Calibration
6.2 Ordinary and Weighted Least Squares Method Formulation
The least squares method provides a mathematical procedure that creates a model which fits better the experimental data. Lets useN sets of aerodynamics components, each set has a response of the 27 arrangements of the sixrj. The quantities are now represented asfi,pandrp,m, wherep= 1, ..., N andm= 1, ...,27.
The least squares searchs the coefficients by minimise the sum of squared errors. Taking Equation (6.2), the least squares function is [30]
Si=
N
X
p=1
i2=
N
X
p=1
fi,p−
27
X
m=1
rp,mci,m
!2
(6.4) To minimise the functionS, the partial derivation is required with respect to each of the coefficients, and these partial derivatives must be zero ∂c∂Si
i,m = 0. This procedure results in 27 equations and 27
unknowns for eachficomponent,
−2
N
X
p=1
fi,p−
27
X
m=1
rp,mci,m
!
rp,1= 0
... (6.5)
−2
N
X
p=1
fi,p−
27
X
m=1
rp,mci,m
!
rp,27= 0
The matrix notation is more convenient to deal with multiple regression models. The model is de- scribe in matrix notation as
f =Rc+⇔
f1 f2
... ... fN
N×1
=
r1,1 r1,2 . . . r1,27 r2,1 r2,2 . . . r2,27
... ... . .. ... ... ... ... . .. ... ... rN,1 rN,2 . . . rN,27
N×27
c1 c2
... ... c27
27×1
+
1 2
... ... N
N×1
(6.6)
where the vectorfcorresponds to the observations, the matrixRhas the sensing bar force in its rows for each calibration loading, the vectorcis the regression coefficients, andrepresents the random errors.
To find the least squares estimator, the previous approach is repeated for the matrix form [30]
Si=
N
X
p=1
i2=T= (f−Rc)T−(f−Rc)
=fTf −cTRTf −fTRc+cTRTRc=fTf−2cTRTf +cTRTRc (6.7) Applying the least-squares criteria that minimisesSholds
∂S
∂c ˆc
=−2RTf+ 2RTRc= 0 (6.8)
Simplifying and reorganise, the least-squares estimator ofCis then
C= (RTR)−1RTf (6.9)
Equation (6.9) is the solution of the least-squares normal equations [30], which completes the system presented in Equation (6.3) and defines the calibration model.
Although the generalised least square is a good data approximation method, the weighted least squares method adds some advantages [33]. The generalised method assume the weighting factors for all the data points are one, an assumption not very appropriate for this type of calibration.
The weighted least squares method considers an individual weighting factor for each data point. This weight can be computed by different approaches and introduces a more accurate experimental analysis.
A simple linear regression, such as Equation (6.4), in weighted least-square function is Si =
N
X
p=1
wp fi,p−
27
X
m=1
rp,mci,m
!2
(6.10) The same approach can be replicated for this new interpretation of data. The created matrixW is a N×N diagonal matrix with diagonal elements ofw1, w2, ..., wN [30].
The solution of the weighted least-squares normal equations results in the weighted least-squares estimator
C= (RTW R)−1RTW f (6.11)
6.2.1 Weighted Scheme
The formulation of the weighted matrix has many approaches concerning the references. The matrix W is an improvement of values precision but its computation methodology will set the complexity of the problem. As any estimation analysis, the weighting factors are always parameters with uncertainty but the lower this uncertainty is, the more trustful the results are.
The simplest approach for theW matrix is assume the identity matrix (I), which leads to the ordinary least squares approach. Most of the times, this approach is the initial step to get a general idea of the scatter of data. After that, variation models could estimate new weighting factor for the model. For instance, using the standard deviation to stipulate a newW matrix for subsequent interpolations [33].
The weighting matrix approach described in [32] is a current and extensive process to minimise the uncertainty of the applied loads and random errors in the measurement procedure. The construction of the weighted matrix considers two contributions to the uncertainties of the applied loads: the contri- butions of the sources of errors due to the application of weights in the calibration systemVW and the uncertainties in the readings of the bridgesVR, expressed as
W = (VW +DVRDT)−1 (6.12)
MatrixVW
The matrixVW is diagonal with dimensions ofN ×N. Its elements are based on the uncertainties of the weights used in the loading sequence and an estimation of the uncertainties caused by the calibration system. This approach leads to the testing of the weights employed to verify its uncertainties.
