On the relation between the accuracy of the ships position and the accuracy and frequency of position fixes

Augmat 1971

LABORATORIUM VOOR

SCHEEPSBOUWKUNDE

TECHNISCHE HOGESCHOOL DELFT

ON THE RELATION BETWEEN THE ACCURACY OF THE SEIP'S POSITION AND THE ACCURACY AND

FREQUENCY OF POSITION FIXES

by T. Koyams.

Report No, 321

Content' s

Summary.

List of Figures. Introduction

System description and poem:manes index.

Optimal control of the system.

Optima]. estimation. Numerical examples. Oonolusion

:41b4tunl,

triaiitional *OOPS of navigation can be derided in three ostagerieel

reuttms, are° oogreotion

keering. An inveetigation of the Deism& co 6 her to make the beet decision about the °sure0 oasmeotion,

interia. In sounestion te thief it use neoemary to Otain the gelation

between the anon:easy of the ehip's position and the aeon:sexy and the fro.

quern' ce 'patios time.

TIQV thief Velcan'e Dynaais and the Kalman Vat= Vero adopted

to rake fully vac of the mailable integration*

Using the results above. automated navigation and traffic obntrol in the future disonseed briefly.

Mat of Fiaures.

The pattern of navigation. Tho state of the Wrimm.

Distributien of chip poeitica.

Perfarleasoe Loden.

50 Optimal control algorithm.

The decrease ef P K s000rding to k. Accuracy of course ON a

e

If

Accuraoy of cource crw = 0.50. Simulation.

11.2111210112a-The navigation of ehipe can be sohematioally repreeented IV thePattern shown in Pig* 1.

/hie patteen chow, that the navisating prooedure oan be divided into theee

stages*

In the fire* stage, the ehip'e trade io determined by the _{caPtain. Ammanff to}

the misaienDef the eonsidered voyage, his desisten is made an

a beets et all

available information it La poesibly eapported by a Shore haced

neteorelo-t

stool =ratan& advioe

).

Ir the moon& etage, the dialp's wanes La decided upen by oomparing the measured

ehip'e mitten and the desired mate* Shia eouree lei held sonatas* uutil the

next deoialon (next seam:cement) will be nade. Ve eheuld remark here,

_{that thie}

position weaeurement ean only be nade in e diserete _{way. In the middle of an} ocean, wheee no eleotronic positioning devises are available, ve can get thie information only pone three times a day. On the other hand, when wailing along

a (mat, ve oen get it much lam frequently, for simple eveey five minstee if neemeary, by seam of viola bearings of land marks or by radar, bat nevexthe-leee the infoemation oemee in at &smote instant° and that le the reacts by

ve need to eeparate the Magee 2 and 3.

In the third stage, the ehip's course is oontrolled primarily by _{application of} e small oorreotion to the refeeenoe °curse. The real of this etage is exesuted

by the auto-pilot or by a human helmamen.

2)

xt will be poseible in the future to combine the egoond and third etages

reeultins la a continuoue oontrol of the ehip'e _{poeition. However this kind of} ocntterloill be reetrieted to meow where a high degree of _{aeouraey La required} eneh as harbour menoeuvring and it will not be necessary in other mere opon placeo. Therefore, even if ve imagine the navigation to be fully automated, in almoet all owes, the pattern of Pig. 1 will be followed and the importance of the moond atage will remain. Thie Le laid to explain why ve ocacentrate cur attention to the second stage in thie paper.

Hy the considerable development of the navigatioaal aide, ouch as Doom, Loran,4 and -00 Omega and navigation satellites, it has beoone possible to easure the

ship'e position whenever _{Wherever ve need. ir aocuracy je excellent. The} women for example, of a navigation satellite fix _{is maid to be 0.4 n.s..}

O

2. In these oiromeetamoee, ve have to change our minds about the wend stage, It is neoweary to ocnsider a poesible relation between the accuracy of the shiple poaition and the assure-ay and frequency of the position firmo.

