A Bricard 6R linkage is a closed spatial chain with six revolute joints. Special relationships between its hinge axes allow motion that a generic spatial mobility count does not predict.
The animation shows a general line-symmetric example. Six hinge shafts join six rigid links, with one link fixed to the support. As the input turns, the calculator solves the remaining joint angles so the chain closes in three dimensions.
Bricard 6R Linkage Interactive Calculator
Explore a solved spatial 6R loop. Scale its dimensions, compare two assembly forms, change speed and rotate the oblique view.
Equation Used
- General line-symmetric example from equation 27 of the cited primary paper.
- Opposite DH spacings, twists and offsets equal in the stated axis convention.
- Ideal rigid links and revolute joints; no clearance, friction, force or collision analysis.
- Assembly forms are separate configurations, not a continuous operating transition.
- Input motion starts at 60 degrees; animation period 60/rpm seconds.
Fixed geometric example, uniformly scaled. Link 6 is fixed; five links move. Angle wrapping does not interrupt the motion. Thickness is illustrative.
Six revolute axes close a three-dimensional loop
A revolute joint permits rotation about one shaft axis. Adjacent axes in this model are skew: their common perpendicular has a fixed length, and the axes have a fixed twist angle. Short hinge shafts show the offsets between neighboring link attachments.
The grey sixth link is fixed. Turning joint 1 moves the other five links while the final connection returns to its original position and orientation. The second view looks down on the same calculated geometry, making its three-dimensional shape easier to inspect.
Opposite joints have equal angles in the chosen line-symmetric convention: θ1=θ4, θ2=θ5 and θ3=θ6. Equality of opposite dimensions alone is not used as a substitute for closure: the entire chain transformation is checked.
The two assembly forms are separate solutions for the same dimensions. Switching the form control compares them; the animation does not imply that a rigid mechanism can jump from one to the other during operation.
Inspect a spatial-linkage example
Use the scale slider to size the illustrated linkage, the form control to compare its two assemblies, and the camera slider to expose connections that overlap in one view. Input speed changes the actual playback period.
The drawing is a kinematic model. Link diameters, hinge sleeves and feet make the construction readable, but their thicknesses do not establish a collision-free physical design. A real build must accommodate joint travel, bearings and interference.
This page demonstrates one general line-symmetric example. It does not claim that every Bricard mechanism folds flat, follows the same path or can replace a specified commercial deployable structure.
Geometry and loop-closure calculation
The dimensional scale is s mm per model unit. Successive common-perpendicular distances are (2.4, 2.9, 1.5, 2.4, 2.9, 1.5)s. Successive axis twists are (40°, 80°, 130°, 40°, 80°, 130°). Offsets along the hinge axes are (0.50, 0.55, 0.42, 0.50, 0.55, 0.42)s.
For joint i, use the Denavit–Hartenberg step Tᵢ=Rz(θᵢ)Tz(rᵢ)Tx(aᵢ)Rx(αᵢ). The six steps must multiply to the identity: T₁T₂T₃T₄T₅T₆=I. This checks both the position and orientation of the final frame.
The solver finds the two possible second-joint angles for the entered first-joint angle. A quaternion half-turn condition determines the third angle on the corresponding branch; the opposite three angles repeat the first three. The resulting full matrix product is verified, rather than assuming the chain closes because its endpoint looks close in a projection.
The generic spatial count M=6(L−J−1)+Σf gives zero for L=6 links including the fixed link, J=6 revolute joints and Σf=6. That count assumes independent constraints. The special geometry here makes some constraints dependent, allowing a continuous one-parameter motion.
Angles are displayed between −180° and 180°. The numerical readout can wrap at that boundary while the mechanism moves continuously. Joint-origin span means the straight-line distance between the defined origins of frames 1 and 4, not a load rating or deployment specification.
Example at 40 mm per scale unit
The three repeated common-perpendicular spacings are 96, 116 and 60 mm. The repeated hinge offsets are 20, 22 and 16.8 mm. The reset position sets the first and fourth joint angles to 60°.
In assembly form 1, joints 2 and 5 are approximately −63.55°, and joints 3 and 6 approximately −151.62°. Form 2 instead gives approximately 161.47° and −20.37° for those pairs. Both close the spatial loop with the same dimensions.
At six rpm, one full input rotation takes ten seconds. Changing the scale changes lengths and the measured span, while the joint-angle relationships remain the same.
What the ideal model does and does not establish
A small numerical closure residual means the modeled centerline geometry returns to the same frame. It is not a prediction of a workshop’s attainable tolerance, bearing clearance or a physical mechanism’s binding risk.
Only uniform scaling is offered for the dimensional example. Arbitrarily changing an individual axis spacing, twist or offset can destroy its special closure relationships. That would require a new synthesis problem, not an extra degree-of-freedom adjustment.
The mechanism has moving members in several planes. The oblique view uses depth ordering, and the top view reveals projection overlaps. These are aids to understanding the geometry, not a structural, force or collision certification.
Bricard 6R questions
Why can it move when the mobility count gives zero?
The generic count assumes independent constraints. This special geometry has a continuous closed configuration family, verified here by the complete six-step transformation.
Are there six moving links?
There are six links in the closed chain. With link 6 fixed, five move.
Are the visible corner angles the axis twists?
No. Twist is the angle between consecutive hinge-axis directions. A projected corner between bars is a different quantity.
Why does an angle jump from 180° to −180°?
Those represent the same orientation. The display wraps its angle convention; the physical animation remains continuous.
What does the camera control change?
Only the view. It leaves all link dimensions, joint angles and closure results unchanged.
Can the two assembly forms switch during normal running?
The control compares distinct closed assemblies. No transition between them is modeled.
Is this the exact straight-line Bricard linkage?
No. The planar straight-line construction uses a different arrangement. This page shows a spatial six-revolute loop.
Primary kinematic reference
Song, C.-Y., Chen, Y. and Chen, I.-M. (2014), Kinematic Study of the Original and Revised General Line-Symmetric Bricard 6R Linkages, Journal of Mechanisms and Robotics, 6, 031002. DOI: 10.1115/1.4026339. The animation uses the geometric example in equation 27 and checks its two closure branches against the full transformation chain. Uniform scaling preserves the example’s joint-angle relationships.
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