METHOD AND SYSTEM FOR DETERMINING THE WEIGHT AND CENTRE OF GRAVITY OF A STRUCTURE

Information

  • Patent Application
  • 20190219437
  • Publication Number
    20190219437
  • Date Filed
    January 11, 2019
    5 years ago
  • Date Published
    July 18, 2019
    5 years ago
Abstract
A method and system for determining the weight and at least a first coordinate of the center of gravity of a structure such as a vehicle, in particular, an aircraft.
Description
RELATED APPLICATION

This application claims priority to European Patent Application No. EP18382018.2, filed Jan. 17, 2018, the entire contents of which is incorporated by reference.


TECHNICAL FIELD OF THE INVENTION

The present invention belongs to the field of structures and relates to a method and system for determining the weight and the centre of gravity of a structure such as a vehicle, in particular, an aircraft.


BACKGROUND OF THE INVENTION

Accurate determination of the weight and the centre of gravity of a structure are crucial for ensuring the stability and correct performance of said structure. In the particular case of a vehicle, determining the weight and the centre of gravity of the vehicle is essential to avoid critical movements which could compromise the stability of the vehicle when said vehicle is operated.


According to the current practice, aircrafts and other vehicles are usually weighed on scales or by lifting on jacks with load cells inside hangars. However, when the vehicles are outside hangars, the cargo weight of a vehicle is usually obtained by estimation of each pallet weight when the vehicle is loaded. After loading the weight in the vehicle, a calculus is performed to estimate the centre of gravity (C.G.) based on the estimation of the weight and the estimation of the weight positioning. When the estimations of weight and centre of gravity are not accurate, the safety of the transport performed by said vehicle can be compromised.


In an attempt to provide a solution for the autonomous determination of the weight and centre of gravity of an aircraft in any situation on ground, it has been proposed to obtain the weight and centre of gravity by locating sensors inside the aircraft, especially in the structure where landing gears are attached.


The landing gears are the undercarriage of an aircraft which support the weight of the aircraft and are, in principle, the best place to instrument the sensors. However, landing gear structures have some disadvantages because their geometry changes with weight and is also dependent on the shock absorbers behavior, which is in turn strongly dependent on temperature, internal frictions, etc. Also, during landing operations the sensors installed on the landing gears support very heavy or even shock loads, which can destroy or damage the installed sensors. Therefore, these behaviors, characteristics, and geometry changes have to be taken into account by correcting the measurements obtained by the sensors. This involves complicated and time consuming calculations in order to provide an estimation of the weight and the position of the centre of gravity, the accuracy of which is not fully guaranteed.


The U.S. Pat. No. 5,548,517 A discloses an on-board system for use in measuring, computing and displaying the gross weight and location/relocation of the centre of gravity, for aircrafts. The system also incorporates a software correction program, along with attached hardware, to correct and compensate for physical changes to strut components due to temperature fluctuations, seal drag and hysteresis within modern day “shock absorbing” aircraft landing gear struts. However, the system implies collecting data from several types of sensors in order to deduce the landing gear movements and changes in the geometry of the landing gear, namely pressure sensor data to obtain the telescopic strut movements, rotation sensor data to obtain the hinged movements of struts components, strain gauge data to obtain the axle deflections, etc. From all these data, a complex calculation method has to be carried out to determine the aircraft weight and the center of gravity. Moreover, because there are many factors to be considered, the accuracy of the results could be affected.


There is thus a need for a method for determining the weight and the position of the centre of gravity of a structure which avoids the above problems and provides accurate measurements in a simple and reliable way.


SUMMARY OF THE INVENTION

The present invention provides a solution for the aforementioned problems, by a method for determining the weight and/or at least a first coordinate of the position of the centre of gravity of a structure according to claim 1, a system according to claim 14 and a vehicle according to claim 15. In dependent claims, preferred embodiments of the invention are defined.


In a first inventive aspect, the present invention provides a method for determining the weight and/or at least a first coordinate of the position of the centre of gravity of a structure, the method comprising the following steps:

    • a) measuring the responses of a first plurality of installed sensors located in sensitive load points of the structure, and
    • b) determining the weight and/or the first coordinate of the position of the centre of gravity of the structure based on the measured responses, as:






W
=


C
w

+





j
w

=
1


k
w









μ

j
w


·

β

j
w











X
=


C
x

+





j
x

=
1


k
x









μ

j
x


·

γ

j
x









wherein W is the weight of the structure, X is the first coordinate of the position of the centre of gravity of the structure on a plane of projection, kw and kx are natural numbers and μ are the responses measured by the first plurality of installed sensors, wherein the subscript in the response denotes the sensor which has performed the measurement,


wherein


for determining the weight a first group of sensors is used which comprises a number of installed sensors kw,


for determining the first coordinate of the centre of gravity a second group of sensors is used which comprises a number of installed sensors kx, and wherein


coefficients βjw, with jw=1, . . . kw, and γjx, with jx=1, . . . kx, and constants Cw and Cx are previously determined applying the following calibration steps:

    • i) locating a plurality of calibration sensors in sensitive load points of the structure, wherein the number of calibration sensors is m, m being a natural number,
    • ii) loading the structure with a probe weight, obtaining a loading state, and measuring at least the first coordinate of the position of the centre of gravity and the weight of the structure in said loading state,
    • iii) measuring for each calibration sensor the response associated to the loading state,
    • iv) repeating steps ii) and iii) for a plurality of loading states i, with i=1, . . . s, beings the number of loading states, s being a natural number,
    • v) selecting groups of at least two calibration sensors and solving for each group an equation system with a number of equations P equal to or greater than Nw+1, being Nw the number of calibration sensors in the group, wherein the equation system is:







W
1

=


C
w

+




n
=
1


N
w









μ

n
,
1


·

β
n
















W
P

=


C
w

+




n
=
1


N
w









μ

n
,
P


·

β
n










    • μn,q, with n=1, . . . , Nw being the measured responses of the calibration sensors and Wq being the weight associated to the respective loading state q, q=1, . . . , P,

    • wherein the equation systems are solved by least squares method, obtaining as a result weight coefficients βn for n=1, . . . , Nw, and weight constant Cw,

    • vi) determining for each group of calibration sensors an estimated weight for each loading state as










We
i

=


C
w

+




n
=
1


N
w









μ

n
,
i


·

β
n










    • wherein μn,i are the responses of the calibration sensors of the group, associated to a loading state i,

    • vii) selecting a first group of calibration sensors which fulfils that a first probable error PEw is minimum, wherein PEw is:










PE
w

=


C
E








i
=
1

s








(


We
i

-

W
i


)

2



s
-

(


N
w

+
1

)









s being the total number of loading states, Nw being the number of calibration sensors in said group of calibration sensors, CE being a real positive number, Wei being the estimated weight and Wi being the measured weight for loading state i,

    • viii) selecting groups of at least two calibration sensors and solving for each group an equation system with a number of equations P equal or greater than Nx+1, being Nx the number of calibration sensors in the group, wherein the equation systems is:







X
1

=


C
x

+




n
=
1


N
x









μ

n
,
1


·

γ
n
















X
P

=


C
x

+




n
=
1


N
x









μ

n
,
P


·

γ
n










    • μn,q being the measured responses of the calibration sensors and Xq being the X-coordinate of the centre of gravity associated to the respective loading state q, q=1, . . . , P,

    • wherein the equation systems are solved by least squares method, obtaining as a result X-coefficients for n=1, . . . , Nx, and X-constant,

    • ix) determining for each group of calibration sensors an estimated X-coordinate of the centre of gravity for each loading state, as










Xe
i

=


C
x

+




n
=
1


N
x









μ

n
,
i


·

γ
n










    • wherein μn,i are the responses of the calibration sensors of the group, associated to a loading state i,

    • x) selecting a second group of calibration sensors which fulfils that a second probable error is minimum, wherein PEX is:










PE
X

=


C
E








i
=
1

s








(


Xe
i

-

X
i


)

2



s
-

(


N
x

+
1

)









s being the total number of loading states, Nx being the number of calibration sensors in said group of calibration sensors, CE being a real positive number, Xei being the estimated X-coordinate and Xi being the measured X-coordinate for loading state i

    • xi) selecting as the first plurality of installed sensors a plurality of calibration sensors including:
      • the first group of calibration sensors selected in step vii), and
      • the second group of calibration sensors selected in step x),


wherein kw is the number of sensors in the first group of calibration sensors and kx is the number of sensors in the second group of calibration sensors. kw and kx are less than or equal to m.


Thus, prior to determining the weight and/or at least a first coordinate of the position of the centre of gravity of the structure based on the measured responses of a first plurality of installed sensors located in such structure, it is determined which sensors are going to be used. The selection of such first plurality of installed sensors is carried out by the calibration steps.


As a result of the calibration steps, the first plurality of installed sensors is selected and coefficients βjw, with jw=1, . . . kw, and γjx, with jx=1, . . . kx, and constants Cw and Cx are computed. In the present invention constant Cw and Cx are real numbers. Based on the computed coefficients and constants, the weight and/or the first coordinate of the centre of gravity of the structure can be determined at any moment as a function of the measured responses of the first plurality of installed sensors using the equations defined on step b) of the method of the present invention.


The present method allows determining at least one of the coordinates of the position of the centre of gravity. In a particular embodiment, the position of the centre of gravity of a structure is defined by two coordinates. These two coordinates indicate the position of the centre of gravity on the plant (floor) of the structure, i.e. the position of the projection of the centre of gravity on the plant of the structure, or on the plant or surface on which the structure is supported. Therefore, through this entire document, the plane of projection will be understood as the plant of the structure or the plant or surface on which the structure is supported. Further, through this entire document, the first coordinate can be any of the two coordinates of the position of the centre of gravity. The position of the centre of gravity is defined in relation to a coordinate system, the origin of which is a predefined point. In an embodiment where the structure is an aircraft the X-axis is preferably selected as the projection on the horizontal plane of the longitudinal direction along the fuselage (roll axis), and the Y-axis is preferably selected as the projection on the horizontal plane of the lateral direction (pitch axis). In a preferred embodiment, the origin of the axes is located at a specified distance in front of the aircraft nose.


In calibration step i) of the present method, a plurality of calibration sensors are installed in sensitive load points of the structure.


In the context of this specification, a sensitive load point of a structure is to be understood as a point of the structure which suffers a substantially linear stress and/or strain in response to the weight of the structure. A substantially linear stress and/or strain in said sensitive load point means that the structural variation in said sensitive load point is only due to a weight loaded in the structure and the weight of the structure itself.


Once a plurality of calibration sensors are installed in the structure, firstly, the structure is loaded with different probe weights, wherein from each probe weight it is obtained a loading state. The first coordinate of the position of the centre of gravity and/or the weight of the structure are measured for each loading state. Measurement of the weight and the first coordinate of the position of the centre of gravity can be performed in a conventional way. In an embodiment a plurality of weighing scales is used for determining the weight and the position of the centre of gravity. In this embodiment the structure is placed on the weighing scales and the weight of the structure is determined as the addition of the weights measured by the weighing scales. The position of the centre of gravity is determined based on the known position of each weighing scale and on the weight measured by each weighing scale. Since a first coordinate of the position of the centre of gravity can be any of the two coordinates of the centre of gravity, any of such coordinates can be determined as described above.


Additionally, the measured response of each calibration sensor is recorded for each loading state. Once the responses of the calibration sensors and the weight and the first coordinate of the position of the centre of gravity are obtained and recorded for each loading state, then the coefficients and constants of the weight and the first coordinate of the position of the centre of gravity of the structure are respectively determined.


The plurality of loading states in the calibration steps can be obtained by modifying the weight and/or the location of the probe weight loaded on the structure.


For determining such coefficients and constants, groups of at least two calibration sensors are selected, and for each group an equation system as defined on steps v) and viii) is solved by the least squares method. In the equation system each equation corresponds to a loading state and the number of equations is equal to or greater than the number of calibration sensors in the group plus one.


To determine which sensors, among the calibration sensors, are going to be the installed sensors used for determining the weight and/or the first coordinate of the position of the centre of gravity of the structure, it is selected:

    • a first group of calibration sensors which fulfils that a first probable error is minimum; and
    • a second group of calibration sensors which fulfils that a second probable error is minimum;


wherein such probable errors are defined in step vii) and x) respectively of the method.


The value of constant CE for calculating the first probable error may be equal to or different from the value of constant CE used for calculating the second probable error. The constant CE is a real and positive number. In a preferred embodiment, for calculating the first and second probable errors, the constant CE is 0.6745.


Thus, for determining the weight it is used a first group of sensors and for determining the first coordinate of the position of the centre of gravity it is used a second group of sensors. Such groups of sensors are previously determined in the calibration steps and define the first plurality of installed sensors which are located in the sensitive load points of the structure and which are used to determine at least one of the weight and the first coordinate of the centre of gravity. One or several of the installed sensors included in one of the groups may be included also in the other group.


Advantageously, since the present invention allows determining at least a first coordinate of the position of the centre of gravity and the weight based on measurements by the sensors located in at least one sensitive load point, the method increases the accuracy and the reliability of the determination of the weight and the centre of gravity.


