Tensegral linkages create controllable joins between tensegrity modules. Constraints on linkages: geometries must match and symmetries must match. Rules to describe how tensegrities meet. Linkage designs include face to face, struts kissing, revolute joint, universal joint, interpenetrating modules, and sliding struts.
The need for tensegral linkages
(April 11, 2015) I’m studying the science of linkages and kinematics in general and realize that a lot of classical engineering will have to be rewritten in light of articulating tensegrities. Compliant joints for example need to be evaluated differently in that global distribution of forces are integral to understanding tensegrity systems. Augmented levers and floating fulcrums are some of the terms I’m playing with – it’s clear that joints in the body are not simple levers and it’s also clear that the strength in the body can’t be accounted simply in terms of 1st 2nd and 3rd class levers. I think that fascia sleeves (which have a tensegrity architecture) wrap muscle/bone complexes and contribute to the ability of our bodies to handle large loads. What we are capable of achieving in robotics at this point are crude low res approximations of anatomy. By necessity our biomimicy is approximate and contingent on the techniques and materials we have at hand. So short cuts are needed – models which are under-actuated and yet can emulate biologic activity requires some innovative thinking. I think the key is solving the linkage problem and that is where I’m putting most of my efforts. I’ll let the roboticists and the computer scientists figure out how to actuate and control the structures – but form finding comes first.
Constraints on linkages
Need matching geometries
Matching geometries are illustrated in this video showing a linkage between tensegrity tetrahedrons.
Form-finding algorithms could use constraints to create matching geometries and matching symmetries, as discussed in Simulation Could Help.
(Oct 13, 2015) As for form finding through genetic algorithms, I may be ignorant here but it seems to me that you are better off to winnow the field of possible forms first before you subject them to an iterative control finding algorithm. Because I spend a fair bit of time building models I know pretty well which geometries fit with each other – how three-fold, four-fold and five-fold patterns with their accompanying rotations line up and how to connect them to good advantage. Why not start with some of the geometries I’ve identified as probable contenders and go from there? I understand it is tempting to build something like Tomohiro Tachi’s bunny and then run it through the algorithm till it hops about, which would be cool… but if the goal is to actually build something real maybe it would be better to limit the attempt to what is possible now. I’ve figured out a lot of the basic components to enact my approach and am willing to have your students have a go at making them link up and seeing what happens.
Need matching symmetries
(March 9, 2016, while discussing Liu et al Structural Transition from Helices to Hemihelices) I think Graham’s idea of knitting hemi-helixes together is interesting. Attached is an image of a dragon tensegrity I built 30 years ago that was 20 feet long. I knitted a number of helical tensegrities together in a similar manner to what Tomohiro Tachi suggested in his 2012 paper Interactive Freeform Design of Tensegrity. It’s really a matter of matching symmetries – in a cylindrical torso facets with three struts can receive a threefold mast appendage for example. Or a neck can expand to fit onto a spherical head tensegrity etc.
Rules to describe how tensegrities meet
(Nov 6, 2015) Re: rules to describe how tensegrities meet… well I’m working on that and will have something to share in the next few days. I think I can lay out the basics in such a way that the rules are inclusive enough to allow new ideas and attempts while at the same time saving everyone a lot of time trying things I’ve found don’t work. There are so many parameters to deal with though… prestress stiffness, shock loadable tension members i.e. elastic vs. fixed length, redundant tension members to add stability, adding additional compression struts for added stability or to alter the shape of the structure etc. etc.
(Dec 4, 2015) Definitions:
Tensegrity – is a three-dimensional non solid structure composed of struts and cables. The struts are always and only ever loaded in pure axial compression held isolated from each other by means of a set of lines connected to the strut ends in a continuous tensioned network.
Struts – chordally pass through the imaginary body of the structure defined by the envelope of tension cables. The larger the number of struts, the more complex the structure and the closer the struts are to the boundary of the envelope.
Cables – connect to struts at their ends and are knitted into a collective tensioned network that defines the boundary of the structure.
Faces – tensegrities by the nature of an inherent geometrical order are constructed having a number of polygonal boundary facets defined by the tensional cables which are not solid surfaces and are planar – these are referred to as faces. They may have any number of sides but 3,4,5, and 6 sides are sufficient to build most tensegrities. Skewed faces are found along the sides of tensegrity prisms. Depending on the number of struts in the prism the skewed faces can have 4 or 5 edges.