In general, it ahead be possible to get enough information frets the meaeurements vift poor accuraey if ve imams° the number of meaeurements. Therefore, ve oan anpost to find acme relation between the aocumney and the interval of measurement

to fullfil the rquiremont of a certain acouracy of the dhip's poeitiem4 Per till° purpose, it io necoosary to speoify the vay in vhioh the decisien about the orreotion of the ohip's wouree is made, aim* otherwiee it is not possible to

mho any ocmparisen. We therefore specify that the decision should be made in an optimal way, mooning that ve vent to minimise a oertain objeot funetion. Sections 5 ond 4 ago devoted to the determination of this optimal control in the seemed stem. The results may be interesting, beocuse ve have had only soaroe investi-gations on this matter until nov.

2. Stratea domiwktou cud 15wgezgat22.6sfiaa...

In case of the amend atage tha a-plea oan be siaplified by negleoting the

transient pnrt of the ship'o cation if the time intervala betimes the

&plosions CaNO not too email.

US describe this stage by sIana of differenee equations.

*k+1) 21(k) + UTIPi(k) + UTU(k) (1)

1(k) m X(h) + v(12) (2)

vile=

I(k) deviation from the deairod route at time instant k,

U ship's apsed,

time interval batman the decisions,

Yq(k) sOrmectien inyat to the global 00U2Q01 Ezdo at time h, remaining oonatent until ties k+1.

W(h) s disturbance of the ohip's *curse.

1(k) massurament of U(k) made at time k,

V(k) measureaent error at time k.

14. (1) Deana that the ship's tn.& error at the nest firing inotout. 3(Tx4), is the sms of the error at the present time, X(k), the distance travailed

during the interval tima the prassat worse correction h(t) and the deviation

frost the desired traok mused by dieleggionove UTU(k).

The third term io considered to be proportional to UT, because, roughiy :meshing, it uill be estimated as

U dOOdt

Mil

d(t)dt

0 0

where

h(t)

m course deviation wooed by disturbances.

By the nature of thia stochastic promos, ths Gapeaaaicat _(frd(Odt,

o

uill be takon to be independent of T (see Fig. 3)

_{and ye mite it t(i).}

Ufa assume V(k) to be a swato-inan causeian _{prooeas. If 11(k) is not sero-mean}

ye can adjuat it uhen we make the decision about _Oi(k).

Bk. (2) means that the poeition fix et time k cannot be exaot, as it inolake

the meeeuremeut error V(k). comma: V(k) to be a sero-mean salmi= posea. again.

We further assume that the stoohastio processes X, W and V are etatiotically independent of eaoh other.

The first thing we have to do inihie paper is to find the optimal decieion for (Pi() at each time instant k* TO get the optimal control, ve have to

epecif)' the performance index. in this papers we adopt as a performanse index the value

J a Ble)

₍₄₎

which meens the variance of the position error from the desired track. There may be oaseekin whieh Ek* (4) is not euitable. Pbr example, it may be possible that the routs A-4C is better than the route A--,313-4C in Pig. 4, because it is ¡Shorter* In that case, ve can me the performanee index

J

le[X2 +a. j.23 ₍₃₎

If me neks the weighting eoefficient X larger, we oan reduce the reste 'o

length, t ve ehall looee 00210 =Jersey. But as long as the deviation from the

deeired ems* ir reetricted to a email range, ear within t 100, the reute

elongation mili not be considerable (lesa than 2g).

Ds it Le net co unreamable to use (4) as a performance index.

Of wars., we oan very easily ethane* the following investigation if the per-formanoe index4 given by (9) turne out to be bettor.

10211181 oontrol of the

systaa)

Our problem is to obtain the sequence of oorreotions

Oi(k), k

0, 1, 20 which minisisea J E[X], utiliaimg the available resulte of asesuremant 8(1),

For this purpose, Sallman's method of pynesio Programming ie most convenient. This method is based an the Principle of Optimality:

An optimal control has the property that whatever the initial state and the initial oentrol are, the remaining oontrol must constitute an optimal one with regard to the state which resulta from the

initial control.

