I haven't made or flown any quadcopter, but I am fascinated by them. But when looking at the frame of a lot of designs, after whachting this video I wondered why a lot the frames are in an X shape. Since the most efficient shape would according to the video be something like this >-<, where each corner is 120°. I also did a quick search on the internet and found this blog which stated the same (however he did not mention the exact angle) and said: "Even though this is not entirely a new idea, it has not yet been widely accepted by the community."
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2$\begingroup$ It is a mistake to assume that a Steiner tree, though it represents the shortest length to connect 4 points in space, is the most material efficient, since each segment may have different thicknesses to accommodate different strengths. In particular, in many cases, the central segment appears to be thicker for greater strength and rigidity. More complete calculations into the strength of each design, and the actual material by weight used, would be needed to conclusively say which is more efficient. $\endgroup$– ronalchnCommented Jun 16, 2013 at 0:54
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If adjacent motors are 1 unit apart, the overall path-length of an X form is 2√2, or about 2.8284, and the overall path-length of a Steiner form is 1+√3, or about 2.732, which is 3.5% less. It is likely that hub mounting will decrease either or both of the total lengths by a few percent, but I will ignore that factor for the moment and merely say that the length improvement provided by using a Steiner form is quite small.
In addition, the Steiner form will require either two hubs or one rather-long hub, and may require mounting hardware for spokes at six points instead of four. Because hub weight often is comparable to the weight of one or more spokes, the Steiner form has a weight disadvantage. (See “Intro to Multirotors, Theory, Build Log and Tips” for a series of pictures that illustrate relative hub and spoke sizes for a variety of quadcopters.)
A central point, as available with an X-frame, is a convenient location for a battery and a controller that serves all the motors. For balance, an extra pad for mounting the battery and controller would be needed on the middle bar of a Steiner frame (unless multiple batteries and controllers are used).
Note, the quacopters-drones-quad-flyers page at wowhobbies.com includes a quadcopter, hexacopter and octocopter buying guide, in which several units are characterized by IAS, meaning there is “Inadequate Structural Support (eg: booms are too long and center frame is too weak to support the size of the quadcopter structure or any substantial payload)”. As size scales up, the crossbar in the Steiner form needs to scale up more than the rest of the arm structure (which will also overpower that form's 3.5% length advantage) else the frame will become structurally inadequate.
answered Jun 16, 2013 at 1:17