In a particular embodiment, the method further comprises determining a second coordinate of the position of the centre of gravity of the structure, wherein the second coordinate of the position of the centre of gravity of the structure is determined based on the measured responses of the first plurality of installed sensors, as:






Y
=


C
y

+





j
y

=
1


k
y









μ

j
y


·

τ

j
y









wherein Y is the second coordinate of the position of the centre of gravity of the structure on a plane of projection, ky is a natural number and μ are the responses measured by the first plurality of installed sensors, wherein the subscript in the response μ denotes the sensor which has performed the measurement,


wherein for determining the second coordinate of the position of the centre of gravity a third group of sensors is used which comprises a number of installed sensors ky, and wherein


Y-coefficients τjy, with jy=1, . . . ky, and Y-constant Cy are previously determined applying the following calibration steps at least after steps i)-iv), wherein the step ii) further comprises measuring the second coordinate of the position of the centre of gravity of the structure in each loading state:

    • viii′) selecting groups of at least two calibration sensors and solving for each group an equation system with a number of equations equal or greater than Ny+1, being Ny the number of calibration sensors in the group, wherein the equation systems is:







Y
1

=


C
y

+




n
=
1


N
y









μ

n
,
1


·

τ
n
















Y
P

=


C
y

+




n
=
1


N
y









μ

n
,
P


·

τ
n










    • μn,q being the measured responses of the calibration sensors and Yq being the Y-coordinate of the centre of gravity associated to the respective loading state q, q=1, . . . , P,

    • wherein the equation systems are solved by least squares method, obtaining as a result Y-coefficients τn for n=1, . . . , Ny, and Y-constant Cy,

    • ix′) determining for each group of calibration sensors an estimated Y-coordinate of the centre of gravity for each loading state as










Ye
i

=


C
y

+




n
=
1


N
y









μ

n
,
i


·

τ
n










    • where μn,i are the responses of the calibration sensors of the group, associated to a loading state i, and

    • x′) selecting a third group of calibration sensors which fulfils that a third probable error PEY is minimum, wherein PEY is:










PE
Y

=


C
E








i
=
1

s








(


Ye
i

-

Y
i


)

2



s
-

(


N
y

+
1

)









s being the total number of loading states, Ny being the number of calibration sensors in said group of calibration sensors, CE being a real positive number, Yei being the estimated Y-coordinate and Yi being the measured Y-coordinate for loading state i,


wherein the first plurality of installed sensors further includes the third group of calibration sensors selected in step x′) and wherein ky is the number of sensors in the third group of calibration sensors. ky is less than or equal to m.


The value of constant CE for calculating the third probable error may be equal to or different from the values of constant CE used for calculating the first and/or the second probable error. The constant CE is a real and positive number. In a preferred embodiment, for calculating the first, second and third probable errors, the constant CE is 0.6745.


Prior to determining a second coordinate of the position of the centre of gravity of the structure based on the measured responses of the first plurality of installed sensors located in such structure, it is necessary to determine which sensors are going to be used.


Thus, as a result of calibration steps, a third group of installed sensor is further selected and coefficients τjy, with jy=1, . . . ky, and constant Cy are computed. In the present invention constant Cy is a real number. Based on the computed coefficients and constant, the second coordinate of the centre of gravity of the structure can be also determined at any moment as a function of the measured responses of the first plurality of installed sensors.


According to this embodiment, the calibration further comprises the steps viii′)-x′). Once a plurality of calibration sensors are installed in the structure and the structure is loaded with different probe weights to obtain a plurality of loading states, the second coordinate of the position of the centre of gravity is also measured for each loading state. Measurement of the second coordinate of the position of the centre of gravity can be performed in a conventional way, such as using a plurality of weighing scales. The second coordinate of the position of the centre of gravity is determined based on the known position of each weighing scale and on the weight measured by each weighing scale.


Once the responses of the calibration sensors and the second coordinate od the position of the centre of gravity is obtained and recorded for each loading state, then the coefficient τjy and constant Cy of the second coordinate of the position of the centre of gravity of the structure is determined.


For determining such coefficient and constant for the second coordinate of the centre of gravity, groups of at least two calibration sensors are selected, and for each group an equation system as defined on step viii′) is solved by the least squares method as for the weight and the first coordinate of the centre of gravity.


To determine which sensors, among the calibration sensors, are going to be installed sensors used for determining the second coordinate of the position of the centre of gravity of the structure, it is selected a third group of calibration sensors which fulfils that a third probable error is minimum, as it is defined in step x′). Thus, in this embodiment the first plurality of installed sensors includes the first group of calibration sensors selected in step vii), the second group of calibration sensors selected in step x) and the third group of calibration sensors selected in step x′). For determining the second coordinate of the position of the centre of gravity the third group of installed sensors is used.


These groups of installed sensors defined the first plurality of installed sensors which are located in the sensitive load points of the structure and which are used to determine the weight and/or the first and second coordinates of the position of the centre of gravity of the structure. One or several of the installed sensors included in one of the groups may be included in one or both of the other groups.


Advantageously, since the present invention allows further determining the second coordinate of the position of the centre of gravity based in measurements by the sensors installed in the at least one sensitive load point, the method increases the accuracy and the reliability of the determination of the position of the centre of gravity.


In a particular embodiment, the method comprises, after steps vii), x) and/or x′), verifying for at least one loading state i that

    • the first group of calibration sensors fulfills that a weight residual value (εwi) is lower than a first predefined threshold,


wherein








ɛ






w
i


=

100
·


(


We
i

-

W
i


)


W
i




,




and/or

    • the second group of calibration sensors fulfills that a X-coordinate residual value (εxi) is lower than a second predefined threshold,


wherein








ɛ






x
i


=

100
·


(


Xe
i

-

X
i


)


X
i




,




and/or

    • the third group of calibration sensors fulfills that a Y-coordinate residual value (εyi) is lower than a third predefined threshold,


wherein








ɛ






y
i


=

100
·


(


Ye
i

-

Y
i


)


Y
i




,




In an embodiment the first predefined threshold, the second predefined threshold and/or the third predefined threshold is 3%, preferably 2%.


Advantageously, this additional step of verifying if the groups of calibration sensors selected fulfill such requirement allows checking the accuracy of the equations.


In a particular embodiment, the present method comprises performing the following steps prior to step a):

    • selecting at least one sensitive load point in the structure, and
    • installing at least one sensor on the at least one sensitive load point.


In a particular embodiment, at each iteration of step ii) of the present method, the structure is loaded with the probe weights Wpi in steps of a percentage of a maximum admissible weight of the structure. For example the structure can be loaded with the 0%, 20%, 40%, 60%, 80% and 100% of the probe weights Wpi. Advantageously, loading the structure in steps allows checking the linearity of the sensors.


In a particular embodiment of the method:

    • a plurality of first groups of calibration sensors are selected at step vii), wherein the first groups of calibration sensors fulfil that the first probable error is minimum, and/or
    • a plurality of second groups of calibration sensors are selected at step x), wherein the second groups of calibration sensors fulfil that the second probable error is minimum, and/or
    • a plurality of third groups of calibration sensors are selected at step x′), wherein the third groups of calibration sensors fulfil that the third probable error is minimum.


In this embodiment more than one group of calibration sensors are selected for determining the weight and/or at least one of the coordinates of the position of the centre of gravity, wherein each group of calibration sensors has an associated equation for determining the weight and/or at least one of the coordinates of the position of the centre of gravity. Thus, once the calibration has been performed for a structure, the weight and/or the position of the centre of gravity of the structure loaded with an unknown weight can be determined using several equations. The redundancy in the determination of the weight and/or the position of the centre of gravity of the structure allows verifying if the method and/or the sensors are working properly. In a preferred embodiment at least two groups of sensors having different number of sensors are selected.


In a particular embodiment, the structure is an on ground vehicle.


In a particular embodiment, the at least one sensitive load point of the structure is located in at least one bulkhead of the vehicle.


In the context of this specification, the bulkhead of an on ground vehicle is to be understood as the area of the vehicle where the support elements are fixed. The support elements of the on ground vehicle are the elements on which the vehicle is seated on the ground. For example, a bulkhead may be the area where such support elements are housed. In a particular embodiment wherein the vehicle is an aircraft, the support elements are understood as landing gears.


In a particular embodiment, the on ground vehicle is an aircraft.


In a particular embodiment, the at least one sensitive load point is located in at least one landing gear bulkhead.


In a particular embodiment, the at least one sensitive load point is located in the nose landing gear pin.


In a particular embodiment, the at least one sensitive load point is located in the main landing gear pin.


As a difference with the state of the art, in the method of the invention the landing gears are not instrumented with sensors. In this way, the measurements taken by the sensors are not dependent on changes in geometry nor on the shock absorbers behavior. In this embodiment, the instrumentation is located in an area surrounding the landing gears or in instrumented pins receiving loads without direction changes or with very small direction changes. Instrumenting in such way there is not dependency on geometry changes neither on temperatures nor sticking influences neither other extraneous influences.


In a particular embodiment, the position of the centre of gravity and/or the weight are measured in calibration step ii) using scales and/or jacks with load cells, preferably, using three hydraulic jacks.


In a particular embodiment, the method further comprises the following steps:

    • c) measuring the responses of a second plurality of installed sensors located in at least one lifting surface of the structure, and
    • d) determining a corrected weight of the structure and/or a corrected position of the centre of gravity of the structure as:







W
corrected

=

W
-




h
=
1

M







V
h











X
corrected

=



X
·
W

-




h
=
1

M







(


X
h


·

V
h



)




W
-




h
=
1

M







V
h












Y
corrected

=



Y
·
W

-




h
=
1

M







(


Y
h


·

V
h



)




W
-




h
=
1

M







V
h









wherein W is the calculated weight of the structure, (X, Y) is the calculated position of the centre of gravity of the structure,


wherein M is the number of lifting surfaces considered at step c),


wherein V′h is the vertical component of a resultant force applied on a lifting surface h of the structure,


wherein X′h and Y′h are the X-coordinate and Y-coordinate, respectively, through which the vertical component of the resultant force passes, wherein V′h, X′h and Y′h are determined based on the measured responses, for each lifting surface as:







V


=


C
v


+





j
v

=
1


L
v









μ

j
v



·

β

j
v













X


=


C
x


+





j
x

=
1


L
x









μ

j
x



·

γ

j
x













Y


=


C
y


+





j
y

=
1


L
y









μ

j
y



·

τ

j
y










wherein μ′ are the responses measured by the second plurality of installed sensors, the subscript in the response denoting the sensor which has performed the measurement, being Lv, Lx and Ly natural numbers,


wherein W, V′ and Wcorrected are expressed in the same axes system; X, X′ and Xcorrected are expressed according to the same coordinate system, and Y, Y′ and Ycorrected are expressed according to the same coordinate system,


and wherein:


for determining V′ a fourth group of sensors is used which comprises a number of installed sensors Lv,


for determining X′ a fifth group of sensors is used which comprises a number of installed sensors Lx


for determining Y′ a sixth group of sensors is used which comprises a number of installed sensors Ly, and wherein


coefficients β′jv, with jv=1, . . . Lv, γ′jx, with jx=1, . . . Lx, and τ′jy, with jy=1, . . . Ly, and constants C′v, C′x, C′y are previously determined applying the following calibration steps:

    • s-i) locating a plurality of calibration sensors in at least one lifting surface of the structure, wherein the number of calibration sensors is m′, m′ being a natural number,
    • s-ii) applying a probe force to the at least one lifting surface of the structure, obtaining a force state, and measuring the vertical component of the resultant force, the X-coordinate and the Y-coordinate respectively through which the vertical component of the resultant force passes in said force state,
    • s-iii) measuring for each calibration sensor the response associated to the force state,
    • s-iv) repeating steps s-ii) and s-iii) for a plurality of force states f, with f=1, . . . s′, being s′ the number of force states, wherein for each force state a different probe force is applied to the at least one lifting surface, s′ being a natural number,
    • s-v) selecting groups of at least two calibration sensors and solving for each group an equation system with a number of equations greater than Nv+1, being Nv the number of calibration sensors in the group, wherein the equation system is:







V
1


=


C
v


+




n
=
1


N
v









μ

n
,
1



·

β
n

















V
P


=


C
v


+




n
=
1


N
v









μ

n
,
P



·

β
n











    • with μ′n,q, with n=1, . . . , Nv being the measured responses of the calibration sensors and V′q being the vertical component of the resultant force measured for the respective force state q, with q=1, . . . , P,

    • wherein the equation systems are solved by least squares method, obtaining as a result vertical force coefficients β′n for n=1, . . . , Nv and constant C′v,

    • s-vi) determining, for each group of calibration sensors an estimated vertical force Ve′f for each force state f as










Ve
f


=


C
v


+




n
=
1


N
v









μ

n
,
f



·

β
n











    • where μ′n,f are the responses of the calibration sensors of the group, associated to a force state f,

    • s-vii) selecting a fourth group of calibration sensors which fulfils that a fourth probable error PE′v is minimum, wherein PE′v is:










PE
v


=


C
E








f
=
1


s















(


Ve
f


-

V
f



)

2




s


-

(


N
v

+
1

)









s′ being the total number of force states, Nv being the number of calibration sensors in said group of calibration sensors, CE being a real positive number, Ve′f being the estimated vertical force and V′f being the vertical component of the resultant force measured for force state f,

    • s-viii) selecting groups of at least two calibration sensors and solving for each group an equation system with a number of equations equal to or greater than Nx+1, being Nx the number of calibration sensors in the group, wherein the equation systems is:







X
1


=


C
x


+




n
=
1


N
x









μ

n
,
1



·

γ
n

















X
P


=


C
x


+




n
=
1


N
x









μ

n
,
P



·

γ
n











    • μ′n,q being the measured responses of the calibration sensors and X′q being the X-coordinate through which the measured vertical component of the resultant force passes, associated to the respective force state q, q=1, . . . , P

    • wherein the equation systems are solved by least squares method, obtaining as a result X-coefficients γ′n for n=1, . . . , Nx and X-constant C′x,

    • s-ix) determining for each group of calibration sensors an estimated X-coordinate Xe′f through which the vertical component of the resultant force passes for each force state f, as










Xe
f


=


C
x


+




n
=
1


N
x









μ

n
,
f



·

γ
n











    • where μ′n,f are the responses of the calibration sensors of the group, associated to a force state f,

    • s-x) selecting a fifth group of calibration sensors which fulfils that a fifth probable error PE′X is minimum, wherein PE′X is:










PE
X


=


C
E








f
=
1


s















(


Xe
f


-

X
f



)

2




s


-

(


N
x

+
1

)









s′ being the total number of force states, Nx being the number of calibration sensors in the group of calibration sensors, CE being a real positive number, Xe′f being the estimated X-coordinate through which the vertical component of the resultant force passes and X′f being the X-coordinate through which the measured vertical component of the resultant force passes for force state f,

    • s-xi) selecting groups of at least two calibration sensors and solving for each group an equation system with a number of equations P equal or greater than Ny+1, being Ny the number of calibration sensors in the group, wherein the equation system is:







Y
1


=


C
y


+




n
=
1


N
y









μ

n
,
1



·

τ
n

















Y
P


=


C
y


+




n
=
1


N
y









μ

n
,
P



·

τ
n











    • μ′n,q being the measured responses of the calibration sensors and Y′q being the Y-coordinate through which the measured vertical component of the resultant force passes, associated to the respective force state, q=1, . . . , P

    • wherein the equation systems are solved by least squares method, obtaining as a result Y-coefficients τ′n for n=1, . . . , Ny and Y-constant C′y,

    • s-xii) determining for each group of calibration sensors an estimated Y-coordinate through which the vertical component of the resultant force passes for each force state, as










Ye
f


=


C
y


+




n
=
1


N
y









μ

n
,
f



·

τ
n











    • where μ′n,f are the responses of the calibration sensors of the group, associated to a force state f,

    • s-xiii) selecting a sixth group of calibration sensors which fulfils that a sixth probable error PE′Y is minimum, wherein PE′Y is:










PE
Y


=


C
E








f
=
1


s















(



Y




e
f


-

Y
f



)

2




s


-

(


N
y

+
1

)









s′ being the total number of force states, Ny being the number of calibration sensors in the selected group of calibration sensors, and CE being a real positive number,

    • s-xiv) selecting as the second plurality of installed sensors a plurality of calibration sensors including:
      • the fourth group of calibration sensors selected in step s-vii),
      • the fifth group of calibration sensors selected in step s-x), and
      • the sixth group of calibration sensors selected in step s-xiii),


wherein Lv is the number of sensors in the fourth group of calibration sensors, Lx is the number of sensors in the fifth group of calibration sensors and Ly is the number of sensors in the sixth group of calibration sensors. Lv, Lx and Ly are less than or equal to m′.


