Caterpillar with Chariot

The Caterpillar with Chariot is 18 feet long, battery powered, and has a top speed of over two miles per week. The chariot has cams that make the caterpillar walk. But it is the caterpillar that pulls the chariot! I made it by hand and it took the better part of a year. Although, I suppose, it would be equally accurate to say that it’s taken me twenty years to make . . .

In 1994 I saw a little green caterpillar that changed my life. I was camping with friends in the Deep Creek Mountains of Western Utah. It had taken several days of driving on dirt roads to get there. One morning we all went off in different directions and I found myself running and leaping over the smooth stones of a narrow dry streambed. I came to a stop in the middle of a wider sandy place. At my feet was a little green caterpillar no longer than an inch. I got down to look more closely. It was translucent, and I could see right through it. A continuous wave flowed through it as it walked. The movement was part of an abstract sine curve that extended past the boundaries of the caterpillar. If I had been a foot to the left, or a foot to the right, or if the caterpillar had set out on its daily journey a few minutes earlier, or a few minutes later, I would not have seen it. But I did, and it altered the course of my life.

I spent the subsequent four years learning how to draw and paint, but in 1998 I turned towards the caterpillar. Partly I wanted to work with wood. Partly I wanted a good math problem. In school we are given problems and we solve them. But it turns out that best part of math is actually the phrasing of the problem. The solution is only the beach at the end of a long river trip where you pull out your canoe for the last time. It is not something to rush towards. Maybe this is just an excuse for why it’s taken me twenty years to figure out the caterpillar math.

For the first three caterpillars I assumed that the motion was best understood as a wave cut in half by the ground, with only the peak of the curve visible. I spent years trying to work out derivatives, thinking that I needed to solve for the speed of each element. It was a tough problem, and in 2003 I left the caterpillar and began making overhead waves. They have been immensely satisfying, but I’ve always known the caterpillar would pull me back. In 2017 I took a fresh look at the math. In a flash it was clear to me that the caterpillar I had seen in the Deep Creek Mountains was not a wave cut in half by the ground, but rather a wave riding on top of the ground! And that instead of finding a derivative and then integrating to find position, I could directly solve for position with respect to time. These two insights made things much easier and in the matter of days I had an elegant solution. But it required being able to find the intersections between circles and sine waves, and it turns out those intersections have no closed-form solution. So while I could find a point by trial and error and check that it was on both equations, I couldn’t solve for it directly. Luckily my friends Perrin Meyer and Dan Torop came to the rescue. They wrote an algorithm in Julia that brute forced millions of intersections and this resulted in a list of Cartesian coordinates that corresponded to a finite moment in the caterpillar cycle. From there I tiled and transformed the points through longhand trig till I had the complete movement.

The cams for the Chariot were particularly challenging. I spent several weeks reading old textbooks on cam design and settled on using 24 conjugate oscillating roller cams. This is an unforgiving arrangement, to say the least, and so I did my best to hold both the math and the woodwork to a thousandths of an inch. After some effort I was able to describe the cam by knowing the distance from the center of the roller to the center of the cam at any given angle. But it turns out there’s no easy way to go from there to the cam profile as you still have to find the curve that is tangent to all the roller positions. I went online and found a company that made software that could do this. I got in touch and they generously took my list of numbers and sent back a DXF. I sent that file to a friend with a CNC router and got back a promising looking piece of plywood. But when I put it in the caterpillar it didn’t work at all! Where was the mistake? It was hard to tell. There were pages of trig calculations I might have messed up, or just as likely I might have made some part of the caterpillar to a different proportion then I had assumed in the math. I had almost given up when I discovered that early on I had mapped the dwell onto ½ a circle instead of 5/12 of a circle. It was a major error, but easy enough to correct and generate a new set of numbers. Since it seemed unlikely it was my only mistake I figured it would be wise to make a cam in the shop and see if it were even close. I updated my excel file and printed out a new set of numbers that I taped under the DRO of the mill. I bolted a piece of plywood to the rotary table and chucked up a hole saw the same diameter as the roller I would be using. I dropped the hole saw down and kissed the wood to make a circular mark. Then I repositioned the wood by changing the x-axis and rotary angle and made another mark. When I had gone all the way around I drew a line tangent to all the hole saw marks and cut out the cam with a band saw. It worked surprisingly well, so I made 23 more of them. I put all the cams into the chariot, took a deep breath, and turned it on. I had the speed turned up a little too fast and the caterpillar practically galloped out the door! I couldn’t believe it!

For more caterpillars please see the Green Stripe and the Early Caterpillars.