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Approved for Public Release
Simulation Educators, LLC (29 June 2011)
Jeffrey Strickland, Ph.D., CMSP
1DISTRIBUTION STATEMENT A. Approved for
public release; distribution is unlimited.
Approved for Public Release
Simulation Educators, LLC (29 June 2011)
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.
Approved for Public Release
Simulation Educators, LLC (29 June 2011)
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
3Copyright© 2010 Jeffrey Strickland, Ph.D.
Approved for Public Release
Simulation Educators, LLC (29 June 2011)
Level of Detail
Conceptual Reference Model
Data Collection
Data Processing
Static Environment
Dynamic Environment
Standardization
4Copyright© 2010 Jeffrey Strickland, Ph.D.
Approved for Public Release
Simulation Educators, LLC (29 June 2011)
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.
Approved for Public Release
Simulation Educators, LLC (29 June 2011)
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
Copyright© 2010 Jeffrey Strickland, Ph.D.
Approved for Public Release
Simulation Educators, LLC (29 June 2011)
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
Copyright© 2010 Jeffrey Strickland, Ph.D.
Approved for Public Release
Simulation Educators, LLC (29 June 2011)
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
Copyright© 2010 Jeffrey Strickland, Ph.D.
Approved for Public Release
Simulation Educators, LLC (29 June 2011)
Storing Environmental Data
9
Triangulated Irregular Network (TIN)
Data point correlation
Surface tiled with hexagons
Copyright© 2010 Jeffrey Strickland, Ph.D.
Approved for Public Release
Simulation Educators, LLC (29 June 2011)
Static Environment
10
Trafficability
Terrain Type
Visibility
Copyright© 2010 Jeffrey Strickland, Ph.D.
Approved for Public Release
Simulation Educators, LLC (29 June 2011)
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.
Restrictions: Ghosted unless LANDCL Reflectivity is selected.
Name: Wooded Hills (radiobutton) Description: If selected, the LANDCL model will assume wooded hill
terrain.
Restrictions: Ghosted unless LANDCL Reflectivity is selected.
Name: Mountains (radiobutton) Description: If selected, the LANDCL model will assume
mountainous terrain.
Restrictions: Ghosted unless LANDCL Reflectivity is selected.
11Copyright© 2010 Jeffrey Strickland, Ph.D.
Approved for Public Release
Simulation Educators, LLC (29 June 2011)
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
Copyright© 2010 Jeffrey Strickland, Ph.D.
Approved for Public Release
Simulation Educators, LLC (29 June 2011)
Classic Problems in Interpretation
13
1
2
3a 3b
1
2a 2b
Terrain Points Building Corners
Copyright© 2010 Jeffrey Strickland, Ph.D.
Approved for Public Release
Simulation Educators, LLC (29 June 2011)
Environmental Standardization
14Copyright© 2010 Jeffrey Strickland, Ph.D.
Approved for Public Release
Simulation Educators, LLC (29 June 2011)
Physical Modeling
15
Detect/Acquire
Engage(other major
combat functions)
Communicate
Move
Start Cycle Here
Copyright© 2010 Jeffrey Strickland, Ph.D.
Approved for Public Release
Simulation Educators, LLC (29 June 2011)
Movement Points Movement
Bald Earth Movement
Terrain and Feature Movement
Physics-based Movement
Automated Route Planning
A* Search
Topology Smart
Grid Registration
Behavioral
16Copyright© 2010 Jeffrey Strickland, Ph.D.
Approved for Public Release
Simulation Educators, LLC (29 June 2011)
Movement Points Movement
17
2
3
6
1
2 6 2
1
Movement
Points =
20
Movement
Points
Remaining =
20 – 11 = 9
Copyright© 2010 Jeffrey Strickland, Ph.D.
Approved for Public Release
Simulation Educators, LLC (29 June 2011)
Bald Earth Movement
18
Set heading, speed, start time
Rate*Time = Distance
20 km/hr * 30 min = 10 km
Copyright© 2010 Jeffrey Strickland, Ph.D.
Approved for Public Release
Simulation Educators, LLC (29 June 2011)
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
Copyright© 2010 Jeffrey Strickland, Ph.D.
