NPS ARCHIVE
1997.09
WONNACOTT, W.
NAVAL POSTGRADUATE SCHOOL
Monterey, California
THESIS
MODELING IN THE DESIGN AND ANALYSIS OF A
HIT-TO-KILL ROCKET GUIDANCE KIT
by
W. Mark Wonnacott
September, 1997
Thesis Advisor:
Second Reader:
Conrad F. Newberry
Louis V. Schmidt
Thesis
W757
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Master's Thesis
4. title and subtitle MODELING IN THE DESIGN AND
ANALYSIS OF A HIT-TO-KILL ROCKET GUIDANCE KIT
6. AUTHOR(S)
Wonnacott, W. Mark
5. FUNDING NUMBERS
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Naval Postgraduate School
Monterey, CA 93943-5000
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The views expressed in this thesis are those of the author and do not reflect the official policy or
position of the Department of Defense or the U.S. Government.
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13. ABSTRACT (maximum 200 words)
This thesis presents several computer models used in the design and analysis of a Hit-to-Kill Rocket
Guidance Kit (HRGK). The HRGK — proposed as an inexpensive add-on kit — has the potential of converting
unguided 2.75" diameter rockets into precision weapons against non-tank targets. A Naval Postgraduate School
design team recently participated in a nation-wide graduate student competition for the design of such a kit. The
design and analysis process led the author to develop and use various computer models and simulations. This
thesis documents three distinct types of computer models found useful in the design.
The first, operational effectiveness modeling, established the cost effectiveness of the NPS HRGK. The
second was related to the preliminary sizing of various design aspects — ensuring the proper flow-down of system
requirements into design specifications. The third was a six-degree of freedom (6DOF) simulation, developed to
perform detailed analyses of the HRGK's performance.
Although the models presented in this thesis pertain to the HRGK, the basic principles apply to the design
or evaluation of other missile systems, and this thesis provides general insights regarding the benefits and
limitations of computer modeling in missile design.
14. SUBJECT TERMS
Rocket Guidance Kit, Missile Design, Missile Modeling and Simulation
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106
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ABSTRACT
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u
Approved for public release; distribution is unlimited
MODELING IN THE DESIGN AND ANALYSIS OF A
HIT-TO-KILL ROCKET GUIDANCE KIT
W. Mark Wonnacott
B.S.M.E., Brigham Young University, 1989
Submitted in partial fulfillment of the
requirements for the degree of
MASTER OF SCIENCE IN AERONAUTICAL ENGINEERING
from the
NAVAL POSTGRADUATE SCHOOL
September 1997
DUDLEY KNOX LIBRARY
NAVAL POSTGRADUATE SCHOOL
MONTEREY, CA 93943-5101
DUDLEY KMOX LIBRARY
^ADUATE SCHOOL
,-5101
ABSTRACT
This thesis presents several computer models used in the design and analysis of a
Hit-to-Kill Rocket Guidance Kit (HRGK). The HRGK — proposed as an inexpensive add-
on kit — has the potential of converting unguided 2.75" diameter rockets into precision
weapons against non-tank targets. A Naval Postgraduate School design team recently
participated in a nation-wide graduate student competition for the design of such a kit.
The design and analysis process led the author to develop and use various computer
models and simulations. This thesis documents three distinct types of computer models
found useful in the design.
The first, operational effectiveness modeling, established the cost effectiveness of
the NPS HRGK. The second was related to the preliminary sizing of various design
aspects — ensuring the proper flow-down of system requirements into design
specifications. The third was a six-degree of freedom (6DOF) simulation, developed to
perform detailed analyses of the HRGK's performance.
Although the models presented in this thesis pertain to the HRGK, the basic
principles apply to the design or evaluation of other missile systems, and this thesis
provides general insights regarding the benefits and limitations of computer modeling in
missile design.
VI
TABLE OF CONTENTS
I. INTRODUCTION 1
A. BACKGROUND 1
B. NPS HRGK DESIGN 2
C. THESIS SCOPE AND OUTLINE 4
H. COST OF OPERATIONAL EFFECTIVENESS ANALYSES 7
A. SIMPLIFYING ASSUMPTIONS 7
B. COST PER KILL 10
C. KILLS PER SORTIE 15
D. COST PER MISSION 15
E. OBSERVATIONS 20
IH. PRELIMINARY SIZING ANALYSES 23
A. LENGTH, WEIGHT, AND CENTER OF GRAVITY 23
B. MANEUVERABILITY 26
C. SEEKER FIELD OF VIEW 39
D. COMMENTS AND OBSERVATIONS 46
IV. SIX-DEGREE OF FREEDOM SIMULATION 51
A. SIMULATION OVERVIEW 51
B. SIMULATION COMPONENTS 54
C. EQUATIONS OF MOTION 55
D. NPS HRGK RESULTS 58
E. OBSERVATIONS 60
V. CONCLUSIONS AND RECOMMENDATIONS 61
APPENDIX A. MISSILE AERODYNAMICS NOMENCLATURE 63
APPENDIX B. NOTES ON MATLAB AND SIMULINK 65
APPENDIX C MATLAB CODE LISTINGS. 67
LIST OF REFERENCES 89
INITIAL DISTRIBUTION LIST 91
vu
Vlll
LIST OF FIGURES
1. Sample Normal Plane View for a 12 mil, One-Dimensional Dispersion at 5 km
Downrange 3
2. Schematic View of the NPS HRGK Design 5
3 . Projected Area for Light Armored Vehicles 9
4. Single-Shot PK and Number of Rockets Required per Target as a Function of
CEP 13
5 . Relative Cost per Kill against Various Targets 14
6. AH-64 Apache Attack Helicopter Sorties Required to Achieve Mission Objective. . . 19
7 . Number of Weapons Fired to Achieve Mission Objective 19
8 . Number of Targets Killed by Category, Weapon, and Loadout Option 20
9 . Physical Size and Center of Gravity Design Space 25
10. Graphical Representation of Maneuverability Model Inputs 27
1 1 . Closed-Form Solution for Maneuverability Specifications with a Stationary Target. . 29
12. Moving Target Scenario for Maneuverability Analysis 31
13. Maximum Turn Rate Required to Hit Target (95% Probability) 33
14. Minimum Turn Radius Required to Hit Target (95% Probability) 33
1 5 . Plane- View, Free-Body Diagram for Steady-Turn Maneuver 35
16. Required Thrust and Angle of Attack for Maximum Turn Rate or Minimum Turn
Radius 38
17. Required Field of View. Results for 95% Probability of Target Staying in Field of
View with Worst Case Target Motion 40
18. Target Distribution Model 42
19. Illustrative Case for Probabilistic Seeker Field of View Analysis 43
20. Possible Target Locations 44
2 1 . Alternative Method for Computing Target Distribution Area within Seeker Field of
View Limits 45
22. Probability Density Functions, (a) Rocket cross- track error, (b) Target cross-track
position, and (c) Probability of acquisition 45
23. Probability of Target in Field of View for Maximum Launch Range (6 km) 47
24. Sensitivity Analysis of the Steady-Turn Maneuver Model for Thruster Sizing 49
25. HRGK 6DOF Simulation Overall Block Diagram 52
26. Detailed View of the Rocket Dynamics Block 53
27. Detailed View of the Parameters and Coefficients Block 53
28. Future 6DOF Architecture 54
29. HRGK-Equipped Rocket Ballistic Trajectories 59
30. Sustained and Pulsed Thruster Maneuvers 59
IX
LIST OF TABLES
1 . Measures of Merit and Their Associated Objectives and Requirements 3
2. COEA Target Set with Assumed Dimensions, Desired PK, and P^ 9
3. Single-Shot Probability of Kill Computer Model Inputs 11
4. Cost per Kill 14
5. Kills per AH-64 Apache Sortie against Light Armor Targets 15
6. Weight, Length, and Center of Gravity Model Inputs 24
7. Maneuverability Model Inputs 26
8. Steady-Turn Maneuver Computer Model Inputs 36
9. Seeker Field of View Analysis Inputs 41
XI
Xll
ACKNOWLEDGMENT
I am grateful for the support of my thesis advisor, Dr. Conrad Newberry, and the
second reader, Dr. Lou Schmidt, for their help and guidance. Dr. Robert Ball, Aeronautics
and Astronautics Department, and Major Vince Tobin and Captain Pat Mason, U.S. Army,
were also very helpful with the cost of operational effectiveness portion of this thesis. I am
very appreciative of the sacrifices of Major Boaz Pomerantz, Israeli Air Force; Major
Silvino L. Silva, Brazilian Air Force; and Lieutenant Nigel A. Nurse, U.S. Navy who
served on the NPS Hit-to-Kill Rocket Guidance Kit Design Team.
I am indeed indebted to my management at the Naval Air Warfare Center Weapons
Division and the NAWCWPNS Long-Term Fellowship Committee for their continued
support of my studies. I wish to specifically acknowledge Tomma Bersie, Doug Savage,
John Freeman, Lee Gilbert, Paul Homer, and Ron Derr for allowing me this time at the
Naval Postgraduate School. Special thanks also go to Ted Fincher and Ken Morton for
inspiring me to pursue a graduate education.
But, my most tender and heart-felt gratitude is reserved for my sweet wife Diane
and our dear children: Andrea, Jared, Nathan, Melissa, Marsha, Jennifer, and Kayla.
Without their love and sacrifice, this thesis would not have been possible.
xiu
XIV
I. INTRODUCTION
During the first half of the 1996-1997 Naval Postgraduate School (NPS) academic
year, a team of aeronautical engineering graduate students was assembled to perform the
conceptual design of a hit-to-kill guided rocket kit (HRGK). (The kit attaches to existing
unguided rockets to give them precision strike capability.) As part of the design team, the
author developed several computer codes or models to facilitate the design and analysis of
the HRGK. Since the completion of the design project, many of these codes have been
expanded or refined.
This thesis documents the theory used in the development of the improved
computer models. It also describes the process of using the models and presents the
modeling results for the NPS HRGK design. These descriptions provide an example of the
type of modeling that can be used in the design and analysis of missiles.
This introductory chapter provides background information about the HRGK and
pertinent details of the NPS HRGK design. It also outlines the remaining chapters of the
thesis.
A. BACKGROUND
In the mid 1990s, US Army Aviation identified the need for a low cost, precision
kill weapon system for use against soft or light-armor targets (such as trucks, armored
personnel carriers, artillery, air defense systems, command posts, amphibious landing
vehicles, or patrol boats). The objective of such a system would be to minimize the cost per
kill for these types of targets. The 2.75-inch rocket (Mark-66) with a unitary, high-
explosive (HE) warhead (M151) and a point detonation fuze (M423) was subsequently
identified as a promising candidate for this low-cost mission.
These unguided rockets cost only about $ 1 ,000 per unit, and large inventories are
already available to the military services. The 2.75-inch rockets are launched from
helicopters, fixed-wing aircraft, surface combat vehicles, and naval vessels. U.S. Army
Helicopters carry the rockets in either a 7- or 19-tube launcher pod as part of the Hydra-70
weapon system. The Hydra-70 system includes the weapon (rocket, fuze, and warhead),
launch pod, and an armament management system. With this system, the rockets can be
fired in single-shot mode or in various sized ripple or salvo shots. The rockets can carry
various warheads or payloads (including HE, kinetic energy flechettes, multi-purpose
submunitions, illumination flare, marking smoke, and chaff) with various fuzing options
(including remotely set timed burst and super-quick or delayed contact fuzes).
The Mark-66 rocket with warhead and fuze is spin stabilized to provide some
degree of accuracy. The rocket uses a fluted single throat nozzle to achieve approximately a
10 Hz spin by the time it exits the launcher. By motor burnout, the spin rate has increased
to approximately 35 Hz. After motor burn out, the rocket's wrap-around fins de-spin the
projectile and cause a significant spin rate (15-20 Hz) in the reverse direction. The spinning
characteristics of the rocket played a significant role in the NPS design as will be discussed
in the following section.
The baseline rocket configuration chosen for the kit (HE warhead and super-quick
contact fuze) is an area suppression weapon. The unguided rocket has a total 12
milliradians (mill), one-dimensional dispersion, meaning that for every thousand meters of
down range fly out a 12 m error results (1 a or 68%). For example, a nominal 5 km range
launch would result in a normal-plane1, radial miss of up to 60 m (5 x 12) — with 39%
probability (see Figure 1). This level of accuracy is far from that required for single-shot,
precision kills. To achieve point target accuracy, the rocket needs to be fitted with some
type of guidance and control kit. The conceptual NPS HRGK, described in the next
section, was designed to serve this purpose.
B. NPS HRGK DESIGN
The NPS HRGK was designed to meet certain requirements. The following
subsections summarize these requirements and the pertinent aspects of the NPS HRGK
design.
1. HRGK Design Specifications
The NPS HRGK design requirements were prepared by the American Institute of
Aeronautics and Astronautics (AIAA) Missile Systems Technical Committee and were used
for the 1996/1997 Graduate Team Missile Design Competition sponsored by the AIAA and
Northrop Grumman Corporation [Ref. 1]. Table 1 summarizes the HRGK's measures of
merit with their associated objectives and pertinent requirements.
1 For the purposes of this thesis, all accuracies are in terms of the plane normal to the rocket trajectory
rather than in the ground plane. This convention is typical for guided missiles.
Elevation Error
(68% of hits)
±60 m
Cross Track Error
(68% of hits)
CE: circular error
CEP: circular error probable
±60 m-
90% CE
(2.15 a)
50% CE-
(1.18 a)
60mR(1a)
(39% CE)
-CEP
Figure 1. Sample Normal Plane View for a 12 mil, One-Dimensional
Dispersion at 5 km Downrange. [(12 m/ 1000 m)5000 m = 60 m (la)].
Table 1 . Measures of Merit and Their Associated Objectives and Requirements.
Measure of Merit and Objective
Specific Requirements
Cost
Maximize Units Bought
• $10,000 Avg. Unit Cost (5,000 Unit LRIF)
• Maximum $1 Billion Total 5 Year Production Buy
Current Systems Computability
Maximize Operability &
Reduce Life Cycle Cost
• Existing Rocket, Fuze, Warhead, & Launch Pod
— 72 inch total system maximum length
— loaded pod center of gravity between lugs
• Laser Designator Compatibility
Accuracy
Provide Hit-to-Kill Accuracy
• 0.5 m CEP
• 60 mph target speed
Weight
Maximize Kits Deliverable via Airlift
• 30 pound Total System Maximum Weight
Maximum Range
Maximize Platform Standoff Survivability
• 6 km (or More) Maximum Range
(Maneuverability at Maximum Range)
Minimum Range
Maximize Launch Acceptable Region
• 1 km (or Less) Minimum Range
(Guidance, Fuzing, and Launch Transients)
Adverse Weather Performance
Maximize Operability and Combat Flexibility
• 99% Worldwide Weather Capable
Time to Target
Minimize Time (Improve Survivability)
• 16 seconds or Less to 5 km Range
' Low Rate Initial Production
2. NPS Design Highlights
The following paragraphs briefly describe the major features of the NPS HRGK
design. The design team's proposal gives a more detailed description of the design [Ref.
2].
The HRGK attaches over the fuze at the front end of a 2.75 inch rocket. It utilizes a
strapped-down, four-quadrant silicon detector for semi-active laser spot homing. The
HRGK employs three hot gas reaction jets (thrusters) to control the rocket' s attitude and
trajectory. A solid propellant burning in an insulated pressure chamber generates the
thruster gas. The kit uses a look-up table scheme to control the thrusters based solely on the
seeker outputs. Fast-cycle solenoid valves open and close the "on-off" thrusters. The
thruster nozzles are inter-spaced between three fixed, wrap-around canards. The canards
provide increased maneuverability (by reducing the rocket's static stability) and help
constrain any jet and air stream interaction effects. The canards also contribute significantly
to the HRGK's roll stability.
Thrust and radial needle rollers effectively isolate the NPS HRGK from the
spinning motion of the rocket. Simulations show the kit's roll rate to be less than 9 degrees
per second after the first seconds of flight. The kit's low spin rate and the simplistic pursuit
navigation scheme eliminate the need for inertial sensors and greatly reduce the kit's signal
processing requirements.
As shown in Figure 2, the basic kit is 19.0 inches long. It has a diameter of 2.75
inches (with canards wrapped in) and weighs 7.0 pounds. The NPS HRGK attaches to the
2.75 inch rocket between the M151 high-explosive warhead and M423 point detonation
fuze. The all-up round is 68.75 inches long and weighs 30 lb. The guided rocket is
compatible with the Hydra-70 rocket system.
C. THESIS SCOPE AND OUTLINE
The NPS HRGK design was achieved and verified with the aid of several original
computer codes and models. The codes and models developed by the author and described
in this thesis fall into the following three major categories:
• cost of operational effectiveness analyses,
• preliminary sizing analyses, and
• six-degree of freedom simulation.
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The following three chapters provide descriptions of the modeling processes in the
respective categories. Each of these chapters (1) includes a description of the modeling
methodology, (2) gives example results specific to the NPS HRGK design, and (3)
discusses observations regarding the general use of the models. The final thesis chapter
summarizes these conclusions and provides recommendations for additional work with the
computer models. Appendices that contain the computer code listings and additional
background have also been included. The appendices include a brief description of missile
aerodynamics nomenclature (Appendix A) and a discussion concerning the use of the
computer software Matlab® and Simulink® (Appendix B).
In general, this thesis is not intended to be read from cover to cover. Instead this
thesis is offered as a reference to possible analysis types and modeling methods for use in
future conceptual design of missile systems.
® Matlab and Simulink are registered trademarks of The Math Works, Inc.
II. COST OF OPERATIONAL EFFECTIVENESS ANALYSES
Cost of operational effectiveness analysis (COEA) must be an important part of the
design process. First, COEAs provide justification for the design based on operational
effectiveness cost savings. But, just as importantly, COEA-type analyses provide the
designer with focus and insight into the truly important aspects of the design. Basic COEA
models with many simplifying assumptions can be used early in the process to focus
design decisions. As the design matures, refined models can be used to evaluate major
design iterations.
The COEA modeling developed for the NPS HRGK provided the justification for a
HRGK-like system. The overall objective of the HRGK was to minimize the total cost per
kill against appropriate targets. From the analysis, guided rockets showed a decreased cost
per target kill and increased target kills per sortie when compared to alternative weapon
systems. Additional modeling showed that guided rockets resulted in a reduction of sorties
and weapon costs required to achieve a specific mission objective. The final NPS HRGK
was an attempt to provide the lowest possible cost per kill design.
The COEA modeling discussed in this chapter provides several different levels of
results. The low level results of the modeling include single-shot probabilities of kill and
the number of weapons required per kill as a function of circular error probable (CEP).
These values are then used to derive the cost per kill and the sorties per mission results.
The number of weapons required per kill results also provide some additional insights that
was critical to the NPS HRGK design.
The following section describes the simplifying assumptions used in the NPS
HRGK COEA models. Subsequent sections provide more detailed descriptions of the
model methodologies and the results. The chapter ends with a discussion of some
observations concerning the use of COEA modeling in design. (Listings of the computer
codes used in the NPS HRGK COEA are included in Appendix C.)
