Tom says that after years of work trying to create a tensegral joint, he feels satisfied with this bistable tensegrity linkage. The design is inspired by a bistable closed chain of four levers. If a fulcrum is removed, the structure continues to operate. This same kinematic structure is built as a tensegrity by coupling four triangular tensegrity prisms in a four fold (rhombic) array. There are no fixed fulcrums left.
Context: Tom Flemons Archive and Tensegral Linkages
A closed chain of four levers is bistable
Tensegrity Levers (July 2015) The video shows two models. First, four levers linked in a closed chain such that the output of one is the input of the next. If the fulcrums are fixed and at a height that allows each lever its full range of motion, it can be shown to be a bi-stable linkage. It is unstable when all the levers are horizontal and want to flip into one of two tetrahedral geometries at either range of its motion. If one of the fulcrums is removed, surprisingly the structure continues to operate without much noticeable impairment. [illustrated at 0:35 in the video] The lever floats in space and does its work as before and yet there is no fulcrum supporting it and therefore no bending moment in that lever. It could be said then, that the entire linkage system performs the function of the missing fulcrum – i.e. the whole substitutes for the part. This is beginning to approach a description of how tensegrities work. But for our purposes it’s enough to note that it appears possible to have a lever working without a fixed fulcrum. Of course if we take away all of the fulcrums it’s just a chain of bars lying on the ground. But surprisingly the structure will do some work if only two adjacent fulcrums are left. As an addendum, if three levers or five levers are linked together they remain locked in the horizontal plane and can do no work. Six levers have some interesting properties as well but four levers suffice to make the point and may even have promise as a useful bistable linkage (albeit in a tensegrity form).
The four lever linkage built as a tensegrity (no fixed fulcrums)
Tensegrity Levers (July 2015) The second model [at 1:15 in the video above] shows the same kinematic structure of a four lever linkage built as a tensegrity. Four triangular tensegrity prisms are loosely coupled in a four fold (rhombic) array that shows identical properties of movement, and geometry. The tensegrity linkage is bi-stable and wants to resolve the connected hub into one of two tetrahedral shapes in the same manner as the connected levers. Points to note here include the fact that there are no fixed fulcrums left, and where before there were linked levers, there are now tension members connected to the ends of four struts which radiate out into the rest of the mechanism. In other words, the elements are reversed – where there were solid bars (held in pure compression) acting as levers there are now tension members that link separate integral complexes (each prism is self supporting). There are no bending moments because no fulcrums bifurcate levers. And yet this tensegrity system does the same work as the four linked levers – so they can be considered equivalent. This cluster of four triangular tensegrity prisms defines a bi-stable rhombic hub that flips from one tetrahedral geometry to its opposite. As it does this it creates two interconnected revolute hinges that fold at 90 degrees to each other. This linkage may prove useful in designing complex tensegrity joints in robotics and prosthetics as well as modelling complex joints in the body.
Significance of this linkage
(Aug 8, 2015) I’ve been trying for years to build joints that I would define as tensegral and I think with this latest model [at 1:15 in the video above] I have come up with something that finally fits the definition. (I have also built a saddle sling between two tensegrity tetrahedrons which I would also consider a tensegrity joint complex.) All the components are in pure compression or tension and the rhombic linking of four tensegrity prisms creates a complex joint that is two revolute joints arranged at 90 degrees to each other. It is a bistable oscillating joint complex that allows ROM and DOF with no torque or bending moments. Of all the structures I’ve built I think this one has the most potential for the field of applied tensegrity.
(Feb 24, 2016) I think I’ve figured out a way to use my four fold tensegrity prisms linkage to create a pelvic analogue that could simply and effectively create a gait pattern in legs using very few control lines. This just came to me yesterday and I’ll need some time to work out the details…
Earlier description of the Jacob’s-Ladder-like linkage
(Feb 23, 2014) You may be interested in viewing two videos I sent out to the other recipients of this conversation about two months ago. They are pursuant to my point about how to animate semi-compliant joints. Below is the text that attached to the videos. The bistable criss cross wiring allows for a joint that is not pin jointed and moves more fluidly between two states. I think this is the way to create joints that look and act plausibly.
(Jan 2014) Hi all, I am sending out these short videos selectively to those who I think might have an interest in them, on the proviso that they be kept confidential for now (I’m not sure what these mechanisms are good for yet… and they may have IP value).