In addition to determining the total weight and the position of the centre of gravity of a structure, the present method also allows determining the external forces acting on lifting surfaces of the structure. Therefore, the results of the weight and the position of the centre of gravity obtained in step b) can advantageously be corrected on the basis of said external forces acting on the lifting surfaces of the structure.


Through this entire document, the coordinates through which the vertical force or the vertical resultant force passes are defined in relation to the same coordinate system of the centre of gravity. The external forces which act on lifting surfaces are defined in relation to an axes system which will be understood as the previous coordinate system.


Prior to determining the corrected weight and the corrected centre of gravity of the structure based on the vertical component of a resultant force applied on a lifting surface of the structure and the X-coordinate and the Y-coordinate through which such vertical component of the resultant force passes, it is necessary to determine such component and coordinates based on the measured responses of a second plurality of installed sensors located in the lifting surface of a structure, and also to determine which sensors are going to be used. The determination of such second plurality of installed sensors is carried out by calibration steps.


The calibration steps defined for determining the weight and the position of the centre of gravity are similar to the calibration steps for determining the resultant force and the X-coordinate and the Y-coordinate through which the vertical component of the resultant force passes according to each lifting surface of the structure.


In the calibration steps s-i) to s-xi) of the present method, a plurality of calibration sensors are installed in at least one lifting surface h of the structure. In the context of the specification, a lifting surface of a structure is to be understood as a structure body shaped to produce a normal reaction to its movement through the air.


Once a plurality of calibration sensors are installed in at least one lifting surface, different probe forces are applied on the lifting surface, obtaining different force states. The vertical component, and the X-coordinate and the Y-coordinate respectively through which such vertical component of the resultant force passes in each force state can be measured in a conventional way. Furthermore, the measured response of each calibration sensor is recorded for each force state. Once the responses of the calibration sensors and the vertical component, and the X-coordinate and the Y-coordinate of the resultant force are obtained and recorded for each force state, then the coefficients β′jv, γ′jx and τ′jy and constants C′v, C′x, C′y are respectively determined.


For determining such coefficients and constants, groups of at least two calibration sensors are selected, and for each group an equation system as defined on steps s-v), s-viii) and s-xi) is solved by the least squares method. In the equation system each equation corresponds to a loading state and the number of equations is equal to or greater than the number of calibration sensors in the group plus one.


To determine which sensors, among the calibration sensors, are going to be the installed sensors used for determining the vertical component of a resultant force, the X-coordinate and the Y-coordinate respectively through which the vertical component of the resultant force passes on lifting surfaces, it is selected:

    • a fourth group of calibration sensors which fulfils that a fourth probable error is minimum;
    • a fifth group of calibration sensors which fulfils that a fifth probable error is minimum; and
    • a sixth group of calibration sensors which fulfils that a sixth probable error is minimum;


wherein such probable errors are defined in step s-vii), s-x) and s-xiii) respectively of the present method.


For determining the vertical component of a resultant force applied on the lifting surfaces it is used a fourth group of sensors; for determining the X-coordinate through which the vertical component of the resultant force passes on the lifting surface it is used a fifth group of sensors; and for determining the Y-coordinate through which the vertical component of the resultant force passes it is used a sixth group of sensors. Such three groups of sensors are previously determined in the calibration steps and define the second plurality of installed sensors which are located in lifting surfaces of the structure and are used to determine a corrected weight and/or a corrected centre of gravity of the structure. One or several of the installed sensors included in one of the groups may be included in one or both of the other groups. In a particular embodiment wherein it is determined only one of the coordinates of the position of the centre of gravity, it is understood that the present method can correct such coordinate of the centre of gravity independently.


Advantageously, since the present embodiment allows determining the position of the centre of gravity and the weight considering the external forces applied on lifting surfaces based on measurements by the sensors located in at least one lifting surface, the method increases the accuracy and the reliability of the determination of the weight and the centre of gravity.


In a more particular embodiment, the structure is an aircraft comprising lifting surfaces, and in step s-ii) and s-iv) of calibration steps, a plurality of probe forces are applied on the at least one lifting surface of the aircraft.


In a second inventive aspect, the present invention provides a system configured for determining the weight and/or at least a first coordinate of the position of the centre of gravity of a structure, comprising:

    • a plurality of installed sensors located on the structure,
    • a data acquisition module configured to register data from the plurality of installed sensors on the structure, and
    • data processing means configured to perform at least step b) of the method according to the first inventive aspect.


The data acquisition module and the data processing means may be implemented as two different elements in data communication or as a single processor.


In a third inventive aspect, the present invention provides a vehicle comprising the system according to the second inventive aspect. Preferably the vehicle is an aircraft.


All the features described in this specification (including the claims, description and drawings) and/or all the steps of the described method can be combined in any combination, with the exception of combinations of such mutually exclusive features and/or steps.





DESCRIPTION OF THE DRAWINGS

These and other characteristics and advantages of the invention will become clearly understood in view of the detailed description of the invention which becomes apparent from a preferred embodiment of the invention, given just as an example and not being limited thereto, with reference to the drawings.



FIG. 1 shows sensitive load points used in an embodiment of the present invention.



FIG. 2 shows sensitive load points used in an embodiment of the present invention.



FIGS. 3a-3b show the installation of a sensor in at least one sensitive load point according to an embodiment of the present invention.



FIG. 4 shows an aircraft loaded with weight probes according to an embodiment of the present invention.



FIGS. 5a and 5b show a method to obtain the weight and the center of gravity by mean of jacks with load cells according to an embodiment of the present invention.





DETAILED DESCRIPTION OF THE INVENTION

Once the object of the invention has been outlined, specific non-limitative embodiments are described hereinafter. In particular, it is described an embodiment of the method for determining the weight (W) and a first coordinate, which is the X-coordinate, of the position of the centre of gravity (X, Y) of an aircraft (6). In this particular embodiment, the position of the centre of gravity is defined in relation to a coordinate system, the origin of which is a predefined point. In this particular example, the origin is located in the X-axis in front of the nose of the aircraft (6) if seen from the side, in particular at a distance of 1587 mm from the nose. In this aircraft (6) the X-axis is selected as the projection on the horizontal plane of the longitudinal direction along the fuselage (roll axis).



FIGS. 1 and 2 respectively show a main landing gear bulkhead (1) and a nose landing gear (2) of an aircraft (6). Selected sensitive load points (3) in the main landing gear bulkhead (1) and in pins (7) of the nose landing gear (2) are indicated in these figures. In the context of the invention the sensitive load points (3) of the structure which weight and/or centre of gravity are to be determined are the points where the structure is seated on the ground. In this particular example four of the sensitive load points (3) are located in the main landing gear bulkhead (1) and three of the sensitive load points (3) are located in the pins (7) of the nose landing gear (2), as shown in FIGS. 1 and 2 respectively.


After selecting the sensitive load points (3), at least one sensor (4) is located in each sensitive load point (3). In this particular example, the sensors (4) are strain gauge bridges. In FIGS. 3a and 3b, some of the strain gauge bridges (4) installed on the structure are schematically shown.


The strain gauge bridges (4) are attached to the structure by abrasion, adhesive application, application of pressure, and thermal curing. In the example shown in FIG. 3a, the strain gauge bridges (4) are attached to the main landing gear bulkhead (1). In FIG. 3b, the strain gauge bridges (4) are attached to the nose landing gear pin (7). In this particular example of FIG. 3b, the sensor (4) is embedded in one of the pins (7) of the nose landing gear pins.


Example

For a particular example, prior to determining the weight (W) and the X-coordinate (X) of the position of the centre of gravity (X, Y) of an aircraft (6), the following calibration steps were performed:


i) 19 calibration sensors were located in sensitive load points (3) of the aircraft (6). Thus, in this particular example m=19. In this particular example, all the calibration sensors located are strain gauges bridges. Further, the calibration sensors 1 to 16 are located in the main landing gear bulkhead, and the calibration sensors 17 to 19 are located in the nose landing gear pin (7).


ii) The structure was loaded with a probe weight Wp1, obtaining a first loading state (i=1), and the X-coordinate of the centre of gravity (X1) and the weight (W1) of the aircraft were measured for the loading state i=1. In this particular example, the Y-coordinate of the centre of gravity is not exemplified below, since it is obtained as for the X-coordinate.


iii) For each calibration sensor r the response μr,1 associated to the loading state i=1 was measured, with r=1, . . . 19.


iv) Steps ii) and iii) were repeated for a plurality of loading states i. In this example, nine different configurations of loaded probe weights were used, different loading states being obtained by modifying the weight and/or the location of the probe weight loaded on the aircraft (6). As schematically shown in FIG. 4, twelve water tanks (8) were arranged at specific positions in the aircraft (6). In this example the loading states were obtained by filling up different combinations of tanks at different water levels. Tables 1A-1B show the filled water tanks and the maximum probe weight in each configuration. In Tables 1A-1B and in FIG. 4 the twelve tanks (8) are identified as T1, T2, . . . T12.











TABLE 1A









Probe weight (kg)















T1
T2
T3
T4
T5
T6
T7


















configuration 1
1000

1000






configuration 2


configuration 3
1000
1000


configuration 4

1000

1000


configuration 5




1000
1000
1000


configuration 6






1000


configuration 7





1000


configuration 8

1000


configuration 9

1000

1000
1000
1000
1000


















TABLE 1B









Probe weight (kg)














T8
T9
T10
T11
T12
Total Kg

















configuration 1





2000


configuration 2

1000

1000

2000


configuration 3





2000


configuration 4



1000
1000
4000


configuration 5
1000




4000


configuration 6
1000


1000
1000
4000


configuration 7
1000
1000

1000

4000


configuration 8

1000



2000


configuration 9
1000
1000
1000
1000
1000
10000









Additionally, for each of the configurations of filled water tanks (8) of Tables 1A-1B, several percentages of filling were considered. For example, for configuration 1 as indicated in Tables 1A-1B increasing percentages of tanks filling of 33%, 66%, 100% and decreasing percentages of tanks filling of 66%, 33% and 0% where considered. As a result, six loading states were obtained for configuration 1 of tanks (8) indicated in Table 1.


In these calibration steps, the weight (Wi) and the X-coordinate (Xi) of the position of the centre of gravity of the aircraft (6) in each loading state i have been measured with jacks with load cells as it is schematically shown in FIGS. 5a and 5b, wherein the load cell (10) is attached to the hydraulic jack (9) in a jacking point (11) in order to measure the real weight value (Wi) and real X-coordinate (Xi) of the centre of gravity of the aircraft (6) in each loading state i. The load cell (10) is connected to a weight indicator system (5) which registers the measures.


