C 09 . roa. \

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NATIONAL BUREAU OF STANDARDS REPORT

10 859

FINAL REPORT

NBS PROJECT WHERE

in support of

USAAMCA PROJECT MNPLS

by

Applied Mathematics Division

and

Computer Services Division

U.S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
NATIONAL BUREAU OF STANDARDS

The National Bureau of Standards1 was established by an act of Congress March 3, 1901. The Bureau’s overall goal is to strengthen and advance the Nation’s science and

technology and facilitate their effective application for public benefit. To this end, the

Bureau conducts research and provides: (1) a basis for the Nation’s physical measure- ment system, (2) scientific and technological services for industry and government, (3) a technical basis for equity in trade, and (4) technical services to promote public safety.

The Bureau consists of the Institute for Basic Standards, the Institute for Materials

Research, the Institute for Applied Technology, the Center for Computer Sciences and

Technology, and the Office for Information Programs.

THE INSTITUTE FOR BASIC STANDARDS provides the central basis within the

United States of a complete and consistent system of physical measurement; coordinates

that system with measurement systems of other nations; and furnishes essential services

leading to accurate and uniform physical measurements throughout the Nation’s scientific community, industry, and commerce. The Institute consists of a Center for Radiation Research, an Office of Measurement Services and the following divisions: Applied Mathematics—Electricity—Heat—Mechanics—Optical Physics—Linac

Radiation2—Nuclear Radiation2—Applied Radiation2—Quantum Electronics3—

Electromagnetics3—Time and Frequency3—Laboratory Astrophysics3—Cryogenics3

. THE INSTITUTE FOR MATERIALS RESEARCH conducts materials research lead- ing to improved methods of measurement, standards, and data on the properties of

well-characterized materials needed by industry, commerce, educational institutions, and

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Radiation—Physical Chemistry.

THE INSTITUTE FOR APPLIED TECHNOLOGY provides technical services to pro- mote the use of available technology and to facilitate technological innovation in industry and Government; cooperates with public and private organizations leading to the

development of technological standards (including mandatory safety standards), codes

and methods of test; and provides technical advice and services to Government agencies

upon request. The Institute also monitors NBS engineering standards activities and

provides liaison . between NBS and national and international engineering standards

bodies. The Institute consists of the following divisions and offices: Engineering Standards Services—Weights and Measures—Invention and

Innovation—Product Evaluation Technology—Building Research—Electronic

Technology—Technical Analysis—Measurement Engineering—Office of Fire

Programs.

THE CENTER FOR COMPUTER SCIENCES AND TECHNOLOGY conducts re- search and provides technical services designed to aid Government agencies in improving cost effectiveness in the conduct of their programs through the selection, acquisition,

and effective utilization of automatic data processing equipment; and serves as the principal focus within the executive branch for the development of Federal standards for

automatic data processing equipment, techniques, and computer languages. The Center

consists of the following offices and divisions: Information Processing Standards—Computer Information—Computer Services —Systems Development—Information Processing Technology.

THE OFFICE FOR INFORMATION PROGRAMS promotes optimum dissemination

and accessibility of scientific information generated within NBS and other agencies of

the Federal Government; promotes the development of the National Standard Reference

Data System and a system of information analysis centers dealing with the broader

aspects of the National Measurement System; provides appropriate services to ensure

that the NBS staff has optimum accessibility to the scientific information of the world,

and directs the public information activities of the Bureau. The Office consists of the

following organizational units: Office of Standard Reference Data—Office of Technical Information and

Publications—Library—Office of International Relations.

1 Headquarters and Laboratories at Gaithersburg, Maryland, unless otherwise noted; mailing address Washington, D.C. 20234.

2 Part of the Center for Radiation Research.

3 Located at Boulder, Colorado 80302.
NATIONAL BUREAU OF STANDARDS REPORT

NBS PROJECT NBS REPORT

6300505

2050505

May 31, 1972 10 859

FINAL REPORT

NBS PROJECT WHERE

in support of

USAAMCA PROJECT MNPLS

by

Applied Mathematics Division

and

Computer Services Division

IMPORTANT NOTICE NATIONAL BUREAU OF STAf accounting documents intended

for use within, thel Government B Approved for public release by the

rbjected to-additional evaluation and review. For this reason, the | r J isting of this Report, either in whole or in part, is not authorizi Director of the National Institute of Office of the Director, National Bureau of Standards, Washington, Standards and Technology (NIST) the Government agency for which

the Report has been specifically p ^ ^ c ties for its own use. on October 9, 2015.

U.S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

List of Contents

1. Executive summary of report

2. Description of general approach used in this report

3. Review of analyses to date

4. Conclusions

5. Appendixes

Working paper no. 6: Error propagation for one-dimensional location

of two units. By M.H. Pearl

Working paper no. 7: Algorithms. By D.J. Sookne

Working paper no. 8: LS, LSS and relocation. By J.A. Lechner

Working paper no. 9: The MNPLS simulation ("WHERESIH")

.

By R.H.F. Jackson

1. Executive summary of report

This report summarizes the work done at the National Bureau of

Standards (NBS) under its project WHERE in support of the US Army Advanced

Materiel Concepts Agency's (USAAMCA) project MNPLS (Micro Navigation and

Position Location System)

.

MNPLS is a system that utilizes a time-division multiplexed frequency

that is shared by n units , all of which are synchronized by a suitable

electromagnetic signal during each time interval AT . Each unit is assigned

one or more time slots within AT during which it emits an electromagnetic

signal. This signal in turn is received by several but not necessarily all

other units in the field which measure the time of flight of the EM signal;

from the knowledge of the time and the assignment of the time slots for each

unit, the distance between the sending unit and the receiving unit can be

inferred

.

For NBS the problem to be solved was twofold:

(1) To determine the feasibility of such a system in the presence of

measurement errors. If all measurements were exact three units in the plane

or 4 units in three-space would suffice to locate another (or all other)units.

In the presence of error more measurements are needed and the "best" solution

that minimizes total error in some mathematical sense has to be found. The

fundamental problem for NBS was to show that the overall error does not

increase unduly as the system develops in time.
.
(2) To develop a computer simulation for such a system that allows the

investigation of real-time scenarios with a sufficiently large number of

units undergoing battle-field manoeuvres, each with prescribed movements

over suitably long times. The simulation program determines the distance

between any two units i and j , perturbs that distance according to

suitable assumptions about the measurement errors, and supplies to the

master computer (the program that determines positions from the distance

measurements) the distances as though they came from actual field measurements .

The output of the program consists of the successive locations of all

units, the errors between true position and calculated position, and a measure of confidence derived from the available measurements but not from

the knowledge of exact position (which is only available to the simulation

program but not to the position location algorithm)

„

NBS has designed a set of mathematical algorithms for position location

determination that allow the determination of successive positions of n

field units, from inputs containing only distances between units,' or perhaps

those distances accompanied by independent measurements of differences in

height above sea level* The experiments based on actual deployments in the

Boston area show that, with a suitable number of fixed units, the average

error of position overall units can be kept well within acceptable limits.

Based on a range-error standard derivation of 6m , and using all 83 units

as reporters, the maximum error during 2 hours of actual time remained under

110m. The average error was 4.5m over all units, with only 3 units "lost"

more than 8 times out of 240 tries. When only the 39 fixed units were used

as reporters, the average error was 4.7m, the maximum was 40m, but 15 units

were "lost" more than 50 times out of 240. (The term "locators" refers to

those users that do actually receive a signal from the locatee, and do

report a time -of -flight to the master computer; the term "reporters" refers

to those users that have been instructed to report time-of -flight measurements to the computer. Because of "line-of -sight" requirements, not all of

the latter may be able to serve as locators for any given location-operation.)

A simulation package WHERSM was designed that simulates the operation of

a Micro-Navigation and Position Location System. This program allows one to

stage a wide variety of movements for any battle-field scenario; it generates

the inter-unit distances, perturbs the ranges according to agreed upon error

distributions and provides the links to the position location algorithms

which were also developed at NBS for this purpose.

3

The position location algorithms fall into three distinct categories:

(1) Two-dimensional algorithms . Where the terrain is flat, no altitude

reference is necessary and the resulting algorithms become two-dimensional.

2 2 2

The sura £w.(r, - d. ) is minimized, where the sum extends over some or

i i l

all reporting units and the weights are determined by the estimates of

variance of each of the reported distances d^

(2) Pseudo three-dimensional algorithms . Where the altitudes are so small

in relation to the horizontal distances that the slant distances are nearly

the same as the horizontal distances it becomes necessary to reduce the

position location problem to the two-dimensional one above. Not only is the

three-dimensional problem ill defined mathematically, but we have also a genuine multiplicity of solutions if all locators are nearly in a plane and

trying to locate a unit outside the plane. The determination of heights that

are small against the horizontal distance is impossible within any reasonable

limits of accuracy and the altitude must therefore be determined either by

barometric reference or a reference to a digitized terrain map. For example,

at a distance of 1km, a range uncertainty of only 6m leads to a height

uncertainity of 110m.

(3) Fully three-dimensional algorithms are being used when the altitudes are

comparable to the horizontal distances. Again, a penalty-function approach

2 2 2

is used that minimizes £ w^(d^ " r

i

) with suitably determined weights

depending on the variance of the measurement and the sum being taken over

some or all reporting units.
V -I
The following results were obtained:

o The test runs conducted on the Boston deployment show that stability

of the position-location process is indeed achievable as long as some units

remain fixed.

o Computer size for position location:

Runs were all carried out on the UNIVAC 1108 at NBS . For this machine,

the formula for computing the number of computer words required by the

position-location algorithms as they are currently programmed is as follows:

26N + 7R + 3874, •

where N » number of units in the field, and R = the maximum number of

ranges being reported.

As a matter of possible interest, the number of additional words

required by the simulation program (into which the position-location

algorithms were inserted) is:

9 IN + N

g

+ 4002,

where N = the number of subcycles in a cycle of the MNPLS (see Working

u Paper No. 9) „ Furthermore,- the preprocessor to the simulation requires

39,714 computer words.

o Timing for position location operation:

Several runs are available from which the time required by the

algorithms can be estimated. One run was done using LSS(3d), with 20 locators

5
I " ] t I
it required approximately 16«5ms per location operation. A run with 15

locators using slant range reduction took 8-12ms ; with 20 locators, the

computation took 14 to 18ms. For the Boston deployment, using 39 reporters

(but probably less than 15 locators on the average because of intervisibility

restrictions), 8 to 14ms were required; using 83 reporters, 17 to 25ms were

required. Since the algorithms are not optimized for running time, it is

reasonable to expect some decrease in these times for the operational system.

2 . Description of general approach used in this report , a) Simulation .

The simulation of the battlefield scenario is done in our NBS-developed

computer program WHERSM. This program provides for a "real life" framework

for MNPLS through its capability for moving all field units along prescribed

paths, or subjecting these to random changes in their direction of movement.

WHERSM generates all the inter-unit distances (ranges) ; checks, if necessary,

for intervisibility (by look up in an intervisibility matrix) ; disturbs the

exact ranges according to known error distribution laws; and transmitts these

"measurements" to the position location algorithms that we developed here at

NBS and which are described below.

The simulation package also provides a facility for monitoring the

operation of such a system under a variety of movement scenarios; and

ultimately it provides for a variety of outputs that are desired for both

checking out the feasibility of the entire system and what might be called

the ultimate output that would be useful to the field commander who is

interested in the whereabouts of his units.- For details of WHERSM, the

reader is referred to working paper No. 9 in the appendix of this report.
- .
(2) Algorithms .

The basic LSS algorithm that we finally used after numerous experiments designed to eliminate competing ones is the so-called "least-squares

squared" algorithm, which minimizes the penalty function

^ ,.2 2 N 2

E = £ w. (d. - r . )

where i is taken over all of the locators and w^ is a weight that

increases with decreasing variance of the observation , the reported

distance of the locatee from the locator with index i . The quantity d^

is the calculated distance between (x^, y^, z ) and (x,y,z) the estimated

position of the locatee. The solution (x,y,z) is obtained iteratively

from the equations BE/dx = 0 , dE/dy = 0 , BE/dz = 0 once a starting

point is known.

As long as a trajectory is known for a given locatee, the initial guess

is simply obtained from extrapolation. If the unit is lost, however, a set

of linear equations is solved (LSL method). Details of the LSL method can

be found in NBS report 10663: First Interim Progress Report on NBS Project

WHERE in support of USA AMCA Project MNPLS.

The appendix contains a list of variables used in the simulation code.

Since this is the part of the NBS system that interfaces with the outside

world, we believe that those are the only variables that have to be listed.

The variables in the algorithm parts are internal and are thus not referencable from outside.

7

3. Review of analyses to date.

3.1, General .

As originally conveived, the testing was to comprise two phases:

initial testing to choose the best position-location algorithm, and final

verification testing to determine how well the chosen algorithm would work

on the Boston deployment. This neat plan did not fit the real-life flow of

events, however: every stone turned over revealed two more yet unturned.

The process of developing a new system is bound to contain such loops. For

example, one does not know how to optimize the algorithm until one knows

the system design, but the system design is sensitive to the effectiveness

of the chosen algorithm. Thus the two phases gradually blurred into each

other. Furthermore, in the limited time allotted for this study, it seemed

that effort would be better spent getting reasonable answers to more of the

real questions, rather than "optimizing” on a smaller subset of the questions while assuming particular (and questionable) answers to the others.

For these reasons, a strictly chronological account of our analyses

would not be very helpful. Instead, it seems reasonable to describe the

test results by first building a framework of the questions to be answered,

and then describing the tests performed and the answers obtained. The next

section consists of a set of questions; the following section presents the

answers to date, as gleaned from the experimental program and from certain

theoretical studies undertaken specifically to support that program.

3.2, Questions pertinent to the MNPL system.

The general question that • constitutes the focus of a feasibility study

is: Will the proposed system perform satisfactorily under the conditions of

2
I ,
its intended use? This question is however a very broad one; it asks

essentially whether the final outcome of the feasibility study is "yes",

"no" or "maybe” , and thus does not give much guidance in conducting the

study. Thus one is led to pose narrower queries:

a. General Questions

(1) Under highly optimistic conditions, will the system work? (If not,

terminate the study.)

(2) Under what reasonable near-minimum conditions can the system be made

to work?

(3) How well does it work?

b . More Specific Questions

The next question leads to a whole set of sub-questions, which are

listed along with the main one.

(4) What position-location algorithm(s) should be used? For each candidate

algorithm, consider:

(a) Computer time required?

(b) One-step accuracy?

(c) Error propagation (sensitivity to locators 1 position errors)?

(d) Sensitivity to occasional very large errors in the range

measurements?

(e) Affected by dimensionality?

(f) Need an initial estimate of position?

The next set of questions refers to restrictions placed on the units in

the system.

(5) How many (if any) units must remain fixed?

(6) Must they be permanently fixed in known positions?

(7) How many reporters are required, to keep the errors small?

(8) Will ground units have to carry altimeters? Instead, will a digitized

map, stored in the computer, suffice?

(9) How accurate must altimeters be (both ground and airborne)?

Questions (10) - (19) refer to design of the algorithm.

(10)

How often should units report?

(11)

How many aircraft are required, in order to use a three-dimensional

algorithm on ground units?

(12)

How should one handle near-planarity (when any aircraft present are

flying very low)?

(13)

When should the algorithm?

(a) ignore height differences?

(b) reduce slant ranges to horizontal distances?

(c) use a 3-dimensional method?

(d) use 3d for aircraft, slant range reduction for ground units?

(14)

How should the locators be chosen?

(15)

How weight the locators?

(a) What is the distribution of range errors?

(b) How handle the occasional occurrence of range measurements with

extremely large errors?

(c) How estimate the accuracy of an estimated position?

(16)

Should the locators be "relocated1

' ? (i.e., should the locators'

positions be adjusted, based on discrepancies between measured

ranges and calculated ranges to the best estimate of-locatee's

position?)

lo

(17) Use ' only fixed units as reporters? How soon after they become fixed?

(18) Can the algorithm determine, (satisfactorily) which units are fixed?

If not, could a moving/stopped indicator bit be included in the

transmission?

(19) After the system has been running for a while, and errors have

accumulated, is it possible to ,, restart M the system i.e. to treat

the set of (currently) fixed units in toto, simultaneously, allowing

all position coordinates to vary, in order to converge to good estimates

for their positions?

3.3. Conduct , progress , and results of the test program .

The first few questions, above, suggest a beginning emphasis on

favorable conditions. The rationale is that, when very little is known

about a system, more is learned faster by starting with conditions where the

system works tolerably well, and then investigating the effects of departures

from these conditions. Accordingly, the first major project was to devise,

develop, program, and test several candidate algorithms for determining

the position of a locatee, using the (perturbed) ranges to a set of locators,

and assuming the locators’ positions to be known exactly. That is, attention

was focussed first on questions (4) . Some of these algorithms were suitable

for 2 or 3 dimensions, others were usable in 2 dimensions only. Methods

considered were: Linear Method (LM) , Smallest Tangent Circle (STC), Least

Squares Linear (LSL) , Minmax, Least Squares(LS), and Least Squares Squared

(LSS). The first two methods use exactly 3 locators in 2 dimensions, or 4

locators in 3 dimensions. The next two are generalizations of the first two.

//
W I

'
respectively , to an arbitrary number of locators. Least squares and LSS

are penalty-function methods. All are fully described in earlier sections,

in Working Papers Nos. 7 and 8, and/or in NBS Report 10663, First Interim

Progress Report on Project WHERE in support of USAAMCA Project MNPLS .

