Patent Description:
further comprising at least one data processing system being prepared to:
(d) receiving measurements from said accelerometers at the measurement-points when periodically moving the structure within said first frequency range by said at least one shaker.

To conventionally analyze the motions of mechanical structures it is known that the structure is instrumented with accelerometers. The orientations and positions of the latter need to be measured and documented, but this is a time-consuming, tedious, and error-prone manual process. Each of the accelerometers applied to such a structure may be at nontrivial angles, may be hard to reach but needs accurate measurements of coordinates and orientations. A normal setup may comprise several <NUM> of such sensors.

Whereas more established analysis techniques like Modal Analysis do not require high precision for this geometry information, more modern analysis techniques (FBS, TPA) have far higher demands on accuracy.

At present, techniques are under investigation to use measurements from the sensors themselves to compute orientations and positions from rigid-body motions of the mechanical structure. But these techniques suffer from several problems:.

From Dmitri Tcherniak [(<NPL>] a procedure is known to cope with uncertainties in the alignment of the sensors.

The <CIT> and <NPL> deal respectively at least in part with the idea to deduce the orientations and positions from preliminary measurements.

The above referenced procedure and algorithms from Dmitri Tcherniak may be summarized as follows:
Step <NUM>. : Structure excitation. In this step, Tcherniak et al. suggest moving the structure long enough in all <NUM> rigid body motion patterns. Tcherniak et al. suggest keeping the excitation at low frequencies and/or to filter the acquired data through a low-pass filter, to suppress motion patterns where the structure is deformed by the excitation.

Problems in this process step are identified to be that:.

Step <NUM>. : Extracting rotations and translations. In this step, each time sample is turned into an immediate rotational/translational acceleration pair. Each of these pairs will be used as separate nonlinear equations in the subsequent step.

One problem in this process step is identified to be that there are often points in time with immediate accelerations close to <NUM>, yielding equations that consist of little more than noise.

Step <NUM>. : Computing the orientations and positions of non-reference accelerometers is done by using a generic constrained nonlinear optimizer (SQP) [see: https://en. wikipe-dia. org/wiki/Sequential_quadratic_programming] to solve the set of nonlinear equations from the previous step. This approach is unstable for the following reasons:.

As an example, the exact solution may be <NUM> degrees. The gradient of the SQP method will probably point in the direction of <NUM> (= <NUM>) degrees then since this is closest to the current <NUM>. But due to the constraints, the iterations cannot break through the artificial wall to reach <NUM>. The iterations will also not follow the gradient in the opposite direction. This results in an end error on this angle of at least <NUM> degrees.

It is one objective of the invention to improve the above explained measurement problem.

It is another objective of the invention to turn the problem into a fixed number of steps preferably involving linear algebra only.

It is another objective of the invention to yielding a unique globally optimal solution, and with a clear separation between identifying the quality of the input data and the identification of positions and orientations.

It is another objective of the invention to cope with inaccuracies in the sensors, too.

Based on the prior art described above and the problems associated, the invention is based on the task of improving the. method of calibrating the accelerometers' positions and orientations in such systems.

The object of the invention is achieved by the independent claims. The dependent claims describe advantageous developments and modifications of the invention.

To enable accurate and quick calibration the invention proposes a system and method of the incipiently defined type wherein said at least one data processing system is further prepared to calibrate the accelerometers' positions and orientations by the following steps:.

One preferred embodiment may provide that step (e) further comprises:.

This feature will enable fast and systematic processing of singular values and vectors using standard algorithm respectively modules resulting in quick determination of rigid body motion patterns.

An alternative preferred embodiment may provide that step (e) further comprises:.

This feature will speed up processing of singular values and vectors.

According to another preferred embodiment step (e) may further comprise a sub-step as:
(e3) comparing the resulting singular values with a predefined singular value lowest threshold and repeating steps (c), (d), (e) in case that the six biggest resulting singular values are smaller than the singular value lowest threshold.

