Various basic research was performed on efficient load measurement estimation techniques for aircraft structure analysis. An overview is presented of the load measurement problem. Two basic equivalent approaches to load measurement evaluations were considered. Under approach 1, the load values are modeled as depending linearly on the measured values. Under approach 2, the measured values depend linearly on the load values. By using the modern Singular Value Decomposition method, it was shown that under all conditions of the number of loads and number of gages, approach 1 is equivalent to approach 2. By using the conventional normal equation (linear regression) approach, approach 1 is only valid when the number of loads is equal to or greater than the number of gages, while approach 2 is the reverse. Furthermore, except for the case of the number of loads equals the number of gages, the load prediction formulas under the two approaches are not equivalent

Balakrishnan, A. V.

Yao, Kung

English

NASA Technical Reports Server

Final Report 
to 
NASA-Ames Research Center 
Moffett Field, CA 94034 
Contract No. NAG 2-433 
January 1,1987 - March 31,1988 
on 
Efficient Load Measurements Using 
Singular Value Decomposition 
(LISA-CR- 1846 14) E P F I C l E l T  LCAD N89-14795 
MEASUBEMEYTS C S I Y G  SIEGULaP YAIUE 
C E C G E F O S I T I C I  F i a r l  Recart, 1 J a n .  1987 - 31 
tar- 1988 ( C a l i f c r n i a  U n i r , )  18 p CSCL 12A Unclas  
G3164 0 I83373 
Pr incip a1 Investigator 
Professor Kung Yao 
Co-Principal Investigator 
Professor A.V. Balakrishnan 
NASA TechnicaI Officer 
Mr. Karl Anderson 
Laboratory for Flight Systems Research 
University of California 
Los Angeles, CA 90024- 1594 
c 
https://ntrs.nasa.gov/search.jsp?R=19890005424 2020-03-20T05:01:16+00:00Z
TABLE OF CONTENTS 
1. Introduction ........................................... 1 
2. Two Equivalent Approaches to Load Measurements 
Evaluations 4 ........................................... 
3. Multi-Stage Load Estimation Technique ................. 12 
4. Conclusions ........................................... 14 
5. References ............................................ 16 
1. Introduction 
There are two fundamental operations in the load measurement 
problem on an aircraft structure. In the first part, we perform 
a system identification by measuring the responses of the strain 
gages mounted on different locations of the structure from a 
series of known applied load at various specified load points on 
the structure during the calibration stage on the ground. 
second part, by using some characterization of the system 
obtained in the first part, we can predict the actual equivalent 
load value and location from the gage measurements during a 
flight. Various known successful approaches and results have 
been reported in the past on the load measurement problem [l-33. 
In the 
There are two fundamental and intuitively equally justifiable 
linear approaches (arbitrarily denoted as Approach 1 and Approach 
2 in Section 2 ) applicable to the load measurement problem. In 
Approach 1, we model the load value matrix L as dependent 
linearly on the influence coefficient value matrix l$ measured by 
the gages. In Approach 2, we model as dependent linearly on &. 
In general these matrices are rectangular, thus it is not immedi- 
ately clear that these two approaches are equivalent. 
cally, all the work in [l-31 were based on that of Approach 1. 
In Section 2, we shall show that these two approaches are indeed 
equivalent in all cases, and can be proved by the use of the 
modern Singular Value Decomposition (SVD) technique. On the 
other hand, if we only use the more conventional and previously 
Histori- 
-1- 
used normal equation technique [l-31 (also called the linear 
regression technique), then the limitation of this analytical 
technique can only show the validity of Approach 1 when the 
number of gages n is less or equal to the number of loads m. In 
addition, by using the normal equation approach, it is only 
possible to handle these problems with n greater or equal to m 
from the Approach 2 point of view. There are several theoreti- 
cal, practical, and computational consequences to these observa- 
tions. 
At the most basic level of understanding, of course, it is 
theoretically important to know the equivalency of these two 
seemingly different approaches that yield the desired result. 
the practical algorithmic operational level, the inadmissibility 
of having the number of gages n greater than the number of 
applied loads m in the calibration stage in Approach 1 is not 
fatal. However, as we shall show in Section 3, for the multi- 
stage load estimation technique (which can yield extremely accu- 