The following tables 2A-2E to 10A-10E show the weight (Wi), the X-coordinate (Xi) of the centre of gravity and the response of each sensor (the sensors are identified as “bridges”) measured for each loading state i:









TABLE 2A







CONFIGURATION 1: six loading states













Percent-


Bridge
Bridge
Bridge
Bridge


age of
Weight
X_CG
01
02
03
04


cargo
(N)
(mm)
(mV/V)
(mV/V)
(mV/V)
(mV/V)
















33%
111022.080
11117.346
−0.590
0.022
−1.503
−0.250


66%
117004.640
10835.995
−0.597
0.024
−1.518
−0.247


100% 
122297.760
10614.656
−0.604
0.027
−1.542
−0.241


66%
115403.360
10948.294
−0.607
0.028
−1.549
−0.243


33%
110888.640
11164.416
−0.603
0.031
−1.570
−0.224


 0%
103527.200
11424.708
−0.586
0.032
−1.537
−0.219
















TABLE 2B







CONFIGURATION 1: six loading states













Percent-


Bridge
Bridge
Bridge
Bridge


age of
Weight
X_CG
05
06
07
08


cargo
(N)
(mm)
(mV/V)
(mV/V)
(mV/V)
(mV/V)
















33%
111022.080
11117.346
−0.033
−0.247
−0.377
−0.397


66%
117004.640
10835.995
−0.038
−0.253
−0.391
−0.407


100% 
122297.760
10614.656
−0.035
−0.259
−0.395
−0.414


66%
115403.360
10948.294
−0.028
−0.253
−0.380
−0.402


33%
110888.640
11164.416
−0.022
−0.263
−0.328
−0.427


 0%
103527.200
11424.708
−0.013
−0.249
−0.311
−0.398
















TABLE 2C







CONFIGURATION 1: six loading states













Percent-


Bridge
Bridge
Bridge
Bridge


age of
Weight
X_CG
09
10
11
12


cargo
(N)
(mm)
(mV/V)
(mV/V)
(mV/V)
(mV/V)
















33%
111022.080
11117.346
−0.607
0.005
−1.433
−0.087


66%
117004.640
10835.995
−0.614
0.005
−1.450
−0.086


100% 
122297.760
10614.656
−0.626
0.012
−1.489
−0.064


66%
115403.360
10948.294
−0.628
0.009
−1.497
−0.069


33%
110888.640
11164.416
−0.611
0.007
−1.443
−0.086


 0%
103527.200
11424.708
−0.593
0.007
−1.418
−0.080
















TABLE 2D







CONFIGURATION 1: six loading states













Percent-


Bridge
Bridge
Bridge
Bridge


age of
Weight
X_CG
13
14
15
16


cargo
(N)
(mm)
(mV/V)
(mV/V)
(mV/V)
(mV/V)
















33%
111022.080
11117.346
−0.068
−0.226
−0.434
−0.347


66%
117004.640
10835.995
−0.071
−0.231
−0.441
−0.356


100% 
122297.760
10614.656
−0.068
−0.245
−0.434
−0.382


66%
115403.360
10948.294
−0.057
−0.238
−0.416
−0.371


33%
110888.640
11164.416
−0.066
−0.219
−0.444
−0.333


 0%
103527.200
11424.708
−0.061
−0.209
−0.432
−0.316
















TABLE 2E







CONFIGURATION 1: six loading states












Percentage
Weight
X_CG
Bridge 17
Bridge 18
Bridge 19


of cargo
(N)
(mm)
(mV/V)
(mV/V)
(mV/V)















33%
111022.080
11117.346
1.915
1.042
1.080


66%
117004.640
10835.995
2.335
1.294
1.289


100% 
122297.760
10614.656
2.895
1.627
1.611


66%
115403.360
10948.294
2.317
1.334
1.372


33%
110888.640
11164.416
1.860
1.012
1.045


 0%
103527.200
11424.708
1.460
0.798
0.882
















TABLE 3A







CONFIGURATION 2: six loading states













Percent-


Bridge
Bridge
Bridge
Bridge


age of
Weight
X_CG
01
02
03
04


cargo
(N)
(mm)
(mV/V)
(mV/V)
(mV/V)
(mV/V)
















33%
108686.880
11509.888
−0.618
0.025
−1.561
−0.250


66%
114157.920
11590.373
−0.661
0.031
−1.665
−0.250


100% 
120162.720
11673.937
−0.727
0.040
−1.826
−0.242


66%
112734.560
11579.179
−0.681
0.043
−1.725
−0.215


33%
108086.400
11509.259
−0.613
0.014
−1.508
−0.282


 0%
103349.280
11420.250
−0.585
0.014
−1.450
−0.270
















TABLE 3B







CONFIGURATION 2: six loading states













Percent-


Bridge
Bridge
Bridge
Bridge


age of
Weight
X_CG
05
06
07
08


cargo
(N)
(mm)
(mV/V)
(mV/V)
(mV/V)
(mV/V)
















33%
108686.880
11509.888
−0.030
−0.253
−0.387
−0.403


66%
114157.920
11590.373
−0.037
−0.278
−0.421
−0.443


100% 
120162.720
11673.937
−0.043
−0.321
−0.454
−0.503


66%
112734.560
11579.179
−0.029
−0.309
−0.418
−0.471


33%
108086.400
11509.259
−0.047
−0.244
−0.460
−0.369


 0%
103349.280
11420.250
−0.038
−0.232
−0.437
−0.341
















TABLE 3C







CONFIGURATION 2: six loading states













Per-


Bridge
Bridge
Bridge
Bridge


centage
Weight
X_CG
09
10
11
12


of cargo
(N)
(mm)
(mV/V)
(mV/V)
(mV/V)
(mV/V)
















33%
108686.880
11509.888
−0.641
0.011
−1.530
−0.067


66%
114157.920
11590.373
−0.683
0.016
−1.625
−0.056


100%
120162.720
11673.937
−0.728
0.017
−1.687
−0.059


66%
112734.560
11579.179
−0.680
0.018
−1.592
−0.037


33%
108086.400
11509.259
−0.647
0.014
−1.551
−0.048


0%
103349.280
11420.250
−0.613
0.015
−1.489
−0.037
















TABLE 3D







CONFIGURATION 2: six loading states













Per-


Bridge
Bridge
Bridge
Bridge


centage
Weight
X_CG
13
14
15
16


of cargo
(N)
(mm)
(mV/V)
(mV/V)
(mV/V)
(mV/V)
















33%
108686.880
11509.888
−0.060
−0.240
−0.404
−0.384


66%
114157.920
11590.373
−0.069
−0.262
−0.435
−0.416


100%
120162.720
11673.937
−0.098
−0.279
−0.575
−0.386


66%
112734.560
11579.179
−0.083
−0.267
−0.540
−0.367


33%
108086.400
11509.259
−0.065
−0.258
−0.373
−0.414


0%
103349.280
11420.250
−0.056
−0.247
−0.353
−0.393
















TABLE 3E







CONFIGURATION 2: six loading states












Percentage
Weight
X_CG
Bridge 17
Bridge 18
Bridge 19


of cargo
(N)
(mm)
(mV/V)
(mV/V)
(mV/V)















33%
108686.880
11509.888
1.334
0.735
0.784


66%
114157.920
11590.373
1.336
0.743
0.788


100% 
120162.720
11673.937
1.296
0.746
0.799


66%
112734.560
11579.179
1.308
0.744
0.802


33%
108086.400
11509.259
1.374
0.771
0.822


 0%
103349.280
11420.250
1.404
0.786
0.836
















TABLE 4A







CONFIGURATION 3: six loading states













Per-


Bridge
Bridge
Bridge
Bridge


centage
Weight
X_CG
01
02
03
04


of cargo
(N)
(mm)
(mV/V)
(mV/V)
(mV/V)
(mV/V)
















33%
109198.400
11162.881
−0.560
0.014
−1.417
−0.270


66%
115136.480
10924.562
−0.553
0.018
−1.404
−0.261


100%
120385.120
10738.966
−0.548
0.017
−1.388
−0.263


66%
115069.760
10940.000
−0.558
0.017
−1.414
−0.266


33%
109554.240
11157.164
−0.563
0.015
−1.431
−0.265


0%
104149.920
11395.553
−0.568
0.016
−1.446
−0.264
















TABLE 4B







CONFIGURATION 3: six loading states













Per-


Bridge
Bridge
Bridge
Bridge


centage
Weight
X_CG
05
06
07
08


of cargo
(N)
(mm)
(mV/V)
(mV/V)
(mV/V)
(mV/V)
















33%
109198.400
11162.881
−0.044
−0.208
−0.449
−0.320


66%
115136.480
10924.562
−0.048
−0.214
−0.460
−0.334


100%
120385.120
10738.966
−0.053
−0.194
−0.481
−0.296


66%
115069.760
10940.000
−0.049
−0.190
−0.484
−0.289


33%
109554.240
11157.164
−0.034
−0.220
−0.398
−0.352


0%
104149.920
11395.553
−0.033
−0.215
−0.400
−0.343
















TABLE 4C







CONFIGURATION 3: six loading states













Per-


Bridge
Bridge
Bridge
Bridge


centage
Weight
X_CG
09
10
11
12


of cargo
(N)
(mm)
(mV/V)
(mV/V)
(mV/V)
(mV/V)
















33%
109198.400
11162.881
−0.634
0.015
−1.549
−0.050


66%
115136.480
10924.562
−0.663
0.018
−1.612
−0.041


100%
120385.120
10738.966
−0.690
0.021
−1.657
−0.037


66%
115069.760
10940.000
−0.670
0.020
−1.628
−0.039


33%
109554.240
11157.164
−0.632
0.014
−1.525
−0.056


0%
104149.920
11395.553
−0.614
0.010
−1.495
−0.066
















TABLE 4D







CONFIGURATION 3: six loading states













Per-


Bridge
Bridge
Bridge
Bridge


centage
Weight
X_CG
13
14
15
16


of cargo
(N)
(mm)
(mV/V)
(mV/V)
(mV/V)
(mV/V)
















33%
109198.400
11162.881
−0.058
−0.249
−0.370
−0.412


66%
115136.480
10924.562
−0.067
−0.270
−0.399
−0.444


100%
120385.120
10738.966
−0.078
−0.285
−0.472
−0.442


66%
115069.760
10940.000
−0.067
−0.276
−0.443
−0.425


33%
109554.240
11157.164
−0.056
−0.252
−0.369
−0.418


0%
104149.920
11395.553
−0.040
−0.236
−0.330
−0.395
















TABLE 4E







CONFIGURATION 3: six loading states












Percentage
Weight
X_CG
Bridge 17
Bridge 18
Bridge 19


of cargo
(N)
(mm)
(mV/V)
(mV/V)
(mV/V)















33%
109198.400
11162.881
1.804
1.004
1.019


66%
115136.480
10924.562
2.243
1.261
1.239


100% 
120385.120
10738.966
2.656
1.517
1.461


66%
115069.760
10940.000
2.305
1.352
1.331


33%
109554.240
11157.164
1.856
1.040
1.044


 0%
104149.920
11395.553
1.494
0.855
0.906
















TABLE 5A







CONFIGURATION 4: five loading states













Per-


Bridge
Bridge
Bridge
Bridge


centage
Weight
X_CG
01
02
03
04


of cargo
(N)
(mm)
(mV/V)
(mV/V)
(mV/V)
(mV/V)
















33%
115381.120
11364.245
−0.662
0.030
−1.679
−0.250


66%
127168.320
11344.276
−0.745
0.042
−1.871
−0.248


100%
138221.600
11320.850
−0.828
0.052
−2.046
−0.250


50%
121719.520
11381.470
−0.735
0.049
−1.835
−0.222


0%
103972.000
11414.744
−0.587
0.029
−1.540
−0.229
















TABLE 5B







CONFIGURATION 4: five loading states













Per-


Bridge
Bridge
Bridge
Bridge


centage
Weight
X_CG
05
06
07
08


of cargo
(N)
(mm)
(mV/V)
(mV/V)
(mV/V)
(mV/V)
















33%
115381.120
11364.245
−0.034
−0.293
−0.398
−0.465


66%
127168.320
11344.276
−0.051
−0.350
−0.479
−0.550


100%
138221.600
11320.850
−0.074
−0.403
−0.577
−0.629


50%
121719.520
11381.470
−0.035
−0.355
−0.495
−0.538


0%
103972.000
11414.744
−0.029
−0.235
−0.348
−0.379
















TABLE 5C







CONFIGURATION 4: five loading states













Per-


Bridge
Bridge
Bridge
Bridge


centage
Weight
X_CG
09
10
11
12


of cargo
(N)
(mm)
(mV/V)
(mV/V)
(mV/V)
(mV/V)
















33%
115381.120
11364.245
−0.650
0.010
−1.527
−0.079


66%
127168.320
11344.276
−0.703
0.016
−1.648
−0.064


100%
138221.600
11320.850
−0.759
0.030
−1.794
−0.014


50%
121719.520
11381.470
−0.694
0.032
−1.672
0.004


0%
103972.000
11414.744
−0.596
0.005
−1.418
−0.096
















TABLE 5D







CONFIGURATION 4: five loading states













Per-


Bridge
Bridge
Bridge
Bridge


centage
Weight
X_CG
13
14
15
16


of cargo
(N)
(mm)
(mV/V)
(mV/V)
(mV/V)
(mV/V)
















33%
115381.120
11364.245
−0.069
−0.244
−0.445
−0.374


66%
127168.320
11344.276
−0.079
−0.275
−0.485
−0.420


100%
138221.600
11320.850
−0.077
−0.316
−0.478
−0.495


50%
121719.520
11381.470
−0.060
−0.294
−0.444
−0.454


0%
103972.000
11414.744
−0.069
−0.198
−0.471
−0.285
















TABLE 5E







CONFIGURATION 4: five loading states












Percentage
Weight
X_CG
Bridge 17
Bridge 18
Bridge 19


of cargo
(N)
(mm)
(mV/V)
(mV/V)
(mV/V)















33%
115381.120
11364.245
1.692
1.010
1.010


66%
127168.320
11344.276
1.885
1.107
1.087


100% 
138221.600
11320.850
2.017
1.161
1.153


50%
121719.520
11381.470
1.692
0.998
1.030


 0%
103972.000
11414.744
1.417
0.823
0.820
















TABLE 6A







CONFIGURATION 5: six loading states













Per-


Bridge
Bridge
Bridge
Bridge


centage
Weight
X_CG
01
02
03
04


of cargo
(N)
(mm)
(mV/V)
(mV/V)
(mV/V)
(mV/V)
















0%
103972.000
11414.744
−0.588
0.029
−1.542
−0.228


33%
115603.520
11301.925
−0.649
0.037
−1.683
−0.233


66%
126567.840
11224.119
−0.702
0.040
−1.773
−0.243


100%
135508.320
11150.887
−0.754
0.049
−1.892
−0.237


50%
119717.920
11267.787
−0.674
0.038
−1.723
−0.239


0%
102370.720
11424.841
−0.601
0.035
−1.551
−0.210
















TABLE 6B







CONFIGURATION 5: six loading states













Per-


Bridge
Bridge
Bridge
Bridge


centage
Weight
X_CG
05
06
07
08


of cargo
(N)
(mm)
(mV/V)
(mV/V)
(mV/V)
(mV/V)
















0%
103972.000
11414.744
−0.029
−0.234
−0.350
−0.379


33%
115603.520
11301.925
−0.035
−0.274
−0.399
−0.440


66%
126567.840
11224.119
−0.036
−0.315
−0.429
−0.503


100%
135508.320
11150.887
−0.048
−0.353
−0.484
−0.562


50%
119717.920
11267.787
−0.037
−0.299
−0.412
−0.476


0%
102370.720
11424.841
−0.010
−0.258
−0.344
−0.408
















TABLE 6C







CONFIGURATION 5: six loading states













Per-


Bridge
Bridge
Bridge
Bridge


centage
Weight
X_CG
09
10
11
12


of cargo
(N)
(mm)
(mV/V)
(mV/V)
(mV/V)
(mV/V)
