The comparison with said singular-value-lowest-threshold will ensure sufficient excitation of the structure preferably by the shaker to enable accurate determination of the rigid body motion. The repetition of steps (c), (d), (e) may be done in several loops to avoid on the one hand vibration excitation involving deformation (modes of the structure) and on the other hand to low singular values possibly leading to inaccuracies. The starting excitation may therefore be done with a low frequency and/or low amplitude and may be stepwise raised to reach the singular-value-lowest-threshold. All these steps are computer-implemented.

According to the invention step (f) further comprises:.

Another preferred embodiment provides that said solver is designed to apply the least-squares-method or the Moore-Penrose-pseudo-inverse-method.

Another preferred embodiment provides that said system or method according to the above explanations further comprises step (g) in turn including the following steps:.

Generally, single value SVL decomposition SVD is a well-defined mathematical concept and is a factorization of a real or complex matrix that generalizes the eigendecomposition.

The singular value decomposition of an m x n complex matrix Ar is a factorization of the form UΣVT where U is a m x m complex unitary matrix, Σ is an m x n rectangular diagonal matrix with non-negative real numbers on the diagonal, and VT is a n × n complex unitary matrix. The diagonal entries of Σ are known as the singular values of Ar. The number of nonzero singular values is equal to the rank of Ar. The columns of U and the columns of VT are called the left-singular vectors and right-singular vectors of Ar, respectively, more details may be found under https://en. org/wiki/Singular_value_decomposition.

Another preferred embodiment provides that step (g) may still further comprise:.

Another preferred embodiment provides that said first frequency range is such a low-frequency that the accelerometers are sufficiently accurate and none of the structural vibration modes of the structure are excited.

Embodiments of the invention are herein described, by way of example only, with reference to the accompanying drawings, of which:.

The illustration in the drawings is in schematic form. It is noted that in different figures, similar or identical elements may be provided with the same reference signs.

<FIG> shows a simplified system setup according to the invention. <FIG> shows a system SYS for analyzing motions of the mechanical structure STR. The system comprises accelerometers ACC which are attached to said mechanical structure STR at measurement-points MPI. Some of these accelerometers ACC are provided as standard accelerometers SAC and others are provided as reference accelerometers RAC. The reference accelerometers RAC are preferably attached to measurement points MPI being remote to each other on the structure STR, preferably on the outer edge(s) of said structure STR. At least three reference accelerometers RAC are to be provided to the mechanical structure STR and are to be arranged not-collinearly.

The accelerometers ACC are connected to an interface IFC which is connected to a data processing system DPS. The connection between the accelerometers ACC and the interface IFC is illustrated simplified wherein the interface IFC indeed is connected to every single accelerometers ACC, receives the data generated, normally changes the data format (e.g., an Outlook/digital) and provides the changed measurement data to the data processing system DPS. The data processing system may comprise one or several computers CMP - here on several computers CMP of a network WWB comprising a cloud CLD. The software installed on the computer(s) CMP is a computer program product CPP which, when executed on at least one computer CMP, enables the user to carry out the method according to the invention respectively to calibrate the accelerometers' ACC positions and orientations. Further, the system SYS and in particular the data processing system DPS enables to carry out the analysis of the motions of the mechanical structure STR preferably after the calibration. Each of the - maybe several hundred - accelerometers ACC may be arranged at nontrivial angles, may be hard to reach but conventionally needs accurate measurements of position-coordinates and orientations.

<FIG> shows a simplified flow diagram illustrating the method according to the invention being. This method illustrated is for calibrating a system in particular for calibrating a system SYS as illustrated in <FIG>. In step (a) accelerometers ACC are provided as standard accelerometers SAC to measurement-points MPI of said mechanical structure STR. In STEP (b) at least three accelerometers ACC are provided as reference accelerometers RAC. During STEP (c) at least one shaker SHK is attached to said mechanical structure STR for moving the structure STR periodically within a first frequency range FR1. The order of the STEPS (a), (b), (c) is arbitrary and a person with ordinary skill in the art understands that these steps can be carried out in any sequence.

As soon as this basic hardware setup of STEPS (a), (b), (c) is done STEP (d) of receiving measurements from said accelerometers ACC at the measurement points MPI when periodically moving the structure STR within said first frequency range FR1 by said shaker SHK may be performed. These measurements are received by said data processing system DPS.

