The Bricard 6R Linkage is a spatial closed-loop mechanism with six revolute joints whose axes obey special geometric conditions, giving it a single degree of freedom even though the standard Kutzbach mobility count says it should be rigid. It moves with one input drive at any speed the joints tolerate — typical demonstrator builds run 5 to 30 RPM. The Bricard linkage exists to fold and unfold a closed 3D loop along one continuous path, which is why it appears in deployable space antennas, foldable architectural canopies, and origami-inspired metamaterials.
Bricard 6R Linkage Interactive Calculator
Vary the spatial mobility terms and symmetry release to see why a 6R Bricard loop calculates as rigid by Kutzbach but moves with 1 DOF.
Equation Used
The spatial Kutzbach count estimates mobility as M_K = 6(L - J - 1) + J*f. For the Bricard 6R example, L = 6, J = 6, and f = 1, so M_K = 0. The special Bricard symmetry makes the loop constraints dependent, represented here by s = 1, giving an effective 1 DOF motion.
- Spatial closed-loop mechanism using the Kutzbach mobility criterion.
- Each revolute joint contributes one joint DOF when f = 1.
- The symmetry release s represents the Bricard constraint dependency that opens the paradoxical DOF.
- Default values reproduce the article's line-symmetric Bricard 6R example.
The Bricard 6R Linkage in Action
The Bricard 6R Linkage, also called the Bricard linkage in most academic and aerospace papers, works by exploiting a geometric coincidence between the six revolute-joint axes that the standard mobility formula cannot see. Run the Kutzbach criterion on a closed 6R spatial loop and you get mobility = 6(6−6−1) + 6 = 0. The loop should be rigid. It is not — Raoul Bricard proved in 1897 that if the six axes satisfy specific symmetry conditions (line-symmetric, plane-symmetric, or trihedral), the constraint equations become linearly dependent and one degree of freedom opens up. That is what engineers call an overconstrained mechanism, and the Bricard family is the canonical 6R example.
The motion happens along a single screw axis path. As you drive any one joint, the other five rotate in a coupled, non-linear pattern dictated by the loop closure equations. The whole structure sweeps from a flat or near-flat folded state to an extended deployed state and back, all without any joint ever leaving its prescribed axis. The line-symmetric Bricard case is the most common in deployable hardware because it folds nearly flat — useful when you need a bundle that fits inside a launch fairing.
If the geometry is off, the linkage simply will not move. This is the part that catches new builders. Joint axes machined 0.5° off the symmetry plane will lock the loop solid, because the mobility paradox depends on exact geometric conditions — there is no slop budget for the symmetry itself. Common failure modes are: pivot bearings drilled with axis misalignment greater than ±0.1°, link lengths fabricated outside ±0.05 mm of the symmetric pair, and bushing wear that lets one axis drift over time. When a Bricard rig binds halfway through its motion, 9 times out of 10 it is one joint axis off-square, not a force or torque problem.
Key Components
- Revolute joints (×6): Six single-axis hinges arranged around the closed spatial loop. Each must hold its axis to within ±0.1° of the theoretical symmetry condition or the loop will not articulate. Bearing radial play above 50 µm starts to introduce wobble in the deployed state.
- Rigid links (×6): The structural members between joints. In the line-symmetric Bricard case, links pair up — link 1 matches link 4 in length, 2 matches 5, 3 matches 6, all to within ±0.05 mm. Asymmetry of more than 0.1 mm in a 100 mm link will jam the loop.
- Common perpendicular offsets (ai): Denavit-Hartenberg distances along each link's common perpendicular. For the line-symmetric case these obey a1 = a4, a2 = a5, a3 = a6. These set the deployed envelope size.
- Twist angles (αi): The angle between consecutive joint axes. For a line-symmetric Bricard, α1 + α4 = π, α2 + α5 = π, α3 + α6 = π. Get one wrong by more than half a degree and the symmetry condition collapses.
- Drive joint: Any one of the six revolutes serves as the input. A small servo or stepper rotating at 5 to 30 RPM is typical for desktop-scale deployable demos. The other five joints follow the coupled motion automatically — you do not drive them.
Who Uses the Bricard 6R Linkage
The Bricard 6R Linkage shows up wherever engineers need a closed 3D loop to fold flat and deploy on a single actuator. It is rare in mass-produced hardware because the geometry is unforgiving, but in spacecraft, architecture, and metamaterial research it solves problems no other 6R loop can.
- Spacecraft deployables: Foldable antenna support rings and solar reflector hoops on small satellites — the line-symmetric Bricard linkage stows flat against the bus and deploys to a rigid hexagonal loop with one motor, similar to the deployable rings studied for the AstroMesh family of mesh reflectors.