Tests were first conducted in two dimensions. STC turned out considerably more accurate than LM, but slower; LS and LSS agreed closely, and

were more accurate than any others, but slower than all but Minmax 9 Since

Minmax was no more accurate, and several times more time-consuming, than

LSS, it was dropped from further consideration. Also, since LM was inferior

to STC, it was dropped. Thus the survivors at this point were: STC, LSL, LS,

and LSS.

The next sequence of tests was directed to question (4)(c) . For these

runs, errors were applied to the positions of the locators, corresponding

to a point in time- after the system has been operating long enough to

accumulate such errors. In these runs, STC fell down markedly. LS and LSS

did better than LSL, but the latter was kept since it alone (of the three

methods still in contention) does not need a starting value — i.e., is not

an iterative method — and therefore can provide a position estimate when

the locatee first joins the net or has been out of touch for some time.

Three-dimensional tests confirmed the pattern of results above, and

also accented the importance of geometry: when the locators are nearly

coplanar (collinear in 2 dimensions), LSL often produces a very poor estimate,

because the planes (lines) that it tries to fit are nearly parallel. LS and

LSS also have problems, but not as severe as LSL. These problems were

!l

accentuated in 3d, simply because locators on the surface of the earth are

relatively coplanar; the corresponding situation in 2d would have the locators

strung out nearly on a single line, a less likely situation.

In terms of computer time LSL was best, with LSS next. Since LSS was

devised to perform like LS but faster, these two were then compared for

accuracy. 72 trials were run on each method, with 18 users scattered over

.an 8km by 16km area, with altitudes to 9km, for each of two different values

of a (the standard deviation of the range error). Each of the 18 units

served as locatee 4 times for each method and each value of a . The results

were as follows; For a = lm , the position estimates were in error by as much as 3.32m , but the difference between the LS estimate and the LSS

estimate was never more than 0.0004m „ For a * 10m , the errors reached

31. 9m, but the two methods gave estimates that differed by no more than 0.045m.

As a result of this test, it was concluded that LS and LSS could be used interchangeably so far as accuracy is concerned. Since LSS was considerably

faster, LS was dropped. The final result thus was; Use LSS; if no starting

value (initial estimate of position) is available, use LSL to obtain one.

(Complications develop when there is a choice between 2- and 3-dimensional

versions, as we will see.)

The only part of Question (4) yet unanswered is (4)(d): sensitivity to

occasional large errors ( outliers) in the measured ranges. This is a severe problem for STC and Minmax, since STC uses only three ranges and

Minmax concentrates on minimizing the maximum discrepancy. For LSL, LS, and

LSS, the matter is treated in a statistical sense, as is commonly done in

(3

fitting regression curves. The true position is estimated, and then the

discrepancies between measured and calculated ranges are examined. Any

discrepancies significantly larger than the average are attributed to faulty

measurements. These measurements are ignored, and the cycle repeated.

This approach is sure to detect all really bad errors, provided they occur

infrequently, unless there is exactly the minimum number of ranges necessary

to produce an estimate. It was been shown to be quite efficient, in the

sense that the occasional rejection of measurements "apparently" but not really

contaminated with large extraneous errors does not appreciably degrade the

accuracy of the estimation procedure. The actual cutoff point for rejection

is a variable that will need further investigation as part of a system development effort.

Questions 5,6, and 7 pertain to the set of reporters. (The term

"locators" refers to those users that do actually recieve a signal from the

locatee, and do report a time-of-f light to the master computer; the term

"reporters" refers to those users that have been instructed to report timeof-f light measurements to the computer. Due to the requirement for a "lineof-sight", not all of the latter may be able to serve as locators for any

given location-operation.) In order to obtain broad(rather than situationspecific) answers to these questions, it was necessary to set up a number of

different scenarios, with different numbers of users and different proportions stopped. This diversity was created by employing several kinds of

"randomness" . Users were assigned to randomly chosen starting points within

a specified rectangle, with aircraft heights specified or randomly chosen;

IV
.
sequences of azimuths were chosen randomly over a 90° range , and azimuth

change times were drawn from an exponential distribution. Stopping and

starting times were drawn from exponential distributions, chosen to achieve

specified values for the expected (long-term average) proportion of users

stopped at any given instant, and for the mean times spent in ’'fixed"

status and in "moving" . Intervisibilities were similarly randomized, to

achieve specified values for the average time two users maintain communication

and the average time they remain out of Communication. These features were

used in various combinations to investigate most of the remaining questions.

The results of those investigations are presented below.

If all the reporters are moving, strange things can happen. Small

(random) errors in locating a given reporter propagate into the location of

other reporters, so that each remains well-located relative to the others

but the whole cluster is displaced(and perhaps rotated) far from its true

position. This effect is reasonably well understood, and has been clearly

demonstrated. One case involved 30 users, moving more or less in the same

direction. 12 users were reporting, with perfect intervisibility (2 dimensions)

By the time 100 location operations had been done on each, they had moved

0.9 to 4.1km . The estimated position of the cluster as a whole was 3.3km

from its true location, but each element was located within 150m of its true

position relative to the others. (Range errors were less than 2m in this

case.) The problem can be seen clearly in the following simple example: If

all of a collection of fixed users suddenly begin to move, each with the

same velocity vector, or all of a collection of users moving with a common
.
velocity vector suddenly stop, no algorithm based on inter-unit ranges can

see the change, because the inter-unit ranges don't change. Therefore, the

algorithm will leave the users fixed in the first case, and will continue to

move them in the second. This is a special case, of course, but the same

principle applies to the average velocity vector when the units begin to

move with different velocities - or more generally, jto the average change in

the velocity vector when they (more or less) all change direction at about

the same time.

These results made it plain that one needs some users fixed. The next

question was, how many?

Several runs were made with different sets of fixed users, under two

conditions: first, that the computer does not know which users are fixed,

and second, that it does know(and can use) this information. The first run,

with range errors distributed uniformly from -3m to +3m , involved 15

motorized and 15 foot units; in each set of 15, 5 were fixed and reporting, 5

were moving and reporting, and the remaining 5 were moving but not reporting.

After 220 cycles (109 minutes), the xy errors ranged from 500m to 10,240m,

and were distributed into all four quadrants (more specifically, 14 users

were placed NW of their true locations, 5 SW,10 SE, and 1 NE ) . For the

second run, the subcycle At was reduced by a factor of 5, to see if locating

the users more often would improve the accuracy. After 613 cycles (60 min),

the errors ranged from 570m to 2200Qnu (The errors grew more slowly per

cycle, but faster in real time.) Next, 4 of the fixed users were assumed

"known" - i.e., no estimation of their positions was allowed - to provide an

anchor o£ sorts. After 239 cycles (116 min.), the average error over all

users was 9.4m , and the maximum, was 23m ; over all cycles, the average was

7.2m , the maximum 82m * But there is some evidence, that the errors fluctuate

considerably with time: in the set of ten cycles prior to the above (i.e.,

cycles 221-230) , the average was 23m ; the maximum 69m !

At this point, relocation (Question 16)was tried. (Refer to Working

Paper No. 8 for technical details.) Tentative conclusions from the dozen or

so runs are: In the beginning, relocation increases the errors. Eventually,

the errors without relocation become larger than those with relocation: how-

\ r

' * ever, this did not show up until about 2 hours of real time had been

simulated, by which time the fast-moving users had separated from the slowmoving and fixed users by about 20km . (At this time, the overall average

and maximum errors were 19m and 557m with relocation, and 34m and 1036m without. Least Squares was also tried; the average errors were comparable, but

the maxima were 570 and 1695 respectively.)

Perhaps more instructive were the shorter runs with differing values of

the fraction mentioned in Working Paper No. 8, which measures the extent to

which relocation is carried. Runs were done with this fraction set at 0„5,

0.2, 0.1, and 0 (i.e., no relocation). The results showed steady improvement as the fraction went down, with smallest errors occurring when no relocation was done. This shows clearly that there is no advantage to relocation

for the first 100 cycles (50 minutes) at least. Still open (and reasonable)

is the possibility that relocation should be instituted for a time, now and

then, with a small fraction, to help remove accumulated errors.

n

However, if there are sufficiently many fixed reporters, one can probably

do better by periodically going into a survey mode wherein all the fixed

units’ positions are simultaneously adjusted, effectively ’’restarting” the

system.

At this point, it seemed clear that relocation was not worth its price

(more than doubling the algorithm’s computer time). This conclusion was

checked once more, in the next set of runs, under more realistic conditions,

and was supported by the results: the errors were smaller (at 100 cycles)

without relocation.

Parallel efforts by ECOM had meanwhile produced a ’’best engineering

judgment" range-error distribution, which can be adequately modeled as

2

"Gaussian with variance 36 m ; mean 0 , except that a random 1% of measure

ments are inflated by about 7 meters." There will also be occasional

extremely large errors; a simple preprocessor can weed most of these out in

the field system, by simply checking whether the range measurement is

compatible with the last known position and the velocity potential of the

user. This error distribution was implemented in the balance of our tests.

Also implemented were moving boundaries, which can be set to advance at a speed appropriate for foot-soldiers; any motorized unit which reaches the

boundary will be reflected off it, so that the motorized units do not stray

far from the foot units. Additional features implemented at this time were:

a moving/stopped indicator; calculation of weights which reflect the

estimated accuracy of a user’s estimated position; the ability to choose as

locators those users with greatest weights, or those which have been stopped
5
for a certain number of location-operations; improvement of estimates for the

positions of fixed users , by combining successive estimates; a functional

representation of terrain; use of an independent measure of the z-coordinate

(the height), with random errors; algorithms to reduce slant-range distances

to planar (xy-plane) distances, using the independent measure of height

difference between locator and locatee; and tests to determine whether a particular location-operation should be treated by slant-range reduction or by a 3 -dimensional method. Again, these features were used in various

combinations for the runs recounted below.

Before these remaining runs are described, the rational used to

"weight" the various range measurements will be explained. No claim is made

that these weights are optimal, but it is felt that they are reasonably close

to the best values, which is sufficient for a feasibility study. Suppose half

the locators had the same x-coordinate as the locatee, and half had the same

y-coordinate . Then (assuming ranges to be large relative to range errors) the

first half of the locators is useless in determining the x-coordinate, and the

second half is useless in determining the y-coordinate. Thus if there are n

locators, n/2 are used for each coordinate, and so we have n/2 estimates

(assumed independent) of each of the two coordinates of the locatee*

s

position. Consider for definiteness the x-coordinate. If the locators' position errors are unbiased (i.e., have mean value equal to zero), the best

estimate of the locatee* s x-coordinate is a weighted average of these n/2

values: weighted by the reciprocals of the variances of these observations.

Since the observational error is the sum of the locator's x-coordinate error

and the range measurement error , its variance is the sum of the position

error variance (determined when the locator was last located) and the

2

measurement error variance (

** 36 m ) 0 Thus one can calculate an estimate

of the x-coordinate error variance, which in turn is used to calculate the

weight for this user when he serves as a locator.

[ There are two considerations which should be mentioned here. First,

locators are not in general split so nicely along perpendicular lines. Does

this matter? Of course it does, in the following sense: if more locators

serve to estimate x than y, then x is likely to be better known - i.e.,

to have smaller variance — and y will be less well known. On the average,

however, things even out: if azimuths (from locatee to locators) are uniformly

distributed from 0 to 360* , the average (or expected) variance in a given

direction turns out to be the value derived above. The second consideration

is: What if the estimated variances for the locators are wrong? Two

consequences follow: (a) To the extent that the variances are different

multiples of the assumed values, the estimate of position is less than

optimum (because the true relative variances should be used to get the

estimate) $ and to the extent that the variances are (as a group) larger (or

smaller) than assumed, the estimated variance of the locatee* s position will

be too small (or large, respectively). Point (a) is not likely to be important

since it would take very large discrepancies to affect the estimate noticeably.

Point (b), on the other hand, should at least be investigated. One technique

is to act as though the true variances are proportional to the estimated values

with an unknown constant of proportionality (say c) . Then c can be

estimated, and used to produce a fair variance estimate. Specifically, if each

range observation has variance co

- ^ (a^ *= location variance + range measurement variance), then the best estimate for the locatee can be found without

considering c. If d^ represents the discrepancy between the calculated

range from this position to the assumed position of the i-th locator and

the corresponding measured range, then [ £ d^/a^ ^ t*ie (multiplicative) correction to be applied to the variance estimate already given. (If

c - 1 , this value should also be ~1,) This is a recent result and the

corresponding modification has not yet been made to the algorithms.]

A new series of 2-dimensional runs was done, with 30 users. An initial

set of fixed users (reporters) was specified. From then on, units became

reporters as soon as they had been fixed for ten location—operations , and

ceased to be reporters as soon as they began to move. Average times moving

and fixed were set at 15 min., so that about 15 (half the units) were stopped

at any one time. A 2%-hour (300 cycle) run was done, and resulted in an overall average error of 3.2m, and a maximum error of 35raj in the last set of 5

cycles, the average was 3.5m, the maximum 11m . Next, a run using longer

cycles (90 sec. instead of 30) produced errors of 3.8 and 50.5 overall, 5.7

and 33.8 for the last 5 cycles. Next, the average times fixed and moving

were set at 45 min. and this run repeated (because with 15 min. times and

\\ min. cycles, many units didn’t stay stopped long enough to be much use as locators). The error figures then came back down toward those of the first

run: 3.4 and 33.3 overall, 5.9 and 22.5 for the last 5 cycles.

At this point, the cycle time was returned to 30 sec., aid a relocation

run was tried, with fraction 0.2. It did poorly: errors 5.8 average and

35 maximum overall, with 8.4 and 17.5 for the last 5 cycles. All errors

were in one quadrant, indicating that some instability had been introduced.

Next, a run with fraction .001 was done. Since this fraction is so small,

the magnitudes of the actual relocations will be essentially negligible;

thus the effect of relocation in this run is simply to allow more iterations

for the algorithms. (Remember that the number of iterations has been kept

small to keep the running time down, with the thought that range errors of

the order of 6m don't justify too much precision in locating the true solution

of the least squares problem.) Only 100 cycles were done, with results equal

(up to roundoff effects) to those without relocation. This supports the

conclusion that generally, it is not worth doing too many iterations. (Of

course, how many is "too many" depends very much on how good the algorithm

is*.)

The algorithm was modified to treat aircraft with a 3-dimensional method,

while continuing to treat ground units 2 -dimensionally . (This is possible

since only ground units were being used as locators.) Ten aircraft were

added, at (random) altitudes ranging from 128 to 2220m. That low-flying aircraft

presented a problem became very obvious, since the algorithm located them where

they belonged at some times, near the ground at others, and below ground

(mirror image) at still other times. One more attempt was made in this series,

using only ranges that were at most 10 times height, to ensure that not all

lines of sight were flat. This helped for the second-lowest aircraft, at 339m:

it had an average (x,y) error of 19m, with a maximum of 113m. However, the

lowest aircraft- generally did not have enough acceptable (short enough) ranges

to estimate its location. It was decided, therefore, to do a separate systematic

study of aircraft location errors. This will be discussed below,. The

conclusion is already clear , however , that some independent height information will be required to locate low-flying aircraft, unless other aircraft

are serving as locators; however, the use of aircraft as locators will

probably require very stable aircraft and good tracking techniques.

A functional representation of terrain - i.e., z = f(x,y), with f(x,y)

chosen to give a reasonable surface-was implemented at this stage, to test

the use of independent pressure references (p.r.'s) as a source of heightdifference information. After consultation with AMCA personnel, these were

modeled as giving height with an error that had zero mean and a - 14m (so

that the difference of 2 heights will have c = 20m ). 40 units were used,

on a terrain that varied 200m in elevation, with 15 units stopped initially)

again only units that had been stopped for 10 location-operations were used

as locators. The x,y coordinates were obtained by reducing slant ranges to

horizontal distances and applying 2d algorithms. In 2% hours, the overall

average error was 3.5m, the maximum 29.2 ) for the last 5 cycles, the

average was 4.8, the maximum 13.7 . Next the p.r. a was set to 0 , with

results identical to another run where the terrain was flat and no heights

were considered. For extraneous reasons, it only went 260 cycles instead of

300) it had errors of 3.2 (avg), 28.2 (max) overall) 3.3, 14.6 for the last

5 cycles. Next, the height estimates were set at 0 ; i.e., the hills and

valleys were ignored by the algorithm. The errors were slightly smaller than

the ordinary run using p.r.'s: 3.35 and 28.2 overall, 4.4 and 14.3 for the

last 5 cycles. This is not surprising, since the errors introduced by the

^3
I
p.r.'s are comparable to the terrain differences. If p.r.'s don't perform

any better, perhaps they should be ignored when only ground units are

concerned. However, in particular situations, the terrain heights may

introduce systematic errors if ignored, which may then propagate j the p.r.

errors at least, have the advantage of being random. More investigation of

this possibility should be undertaken when more is known about p.r.'s . For the next run, terrain roughness was tripled: with the p.r.'s the errors

were essentially the same as on the flatter terrain.

Only two sets of runs are yet to be described: the aircraft error study

and the runs on the Boston terrain. The Boston scenario had no aircraft, so

it was treated using slant-range reduction with p.r.'s. Two runs using the

1st Brigade constitute the final trials: one with only the fixed users

reporting, and one with all units reporting. In preliminary trials, the

effect of dropping those reporters which are in another brigade was seen to

be minimal, except for a few front-line users which had severe intervisibility

problems. The errors for the two primary runs are summarized in Table 1 .

