One of the things that I love about biomechanics is that it’s intuitive. For example, a knee joint angle is the angle between the shank/tibia and the thigh. This is easy to picture in our mind and understand and since many people have knees, this concept makes intuitive sense right off the bat. However, since there are so many parts of biomechanics that we learn by doing – there are certain topics that are important or useful to know and get glossed over when we learn biomechanics “in the wild”. I will be writing a few blogs to cover some topics that I think often get overlooked, forgotten over time, or glossed over when being taught because we assume they’ll get picked up later, but I really recommend spending time to review or learn these topics.
The equations to calculate a joint angle are provided in numerous textbooks, but in order to understand the end equations, we should know where they came from, and I think this often gets glossed over. In truth, the equations are actually very simple. I think reviewing the math is very important because it shows that:
Rotation sequence matters
Joint angles are not magic
The math appears daunting, but it’s just trigonometry
For a rotation sequence R3R2R1:
The first rotation is about Axis1
The second rotation is about Axis2’
The third rotation is about Axis3’
This is important for interpretation – stay tuned for this future blog!
The joint angle is the angle between the coordinate systems – and is not normalized to the posture during the static trial – stay tuned for this future blog!
Also, I will take the time now to apologize for the length of this blog post, but I really wanted to work through the joint angle calculation from start to finish.
A little background on joint angles:
Chances are, if you hear about biomechanics, you will also hear about joint angles. In biomechanics you will often see an angle plotted as a percent of the gait cycle (“gait” meaning walking and “cycle” meaning heel strike to subsequent heel strike).
The graph contains the joint angle (Y-Axis) at each point in the gait cycle (X-Axis). So, at right heel strike (when the foot first hits the ground), the angle between the thigh (femur) and the shank (tibia/fibula) is usually around zero since the leg is straight. The leg tends to bend some around mid-stance, and then straightens around left heel strike. During swing (when the right foot is not in contact with the ground) the knee bends the most (we call this flexion) which allows the foot to clear the ground. Then the leg is straightened again close to right heel strike (the end of the cycle).
From a 2D perspective, this is pretty straight forward – however, we live our life in 3D and joint angles are 3D angles. The math to calculate a joint angle is a little more complex than drawing with a protractor, but nothing you can’t do!
There’s a lot of information online about Euler angles, even Wikipedia will give you a good overview of what a Euler angle is, so if you are unfamiliar with Euler angles, I would recommend reading some introduction material on Wikipedia or in a biomechanics textbook. Euler angles can be described by different rotation sequences. In biomechanics, we tend to use the sequence mediolateral (ML) – anterior/posterior (AP) – axial (AX), or ML-AP-AX. If your coordinate systems are defined X along the ML axis, Y along the AP axis and Z along the Axial axis, your rotation sequence would be X-Y-Z.
I’m going to go over the math for calculating an X-Y-Z rotation sequence. You will see these equations in any textbook – I tend to reference Research Methods in Biomechanics 2nd Edition (page 52).
Recap
Although that’s a lot of math - hopefully you can see that:
If you learned sine, cosine and tangent, you can work through joint angle calculations
The rotation sequence matters since a different rotation sequence will result in a different 3x3 matrix
The first rotation is about the first axis (in this case Rx), then about the second (in this case Ry’), and the last is about the third axis (in this case Rz’)
Understanding these calculations will be beneficial when interpreting joint angles, and when understanding models and markersets. Before jumping into interpreting joint angles, the next step is to understand the sign of joint angles (what is positive or negative). The right hand rule is used to determine the sign of joint angles, and is how we determine if flexion or extension is positive when interpreting a joint angle. A blog reviewing the right hand rule is coming soon!
Additional Resources
This link has a great tool which allows you to define the rotation sequence and change the angles to see how it affects the coordinate system rotation.
I highly recommend C-Motion’s wiki page and Tom Kepple’s Joint Angle lecture because both were written by Tom, and everything I know about joint angles, I learned from Tom and he is hands down, the expert on joint angles
Biomechanics Keywords
Proximal – closer the body/pelvis
Example: At the knee, the thigh is proximal, the shank is distal
Distal – further away from the body/pelvis
Example: At the knee, the thigh is proximal, the shank is distal
Mediolateral (ML): Axis perpendicular to the sagittal plane, and runs medial/lateral (left/right)
Anterior/Posterior (AP): Axis perpendicular to the coronal plane, and runs front/back
Axial (AX): Axis perpendicular to the transverse plane, and runs up/down
Planes of Motion: Sagittal (Rot ML), Coronal (Rot AP), Transverse (Rot AX)
Coordinate System (CS)
Made up of three orthogonal unit vectors (X, Y and Z axes)
Orthogonal: at right angles
Unit vectors: lines that have a direction and a length of one
In biomechanics, two types of CS:
Segment CS – A coordinate system that is fixed to a bone or segment
Lab or Global CS – A coordinate system that is fixed in space, it doesn’t translate or rotate
Joint Angles: Angles between two coordinate systems
Segmental Angle – angle of a segment relative to the Lab CS (a fixed location in space)
Joint Angle – angle of a segment relative to another segment
Also called: Inter-segmental angle
Euler Angle: Repeating sequences (Z-Y-Z)
Cardan Angle: No repeating sequences (X-Y-Z), also called Tait-Bryan Angles
Cardan angles are often seen as a type of Euler angle
Contralateral – Opposite leg
Example: Heel strike to Contralateral Heel Strike; Right Heel Strike to Left Heel Strike
Ipsilateral – Same leg
Example: Heel strike to Ipsilateral Heel Strike; Right Heel Strike to Right Heel Strike