Approved for Public Release
Simulation Educators, LLC (29 June 2011)
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
Copyright© 2010 Jeffrey Strickland, Ph.D.
Approved for Public Release
Simulation Educators, LLC (29 June 2011)
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
21Copyright© 2010 Jeffrey Strickland, Ph.D.
Approved for Public Release
Simulation Educators, LLC (29 June 2011)
Topology Smart
22
Set Objective: Position or Vector
Movement model selects path from topological map
Maintain objective
Route traveled is function of topology
Copyright© 2010 Jeffrey Strickland, Ph.D.
Approved for Public Release
Simulation Educators, LLC (29 June 2011)
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
Copyright© 2010 Jeffrey Strickland, Ph.D.
Approved for Public Release
Simulation Educators, LLC (29 June 2011)
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
nnn
DistGNest
TargetGTargetRF
Copyright© 2010 Jeffrey Strickland, Ph.D.
Approved for Public Release
Simulation Educators, LLC (29 June 2011)
Perfect Detection
Gridded Probability Areas
Detection Range
3D Detection Range
Target Acquisition Process
Line-of-Sight
NVEOL Model
25Copyright© 2010 Jeffrey Strickland, Ph.D.
Approved for Public Release
Simulation Educators, LLC (29 June 2011)
Perfect Detection
26
Every object knows the true location of every other object.
There are no models of sensors or processors.
Copyright© 2010 Jeffrey Strickland, Ph.D.
Approved for Public Release
Simulation Educators, LLC (29 June 2011)
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%
Copyright© 2010 Jeffrey Strickland, Ph.D.
Approved for Public Release
Simulation Educators, LLC (29 June 2011)
Detection Range
28
Complete circle—no field of view/field of regard Terrain line-of-sight (LOS) is separate
Copyright© 2010 Jeffrey Strickland, Ph.D.
Approved for Public Release
Simulation Educators, LLC (29 June 2011)
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
sin𝜋𝑎𝜆sin 𝜃
𝜋𝑎𝜆sin 𝜃
sin𝑁2
2𝜋𝑑𝜆
sin 𝜃 + 𝜙
sin𝜋𝑑2sin 𝜃 + 𝜙
𝐼 = 𝐼0sin
𝜋𝑎𝜆sin 𝜃
𝜋𝑎𝜆sin 𝜃
2 sin𝑁2
2𝜋𝑑𝜆
sin 𝜃 + 𝜙
sin𝜋𝑑2sin 𝜃 + 𝜙
2
Copyright© 2010 Jeffrey Strickland, Ph.D.
Approved for Public Release
Simulation Educators, LLC (29 June 2011)
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
30Copyright© 2010 Jeffrey Strickland, Ph.D.
Approved for Public Release
Simulation Educators, LLC (29 June 2011)
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
Copyright© 2010 Jeffrey Strickland, Ph.D.
Approved for Public Release
Simulation Educators, LLC (29 June 2011)
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 -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.
Copyright© 2010 Jeffrey Strickland, Ph.D.
Approved for Public Release
Simulation Educators, LLC (29 June 2011)
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)
Copyright© 2010 Jeffrey Strickland, Ph.D.
Approved for Public Release
Simulation Educators, LLC (29 June 2011)
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
Copyright© 2010 Jeffrey Strickland, Ph.D.
Approved for Public Release
Simulation Educators, LLC (29 June 2011)
Comms Model Effects
Perfect Communications
Direct Message Passing
Broadcast Messages
Virtual Cell Layout
Physics Modeling
35Copyright© 2010 Jeffrey Strickland, Ph.D.
Approved for Public Release
Simulation Educators, LLC (29 June 2011)
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
Copyright© 2010 Jeffrey Strickland, Ph.D.
Approved for Public Release
Simulation Educators, LLC (29 June 2011)
Perfect Communications
37
Targets
~~~~~
Orders
~~~~~
Reports
~~~~~
Shared information, no representation of comms
Software-to-software message delivery
Copyright© 2010 Jeffrey Strickland, Ph.D.
Approved for Public Release
Simulation Educators, LLC (29 June 2011)
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
… …
Copyright© 2010 Jeffrey Strickland, Ph.D.
Approved for Public Release
Simulation Educators, LLC (29 June 2011)
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
Copyright© 2010 Jeffrey Strickland, Ph.D.
