Written by Glen Whitney for the Museum of Mathematics

In this installment we are branching out into the wider world of more complex linkages. See the Linkages series introduction for the MoMath Linkage Kit, an introduction, and general instructions.

Given the incredible flexibility of the seemingly simple four-bar linkage, it should come as no surprise that more complex linkages can produce an essentially limitless array of different behaviors. Today, we’re going to look at some linkages designed to mimic the walking behavior of living creatures. They are designed to lift a strut up off the ground, plant it farther ahead, and pull it back (which is the power stroke that actually moves an object — or animal — forward.) One inventive person in this area is Joe Klann, creator of this linkage:

You can find lots of information about the Klann linkage at Klann’s website — and here’s a great video of an operating vehicle based on this concept:

Now, compare that concept to the work of Theo Jansen, whose wind-powered linkages move entirely of their own accord:

For this week’s recipe, we have a

**Simplified Klann linkage**

Ingredients: An 11-bar (A), 13-bar (B), 18-bar (C), 51-bar with a hole at 29 (D), and a 54-bar (E) with a hole at 29; an extra 60-bar (E); and four linkers.

Directions: Link A to D0; B to D29; C to E54, and D51 to E29. That’s all. Note this leaves ends of A, B, C, and E free. The free end of E is the “foot.” The other free ends need to be anchored to the table, as follows:

To use: In order to see the walking motion, the free ends of A, B, C need to be anchored to the table but free to rotate in place, as follows: Fix the free end of B wherever you like; fix the free end of C about 18 units above that; and about 25 units to the right of that and five units down, fix the free end of A. Now, one rotation of A yields one “step” of the machine. Here’s a short clip of ours in action:

It’s also instructive to put a pen in the “foot” of the walker and get a trace of what the linkage’s stride looks like.

**More:**

- Linkages, Introduction
- Linkages, Part 2: Four Bars, One Freedom
- Linkages, Part 3: Four Bars, Two or Three Positions
- Linkages Part 4: Four Bars, Four Positions
- See all of our Math Monday columns

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