Tensegrity Levers

Tom Flemons, July 2015; revised 2018

There are several differences between classical biomechanical theory and the alternative theory proposed by the concept of biotensegrity. The following discussion discusses one of these differences and proposes a synthesis that encompasses both bodies of knowledge.

 A key point in biotensegrity theory is that all of life is tensegrity based and most engineered objects are not. There is a fundamental difference in how life is created and organized, from, for example, how a car is manufactured. From a tensegrity perspective, life is an organic process proceeding from cells to tissues to organs to structural anatomy in an unbroken cascade of nested tensegrities. If life is better described as tensegrity based then mechanical theories about the body  based on engineered objects are necessarily incorrect or incomplete.  Steve Levin discusses this in detail in his papers found here-  biotensegrity.com. Also my papers Geometry of Anatomy and Bones of Tensegrity look into how tensegrity might be operating in the body. Further basic texts can also be found on Graham Scarr’s site as well.

Levin has clearly pointed out the flaws in the standard biomechanical model which assumes the body is full of levers pivoting around fixed fulcrums (joints) with individual muscles pulling across the joint from discrete bony attachments on either side. If the arm for example hinged at the elbow was just a simple third class lever, the mechanical advantage is considerably less than 1.0 and it’s hard to see how any significant weight could be held in the hand without tearing muscles or wearing down the joint due to excessive forces concentrated at the joint. Borelli, the father of biomechanics noted that the (assumed) levers in our body for the most part enable us to achieve a wide range of motion at the expense of a reduced mechanical advantage. 

Furthermore it’s axiomatic of a lever that there be a fixed fulcrum for it to operate against. We can secure a joint (e.g. elbow) and  (supposedly) isolate a specific muscle (bicep) to lift a weight, but most of the time our joints are fluid and unfixed in space, so how can they be acting as fulcrums for bone levers? This make it hard to accept the standard bio-mechanical explanation. Levin’s point is simple, no fulcrums means no levers. So he’s asking us to throw out 500 years of biomechanics and replace it with a better biotensegrity explanation. 

Which is fine I suppose, except biotensegrity doesn’t yet have a body of math attached to it that we can work with to predict and solve kinematic equations. (we’re getting there- the NASA TensegrityRobotics Toolkit software -NTRT is helping model complex tensegrity systems) lf people involved in clinical and scientific research are asked to throw out a bio-mechanical model which is outdated, there has to be something useful to replace it with. Besides, it is still difficult to understand how the body operates without invoking levers. Even if the fulcrum side of the equation is problematic, the way our limbs fold and extend certainly look a lot like levers. The way we rise up on our toes certainly seems to be a second class lever operating. And yet while muscles may insert via the periosteum across a distance and not at a single point, the mechanical advantage still seems inadequate to the task of lifting a significant weight any distance. 

So what is another way of looking at this? Over the years I’ve used the term floating fulcrums occasionally without giving it much thought. How can there be a lever operating without a fixed fulcrum that provides the resistance? The body is structurally a closed system – there is no fixed point that provides leverage to do work and any point in the body can contract to generate movement. And yet we are much stronger that we should be given what we know about simple machines. Rather than discard levers I want to suggest something different. I want to describe a type of lever that doesn’t require fixed fulcrums. If we could then also show a way that a tensegrity linkage could do the same work as a traditional lever or a mechanical linkage then we are describing equivalent structures that answer to the same kinematic equations.

This video illustrates a possible way forward. Briefly what it shows are 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 wants 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. The lever floats in space and does it’s 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 second model 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 it’s 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.

Tensegrity systems can be described as floating interconnected levers operating without fixed fulcrums in networked arrays. Looking for equivalent structures in the body, we find bones coming together with muscles, ligaments and tendons crossing and securing the joint, and with fascia wrapping each component and the entire joint latitudinally and obliquely in a complex multi-tiered manner.  Something very similar to a tensegral weave pattern connects everything via the periosteum to the bones. A simple model of this is a double spiral tensegrity mast (T-spiral mast jpg.) You can see how each strut is linked via slings such that the entire structure supports and transmits forces as a collection of first class levers. The fascial weave that wraps and passes over each joint could be describes in the same way and account for why our joints are so much stronger than third class levers should be. 

