Re: [Algorithms] Quaternions in IK

[Auday <au...@so...>]

Actually the main problem with Eulers is the sequence dependency , so if 
you know the sequence you want to get the Eulers in ( from a quaternion or 
a matrix )  then you will usually have only 2 solutions , for example if 
your sequence is ZYX then you will get 2 solutions for Y , one less than 
180 and the other is greater then 180 , and selecting 1 of them will give 
you 1 solution for each of X and Z .
I agree with Ron that Euler angles are not a natural representation of 
rotation, they are just like applying 3 rotations in quaternions or 
matrices, and obviously you have the same problem of dependency if you 
apply any 3 matrices or quaternions. We use the axis aligned Eulers just 
because they make sense and they give us better imagination of the rotation .
For your question Ron , sorry I didn't understand what you meant , maybe if 
you explain I can answer :)

Thanks

Auday




>It is not a question of loss of information, but of absence of
>information from the start.  What I mean by this is that the real
>geometric thing is the rotation (or orientation) not an Euler angle
>construction of it.  The rotation does not uniquely determine the
>Euler angles.  If FREE rotation is allowed (think a ship on the sea or
>an aircraft or spacecraft) , then a given rotation can be decomposed
>into Euler components in infinitely many ways.
>
>So if you start with an Euler construction to get a rotation matrix
>(or a quaternion) then use a routine such as one that Dave Eberly
>points to on his site to get Euler angles from that quaternion or
>matrix, then there is no guarantee that they will be the same Euler
>angles you started with....The rotation does not determine the Euler
>angles.
>
>On the other hand, for the systems at issue in IK problems, viz.,
>linked structures with constrained joints, such as robots and animated
>characters, the rotation of the end effectors are NOT free, but
>constrained. Typically, the Euler axes DO have a mechanical reality in
>the construction of your robot as a linkage of defined 1-DOF joints,
>and the range of rotation of each joint is limited.  In many such
>cases, no doubt, the Euler angles ARE uniquely determined by the
>orientation of the end effector, and in principle one could find an
>algorithm to determine them from the rotation.  But it would be
>fortuitous if one of Dave's routines gave the correct decomposition in
>your particular case.
>
>These considerations lead to a question that keeps coming up for me,
>but for which I have not found the answer.  I've posted it on various
>forums and lists from time to time, without response.  I'm going to
>ask it again here, because at least a few will understand its
>statement, and maybe Dave or Jeff's grad student or someone will
>remember seeing it discussd somewhere:   In the rotation group SO(3),
>how can one characterize a maximal neighborhood of the identity that
>has a unique analytic Euler angle parametrization?
>
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