Using math to defeat the enemy (7 7-2011)

Approved for Public Release
Simulation Educators, LLC (29 June 2011)
Jeffrey Strickland, Ph.D., CMSP
1
DISTRIBUTION STATEMENT A. Approved for
public release; distribution is unlimited.

Learning Objectives
1. Describe the scope of mathematical and heuristic
combat models.
2. Compare and contrast different representations of
combat phenomenon.
3. List combat behaviors that can be represented by
mathematical & heuristic models.
4. State the various types of mathematical and
heuristic combat models.
5. Identify examples of mathematical and heuristic
combat models.
2Copyright© 2010 Jeffrey Strickland, Ph.D.

Tutorial Outline
 Environmental modeling
 how to model the
environment
 level of detail
 entity interaction
 Physical modeling
 how to move
 how to sense or detect
 how to shoot (or create
other effects)
 how to communicate
 Simulation scenario
development
 what are the elements of a
scenario
 how to develop scenarios
 Missile Flight Modeling
 Missile dynamics
 Sensor dynamics
 Racking error
 Coordinate systems
 Simulation
 Results
 Simulation scenario
development
 what are the elements of a
scenario
 how to develop scenarios

Level of Detail
Conceptual Reference Model
Data Collection
Data Processing
Static Environment
Dynamic Environment
Standardization

Level of Detail
 Perceived details
 bitmaps over data points
 hills, trees, rivers, rocks
 No interaction
 simulated system does not
interact directly with terrain
details.
 Visual detail
 polygon color & lighting
 bit mapped surfaces
 hard surfaces
 Modeling detail
 surface trafficability
 foliage density
 tree trunk diameter
5
Air Combat Terrain Ground Combat Terrain
Copyright© 2010 Jeffrey Strickland, Ph.D.

Conceptual Reference Model
6
Component
Models
Environmental
State
Behavior
Models
Environmental
Models
Synthetic Natural Environment
Behaviors (e.g.)
• Maneuver
• Sustainment
• Force
Protection
• Intelligence
• Command &
Control
• Fires
Military System Model
Effects (e.g.)
• Attenuation
• Propagation
• Mobility
Internal Dynamics
Impacts (e.g.)
• Obscurants/
Energy (smoke,
chaff, spectral,..)
• Damage
(engrg, craters,..)
Data (e.g.)
• Terrain
(surface, hydro,..)
• Atmosphere
(aerosols, clouds,..)
• Ocean
(sea state, SVP,..)
• Space
(particle flux,..)
• Cultural
(roads, structures,..)
• Military
(engrg. works,..)
Passive
Sensors
Active
Sensors
Weapons &
Countermeasures
Units/Platforms
SOURCE: Paul A. Birkel, "SNE Conceptual Reference Model", 1999 Fall SIW Conference, September 1999.
http://www.sisostds.org/siw/98Fall/view-papers.htm

Data Processing
7
 Collection
• survey the
environment
(satellite, maps, etc.)
• store the results
• vector, grid, and
model data
 Cleaning
• remove collection
process
discontinuities
• synchronize vector
and grid data
 Organizing
• index and archive
 Integration
• merge vector, grid,
model
• generate terrain
skin with embedded
features and
surface data
 Transmission
• move data to the
host system
 Compilation
• create
performance-
optimized runtime
databases
• cut into sheets

Landclass Terrain in AVSIM FSX
 The scenery engine in FSX, as with previous versions, uses the Olson Global Ecosystem
Legend, a table of terrain coverage types created by the USGS Earth Resources
Observation and Science Center (EROS). This data, called landclass data, is used by the
simulator to associate up to 255 types of terrain to map the entire surface of the globe. The
smallest level of detail is 1.2 square kilometers (0.46 square miles).
 The landclass data is used by the simulator to select textures and objects to render the
scenery. The table below shows an example of the Olson classes used in FSX:
8
0 Ocean, Sea, Large Lake 40 Cool Grasses And Shrubs 131 Dirt
1 Large City Urban Grid Wet 41 Hot And Mild Grasses And Shrubs 132 Coral
2 Low Sparse Grassland 42 Cold Grassland 133 Lava
3 Coniferous Forest 43 Savanna (Woods) 134 Park
4 Deciduous Conifer Forest 44 Mire Bog Fen 135 Golf Course
5 Deciduous Broadleaf Forest 45 Marsh Wetland 136 Cement
20 Cool Rain Forest 46 Mediterranean Scrub 137 Tan Sand Beach
27 Conifer Forest 53 Barren Tundra 143 Glacier Ice
29 Seasonal Tropical Forest 54 Cool Southern Hemisphere Mixed Forests 144 Evergreen Tree Crop
33 Tropical Rainforest 60 Small Leaf Mixed Woods 146 Desert Rock

Storing Environmental Data
9
Triangulated Irregular Network (TIN)
Data point correlation
Surface tiled with hexagons

Static Environment
10
Trafficability
Terrain Type
Visibility

Landclass Terrain in EADSIM
 Name: Desert (radiobutton)
 Description: If selected, the LANDCL model will assume desert
terrain.
 Restrictions: Ghosted unless LANDCL Reflectivity is selected.
 Name: Farmland (radiobutton)
 Description: If selected, the LANDCL model will assume farmland
terrain.
 Name: Wooded Hills (radiobutton)
 Description: If selected, the LANDCL model will assume wooded hill
terrain.
 Name: Mountains (radiobutton)
 Description: If selected, the LANDCL model will assume
mountainous terrain.