- Architecture: Chuck Hoberman's deployable structures and follow-on academic work at Tianjin University use Bricard 6R loops as building blocks for transformable canopies and pavilions that retract on a single drive.
- Origami metamaterials: Research groups at Oxford and Tsinghua tile arrays of Bricard linkages to build 3D folding metamaterials with programmable Poisson ratios — each Bricard cell acts as a single-DOF unit.
- Robotics research: Mobile assembly robots use trihedral Bricard cells as reconfigurable end-effector skeletons, where one joint drive collapses the entire wrist into a packed shape for tool change.
- Education and demonstrators: University kinematics courses (MIT 2.72, University of Sheffield mechanism design) build acrylic line-symmetric Bricard rigs to show students that the Kutzbach formula has exceptions — overconstrained mechanisms are not a textbook curiosity, they fly on satellites.
- Surgical instruments: Single-DOF deployable retractors and biopsy graspers in minimally invasive tools, where a small Bricard loop expands inside a body cavity once past the trocar.
The Formula Behind the Bricard 6R Linkage
The closure condition for the line-symmetric Bricard case is what tells you whether your geometry will actually move. It is not a speed or force formula — it is a go/no-go check on the joint twists and link lengths before you cut metal. At the low end of practical builds (small twist angles near 30°) the deployed-to-folded ratio is modest, around 2:1, and the linkage feels stiff through its motion. At the nominal sweet spot (twists near 60°) you get clean 4:1 stowage with smooth articulation. Push twists toward 90° and the linkage folds nearly flat — 8:1 or better — but the motion gets jerky near the singular configurations at each end of travel.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| αi | Twist angle between joint axis i and joint axis i+1 (Denavit-Hartenberg twist) | radians | degrees |
| ai | Common perpendicular distance between joint axis i and joint axis i+1 (link length) | millimetres | inches |
| π | Half-turn constant — the symmetry condition for line-symmetric Bricard requires opposite twists to sum to π (180°) | radians | degrees (= 180°) |
Worked Example: Bricard 6R Linkage in a deployable CubeSat antenna hoop
A 6U CubeSat needs a hexagonal antenna support hoop that stows inside a 100 mm × 100 mm footprint and deploys to a 600 mm-diameter ring on a single 12 V gearmotor. You pick a line-symmetric Bricard 6R Linkage. The team has settled on equal link lengths a = 300 mm and is choosing the twist angle α to balance stowed compactness against deployment smoothness. Check three candidate twists — 30°, 60°, and 85° — and tell the mechanical lead which one to build.
Given
- a1 = a4 = 300 mm
- a2 = a5 = 300 mm
- a3 = a6 = 300 mm
- α (candidate twist) = 30, 60, 85 degrees
- Drive speed = 10 RPM
Solution
Step 1 — confirm the symmetry conditions are satisfied. All three opposite twist pairs must sum to 180°:
The link-length pairs are equal by construction. Geometry passes the line-symmetric test for any α between 0° and 90°.
Step 2 — at the low end of the typical operating range, α = 30°. Estimate the stowed-to-deployed ratio using the line-symmetric envelope approximation R ≈ 1 / sin(α):
A 2:1 stowage ratio means a 600 mm deployed ring collapses to roughly 300 mm — way too big for the 100 mm CubeSat slot. The motion is stiff but predictable. Reject.
Step 3 — at the nominal sweet spot, α = 60°:
This collapses the 600 mm ring to about 150 mm packed footprint when you include link nesting. Still over the 100 mm budget but close. Smooth articulation throughout the deployment sweep.
Step 4 — at the high end, α = 85°:
The 600 mm ring now packs to roughly 75 mm — fits the 100 mm slot with margin. The catch: at α near 90° the linkage approaches a singular configuration at full deployment, and the gearmotor sees a sharp torque spike in the last 5° of travel. You need either a soft-stop or an over-centre latch.
Result
Pick α = 85° and add a deployment latch — the nominal 60° twist gets you 4:1 stowage and clean motion but misses the 100 mm CubeSat budget, while 85° hits 8:1 stowage and packs to ~75 mm. The 30° option folds to only 300 mm and is non-viable. The trade is real: 85° gives the compactness but you live near a singularity, so the deploy torque rises sharply in the last few degrees. If your built rig refuses to move past about 70% deployed, the cause is almost always (1) one twist angle machined off by more than ±0.5° breaking the αi + αi+3 = 180° condition, (2) link-pair length mismatch above 0.3 mm forcing the loop into an over-determined state, or (3) bearing preload too tight at the drive joint stalling the gearmotor before the singularity is cleared.
