A catenary is the result of an equilibrium of forces among the chain links. A few important property of the uniform density catenary are:

- vertical tension is null at the apex.
- horizontal tension is constant through the catenary
- tension accross a chain link in only determined by the horizontal tension, and by the vertical tension due to the catenary segment that lies below that link down to the catenary apex.

Now the Maslow sled and ring kit rollers are lumps of mass. At a first degree, we could say it is not a uniform density catenary. But it can be simplified by:

- first considering the whole sled, including rollers, as a mass lump.
- Divide the lump in two parts reflecting the part of the weight held by each chain (this depends on the sled horizontal position).
- Replace the mass lump at the tip of each chain by an equivalent chain length of same weight.
- Combine each real chain length with the equivalent chain length it holds, and from that point on, stop assuming these two long chain connect together: Consider those as two separate catenaries having the same horizontal tension.
- Calculate a horizontal tension on the sprockets based on the sled weight and position. It is a valid approximation.
- Use the simple catenary calculations to find its shape, knowing that one part hangs from the sprocket, end the other is flat horizontal where vertical tension is null (the apex).
- Now you want to compute the relative coordinates of two points on each of these catenaries:

- the position of the highest point (where the sprocket tengeant contact occurs
- the position of the chain tip that hangs at the length of the real chain minus the part the wraps around the sprocket.

- Also compute the angle of each to figure out the tangeant points on sprocket and the connection point on the ring,
- Now you might get something that does not exactly fit between the sprockets. That is not an error. Remember your horizontal tension estimation was assuming a sled position, but the catenary equilibrium is moving away from that. So use this first catenary calculation to better estimate the position and horizontal tension, and iterate until you get a stable result that fit between the sprockets for the given chain lenghts.

And there you get a catenary calculation of the sled position.

Is that the optimal way to compute the solution? I don’t know. But you can **play with it**

I did implement that catenary calculation here to analyse the MaslowCNC errors and document the list of sources of errors.