Dynamic Environment
12
 Independent
• weather movement –
clouds, rain, wind
• sea state – storms, daily
tide
• daylight – sunrise, sunset,
dark
• smoke & dust – clouds,
raising, dispersing
 Interaction
• holes – artillery craters,
engineering artifacts
• tank treads – tracks,
destruction
• terrain morphing –
engineering, construction
• feature modification –
building damage, trees
burned

Classic Problems in Interpretation
13
1
2
3a 3b
1
2a 2b
Terrain Points Building Corners

Environmental Standardization

Physical Modeling
15
Detect/Acquire
Engage
(other major
combat functions)
Communicate
Move
Start Cycle Here

Movement Points Movement
Bald Earth Movement
Terrain and Feature Movement
Physics-based Movement
Automated Route Planning
A* Search
Topology Smart
Grid Registration
Behavioral

Movement Points Movement
17
2
3
6
1
2 6 2
1
Movement
Points =
20
Movement
Points
Remaining =
20 – 11 = 9

Bald Earth Movement
18
Set heading, speed, start time
Rate*Time = Distance
20 km/hr * 30 min = 10 km

Terrain and Feature Movement
19
Set Objective: position or vector
Terrain & features modify instantaneous heading & speed
Speed = min(order_speed, max_speed*trafficability*slope_factor)*
weather_factor*lighting_factor*fatigue_factor*supression_factor

Physics-based Movement
20
Proportional Force
Calculation
Resistive Force
Calculation
Braking Force
Calculation
main force calculations
Dynamic
Equation
Calculations
net force
new vehicle state
(pos, vel, acc)
Vehicle type, terrain
type, slope, controls,
current platform state
 The CCTT ground vehicle mobility
model is based on a general first-
principle dynamics model.
 The model integrates explicit
driver inputs (e.g., throttle, brake)
with vehicle class-specific velocity,
resistance force, and deceleration
pre-computed curves.
Simple View of a Dynamic
Movement Model
CCTT Vehicle Dynamics Block Diagram

Automatic Route Planning
 CONCEPT: provide an algorithm by which units
can automatically find their own routes.
 allows the analyst to focus on higher issues such as the
overall scheme of maneuver
 reduces the intrusion of the analyst into C2
 units can still be given explicit routes if desired, or closely
grouped intermediate objectives
 ALGORITHMS: based on graph theory
 could be a satisfying algorithm (not guaranteed to find an
optimal route)
 might be an optimal algorithm
 “optimal" may mean fastest, or shortest, or safest, etc.
 EXAMPLES
 A* search, Johnson’s algorithm, Dijkstra's algorithm, hill
climbing

Topology Smart
22
Set Objective: Position or Vector
Movement model selects path from topological map
Maintain objective
Route traveled is function of topology

Beyond 2-D Movement
 3 Dimensional—aircraft rotation axes
 yaw - vertical axis rotation
 roll - longitudinal axis rotation
 pitch- lateral axis rotation
 3-D Mathematics
 Euler angles
 axis angle
 rotation matrices
 quaternions
 Other degrees of freedom: 3+3 DOF, 6
DOF
23
Pitch
Yaw
Roll

Behavioral—Agent Based
 Behavioral evolution
and extrapolation
 Each avatar generates
(a) a stream of ghosts
samples the personality
space of its entity.
 They evolve (b, c) against
the entity’s recent observed
behavior.
 The fittest ghosts run into the
future (d),
 and the avatar analyzes their
behavior (e) to generate
predictions.
24
a
b
e
d
Prediction Horizon
Observe Ghost prediction
Insertion Horizon
Measure Ghost fitness t=τ
(Now)
Ghost time τ
c
Real-World
Entity
Avatar
Ghosts
    





  1nRThreat
nn
nn
n
DistGNest
TargetGTargetR
F

Perfect Detection
Gridded Probability Areas
Detection Range
3D Detection Range
Target Acquisition Process
Line-of-Sight
NVEOL Model

Perfect Detection
26
 Every object knows the true location of every other
object.
 There are no models of sensors or processors.

