Plug-in Gait uses Cardan angles, modified in the case of the ank= le angles, to represent both

- Absolute rotations of the pelvis and foot segments and
- Relative rotations at the hip, knee, and ankle joints

These angles can be described either as a set of rotations carried out o= ne after the other (ordered).

For more information about the use of Cardan angles to calculate joint k= inematics, refer to Kadaba, Ramakrishnan and Wooten (1990) and Davis, =C3= =95unpuu, Tyburski and Gage (1991) (see Plug-in Gait referen= ces).

The rotations are measured about anatomical axes in order to simplify th= eir interpretation.

For more information on joint angle descriptions, including the issues o= f gimbal lock and Codman's Paradox, see:

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To describe an angle using ordered rotations, the following are true:

- One element is 'fixed'. For absolute rotations the laboratory axes are = fixed. The proximal segment axes are fixed for relative rotations.
- The second element 'moves'. This means the segment axes move for absolu= te rotations and distal segment moves for relative rotations.

A joint angle is then defined using the following ordered rotations:

- The first rotation (flexion) is made about the common flexion axis. The= other two axes, abduction and rotation, are afterwards no longer aligned i= n the two elements.
- The second rotation (abduction) is made about the abduction axis of the= moving element. The third rotation (rotation) is made about the rotation a= xis of the moving element.

In addition to using ordered rotations, joint angles can also be describ= ed using goniometric information. Using goniometric definitions, a joint an= gle is described by the following:

**Flexion**is about the flexion axis of the proximal (or = absolute) element.**Rotation**is about the rotation axis of the distal elem= ent.**Abduction axis**'floats' so as always to be at right an= gles to the other two.

Cardan angles work well unless a rotation approaching 90 degrees brings = two axes into line. When this happens, one of the possible rotations is los= t and becomes unmeasurable. Fortunately, this does not frequently occur in = the joints of the lower limbs during normal or pathological gait. However t= his may occur in the upper limb and particularly at the shoulder. For more = information, see Gimbal lock a= nd also Codman's Paradox below.

Gimbal lock occurs when using Cardan (Euler) angles and any of the rotat= ion angles becomes close to 90 degrees, for example, lifting the arm to poi= nt directly sideways or in front (shoulder abduction about an anterior axis= or shoulder flexion about a lateral axis respectively). In either of these= positions the other two axes of rotation become aligned with one another, = making it impossible to distinguish them from one another, a singularity oc= curs and the solution to the calculation of angles becomes unobtainable.

For example, assume that the humerus is being rotated in relation to the= thorax in the order Y,X,Z and that the rotation about the X-axis is 90 deg= rees.

In such a situation, rotation in the Y-axis is performed first and corre= ctly. The X-axis rotation also occurs correctly BUT rotates the Z axis onto= the Y axis. Thus, any rotation in the Y-axis can also be interpreted as a = rotation about the Z-axis.

True gimbal lock is rare, arising only when two axes are close to perfec= tly aligned.

The second issue however, is that in each non-singular case there are tw=
o possible angular solutions, giving rise to the phenomenon of "Codman's Pa=
radox" in anatomy (Codman, E.A. (1934). *The Shoulder. Rupture of the Su=
praspinatus Tendon and other Lesions in or about the Subacromial Bursa*=
. Boston: Thomas Todd Company), where different combinations of numerical v=
alues of the three angles produce similar physical orientations of the segm=
ent. This is not actually a paradox, but a consequence of the non-commutati=
ve nature of three-dimensional rotations and can be mathematically explaine=
d through the properties of rotation matrices (Politti, J.C., Goroso, G., V=
alentinuzzi, M.E., & Bravo, O. (1998). *Codman's Paradox of the Arm =
Rotations is Not a Paradox: Mathematical Validation*. Medical Engineeri=
ng & Physics, 20, 257-260).

Codman proposed that the completely elevated humerus could be shown to b= e in either extreme external rotation or in extreme internal rotation by lo= wering it either in the coronal or sagittal plane respectively, without all= owing any rotation about the humeral longitudinal axis.

**To demonstrate Codman's Paradox, complete the following steps:**

- Place the arm at the side, elbow flexed to 90 degrees and the forearm i= nternally rotated across the stomach.
- Elevate the arm 180 degrees in the sagittal plane.
- Lower the arm 180 degrees to the side in the coronal plane.

Observe = that the forearm now points 180 degrees externally rotated from its origina= l position with no rotation about the humeral longitudinal axis actually ha= ving occurred. - Note the difficulty in describing whether the fully elevated humerus wa= s internally or externally rotated.

This ambiguity can cause switching between one solution and the other, r= esulting in sudden discontinuities. A combination of gimbal lock and Codman= 's Paradox can lead to unexpected results when joint modeling is carried ou= t. In practice, the shoulder is the only joint commonly analyzed that has a= sufficient range of motion about all rotation axes for these to be an issu= e. Generally, if you are aware of the reasons for the inconsistent data, yo= u can manipulate any erroneous results by adding 180 or 360 degrees.

As Plug-in Gait uses Cardan (Euler) angles in all cases to calculate joi= nt angles, they are subject to both Gimbal Lock in certain poses, and the i= nconsistencies that occur as a result of Codman's Paradox.

Plug-in Gait includes some steps to minimize the above effects by trying= to keep the shoulder angles in consistent and understandable quadrants. Th= is is not a complete solution however, as the above issues are inherent whe= n using Cardan (Euler) angles and clinical descriptions of motion.