Can Tensegrities Model Articulations?

Articulations can be created as breaks in the triangulation. Definition of articulation. The spine articulates in graduated steps. The octopus has sub-visual articulations mediated by hydrostatic forces. Muscles and ligaments alone are insufficient for allowing a vertebral skeleton to move and handle loads. In order for the skeletal framework to articulate effectively, the joint complexes must be assisted by fascial envelopes which act as tensegral structures equivalent to invertebral hydrostatic bodies.

Context: Tom Flemons Archive and Advantages of Tensegrity

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Difficult to use tensegrity to describe articulating anatomical structure

(Oct 13, 2015) I remember a conversation I had with Snelson years ago when I approached him about the possibilities of employing tensegrities to describe anatomical movement. He couldn’t see how it could be done given the enormous prestress a tensegrity of any size or weight is experiencing. Oddly, he felt that vertebrate activity was more akin to a marionette’s movement… He was expressing the same reservations I had regarding importing wholesale tensegrity concepts into the domain of articulating anatomical structure.

It bears repeating… tensegrities model envelopes, and as such any joint has to be understood as a failure in the tension system to constrain all degrees of freedom and ranges of motion. In a tensegrity, a joint is a ‘disjoint’. Because of this fundamental constraint there has been a great deal of misunderstanding in the biotensegrity community as to how the body functions as a tensegrity structure.  I’ve tried to come up with a plausible description of how a myofascial sleeve similar to the arm video I sent you could account for this but I don’t get the feeling that it was understood well. (See my paper The Bones of Tensegrity). Some in the biotensegrity community seems to think that there are no levers in the body because there are no fixed fulcrums. This is plainly wrong – in a tensegrity system levers are contingent (as long as they need to be, including crossing joints which can be stiffened) and fulcrums are provisional (provided by the entire structure as needed).

Articulations can be created as breaks in the triangulation

Tensegrity Levers (July 2015) 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 and DOF are permitted by partially dis-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.

Definition of articulation in a tensegrity

(July 26, 2015) I suggest that an articulation could be defined as a node where two or more tensegrities interact to alter a force trajectory. Appendages have distinct (discrete) joints which possess various ranges of motion and degrees of freedom and allow forces to be transmitted across joints.

The spine articulates in graduated steps

(July 26, 2015, continued) The spine is a more subtle articulation in that the redirection of forces proceeds in graduated steps – the range of motion and degrees of freedom are restrained between any two vertebrae but cumulatively they add up to significant redirections.

The octopus has sub-visual articulations mediated by hydrostatic forces

(July 26, 2015, continued) It may appear that an invertebrate like an octopus has no discrete joints and thus no articulations but this goes against what our eyes tell us. For an octopus to bend, stretch and coil its tentacles there are sub-visual articulations happening – hydrostatic forces allow muscles arranged longitudinally and transversally to push against the internally generated pressures to generate movement. See Dynamic model of the octopus arm. I. Biomechanics of the octopus reaching movement (PMID 15829594)

This can be modelled exactly as a spiral tensegrity mast which contains transverse spirals of compression which are equivalent to the internal hydrostatic pressure in the tentacle.  Each of the struts that spiral around the column represents the hydrostatic pressure which the muscles push against. In the model it can be seen that there are discrete angles each strut assumes tangential to its spiral. It is, if you will, the averaging out of all the forces in a tangential spiral into discrete steps or struts. If all the transverse muscles contract at the same time they try to assume a shorter distance from one end to the other resulting in the arm elongation and narrowing but the interior volume remains the same. Longitudinal muscles pulling the tentacle in one direction or another are equivalent to longitudinal control lines introduced into the tensegrity mast to cause it to bend. The articulations happen within the flexible hydraulic structure at the level of individual atoms.

Why we need bones

(July 26, 2015, continued) As to why we need bones i.e. a vertebrate structure, consider that the largest land invertebrate is the coconut crab which is about a foot in size with claws that reach out another foot. The only larger invertebrate was the extinct arthropleura which got to six feet long.   Notice that neither get very far off the ground – issues around breathing and maintaining internal pressures which generates considerable heat are why we need an internal skeleton. This reduces the need for some of the internal pressure which allows us to be larger and more vertical. The fascial musculature is our invertebral wrap that overlays and augments the internal vertebral skeleton. This is why I believe the body needs to be modelled at two levels as a complex tensegrity structure. The internal skeletal structure is composed of discrete articulations that are visible fulcrumless levers surrounded and multi-level wrapped with a tensegrity like spiral or braided mast where the discrete articulated steps are at the atomic level. But I’m suggesting it can be modelled as a tensegrity mast where each strut (or section of braid) acts like both a first class lever and part of the fulcrum as needed.

