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Joint angles represent a conceptual challenge to many users. Part of the challenge is related to the fact that joint angles are not vectors (unlike every other Kinematic variable in Visual3D). This means that they can't be added or subtracted, which makes the specification of a reference angle awkward. Another part of the challenge is related to the fact that there are a number of clinical/sports related conventions that specify a reference angle that is often not consistent with the definition of the segment coordinate systems in Visual3D, which usually requires Visual3D users to create Virtual Segments whose segment coordinate systems are consistent.
A more insidious problem is that there are joints, such as the shoulder, for which there is no single definition of the joint angle that is anatomically meaningful for the full range of motion of the joint. The consequence is that there is no standard sequence of rotations for describing the shoulder motion, despite the recommendation by the International Shoulder Group. The default convention that Visual3D uses to define joint angles
The default segment coordinate system in Visual3D is defined as follows:
- x-lateral, y-anterior, z-up
- x-medial, y-anterior, z-up
The default Cardan sequence for the calculation of joint angles is x-y-z, which is equivalent to flexion/extension - abduction/adduction - axial rotation. The default sign conventions for describing the joint angles are as follows:
- Ankle (DFL+) (FFADDUCTION+) (EVER+)
- Knee(EXT+) (ADD+) (INTROT+)
- Hip(FL+)(ADD+) (INTROT+)
- Ankle (DFL+) (FFABDUCTION+) (INVER+)
- Knee(EXT+) (ABD+) (EXTROT+)
- Hip(FL+)(ABD+) (EXTROT+)
Note: Conceptually, the foot's long axis is in the Anterior-Posterior direction. The result will be that the foot's coordinate system has a different axis convention compared to the other segments, with X = Medio-Lateral, Y = Up and Z = Anterior-Posterior. The above-mentioned sign convention reflect this coordinate system. However, to avoid confusion, and to maintain sign conventions through all segments, we recommend to rotate the foot's coordinate system so that X = Medio-Lateral, Y = Anterior-Posterior, Z = Up.
This can be done using the Modify Segment Coordinate System option.
Joint Angle Lecture
Joint angle lecture prepared by Tom Kepple when he was teaching at the University of Delaware. [Joint Angle Lecture]
A Joint Angle is the transformation between two coordinate system that is described by a rotation matrix.
This rotation matrix is rarely used to report the joint angle. From the rotation matrix several different representations are possible, such as Cardan/Euler angle, Helical angle, Quaternion. All representations of the transformation are equivalent but the interpretation of the joint angle relative to the anatomy varies with the description.
For example, the Cardan sequence XYZ as represented in the following transformation matrix abg can be computed as follows:
Step 1: compute b
Step 2: compute a
Step 3: compute g
For example, represent the rotation matrix as:
Resolve the joint angle as follows:
Units for Joint Angles
Joint Angles, Joint Angular Velocity, and Joint Angular Acceleration are the only signals in Visual3D that don't use standard MKS units. Visual3D uses degrees, degrees/sec, and degrees/sec^2. These are traditional units in Gait Analysis and we have followed the trend.
Right Hand Rule
The sign convention for joint angles doesn't seem to be consistent between left and right sides. For example, inward rotation of the right leg is positive, but inward rotation of the left leg is negative.
All joint angles are treated the same in Visual3D, so the direction of positive angle is determined with respect to the segment coordinate system of the reference segment; using the Right Hand Rule.
The result of this decision is that flexion/extension has the same sign for left and right legs, but inward/outward rotation and abduction/adduction have opposite signs.
The user can change this by negating the necessary terms when the joint angle is defined in the Compute Model Based Items Dialog
The Joint Coordinate System
Visual3D calculates Joint Angles using a Cardan sequence of rotations. For the Visual3D default segment coordinate system (z-up, y-anterior) the cardan sequense x-y-z is equivalent to the Joint Coordinate System.
If the user has modified the Segment Coordinate System from the Visual3D default, then more care must be taken in determining the sequence of rotations that is equivalent to the Joint Coordinate System. The sequence should be selected as flexion/extension - abduction/adduction - axial rotation. Note that Visual3D will only allow a right hand coordinate system so the positive direction of rotation about an axis will need to be verified by inspecting the Segment Coordinate System visually in Visual3D's Model Builder mode.
