(beginner) calculating orientation change with eulers

[astull]

I'm capturing Euler angles before and after a user manipulation. The object is fixed in location and only changes in orientation. What I want to measure is the error between a target orientation and the user's response orientation for the object. I've calculated the difference for each of the 3 axis but, if I understand correctly, this doesn't give a measure of the orientation change in 3D space. What is the proper method/strategy to attack this question? I'd like one measure of orientation error but I'm not sure how to get it.

[farshizzo]

Using the euler angles, you can find the look vector of each orientation, then compute the angle difference between these vectors. However, this method will not take roll into account. For example, if both orientations are looking straight ahead, but one has the head rolled 90 degrees, the difference will still be 0 degrees. If you are interested in the roll, then you can also extract the up vector of each orientation and compute the angle difference between these. It would then be up to you to figure out how to convert these two angle differences into some orientation error metric.

Let me know if you are interested in this method and I will supply you with some sample code.

[astull]

Yes, this method makes sense and sample code would be welcomed.

[farshizzo]

Here you go

import viz
viz.go()

euler1 = [90,0,0]
euler2 = [91,0,5]

def ComputeAngleDifference(euler1, euler2):
	m = viz.Transform()
	
	#Extract forward/up vector from euler1
	m.setRot( vizmat.EulerToAxisAngle(euler1) )
	d = m.get()
	forward1 = d[8:11]
	up1 = d[4:7]
	
	#Extract forward/up vector from euler2
	m.setRot( vizmat.EulerToAxisAngle(euler2) )
	d = m.get()
	forward2 = d[8:11]
	up2 = d[4:7]

	#Compute angle difference between forward/up vectors
	forward_diff = vizmat.AngleBetweenVector(forward1,forward2)
	up_diff = vizmat.AngleBetweenVector(up1,up2)

	return forward_diff,up_diff
	
print ComputeAngleDifference(euler1,euler2)

[astull]

Jerry pointed out that the script doesn't always give a unique solution. For example, both of the following sets of eulers give the same forward_diff and up_diff solution, which is (90, 90).

euler1 [0, 90, 0]
euler2 [0, 0, 0]

and

euler1 [0, 90, 30]
euler2 [0, 0, 0]

Is there a way to address this?

[farshizzo]

I'm not sure what the problem is. The solution you are getting for those sets of euler angles is correct. The problem with getting non-unique solutions is when extracting euler angles from a matrix. In this example we are creating a matrix from euler angles, so there should be no problem.

[astull]

Is the non-unique solution due to the selection of 90 degrees as the pitch angle? Ultimately, I need a unique solution. Is it reasonable to assume that this might be addressed by calculating the difference for a 3rd vector? I don't know the math so I'm only assuming the following modified script would retrieve the necessary values from the matrix to calculate a 3rd vector:

#Extract forward/up vector from euler1
m.setRot( vizmat.EulerToAxisAngle(euler1) )
d = m.get()
forward1 = d[8:11]
up1 = d[4:7]
side1 = d[12:3]

#Extract forward/up vector from euler2
m.setRot( vizmat.EulerToAxisAngle(euler2) )
d = m.get()
forward2 = d[8:11]
up2 = d[4:7]
side2 = d[12:3]

Am I wondering in the dark?

Alternatively, wouldn't a unique solution also result if I calculated the displacement from the facing direction AND the angle of rotation about that axis (the facing direction)?

[farshizzo]

Yes, you could also compute the angle difference between the side vectors as well. Here is a modified version of the sample code that also returns the difference between the side vectors:

import viz
viz.go()

euler1 = [0,0,0]
euler2 = [0,90,30]

def ComputeAngleDifference(euler1, euler2):
	m = viz.Transform()
	
	#Extract forward/up/side vector from euler1
	m.setRot( vizmat.EulerToAxisAngle(euler1) )
	d = m.get()
	forward1 = d[8:11]
	up1 = d[4:7]
	side1 = d[:3]
	
	#Extract forward/up/side vector from euler2
	m.setRot( vizmat.EulerToAxisAngle(euler2) )
	d = m.get()
	forward2 = d[8:11]
	up2 = d[4:7]
	side2 = d[:3]

	#Compute angle difference between forward/up/side vectors
	forward_diff = vizmat.AngleBetweenVector(forward1,forward2)
	up_diff = vizmat.AngleBetweenVector(up1,up2)
	side_diff = vizmat.AngleBetweenVector(side1,side2)

	return forward_diff,up_diff,side_diff
	
print ComputeAngleDifference(euler1,euler2)

Alternatively, wouldn't a unique solution also result if I calculated the displacement from the facing direction AND the angle of rotation about that axis (the facing direction)?