Approved for Public Release
Simulation Educators, LLC (29 June 2011)
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
40Copyright© 2010 Jeffrey Strickland, Ph.D.
Approved for Public Release
Simulation Educators, LLC (29 June 2011)
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
41Copyright© 2010 Jeffrey Strickland, Ph.D.
Approved for Public Release
Simulation Educators, LLC (29 June 2011)
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
Copyright© 2010 Jeffrey Strickland, Ph.D.
Approved for Public Release
Simulation Educators, LLC (29 June 2011)
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
Ta
rge
t
…
Phit is rolled into the overall Pkill
Damage = 1, where Random Number <= Pk
= 0, where Random Number > Pk
Copyright© 2010 Jeffrey Strickland, Ph.D.
Approved for Public Release
Simulation Educators, LLC (29 June 2011)
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
Copyright© 2010 Jeffrey Strickland, Ph.D.
Approved for Public Release
Simulation Educators, LLC (29 June 2011)
Pk’s and Random Numbers
45
Kill Area No-Kill Area
0% 75% 100%
Random Number = 0.63
Pk = 75% = 0.75
Copyright© 2010 Jeffrey Strickland, Ph.D.
Approved for Public Release
Simulation Educators, LLC (29 June 2011)
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
Copyright© 2010 Jeffrey Strickland, Ph.D.
Approved for Public Release
Simulation Educators, LLC (29 June 2011)
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
Copyright© 2010 Jeffrey Strickland, Ph.D.
Approved for Public Release
Simulation Educators, LLC (29 June 2011)
Single-Shot Accuracy1D 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, 10 m, 0664.376745.025 0,
z
z
x
PSSH
0
-z +z
Copyright© 2010 Jeffrey Strickland, Ph.D.
Approved for Public Release
Simulation Educators, LLC (29 June 2011)
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 = 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)
Copyright© 2010 Jeffrey Strickland, Ph.D.
Approved for Public Release
Simulation Educators, LLC (29 June 2011)
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
2exp1
RRPd
2
1
2exp1
2
2
00
RRP
Target
Simplified Vehicle
Assembly Area
Cluster of Soldiers
Copyright© 2010 Jeffrey Strickland, Ph.D.
Approved for Public Release
Simulation Educators, LLC (29 June 2011)
Kill CategoriesK-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
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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).
52
2
1 44
3
0.6
1.6
1.0
1.4 2.6
0.6
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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 Ycoordinate.
Interpolating between 0.77 and 0.78, gives Z-1 = 0.743.
Solve for y in −0.743=(y − 0)/0.5 gives y=−0.37154) 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
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Lanchester Equations
Aggregated Combat Groups
Epstein’s Equations
Quantified Judgment Model (QJM)
Force Ratio Approach
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Lanchester Equations
dx
dtf x y
dy
dtf 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.
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
57
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
58
aTa
aT
gaT
gg
dTd
dT
tt
tt
ttWW
tWtW
11
1
11
11 max
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Missile Dynamics
Sensor Dynamics
Coordinate Systems
Missile Flight Simulation
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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
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Missile Dynamics
Axis
Force
Along
Axis
Moment
About
Axis
Linear
Velocity
Angular
Displacement
Angular
Velocity
Moment
of Inertia
𝑥 𝐹𝑥 𝐿 𝑢 𝜙 𝑝 𝐼𝑥
𝑦 𝐹𝑦 𝑀 𝑣 𝜃 𝑞 𝐼𝑦
𝑥 𝐹𝑧 𝑁 𝑤 𝜓 𝑟 𝐼𝑧
61
𝑀, 𝑞
𝑦𝑏
𝑥𝑏
𝑧𝑏
𝑁, 𝑟
𝐿, 𝑝
𝐹𝑥, 𝑢
𝐹𝑦, 𝑣
𝐹𝑧, 𝑤
𝑢 =𝐹𝑥𝑏𝑚
− 𝑞𝑤 − 𝑟𝑣 , m/s2
𝑣 =𝐹𝑦𝑏𝑚
− 𝑟𝑢 − 𝑝𝑤 , m/s2
𝑤 =𝐹𝑧𝑏𝑚
− 𝑝𝑣 − 𝑞𝑢 , m/s2
𝑝 = 𝐿 − 𝑞𝑟 𝐼𝑧 − 𝐼𝑦 𝐼𝑥 , m/s2
𝑞 = 𝑀 − 𝑝𝑟 𝐼𝑥 − 𝐼𝑧 𝐼𝑦 , m/s2
𝑟 = 𝑁 − 𝑝𝑞 𝐼𝑦 − 𝐼𝑥 𝐼𝑧 , m/s2
ROTATIONAL EQUATIONS
TRANSLATIONAL EQUATIONS
Copyright© 2010 Jeffrey Strickland, Ph.D.