A. SIMPLIFYING ASSUMPTIONS
Several simplifying assumptions were made to facilitate the early NPS HRGK
COEA modeling. These assumptions included defining guided and unguided weapon
trajectories and costs, establishing a target set with desired probabilities of kill (PK) for each
target, and estimating each target's vulnerability in terms of the conditional probability of
kill given a hit (P^w).
1. Weapon Accuracies
The accuracies for the unguided rocket were taken as a function of launch range and
were based on a 12 milliradian accuracy [Ref. 1]. As a baseline for the comparison, a mid-
range, 3-km shot was used with a resulting 42.4 m normal-plane CEP (36 m 1-
dimensional, standard deviation multiplied by 1.1774 — see Figure 1, for example). For the
guided rockets, three different normal-plane CEP accuracies was assumed for comparison
purposes — 0.5 m, 1.0 m, 1.5 m, and 3.0 m.
For both the unguided and guided rockets, a 20° terminal dive angle (relative to the
horizontal) was assumed as a best estimate1 . The shallow dive caused an increase in the
rocket's ground plane down-track error but did not affect the cross-track error. These
effects were accounted for implicitly in determining the probability of hitting the target.
For simple comparison purposes, the AGM-114 laser Hellfire's accuracy was
assumed to be such that if it was shot at a target it would achieve the desired killed.
2. Target Set
The effectiveness analysis used four representative targets (mobile air defense unit,
armored personnel carrier, support vehicle, and patrol boat). These targets were each
assumed to be simple blocks of appropriate length, width, and height dimensions (see
Figure 3). Table 2 lists the four targets with their associated dimensions [Ref. 3, 4, 5, and
6]. The dimensions are representative for targets found in the references. Table 2 also lists
a desired PK for each of the targets. These desired probabilities were assumed, based on the
typical importance of the target.
The probability of each target being hit (PH) was computed as the joint probability
of the rocket being within both the top-to-bottom and the side-to-side dimensions of the
target. The projected dimensions of the targets (normal to the flight trajectory) were
computed for both a head-on and a broad-side attack. The two attack approaches gave the
extremes in the target's presented area as shown in Figure 3. The Pw's for the two extreme
cases were computed and then equal-weight averaged as an estimate of the PH for uniformly
distributed attack approaches (0°-360°).
' Later analysis showed that unguided rockets may have even more shallow dive angles, making them
even more inaccurate in the ground-plane.
1
2m
Ss£L 7?> sn*^«
^j\9.4m2\
-^ 2.5 m |__
'".■
15.0 m2
J""^5:
5.5 m
Figure 3. Projected Area for Light Armored Vehicles (head-on,
45°, and broad-side attacks with 20° dive angle).
Table 2. COEA Target Set with Assumed Dimensions, Desired PK, and P^.
Target Description
Dimensions
L x W x H
Desired PK
P KIH
Mobile Air Defense Unit (Light Armor)
5 x 2.5 x 2 m
95%
90%
Armored Personnel Carrier (APC)
5 x 2.5 x 2 m
60%
80%
Support Vehicle (Large Truck)
8 x 2.5 x 2.5 m
50%
100%
Patrol Boat
14 x 4.5 x 1.5 m
75%
70%
3. Target Vulnerabilities
For each target a "cookie-cutter" model [Ref. 7] was used in determining the single-
shot PK (PKss). With the cookie cutter model, the target suffers a kill (according to an
assumed P^) if and only if it is actually hit. "Near misses" do not provide any PK. Thus
the value for PK was the simple product of P^ and PH . The assumed values of P^H are
best estimates and are shown in the last column of Table 2. These high values of PKm
reflect the facts that (1) the targets were relatively soft, (2) the rocket is actually impacting
the target to score a hit, and (3) the desired type of kill (firepower, mobility, catastrophic,
etc.), which was not specified, may only require minor damage to the target.
4. Weapon Costs and Additional PK's
The per unit cost of the guided rocket concepts was assumed to be $10,000 — the
upper threshold cost from the AJAA design contest, and the unguided, unitary warhead
round (M151) was assumed to cost $1,000 [Ref. 1]. A multipurpose submunitions
(MPSM) round was assumed to cost $2,000 and have a 10% PK against non-tank targets.
The Hellfire missile was assumed to have a per unit cost of $40,000 [Ref. 8] and a single-
shot kill capability against all four representative targets (PK - 80% for tanks and PK =
100% for all others).
5. Additional Assumptions
Additionally, the assumption was made that all weapons functioned during launch,
flyout, and fuzing; with 100% reliability. The aimpoint was assumed to be the centroid of
the target block without any bias errors in designation or flyout. It was also assumed that
all rocket accuracy errors had a bivariate normal distribution with independent, equi-
variance (a2) distributions along any two orthogonal axes normal to the flyout.
Furthermore, the errors were assumed independent from rocket to rocket. Thus, the CEP
was 1.1774 a and the PK for n multiple rockets was 1 - (l-PKsingieshot)D-
The simplifying assumptions discussed in this section are reasonably valid under
many conditions. The last section of this chapter (II.E. Observations) includes a brief
discussion on the affect of some of the assumptions.
B. COST PER KILL
The overall objective of the HRGK was to rninimize the total cost per kill against
appropriate targets. To determine the cost per kill, the probability of kill per single weapon
had to be established using the assumptions discussed above. This single-shot probability
of kill, PKss, with the additional assumptions above, could then be used to determine the
number of weapons required to achieve the desired PK. From the number of weapons
required per kill and the assumed cost per weapon, the estimated cost per kill could then be
established. The process for determining the cost per kill and the results for the HRGK
analysis are discussed in the following subsections.
1. Methodology
The methodology for determining the cost per kill involves determining the PKss for
each weapon against each target and then determining the number of weapons required to
achieve the desired PK. A computer program was written in Matlab to compute the PKss
and number of weapons required per kill as a function of CEP. This program, which also
computes a relative cost per kill tradeoff plot, is described in the following paragraphs.
a. Model Inputs
Only a small number of inputs are required for the single-shot probability of
kill computer model. These inputs can be divided into descriptions of the target and of the
weapon trajectory. Table 3 lists these two types of required inputs.
10
Table 3. Single-Shot Probability of Kill Computer Model Inputs.
Input Name
Input Description
Target
L, H, and W
Effective Target Length, Height, and Width (unit of length)
Pkh
Probability of Kill given a Hit within the Effective Target Dimensions
PkD
Desired Probability of Kill for the Target
Weapon Trajectory
dive
Dive Angle Relative to Horizontal (degrees) — 9
CEP min
Minimum CEP to Be Used in Computations (unit of length)
CEP max
Maximum CEP to Be Used in Computations (unit of length)
b. Single-Shot Probability of Kill
The first step in estimating the single-shot probability of kill is to establish
the end-on and broad-side projected edges of the target normal to the weapon trajectory.
The side-to-side dimensions for the two cases are simply X, =W and X2 = L, for the end-
on and broad-side cases, respectively. The top-to-bottom dimensions are given by the
following equations:
K = //cos0 + Lsin0
(end-on)
Y2 = tfcos0 + Wsin0
(broad-side)
Next, the probability of hit, PH, was determined by estimating the
probability of the weapon striking within the projected edges of the target — the joint
X Y
probability of being within ±— and±— of the aimpoint, where i- 1 or 2 depending on
the approach case. Given the assumed, independent bivariate nature of the weapon errors,
the joint probability could be computed as the product of the two — cross-track (X-
direction) and elevation (y-direction) — probabilities. The erf function in Matlab is
defined as the following:
P = -%=\e-,2dt,
VtJo
so the joint probability described above is given by the following Matlab expression
where the argument to erf is normalized by V2 :
11
X: .Y\ JXI 1
PH\ ±-±,±-t =erf -±-f=- -erf
K
2 ' 2 J i, 2 V2crJ 1,2 V2<T
and where i = 1 or 2 and cr is the CEP I \ All A. The total Pw is then the equal weighted
average of />„ and P„ . In other words, PH = (PH + PH )/2.
With the probability of hit, PH, and the input P^H, the single-shot
probability of kill is estimated as the product of the two.
P = P ■ P
c. Number of Weapons per Kill
With the Pfc5 and the assumption that flyout errors for each weapon shot are
unbiased and independent, the number of weapons required to achieve a desired PK can be
calculated.
The probability of the target surviving a single shot is given as the
complement of PKss, \-PKss. The probability of surviving n, independent shots is that
quantity raised to the nth power, (l-PKssT- Thus, the probability of kill for n, independent
shots is the complement to the latter probability of survival, namely:
The number of shots, n, to achieve a desired PK can be determined by
solving the above equation for n as is shown in the following equation.
ln(l-PK)
n =
HI-PkssY
d. Accuracy and Relative Cost Trade-off Chart
Knowing the number of weapons, n, required to achieve a desired
probability of kill (PK) against a specific target, the relative weapon costs to give equal cost
per kill can be computed. For example, if two weapons (A) with a 0.5 m CEP or six
weapons (B) with a 2.0 m CEP would be required to achieve a desired PK, then if the cost
of weapon B was one-third the cost or less of weapon A , weapon B would have a lower
cost per kill against the specified target.
In the computer model, weapon costs are normalized relative to the cost of
the minimum input CEP, CEP_min, weapon. Also, the number of weapons required to
achieve a desired PK are considered only as integer values.
12
e. Cost per Kill Calculations
The absolute cost per kill can easily be computed from the results of the
computer model described above. The cost per kill for a specified target and weapon is
simply the integer number of weapons required to achieve the desired PK against that target
multiplied by the assumed cost of that weapon.
2. Results
The results of the computer model described above are shown for the targets and
weapons described above in section II.A. Simplifying Assumptions. The results include
both the PKss and the cost per kill.
a. Direct Model Outputs
The combined single shot probability of kill and the number of weapons
required to achieve a desired PK are shown as a function of CEP in Figure 4. Various
assumed accuracies for the unguided rockets are also marked on the plot with their
corresponding launch ranges.
Figure 4. Single-Shot PK and Number of Rockets Required per Target as a Function of CEP.
(PKss curves slope down from upper left; number of rockets curves slope up to upper right.)
13
Figure 5 shows the relative weapon costs for equal cost per kill as a
function of CEP. In this chart the weapon costs are relative to the 0.5 m CEP weapon cost.
From the chart it appears that a weapon with a 2 m CEP (B) would have an improved cost
per kill against all targets if the weapon cost were less than 33% of the 0.5 m CEP weapon
(A). Weapon C, at half the cost of A, has the highest cost per kill at 4 m CEP.
2 3 4
Normal-Plane Circular Error Probable (m)
Figure 5. Relative Cost per Kill against Various Targets. Lettered design points (A, B,
and C) are shown for illustration only.
b. Cost per Kill Results
From the number of weapons required per target kill, the costs per kill for
each of the weapons can be computed. The results for the HRGK analysis based on the
single assumed guided rocket cost are shown in Table 4.
Table 4. Cost per Kill.
Unguided
Rocket
Guided
Rocket
Target
0.5 m
1.0 m
1.5 m
3.0 m
Hellfire
Mobile Air Defense Unit
$2,219k
$20k
$20k
$40k
$120k
$40k
Light Armor
$764k
$10k
$10k
$20k
$50k
$40k
Large Truck
$294k
$10k
$10k
$10k
$20k
$40k
Patrol Boat
$467k
$20k
$20k
$20k
$40k
$40k
14
C. KILLS PER SORTIE
A shortcoming of the cost per kill analysis previously presented is illustrated in the
following example. A weapon with a 2.0 m CEP (such as B in Figure 5) and a quarter of
the cost of a 0.5 m CEP weapon has a lower cost per kill than the 0.5 m CEP weapon
(such as A in Figure 5). However, it would take three times as many of the less accurate
(but inexpensive) weapons to achieve the same number of kills against mobile air defense
targets. The number of weapons available for a given mission would be limited by the
number of launch platforms sorties. The importance of kills per sortie is completely ignored
in the cost per kill data. A kills per sortie analysis helps to illustrate the importance of
precision in strike warfare.
The kills per sortie analysis performed for the HRGK was based on AH-64 Apache
attack helicopter sorties. The helicopters were assumed to have either pure or mixed
weapon payloads or loadouts. The pure loadouts were either 16 Hellfires or 76 rockets
(four 19-tube rocket launchers). The mixed loadouts combined eight Hellfires with a
limited number of unitary warhead (M151) rockets (guided or unguided) and additional
MPSM-equipped rockets. The number of kills per sortie were calculated based on the PKss
data in Figure 4 and the assumed P^'s from the sub-section II.A.4. Weapon Costs and
Additional PK's. The results are tabulated in Table 5 for the light armor target case.
Table 5. Kills per AH-64 Apache Sortie against Light Armor Targets.
Weapon
Maximum Load
Mixed Load
Hellfire
16 Hellfires
16 Kills
8 Hellfires & 38 MPSM
9.90 Kills
HRGK (0.5m)
76 HRGKs
60.7 Kills
8 Hellfires, 24 MPSM, & 14 HRGKs
20.4 Kills
HRGK (1.0m)
76 HRGKs
52.5 Kills
8 Hellfires, 24 MPSM, & 14 HRGKs
18.9 Kills
HRGK (1.5m)
76 HRGKs
38.7 Kills
8 Hellfires, 24 MPSM, & 14 HRGKs
16.3 Kills
HRGK (3.0m)
76 HRGKs
15.1 Kills
8 Hellfires, 24 MPSM, & 14 HRGKs
12.0 Kills
Unguided
76 Rockets
0.09 Kills
8 Hellfires, 24 MPSM, & 14 Rockets
9.22 Kills
D.
COST PER MISSION
A final type of modeling performed by the author for the NPS HRGK design
combines the attributes of the cost per kill and the kills per sortie analyses. This analysis
assesses the cost (in terms of sorties and cost of weapons fired) to achieve a mission
15
objective. The following subsections outline the analysis and present the results for the
HRGK.
1. Mission Definition
The cost per mission analysis was based on a specific assumed mission scenario.
This scenario defined the target matrix with a set priority on the target types as well as a
mission objective. The scenario also defined the attacking helicopter loadouts. The assumed
scenario in defined in the following paragraphs.
a. Target Matrix
The assumed target was an assembly area with the following targets types
listed by priority and with the target quantities in parentheses:
• Mobile Air Defense Units (8)
• Main Battle Tanks (40)
• Light Armored Vehicles ( 1 50)
• Support Vehicles (250)
The mission objective was to destroy half of the vehicles in each combatant category.
b. Weapon Loadouts
The assumed initial strike force was a flight of five AH-64 Apache attack
helicopters. As additional helicopters were needed they were added to the attack force one
at a time. Each Apache carried a nominal load of eight Hellfires and two 19-rocket launcher
pods. Each pod (which could be divided into zones) was loaded with twelve M261 MPSM-
equipped rockets and seven guided or unguided M151 unitary warhead equipped-rockets.
[Ref. 9]
The limited number of HRGK-equipped weapons (14) per helicopter was
based on the assumption that only six to eight laser-designated flyouts could be made from
a firing position before the helicopters would be forced to relocate. With eight Hellfires and
14 HRGKs, three firing positions would be required for the attack. This was considered to
be the upper limit for a deep strike mission. The MPSM rockets would not require laser
designation, yet they would still be effective against clusters of light armor, material, or
personnel. [Ref. 9]
2. Example Calculation
The following provides an example of the process used to model the mission
engagement. The 1 .0 m CEP case is illustrated.
16
a. First Five Helicopters
The weapons loadout of the first five helicopters consists of the following:
• 40 Hellfire Missiles (each at a tank)
• 40 x 0.8 = 32 tank kills
• 70 HRGK Rockets (16 at air defense units— 2 per unit, 54 at light armor)
• 1 6 — 8 air defense unit kills with PK = 95%)
• 54 x 0.6909 = 37.31 light armor kills
• 120 MPSM Rockets (120 x 0.1 = 12 kills divided between non-tank targets)
• 6 light armor kills
• 6 support vehicle kills
In summary, after five helicopters, 8 air defense units are killed (exceeding objective), 32
tanks are killed (exceeding objective), 43.31 light armor vehicles are killed (31.69 short of
the objective), and 6 support vehicles are killed (exceeding objective).
b. Sixth Helicopter
The following delineates the use of the sixth helicopter's weapons:
• 8 Hellfire Missiles (each at a tank)
• 8 x 0.8 = 6.4 tank kills
• 14 HRGK Rockets (each at light armor)
• 14 x 0.6909 = 9.67 light armor kills
• 24 MPSM Rockets (24 x 0.1 = 2.4 kills divided between non-tank targets)
• 1 .2 light armor kills
• 1 .2 support vehicle kills
In summary, after the sixth helicopter, 8 air defense units are killed (exceeding objective),
38.4 tanks are killed (exceeding objective), 54.18 light armor vehicles are killed (20.82
short of the objective), and 7.2 support vehicles are killed (exceeding objective).
c. Seventh Helicopter
The following is for the seventh helicopter:
• 8 Hellfire Missiles (2 — each at a tank, 6 at light armor)
• 2x0.8 = 1.6 tank kills
• 6 x 1 .0 = 6 light armor kills
• 14 HRGK Rockets (each at light armor)
• 14 x 0.6909 = 9.67 light armor kills
• 24 MPSM Rockets (24 x 0. 1 = 2.4 kills divided between non-tank targets)
• 1 .2 light armor kills
• 1 .2 support vehicle kills
After the seventh helicopter, 8 air defense units are killed (exceeding objective), 40 tanks
are killed (exceeding objective), 71.05 light armor vehicles are killed (3.95 short of the
objective), and 8.4 support vehicles are killed (exceeding objective).
17
d. Subsequent Helicopters
For any subsequent helicopters the weapons would be used as follows:
• 8 Hellfire Missiles (each at light armor)
• 8 x 1 .0 = 8 light armor kills
• 14 HRGK Rockets (each at light armor)
• 14 x 0.6909 = 9.67 light armor kills
• 24 MPSM Rockets (24 x 0. 1 = 2.4 kills divided between non-tank targets)
• 1.2 light armor kills
• 1.2 support vehicle kills
This gives 18.87 light armor kills per sortie; therefore, the number of sorties, after the
seventh, to reach the objective of 75 light armor kills is 3.95/18.87 = 0.21, and the total
number of sorties required for the mission is 7.21.
e. Summary
After 7.21 sorties the following number of weapons have been fired with
the indicated assumed weapon cost:
• 57.67 Hellfires ($2.3M)
• 100.9 HRGKs ($1.0M)
• 173.0 MPSMs ($0.35M)
for a total weapon cost of $3.66M. The total targets killed were:
• 8 air defense units
• 40 tanks
• 75 light armor vehicles
• 8.65 support vehicles.
3. Results
Figures 6, 7, and 8 illustrate the results for the HRGK mission analysis. Figure 6
shows the number of sorties required to achieve the mission objective. With an assumed
cost per sortie, these numbers could be directly applied to the weapon's cost per kill. The
inclusion of a helicopter attrition rate would also make the cost per kill more realistic. Note
that the 1 .0 m CEP rocket does not require many more sorties than does the 0.5 m CEP.