Rhombic faces – In addition to planar polygons, there are also naturally occurring dihedral rhombic shapes formed by tension lines travelling from either end of a chordal strut to the ends of two adjacent struts that are on the periphery. This creates a dihedral angled diamond shaped facet composed of four cables and one strut.
Membranes – a potential network of prestressed lines can be replaced in all or part by a membrane which attaches to all of the nodal ends of all of the struts. Membranes disperse forces through many pathways not available to a line based tensegrity. Generally for ease of construction and clarity membranes are left out of models but they should not be discarded as in some cases it may be important that the tension network is expressed by membranes and not the cables.
(Dec 4, 2015, continued) Articulating Tensegrities
There are two primary ways to employ tensegrity structures as articulating robots. The first way is to actuate a large proportion of the prestressed tension lines that make up the tension envelope. By altering the lengths of the lines the shape can be distorted in ways that allow the structure to move. While there is nothing to prevent this strategy creating movement in a quadruped or biped tensegrity it does require a large number of control lines, a significant amount of energy, and a sophisticated command and control system. Undoubtably animal life is a tensegrity structure that has solved the problem of control using this method to pull on over 600 muscles but the present state of tensegrity robotics won’t be able to simulate this level of control for a long time to come.
But there is another way to go about it…The second way is to join individual tensegrities together by means of separate tensioned connecting slings and causing them to move relative to each other by a third set of control lines. This requires hinges that are unique to tensegrities. The advantages to this method include simpler actuation, less energy and a simpler algorithm to control the tensegrity linkage. For asymmetrical forms such as quadrupeds, a cluster of tensegrity modules separately actuated is likely the only way to achieve significant movement for the foreseeable future.
There are a few basic ways to hook up separate tensegrities. A general rule of thumb is, choose linkages that geometrically match each other. In other words it is not elegant in form or function to attempt to join differently numbered polygon faces together. Triangular faces mate with triangular faces and squares mate with squares. A triangle mated to a square or pentagon means the forces that pass through the faces are not symmetrically distributed and failures are possible.
Face to face – This is the most likely way to hook tensegrities together. Snelson’s famous floating needle sculpture is composed of a face stacked series of tensegrity threefold prisms that diminish in size as it climbs. Each prism is rotated 60 degrees from the adjacent prism so the struts don’t line up. Their face ends are connected by means of a hexagonal tensional sling that replaces the two triangular ends of the individual prisms. This creates a stable cylindrical mast that oscillates very slightly like a grass stalk in a wind but does not articulate. There are two versions this mast can take – because each prism is either clockwise or counterclockwise a stack of them can either all have the same chirality or they can alternate chirality which makes a mast that is more balanced as the rotations cancel out.
Struts Kissing – If the struts are aligned so that they touch, it is possible to build a mast that looks and can act like a three legged scissor jack. If a compression load does not pass across from one strut to the other it can be considered a class 2 tensegrity. This refers to the number of struts that contact each other. The struts meet end to end and hinge inward and outward from the axial centre. Vertical and lateral control lines can be used to telescope the mast longer or shorter. However if the two faces are completely connected to each other, there is no articulation or movement from side to side.
Revolute joints – a hinge between two tensegrities is created when 2 struts from one tensegrity are allowed to revolve around a cable sling connecting them together. This is equivalent to an edge to edge bond between two solid objects. In a properly constrained revolute joint there is one degree of freedom.
Universal joint – Two tensegrities can be joined such that there are multiple degrees of freedom in revolution. This can involve two X or Y modules linked with a saddle sling.
Interpenetrating modules – Two modules e.g. 4prisms can be interlinked to form a revolute hinge. Other interpenetrating linkages are possible.
Sliding struts – It is possible to design a telescoping mast by allowing struts to slide along cables. This creates the possibility of extruding appendages from a central tensegrity body. (Feb 6, 2017) Adding additional tensegrity prisms that slide along a tensegrity module’s tension network creates telescoping prismatic joints. These joints function somewhat like the iris in a camera lens to form contingent protrusions that can walk or even propel the structure forward. Tom illustrates telescoping appendages in this video:
There are more possibilities for linking tensegrity modules but these cover the basics for now. The NTRT software should be able to emulate all of these methods in the construction of complex tensegrity linkages.
Further discussion and illustration of tensegral linkages in New Approaches to Mechanizing Tensegrity Structures