From a Certain Point of View

[Title from Obi-Wan's BS explanation for lying to Luke]

When I first put up the post about my stair-climbing cart, my father Steve was disappointed there was no video of it in action. At the time, I had hoped to create an animation of the movement, but I couldn't get my head around the equations involved. This week, I had an idea though: The system is fairly simple to describe if we only consider the cart rotating around one of its wheels. Bringing back the diagram from last time,

we can imagine holding the front wheel fixed, and letting the other two rotate around it.

In Physics, this is called choosing the reference frame. Essentially, we're deciding where the x- and y-axes are, and what we're calling zero. Reference frames can also move relative to each other. As long as that velocity difference is constant, all the same physical laws apply. In accelerating frames, things can get a bit crazy, but we don't have to worry about that here.

In this case, we can put the corner of the front step at (0, 0). We can also rotate 45° to make the stairs horizontal, which makes the motion of the cart symmetric. Here's an animation of what the system looks like going through one step:

Now that we have that simpler motion defined, we can apply a couple transformations to bring it back to the view we're interested in: We can rotate back 45° to line up the stairs, and then follow the cart as it moves, going up one step for every turn.

If you'd like to see the equations that define this, you can dig into the code.