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900MAN2252B DISTRIBUTION 21.2.3

VOLUME 5

Special Report

Optical Signatures Code

Volume 5

BALIS3 Users’ Manual

June 2016

SPONSORED BY THE MISSILE DEFENSE AGENCY

NOTICE – The views, opinions, and/or findings contained in this document are those of the author(s) and should not be construed as an official Department of Defense position, policy, or decision unless designated by other official documentation.

DISTRIBUTION STATEMENT C – Distribution authorized to U.S. Government Agencies and their contractors; Critical Technology, Export Controlled; 30 November 2015. Other requests for this document shall be referred to MDA/ESP (Attn: Mark Underwood or active program COTR), 5224 Martin Road, Redstone Arsenal, AL 35898 or higher DoD authority.

WARNING – This document contains technical data whose export is restricted by the Arms Export Control Act (Title 22, U.S.C. sec 2751 et seq.) or the Export Administration Act of 1979, as amended, Title 50, U.S.C., App. 2401 et seq.

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Disseminate in accordance with the provisions of DoD Directive 5230.25.

DESTRUCTION NOTICE – For classified documents follow the procedures in DoD 5200.22-M, Industrial Security Manual, Section 11-19 ORD or 5200.1-R, Information Security Program Regulation Chapter IX. For unclassified, limited documents destroy by any method that will prevent disclosure of contents or reconstruction of the document.

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Optical Signatures Code Volume 5 – BALIS3 Users’ Manual OSF Contract No.:

HQ0147-11-D-0015

Task Order 9

CDRL A031

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900MAN2252B

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Missile Defense Agency

MDA/ESP

5224 Martin Road

Redstone Arsenal, Alabama 35898

Attention:

11. SUPPLEMENTARY NOTES

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DISTRIBUTION STATEMENT C – Distribution authorized to U.S.

Government Agencies and their contractors; Critical Technology, Export

Controlled; 30 November 2015. Other requests for this document shall be referred to MDA/ESP (Attn: Mark Underwood or active program COTR), 5224

Martin Road, Redstone Arsenal, AL 35898 or higher DoD authority.

13. ABSTRACT (Maximum 200 words)

BALIS3 is a ballistic trajectory propagator for endoatmospheric and exoatmospheric trajectories of targets and sensors. The program is accessed through the OSC (Optical Signatures Code) Basic Option, and it provides common, consistent trajectory data to the other OSC Basic Option programs. This manual provides users with a technical discussion of BALIS3 and it provides tips for composing input data and troubleshooting the program execution. (Portions of this manual that deal with input options are not necessarily updated with each OSC release, so users are advised to review the latest OSC Basic Option

Reference Manual and OSC Basic Option Example Case Manual for a complete set of input options and example cases).

14. SUBJECT TERMS 15 NUMBER OF PAGES

ballistic, trajectory, 3-DOF, 6-DOF, aerodynamic, reentry, missile 53

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REPORT

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Standard Form 298 (Rev. 2-89) Prescribed by ANSI Std. Z39-18

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SPECIAL REPORT

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900MAN2252B

Optical Signatures Code

Volume 5

BALIS3 Users’ Manual

Edited by

June 2016

NOTICE – The views, opinions, and/or findings contained in this document are those of the author(s) and should not be construed as an official Department of Defense position, policy, or decision unless designated by other official documentation.

DISTRIBUTION STATEMENT C – Distribution authorized to U.S. Government Agencies and their contractors; Critical Technology, Export Controlled; 30 November 2015. Other requests for this document shall be referred to MDA/ESP (Attn: Mark Underwood or active program COTR), 5224 Martin Road, Redstone Arsenal, AL 35898 or higher DoD authority.

WARNING – This document contains technical data whose export is restricted by the Arms Export Control Act (Title 22, U.S.C. sec 2751 et seq.) or the Export Administration Act of 1979, as amended, Title 50, U.S.C., App. 2401 et seq.

Violations of these export laws are subject to severe criminal penalties.

Disseminate in accordance with the provisions of DoD Directive 5230.25.

DESTRUCTION NOTICE – For classified documents follow the procedures in DoD 5200.22-M, Industrial Security Manual, Section 11-19 ORD or 5200.1-R, Information Security Program Regulation Chapter IX. For unclassified, limited documents destroy by any method that will prevent disclosure of contents or reconstruction of the document.

