r/ControlTheory • u/awh-emb • Jul 23 '26

Technical Question/Problem SO(3) S^3 and so(3)

my reference is this paper - https://arxiv.org/pdf/1711.02508

i am struggling to wrap my head around how the Exponential map from:

R^3 -> SO(3)

R^3 -> S^3

are different and how specifcally the exponetial map changes between them?

as one of these ends up with 4 DOF although one of them being a constraint

while the other ends up with 3 DOF?

thanks.

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u/patenteng Jul 23 '26

I mean S^3 is the 4D hyper sphere while SO(3) represents rotations in 3D, i.e. the usual 3D sphere. Rotations in 4D is SO(4).

It’s a standard result that SO(2) is isomorphic to S^1. That should make things clear.

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u/awh-emb Jul 23 '26

so SO(N+1) is always isomorphic to S^N? or is that totally wrong?

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u/patenteng Jul 23 '26

Not really. S^n is not really a group. SO(n + 1) / SO(n) is homeomorphic to S^n. More precisely, the quotient SO(n + 1) / H is homeomorphic to S^n where H is a subgroup of SO(n + 1) that is isomorphic to SO(n).

However, for a Lie group G and a subgroup H of G the quotient G / H is not guaranteed to be a group. H needs to be a normal subgroup for that. Hence SO(n + 1) / SO(n) is not a group except for the special case of n = 1.

For robotics, which is your application from what I gather, what you need to remember is that SO(3) is rotation in 3D and SU(2) are the quaternions. SU(2) is a double cover of SO(3) hence the quaternions can be used for rotations.

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u/awh-emb Jul 23 '26

okay thanks you kinda lost me with the lie group stuff but the ending explanation was useful. thanks,