Table Of Contentr~\
POOR MAN'S EXPLANATION OF KALMAN FILTERING
OR
HOW I STOPPED WORRYING AND LEARNED
TO LOVE MATRIX INVERSION
-By
Roger M. du Plessis
June 1967
V*II .^j North American Rockwell
Electronics Group
3370 Mirsloma Avenue / Ar,3h».rn. Californi* 92803
o
CONTENTS
Page
Introduction and Summary I
The Elements of Kalman Filtering - 3
Least Squares and Maximum Likelihood Estimation 3
Least Squares Method 4
Maximum Likelihood Method and the Kalman Concept 4
Basic Maximum Likelihood Technique 4
The Iterative Nature of the Kalman Filter " 6
3
Effect of Erroneous Statistics for Resistor Example 7
Application of Kalman Filter to Dynamic Systems 9
Some Salient General Facts Regarding Kalman Filters • 17
Appendix 21
ILLUSTRATIONS
Figure Page
I Comparison of Least Squares vs Maximum Likelihood Er'imatibns of the
Value of a Resistor with 1 Ohm RMS Tolerance, Using an Q'hmmeter
with 3 Ohms RMS Random Error
Comparison of the Effects of Erroneous Statistics on Least Squares and
Maximum Likelihood Methods for Estimating Resistor Value - 8
3 Simp''"-cd Schulcr Loop _ 9
4 Comparison of Straight Position Resets with Kalman Resets 10
o
INTRODUCTION AND SUMMARY
Dr. R. E. Kalman introduced his concept of opti- are explained by means of a very simple example,
mum estimation in 1960. Since that time, his tech- entirely free from matrix notation and complicated
nique has proved to be a powerful and practical dynamics. A second illustrative example is taken
tool. The approach is particularly well suited for from the field of inertial navigation, since this is
optimizing the performance of modern terrestrial one of the most fruitful areas of application for
and space navigation systems. Kalman estimation. This slightly more complicated
second example is used to describe more of the
Many people not directly involved in systems
concepts and terminology of Kalman filters (e.g.,
analysis have heard about Kalman-filtering and
"state vector," and so forth}. At the same time, it
have expressed an interest in learning more about
permits a closer look at the actual operations in-
it. Although attempts have been made to supply
volved in applying Kalman theory. Not all readers
such people with simple, intuitive explanations of
will have the familiarity with matrix notation or
Kalman filtering, it is this writer's opinion that
the patience to unravel evrry single equation in the
none of these explanations has been completely suc-
discussion of the second example. The writer feels,
cessful. Almost without exception, they have tended
however, that the gist of this discussion will be
to become enmeshed in the jargon and state-space
grasped by the majority of those readers who are
notation of the "cult." But matrix notation, regard-
analytically inclined. *
less of how useful and efficient it may be, does not
- assist the uninitiated reader to understand the con- The main body of the paper concludes v/ith
cepts. qualitative comments concerning the practical ad-
Surprisingly, in spite of all the obscure-looking vantages and disadvantages of the Kaiman tech-
mathematics (the most impenetrable of which can nique, and the difficulties of applying it in actual
be found in Dr. Kalman's original paper), Kalman systems. Again, navigation systems are particularly
filtering is a fairly direct and simple concept. The cited.
full-blown matrix equations can be made intelligi-
Finally, an appendix deF.nes a criterion for opti-
ble by being presented in a way that appeals to the
mum estimation and derives the optimum estima-
intuition, and a statistical error analysis or actual
tion equations used In the examples. As is the case
system performance data can be viewed with an
with the main body of the paper, the appendix is
intuitive understanding of the results.
intended to be tutorial and to provide, for those
The present paper presents a straightforward ex- readers who can rccaiJ some of their probability
planation of Knlman filtering that will be accessible theory, a thumbnail derivation of the estimation
to virtually all readers. The most salient features eouations.
o
THE ELEMENTS OF KALMAN FILTERING
LEAST SQUARES AND MAXIMUM LIKELI- Suppose, further, that from this bucket we select
HOOD ESTIMATION a single resistor and that we want to estimate its
actual value. We shall use an ohmmeter with an
In understanding the concept of Kalman filter-
accuracy of 3 ohms RMS random error on each
ing, it is helpful to examine the differences between
reading. The error in each reading is statistically
two basic techniques of estimation. These are the
uncorrelated with any other. The ohmmeter has
so-called "least squares" and the "maximum likeli-
no systemic error or biases.