Gridded Probability Areas
27
 Perfect detection within the
same grid area
• (Pdet = 1.0)
 Probability of detection
within adjacent areas
• Adjacent Pdet =F(terrain)
• Non-Adjacent Pdet = 0.0
60%
30%
100%
0%

Detection Range
28
 Complete circle—no field of view/field of regard
 Terrain line-of-sight (LOS) is separate

3D Detection Range
29
 Probability of detection
based on range of spheres
 Concentric areas
• Different Pdet for each ring
• For some sensors, Pdet of
inner ring is 0.00
𝜓 = 𝜓0
sin
𝜋𝑎
𝜆
sin 𝜃
𝜋𝑎
𝜆
sin 𝜃
sin
𝑁
2
2𝜋𝑑
𝜆
sin 𝜃 + 𝜙
sin
𝜋𝑑
2
sin 𝜃 + 𝜙
𝐼 = 𝐼0
sin
𝜋𝑎
𝜆
sin 𝜃
𝜋𝑎
𝜆
sin 𝜃
2
sin
𝑁
2
2𝜋𝑑
𝜆
sin 𝜃 + 𝜙
sin
𝜋𝑑
2
sin 𝜃 + 𝜙
2

ALARM—Advanced Low
Altitude Radar Model—4.2
 ALARM is a generic digital computer simulation designed to evaluate the performance of a
ground based radar system attempting to detect low altitude aircraft.
 The purpose of ALARM is to provide a radar analyst with a software simulation tool to evaluate
the detection performance of a ground-based radar system against the target of interest in a
realistic environment.
 Used in EADSIM
 ALARM can simulate
 pulsed/Moving Target Indicator (MTI)
 pulse Doppler (PD) type radar systems
 limited capability to model continuous
wave (CW) radar.
 Radar detection calculations are based on the
signal-to-noise (S/N) radar range equations
commonly used in radar analysis.
 ALARM has four simulation modes:
 Flight Path Analysis (FPA) mode,
 Horizontal Detection Contour (HDC) mode
 Vertical Coverage Envelope (VCE) mode
 Vertical Detection Contour (VDC) mode

Target Acquisition
Glimpse models
 Intermittent glimpses: E[N] = Σn np(n)
 Continuous looking model = PROBDETECT in time t = 1 - e-Dt
 DYNTACS curve fit model = D = PFOV (α/(β + t(δ + ζR2 – ξVc)))
 NVEOL acquisition algorithm
 Factors
 Sensor
characteristics
 Target characteristics
 Line-of-sight
31
Glance/
Glimpse
Target
Found?
No
Yes
tg tg tg
Pacq Pacq Pacq

NVEOL Acquisition Algorithm
32
Joint
Conflicts
And
Tactical
Simulation
Developed by US Army's Night Vision
and Electro-Optical Laboratories
In Time-Stepped Model:
PROBDETECT in time T = PINF (1 - e -CT)
Use this as success probability for a Bernoulli trial.
In Event-Stepped Model:
Compute PINF and draw a random number to determine if
detection would occur in infinite amount of time
Sample from an exponential distribution with mean C to
determine time till detection given that a detection will occur.

Line-of-Sight Models
 EXPLICIT: combat model stores a terrain representation and uses it
to compute line-of-sight
 Grid: covers the battlefield with regular polygonal grid, each grid having
associated terrain attributes (e.g., elevation, vegetation, etc.)
○ Look at intervening grids between observer and target to see if any grid is higher than the
line between them.
○ Discontinuity is a disadvantage in high-res models.
○ Simplicity and speed are advantages.
 Surface
○ Triangulate the terrain data grids, then interpolate for a point between grid points.
○ Greater accuracy is an advantage in high-res models.
 IMPLICIT: combat model stores expected results of line-of-sight and
looks up the result when required
 probability of LOS
 intervisibilty segment length
33
. . . . . . . . . . . . . . . . .Primary Direction of
view (white)
Max Range
of view
LOS does not
exist
LOS exists
Orange lines
Left Limit
of View (white)
Right Limit
of View (white)

Line-of-Sight Elevation
Calculation in EADSIM
34
 The equation known as the law of cosines can be written for the
triangle ABC as:
 b = a2 + c2 − 2ac * cos(∂) . (1)
 This equation can be solved as:
 Cosψ=(a2+c2-b2)/2ac (2)
 The law of cosines equation can also be written for the triangle BCQ as:
 b’2=a2+c’2-2ac’*cosψ (3)
 Substituting equation (2) into (3) and simplifying yields:
 b’2=c’2+(((b2-a2-c2)*c’)/c)+a2

Comms Model Effects
Perfect Communications
Direct Message Passing
Broadcast Messages
Virtual Cell Layout
Physics Modeling

Comms Model Effects
 Information exchange
 process info
 process data
 Intelligence collection
 ISR sensors
 target sensors
 fire control sensors
 Comms system overload
 network, sender, receiver
 Interference
 environment, electronic
warfare
 Time delay
36
Evaluate Target's Intent
Evaluate Target's Geometry
Recognize Target
Update Target's Knowledge
Notify Knowledge Processing
Activity Diagram: Process Info Use Case
Process Info
Get Data from Fire
Control Sensor
Get Data from
Target Sensor
Get Info from Data
Processing