Several points deserve to be mentioned with regard to these results. It

will be noted that for 30 of the 44 units, the average error is less when all

units serve as reporters. Furthermore, in 10 of the remaining 14, the higher

average using all units seems to be due to poor estimates where otherwise

(with only the fixed units reporting) there would be no estimates. For most

of the units with maximum error above 20m, those maxima occurred with less

than 2 fixed locators (i.e., where no estimate would be available using only

fixed locators). The greatest error occurred for unit 28, and was 108m j it

(as well as the two unsuccessful location-operations for this unit) involved

21 locators, including 2 fixed locators. (It is interesting to note that

relatively large errors will occasionally occur even with a system " in

control." ) Using only the fixed locators, that unit never got lost and

never had an error greater than 18m . This anomalous behavior suggests that

perhaps one should use only the ranges from fixed units, if there are enough,

and should use at most a small number of ranges from moving units in any case.

Also worthy of note is the fact that apart from this unit, every other

maximum error greater than 31m occurred with 4 or fewer locators.

As mentioned earlier, it became obvious very quickly that low-flying

aircraft could not be tracked from the ground. To study the effects on

accuracy of such variables as number of locators and height, a separate

study was done on aircraft. 20 reporters were randomly positioned on a

10 X 10km piece of terrain, and kept fixed. (See Fig. 1.) Ten( simulated)

aircraft were started at the southwest corner, at altituted of 100, 250,

400, 550, 700, 1000, 1500,2000, 2500, and 3500m. Each flew a level course,

with constant x-velocity; the course reflected several times off both north

and south boundaries, terminating at the eastern boundary, systematically

covering the entire terrain. (Shown on Fig. 1.) Each aircraft went through

either 1230 or 2790 location-operations. The average error and average

squared error were tabulated, both in the x,y-directions and in z, by

number of locators, for each aircraft (i 9 e., each height). These tabulations were then studied, and regression techniques applied.

The first run counted locators for which the range was less than

8 X height, 12 X height, 16 X height, etc., up to 80 X he ight, stopping as • soon

as 4 or more locators were found. It then used LSS (3d) . Table 2 shows

the numbers of trials for which the algorithm was and was not successful by

aircraft height and number of locators. There are few cases where an aircraft at 700m or higher was not located, but things are not too good below

that height. The error statistics bore this out; the maximum z-errors for

the 10 aircraft were 1880, 192, 97, 83, 39, 22, 14, 15, 16, 19m (in order of

increasing height), while the maximum x,y-errors were 26339, 2373, 337,600,

72, 36, 29, 23, 27, 18m respectively. Since 3d obviously does not work with
.
low-flying aircraft, an examination was made of the dependence of average

error on number of locators and height, for aircraft heights of 550m and

above. Regression analyses showed that the average x,y-error could be — 2500 2 represented adequately by the expression E = 16 - ,9n - —-— + ,017n

x,y , ax t

— 2

and average z-error by E ~ 31 - 3n + .08 n , where n is the number of

z locators used. (Note that these expressions were developed for aircraft

higher than about 500m.) The values given by these expressions agree with

the values obtained in the experiment within an average deviation of 0 o 5m

for x,y-error, and 2m for z-error. (The experiment values for x,y-error ranged

from 3 to 10m; for z-error, from 4 to 29m .)

Finally, a similar run was done using slant range reduction, with aircraft at 100, 250, 400, and 550m altitudes, using p.r.'s, and only those

reporters with elevation angles < 20". The average x,y-errors were 3.2,

3.3, 3.4, and 3.7m respectively; the maxima were 15, 15, 15, and 17m

respectively. This suggests that one should use slant range reduction up to

at least 600m 9 and use LSS 3-d or slant range reduction above that value

depending on whether there are many good ranges at large elevation angles

or many at small elevation angles, and on the quality of altitude information

available at different heights. (It is understood that p.r.’s of the

accuracy contemplated herein are limited in altitute, but will cover at least

0 to 600m .)

Table 1 Error surrctmrlos, Boston runs.

All reporting Fixed units reporting

Unit no. Errors (m)

Average Max. No . of times not found Errors (m)

Avera3e Max.

No. of times

not found

7 3.8 16.6 49 5-0 21.7

8 5-9 39.8 7-5 33-4 105

9 8.6 62.1 7-0 39-6 77

10 3.7 12.0 • 5-9 20.2 11 3-4 9.3 20 5.1- 22.2 12 4.2 16.8 4.7 14.2

13 • 3-7 11.2 104 5-7 21.1 104

14 3-5 10.0 6.8 32.4

15 3-7 18.7 • 4.8 18.5

16 3-7 23o 1 4.8 16.6 17 5.1 27.

• * 5-1 25.4 18 4-9 20.4 5-3 29.5 19 5-0 18.0 5-1 23-3 20 3.9 15.4 5-8 30-7

24 3.6 24.1 4-3 25.8 25 4.3 13-3 5-0 16.3 73

26 2.9 10.0 4.1 11.8 27 3-4 10.1 4.6 21.9 28 6.3 108.1 2 4.5 17.7

29 3-4 10-9 3.4 15.0

30 5-9 51.6 4.3 12.9 79

31 7-7 79-9 8 4.2 12-3 61

32 3-7 14.1 5-1 33-9

33 3-3 12. 2 3.6 12.5 34 3-7 12-5 3-5 ' 20.4

35 3-2 11.4 1 3-4 10.5

36 3-6 ll.l 1 3-1 12.8

37 3.6 11.6 3.3 13-4

38 . 2.7 10.1 3.4 10-7 42 3-9 12.8 3-.4 8.5 102

43 3.6 10.9 67 4-5 16.8 67

44 5-7 54.8 3 4.3 12.9 57 45 4-9 23.6 4.7 12.5 69 46 3-5 13.8 3.8' 11-7 67 47 9-4 101.3 4.0 11.9 55 48 4.8 . 27.7 3 4.4 14.2 77

49 3-9 .12.8 2 4.2 13.4 2 50 8.3 88.3 5.9 26.7 30.

51 5.0 57-5 4.4 19.2 20

52. 3-7 14.4 . 5-2 17-5

53 3-8 13-5 4.7 23.1 77

54 3-8 13-1 4.6 25.9 105

55 3-9 17.2 5.4 25.5

56 4.2 16.4 5.7 22.0

All 4-5 108.1 4.7 39-6 •

Table. 2

Lost and found summary aircraft study

Aircraft height

Number of Locators 100 250 400 550 700 1000 1500 2000 2500 3500

3 4/0

4 136/640 84/396 44/175 18/80 - 7/52

5 40/255 48/233 28/189 18/92 3/20

6 18/98 18/163 8/131- 9/103 0/11

7 7/24 14/98 22/192 16/128 4/32 0/1

8 1/5 6/57 4/122 2/119 0794 0/2

9 0/2 6/83 3/189 5/159 1/57 0/1

10 1/22 2/95 1/115 1/70 0/3

11 0/1 0/16 0/85 1/98 0/0

12 0/10 0/94 1/130 1/28

13 0/90 0/44 0/13

14 0/70 0/65 0/36

15 0/16 2/228 0/184

16 0/10 0/133 0/154

17 0/44 0/212

18 0/33 0/231

19 0/232 0/67

20 0/132 1/1162 2/1228 0/1230 0/1230

Total

Lost 206 177 111 69 19 1 1 2 0 0

Fig. 1

Locators 1 positions and aircraft path. Aircraft error study.

4. Conclusions

The following represent conclusions with, varying degrees of

certainty; some only say what might be true^ or what should be checked

out. They are organized to parallel the earlier list of questions.

(1) The system will work under reasonable conditions, such as

outlined below.

(2) The system will generally locate ground units within reasonable

error bounds whenever such units have at least 4 fixed locators

or well-located moving locators (which may be airplanes, as long

as they are not too nearly overhead)

.

(3) x, y-errors will be the same order of magnitude as range errors,

for about 4-6 good locators; the accuracy (error) should improve as

l//n-2 where n is the number of locators, up to the point where

the computer word length begins to limit the algorithm ! s accuracy.

The z-errors will be comparable for genuine 3-d situations; they

will depend solely on the accuracy of the independent pressure

reference when that is used.

(4) Use LSS; to find a starting value when extrapolation from previous estimates is not to be trusted, use LSL. Use an outlier^

rejection scheme in each, in addition to a preliminary screening

of the incoming ranges to eliminate gross (10%) errors. Use the

2-dimensional version on ground units, with slant^range reduction

if the terrain is not smooth (unless all but 1 or 2 of the locators

are aircraft at high elevation angles, say 60° or more, in which

case 3-d treatment is required) ; also use 2-d with slant-range

reduction for aircraft with altitudes less than about 1000m,

possibly in conjunction with 3d for aircraft between (say) 600

and 1000m. Use 3d for aircraft over 1000m . (Note; These numbers

depend on the spread of ground units ; the values given are appropriate for the Boston deployment.)

(5)

( 6 )

(7)

( 8 )

(9)

( 10 )

( 11 )

(12)

Fixed units should constitute a large fraction (at least 1/2) of the

set of reporters. The system will work with fewer. But the possibil- ity exists that the set of users' estimated positions will "break its moorings" and wander away, as one entity, from the true position

It is not necessary that the same units be fixed throughout the period

Of time that the system is in use. (In which case it follows that

the positions in which they stop will not be "known", only estimated.) This question, concerning the number of reporters needed, is answered

under (3) and (4) above.

It is understood that maps will not be generally available, but this

is fortunate in a way, since their use extracts a very heavy penalty m computation time. Altimeters (or more accurately, "pressure references", or p.r.'s) will be necessary when aircraft at low elevation

angles are working with ground units, either as locatee or as locators

and will be helpful whenever the terrain is rough enough that altitude

differences among ground units are as large as the error a of the p.r

accuracy .of 14m (a) in p.r. values (that is, a = 20m for the

difference between the heights of two units) will enable the system

to maintain position errors comparable to range-measurement errors.

Better p.r. 's will of course permit more accurate position estimates.

eporting once per 30sec. seems reasonable, except: a) Moving

locators should report more frequently, especially aircraft; and

b) if the system is to be used for aircraft guidance, close support

or fire control, then all aircraft should report more often, say once

every second or two.

To use genuine 3d, a minimum of 3 locators is required; only one need

be an aircraft. However, at least 5 (including 2 aircraft) is advisable

so that bad measurements may be detected. Note that aircraft at low

elevation angles count as ground units here,

and (13) - See answer to (4).

(14) Use as many locators as possible, but properly weighted.

(15) Weight fixed, well-known locators considerably higher than others.

(See discussion of weights earlier in this section.)

(16) Continual relocation is not advisable. Periodic relocation may be

worthwhile; compare with (19). below.

(17) Do not use only fixed units as reporters, except in situations with

unusually good intervisibility. (There are too many times when a unit won’t have enough fixed locators.)

(18) There are indications that relatively large errors will occur when

a locator begins to move, if the system continues to treat his position as fixed. Also, whatever time it takes to establish that a unit

has stopped becomes a delay in "fixing" that unit’s position so that

he can become a good locator. Therefore a definite penalty is attached

to not having a moving/stopped indicator. The size of this penalty

can only be established after the appropriate filtering techniques are

implemented. Thus a complete answer awaits a development study.

(19) Indications are that it is possible to resurvey the set of fixed units.

In the course of system development, attention should be paid to developing, implementing, and testing a multivariable least-squares approach,

allowing the adjustment of all but a few of the fixed reporters’ position estimates simultaneously. This technique would be applied periodically during long battles, when the set of currently-fixed reporters

becomes considerably different from the set of initially-fixed reporters.

(This technique might be either a supplement to, or a replacement for,

the periodic use of relocation; see (16) above.)

Appendix

PROJECT "WHERE” : WORKING PAPER NO. 6

ERROR PROPAGATION POR ONE -DIMENSIONAL LOCATION OF TWO UNITS

M.H. Pearl

March ,1972

ABSTRACT

The analysis of error propagation for position location in a onedimensional geometry, assuming Gaussian error distributions and maximumlikelihood estimation (i.e., location by minimizing a particular quadratic

penalty function), is mathematically tractable. Although this scenario is

unrealistic, "solving" it in closed form may yield insights applicable to

more general situations. Working Paper No. 4 developed formulas for the

law of error propagation through a single location-operation in such a situation. The present note proceeds to treat a series of location-operations for the (simplest) case where only two units are involved, and shows

that in this case the precision of location stabilizes over time. It is • planned next to seek to extend this encouraging result to more than two

units.

Note : Project working papers are informal documents prepared to facilitate

discussion and communication; they may contain tentative or relatively

unchecked material.

0. INTRODUCTION

Following the terminology of other papers in this series , we shall

use the term location-operation to refer to the process of imputing a position to one unit (the locatee ) , on the basis of its measured distances

from a set of other units (the observers ) whose estimated positions are at

hand. Both the distance measurements ; and the estimates of the observers'

locations, are subject to error. The purpose of Project "WHERE" can be

roughly described, from a technical viewpoint, as that of determining (for

a variety of scenarios) the propagation of location-error over a series of

such operations, in which the role of "locatee" varies cyclically among a large set of units, many of which may be in motion during part or all of

the series. As a principal tool for this purpose, project staff are

developing the various components of a computerized simulation ("WHERESIM")

.

While such a simulation is needed for the examination of error-propagation in "general" or "unpatterned" situations, it is plausible that valuable

insights can be obtained from the study of special situations which are

simple enough to admit direct mathematical analysis. Working Paper No. 4^^

initiated one such study, whose key simplifying assumptions are those of

(a) one -dimensional geometry, and

(b) Gaussian error distributions.

Specifically, WP4 derived the error-propagation formula under those assumptions for a single location-operation, leaving for a next phase of work the

recursive application of this formula to determine error build-up through a sequence of location-operations, with various units serving successively as locatee. The present paper carries out this further analysis for an especially

simple situation, that in which only two units are involved. The units thus

alternate between the roles of locatee and observer.

Project "WHERE" Working Paper No. 4, Error Propagation for Position

Location in One Dimension , M.H. Pearl, December, 1971. To be referred to

in what follows as "WP4."

The analysis in WP4 was divided into two cases, according as the

single location-operation considered there did or did not make use of a prior estimate of the locatee's position. From the results in WP4 (see

its last page) and a variety of additional evidence developed in other

parts of Project "WHERE" , it now seems clear that such prior estimates

should indeed be employed, and so their use is assumed in what follows.

Specifically, each such prior estimate is obtained from (i) the unit's

estimated position at the end of the last previous location-operation and

( ii) its velocity of motion(assumed known exactly) in the interim.

With the decision between "prior-using" and "prior-ignoring" thus

made in favor of the former, the analysis of the present paper is split

( 2 ) into two cases along a different line of cleavage. In Working Paper No. 2 , it was suggested that the measurements and calculations taking place during

a location-operation might be exploited not only to locate the locatee, but

also to revise the currently assumed positions of the observers, so that

one would in effect have a "location-and-revision" operation. Accordingly,

both "with revision" and "without revision" scenarios will be analyzed

below. The most significant finding is that- (in both cases) the precision

of location stabilizes over time . (The natural next step is to investigate

whether this encouraging result combines to hold for more than tv?o units.)

Although familiarity with WP4 is not essential, it will be quite helpful

in reading the present text, which is organized as follows. Section 1,

appearing next, describes the underlying mathematical models involved and

sets up most of the necessary notation. Section 2 presents the error-propagation analysis for a sequence of "location-with-revision" operations, while

Section 3 does the same for the "without revision" case.

( 2 ) Project "WHERE" Working Paper No. 2, An Iterative Approach to the

Location-Operation , A.J. Goldman, October, 1971.
.
1. MATHEMATICAL FORMULATION

The analysis deals with 2 units located on a line (i.e., the geometry

is one-dimensional),, The relevant time period begins at time t * 0 , and

the successive location-operations occur at times t. (j - 1 ,2 , ...) where

0 = t < t. < t 0 < . . . . o 1 2

For i = 1,2 and j = 0,1,2,..., we set

so that

U. , = true position of unit i at time t.

ij J

D.

J true (signed) distance between

the units at time t

( 1 . 1 )

The 1 s are unknown to the location-system, which jls however "given”

at time t. quantities.

u.. — prior estimate of U. . at time t. , ij iJ J

to be discussed further below. The

tion-operation is based consists of

information on which the

u

2 j

3 and

j-th locad. = measured value of D

.

J J

This measurement is assumed unbiased, i.e. E(D.)=d, , and also normally

J J distributed, so that the random measurement error

S . *= d . - D . J . J J

( 1 . 2 )

is Gaussian and satisfies (for some a > 0 )

E(S ) = 0 , E(5

2

) = a

2

. (1.3)

Note that our notation involves a and not O. ; i.e., we assume no

J systematic change over time in the quality of the distance measurements.

The initial position-estimates, u^q and u^q , are assumed to be

unbiased. i„e. E(u. ) = U. , and also to be normally distributed; in 3 10 10 3 3 terms of the vector u defined by

o

Tu

o

(u

10> u20^ ’

we set

Q = var(u ) o

N o

var ( u

10> COV ^ U

10

,U20^

COV( U

10'U

2(P

var ( u

2o^

(1.4)

the matrix Qq

(describing the quality of our start ing position-data) , as well as a (which describes the quality of distance measurements), are the

basic inputs to our analysis of error propagation.