hood" methods.* Toward this end, let us consider
the following example of an estimation problem: This simple example has most of the elements of
a complete Kalman filter, except that (1) it is a
Suppose that'we have a bucketful of nominally
one-dimensional problem and hence docs not re-
100-ohrn resistors, accurate to 1 percent RMS.
quire matrix notation, mnd (2) there are no dy-
namics in the problem. If the concepts in the
example are understood, t'heir extension to the gen-
*These terms involve some scmnntrc difficulties which are
briefly mentioned in the appendix. eral case is not difficult.
o
1 1
RMS ERROR IN
ESTIMATING RESISTANCE,
OHMS
3.0
•LEAST SQUARES (SIMPLE AVERAGE IN THIS CASE)
2.0
1.0
MAXIMUM LIKELIHOOD
NUMBER OF OHMMETER READINGS
Figure 1. Comparison of Least Squares vs Maximum
Likelihood Estimations of the Value of a Resistor with
'1 Ohm RMS Tolerance, Using an Ohmmeter with 3 Maximum Likelihood Method and the Kalman
Ohms RMS Random Error Concept
BASIC MAXIMUM LIKELIHOOD TECHNIQUE
Maximum likelihood, estimation introduces the
new factor of "weighting" into the estimation
process, in order to mai>= allowances for the accu-
racy of the measuring instrument (in our example,
an ohmmeter), as well as the accuracy of the thing
Least Squares Method
being measured (in our -example, a resistor). The
Least squares estimation involves fitting a curve precision of the meter imay be known, possibly,
to the available data in such a way that the sum from calibration data; ttcit of the resistor may have
'square of the residuals to the data points is mini- been learned from vendor's specifications, sampled
mized. For our bucketful-of-resistors example, the or 100-pcrcent-inspection data, or engineering judg-
curve to be fitted to the ohmmeter readings is sim- ment based on experience or analysis of similar
ply a constant. It is well known that the least resistors. We thus have a-vailable to us statistics con-
squares curve fit in this case is simply the average cerning both the qucntby being estimated and the
of the ohmmeter readings. The RMS error in the errors in measurement.
estimate is 3/V N, where N is the number of read-
Now, suppose that, m our buckctful-of-resistors
ings. Until at least ore ohmmeter reading is ob-
example, a reading of P-5 ohms is indicated on the
tained, no curve cr.- ~t and no estimate can be
face of the ohmmeter. In is extremely unlikely that
made. In general, no >:ast squares cui-vc fit is pos-
this quantity can actually be the exact value of the
sible until there are a: least as many equations as
resistance. But a reading of "95" for the nominally
there arc unknowns.
"100-ohm" resistor conic: well be expected with this
The RMS error vs the number of readings is meter, in view of its "-ohm ft MS error. In maxi-
shown in the upper curve of Figure 1. mum likelihood cstirrtarion, mcicr readings arc
Q
weighted with a factor-which takes into account - as -the condition in which RMS -error -in the esti-
the relative quality of the meter, and obtains an mate is minimized. T.it-r acy other statistically op-
optimum estimate of the actual resistance. timum policy, whether in estimation or at the
-blackjack table_the maximum likelihood method
' In explaining how the weighting factor "works," r
does not yield a uniformly excellent result each and
a, few definitions are necessary.
every time. But b "plays tine odds" in such a way
Let that, on the average, the- best possible estimate is
x = value of the resistance for the particular obtained.
-I - resistor in question (the quantity to be esti-
We should note that there is a fundamental dif-
mated)
ference between the two er-timation methods— least
e = error in ohmmeter reading
squares and maximum Ilkeliihood— in the latter's in-
M = ohmmeter scale factor, i.e., output/input
troduction of the concept of "weighting," which
(in the bucketful-of-resistors example,
takes into account trie accuracies of both the mea-
-„'__. M = l) __ suring instrument and the'-object being measured.
yr= ohmmeter reading -•—-•.-__», ___
Least squares, as" we hive-resn, works by the fitting
= MX -ft ~
» of data to a curve by mirrrmizmg the squared dif-
The maximum likelihood weighting factor is ference between the Apr* a.nd the curve fit— but a
given by good data fit does not necessarily imply a good esti-
mate. Maximum likelihood obtains an ^'optimum"
estimate by minimizing- die mean squared error in
the estimate— but with, baic data there can be a
where
poor fit.