Perfect Communications
37
Targets
~~~~~
Orders
~~~~~
Reports
~~~~~
Shared information, no representation of comms
Software-to-software message delivery

Direct Message Passing
 Consult command
status
 If sender and receiver
are alive, then pass
message.
 If sender health is
degraded, add error to
target location.
38
… …

Broadcast Messages
 Receiver determines
whether signal is accessible
to them based on
 range
 terrain degradation
 earth curvature
 jamming environment
 communications contention
 quality of receipt
 etc.
39
…
…Success
Lost
Degraded
Delayed

Communications
Connectivity Modeling by
Propagation Network Types and their number of
participating platforms are as follows:
 Duplex communication occurs in both
directions and is limited to two
participants.
 Simplex communication occurs in
only one direction between two
participants. The first platform on list
is the transmitter.
 Broadcast communication occurs
from one platform to several other
platforms. The first platform on the
list is the transmitter.
 N-to-N serial communication occurs
between all the participant platforms.
 N-Broadcast simultaneous
communication occurs between all
the participant platforms.
 Land Line communication occurs
between two participants (not affected
by Jamming).
 Links Exist if two Conditions are met:
 Receiver signal power level must be
equal to or greater than user-specified
minimum discernible signal level
 Signal-to-noise level (received signal
power level received jam power level)
must be equal to or greater than user-
specified signal-to-noise threshold

Point System
Markov Pk Tables
Random Numbers
Pk’s and Random Numbers
Precision Engagements
Linear Target Phit
Rectangular Target Phit
Circular Target Phit
Kill Categories

Point System
42
New Health = (Health + Armor) – (Weapon Power – Path Degrade)
New Health = (18 + 8) – (20 – 4) = 10
New Armor = Armor – ABS[( Weapon Power – Path Degrade) *0.25]
18
4
20
8
Weapon Power
Path Degradation
(range, shelters, obstructions)
Health
Armor

Markov Pk Table
43
Pk
Weapon
W1 W2 W3 W4 …
T1 0.5 0.7 0.8 0.92
T2 0.4 0.45 0.76 0.99
T3 0.31 0.34 0.56 0.85
T4 0.27 0.55 0.67 0.81
Target
…
Phit is rolled into the overall Pkill
Damage = 1, where Random Number <= Pk
= 0, where Random Number > Pk

Random Numbers
44
0.002589 0.709121 0.688907 0.23241 0.248291 0.279792 0.099733
0.672374 0.177176 0.5124 0.253238 0.885889 0.08127 0.337699
0.967582 0.11894 0.917944 0.691778 0.377643 0.167685 0.23337
0.821207 0.775446 0.94055 0.916313 0.342373 0.494679 0.83171
0.76565 0.300179 0.081692 0.212297 0.323383 0.088898 0.976731
0.826355 0.633324 0.390983 0.559808 0.032313 0.337002 0.429531
0.284963 0.978167 0.177686 0.39425 0.729517 0.196937 0.053272
0.537055 0.753125 0.189256 0.790979 0.437795 0.757163 0.953741
0.714325 0.899821 0.139968 0.139168 0.803138 0.274158 0.226658
0.151101 0.555232 0.533085 0.327454 0.753654 0.268759 0.307099
0.21175 0.644434 0.011707 0.809213 0.3742 0.38085 0.412449
0.425525 0.346873 0.490443 0.397201 0.114504 0.831309 0.291209
0.157902 0.994106 0.22623 0.215775 0.503133 0.544428 0.05825
0.173804 0.322742 0.984154 0.512732 0.340096 0.626067 0.746717
0.391907 0.168648 0.606554 0.280939 0.804009 0.290058 0.550802
0.743599 0.108666 0.557355 0.850634 0.908114 0.209818 0.600702
0.682586 0.265387 0.792137 0.241523 0.077536 0.282332 0.244388
0.688018 0.607142 0.296545 0.583956 0.652407 0.773843 0.801856
0.037354 0.516678 0.27669 0.360097 0.700107 0.821834 0.912564
0.914889 0.18311 0.164431 0.880446 0.527801 0.887302 0.209683
 Generated by a recursive function
 Evenly distributed between 0 and 1 ~ Unif(0,1)