rate load predictions), we will always use more gages than the 
number of loads in the prediction stage. Conventional normal 
equation approach (i.e., Approach 1) is not possible since a 
crucially needed matrix involved in the processing is singular. 
At 
When the data from the gages are quite linearly independent, 
then there is no significant numerical difference between the use 
of the SVD technique or the normal equation technique. However, 
f o r  highly dependent data, there can be significant advantages 
for the SVD technique. Detailed numerical computations based on 
-2- 
practical observed gage measurements and load values are neces- 
sary to verify their differences. The crucial point is that in 
all cases, the SVD approach is always computationally more costly 
as well as numerically more stable. For dimensions encountered 
in the load measurement problems, the additional computational 
cost of the SVD approach is not significant to be of concern. 
-3- 
2. Two Equivalent Approaches to Load Measurement Evaluations 
Now, we consider two possible basic equivalent approaches for 
load measurements. These approaches are denoted as Approach 1 
and Approach 2. 
2.1 Approach 1 - Linear Dependency of Load Values on Gage Val- 
ues 
Consider 
(1) 4 = 141, 421 431, 
the m x 3 load matrix, where 
- (2) 21 - -Ls = [Sir S2,*.*r S,I', 
is the shear vector at m load locations (Xi, Yi), 
B m l ' r  - (3) 42 - LB = [Blr B ~ I * = * I  
is the bending moment vector with its i-th component located at 
yi given by 
( 4  1 Bi = Siyi i=l,...,m, 
and 
( 5 )  4 3  = LT = [Tj,*-*i Tml', 
is the torque vector with its i-th component located at Xi given 
by 
( 6 )  Ti = SiXi, i=l,...,m. 
Let the m x n influence coefficient matrix M denote the 
response of the n gages to the m loads in the calibration pro- 
cess. Specifically, let 
-4- 
( 7 )  l! = [ulr***r~nl = 
where each Uit i=l,...,n, represents the normalized response of 
the i-th gages to the m loads. 
cient matrix b consists of 
Let the n x 3 dependency coeffi- 
( 8 )  h = [bp b2, b,l, 
where 
(9) hi = E bj, i=l,2,3, 
or in matrix form 
(10) & =  Mb. 
For i=1, the n x 1 vector hi yields the dependency of 
L,, the shear vector, to the linear combinations of the influence 
coefficient vectors {u l ,  ..., un) of in (7). Similarly, for i = 
2 and 3, b2 and b3 are related to the bending moment vector &2 = 
LB and the torque moment vector L3 =,&, respectively. 
= 
In the calibration process, the matrix as well as 41, &2, 
and L3 are available. 
matrix from the %orma1 equation" point of view as 
(11) B+ = @?'MI -lM/. - 
We note, in (11) is defined if and only if m 2 n and all columns 
of M are linearly independent. In particular, if n > m, then in 
(11) is not defined. 
Define the pseudo-inverse of as a nxm 
I 
I 
Then (9) becomes , 
(12) Bi = a+ Li i=1,2,3. 
By using the notation of in (8) and & in (1) I (12) can be 
written in matrix form as 
-5- 
In the prediction process, we observe one 1 x n dimensional - 
gage measurement vector (corresponding to the first row vector 
of H in From the predicted 1 x 3  load vector is 
given by 
(14) L = [St B, T] = H b = &f H-l L = M (H'H)'l H'J. 
The first component of L yields the predicted shear, 
- - - -  - - - 
A 
- - 
(15) s = M+ Ll, 
the second and third components of 
moment B = Sy and predicted torque T = Sx, 
yield the predicted bending - -- - -- 
- - -  - 
(16) B = S y = M 142, 
(17) T = S X = H M+ L3. 
- 
- 
From (15) and (16), we can solve for y as 
(18) - H M+ L2 
M H+ Id1 
- M M+ L3 
M M+ L1 
Y =  I - 
- 
and x as - 
(19) x =  - 
Thus, (15), (18) and (19) represent the predicted equivalent net 
shear, bending moment location, and torque location of the 
applied load that yielded the measured gage vector 
- 
using the 
normal equation approach. 
Now consider the use of the SVD technique via Approach 1. 
Consider a general form of the SVD of the matrix with rank p 
-6- 
S min(m,n) as given by 
(20) = UM ZM V'M I 
where UM is a m x m orthogonal matrix, p ' ~  is a n x n orthogonal 
matrix, and EM is a m x n matrix where the upper left p x p sec- 
tion is a diagonal matrix with positive singular values (S.V.) 
denoted by u1 2 a2 2 ... 2 up > 0, and the remaining section is 
the zeroth matrix. 
m matrix denoted by E++, and from the SVD point of view is then 
given by 
The pseudo-inverse of the matrix H, is a n x 
(21) @+ = IM ZM+ UM', 
where EM+ is a n x m matrix where the upper left p x p section is 
a diagonal matrix with reciprocal of the positive singular values 
of E, and the remaining section is the zeroth matrix. Then by 
using (21) in (10) , we have 
(22) - M++ L = . 
We note, (22) corresponds to (13) in the calibration stage of the 
previously considered normal equation technique. Then in the 
prediction stage, we have 
