Subject: TDC Author: MathscinotesTitle: Position Keeper Modeling Checked By: SelfKeywords: TDC, Submarine, Torpedo, World War II Checked By:

Position Keeper ModelingAbstract

During World War II, all the submarines of all warring nations used analog computers toassist submarine commanders with aiming straight-running torpedoes (e.g. UK used their"fruit machine."). US submarines used a computer called the Torpedo Data Computer(TDC) that was considered the best of these devices.The Torpedo Data Computer(TDC) included a function known as Position Keeping that was a mechanical solver for asystem of differential equations .This technical note compares the output from a simpleMathcad model of the Position Keeper with the output from a kinematic model for asimple test case. I solve the system of ODEs using Mathcad's standard ODE solver and ahomebrew routine that may be more appropriate for people who are implementingsoftware versions of the TDC's position keeper function.

Table of ContentsUpdate TOC

AbstractTableofContentsTOCIntroductionAnalysisofBearingAngleandRangeVersusTimeSimulationApproachFieldofBattleKinematicModelDifferentialEquationModelEulerMethodSolutionGraphicalDisplayStandardDifferentialEquationSolverConclusionReference

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IntroductionThe model I am using here comes from following web site.

http://www.hnsa.org/doc/tdc/pg011.htm See reference section for excerpt

I am going to focus on the Position Keeper function of the TDC, which is made up of twocoupled differential equations and two subsidiary equations.

The key function of the Position Keeper is to provide estimates of the target's bearingand range. My simulation will focus on these variables, which are critical to obtaining afire control solution , which I will not cover here.

A more thorough modeling would include more complex scenarios, but this illustratesthe basic approach used. Tactically, the submarine would estimate the target course,range, and speed from periscope and hydrophone readings.

Position Keeper Model for the TDC 1 of 6 22-February-2012

Subject: TDC Author: MathscinotesTitle: Position Keeper Modeling Checked By: SelfKeywords: TDC, Submarine, Torpedo, World War II Checked By:

The TDC would put out a constantly updated position for the target, which the sub skipperwould compare against further periscope readings. The TDC target model would be updateduntil it could accurately predict the target's motion.

With the TDC, the submarine could maneuver and the TDC would keep track of the target'sposition relative to the submarine. This greatly improved the accuracy of the whole firecontrol operation.

Analysis of Bearing Angle and Range Versus Time

Simulation Approach

I am going to work this problem assuming the simple case of a target and submarine(referred to as "Own Ship") both pursuing constant velocity courses. This will allowme to determine the exact distance and bearing numbers using a simple kinematicmodel and the differential equation model. This way, I can verify that my differentialequation solution is reasonable.

I could use the Mathcad differential equation solvers, but I decided to put together asimple solution using Euler's method (i.e. the simplest possible way). This would beeasy for someone to code using any number of programming languages.

Field of Battle

Assume that we are going to place our submarine and target on a 1000 meter by 1000meter grid indexed as shown in Figure 1.

Y A

xis

X Axis 1000 meters0 meters

1000 meters

0 meters

Figure 1: Torpedo Data Computer Simulation Grid

Position Keeper Model for the TDC 2 of 6 22-February-2012

Subject: TDC Author: MathscinotesTitle: Position Keeper Modeling Checked By: SelfKeywords: TDC, Submarine, Torpedo, World War II Checked By:

Kinematic Model

I am going to work this problem assuming the simple case of a target and submarine bothpursuing constant velocity courses. This will allow me to determine the exact distance andbearing numbers using a simple kinematic model and the differential equation model.

Target Own Ship

x coordinate x coordinateRT0

10

900

m RO0900

10

my coordinate y coordinate

x component x componentVT0

10

10

ms

VO010

5

ms

y component y component

RT t( ) RT0 VT0 t Target Position as a function of time.

RO t( ) RO0 VO0 t Submarine Position (i.e. Own Ship) Position as afunction of time.

R t( ) RT t( ) RO t( ) Distance between target and submarine.

t( ) 180deg atan2 RT t( ) RO t( ) 0 RT t( ) RO t( ) 1 Bearing of target fromsubmarine.