 Perfect for Pk evaluations

Pk’s and Random Numbers
45
Kill Area No-Kill Area
0% 75% 100%
Random Number = 0.63
Pk = 75% = 0.75

Precision Engagements
46
Round Impact Point
 PROBLEM: Find point of impact (if any) of round on its target.
 ASSUMPTION: The projectile impact point is a random variable with a
normal probability distribution (empirically shown to be a good assumption).
Actual Target Location
Doctrinal Aim Point
Aim Point
“Bias” : Systematic Errors
“Dispersion” : Round-to-Round
Independent Errors
Perceived Doctrinal
Aim Point
Perceived Target Location

Linear Target Phit
47
 Normal parameters for 1D target:
• “Front view" (i.e., direct-fire weapon)
○ Deflection error
• "Top view" (i.e., indirect-fire weapon)
○ Range error
• DEFINE:
○ Bias = μ
○ Dispersion = σ
 Error Probable - distance in deflection (for x) within
which half of rounds will land.
 Linear Error Probable (LEP) - linear distance from aim
point within which half of rounds will land, based on the
error probable (details to follow).
x
p(x)
25 m

Single-Shot Accuracy
1D Target Example 1
 Assume no systematic error.
48
    2126.03937.06063.0  zzPSSH
NOTE: “” is available in
tabular form in any Statistics
text: see Normal Distribution.
 
  3937.00644.37010
6064.00644.37010
then,m,10m,0664.376745.0250,



z
z
x
PSSH
0

-z +z

Rectangular Target Phit
 Normal parameters for 2D target:
 "Side view" (i.e., direct-fire weapon)
○ Elevation error
 "Top view" (i.e., indirect-fire weapon)
○ Range error
 DEFINE:
 Bias = μx , μy
 Dispersion = σx , σy
 Range Error Probable (REP) – linear distance from aim
point within which half of rounds will land, x-coordinate
 Cross-range Error Probable (CREP) – linear distance
from aim point within which half of rounds will land, y-
coordinate
49
x
y
p(y)
p(x)

Circular Target Phit
 P(destruction of a point target) = P(hit within a circle of radius
R), i.e., Pd = P.
 When x0 = y0 = 0 and x2 = y2 = 2,
 If R0 is the radius of a circle for which
then 50% of all impacts points for the probability distribution P(r) will fall
within this radius r ≤ R0.
 R0 is called the circular error probable (CEP), and R0 =
1.1774.
50












2
2
2
exp1

RR
Pd
 
2
1
2
exp1 2
2
0
0 







R
RP
Target
Simplified Vehicle
Assembly Area
Cluster of Soldiers

Kill Categories
K-Kill: catastrophic kill
F-Kill: firepower kill
M-Kill: mobility kill
MF-Kill: mobility & firepower kill, usually => K-Kill
P-Kill: personnel kill (crew and passengers)
No-Kill: no damage due to hit.
51
ranx = random(seed)
if (ranx < PkN)
{No Kill}
else if (ranx < PkN + PkM)
{Mobility Kill}
else if (ranx < PkN + PkM + PkF)
{Firepower Kill}
else if (ranx < PkN + PkM + PkF + PkMF)
{Mobility & Firepower Kill}
else
{Catastrophic Kill}
Single random number draw can result
in more than just “Miss/Hit”
Engagement outcome has at least 5
states

Direct-Fire Accuracy Example (1)
 An infantry fighting vehicle (IFV) has the following frontal profile:
 A hit in area 1 will
produce a firepower kill.
produce a catastrophic kill.
produce a mobility kill.
 A hit in other areas will
produce no permanent effect.
 Assess the IFV’s vulnerability when engaged with a frontal shot whose impact
point is modeled as a random variable pair (X,Y) ~ BVN(0,0,.5,.5,0).
 Using the below list of pseudo random numbers as needed, simulate the first
round to determine which type of kill, if any, occurs (.8554, .2287, .6659, .8243,
.6840, .0430, .8598, .2381, .5035, .2723).
52
2
1 44
3
0.6
1.6
1.0
1.4 2.6
0.6

Direct-Fire Accuracy Example (2)
1) Do a Monte Carlo simulation of impact
point with origin centered on the target,
