quote:Original post by Timkin From a mathematicians perspective, I can say most definitely that you cannot uniquely identify any point in a 3-dimensional Euclidean space with 3 angles. By definition, an angle identifies a unique sub-manifold (in this case a line) by considering the sub-manifold to be an affine transformation (rotation) of a reference sub-manifold (line).
In lay terms, since angles only define lines, you cannot use a combination of angles to identify a unique point, only a unique set of co-linear points!
Timkin, Thanks for putting that into words, the translation especially. I never heard the term manifold used in that context before.
Quaternions are not my strong point... I''m not really a fan of hypercomplex numbers! Having said that though, to my understanding, quaternions do not represent a single point, but rather a noncommutative division algebra. One of the properties of the quaternion group is that it can be used to represent a rotation. The components of the quaternion are the Euler parameters of the rotation.
This leads to the fact that rotations are noncommutative (to respond to the insistance that they are). Imagine an aircraft undergoing the following rotation set (90 deg yaw right, 90 deg pitch up, 90 degree roll right). Now re-order these operations... (90 deg roll right, 90 deg yaw right, 90 deg pitch up). The aircraft does NOT end up in the same position, making the rotational operations noncommutative (you can think of each individual rotation in the set as a quaternion if you like, or all three together as a concatenated quaternion). Mathematically speaking, this means that rotations alone cannot form the basis set for a closed manifold, which would be necessary for them to be used as a coordinate system for that manifold.
By god's sake, read carefully before replying!!!!!!!!!!!!!! This is the second time you put words in my mouth I didn't use!!
I never said that rotations are commutative!!!!!!! Of course they are not! I talk about the case of a standard robot arm (I TALK ABOUT A ROBOT ARM!!!) with 3 rotational degrees of freedom! The ordering of the rotations is given in the case of a ROBOT ARM by its mechanical configuration. Now if you want to change the position of the endeffector there is no difference in saying:
alpha=alpha+3 beta=beta+4 gamma=gamma+5
or
gamma=gamma+5 beta=beta+4 alpha=alpha+3
since "all transformations are specified relatively and not absolutely!!" in the case of a ROBOT ARM for the rotations. (or do you really think that sending following commands to a robot arm will result in different positions/orientations of any part:
"turn 40 degrees around joint 1 then turn 35 degrees around joint 2"
or
"turn 35 degrees around joint 2 then turn 40 degrees around joint 1"
????
The position of the end-effector can be found with the absolute transformation:
As you can see in the above transformation, the ordering is fixed! (unless you want to do some soldering work). "TjointRelative" are the relative transformations linking one joint to another.
Well, I will not reply to this thread anymore, but please Timkin don't reply anymore unless you learn to read carefully.
The alpha,beta,gamma specification for the state of a robot arm would have to be the silliest way of representing a point in 3d space there is. Not only do you need the position of the robot''s shoulder, but you also need orientation of the rest of the robot and the lengths of the limbs as well.
After this mistake and taking into account the context it''s no wonder Timkin misinterpreted you. And I got a lot out of his/her posts regardless so you''re being pretty selfish in asking him not to reply.
Getting frustrated and angry does not help clear up what you believe is a misunderstanding. If you don't wish to read this thread any further, then that is your choice. I would prefer though that we clear the air so that hostilities do not develop between us and muddy these forums.
As to the situation at hand... you entered into a discussion about coordinate systems and used an example of a robot arm to attempt to justify the use of 3 rotations as a coordinate system; you gave a robot arm as an example for locating points in 3-space. Firstly, as I said earlier, robot arms have nothing to do with coordinate systems... you yourself have since affirmed this when you say that rotations are relative to the orientation of segments of the arm. You even highlight this by giving an example of a robot arm with two joints and talk about the final position of the end of the arm. Clearly the joints are not co-located, making this an inappropriate comparison to a 3 angle coordinate system. Any coordinate system requires an origin about which ALL basis vectors are defined.
By attempting to use a robot arm to justify the use of 3 rotations for a coordinate system, it appeared that you were trying to affirm that 3 rotations were sufficient to locate any point in 3-space. Indeed in your earlier post you say exactly this. As I pointed out, rotations are noncommutative. I did not say that robot arm manipulations are noncommutative. You seem to have taken offense to something that was never said, nor even implied. It is quite possible that I incorrectly assumed that you were insisting that 3 rotations were sufficient to define a coordinate system, however, I only have what you wrote to go on.
As to the statement that I should read posts carefully... I do. Like anyone else, I do make mistakes and misinterpret things occasionally, however I do not think this is the case in this thread. I think that you have taken offense that your example is not being considered as a justification of a 3-rotation coordinate system and you have singled me out since I was the one to insist that mathematically, it is impossible to define a 3-rotation coordinate system for a manifold.
I would hope that you realise that by offering an example that bears little or no relationship to the problem at hand you have appeared to make a claim that is not true - that 3-rotations can define a coordinate system - and have hence confused the issue.
If you feel that this is an incorrect assessment of the situation and you wish to discuss this matter further, I think that we should take it to private email and not further distract the discussions taking place in this thread.