Differential Equation Model

SO VO0 11.18ms

Speed of the Submarine (Own Ship)

ST VT0 14.142ms

Speed of the Target

B' Br A R( )SO sin Br( ) ST sin A( )

R Bearing Angle Differential Equation

R' Br A( ) SO cos Br( ) ST cos A( ) Range Differential Equation

CO atan2 VO00VO01

180deg 26.565 deg Own Ship Course (sub knows itscourse)

CT atan2 VT00VT01

180deg 225 deg Target Ship Course (subestimates this from periscopereading of angle on the bow)

Position Keeper Model for the TDC 3 of 6 22-February-2012

Subject: TDC Author: MathscinotesTitle: Position Keeper Modeling Checked By: SelfKeywords: TDC, Submarine, Torpedo, World War II Checked By:

Euler-Method Solution

It is a bit crude, but it appears to work.B

Br

R

A

dt 0.1s

B0 0( )

R0 1259m

A0 B0 180deg CT

Br0 B0 CO

Bi dt B' Bri 1 Ai 1 Ri 1 Bi 1Bri Bi CO

Ai Bi 180deg CT

Ri dt R' Bri Ai Ri 1Bi dt B' Bri Ai Ri Bi 1

i 1 900for

B

Br

Rm

A

Algorithm time step

Initialize Bearing

Initialize Range

Initialize Target Angle

Initialize Relative Bearing

Main Loop

Predict the Bearing

Compute the Next Relative Bearing

Compute the Next Target Angle

Compute Next Range

Corrector for Bearing

Return all the valuesI needed to remove the unitsfrom the range vector (a Mathcad15 limitation)

Graphical Display

Here is a graph comparing my kinematic model with the model from the differentialequation solution. They are identical, which they should be. The nice thing about thedifferential equation solution is that it can handle changes in course by the submarine. Thesecourse changes were automatically fed into the TDC by the sub's gyrocompass.

Position Keeper Model for the TDC 4 of 6 22-February-2012

Subject: TDC Author: MathscinotesTitle: Position Keeper Modeling Checked By: SelfKeywords: TDC, Submarine, Torpedo, World War II Checked By:

i 0 900 Each time increment corresponds to 0.1 seconds.

0 200 400 600 800 1 1030

500

1 103

1.5 103

0

50

100

150

200

250

Exact Target Range (Kinematic)TDC Target RangeExact Target Bearing (Kinematic)TDC Target Bearing

Kinematic Model vs Simulated TDC Output

Time Increment (0.1 seconds)

Ran

ge (m

eter

s)

Bea

ring

Ang

le (

)

Standard Differential Equation Solver

Mathcad has excellent ODE solvers. Here is the same problem worked using one of theirODE solver routine.

T1 90 Maximum time -- the solver does not like units

SOSOm

s

STSTm

s

The solver does not like units

Given

uB u( )d

d

SO sin B u( ) CO ST sin B u( ) CT R u( )

= B 0( ) 0.785=

uR u( )d

dSO cos B u( ) CO ST cos B u( ) CT = R 0( ) 1259=

f

g

OdesolveB

R

u T1 900

z1 0 0.1 90

Position Keeper Model for the TDC 5 of 6 22-February-2012

Subject: TDC Author: MathscinotesTitle: Position Keeper Modeling Checked By: SelfKeywords: TDC, Submarine, Torpedo, World War II Checked By:

0 20 40 60 80 1000

250

500

750

1000

1250

1500

0

50

100

150

200

250

300

RangeBearing

Solution using the Mathcad Standard Solver

Time (sec)

Ran

ge (m

)

Bea

ring

Ang

le (

)

My crude solver got the same result.

Conclusion This model appears to provide a reasonable example for the operation of the TDC, at leastas I read it in the old manual.

Reference

Old Navy TDC manual excerpt

Position Keeper Model for the TDC 6 of 6 22-February-2012

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Historic Naval Ships Association - Torpedo Data Computer Mark 3 http://www.hnsa.org/doc/tdc/pg011.htm

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