Figure 7 shows the number of weapons fired during the mission for each of the
different weapon cases. The total weapon cost in millions of dollars is also indicated on the
graph.
Figure 8 summarizes which weapons killed which targets. It is interesting to note
that with accurate HRGKs, the more expensive Hellfire missile could be used almost
exclusively on tanks. On the other hand, when unguided rockets were used, nearly all the
18
target kills were from the Hellfire, and the unguided unitary (M151) rocket only registered
0.242 light armor and 0.474 support vehicle kills.
Sorties Required
Ol O Ol
—^^m
^ CD
JkJkb
■
0.5m 1.0m 1.5m 3.0m unguided
Rocket Type
Figure 6. AH-64 Apache Attack Helicopter Sorties Required to Achieve Mission Objective.
0.5m
1.0m 1.5m 3.0m
Rocket Type
unguided
Figure 7. Number of Weapons Fired to Achieve Mission Objective. Total weapon costs
indicated on graph in millions of dollars..
19
0.5m
1.0m
1.5m
3.0m
unguided
Rocket Type
Figure 8. Number of Targets Killed by Category, Weapon, and Loadout Option.
E.
OBSERVATIONS
Simple COEA type modeling is essential to the design process. Besides assuring
that the design can be cost effective, the results from early analyses provide focus for the
design team. COEA results can show general trends that might otherwise go unnoticed.
COEA type modeling continues to be useful throughout the design process as refined-
model analyses can help in verifying design decisions.
An example of how unexpected information can come from a COEA study occurred
in the NPS HRGK analysis. Figure 4 (page 13) shows a "knee in the curve" around 1 m
CEP. That is the approximate CEP where the P^/s begin to drop off significantly and the
number of weapons needed per kill begins a steep increase. From the figure, it appears that
weapons achieving a CEP much smaller than 1 m are not needed against the assumed
targets. Therefore, as a result of the COEA modeling, designs that achieve tighter than 1 m
CEPs at significantly higher costs are eliminated from consideration.
On the other hand, COEA results can be very sensitive to underlying assumptions.
The possibility of invalid assumptions can make COEA modeling too unreliable for
confident design decisions. For example, small changes in the low-level target set
assumptions can result in important changes in PKss values. These values, in turn, will
20
impact sortie rates. Likewise, high-level assumptions like the mission definition used in the
sorties per mission analysis will also impact the final COEA results. The validity of the
assumptions involved in both low- and high-level models must be continuously assessed.
Specific examples of COEA sensitivities to underlying assumptions in the NPS
HRGK models include two opposing possibilities. If the assumed target set were expanded
to include a smaller point target, the "knee" of the P^ curve would shift to a smaller CEP.
Then a more expensive, 0.5 m CEP weapon might become more cost effective then less
expensive, 1 m CEP rockets. Conversely, if the centered, unbiased aimpoint assumption
(discussed on page 10) were not valid, the general cost effectiveness advantage of the high
precision rocket would become smaller, and guided rockets with slightly larger dispersion
could become more cost efficient.
Ideally, COEA studies will include sensitivity analyses to determine the range of
results depending on changes in the assumed inputs. With these sensitivities in mind, the
designer can use the COEA models as powerful design tools.
21
22
III. PRELIMINARY SIZING ANALYSES
Several aspects of any design require preliminary sizing. For the NPS HRGK
design, the physical dimensions (length, weight, and center of gravity), the thruster, and
the seeker's field of view each needed to be sized early in the design process.
The author developed several computer codes to allow the sizing of the kit, its
control mechanisms, and the seeker's field of view. These codes each use top-level
requirements to generate lower-level design specifications. This flow down of
specifications helps to assure that the final design meets the system requirements.
The following sections describe the methodology used in developing the computer
models and describe some results specific to the NPS HRGK design. The chapter ends
with observations regarding the models and the flow-down of design requirements.
A. LENGTH, WEIGHT, AND CENTER OF GRAVITY
The HRGK design weight and center of gravity affects the overall center of gravity
of the guided rocket. The guided rocket's center of gravity is a critical design consideration
in that it affects the rocket's stability in flight. Additionally, the guided rocket's center of
gravity must be such that a fully loaded launcher pod's center of gravity is kept between the
pod's two mounting lugs. Furthermore, the length and weight of the HRGK are
constrained by limits specified for the overall length and weight of the guided rocket.
A Matlab code was written by the author to help visualize the relationships
between the HRGK size and the rocket's center of gravity. The code also maps the useable
design space bounded by the maximum length and weight as well as the launcher's center
of gravity constraints. This code is briefly described and some results are presented in the
following subsections. The code listing is included in Appendix C.
1. Methodology
The computer code used for the length, weight, and center of gravity analysis is
based on the simple equation defining a composite center of gravity, XCG, based on the
component weights, Wt, and component centers of gravity, xCG . Namely,
XCg ~
23
a. Inputs
The user-defined parameters in the code are either design or constraint
parameters as listed in Table 6. Additional model parameters needed for the model include
the weight and center of gravity locations for the live and fired rocket and for the empty
launcher. The length of the rocket and the location of the pod's front lug are also required.
These parameters for the Hydra-70 rocket system were available from [Ref. 10 and 11].
Table 6. Weight, Length, and Center of Gravity Model Inputs.
Input Name
Input Description
Design Space Parameter
Wkl
Range of kit weights (evenly-spaced vector) (lb)
Lkl
Specific kit lengths to be plotted (vector) (in)
Dkl
Specific kit densities to be plotted (vector) (lbm/in3)
Cgkl
Assumed kit center of gravity (fraction of kit length back from nose)
dia
Assumed kit diameter (in)
Constraint Parameters
Lmax
Maximum total length (in)
Wmax
Maximum total weight (lb)
CgL
Pod forward Cg limit (fraction of lug spacing back from front lug)
The kit's length is defined as the length extending forward from the tip of
the rocket's fuze to the forward tip of the kit. The kit's center of gravity is based on that
length; therefore, if the kit's center of gravity were behind the tip of the fuze, the value for
Cgkl would be greater than unity.
b. Outputs
The computer code creates a plot of the rocket's center of gravity1 as a
function of guidance kit weight. The design space is marked with various curves indicating
user specified guidance kit densities and lengths. With any two parameters defined (kit
weight, density, length, or guided rocket center of gravity), the other two parameters are
fixed on the plot. The usable design space is bounded by the maximum total length and
weight as well as the loaded launcher's center of gravity constraints. The code generates the
described plots for both the live and the fired rocket cases.
Measured in calibers (missile diameters) from the forward tip of the guidance kit.
24
2. Results
Figure 9 shows the output from the computer model for the live rocket case. The
solid lines represent the case where the kit's center of gravity is located at the kit's defined
mid-length. The cross-hatched markings around the lines allow for the kit's center of
gravity to travel within the middle third of the kit's defined length. The upper edge of the
cross-hatched regions represents the kit's center of gravity being two-thirds of the length
back from the nose, and the lower edge is for the center of gravity at the one-third mark.
The bold lines represent the indicated design constraints. The NPS HRGK design point (7
pounds and 13.625 inches longer than the fuze) is marked on the plot as a reference.
CD
^vKwwouu^1^.'
0.15 lbm/in3
4 5
Guidance Kit Weight (lb)
Figure 9. Physical Size and Center of Gravity Design Space. Kit length is defined from the
tip of the rocket fuze forward. The launcher center of gravity constraint is tighter than
specified for illustrative purposes.
From the plot it appears that the loaded launcher's center of gravity requirement
(keeping the pod's center of gravity behind the launcher's front mounting lug) is not
binding for most conditions. Indeed, for Figure 9, the constraint was tightened to keeping
the pod's center of gravity within the middle third of the 14 inch lug spacing so that the
constraint could even be seen in the plot.
25
B.
MANEUVERABILITY
Another computer model was developed in Matlab to defme — based on the system
requirements — the HRGK's maximum maneuverability specifications. An additional
computer program uses a steady maneuver to determine the angles of attack and thruster
forces needed to achieve the specified maneuverability. This latter code sizes the HRGK's
thrusters. The following subsections describe these computer models and their results.
1. Turn Rate and Turn Radius Methodology
The maneuverability analysis determines the maximum turn rate in g's or minimum
turn radius in kilometers needed to hit a target over various launch and seeker acquisition
ranges. The two-dimensional model used in this analysis is described in this subsection.
a. Model Inputs
The maneuverability model requires several user-defined inputs. These
inputs are listed and described in Table 7 and are illustrated graphically in Figure 10.
Table 7. Maneuverability Model Inputs.
Input Name
Input Description
Launch and Seeker Acquisition Ranges
Rmin, Rmax
Minimum and maximum range to target at launch (km)
n
Ivacq
Specific seeker's maximum target acquisition range (vector) (km)
Unguided Errors
Pnt
Launcher pointing error at time of launch — la (degrees)
B
Ballistic unguided flyout error — la (milliradians)
Weapon System Delays and Limitations
GD
Guidance delay (time after launch before guidance is possible) (sec)
k>
Target identification delay (from initial acquisition until start of turn) (sec)
sF
Straight final flight distance (terminal non-maneuvering zone) (m)
Missile and Target Motion
Time
Time values for missile range and velocity time profiles (vector) (sec)
Dist
Missile range profile (distances corresponding to Time) (vector) (m)
VM
Missile velocity profile (velocities corresponding to Time) (vector) (m/s)
VT
Target speed (mph)
Statistical Confidence
sig
Number of sigma's confidence (e.g., sig=2 gives ±2a or 95% confidence)
26
VT
First Time Seeker
Detects Target
$ = sig ■ ^JPnt2 + B2
No Guidance Possible
(guidance time delay, Go)
1
Figure 10. Graphical Representation of Maneuverability Model Inputs.
27
The two-dimensional model assumes that the rocket flies a straight path
toward the initial target location but with an azimuth error angle, 6. This angle is a
statistical combination1 of the launcher pointing error and the ballistic fly out dispersion
error of the rocket. After both the initial guidance delay2, GD, and after the rocket is within
the assumed seeker acquisition range, R , of the target; the rocket can begin a turn toward
the target — correcting any cross-track errors due to 6 or target motion.
A target identification delay time, tD, between the first detection of the target
and the start of the turn can also be included in the model. The distance associated with tD,
SD, is based on the rocket velocity at the time of first detection (or the end of guidance
delay, whichever is later). The model also allows for a terminal non-maneuver zone — some
radius, SF, from the target. For modeling purposes, this zone around the target forces the
rocket to have completed its turn to the target and align itself for a final straight approach.
b. Stationary Target — Closed-Form Solution
With a stationary target, the inputs described above can provide a closed-
form solution for the maneuverability specifications. This process is laid out in Figure 1 1 .
The equations illustrated in Figure 11 provide the maneuverability
requirements in terms of both required turn radius, rT, and turn rate, nmax. (They also
provide the solution to the minimum required seeker field of view — FOV. This part of the
analysis will be described more fully in a later section.)
For the ±2o unguided error case (sig = 2), the resulting maneuverability
values (rT and nmai) could be interpreted as the biggest turn radius or slowest turn rate that
would still ensure a 95% probability of hitting a stationary target. In other words, any
larger or slower turns, than those specified, would not give a 95% confidence with the
provided inputs. On the other hand, tighter or faster turns would give a higher than 95%
confidence. Therefore, if the system is specified to 95% confidence, the computed values
are the maximum turn rate and minimum turn radius required.
c. Moving Target
The basic equations shown in Figure 1 1 can be applied to the moving target
1 The standard deviation, a, of 6 is the square root of the sum of the variances Pnf and B2 —
I 2 2
a = ~4Pnt + B — where Pnt and B are the standard deviations for the pointing error and the ballistic
dispersion errors, respectively. The angle 6 is some user-defined multiple {sig) of a.
2 The range associated with GD is calculated based on the user-supplied range time profile for the rocket.
28
n
Field of View and Turn Rate Calculations
x = R sin 0 y = R cos 6
Rd = y~ ^Racq ~ x* => VMD
2 2
acq ~ x ~
sd^(vMD)(td)
z = Jx2+(y-Rr)'
FOV = sin-1 -
0 = --FOV
y 2
law of cosines:
d2 = (sf +r?\ = r2 +z2-2rt zcos</>
2 2
z ~sf
rt = —
2 z cos 0
n
'M7
max
^00
Figure 1 1 . Closed-Form Solution for Maneuverability Specifications with a Stationary
Target.
29
scenario shown in Figure 12. The moving target scenario assumes a "worst-case" target
motion. The target moves with a crossing velocity in the direction opposite the flyout error.
Then, after it is detected, the target turns in at 45° making the rocket's turn tighter.
The moving target scenario precludes a closed-form solution of the
maneuverability specifications. However, the equations in Figure 1 1 can be used with an
iterative step at each of the flight milestones (target detection, start of turn, and end of turn).
The Matlab code listing in Appendix C uses this approach. The following description
provides an example of the iterative process used in the computer code.
After RD is calculated using the appropriate stationary target equations (see
Figure 11), the time required to fly that distance is available from the range time profile.
Using the time and the target speed, the new target location can be computed. The value for
RD can then be re-calculated based on the updated target location. These iterations continue
until the change in RD between iterations is less than some set tolerance (such as 1 m).
2. NPS HRGK Turn Rate and Turn Radius Analysis
This subsection discusses the maneuverability analysis performed specifically for
the design of the NPS HRGK. The following paragraphs describe the inputs used for the
computer model and present the analysis results.
a. NPS HRGK Model Inputs
The NPS HRGK analysis used the full gamut of specified launch ranges ( 1
to 6 kilometers) and various seeker acquisition ranges (0.75, 1.0, 1.15, 2.0, 3.0, and 6.0
km). The assumed la values for launcher pointing error and ballistic dispersion were 3?
and 12 mils, respectively. (Using a 95% confidence level — sig = 2, the angular flyout error
was internally computed to be 6.16°.) The target speed was 60 mph per the system
requirements. Initial guidance was delayed as a linear function of launch range (0.5
seconds for 1 km, up to 1.0 second for 6 km). The target identification delay was 0.625
seconds (arbitrarily chosen to correspond to 5 laser pulses at 8 Hertz), and the terminal
non-maneuvering distance was arbitrarily chosen to be 200 meters.
The rocket's velocity and distance time profiles used in the analysis were
based on an early simulation of a HRGK-equipped rocket using a ballistic trajectory.
Specifically, the simulation used the 150°F propellant thrust profile [Ref. 1 1] and a generic,
hemispherical-nosed guidance kit with three canards. The aerodynamic coefficients and
derivatives used in the simulation were determined using the software code, Advanced
Design of Aerodynamic Missiles — ADAM [Ref. 12] .
30
Target at Detection
(target begins 45° turn)
Target at
Rocket
Launch
Target Position
(when rocket
starts turn)
Start of
Rocket
Turn
Assumptions:
•Target has a crossing velocity
•After target detection, target turns 45c
Unguided Flyout Azimuth Error
(Initial Pointing and Dispersion)
Launch Helicopter
Figure 12. Moving Target Scenario for Maneuverability Analysis.
b. NPS HRGK Analysis Results
Based on the inputs described above, the maximum turn rate and minimum
turn radius specifications for the NPS HRGK are shown in Figure 13 and Figure 14,
respectively. The results are shown as a function of launch range for several seeker
acquisition ranges. The analysis results show that as long as the seeker can initially detect
31
the target at 1.15 kilometers or more, the hardest g-loading maneuver for the rocket would
be less than 15 g's and tightest turn radius would be greater than 1.83 kilometers.
The maximum turn rate is a strong function of launch range with short range
launches requiring higher g-loading maneuvers. This is primarily due to higher rocket
speeds at shorter ranges. For example, after flying 0.5 km, the rocket's velocity is 1.8
times higher than it is after coasting out to 5 km. For equal turn radii, a 1.8 factor increase
in velocity yields a 3.2 (or 1.82) factor increase in turn rate. The guidance delay also
increases the turn rate at shorter launch ranges. Target identification delays primarily impact
the shorter seeker acquisition range cases — longer delays causing higher turn rates.
Unlike the maximum required turn rate, the minimum required rum radius is
almost independent of rocket speed1. Once the launch range places the target initially
outside the seeker's acquisition range, the turn radius curves remain fairly flat with
increasing launch range. For example, with a 1.15-km-acquisition-range seeker, the
minimum required turn radius changes less than 59% across the spectrum of launch ranges
(1.83 km at 1 km launch to 2.90 km at 6 km launch). Conversely, if the seeker has
sufficient detection range to take advantage of it, the longer range launches allow
significantly wider turns. For example, only a gentle 12.9 km turn radius is required with a
6 km seeker acquisition range and a 6 km launch range.
By either measure (turn rate or turn radius), the most stressing case for the
maneuverability requirement is the short range launch, because of rocket speed and the lack
of reaction time. The seeker's acquisition range can also significantly impact the HRGK
maneuverability requirements — shorter range seekers forcing higher maneuverability
requirements. Conversely, long range seekers can alleviate some maneuverability
requirements by detecting the target earlier and allowing gentler turns.
An additional benefit of the maneuverability analysis is that the minimum
required seeker acquisition range is shown to be 1.15 km. With shorter acquisition ranges
and long launch ranges, the target's cross-track position relative to the rocket (due to 6 and
target motion) exceeds the seeker's range. For example, with a 4.1 km launch range, the
rocket's closest unguided approach to the target is 0.75 km (2a). For longer launch ranges,
the closest approach distance would be greater than 0.75 km; therefore, a 0.75-km-
detection-range seeker would not have a 95% probability of detecting the target.
1 Rocket speed indirectly affects the turn radius since it partially determines the distance the target can
travel before it is detected or impacted.
32
16
w
3I2
a>
"co
QC
c
§ 8
h-
E
£
cc 4
t ,,,,,,
I. J
" ♦
i ' ", ♦ •
V/ *. ♦ :
V '. ♦'
X V *» •
X *k •
,..X ;...>« .'.. * ..• ;
Seeker
Acquisition
Range (km)
•""0.75
//////// 1 _Q
■■■■■•■■a 1.15
Xv ! '>. '., *.
i 1
1 1
3 4
Launch Range (km)
Figure 13. Maximum Turn Rate Required to Hit Target (95% Probability). Only with a
seeker able to detect a target from 1.15 km or more (Racq > 1.15) can the maximum
range requirement (6 km) be meet.
0.75 km
3 4
Launch Range (km)
Figure 14. Minimum Turn Radius Required to Hit Target (95% Probability). Seeker
acquisition ranges are indicated with the corresponding curves.
33
3. Steady-Turn Maneuver Analysis
Once maneuverability specifications are set, a steady-turn maneuver model can be
used to size the control mechanism needed to achieve the needed turn rate or turn radius.
The following paragraphs describe a computer code developed by the author to perform
this task. Later a specific example for a guided rocket is presented. The Matlab code
listing with the NPS HRGK parameters is included in Appendix C.
a. Mathematical Steady-Turn Model
The purpose of this mathematical steady-turn model is to determine the
thrust force and angle of attack (or sideslip angle) necessary to generate a constant
acceleration turn in the horizontal. Throughout the description, the more familiar
longitudinal nomenclature (loosely associated with the vertical plane) is used even though
the turn is described in the directional case (loosely associated with the horizontal plane).
