage right into one.
Joint slop stacks up multiplicatively along the kinematic chain. With 13 joints each contributing 0.05 mm of radial play, the worst-case accumulated error at the coupler point can be 0.6-0.8 mm depending on geometry — even though no single joint feels loose. The links closest to the input contribute most because their slop gets amplified through every downstream link.
Two fixes: tighten the joints nearest the input crank to ±0.02 mm and accept ±0.05 mm at the output end (graded tolerancing), or pre-load the chain with a light torsion spring at one floating link so all joints are loaded against the same side of their bushings, which removes the directional component of slop.
Three usual suspects beyond link length. First, pivot hole position — link length is the centre-to-centre distance between two holes, and if the holes are 0.1 mm out of position on the link body, that contributes the same as length error. Check hole positions on a CMM, not just calipers across the link.
Second, ground-frame deflection. The frame carrying the fixed pivots flexes under load, and even 0.1 mm of frame flex moves a fixed pivot, which moves the coupler curve. Stiffen the frame or measure deflection under operating load. Third, pin-bore clearance taken up asymmetrically — if your H7/g6 fits are at the loose end of the band on some joints and the tight end on others, effective link lengths shift unpredictably during operation.
Yes, and it's common in deployment and seating applications. You replace the input revolute pair with a prismatic (slider) joint, which still counts as a single-DOF joint in the Grübler equation, so mobility stays at DOF = 1. The actuator becomes one of the 10 links — typically the ground-to-piston interface counts as the prismatic joint, and the piston rod itself is one of the floating links.
Watch out for one thing: a linear actuator can't pass through dead-centre the way a rotating crank can. If your synthesis assumed a continuously rotating input, you can't just swap in a linear drive — you need to resynthesize with the prismatic input from the start, or you'll find the linkage refuses to complete its cycle.
References & Further Reading
- Wikipedia contributors. Linkage (mechanical). Wikipedia
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