In this mootion ve use e *lightly °hanged performance index, e.g.

E [

X2(i)j

₍₆₎

be-algae Amanda Programming del a with 0(1) _{stages, inoluding the transient}

stales..

Nb start with the last Dingo of control in our probles. Lot ua osnaidor tbe simple problem to determine a control whioh optimises the last atage.

We define

Vi 0 min M(XF(11))

iPi (iffl)

PUCE aquatics (1) we see that

X(Z) X(11-1) + UT Oi(S..1) +

so equation (7) bocones

min E[:2 + U2T2f12

Oi (11-1)

NM 3

(In this expression the vs:Males 1(.1) _{en d V(1-1) are briefly denoted as}

X and V reapeotively).

Ve recall that x(i) and V(i) are statistically

_{independent for all i, while}

V(i) is zero mean.

(T)

DTV(2.1),

etr2v2 + (Pi + 2u2T2(friw +

So the sixth term of equation (8) vanishes:

EDW.] EEX1.2[W3 a o

In addition, sinos kiPi(N-1) ie required to be physically realisable, it means that ehould be a function of W*(111-1) and X(0), where Z* (5-1) is

the set of measurensats of Z(i) for i a 1(1)E4 and X(0) is the ship's initial

traok error, which has a gauseian diatribution with :681.0 mean.

Therefore W(j) and Z(i) are statistically independent for all j so we

see that

Efu

-

ENla ((q)

-

0-consequaitly ve have

V, .

min XfX2 + U2T2 0 i2 + U2T2W2 + SUIPX 3

(N-1)

From the propartiee of oonditional expectations, the equality E[X) *DC:1'33

follows. where the euterly expected value is ever Y. Utiliaing thie resultt we pan rewrite equation (11):

V a

niri E[E + U2T2 4/ + 112112112 + 2trrx ¡a* (N-1),x(o)3]

(13)

Where the euterly expected value is over Z*-(w-1) and X(0).

We can eee now that the performance index is minimised by minimising the inner expeeted value in equation (13). The physical realisability condition requires that oi be some daterministio function of X(0) and of the random miter

Z*(3-1), instead of being a statistic variable.

Consequently, in Ng. (13) ve oan put

E [2UTX LP a 2172E1x)

Then, differentiating the inner expeeted value in 4. (13) with respect to (pi and setting the reeulti equal te seret we obtains

oi(N-1) tie[xse (N-1), x(o)3 ₍₁₄₎

Kalman has proved that this oonditienal expected value _{is the optimal filtered} estimationt,wrjttean as 1(w-1), se

qq(Eal) 4

41(w-)

(15) 6.

2v indention Ile get the resulto for j 1(1)2-1 s

i(11-i

IN-i)

7.

This result will to incertod in the next part.

We evaluato Vi by substituting eq. (15) in eq. (11) and we obtains

Vi = 2[12(1-1)3+ eT2EN2(11-1)) _(1d)

uhore X - _{is tho eotication error cine 2[X2(0-1)3 is o term of the}

admisaible lose funetion.

Ue new turn to the quoetion of the optical control procese for a 2-otage preoesa.

'de Wine

2 NU. E

x( .i)

X2 (N) _(17).

qyz..2) 01.(04)

However, (roe the principle of optim.13tty it follows that thie _{equation Gan}

be imp:mama as

V2 0

min Ege(134) + vi)

piano the ohoioe of 41(14-1) does not affect X(2-1).

