How to find the angle between two quaternions?

If quaternions represent an orientation in space, there is an axis between any two orientations and an angle between them. I'm looking for the procedure to find that angle.

[James Tursa]

See this link for a discussion of the MATLAB quaternion convention:

https://www.mathworks.com/matlabcentral/answers/465053-rotation-order-of-quatrotate

Answer 1

[James Tursa]

For example purposes I am using the coordinate frames as ECI and BODY

Q1 = quaternion from ECI->BODY1

Q2 = quaternion from ECI->BODY2

Then perform the following calculation

Q12 = conj(Q1) * Q2  % <- quaternion conjugate and quaternion multiply

Q12 = quaternion from BODY1->BODY2

There may be MATLAB functions to do the conjugate and multiply, but I don't know at the moment. The conjugate of Q1 is simply [Q(1),-Q(2:4)] of course assuming the scalar is the first element.

If we assume the scalar is the first element of the quaternion, matching the MATLAB quaternion functions convention, then you have

Q12(1) = cos(angle/2)

and

Q12(2:4) = sin(angle/2) * e

where e is the unit axis of rotation

From these you can solve for the angle

angle = 2 * atan2(norm(Q12(2:4)),Q12(1))

See also this post:

https://www.mathworks.com/matlabcentral/answers/283830-why-do-we-use-q-1-r-q-when-convert-quaternion-to-dcm?s_tid=answers_rc1-1_p1_MLT

and this post:

https://www.mathworks.com/matlabcentral/answers/415936-angle-between-2-quaternions?s_tid=answers_rc1-2_p2_MLT

[Silas Waxter]

angle = 2 * atan2(norm(Q12(2:4)),Q12(1))

In this code, the "norm(Q12(2:4))" is a vector correct? How does the atan2() function handle the vector?

Note: I'm actually developing the code in c#, in which the atan2() function only has double types for the inputs. Also, sorry for my lack of knowledge. My highest level of math is math 112. Thanks.

[Bruno Luong]

In this code, the "norm(Q12(2:4))" is a vector correct?

No, norm returns a scalar

[Silas Waxter]

Using this method with the quaternions from this post, the returned angle is 118.252098940265 deg, whereas the mehtod used in that post returns 127.7227 deg. Is my question and the post's question the not the same?

[James Tursa]

The confusion is caused by which element is the scalar part. The scalar part can be either the 1st element or the 4th element ... both are widely used in industry and both are "correct". If your quaternions are defined with the scalar as the 1st element (matching MATLAB toolboxes), then use the method above. If your quaternions are defined with the scalar in the 4th element (not matching MATLAB toolboxes), then use the other post.

[Silas Waxter]

Ok thanks.

Answer 2

[Jim Riggs]

I am more comfortable working with direction cosine matrices, so the way I would do this is to first convert the quaternions to DCM's;

Assume Quaternion A represents the orientation of body A in the I frame

Quaternion B represents the orientation of Body B in the I frame.

The direction cosine matrix, C, that transforms from I to A is defined as:

If the Quaternion is defined as [a, b, c, d], (where a is the scalar part and b, c, d is the vector part) then the direction cosine matrix in terms of the quaternion is

So, first compute the direction cosine matrix from quaternion A (DCMA) and from quaternion B (DCMB).

Now the direction cosine matrix for the transformation from A to B is

DCMAB = DCMB * transpose(DCMA).

Now that you have the transformation matrix from A to B, you can get the Euler angles or rotation vector from this DCM.

Using DCMAB, this gives the rotation vector from A to B.

[Andrei Neboian]

Hello Jim,

thanks for a detailed guideline using the DCM's.

Will this method work for the following setup/problem and do you think that there are additional issues that could arise from applying the DCM method you describe?

Setup: I have two IMU Sensors, each delivering Quaternion data (w1,x1,y1,z1) and (w2,x2,y2,z2) for their absolute orientation in space in real-time (these sensors produce magnetic compass, gyroscope, and accelerometer data, which are fused together inside the sensor to provide Quaternion data).

Problem: I want to know how both sensors are oriented to each other. In particular, I want to know the angle (-180deg to 180deg) between the y-axis of the first sensor and the y-axis of the second sensor - lets call this angle Theta.

I have tried two methods:

• Quaternions -> convert to Euler Angles -> Pitch2-Pitch1 should (?) correspond to the angle Theta between the y-axes of both sensors (as far as my understanding goes),
• Euler Angles -> determine Rotation matrices R1 and R2 for each sensor -> rotate unit-vector (0,1,0) along y-axis using R1 and R2 -> angle between both rotated unit-vectors should (?) correspond to the angle Theta I look for.

But both methods yielded an angle between both sensors. But at certain orientations of both sensors, the value of Theta was sporadically jumping like +-90 or +-180deg or so. I assume the reason for these jumps was the Gimbal Lock.

[Jim Riggs]

It sounds like the problem you are working is exactly the same as the original question in this post.

I offered a solution based on converting quaterions to DCM's and then manipulating the DCM's to get the relative DCM from sensor 1 to sensor 2. Then, from this relative DCM, extract the Euler angles, or rotation vector. I suggested this approach because I am less comfortable working with quaternion math, as in James Tursa's answer. His approach is, no doubt, the most direct.

I would advise against working with Euler angles until the very last step, because they are fraught with potential problems, like the indeterminant angles (+-90 or +-180 situation). This is not a sign of gimbal lock, but of an indeterminant solution in the Euler angle. Quaternions and direction cosine matrices do not have these issues, so you should calculate the relative orientation between the two frames using quaternion math (per James Tursa's answer) or using DCM's (as I suggest) and then, in the final step, you can get the relative Euler angles from the relative quaternion or DCM.