then compare impact point with target
profile to calculate where it hit.
2) Determine X coordinate of impact point:
 Enter the Normal Table with 0.8554
 Find Z-1 = 1.06
 Note that Z-1 = ((x − x)/x
 Solve for x in 1.06 = (x − 0)/0.5
 x = 0.53
3) Determine the Y coordinate of the impact point (using RN .2287):
 Normal Table goes from 0.5000 to 0.9999, but Normal Dist. is symmetric,
so compute 1.0 − 0.2287 = 0.7713, and change sign of resulting Y
coordinate.
 Interpolating between 0.77 and 0.78, gives Z-1 = 0.743.
 Solve for y in −0.743=(y − 0)/0.5 gives y=−0.3715
4) Round hits area 4, so no kill is assessed.
53
2
1 44
3
0.6
1.6
1.0
1.4 2.6
0.6
Y
X
−0.3715
. 53

Lanchester Equations
Aggregated Combat Groups
Epstein’s Equations
Quantified Judgment Model (QJM)
Force Ratio Approach

Lanchester Equations
   
dx
dt
f x y
dy
dt
f x y 1 2, ,... , ,...
55
CONCEPT: describe the rate at which a force loses
systems as a function of the size of the force and
the size of the enemy force. This results in a system
of differential equations in force sizes x and y.
The solution to these equations as functions of x(t)
and y(t) provide insights about battle outcome.
ay
dt
dx

bx
dt
dy


Aggregated Combat Groups
 Contiguous
pistons
 Aggregated force
attrition
 Distance from
middle affects
power and attrition
 Units accumulate
as piston moves
 Explicit withdrawal
required

Force Ratio Attrition Models
 CONCEPT:
 Summarize effectiveness in combat with a single scalar
measure of combat power for each unit.
 When combat occurs, use the ratio of attacker's to defender's
measures to determine the outcome.
 Assign a firepower score to each weapon system and sum these
scores for each weapon system on hand in a unit.
 DEFINITIONS:
 n = number of distinct types of weapon systems in a unit
 Xi = number of systems of type i (I =1,2,...,n) in a unit
 Si = firepower score for each weapon of type i
57
  unitofindexfirepowerFPI
1
 
n
i
iisx
battleainforce
FPI
FPI
FR
defender
attacker


Other Aggregated Models
 Epstein equations
 Defender’s withdrawal rate:
 Attacker’s Prosecution rate:
 Quantified Judgment Model (QJM)
 T.N. Dupuy created the QJM to transform Clausewitz’s Law of Number to
a combat power formula.
 Multi-agent models
 The environment takes the form of a distributed network of place agents.
 Aggregate state-space models
 Represented by aggregate state variables, rather than the locations and
current behaviors of individual entities
58
        
   
 
  aTa
aT
gaT
gg
dTd
dT
t
t
tt
t
tWW
tWtW











 










1
1
1
1
1
1
1 max

Missile Dynamics
Sensor Dynamics
Coordinate Systems
Missile Flight Simulation

Components of Missile Flight
Simulation
 MISSILE SYSTEM DESCRIPTION
 MISSILE
 GUIDANCE
 LAUNCHER
 MISSILE DYNAMICS
 MISSILE AERODYNAMICS
 MISSILE PROPULSION
 MISSILE AND TARGET MOTION
 GUIDANCE AND CONTROL MODELING
 SCENE SIMULATION
 IMPLEMENTATION

Missile Dynamics
Axis
Force
Along
Axis
Moment
About
Axis
Linear
Velocity
Angular
Displacement
Angular
Velocity
Moment
of Inertia
𝑥 𝐹𝑥 𝐿 𝑢 𝜙 𝑝 𝐼 𝑥
𝑦 𝐹𝑦 𝑀 𝑣 𝜃 𝑞 𝐼 𝑦
𝑥 𝐹𝑧 𝑁 𝑤 𝜓 𝑟 𝐼𝑧
61
𝑀, 𝑞
𝑦 𝑏
𝑥 𝑏
𝑧 𝑏
𝑁, 𝑟
𝐿, 𝑝
𝐹𝑥, 𝑢
𝐹𝑦, 𝑣
𝐹𝑧, 𝑤
𝑢 =
𝐹𝑥 𝑏
𝑚
− 𝑞𝑤 − 𝑟𝑣 , m/s2
𝑣 =
𝐹𝑦 𝑏
𝑚
− 𝑟𝑢 − 𝑝𝑤 , m/s2
𝑤 =
𝐹𝑧 𝑏
𝑚
− 𝑝𝑣 − 𝑞𝑢 , m/s2
𝑝 = 𝐿 − 𝑞𝑟 𝐼𝑧 − 𝐼 𝑦 𝐼 𝑥 , m/s2
𝑞 = 𝑀 − 𝑝𝑟 𝐼 𝑥 − 𝐼𝑧 𝐼 𝑦 , m/s2
𝑟 = 𝑁 − 𝑝𝑞 𝐼 𝑦 − 𝐼 𝑥 𝐼𝑧 , m/s2
ROTATIONAL EQUATIONS
TRANSLATIONAL EQUATIONS

Sensor Dynamics
 Pseudo-imaging
 Imaging
 Radio Frequency
Seekers
 Pulse Radar
 Continuous Wave
Radar
 Pulse Doppler Radar

Projection of Tracking
Error on Reticle Plane
63
Boresight
Axis
Angular
Tracking Error
Tracking Error Vector
Field of View
Plane of Reticle
Detector
Arbitrary
Reference