Polleuing tho mama prooedure ao tm did in the aingle stage problaa we get

the following recultss

qii(2-2)

_(u)

,

2 V2 2 1. X6(L1-i)

r

eT41014(2-i) _J

(t)

i01 1.01 (20)

r

J o _{o o o} L

r(3.4)

E ev&te(z-4))

₍₂₁₎ ti ₁₀₁ i=1

Tho invorso proportionality of _{(pi to UT im rather trivial. HOUGVOW, those} restate aleo indieato that the oaquanee ef optical _{gentrels een be obtained} by the optical ostimationse These last objoots will be considered in tho nort ovation.

4. Op estimatim.3)

Thie section is based upon a theorem by Kalman on optimal filtering for disrarets linear systems. Ow& the strong points of this theorem is, that it paye attention to the fact that the signal is the output of a gysten whose

charaoterietics are known.

It deals with the following linear system:

X(k+1) ₀ (k+1,k)gle) + r(k+10k)W(k) (22) Z(k+1) H(k+1)X(k+1) + V(h+1) (23) Notation: V T' 8. Mimensien: Name:

nal state vector

pal dieturbance 'rooter

sal output measurement vector sal measurement error vector nmn atate transition matriz

nap disturbance transition matriz man measuraementiattrix

now reads:

a, The optimal filtered estimate it(k+lsk+1) - which memos the optimal estimate

of the state gh+1) based on the measurimants Z(i), i 1(1)k+1 is given by the recursive relation

i4+11k+1) 0 4)(k+1,h)Uktk) + K(k+1)[z(k+1)-11(h+1)Ck+1,k)k(kik)] ₍₂₄₎ for k 0, 1, 2, ... where il(020) . O.

b. K(k+1) is an nimeatrix, speoified by the relations

gk+1) 13(k+1:k)Hqk+1)(gh+1)P(k+ltk)Hqh+1)+0+1)J-1 (25) P(k+1:10 0 4i (k+1,k)P(Irsk) (k+i,k) + r7(k+1,k)q(k) 171(k+1,k) (26)

ro(k+isk+1) _Tok+3.)10+1) 3 p(cia,k) ₍₂₇₁

for k = 0, 1, 2/ ... where I is the non identity matrix, while P(0:0), Q(k) and R(k) are oovarianee matriees for X(0), W(k) and V(k)

respectively.

o. The stochastio process 01C(k+1:k+1), k 0, 1, 2, .... j whioh is defined by the filtering error relation

i(1+1:k+1) = X(k+1) _i(k+lsk+1)

is a zero mean Gause-Markov sequence, whose _{oovariance matrix is given by}

9.

Applying these resulta to our problem, where H = 1 and r= UT, we get k(k+1:10.1)

lottio

+ 1C(k+1)[Z(k+1) ii(uk) 3 (28) K(k+1) P(4+140 P(k+lsk) + R(k+1) P(k+1:k) P(k:k) + U2T2Q(k) P(k+lsk+1) = Cl

-

X(k+1)JP(k+lsk)

In this expression, only the present measurement Z(k+1) and the estimation of

the last state

i(ksk) appear explicitly, while the other measurements Z(i),

m 1(1)k are integrated in the last estimation.

It mane that we can use this filter on line and save the memory eapaeity of a computer.

The qualitative meaning of this theorem is quite olear, i.e. when the

measure-ment is accurate (small R), the value of K is near to 1, meaning that we have to rely on Z(k+1) more than en it(ksk). On the ether hand, vhen the measure-ment's socuracy is poor, the value of K becomes near to sere, meaning ve

;should rely more on ¡(ksk) than on Z(k+1). In ether words, we should rely on the feet that the signal ve are estimating iB the output of the known system and it should be constrained by the system's construction.

When we combine this resulat with the result of the previous section, _{we get an} algorithm for the optimal estimation and control system which is shown in figure 5.

V

laPos

Let us recall tho system win.

11(k4.1) = x(k) UTIPi(k) UTU(h) Z(k) I(k) v(k)

ubere V(k) an V(k) ors zero man gausaian pr000mes uith standard variations, a'iu and G`v noraNetivar (Et es 04w,a 0Tu2).

