difference between joint coordinate system and cardan angle

[jean jeuvrey]

Hello,
i'am a beginner in biomechanics, i try to understand the difference between the calcul of joint angle with cardan method and joint coordinate system method.
if we consider the calcul of the knee joint with the thight LCS (e.g. proximal segment) and the leg LCS (e.g. distal segment), with cardan method and Xyz sequence, we find a flexion/extension angle around the X axis of the proximal segment (e.g. thight LCS), an abduction/adduction around a nodal axis (y) and an axial rotation around the z axis of the distal segment (e.g. leg LCS). Up to now, it is clear for me, but it is the joint coordinate system i don't understand. the sequence of the JCS is Xyz with first the rotation around the X axis of the proximal segment (e.g. thight LCS), an abduction/adduction around a floating axis (FA) which is orthogonal with the X axis of the proximal segment and the z axis of the distal segment (so not the same axis -nodal axis of the cardan sequence) and an axial rotation around the z axis of the distal segment (e.g. leg LCS). So the angle with cardan method and JCS method should not be the same!!! Why did i read the angle are the same?
thanks

[Ton van den Bogert]

Re: difference between joint coordinate system and cardan angle

They are the same.

The cardan model obtains the orientation of the distal segment by rotating first about X, then about Y, and then about Z. The second rotation does not happen about the original Y axis, but the Y axis that results from the first rotation about X. So this "nodal axis" is still perpendicular to X of the proximal segment. Similarly, the nodal axis is also the Y axis of the distal segment, before it has undergone its final rotation about Z. So the nodal axis is also perpendicular to the Z axis of the distal segment. This is also exactly how the JCS is defined.

Also of you look up the symbolic form the rotation matrix for the JCS (Grood & Suntay, J Biomech Eng 1983), you can see that the expressions are identical to the rotation matrix that results from the cardan sequence in which you multiply three matrices: R = Rx*Ry*Rz.

I hope this helps.

Ton van den Bogert

[jean jeuvrey]

Re: difference between joint coordinate system and cardan angle

They are the same.

The cardan model obtains the orientation of the distal segment by rotating first about X, then about Y, and then about Z. The second rotation does not happen about the original Y axis, but the Y axis that results from the first rotation about X. So this "nodal axis" is still perpendicular to X of the proximal segment. Similarly, the nodal axis is also the Y axis of the distal segment, before it has undergone its final rotation about Z. So the nodal axis is also perpendicular to the Z axis of the distal segment. This is also exactly how the JCS is defined.

Also of you look up the symbolic form the rotation matrix for the JCS (Grood & Suntay, J Biomech Eng 1983), you can see that the expressions are identical to the rotation matrix that results from the cardan sequence in which you multiply three matrices: R = Rx*Ry*Rz.

I hope this helps.

Ton van den Bogert

Hello,
Thanks for your response.
I indeed make a mistake when i look at the y axis of cardan angle and the floating axis of the JCS method, i find there aren't the same, but i have a look again and there are indeed the same...
Thanks for your help!

[Frances Sheehan]

Re: difference between joint coordinate system and cardan angle

Just a point of clarification. Ton's email is correct, but the manner in which Grood & Suntay set up their axes creates a set of equations that are slightly different than the classic XYZ cardan rotation angles (Mitiguy & Sheehan, J Biomech. 1999 Oct;32(10):1135-6). Also, if you do not assign the axes (x, Y, and Z) in the same exact manner, the Grood & Suntay equations do not work.

Fran

[jean jeuvrey]

Re: difference between joint coordinate system and cardan angle

Just a point of clarification. Ton's email is correct, but the manner in which Grood & Suntay set up their axes creates a set of equations that are slightly different than the classic XYZ cardan rotation angles (Mitiguy & Sheehan, J Biomech. 1999 Oct;32(10):1135-6). Also, if you do not assign the axes (x, Y, and Z) in the same exact manner, the Grood & Suntay equations do not work.

Fran

Ok, It is good to know...thanks

[Ton van den Bogert]

Re: difference between joint coordinate system and cardan angle

Thanks, Fran, for pointing this out.