The interchange of nomenclature is justified since, for the guided rocket or missile case, the
longitudinal and directional dynamics are equivalent. A useful listing of missile
aerodynamic nomenclature is included in Appendix A.
Figure 15 shows a free-body diagram for a horizontal maneuver. For a
constant acceleration turn, the sum of the forces and moments can be expressed by the
following equations:
£ Forces = « W = N + T = qACNa a + kT
(
1
A
2&
pV2AC„
a + kT
£ Moments = -NXCP + TXT - Mq = 0
= -{qACNa)Xcp + kTXT - qAd{cMq ^j
= -{qACNaa)xcp + kTXT - qAd
= -(qACNa a)Xcp + kTXT - qAd[cMq ^j
, Mq 2V ,
- a
1 ^ 1
—pV2ACNa XCP + TkXT-—pACMd2ng
34
where k is a thrust multiplication factor (accounting for misalignment of the thruster and the
effect of jet and free air steam interaction) and the other terms are either defined by Figure
15 or are standard terms defined in Appendix A. The moments equation uses the fact that
rT
V
rV2\
\azJ
V " V
Figure 15. Plane- View, Free-Body Diagram for a Steady-Turn Maneuver (horizontal, left
turn — gravity force vector pointed into the page).
The sum of forces and sum of moments equations can be rewritten as the
following matrix equation:
1
2£0
pV2AC„
KA-t
\
2*o
pV2AC„
LC/>
a
4*o
nW
pACMd2ng
which gives the thrust magnitude and angle of attack in terms of guided rocket parameters
and the turn rate n. Using the relationship rT = V2/ng, the matrix equation can be expressed
in terms of the turn radius as the following:
V2W
1
2g0
pV'AC„
kX7
2£0
pV2AC.
Lc/>
a
4rTg0
grT
pV2ACMd2
35
b. Steady-Turn Computer Code
The author has developed a computer code to implement the mathematical
model described above. This code solves for the thrust and angle of attack required to
achieve the required turn rate or turn radius over a range of Mach numbers. The code
requires the user-supplied inputs listed and described in Table 8.
Table 8. Steady-Turn Maneuver Computer Model Inputs.
Input Name
Input Description
Scenario Definition
n or rT
Desired turn rate (g's) or turn radius (ft)
type
Indicator for computations based either on turn rate (1) or turn radius (2)
alt
Pressure altitude (ft)
Mmin, Mmax
Minimum and maximum Mach number for the calculations
Rocket Configuration
Wght
Rocket total weight at maneuvering time (lb)
Dia
Rocket diameter (for reference area and length) (in)
Xcg
Rocket center of gravity (calibers from nose)
Xt
Thruster location (calibers from nose)
Aerodynamic Parameters (functions of Mach)
Mach_dat
Mach values for corresponding aerodynamic parameters (vector)
CNa_dat
Rocket normal force curve slope (corresponding to Mach) (vector) (rad1)
Xcp_dat
Center of pressure (corresponding to Mach) (vector) (calibers from nose)
CMq_dat
"Pitch" damping (corresponding to Mach) (vector) (rad1)
c. NPS HRGK Example
The following provides example solutions to the equations previously
developed for the steady-turn model. The examples are specific to the NPS HRGK with the
guided rocket's maneuverability specifications set at 15 g's turn rate or 1.83 km (6,000 ft)
turn radius. The manual calculations are for the NPS HRGK design at Mach 1.0. These
calculations are done first for the turn rate specification and then for the turn radius
specification. The aerodynamic coefficients are from ADAM [Ref. 12]. All computations
are for 500 feet pressure altitude and use the fired rocket center of gravity (most
maneuvering takes place after the 1.0 sec rocket motor burn). Because no additional data is
available, the value of k (an adjustment factor for the thruster force) is assumed to be unity.
36
The manual calculations are followed by sample output from the steady-turn computer
code.
( 1 ) Sample Manual Calculation Based on Maximum Turn Rate.
(0.0754^)(lll4f)2(0.0412 ft2)(20.05 rad1)
zr L-L ibf s2 /
(0.0754 ^)(ll 14 f)2 (0.0412 ft2)(20.05 rad1)
1.79ft
ZrZ,Z Ibf s2 /
15 (22.8 Ibf)
0.0824 ft
a
(0.0754 ^)(0.04 12 ft2)(-5520 rad_1)(0.229 ft)2 15 (32.2 f)
4(32.2^)
The solution to the set of equations is:
1
1200^"
~T
' 342 Ibf
~T~
" 13.2 Ibf "
"13.2 Ibf
1.79 ft
_QQ 1 'W ft
771 rad _
a
-3.38 Ibf ft
— >
a
0.273 rad
15.6°
1 1200^
1.79 ft -99.1^
(2) Sample Manual Calculation Based on Minimum Turn Radius.
(1114^)2 (22.8 Ibf)
(32.2^-)(6000 ft)
(0.0754 ^)(H14f)2 (0.0412 ft2)(-5520 rad_1)(0.229 ft)2
4(32.2 «-)(6000 ft)
The solution to the set of equations is:
1
1200^ "
~T~
" 147 Ibf
~T
" 5.68 Ibf "
"5.68 Ibf
1.79 ft
QQ 1 Ibf ft
77,1 rad .
a
-1.45 Ibf ft
— >
a
0.117 rad
6.71°
37
(3) Sample Output from Steady-Turn Computer Code. Figure
16 shows the output from a computer code developed by the author. The code computes
the solution to the steady-turn equations that were manually solved in the previous
paragraphs.
CD
cc
Q.
CO
-a
o
O"
CD
DC
1
3-4 0.6 0.8 1.0
1.2
1.4
1.6
Af\
4U
20
3^ J
0
0-4 0.6 0.8 .1.0 1.2
Mach Number
1.4
1.6
Figure 16. Required Thrust and Angle of Attack for Maximum Turn Rate (15 g's —
dashed line) or Minimum Turn Radius (1.83 km — solid line) (NPS HRGK design).
From the data plotted in Figure 16, the difference in the two
methods of specifying maneuverability requirements is apparent. For example, at subsonic
speeds, the model predicts that the rocket could only achieve a 15 g maneuver with very
high thrust and extremely high angle of attack. On the other hand, the requisite 1.83 km
turn radius can be achieved at subsonic speeds with a moderate angle of attack and low
thruster force.
Fortunately, 15 g maneuvers are not required when the missile has
coasted down to subsonic speeds (see long range side of Figure 13, page 33). Clearly, the
reasonable curve for sizing the thruster is the rninimum turn radius curve (solid line). This
seems reasonable since the turn radius requirements remained fairly flat over the entire
gamut of launch ranges (see Figure 14).
38
Designing to the minimum turn radius and given the fact that the
rocket has slowed down to nearly Mach 1.3 by the first kilometer of flight, the thruster can
be sized to provide 10 to 15 pounds of thrust. A design with higher thrust early in the flight
would be ideal, since the short-range, high-speed scenario is the most demanding.
4. Probabilistic Approach
The maneuverability analysis performed in the previous subsections was based on
the statistical (probabilistic) fly out of the rocket but a deterministic, worst-case target
motion. Another approach to determining the needed level of maneuverability would be to
apply a statistical model to the target motion as well as the fly out. This latter approach
would give valuable insight into the reasonableness of the worst-case approach taken
above. This type of purely probabilistic approach was needed for the seeker field of view
analysis, which is described next.
C. SEEKER FIELD OF VIEW
The seeker's field of view1 was another early consideration for the NPS HRGK
design. Again, this design specification could be derived from top-level system
requirements and simple mathematical models. A possible approach to determining the
needed seeker field of view was alluded to in the maneuverability analysis section. This
section will briefly discuss that approach and its results. Then a probabilistic approach with
its results will be discussed in greater detail.
1. Analysis with "Worst-Case" Target Motion Model
The two-dimensional scenario described in subsection EQ.B.l. Turn Rate and Turn
Radius Methodology provides a means for calculating the needed HRGK seeker field of
view. The equations shown in Figure 1 1 (page 29), can be used to determine the field of
view that keeps a stationary target within the seeker's field of view with some specified
level of confidence (for example, 95%) up until the target can be detected and a turn toward
the target can be initiated. As discussed in the earlier subsection, the equations shown in
Figure 1 1 can be solved iteratively to determine the required field of view for a moving
target (see Figure 12, page 31).
1 The seeker's field of regard — meaning the total area covered by a fixed or scanning seeker — is the actual
consideration, but for a strapped-down seeker, the field of view and field of regard can be used
interchangeably. This thesis uses the term field of view.
39
Unfortunately, the target motion model used for the maneuverability analysis is a
worst-case example. (The target crosses at maximum velocity in the opposite direction of
the statistical flyout and pointing errors. Furthermore, after detection, the target turns in at
45° which creates the need for a still larger field of view. See Figure 12.)
The required field of view results shown in Figure 17 for the NPS HRGK were
obtained using the worst-case target motion model and the computer code inputs discussed
in subsection III.B.2. NPS HRGK Turn Rate and Turn Radius Analysis (page 30). The
values are for half field of view and are shown as a function of launch range and for
various seeker acquisition ranges. For example, with a maximum launch range and a seeker
acquisition range of 1.5 km, a ±48° seeker field of view would be needed to assure (95%)
that the target stays in the field of view up until a turn can be initiated. (With a stationary
target, only a ±29° seeker field of view would be needed.)
Racq = 6 km
3 4
Launch Range (km)
Figure 17. Required Field of View Results for 95% Probability of Target Staying in Field
of View with Worst Case Target Motion. (Corresponding field of view requirements for a
stationary target are indicated with dashed lines.)
From the curves in Figure 17, it is clear that longer launch ranges require larger
fields of view (after longer fly outs, the rocket's cross-track error and the target's motion
40
away from the rocket will be larger). Shorter acquisition range seekers also require larger
fields of view.
Unfortunately, the indicated field of view sizes for long range launches and short
range seekers are unattainably large, especially for a strapped-down, semi-active laser spot
homing seeker like that on the NPS HRGK. The worst-case target motion model is
unacceptable, and so, a probabilistic approach must be taken.
2. Analysis with Probabilistic Target Motion Model
The probabilistic target motion model used for the seeker field of view analysis is
based on a distribution of target positions. In this analysis, the probability of the seeker
acquiring a target is derived from an assumed seeker field of view and the probabilistic
distributions for the rocket and the target. The methodology for defining a seeker field of
view specification with this approach is developed in this subsection. The resulting
specifications for the NPS HRGK are also presented. The Matlab code developed by the
author to perform this analysis is included in Appendix C.
a. Computer Model Inputs
The inputs required for the probabilistic seeker field of view analysis are
listed and described in Table 9.
Table 9. Seeker Field of View Analysis Inputs.
Input Name
Input Description
Seeker Descriptions
FOVmin, max
Minimum and maximum seeker half field of views (deg)
Specific seeker's maximum target acquisition range (vector) (km)
Unguided Errors and Weapon System Delays
Pnt
Launcher pointing error at time of launch — la (degrees)
B
Ballistic unguided flyout error — lo (milliradians)
GD
Guidance delay (time after launch before guidance is possible) (sec)
tD
Target identification delay (from initial acquisition until start of turn) (sec)
Missile and Target Motion
RL
Range to target at launch (single value) (km)
Time
Time values for missile range and velocity time profiles (vector) (sec)
Missile range profile (distances corresponding to Time) (vector) (m)
Vx V-r
" Tmin' * Tmax
Minimum and maximum target speed (mph)
41
b. Target Motion Model
The probabilistic seeker field of view analysis uses a simple distribution
model for target motion. The target's speed is assumed to be constant and uniformly
distributed between the user-supplied inputs, VTmin and VTmax. Its bearing, <f), is also
assumed to be constant and to be uniformly distributed between 0° and 360°. Figure 18
shows a plane view of possible target locations after time, tT (the flyout time before the
rocket starts its turn to the target). The figure also includes the corresponding probability
density function for the target's cross-track position, x. This function provides the
probability of the target being at some specific cross-track position, xT, or between two
cross-track limits, x, and x2. (The v-axis indicates the direction of the rocket flyout.)
original
target
location.
trVm
illustrative target location
Pt(xt)
(probability: target
at cross-track Xi)
iPr{x)
tr Vjmax
area of possible
target locations
(b)
Pt (xuxs)
(probability: target
between X2 and X2)
Figure 18. Target Distribution Model, (a) Plane view of possible target positions and (b)
Cross-track probability density function, Pj(x)
c. Analysis Methodology
The methodology used to compute the probability of the target being within
the HRGK field of view, has previously been documented by the author. The following
paragraphs briefly summarize the methodology. In these paragraphs, the assumption is
made that if the target is within the HRGK's field of view, it will be acquired; therefore, the
desired quantity is the probability of acquisition, Pacq. The referenced technical
memorandum should be consulted for a more complete description of the theory and the
limitations of the simplifying assumptions. [Ref. 13]
Figure 19 shows a sample case for the probabilistic model and shows
several of the variables used in the model computations. To obtain the probability of the
target being within the seeker field of view (x, < xT< x2), the independent probabilities of
42
the rocket and target cross-tracks must be computed. The rocket cross-track computation
assumes a normal distribution of 9 — which depends only on the independent, normally
distributed pointing and ballistic flyout errors, Put and B, and the target's position depends
on the target motion model described above. The cross-track distribution for the rocket is
evaluated for the assumed range to detection (RD), and the target distribution is evaluated
for the assumed time at start of turn (f^ time required for the rocket to fly to RT).
Figure 19. Illustrative Case for Probabilistic Seeker Field of View Analysis. The rocket is
shown at the assumed position RD.
The probability of the rocket being at cross-track position xR is
1
p*w =
-x2
2a2
<7V2tt
where a = RDtan0 = RD^]tan2(Pnt) + B2 . Because of the assumed target distribution, the
target has equal probability of being anywhere in the possible region (ATOT) shown in
Figure 20. Therefore, the probability of the target being within the field of view limits (jc;
and x2)] — the shaded area, A, in the figure — is the ratio of the seeker acquired area, A, to
the total possible area, AT0T. The seeker acquired area, A, can be expressed as the area
between the limits (*; and x2) and inside the outer circle (radius = r2) minus the area
between the limits and inside the inner circle (radius = r,). The total possible area, AT0T, is
K{r2 ~ r\\> tnus' tne desired probability is given as:
The field of view limits are x, = xR - F and x2 = xR + F where the offset, F = Racq sin(FOV).
43
rTyXl,X2) -
"n
OT
sign(x2)min(r2\x2\) sign(x2)min(rl,\x2\)
2 j-sjr2-x2dx - 2 j^r2-x2dx
sign(x, ) min(r2 ,| xx \ ) sign(x^ ) min( r, ,| x, | )
where the limits of the two integrations are set to never exceed the radius of the outer or
inner circle, respectively.
possible target locations
within field of view
n = tr VT>
min
Atot
r2= trVjmax
all possible target locations
Figure 20. Possible Target Locations. The ratio of the area (A) bounded by
the field of view limits — x, and x2 — and the total possible area (ATOT) is the
probability of the target being within the two limits.
An alternative calculation of the seeker acquired area, A, is illustrated in
Figure 21. The calculation method relies on the fact that since the probabilities are
sy metrical about the y-axis, the calculations only need to be made for positive values of xR ;
thus x2 will always be positive. The method involves calculating the area, A , , bounded by
the circles, the lower field of view limit (x2), and the y-axis as given by the equation:
/] =min(r2,£2) L2=nuB(,r1,x2 )
A, =2 j^jr22-x2dx-2 j^jr2-x2dx
(
= L,Vr22-Z12+r22sin-ip -Z^^-^+l
(i \
2sin-!
Vr2
Li
Kr\J
Next, the area A2 is calculated. This area is bounded by the circles, the lower field of view
limit (jc,), and the y-axis, and is given by the equation
44
l} =min(r2,|j||) Lt =min(r, ,| X| | )
A2 = 2 \^r2-x2dx- 2 \ -^r2 - x2 dx
= L^^jr2 -L] + r2 sin ' — - L4 s\r2 - L24 + r2 sin '
' L ^
Finally, the area A2 is subtracted from A, if jc, is greater than zero (Figure 21 (a)) or added
to A, if*, is less than zero (Figure 21 (b)) — A = A, - sign(x,) A2.
£z
*<&&
^/r
®
;/ *
(b)^
1
r / /
r"Xi
Figure 21. Alternative Method for Computing Target Distribution Area within Seeker
Field of View Limits. (A = A, - sign(x,)A2.)
Now, the probability of the rocket being at a specific cross-track and the
target being within the HRGK's field of view can be expressed as the joint probability
Pacq(xR) = P(xR, x,<xT< x2) = PR{xR)P1{x1,x2). The total probability of the rocket having
the target in its field of view for any flyout is the integral of this joint probability. These
probabilities are illustrated in Figure 22. The actual method for computing the probability of
acquisition is described in the following paragraphs.
min. target
position, -rz
Extreme values
at x = ±(i2+F)
Pacq(Xfi) =
Pfl(Xfl)-Pr(Xi,X2)
• acq\X)
•acq = J 'acq \X)
(c)
Figure 22. Probability Density Functions, (a) Rocket cross-track error, (b) Target cross-
track position, and (c) Probability of acquisition.
45
d. Computer Code Description
The Matlab computer code developed to determine the probability of target
acquisition1 includes a Matlab script file and a Matlab function. (Appendix B briefly
describes the difference between script files and functions.) The script file sets the user
inputs, determines the rocket's cross-track error standard deviation, sets the seeker offset
(F), calculates the integral of the joint probability \P R{x) P j{x±F)dx from zero to an upper
bound, and plots the results. The Matlab function is called by the integration scheme in the
script file and returns values for the joint probability (Pacg(x) = P/?(x)P7(;t±F)). These
functions are all performed over a range of fields of view and for several seeker acquisition
ranges. Both the script file and the function are included in Appendix C.
e. NPS HRGK Results
The computer code results for the NPS HRGK design case are shown in
Figure 23. The results are for the maximum launch range case (6 km) as this is the most
stressing case. All other input parameters are set as in previous analyses. The target speed
is uniformly distributed between 20 to 60 miles per hour. The results show that a 95%
probability of having the target in the field of view at time, ?p can be obtained with fields of
view slightly smaller than those that would be required for the same results against
stationary targets. For example, with a seeker acquisition range of 1.5 km, a ±25° seeker
would provide a 95% probability of acquisition against the probabilistic target model. But,
against a stationary target, a ±29° seeker would be required for the same confidence (Figure
17, page 40). The probabilistic moving target model results in a smaller field of view
because of the possibility of the target moving in the same direction as the rocket cross-
track error is allowed. This possibility increases the probability of having the target within
the field of view, even with a slightly smaller field view.
D. COMMENTS AND OBSERVATIONS
In concluding this chapter on preliminary sizing analyses, several observations are
appropriate. These include (1) a discussion of the benefits and data that came from the
computer models, (2) a word of caution about the limitations of the models, and (3) a
listing of additional work that could be done to expand the usefulness of the preliminary
sizing models.