It is assumed that between t.. , and t. , unit i moves with known

j-i j

velocity v„ ; it therefore covers a known distance

A. . v

ij

(t

j

(1.5)

Thus the prior estimates of the two units' positions, just before the first

location-operation, are

u

il u . + A.10 . 10 (i = 1,2) ;

it follows from (1.4) that

Qq = var( Ul ) , ( 1 . 6)

where we have used for j = 1 the notation

u. = (u _,u 0 . ) . J Ij 2j

Let us define the vector by

U <W •

The calculations of the j-th location-operation, in our present model

,

yield

U. — maximum- likelihood estimate of U.

J J based on the data u. and d. ; j j

’

this estimate is known (cf . WP4) to be unbiased, and we put

Q. = var (U.

)

J

N J (1.7)

It is at this point that the "with revision" and "without revision"

-k cases diverge. In the first case, U. .is accepted as the vector of loca1 * tion estimates after the j-th location-operation; thus represent the

quality of our position- locations at time j , so that our aim is to

•k determine these matrices (and in particular, their behavior as j

increases). Moreover, in this case the "next" set of prior estimates is

given by

u .,i = U + A. , j+i j i

’ ( 1 . 8 )

where we have defined A. by

J

In the second ("without revision")

after the j-th operation, denoted

is taken as

case, the vector of

kk

U. with components

location estimates

Q

kk

ij

and U

2j

kk kk

U

lj

U

lj J U

2j

kk kk

U

lj = U

lj ’ U

2j

k

if j odd (unit 1

if j even (unit 2

the locatee) ,

(i.

the locatee) ,

so that the prior estimate of the observer’s position indeed goes unrevised.

The "updating" formula (1.8) is of course replaced by

And now the matrices

kk

u.-t = U. + A.

J+l J J

kk _ kk % Q. = var(U. ) J J

( 1 . 10)

( 1 . 11 )

are the matrices which are to be- investigated, and whose limiting behavior

(for large j ) is to be investigated.

The analyses of the two sequences [Q^ J and (Q^. J of matrices, are

the topics of Section 2 and 3 respectively.
c?
2. ANALYSIS ASSUMING REVISION

It is convenient to define the random error vectors

e. = u. - U.

J J J ( 2 . 1 )

Then the likelihood function for the first location-operation is

L = K exp{-%(ej[ Qg

1

e x + a'

2

&

2

) }

where K is a positive constant, so that maximizing L is equivalent to

minimizing

"k T — 1 — 9 9

L = %(«£ Q 0 \ + a

z Sp . ( 2 . 2 )

From (1.1) and (1.2) ,

b, = d. + [1, -1 ]U- ,

and from this, (2.1), and (2.2), it follows that the maximum- likelihood

* estimate is the value of which minimizes

L* = %{(u

x

- Up1 Q" 1 ^ - U

x

)

+

^‘ 2

(d

1

+ [1,-1] u

1

)

T

(d

1

+ [1,-1] Uj_) } .

(2.2a )

We next apply the (straightforwardly verified) formula

3 [(AX + B)

T

S(AX + B)]/3X = 2A

XS(AX + B) (2.3)

where X is a column vector and S is a symmetric matrix. Taking A - I

(the 2 X 2 identity matrix), B = u^ and S = Q“^ , we obtain
'
3{ VV (2.4)

T

Qo

1(u rU

l )}/5Ul = 2Q0 1(Ul"u

l

) 5

taking A 55 [1,-1] , B = d^, and S = I ; we obtain

3{(d

x+ [l,-l]U

1

)

T

(d

1

+[l,-l]U

1 )}/3U1 = 2[l,-l]T

(d

] +[l,-l]U

= 2{ [1,-1 ] T dj+ NU

X

}

where we have put

N = [1,-lf [1,-1] 1

-1

Thus the equation dL /dU^ ** 0 , to be satisfied by **

,

Qo1(ur

u

i>

+ a

’ 2

t 1 ^- 1 ] di

+ a

' 2

nu

x = 0 ;

setting

H *= a”

2

Q N , o o’

we see that this is equivalent to

(I + H

q

)U* = u

x + a'

2

Qq[ 1,-1] d

x ,

so that

U* = (I + H

o )' 1

(u

x + a'

2

Q

o [l,-l] d

x

) .

Since cov(u^d^) = 0 y we have by (1.6) and (1.8)

var (.u

X + CT " 2

d p

- Q0 + o'

4

Q

0 [l,-l]a

2

[1,-1 f Qc

)

, (2.5)

( 2 . 6 )

reads

(2.7)

( 2 . 8 )

^VV (2.4)

1 Q^CvVJ/aUjL “ ;

taking A a [1,-1] , B - d^, and S = I , we obtain

3f(d

1

+ [l,-l]U

1

)

T

(d

1+[l,-l]U

1 )}/SU

1 = 2[l,-l]T

(d

1

+[l,-l]U

] = 2{ [1,-1 ] T dj+ nu

x

5

where we have put

n = [i,-ir [i,-i] « i -i

-i i

"k . ~k Thus the equation dL /dU^ = 0 } to be satisfied by *=

,

Q" 1

(U

1

-U

1

> + a'

2

[l,-l] d

x + a"

2

HU

X - 0 ;

setting

H = a“

2

Q N . o o

7

we see that this is equivalent to

(I + H

q

)U* = u

x + a* 2

Qq[ 1,-1] d

L ,

so that

U* = (I + H

o

)

_1

(u

x + a

-2

Q0

[l,-1] dp .

Since cov(u^^d^) = 0 } we have by (1.6) and (1.8)

var(u

x + a'

2

Q

o [l,-l] d^

“ Qo + CT ’ 4

^o [1 '" 1]a2 [1 ^

1]T

Qo

)

, (2.5)

( 2 . 6 )

reads

(2.7)

( 2 . 8 )

= Q + cr (2.9)

o -2

Q N Q = (I + H )Q

o o o o

Since (2.8) yields

Q 1 = var(U

1

)

= (I + H

o

)

_1

var(Ul + a

' 2

Q

o [l,-l]d1

)(I + H

o )"T

,

it follows from (2.9) that

•k -T

Q 1 - VI + V >

which is equivalent to

<

= (I+H0

) _ 1

Q

0

.

We turn now to the second location-operation. By (1.8),

"k k

var(u

2

) = var(U

1

) = Q x ,

k

i.e. plays the same role now that was played by previously,

if by analogy with (2.7) we put

H

x = a

2

N ,

then by analogy with (2.10) and (2.11) we will obtain

Q 2 = V1 +

k -T * -1

H,) «* (X + HO

( 2 . 10)

( 2 . 11 )

Thus

,
.
More generally, the transition from to is given by the two

equations

Q

kk

j+1

*

H.

J

-2 * * q

j

N

•k k , * -T

Q. = Q.(I + H.)

J+l J J

* -1 k

(I + H.) Q.

J J

( 2 . 12 )

(2.13)

k

Q - Q

o o with initial conditions 0 =0 and

Using these results, we will prove by induction on j

*H = H

o o that

Q* = (I + jH )' 1

Q = Q (I + jH )" T

J

v o o o o (2.14)

By (2.11), it is true for j = 1 . Suppose it is true for some particular

value of j ; we will prove it correct for j + 1 . First, by (2.12) and

(2.14),

JL A i i i H. = a"

z (I + jH ) Q N = (I + jH ) H = H (I + jH ) . (2.15)

J

v o o o o o o

If we set

Z. = (I + H* )

_1

, J J

then

I

so that

(I + jH ) « Z . (I + jH ) + Z ,H - Z . [I + ( j+l)H ] O J O JO J O

and thus = (I + jH

o )[I + (j+DHj

-1 = [I + (j+l)H

0

] _1

(l + jH

o

);

= Z . (X + H* ) , J J- = Z. + Z.H (I + jH ) J jo o

y

Z.

J

it follows from (2.13 - 15) that

*Vi*

Z.Q.

J J

[I+(j+l)H

o

] _1

[I + jH

Q

][l + jH

o

] -1

Qo [X+(j+l)H

o

] -1

Qq ,

so that (2.14) also applies for j-fl . The induction proof is complete.

Note that with (2.6), (2.7) and (2.14), we have succeeded in

k

determining the matrices explicitly in terms of the basic data

and ex . We now prove that the limit

= lim

CO *Q

j

(2.16)

exists, i.e. that the precision of position-location stabilizes over time , k

and moreover we shall evaluate explicitly.

Let

then by (2.6-7)

I + jH * J o

1 + cr 2

j(qx -q) o

2

j (q -

- 2 . <7 J

-2 . / N o J(q ~ q 2

>

(a /j) + (q

x -q)

(q ~ q2

)

-2

1 + O j(q 0 - q)

q - qi

(o /j) + (q2 -q)

(2.17)

The coefficient of a is a matrix with determinant

(a

4

/j

2

) + (a

2

/j)(q

1 + q

2

“ 2ci) = (tf

2

/j)[(a

2

/j) + (q

x+ q2

~ 2 q)]

and thus with inverse matrix

(a

2

/j)

1

[(a

2

/j) + (q x+ q2

~ 2q) ] 1

(a

2

/j) + (q 0

“ q) q

x - q

q2

“ q (<? /j) + (q^ q)

It follows from (2.17) that

(I + jH

Q

)

1 = [(cf

2

/j) + (q^ + q2

- 2q)]

1

(c /j) + (q2 - q) q

x -q

q2

- q (a

2

/j) + (q r q)

In particular.

(2.18)

lim . (1 + jH )

_1 = H*

J

-» CD N O (2.19)

exists and is given by

H s (q

x + q 2

~ 2 q)

1

q 9 - q qi - q

q 9

- q «i

- q

( 2 . 20)
.
thus by (2.14) the limit exists and is given by

= (q 1 +q2

‘ 2 <i >' 1 (q 1 q2

‘ c

i 2) J > ( 2 - 21 )

where J is a 2X2 matrix of ones. It is somewhat surprising that the

limit is independent of a

A natural question is whether the position-locations grow increasingly

* * precise over time, i.e. whether the diagonal entries of = var(tL ) are

decreasing functions of j .By (2.14) and (2.18), the (1,1) entry of

*

var(U^j) = q1[(a

2

/j) + q^Uj^" q

2

) 3/ [ Cct 2

/ j ) + (q

1+q

2

" 2 q)] • ( 2 . 22 )

Now it is readily verified that the function

f (x) s» (x+a)/(x+b) = 1 - (b-a)/(x+b)

is increasing or decreasing according as (b-a) is positive or negative.

Applying this with

2 -1 2

x = a /j , a = q x (q^ - q ) , b 88 qx + q2

“ 2q ,

ql

+ q2

“ 2q ~ ql

1

^ q l q2

" q2

^ ^ • (2.23)

2 which indeed holds since it is equivalent to (q. - q) > 0 , Analogously,

* var(U

2j) is a non- increasing function of j
' I : m
3. ANALYSIS FOR LOCATION WITHOUT REVISION

We now turn to the matrices

var(U. ) J (3.1)

corresponding to the location-without-revision scenario. The analysis is

more complicated here, since the two units no longer figure symmetrically

in each location-operation (recall that unit 1 is the locatee in odd-numbered

operations, unit 2 the locatee in even-numbered ones).

It is convenient to repeat from Section 2 the definitions

column k ) position, and all other entries zero. Thus, for example, the

consequence

N = [1,-1]

T

[1,-1]

1 -1

(3.3)

-1 1

H

o (3.3)

of (1.9) can be written
• !
From this and (3.1) it follows that

irk

Q x = E n

var(U

x

) E n + E 22 va^u^E^

+ E

X1 cov(U1

, ^

1

)E

22 + E 22 cov(u

i;

U

x

) E

1X

(3.5)

By (2.8) and (2.11)

cov(U

1 ^u

1

) (I + H

)“ 1Q o o - Q, >

so that (3.5) yields

Q 1

E

11 Q 1 E

11

+ E22 Q

o E

2:

+ E

X1 Q l

E

22 + E22 Q x E

1X

= <*1 + E

22 <V ^ 1> E22

By use of (2.11), this reads

Q** = (I + H

o

)' 1 Qo + E

22 [I - (I + H

o )' 1

]Q„ E

o 22

' (3.6)

The second location-operation follows the same logic as the first,

irk

except that plays the part of Qq

and that the roles of the two units

are interchanged. Thus, by analogy with (3.3) and (3.6), we have

H.

irk

a’

2

Q, N

irk irk^ - ] irk , irk -1 _ irk >2 = (X + Hj_ ) Q x + E u tl - (X + H

1

) ] Q x En

(3.7)

(3.8)
tT
More generally, the appropriate recursion equations are

kk - 9 **

H. = cr Q. N

J J kk

Q

2j+1

kk -1 kk kk^ -1 kk

(I + H

2j ) Q2j

+ e 22 [i - (I + H

2j ) ]Q 2

. e 22 ,

(3.9)

(3.10)

Q 2j+2 = (1 + H

2J+1

) Q 2j+1 + E n [X - (I + H^)-1

] Q2j+1

E u ; (3

these are accompanied by the initial conditions Qo^ooo =0 and H ~ H

In view of the way in which the analysis "alternates" between the two

units, it would be unreasonable to expect the type of convergence given by

the existence of a single limiting matrix

kk kk

Q = 1 im . Q

.

^00

J

•> 00 ^ j

Rather, one might hope for "convergence with period 2" , i.e. for the

existence of a pair of matrices

Q° = lim. Q 0 . - , Q

6 ~ lim Q 0 . , (3.12)

\» j

-> CD X 2j+1 ^ OO j

-> CD ^2j 3

where the superscripts "o" and "e" stand for "odd" and "even"

respectively. 1 Furthermore, in view of the results obtained in Section 2, one

would not be surprised if these two limiting matrices turned out to be

independent of a .

At present we have not succeeded in analyzing this situation with anything like the completeness achieved in Section 2 for the "location with

revision" case. However, we have succeeded in proving that in the present
F

$X
case, like that of Section 2, the precision of position-location stabilizes

over time . To make this precise, let

Q

j

(j) (j)

1 q

(j) (j)

then = q and q^°^ = q^ (i = 1,2) . The precision with which the j-th

location-operation estimates the position of the locatee is given by

a)

2 = var (U* ) if j is even ,

(j>

1 = var(U, .) if j is odd

It will be shown below that

sequences {q^^} and are convergent, (3.13)

thus proving the "stabilization" mentioned above.

The logic of the proof will run as follows. It will be shown that

sequences {q^*^ } and {qjj^*^ ] are convergent . (2j) (3.14)

This is a property of the sequence {q } of even-index matrices. Since the

sequence {Q

2

j-j-i

5 °f odd-index matrices obeys the same type of recursion,

and since (3.14) is symmetric as between the two units, it follows by analogy

that

sequences { qi

(2J+1)

} and r ( 2 j+l)->Uo i are convergent

.

(3.15)

The desired result (3.13) then follows from (3.14) and (3.15).

Thus it suffices to prove (3.14). Since the variance terms q

(2j) and

(2j) are nonnegative, i.e. bounded below by 0 , it suffices to prove that

the sequences in (3.14) are non- increasing . These statements.

q

(2j +2) < q

( 2j ) (2j+2) < q

(2j)

are to be proved by induction on j , but it suffices to establish the

prototype induction step, namely

( 2 ) . „ q l ^ q l

(2) < n q 2 - q 2 (3.16)

The proof of the desired result (3.13) has now been reduced to the

demonstration of the two statements (3.16) . For proving them, it is

convenient to introduce the auxiliary quantities

s = (q

x -q) + (q 2 -q) A

2

ql q 2" q (3.17)

Since Q is a variance-covariance matrix, its determinant A > 0 , and

it follows by (2.23) that S > 0 . The following calculations will make use

of the easily-checked identities

A - qs = (q 1 -q)(q

2

“ ci) * 2 A - = -(q^q)

(3.18)

(3.19)
I
Direct calculation from (3.6) yields

** ,2 N -1

Q n - (a + S)

2 2

a + A a q + A

2 2

a q + A (a + S)q,

(3.20)

From (3.7 ), we obtain

**.-1 r 4 2 N 2,-1

(I + H

x

)

1 - [c + 2Sa + (q2 «q) ]

a

4

+ (S + q2 -q)a

2

+ (q2

*-q)

2

^(q^^-q)

ff 2

( q

2

“ q ) + ( q2

" q ^ 2

a +(S+q-- q)cr

Application of (3.8) and (3.20) yields

= (a

2

+ S)‘

1

(a

2

q 1 + A) ,

so that the first of the desired statements (3.16) is equivalent to

cr q^+ A < (a + S)q^

and hence to A - q^S < 0 , which is true by (3.19). Next, substitution

into (3.8) yields

qo

2^ = (cr + S) [a

4

+ 2Sa

2

+ (q0 -q)

2

] 1 {q9 a^ + [A + 2q_S ]a

4

+ [2q S

2

- (q 9 -q)

2

(q 9 -2q) ]a

2

+ A(q9 -q)

2

} ,
.
so that the second of the desired statements (3.16) is equivalent to

q2

a

6

+ [A + 2q2

S]a4

+ [2q,,S

2

- (q 2 -q)

2

(q 2 -2q) ]a

2

+ A(q

2 -q)

2

< q 2

^

CT ^ + ^sa + Cq2

-<i)

2

1 i

which is true since it is in turn equivalent to

(q2 -q)

2

[C

2

+ (q2 -q)]

2

> 0 .

This completes the proof of (3.13), so that the existence of the

"variance terms" in (3.12), i.e. entries q° and q° in Q° and entries

e e e q^ and q^ in , is established. We have not as yet succeeded in proving

the existence of the covariance limits

oq(2j+l)

-> CD ^ lim.

J (2j)

-> OO (3.21)

i.e. the off-diagonal terms in (3.12), except under the additional hypotheses

q x > q ; q2 > q * (3.22)

This assumption is probably not very restrictive for the situations of

interest here; it asserts that the errors in the initial estimates (of the

two units’ positions), if positively correlated, have covariance less than

either of their individual variances. Because A > 0 , (3.22) will automatically be satisfied if q^ = .

