Note: This is what I know almost wrote in the article, it is handling so far.
1. "14 say" p51 page mentioned "In fact, we could not find a Singular three-dimensional vector describing the way." So Lie algebra we usually use, for example, so (3), expressed as a three-dimensional vector, whether the singularity? If there is singularity Lie algebra, Lie algebra why use it? (On behalf of the view that individuals would like to think, does not mean the right, welcome to the exchange)
A: SLAM backend Lie algebra representation reasons pose is constrained rotation matrix unit orthogonal Lie algebra can be constructed using unconstrained optimization problems. The relationship between (3) and the Lie algebra SO (3) is full Lie SO exit but not injective, see <state estimation> p191 last line of the page, text copied thereto:
From a mathematical point of view, from so (3) to SO (3) index mapping is a full shot. This shows that elements (3) may correspond to each element of SO (3) a plurality of so.
Indeed Lie singularity, see <robot state estimation> p191, footnote 10:
The concept Surjectivity parameterized in the rotation, with the singularity (or uniqueness) related, we know that every argument with the three representations of the rotation can not guarantee the global and unique. Non-uniqueness of the show given rotation matrix C, we could not find a singleto produce it, since the corresponding
there are numerous.
But if we limit the angle of rotation , then with Li Lie algebra is one to one. So, in our SLAM system to limit the scope of the Lie algebra, then do not consider the singularity problem.
2. Lie algebra derivation, the disturbance model and the direct derivation What is the difference? Iteration of our argument is not a disturbing attitude, why was the use of derivative disturbance? (On behalf of the view that individuals would like to think, does not mean the right, welcome to the exchange)
A: First Incidentally, the purpose of our derivation of the iteration is to find a direction and step size, to make the re-projection error is minimized. Direct derivation and disturbance model is indeed very different, the details refer to <state estimation> p215, calculus and optimization . We are briefly stated as follows:
1. Optimization of the rear end, i.e., the direct derivation of the direct derivation Lie algebra, Lie algebra argument as the gesture represented.
2. For the disturbance model in our derivation of the argument is not a gesture. Instead disturbance, in other words, we pick a disturbance, so that the weight of the added disturbance projection error is reduced. More particularly, we first solve the perturbation direction, and then select a step size, so that decrease maximize reprojection error. For example, for in a rotation , which is expressed as a Lie algebra
, we in
the vicinity of the
application of a disturbance
. Then we are
close to unfold:
After the disturbance rotation and points
can be (loosely) when multiplied by writing to:
So now, we have set up on a non-linear optimization function
, we will bring the disturbance, and then expand in the near R0:
If we can find and
, so long as we choose a function of the reduced value of the disturbance can be completed iterative nonlinear function, so that its value continues to decrease. For more information please see <state estimation> p215-p219 page, which also has Gauss-Newton method for another explanation.
I see a good article, but also on Lie algebra disturbances, recommend it to everyone!