Cole GK, Nigg BM, Ronsky JL, Yeadon MR (1993) Application of the Joint Coordinate System to Three-Dimensional Joint Attitude and Movement Representation: A Standardization Proposal. J Biomech Eng, 115, 344-9 If I change the order of the rotation sequence for the Cardan representation of a Joint Angle, what are the 3 components of the resulting signal?
In early versions of Visual3D (version 2 and lower), we output the 3 components of the Cardan angle in terms of the first, second and third rotation. This was confusing to people because the edit report dialog refers to the first three components as XYZ. In Version 3.00.3 we changed the output so that the X component of the joint angle refers to the rotation about the X axis regardless of the sequence, similarly for Y and Z rotations.
In version 3.28 we introduced the Euler sequences (e.g. ZYZ), in which the first and third component are the same. In this case instead of referring to the components as XYZ, we revert to our original scheme, in which the output angle is defined with respect to the first axis, second axis, and third axes.
The helical angle calculation is output as vector rather than a unit vector and an angle of rotation.
Visual3d is simply multiplying the angle of the rotation by the three components of the unit vector.
Is possible to calculate a helical axis (vector and angle) in Visual 3D?
The answer yes by going the opposite way. To do this you would have to first compute the magnitude of the output of the “Compute Model BasedàHelical_Angle” and then divide each of the components of this vector by the magnitude.
To actually do this inside Visual3d you would:
Step 1) Use “Compute Model BasedàHelical_Angle” to get the helical angle as a vector.
Step 2) Use the Pipleline command “Signal Magnitude” (which is under Signal Math in the Pipeline) to get the magnitude of the “Compute Model BasedàHelical_Angle”. This would give you the angle you are looking for.
Step 3) Use the pipeline command Divide_Signal_BY_Constant to divide the result of “Compute Model BasedàHelical_Angle” by the result of the Signal Magnitude (Divide_Signal_BY_Constant is also under Signal Math in the Pipeline). This will give you the unit vector you are looking for.
Joint Angle Normalization
Normalization is the process of referencing a joint (or segment) angle to the reference posture. In Visual3D the reference posture is the static trial used for the link model.
Joint angles aren't vectors, which means they can not be added or subtracted. A normalized joint angle is, therefore, not computed as 3 offsets from a reference posture.
The reference posture is the static posture (often referred to as the standing pose) captured as the model posture (e.g. the c3d file used to define the link model).
For example, the following definition of the knee causes the RKNEE_ANGLE signal to be zero in all 3 components when the subject is in the standing posture. This has not affect on the segment coordinate system definitions, so normalizing the joint angle does not "clean up" any errors in the definition of the segment. It simply sets the posture in which the joint angle is zero.
As a rule I don't recommend using the default normalization because there are situations in which the intermediate pose in the calculations will be in gymbal lock and the normalized angle will be undefined.
I usually recommend the use of virtual segments, which can be defined so that they explicitly cause the static trial to be a reference posture. For example, see the following topics for normalizing the ankle joint angle.
Note: In Visual3D joint angles are simply the transformation from one segment coordinate system to another segment coordinate system, so it follows that the definition (e.g. orientation) of the segment coordinate systems matters. For example, if two segment coordinate systems are aligned perfectly (even though they may be displaced from each other) the resulting joint angle signal is zero.
- Normalizing the Pelvis Segment Angle. The Coda and Helen Hayes pelvis are tilted forward approximately 20 degrees from the horizontal. In describing the orientation of the pelvis or for describing the hip joint angle, it is often convenient to define a pelvis angle that has a coronal plane parallel to the floor (eg a vertical segment with zero tilt).
There is no single definition of the shoulder joint angle that is anatomically meaningful for the full range of motion of the shoulder joint.
The consequence is that there is no standard sequence of rotations for describing the shoulder motion, despite the recommendation by the International Shoulder Group.
A good description of the challenges of the shoulder angle by Jim Richards can be found [here]