The second contribution is based on the quantification of the sources of error that affect the resolution of the calibration system, such as frictional forces and misalignment between cables and pulleys. To identify the second contributions, an additional experimental calibration with great precision is required to characterise the supplementary structures used in the main calibration. This extra experimental test makes the calibration more extensive and dependent of many tools and instruments.
The replacement method considers the standard deviation of the fitσi, expressed as σ2i = 1
N−m
N
X
p=1
(fi,p−fˆi,p)2 (6.13)
wheremis the number of parameters to be fitted (number of coefficients in Equation (6.2)). The standard deviation is initially computed by considering the identity matrix asW to obtain the first iteration. The matrixVW is build up by adding the standard deviations corresponding to each aerodynamic component to the main diagonal, taking into account the number of loads for each component.
MatrixVR
The uncertainty of the reading values can be mitigated by repeating the same calibration several times to buildVR that is diagonal symmetric. Several calibrations under repeatability conditions should be carried out over a short period of time. The diagonal elements ofVRare represented by the variances and the off-diagonal elements are the covariances between the readings of the six strain bridges. It
has dimensions 6N ×6N, where N lines are grouped corresponding to each force transducer. The formulation of matrixVRis presented in Appendix E, Equation (E.1).
This matrix is calculated from the matrixRglobal, of size6N ×z, where N is the number of sets and z the number of repetitions of the sets. The firstN elements of column 1 correspond to the readings of the load cellr1from the first calibration loading condition, followed by theN elements of the readings of the load cellr2 and so on. The same procedure is repeated for the remaining columns, using the data set from the other calibrations. In the present work, the calibration was repeated three times, thusz= 3.
The matrixVRis obtain by transpose and apply function ”cov” in MatlabR.
MatrixD
The elements of matrixD (N ×6N) correspond to the sensitivity coefficients, evaluated by taking the partial derivatives of Equation (6.2). For the aerodynamic force, f1, the formulation of matrixD is presented in Appendix E, Equation (E.2).
In this computation, only the sensitivity coefficients whose second subscripts match, i.e. correspond to the same loading, have values different from zero. The matrixDis reduced to several block subma- trices, being each block diagonal not null. Each element of the diagonal corresponds to the coefficients of the corresponding aerodynamic force component,
∂fp,1
∂Rp,1 =c1,1
∂fp,1
∂Rp,2 =c2,1 ... ∂fp,1
∂Rp,6 =c6,1 (6.14)
Therefore,DVRDT can only be obtained after the first iteration in the calibration procedure, where it is assumed thatW equals toVW−1.
6.2.2 Goodness of Fit
The least squares method settles the hypothesis that by minimising the sum of squared errors (ssE), the optimum fitting of the data is obtained. The errors, in turn, can be described as the deviation of the data observations (fi) and the estimated values from the fitting function (fˆi) [34]. Therefore, the sum of squares errors is equal to
ssE=
N
X
p=1
(ˆi,p)2=
N
X
p=1
(fi,p−fˆi,p)2 (6.15)
A good measurement of goodness of fit is the chi-square, χ2, which provide a ratio between the computed difference and the varianceσ2[33]. Theχ2is described as [33][34]
χ2= ssE σ2 =
N
X
p=1
1
σ2i(fi,p−fˆi,p)2 (6.16) This equation can be written in matrix form as
χ2= (F−Fˆ)TW(F−Fˆ) (6.17)
The relation between quantities can be applied for the number of degrees of freedom for fitting the data point,v=N−m. This leads to the reduced chi-square,
χ2v= χ2
v =(F−Fˆ)TW(F−Fˆ)
N−m (6.18)
The fitting function is a good approximation to the parent function ifχ2vis approximately unitary. That means the estimated variance of the fit should agree with the parent variance. Values ofχ2vgreater than one translate large deviations and consequently large estimated variance, so the selected fitting function is not appropriate for describing the data. On the other hand, values ofχ2v less than one do not surely indicate a better fit, however indicate the existence of uncertainty in the estimation and theχ2v will vary from experiment to experiment [33].