Biomedical Engineering Reference
In-Depth Information
2.4.1
Example: knee rotation
The knee joint is a pivotal hinge joint, which permits flexion and extension as
well as a slight medial and lateral
rotation.
For the purpose of this example we
shall consider it as purely a hinge joint with one degree of freedom (DOF). We
seek to represent the position vector of a point specified on the foot as the lower
limb undergoes a rotational motion at the knee. A point
Q
on the foot is shown in
Figure 2.7
and is represented by the vector
T
with respect to
x
Q
5
½
3
701
2
the coordinate system located at
the knee. The knee is constrained to allow
motion only about the axis
. It is required to calculate the final
position of the foot point
Q
as the joint rotates (i.e., as a function of
z
by an angle
θ
θ
).
The lower limb rotates about the axis
z
, thus the pure rotational homogenous
matrix is
2
4
3
5
cos
θ
2
sin
θ
00
sin
θ
cos
θ
00
T
z;θ
5
(2.18)
0
0
1
0
0
0
0
1
Note the translation vector is zero. To determine the location of
Q
at any joint
displacement
0
Q
θ
, we multiply
x
Q
by
T
z;θ
to calculate the rotated vector
x
0
Q
T
x
5
T
z;θ
x
Q
5
½
3cos
θ
1
7sin
θ
3sin
θ
2
7cos
θ
01
(2.19)
With this expression, it is possible to calculate the position of the lower limb
at any specified value of
90
, then the rotated limb's new
θ
. For example let
θ
5
position is
0
Q
90
Þ
5
T
x
ð
7301
(2.20)
which is shown in
Figure 2.7
as the lower limb is extended.
y
Q
θ
y
θ
z
3
x
x
7
7
Q
3
FIGURE 2.7
Rotation of the lower limb about the z-axis by an angle θ.
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