ZigzaggingBy

Brian McCue

An Exploration of

This is not a product of the Center for Naval Analyses

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Problem Statement- Under what circumstances is, and is

not, zigzagging of benefit?

This question has proven difficult to answer analytically or empirically,

so let us experiment to ask:

- Can we find cases in which zigzagging is, and is not, of benefit?

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Background: Intercept Kinematics- Target’s speed is triple that of the interceptor.

- The interceptor is able to detect the target a long way off.- The target has no capability to make a counter-detection.- Find the intercept solution.

Ut

VtWt

aa

Vt

Detection radius R

U = speed of interceptor; V = 3U = speed of targetW = interceptor’s velocity in target’s frame; Wt = R , the detection radius t = time needed to intercept

(Ut)2 = (Vt/3)2 = (Wt)2 + (Vt)2 - 2 Wt Vt cos(a)

cos(a) = -Wt Ut / || Ut ||

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Intercept Kinematics: Alternate Solution

Detection radius R

Mariner’s rule-of-thumb is: “Constant bearing, decreasing range.”

Consequence of alternate choice of root in Law of Cosines.Wt = R (still), but with a lesser W and greater t, Ut, and Vt.

Alternate intercept solution for target whose speed is triple that of interceptor

Vt

WtUt

aa

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An Important Degenerate Case

If the Law of Cosines has repeated roots, the triangle is a right triangle, leading to the largest possible angle a for which an intercept can be made; this defines the amount of frontage that the interceptor can defend.

You can think of 2h as the “capture cross section” of the interceptor for targets of speed V.

2h = 2 RU / Vis the size of the “front” along which targets of this speed V can appear and be intercepted.

Detection radius R

Vt

Wt

Ut

aVt

h = RU / V

T2 = R2/(V2-U2) at the extreme

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Rationale for Zigzagging- The preceding kinematic analysis

suggests that a blind target with a speed advantage can, by course alterations, spoil an on-going intercept, should one be occurring.

- This is the basis for zigzagging.- Zigzagging strategies are therefore

distinguishable by their timescales.

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Types of ZigzaggingZigzag so as to avoid Scale of Time

Torpedoes MinutesSubmerged intercept Hours

Surface intercept(Sometimes hard to tell from

evasive routing)

Days

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Zig-Zag Diagrams For Single Ships and Convoys,

Royal Admiralty, 1940- Threat levels characterized as

• "Open waters where submarines have not previously been operating, but where they may appear,"

• "Submarine areas," and• "Specially dangerous waters.”

- Changes of course every 5-10 minutes, by amounts of 10o to 80o.

- Patterns given for slow ships, fast ships, and convoys

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Example: Admiralty Pattern #16

- “For general use in submarine areas.”- “Suitable for convoys of all speeds.”- Chart shows application of #16 to base course 90o, speed 10 knots.- Course-made-good is 17.85 nm in the pattern’s 2-hour cycle.- Purpose is evidently the avoidance of submerged intercept.

Nautical Miles

-2-1012

2 4 6 8 10 12 14 16 18

N

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Zigzagging Went Un-analyzed

- Zigzagging has benefits, but it also has costs.– Delayed arrival, added difficulty of navigation.– Increased path length increases potential

encounters.- Zigzagging’s overall worth has gone un-

analyzed:– Wartime operations researchers couldn’t get data.– It’s difficult to frame analytically.– Until lately, it was too much to do in Monte Carlo.

- So we will look at a model of zigzagging, try analyzing it, and then use Monte Carlo.

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Set-up of Zigzagging Model

Nautical miles

0

250

500

750

1000

0 250 500 750 1000 1250 1500 1750 2000

One transiting 30-knot “target” headed Eastwards

25 10-knot “submarines” and their 20-mile detection radii

Nautical miles

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Comments on RealismThis is an exploration of zigzagging, not a

simulation of World War II, but, ...- 10 knots is about right for U-boats.- 30 knots might be fast even for a liner-

troopship.- 1000 x 2000 nm is about right for Atlantic

arena- Real U-boats were coordinated, but tried to

attack in packs rather than ASAP.

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Transitor “Doctrine” in the Model- Zig or zag a given number of degrees North or

South of the due-East base course.- Times between zigs are exponentially

distributed. (The transitor doesn’t know he’s in a discrete time-step model and that the distribution is geometric.)

- Bounce off North and Sout boundaries elastically.- Figure shows example with average time of 7.5

minutes and angles of +/- 21.25o --same as #16, shown again for comparison. (Model returns to baseline only by chance.)