0%
103972.000
11414.744
−0.597
0.005
−1.420
−0.095


33%
115603.520
11301.925
−0.659
0.012
−1.557
−0.081


66%
126567.840
11224.119
−0.721
0.022
−1.685
−0.046


100%
135508.320
11150.887
−0.766
0.029
−1.782
−0.028


50%
119717.920
11267.787
−0.686
0.020
−1.620
−0.053


0%
102370.720
11424.841
−0.600
0.018
−1.445
−0.035
















TABLE 6D







CONFIGURATION 5: six loading states













Per-


Bridge
Bridge
Bridge
Bridge


centage
Weight
X_CG
13
14
15
16


of cargo
(N)
(mm)
(mV/V)
(mV/V)
(mV/V)
(mV/V)
















0%
103972.000
11414.744
−0.070
−0.197
−0.472
−0.284


33%
115603.520
11301.925
−0.078
−0.234
−0.516
−0.339


66%
126567.840
11224.119
−0.080
−0.295
−0.509
−0.445


100%
135508.320
11150.887
−0.092
−0.325
−0.547
−0.491


50%
119717.920
11267.787
−0.076
−0.264
−0.480
−0.400


0%
102370.720
11424.841
−0.057
−0.230
−0.427
−0.339
















TABLE 6E







CONFIGURATION 5: six loading states












Percentage
Weight
X_CG
Bridge 17
Bridge 18
Bridge 19


of cargo
(N)
(mm)
(mV/V)
(mV/V)
(mV/V)















 0%
103972.000
11414.744
1.422
0.760
0.822


33%
115603.520
11301.925
1.756
0.949
0.988


66%
126567.840
11224.119
2.012
1.111
1.131


100% 
135508.320
11150.887
2.283
1.265
1.266


50%
119717.920
11267.787
1.901
1.069
1.129


 0%
102370.720
11424.841
1.411
0.778
0.867
















TABLE 7A







CONFIGURATION 6: six loading states













Per-


Bridge
Bridge
Bridge
Bridge


centage
Weight
X_CG
01
02
03
04


of cargo
(N)
(mm)
(mV/V)
(mV/V)
(mV/V)
(mV/V)
















0%
102370.720
11424.841
−0.599
0.034
−1.552
−0.212


33%
114091.200
11544.014
−0.705
0.044
−1.827
−0.223


66%
125878.400
11630.835
−0.804
0.058
−2.047
−0.217


90%
135019.040
11692.424
−0.900
0.070
−2.245
−0.223


50%
120184.960
11586.200
−0.779
0.065
−1.964
−0.183


0%
102192.800
11423.009
−0.570
0.022
−1.480
−0.245
















TABLE 7B







CONFIGURATION 6: six loading states













Percent-


Bridge
Bridge
Bridge
Bridge


age of
Weight
X_CG
05
06
07
08


cargo
(N)
(mm)
(mV/V)
(mV/V)
(mV/V)
(mV/V)
















 0%
102370.720
11424.841
−0.009
−0.257
−0.344
−0.408


33%
114091.200
11544.014
−0.036
−0.311
−0.414
−0.502


66%
125878.400
11630.835
−0.058
−0.383
−0.518
−0.611


90%
135019.040
11692.424
−0.077
−0.444
−0.635
−0.698


50%
120184.960
11586.200
−0.034
−0.384
−0.524
−0.579


 0%
102192.800
11423.009
−0.024
−0.223
−0.346
−0.366
















TABLE 7C







CONFIGURATION 6: six loading states













Percent-


Bridge
Bridge
Bridge
Bridge


age of
Weight
X_CG
09
10
11
12


cargo
(N)
(mm)
(mV/V)
(mV/V)
(mV/V)
(mV/V)
















 0%
102370.720
11424.841
−0.599
0.019
−1.446
−0.036


33%
114091.200
11544.014
−0.632
0.004
−1.463
−0.109


66%
125878.400
11630.835
−0.650
0.006
−1.509
−0.111


90%
135019.040
11692.424
−0.732
0.027
−1.709
−0.043


50%
120184.960
11586.200
−0.678
0.029
−1.606
−0.016


 0%
102192.800
11423.009
−0.585
0.003
−1.373
−0.100
















TABLE 7D







CONFIGURATION 6: six loading states













Percent-


Bridge
Bridge
Bridge
Bridge


age of
Weight
X_CG
13
14
15
16


cargo
(N)
(mm)
(mV/V)
(mV/V)
(mV/V)
(mV/V)
















 0%
102370.720
11424.841
−0.056
−0.228
−0.427
−0.339


33%
114091.200
11544.014
−0.087
−0.204
−0.571
−0.252


66%
125878.400
11630.835
−0.086
−0.214
−0.577
−0.270


90%
135019.040
11692.424
−0.079
−0.256
−0.562
−0.363


50%
120184.960
11586.200
−0.065
−0.244
−0.532
−0.337


 0%
102192.800
11423.009
−0.058
−0.194
−0.445
−0.288
















TABLE 7E







CONFIGURATION 6: six loading states












Percentage
Weight
X_CG
Bridge 17
Bridge 18
Bridge 19


of cargo
(N)
(mm)
(mV/V)
(mV/V)
(mV/V)















 0%
102370.720
11424.841
1.410
0.796
0.870


33%
114091.200
11544.014
1.359
0.753
0.794


66%
125878.400
11630.835
1.481
0.821
0.859


100% 
135019.040
11692.424
1.349
0.725
0.805


50%
120184.960
11586.200
1.350
0.743
0.816


 0%
102192.800
11423.009
1.363
0.749
0.797
















TABLE 8A







CONFIGURATION 7: six loading states













Percent-


Bridge
Bridge
Bridge
Bridge


age of
Weight
X_CG
01
02
03
04


cargo
(N)
(mm)
(mV/V)
(mV/V)
(mV/V)
(mV/V)
















 0%
102192.800
11423.009
−0.575
0.017
−1.478
−0.247


33%
113735.360
11509.838
−0.652
0.026
−1.655
−0.255


66%
124988.800
11572.303
−0.732
0.040
−1.833
−0.250


100% 
136397.920
11632.690
−0.810
0.052
−2.008
−0.246


50%
119740.160
11552.059
−0.694
0.030
−1.730
−0.269


 0%
102392.960
11412.718
−0.594
0.024
−1.498
−0.234
















TABLE 8B







CONFIGURATION 7: six loading states













Percent-


Bridge
Bridge
Bridge
Bridge


age of
Weight
X_CG
05
06
07
08


cargo
(N)
(mm)
(mV/V)
(mV/V)
(mV/V)
(mV/V)
















 0%
102192.800
11423.009
−0.030
−0.227
−0.347
−0.365


33%
113735.360
11509.838
−0.037
−0.276
−0.401
−0.436


66%
124988.800
11572.303
−0.046
−0.322
−0.468
−0.503


100% 
136397.920
11632.690
−0.061
−0.373
−0.538
−0.583


50%
119740.160
11552.059
−0.052
−0.282
−0.501
−0.441


 0%
102392.960
11412.718
−0.023
−0.233
−0.418
−0.355
















TABLE 8C







CONFIGURATION 7: six loading states













Percent-


Bridge
Bridge
Bridge
Bridge


age of
Weight
X_CG
09
10
11
12


cargo
(N)
(mm)
(mV/V)
(mV/V)
(mV/V)
(mV/V)
















 0%
102192.800
11423.009
−0.587
−0.001
−1.372
−0.102


33%
113735.360
11509.838
−0.663
0.006
−1.546
−0.090


66%
124988.800
11572.303
−0.749
0.021
−1.743
−0.045


100% 
136397.920
11632.690
−0.826
0.032
−1.905
−0.017


50%
119740.160
11552.059
−0.721
0.027
−1.721
−0.017


 0%
102392.960
11412.718
−0.607
0.022
−1.492
−0.010
















TABLE 8D







CONFIGURATION 7: six loading states













Percent-


Bridge
Bridge
Bridge
Bridge


age of
Weight
X_CG
13
14
15
16


cargo
(N)
(mm)
(mV/V)
(mV/V)
(mV/V)
(mV/V)
















 0%
102192.800
11423.009
−0.068
−0.201
−0.447
−0.287


33%
113735.360
11509.838
−0.078
−0.240
−0.492
−0.350


66%
124988.800
11572.303
−0.085
−0.291
−0.536
−0.428


100% 
136397.920
11632.690
−0.101
−0.339
−0.596
−0.497


50%
119740.160
11552.059
−0.072
−0.282
−0.457
−0.442


 0%
102392.960
11412.718
−0.046
−0.233
−0.387
−0.362
















TABLE 8E







CONFIGURATION 7: six loading states












Percentage
Weight
X_CG
Bridge 17
Bridge 18
Bridge 19


of cargo
(N)
(mm)
(mV/V)
(mV/V)
(mV/V)















 0%
102192.800
11423.009
1.367
0.729
0.793


33%
113735.360
11509.838
1.427
0.755
0.817


66%
124988.800
11572.303
1.436
0.762
0.831


100% 
136397.920
11632.690
1.481
0.787
0.851


50%
119740.160
11552.059
1.422
0.741
0.824


 0%
102392.960
11412.718
1.363
0.724
0.808
















TABLE 9A







CONFIGURATION 8: five loading states













Percent-


Bridge
Bridge
Bridge
Bridge


age of
Weight
X_CG
01
02
03
04


cargo
(N)
(mm)
(mV/V)
(mV/V)
(mV/V)
(mV/V)
















 0%
102392.960
11412.718
−0.594
0.026
−1.497
−0.236


50%
111555.840
11324.863
−0.589
0.019
−1.489
−0.262


100% 
119540.000
11263.699
−0.621
0.026
−1.583
−0.246


50%
111066.560
11341.331
−0.609
0.028
−1.568
−0.237


 0%
102170.560
11411.070
−0.568
0.011
−1.435
−0.266
















TABLE 9B







CONFIGURATION 8: five loading states













Percent-


Bridge
Bridge
Bridge
Bridge


age of
Weight
X_CG
05
06
07
08


cargo
(N)
(mm)
(mV/V)
(mV/V)
(mV/V)
(mV/V)
















 0%
102392.960
11412.718
−0.021
−0.232
−0.417
−0.354


50%
111555.840
11324.863
−0.034
−0.230
−0.393
−0.362


100% 
119540.000
11263.699
−0.030
−0.258
−0.361
−0.410


50%
111066.560
11341.331
−0.023
−0.251
−0.352
−0.388


 0%
102170.560
11411.070
−0.032
−0.228
−0.356
−0.362
















TABLE 9C







CONFIGURATION 8: five loading states













Percent-


Bridge
Bridge
Bridge
Bridge


age of
Weight
X_CG
09
10
11
12


cargo
(N)
(mm)
(mV/V)
(mV/V)
(mV/V)
(mV/V)
















 0%
102392.960
11412.718
−0.607
0.023
−1.492
−0.012


50%
111555.840
11324.863
−0.668
0.011
−1.577
−0.071


100% 
119540.000
11263.699
−0.732
0.018
−1.707
−0.048


50%
111066.560
11341.331
−0.673
0.020
−1.587
−0.033


 0%
102170.560
11411.070
−0.595
0.004
−1.423
−0.082
















TABLE 9D







CONFIGURATION 8: five loading states













Percent-


Bridge
Bridge
Bridge
Bridge


age of
Weight
X_CG
13
14
15
16


cargo
(N)
(mm)
(mV/V)
(mV/V)
(mV/V)
(mV/V)
















 0%
102392.960
11412.718
−0.044
−0.232
−0.384
−0.362


50%
111555.840
11324.863
−0.066
−0.259
−0.432
−0.407


100% 
119540.000
11263.699
−0.093
−0.309
−0.516
−0.462


50%
111066.560
11341.331
−0.072
−0.290
−0.466
−0.430


 0%
102170.560
11411.070
−0.057
−0.225
−0.334
−0.374
















TABLE 9E







CONFIGURATION 8: five loading states












Percentage
Weight
X_CG
Bridge 17
Bridge 18
Bridge 19


of cargo
(N)
(mm)
(mV/V)
(mV/V)
(mV/V)















 0%
102392.960
11412.718
1.376
0.724
0.819


50%
111555.840
11324.863
1.640
0.874
0.932


100% 
119540.000
11263.699
1.863
1.017
1.043


50%
111066.560
11341.331
1.649
0.918
0.970


 0%
102170.560
11411.070
1.389
0.729
0.818
















TABLE 10A







CONFIGURATION 9: five loading states













Percent-


Bridge
Bridge
Bridge
Bridge


age of
Weight
X_CG
01
02
03
04


cargo
(N)
(mm)
(mV/V)
(mV/V)
(mV/V)
(mV/V)
















 0%
102122.100
11412.858
−0.562
0.016
−1.443
−0.259


50%
147150.000
11409.637
−0.841
0.058
−2.068
−0.251


100% 
185801.400
11399.015
−1.075
0.103
−2.583
−0.189


50%
145580.400
11431.630
−0.856
0.074
−2.081
−0.187


 0%
102612.600
11400.263
−0.602
0.036
−1.505
−0.222
















TABLE 10B







CONFIGURATION 9: five loading states













Percent-


Bridge
Bridge
Bridge
Bridge


age of
Weight
X_CG
05
06
07
08


cargo
(N)
(mm)
(mV/V)
(mV/V)
(mV/V)
(mV/V)
















 0%
102122.100
11412.858
−0.028
−0.228
−0.361
−0.366


50%
147150.000
11409.637
−0.073
−0.419
−0.595
−0.650


100% 
185801.400
11399.015
−0.138
−0.605
−0.840
−0.923


50%
145580.400
11431.630
−0.049
−0.460
−0.608
−0.680


 0%
102612.600
11400.263
−0.006
−0.253
−0.369
−0.383
















TABLE 10C







CONFIGURATION 9: five loading states













Percent-


Bridge
Bridge
Bridge
Bridge


age of
Weight
X_CG
09
10
11
12


cargo
(N)
(mm)
(mV/V)
(mV/V)
(mV/V)
(mV/V)
















 0%
102122.100
11412.858
−0.593
0.009
−1.430
−0.072


50%
147150.000
11409.637
−0.876
0.049
−2.036
0.036


100% 
185801.400
11399.015
−1.111
0.094
−2.515
0.170


50%
145580.400
11431.630
−0.871
0.064
−2.025
0.105


 0%
102612.600
11400.263
−0.606
0.026
−1.486
−0.015
















TABLE 10D







CONFIGURATION 9: five loading states













Percent-


Bridge
Bridge
Bridge
Bridge


age of
Weight
X_CG
13
14
15
16


cargo
(N)
(mm)
(mV/V)
(mV/V)
(mV/V)
(mV/V)
















 0%
102122.100
11412.858
−0.049
−0.223
−0.338
−0.377


50%
147150.000
11409.637
−0.102
−0.408
−0.582
−0.629


100% 
185801.400
11399.015
−0.167
−0.580
−0.840
−0.852


50%
145580.400
11431.630
−0.084
−0.432
−0.614
−0.647


 0%
102612.600
11400.263
−0.033
−0.241
−0.378
−0.404
















TABLE 10E







CONFIGURATION 9: five loading states












Percentage
Weight
X_CG
Bridge 17
Bridge 18
Bridge 19


of cargo
(N)
(mm)
(mV/V)
(mV/V)
(mV/V)















 0%
102122.100
11412.858
1.402
0.746
0.824


50%
147150.000
11409.637
1.993
1.088
1.117


100% 
185801.400
11399.015
2.415
1.330
1.335


50%
145580.400
11431.630
1.979
1.109
1.168


 0%
102612.600
11400.263
1.422
0.786
0.891









Considering all the filling percentages, a total of 51 loading states were obtained.


