How to solve gimbal lock logically?

Hello.

I am analyzing the three-dimensional motion data of a glenohumeral joint with MATLAB, and I think the analyzed results seem to have an error.

I am told that this kind of error is called 'gimbal lock,' but I don't know well about it as I am new to this area.

So my question is, how to logically solve this gimbal lock? Is it appropriate to remove the error data and interpolate it?

Thank you.

Answer 1

[William Rose]

@Kang Geonhui,

Yes, your plot illustrates what happens when you are near "gimbal lock". Interpolation is not a good idea.

Background: From motion caputure, you obtain a 3x3 rotation matrix at each instant. The rotation matrix is always well defined. Nonetheless, the determination of flexion/extension, ab/adduction, and internal/external rotation can become problematic, as your plot shows, when the middle angle is approximately 90 degrees. Sometimes the problem can be avoided by choosing a different rotation sequence, so that the middle angle in the sequence is not near 90 degrees. Another solution is to represent rotations with helical angles. Helical angles never suffer from gimbal lock.

Consult the large literature on the subject. Start with the recommendations for analysis of shoulder joint motion of the International Society of Biomechanics, here. For better or worse, not many researchers follow the ISB recommendations for reporting shoulder joint angles. Here is an example paper (one of many) about how to avoid gimbal lock issues when analyzing shoulder motion.

Good luck with your work.

[Paul]

@William Rose

Do you have links to a good reference or two that explains helical angles? I'd like to read more about that concept.

[Kang Geonhui]

@William Rose

Thank you for your detailed reply!

I am analyzing shoulder motion as ISB recommended, calculating the angle between the coordinate systems of both the shoulder and humerus. After that, I made two rotation matrices from coordinate systems and converted these to quaternions using 'rotm2quat' for quaternion multiplication (quaternion of one segment * quaternion conjugate of the other segment using 'quatmultiply') to avoid gimbal lock issue. But after the conversion of calculated quaternions to Euler angles, the gimbal lock issue was still present. In this calculation process, how to choose a different rotation sequence?

I can not understand what "the middle angle" means. Could you explain it more?

I learned about the helical angle for the first time. I will read the paper you attached. Thank you!!

[Kang Geonhui]

@William Rose My MATLAB code is here. It won't run because it's part of the whole code.

[William Rose]

@Kang Geonhui:

The middle angle:

When trying to describe a 3D rotation, you can do it with the rotation matrix. This 3x3 matrix has some reducndancy in it, since it has 9 elements, but only three independent quantities. There are various versions of the three quantities that can be extracted from the rotaiton matrix. In biomecha ics, we have the proximal and sital segment. Each has a coordinate system defined by an i,j,k set of vectors. The roation matrix transforms the proximal ijk to the distal ijk. THis transformation could be acocmplished in various ways. I could be done by 3 successive rotaitons, and you could do them in different orders. But if you change the order, the amount of rotaiton on each axis will tend to differ. This is a consequence of the non-commutativity of the 3D rotaiton matrices when you multiply them. So you could do flex/extend, then internal/external rotation, then ab/adduction. In that case, Internal/external rotaiton is the "middle angle". If you do ab/adduct, then flex/extend, the IR/ER, then flex/extend is the middle angle.

Any rotation can be accomplisheed by just one rotation, if you choose the axis of rotaiton, and the amount of rotation, properly. This rotation is called the helical rotation. The axis of rotation is called the helical axis. The amount is called the helical angle. You can represent a helical rotation by a vector pointing along the helical axis, with a length equal to the helical angle. The projection of this vector onto the three axes can be used to express the amount of flex/extend, abaddcut, and internal external rotaiton. In the special case of a rotation about one of the standard anatomical axes (for example, a pure abduction or adduction, or a pure flexion or extension), the helical approach gives the answer you would want to get.

There are formulas and scripts (see the notes attached) which compute the helical angles from the rotation matrix.

[Kang Geonhui]

Thank you very much for your reply and explanation of the middle angle. I should calculate the helical angle instead of using quaternion! Have a nice day!!

Answer 2

[William Rose]

@Kang Geonhui,

My lecture notes on rotation matrices are attached. Lectures 2 and 3 include discussion of helical angles.

The lecture notes of Young-Hoo Kwon are nice. Here is the top page. I recommend reading his notes on rotation matrix, here. Then see his notes on Euler and Cardan angles, here. Then see his notes on helical axis, here.

Here is another paper that might interest you: "The Appropriateness of the Helical Axis Technique and Six Available Cardan Sequences for the Representation of 3-D Lead Leg Kinematics During the Fencing Lunge", 2013, https://sciendo.com/article/10.2478/hukin-2013-0020

[Kang Geonhui]

There's a lot to study! I appreciate your providing references.