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

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T on 9 Aug 2016
Commented: Justin Boyd on 8 Apr 2020
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


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Dhruvesh Patel
Dhruvesh Patel on 9 Aug 2016
Calculating roll, pitch and yaw angles for a body amounts to finding the orientation of the XYZ-axes, rigidly attached to the body, w.r.t the reference frame (in this case it is floor). Trying to get the orientation of the body fixed XYZ-axes just from 3 coplanar points presents one ambiguity i.e., the direction of the axis perpendicular to the plane formed by these three points cannot be determined, as the plane has two faces and this axis can point to either of the two directions. This can be resolved if the points have a sense of direction in their ordering i.e., if it is known that the points are listed in clockwise or any clockwise direction. If you can mention that then this problem can be solved.
T on 9 Aug 2016
The body-fixed XYZ-axes are represented by the red, blue and green lines in the image. All three points lie on the XY plane (assuming Roll axis (red) is X and pitch axis (blue) is Y). The points form a 2.2mm equilateral triangle. P1 is forward left, P2 is forward right and P3 is the rearmost point on the body (points are ordered clockwise). The direction of the axis perpendicular to the plane should be up or away from the body.
Is this information sufficient to resolve my problem?
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.


T on 12 Aug 2016
Thank you Dhruvesh. This is exactly what I was looking for. Really appreciate your clear answer.
Justin Boyd
Justin Boyd on 8 Apr 2020
For this part:
R=[X1,Y1,Z1; X2,Y2,Z2; X3,Y3,Z3],
What is X1, Y1 Z1 - the xyz components of point 1?

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