crj = the variance or mean square deviation of
the resistance being estimated The maximum likalihocd estimate (see the ap-
«rj — mean square error in the ohmmeter reading pendix for' a derivation) ss the -expected value of
the quantity beine estimzned, given the measure-
As described in more detail below, the raw ohm-
ment data. With no measurements, the expected
meter reading y is weighted by the factor b in order
value 't>f the resistance of eur particular resistor is
to get an estimate of the true value of the resist-
nominal, or 100 ohms. We have no basis for stat-
ance x. While the derivation of this factor is rele-
ing that it is above or below nominal, and our best
gated to the appendix, one can see that b has at
estimate is, indeed, 100 ah-'ms. There is an error in
least one desirable attribute: it takes into account
this estimate due to the d'.sviation of the resistor
the relative accuracy of the ohmmeter measure-
from nominal ; the RMS value of the error in esti-
ments. Assuming the scale factor of the ohmmeter
mation is the resistor's "m.-.ird" accuracy— 1 ohm.
to be 1, if the ohmmeter were very accurate relative
to the precision of the resistors in the bucket (that Now suppose a reading 1:3 taken of the resistance
is, <rf much less than v\), then the weighting factor using the ohmmeter. This sreading will be weighted
would approach 1. In other words, we simply would by the factor
use the raw value of the ohmmeter reading in esti-
mating the resistance. If, on the other hand, the .
i +
resistors were very precise, say 0.1 percent from
nominal, and the ohmmeter were very inaccurate
and unreliable, the b would approach zero. In this ~ 1* • 1* + 3=
instance, we would reject the ohmmeter readings
= 0.1
and simply use the nominal value of the resistance
- The estimate, based oc Erst reading y, is given by
as our estimate of x. t
. x, = 100 + b(y - 10T-)
But we have more to go on here than just intui- r
tion. The value of b is derived on the bask that where
it renders the estimate optimum. Because of the x = the estimate oT tlhe resistance x based on
}
random errors in the resistors and ohmmeter, "op- one ohmrmcttr rr.ading. (The A symbol is
timum" is taken in a statistical sense and is defined frequently reserve: cl for statistical estimates. )
o
Suppose, for example, the reading on the ohm- THE ITERATIVE NATURE OF THE KALMAN FILTER
meter was 95 ohms. This value of resistance would
-Now suppose that a second measurement is ob-
be highly unusual from a bucket of 1-percent RMS
tained on the ohmmeter. It is at this point that the
fesbtors. It would represent a 50 deviation from
second fundamental concept of the Kalman Filter—
normal. However, the maximum likelihood esti-
•its recursive or iterative nature—rears its head.* It
mate for this particular situation would be
'is this concept, brought forth by Dr. R. E. Kalman
x, = 100 + 0.1(95- 100) in 1960, that has rendered maximum likelihood
estimation a feasible and practical technique for
= 99.5 ohms
use on small digital computers in real-time appli-
a result which is not at all unreasonable. Least
cation. As a result of our first reading, we have an
squares, on the other hand, would minimixe the
estimate of x, which we call x,, and can calculate
sum square of the residuals by estimating 95 ohms.
the mean square error in the estimate, or 0.9 ohm=.
This gives a good data fit, but most probably is not
-Now, when the second reading is made, we recog-
a very accurate estimate.
•nize similarity between this situation and the pre-
Naturally, there are some errors in maximum vious situation, when our estimate was a nominal
likelihood estimates. The formula for the mean 100 ohms with a mean square error of 1 ohm1'. We
square error in a maximum likelihood estimate is proceed to weight the data as before, but realizing
given by that we have a better handle on the resistor than
' we did before. The new weighting factor is
M«r« '
Mt• trj« +t ol•> '
b =
' = (1 -bM)cr;
1 (0.9)
This equation is derived in the appendix, but it
1(0.9)+3s
can.be observed that at least in. one regard . the
formula seems reasonable. Assuming again that = 0.091
M = 1,- and •supposing that we have a relatively
which is slightly less than the previous weighting
precise. measurement (that is. al is much less than
facior of OTI. This is because, as a result of the first
«r;), then the RMS error reduces to <r, or the RMS
t ohmmeter reading, we have a more accurate esti-
error in the estimate is simply the RMS error in
mate of x, and we will tend to weight the second
the -measurement. At the other extreme, suppose
ohmmeter reading more lightly. Using this new
the measurement was very inaccurate relative to the
value of b. the estimate based on the two ohmmeter
tolerance. on the resistors; then the RMS error in
readings is
the estimate is <r,, the RMS tolerance of the re-
sistance. In this extreme case, we have rejected the x, = xt + b(y=-Mx1)
data (b = 0), and we are no smarter after the where b = 0.091. In this equation, (y» — MX,) is
measurement than we were before. the difference between the actual reading y and the
:
value of the ideal reading ivhich would have been
For this example, the mean square error in the
obtained if x, had been perfect. It is very important
estimate is given by
to note that the form of this equation is identical
to the form of the equation used to make the esti-
= (1 -0.1)P mate wkh the first reading. It is not necessary to
save the first reading y,; the only information re-
= 0.9 ohm'
aJ = 0.95 ohm RMS
i •This is not to imply thnt least squares cannot be made
iterative—ii can. In the example given here,'a new least
The RMS error has been reduced from 1 ohm with
squares curve fit can be dorivrd by updating the old
no measurements, to 0.95 ohm after one measure-
average to include the new data point:
ment. This is not much of an improvement, and
I
— y-
is a reflection of the relative inaccuracy of the ohm- n
meter. The result can be generalized.
