Euler Angles Transform: Rotating a Body in 3D Space

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• aydos
In summary, the conversation discusses understanding Euler angles and rotation matrices, as well as using them to rotate between the inertial frame and body frame. The problem at hand is figuring out how to calculate the equivalent rotation with respect to the inertial frame when the body is rotated in its own frame. Further clarification is provided on the specific application of this problem in predicting the pose of a vehicle using a 2D kinematic model.

aydos

Only recently started to understand Euler angles and rotation matrices, and I am reasonably comfortable with the concepts already posted here. I am pretty sure I am missing something obvious, but I cannot figure out the way to solve this problem:
A body in 3D space with a orientation defined by yaw, pitch, roll angles. I know how to rotate any (X,Y,Z) point between the inertial frame and body frame using the rotation matrices. The problem I have is:
If I rotate the body with respect to the body frame, how do I calculate the equivalent rotation with respect to the inertial frame?

aydos said:
Only recently started to understand Euler angles and rotation matrices, and I am reasonably comfortable with the concepts already posted here. I am pretty sure I am missing something obvious, but I cannot figure out the way to solve this problem:
A body in 3D space with a orientation defined by yaw, pitch, roll angles. I know how to rotate any (X,Y,Z) point between the inertial frame and body frame using the rotation matrices. The problem I have is:
If I rotate the body with respect to the body frame, how do I calculate the equivalent rotation with respect to the inertial frame?
That is not often done. After the yaw, the rotations are not in the same inertial frame.
Here is something you might try to see if it is what you want:
Reverse the order of the angle rotations. The first rotation should be to roll the plane, the second should be to pitch the plane up, and the third should be to yaw the plane over to the correct position.

EDIT: I changed the sign of the angles.

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Yes, ok. I think this is how to rotate back from body to frame given the original yaw, pitch and roll. I think I was not 100% clear on the problem. Let's say in the body frame, I have a yaw angle with respect to the body frame. What is the yaw, pitch and roll angles with respect to the inertial frame?

Perhaps I am not using the tools correctly, so I will explain the larger application I need this for. I have a road vehicle whose pose is described by X, Y, Z, Yaw, Pitch, Roll in a global coordinate system at T0. I need to predict the pose of the vehicle at T1 in this global coordinate system by using dead-reckoning based on given sensor information: wheel speed and steering angle. The way I set about solving the problem was to use a simple 2D kinematic model based on ackerman steering geometry. This model allows me to predict a new pose at T1 with X,' Y', Yaw' in a 2D plane in the body coordinate system. It seem very straightforward to me to update X, Y and Z by rotating back. However, I do not know how to map the Yaw' back to the global coordinate system. Am I going into a dead end here? Is there perhaps a different way of doing all of this?

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Sorry, I changed the sign of the angles as you were responding to my original version of post #2. I don't know if that makes a difference.

Hi FactChecker, you were on to it. I found the solution by searching for "euler angle rates". Section 1.3 of this link.

FactChecker

What are Euler angles and why are they used?

Euler angles are a set of three angles that can be used to describe the orientation of a rigid body in three-dimensional space. They are commonly used in physics, engineering, and computer graphics to represent rotations in 3D space.

What is the Euler rotation sequence?

The Euler rotation sequence, also known as the Tait-Bryan angles, refers to the order in which the three angles are applied to rotate a body. The three axes of rotation are typically labeled as roll, pitch, and yaw, and the order of rotation can be either XYZ, ZYX, or any other combination.

How do Euler angles differ from other methods of representing rotations?

Euler angles are just one way of representing rotations in three-dimensional space. Other methods include rotation matrices, unit quaternions, and axis-angle representations. Each method has its own advantages and disadvantages, and the choice depends on the specific application.

What are some common problems or limitations associated with using Euler angles?

One of the main limitations of Euler angles is the issue of gimbal lock, which occurs when two of the axes align and the third axis becomes redundant. This can lead to ambiguity and difficulty in interpreting the rotations. Another problem is that the angles themselves are not always intuitive and can be difficult to visualize.

Are there any alternative methods to Euler angles for rotating a body in 3D space?

Yes, there are several alternative methods that can be used to represent rotations, such as rotation matrices, quaternions, and axis-angle representations. Each method has its own advantages and disadvantages, and the choice depends on the specific application and the desired level of precision and complexity.