Actually the probability of the target being within the cross-track limits of the seeker's field of view.
46
Figure 23. Probability of Target in Field of View for Maximum Launch Range (6 km).
1. Additional Data from Modeling
The primary purpose of the computer models discussed in this chapter was to give
the designer a better understanding of how the system requirements affect different design
specifications. However, the codes not only facilitate the "flow-down" of requirements, but
they also provide additional insights into the design of the system. In the case of the NPS
HRGK design, several observations were made from the use of the preliminary sizing
codes. The following paragraphs site some examples.
From the length, weight, and center of gravity analysis it was determined early in
the design that the launcher center of gravity constraint is not binding for any design within
the maximum weight and length constraints. Thus, the kit's center of gravity can be placed
to provide the best combination of performance and packaging without concern for the
launcher's center of gravity. In a general case, a designer can use this type of knowledge to
move through design iterations without making calculations regarding the non-binding
constraints. Then, as the final design is approached, calculations can be made to verify the
preliminary assessments.
Another good example of additional information gained from the models is the
minimum required seeker acquisition range. The required seeker range was inadvertently
47
discovered by trying several seeker acquisition ranges in the maneuverability analyses. The
final seeker range specification could be derived based on any of the three target motion
types (stationary, worst-case, or uniformly distributed).
The maneuverability and field of view analyses provided other general insights into
the effect of the system requirements. For example, from the analyses it became apparent
that the minimum launch range requirement defines the maneuverability specification and
that the maximum launch range requirement drives the seeker field of view specification.
This insight is useful in determining design tradeoffs with relaxed system requirements.
For example, if the minimum range requirement for the NPS HRGK were increased to 2
km, the effect on the seeker would be minimal, but the affect on the thruster design would
be significant. Knowing this the designer can determine the value of relaxing the system
requirement.
2. Limitations of Models
Despite the usefulness of the codes discussed in this chapter, some caution must be
applied when using these and other mathematical or computer models. Model outputs are
sometimes skewed or erroneous because of (1) the nature of the model with its simplifying
assumptions or (2) because of the model's sensitivity to incorrect inputs. Examples of both
these error sources are abundant in the NPS HRGK analyses.
For example, both the maneuverability and the field of view analyses used a two-
dimensional model for a three dimensional problem. Implied in these models is the
assumption that the azimuth-related maneuverability and field of view requirements
outweigh the elevation-related requirements. In reality, the HRGK must execute significant
maneuvers in both yaw and pitch (at the same time) to prosecute a target as it appears in the
seeker's field of view. Furthermore, the field of view sizing may well be defined by the
requirement to look down at a target rather than the requirement to look to the left or right
for a target. These possibilities are not accounted for in the two-dimensional model.
Another example of potentially incorrect modeling concerns the appropriateness of
the target motion model. The constant heading and speed with uniform distributions may be
an appropriate target motion model for short time of flight attacks against sluggish ground
targets. However, this type of target motion model would not be appropriate in a surface-
to-air scenario where the target may have evasive maneuver capability. In general, any
target motion model must allow for a reasonable, worst-case evasive maneuver.
48
I I I
' ''-,!.. '
0.4 0.6 0.8 1.0 1.2 1.4 1.6
Mach Number
0.4 0.6 0.8 1.0 1.2 1.4 1.6
Mach Number
Figure 24. Sensitivity Analysis of the Steady-Turn Maneuver Model for Thruster Sizing,
(a) CM ± 100% and (b) XCP ± 3%.
Finally, Figure 24 illustrates the importance of sensitivity analysis and the potential
for erroneous results due to incorrect inputs. The figure shows data from the control
thruster sizing analysis with variations in the steady-turn model parameters. The plot on the
left (a) shows the effect of ±100% changes in CM . Doubling or zeroing the parameter
produces only minor changes in required control thrust. Clearly, the thrust required is very
insensitive to changes in CM . On the other hand, the plot on the right shows the impact of
±3% changes in xCP. The small changes in this input parameter affect the required control
thrust significantly.
Another point of interest illustrated in the above example is the importance of
parameter selection. In the NPS HRGK case, the parameter xCP is measured from the nose
of the guided rocket and varies between 28.5 and 31.8 inches as a function of Mach
number. Small percentage changes in xcp result in large percentage changes in the rocket's
static margin (the small distance between the xcp and the xCG). In fact, the ±3% change in
xcp produces between a ±22.7% and ±86.5% change in static margin. Although either xCP
or static margin could be used as model inputs, the model sensitivity to percent changes in
the two parameters would vary significantly.
3. Additional Work
The computer models developed by the author could be modified to produce new or
refined models. The addition of probabilistic target motion to the turn rate and turn radius
code would provide useful data. The analysis of elevation plane considerations would also
be helpful in the guided rocket design. A model that could estimate both the required
vertical field of view and pitch axis maneuverability would help in verifying the thruster
49
50
IV. SIX-DEGREE OF FREEDOM SIMULATION
This chapter addresses the use of detailed computer modeling in the missile design
process. The chapter describes a six-degree of freedom (6DOF) simulation developed by
the author for the analysis of the NPS HRGK. The HRGK 6DOF is a major modification
to a simulation developed by Professor Robert (Gary) Hutchins, Electrical and Computer
Engineering Department, Naval Postgraduate School, and his 1996 "Navigation, Missile,
and Avionics Systems" class. The simulation was developed in Simulink using Matlab
functions to perform the simulation's computations. The listings of the Matlab functions
are included in Appendix C.
Currently, the 6DOF code requires more than 20 minutes to simulate a 15 second
rocket flight1. To speed up the simulation, some of the simulation features are not yet fully
implemented, but the author is in the process of converting the simulation to a more
efficient computation structure. This structure uses the native Simulink "S-function" and
was introduced to the author by Mr. Allen Robins; Dynamics and Controls Section, Naval
Air Warfare Center Weapons Division, China Lake, California. (Appendix B briefly
describes the different computation structures the can be used in Simulink.) Preliminary
tests by the author have shown that the use of the S-function structure can cut simulation
run times by a factor of nearly 30.
This chapter begins with an overview of the simulation architecture and then
addresses the functions of the major simulation components. A section of the chapter is
devoted to the development of the guided rocket equations of motion. And another,
presents simulation results used in the design of the NPS HRGK. The chapter ends with a
few observations concerning the use of detailed simulations in the missile design process.
A. SIMULATION OVERVIEW
The HRGK 6DOF simulation is designed to be used with an initialization script file.
This Matlab file is executed prior to running the simulation. It sets up simulation
parameters such as vectors with the rocket's aerodynamic coefficients (functions of Mach)
and time profiles for mass properties and the main rocket thrust. The file also sets the initial
conditions for all the simulation states. When run, the simulation uses a user-selectable
1 Silicon Graphics, Inc. Indigo II workstation
51
integration scheme to simulate the rocket flight. The simulation writes several output
vectors to the Matlab workspace. These outputs include the flight time, target miss
distance, the guided rocket's states (U, V, W, P, Q, R, North, East, Altitude, (p, 6, and
y/), and their time derivatives with respect to the appropriate coordinate frames.
Figure 25 shows a top-level block diagram of the HRGK 6DOF. Figure 26 and
Figure 27 respectively show detailed views of the "Rocket Dynamics" and the "Parameters
and Coefficients" blocks as they are currently implemented.
0-
Clock
55
time
flight time
2.75" GUIDED ROCKET SIMULATION
TgtN
on
0*Rlce
timer noise timer
foM-'
Guidance
Delay
Switch
off
WsT"*^. : phi
SkVNofses ; v\
r... ,,
angle
Off Bore
TgtE
Tgth
Target
Dynamics
' Sj
Angle
'•j
Thruster
Cmds
Controller
Seeker /
Filter
's>
MslN»
MslE
Mslh,
phi skii
theta ,
psi
V
LOS's
True Horiz Line of Sight
Wind Gust Models
Rocket True Geometry
Dynamics
True Vert Line of Sight
| Dmiss |
Miss Dist
Demux
Figure 25. HRGK 6DOF Simulation Overall Block Diagram. (Dashed blocks are not yet fully
implemented.)
The blocks in Figure 25 with dashed lines are not yet fully implemented. For
example, the thruster control inputs are not currently generated based on seeker inputs, but
rather, the thruster commands must be "hard coded" into the conttoller block.
Inside the Rocket Dynamics block, the time derivatives of the rocket's states are
computed by a Matlab function. The state derivatives are then integrated and feedback into
52
the function and other parts of the simulation. The rocket's aerodynamic and mass property
parameters are determined using lookup tables in the Parameters and Coefficients block.
Thruster 1
1
Thruster 2
T1
Thruster 3
T2
T3
Thrust dat
Thrust History
Rkt X
Rocket State Vector
Rocket Rate Vector
Rkt Xdot
Matlab
Function
1/s
ROCKET DYNAMICS
FlatEarth
MotionEqns
Integrator
Mach
■£:
*>-T-
Matlab
Function
Mach No.
Params & Coefs
u.v.w,
P.Q.R
phi
c
theta
X
E
CD
Q
* 6
-^.P_rocket
1
Pn
2
Pe
3
Rocket
States
Figure 26. Detailed View of the Rocket Dynamics Block.
X
Z3
5
* — I Mass_dat
I IME & MACH
1
Mass History
*-| CG.dat |
CG History
DEPENDENT
— DADAft/ICTCDC
Mach
V
rArlAlVIt I tno
*— | JxR_dat
Rocket Jx
« — | JxK_dat
Xcp
h
/
Guidance Kit Jx
*^ C
IdO-Bun
ling
1
4 — I Jy_dat
^A
Param&
Coefs
Jy History
«a — | R_Pdot
*^
Thrust
X
rocket roll accel
A
c
dO-Coas
sting
CNa
/
/
CMq
Clp
Figure 27. Detailed View of the Parameters and Coefficients Block.
53
Figure 28 shows the overall architecture for the simulation currently under
development. Most of the components of this architecture are described in the following
section.
*— »
target motion
Geomeiry
Environment |
• •• i
—
relative
LOS
translation
matrix
absolute
positions
clock
atmosphere
gravity
wind
Missue r-
parameters &
coefficients
non-linear
equations
of motion
^
Mach
i
i
I
i i
I
-
seeker
forces &
moments
i
control
logic
IMU
ii
i
Figure 28. Future 6DOF Architecture.
B.
SIMULATION COMPONENTS
The simulation shown schematically in Figure 28 is made up several components.
These components are very briefly described in the following subsections.
1. Environment
The environment block simulates the environment in which the rocket operates.
This block produces the simulation time and provides constant gravity. It also computes the
atmospheric properties needed to calculate the Mach number and dynamic pressure. The
block uses the Dryden wind gust model to produce wind gusts in pitch and yaw.
2. Missile
The missile block simulates the dynamics of the guided rocket. The block has
several sub-components. These blocks are divided to align as closely as possible with
actual subsystems. The missile block uses external signals and data to produce realistic
seeker and IMU signals. From these signals control inputs are determined and applied to
54
the forces and moments sub-component. The forces and moments component computes all
the body frame forces on the rocket. From these forces and moments the dynamics of the
body can be determined by the flat-earth equations of motion.
3. Target Dynamics
The target block provides the target location and motion to the relative geometry
block. This data allows the line of sight (between the rocket and the target) and the line of
sight rate data to be calculated.
4. Relative and Absolute Geometry
The relative geometry block uses the target and rocket dynamics to solve the
navigation equations (described in the following section). From the navigation equations,
the absolute and relative position of the rocket and target can be can be computed. The line
of sight and line of sight rates are also provided by the geometry block.
C. EQUATIONS OF MOTION
This section contains a brief derivation of the flat-earth equations of motion as they
apply specifically to the HRGK 6DOF. The derivation is adapted from Stevens and Lewis
[Ref. 15].
1. General Matrix-Form Equations
Starting with generic flat earth equations of motion:
F
vB = -£2Bvg + BBg'Q + — (Force Equation)
m
cog = —J~ QgJ(dg + J~ TB + HR (Moment Equation)
6 = E(Q>)(£>B (Attitude Equation)
T
Pned = ^ByB (Navigation Equation)
where the vB is the velocity vector, [U V W]T; gg is the gravity vector at the surface of the
earth (adjusted for the earth rotation), [0 0 9.81m/s]T; FB is the force vector, [Fx Fy FJT;
coB is the angular rate vector, [P Q R]T; TB is the torque vector, [L M N]T; HRi to be
defined later, is a vector to account for the gyroscopic effects of the spinning rocket motor;
O is the Euler angle vector, [<J) 6 V|/]T; pNED is the north-east-down frame position vector,
[pN pE pD]T; Qq is the body frame angular rates cross-product matrix,
55
nB =
BB is the rotation matrix from the North-East-Down frame to the body frame,
0
-R
Q~
R
0
-p
-Q
p
o _
BB =
10 0
0 cos0 sin</>
cos0 0 -sin0
0 1 0
0 -sin0 cos0_||_sin0 0 cos0 J 0
J is the inertia matrix (with symmetry assumed about the x-axis),
cosi/a sini/f 0
-sini/f cosy/ 0
0 1
J =
0
0
0
0
0
0 /,
and the Euler angle function is
£(0) =
1 tan0sin0 tan#cos0
0 cos0 -sin<p
sin0 cos(f>
cosO cos 6
The state vector is [vBT coBT FT pNEDT]T. Now the forces and moments need to be defined
for the HRGK case.
2. Forces and Moments
The aerodynamic forces and moments are functions of Mach number and can be
expressed by the coefficients and their derivatives. The following summarizes the forces
and moments computed in HRGK 6DOF:
/
Fx=T-qA
r2
CD +
V "a
Fy = Ty+qA(-CNaP)
(Axial Force)
(Side Force)
(Down Force)
(C, d \
L = qA d
\ 113
+ TR (Rolling Moment on HRGK)
56
M = TZXT + qAd
N = TyXT + qAd
Cm OCXrp +
'N,
CP
CMad
— —Q
w
\
Cm d
Cm uXcp H R
\
(Pitching Moment)
(Yawing Moment)
where T is the rocket motor thrust; Ty and Tz are the y and z components of the reaction
thruster forces, Tz = r,cos0 + T2cos(<(h-2/3k) + r3cos(</H-4/37c) and T = T^siiKp +
r2sin( 0+2/371) + r3sin(0+4/37t); and TR is the rolling torque exerted on the HRGK by the
spinning rocket motor,
TR = idXRPR
where fi is the coefficient of friction for the bearings, Jx is the axial moment of inertia for
the rocket alone, and PR is the rocket motor's spin axis angular acceleration (adapted from
Ref. [11]). The state PR has to be added to the state vector, and the gyroscopic affect of the
spinning rocket is added to the moment equation. The effect can be accounted for by adding
the vector HR to the right-hand side of the moment equation, where HR is
T
H.=
0
,RJxBPR
Q'x.Pr
3. HRGK Scalar Flat-Earth Equations
From the general matrix equations, the HRGK-specific scalar form of the equations
can be written in terms of the defined forces and moments.
a. Force Equations
U = RV-QW-g0sine + ^L
m
V = -RU + PW + g0sm<l)cosd + ^-
m
W = QU-PV + g0 cos0cos0 + -^
m
Moment Equations
57
Q =
( j)
R =
'LA
A )
PR
J.. 7.
RPD
N Jx
Jy Jy
Kinematics Equations
0= P + tan 0(0 sin 0 + /? cos 0)
0 = QcosQ- R sirup
,COS0
i// = Qsirut) + R
cos 6
d. Navigation Equations
PN = £/cos#cosyf + V (sin 0 sin 6 cost// -cos 0 sin y/) + W(cos0sin0cosy/, + sin0siny/')
/^ = £/cos0siny/' + V(sin0sin0sini// + cos0cosy/) + W(cos0sin0siny/-sin0cosi//')
PD = -Usin6+ Vsin0cos# + Wcos0cos0
D.
NPS HRGK RESULTS
The force and moment equations, as well as the equations of motion, are
implemented in a Matlab function which is used by the HRGK 6DOF simulation. The
following subsections show two examples of how the simulation was used in the design.
1. Ballistic Flight and Maximum Range (3DOF)
Early in during the NPS HRGK design process, the HRGK simulation was
simplified to only include the three dimensions of the rocket's longitudinal dynamics. (The
gyroscopic effects of the spinning rocket were also ignored.) Using the simplified 3DOF,
the results shown in Figure 29 were obtained. These results confirmed that the HRGK-
equipped rocket could meet the time to target and the maximum range requirements.
2. Pulse Width Modulated Turn Rate (6DOF)
Later in the design, the affect of thruster pulse width modulation was investigated
using the HRGK 6DOF. Two simulations were run. The first simulated the missile flight
58
Figure 29. HRGK-Equipped Rocket Ballistic Trajectories. Ground launch at sea
level with various launch elevation angles. Solid lines are for cold rocket motor,
dashed lines are for hot rocket motor, and flight times are indicated.
with a constant thruster force from the appropriate side only. The second used a pulsed
sequence of all three thrusters, but with a longer pulse on the appropriate side. The results
for the two runs are shown in Figure 30. From the analysis, it was determined that constant
thrust does not generate satisfactory maneuvers. (In fact, in the simulation, the rocket
began "cartwheels" without making substantial changes in cross-track). On the other hand,
the pulsed control generated an adequate maneuver while maintaining a somewhat
reasonable side slip angle, /?.
X-Trk=12.1 m
Constant Thruster Maneuver
$ "$
// 11
to
c
n r\ r\ j.
it -^i— ^— ^
a
'a
0
"t, C?'"' W "»
CO
CL
p = 9.7° p = 29.2°
p = 52.1°
P = 79.7°
o
CO
h-
co
o
Pulsed Thruster Maneuver
X-Trk = 13.6 m
O
(with short opposing pulses)
P =
= 32.3°
p = 3.6° P = 12.1° P = 22.3°
0 50 100
1 50
200
Down -Track Range (m)
Figure 30. Sustained and Pulsed Thruster Maneuvers. Flight paths and headings indicated for
a 0.8 second maneuver.
59
E. OBSERVATIONS
As designs become more refined and leave the conceptual phases, high fidelity
simulations becomes more important. In the case of the NPS HRGK, the 6DOF simulation
was helpful in performing some detailed analyses. Nevertheless, the time required per
simulation run precluded the use of the simulation in many cases.
Fast, modular simulations that can be easily modified for various designs are
desirable design tools. In many cases, lower-order simulations (3DOF or 5DOF) meet all
the needs of the designer. For example, the 3DOF version of the HRGK simulation had
sufficient fidelity to be used in the maximum range and time to target analyses.
Unfortunately, other analyses required a 6DOF so that the roll of the seeker and the
gyroscopic effects of the spinning rocket could be observed.
If the restructuring of the HRGK simulation provides the expected improvement in
run time, this simulation would become a much more useful design tool. With run times of
only a minute or so, several design iterations could be evaluated quickly.
60
V. CONCLUSIONS AND RECOMMENDATIONS
The purpose of this thesis is to provide documentation for the several computer
codes developed by the author during the design and analysis of the NPS HRGK. The
thesis is also intended to provide insights concerning the use of modeling in the design and
analysis processes. Several germane points can be made in conclusion.