quired to make the second optimum estimate is the more accurate than the least squares method. Of
first optimum estimate, the computed mean square course, the bucketful-of-resistors example which we
error in the estimate tr* , and the mean square ohm- •have cited here is a highly artificial one. It is in-
meter error v\. Again,'a mean square error in the tended only to serve to bring out the basic concepts
estimate can be computed thus: of Kalman estimation, and no general quantitative
conclusions should be inferred from the information
presented in Figure 1.
"=(1 -0.091) (0.9)
Effect of Erroneous Statistics for Resistor Example
= 0.82 ohm=
or 0.90 ohm RMS. In the preceding discussion, it was noted that
three quantities had to be known before the cyclic
It can be seen that a recursive or iterative proce-
application of the Kalman filter equations could
dure is developing. Each time a reading is taken
be initiated. For the bucketfu!-of-resistors example,
a new weighting factor h is computed. It is applied
1. The expected value of the resistance x prior
against the data to yield a new estimate x, and a
to the taking of any readings—namely, 100
mean square error in the estimate is computed. The
latter quantity is necessary in computing b for the ohms
next reading, but past data need never be saved. 2. The mean square deviation about nominal
The equations involved are summarized as shown —namely, 1 ohm3
here. 3. The mean square error in the ohmmeter
1. Compute weighting coefficient: readings—namely, 3 ohms:
Mo* In any actual situation, these three statistics are
A
only approximately known, and arc often, at best,
only estimates or educated guesses. Let us now take
a look at the consequences of using vague or errone-
2. Use weighting coefficient to make new esti- ous statistics in the estimation process.
mate:
A A 'A For simplicity, we will assume that only the sec-
X = x -, -h bnfy,, — Mx .)
B n nt ond of the three quantities listed above is in error;
3. Update mean square error in estimation:
that is, the RMS deviation of the resistors which
«"! =(1 ~~ b,,M)<7,i is assumed in the filter equations is 1 ohm, but the
rt * •• • r\
'« *n-i
To start this cyclic process, three quantities are actual RMS deviation of the resistors in the bucket
needed: 1) the expected value of the item to be is some value other than 1 ohm. The effect on the
measured, prior to the taking of any readings; errors in estimation for various values of the true
2) the mean square deviation about nominal value RMS deviation will now be examined. .
of the item; and 3) the mean square error in the
For illustration, let us assume that the true RMS'
measuring instrument's indication.
deviation of the resistors is 2 ohms. Using the
The importance of the Kalman process' recursive formula previously presented for b, the optimum
nature cannot he overcmphasixcd. Review of the weighting coefficient is
equations presented above shows that no past dr a
need ever be stored. Each estimation is identical b =
Mr«rj 4-
in procedure to all those which took place before
it, but each has a new weighting factor b computed
to take into account the sum total effect of ail the 1s • 2--r3-
previous estimates.
= 0.308
If our resistor example is carried out for succes- This weighting coefficient will not actually be ap-
sive measurements ami csti . uions and the RMS is plied, however. Under the erroneous assumption
•imputed, the lesults point out (Fisjurc 1) that th.it the bucket of n-Mviors tolerance is 1 percent
maximum likelihood Kalm.in estimation is alwnvs RMS. rather than the actual 2 percent RMS, it was
calculated in the previous section of this paper that Case 1. In this case, thr resistors arc much more
the coefficient to be used in the estimation process out-of-tolcrance firelativc to the assumed
was 0.100. This coefficient is too small and would ohmmetcr accuracy) than assumed, and it
result in the placing of too little weight on the ohm- is possible for thy? Kaima;i procedure to
meter reading. The RMS error in estimating the yield estimation errors worse than those
resistance is computed to be 1.02, in contrast to obtainable with oxeJinary least squares esti-
the error of 1.66 which would have resulted if the mation. In other xwords, the weighting co-
optimum weighting of 0.308 had been used. efficients were too small, and we should
have given more credence to the ohmmetcr
readings.
