# Euler angle transform

• A
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?

## Answers and Replies

FactChecker
Science Advisor
Gold Member
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. Lets 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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FactChecker
Science Advisor
Gold Member
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