STEPS (e), (f), (g) predominantly refer to the preparation of said data processing system DPS to calibrate the accelerometers' ACC positions and orientations and therefore include algorithmic features requiring the following definitions:.

Before explaining STEPS (e), (f), (g) some introductory remarks are given below.

For infinitesimal rotations around a rotation axis - small rotations - the rotational motion xi may be approximated as a cross product with the Rodrigues vector that describes the rotation (e.g. https://en. org/wiki/Rodrigues%27_rotation_formula). This may be written as a rotation around a Rodrigues vector with unit direction k and magnitude θ expressed as a matrix multiplication with matrix R defined as: <MAT>.

A point pi may rotate to Rpi such that rotational motion xi can be computed as: <MAT>.

Substitution of eq. (<NUM>) yields: <MAT>.

Differentiating twice results in the accelerations: <MAT>.

For small rotations θ this can be approximated (e.g. using a Taylor series of sin and cos truncated to <NUM>st degree) as: <MAT>.

This is the sum of respectively a tangential and radial (= centripetal) acceleration component. The radial acceleration will vanish to zero for small rotations because of the square in θ̇<NUM>. This may be illustrated by assuming a sinusoidal rotation with infinitesimally small amplitude: θ = θ<NUM>sinφ with φ<NUM> very small. Then: <MAT>.

For low frequencies and small θ<NUM>, θ̇<NUM> will vanish against θ̈. With the radial component vanished, the approximation may be written as: <MAT>.

With the direction of rotation assumed constant, this is equivalent to: <MAT>.

This approximation (<NUM>) may be used for further thoughts underlying the invention.

STEP (e) of the method according to the invention provides determining rigid body motions RBM from said measurements of said at least three reference accelerometers RAC. This may be done as follows. STEP (e) may be done stepwise to extract <NUM> independent rigid body motions from the time data of the measurements. These steps may include STEPS (e1), (e2) and preferred STEP (e3).

With all acceleration samples re-arranged in the matrix Ar, Singular Value Decomposition may be applied to the measurements, yielding: <MAT>.

Since all possible motions are a linear combination of rigid body rotations and translations, and there are only <NUM> degrees of freedom, the rank of Ar cannot be higher than <NUM>.

If the rank is less than <NUM> not all rigid body motions were excited by said shaker SHK. This would require repetition of STEPS (c), (d) to obtain sufficient excitation. This loop may be repeated with stepwise increasing the frequency or amplitude of the excitation as long until sufficient excitation is guaranteed.

The right singular vectors of Ar belong to the biggest six singular values and therefore hold six independent acceleration patterns. These singular values and their vectors dominating all others represent six independent rigid body motions. The first six columns of V may be called V'.

Since V' is a 3ni × <NUM> matrix column k may be rearranged into a <NUM> × ni matrix Ark comprising the relative vector accelerations of all triax accelerometers of the k'th experiment. Alternatively, each group of <NUM> rows (3i. 3i+<NUM>) may be sliced into a <NUM> × <NUM> matrix Ari holding the relative vector accelerations of all experiments of the i'th triax accelerometer.

These columns are independent, and they are also orthogonal to each other, which provides an additional filtering effect against intrusion of deformation modes. Indeed, deformation modes tend to alternate more (i.e. deformations will change direction over space more frequently) than rigid body motions. Since this effect is only approximative it cannot be successfully applied in completely suppressing deformation modes.

The matrix Ar may be rather sizable and single value decomposition SVD may take a long time.

As an alternative to this approach a preferred embodiment provides that:
STEP (e1) may be carried out as: providing a reference-measurement-matrix RMM composed of said measurements from said at least three reference accelerometers RAC, and that STEP (e2) may therefore performed as: determining from said reference-measurement-matrix RMM said rigid body motions RBM as singular values SVL and the accompanying singular vectors SVC by single value decomposition SVD.

Accordingly, the single value decomposition SVD may be performed using only the measurements of said reference accelerometers RAC. When <NUM> reference triax accelerometers are used, the maximal rank of Ar may be reduced to <NUM>, resulting in a huge speed-up for the single value decomposition SVD. This alternative may be used in case of low probability of deformation since this alternative does not benefit from the filtering effect due to orthogonality in the same magnitude.