Once the measurements of each sensor (4) were obtained according to each loading state i, the following calibration steps were performed:


v) Groups of four calibration sensors (Nw=4) were formed and for each group an equation system with a number of equations P=51 was solved by least squares. Also, groups of three calibration sensors (Nw=3) were formed and for each group an equation system with a number of equations P=51 was also solved by least squares. In this example, all the combinations of 3 sensors and 4 sensors were considered. For each group of sensors the weight coefficients βn for n=1, . . . , 4 and for n=1, . . . , 3 were obtained, respectively.


vi) The estimated weight (Wei) for each loading state i and the first probable error (PEw) were determined for each group of 4 and 3 sensors (Nw=4, Nw=3) using the measured responses and the obtained weight coefficients.


Below are shown several (not all) equations obtained in steps v) and vi) for some of the groups of four sensors (Nw=4) and the first probable error (PEw) obtained for said groups:






We
Eq1=(−86556.3×μbridge1)+(−73945.4×μbridge2)+(−63713.4×μbridge9)+(11350.0×μbridge17)  Equation 1:


First probable error for Equation 1: PEw=736.0






We
Eq2=(−72791.4×μbridge1)+(−75957.2×μbridge9)+(−65926.8 65926.8×μbridge10)+(11083.7×μbridge17)  Equation 2:


First probable error for Equation 2: PEw=684.3






We
Eq3=(−29698.7×μbridge3)+(−75203.4×μbridge9)+(24293.0×μbridge17)+(−25100.2×μbridge19)  Equation 3:


First probable error for Equation 3: PEw=713.8






We
Eq4=(−28387.9×μbridge3)+(−44809.5×μbridge5)+(−72570.8×μbridge9)+(11039.6×μbridge17)  Equation 4:


First probable error for Equation 4: PEw=704.4






We
Eq5=(−85744.5×μbridge1)+(−81280.3×μbridge2)+(−67149.1×μbridge9)+(18817.2×μbridge18)  Equation 5:


First probable error for Equation 5: PEw=935.3






We
Eq6=(−80364.5×μbridge1)+(−57544.1×μbridge2)+(−67523.6×μbridge9)+(20592.7×μbridge 19)  Equation 6:


First probable error for Equation 6: PEw=1036.4






We
Eq7=(−156095.0×μbridge1)+(−335755.3×μbridge2)+(181157.1×μbridge10)+(23123.7×μbridge18)  Equation 7:


First probable error for Equation 7: PEw=1749.8


wherein WeEq1 denotes an estimated weight obtained with a first combination of sensors, namely sensors 1, 2, 9 and 17, and wherein the first probable error (PEw) has been obtained for each group of sensors according to the following expression







PE
w

=


C
E








i
=
1

s








(


We
i

-

W
i


)

2



s
-

(


N
w

+
1

)









with s=51, Nw=4, and CE=0.6745, i denoting the loading state. In this particular example, the obtained weight constant Cw in the above equations is 0.


From the above results, equations 2 and 4 were selected due to their lower values for the first probable error. The sensors corresponding to said equations are sensors 1, 9, 10 and 17 for equation 2 and sensors 3, 5, 9 and 17 for equation 4.


Below are shown several (not all) equations obtained in the steps v) and vi) for some of the groups of three sensors (Nw=3) and the first probable error (PEw) obtained for said groups:






We
Eq1=(−67486.3×μbridge1)+(−141344.7×μbridge2)+(−114424.1×μbridge9)  Equation 1:


First probable error for Equation 1: PEw=2989.5






We
Eq2=(−67642.3×μbridge1)+(−74448.0×μbridge2)+(−46822.8×μbridge11)  Equation 2:


First probable error for Equation 2: PEw=3142.9






We
Eq3=(−74227.1×μbridge1)+(−71078.6×μbridge9)+(11693.7×μbridge17)  Equation 3:


First probable error for Equation 3: PEw=899.9






We
Eq4=(−42357.3×μbridge1)+(−132662.4×μbridge9)+(9092.1×μbridge13)  Equation 4:


First probable error for Equation 4: PEw=3159.1






We
Eq5=(−125553.4×μbridge1)+(−87136.1×μbridge7)+(−253449.0×μbridge10)  Equation 5:


First probable error for Equation 5: PEw=5497.9






We
Eq6=(−28267.7×μbridge3)+(−75931.5×μbridge9)+(10896.8×μbridge17)  Equation 6:


First probable error for Equation 6: PEw=856.5


wherein the first probable error (PEw) has been obtained for each group of sensors according to the following expression







PE
w

=


C
E








i
=
1

s








(


We
i

-

W
i


)

2



s
-

(


N
w

+
1

)









with s=51, Nw=3, and CE=0.6745, i denoting the loading state.


From the above results, equations 3 and 6 were selected due to their lower values for the probable error. The sensors corresponding to said equations are sensors 1, 9 and 17 for equation 3 and sensors 3, 9 and 17 for equation 6.


vii) Four first groups of sensors were selected from the plurality of calibration sensors (4) according to the first probable errors (PEw) previously determined for each group of sensors (Nw=4, Nw=3). The sensors included in said four groups are sensors 1, 3, 5, 9, 10, and 17. Said six sensors (4) are thus selected as the installed sensors usable for determining the weight (W) of the aircraft (6). Once performed the calibration steps, the determination of the weight (W) of the aircraft (6) can be performed based on the responses of the sensors according to one or several of the following expressions:






We
App1=(−72791.4×μbridge1)+(−75957.2×μbridge9)+(−65926.8×μbridge10)+(11083.7×μbridge17)  Approach 1:






We
App2=(−28387.9×μbridge3)+(−44809.5×μbridge5)+(−72570.8×μbridge9)+(11039.6×μbridge17)  Approach 2:






We
App3=(−74227.1×μbridge1)+(−71078×μbridge9)+(11693.7×μbridge17)  Approach 3:






We
App4=(−28267.7×μbridge3)+(−75931.5×μbridge9)+(10896.8×μbridge17)  Approach 4:


In a preferred embodiment at least two groups of sensors having different number of sensors are selected.


The following calibration steps were performed to obtain the X-coordinate of the centre of gravity:


viii) Groups of four calibration sensors (Nx=4) were formed and for each group an equation system with a number of equations P=51 was solved by least squares. Also, groups of three calibration sensors (Nx=3) were formed and for each group an equation system with a number of equation P=51 was also solved by least squares. For each group of sensors, the X-coordinate coefficients γn for n=1, . . . , 4, and for n=1, . . . , 3, and the constant Cx were obtained, respectively.


ix) The estimated X-coordinate (Xei) for each loading state i and the second probable error (PEx) were determined for each group of 4 and 3 sensors (Nx=4, Nx=3) using the measured responses, the obtained X-coordinate coefficients and the obtained X-constant Cx.


Below are shown several (not all) equations obtained in steps viii) and ix) for some of the groups of four sensors (Nx=4) and the second probable error (PEx) obtained for said groups:






Xe
Eq1=(−2501.7×μbridge5)+(−299.5×μbridge7)+(2695.3×μbridge10)+(−1031.6×μbridge19)+12040.8  Equation 1:


Second probable error for Equation 1: PEx=48.0






Xe
Eq2=(−498.6×μbridge5)+(−569.2×μbridge8)+(−654.5×μbridge9)+(−979.6×μbridge19)+11555.4  Equation 2:


Second probable error for Equation 2: PEx=31.3






Xe
Eq3=(−2330.6×μbridge13)+(−580.9×μbridge15)+(−953.6×μbridge17)+(616.9×μbridge19)+11934.0  Equation 3:


Second probable error for Equation 3: PEx=38.4






Xe
Eq4=(−17814.8×μbridge1)+(4114.3×μbridge7)+(12056.8×μbridge8)+(1118.8×μbridge11)+8539.4  Equation 4:


Second/probable error for Equation 4: PEx=98.4






Xe
Eq5=(−302.2×μbridge3)+(−2523.8×μbridge10)+(−393.1×μbridge11)+(−570.3×μbridge17)+11223.3  Equation 5:


Second probable error for Equation 5: PEx=16.6






Xe
Eq6=(−272.6×μbridge3)+(−807.4×μbridge9)+(−1508.3×μbridge10)+(−571.7×μbridge17)+11341.5  Equation 6:


Second probable error for Equation 6: PEx=16.7


wherein XeEq1 denotes an estimated X-coordinate obtained with a first combination of sensors, namely sensors 5, 7, 10 and 19, and wherein the second probable error (PEX) has been obtained for each group of sensors according to the following expression







PE
X

=


C
E








i
=
1

s








(


Xe
i

-

X
i


)

2



s
-

(


N
x

+
1

)









with s=51, Nw=4 and CE=0.6745, i denoting the loading state.


From the above results, the equations 5 and 6 were selected due to their lower values for the probable error. The sensors (4) corresponding to said equations are sensors 3, 10, 11 and 17 for equation 5 and sensors 3, 9, 10 and 17 for equation 4.


Below are shown several (not all) equations obtained in the steps viii) and ix) for some of the groups of three sensors (Nx=3) and the second probable error PEX obtained for said groups:






Xe
Eq1=(−274.9×μbridge3)+(−632.6×μbridge9)+(−570.8×μbridge17)+11428.6  Equation 1:


Second probable error for Equation 1: PEx=17.6






Xe
Eq2=(−6343.1×μbridge1)+(−21746.6×μbridge2)+(15172.6×μbridge5)+8477.4  Equation 2:


Second probable error for Equation 2: PEx=128.1






Xe
Eq3=(−924.9×μbridge3)+(6513.9×μbridge5)+(80.9×μbridge11)+10182.0  Equation 3:


Second probable error for Equation 3: PEx=145.2






Xe
Eq4=(−688.2×μbridge1)+(−559.8×μbridge9)+(—570.2×μbridge17)+11481.6  Equation 4:


Second probable error for Equation 4: PEx=18.2






Xe
Eq5=(−667.9×μbridge8)+(−2024.6×μbridge13)+(−575.1×μbridge17)+11872.0  Equation 5:


Second probable error for Equation 5: PEx=25.1






Xe
Eq6=(−801.3×μbridge5)+(−834.1×μbridge7)+(−1015.7×μbridge19)+11903.2  Equation 6:


Second probable error for Equation 6: PEx=48.3


wherein the second probable error PEX has been obtained for each group of sensors according to the following expression







PE
X

=


C
E








i
=
1

s








(


Xe
i

-

X
i


)

2



s
-

(


N
x

+
1

)









with s=51, Nw=3 and CE=0.6745, i denoting the loading state.


From the above results, the equations 1 and 4 were selected due to their lower values for the second probable error. The sensors (4) corresponding to said equations are sensors 3, 9 and 17 for equation 1 and sensors 1, 9 and 17 for equation 4.


x) Four second groups of sensors were selected from the plurality of calibration sensors (4) according to the second probable errors (PEX) previously determined for each group of sensors (Nw=4, Nw=3). The sensors included in said four second groups are sensors 1, 3, 9, 10, 11 and 17. Said six sensors (4) are thus selected as the installed sensors usable for determining the X-coordinate of the centre of gravity of the aircraft (6). Once performed the calibration steps, the determination of the X-coordinate of the centre of gravity of the aircraft (6) can be performed based on the responses of the sensors (4) according to one or several of the following expressions:






Xe
App1=(−302.2×μ+bridge3)+(−2523.8×μbridge10)+(−393.1×μbridge11)+(−570.3×μbridge17)+11223.3  Approach 1:






Xe
App2=(−272.6×μbridge3)+(−807.4×μbridge9)+(−1508.3×μbridge10)+(−571.7×μbridge17)+11341.5  Approach 2:






Xe
App3=(−274.9×μbridge3)+(−632.6×μbridge9)+(−570.8×μbridge17)+11428.6  Approach 3:






Xe
App4=(−688.2×μbridge1)+(−559.8×μbridge9)+(−570.2×μbridge17)+11481.6  Approach 4:


xi) In this example, the first plurality of installed sensors includes the calibration sensors in the first groups of sensors selected in step vii) and the calibration sensors in the second groups of sensors selected in step x), specifically calibration sensors 1, 3, 5, 9, 10, 11 and 17.


Once the approaches for determining the weight (W) of the aircraft (6) have been obtained in calibration step vii), it is shown below a check that the weight approaches are satisfied for all the loading states i. For this, it is determined a weight residual value (εwi) to verify that said obtained weight residual value (εwi) is small, wherein εwi=100×(Wei−Wi)/Wi.


