Hoberman linkage expanding geodesic dome designed and modeled by Michael DiNardo.
We've all seen the famous Hoberman sphere, and its popularity is a testament to its beauty and ability to spark the curiosity within people of all ages.
Some History
The first Geodesic Dome was built in 1924 by German engineer Walther Bauersfeld for the housing of the projection Zeiss-Planetarium in Jena, Germany and proved to be such great method for building large and strong structures with minimum material requirements that it inspired American architect Buckminster Fuller to extensively study and characterize the geometry of geodesic spheres and their structural applications. Geodesic spheres provide an extremely high strength to weight ratio compared to structures of comparable internal volume making them ideal for space exploration applications and for creating large structures at low cost.
The construction of the first geodesic dome, the Zeiss-Planetarium, designed by German engineer Walther Bauersfeld in 1924.
The projection Zeiss-Planetarium in Jena, Germany
Expanding Geodesic Structures
The first example of an expanding geodesic dome was made by American engineer and inventor Chuck Hoberman in the early 90's. Hoberman studied the work of Buckminster Fuller and invented the Hoberman linkage to solve the problem of making an expanding version of the Geodome. Having the ability to expand and collapse a structure as strong and light as a geodesic dome with a single degree of freedom makes it even more appealing as a deployable structure for space applications, however, the Hoberman linkage isn't without its limitations. For instance, the expansion ratio of a structure made with Hoberman links is directly related to the number of links in the structure. The more links there are the greater the expansion ratio, and the more complex as well.
"Everything should be made as simple as possible but no simpler." - Albert Einstein -
How does the Hoberman Link work?
How does one make a single degree of freedom expanding structure using only scissoring mechanisms? I spent some time looking online for resources that could explain how the linkage works and how to use it in design, and after finding nothing of help, I decided to spent some time thinking about how it works myself. In my exploration of the subject, I found the characteristics of the Hoberman Link that allow it to be geometrically programed to follow a desired path, and a method to create expanding structures of various polyhedral configurations using any CAD program. Some of my peers from Ventura Community College and from UCSB would like to know how it works, so I decided to share what I learned here.
Geometric Observations
I used the CAD program Solid-Works to discover the basis of the Hoberman link by first experimenting with regular two dimensional polygons and a single link as shown below. The triangle represents a rigid link and its vertices represents a linkage pin, Furthermore, the lines connecting the vertices of the hexagon to its center represent the path each pin will be programed to follow.
You may notice right away that the internal obtuse angle of the triangle is the same as the internal angle of the hexagon. This was my first observation. Another important observation is that the acute angles on the triangle match the angle of the wedge formed by adjacent paths. This is enough information to create an expanding linkage system based on any regular polygonal geometry.
Simple Relations
The angles of the triangle formed by the three pins of a Hoberman Link have the following relationship for all regular polygonal path systems:
Let n = the number of scissoring pairs in the regular polygonal linkage system:
We have,
α = 360/n
α = 2β
180 = α + ϕ
ϕ = 180(1 - 2/n)
This is enough information to create an expanding regular polygon of any order.
The magic of the Hoberman link is that the angle between the pins remains constant throughout its range of motion making it possible to program the paths of a system of links through a plane. Furthermore, we can analyze the pair of scissoring links with Grubler's equation for planar linkage systems to find that it has one degree of freedom.
Grubler's equation:
DOF = 3(L-1) - 2P
where,
DOF = Degrees of freedom of the system
L = Number of links
P = Number of pins
We have 1 Pin and 2 Links which gives: 3(2-1) - 2(1) = 1DOF
Once the path of the pins is known it is easy to construct an expanding geometry.
Knowing this we can easily construct an expanding planar geometry like the one below:
This linkage system is based on a hexagon with 12 links and 18 pins. If we analyze it using Grubler's equation as we did with the two link case we can see it is over constrained.
Grubler's gives: 3(12-1) - 2(18) = -1
This is because one of the two pins that close the loop is theoretically unnecessary and the linkage system will move with a single degree of freedom without the extra pin.
Analyzing the system again without the extra pin we get: 3(12-1) - 2(17) = 1 degree for freedom.