As a STEP (e1) a slice ArA of matrix Ar may be composed, containing only the columns of reference accelerometers RAC.

For easier notation, it is assumed that the reference accelerometers RAC precede the standard accelerometers SAC in the columns of A. Hence:
Ar = [ArA | ArB] will be a nj × 3ni matrix.

As a STEP (e2) the single value decomposition SVD on ArA may be applied, yielding UAΣAVAT =ArA. Assuming that nj is bigger than <NUM> niA:.

Preferably - as a STEP (e3) - a check may be performed whether ΣA has <NUM> sufficiently dominant singular values SVL. As long as this is not the case, additional excitation may be performed (repeating STEPS (c), (d), (e) and potentially involving additional shakers SHK).

Continuing STEP (e2) six dominant singular values SVL are determined, the six left singular vectors with the biggest singular values SVL from UA into U'A as columns may be extracted. U' A is a nj × <NUM> matrix.

Multiplying the left Ar with U'AT, yielding <MAT>.

Here, six different linear combinations of the time samples have been taken. As the rigid body motions RBM are linear at an infinitesimal scale, any linear combination is a valid rigid body motion RBM, too. U'ATAr has rank <NUM>.

Having determined from said measurements (maybe only by said at least three reference accelerometers RAC) rigid body motions RBM the next STEP (f) is performed. This step may be carried out stepwise (f1), (f2), (f3), (f4).

From the previous STEP (e) six independent rigid body motion RBM accelerations <MAT> for all measurement points MPI were obtained, including reference accelerometers RAC. Since the orientations of reference accelerometers RAC are known from STEP (e), the absolute accelerations of all reference accelerometers RAC may be obtained: <MAT>.

STEP (f1) provides decomposing the six rigid body motions RBM applying an infinitesimal rotation model into six infinitesimal rigid body translation-rotation pairs RTR by solving a resulting set of linear equations using a solver LSQ.

Here, the invention makes use of the fact that for translational accelerations, the respective acceleration is identical for all measurement points MPI.

The acceleration of a measurement points MPI, here point pi in the k'th rigid body motion RBM is a combination of a small position-dependent rotational acceleration and a small position-independent translational acceleration: <MAT>.

This may be a basis to compute the unknowns <MAT> and <MAT> for all <NUM> independent rigid body motion RBM.

Equation (<NUM>) is linear in both <MAT> and <MAT> and may possibly be overdetermined, yielding to: <MAT>.

This may be expressed as a matrix equation: <MAT>.

Which can be solved by said solver LSQ using least-squares or Moore-Penrose pseudo-inverse: <MAT>.

The rank of Bk needs to be six to have a unique solution. In order for the solution to be unique, the three measurement points MPI of said reference accelerometers RAC must be arranged non-collinearly (STEP (b)). More reference measurement points MPI than three are possible and may increase the accuracy but are not strictly necessary to guarantee a unique solution.

At that stage, when the rotation axes for each measurement and frequency line are known, the positions and orientations may be derived.

Using the same equations as above, the unknowns and the known variables are now exchanged, meaning that pi and Ei are now unknown.

In matrix notation, combining for all <NUM> possible values of k, this may be expressed as: <MAT>.

In STEP (f2) said six infinitesimal rigid body translation-rotation pairs RTR are linearly recombined into three pure translations PTR. Relative accelerations may be translated into absolute accelerations through the unknown transformation matrix Ei: <MAT>.

Combining with equation (<NUM>) results in: <MAT>.

Z may be a matrix holding the null space of R. Z has at least <NUM> independent columns since R is a 3x6 matrix.

If all rotational degrees of freedom are available in the measurements of said accelerometers ACC Z has not more than <NUM> columns in the null space - so Z is a regular 3x3 matrix for which: <MAT>.

The null space Z may be computed using a full single value decomposition SVD on R and taking the <NUM> right singular vectors with the lowest singular values SVL.

Post-multiplying eq. <NUM> with Z yields: <MAT>.