First, models with differing levels of complexity are appropriate during different
phases of the design. Models used in the analysis of conceptual and preliminary designs are
usually the simplest while those models used in design refinement are more complicated.
COEA-related modeling is essential throughout the design process to ensure that the design
meets the system's cost effectiveness goals.
Second, simplistic models can be used to ensure the flow down of system
requirements into design specifications. In this way, the models help ensure a design
driven by the requirements. Often these simple models provide great insight into additional
aspects of the design problem allowing several critical specifications to be defined in the
earliest phases of the design.
Third, modeling in the design process should include the appropriate mix of
probabilistic and worst-case conditions. These two types of models can provide very
different results which must be judiciously evaluated.
Finally, knowing the sensitivity of a model to variations in parameters is critical to
the proper use of the model in decision making. Sensitivity analyses or "what-if ' studies
must be performed to ensure the modeling is providing reliable data.
In addition to these concluding points several recommendations for follow-on work
are also appropriate. First, the design and analysis of the NPS HRGK would have been
facilitated by simple Matlab codes that allow the early aerodynamic sizing of the kit. Such
a program would be helpful in evaluating the effect of nose shape and canard size and
shape on the range of the HRGK-equipped rocket.
Additional work is also needed to include a probabilistic target motion model in the
maneuverability analysis code. The development of models that evaluate the required pitch
maneuverability and the required elevation field of view would also be useful.
Finally, the modification of the HRGK simulation to make it a generic 6DOF
simulation would be very useful for future missile design work. The Simulink architecture
can offer fast modular simulations that can be easily modified to meet future users' needs.
61
62
APPENDIX A. MISSILE AERODYNAMICS NOMENCLATURE
Many of the aerodynamic coefficients, stability derivatives, and various other terms and
symbols used in this thesis are defined below. The nomenclature closely follows that used
by Chin [Ref. 16] with some adaptations like that used by Blakelock [Ref. 17] and by
Howard [Ref. 18]. It is worth noting two significant differences between terms associated
with missiles and those associated with traditional aircraft. First, most missile coefficients
are referenced to the missile's diameter or its cross-sectional area (normal to the
longitudinal axis) rather than a wing chord or a planform area. Second, normal and axial
force coefficients, C^ and C^, (aligned to the body-axis system) are used rather than the
lift and drag force coefficients, Cl and Cr> (in the stability axis system). When the
missile's angle of attack (a) is small there is very little difference between the two systems,
but for significant angles of attack, the relationship given in the following equations should
be used.
Ca = Cd cos a - Cl sin a
CN = CL cos a - CD sin a
Reference Dimensions
A or Sn Cross-sectional area of missile body (5.94 in2)
d Diameter of missile body (2.75 in)
L Missile length (68.75 in)
Xcg Distance from nose tip to center of gravity on missile (27 to 3 1 .7 in)
Flight Conditions
a angle of attack
P side slip angle
M Freestream Mach number
Vt Freestream air speed
p Freestream air density
q Dynamic pressure ( 4 pVt )
63
Aerodynamic Coefficients*
CT Coefficient of lift,
qsn
N
CN Normal force (N) coefficient,
qsn
M
Cm Pitching moment (M) coefficient,
qSnd
C a Axial force (A) coefficient,
qsn
CDn Drag force coefficient at zero lift
Cjxj Coefficient of drag due to normal force (DN), — —
Static and Dynamic Derivatives*
op
Cw Normal force coefficient per radian of angle of attack, — —
a da
op
CM Pitching moment coefficient per radian of angle of attack, — —
a da
— — -M
d J da
Missile "States"
£/, V, W Body axis velocities (jc, v, and z-axes)
P, (>,/? Body-axis angular rates (about the x, y, and z-axes)
0, 9, i// Euler Angles (roll, pitch, yaw)
pn>pbpd Flat Earth Positions (North, East, Down)
h height {-PD)
Directional coefficients and derivatives {e.g.; CY, C^, CY , C ^ , C ^ ) are assumed equal to their
longitudinal counterparts {e.g.; CN, CM, CNot, CMa, CMq; respectively).
64
APPENDIX B. NOTES ON MATLAB AND SIMULINK
A. MATLAB
Matlab is a high-performance numeric computation and visualization computer
software package. It is essentially a high-level programming language that is especially
tailored for the manipulation of matrices, (see Matlab Reference Manual and Matlab
User 's Guide)
Matlab commands can either be invoked in an interactive mode at the keyboard or
through Matlab script files. Another type of file is the Matlab function. Both the script
and function files are labeled with an "m" extension at the end of the file name and are
commonly called M-files.
A script M-file can run as a stand alone "batch" file invoking Matlab commands
when the file is executed. On the other hand, a function M-file is called from within script
files or from the keyboard. A function can be called with or without arguments and can
return computed values when completed.
Variables declared or set in a script file or at the keyboard are available throughout
the Matlab "workspace." These variables are common and can be accessed by other script
files. On the other hand, variables declared in a function M-file are local variables and are
not available in the workspace. Furthermore, functions do not have access to workspace
variables unless the variables are passed to the function or are declared as "global" in both
the workspace and in the function.
The standard syntax used in the first line of a function M-file is
function y = funciu)
where "function" is the Matlab function indicator, y is the variable that will be passed
back, u is the input, and func is the name of the function (func.m).
Files that are written in C or FORTRAN can be converted into executable Matlab
files with a ".mex" extension by using the "cmex" or "fmex" commands, (see Matlab
External Interface Guide) Mex-files have the advantage of running faster than normal M-
files.
65
B. SIMULINK
Simulink is a dynamic system simulation software package and an extension to
Matlab. Simulink offers many ready-made blocks that can be assembled graphically to
model a dynamic system. Two special Simulink blocks allow tremendous flexibility in
modeling systems, (see Simulink User's Guide)
The first is the Matlab function block. This block when placed in a block diagram
will execute the commands in the indicated M-file function. So for example, the equations
of motion for a system could be placed in a M-file function. This function would have
inputs from simulink that include the states and the control input. The function would then
output the states' time derivatives which could then be integrated and then passed back the
Matlab function.
This method runs extremely slow!
The second more useful block is the S-function block. This block calls either a M-
file or a Mex-file that is specifically structured to run efficiently in Simulink (see Simulink
1.3 Release Notes). Chapter 4 of Simulink 1.3 Release Notes outline the procedure for
creating Mex-files from C codes for use in fast running Simulink models.
Additionally, simulink models that do not contain M-file references (that use Mex-
files in the S-function blocks) can be turned into stand-alone, accelerated, executable code
using Simulink's Real-Time Workshop (see Real-Time Workshop User's Guide).
66
APPENDIX C. MATLAB CODE LISTINGS
This appendix provides listings of the following Matlab computer codes which
were discussed in this thesis:
• Single-Shot Probability of Kill
• Length, Weight, and Center of Gravity
• Turn Rate and Radius
• Steady-Turn Maneuver
• Seeker Field of View
• 6DOF Codes
-Initialization Code
-Aerodynamics Data
-Equations of Motion
-Line of Sight
-Mach
The user-defined inputs in the codes are denoted with bold-face type.
C. SINGLE-SHOT PROBABILITY OF KILL
% Single Shot Pk using "Cookie Cutter" methodology
% W. Mark Wonnacott, Naval Postgraduate School, May 1997, corrected/revised Aug 1997
n%%%%nnHn%%iNPOTs
% Terminal Dive Angle (deg relative to horizontal)
dive « 20;
% CEPs to be used in computations (for a single CEP set CEP_min egual to CEP_max)
CEP_min = .5; % minimum CEP (m) to be used
CEP.max = 10 0; % maximum CEP (m) to be used
% Target Size (m) , Prob. of Kill given Hit, & Desired Prob. Kill
trgt='Light Arm. (AirDef)'
trgt=' Light Armor (APC) ' ;
trgt= ' Large Truck ' ;
L = 5.5; H = 2.0; W = 2.5; Pkh =.90; PkD =.95
L = 5.5; H = 2.0; W = 2.5; Pkh =.80; PkD =.60
L = 8.0; H = 2.5; W = 2.5; Pkh = 1; PkD =.50
L = 14.; H = 1.5; W = 4.5; Pkh =.70; PkD =.75; trgt='Patrol Boat'
%%%%%%%%%%%%%%%!. NP UTS
% Elevation and Crossrange Projected Dimensions (l:Head on & 2:Broadside)
Yl = H*cos(dive*pi/180) +L*sin(dive*pi/180) ;
XI = W;
Y2 = H*cos(dive*pi/180)+W*sin(dive*pi/180) ;
X2 = L;
%CEPs in desired range
if CEP_min<=0, CEP_min=eps; end %CEP_min set to allowable small number
if CEP_max<=0, CEP_max=eps ; end %CEP_max set to allowable small number
if CEP_min>CEP_max, temp=CEP_min; CEP_max=CEP_min; CEP_min=temp; end
67
CEPs = logspacedoglO (CEP_min) , loglO (CEP_max) ) ;
sigmas = CEPs/1 .1774; %corresponding sigmas, 1-D miss distances (m)
S = sigmas*sqrt (2) *2; %normalizd half sigmas for Matlab
% Probability of Hit (equal (50%) prob. of broadside or head-on encounter)
Ph = 0.5*(erf (Yl./S) .*erf (Xl./S) ) + 0.5* (erf (Y2. /S) . *erf (X2 . /S) ) ;
% Probability of Single Shot Kill
PkSS = Pkh*Ph;
% Number of Weapons Needed to Achieve Desired Prob. of Kill
N = log(l-PkD) ./log(l-PkSS) ;
figure (1) % Single-Shot Probability of Kill
semilogx (CEPs, PkSS, '-') , grid on
axis ( [CEP_min, CEP_max ,0,1])
xlabeK'CEP (m) ' ) , y label (' Single Shot Probability of Kill')
figure (2) % Number of Weapons per Kill
loglog (CEPs,N, ' -' ) , grid on
axis( [CEP_min,CEP_max, 1,1000] )
xlabeK'CEP (m) ' ) , ylabel ([' Number of Rockets for Pk = ' ,num2str (PkD) ] )
figure (3) % Relative Cost for Equal Cost/Kill
Nc = max (1, ceil (N) ) ;
[X,Y]=stairs(CEPs,Nc(D ./Nc) ;
plot (X,Y, '-' ) , grid on
axis ( [CEP_min, CEP_max, 0,1])
xlabel ( 'CEP (m) ' ) , ylabel ( 'Relative Weapon Cost' )
D. LENGTH, WEIGHT, AND CENTER OF GRAVITY
% GUIDED Rocket Center of Gravity as a Function of Kit Weight
%============================================================
% W. Mark Wonnacott, Naval Postgraduate School, May 1997
%%%% Design Space Paramters
Wkl = linspace ( 1, 8 ) ; % assummed kit mass range (lbm)
Lkl = [5,10, 13.625,Lmax-Lr] ; % specific kit lengths (in)
Dkl = [ . 05, . 07, . 0865, . 1, . 15] ; % specific kit densities (lbm/in~3)
Cgkl= 1/3; % kit Cg from nose (fraction of kit length)
dia = 2.75; % kit diameter (in)
area> pi* (dia/2 ) A2; % kit cross section area (in~2)
68
%%%% Constraint Parameters
Lmax= 72; % max total length
Wmax= 30; % max total weight
CgL = 1/3; %pod forward Cg limit (frac of lug spacing back from front lug)
%%%% Rocket Parameters
Lr = 55.125;
CgR_live = 29.96;
CgR_burn = 33.55;
WR_livo = 22.95;
WR_burn = 15.73;
% rocket length
% live rocket Cg (inches from base)
% fired rocket Cg (inches from base)
% live rocket mass (lbm)
% fired rocket mass (lbm)
%%%% Launcher Pod Parameters
Llug7 = 43.2;
Llugl9= 43.4;
Cg7 = Llug7-(CgL)*14;
Cgl9=Llugl9- (CgL) *14;
Cgpod = 32.6;
Wpod7 = 196.2,
Wpodl9= 516.0;
% front lug from back of launcher— 7 (in)
% front lug from back of launcher— 19 (in)
% constraint cg (e.g. within middle half of lug space)
% constraint cg (e.g. within middle half of lug space)
% loaded pod Cg from back w/o kits (in)
% loaded pod (7) mass w/o kits (lbm)
% loaded pod (19) mass w/o kits (lbm)
% (VARIOUS DENSITY CURVES PLOTTED)
% ================================
Wk = Wkl' *ones(l, length (Dkl) ) ; % matrix of kit masses
Lk = Wk(: ,l)*(l./(area*Dkl) ) ; % matrix of kit lengths (in)
Cgk= Lr+(1-Cgkl) *Lk; % kit Cg locations (inches from base)
Mk = Wk.*Cgk;
% Moment about base due to kit (lbm in)
% centers of gravity are calculated from base in inches
Cg_live = (WR_live*CgR_live+Mk) ./ (WR_live+Wk) ;
Cg_burn = (WR_burn*CgR_burn+Mk) . / (WR_burn+Wk) ;
% centers of gravity are changed to calibers from nose
Cg_live = ( (Lr+Lk) -Cg_live) /dia;
Cg_burn = ( (Lr+Lk) -Cg_burn) /dia;
%figure
figure (1) ,
plot (Wk(:,l), Cg_live) , grid on
title ('Live Round with Kit')
ylabel ('Cg (Calibers from nose) ' )
xlabel ('kit weight (lbm)')
hold on
69
figure (2) ,
plot (Wk( : , 1) , Cg_burn) , grid on
title ("Fired Round with Kit')
ylabel ('Cg (Calibers from nose) ' )
xlabel ('kit weight (lbm)")
hold on
% (VARIOUS LENGTH CURVES PLOTTED)
% ===============================
Lk = Lkl;
Dmin = 0. 8*min(Dkl) ;
Dmax = 1.2*max(Dkl) ;
Dk = linspace (Dmin, Dmax) ;
Dk = Dk ' *ones ( 1 , length ( Lk) ) ;
Wk = area*Dk( : , l)*Lk;
Cgk= Lr+(l-Cgkl)*Lk;
Cgk= ones ( length (Wk) , 1 ) *Cgk;
% specific kit lengths (in)
% lower limit on assumed density range
% upper limit on assumed density range
% assummed kit density range (lbm/in"3)
% matrix of kit densities
% matrix of kit masses (lbm)
% kit Cg locations (inches from base)
% matrix of Cg locations
Mk = Wk.*Cgk;
% Moment about base due to kit (lbm in)
% centers of gravity are calculated from base in inches
Cg_live = (WR_live*CgR_live+Mk) . / (WR_live+Wk) ;
Cg_burn = (WR_burn*CgR_burn+Mk) . / (WR_burn+Wk) ;
% centers of gravity are changed to calibers from nose
Cg_live = ( (Lr+ones (length (Wk) , 1) *Lk) -Cg_live) /dia;
Cgjourn = ( (Lr+ones ( length (Wk) , 1) *Lk) -Cg_burn) /dia;
figure (1) ,
plot (Wk, Cg_live)
plot ([ (Wmax-WR_live) *ones(l,2) ], [1,20] ) %max weight constraint
figure (2),
plot (Wk , Cg_burn)
plot ( [ (Wmax-WR_live) *ones(l,2) ] , [1,20] ) %max weight constraint
% (POD CENTER OF GRAVITY CONSTRAINT)
% ==================================
Wk = Wkl ' *ones (1, 2) ; % matrix of kit masses
%length from loaded pod (w/o kits) cg to constraint Cg (in)
LI = ones ( length (Wk) , 1) * ( [Cg7,Cgl9] -Cgpod) ;
% loaded pod weights (lbm)
Wp = ones ( length (Wk) , 1) * [Wpod7,Wpodl9] ;
X = ones (length (Wk) ,1) * [7, 19] ; % number of kits per pod
% length from rocket nose back to front lug
% (assume rocket base 4" from back of pod)
Lt = ones ( length (Wk) , 1) * [Lr-Llug7+4,Lr-Llugl9+4] ;
70
% Solve for allowable kit length
Lk = (1/ (1-Cgkl) ) * ( (LI. *Wp) ./ (X.*Wk)-Lt) ;
Cgk= Lr+ (1-Cgkl) *Lk; % kit Cg locations (inches from base)
% center of gravity calculated from base in inches
Cg_live = ( (WR_live*CgR_live) + (Cgk. *Wk) ) . / (WR_live+Wk) ;
Cg_burn = ( (WR_burn*CgR_burn) + (Cgk . *Wk) ) . / (WR_burn+Wk) ;
% center of gravity changed to calibers from nose
Cg_live = ( (Lr+Lk) -Cg_live) /dia;
Cg_burn = ( (Lr+Lk) -Cg_burn) /dia;
figure ( 1) ,
plot(Wk,Cg_live)
axis([1.8,7.2,7.8,14])
%hold off
figure (2) ,
plot (Wk,Cg_burn)
axis( [1.8,7.2,6.8,13] )
%hold off
E. STEADY-TURN MANEUVER
% This m-file computes thrust required, T, for certain load factor, n
% (g's) , in a steady turn for various Mach numbers. The load factor is
% achieved using a thruster forward of the CG.
%
% The inputs include:
% Scenario Definition:
% n: desired load factor (g's)
% rt : desired turn radius (ft)
% alt: altitude (ft) (std day assumed)
% Mmin: min Mach number
% Mmax : max Mach number
% Rocket Configuration:
% Wght : rocket mass (lbm)
% Dia: rocket cross-section diameter (in) (for ref area and length)
% Xcg: rocket center of gravity in calibers from nose
% Xt : thruster location in calibers from nose
% Aerodynamic Parameters:
% CNa : normal force coefficient per radian of alpha [f(Mach)]
% Xcp : center of pressure location in calibers from nose [f(Mach)]
% CMq: moment coefficient per rad/sec of pitch rate [f(Mach)]
% (Aero Paramaters are linearly interpolated between points in table)
71
% Code uses normal force and pitch moment equations:
% Fz=nW=N+T= (Qbar) *S*CNa*alpha + T, and
% m = 0 = -N*(Xcp-Xcg) + T*(Xcg-Xt) - CMq*q* (Dia/2V) *Qbar*S*Dia
% {since, q = theta-dot = V/R = V/(V/v2/n) = n/V
% and N = (Qbar)*S*CNa*alpha}
% M =(Qbar*S*CNa*alpha) (Xcp-Xcg) + T* (Xcg-Xt) - CMq*n* (Dia~2/V~2) *Qbar*S
% The two equations are solved for the unknowns, alpha and T, using several
% values of Mach number.
