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?
 
Engineering news on Phys.org
  • #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

What is a rotation matrix of three intrinsic rotations?

A rotation matrix of three intrinsic rotations is a mathematical representation used to perform rotations in three-dimensional space using three angles, often referred to as Euler angles. These rotations are applied about the axes of a coordinate system that moves with each rotation, unlike extrinsic rotations which are applied about the fixed axes of the original coordinate system. The matrix is a product of three simpler rotation matrices, each representing a rotation around one of the principal axes (X, Y, or Z).

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

To construct a rotation matrix for three intrinsic rotations, you start by determining the sequence of rotations around the axes, which might be around the X, Y, or Z axis. For each rotation, a basic rotation matrix is created. For example, rotating by an angle θ around the Z-axis would use the matrix:\[ R_z(\theta) = \begin{bmatrix} \cos \theta & -\sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{bmatrix} \]These individual matrices are then multiplied together in the order of the rotations. The order of multiplication is crucial because matrix multiplication is not commutative.

What are Euler angles?

Euler angles are a set of three angles that describe the orientation of a rigid body with respect to a fixed coordinate system. They represent successive rotations around the coordinate axes. There are different conventions to define the sequence of rotations, but one common example is the Z-Y-X convention, where the body is first rotated about the Z axis, then the Y axis, and finally the X axis. Each angle in the sequence is responsible for rotating the system around one of the axes of the coordinate system.

Why is the order of rotations important in a rotation matrix?

The order of rotations is crucial because matrix multiplication, which is used to combine individual rotation matrices, is not commutative. This means that the result of multiplying two matrices depends on the order in which they are multiplied. Consequently, changing the order of rotations will result in a different overall rotation. This characteristic requires careful attention to ensure that the intended orientation is achieved in applications like computer graphics, robotics, and aerospace.

What are gimbal lock and its implications in using rotation matrices?

Gimbal lock is a phenomenon that occurs when two of the three gimbals (rotational axes) align in such a way that one degree of freedom is lost. In terms of rotation matrices and Euler angles, this occurs when the pitch angle approaches ±90 degrees, causing the first and third axes to align. As a result, it becomes impossible to distinguish between rotation about the yaw and roll axes. This loss of one degree of freedom can cause issues in applications like flight control and 3D computer graphics, where maintaining all three degrees of freedom is crucial. Using quaternion-based rotations is one common method to avoid gimbal lock.

Similar threads

X,y,z new coordinate calculation for joint rotations in CoppeliaSimEdu
    • Mechanical Engineering
Replies
2
Views
648
A Order of rotations due to torque in 3DOF in simulations
    • Mechanics
Replies
2
Views
773
FEA - Rotation Matrix of Angular Deflection
    • Mechanical Engineering
Replies
1
Views
2K
MCNP6.2 - Combination of transformations
    • Nuclear Engineering
Replies
1
Views
1K
Observation about the rotation of a disc
    • Mechanics
Replies
4
Views
730
A Corresponding case of steady precession but for Tait-Bryan angles
    • Classical Physics
Replies
1
Views
544
I Intuition Behind Intermediate Axis Theorem in an Ideal Setting
    • Classical Physics
Replies
6
Views
616
A Can the orientation of a body be defined by Euler angles?
    • Classical Physics
Replies
18
Views
1K
A Algebraic proof that Euler angles define a proper rotation matrix
    • Linear and Abstract Algebra
Replies
6
Views
1K
Rotate IMU data to obtain correct measurement data
    • General Engineering
Replies
2
Views
2K

Hot Threads

Recent Insights

Back
Top