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See You can absolutely use "flat" Euclidean space based optimizers while also optimizing on the manifold, but I agree the default scipy solvers don't give you an easy way to do that. Perhaps you can use pymanopt? See Although I wouldn't be scared of ...


The simplest way to have values between [0, 2$\pi$] is using the modulo operator. You can simply do $\hat{e}_k = (\hat{\theta}_k - \theta_k) \% 2\pi$.


As mentioned by @long-smith the standard solution is to use quaternions. However if you specifically are asking how to deal with errors using angles that are modulo $2\pi$ you are going to want to add logic to compute the smallest angle between two angles which take into account wrapping. An example is shorted_angular_distance from the ROS angles package ...

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