Choosing the Bricard 6R Linkage: Pros and Cons
The Bricard linkage is one of several spatial overconstrained loops, and choosing between them comes down to how flat you need it to fold, how much torque you can spend, and whether your shop can hold the tolerances. Here is how it stacks against the two closest alternatives most engineers consider.
| Property | Bricard 6R Linkage | Bennett 4R Linkage | Myard 5R Linkage |
|---|---|---|---|
| Degrees of freedom | 1 | 1 | 1 |
| Number of joints | 6 | 4 | 5 |
| Typical stowage ratio | Up to 8:1 (line-symmetric, α near 90°) | 2:1 to 3:1 | 3:1 to 4:1 |
| Geometric tolerance on joint axes | ±0.1° (very tight) | ±0.2° | ±0.15° |
| Deployed footprint shape | Hexagonal closed loop | Skew quadrilateral | Pentagonal loop with plane symmetry |
| Drive torque profile | Spikes near singularity at full deploy | Smooth, near-sinusoidal | Smooth with mild end-of-travel rise |
| Manufacturing complexity | High — 6 axes must hold symmetry | Low — only 4 joints | Medium |
| Best application fit | Deployable rings, antenna hoops, 3D metamaterials | Foldable booms, scissor pairs, simple deployables | Specialty single-loop deployables, research |
| Cost (relative) | 3× | 1× | 2× |
Frequently Asked Questions About Bricard 6R Linkage
Kutzbach counts constraints as if every joint imposes 5 independent restrictions on a rigid body in 3D space. In a Bricard loop, the special geometry — opposite twists summing to 180°, paired link lengths — makes some of those constraint equations linearly dependent. They double-count the same restriction. When you remove the dependency, one degree of freedom appears.
This is why Bricard, Bennett, and Goldberg loops are called overconstrained mechanisms. They are real exceptions to a counting formula that assumes generic geometry. The formula is not wrong — it is an inequality, and the special-geometry cases sit on the equality boundary.
Mid-travel lockup almost always points to a non-symmetric pivot location rather than an axis-angle error (axis errors typically prevent ANY motion). Check that your three link-length pairs (a1=a4, a2=a5, a3=a6) are matched to within 0.1 mm. A common cause is using nominally identical machined parts where one was reworked or shimmed and the others were not.
The diagnostic: rotate the loop to the lockup point and measure the strain on each link with a dial indicator. The link going into bending is the one whose length-pair partner is mismatched. Fix that pair and the motion sweeps clean.
Pick by what shape you need deployed and how flat you need to fold. Line-symmetric is the workhorse for closed hoops and rings — it folds the flattest and is the easiest to analyse. Plane-symmetric suits structures that need a planar deployed footprint, like flat panels with hinged perimeters. Trihedral (the rarest) gives you three orthogonal axes and shows up in research wrist mechanisms and reconfigurable robot bases.
For 90% of deployable hardware, start with line-symmetric. Only move to the other cases if the deployed geometry demands it.
Mathematically any joint can be the input — the loop has one DOF, so driving any joint determines all the others. In practice some joints are easier to drive than others because the torque-versus-angle curve looks different at each one.
For a line-symmetric Bricard, joints on the symmetry axis tend to see the highest mechanical advantage near the folded state, which means lower motor torque to break out of stow. If you have a small CubeSat-class motor and you are watching every milli-newton-metre, instrument all six joints in CAD with a constant deployment-rate simulation and pick the joint with the lowest peak torque.
Springiness in the deployed loop comes from the loop sitting near a singular configuration, not from link compliance. Near a singularity the instantaneous mobility direction has a small projection onto the joint torques, so external loads on the loop produce large internal forces that the joints absorb as elastic deformation.
If you measured negligible deflection in a single link under the same load but the whole loop deflects 5 mm, you are operating too close to the singular pose. Back the deployed angle off by 3° to 5° with a hard stop, or move to a smaller twist α (60° instead of 85°). You trade stowage ratio for stiffness.
No, though they are cousins. Both are spatial overconstrained mechanisms with one degree of freedom. The Sarrus linkage uses six revolute joints arranged in two perpendicular planar four-bar groups and produces pure straight-line translation between two platforms. The Bricard 6R Linkage is a single closed loop of six revolutes whose motion is generally a complex coupled rotation, not pure translation.
Use Sarrus when you need linear motion without prismatic guides. Use Bricard when you need a closed 3D loop to fold and deploy.
References & Further Reading
- Wikipedia contributors. Bricard octahedron. Wikipedia
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