This is one route from lever linkages to tensegrity linkages. Another route is illustrated in the attached diagram entitled T-levers. In it I diagram a progression from a simple first class lever with a fixed ground based fulcrum, through three suspension based fulcrums (the second and third causing no bending moments)  to a structure where the lever is stabilized in an interconnecting tensional network such that the ‘output’ of one lever inputs as the ‘fulcrum’ for another lever. This interconnected linkage is the archetypal tensegrity system known as an expanded octahedron tensegrity. All tensegrities involve compression members connected through tension slings  and the diagram can be seen as 6 levers suspended from fulcrums each of which are composed of two lever ends and one suspension sling. The elegance of such a system bears testament to its strength and integrity. Each lever is held in position by four tension members at each end and cannot move. A force acting on one lever is quickly transmitted through the system distributing the load equally to all parts.

Tensegrities achieve their stability through the prestressing of their components- the tension net is taut and the compression members are under pure compression- no shear and no bending moments. A minimal energy tensegrity system (and all tensegrities have to be seen as structures where all parts act systemically) contains just enough prestress to maintain it’s maximal possible volume. Any less and it is a deflated system- any more and the overall tension of the system rises. The higher the prestress the more rigid the body. A force acting on a tensegrity system whether endogenous or exogenous, propagates through the system at speeds and effect proportionately to it’s tension state. A visco elastic system or a loosely coupled complex tensegrity like the 4-prism system (or a human body) has a certain amount of slack built in which dampens and absorbs some of the force as it propagates outward from the source. Because all tensegrities have multiple lines of tension radiating from each node, propagation trees are complex and non-linear because the network is reiterative. 

One of the reasons tensegrity systems are so strong and resilient is because force propagation is distributed efficiently through multiple paths of redundancy. It’s possible for a significant section of a tensegrity to be damaged and yet still be able to maintain structural integrity. This seems to be a property of all complex systems- the four lever/ three fulcrum model demonstrates similar integrity.

To sum up- I’m suggesting that tensegrities can possibly be modelled as complex networks of linked levers. Any strut under force can act as a lever and the rest of the structure then acts as its fulcrum. Progressively uncoupling struts from the fixed system allows increased ranges of motion and degrees of freedom in that segment of the system. Alternatively linking fixed components in an unstable relationship (a rhombic linkage) will net some features very similar to the kinematics found in the body. The anatomist Jaap Van der Wal noted that a hinge or joint in a  tensegrity structure is more of a disjoint- a locus where some ROM (range of motion) and DOF (degrees of freedom) are permitted by partially de-coupling the tensegrity truss. Tensegrity disjoints can be constrained in their range of motion and degrees of freedom (e.g. limiting hyperextension) by additional tension lines which can serve as actuators in robotic and prosthetic applications.

Articulating systems, that is, systems which have loosely coupled properties that include flexible spines, and jointed appendages fall outside of traditional definitions of tensegrity structures. For the most part the discoverers of tensegrity, the inventor Buckminster Fuller and the sculptor Kenneth Snelson concerned themselves with closed systems of fixed or frozen geometries either spherical or asymmetrical (though technically they oscillated a tiny amount). Nothing in the definition, however, prevents the design of articulating tensegrity systems composed of linked tensegrities, that can emulate the complex movements and rotations of  living bodies in motion. 

Based upon the above argument I think it conceivable to posit that the structure of (any) living body consists of levers (of all types) embedded in an enveloping tensegrity system that allows them to operate without fixed fulcrums”. And a corollary is that it is conceivable that there exists an equivalent unique tensegrity system for any complex linkage. Perhaps this approach may allow a new mathematical approach to analyzing tensegrity structures and tensegrity systems for a variety of purposes including the depiction of vertebrate life.