Target Projection on Reticle
Line of
Sight
Target

Coordinate Systems
64
Target Coordinate System (𝑥 𝑡, 𝑦𝑡, 𝑧𝑡)
Body Coordinate System (𝑥 𝑏, 𝑦 𝑏, 𝑧 𝑒)
Earth Coordinate System (𝑥 𝑒, 𝑦𝑒, 𝑧 𝑒)
Guidance Coordinate System (𝑥 𝑔, 𝑦 𝑔, 𝑧 𝑔)
Tracker Coordinate System (𝑥 𝑠, 𝑦𝑠, 𝑧 𝑠)
Wind Coordinate System (𝑥 𝑤, 𝑦 𝑤, 𝑧 𝑤)

Missile Simulation
65
𝑋 1 = 𝑃 𝑀 𝑖
𝑋 2 = 𝑃 𝑀 𝑗
𝑋 3 = 𝑃 𝑀 𝑖𝑘
𝑋 4 = 𝑢
𝑋 5 = 𝑣
𝑋 6 = 𝑤
𝑋 7 = 𝑢
𝑋 8 = 𝑣
𝑋(9) = 𝑤
Read and Initialize
Input Data
Atmosphere,
Mach Number,
Dynamic Pressure
Relative Velocity,
Range,
Range Rate
Closest
Approach
?
Guidance and Control
Forces on Missile
Missile Accelerations
Update Missile and Target
Positions and Velocities
Update Time, Missile
Mass, CM Location, and
Moments of Inertia
T > Tmax
Or
Crash?
End
Miss Distance
Yes
YesNo
No
𝑃 𝑀 𝑖 = 𝑋𝑂𝑈𝑇 1
𝑃 𝑀 𝑗 = 𝑋𝑂𝑈𝑇 2
𝑃 𝑀 𝑘 = 𝑋𝑂𝑈𝑇 3
𝑢 = 𝑋𝑂𝑈𝑇 4
𝑣 = 𝑋𝑂𝑈𝑇 5
𝑤 = 𝑋𝑂𝑈𝑇 6
𝑢 = 𝑋𝑂𝑈𝑇 7
𝑣 = 𝑋𝑂𝑈𝑇 8
𝑤 = 𝑋𝑂𝑈𝑇(9)

FlatEarthMissileEqns(u)
% Define Control Variables from Inputs
T = u(1); % thrust along missile velocity
wel = u(2); % turn rate in elevation
waz = u(3); % turn rate in azimuth
% Define State Variables from Inputs
x = u(4:12);
% Location Variables
Px = x(1); % Position in Direction of North Pole
Py = x(4); % Position At Equator in y
Pz = x(7); % Position At Equator in z
% Body_Axes Velocities
U = x(2); % velocity in Px direction
V = x(5); % velocity in Py direction
W = x(8); % velocity in Pz direction ("Up")
% Body Axes Acceleration
%Accx = x(3);
%Accy = x(6);
%Accz = x(9);
% Speed, Atmospheric Density and Drag
Vxy2 = U^2 + V^2;
Vxy = sqrt(Vxy2);
Vxz2 = U^2 + W^2;
Vt2 = Vxz2 + V^2;
Vt = sqrt(Vt2);
az = atan2(V, U);
el = atan2(W, Vxy);%
Atmospheric Density in kg/meterA3
if Pz < 0 % Travel inside the Earth is Viscous
rho = 10^2;
elseif Pz < 9144 % Altitudes below 9144 meters
rho = 1.22557*exp(-Pz/9144);
else % Altitudes above 9144 meters
rho = 1.75228763*exp(-Pz/6705.6);
end
beta = cfric*rho;
Tacc = T/Vt;
% Compute the Derivatives
dPx = U;
dPy = V;
dPz = W;

Simulation Results
67
-500 0 500 1000 1500010002000
-100
0
100
200
300
400
500
600
X (km)
Three Dimensional Missile Trajectory in kilometers
Y (km)
-500 0 500 1000 1500
0
200
400
600
800
1000
1200
1400
X (km)
Three Dimensional Missile Trajectory in kilometers
Y(km)
0 50 100 150 200 250 300 350
0
1000
2000
3000
4000
5000
6000
7000
8000
MissileSpeed(m/s)
Time (seconds)
Missile Speed vs Time
0 50 100 150 200 250 300 350
-200
-150
-100
-50
0
50
100
150
200
Missile Azimuth Heading vs Time
Tome (seconds)
Missile Azimuth Heading
vs. Time
Missile Speed vs. TimeY vs. X in kmZ vs. X in km
-500
0
500
1000
0
500
1000
1500
0
100
200
300
400
X (km)
Intercept Time = 209.2 seconds
Miss Distance = 0.54057 meters
Y (km)
Z(km)
Blue Interceptor
Red TBM
Blue Launch Pt.
Red Launch Pt.
Intercept Pt.
Sensor Track w/o noise
-500
0
500
1000
1500
0
500
1000
1500
-200
0
200
400
600
X (km)
Bal1istic Missile Base Trajectory with Measurement Noise
Y (km)
Z(km)
Threat TBM
Threat TBM Noise
Launch Position
Interceptor
EFK Sensor

Elements of a Scenario
Scenario Development
Scenario Generation Tools

Elements of a Scenario
 Settings
 environment, terrain, etc.
 Actors
 Blue/Red forces, weapons, sensors, etc.
 Task Goals
 missions, objectives, etc.
 Plans
 overlays, control measures, etc.
 Actions
 move, shoot, communicate, etc.
 Events
 contact, engagements, etc.

Scenario Considerations
 Resolution (high or low)
 Aggregated-disaggregated
 Terrain data
 Weapon/Sensor data
 Virtual or constructive
 Interfaces
 Distributed/federated

Scenarios in EADSIM
71
ELEMENT DATA
LAYDOWN
SCENARIOS
PLATFORMPLATFORMPLATFORMPLATFORM
NETWORKS
ROUTES
AOIs
MAP
ENVIRON
