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Approved for Public Release
09-MDA-4814 (2 SEPT 09)
Approved for Public Release
09-MDA-4814 (2 SEPT 09) 1
Mathematical and
Heuristic Models of Combat with
ExamplesJeffrey Strickland, Ph.D., CMSP
Missile Defense Agency
DISTRIBUTION STATEMENT A. Approved for
public release; distribution is unlimited.
Approved for Public Release
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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.
2
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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
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Environment Modeling
Level of Detail
Conceptual Reference Model
Data Collection
Data Processing
Static Environment
Dynamic Environment
Standardization
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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
Air Combat Terrain Ground Combat Terrain
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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
Conceptual Reference Model
SOURCE: Paul A. Birkel, "SNE Conceptual Reference Model", 1999 Fall SIW Conference, September 1999.
http://www.sisostds.org/siw/98Fall/view-papers.htm
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Data Processing
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
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Storing Environmental Data
Triangulated Irregular Network (TIN)
Data point correlation
Surface tiled with hexagons
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Static Environment
Trafficability
Terrain Type
Visibility
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Dynamic Environment
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
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Classic Problems in Interpretation
1
2
3a 3b
1
2a 2b
Terrain Points Building Corners
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Environmental Standardization
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Physical Modeling
Detect/Acquire
Engage(other major
combat functions)
Communicate
Move
Start Cycle Here
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Movement Modeling
Movement Points Movement
Bald Earth Movement
Terrain and Feature Movement
Physics-based Movement
Automated Route Planning
A* Search
Topology Smart
Grid Registration
Behavioral
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Movement Points Movement
2
3
6
1
2 6 2
1
Movement
Points =
20
Movement
Points
Remaining =
20 – 11 = 9
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Bald Earth Movement
Set heading, speed, start time
Rate*Time = Distance
20 km/hr * 30 min = 10 km
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Terrain and Feature Movement
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
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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
Physics-based Movement
• 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
18
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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
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Topology Smart
Set Objective: Position or Vector
Movement model selects path from topological map
Maintain objective
Route traveled is function of topology
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Grid Registration
Grid 14 → V71, V109, V1212, V10101
Vehicles registered into geographic grid during movement
Improve LOS, sensor, and interaction performance
11 12 13 14 15 16 17
21 22 23 24 25 26 27
31 32 33 34 35 36 37
41 42 43 44 46 4745
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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
Pitch
Yaw
Roll
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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.
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
nnn
DistGNest
TargetGTargetRF
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Detection Modeling
Perfect Detection
Gridded Probability Areas
Detection Range
3D Detection Range
Target Acquisition Process
Sensor & Target Characteristics
Line-of-Sight
NVEOL Model
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Perfect Detection
Every object knows the true location of every other
object.
There are no models of sensors or processors.
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Gridded Probability Areas
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%
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Detection Range
Complete circle—no field of view/field of regard
Terrain line-of-sight (LOS) is separate
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3D Detection Range
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
2
sin2
sin
sin2
2sin
2
sin
sinsin
0
sin2
sin
sin2
2sin
sin
sinsin
0
d
dN
a
a
II
d
dN
a
a
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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
Glance/
Glimpse
Target
Found?
No
Yes
tg tg tg
Pacq Pacq Pacq
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NVEOL Acquisition Algorithm
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 -CT)
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.
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Sensor & Target Characteristics
Sensor characteristics Maximum range
Sensor footprint
Frequency, pulse rate
EO, IR, RF, mag, sonar
Geometry Range
Off-set angle
Terrain & weather effects Line-of-sight (LOS)
Obscurants
Earth curvature
Target characteristics Camouflage
Color & pattern
Radar cross section
IR signature
Movement
Cavitations
Magnetic mass
Obscurants
Earth curvature
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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
intervisibility segment length
. . . . . . . . . . . . . . . . .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)
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Communications Modeling
Comms Model Effects
Perfect Communications
Direct Message Passing
Broadcast Messages
Virtual Cell Layout
Physics Modeling
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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
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
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Perfect Communications
Targets
~~~~~
Orders
~~~~~
Reports
~~~~~
Shared information, no representation of comms
Software-to-software message delivery
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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.
… …
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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.…
…Success
Lost
Degraded
Delayed
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Physics-Based Communication Networks
38
Packet-based model:
network traffic flow: model packets in flow
# sources, data rates increase, so too does simulation workload
Fluid-based model:
network traffic flow: continuous fluid
rate changes at discrete points in time
rate constant between changes
can modulate rate at different time scales
single modeling paradigm for many time scales
abstract out fine-grained details: simulation efficiency
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Virtual Cell Layout (VCL)
The real cells are mobile and created by the mobile base stations, which are either:
radio access points (RAPs) or
cluster head man packed radios (MPRs).