To prove that (3.22) implies (3.21),-

ilization

we first prove that it implies the

(3.23)
i?f o
of itself. The proof is by induction on j , and since (3.23) is symmetric

as between the two units , it suffices to treat a prototype induction step,

i.e. to show that

> q

(i)

(3.24)

follows from (3.22). From (3.20), we see that the first part of (3.24) is

an immediate consequence of the first part of (3.22). Also from (3.20), we

see that the second part of (3.24) is equivalent to

(a

2

+ S)q 9 > a

2

q + A ,

hence to

2 2

o’ (q 9 - q) > A - q 9 s - (q 9 - q) ,

which follows from the second part of (3.22) .

We will show below that (3.22) implies

/ n • \ sequence {q'' J

} is non-decreasing (3.25)

ii'k Because the variance-covariance matrix Q must have a non-negative

^J determinant

,

r (2j) 2 . (2j ) (2j)

Lq . J <

q^_ q 2 *

it follows that the sequence in (3.25) is ultimately bounded above by a

g 0 quantity [ q^q2 3 2 + G where e > 0 , and so this sequence must have a limit

e r ** -s c * * 1 q . This pertains to the sequence 1^2j

> since the sequence j_Q 2 j+i

is generated by the same type of recursion, and since by (3.24) its initial

matrix satisfies the analog of (3.22) , the same logic proves the

existence of the limiting covariance q°

It only remains to prove that (3.22) implies (3.25). Since (3.22)

implies (3.23), it suffices to establish the prototype induction step,

namely that (3.22) implies

> q • (3.26)

Application of (3.8) yields

q

(2) = (a

2

+ S)“

1

[a

4

+ 2Sa

2

+ (q^q)

2 ]" 1 {qa6

+ [(q 2 ”q)q-j+ A + q(s + q

1

- q)]cr

4

+ [(q 2 -q) A + q

1 (q2

- q)

2+

( s + q 1 "q)A]a

2

+ (q 2

- q )

2 A 3

« (a

2

+ S)**

1 [a

4

+ 2Sa

2

+ (q 2 -q)

2 ]“ 1 {qa6

+ [2A + qS]a4

+ [2SA + q^ (q 2 ~q)

2

]a

2

+ (q^-q)^} .

Thus (3.26) is equivalent to

qa

6

+ [2A + qS ]a

4

+ [2SA + q^(q2 ~q) ]a

2

+ (q 2 ~q)

2 A

> q(^

2

+ S)[a4+ 2Sa

2

+ (q2 ~q)

2

] ,

or in turn to

(A - qS){2a4

+ [2S + (q2 ~q) ]a

2

+ (q2 ~q)

2

] > 0 ,

which is true by virtue of (3.18) and (3.22). This completes the proof

that the limiting covariances (3.21) exist.
I
matrix satisfies the analog of (3.22) , the same logic proves the

existence of the limiting covariance q°

It only remains to prove that (3.22) implies (3.25). Since (3.22)

implies (3.23), it suffices to establish the prototype induction step,

namely that (3.22) implies

q

(2) > q . (3.26)

Application of (3.8) yields

= (a

2

+ S)

1

[cj

4

+ 2Sa

2

+ (q 2 ~q)

2

] 1{q^

6

+ [(q 2

“ ci) c

i 1

+ A + <i( s + q 1 ~q) t(q 2 -q) a + q^(q2 "q) 2+

( s + q 1 ~q)^]cr

+ (q2 -q)

2

a }

= (a

2

+ S)"

1

[a

4

+ 2Sa

2

+ (q^q)

2 ]" 1 {qa6

+ [2A + qS]a4

+ [2SA + q

1 (q2 ~q)

2

]a

2

+ (q 2 -q)

2A} »

Thus (3.26) is equivalent to

qa

6

+ [2A + qS ]a

4

+ [2SA + q1 (q 2 ~q)

2

]a

2

+ (q2 ~q)

2 A

> q(a

2

+ S)[ct

4

+ 2Sa

2

+ (q2 -q)

2

] ,

or in turn to

(A - qS){2a4

+ [2S *f (q 2 “q) ]a

2

+ (q2 ~q)

2

3 > 0 ,

which is true by virtue of (3.18) and (3.22). This completes the proof

that the limiting covariances (3.21) exist.
-

.
Although the main objectives in this Section have been achieved, we

should note several items of "unfinished business" which need to be

settled to obtain a thorough analysis :

o e

* (a) Determine whether the limiting covariances q and q (and

o e hence, the limiting matrices and ) exist in general, and not only

under the extra hypotheses (3.22).

(b) Evaluate the various limits in closed form.

(c) If (b) is too ambitious, at least determine whether the

limits are independent of a

PROJECT "WHERE" : WORKING PAPER NO. 7

Algorithms

D. J. Sookne

1. Description of the Algorithms used by WHERSM

The algorithm used to locate each unit is the least squares squared

(LSS) algorithm. Both two- and three-dimensional subroutines are included

in the WHERSM system. The three-dimensional program is described in

detail below.

For each locator i, i = l,2,...,n, let (x^, y , z ) be its estimated

position, r^ its reported distance to the locatee, and w^ the reciprocal

of its variance, the calculation of which is described below. The w.’s

act as weights; the smaller the variance of a locator, the more accurately

we know its position, and the higher its weight will be. Then LSS minimizes

E =

E. =

l

n

T

i=l

w.E.

i l

where

7

(1)

( 2 )

here d_^ is the calculated distance . between (x^, y , z ) and (x, y, z) , the estimated position of the locatee,, so

d? = (x- xp

2

+ (y- y

i

)

2

+ (z- z^2

(3)

Note that

2 2

2 2
1II
When the estimated positions of locators and locatee are near their true

2 2

positions, r. is very near d. , so the factor (d.+ r.) /4r . is very

i i

2

1 1 1

near 1 , and behaves very much like (d^-r^) * Thus we expect this

algorithm to produce answers nearly identical to those produced by the

2

least squares (LS) algorithm, for which E^ = (d^- r^) . Indeed, computer

tests indicate that there is no significant difference between the results

of these two algorithms. Thus LSS is used because it is more economical,

since LS requires the calculation of d^ , which in turn requires a square root to be taken (see (3) ) . LSS avoids the square root, since only

d? is needed,

l

Where E is a minimum, its partial derivatives with respect to x, y,

and z will be zero. Thus the program calculates

F

G

H

BE

Bx

be

By

BE

Bz

2 2

n w. (d . - r . ) „ i i l , £ ( x

i=l rT

l

2 2 . ^ii n w.(d. - r. ) i / £ (y

i=l • r

V

X.)

*i>

2 2

n w (d t r )

S 2

(z

i=l r.

Z

i>

(4)

Since d_^ depends on x, y, and z (see (3) ), F, G, and H depend nonlinear ly on these variables. Then the system (4) cannot be solved directly

for x, y, and z . Instead, LSS does successive Newton-Raphson iterations

( [1], pp. 201-223) , beginning with a starting point .p = (x,y,z) . The

first iteration yields a new point p** = (x + Ax, y +Ay, z + Az) ) this

point is used as the starting point for the next iteration, and so on. The

process begins with the first order approximation

__ . bF A . 3F a . c>F A F = F + r— Ax + — Ay + — Az

p p dx dy dz

G * BG . . 5G a , dG A p = G + — Ax + — Ay + — Az

^ p Bx dy dz

„ „ . dH ' ,dH A , dH A H ^ = H + — Ax + — Ay + — Az

p p dx dy dz

Here F denotes the value of F at point p , and all partial derivatives

are evaluated at p . Thus this system represents the expansions of the

functions F, G, and H about p . Now we want F / , G

'* and H > all to 9 P P P

be zero, so the problem reduces to that of finding Ax, Ay, Az which satisfy

the three simultaneous equations

|^Ax + |^Ay + —-Az+F = 0

dx dy dz p — Ax + Ay + ~ Az + G - 0

dx dy dz p

( 5 ) —dH A , dH A , dH A , n Ax + T— Ay+^-Az + H = 0 . dx dy dz p

The program calculates F, G, and H (see (4) ), and

2 2 2

r.(df - rT + 2(x - x.) , i

v i i

v a/ ' 2L »

n

dx Lj

i-1

n

dy

Lj

i=l

dH

dz

n

£

i-1

2 r

.

l

r.(d

2

1

N 1

2

r

.

i

, ,2 2

r i(

d

i

- 2 / r .

.

i

( 6)
'
( 6 ) bG b bE b bE bF

n 2w . (x

l - x

i )(y -yp

bx bx by by bx by 2j

i=l

2 r

.

l

bH b bE b bE bF

n

E

i=l

2w. (x

l

- x

i

)(z - z.)

bx bx bz bz bx bz 2 r

.

L

bH b bE b bE bG

n 2w

i Cy - y

i

)(z - z.)

by by bz bz by bz 2j

i=l

2 r

.

then solves (5) for Ax, Ay, Az . This determines p*= (x + Ax, y + Ay,

z + Az) , which is used as the starting point for the next iteration. For

a given set of locators, the program will do up to 4 iterations, stopping

if an iteration converges, namely

n

max( | F | , | G | , | H | )< S w , i=l

or if an iteration diverges (which see below) . I.e. convergence occurs

when F, G, and H are all small.

Computer tests indicated that for both the two- and three-dimensional

versions of LSS , 2 to 4 Newton-Raphson iterations are generally sufficient

for convergence, even when the starting point is 100 meters from the true

position.

The LSS algorithm allows for easy elimination of outliers. These

are locators whose reported distance (r^) to the locatee contains a greater

error than do the reported distances of the other locators. Let

n n

* ( r we)/ d w,

• i=i 11 i=i 1

E (7)
I
V

Now any locator i is eliminated for which

E

± > 4eT . (8)

Several other constants greater than 1 were tried in place of 4, but

computer tests showed that 4 was best.

The algorithm iterates on all locators until convergence occurs , up

to 4 iterations (for divergence, see below). Then E is calculated and

some locators may be eliminated via (8) . (If no locators are eliminated,

we're through). Then more iterations are performed on the reduced set of

locators. After convergence, (7) and (8) are again applied to eliminate

outliers . This iteration-elimination process continues until either (1)

no locators are eliminated in a step^ (2) 1/4 of the locators have been

eliminated, or (3) the algorithm diverges (see below) „ In cases (1) and

(2) the algorithm terminates.

There are two types of divergence^ an increase in the objective function, E , from one iteration to the next and a badly scaled determinant

-8

less than 10 in absolute value which makes equations (5) impossible to

solve accurately. In each case, we halve the step size of the previous

iteration, and try again. This can be done unless the iteration at which

divergence occurs is the first iteration for this locatee . Then we must

quit without finding a solution.

For any locatee and set of locators, the size of successive iteration

steps should decrease rapidly. If not, then the problem is ill-behaved,

and perhaps dozens of iterations will be necessary to achieve convergence.

To avoid this, the algorithm attempts to " home in" on the solution

faster. If the step size in the x-direction (or y or z) is Ax^ for one

iteration and Ax

2

for the next, and if | Ax^ | > % | Ax^ | , then Ax

2

replaced by

2min( |Ax^|, | Ax

2

| )

® sgn Ax

2

if Ax^ and Ax^ have the same sign, and if they have opposite signs,

AX

2

is replaced by

AX2 • Ax^ • Ax

1 - Ax

A further refinement of the algorithm was tried; this involved relocating the locators. More concerning this may be found in sec* 3 and Working Paper

No. 8. Once the Newton iterations have converged, there is an error between

the reported distance r_^ and the calculated distance d^ , for each locator

i in the set. By changing the estimated position of each locator, i.e. by

moving it along the straight line connecting it with the locatee, this error

d - r_^ may be reduced, even eliminated. After relocating, more NewtonRaphson iterations may be done. Several runs were made, using different values

for the fraction, f , of the error eliminated by relocation, with 0 < f < 1 . The best such f was the smallest, and that was not quite as good as no

relocation (f = 0) . Relocation remains an option in the LSS routines, but

one which is not now used.

The 2-dimensional LSS program is analogous to the 3-dimensional one. It

solves the equations

dF

dx

Ax +

dF

dy Ay + F ** 0

dG

dx

Ax + ~ Ay +

dy

G = 0

Here, F and G and their partial derivatives are as described above, except

that all z-components are zero. The 2-dimensional program is less likely to

result in divergence of an iteration, so the special steps to handle divergence

(see previous page) are omitted.

The weights w (see (1)) are again taken as the reciprocals of the

2

1

variances o\ . When a unit has been located, the variance of its location

is calculated as

n

( £ w.E.)

- l l

i=l

n

(n-m) £ w .

)

i=l

1

C9)

when n is the number of locators, m the dimensionality (2 or 3) of the

algorithm*

When the locatee is lost, so that no starting point exists for the LSS

routine, the least squares linear (LSL) routine is called to provide a starting point. First, a slant-range reduction is done to convert 3-D reported

distances to 2-B . Then the 2-D LSL algorithm is called. This is the same

basic algorithm which is described in sec. 2 of [2] * The modifications are

these: all pairs of circles are used, and outliers are eliminated. Consider

the M radical axis lines each formed by a pair of circles ) the equation

for such a line is found by subtracting the equation of one circle from the

equation of the other as in [2] , p » 10„ For each such line k, k — 1,2, ...M,

let d, be the calculated distance from the locatee to the line. Then LSL

k

minimizes

D

M

L

k=l

After a locatee has been found using all M lines, the routine calculates

d

2 = D/M ,

the average squared distance to the lines. Then any line k is eliminated

for which

jThe reduced set of locators is used to calculate a position, d is

recalculated, and a line k is eliminated for which

2 2 df> 4 • d

Z

k

The process is repeated twice more, once with the elimination condition

d. > 2

k

2 2

df > d

k

and finally with

The LSL algorithm will fail only when the equations at the bottom of

p. 13 of [2] are ill-conditioned. This will happen when the M lines are

nearly parallel, i 0 e 0 the locators all lie near one straight line. In this

case, no starting point can be found for LSS

The routine FINDIT calls LSS (2-D or 3-D) and LSL when necessary.

If the locatee is not lost, two previous locations are used to extrapolate

in the x- and y- directions to get a starting point for LSS. The last

z-value is used, with no extrapolation. Then LSS is called; the 2-dimensional routine for ground units, 3-dimensional for airplanes.

If the unit is lost, LSL is called after doing a slant-range reduction

(using the last known z-value) to convert 3-dimensional distances to planar

distances. Then using the planar distances, the 2-dimensional LSS is

called. If the unit is an airplane, the x- and y-coordinates returned by

LSS (2-D) plus the last known z-value are used as a starting point for

LSS (3-D)

.

Control is returned to FINDIT with the variance defined by (9) . This

represents the variance of the position estimate, based on the reported
.
distances. The total variance is computed by adding RVAR, the variance

of the distances. If the locatee is known to be moving, its weight is

calculated by inverting the total variance.

If the unit is fixed, the current position estimate is taken to be

a weighted average of the old estimate and the just-calculated estimate.

Then the weight of the locatee is computed as a weighted average of the

old weight and the just-calculated weight. A further description of

these calculations may be found in Section three of this report.

2. Description of Algorithms Programmed for WHERSM But Not Now Used

The linear method (LM) and smallest tangent circle (STC) method

are described in [2], They were both found to be less accurate than

the LSS and LSL algorithms.

The least squares method mentioned in §1 is the same as LSS,

except equations (2), (4), and (6) are replaced by

2

(20

(d

i-r

i

) (y-y

±

) \ (40

d.

l

/

and

n .* d.-r.

9F _ y

i i

9x -iK d. i=l l

r (x-x ) x

d.

l

3 J

9 G

n /d.-r.

i l

3y

= i-ll d

x

r

i (y~yi

)

d.’

i

9H

9z

9

G

9x

l l

n / d.-r. r . (z-z .

)

JA

' ' ' '

9F

9y

d.

l

+

±

KZ-Z ±

)

) 3 J

l

i=l

d.

i

n r (x-x ) (y-y )

(60

9H

9x

9F

9z

n r . (x-x. ) (z-z .

)

l l i

l

i=l d.

3

1

9H

9y

9G

9z

n r. (y-y.) (z-z.

)

l w J i i

l

i=l d.*

l

The other algorithm programmed was MINMAX which minimizes the maximum

of the errors d^ - r' . This algorithm proved

to be less accurate than STC and much less accurate than LSS. Further,

it uses 10 to 100 times the computer time the other algorithms use,

since (in the two-dimensional case) it considers all pairs and triples

of locator circles.
.
References

1. Scarborough, James B., Numerical Mathematical Analysis , John

i Hopkins Press, Baltimore, 1930.

2. First Interim Report on NBS Project WHERE.

LISTING OF POSITION LOCATION PROGRAMS

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PROJECT " WHERE " : WORKING PAPER NO. 8

LS , LSS , AND RELOCATION

James A. Lechner

April , 1972

ABSTRACT

In Working Paper No. 2 it was suggested that since the locators' positions are not known exactly, it might prove profitable to re-estimate these

positions as an adjunct to estimating the position of the "locatee" unit.

This paper presents an iterative method, along the lines proposed in Working

Paper No. 2, for handling the computations of such a "relocation" process.

The discussion is carried out in the context of the LS(least squares) and

LSS(least squares, squared) position-location methods. As background,

evidence is given which shows that these two criteria behave very similarly

so long as range measurement errors remain small.

Note : Project working papers are informal documents prepared to facilitate

discussion and communication; they may contain tentative or relatively

unchecked material.

CONTENTS

0 o INTRODUCTION - 1

1. RELATION BETWEEN LS AND LSS APPROACHES 3

2. THE RELOCATION PROCESS - 7

0. 'INTRODUCTION

In this paper the term location- operation will be used for a process

of attributing a position to one unit (the locatee ) on the basis of its

recorded distances from n other units (the locators ) . These recorded

distances are of course subject to measurement error.