OBJECT REF
PROTOCOLS
SYSTEMS
WEAPONS
EMP
COMM DEV
JAMMERS
SENSORS
RULESETS
MANEUVERS
FORMATIONS
PP TABLES
FLYOUT
TABLES
PK TABLES
ICONS
IR SIG
RADAR SIG
AIRFRAMES
SPECIFICATION OF A SCENARIO
SCENARIOS ARE THEN A
FURTHER COMBINATION
OF LOWER LEVEL DATA
SYSTEMS ARE DEPLOYED
ELEMENTS
COMBINE TO MAKE
SYSTEM ELEMENTS
INDIVIDUAL
COMPONENTS ARE
SPECIFIED AS
ELEMENTS

Simulation Educators, LLC (29 June 2011) 72
Provide users the ability to:
• Create, modify, and verify
scenario files.
• Specify entities,
tactical overlays,
and environment
parameters.
Scenario Generation Tools are typically developed to be utilized as an off-
line pre-runtime tool that can be run on a laptop and provide a modular
scenario development environment
Ability to translate legacy scenario files
into the new scenario file format & able to
translate the new scenario files back into
the legacy format
Simulation
System
Scenario Generation Tools (SGTs)

Summary
 The are several types of combat models driving
simulations for combat training, research &
development, and advanced concepts requirements:
 Environmental models
 Physical models (engagement, target acquisition,
communications, etc.)
 Behavioral models
 In addition, simulations require some means of
scenario development, and these are often separate
components.
 Understanding the underlying concepts and methods of
combat models embedded in simulations, enhances
our ability to choose the right simulations for our
training or analysis requirements.

References
Ancker, C.J., Jr. and Gafarian, A.V., Modern Combat Models: A Critique of Their Foundations, Operations Research
Society of America, 1992.
Bracken, J., Kress, M. and Rosenthal, R.E., Eds., Warfare Modeling, MORS, 1995.
Caldwell, B, Hartman, J., Parry, S., Washburn, A., and Youngren, M., Aggregated Combat Models. NPS ORD, 2000.
Davis, P.K., Aggregation, Disaggregation, and the 3:1 Rule in Ground Combat. MR-638
DuBois, E.L., Hughes, W.P., Jr., Low, L.J., A Concise Theory of Combat, Institute for Joint Warfare Analysis, NPS, 2000.
Dupuy, T.N., Understanding War: History and Theory of Combat, Falls Church, VA.: Nova 1987.
Epstein, J.M., The Calculus of Conventional War: Dynamic Analysis without Lanchester Theory, Washington, D.C.,
Brookings Institute, 1985.
Fowler, B.W., De Physica Beli: An Introduction to Lanchestrial Attrition Mechanics, 3 Vols. IIT Research
Institute/DMSTTIAC, Rept. SOAR 96-03, Sep. 1996.
Hillestad, R.J., and Moore, L., The Theater-Level Campaign Model: A New Research Prototype for a New Generation of
Combat Analysis Model, RAND, 1996. MR-388
Koopman, B.O., Search and Screening, MORS, 1999.
Reece, D.A., Movement behavior for soldier agents on a virtual battlefield, Teleoperators and Virtual Environments ,
Volume 12 , Issue 4 (August 2003). MIT Press Cambridge, MA, USA
Smith, R. Military Simulation, http://www.modelbenders.com/
Strickland, J. S. Missile Fight Simulation. Lulu.com, 2011.
Strickland, J. S. Using Math to Defeat the Enemy. Lulu.com, 2011.
Strickland, J. S., Fundamentals of Combat Modeling, Lulu.com, 2010.
Taylor, J.G., Lanchester Models of Warfare, 2 Vols, Defense Technological Information Center (DTIC), ADA090843 (Naval
Post Graduate School, Monterey, CA), October 1980.
Taylor, J.G., Force-on-Force Attrition Modeling, Operations Research Society of America, Military Applications Section,
1981.
Washburn, A.R., Search and Detection, 4th Ed., Operations Research Section, INFORMS, Baltimore, MD, 2002.
Washburn, A., Lanchester Systems, NPS, April 2000.

Using math to defeat the enemy (7 7-2011)

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