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I'm trying to calculate the position of a multi-constellation GNSS receiver using GPS and GLONASS satellites using least-squares optimization. Sparing the details, I have 5 equations to solve for 5 unknowns: (x, y, z, r_gps, r_gal).

4 equations are for GPS satellites, taking the form:

sqrt((x - x1)**2 + (y - y1)**2 + (z - z1)**2) - (pr - r_gps) = 0

And 1 equation is for the sole GLONASS satellite, taking the form:

sqrt((x - x1)**2 + (y - y1)**2 + (z - z1)**2) - (pr - r_gal) = 0

When implemented in python, the LS method converge to the same x,y,z solution no matter whether all 5 equations are taken, or only the 4 GPS equations or used. I.e, the 1 equation having the term r_gal is ignored. Mathematically, why is this the case?

Shawn Lim
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  • To those interested the actual code can be found here – Shawn Lim Jan 04 '22 at 14:21
  • Isn't it simply because the last equation can be rewritten as sqrt((x - x1)**2 + (y - y1)**2 + (z - z1)**2) - pr = -r_gal? The LS method will just find the solutions for the other four variables, then use the values of x, y, and z to compute r_gal? – VTand Jan 04 '22 at 15:19
  • That’s a good point. I realise with another GLONASS equation the x,y,z solution will not longer face this problem. Thanks for your help, should have noticed this earlier – Shawn Lim Jan 04 '22 at 15:29

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