% W. Mark Wonnacott, Naval Postgraduate School, May 1997
clear, close all
% INPUTS
%
% Scenario Definition
n = 15;
Rt = 6000;
req_type =
alt = 500;
Mmin= 0.4;
Mmax= 1.6;
2;
% desired load factor (g's)
% desired min turn radius (ft)
% requirement (l:turn rate, 2: turn radius)
% std day pressure altitude (ft MSL)
% minimum Mach number for calculations
% maximum Mach number for calculations
% Rocket Configuration (based on burned-out, 30 lbm, 68.75" rocket)
Wght = 22.78; % rocket mass (lbm)
Dia = 2.75; % reference length (in)
Xcff =10.0; % CG in calibers from nose
Xt = 2.2; % thruster location in calibers from nose
% Aerodynamic Parameters (based on final config 68.75"-w/ 3 case#8 strakes)
%
aero table =
Mach#
[0
0 .6614
866
9682
0308
118
1.25
1.4142
1.6008
1.8028
2.0156
Mach_dat= aero_table ( : , 1 ) ;
CNa_dat = aero_table ( : , 2 ) ;
Xcp_dat = 1 . 03 *aero_table (
CMq_dat = aero_table ( : , 4 ) ;
CNa Xcp CMq
15.0616 11.52547 -4166.814
16.41589 11.32152 -4556.4946
18.10928 11.01214 -5041.0303
20.04303 10.36514 -5518.2495
20.05156 10.35951 -5519.5352
20.05156 10.35951 -5519.5352
18.91632 10.45431 -4859.3901
17.51701 10.5577 -4376.4146
16.10659 10.68451 -3938.8662
14.68889 10.88046 -3541.0081
13.47025 10.99184 -3202.1553
12.38573 11.05396 -2916.2678];
% Mach numbers
% Body Normal Force per Radian Alpha
,3); % Center of Pressure in calibers from nose
% Pitching Moment per rad/sec of Pitch Rate
72
% CONSTANTS
gO = 32.17;
TempO = 518.67;
PresO = 14.696*144;
B_atm = 0.003566;
R_air = 53.35;
k_air = 1.4;
% Gravity Constant.
%surface temperature (°R)
%surface pressure (lbf/ft'>2)
%temp lapse rate (°R/ft)
%gas constant (ft lb/(lbm °R) )
%specific heat ratio
% INTERMEDIATE CALCULATIONS
% Speed of Sound
temp = TempO -B_atm*alt;
pres = PresO* ( 1-B_atm*alt /TempO )" (1/R_air/B_
rho = pres/ (R_air*temp) ;
Vs = sqrt(k_air*R_air*temp*gO) ;
% Rocket Config
d = Dia/12; %
S = pi*d~2/4; %
XT = (Xcg -Xt)*d; %
XP_dat = (Xcp_dat-Xcg)*d; %
Fz = n * Wght; %
% Mach numbers and speeds
% adjust min and max Mach to table values
Mmin = max (min (Mach_dat ), Mmin) ; %
Mmax = min(max(Mach_dat) ,Mmax) ; %
kmax =50; %
Mach = linspace (Mmin, Mmax, kmax) ; %
% temperature at altitude (°R)
atm) ; %pressure @ alt (lb/ft"2)
% density at altitude (lbm/ft~3)
% speed of sound (ft/sec)
reference length (ft)
reference area (ft~2)
thruster location from CG (ft)
center of pressure location from CG (ft)
normal force (lb)
min Mach number to be used
max Mach number to be used
number of Mach numbers to be used
vector of Mach numbers to be used
% THRUST AND ALPHA CALCULATIONS FOR EACH MACH NUMBER
for k = 1 : kmax
V = Mach(k)*Vs;
OS = S * l/(2*gO)*rho*V"2;
CNa = interpl(Mach_dat,CNa_dat,Mach(k) )
Xcp = interpl(Mach_dat,XP_dat,Mach(k) ) ;
CMq = interpl(Mach_dat,CMq_dat,Mach(k) )
A = [QS*CNa, 1;
-QS*CNa*Xcp, XT] ;
if req_type == 2,
B= [(V~2*Wght)/(gO*Rt) ;
(1/2) *QS*CMq*d~2/Rt] ;
else
B= [Fz; (l/2)*QS*CMq*n*gO*(d/V)"2];
end
X = inv(A)*B;
a(k) = X(l)*180/pi;
T(k) = X(2);
end
% rocket speed (ft/s)
% (dyn pres-lb/ft"2) (ref area-ft"2) (lb)
%CNa for Mach#
%Xcp for Mach#
%CMq for Mach#
% RHS vector for turn radius
% RHS vector for turn rate
% solution vector
% alpha in degrees
% required thrust in pounds
73
subplot (2,1,1)
plot (Mach,T, 'r ' ) , grid on
xlabel ( ' Mach Number ' )
ylabeK 'Required Thrust (lbf) ' )
hold on
subplot (2,1,2)
plot (Mach, a, 'r ' ) , grid on
xlabel ( 'Mach Number' )
ylabeK 'Required alpha (deg) ' )
hold on
F. SEEKER FIELD OF VIEW
% This m-file calculates the probability of a target being in the
% seeker field of view, FOV (deg) , for for several seeker acquistion
% ranges (Racq) and fields of view. Only single launch range is used.
%
% inputs include: Launcher Pointing Error, Pnt (deg) - 1 sigma
% Missile unguided ballistic error, B (mils) - 1 sigma
% Missile distance time profile, (km) (function of fly out time)
% Guidance time delay, Gd (sec)
% Time delay for target ID, td (sec)
% Min and Max Target Speed (mph)
% also required are:
% Seeker acquistion ranges (km) - (vector)
% Min and Max FOV (deg)
% Launch Range (km)
% Mark Wonnacott, April 1997
% Naval Postgraduate School
clear, close all
%% Target Description
VTmin =20; % (mph) min target speed
VTmax =60; % (mph) max target speed
%% Missile Configuration Inputs
% Launcher Pointing Error
Pnt =3; % (deg- 1 sig) initial pointing error
% Unguided Missile Ballistic Error
B = 12; % (mills- 1 sig) meters error per km flight
% Guidance Delay
Gd = 1.0; % (sec) max range delay (no guidance for 1 sec)
% Delay time to ID target
td = 5*1/8; % (sec) 5 laser pulses received at 8 Hz
74
% Seeker Field of View
FOVmin = . 5;
FOVmax = 45;
% Seeker Acquisition Range
Racq = [1.15,1.5,2,3,6] ;
% Launch Range
RL = 6;
% Missile Speed Table
Time = [0,0.22,0.89,1.02,1.67,... %flight time (s)
2.52,4.96,7.91,11.24,14.88,18.8,22.99,27.3] ;
Dist = [0,0.01,0.2,0.27,0.6,1,2,3,4,5,6,7,8]; % flight dist (m)
% (deg) min half FoV
% (deg) max half FoV
% (km) parametric seeker acquistion ranges
% (km) maximum launch range
%% Inputs converted to usable variables
VTmin2 = VTmin/2237;
VTmax2 = VTmax/2237;
PntR = Pnt*pi/180;
Bkm = B/1000;
kmax = length (Racq) ;
mmax = 30;
FOV = linspace (FOVmin, FOVmax.mmax) *pi/180;
Rg = interpl (Time, Dist, Gd) ;
% min target speed (mph->km/s)
% max target speed (mph->km/s)
% initial pointing error (deg->rad)
% X-trk error (km per km flight)
% number of seeker acq. ranges
% number of FOV's
% FOV's (deg->rad)
% unguided flyout range (km)
Pacq = zeros (kmax, mmax) ;
% pre-set Pacq matrix size
for k = 1 : kmax
for m = l:mmax
D = Racq(k)*cos(FOV(m) ) ;
F = Racq (k)*sin (FOV (m) ) ;
if RL-Rg > D
Rd = RL - D;
else
Rd = RL - Rg;
end
tD = interpl (Dist, Time, Rd) ;
tT = tD + td;
Rt = interpl (Time, Dist, tT) ;
s = Rt*sqrt (tan (PntR) A2+(BkmP2)
rl = tT * VTmin2;
r2 = tT * VTmax2;
% calculations for each Racq
% calculations for each FOV
% detection range (trgt @ FOV limit)
% max x-trk offset for FOV
% range flown before detection
% range flown when guided flight begins
% time at detection
% time at start of turn
% distance flown at start of turn
%1 sig X-trk error at start of turn
% min target motion at start of turn
% max target motion at start of turn
L5 = min(5*s, r2+F) ; % integration upper limit
%%% Integration of Pr (Xr) *Pt (Xr-F,Xr+F)
%%% from 0 to L5 done using 'quad8' (Pacq function in 'ProbAcq')
Pacq(k,m) = 2*quad8 ( 'ProbAcq' , 0,L5, 0.01, [] ,F,s,rl,r2) ;
end
end
75
FOV = FOV*180/pi; % field of view changed back to degrees
figure
plot (FOV, Pacq), grid on
title (['Vt = [ ' ,num2str (VTmin) , ' -> ' , num2str (VTmax) , ' ] mph'])
legend (num2str (Racq(l) ) ,num2str (Racq(2) ) ,num2str (Racq(3) ) , num2str (Racq(4) ) , num2str (Rac
q(5)))
ylabel ('Probability of Target in FOV),
xlabeK'half FOV (±deg) ' )
ProbAcq Function
function Pacq = ProbAcq(Xr,F,s,rl,r2)
% Returns the product of the assumed probability of the rocket being
% approximately at a certain cross-track position, Xr, and the
% probability of the target being between ±F to either side of
% the rocket position (seeker's FOV limits).
% The probabilities are based on the 1-sigma rocket cross-track error— s
% and the min and max possible target cross-track motion— rl and r2 .
%
% This function has to take Xr as a vector and output Pacq as a vector! ! !
% W. Mark Wonnacott, Naval Postgraduate School, 1997
if rl == 0; rl = 0.0001; end % avoid division by zero
Prkt = l/(s*sqrt(2*pi) ) *exp(-Xr.~2. / (2*s~2) ) ; %normal distribution of rocket
% Find probability of target in FOV extremes (Pt(xl,x2)) for each Xr
for k = 1: length (Xr)
x = Xr(k); % step through each rocket x-trk position
XI = x - F; % extreme bound of FOV
X2 = x + F; % extreme bound of FOV
if XI > r2 % weapon farther out than possible for target
Ptgt(k) = 0;
elseif XI < -r2 % all possible target locations inside FOV
Ptgt(k) = 1;
else
% compute area bounded rl & r2 circles, fly-out axis, and X2
LI = min(X2,r2); % limit of intgrt'n for r2 circle
L2 = min(X2,rl); % limit of intgrt'n for rl circle
Ala = Ll*sqrt(r2~2-Ll~2) + r2~2*asin(Ll/r2) ; %bounded area in r2 circle
Alb = L2*sqrt(rl~2-L2~2) + rl~2*asin(L2/rl) ; %bounded area in rl circle
Al = Ala-Alb;
76
% compute area bounded rl & r2 circles, fly-out axis, and XI
L3 = min(abs(Xl) ,r2) ;
L4 = min(abs(Xl) ,rl) ;
A2a = L3*sqrt(r2/S2-L3~2) + r2A2*asin(L3/r2) ;
A2b = L4*sqrt(rl"2-L4"2) + rl"2*asin(L4/rl) ;
A2 = A2a-A2b;
A = Al-sign(Xl)*A2;
Ptgt(k) =A/ (pi*(r2"2-rl"2) ) ;
end
end
Pacq = Prkt.*Ptgt*100;
return, end
G. 6DOF CODES
1. Simulation Initilization
% File Initializes Simulation
clear
data_out = [2001,5] ;
%%% Missile INITIAL CONDITIONS (6 DOF Flight)
%%% (u,v,w,p,q,r,phi, theta,psi,p_rocket,pn,pe,ph)
% ground launch initial conditions
Msl_Xo = [0;0;0;0;0;0;0;10*pi/180;0;0;0;0;0] ;
% target pointing after 12 sec flight (10° elev launch) trgt § 5000m
%Msl_Xo = [285;0;0;0.03; . 01; 0;0; -.383 ; 0; -125 . 6;4230;0;310] ;%theta =0.0246
% straight and level cruise after 12 sec flight
%Msl_Xo = [300;0;0;0;0;0;0;0;0;-125.6;0;0;300];
%%% Target INITIAL CONDITIONS (6 DOF Flight)
%%% (Pn, Pe)
Tgt_Xo = [5000, -00];
T_Vn = 0;
T_Ve = 0
Talt = 0
Aerodat % weight and aero data file for 2.75" HRGK-equipped rocket
77
2. Aerodat
% This File Contains Look-up Table Data
% 2.75" Rocket with Final Design Guidance Kit
global gO TempO PresO B_atm R_air S d Sd Xt mu rho c;
%CONSTANTS & CONVERSIONS
%%% Conversions
in2m = 39.37; % inches per meter
lb2kg = 2.2046; %pounds per kilogram
%%% Gravity
gO = 9.81; % Gravity Constant, g'0, in m/s^2.
%%% Atmospheric Data:
TempO = 288.15; %surface temperature (K)
PresO = 101330; %surface pressure (Pa)
B_atm = 0.00650; %temp lapse rate (°C/m)
R_air = 287.0; %gas constant (N m/ (kg K) )
% =====================================================
% Guidance Package Parameters
% =====================================================
radius = 2.75/2/in2m; %Guidance Package Radius (m)
GP_Mass0 = 7/lb2kg; %Initial Guidance Package Mass (kg)
GP_Massf = 7/lb2kg; %Final Guidance Package Mass (kg)
GPJLngth = 13.625/in2m; %Guidance Package Length (m)
Xt = 6.6/in2m; %Thruster Location from Nose (m)
mu = 0.006; %Rocket to Kit Coefficient of Friction
GP_CG0 = 8.5/in2m; %Initial Guidance Package CG from nose (m)
GP_CGf = 8.5/in2m; %Final Guidance Package CG from nose (m)
GP_Jxx0 = ... %Aprox. Initial Axial Moment of Inertia (kg m^2)
GP_MassO*radiusA2/2;
GP_Jxxf = ... %Aprox. Final Axial Moment of Inertia (kg m~2)
GP_Massf *radius"2/2 ;
GP_Jyy0 = ... %Aprox. Initial Transverse M. of I . (kg m~2)
GP_Mass0*(GP_CG0-GP_Lngth/2)~2 + GP_MassO*GP_Lngth~2/12;
GP_Jyyf = ... %Aprox. Final Transverse M. of I. (kg m"2)
GP_Massf*(GP_CGf-GP_Lngth/2) A2 + GP_Massf*GP_Lngth'N2/12;
FlyTime = 30; %Approximate Flyout Time Bound (sec)
78
% =====================================================
% Rocket Parameters from IHSP 89-289, Mk-66 Data Book
% =====================================================
d = 2.75/in2m; %Rocket Diameter (m)
R_MassO = 22.95/lb2kg;
R_Massf = 15.73/lb2kg;
R_Lngth = 55. 125/in2m;
%Initial Rocket Mass (kg)
%Final Rocket Mass (kg)
%Rocket Length (m)
R CGO
R_Lngth-29.96/in2m + GP_Lngth;
R CGf
%Initial Rocket CG from nose (m)
%Final Rocket CG from nose (m)
R_Lngth-33.55/in2m + GP_Lngth;
R_JxxO = 26.2/(lb2kg*in2nT2) ;
R_Jxxf = 1 9 . 7 / ( lb2 kg* in2m~ 2 ) ;
%Initial Axial Moment of Inertia (kg m^2)
%Final Axial Moment of Inertia (kg m/v2)
R_JyyO = 6248/(lb2kg*in2nT2) ;
R_Jyyf = 5008/(lb2kg*in2nr2) ;
%Initial Transverse M. of I. (kg m^2)
%Final Transverse M. of I. (kg m"2)
BurnTime = 1.05;
%Average Rocket Motor Burn Time (sec)
Thrus
tHot_dat= .
[0.00
0
0.05
1600
0.07
1700
0.10
1690
0.15
1600
0.20
1530
0.25
1500
0.30
1490
0.35
1510
ThrustCold_dat=
[0.00 0
0.05
1150
0.07
1300
0.1
1380
0.15
1400
0.2
1400
0.25
1400
0.3
1390
0.35
1400
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
%Thrust Time History (sec & lbf for 150°F)
1520 0.85 1800
1560
1610
1650
1700
1730
1750
0.90
1700
0.95
600
1.00
200
1.05
40
1.10
10
1.15
0
FlyTime 0] ;
1780
1800
%Thrust Time History (sec & lbf for -50°F)
0.4 1380 0.85 1680
0.
.45
1350
0.
.5
1340
0
,55
1350
0
.6
1350
0
.65
1400
0
.7
1440
0
.75
1500
0
.8
1590
0.9
1760
0.95
1750
1.00
1600
1.05
700
1.1
100
1.15
0
FlyTime 0] ;
ThrustHot_dat ( : , 2 ) = ThrustHot_dat ( : , 2 ) *g0 /lb2kg;
ThrustCold_dat ( : , 2 ) = ThrustCold_dat ( : , 2 ) *g0/lb2kg;
79
% Reference Area/Length and Mass & Moment Time Histories
% =====================================================
S = pi* (d~2) /4; % Cross-Sectional Area (Ref Area-nT2)
Sd = S*d; % Ref Area x Ref Length (m~3)
t = [0; BurnTime; FlyTime] ;
%time vector for time histories
Mass_dat = [t, [R_MassO+GP_MassO
R_Mas s f +GP_Mas s 0
R_Massf+GP_Massf ] ] ;
% Mass Time History (sec & kg)
cg0= (R_CG0*R_Mass0+GP_CG0*GP_Mass0) / (R_MassO+GP_MassO)
cgl= (R_CGf *R_Massf+GP_CG0*GP_Mass0) / (R_Massf+GP_MassO)
cg2= (R_CGf *R_Massf +GP_CGf *GP_Massf ) / (R_Massf +GP_Massf )
CG_dat = [t, [cgO % CG History (sec & m from nose)
cgl
cg2]];
JxO = R_JxxO+GP_JxxO
Jxl = R_Jxxf+GP_JxxO
Jx2 = R_Jxxf+GP_Jxxf
Jx_dat = [t, [JxO
Jxl
Jx2]];
% Axial M. of I. History (kg m~2)
JxR_dat = [t, [R_JxxO; R_Jxxf; R_Jxxf ] ] ;
JxK_da t = [ t , [ GP_ JxxO ; GP_ JxxO ; GP_Jxxf ] ] ;
JyO = R_Jyy0+R_Mass0*(R_CG0-cg0)~2+GP_Jyy0+GP_Mass0*(GP_CG0-cg0)'N2
Jyl = R_Jyyf+R_Massf*(R_CGf-cgl)~2+GP_Jyy0+GP_Mass0*(GP_CG0-cgl)"2
Jy2 = R_Jyyf+R_Massf*(R_CGf-cg2)"2+GP_Jyyf+GP_Massf*(GP_CGf-cg2)A2
Jy_dat = [t, [JyO % M. of I. History (kg m~2)
Jyl
Jy2]];
% Rocket Roll Acceleration History
R Pdot = . . .
[0
0
0.5
45.0
0.6
100.0
0.9
15.0
1.0
-50.0
1.2 ■
-120.0
1.4
-92.0
1.6
-34.0
R_Pdot (
:,2) =
% Rocket Roll Accel (rev/s/s)
1.8
-8.0
2.0
0.0
2.4
2.0
3.0
7.5
3.6
8.5
4.0
9.5
4.4
7.5
5.6
0.0
5.8
-4.5
6.2
-12.5
6.6
-10.0
7.0
-7.0
8.0
-4.0
9.5
0.0
12.