The following tables 11A-11B to 19A-19B show the estimated weight (Wei) obtained in the calibration step vi), the subtraction of Wei minus Wi and the weight residual value (εwi) for each weight approach and each loading state i respectively. In these tables the weight residual value (εwi) has been denoted (Error W)i.









TABLE 11A







CONFIGURATION 1: six loading states








Approach 1
Approach 2












Wei
Wei − Wi
(Error
Wei
Wei − Wi
(Error


(N)
(N)
W)i (%)
(N)
(N)
W)i (%)















109948.564
−1073.516
−0.967
109336.959
−1685.121
−1.518


115644.949
−1359.691
−1.162
115131.437
−1873.203
−1.601


122811.347
513.587
0.420
122731.325
433.565
0.355


116973.050
1569.690
1.360
116380.646
977.286
0.847


110457.225
−431.415
−0.389
110429.150
−459.490
−0.414


103419.071
−108.129
−0.104
103366.964
−160.236
−0.155
















TABLE 11B







CONFIGURATION 1: six loading states








Approach 3
Approach 4












Wei
Wei − Wi
(Error
Wei
Wei − Wi
(Error


(N)
(N)
W)i (%)
(N)
(N)
W)i (%)















109332.062
−1690.018
−1.522
109444.150
−1577.930
−1.421


115260.537
−1744.103
−1.491
114976.328
−2028.312
−1.734


123181.516
883.756
0.723
122668.120
370.360
0.303


116787.422
1384.062
1.199
116719.527
1316.167
1.140


109938.178
−950.462
−0.857
111042.493
153.853
0.139


102719.440
−807.760
−0.780
104384.185
856.985
0.828
















TABLE 12A







CONFIGURATION 2: six loading states








Approach 1
Approach 2












Wei
Wei − Wi
(Error
Wei
Wei − Wi
(Error


(N)
(N)
W)i (%)
(N)
(N)
W)i (%)















107734.091
−952.789
−0.877
106902.448
−1784
−1.64


113746.857
−411.063
−0.360
113238.508
−919
−0.81


121459.891
1297.171
1.080
120901.919
739
0.62


114532.618
1798.058
1.595
114056.485
1322
1.17


108071.443
−14.957
−0.014
107036.658
−1050
−0.97


103717.322
368.042
0.356
102850.655
−499
−0.48
















TABLE 12B







CONFIGURATION 2: six loading states








Approach 3
Approach 4












Wei
Wei − Wi
(Error
Wei
Wei − Wi
(Error


(N)
(N)
W)i (%)
(N)
(N)
W)i (%)















107033.080
−1653.800
−1.522
107334.331
−1352.549
−1.244


113233.535
−924.385
−0.810
113485.094
−672.826
−0.589


120863.316
700.596
0.583
121017.248
854.528
0.711


114177.419
1442.859
1.280
114648.254
1913.694
1.698


107556.163
−530.237
−0.491
106727.600
−1358.800
−1.257


103411.940
62.660
0.061
102833.303
−515.977
−0.499
















TABLE 13A







CONFIGURATION 3: six loading states








Approach 1
Approach 2












Wei
Wei − Wi
(Error
Wei
Wei − Wi
(Error


(N)
(N)
W)i (%)
(N)
(N)
W)i (%)















107926.108
−1272.292
−1.165
108122.524
−1075.876
−0.985


114287.281
−849.199
−0.738
114883.642
−252.838
−0.220


120353.946
−31.174
−0.026
121172.235
787.115
0.654


115738.273
668.513
0.581
116404.779
1335.019
1.160


108634.846
−919.394
−0.839
108500.776
−1053.464
−0.962


103882.990
−266.930
−0.256
103579.188
−570.732
−0.548
















TABLE 13B







CONFIGURATION 3: six loading states








Approach 3
Approach 4












Wei
Wei − Wi
(Error
Wei
Wei − Wi
(Error


(N)
(N)
W)i (%)
(N)
(N)
W)i (%)















107726.375
−1472.025
−1.348
107853.734
−1344.666
−1.231


114401.579
−734.901
−0.638
114471.947
−664.533
−0.577


120779.046
393.926
0.327
120570.177
185.057
0.154


115995.272
925.512
0.804
115961.744
891.984
0.775


108414.969
−1139.271
−1.040
108664.251
−889.989
−0.812


103273.587
−876.333
−0.841
103776.872
−373.048
−0.358
















TABLE 14A







CONFIGURATION 4: five loading states








Approach 1
Approach 2












Wei
Wei − Wi
(Error
Wei
Wei − Wi
(Error


(N)
(N)
W)i (%)
(N)
(N)
W)i (%)















115654.412
273.292
0.237
115036.758
−344.362
−0.298


127465.421
297.101
0.234
127225.883
57.563
0.045


138300.782
79.182
0.057
138745.570
523.970
0.379


122859.913
1140.393
0.937
122703.194
983.674
0.808


103374.989
−597.011
−0.574
103912.090
−59.910
−0.058
















TABLE 14B







CONFIGURATION 4: five loading states








Approach 3
Approach 4












Wei
Wei − Wi
(Error
Wei
Wei − Wi
(Error


(N)
(N)
W)i (%)
(N)
(N)
W)i (%)















115125.109
−256.011
−0.222
115254.350
−126.770
−0.110


127310.001
141.681
0.111
126809.203
−359.117
−0.282


138994.816
773.216
0.559
137446.595
−775.005
−0.561


123671.147
1951.627
1.603
123005.105
1285.585
1.056


102504.076
−1467.924
−1.412
104228.222
256.222
0.246
















TABLE 15A







CONFIGURATION 5: six loading states








Approach 1
Approach 2












Wei
Wei − Wi
(Error
Wei
Wei − Wi
(Error


(N)
(N)
W)i (%)
(N)
(N)
W)i (%)















103579.156
−392.844
−0.378
104096.635
124.635
0.120


115969.240
365.720
0.316
116554.789
951.269
0.823


126714.684
146.844
0.116
126480.027
−87.813
−0.069


136460.101
951.781
0.702
136653.308
1144.988
0.845


120919.588
1201.668
1.004
121340.072
1622.152
1.355


103774.348
1403.628
1.371
103597.023
1226.303
1.198
















TABLE 15B







CONFIGURATION 5: six loading states








Approach 3
Approach 4












Wei
Wei − Wi
(Error
Wei
Wei − Wi
(Error


(N)
(N)
W)i (%)
(N)
(N)
W)i (%)















102707.850
−1264.150
−1.216
104415.173
443.173
0.426


115548.258
−55.262
−0.048
116748.198
1144.678
0.990


126882.744
314.904
0.249
126789.620
221.780
0.175


137110.072
1601.752
1.182
136523.422
1015.102
0.749


121018.638
1300.718
1.086
121509.089
1791.169
1.496


103757.408
1386.688
1.355
104777.513
2406.793
2.351
















TABLE 16A







CONFIGURATION 6: six loading states








Approach 1
Approach 2












Wei
Wei − Wi
(Error
Wei
Wei − Wi
(Error


(N)
(N)
W)i (%)
(N)
(N)
W)i (%)















103475.797
1105.077
1.079
103496.991
1126.271
1.100


114121.910
30.710
0.027
114345.336
254.136
0.223


123915.847
−1962.553
−1.559
124229.581
−1648.819
−1.310


134284.806
−734.234
−0.544
135195.347
176.307
0.131


121254.584
1069.624
0.890
121383.759
1198.799
0.997


100835.340
−1357.460
−1.328
100590.354
−1602.446
−1.568
















TABLE 16B







CONFIGURATION 6: six loading states








Approach 3
Approach 4












Wei
Wei − Wi
(Error
Wei
Wei − Wi
(Error


(N)
(N)
W)i (%)
(N)
(N)
W)i (%)















103526.181
1155
1.13
104718.952
2348
2.29


113143.473
−948
−0.83
114442.587
351
0.31


123197.998
−2680
−2.13
123357.663
−2521
−2.00


134608.684
−410
−0.30
133742.688
−1276
−0.95


121800.653
1616
1.34
121710.046
1525
1.27


99828.893
−2364
−2.31
101108.485
−1084
−1.06
















TABLE 17A







CONFIGURATION 7: six loading states








Approach 1
Approach 2












Wei
Wei − Wi
(Error
Wei
Wei − Wi
(Error


(N)
(N)
W)i (%)
(N)
(N)
W)i (%)















101659.254
−533.546
−0.522
100991.735
−1201.065
−1.175


113240.473
−494.887
−0.435
112507.813
−1227.547
−1.079


124706.957
−281.843
−0.225
124304.588
−684.212
−0.547


136006.963
−390.957
−0.287
136029.344
−368.576
−0.270


119263.349
−476.811
−0.398
119462.956
−277.204
−0.232


103000.784
607.824
0.594
102653.084
260.124
0.254
















TABLE 17B







CONFIGURATION 7: six loading states








Approach 3
Approach 4












Wei
Wei − Wi
(Error
Wei
Wei − Wi
(Error


(N)
(N)
W)i (%)
(N)
(N)
W)i (%)















100388.960
−1803.840
−1.765
101247.400
−945.400
−0.925


112208.041
−1527.319
−1.343
112675.391
−1059.969
−0.932


124364.214
−624.586
−0.500
124335.231
−653.569
−0.523


136153.196
−244.724
−0.179
135619.167
−778.753
−0.571


119389.671
−350.489
−0.293
119145.016
−595.144
−0.497


103174.073
781.113
0.763
103287.798
894.838
0.874
















TABLE 18A







CONFIGURATION 8: five loading states








Approach 1
Approach 2












Wei
Wei − Wi
(Error
Wei
Wei − Wi
(Error


(N)
(N)
W)i (%)
(N)
(N)
W)i (%)















103078.945
685.985
0.670
102678.592
285.632
0.279


111065.588
−490.252
−0.439
110375.276
−1180.564
−1.058


120266.346
726.346
0.608
119970.855
430.855
0.360


112407.615
1341.055
1.207
112587.226
1520.666
1.369


101671.578
−498.982
−0.488
100684.112
−1486.448
−1.455
















TABLE 18B







CONFIGURATION 8: five loading states








Approach 3
Approach 4












Wei
Wei − Wi
(Error
Wei
Wei − Wi
(Error


(N)
(N)
W)i (%)
(N)
(N)
W)i (%)















103326.090
933.130
0.911
103401.188
1008.228
0.985


110377.875
−1177.965
−1.056
110683.614
−872.226
−0.782


119909.858
369.858
0.309
120630.377
1090.377
0.912


112323.053
1256.493
1.131
113394.494
2327.934
2.096


100695.260
−1475.300
−1.444
100879.068
−1291.492
−1.264
















TABLE 19A







CONFIGURATION 9: five loading states








Approach 1
Approach 2












Wei
Wei − Wi
(Error
Wei
Wei − Wi
(Error


(N)
(N)
W)i (%)
(N)
(N)
W)i (%)















100897.369
−1224.731
−1.199
100730.350
−1391.750
−1.363


146615.444
−534.556
−0.363
147551.129
401.129
0.273


183209.184
−2592.216
−1.395
186796.344
994.944
0.535


146183.456
603.056
0.414
146327.336
746.936
0.513


103897.388
1284.788
1.252
102668.802
56.202
0.055
















TABLE 19B







CONFIGURATION 9: five loading states








Approach 3
Approach 4












Wei
Wei − Wi
(Error
Wei
Wei − Wi
(Error


(N)
(N)
W)i (%)
(N)
(N)
W)i (%)















100259.757
−1862.343
−1.824
101095.005
−1027.095
−1.006


147995.318
845.318
0.574
146690.950
−459.050
−0.312


187002.658
1201.258
0.647
183691.178
−2110.222
−1.136


148589.621
3009.221
2.067
146526.219
945.819
0.650


104386.737
1774.137
1.729
104052.650
1440.050
1.403









Furthermore, once the approaches for determining the X-coordinate of the centre of gravity of the aircraft (6) have been obtained in the calibration step x), it is shown below a check that the approaches for determining the X-coordinate are satisfactory for all the loading states i. For this, it is determined an error (Error X)i to verify that said obtained error is small, wherein (Error X)i=100×(Xei−Xi)/2561, wherein such error is calculated according to the overage chord length of the aircraft (6) that is 2561 mm in this example.


The following tables 20A-20B to 28A-28B show the estimate weight Xei obtained in the calibration step ix), the subtraction of Xei minus Xi and the (Error X)i for each weight approach and each loading state i respectively.