Sensor Dynamics
Pseudo-imaging
Imaging
Radio Frequency
Seekers
Pulse Radar
Continuous Wave
Radar
Pulse Doppler Radar
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Projection of Tracking
Error on Reticle Plane
63
BoresightAxis
AngularTracking Error
Tracking Error Vector
Field of View
Plane of ReticleDetector
ArbitraryReference
Target Projection on Reticle
Line ofSight
Target
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Coordinate Systems
64
Target Coordinate System (𝑥𝑡, 𝑦𝑡, 𝑧𝑡)
Body Coordinate System (𝑥𝑏, 𝑦𝑏 , 𝑧𝑒)
Earth Coordinate System (𝑥𝑒 , 𝑦𝑒 , 𝑧𝑒)
Guidance Coordinate System (𝑥𝑔, 𝑦𝑔, 𝑧𝑔)
Tracker Coordinate System (𝑥𝑠, 𝑦𝑠, 𝑧𝑠)
Wind Coordinate System (𝑥𝑤, 𝑦𝑤, 𝑧𝑤)
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Missile Simulation
65
𝑋 1 = 𝑃𝑀 𝑖
𝑋 2 = 𝑃𝑀 𝑗
𝑋 3 = 𝑃𝑀 𝑖𝑘
𝑋 4 = 𝑢𝑋 5 = 𝑣𝑋 6 = 𝑤𝑋 7 = 𝑢𝑋 8 = 𝑣𝑋(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
𝑃𝑀 𝑖 = 𝑋𝑂𝑈𝑇 1
𝑃𝑀 𝑗 = 𝑋𝑂𝑈𝑇 2
𝑃𝑀 𝑘 = 𝑋𝑂𝑈𝑇 3
𝑢 = 𝑋𝑂𝑈𝑇 4𝑣 = 𝑋𝑂𝑈𝑇 5𝑤 = 𝑋𝑂𝑈𝑇 6 𝑢 = 𝑋𝑂𝑈𝑇 7 𝑣 = 𝑋𝑂𝑈𝑇 8 𝑤 = 𝑋𝑂𝑈𝑇(9)
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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;
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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)
Z (
km
)
-500 0 500 1000 15000
200
400
600
800
1000
1200
1400
X (km)
Three Dimensional Missile Trajectory in kilometers
Y (
km
)
0 50 100 150 200 250 300 3500
1000
2000
3000
4000
5000
6000
7000
8000
Mis
sile
Speed (
m/s
)
Time (seconds)
Missile Speed vs Time
0 50 100 150 200 250 300 350-200
-150
-100
-50
0
50
100
150
200Missile 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 secondsMiss Distance = 0.54057 meters
Y (km)
Z (k
m)
Blue Interceptor
Red TBM
Blue Launch Pt.
Red Launch Pt.
Intercept Pt.
Sensor Track w/o noise
-500
0500
1000
1500
0
500
1000
1500-200
0
200
400
600
X (km)
Bal1istic Missile Base Trajectory with Measurement Noise
Y (km)
Z (k
m)
Threat TBM
Threat TBM Noise
Launch Position
Interceptor
EFK Sensor
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Elements of a Scenario
Scenario Development
Scenario Generation Tools
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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.
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Scenario Considerations
Resolution (high or low)
Aggregated-disaggregated
Terrain data
Weapon/Sensor data
Virtual or constructive
Interfaces
Distributed/federated
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Scenarios in EADSIM
71
ELEMENT DATA
LAYDOWN
SCENARIOS
PLATFORMPLATFORMPLATFORMPLATFORMNETWORKS
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
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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)
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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.
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ReferencesAncker, 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.
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