Solving Rotational Matrix: 6 Parameters from 3 Angles

[mundo44]

The Scenario:

We have a frame 1 and a frame 0. The frame 0 is rotated in some manner, but not translated so their share an point of origin. I have 3 angles between different axes of the different frames. I am suppose to describe the orientation of frame 0 in frame 1 with an 3x3 rotation matrix. I want to solve for the other 6 parameters in the Rotation Matrix.

Hint: you can get 9 parameters from only having 3 parameters of the 3x3 rotational matrix.


Solutions:

- I believe that when the angles of the rows or columns in each rotation matrix must follow this equation since it the matrix is based off of an axis of one from projected onto each of the other ones. : (cos(a))^2 + (cos(b))^2 + (cos(c))^2 = 1

- Also:

R = Rotational Matrix
RT = Transpose of Rotational Matrix
I = Identity Matrix

R*RT = I

I can create equations using this and solve for different parameters...


Are these valid claims? And will they get my answer?

 

[D H]

I have 3 angles between different axes of the different frames. I am suppose to describe the orientation of frame 0 in frame 1 with an 3x3 rotation matrix. I want to solve for the other 6 parameters in the Rotation Matrix.

Whether you can answer this question depends on what you mean by "I have 3 angles between different axes of the different frames." You don't have enough information if, for example, you mean that you know the angles between the x and x' axes, the y and y' axes, and the z and z' axes. There are up to eight transformation matrices that fit the bill here.

If on the other hand you mean that you know that (for example), frame 1 was constructed by rotating about the x-axis by some angle, then rotating about the rotated y-axis by some other angle, and finally rotating about the twice-rotated z axis by yet another angle, then you do have enough information.

 

[mundo44]

So I cannot solve for the 6 other parameters from just 3 entries of the rotational matrix? I have the angle between x' and x, the angle between y' and y and at last the angle between z' to y.

The method you described is a description of what I learned as the fixed angle rotation which produces and a matrix dependent on the individual axes rotations. For the fixed angle rotation, does the order of the rotations about certain axes matter? If I am just given a picture, is there some way of figuring out the order of these rotations in order to use one of these matrices?


BTW thank you so much. You are helping me clear things up!

 

[D H]

So I cannot solve for the 6 other parameters from just 3 entries of the rotational matrix?

While there are obviously some special cases where you can do this (e.g., you are given that all three diagonal elements are one), in general, you can't. Consider the case of being being given the angles between x' and x, x' and y, and x' and z. This leaves a plane completely unspecified. Or suppose you are given the three diagonal elements. This leaves three underlying values for which you know the magnitude but not the sign. So, except for special cases involving one or more of those underlying values being zero, there are eight possibilities here.

I have the angle between x' and x, the angle between y' and y and at last the angle between z' to y.

I think this corresponds to the first example, where you have a plane completely unspecified. In any case, knowing only three elements of the transformation matrix does not provide enough information.

The method you described is a description of what I learned as the fixed angle rotation which produces and a matrix dependent on the individual axes rotations. For the fixed angle rotation, does the order of the rotations about certain axes matter?

Order of rotations absolutely does matter. Rotations are not commutative. Grab a book and set it down on the table so you can see the book cover face-on. Designate the direction the text reads, left to right, the xhat axis, the direction from the bottom of the page to the top the yhat axis, and the outward normal (pointing at your eyes) the zhat axis. Pick up the book, keeping it in this orientation. Now rotate +90 degrees about x and then +90 degrees about y. You should be looking at the binding. Next let's reverse the order of the rotation. Restore the book to its original orientation, then rotate 90 degrees about y followed by +90 rotation about x. You should be looking at the front of the book, with a line of text going up to down rather than left to right.

Bottom line: Rotations don't commute.

 

[mundo44]

Hm... So if I am given 3 angles between dif. axes and no order of rotation how would one go about solving the rotation matrix?

 

[D H]

Hm... So if I am given 3 angles between dif. axes and no order of rotation how would one go about solving the rotation matrix?

You can't.