A Bistable Four-Lever Linkage with Floating Fulcrums

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…