Computer aided exercise interacted tactical communications simulation (CITACS)
A scenario with 153 units are simulated over an area of 115 km ×170 km
Location manager deployed 77 RAPs and 18529 MPRs for this scenario based on the unit types and sizes.
kr
kr r
r
CITACS interacts with Joint Theater Level Simulation (JTLS)
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Engagement Modeling – Entity
Level
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
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Point System
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
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Markov Pk Table
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
Ta
rge
t
…
Phit is rolled into the overall Pkill
Damage = 1, where Random Number <= Pk
= 0, where Random Number > Pk
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Random Numbers
Generated by a recursive function
Evenly distributed between 0 and 1 ~ Unif(0,1)
Perfect for Pk evaluations0.002589 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
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Pk’s and Random Numbers
Kill Area No-Kill Area
0% 75% 100%
Random Number = 0.63
Pk = 75% = 0.75
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Precision Engagements
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
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Linear Target Phit
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 =
Dispersion =
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
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Assume no systematic error.
2126.03937.06063.0 zzPSSH
NOTE: “” is available in
tabular form in any Statistics
text: see Normal Distribution.
Single-Shot Accuracy1D Target Example 1
3937.00644.37010
6064.00644.37010
then,m, 10 m, 0664.376745.025 0,
z
z
x
PSSH
0
-z +z
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Rectangular Target Phit
Normal parameters for 2D target: "Side view" (i.e., direct-fire weapon)
Elevation error
Deflection error
"Top view" (i.e., indirect-fire weapon)Range error
Deflection error
DEFINE:Bias = x , y
Dispersion = x , y
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
x
y
p(y)
p(x)
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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.
Circular Target Phit
2
2
2exp1
RRPd
2
1
2exp1
2
2
00
RRP
Target
Simplified Vehicle
Assembly Area
Cluster of Soldiers
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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. 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
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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.
A hit in area 2 will
produce a catastrophic kill.
A hit in area 3 will
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).
2
1 44
3
0.6
1.6
1.0
1.4 2.6
0.6
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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 0.75 and 0.74, gives Z-1 = 0.743.
Solve for y in −0.743=(y − 0)/0.5 gives y=−0.37154) Round hits area 4, so no kill is assessed.
2
1 44
3
0.6
1.6
1.0
1.4 2.6
0.6
Y
X
−0.3715
. 53
Direct-Fire Accuracy Example (2)
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Engagement Modeling –
Aggregate Level
Lanchester Equations
Aggregated Combat Groups
Epstein’s Equations
Quantified Judgment Model
(QJM)
Force Ratio Approach
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Lanchester Equations
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.
dx
dtf x y
dy
dtf x y 1 2, ,... , ,...
aydt
dx
bxdt
dy
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Aggregated Combat Groups
Contiguous pistons
Aggregated force
attrition
Distance from
middle affects
power and attrition
Units accumulate
as piston moves
Explicit withdrawal
required
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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
unit ofindex firepower FPI1
n
i
iisx
battle ain forceFPI
FPIFR
defender
attacker
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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
57
aTa
aT
gaT
gg
dTd
dT
tt
tt
ttWW
tWtW
11
1
11
11 max
Approved for Public Release
09-MDA-4814 (2 SEPT 09)
Approved for Public Release
09-MDA-4814 (2 SEPT 09) 58
Scenarios
Elements of a Scenario
Scenario Development
Scenario Generation Tools
Approved for Public Release
09-MDA-4814 (2 SEPT 09) 59
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.
Approved for Public Release
09-MDA-4814 (2 SEPT 09) 60
Scenario Development
Resolution (high or low)
Aggregated-disaggregated
Terrain data
Weapon/Sensor data
Virtual or constructive
Interfaces
Distributed/federated
Approved for Public Release
09-MDA-4814 (2 SEPT 09) 61
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)
Approved for Public Release
09-MDA-4814 (2 SEPT 09) 62
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.
Approved for Public Release
09-MDA-4814 (2 SEPT 09) 63
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Strickland, J.S., Fundamentals of Combat Modeling with Microsoft Excel, USALMC, 2004.
Taylor, J.G., Lanchester Models of Warfare, 2 Vols, Defense Technological Information Center (DTIC), ADA090843 (Naval Post Graduate School, Monterey, CA), October 1980.
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Volluz, R.J. and Volluz, R.M., The Anatomy of Combat, 17th ISMOR Symposium, 28 Aug – 1 Sep 2000.
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