NOTICE – This document contains information that may be exempt from mandatory disclosure under the Freedom of Information Act.

Prepared For

Missile Defense Agency

Redstone Arsenal, Alabama

Under Contract Number

HQ0147-11-D-0015, Task Order 9

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All

This document was prepared and released in accordance with CDRL A031.

A 19 May 2014 2-3 Corrected exotmospheric to exoatmospheric.

B 27 June 2016 Cover, i-vii, 1-1, 2-

2, 4-18, 7-1

Updated distribution markings, task order number, CDRL number, sponsor, heading 1.2, discussion of NUMTIM, Table 4-8, and Reference 1.

v

ABSTRACT

BALIS3 is a ballistic trajectory propagator for endoatmospheric and exoatmospheric trajectories of targets and sensors. The program is accessed through the OSC (Optical

Signatures Code) Basic Option, and it provides common, consistent trajectory data to the other OSC Basic Option programs. This manual provides users with a technical discussion of

BALIS3 and it provides tips for composing input data and troubleshooting the program execution.

Portions of this manual dealing with input options are not necessarily updated with each

OSC release, so users are advised to review the latest OSC Basic Option Reference Manual and OSC Basic Option Example Case Manual for a complete set of input options and example cases.

APPROVED BY:

vi

FOREWORD

The ballistic trajectory propagator, BALIS3, discussed herein, is part of the Operation, Maintenance, and Sustainment of the Hardbody Toolset of the MDA Electro-Optical/Infrared

(EO/IR) Toolkit for Contract Number HQ0147-11-D-0015, under the technical direction of Dr.

Mark Underwood (MDA/ESP) for the Missile Defense Agency. Recent previous releases were issued under the technical direction of Mr. Cliff Burson (MDA/ESP) and Ms. Carol Barclay

(USASMDC/ARSTRAT). The following authors have contributed to this manual and the

BALIS3 software: J.V. Beaupre, L.W. Wright-Teats, J.L. King, and R.D. Tippets.

vii

Table of Contents

1. Introduction .......................................................................................................................... 1-1

1.1 Description of Manual .................................................................................................. 1-1

1.2 Purpose of Balis3 ......................................................................................................... 1-1

2. OSC/BALIS3 Description ..................................................................................................... 2-1

2.1 OSC Overview................................................................................................................ 2-1

2.2 BALIS3 Parameterization ............................................................................................. 2-2

3. BALIS3 Functional Overview ............................................................................................... 3-1

3.1 General Capabilities/Options ...................................................................................... 3-1

3.2 Program Description .................................................................................................... 3-5

3.3 Technical Description ................................................................................................... 3-6

3.3.1 Definition of Forces and Torques ......................................................................... 3-6

3.3.2 Treatment of the Numerical Integrations ............................................................ 3-9

4. BALIS3 Input Instructions.................................................................................................... 4-1

4.1 BALIS3 Access Commands .......................................................................................... 4-1

4.2 BALIS3 Input Data Specification ................................................................................. 4-1

5. BALIS3 Output Data ............................................................................................................. 5-1

5.1 BALIS3 Output .............................................................................................................. 5-1

5.2 Example Cases ............................................................................................................. 5-3

5.3 Post-Processing Programs ........................................................................................... 5-5

6. Diagnostics and Troubleshooting ....................................................................................... 6-1

6.1 System-Generated Messages ...................................................................................... 6-3

6.1.1 Input Format Errors ............................................................................................... 6-3

6.1.2 Other Errors ........................................................................................................... 6-3

6.2 Computer-Program Generated Messages .................................................................. 6-4

7. References ........................................................................................................................... 7-1

A. Nose Initialization and Range Invariant Impulses ............................................................. A-1

A.1. Nose Initialization ......................................................................................................... A-1

A.2. Range Invariant Deployment ....................................................................................... A-3

B. Concise Definition of OSC Euler Angles ............................................................................. B-1 viii

List of Tables

Table 3-1: Vehicle Types ............................................................................................................. 3-3

Table 3-2: Trajectory Options ..................................................................................................... 3-4