Frown the results of the prom/sun motiono, the optical oontrol oequenoe can be obtained:

i(h)

itatah)

₍₂₀

X(ktk) D8(k-Ish-1) + 1011)[3(1s) kat-ls/s-1)3 (28)

Por the perforsanoe Laden

di

[ E

X2 (1)

Leal

tro got tho oinitatia Yaw 3-1

leal { x2(r-i)

erev2(r-i)i 3

Frets eq. (50) us me that this eas be aetritten _aa 11.1

vi

P(tI-Ask)

ho0

Ao uo ahall GOO latore the value of P(h+lsh) dooreases a000rding te _tho

Lmore-meat of k and saturate° to a pertain value. So, if the rate of this oatura.

tion is not too snail, us can negloot tho tranoient °togs and *valuate _our motes only in the stationary stage. ln that ease us can return to tho original

porformanoo index

J

EV)

₍₄₎

and thio fia a fUnotion of Tu, Tv and UT oar.

First ue ahall see how the Zalman _{filter =rho. In Pig. 6 the doors-sae of the} varianoo of the estimation _{error, P(ksk), and the gain of the &slam}

_filter,

gk), ame illuotrated for

_{the case of}

Crw, w 101

(rv 0 1.5 nautioal riles.

Us.

(21)

FIT . 5 nautioal miles.

We can see that on/y 0 (ic(oo)) of the measurement information is used

for the prevent stage information after a sufficient number of poeition firms is accumulated. The rete of oaturation varies considerably according te the

valuos of UT and Cit'Ir. Por exanplo, When UT is large enoagh or Ge. io

enomgh, thio pronws saturates afterT. oeveral stepo. So we sball only

oon-sider the stationary stage hereafter.

Next the relations between J, 17-/a, ; and D? are shown in Pigs. 7 and 8. Fran

theae figureo ne can find the necessary UT for the opeoified J when no know

Tw and 0-v, For le, if 2 n,13. standard deviation from the deaired mate a 3 Ce ! 6 n.n,;) is accurate enough for (moan navigation and if 0-/1 0 1°0

then J a (r2 4 luine29 as, frem fig. 7, the nooessary UT is 104 U.S6 for 0)r . 1 n.m. and 63 1201B. for TT . 3 non., which means we need a potation fin every 5.8 and 3.6 hours respectively at a epeed of 18 knots. These values oan be qualified as moderate if modern poaitionimg devioes are °mailable.

Some simulation remits are shown in Fig. 9 to see the effeot of redmaing the value of DT. The ailaslation ts exoeuted on a digital cewater, produaLtg the owed= random number for the value of W(k) and V(k). As we can olearly smoD

the affeet of UT is very large,

On the other hand, when a ship j8 en9ing in an area of high traffic, knotty

ouch as the English Channel or u big harboWo approsoh, the allowanao of ths poaition error from the desired route ahead be reetrioted to a verr

value in cemparison with ocean navigation. If the allowed width of vaimms4r

for a ship is 500 meters, we have te restrict the standard deviation fran the dosirod route within 0.05 n.n. This means, even if we can expeot Cry 0 0.25 n.m.

and 0-11 . 0.5°, we need staourcesnta every 3 minutes (U 0 18 kn.). Pe w such a

ehort interval we have to use a core sophistioated wyetememdel to inalado the true:lent etage.

Eut in this oese we can use sere Gown:ate poeitioning vetoes than in the ocean, euoh ea Deem, bearinge on radar, bconane ouch placoo are nearer to tho °here. Wo can expect than te be very acourato, for example a`v = 0.01 n.s.. With thie acouraoy, the mammeastit intervals required are about 20 minutes for Tv a 0.5e

and 10 minute° for air = e to fulfill our requirevent (too Elg. 18) and our

cyst= will work again.

Thie point is very important fees the vim point of traffic control.

12.

