How Can I Decompose a 3x3 Rotation Matrix into a Product of 3 Rotations?

[Cristi-Tota]

Hi,

How can I decompose a 3x3 rotation matrix R, into a form:

R = rot(v3,c) X rot(v2,b) X rot(v1,a)

where v1,v2,v3 are known unit length axes (with angles a,b,c unknowns)?

Thank you,
Cristian

 

[Timbuqtu]

I don't think it can be solved generally. Have a look at:

http://mathworld.wolfram.com/EulerAngles.html

 

[tacman]

Hi Cristian,

Decomposing a 3x3 rotation matrix into a product of 3 rotations is known as the Euler angle decomposition method. This method involves finding the Euler angles, which are the rotations around each axis, that make up the original rotation matrix.

To decompose the rotation matrix R, we can use the following steps:

1. Find the rotation angle a: The first step is to find the rotation angle a around the first axis, which is v1. This can be done by using the formula: a = atan2(R(3,2), R(3,3)).

2. Find the rotation angle b: The second step is to find the rotation angle b around the second axis, which is v2. This can be done by using the formula: b = atan2(-R(3,1), sqrt(R(3,2)^2 + R(3,3)^2)).

3. Find the rotation angle c: The final step is to find the rotation angle c around the third axis, which is v3. This can be done by using the formula: c = atan2(R(2,1), R(1,1)).

Once we have the Euler angles a,b,c, we can construct the rotation matrix R by using the following formula:

R = rot(v3,c) X rot(v2,b) X rot(v1,a)

where rot(v3,c) represents the rotation around the third axis by angle c, rot(v2,b) represents the rotation around the second axis by angle b, and rot(v1,a) represents the rotation around the first axis by angle a.

I hope this helps. Let me know if you have any further questions.