User guide
Charnwood Dynamics Ltd. Coda cx1 User Guide – Advanced Topics III - 4
CX1 USER GUIDE - COMPLETE.doc 26/04/04
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Ankle Moments
Centre of mass of foot: G
f
= A + µ
f
(T - A)
where A
& T are the position vectors of the Ankle joint centre and Toe respectively.
Linear acceleration of centre of mass of foot: a
G
f
= a
A
+ µ
f
(a
T
- a
A
)
Ankle joint forces: R
A
x
= m
f
a
G
fx
- F
P
x
; R
A
y
= m
f
a
G
fy
- F
p
y
R
A
z
= m
f
(a
G
fz
+ g) - F
P
z
where F
P
is the reaction force vector from the foot-plate and a
G
fx
is the x-component of the
acceleration of the centre of mass of the foot.
Then ankle force vector : R
A
= [R
Ax
, R
Ay
, R
Az
]
T
Treating the foot as a free body, sum the applied torques (about G
f
) and combine with
inertial aspects about the principal axes to obtain moments equations.
Let P be the position vector of the centre of pressure on the foot-plate as determined
from force-plate data. (See, for example, Kistler Force Plate documentation).
Torque due to ground reaction: Q
P
= -F
P ^
d
p
where d
p
= P - G
f
Torque due to ankle force:
f
Q
R
A
=
R
A ^
d
A
where d
A
= A - G
f
= µ
f
(T - A)
Total torque on foot: Q
f
= Q
P
+
f
Q
R
A
= [ Q
f x
, Q
f y
, Q
f z
]
T
with respect to Coda x, y, z co-ordinates.
This torque vector must now be transformed into a torque vector localized into the
co-ordinate system defined by the principal axes of the foot segment. The new
components are given by the following transformation:
foot
Q
0
= Q
f
.
foot
e
0
where
foot
e
0
is the unit vector aligned with the first principal axis of the segment.
Similarly,
foot
Q
1
= Q
f
.
foot
e
1
and
foot
Q
2
= Q
f
.
foot
e
2
Then
foot
Q = [
foot
Q
0
,
foot
Q
1
,
foot
Q
2
]
T
The three components of the ankle-moment upon the foot are calculated by means of
Euler’s Equations thus:
M
A 0
=
foot
I
0
foot
α
0
+ (
foot
I
2
-
foot
I
1
)
foot
ω
1
foot
ω
2
+
foot
Q
0
M
A 1
=
foot
I
1
foot
α
1
+ (
foot
I
0
-
foot
I
2
)
foot
ω
0
foot
ω
2
+
foot
Q
1
M
A 2
=
foot
I
2
foot
α
2
+ (
foot
I
1
-
foot
I
0
)
foot
ω
1
foot
ω
0
+
foot
Q
2
where subscripts 0, 1, 2 denote components corresponding to the principal axes of the
segment, the zeroth axis being the longitudinal or main segmental axis whereas the
second axis is in the local lateral direction and axis 1 is perpendicular to 0 and 2.