The Dynamics of “Simple” Walking

Trent Dye
Trent’s Blog
Published in
3 min readNov 19, 2016

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As a final project in our dynamics course, I worked on a project that explored passive dynamic walking motion.

To construct our mathematical and physical models, we relied heavily on two research papers, Actuating a Simple 3D Passive Dynamic Walker and The Simplest Walking Model: Stability, Complexity, and Scaling.

The process

Up to now, the course had revolved around the use of MATLAB’s ode45 tool to solve sets of ordinary differential equations. Therefore, we were not strangers to using MATLAB to create a simulation based on the mathematical model proposed in the research papers.

A mathematical model using ode45 in MATLAB.

As its name suggests, the model is really simple, and therefore makes some unrealistic assumptions. Most notably, the model assumes that all of the walker’s mass is concentrated at the pivot.

We knew from the start of the project that we wanted to build a walking machine to demonstrate the proposed motion, and requested a budget to afford materials for it.

Tedrake et. al.’s “3D Passive Dynamic Walker” achieves a walking motion using an inverted pendulum. The walker is agitated initially so that it is rocking side to side; its curved feet encourage this motion. This rocking allows its feet, one at a time, to be lifted from the ground. Since the walker is on a slope, the free foot swings forward.

We decided to design the walker’s feet as spherical sections, since they needed curvature in two directions. It was imperative that the curvature be just right, so the walker would be stable yet still able to walk. Wanting to build a fairly large model, we chose to use aluminum tubing and aluminum rod for the skeleton, joined simply with bearings and shaft collars.

3D printing was an incredibly useful tool for this project. First and foremost, it allowed us to create and tweak the specific geometry that our models required. Secondly, it made the fabrication process a breeze, allowing us to assemble the model in a few minutes, with just fasteners and press fits.

Results

Using the simple mathematical walking models proposed in the research papers, we successfully built our MATLAB model. The hardest part of the model was setting up the initial conditions that would give the walker the correct “push” in both directions to keep it from falling. This was no surprise since it wasn’t easy to get our physical model walking.

Taking cues from the designs of the researchers, we were able to construct our own successful walking machine. In our first iteration, the radius of curvature was much too small and the walker fell forward. By sheer luck, we tweaked the design of the feet just enough that our second iteration worked perfectly. We also added a lot of weight to the top of the model — to move the center of gravity more toward the pivot — which further contributed to the stability of the walker.

If you would like to read more about this project, our mathematical findings are compiled in this paper. The GIFs in this article come from three YouTube videos on my channel, this one, this one, and this one.

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