When the traffic density inereases in the future, it will be necessary to oentrel all ships in a cartianarea by one single control centre. This centre has te sean all ships in its ares and give them instructionao so the scanning speed is a very impottant problem and we have to make the instructions as simple as possible. If our conventional system fits that rposeo the centre'8 jobs are to measure the ship's position, to solve the interference problem of them to prevent collisions and to instruct a proper course and speed for each ship. These jobs would not be difficult if the scanning interval is net

too short.

Another strong point of this traffic control is its flexibility. _{Beesone the} instructions fran the centre to each ship are only _{course and speedo then, if} the anto-pilot and engine telegraph of a ship are linked with the ineimuction receiver, the ship can be guided automatically to a harbour entrance. Tha instructions can also be followed neumally;seaVfor that case, the required instrument is only a wireless tranemittor/receiver _{on the ship.}

6, Coneluaion.

The relations between the accuracy of the ship's route and system dirbe,

measuring error and meaeuring interval are obtained when the ship is eontrolled to miniadme _{the mean square of the route error with the availablo information.}

The effect of the measuring intervml is very large and me can gel enomgh imfor-nation by inoreasing the number of messurenents even if each of them has poor accuracy, as long as our grams modal, Eke. (1)81(2) is valid.

Ftea the above results, we can expect that the conventional navigation psttern vill fit the purpose of automatic navigation, if ve provide two kinds cf

navigation control centres, one for ocean navigation in connection with the optimum reuteing system and the other one for the local navigation vith high measurement accura070 guiding the ship as close as possible to its mooring berth.

References:

Narks, W. Goodman, P.R. & Pierson Jr., W.J.:

"An Automated System Por Optimum Routing"

TSEAKR, Vol. 760 1968.

de Wit0 C.:

n0Prtima1 Zetheorological Ship Routing°

TNO Report 142S, 1970,

Uttanabe0 I., Finea, T., Loma, T. & Moteica, S.:

"The Automatic Control of Ships"

J.S.R.A. Japan, Vol. 124, 1968.

Zuidueg, J.K4:

"Autematic Guidance of Ships as a Control Problem" Doctoral Thesis, Delft University of Technology, 1970.

Itleaitch,

"Stochastio Optimal Linear Retimation and Control"

Chapters 3 & 9 ¡4cGasu-Hill0 1969.

1st stage 2nd stage Decision of route Decision of course ---1Measurementof position

Fig. 1 The pattern of navigation

Fig. 3

Frample; Hold

Fig. 2 The state of system

3rd stage Course cotroller Measurement of course time

Distribution of ship position

A

Fig. 4 Performance Index

f(k/k-1) UT 1(kik) delay 2(k-1/k-1) delaylc Vk-1)

Fig. 5 Optimal control algorithm

U T

0(k)

z(k)

K(k)

F g.6

The decrease of P&K according to k

n- _i

Tv=1.5 n.m.

UT 5.0

n.m.

P :

variance

.

variable

of estimat

gain of the

on error

n.rn?

Kalman Filter

P(k1k)

K(k

) ₁₀

₂₀

₃₀

₄₀

₅₁

Fig. 7 Accuracy of

_course

_Tw=1°

150

₂₀₀

n.

2

o

Fig. 8 Accuracy of

course

25

50

75

100

-2.5 2.5 o -2.5 2.5 o -2.5 2.5 o -2.5 Fig. 9 Simulation i rse Tor miles) UT= 200 n.mil:

7

I 0-w=1° Crv=1 n.m. measured imated position

---reat

/

10 , . \vi UT=50n.m. r"

,

AL

_A

,

N./

. _. , ..

/

i

UT=10n.m 5C0 1000

iStidk. LILA

i frot...

111 w

!

' ! UTi= 5 n.m.

.,_

( i ,Amallill 4 i 414.4.0.,, 1I 4, I I

li

I 111' ! l' 1

11

-1 rikOPI 7trril' - i Yliy !! -o 250 500 750 1000 Distance traveled nm (n. 2.5

Fig.10 Posftionining with

high accuracy