Yes, this should provide a unique solution as well. I'll write up a sample script that does this and post it shortly.

[astull]

I don't mean to be a bother but the modified script still yields a non-unique solution. For example, both of the following sets of eulers yield the same solution:

euler1 = [30,90,0]
euler2 = [0,0,0]

euler1 = [0,90,30]
euler2 = [0,0,0]

solution: (90, 90, 30)

I don't understand why this happens. Maybe the vector/rotation alternative may be a more practical solution.

Thank you again for your help!

[farshizzo]

Those values are correct. I'm not sure what you are expecting. There is always going to be multiple euler sets that will yield the same angle difference. The vector/rotation alternative will also yield the same value for multiple sets. For example, the vector/rotation difference between the following sets will be the same:

euler1 = [0,0,0]
euler2 = [90,0,0]

#Vector difference will be 90
#Rotation difference will be 0

euler1 = [90,0,0]
euler2 = [180,0,0]

#Vector difference will be 90
#Rotation difference will be 0

From your first post it seems you are interested in the relative error between the two orientations. I don't see how having non-unique solutions affects this. Maybe I'm misunderstanding your original request.

[astull]

Yes, indeed! There are likely to be multiple euler sets that yield the same displacement errors. Sorry for being so bone-headed.

[farshizzo]

No worries. I know how confusing all this 3d math can get.

Would you still like me to provide you with the vector/rotation alternative?

[astull]

Yes, it would be helpful if you could offer a script for calculating the vector/rotation. Fewer variables makes the analysis much easier.

Thank you.

[farshizzo]

Here is the vector/rotation code:

import viz
viz.go()

euler1 = [0,0,30]
euler2 = [0,5,-30]

def ComputeAngleDifference(euler1, euler2):
	m = viz.Transform()
	
	#Extract forward/up/side vector from euler1
	m.setRot( vizmat.EulerToAxisAngle(euler1) )
	d = m.get()
	forward1 = d[8:11]
	up1 = vizmat.Vector(d[4:7])
	
	#Extract up vector from euler1 without roll
	m.setRot( vizmat.EulerToAxisAngle(euler1[0],euler1[1],0.0) )
	d = m.get()
	up1_no_roll = vizmat.Vector(d[4:7])
	
	#Compute rotation around forward vector of euler1
	roll1 = vizmat.AngleBetweenVector(up1,up1_no_roll)
	if (up1_no_roll.cross(up1) * forward1) < 0.0:
		roll1 *= -1.0
	
	#Extract forward/up/side vector from euler2
	m.setRot( vizmat.EulerToAxisAngle(euler2) )
	d = m.get()
	forward2 = d[8:11]
	up2 = vizmat.Vector(d[4:7])
	
	#Extract up vector from euler2 without roll
	m.setRot( vizmat.EulerToAxisAngle(euler2[0],euler2[1],0.0) )
	d = m.get()
	up2_no_roll = vizmat.Vector(d[4:7])
	
	#Compute rotation around forward vector of euler2
	roll2 = vizmat.AngleBetweenVector(up2,up2_no_roll)
	if (up2_no_roll.cross(up2) * forward2) < 0.0:
		roll2 *= -1.0

	#Compute angle difference between forward/up/side vectors
	forward_diff = vizmat.AngleBetweenVector(forward1,forward2)
	rotation_diff = abs(vizmat.AngleDiff(roll1,roll2))

	return forward_diff,rotation_diff
	
print ComputeAngleDifference(euler1,euler2)

[astull]

I met with Andy B. today and he pointed me to the quatdiff function as a potential single metric solution to the error calculation. I know the function is available through Vizard but I'd like to verify it on paper. He said that you might be able to send me the C code.

Thanks again!

[farshizzo]

The QuatDiff function simply uses the above method of computing the angle difference between the forward and up vectors, and returns the maximum of those two values.

You can modify the sample code I gave you to replicate this behavior.

[farshizzo]

Here is some code that calculates the difference between two angles based on the method steve described:

import viz
import math
viz.go()

euler1 = [90,0,0]
euler2 = [0,0,90]

def ComputeAngleDifference(euler1, euler2):
       q1 = viz.Quat( vizmat.EulerToQuat(euler1) )
       q2 = viz.Quat( vizmat.EulerToQuat(euler2) )

       delta = q1 * q2.inverse()

       return viz.degrees( math.acos( delta.w ) ) * 2.0

print ComputeAngleDifference(euler1,euler2)

I'll updated Vizard's QuatDiff function to use this method in the next release.