Roughly speaking , the situation studied in Project "WHERE" is one in

which the role of "locatee" varies over some totality of units , with many

of these units in motion at any time. From this it is apparent that for

any one location-operation, the positions of the locators (which may have

been the locatees of previous operations) are not in general known exactly.

This point was emphasized in Section 5 of Working Paper No. 2 [" An Iterative Approach to the Location-Operation" , A P J. Goldman, 10/71], where it

was suggested that the location-operation be expanded to include re-estimation (i.e., relocation) of the locators' positions, based on the latest set

of range measurements.

The present paper has as its main purpose the computational specification of such a relocation procedure. This is given in Section 2, which also

discusses the respective merits of the LS (least squares) and LSS (least

squares, squared) position-location methods in this context. Section 1 presents some relevant background information concerning the relationship

between these two algorithms.

We conclude this Introduction by introducing some necessary mathematical

notation:

0 •» (arbitrary) origin of coordinate system , X ** position to be atrributed to locatee,

x = vector OX

A^ *= position to be attributed to i-th observer

(i 1, 2,o.e, n) >

‘
*

'
vector OA.

1

a.

1

initial estimate of a.

1

— |jx - a^ II 55 estimated distance between locatee

and i-th observer

,

P i

measured value of this distance .

If the distinction between a^ and is dropped (i.e., the initial

estimates of the locators' position are accepted without revision), then we

have the "pure" location-operation, which can be expressed as the problem of

making a "good" choice of x , given the and a^ . The LS method

expresses this problem as that of minimizing the quadratic "penalty function"

f(x) = D w^(r

j

, - pp

2

(0.1)

i

where the w^ are positive "weights" and x enters via the r^s $

the LSS approach involves the minimization of

g(x) = S - P?)

2

, (0,2)

with "weights" w^ . The modification of these formulations, to encompass

relocation, will be taken up in Section 2 .

1„ RELATION BETWEEN LS AND LSS APPROACHES

In this section, as background for the discussion of relocation in

Section 2, we discuss the relation between the LS and LSS approaches

to the "pure" location-operation. Three pieces of evidence will be given

to show that if the "weights" in these two criteria the w^ of (0*1)

and the w^ of (0 o 2) are properly related, then the two methods

will not differ significantly in accuracy so long as the tracking process

is under control 0 (The last phrase means that the units' positions remain

well enough known that the discrepancies between measured ranges p_^and

calculated ranges r^ are small relative to the magnitudes of the ranges

themselves .)

The first piece of evidence begins with the observation that the

"under control" assumption, r^/p^ ~ 1 > yields

,2 2.2 , , .2 , .2

< r

i

- p.) = <r + PjL ) (r. - p.)

(1 + r^p.) 2

P J(r. - p.)

2

2 2

4p7(r . -r p.) , i

s l

r x

9

or equivalently

2 2 2 2

(l/4p

2

)(r^ - pp

Z

(r

i

" p i>

2 * Thus, if we set w^ « (l/4p^) w^ , then comparison of (0 0 1 ) with (0 o 2) shows

that g(x) ex f(x) ; equivalently, if we set

w

i

2

i

( 1 . 1 )

then comparison of (

0

0 1 ) yields- g(x) « 4f(x) , so that minimization of

f(x)--- i.e., the LS approach --- is "approximately equivalent" to the

LSS approach (minimization of g(x) ) .

Equation (1*1) gives the appropriate relation between the two sets of

weights. As yet, no rationale has been found for assigning to the w* any

value other than unity. In fact, since the distribution of errors in

(unrejected) ranges seems (by best engineering judgement at this time) to

be Gaussian, with range- independent variance and zero mean (except for a small percentage of ranges biased by about 8m), maximum likelihood estimation would strongly recommend^^ use of LS with w^ - 1 (together with

some technique for removing "outliers" ) , which by the above is approximately

equivalent to use of LSS with

w. = 1/p* . (1.2)

This choice of w* and w^ will be used throughout the rest of this paper.

The preceding analysis shows that the penalty functions for LS and

LSS are approximately equal (when measured and calculated ranges are close

together), but one would like some direct assurance that the position-estimates

obtained by minimizing these two functions are also nearly equal. This is

provided by our second piece of evidence, an analysis for a simple 1-dimensional configuration in which the locators are situated at the points with

coordinates R and (~R) with R large , while the locatee is actually

situated at the origin. The LS approach calls for the minimization of

f(x) = [(R-x) - Q 1 ] 2

+ [(R-fx) - P 2

] 2

2 = 2 {x + (p^- p^)x } + const..

See Section 1 of Working Paper No. 2 * ( 1 )
.
which is readily found to be given by

x = (P

2

»

p ^ ) /2 . (LS solution) (1 0 3)

The LSS approach calls for the minimization of

/ x -2 r/T. .2 2.2, -2 r/ „ , .2 2.2

g(x) = P

x [(R-x) - P x

] + p 2

[(R+x) - p 2

]

-2 4 -2 4 2 -

p^ (R-x) + P 2

(R+x) - 4x + const, j

setting dg/dx = 0 leads to the cubic equation

, 2 , 2. 3 OT>/ 2 2. 2 ro 2, 2,2. 0 2 2 . , 3, 2 2 . „

(P 1 + P 2

)x + 3R(p

L

“ P 2

)x + [3R (P

x + P 2

) - 2p

x

P 2

] x + R (p 1 » P 2

) * 0 . (1

Let E denote a measure of maximum range-error. In the worst-case scenario

described by p^ = R-E and p 2 = R + E , the LS and LSS approaches yield

exactly the same answer, both (1„3) and (1„4) giving x « E . In the alternative scenario described by p^ = R and p 2 = R + E , the respective solutions

are

*LS = !

E » "LSS - IE [1 - |(E/R) 2

] (1.5)

where the expression for x^gg omits higher-order terms in the small quantity

E/R ; if for example R is at least' 1 km and E is at most 20m , then the

two position-estimates differ by at most 0 o 0015 m .

Our third piece of evidence is empirical in nature. For each of two

distributions of range -measurement errors (’’small" and "large", respectively)

72 trial 2-dimensional location-operations were carried out using both LS

and LSS . Each of 18 units served as locatee 4 times, (for each error distribution and each method), with the other 17 units serving as locators. For

the "small" case, the locations were as much as 3.32m in error, but the LS

and LSS results differed from each other by no more than 0„ 0004m ; for the

"large" case, despite location errors up to 31.9m, the two methods yielded

locations no more than 0 o 045m apart.

2. THE RELOCATION PROCESS

When the location-operation is enlarged to include relocation (of the

locators) , it can be formulated as making a ’‘good" choice of x and the

a^ , given the and . The approach given in Section 4 of Working

Paper No. 2 is based on two notions, one of which we have found it is useful

to modify in a manner described below.

The first notion is that the choices of x and a!s should be made so

i as to minimize a penalty function which generalizes those given earlier [see

(0.1) and (0.2)] for "pure" location. Specifically, this function is

f (x$ a

1 ,..., an

) = 2. w'(r. - p.)^ + D. W. I! a.- o'. I! (4.1a) iiN i l i l

11 l i' v '

for the LS approach, and

g(x; , . . . , a^)

2 2 2

£. w.(rT - PV ) + £. W.

l i

N i i li (4.1b)

for the LSS approach ; the are positive numerical weights.

The second notion is to accomplish this minimization by a process

which alternates between location steps (in which the a^ are held fixed

and an appropriate x is chosen) and relocation steps (in which x is held

fixed and appropriate a^'s are chosen. This process can be formalized as

follows

:

INITIALIZATION: Set k « 1 , and = Oi

.

(all i ) . 9 l l

'

(k) STEP 1 (location, k-th pass) : Given the current estimates a;

(k)

1

of the locator positions, choose x - x

v ' to minimize the first sum in

(4.1), i.e. the part of (4.1) which depends on x

STEP 2 (relocation, k-th pass) : For each i, keep x fixed at

(\r\

x

v ' and choose a. to minimize the sum of those terms in (4.1) which

1

(k) depend on a^ . Increase k by 1 , and set each a^ J equal to the

a_^ just found. Return to Step 1 .

For Step 1, the material in Section 1 shows that there is little

reason to prefer one of LS or LSS over the other so far as locationerrors are concerned. We therefore select LSS, since it seems likely

to converge more quickly and requires fewer time-consuming calculations

per iteration. Thus Step 1 employs (4.1b) rather than (4.1a), and so according to the description above, the treatment of the i-th locator in

the k-th relocation step would require choosing a^ to minimize

w.

l

- a

.

l (4.2)

For Step 2, however, it is more convenient to minimize instead the sum of

corresponding summands from (4.1a_), namely

Hap (4.3)

The over-all method becomes a kind of LS-LSS hybrid ; Steps 1 and 2 in

effect involve different penalty functions, so that the "first notion"

described above has indeed been modified.

In Working Paper No. 2 it was envisiged that the alternating process

would be iterated until it had "settled down" satisfactorily. This idea,

too, has been modified after further reflection. The relocation of the

i-th locator is based mainly upon a single new datum, namely ; it seems

unwise to put too much reliance -or to base too much calculation on this one

datum (which is subject to measurement-error). Besides, from the viewpoint

of reducing total computation time it is desirable to limit the number of

iterations. Thus a sensible compromise seems to be: locate the locatee

(Step 1); relocate the locators (Step 2) ; then locate the locatee again

(second pass through Step 1), and stop. This truncated version of the full

process is assumed below.

We turn now to the execution of Step 2. With superscripts (k) and

subscripts (i) dropped for simplicity, the function (4.3) to be minimized

can be written

h(a) = w'{|jx-a || - p } ^ + W || a-a || 2

. (4.4)

Consider any trial position a' of a .It lies on a sphere of radius

|| x-a^ || centered at x . Suppose a

* is replaced by the point of that

same sphere which is closest to (X

; then the first summand in (4.4) is

unchanged, while the second summand is reduced- Hence the minimum can only

occur for such a "closest point". Simple geometric considerations show

that such a point lies either on the line segment [x, <x ] or on its extension

through a . Thus, for some scalar s > 0 , we have

a = x + s(o? - x) ,

leading to

x - a = s ( x-a ) , a - cx = (1-s) (x-o) .

Substitution of these results into (4.4) converts h(a) into a function

of s \ with the abbreviation

r I! x - a (4.5)

this function reads

H(s) = w'{sr-p)^ + Wr^(l-s)^

= (w^d- W)r^s^ - 2r(w^p + Wr)s + (w^p^ + Wr^ ) ,

(4.6)

and is minimized by setting

s *» (w^p + Wr)/(w** + W)r . (4.7)

(The availability of such a simple explicit solution to the minimization

problem justifies our earlier assertion that use of (4.4) rather than (4.2)

is more convenient.) If p > r then s > 1 , signifying that a lies on

the extension of segment [x, O'] through a (i.e., relocation has moved

the locator farther from the locatee) ; if p < r , then s < 1 and a lies on the segment (so that the locator has been moved closer to the locatee).

These results, which do not depend on w and W ', seem quite consistent with

what one would expect intuitively.

The only remaining question is the selection of numerical values for the

weights W^ . We have at present no definitive answer to this problem.

There are two distinct considerations to be taken into account:

(a) If we consider the current location (and relocation) operation in

isolation, then the weight W_^ should depend primarily on our confidence in

the accuracy of the initial estimate Oi

of the i-th locator's position

(relative to the accuracies of the other ct ' s , and also of the range measurements). In an idealized Gaussian-error case, for example, maximum- likelihood

estimation would dictate weights inversely proportional to the respective

variances of the associated random errors. For our practical situation, how

can we quantify the degree of confidence merited by the initial estimate Q\^ ?

It may well prove possible to concoct an appropriate measure based on the

sizes of the discrepancies | r_. - p | obtained during the last previous

location-operation in which the i-th of the current locators figured as locatee probably modified by some "decay factor" to take account of how

much time has elapsed since this earlier operation.

(b) However , one needs to consider the "current" operation in the

light of the need to maintain stability of the entire location process; the

selection (or least the effect) of the must reflect not only the oneshot independent -measurement case as in (a), but also the sequential correlated-error many-locator nature of the real problem. If a relocat ion(based

mainly on a single and possible "bad" range measurement ) is allowed

to be too large, its influence would persist through a number of operations

until the unit in question again took its turn as locatee, and so such

instabilities in estimation of positions could result in large errors.

The following tactic may prove an acceptable compromise. Weights

will be selected according to the considerations in (a) above. The

resultant solutions (4.7) of the n minimization problems for relocation

(1 problem for each of the n locators) will dictate moving the estimated

position of the i-th locator a certain distance (from a. ) in a certain

(k)

1

direction along the line through x

v 7 and & . To "temper" this solution in accordance with the considerations in (b), we will "damp" that

distance by a multiplicative factor (say, between 0.1 and 0.6) before

actually revising the estimate of the locator's position. Numerical experimentation will be required to test the performance of this approach and to

determine what values of the adjustment factor are preferable.

PROJECT "WHERE"; WORKING PAPER NO. 9

THE MNPLS SIMULATION ("WHERESIM")

R.H.F. Jackson

May, 1972

Abstract : This paper documents the present version of the MNPLS

simulation. The documentation is presented at three levels: the

"executive" level (an overview) , the "user" level (description of input

and output) , and the "analyst" level (detailed discussion of the logic

flow) . It focuses on the concepts indigenous to such a simulation

study, namely: movement of units, range computation (including the

occurrence of measurement error), and position estimation. A dictionary

of variables used in the program, as well as a listing and a detailed

flowchart of the program, are also included.

Note : Project working papers are informal documents prepared to

facilitate discussion and communication; they may contain tentative or

relatively unchecked material.

TABLE OF CONTENTS

Page No.

1. Introduction .

2. Boxes 2 and 3: The Movement Modules

3. Box 4: The Range Computation

4. Boxes 5, 6, and 7: The Location Attempt, Statistics Accumulation

and "End of Cycle" Test

5. Boxes 8 and 9: The "Intervisibility Table Update"

6. Boxes 10 and 11: The Production of Intermediate Output

7. Box 12: Update the Stationary Unit List

8. Boxes 13 and 14: Produce the X-Y Plots

9 . Box 15 : The End of Simulation

10. The Input to the Simulation .

11. The Output from the Simulation

Appendix A: A Dictionary of Variables Used in the Simulation

Programs

Appendix B: Listing of the Programs

Appendix C: Flowcharts of the Main Programs
I
1. INTRODUCTION

(*) WHERSM is a computer code that simulates the operation of a Micro-Navigation Position Location System (MNPLS) . It provides a "real

life" framework within which the propagation over time of position location errors, resulting from the operation of such a system, can be determined. It further provides a facility for monitoring the operation of

such a system under a variety of movement scenarios, and using a variety

of algorithms for accomplishing the location operation.

This paper documents the present version of the WHERSM code at three

levels: the "executive" level, the "user's" level, and the "analyst's"

level. This introductory section, which serves as the executive level

of documentation, is aimed at the person who is interested only in an

overview of what WHERSM is and what it does. The user's level of documentation is meant for one who is concerned with making a computer run

with the code as it now stands; i.e., with no major program modifications.

Sections 10 and 11 provide documentation of all the inputs and outputs —

the information needed to make a run. The remainder of the paper,

Sections 2 through 9 and appendices A, B, and C, is designed with the

analyst in mind. By an "analyst" is meant one who needs not only to run

the program, but also perhaps to modify it to handle somewhat different

situations. This paper should include enough information to start him

on his way.

Due to the nature of the MNPLS under study, WHERSM is a "fixed time

increment" simulation; i.e., simulated time is advanced by a constant

increment. At, rather than by variable increments dictated by the

occurrence of events, as would be the case in the type of simulation

classified as "next event".

(*) Full name: WHERESIM. Compressed because of computer restrictions

to 6 characters per name.

The program is written in FORTRAN V, but with very little conversion

it can be compiled with a FORTRAN IV compiler. It is liberally annotated

with comments, and although the code documented here employs program overlays, parallel processing, and drum usage, every attempt has been made to

keep machine dependence minimized. Furthermore, the WHERSM code is modular

in nature, so that the logic flow is kept simple and easy to understand,

and so that different concepts in deployment, movement, intervisibility,

position location method, etc., can with a minimum of reprogramming be

implemented and tested. This modularity also allows extreme flexibility

in the type and quantity of inputs required to make a run.

The basic ideas involved in an implementation of the MNPLS are that

there are n units in the field, any of which at each moment might (in

the most general case) be either moving or fixed. Every At seconds, a master computer receives information in the form of ranges, subject to

measurement error, from one of these units to some or all of the others.

The master computer must then, using position location algorithms, estimate

the position of that unit and also prepare to receive the set of ranges

for the next unit in sequence. The elements of this description that

are relevant to a simulation study of an MNPLS are: movement, range

computation (including the occurrence of measurement error) , and position

estimation.

In particular, the. simulation must have some facility for determining

the true positions of all the units at all relevant times. It Is easy to

provide a movement scenario in terms of movements in the xy-plane (longitude-latitude) . A starting x and y, an azimuth of movement, and a

(*) speed will suffice for this . It is the z coordinate, the height above

sea level, that causes difficulty.

(*) As will be explained in Section 2, the movements are assumed to be

piecewise linear, so that in fact a set of azimuths, speeds, and

change times are required.

There are at least two methods of providing the z coordinate in

a simulation. One involves a continuous function which yields a value

of z for any values of x and y; the other is by means of a discrete

function which specifies values of z for particular values of x and

y. In the discrete case, values of x and y other than the prescribed

ones are assumed to coincide with the nearest available grid point, whose

(*) specified height is then used . Both of these techniques are explained

in detail in section 2.