Due to associativity of the matrix product, the first right hand term vanishes due to RZ = <NUM>: <MAT>.

It is important to notice that TZ represents the <NUM> pure translations extracted from the <NUM> rigid body motions RBM.

In STEP (f3) orientations of reference accelerometers ACC are derived from three pure translations PTR.

During the accelerometer ACC measurement <NUM> independent rigid body motions RBM are identified. This will result in three independent pure translations PTR. Moreover, since Ei is (close to) an orthogonal matrix, it is rank <NUM>. If the rank of Ei and TZ is both <NUM>, then <MAT> will have rank <NUM>. Hence, matrix inversion is possible, yielding: <MAT>.

According to a preferred embodiment the fact that the matrix Ei is orthogonal may be used as a measure for the quality of the data. The data processing system DPS may calculate an orthogonality index being a measure for data quality DQU and output this information, e.g. display this information through a human machine interface HMI to an operator. The resultant matrix could easily be forced to be orthogonal, but it may also be used to check orientation respectively orthogonality of any transducer mount.

The lengths of the vectors in Ei might not equal <NUM> because of improper calibration. According to another preferred embodiment, the accuracy of said rigid body motions RBM may be improved by using estimated Ei to transform the deflections from the local coordinate system to the global coordinate system. This may also make calibration of the standard accelerometers SAC superfluous.

STEP (f4) provides deriving positions of accelerometers ACC from said <NUM> rigid body motions RBM.

Once Ei is known, the absolute accelerations in all standard accelerometer SAC measurement points MPI may be determined as: <MAT>.

This may be converted into an over-determined set of linear equations in pi: <MAT>.

pi may not be computed from a single equation because <MAT> is singular. Having at least two independent Rodrigues rotation vectors, sufficient information is available because the null spaces of each <MAT> correspond to the Rodrigues vectors themselves, and they are not collinear but are independent. The solution can be computed using least-squares or Moore-Penrose pseudo-inverse.

STEP (g) may be carried out stepwise as STEPS (g1), (g2), (g3) and preferably further including at least one of the STEPS (g4), (g5), (g6).

In STEP (g1) a standard-measurement-matrix SMM composed of said measurements from said standard accelerometers SAC may be provided.

(g2) applying the linear combinations of said single value SVL decomposition SVD of STEP (e2) to standard-measurement-matrix SMM to obtain compatibility with said singular vectors SVC.

(g3) linearly recombining said standard-measurement-matrix SMM to match the rigid body motions, using the left singular vectors of the single value SVL decomposition SVD as a transformation matrix.

Claim 1:
System (SYS) for analyzing the motions of a mechanical structure (STR), comprising:
(a) accelerometers (ACC) provided as standard accelerometers (SAC) to measurement-points (MPI) of said mechanical structure (STR),
(b) at least three accelerometers (ACC) provided as reference accelerometers (RAC) to measurement-points (MPI) of said mechanical structure (STR),
(c) at least one shaker (SHK) being attached to said mechanical structure (STR) for moving the structure (STR) periodically within a first frequency range (FR1),
further comprising at least one data processing system (DPS) being configured to:
(d) receiving measurements from said accelerometers (ACC) at the measurement-points (MPI) when periodically moving the structure (STR) within said first frequency range (FR1) by said at least one shaker (SHK),
said system (SYS) being characterized by said at least one data processing system (DPS) further is configured to calibrate the accelerometers' (ACC) positions and orientations by the following steps:
(e) determining from said measurements of said at least three reference accelerometers (RAC) rigid body motions (RBM),
(f) determining positions and orientations of accelerometers (ACC) from said rigid body motions (RBM), wherein step (f), further comprises:
(f1) decomposing the <NUM> rigid body motions (RBM) applying an infinitesimal rotation model into six infinitesimal rigid body translation-rotation pairs (RTR) by solving a resulting set of linear equations using a solver (LSQ),
(f2) linearly recombining said six infinitesimal rigid body translation-rotation pairs (RTR) into three pure translations (PTR),
(f3) deriving orientations of accelerometers (ACC) from said three pure translations (PTR),
(f4) deriving positions of accelerometers (ACC) from said <NUM> rigid body motions (RBM).