Recently, I’ve been diving into a design for a motion system, and the heart of that system is going to be a set of four-bar linkage mechanisms.

The above animation likely doesn’t resemble what the actual system will look like, but it was a first concept sketch I tossed together just to sanity check some approximate pivot locations.

But the orientation of the linkages, and therefore the overall motion, is in line with the direction I’m thinking.

I wanted a way to quickly look at how trade offs in design parameters will change the range of motion of the assembly.

This will let me look at how changing things like the linkage lengths and angular ranges of motion for each pivot will impact what volume of positions the arm can reach.

So first I made a Van Gogh-level sketch of the mechanism, and laid out the vectors for the chain of linkages (aka the kinematics).

This gave me an equation for the position of E (my end effector) as a function of the angles of the two linkages.

So next I took these equations and tossed them into a Jupyter notebook in Google Colab. I can swap out parameters up at the top, and the plot is showing all of the X,Y positions that E can reach in that configuration.

While the range of motion is of course important, it’s not of much use if the thing breaks or collapses, so next up was to start taking a look at the expected loads.

One important aspect of this design that I haven’t mentioned yet is that my plan for controlling the position of the link arms. My plan is to drive one of the pivots directly for each four bar segment. Below is one such concept I put together where that pivot is being driven by a worm gear.

To ensure this will work, I need to have a reasonable estimate for the load that will be carried by the worm gear teeth.

So time for some more lovely sketches and a whole bunch of algebra.

This gave me equations for the reaction loads at the pivots, as well as the drive force needed. As with above, I tossed this into that Jupyter notebook to get me some visual aides.

Here is an example of a plot for the drive force (FT…for tension, long story), and the reaction forces R1 and R2.