1.0
21.
0.1]
R_Pdot(: ,2)*2*pi;
% converted to rad/s/s
80
% Aerodynamics Data from ADAM ' s MADR
Mach_dat = ...
[0 1.0000 1.4142
0.6614 1.0308 1.6008
0.8660 1.1180 1.8028
0.9682 1.2500 2.0156];
%Static Coefficient /Derivative Components Based on Mach Number Vector
%
CdOCoast_dat = ... % Skin, Wave, and Base Drag (rocket coasting)
[0.52292 0.83218 1.09629
0.48521 1.0067 1.13922
0.51417 1.00424 1.16393
0.75708 1.02549 1.17799];
Cd0Burn_dat = . . . % Skin and Wave Drag (rocket burning)
[0.42291 0.73219 0.90267
0.38521 0.81307 0.95226
0.41417 0.81061 " 0.99252
0.65708 0.83187 1.02295];
CNa_dat = . . . % Body Normal Force per Radian Alpha or Beta
[15.0616 20.05156 16.10659
16.41589 20.05156 14.68889
18.10928 18.91632 13.47025
20.04303 17.51701 12.38573];
CP_dat = . . . % Center of Pressure in Calibers
[11.52547 10.35951 10.68451
11.32152 10.35951 10.88046
11.01214 10.45431 10.99184
10.36514 10.5577 11.05396];
CP_dat = CP_dat*d;
%Dyanamic Coefficients /Derivatives Based on Mach Number Vector
%
CMq_dat = . . . % Pitching damping per Radian/sec of Pitch Rate
[-4166.814 -5519.5352 -3938.8662
-4556.4946 -5519.5352 -3541.0081
-5041.0303 -4859.3901 -3202.1553
-5518.2495 -4376.4146 -2916.2678];
Clp_dat = . . . % Roll damping (HRGK only) /Radian/sec of Roll Rate
[-7.513 -7.513 -6.7529
-7.513 -7.513 -6.453
-7.513 -7.2767 -5.5802
-7.513 -7.0348 -4.7831];
81
3. Equations of Motion
function xo = MotionEqns (uo)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% This Function computes the Flat-Earth, Body Axes,
%%% 6-DOF Dynamics Equations for a
%%% Thruster -Controlled Missile
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% Adapted for rolling airframe, thrust control, and
%%% variable aero coefficient and missile parameters by
%%% W. Mark Wonnacott, March 1997
%%% Naval Postgraduate School
%%% Adapted from the G. Hutchins' Code for EC 4330.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% Original Code by R. G. Hutchins, 10 December 1996
%%% Thanks to
%%% Rob King and Mark Wonnacott for aero assistance
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% The State Vector is defined as:
%%% Body-Axes Velocities
%%% x(l) = U, speed along missile axis, x direction
%%% x(2) = V, speed of sideslip, y direction
%%% x(3) = W, speed along z direction ("Down")
%%% Body-Axes Angular Velocities
%%% x(4) = P, roll rate
%%% x(5) = Q, pitch rate
%%% x(6) = R, yaw rate
%%% Euler Angles
%%% x(7) = phi, roll relative to vertical down
%%% x(8) = theta, pitch
%%% x(9) = psi, yaw
%%% Rocket Rolling
%%% x(10) = Pr, rocket roll rate
%%% Location Variables
%%% x(ll) = Pn, Position North of (0,0,0)
%%% x(12) = Pe, Position East of (0,0,0)
%%% x(13) = h, Height
%%% The input vector is defined as :
%%% u(l) = Tl, thrust from thruster #1
%%% u(2) = T2, thrust from thruster #2
%%% u(3) = T3, thrust from thruster #3
%%% u(4) = T, thrust through the missile x axis
82
%%%Other Missile Parameter Inputs (functions of Mach or time):
%%% P(l) = mass, missile mass (kg) [f(t)]
%%% P(2) = Xcg, missile Cg (m from nose) [f(t)]
%%% P(3) = Xcp, center of pressure (m from nose) [f(M)]
%%% P(4) = JxR rocket MofI about long axis (kg m~2)
%%% P(5) = JxK kit MofI about long axis (kg m~2)
%%% P(6) = Jy=Jz, missile Mofl-y or z axes [f(t)]
%%% P(7) = PdotR, rocket roll accel (rad/s/s) [f (t) ]
%%% Aero Coefficients /Derivatives (functions of Mach or time):
%%% C(l) = CdO,zero alpha drag coefficient
%%% C(2) = CNa, normal force /rad alpha (lift curve slope)
%%% C(3) = CMq, pitch damping per rad/sec pitch rate
%%% C(4) = Clp, roll damp per rad/sec roll rate (HRGK)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% Related Quantities
Angle of Attack
Sideslip Angle
Resultant Vert. Thrust (up, -z dir.)
Reultant Side Thrust (y direction)
Missile Speed
Air Temperature (K) [f (altitude) ]
Air Pressure (Pa) [f (altitude) ]
Air Density (Kg/m~3) [f (altitude) ]
Dynamic Pressure = 1/2 rho(h) Vt*2
missile length (m)
missile diameter (m) (ref length)
missile cross-sectional area(m^2) (ref area)
rocket to kit coefficient of friction
%%% Aerodynamic Forces
%%% CN = CNa*alpha => N=CN*qbar*S+Tz Normal Force (=Lift)
%%% CY = -CNa*beta => Y=CY*qbar*S+Ty Sideslip Force
%%% CD = CdO + or 2 /CNa => D=CD*qbar*S Drag Force
%%% Aerodynamic Moments
%%% CLbar = Clp*P*d/ (2*U) ,
%%% => Lbar = CLbar*qbar*S*d Kit Roll Moment
%%% CM = CNa*alpha(Xcg-Xcp) + CMq*Q*d/ (2*U)
%%% => M = CM*qbar*S*d + Tz(Xcg-Xt) Pitch Moment
%%% CNbar = CNa*beta(Xcg-Xcp) + CMq*Q*d/ (2*U)
%%% => Nbar = CNbar *qbar*S*d + Ty(Xcg-Xt) Yaw Moment
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% Declare Global Variables
global gO TempO PresO B_atm R_air S d Sd Xt mu;
%%%
alpha,
%%%
beta,
%%%
Tz,
%%%
Ty,
%%%
vt,
%%%
Temp,
%%%
Pres,
%%%
Rho,
%%%
qbar,
%%%
L,
%%%
d,
%%%
S=pi*d*2/4,
%%%
mu,
83
%%% Define State Variables from Inputs
x = uo(l:14) ;
%%% Body-Axes Velocities
U = x(l)
V = x(2)
W = x(3)
% speed along missile axis, x
% speed of sideslip, y direction
% speed along z direction ("Down")
%%% Body-Axes Angular Velocities
P = x(4)
Q = x(5)
R = x(6)
%%% Euler Angles
phi = x(7) ;
theta = x ( 8 ) ;
psi = x(9) ;
%%% Rocket Rolling
Pr = x(10) ;
% roll rate
% pitch rate
% yaw rate
% roll
% pitch
% yaw
% rocket roll rate
%%% Location Variables
Pn = x(ll); % Position North of (0,0,0)
Pe = x(12); % Position East of (0,0,0)
h = x ( 13 ) ; % Height
%%% Define Control Variables from Inputs
u = uo(14:17);
Tl = u(l); % thrust from thruster #1
T2 = u(2); % thrust from thruster #2
T3 = u(3); % thrust from thruster #3
T = u(4) ; % thrust through the missile long, axis
%%% Other Missile Parameter Inputs (time or Mach dependent) :
Param = uo(18:24) ;
%missile mass (kg)
%missile Cg (m from nose)
%missile CP (m from nose)
%rocket M of I about long axis (kg m^2)
%kit M of I about long axis (kg m/N2)
%missile MofI about y or z axes (kg m~2]
%rocket roll acceleration (rad/s/s)
%%% Define Mach Number Dependent Coefficients/Derivatives
C = uo(25:28)
%zero alpha drag coefficient
%normal force /rad alpha (lift curve slope)
%pitch damping per rad/sec pitch rate
%roll damping (kit only) per rad/sec roll
mass = Param(l) ;
Xcg = Param(2)
Xcp = Param (3)
JxR = Param(4)
JxK = Param(5)
Jy = Param ( 6 ) ;
PdR = Param(7)
CdO = C(l)
CNa = C(2)
CMq = C(3)
Clp = C(4)
84
%%%%%%% Define the Forces and Moments
%%% Related Quantities
% speed quantities
Vxz2 = UA2 + VT2;
Vt2 = Vxz2 + VA2;
Vt = sqrt ( Vt2 ) ;
% atmospherics
temp = TempO-B_atm*h;
pres = PresO*(l-B_atm*h/TempO) A (gO/R_air/B_atm) ;
rho = pres/ (R_air* temp) ;
qbar = . 5*rho*Vt2;
% control thrusts
Ty = Tl*cos(phi) + ...
T2*cos(phi+2/3*pi) + T3*cos (phi+4/3*pi) ;
Tz = Tl*sin(phi) + ...
T2*sin(phi+2/3*pi) + T3*sin(phi+4/3*pi) ;
% angles of attack
alpha = atan2(W,U);
beta = atan2 (V, sqrt (Vxz2 ) ) ;
%%% Forces
CN = CNa*alpha; % normal force coefficient
Fz = -qbar*S*CN - Tz; % normal force
CY = -CNa*beta; % side force coefficient
Fy = qbar*S*CY + Ty; % side force
CD = CdO + (CNA2+CYA2) /CNa;% drag force coefficient
Fx = T - qbar*S*CD; % drag force
%%% Moments
CI = Clp*P*d/(2*U+.001) ; % roll moment coefficient
Lbar = qbar*Sd*Cl + mu*JxR*PdR; % roll moment on kit
CM = CNa*alpha* (Xcg-Xcp) + ... %pitch moment coefficient
CMq*Q*d/(2*U+.00D ;
M = qbar*Sd*CM + Tz*(Xcg-Xt); % pitch moment
Cn = -CNa*beta* (Xcg-Xcp) + . . . % yaw moment coefficient
CMq*R*d/(2*U+.001) ;
Nbar = qbar*Sd*Cn + Ty*(Xcg-Xt); % yaw moment
85
%%% Compute the Time Derivatives from Flat-Earth Equations
%%% Force Equations
Ud = R*V - Q*W - gO*sin (theta) + Fx/mass;
Vd =-R*U + P*W + gO*sin(phi) *cos(theta) + Fy/mass;
Wd = Q*U - P*V + gO*cos(phi)*cos(theta) + Fz/mass;
%%% Moment Equations
Pd = Lbar/JxK;
Prd = PdR;
Qd = (l-JxK/Jy)*P*R + M/Jy - ( JxR/Jy) *R*Pr;
Rd = (JxK/Jy-l)*P*Q + Nbar/Jy + (JxR/Jy) *Q*Pr;
%%% Kinematic Equations
phid = P + tan(theta) * (Q*sin(phi) + R*cos (phi) ) ;
thetad = Q*cos(phi) - R*sin(phi);
psid = (Q*sin(phi) +R*cos (phi) ) /cos (theta) ;
%%% Navigation Equations
Pnd = U*cos (theta) *cos (psi) + ...
V* (sin(phi) *sin(theta) *cos (psi) - cos (phi) *sin(psi) ) +..
W* (cos (phi) *sin(theta) *cos(psi) + sin(phi) *sin(psi) ) ;
Ped = U*cos (theta) *sin (psi) + ...
V* (sin(phi) *sin(theta) *sin(psi) + cos(phi) *cos (psi) ) +..
W* (cos (phi) *sin( theta) *sin (psi) - sin(phi) *cos (psi) ) ;
hd = U*sin(theta) - V*sin (phi )* cos (theta) - ...
W*cos (phi) *cos (theta) ;
%%%%%%%%%% Define the output vector
%%% Body-Axes Accelerations
xo(l) = Ud
xo(2) = Vd
xo(3) = Wd
% ace along missile axis,x direction
% ace of sideslip, y direction
% ace along z direction ("Down")
%%% Body-Axes Angular Accelerations
% roll ace
% pitch ace
% yaw ace
xo(4) = Pd;
xo(5) = Qd;
xo(6) = Rd;
%%% Euler Angle Rates
xo(7) = phid; % d phi/dt
xo(8) = thetad; % d theta /dt
xo(9) = psid; % d psi/dt
%%% Rocket Rolling Equations
xo(10) = Prd; % rocket spin acceleration
%%% Location Variable Rates
xo(ll) = Pnd; % Position North of (0,0,0) rate
xo(12) = Ped; % Position East of (0,0,0) rate
xo(13) = hd; % Height rate
86
4. Line of Sight
function yo = Lineof Sight (uo)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% This Function computes the seeker frame LOS,
%%% from the tangent plane coordinates
%%% W. Mark Wonnacott, May 1997
%%% Naval Postgraduate School
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% INPUTS
%%% Position Differences Xt-Xm
%%% u(l) = N, delta north
%%% u(2) = E, delta east
%%% u(3) = H, delta height
%%% Euler Angles
%%% u(4) = phi, seeker roll relative to vertical down
%%% u(5) = theta, rocket pitch relative to horizontal
%%% u(6) = psi, rocket yaw relative to north
%%% OUTPUTS Seeker LOS
%%% yd) = alpha, off-boresight angle
%%% y(2) = beta, angle from seeker's ref. roll position (down)
%%% y(3) = R, slant range to target
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% Define Tangent Plane LOS Offsets
N = uo(l); % north distance to target
E = uo(2); % east distance to target
D = uo(3); % down distance to target
%%% Define Rocket Euler Angles
phi = uo(4)
theta = uo(5)
psi = uo(6)
% seeker roll relative to vertical down
% rocket pitch relative to horizontal
% rocket yaw relative to north
%%% Transformation Matrices (NED to XYZ-body)
CI = [cos (psi)
-sin (psi)
0
C2 = [cos (theta)
0
sin (theta)
C3 = [ 1
0
0
X=C3*C2*C1* [N;E;D] ;
Offbore = atan2 (sqrt (X(2) A2+X(3) "2) ,X(1) ) ;
pointing= atan2 (X(3) ,X(2) ) ;
R = sgrt(X(l)'v2+X(2)A2+X(3)"2) ;
yo(l) = Offbore; yo(2) = pointing; yo(3) = R;
87
sin (psi)
0;
cos (psi)
0;
0
1];
0
-sin (theta) ;
1
0;
0
cos (theta) ] ;
0
0;
cos (phi)
sin (phi) ;
-sin (phi)
cos (phi) ] ;
5. Mach Number
function M = Mach(uo)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% This function computes the Mach Number as a function of
%%% component velocities and altitude
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% W. Mark Wonnacott, March 1997
%%% Naval Postgraduate School
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% INPUTS:
%%% Body-Axes Velocities
%%% x(l) = U, speed along missile axis, x direction
%%% x(2) = V, speed of sideslip, y direction
%%% x(3) = W, speed along z direction ("Down")
%%% x(13)= h, altitude (m)
%%% Related Quantities
%%% Vt, Missile Speed (m/s)
%%% Temp, Air Temperature (K) as function of altitude
%%% Pres, Air Pressure (Pa) as function of altitude
%%% Rho, Air Density (Kg/mA3) as function of altitude
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% Declare Global Variables
global gO TempO PresO B_atm R_air S d Sd Xt;
%%% Define Variables from Inputs
%%% Body-Axes Velocities
U = uo(l); % speed along missile axis, x direction
V = uo(2); % speed of sideslip, y direction
W = uo(3); % speed along z direction ("Down")
h = uo ( 13 ) ; % height
Vt = sqrt(lT2 + V"2 + VT2);
% atmospherics
k = 1.4;
temp = TempO -B_atm*h;
pres = PresO* ( 1-B_atm*h/Temp0 ) A (gO /R_air/B_atm) ;
rho = pres/ (R_air*temp) ;
c = sqrt (k*R_air*temp) ;
M = Vt/c;
88
LIST OF REFERENCES
1. AIAA Missile Systems Technical Committee. "Graduate Team Missile Design
Competition, RFP: Hit-To-Kill Guided Rocket (HGR)." American Institute of
Aeronautics and Astronautics; Reston, VA: June 1996.
2. Wonnacott, W.M; Pomerantz, B; Silva, S.L; and Nurse, N.A. "The Three and a Half
Rocketeers Hit-to-Kill Rocket Guidance Kit" (design proposal for AIAA graduate
student competition). Naval Postgraduate School; Monterey, CA: June 1997.
3. Foss, Christopher F. and Gander, Terry J., eds. Jane's Military Vehicles and
Logistics, 1996-97. Jane's: 1996.
4. Foss, Christopher F., ed. Jane's Armour and Artillery, 1996-97 (7th ed). Jane's:
1996.
5. Cullen, Tony and Foss, Christopher R., eds. Jane's Land-Based Air Defense, 1996-
97 (9th ed). Jane's: 1996.
6. Sharpe, Richard, Captain RN, ed. Jane's Fighting Ships, 1995-96. Jane's: 1995.
7. Washburn, Alan. "Notes on Firing Theory." Naval Postgraduate School; Monterey,
CA: 1983.
8. Nicholas, Ted and Rossi, Rita, eds. US Weapon Systems Costs, 1994 (14th ed).
Data Search Associates; Fountain Valley, CA: 1994.
9. Tobin, Vince, Maj. U.S. Army and Mason, Pat, Capt. U.S. Army — AH-64 pilots,
personal communications. Naval Postgraduate School; Monterey, CA: January
1997.
10. HYDRA-70/2.75 Inch Rocket Management Office. "Hydra-70 2.75 Inch Rocket
System Information Handbook." US Army Armament, Munitions and Chemical
Command; Rock Island, IL: April 1994.
11. Ferguson, John H., Jr. and Garvey, Paul B. "2.75-Inch Rocket Motor Mark 66 Data
Book: Characteristics and Performance Data," IHSP 89-289. Naval Ordnance
Station; Indian Head, MD: June 1989.
12. Hindes, John W., Jr. Using ADAM (3rd ed). Madison, AL: October, 1993.
13. Wonnacott, W.M. "Initial Version of the Medium Range, Antiship Weapon Analysis
Program— SeaHit," NWC TM 6823. Naval Air Warfare Center Weapons Division;
China Lake, CA: September 1990.
14. Advanced Systems Concepts Office. "Missile Sizing Program User's Manual." U.S.
Army Missile Command; Huntsville, AL: March 1989.
89
15. Stevens, Brian L. and Lewis, Frank L. Aircraft Control and Simulation. Wiley; New
York: 1992.
16. Chin, S.S.; Missile Configuration Design; 1961: McGraw-Hill Book Company; New
York.
17. Blakelock, John H.; Automatic Control of Aircraft and Missiles; 2nd Ed; 1991: John
Wiley & Sons, New York.
18. Howard, Richard; AA3701: Missile Aerodynamics Class Notes; 1995: Naval
Postgraduate School; Monterey, CA.
90
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Code 471300D (Attn: Mr. W. Mark Wonnacott)
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