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

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

[José-Luis]

By planar you mean colinear? 3 planar points is redundant as three points are the minimum required to define a plane. And if they are colinear then you can only calculate two out of the three. Some goes if they define a plane: you still need to define axes of reference.

[T]

No, I mean planar as they are all on the same plane. Does that make sense? I need to calculate the Euler angles between that plane and the reference plane (which is a horizontal cross-section of the calibrated space ie. the floor). I hope the image is clear. It's basically a question of: what are the Euler angles between the global coordinate system and the fixed coordinate system on the body? But finding these angles from coordinates of 3 points on the plane is proving difficult. Thanks so much for the help :)

[Dhruvesh Patel]

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]

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]

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. [e-mail] this is my email plaese help me to solve my problem.

thanks

Answer 1

[Dhruvesh Patel]

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]

Thank you Dhruvesh. This is exactly what I was looking for. Really appreciate your clear answer.

[Justin Boyd]

For this part:

R=[X1,Y1,Z1; X2,Y2,Z2; X3,Y3,Z3],

What is X1, Y1 Z1 - the xyz components of point 1?

Thanks

[alper yeldan]

@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.