Page 166 - Biomedical Engineering and Design Handbook Volume 1, Fundamentals

BIOMECHANICS OF HUMAN MOVEMENT 143

Euler angles described here are commonly reported in biomechanical studies (e.g., Grood and
Suntay, 1983) because the rotations take place about clinically meaningful axes corresponding
to joint flexion-extension (Z), abduction-adduction (Y′), and internal-external rotation ( ′′ ). The
X
, ′
ZY X′′ sequence of rotations is expressed using matrix notation in Eq. (6.32), with the elements
,
of the individual matrices shown in Eq. (6.33).
R = [R z ][R ' y ][R x '' ] (6.32)
s
⎡1 0 0 ⎤ ⎡ cosθ 0 sinθ ⎤ ⎡ cosφ − sinφ 0⎤
0
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
R = 0 cosψ − sinψ ⎥ R = ⎢ 0 1 0 ⎥ R z = sinφ cosφ 0 ⎥ (6.33)
⎢
⎢
x''
y'
⎢0 sinψ cosψ ⎥ ⎦ ⎣ − ⎢ sinθ 0 cosθ ⎦ ⎥ ⎢ ⎣ 0 0 1 ⎦ ⎥
⎣
where φ, θ, and ϕ are the Euler angles about the ZY X, ′ , ′′ axes, respectively. Expanding Eq. (6.33)
using the matrices from Eq. (6.33) leads to the rotation matrix R in Eq. (6.34).
ψ
⎡ cos( )cos( ) cos( )sin( )sin( ) − sin( )cos(ψψ) cos( φ)sin( cos( ψ) sin( φ)sin( ψ)⎤
+
θ
θ
φ
φ
φ
θ)
⎢ ⎥
ψ
θ
−
ψ
ψ
φ
θ
φ
+
φ
ψ
θ
φ
φ)
R = sin( cos( ) sin( )sin( )sin( ) cos( )cos( ) sin( )sinn( )cos( ) cos( )sin( ) ⎥
(
⎢
⎢ ⎣ − sin( ) cos( )sin( ) c cos( )cos( ) ⎥ ⎦
θ
ψ
ψ
θ
θ
(6.34)
It is easy to show that a different sequence of rotations can be used to move the ACS shank from its
initially aligned position to its final orientation relative to the ACS thigh . Because matrix multiplica-
tion is not commutative in general, the terms of R in Eq. (6.34) will differ depending on the sequence
of rotations selected. Equations (6.35) through (6.37) can be used to determine the Euler angles for
this particular sequence of rotations:
⎛ r ⎞
φ= arctan ⎜ 21 ⎟ (6.35)
⎝ r 11 ⎠
⎛ −r ⎞
θ= arctan ⎜ 31 ⎟ (6.36)
⎜ 2 2 ⎟
⎝ r 11 + r 21 ⎠
⎛ r ⎞
ψ= arctan ⎜ 32 ⎟ (6.37)
⎝ r 33 ⎠
where r is the element in the ith row and jth column of matrix R.
ij
The methods outlined above can also be used to calculate segmental kinematics. For example,
rather than calculating the relative orientation between the shank and thigh at time 1, we can use
Euler angles to determine the relative orientation between the ACS at times 1 and 2.
shank
6.4.4 Body Segment Parameters
Reexamining Eqs. (6.17) and (6.18), we see that estimates for mass (m) and the inertia tensor (I ) for
each segment are required to determine the right-hand side of the equations. Several other terms,
including the location of the center of mass and the distances from the distal and proximal joint
centers to the COM, are embedded in the left-hand side of Eq. (6.18). The term body segment
parameters (BSP) is used to describe this collection of anthropometric information.