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    $\begingroup$ Welcome to CGSE! Just had a quick look at your question without paying much attention to details but it sounds like it could be beneficial for you to take a look at quaternions. It is a different way to describe rotations. At first, they seem quiete complicated, but once you tried working with them, they are rather easy to use. $\endgroup$ Commented Oct 22, 2021 at 22:26
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    $\begingroup$ Don't try to understand the theory behind them. Just search for some tutorials on how to use them. The equations you will need are pretty easy to implement. The theory is a bit more complicated and you don't need to understand it to work with quaternions. Once you feel compfortable using them, you can try to understand why it works ;) $\endgroup$ Commented Oct 22, 2021 at 22:38