The next item with which the simulation of the MNPLS must be concerned

is the computation of ranges. Of course, given the ability to compute the

true positions of all units at any instant of time, the true ranges can

easily be calculated, since they are simply Euclidean distances. But, in

the "real life" situation, true ranges are never reported to the master

computer due to measurement errors intrinsic to the particular method of

measuring those ranges. This phenomenon is easily simulated once a probability distribution of measurement errors is specified, since the

simulation program can simply compute pseudo-random numbers which satisfy

such a distribution law.

Another point relevant to range computation is that a simulation must

he capable of determining whether a range should be computed; i.e.,

whether, for a specified pair of units at a specific moment, a radio line

(**) of sight (LOS) exists. There are many ways of providing a simulation

program with the ability to determine the existence or non-existence of

these intervisibilities. The method used in WHERSM is to maintain a set

(***)

of "intervisibility snapshot matrices", I.. (i = l,2,...,n; j = l,2,...,n;

iJ t •

(*) Four-point interpolation could have been used here, but is very time

consuming. And since it was determined that no significant further

loss of accuracy occurs with the "nearest point" method, that one

was implemented.

(**) See p.28 of "First Interim Progress Report on NBS Project WHERE in

Support of USAAMCA Project MNPLS", NBS Report 10663,’ December 1971.

(***) Obtained from the Electromagnetic Compatibility Analysis Center,

Annapolis, Md.
•?il
t = 0,. 20, 40, ..., 120) where' I. = 1 if unit j can receive and report

•t J t a ranging signal from unit i at time t minutes into the simulation, and

I. = 0 if it cannot. Furthermore, in order to avoid drastic changes

ijt

in the intervisibility pattern at the discrete times indicated by values

of the subscript t, each row of a particular intervisibility matrix defined

by a specific value of t, say is overlaid with the corresponding row

of the next-in-sequence matrix (defined by t' + 20) at some randomly chosen

time between t' and t' +20. (These times are chosen independently

for each row and for each value of t.) More detail regarding each aspect

of range computation will be found in sections 3 and 5.

The last item of concern for the MNPLS simulation is that of position

estimation. Although that item is an integral part of any simulation study

of this nature (indeed, the most important part), no discussion of the

various algorithms used in WHERSM is provided here, since the project staff

members responsible for the development and implementation of position

location algorithms have presented such a discussion in Working Paper

Number 7.

In order to provide the reader with a better understanding of the

flow of events within WHERSM, and to facilitate further discussion of the

WHERSM program, a flowchart is given in figure 1 which indicates the basic

logic of the simulation. With one exception (see section 9), each box in

the flowchart corresponds to an easily recognizable module within the

WHERSM code listed in appendix B. In every case, entry to box 2 is made

just after the simulation’s current time is incremented by a constant

value. That constant. At, is referred to as the sub cycle time and is the

time during which all the operations required to locate a given unit Cone

completion of the loop formed by boxes 2 through 7 of the flowchart) must

be performed. Cycle time is the time required to locate each unit a+ least

once. This may be larger than n • At, where n is the total number of

units, since certain faster moving units may need to be located more often

than do slow moving units. Locatee refers to that unit which, at any

instant of simulated time, is in the process of being located, and

locator refers to a unit whose range to the locatee is used to estimate

the position of the locatee.

The remainder of this report discusses the workings of WHERSM at a more detailed level. Section 2 explains Boxes 2 and 3 of the flowchart

m figure 1, while Section 3 discusses Box 4. Section 4 explains Boxes

5, 6, and 7; Section 5, Boxes 8 and 9; Section 6, Boxes 10 and 11;

Section 7, Box 12, Section 8, Boxes 13 and 14; and Section 9, Box 15.

Section 10 contains a discussion of the input needed to make a run with

the present version of the code; and Section 11 contains a discussion,

with examples, of the outputs that will result from such a run. Appendix

A is a dictionary of variables used in the program; Appendix B, a listing

of the program; and Appendix C, a very detailed flowchart of the program.

Although the remaining section headings refer directly to the flowchart in figure 1, the content of those sections can be better understood

by referring to Appendices B and C.
.


2. BOXES 2 and 3; THE MOVEMENT MODULES

The movement modules are responsible for computing the true positions

of each unit every At seconds. In the current version of WHERSM, this

is a two-stage process; i.e., x and y are computed, and then z is

computed using x and y. The method by which x and y are computed

is explained first.

In the xy-plane, units are assumed to move in a piecewise linear

manner (the projections of the three-dimensional curves of motion onto

the xy-plane are piecewise linear) . The piecewise linear motion is

effected by maintaining a matrix of current true positions (TP) and a matrix of current velocity vectors (VV) , both of which are initialized

by card input, and by updating TP linearly as follows:

TP (1,1) = TP (1,1) + W (1,2)

TP (1,2) = TP (1, 2) + VV(I,2)

for each unit I = 1,2,..., NU, where NU is the total number of units.

See Appendix A for a detailed explanation of TP and W.

After each unit is moved, a check is made to determine whether it

is time for that unit to change its direction of motion (change times are

stored in W(I,.3)). If so, the new velocity vector is retrieved from the

matrix W1 where all velocity vectors are stored, and replaces the existing

one in VV.

The second stage of the movement computation, that of computing z,

has been implemented in two different ways in WHERSM, although only one

appears in appendix B. (The other is too specific to include here and,

furthermore, is trivial to implement.) The first one, the one not

included here, is simply a matter of coding up a continuous function

) representation for the z coordinate and computing a value of z once x and y have been determined.

The method of determining z that is_ included in Appendix B is

somewhat complicated, in that it involves a preprocessor, overlays, a mass storage retrieval routine, and parallel processing. The preprocessor

is a routine called BUILD, whose function is to read a tape containing

topographical data in the format designed by the Electromagnetic Compati-

((*)

**) bility Analysis Center , decode and unpack the individual altitudes

above sea level that are given on that tape, write the individual

elevations out on the drum, create an index (NDX) to the information on

the drum, and initiate the overlaying of itself by the rest of the WHERSM

code

.

The subroutine that actually provides the z coordinate, HEIGHT,

is mainly a drum retrieval routine. HEIGHT takes the x and y that

are input to it, computes the nearest available grid point to that x

and y, determines the position on the drum of the altitude associated

with that grid point, and initiates the retrieval of that altitude. This

last point is important — HEIGHT only initiates the read. It does not

wait for the completion of that operation, but instead immediately returns

control to the calling routine, which has the option of waiting for the

completion of the retrieval or of performing other tasks that are not

dependent on knowledge of the new z coordinate. In view of the fact

that the time required to retrieve one elevation from the drum is approximately 4.254 milliseconds, the above parallel processing capability

provided by the HEIGHT routine allows considerable time savings to be

(*) Note that obtaining such a function for "f ictitious" terrains is

rather easy, but that surface fitting for "real" terrains may not

be trivial. See, e.g., Pavilidis, T., "Piecewise Approximations

of Functions of Two Variables and its Application in Topographic

Data Reduction", Princeton University, Tech. Report No. 86, Sept.

1970, and "A Program Implementing Piecewise Linear Approximations

of Functions of Two Variables", Princeton University Tech. Memo.

No. 3, March 1971.

(**) Encoded altitudes above sea level at points that are three seconds

of latitude and longitude apart.
4
achieved through the thoughtful organization of the main routine relative

to height retrieval.

The previous comment regarding organization serves to introduce the

last point to be discussed in this section — the reason for including

two movement modules — one (Box 2) for the current locatee, and- one

(Box 3) for all other units in the field. Although all units move every

subcycle, a new height is determined only when a unit serves as locatee.

(In between, heights are treated as remaining constant.) Hence, while

the height for the locatee is being retrieved from the drum, the program

can continue on with the moving of all other units, thus taking advantage

of the parallel processing provided by HEIGHT. There is very little loss

of "realism" with this approach, since due to the design criteria no unit

moves farther than 44 meters between times when it is a locatee, and further,

since the topographical grid points define a rectangle with dimensions

approximately 68m x 92m. Hence, only a small distortion of an already

grossly simplified terrain structure occurs.

There is yet another reason for separating movement into two modules.

Namely, it reduces the effort involved in implementing a very specific

type of movement scenario. For example, the code in Appendix B includes

a technique for keeping units from spreading out too much, by disallowing

movement past a boundary that moves at a speed determined by input. (This

is convenient when a randomly generated scenario that includes foot soldiers

and ground vehicles is used. By letting the boundary move at foot soldier

speed, the ground vehicles can be kept "close by".) In this case, it is

wasteful to monitor a unit’s movements all the time. To check only when

a unit serves as locatee is sufficient. And this of course is easier and

faster when there is a separate movement module for the locatee.

3. BOX 4: THE RANGE COMPUTATION

The range computation module is responsible for computing reported

ranges by computing true ranges and perturbing them, and for determining

whether each range should be reported. The computation of true ranges

involves only Euclidean distance calculations, and the perturbation values

are obtained by a call to subroutine RG. Subroutine RG generates pseudorandom numbers having a Gaussian distribution with mean zero and variance

2

36m , and then randomly chooses 1% of them to which it imputes a. positive

bias by adding 7 meters to each. Except for the biasing procedure, the

method used to generate these numbers is the same as the one proposed by

(*)

A. Rotenburg . RG calls RGU which provides uniformly distributed pseudo

random numbers, generated by the multiplicative congruential method. Qther

distributions, or other parameter values in these, could easily be substituted in these routines. In order to save time, since RGU is called so frequently during the simulation, Rit is written in SLEUTH, the machine

language for the UNIVAC 1108. However, it is very short and can easily

be rewritten in FORTRAN.

There are two items that determine whether a particular range should

be reportad. Tha first of these is the logical matrix RPT, which is

initialized at input time, and which, for each unit, has the logical

value ’’TRUE" if that unit is one that reports received ranges, and "FALSE"

if it is not. Note that the maintenance of such a matrix allows the testing of scenarios in which the set of reporting units changes during a run,

and where such changes are either exogenous jor endogenous events

.

The second item that determines whether a particular range should he

reported is the matrix of intervisibilities (IT) , which records whether

or not a radio line of sight (LOS) exists between each pair of units

(*) Rotenburg, A., "A New Pseudo-Random Number GeneratQr", J. ACM 7_. (1960), pp. 75-77.

identified by the subscripts for IT. Also, each, row of IT is condensed

so that 36 row entries fit into one computer word, thereby greatly reducing

storage use. Therefore, the only concern of the range computation module

regarding intervisibilities is that of locating and retrieving a particular

entry in IT from a block of 36 entries, which informs it whether the range

between the defining units should be calculated.

In summary, then, for each subcycle the range computation module

steps through a list of the units (excluding, of course, the one currently being located) and checks RPT to see first if a particular unit

is a reporter. If so, IT is checked to determine whether that unit can

"see" the locatee. If that also is true, then the true distance is

calculated, RG is called to obtain a perturbation value, and that value

is added to the true distance yielding a reported range. Reported ranges

are stored in the matrix RANGE, with the unit numbers of the units that

reported each range in matrix LOCATR.
. V
4. BOXES 5, 6, and 7; THE LOCATION ATTEMPT, STATISTICS

ACCUMULATION, AND "END OF CYCLE" TEST

The discussions of Boxes 5, 6, and 7 have been combined into this

one section simply because there is very little to say about any of them.

The location attempt consists of a call to subroutine FINDIT, which

is a preprocessor for the actual location algorithms. As was mentioned

in Section 1, no discussion of the algorithms is provided here; nevertheless, some discussion of the input to the algorithms from WHERSM

should be given. All input to the position location algorithms is

achieved via the labeled common block CB1, which consists of the following variables: EP, LOCATR, RANGE, S, TLOC, W, IU, CLK, XNEW, YNEW, ZNEW,

FXD, RPT, and NTYPE. For a description of each of these variables, see

Appendix A.

The error statistics that are accumulated are: the current coordinate errors (coordinate-wise subtraction of true from estimated positions) , the locatee’s position location error (Euclidean distance between

true and estimated positions in xy-plane) , the error in the z coordinate, the max error in the xy-plane over all cycles and over the current

block of KIC cycles (KIC is an input parameter) , the maximum error in

z over all cycles and over the current KIC cycles , and the time to perform the location operation (obtained by interrogating the computer’s

internal clock) . The statistics accumulated hare are used to produce

the intermediate output explained in Section 11.

The "end of cycle" test CBox 7 of the flowchart In figure 1) is

nothing more than the end of a DO loop which steps the simulation program

through each one of the subcycles in a cycle.
•k
5. BOXES 8 and 9: THE" INTERVISIBILITY TABLE UPDATE"

The "intervisibility update" module is concerned with maintaining an

LOS pattern that is checked by the WHERSM program during range computation

(see Section 3). The "current" intervisibility pattern is kept in the

matrix IT, which is automatically updated in a manner determined by parameters that are input to the simulation program. This section explains the

manner in which IT is updated.

The intervisibility snapshot matrices that have been discussed in

Sections 1 and 3 are read in from tape at input time and stored in the

matrix ITB. (The format of the tape is explained in Section 10.) At

the start of a simulation run, IT is initialized to the first of the

snapshot matrices —the matrix that corresponds to a line of sight snapshot taken at time zero. Every K2C cycles thereafter, K3C randomly

chosen rows of IT are overlaid with the corresponding rows of the snapshot matrix that is next in sequence. (K2C and K3C are input parameters;

see Section 10 ) This overlaying process continues until simulated time

has been advanced to the time at which that next snapshot was taken, at

which point the intervisibility update module completes the overlaying

of all rows not previously updated, and prepares to start over again

with the new matrix and the one after that.

The choice of a row to be updated is achieved by choosing a pseudorandom number, call it I, between 1 and NU, where NU is the number of

units in the field and also the number of rows and columns in IT. If the

corresponding row of IT has not previously been updated (determined by

a check of the logical matrix ITIND) , it is then updated. If that row

has been updated, try row 1+1. If that row has been updated, try row

1+2. This search process continues until either an "updatable" row is

found or until the checking process circles back to row I. (This last

comment implies, of course, that row NU+1 is considered to be row 1.)

If no updatable row is found, nothing changes and the "intervisibility

update" module has completed its task.
i

1
6. BOXES 10 and 11: THE PRODUCTION OF INTERMEDIATE OUTPUT

This section of code controls the production of intermediate error

output as portrayed in figure 3, and which is discussed in Section 11.

Basically, this module’s function is to convert the statistics

accumulated in Box 6 to a form that is more meaningful for output. For

example, sums of previous errors are divided by the appropriate number

to obtain averages, coordinate error counts are made to produce quadrant

counts, etc.

The frequency with which the intermediate error output is produced

is controlled by an input parameter called K1C. The output summary is

provided at the end of every K1C cycles.

7. BOX 12: UPDATE THE STATIONARY UNIT LIST

The stationary unit list (logical matrix FXD) is a list that contains

''known" information regarding which units are moving and which units are

not moving throughout the simulation. By "known" is meant that this

information i_s_ available to the position location algorithms.

There are two types of real world scenarios that the matrix FXD has

been instrumental in simulating. The first of these is the situation

where units stop and start throughout the "battle", and send a signal

back to the master computer indicating that they have just stopped or

that they have just started moving. The second scenario is one in which

certain units are "surveyed in" and never move, whereas no other units

are ever, treated as fixed. Knowledge of these facts is very helpful to

the position location algorithms , but the "how" and the "why" of that

helpfulness will not be discussed here. For that information, the reader

is referred to Section 3 of the parent document to which, this Working

Paper is an appendix.

In the current version of WHERSM, the starting and stopping of units

mentioned above is achieved by maintaining the matrix FXD, and by updating

it in a random manner that is controlled by parameters which are input to

the simulation. For each unit, FXD is "TRUE" if that unit is currently

fixed, and "FALSE’' if that unit is currently moving. The input parameter

PSS(2) is the average time a unit remains fixed once it stops moving,

while PSS(l) is the average time it keeps moving once it starts moving.

These times are converted to probabilities, stored back into PSS(l) and

PSS(2), and those probabilities in turn are used to determine, at each

cycle, whether a unit should have its state ("fixed" or "moving") changed.

The determination of state changes is precisely the purpose of the

"stationary list update" module. It is achieved by obtaining a uniformly

distributed (0,1) pseudo-random number from the internal subroutine RU,

and, if that number is less than the appropriate PSS probability value,

changing the state of that particular unit.

It should be noted that in the second scenario mentioned above,

one of course does not want the stationary unit list to be changed at

all during the simulation. This can be guaranteed simply by letting

the values of PSS(l) and PSS(2) at input time be very large numbers.

It should also be noted that FXD is. initialized at input time.

8. BOXES 13 and 14: PRODUCE THE X-Y PLOTS

This module is very small, consisting mainly of two calls to subroutine GRAPH. The first call produces' x-y plots of the true positions

of all the units, and the second call produces x-y plots of the estimated positions of all the units.

There are eight parameters in the call statement for subroutine

GRAPH: N, X, Y, IWRD, XMAX, XMIN, YMAX, and YMIN. N is the number of

points to be plotted, X and Y are vectors of the x and y coordinates of the points to be plotted, IWRD contains either ’TRUE’ or ’EST.'

depending on whether the points to be plotted are true or estimated positions, and XMAX, XMIN, YMAX, and YMIN are the upper and lower limits of

the axes. Those last (XMAX through YMIN) are part of the input to the

simulation program.

For a description of the plots that are produced, see Section 11.

