Slerp gives wrong values when fed an euler matrix [0,90,0]

Hello,

I am trying out the slerp function to create an interpolation function so i made a simple program to test it on different values :

a = [0,90,1];
b = [1,1,1];

rotationSequence = 'XYZ';
rotationType = 'frame';

% conversion to quaternions
aa = quaternion(a,'eulerd',rotationSequence,rotationType);
bb = quaternion(b,'eulerd',rotationSequence,rotationType);

% Apply Spherical linear interpolation
cc = slerp(aa, bb, 0);

% Conversion back to euler
c = eulerd(cc,rotationSequence,rotationType);

A 0 is supposed to give me back the exact same a. But I get weird values everytime I put 90° in the second element of my matrix.

If I put :

• a = [0,90,1] --> I get c = [1.0000,90,0]
• a = [0,90,2] --> I get c = [2.0000,90,0]
• a = [0,90,3] --> I get c = [3,90,0]
• a = [0,90,4] --> I get c = [4.0000,90,0]
• a = [0,90,5] --> I get c = [48.814074834290350,89.999999146226360,48.814074834290350]
• a = [1,90,1] --> I get c = [2.0000,90,0]
• a = [5,90,5] --> I get c = [10,90,0]

Changing the rotation sequence or the rotation type will give wrong results as well.

Can anyone help me understand why ?

Answer 1

[Brian Fanous]

What you are seeing is known as gimbal lock. You have set your a variable to a Euler angle singularity. Specifically in this case, because the Y rotation is set to 90 degrees, the X and Z rotations effectively do the same thing. You've lost a degree of freedom in your math. Wikipedia has a nice description.

Once you've converted from Euler angles to quaternions, you've lost the (notational) allocation of the non-Y rotation to either X or Z. Going back to Euler angles after slerp cannot rediscover where you had originally put the non-Y rotation.

Two things might help illustrated this:

• Try this same experiment but not at gimbal lock. For example, setting a = [0 80 1] will work as you expect. The closer the middle angle gets to 90 degrees, the more likely you'll hit the gimbal lock condition.
• Comparing Euler angles is a poor way to compare rotations for exactly reason you've found. If you want to compare two orientations use the quaternion dist method instead: dist(aa,cc). This will give you the angular distance between two orientations. Even at your gimbal lock example, you'll see the angular distance is 0.