It is most interesting to note, that the predicted load vector in 
(23) based on the SVD technique has the same form as the pre- 
dicted load vector in (14) based on the normal equation tech- 
nique. Indeed, when m 2 n (i.e., the number of loads is greater or 
equal to the number of gages), and when the gage measurements are 
quite linearly independent, the pseudo-inverse given by E++ in 
-7- 
(21) is equal to the pseudo-inverse given by & in (11). 
those cases, either the conventional normal equation or the SVD 
methods will yield the same predicted load values. Of course, when 
the measurement values are quite linearly dependent, then the SVD 
approach will be better from the numerical stability point of view. 
As mentioned earlier, when n > m, the normal equation method is 
not applicable for Approach 1 since & in (11) is not defined. 
However, the results of (21)-(23) under the SVD method for 
Approach 1 are valid in all cases including n > m. 
Thus, in 
2.2 Approach 2 - Linear Dependency of Gage Values on Load Values 
From a physical cause and effect point of view, it makes 
sense that the responses of the first gage to the m loads are 
given by, 
In (24), we are describing the gage measurement Uli as a 
linear combination of sicll + siyic12 + S ~ X ~ C I ~ ,  which depends 
linearly on the shear, bending moment and torque. In general, 
for all n gages, we have, 
- (25) H = [ul,-*-r U,] - & [ ~ l r - * * ~ n l  
- - & c, 
where the 3 x n dependency matrix C is denoted by, 
( 2 6 )  C = [GI, ,Gnl* 
-8- 
In the calibration process, as before, and I! are available. In 
the prediction process, as before, we have a measured H, given 
- 
from (25) as - 
(27) M = L C .  
In order to solve for the 1 x 3 predicted load vector - - - -  
( 2 8 )  L = [ s, B, 2. I ,  
we need to use (25) and (27). First, consider the use of the 
normal equation technique. Let the psudeo-inverse of h be 
denoted by 
(29) - L+ = (L'L)'lL'. 
By using (29) in ( 2 5 ) ,  we have 
( 3 0 )  - c = L+ M. 
Substituting (30) in (27), we obtain 
(31) - M = Z L + M .  
Now, in order to solve for L, we need to multiply both sides of 
- 
- 
(31) by M' from the right and try to take inverse. Unfortu- 
nately, since M is a m x n matrix, if m > n (which is often the 
case), then (m') is singular and then does not exist. 
Thus, in this case, it is not Dossible to use the normal equation 
technique based on Approach 2. When n 2 m, then direct solution 
of (31) yields - 
(32) L = M M  ' (- MM,)-1 - L . 
By comparing (32) of Approach 2 (valid for n 2 m) to (14) of 
Approach 1 (valid for m 2 n), we see that only when m = n are 
these two approaches yield identical results. 
-9- 
Now, consider solving for in (25) by using the pseudo- 
inverse of & based on the SVD representation of &. Specifically, 
consider the SVD of L as given by 
(33) - L = v, c, YL8, 
where IJL is a m x m orthogonal matrix, YL is a 3 x 3 orthogonal 
matrix, and 2, is a m x 3 matrix of the form of 
where the top 3x3 sub-matrix is a diagonal matrix of singular 
values aL1 2 aL2 2 aL3 > 0 and the remaining (m-3) x3 sub-matrix 
is an all zero matrix. Then (25) becomes 
(35) - M = VL CL XL' C. 
By using (30) and (33) in (35), we have 
(37) 
0 l/a3 : " I .  . 0 
In particular, we note 
(38) HL+ c, = x3 
By using (36) in (27), we have - - 
(39) I!! = L YL CL+ UL' H 
Now consider the use of the SVD representation of M given by (20) 
-10- 
Direct solution of in (42) yields - 
(43) L = i & p + L .  
By comparing (43) to that of (23), we see that by using the SVD 
technique, both Approaches 1 and 2 yield the same predicted load 
vector, for all cases of m and n (i.e., m 2 n or n > m). But as 
discussed earlier, by using the normal equation technique, 
Approach 1 is applicable only for m 2 n, while Approach 2 is 
applicable only for n 2 m. Furthermore, the predicted & in (14) 
under Approach 1 and (32) under Approach 2 are equivalent only 
for m = n. 
-11- 
3. Multi-Stage Load Estimation Technique (MUSLET) 
Now, we consider a new proposed MUlti-Stage Load EsTimation 
technique (which we shall denote as MUSLET) for a more accurate 
prediction of the load value. As considered in Sections 1 and 2, 
the load matrix & in equation (1) and the gage measurement matrix 
in (7) are needed to estimate the dependency matrix B in 
Approach 1 and the dependency matrix c in Approach 2. 
case, if we actually know more accurately the true value of the 
load location, then we need to use only those load locations that 
are close (or closest) to the true load location in the calibra- 
tion process to yield a more appropriate 
yield a more accurate predicted load value and location. 
analogous argument to this technique is that in numerical analy- 
sis, only the relevant near by x-y points to the desired x loca- 
