How to calculate roll, pitch and yaw from XYZ coordinates of 3 planar points?

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T on 9 Aug 2016
Commented: alper yeldan on 5 Sep 2021
I have a body moving through a calibrated space. The body has 3 co-planar points on it. I know the global x, y and z coordinates of each of those points.
How can I calculate the roll, pitch and yaw angles of the plane on the body?
Thanks, Sophia
Arash Jouybari
Arash Jouybari on 12 Feb 2017
Hi Sophia S I exactly want to know about your GPS antenna configuration... and I didn't understand about body-fixed axis that Dhruvesh Patel said.
I have your problem as you wrote above. this is my email plaese help me to solve my problem.

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Accepted Answer

Dhruvesh Patel
Dhruvesh Patel on 10 Aug 2016
I assume that you are following Euler Angle convention of roll-pitch-yaw in the order of X-Y-Z. You have three coplanar points P1, P2 and P3 on the body in clockwise order (looking from the top) and that the X-axis of the body-fixed frame can be taken along the vector starting from P3 passing through the midpoint of the segment joining P2 and P3 i.e., x = (P1+P2)/2 -P3.
First we need to obtain the unit vectors along the XYZ-axes of the body-fixed frame. To do this:
  1. Construct vectors v1=P2-P1 and v2=P3-P1.
  2. Now Z is along the vector cross(v1,v2). Normalize it to get the unit vector Z.
  3. The unit vector (X), along X-axis is obtained by normalizing the vector x obtained above.
  4. Unit vector (Y) along Y-axis can be obtained as cross(Z,X). This will be a unit vector as Z and X are unit vectors.
Now, the Rotation Matrix representing the orientation of the body-fixed frame can be written as R=[X1,Y1,Z1; X2,Y2,Z2; X3,Y3,Z3], where X=[X1; X2; X3;] and so on for Y and Z. Assuming roll, pitch and yaw angles are called alpha, beta and gamma respectively. These can be obtained as:
  • alpha= atan2(-Z2, Z3)
  • beta= asin(Z1)
  • gamma = atan2(-Y1,X1)
I hope this is what you were looking for. More about Euler angles and finding them from rotation matrix, can be found here. If you have access to MATLAB's Robotic System Toolbox, you can use rotm2eul function to get Euler Angles from rotation matrix.
alper yeldan
alper yeldan on 5 Sep 2021
@Justin Boyd He put unit vectors(column vector) side by side to get thr rotation matrix. This one is XYZ rotation matrix. In wikipedia it explains how to construct the rotation matrix @Dhruvesh Patel is doing the exact same thing.

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