TABLE 20A







CONFIGURATION 1: six loading states








Approach 1
Approach 2












Xei
Xei − Xi
(Error X)i
Xei
Xei − Xi
(Error X)i


(mm)
(mm)
(%)
(mm)
(mm)
(%)















11136.066
18.720
0.731
11139.023
21.677
0.846


10907.772
71.777
2.803
10908.658
72.663
2.837


10593.342
−21.314
−0.832
10594.190
−20.466
−0.799


10935.783
−12.510
−0.488
10932.669
−15.624
−0.610


11186.558
22.143
0.865
11188.943
24.527
0.958


11394.864
−29.844
−1.165
11394.086
−30.622
−1.196
















TABLE 20B







CONFIGURATION 1: six loading states








Approach 3
Approach 4












Xei
Xei − Xi
(Error X)i
Xei
Xei − Xi
(Error X)i


(mm)
(mm)
(%)
(mm)
(mm)
(%)















11132.637
15.291
0.597
11135.413
18.067
0.705


10901.449
65.454
2.556
10904.648
68.654
2.681


10595.984
−18.672
−0.729
10596.849
−17.806
−0.695


10929.102
−19.192
−0.749
10929.632
−18.662
−0.729


11184.980
20.565
0.803
11177.962
13.546
0.529


11392.846
−31.862
−1.244
11384.282
−40.426
−1.579
















TABLE 21A







CONFIGURATION 2: six loading states








Approach 1
Approach 2












Xei
Xei − Xi
(Error X)i
Xei
Xei − Xi
(Error X)i


(mm)
(mm)
(%)
(mm)
(mm)
(%)















11507.900
−1.989
−0.078
11505.384
−4.505
−0.176


11562.908
−27.465
−1.072
11558.962
−31.411
−1.227


11656.214
−17.723
−0.692
11660.544
−13.393
−0.523


11578.985
−0.193
−0.008
11585.886
6.708
0.262


11469.759
−39.501
−1.542
11468.388
−40.871
−1.596


11408.230
−12.020
−0.469
11406.466
−13.785
−0.538
















TABLE 21B







CONFIGURATION 2: six loading states








Approach 3
Approach 4












Xei
Xei-Xi
(Error X)i
Xei
Xei-Xi
(Error X)i


(mm)
(mm)
(%)
(mm)
(mm)
(%)















11501.730
−8.158
−0.319
11505.024
−4.864
−0.190


11555.748
−34.625
−1.352
11556.988
−33.385
−1.304


11651.306
−22.631
−0.884
11650.408
−23.528
−0.919


11586.326
7.148
0.279
11585.039
5.860
0.229


11468.124
−41.135
−1.606
11482.133
−27.127
−1.059


11413.547
−6.703
−0.262
11426.723
6.472
0.253
















TABLE 22A







CONFIGURATION 3: six loading states








Approach 1
Approach 2












Xei
Xei-Xi
(Error X)i
Xei
Xei-Xi
(Error X)i


(mm)
(mm)
(%)
(mm)
(mm)
(%)















11193.740
30.860
1.205
11185.753
22.873
0.893


10956.661
32.100
1.253
10950.132
25.570
0.998


10726.427
−12.539
−0.490
10726.941
−12.025
−0.470


10925.569
−14.431
−0.564
10920.049
−19.951
−0.779


11161.407
4.242
0.166
11159.736
2.571
0.100


11370.676
−24.878
−0.971
11362.273
−33.280
−1.300
















TABLE 22B







CONFIGURATION 3: six loading states








Approach 3
Approach 4












Xei
Xei-Xi
(Error X)i
Xei
Xei-Xi
(Error X)i


(mm)
(mm)
(%)
(mm)
(mm)
(%)















11189.436
26.555
1.037
11193.178
30.298
1.183


10953.622
29.061
1.135
10954.260
29.699
1.160


10730.560
−8.406
−0.328
10730.425
−8.541
−0.333


10925.409
−14.591
−0.570
10926.265
−13.735
−0.536


11162.337
5.173
0.202
11164.471
7.307
0.285


11361.707
−33.846
−1.322
11364.262
−31.291
−1.222
















TABLE 23A







CONFIGURATION 4: five loading states








Approach 1
Approach 2












Xei
Xei-Xi
(Error X)i
Xei
Xei-Xi
(Error X)i


(mm)
(mm)
(%)
(mm)
(mm)
(%)















11340.745
−23.500
−0.918
11341.664
−22.581
−0.882


11321.121
−23.155
−0.904
11317.413
−26.862
−1.049


11320.782
−0.068
−0.003
11313.757
−7.092
−0.277


11389.357
7.887
0.308
11386.535
5.065
0.198


11425.339
10.595
0.414
11424.925
10.181
0.398
















TABLE 23B







CONFIGURATION 4: five loading states








Approach 3
Approach 4












Xei
Xei-Xi
(Error X)i
Xei
Xei-Xi
(Error X)i


(mm)
(mm)
(%)
(mm)
(mm)
(%)















11335.512
−28.733
−1.122
11336.197
−28.048
−1.095


11311.654
−32.622
−1.274
11312.930
−31.346
−1.224


11319.840
−1.009
−0.039
11326.127
5.277
0.206


11406.231
24.761
0.967
11411.066
29.596
1.156


11420.113
5.369
0.210
11411.170
−3.574
−0.140
















TABLE 24A







CONFIGURATION 5: five loading states








Approach 1
Approach 2












Xei
Xei-Xi
(Error X)i
Xei
Xei-Xi
(Error X)i


(mm)
(mm)
(%)
(mm)
(mm)
(%)















11423.878
9.134
0.357
11423.419
8.675
0.339


11312.203
10.277
0.401
11310.417
8.492
0.332


11218.488
−5.632
−0.220
11223.579
−0.541
−0.021


11120.367
−30.519
−1.192
11126.869
−24.017
−0.938


11246.176
−21.611
−0.844
11248.162
−19.625
−0.766


11409.889
−14.952
−0.584
11414.976
−9.865
−0.385
















TABLE 24B







CONFIGURATION 5: five loading states








Approach 3
Approach 4












Xei
Xei-Xi
(Error X)i
Xei
Xei-Xi
(Error X)i


(mm)
(mm)
(%)
(mm)
(mm)
(%)















11418.442
3.698
0.144
11409.566
−5.178
−0.202


11305.773
3.848
0.150
11295.794
−6.132
−0.239


11223.608
−0.512
−0.020
11220.994
−3.125
−0.122


11130.099
−20.788
−0.812
11127.436
−23.451
−0.916


11251.082
−16.705
−0.652
11245.428
−22.358
−0.873


11429.092
4.251
0.166
11426.465
1.624
0.063
















TABLE 25A







CONFIGURATION 6: six loading states








Approach 1
Approach 2












Xei
Xei-Xi
(Error X)i
Xei
Xei-Xi
(Error X)i


(mm)
(mm)
(%)
(mm)
(mm)
(%)















11408.630
−16.211
−0.633
11413.504
−11.337
−0.443


11565.346
21.332
0.833
11566.895
22.881
0.893


11575.284
−55.552
−2.169
11568.640
−62.195
−2.429


11736.004
43.580
1.702
11732.613
40.189
1.569


11604.991
18.791
0.734
11608.822
22.622
0.883


11425.362
2.353
0.092
11433.574
10.565
0.413
















TABLE 25B







CONFIGURATION 6: six loading states








Approach 3
Approach 4












Xei
Xei-Xi
(Error X)i
Xei
Xei-Xi
(Error X)i


(mm)
(mm)
(%)
(mm)
(mm)
(%)















11429.305
4.464
0.174
11425.099
0.258
0.010


11554.890
10.876
0.425
11545.602
1.588
0.062


11557.115
−73.720
−2.879
11554.240
−76.595
−2.991


11738.766
46.342
1.810
11741.482
49.058
1.916


11626.788
40.588
1.585
11627.411
41.212
1.609


11427.484
4.475
0.175
11424.106
1.096
0.043
















TABLE 26A







CONFIGURATION 7: six loading states








Approach 1
Approach 2












Xei
Xei-Xi
(Error X)i
Xei
Xei-Xi
(Error X)i


(mm)
(mm)
(%)
(mm)
(mm)
(%)















11432.179
9.170
0.358
11438.390
15.381
0.601


11502.177
−7.661
−0.299
11503.147
−6.691
−0.261


11590.410
18.107
0.707
11593.340
21.037
0.821


11653.545
20.855
0.814
11660.901
28.211
1.102


11543.481
−8.578
−0.335
11541.608
−10.452
−0.408


11429.627
16.909
0.660
11427.587
14.870
0.581
















TABLE 26B







CONFIGURATION 7: six loading states








Approach 3
Approach 4












Xei
Xei-Xi
(Error X)i
Xei
Xei-Xi
(Error X)i


(mm)
(mm)
(%)
(mm)
(mm)
(%)















11425.917
2.907
0.114
11426.385
3.376
0.132


11488.403
−21.435
−0.837
11487.706
−22.132
−0.864


11586.601
14.298
0.558
11585.772
13.469
0.526


11657.733
25.042
0.978
11656.894
24.204
0.945


11548.565
−3.494
−0.136
11551.930
−0.130
−0.005


11446.350
33.632
1.313
11452.938
40.220
1.570
















TABLE 27A







CONFIGURATION 8: five loading states








Approach 1
Approach 2












Xei
Xei-Xi
(Error X)i
Xei
Xei-Xi
(Error X)i


(mm)
(mm)
(%)
(mm)
(mm)
(%)















11419.388
6.670
0.260
11418.375
5.657
0.221


11330.120
5.257
0.205
11332.622
7.759
0.303


11264.790
1.091
0.043
11271.879
8.180
0.319


11330.075
−11.256
−0.440
11339.475
−1.856
−0.072


11414.069
2.999
0.117
11413.009
1.939
0.076
















TABLE 27B







CONFIGURATION 8: five loading states








Approach 3
Approach 4












Xei
Xei-Xi
(Error X)i
Xei
Xei-Xi
(Error X)i


(mm)
(mm)
(%)
(mm)
(mm)
(%)















11438.654
25.937
1.013
11445.525
32.807
1.281


11324.350
−0.513
−0.020
11325.688
0.825
0.032


11263.387
−0.313
−0.012
11256.374
−7.325
−0.286


11344.093
2.762
0.108
11337.119
−4.212
−0.164


11406.599
−4.471
−0.175
11413.501
2.431
0.095
















TABLE 28A







CONFIGURATION 9: five loading states








Approach 1
Approach 2












Xei
Xei-Xi
(Error X)i
Xei
Xei-Xi
(Error X)i


(mm)
(mm)
(%)
(mm)
(mm)
(%)















11399.206
−13.652
−0.533
11398.602
−14.256
−0.557


11388.292
−21.345
−0.833
11399.287
−10.350
−0.404


11377.973
−21.042
−0.822
11420.301
21.286
0.831


11358.023
−73.608
−2.874
11384.173
−47.458
−1.853


11385.643
−14.620
−0.571
11388.926
−11.337
−0.443
















TABLE 28B







CONFIGURATION 9: five loading states








Approach 3
Approach 4












Xei
Xei-Xi
(Error X)i
Xei
Xei-Xi
(Error X)i


(mm)
(mm)
(%)
(mm)
(mm)
(%)















11400.112
−12.745
−0.498
11400.839
−12.019
−0.469


11413.602
3.965
0.155
11414.256
4.619
0.180


11462.954
63.939
2.497
11466.204
67.189
2.624


11422.004
−9.627
−0.376
11429.763
−1.868
−0.073


11413.964
13.701
0.535
11424.239
23.976
0.936









Therefore, this example performed on an aircraft shows that the method of the present invention provides valid estimations for the values of the weight and the X-coordinate of the position of the centre of gravity of the aircraft. Also, it has been shown that the method allows obtaining several mathematical expressions for determining the values of the weight and of the position of the centre of gravity. The fact that different mathematical expressions provide very similar results shows that the results are accurate and reliable.


While at least one exemplary embodiment of the present invention(s) is disclosed herein, it should be understood that modifications, substitutions and alternatives may be apparent to one of ordinary skill in the art and can be made without departing from the scope of this disclosure. This disclosure is intended to cover any adaptations or variations of the exemplary embodiment(s). In addition, in this disclosure, the terms “comprise” or “comprising” do not exclude other elements or steps, the terms “a” or “one” do not exclude a plural number, and the term “or” means either or both. Furthermore, characteristics or steps which have been described may also be used in combination with other characteristics or steps and in any order unless the disclosure or context suggests otherwise. This disclosure hereby incorporates by reference the complete disclosure of any patent or application from which it claims benefit or priority.

Claims
  • 1. A method for determining the weight (W) and/or at least a first coordinate (X) of the position of the centre of gravity of a structure, the method comprising the following steps: a) measuring the responses (μ) of a first plurality of installed sensors located in sensitive load points of the structure, andb) determining the weight (W) and/or the first coordinate (X) of the position of the centre of gravity of the structure based on the measured responses (μ), as:
  • 2. The method according to claim 1, further comprising determining a second coordinate (Y) of the position of the centre of gravity of the structure, wherein the second coordinate (Y) of the position of the centre of gravity of the structure is determined based on the measured responses (μ) of the first plurality of installed sensors, as:
  • 3. The method according to claim 2, further comprising the following steps: c) measuring the responses (μ′) of a second plurality of installed sensors located in at least one lifting surface (h) of the structure, andd) determining a corrected weight of the structure (Wcorrected) and/or a corrected position (Xcorrected, Ycorrected) of the centre of gravity of the structure as:
  • 4. The method according to claim 1, comprising, after steps vii), x) and/or x′), verifying for at least one loading state i that the first group of calibration sensors fulfills that a weight residual value (εwi) is lower than a first predefined threshold,wherein
  • 5. The method according to claim 1, wherein at each iteration of step ii) the structure is loaded with the probe weights Wpi in steps of a predetermined percentage of a maximum admissible weight of the structure.
  • 6. The method according to claim 1, wherein the structure is an on ground vehicle.
  • 7. The method according to claim 6, wherein at least one sensitive load point is located in at least one bulkhead of the vehicle.
  • 8. The method according to claim 6, wherein the on ground vehicle is an aircraft.
  • 9. The method according to claim 8, wherein at least one sensitive load point is located in at least one landing gear bulkhead.
  • 10. The method according to claim 8, wherein at least one sensitive load point is located in the nose landing gear pin.
  • 11. The method according to claim 8, wherein at least one sensitive load point is located in the main landing gear pin.
  • 12. The method according to claim 1, wherein the first coordinate (Xi) of the position of the centre of gravity and/or the weight (Wi) is measured in calibration step ii) using scales and/or jacks with load cells, preferably, using three hydraulic jacks.
  • 13. The method according to claim 1, wherein at step vii) a plurality of first groups of calibration sensors are selected, which fulfil that the first probable error is minimum, and/or whereinat step x) a plurality of second groups of calibration sensors are selected, which fulfil that the second probable error is minimum, and/or whereinat step x′) a plurality of third groups of calibration sensors are selected, which fulfil that the third probable error is minimum.
  • 14. A system configured for determining the weight (W) and/or at least a first coordinate of the position of the centre of gravity of a structure, comprising: a plurality of installed sensors located on the structure,a data acquisition module configured to register data from the plurality of installed sensors located on the structure, anda data processor configured to:determine the weight (W) and/or the first coordinate (X) of the position of the centre of gravity of the structure based on the measured responses (μ), as:
  • 15. A vehicle comprising the system according to claim 14.
Priority Claims (1)
Number Date Country Kind
18382018.2 Jan 2018 EP regional