tion are used to perform a numerical interpolation. The set of 
far away x-y points may introduce more errors rather than provide 
the desired smoothing effects in the interpolation process. Ini- 
tial numerical experiments based on the HWTS Loads Calibration 
Data of 8/29/76 showed that suppose the 10th calibration load is 
removed from the actual system identification calibration pro- 
cess. 
measurements, the initial predicted bending moment y location was 
given as 80.3 and the predicted torque x location was given as 
60.2. The true locations were 81.3 and 60 respectively. How- 
In either 
or E which in turn can 
An 
By using the remaining 17 load conditions and 15 gage 
-12- 
ever, from this initial prediction result, we can use only these 
calibration points with similar or close y values to 80.3. In 
this example, we can use only loads labeled as (7;8;9;11} with 
calibration load values situated at y = 81.3. The newly predicted 
y value now becomes 81.29992. Similarly, the closest x values in 
the calibration stage are given by loads labeled as (3;4;5;9;11; 
15;17;18). 
We note the original predicted values have errors of have errors 
of 1.23% in y and 0.34% in x, while the iterated second stage 
prediction has errors of only 0.0001% in y and 0.19% in x. Thus, 
this MUSLET technique appears to be able to yield significant 
improvement in load predictions. Many variations of the above 
simple proposed iteration schemes are possible and will be 
further investigated. 
The newly predicted x value is now given by 59.84374. 
-13- 
4. Conclusions 
In this research period from January 1987 to March 1988, we 
have performed various basic research on efficient load measure- 
ment estimation techniques for aircraft structure analysis. In 
Section 1, we presented an over view of the load measurement 
problem. In Section 2, we considered two basic equivalent 
approaches to load measurement evaluations. Under Approach 1, 
the load values are modeled as depending linearly on the measured 
values. Under Approach 2, the measured values depend linearly on 
the load values. By using the modern SVD method, we showed that 
under all conditions of the number of loads m and number of gages 
n, Approach 1 is equivalent to Approach 2. By using the conven- 
tional normal equation (or linear regression) approach, Approach 
1 is only valid for m 2 n (which is commonly encountered case), 
while Approach 2 is valid only for n 2 m. Furthermore, except 
for the case of m = n, the load prediction formulas under the two 
approaches are not equivalent. 
In many practical flight testing situations, we may not be 
able to use as many gages as those used in the calibration pro- 
cess. Thus, there is much interest in finding the most efficient 
set of n gages to be used for predictions. 
formed exhaustive tests on various subsets of the available gages 
and associated load measurements. Preliminary investigations 
show this approach also to be meaningful. 
based on DIStributed Load EsTimation (DISLET) performs a series 
We have also per- 
Another approach 
-14- 
expansion of the fight measured gage data vector in terms of the 
calibration measured gage data vectors. 
expansion indicates the amount of load distributions during 
flight relative to the m calibration load points. 
analytical and computational investigations are under study. 
Another multi-dimensional cluster analysis relevant in the effi- 
cient reduction of gages, also based on the use of SVD technique, 
has also been under consideration and will be reported on in the 
future. 
Coefficients of this 
Further 
Finally, in [ 4 ] ,  we have published a basic research paper on 
the effective use of singular values in estimation problems for 
data contaminated by Gaussian noise. The results in [ 4 ]  yielded 
tighter performance bounds as compared to all previously known 
results in this field. 
-15- 
' .  . .  
5. References 
1. T.H. Skopinski, W.S. Aiken Jr., and W.B. Huston, I8Calibration 
of Strain-Gage Installations in Aircraft Structures for Measure- 
ment of Flight Loads," NACA Rept. 1178, 1954. 
2. M.H. Tang and R.G. Sheldon, "A Modified T-Value Method for 
Selection of Strain Gages for Measuring Loads on a Low Aspect 
Ratio Wing," NASA Tech. Paper-1748, 1980. 
3. J.M. Jenkins, A.E. Kuhl, and A.L. Carter, "Strain Gage Cali- 
bration of a Complex Wing,1t Journal of Aircraft, Vol. 14, Dec. 
4. K. Konstantinides and K. Yao, "Statistical Analysis of Effective 
Singular Values in Matrix Rank Determination," IEEE Trans. on 
Acoustics, Speech, and Signal Processing,May 1988, pp.757-763. 
1977, pp. 1192-1196. 
-16- 


Efficient load measurements using singular value decomposition

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