Rotation matrix of three intrinsic rotations

  • #1
dbeckam
8
1
TL;DR Summary
Find the rotation matrix
I have three frames. The first is the fixed global frame. the second rotates an angle PHIZ with respect to the first. And the third first rotates a PHIX angle with respect to the x axis of the second frame, and then rotates a PHIY angle with respect to the last y axis. That is, there are a total of three intrinsic rotations Z, X, Y. According to Wikipedia, the final rotation matrix results from the image. I would appreciate if you could help me verify two things:
1704500949449.png

1) The angular velocity of the rotation matrix. According to my calculations the result is:
wpx=dphix.*cos(phiz)-dphiy.*cos(phix).*sin(phiz);
wpy=dphix.*sin(phiz)+dphiy.*cos(phix).*cos(phiz);
wpz=dphiz+dphiy.*sin(phix);

2) If I put an IMU sensor in the last frame, it measures the angles in a different format: yaw, pitch, and roll rotations. How can I use these angles to get my initial PHIZ, PHIX and PHIY angles?
 
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  • #2
Hello,

I find your post very confusing. None of your ##\phi_z##, ##\phi_x##, ##\phi_y## appears in the picture you picked up from wikipedia (you forgot to mention where - a simple link would have been useful! ). No axis is mentioned for ##\phi_z## -- we must guess the ##z##-axis I ass-u-me ?

dbeckam said:
The angular velocity of the rotation matrix
What is that ?

dbeckam said:
According to my calculations the result is:
wpx=dphix.*cos(phiz)-dphiy.*cos(phix).*sin(phiz);
wpy=dphix.*sin(phiz)+dphiy.*cos(phix).*cos(phiz);
wpz=dphiz+dphiy.*sin(phix);
I don't see no calculations and again have to guess what variables stand for. Left hand sides may be ##\omega##'s but then right hand sides can not be differentials.

IMO: a mess ! Back to the drawing board (yes: a few sketches might help :smile: )

IMU ?

##\ ##
 
  • #3
  • #4
Yes, I can google too. That's not the point of my rant.

##\ ##
 
  • Like
Likes Filip Larsen

FAQ: Rotation matrix of three intrinsic rotations

What is a rotation matrix in the context of three intrinsic rotations?

A rotation matrix in the context of three intrinsic rotations is a mathematical representation used to describe the orientation of a rigid body in three-dimensional space. It is constructed by sequentially applying three rotations about the axes of the body's coordinate system, which are referred to as intrinsic rotations.

How do you construct a rotation matrix from three intrinsic rotations?

To construct a rotation matrix from three intrinsic rotations, you multiply three individual rotation matrices, each representing a rotation about one of the coordinate axes. For example, if the rotations are about the x, y, and z axes, the overall rotation matrix R can be expressed as R = Rz * Ry * Rx, where Rz, Ry, and Rx are the rotation matrices for rotations about the z, y, and x axes, respectively.

What are the common sequences of intrinsic rotations used in constructing rotation matrices?

Common sequences of intrinsic rotations include the Euler angles (such as the ZYX sequence) and Tait-Bryan angles. Each sequence represents a different order of rotations about the coordinate axes. For example, the ZYX sequence involves a rotation about the z-axis, followed by a rotation about the y-axis, and finally a rotation about the x-axis.

How do intrinsic rotations differ from extrinsic rotations?

Intrinsic rotations are rotations about the axes of the rotating coordinate system, which means the axes themselves move with each rotation. In contrast, extrinsic rotations are about the axes of a fixed coordinate system, meaning the axes remain stationary. This distinction affects the order and result of the rotations.

Can you provide an example of a rotation matrix for a specific sequence of intrinsic rotations?

Sure! For a ZYX sequence of intrinsic rotations, the rotation matrix R can be written as:R = Rz(γ) * Ry(β) * Rx(α), where γ, β, and α are the rotation angles about the z, y, and x axes, respectively. The individual rotation matrices are:Rz(γ) = [[cos(γ), -sin(γ), 0], [sin(γ), cos(γ), 0], [0, 0, 1]],Ry(β) = [[cos(β), 0, sin(β)], [0, 1, 0], [-sin(β), 0, cos(β)]],Rx(α) = [[1, 0, 0], [0, cos(α), -sin(α)], [0, sin(α), cos(α)]]. Multiplying these matrices in the specified order gives the final rotation matrix.

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