Can tensegrity be used to describe vertebrates?

(Aug  8, 2015) My latest thinking revolves around pondering if it really possible to use tensegrity descriptions when we talk about simple jointed or articulated rigid bodies. Rigid bodies in vertebrate biology must mean bones – no matter how elastic or plastic they are in some circumstances they are still much much stiffer than other tissues and as such support significant compression loads when forces move through the body. It is significant that the largest land invertebrates have exoskeletons (Coconut crab) or the prehistoric arthropleura and do not get more than a few inches off the ground. It seems that land animals of any size needed an interior skeleton to help support their weight and movement. But it also seems clear to me that a simple vertebral skeleton can’t handle loads imposed on it or to move its segments by means of muscles and ligaments alone – as Borelli pointed out our lever arms emphasize range of motion and degrees of freedom at the expense of mechanical advantage. In other words the fascial matrix which is our hydrostatic skeleton is necessary to augment the vertebral skeleton. I’ve struggled for a long time (decades) with trying to understand how it is possible to use tensegrity descriptions to describe articulating life forms i.e. vertebral quadrupeds and bipeds. There is a good reason that initial definitions of tensegrities do not talk about articulations or joints – this is because by and large tensegrities best represent envelope structures i.e. invertebrates. Almost all tensegrities have discrete compression members that follow geodesic tangents cutting across an interior space defined by a tensional membrane composed of many tensional force vectors. The more complex the assembly (the greater the number of components) the closer the compression members are to the peripheral tensional membrane.

Tensegrity Anatomy

(May 2018) In the 1980’s I was designing and building tensegrity spines based on some intriguing similarities I had noticed between cow vertebrae and stacked stellated tetrahedrons. I found I could link stacked tetrahedrons together to form a tensegrity mast that resembled a spine but I was stumped as to whether this was a plausible way to describe how spines actually worked. I spent days in a medical library at the University of British Columbia trying to unravel the mystery and see if anyone else had noticed this similarity. I came across a reference to an orthopaedic surgeon, Dr. Steve Levin who was doing work in this area but for several reasons I didn’t follow up on this for years. It wasn’t until 1999 that I looked him up and tracked down an internet address and we began to correspond. He encouraged me to build models of the body and I began to experiment with building tensegrities that were investigated this in more depth. I’ve chewed on this problem now for over 30 years and have discovered some interesting correlations and possibilities but also some fundamental problems that still need to be addressed. From the vantage of those years it looks to me that tensegrity better serves as a complex metaphor than as a claim that anatomy is literally tensegral.

The tensegrity model of individual cells was proposed by Dr. Donald Ingbar in his seminal paper on cells and tensegrities in 1998. In it he notes the comparative similarities between cell membranes and the microtubules that travel through them and the basic components of a tensegrity system – a tension network that acts like a membrane held suspended by a series of discontinuous compression members. Observations of how cells behave mechanically seem to correspond well with tensegrity principles and he was the first to propose that cells can send signals across networks of cells by means of mechanotransduction i.e.  a mechanical force exerted on one cell is passed onto surrounding cells by means of a simple transfer of energy that looks  and acts the same as clusters of tensegrities do.

Tissues and organs could then easily be modelled as arrays of basic cellular units, and nested cascade of tensegrities at different scales describes a key point in the theory of biotensegrity. As a model that describes the architecture of biology systemically tensegrity is a pretty good fit for describing microscopic behaviors of cells all the way up to the behavior of individual organs and systems of organs.

However it is not clear at all what we mean when we say that the structural anatomic body is a tensegrity or is tensegrity like. Trying to match basic geometric tensegrity architecture to the body demonstrates some intriguing clues and reveals some surprising correlates, but the closer one looks the more problems arise as well. When the kinds of movements found in cells, tissues and organs are described, distortions of a basic tensegrity as a nonlinear auxetic system answers well. But it’s important to realize that nowhere in the original description of tensegrity structures was there a definition that included joints and movement that we find happening at the macro-scale in the body. Tensegrities were never seen as mechanisms, that is, they were not kinematic chains connected by means of revolute or prismatic linkages, rather they derived their stability from a continuous network of tensional triangles. A failure in the triangulation could create movement but the goal was the opposite – to maintain stability by ensuring that the tensional triangulation was continuous and unbroken.

Related discussion: The Need for Levers with Floating Fulcrums