The frequency with which the plots are produced is controlled by

the input parameter K4C ; the plots are produced after every K4C cycles.
.
9. BOX 15; THE END OF SIMULATION

Box 15 of the flowchart in figure 1 is the only one that does not

correspond to an easily recognizable module in the WHERSM code. Rather,

its function is performed by a set of IF statements scattered throughout

the other modules. The IF statements are all identical; they test

whether the number of cycles required to run the simulation for the

length of time indicated by the input parameter SIMTIM have been completed.

If so, then a final set of "intermediate" output is produced before

termination.

10. THE INPUT TO THE SIMULATION

There are three types of input used in the WHERSM program: input

from cards, tape input, and "hardwired" input in the form of DATA and

PARAMETER statements inserted in the program deck. This section will

discuss each type of input with regard to input values and formats.

Card input is explained first.

The "Catch-All" Cards : These cards contain the various parameters

and constants needed by the program* there are two of these cards , and

the table below explains their contents. For a definition of the variables mentioned, see Appendix A.

Columns Format Variable Name

Card No. 1: 1-5 15 NU

6-10 15 NS

11-15 15 K1C

16-20 15 K2C

21-25 15 K3C

26-30 15 K4C

31-35 15 INR . 36-40 . 15 NSP

41-50 F10.5 PSS(l)

51-60 F10.5 PSS (2)

61-70 F10.5 BND

71-80 F10.5 BNDSP

Card No. 2: . 1-10 F10.5 CYCTM

11-20 F10.5 SIMTIM

21-30 F10.5 > XMAX

31-40 F10.5 XMIN

41-50 F10.5 YMAX

51-60 F10.5 YMIN

The "Reporting Units" Cards : These cards contain the unit numbers

of the units that will be used at the start of the simulation as reporters

More specifically, the elements of the matrix RPT that correspond to the

numbers that appear on the Reporting Units Cards are the only elements

that are initialized to the logical value "TRUE". All others, are' initialized to "FALSE". There are 20 numbers per card, each with a format

of 14. The number of unit-numbers that appear on these cards is the value

of INR (see Catch-all cards above). Consequently, there will be

[(INR-l)/20] + 1 Reporting Units cards.

The "Fixed Units" Cards : These cards contain the unit numbers of

the units that will be used at the start of the simulation as fixed units.

The elements of the matrix FXD are initialized in the same manner as are

the elements of the matrix RPT above. The format for each card again is

2014, and there will be [(NSP~l)/20] +1 of these cards.

The "Type" Cards : The type cards contain the type number for each

unit in the simulation. There currently are four unit types:

Type Number TyPe Speed

1

2

3

4

Foot Soldier

Ground Vehicle

Low Performance Aircraft

High Performance Aircraft

11/2-2 km/h

8-10 km/h

161 km/h

322 km/h

There will be exactly NU of these type numbers each with a format of 14,

and therefore [ (NU - l)/20] + 1 Type cards.

The "Movement" Cards : These cards contain all the information needed

by the program so that it can provide for the movement of units. There

will be one of these cards for each unit in the scenario, hence NU cards

in all. Their contents are as follows:

Card Columns Format Meaning

1-3 13 Unit number

4-21 3A6 Unit ID (anything)

Starting latitude:

22-24 13 Degrees

25-26 12 Minutes

27-28 12 Seconds

Starting longitude:

29-31 13 Degrees

32-33 12 Minutes

34-35 12 Seconds

37-40 F4.0 Time ^

^ (hrs .

)

41-44 F4.0 Azimuth (deg.)

45-48 F4.0 Speed (km/h)

50-53 F4.0 • (it)

Time (hrs.)

54-57 F4.0 Azimuth (deg.)

58-61 F4.0. Speed (km/h)

63-66 F4.0

(it)

Time (hrs .

)

67-70 F4.0 Azimuth (deg .

)

71-74 F4.0 Speed (km/h)

There are two kinds of tape input used in the WHERSM program. The

first of these is the tape that contains the height information in the

form of a digitized terrain map. The format of this tape is as appears

in figure 2 a but no further discussion of format will be given here since

the. terrain tape was obtained directly from the Electromagnetic Compatability

(*) By "time" is meant the length of time during which the immediately

following azimuth and speed are used. Control is passed to the

next set when "time runs out" for this set.

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(*)

Analysis Center which has published a report that documents such tapes

in detail. The terrain tape must be mounted on logical unit number 8.

The other tape used as input to WHERSM is one that is mounted on

logical unit number 7 and contains the intervisibility snapshot matrices

discussed in Sections 1 and 3. It is a binary tape, created as follows:

Let ITB be a three-dimensional matrix that contains the set of "packed”

intervisibility snapshot matrices;

ITB (I , J ,1) , I = 1,2,. ..,NU; J = 1,2,..., [ (NU - l)/36] + 1

is the snapshot matrix taken at the first discrete, time value; ITB(I,J,2)

is the matrix taken at the second discrete time value; etc. Then the

tape should have been created using a write statement of the form;

WRITE (7) ( ( (ITB (I , J ,K) , J = 1, NITCOL) , I = 1,NU), K = 1,K1)

where NITCOL = [ (NU - 1 ) / 36 ] + 1 (implying of course that 36 entries

of each row of the 2 dimensional matrices are packed into one computer

word), and K1 equals the number of snapshot matrices that exist.

The third and last type of input required to make a simulation run

is the type that must physically be inserted in the program deck - the

PARAMETER and DATA statement cards

.

The table below lists these. The reader is referred to Appendix A

for their meanings.

(*) Cleramitt, M. and Stone, R. , "The Topographic Information System",

Tech. Note No. ECAC-TN-007-222 , June 1969, Electromagnetic Compatibility Analysis Center, Annapolis, Md.
f '

,
Type of Statement Routine

Variable Name Contained In Contained In

MNLAT DATA BUILD

MXLON DATA BUILD

TPOUT DATA WHERSM

IFREQ DATA WHERSM

CNX DATA HEIGHT

CNY DATA HEIGHT

W DATA WHERSM

FXDINI DATA WHERSM

LI PARAMETER WHERSM

L2 PARAMETER WHERSM

L6 PARAMETER BUILD

WHERSM

HEIGHT

L7 PARAMETER BUILD

WHERSM

HEIGHT

11. THE OUTPUT FROM THE SIMULATION

Figure 3 contains a sample of the intermediate output produced by

WHEPvSM. This section explains the contents of that output sample.

The first line of output contains the cycle number of that cycle

after which the summary was produced, the simulation clock time, the

average real clock time the program took to provide for the processing

of one sub cycle, and the average time taken by the algorithms to perform

one location operation.

Each row of the table contains statistics for the unit whose unit

number appears in column 1. Columns 2-7 contain statistics accumulated

over all cycles previous to the current one. They are, in order:

the average error in the xy-plane,

the average error in the z coordinate,

the maximum error in the xy-plane,

the cycle during which that maximum error occurred,

the maximum z error that occurred, and

the cycle during which that maximum z error occurred.

Columns 8-11 contain statistics accumulated over the previous K1C

cycles. Those statistics are: the average x-y error, the average z error, the maximum x-y error that occurred, and the cycle during which

that error occurred.

Columns 12-14 contain the individual coordinate errors (estimated

minus true) that occurred the last time each unit was located prior to

the printing of the intermediate output. Columns 15, 16, and 17 list

the current values of the S,RPT, and FXD vectors respectively; and

column 18 gives the current value of the W vector, the weight.

The last row of the table, which begins with the word "ALL", gives

for a column of averages the average of the numbers appearing above it;

and for a column of maximums, the maximum of the maximums in the column

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The last items of output are the xy-plots, an example of which is

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the graph is of true or estimated positions, list the unit numbers of

the units that appear on that graph, and list the plotting symbols used

to represent those units. The letters of the alphabet are used as plotting

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APPENDIX A: A DICTIONARY OF VARIABLES USED IN THE SIMULATION PROGRAMS

In this appendix, definitions of all non-trivial variables used in

the three main simulation routines (BUILD, WHERSM, HEIGHT) are given.

In each case,. the variable being defined is underlined, and if it is a dimensioned variable, it is written exactly as that variable should appear

in a DIMENSION statement. Immediately following the variable name will

appear either a B,W, or H (or any combination thereof). By this is

meant that the variable being defined is a variable of BUILD, WHERSM

or HEIGHT.

Note: All variables conform to the FORTRAN implicit naming conventions regarding mode.

*****

AVASBT W. The sum of real times used to perform position estimations.

AVKTR W. The number of time intervals accumulated in AVASBT and AVSUBT.

AVSUBT W. The sum of real times used to simulate a "real world" subcycle (includes AVASBT)

.

AZ (4) W. Buffer used in reading in movement azimuths.

BNP W. Boundary beyond which units are not allowed to move. Unit of measurement is the kilometer. If this option is not desired, set BND to

some large number.

BNPSP W. On input, the speed (km/h) with which BND moves in both directions

Immediately converted to a value that is added to BND at every sub cycle.

0(14641) B. An output buffer that contains decoded elevations above

sea level.

CLK W. The simulation’s clock. Contains simulated time in seconds.

CNX H. The inverse of the average longitudinal distance (in meters)

between grid points on the terrain tape.

CNY H. Same as CNX but for latitude,

CT (4) W. Input buffer used to read in the length of time a unit is

to move according to a particular azimuth and speed.

CYCTM W. The "real time" length of a cycle in seconds.

D (14641) B. Output buffer that contains decoded altitudes above- sea

level

.

DELTA W. "Real time" length of a subcycle.

DX(L1) W. List used to store the difference in the x-coardlnates of

the true and estimated positions. Computed immediately after a unit

has had its position estimated.

DY (LI) W. Same as DX, but for y-coordinates

.

DZ (LI) W. Same as DX, but for z-coordinates

.

EP (LI ,14) W. Contains estimated positions for each unit, in addition

to some data relevant to those positions. EP(I,l-3) is the estimated

position of unit I which was determined the next-to-last time it served

as locatee; EP(I,4-6), the last time it served as locatee. The rest

of the EP matrix is used to manipulate positions when the relocation

option is used (see Working Paper No. 8).

ERR W. The error in position estimation in the xy-plane and then in

the z-coordinate. The Euclidean distance between true (TP) and estimated

(EP) positions is used as the error.
"
IFY H. Same as IFX, but for the y-coordinate. Units of measurement

here are 3 seconds of latitutde.

IGP W. Pointer to the unit with the largest xy-error over all cycles.

IGT W. Pionter to the unit with the largest xy-error over the last K1C

cycles

.

INR W. Initial number of units allowed to report.

IP W. Pointer used in constructing Wl.

IPX H. The x-coordinate of some other position (for which a height is

already available) , after it has been converted to the units of the

terrain.

IPY H. Same as IPX, but for y-coordinate.

IQO W, Number of units whose estimated position is exact.

IQ1 W. The number of units whose estimated position is in error In

the direction of the first quadrant.

IQ2 W. Same as IQ1, but for quadrant 2.

IQ3 W. Same as IQ1, but for quadrant 3.

IQ4 W. same as IQ1, but for quadrant 4.

IQUAN B. Quantization factor for the block of terrain grid points

currently being decoded. (See ECAC^TNt'007t-222 , op . cit .

)

IR W. Counter for the number of ranges computed in a particular sub cycle

IT (L1,L3) W. The intervisibility table currently- being used.

ITB(L1 ,L3 ,7) W. The set of all intervisibility tables.

ITIND (LI) W. Logical matrix which indicates for each row of IT, whether

that row has been updated yet.

IU W. The unit number of the unit currently serving as locatee.

IV W. The CO-1) answer to the. question of existence of an LQS between

the pair of units in question.

IVMAT W. Pointer to the intervisibility' matrix currently being used to

update IT

.

IZT W. Pointer to the unit with the largest z-error over all cycles.

K1C W. The number of cycles after which intermediate output is to be

produced.

K1CC W. Counter used with K1C.

K2C W. The number of cyeles after which an intervisibility update

is to occur.

K2CC W. Counter used with K2C.

K3C W. The number of rows of IT to be updated when an intervisibility

update occurs.

K4C W. The number of cycles after which xy plots are to be produced.

K4CC W. Counter used with K4C.

KUF (LI) W. For I = 1,2,..., NS, KUF(I) is the unit number of the

unit which is to serve as locatee during the Ith subcycle.

KZ W. The seed value for the internal uniform psuedo -random number

generator, RU.

B. Status parameter for I/O processor NTRAN. L equals^ -1 until

the completion of the I/O command.

LI W. Parameter variable used in dimensioning. LI must be greater

than or equal to NU.
'
L2 W. Parameter variable us.ed in dimensioning. L2 NS.

L3-L5 W. Parameter variables used in dimensioning, but since they are

strictly dependent on LI, they are automatically set.

L6 B,W,H. Conceptually organize all the 6’ x 6’ terrain blocks on

the terrain tape according to the latitude-longitude of their southwest

corner. L6, then, is the number of these blocks when counting from

west to east.

L7 B,W,H. Same as L6, but count from south, to north.,

LQCATR (LI) W. The unit numbers of the units whose ranges are being

reported in a given subcycle.

M W. Status parameter for I/O processor. See L.

Ml B. Status parameter for I/O processor

.

See L.

M2 B. Status parameter for I/O processor

.

See L.

MAXCYC W. The length of the simulation in terms of cycles.

MNLAT B. The minimum latitude (in seconds) for which, a height is

available on the terrain tape.

MXLON B. The maximum longitude Cin seconds) for which, a height is

available on the terrain tape.

NBITS B. .The number of bits into which the elevation in question is

"packed".

NCYC W. The cycle number of the cycle currently being processed.

NDX(L6,L7) B,H. An index to terrain blocks that are written on the

drum by BUILD. NDX(1,1) is the position of the 6

T x 6* terrain block

whose southwest corner is identified by (JINLAT, MXLON) ; NDXClj2)‘ is the

position of -block (MNLAT, MXLON + 3600); NDX(2,1) is block (MNLAT + 3600,

MXLON); etc.

NITCOL W. The number of computer words in a packed row of the inter—

visibility matrices.

NS W. The number of subcycles in a cycle.

NSP W. The number of units which, at the start of the simulation, will

be identified as being stationary.

NTYPE (4) W. The unit type of each unit. If NTYPE(I) is 1,2,3, or 4,

then unit I is respectively a foot soldier, a ground vehicle, a low

performance aircraft, or a high performance aircraft.

NU W. The number of units in the simulation.

NUMEL B. The number of elevations per word for a given terrain block

represented by a logical record on the terrain tape.

PSS(2) W. At input time, PSSCl). is the average time a unit keeps moving

once it starts moving, and PSS(2) is the average time it remains fixed

once it stops. Subsequently, they are the probabilities of changing

state. If no random starting and stopping is desired, set botlrPSSCl)

and PSS(2) to very large numbers on the catch-all-cards.

RADS W. Constant representing the number of degrees in a radian.

RANGE (LI) W. . The ranges being reported for a given subcycle.

RCON W. Constant used in converting a speed and azimuth. combination

into direction cosines.

RPT (LI) W. Logical matrix that indicates for each unit, whether or

not that unit is a unit that is allowed to report ranges.

S (LI) W. Logical matrix that indicates for each unit whether or not

that unit was "successfully" located the last time a position estimation

was attempted. S is set by the position location algorithms.

SIMTIM W. The "real time" length, of the simulation, in minutes.

SP (4) W. Input buffer used in reading in speeds for each unit.

STAT1 (LI ,10) W. For each row: column 1 is the sum of position

estimation errors (in xy-plane) over all cycles for the unit associated

with that row; column 2, the maximum xy-error that occurred over all

cycles for that unit; column 3, the cycle number of the cycle during

which that maximum error occurred; 4, the partial sum of xy-errors

over the current KlC cycles; 5, the maximum xy-error over the current

* KlC cycles; 6, the cycle number associated with, that error; 7, the

sum of the z-errors over all cycles for that unit; 8, the maximum

z-error over all cycles; 9, the cycle number associated with that

z-error; and 10, the partial sum of z-errors over the current KlC

cycles

.

STAT2 (LI, 2) W. Column 1 contains, for each unit, the number of times

that unit was "successfully" located since tire start of the simulation;

and column 2, the number since the start of the current KlC cycles.
'
TLOC (LI ,4) W. The clock time at which the estimated positions, in

EP were obtained.

TP (LI ,3) W. The current true position of each unit. TP (1,1) is the

x-coordinate; TP (1 ,2), the y-coordinate ; and TP (.1,3), the z-coordinate

.

TPOUT W. Logical variable that allows for the writing of all intermediate output on logical unit 9, in addition to the standard output

unit.

T1-T4 W. Parameters in calls to the system’s clock. Used in determining

AVASBT and AVSUBT.

W(LI ,4) W. Currently used velocity vector for each unit. Column 1

is the x component; column 2, the y component; 3, the time at

which that vector should no longer be used; and 4, a pointer into VV1

where the next velocity vector to be used resides.

VV1 (L5 ,4) W. The set of all velocity vectors for all units. The

format of each vector is the same here as in VV, but the h (h 3

currently) vectors for unit number 1 are in rows 1 through h of Wl,

followed by the vectors for unit number 2, etc.

W(L1) W. Vector used by the position location algorithms to weight

the various estimates (see Working Papers No. 7 and 8). W should be

initialized to the weight to be assigned to non-fixed units at the

start of the simulation.

XMAX, XHIN, YMAX, YMIN W. The upper and lower bounds of the x-and

y-axes (in meters) to be used in producing the xy-^plots.
.

.

.
ZPSUM W. The sum of the position estimation errors (in the z-coordinate)

over all units over the last K1C cycles..

ZTSUM W. The sum of the position estimation errors (in z-coordinate)

over all units over all cycles.
v

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