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Euler sequences in joint angles & Gimbal lock

Euler sequences in joint angles

The Euler representation of the joint angles in MVN is generally calculated using the Euler sequence ZXY. An Euler sequence explains in which order the angle is calculated. In this case, the calculation is started with a rotation around the Z-axis, followed by a rotation around the X-axis and finished with a rotation around the Y-axis. This convention for specifying an orientation is, in this example, called an orientation ZXY fixed angles. The word "fixed" refers to the fact that orientations are specified about the fixed (i.e., non-moving) reference frame. In this case, the reference frame is based on the coordinate system established by ISB.

In MVN you will find that you can choose between the shoulder joint and the shoulder joint XZY. Even though the shoulder joint is the only joint with that sequence that can be shown in MVN, users have the option to export all the joint angles using the Euler sequence XZY. The reason why the shoulder joint is the only joint that shows the XZY sequence in MVN is because the rotation with the expected largest range of motion should be calculated first in the sequence. For most joint angles this corresponds to flexion/extension, which is a motion about the Z-axis (pointing to the right). For the shoulder joint however, the largest range of motion could also be abduction/adduction. This motion occurs about the X-axis (pointing anteriorly).

 

Gimbal lock

It's important to realize that the order of sequence can effect rotations. Example; in figure 1 the arm of a subject is rotated according to the shown rotation vectors. This corresponds to respectively 90 degrees rotation around a frontal, sagittal and longitudinal axis. The example in figure 2 uses the exact same rotations, only in a different order. The comparison between figure 1 and 2 shows us the order of rotations has an effect on the final position.

pos1.JPG

Figure 1. The final position after a combination of 90 degrees about a, respectively, frontal, sagittal and longitudinal axis.

pos2.JPG

Figure 2. The final position after a combination of 90 degrees about a, respectively, frontal, longitudinal and sagittal axis.

 

The method that we just described, to analyze movements, is different then the method that is used in kinematics and MVN. In kinematics, we use a 'moving coordinate system' instead of fixed axes. This moving coordinate system fixes the problem that we just encountered in the example of figure 1 and 2, but brings up a different problem: Gimbal lock.

Gimbal lock is the loss of one degree of freedom in a three degrees-of-freedom mechanism that occurs when two axes are driven into a parallel configuration, resulting in no available rotation about one axis (Wiki: Gimbal Lock). The shoulder joint can be seen as a 'ball and socket joint', which is a three-gimbal mechanism, in contrast to for most joints which can be considered a 'hinge joint'. Therefore, an example of the shoulder joint is chosen to clarify the phenomenon Gimbal lock when the 'regular' sequence ZXY is chosen:

gimbal_lock.JPG

Figure 3. The first position shows a 90 degrees rotation abound the Z-axis (flexion), followed by 90 degrees horizontal abduction about the X-axis. This results in a Gimbal Lock because the Y- and Z- axis are aligned.

 

In the first position of the figure 3, the axes of the shoulder joint are orientated in the regular way. When the subject performs a rotation around the Z-axis (flexion), the other axes rotate along with the arm. The Z-axes itself doesn't move. The result is that the X-axis (for ab- and adduction) is now in the longitudinal direction. For this reason the movements around this axis, in this position of the arm, will be called horizontal ab- and adduction. The axis for internal- and external rotation (Y-axis) is no longer pointing in the longitudinal direction, but sagittal. The moving nature of this coordinate system results in the fact that the rotation of the arm around it's 'length-axis' will always be called internal- or external rotation. The subject performed a horizontal abduction in the second pose of figure 3. The problems from these movements are shown in pose 3. In this pose, obtained by doing flexion and horizontal abduction respectively, the Y- and Z- axis are aligned. The result is that a forward- or backwards rotation of the arm can be called both flexion and external rotation. Furthermore, there is no axis available on which adduction of the arm (back to the initial position) can be described. This problem, of every three degrees-of-freedom mechanism with a moving coordinate system, is called Gimbal lock. Therefore, it is really important to determine whether flexion or abduction had the bigger range of motion. When abduction/adduction had a bigger range of motion then flexion/extension, the XZY sequence should be chosen to prevent Gimbal lock from happening. Otherwise, the default ZXY sequence should be used.

 

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