Thanks to Bo Xin [X] for the following data sources:

Compensation for elevation:

• M2 hexapod: https://github.com/bxin/hexrot/blob/master/LUT/M2%20Hexapod%20Motions%20in%20Elevation%20Axis%202020%2007%2023.xlsx
• Camera hexapod: https://github.com/bxin/hexrot/blob/master/LUT/Camera%20Hexapod%20Motions%20in%20Elevation%20Axis%202020%2007%2023.xlsx

Compensation for temperature:

• Document 7334 (a powerpoint presentation) page 9, which lists the fit coefficient of expansion.

I used the elevation spreadsheets and the attached code to fit various models for elevation. I found:

• A Fourier series worked best: f(el) = C0 + C1 sin(el) + C2 cos(el0 + C3 sin(2 el) + C4 cos(2 el) + .... This model can also be used for azimuth, since it wrap around at 0/360.
• A standard polynomial worked well: f(el) = C0 + C1 el + C2 el^2 + .... This is not suitable for azimuth compensation.
• A "cosine polynomial" did not work very well: f(el) = C0 + C1 cos(el) + C2 cos(el)^2 + .... This can be used for azimuth compensation.

Tiago Ribeiro decided we should use the standard polynomial for now so I did the following on DM-25856:

• I coded the standard polynomial for elevation compensation and removed azimuth compensation. (Before this ticket I had coded a cosine polynomial, as per Tiago Ribeiro's initial request).
• Added the fit coefficients to the default.yaml config file in ts_config_mttcs on the same ticket. I am not positive I got the sign of the corrections correct.
• Added support for the Fourier series to ts_hexapod, so we have it available. I personally think we should use it.

Note: I found that scipy.optimize.curve_fit did a far better job than numpy.polyfit so I used that for all fitting.

I also attached 3 of the graphs – all for the Camera since it is a bit harder to fit than M2.

In addition to the graphs in the attachment, please see code in the "fitter" directory of ts_hexapod, which was added as the 2nd-to-last commit of: https://github.com/lsst-ts/ts_hexapod/pull/20

I would also appreciate a sanity check of the coefficients in Hexapod/v1/default.yaml in pull request: https://github.com/lsst-ts/ts_config_mttcs/pull/8

It may help to know that:

• compensated Hexpod position = uncompensated (user-specified) position + compensation_model(elevation, azimuth, temperature).
• There is no compensation for azimuth at this time. It will be added later if we find a need (once we can study the data an choose a suitable model, though I expect that the Fourier model will do fine).

Thanks for the good work!