-4

-2

0

2

4

2 4 6 8 10 12 14 16 18

Admiralty # 16

Model

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Submarine “Doctrine” in the Model

- Uncoordinated case:– Move to intercept the target if you detect it.– If you can’t make an intercept, move to minimize

the miss distance (CPA) and hope target zigs to you.

- Coordinated case:– Move to intercept the target if anybody detects it.– If you can’t make an intercept, move to minimize

the miss distance (CPA) and hope target zigs to you.

– If contact is lost, stay on course. (Bounce off walls.)

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But 1st, how far can we get analytically?- “Capture cross section” is 2 RU / V

= 40/3 miles- There are 25 submarines in 2

million sq. nm.- Path length is 2000 / cos(zig angle).- Probability of transiting without any

encounter ise-(40/3) x 25 x ( 2,000 / cos(zig angle) ) / (2,000,000)

- About 70% for 0o < zig angle < 30o, degrading to 47% for 64o.

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How far can we get analytically? (cont)- Engagement time is something like

T2 = R2/(V2-U2) = 400/(900-100) = 1/2 hour = 30 min.- Probability of zigging during an

engagement is something like1-e-30/avg zig time

- Probability of transiting without fatal encounter is therefore something likee-(40/3) x 25 x ( 2,000 / cos(zig angle) ) / (2,000,000) (1-exp(30/avg zig time))

- This is a lot of “something like“s!- Probability is relative insensitive to zig

time.

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Defects in Analytic Approach

- “Engagement time” was an upper bound, so comparing zig time to it favors the transitor.

- Analysis credited all zigs with being evasions, which they won’t be. So small zig angles get too much credit for evasion, while big zig angles’ disadvantage of long path length is taken fully into account.

- No treatment of coordination of interceptors.

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Cases to Run in the Model- All combinations of:

– 3, 9, 30, 90, 300, 1000, 3000 minute average times between zigs and

– 0o, 1o, 2o, 4o, 8o, 16o, 32o, 64o zig angles.- Each with:

– Uncoordinated submarines.– Coordinated submarines.

- 100 times each.- The target is “intercepted” if the

submarine gets within a mile--because of torpedo range and time granularity (two minutes).

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Demonstration

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Zigzag Fitness Landscape(v. uncoordinated submarines)

Angle of the zig-zag (degrees)

Average time between zigs

(Minutes)

0 1 2 4 8 16 32 643

10

30

90

300

1000

3000

65-7065-70

70-75

60-65

55-60

75-80

Number of safe passages out of 100

Results smoothed byaveraging neighbors

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Zigzag Fitness Landscape(v .coordinated submarines)

Angle of the zig-zag (degrees)

Average time between zigs

(Minutes)

0 1 2 4 16 32 643

10

30

90

300

1000

3000

75-80

70-7565-70

60-65

55-6050-55

45-50

40-45

35-40

30-3540-45

Number of safe passages out of 100

Admiralty plansfor individual fastships

Results smoothed byaveraging neighbors

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Observations- Zigzagging can help.

– High-frequency, medium-high angle, helped against the coordinated submarines

– Our scenario was WW II-like, and WW II-like zigzagging worked well in it.

- Zigzagging can hurt.– Low-frequency, high angle, against the

coordinated submarines– High-angle, low frequency zigzagging seems to

be worse than none at all against the uncoordinated subs, and no form of zigzagging is particularly better than none at all.

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So is Coordination Bad? NO.

- Zigzagging helps more if the submarines are coordinated, but– This is not to say that the submarines ought

not to coordinate: even when degraded by zigzagging, their coordinated performance is no worse than their uncoordinated performance.

- High-frequency, high-angle zigzagging can reduce the effectiveness of the coordinated submarines to that of the uncoordinated submarines, but not below.

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So Is Zigzagging Good? - IT CAN BE.- We found a regime where it helped a lot:

– High frequency, medium-high angle, against coordinated submarines.

- It doesn’t seem to hurt unless reduced to the absurd extreme of low frequency, high angle.

- There’s a cost we haven’t considered—increased transit time—that may or may not be a big consideration.

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Questions- How much do our findings depend upon our

choices of U, V, and R ?- How real is the local maximum at 2o , 300

minutes in the uncoordinated case?- What would happen if interceptors used the

long intercept solution instead of the short one?

- And now for your questions ...

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Zigzag Fitness (Analytic)(v. uncoordinated submarines)

Average time between zigs

(Minutes)

60-65 55-6050-55

Angle of the zig-zag (degrees)0 1 2 4 8 16 32 643

10

30

90

300

1000

3000

Number of safe passages out of 100

70-75 65-70

45-50