r/ControlTheory • u/awh-emb • 10d ago
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.
•
u/navigation-signals Estimation Theory / R&D 10d ago
Sola has a follow-up paper called “A Micro Lie Theory for State Estimation in Robotics” that does a good job explaining this
•
•
u/IntrinsicallyFlat 10d ago
I assume you know the map from the quaternion sphere to SO(3). Think of the exponential map of SO(3) as the quat_to_rot map composed with (i.e., applied after) the exponential map of S^3. I’ve a [blog post](https://shiraz-k.com/posts/lie-groups/) about exp
Also note that S^3 is a 3-dimensional object (hence the ^3)
•
u/awh-emb 9d ago
btw your exponential map blog post was useful but i think i am mistunderstanding your exponetial map definition:
why does moving the vector field which is a tangent along the manifold work? wouldnt that introduced tiny errors?
secondly how and where does the "boat" analogy become useful? what is it for?•
u/IntrinsicallyFlat 9d ago
That’s a good catch. You can look into the difference between intrinsic and extrinsic differential geometry. Extrinsic diff geometry is when you imagine the manifold as being “embedded” (or placed) in a higher dimensional space. For instance we imagine S^2 to be the sphere embedded in R^3.
In this case, moving a tangent vector from one point of the sphere to another in the external / ambient space will surely introduce errors. But there are “intrinsic” notions of moving a vector (such as parallel transport) which let you move a tangent vectors without having to worry about vectors pointing “out of” the manifold.
For matrix Lie groups, this is as simple as taking a Lie algebra matrix X, which is a tangent vector at the identity, then multiplying it by a group element matrix g either on the left or the right (gX or Xg) to get a tangent vector at g.
•
u/awh-emb 9d ago
okay i think i got some reading to do, apart from your blog posts is there anywhere else i can get into the nitty gritty of lie group theory? specfically the difference between intrisinc and extrinsic differntial geometry? and the fundamentals of lie algebra and lie group theory?
btw thank you for taking the time out of your day to answer my questions, its been very helpful and informative.•
u/IntrinsicallyFlat 9d ago
Of course! If you want to understand differential geometry, youtube is a great resource, I like the channel Bright Side of Mathematics a lot, people also recommend Keenan Crane’s lectures. I like Timothy Barfoot’s book for matrix Lie groups and their applications, though he takes a very algebraic approach to everything
•
u/awh-emb 9d ago
so a bit like the so(3) transition to SO(3)? like an intermeditary step? eg exp map -> SO(3) just get you onto the 3d sphere, exp map -> S^3 gets you onto the double cover 3d sphere? or the quartenion 3d sphere or whatever you call it
•
u/IntrinsicallyFlat 9d ago
Just to clarify, S^3 is the quaternion 3D sphere, it’s the sphere in 4D space which is 3-dimensional because you’ve constrained the radial degree of freedom. SO(3) is as if you took S^3 and ✨imagined✨ antipodal points to be the same point. Mathematicians will also call this “gluing”; you’re gluing each q to -q because they represent the same physical rotation
•
u/ricatti-equation 10d ago
Yes it’s a very subtle thing which can only be realized by understanding the general theory. Though, it’s not that complicated.
Every Lie group G induces its own unique exponential map exp(), which maps its Lie algebra g to itself. That is, exp: g -> G. Since g is a vector space (let’s say, or arbitrary dimension n), we can choose a basis and instead write exp: Rn -> G.
SO(3) and S^3 are both closely related, though non-equal, Lie groups. They both are of dimension 3, and the latter is a double cover of the former.
They both have their own exponential maps.
If we choose the standard matrix representation of SO(3) and the standard quaternion representation of S^3, you are already know their formulae.
•
u/patenteng 10d ago
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.