What you’re looking at is… I have shot two videos of a tensegrity joint/hinge I have been working on for a long time. They are two variations on a series of segmented components linked by passive tension slings but also crosslinked in the manner of a Jacob’s Ladder toy (JT). It was Steve Levin who pointed to the possibility that this is how joints can move with minimal energy in the body. One version is based upon stacked tetrahedrons; the other is stacked octahedrons.
To begin with, a simple tension sling that suspends two ‘Y’ or ‘X’ struts in the form of a tensegrity version of a universal joint is quite well known by now. e.g. Robin Skelton mentioned to me over a year ago that he employs such a hinge ocasionally in his work, and all of my early attempts at biotensegrity linkages were based on such a structure. They are inherently unstable and must be constrained with further tension lines in two axis to stabilize them. Varying the tension of these secondary lines can control the movement in one plane and allow it in the other to create a simple hinge. But it is an unstable linkage and these secondary support guy lines may impede the joint function. (cf Snelson’s original ‘X’ sculpture which froze the two ‘X’ components into rigidity.) This may or may not be a feasible way to build articulating tensegrity structures depending on how much of a load they can handle. I think it isn’t sufficient as the control lines have to handle large forces to control the movement and the angles of connection of these control lines mitigate against precise adjustments.
And it also doesn’t satisfy the necessary conditions required to give it complete descriptive power of anatomical joint. It was Tom Myers who pointed out not unreasonably that anatomic structures did not directly exist to make this a plausible explanation of how any joint in the body is suspended. To answer this objection I wrote my paper The Bones of Tensegrity in 2012. In it I suggested that the fascia that wraps the joints between the bones must play a part in ‘floating’ the surfaces of the bones so that they did not under normal circumstances come into contact. Specifically I proposed that the fascial wrap could create a compression column when necessary. Since then, in conversation with Mark Finch it was suggested to me (for example) that it was likely the erector spinae muscle bundles and accompanying fascial structures that acted like compression elements between the processes and helped keep the spine uncompressed.
These videos add the next element to the puzzle. I’m not qualified to suggest what if any of the necessary structures exist in the body to emulate the action these videos show. But I thought it was of sufficient interest to show what is possible in a tensegrity joint. I have essentially built a self supporting tensegrity version of Jacob’s Ladder.
In any case my interest here is in seeing if this kind of linkage could be useful in designing robotic, prosthetic and exoskeletal platforms that can be manipulated with minimal energy inputs. The Jacob’s Ladder linkage can be said to be bistable, that is, there are one of two positions any two components can assume with respect to each other that are stable. All other orientations are by definition unstable. The linkage traversing between the two resting states has potential energy that resolves very quickly with almost no energy input. There is a threshold the linkage must cross before it will slip easily to its opposite state which is the necessary energy input required. At that point the system is de-powered. From my experiments I think this mechanism can be controlled in a number of ways that will make it useful for articulating tensegral constructions.
The octahedral components stack linearly with simple tension slings as a first order of connectivity. The second order are two tension lines that cross between the struts of two elements such that they can ‘see saw’ easily between two extreme angles. In this sense I call them ‘cross linked’. Note that I have cross linked them in series (or one dimensionally) in a single plane. They could also be additionally linked in a second plane oriented 90 degrees. This would allow them to undulate in both planes at the same time. (See the tetrahedral video e.g.)
One advantage of using octahedral units rather than tetrahedral is the ease at connecting further units off the sides of the mast at right angles. This goes back to a historical dispute between Snelson and Fuller as to whether the ‘X’ element was the key building block of masts or whether the stellated tetrahedral module was superior. Fuller went with the tetrahedron but it lacks the easy ability in this orientation to spread out into two and three dimensional arrays. Octahedral based masts are X-masts with a third strut added to completed the octahedron. They can be expanded in any direction by sling linkages and additional cross linkages. Imagine a wall composed of an array of tensegrity octahedrons cross linked in this manner distorting and undulating. Or perhaps a walking tensegrity quadruped like ‘Big Puppy’…
The tetrahedral stack bears a better resemblance to the spine and in the video is cross linked in two axes so as to emulate all the complex motions a spine can generate. I have added this spine to a model of my tensegrity pelvis. And for contrast, I have included a similar model using the octahedral stack. What is missing from these models is a third set of control lines which generally run vertically down the mast and help stabilize it. It’s not clear to me yet whether these are needed as much when these structures are cross linked or whether they could be tied into the cross linked tension system. Will build that next… I think this mechanism should find a use in the robotic community whether or not it has any bearing on anatomy.