- To explore the situation more completely, Fig-
ure 2 was Generated. Based on the best available Cane 2. In this case, the actiual variation of the re-
O
data or engineering estimates, the resistor tolerance sistors is smaller rihan assumed, and the
was assumed by the filter to be 1 ohm RMS, and estimation errors arre still better than the
the ohmmcter accuracy to be 3 ohms RMS. The "optimum." Hovtevcr, by having been too
effect of the real world's difference from the arti- pessimistic ia the assumed tolerance on the
ficial world assumed in the model is shown by the resistors, we too£. :too little advantage of
various curves in Figure 2, plotted for the five con- their inherent precision. \Vc would have
ditions that actual variation in the bucket of re- been better off just to use the nominal
sistors is 5/3, 1, 1/2, 2, and 3 ohms RMS. Two value of 100 oFazs.and to forget the ohm-
general cases can be observed in Figure 2: metcr readings- aJtmgcther.
I I
RMS ERROR IN
ESTIMATING RESISTANCE, OHMS
OHMMETER ACCURACY= 3 GHWS RMS RANDOM
ASSUMED ACCURACY OF RESISTORS = 1 CK.Y.1 RMS
ACTUAL ACCURACY OF RESISTORS: 1 OHM ;.=MS (CASE A)
1.5 DKM-tS RMS (CASE B)
2.0 OKITIS RMS (CASE C)
3.0 OHV.IS RMS (CASE D)
0.33 dH?.M RMS (CASE E)
NUM8B- C= 'OHMMETER READINGS
0.0
24
Figure S. Comparison of the Effects of Erroneoi's Statistics on Let?-- Scares
and Maximum Likciihooa Methods for Estimating Resislcr '•-'£:;ue
o
INITIAL
*, (0) TILT
*x(0)
INITIAL VELOCITY u
ax s)
ERROR \ 1 1 TILT
/ s R
1' + »x(s)
«X (0) INITIAL POSITION
POSITION ERROR ERROR
Fifjure -7. Simplified Schuler Loop
The moral to our story: Real-world statistics and problem. The following example illustrates appli-
dynamics will generally differ somewhat from the cation of the Kalman technique to a dynamic,
conditions assumed in the Kahnan filter coefficients. multi-dimensional system.
Before the Kahnan procedure is used in a particu-
The example under consideration involves a sim-
lar application, a sensitivity analysis of the sort
plified incrtial navigation system.* The navigation
described above should be made to assess the risks
system is described by the loop shown in Figxtre 3
involved in' the event the real world should happen
and is subject to only three error sources: initial
to differ appreciably from that assumed in the
position error, initial velocity error, and initial tilt.
model. (If one is concerned about the possibility
\Ve will assume that the system has available ex-
that the real world has statistics other than those
ternal position fixes every 10.5 minutes (or every
assumed, one might be tempted to consider "bias-
one-eighth Schuler period^, and these position fixes
ing'' the statistics to be employed in the filter,
can be used to update system accuracy. The total
thereby hedging against .this possibility. To accom-
assumed RMS errors acting on the system are thus:
plish this biasing, it would be necessary to alter
a priori statistics in the proper direction to reduce Error Magnitude
the risk that would be involved if the real world
Sx(OK initial 1.000 ft RMS'
were different from the model. However, it can be
position error
shown that tricks of this sort are pretty futile. The
Sx(0). initial 6 ft/sec RMS
l>est jwlicy is lo use the expected values of the cr's, All uncorre-
velocity error
based on the available data, and to let it go at that.) latt'l
^(0r. initial tilt C.I inrRMS
r
r, position-fix 1,000 ft RMS
APPLICATION OF KALMAN FILTER
error
TO DYNAMIC SYSTEMS
The l>uckct-of-iest«ors example illustrates some *Thc reader who is unfamiliar with inertial navigation
ran ignore the physical interpretation of this example,
salient features of the Kahnan procedure, but in-
yet benefit just as well. Figure 3 ean be ta»cn at fnre
cludes no tivnamics. Also, it is a one-dimensional value.
