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B.5.3.3.
Measurement Function
The COP location in the X axis5 was calculated using the measurement function
COP =
Mx
Fz
(B.10)
where M x = (Fz3 + Fz4 )L is the magnitude of the moment of force and L is the constant distance
factor (no uncertainty assumption), Fz = Fz1 + Fz2 + Fz3 + Fz4 is the magnitude of the vertical
component of the GRF vector, and Fz1 to Fz4 are the magnitudes of the vertical components of the
GRF vector measured by each of the four load cells. In terms of the GUM annotations
COP ≡ f (Fz1 , Fz2 , Fz3 , Fz2 , L)
B.5.3.4.
Error Model
It is assumed that Eq. (B.10) is continuous and it has also continuous derivatives in the domain
of interest. The error model is determined from Eq. (B.10) by applying a first-order Taylor series
approximation. The COP error equation is expressed as
" y = S x " xT
where S x is the sensitivity vector assuming that L has no associated uncertainty
Sx =
”
∂ COP
∂ Fz1
∂ COP
∂ Fz2
∂ COP
∂ Fz3
∂ COP
∂ Fz4
—
and " x is the error vector associated with the inputs quantities
”
" x = " Fz1
B.5.3.5.
" Fz2
" Fz3
" Fz4
—
Uncertainty Model
The associated u2y in the COP measurements is the variance in the propagated error resulted from
the combination of the components used to compute COP. Therefore, if u2F 1 to u2 4 are the noise
z
Fz
variances of the Fz1 to Fz2 components respectively, the uncertainty in COP measurement is then
expressed by Eq. (B.9) where U x is the 4 × 4 symmetric uncertainty — variance-covariance —
matrix associated with x


u(Fz1 , Fz2 ) u(Fz1 , Fz3 ) u(Fz1 , Fz4 )

2
1
2
2
2
3
2
4 
u(Fz , Fz ) u (Fz ) u(Fz , Fz ) u(Fz , Fz )
Ux = 

u(Fz3 , Fz1 ) u(Fz3 , Fz2 ) u2 (Fz3 ) u(Fz3 , Fz4 )
u2 (Fz1 )
u(Fz4 , Fz1 ) u(Fz4 , Fz2 ) u(Fz4 , Fz3 )
u2 (Fz4 )
The diagonal terms of U x are the uncertainty (variance) of the input quantities and the off-diagonal
terms are their mutual uncertainties (covariance).
5
The same procedure is used for the Y axis.
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