In reality, when these linkage systems are physically built, they will move with 1 DOF even when 18 pins are used. However, when modeling and assembling a system like this in a CAD Program like Solid-Works or Creo the extra pins should be left unconstrained to prevent binding - this will save you a lot of headaches if you want to create animated models.
If you would like to create your own expanding linkage system based on a regular polygonal geometry and don't have access to a CAD program, you can use my Hoberman Linkage Calculator to calculate the distances and angles between the pins of a linkage system programed to follow the geometry of any polygon with anywhere between 3 and 100 sides.
You'll notice if you play around with it that the expansion ratio of the geometry (expanded diameter)/(contracted diameter) increases as the number of links increases.
All of the relationships discussed thus far hold for regular planar expanding polygons, but in order to create more complex expanding 3-D structures with irregular path angles more information is needed. I've found some more general relationships between the pins and paths of a Hoberman link that have helped me create some pretty cool structures - I'll share these relationships with you now.
The General Case
In the image below the grey lines are coplanar and represent the desired path of the pins of a single Hoberman link and the black triangle represents the triangle formed by the pins of said link.
Beautifully simple... In order to program a scissoring linkage to follow any desired planar path of angles α and β, the triangle formed by the pins must have this relationship:
180 = α + β + ϕ
Using this relationship one can create a scissoring linkage pair that can follow a desired path while keeping the angles between the pins constant as shown below. There's was one more realization that helped me simplify the design of these linkage pairs, and that was to realize that the lines connecting the pins of adjacent paths are equal for both links and by setting these lines to equal lengths in a CAD program the above angular relationships are automatically satisfied.
The simplest way I've found to create a Hoberman Linkage pair in any CAD program is to first define the desired paths of the link pins, draw two triangles representing the distances between those pins and set the distances between pins on adjacent lines to be equal. See the demo below:
This method can be used to easily find the dimensions of a linkage pair that will follow a desired path and allows for the creation of a wide range of expanding structures.
Creating 3-D Expanding Linkage Systems
3-dimensional linkage systems can be created by intersecting two or more dimensional planar linkage systems with the addition of an intersection link. This is generally straight forward when intersecting planar linkage systems based off or regular polygonal geometries because all of the links are the same. Take the geometry below for instance.
Looking at this linkage system we can see it is comprised of four planar linkage subsystems each based on a dodecagonal geometry (12 sided polygon) all connected with intersection links.
However, in practice, when creating a linkage systems like the one above you want to design from a 3-D perspective. This linkage system is actually based on the geometry of a cuboctahedron where the path between the vertices and the center represents the paths of the intersection links and the planes formed by the point at the center and adjacent vertices represent the planes the links follow throughout their travel. I will try to illustrate this below.
To create a 3-D expanding geometry based on a polyhedral geometry like the above cuboctahedron. Start by drawing the geometry you'd like your linkage pins to follow. Then draw lines connecting any vertices to the center of the desired point of contraction which is in the center of volume in the above example.
Next you want to analyze the wedges formed by the lines connecting the vertices to the center of contraction and note if the wedges have various angles or if they are all the same. If the wedges have various angles like in the case of a geodesic sphere more thought needs to go into how to get all the link pins to follow the same path toward the center of contraction at the same radial rate while having different angular rates of travel. I'll leave that for another time, but for now lets look at a method to create an expanding Cuboctahedron which a has the same angle between all of its intersection link paths.
Since the cuboctahedron has sections between intersection links that are all the same we only need to design for a single scissoring linkage pair and a single intersection link. See the video below:
By creating a plane within the cuboctahedron we can draw a scissoring link pair that travels along the desired path just as we did in the 2-D case. The extra parallel lines added to the Linkage system represent the pats of the intersection links that join the four planar linkage subsystems. The lengths you specify for the joining link and the base of the linkage triangles are arbitrary and can be set to any size that fits you application. The angles of the the linkage triangles are all that really matters.
As discussed, all the linkages have the same dimensions in this system, so there is no need to draw any more of the linkages once the dimensions of one is found.
Once you've given the scissoring linkage triangles their required relationships and set the lengths of a single link and the intersection link you can use those dimensions to create your cuboctahedron out of any material you like.
Using the above methods in Solid-Works and my 3-D printer I was able to design and prototype the expanding geometries below.
Thanks Mario, I appreciate the feedback!
Well done! Great work!