Axial rotation coordinate coupling coefficients update

[Matthew Millard]

The axial rotation of the rib cage with respect to the pelvis does not follow the generalized coordinate axial_rotation. In contrast, flexion/extension have been implemented so that when flex_extension is set to 45, the rib cage rotates with respect to the pelvis by 45 degrees. Ditto for lateral bending. As such, I believe Christophy et al. had the design intent of also having the axial rotation of the rib cage with respect to the pelvis be equal to the generalized coordinate axial_rotation. Please note that this error does not mean that the model is broken, but it does mean that the numerical value of the axial rotation coordinate does not correspond to the rotation between the pelvis and rib cage which is confusing.

Christophy entered these coefficients in Lumbar_C_210.osim

L1/L2: 0.02888888890000000000 (line 15915)
L2/L3: 0.03111111111100000100 (line 15882)
L3/L4: 0.03777777799999999800 (line 15849)
L4/L5: 0.03777777699999999800 (line 15816)
L5/S1: 0.030329999999999999 (line 14674)
Sum : 0.1711

Since the sum of these coefficients is only 0.1711, if you apply an axial_rotation of say pi/4 radians (45 degrees) you see that the torso rotates by only 0.0428 pi radians (7.7 degrees).

If we look at the orginal research, Fugii et al. reports that during their MRI study each lumbar joint twisted by

L1/L2: 1.3 degrees
L2/L3: 1.4
L3/L4: 1.7
L4/L5: 1.7
L5/S1: 1.6

when there was a 45 degree rotation between the trunk and the pelvis. If you scale these coordinates by 45 degrees you get the coefficients used by Christophy et al. However, we are only concerned with what the lumbar spine is doing, and so it does not matter that the trunk rotated by 45 degrees: the additional axial rotation is taking place at other joints (the thorasic spine and scapulothorasic joints perhaps) that have nothing to do with the lumbar spine. Instead we normalize these coefficients so that the final result sums to 1 which yields

L1/L2: 0.168831168831169
L2/L3: 0.181818181818182
L3/L4: 0.220779220779221
L4/L5: 0.220779220779221
L5/S1: 0.207792207792208
Sum : 1

With these coefficients when the axial_rotation is set to 45 degrees, for example, the sum total of the axial rotation applied to the lumbar spine is 45 degrees. The attached files contain a version of Lumbar_C_210.osim in which the coefficients associated with axial rotation have been corrected.