Grood/Suntay have a different sign convention, and they have an offset of pi/2 in the second rotation (ab-adduction). Once you account for those differences, the rotation matrix in their appendix is indeed equal to the cardanic rotation matrix R = Rx(alpha)*Ry(beta)*Rz(gamma). The Femur X axis is the flexion axis, and the tibia Z axis is the internal rotation axis.

The ISB standard (Wu & Cavanagh, J Biomech 1995) had different segment coordinate systems: Zfemur was the flexion axis and Ytibia was the internal rotation axis. If you construct the rotation matrix that way, you still get the same matrix, with some rows and columns switched. So you still get the same relationship between joint angles and 3D rotation.

I would actually prefer not be limited to using segment coordinate axes as cardanic rotation axes. Regardless of the segment coordinate systems, you can define a flexion axis and an internal rotation axis, and define a mechanical linkage as Grood & Suntay did. There is no reason why these axes need to be segment coordinate axes. Modern software like Opensim allows this.

Ton

[Ton van den Bogert]

Re: difference between joint coordinate system and cardan angle

Thanks, Fran, for pointing this out.

Grood/Suntay have a different sign convention, and they have an offset of pi/2 in the second rotation (ab-adduction). Once you account for those differences, the rotation matrix in their appendix is indeed equal to the cardanic rotation matrix R = Rx(alpha)*Ry(beta)*Rz(gamma). The Femur X axis is the flexion axis, and the tibia Z axis is the internal rotation axis.

The ISB standard (Wu & Cavanagh, J Biomech 1995) had different segment coordinate systems: Zfemur was the flexion axis and Ytibia was the internal rotation axis. If you construct the rotation matrix that way, you still get the same matrix, with some rows and columns switched. So you still get the same relationship between joint angles and 3D rotation.

I would actually prefer not be limited to using segment coordinate axes as cardanic rotation axes. Regardless of the segment coordinate systems, you can define a flexion axis and an internal rotation axis, and define a mechanical linkage as Grood & Suntay did. There is no reason why these axes need to be segment coordinate axes. Modern software like Opensim allows this.

Ton

[Gennaro Arguzzi]

Re: difference between joint coordinate system and cardan angle

Hello Ton,

can you tell me if it is correct to say that the nodal axis is an axis perpendicular to both the flexion-extension and axial rotation axes?
I found the above sentence in the book of Gordon Robertson, but I do not know if it is a general definition.

Thank you so much for your time,
Gennaro Arguzzi

[Ton van den Bogert]

Re: difference between joint coordinate system and cardan angle

I had not heard of the term "nodal axis", and I don't own the book, so I can't confirm that this is correct.

Grood and Suntay (1983) uses the term "floating axis" for this axis. The axis is not fixed in either body segment (hence "floating") and it is perpendicular to the first axis (fixed in the first body segment) and the last axis (fixed in the second body segment). In the standard Joint Coordinate System, the first axis is the flexion axis and the last axis is the axial rotation axis.

Robertson may have borrowed the term from astronomy. In astronomy, Euler angles are used to describe the orientation of a planet. The first rotation is the orbit around the sun. The last is the rotation about its own axis. The middle rotation is a constant and represents the tilt of the planet. See https://www.astro.com/astrowiki/en/Moon%27s_Nodes.

Ton

[Gennaro Arguzzi]

Re: difference between joint coordinate system and cardan angle

I had not heard of the term "nodal axis", and I don't own the book, so I can't confirm that this is correct.

Grood and Suntay (1983) uses the term "floating axis" for this axis. The axis is not fixed in either body segment (hence "floating") and it is perpendicular to the first axis (fixed in the first body segment) and the last axis (fixed in the second body segment). In the standard Joint Coordinate System, the first axis is the flexion axis and the last axis is the axial rotation axis.

Robertson may have borrowed the term from astronomy. In astronomy, Euler angles are used to describe the orientation of a planet. The first rotation is the orbit around the sun. The last is the rotation about its own axis. The middle rotation is a constant and represents the tilt of the planet. See https://www.astro.com/astrowiki/en/Moon%27s_Nodes.

Ton

Thank you very much for your explanation Ton.