Table 3-3: Aerodynamic Forces and Torques ............................................................................ 3-6

Table 3-4: Trajectory Equations ................................................................................................. 3-7

Table 4-1: BALIS3 Run Data and Title ....................................................................................... 4-2

Table 4-2: Namelist TRAJOP Input for BALIS3 .......................................................................... 4-3

Table 4-3: Namelist STATE Input for BALIS3 ............................................................................. 4-4

Table 4-4: ZSTATE Definition by Vehicle Designation ............................................................... 4-6

Table 4-5: Variable Ballistic Coefficients ................................................................................... 4-7

Table 4-6: Airplane Positions (TGI = AF) .................................................................................... 4-8

Table 4-7: Powered Flight Positions .......................................................................................... 4-8

Table 4-8: Real Aerodynamic Coefficient Input......................................................................... 4-9

Table 5-1: Trajectory File (TRAJ, Unformatted, Unit 7; ATRAJ, Unit 1) ..................................... 5-2

Table 6-1: Defaults Values ......................................................................................................... 6-3

Table B-1: Body Axis and Ballis Axis Mapping ........................................................................... B-2 ix

List of Figures

Figure 2-1: General OSC Program Flow ..................................................................................... 2-2

Figure 3-1: Deployment of Objects Counting as a Trajectory ................................................... 3-2

Figure 3-2: Independent Trajectory Required Inputs................................................................ 3-3

Figure 3-3: Impulse Deployment Parameters ........................................................................... 3-3

Figure 4-1: Body Axis Initialization for IDOF = 2 ..................................................................... 4-11

Figure 4-2: Body Axis Initialization for IDOF = 3 or IDOF = 4 ................................................ 4-12

Figure 4-3: Body Axis Quantities ............................................................................................. 4-12

Figure 4-4: Sample Aero-coefficient Input .............................................................................. 4-13

Figure 5-1: Target-Sensor Quantities ......................................................................................... 5-2

Figure 5-2: XBALLIS Example Input ........................................................................................... 5-4

Figure 5-3: XBALLIS Results for Acceptance Testing ................................................................ 5-5

Figure 6-1: Erroneous Drag Coefficients Generated by Limited Input Data ........................... 6-2

Figure A-1: Stable Reentry Nose Alignment .............................................................................. A-1

Figure A-2: FORTRAN Program for BALIS3 Attitude Initialization ............................................. A-2

Figure A-3: Range Invariant Deployment ................................................................................... A-3

Figure A-4: BASIC Language Program for Range Invariant Deployment ................................. A-4

Figure B-1: Alpha, Beta, Gamma Defined ................................................................................. B-1

1-1

1. Introduction

1.1 Description of Manual

BALIS3 is a computer program for providing ballistic trajectory data to the rest of the OSC.

This manual is intended to provide the user with sufficient documentation to use BALIS3 either as a part of the OSC or independently. Topics discussed in this manual are the following:

Relationship of BALIS3 to the rest of the OSC

Principal options and associated processing sequences

Technical summary of principal algorithms and equations

Input and output data including example cases

1.2 Purpose of BALIS3

BALIS3 provides the user with trajectory and ranging aspect data between up to 20 ballistic objects categorized as targets or sensors. Information from one user designated target-sensor pair may be additionally written to a file for OSC usage.

BALIS3 is intended to provide fast, reasonably accurate solutions to 3 DOF or 6 DOF ballistic trajectory equations for a variety of vehicle types and data specifications.

2-1

2. OSC/BALIS3 Description

2.1 OSC Overview

The BALIS3 trajectory model is used to provide trajectory and ranging/aspect data to a number of models encompassed by the OSC BASIC option. Every model in the OSC BASIC option with the exception of the material properties selection program (SELECT) as well as the standalone wake model programs (WFM and HAW) may process data initially supplied by

BALIS3. BALIS3 calculates ballistic trajectories, 3 DOF and 6 DOF, and can ingest powered flight data that precedes ballistic flight or telemetry data that could include powered and ballistic flight.

BALIS3 uses the atmospheric profile (temperature, pressure, and density as a function of height above sea level) so ATMOS, the common atmospheric model in OSC will be called before BALIS3 is called. ATMOS requires 2 databases that are used in modeling earthshine:

ESHINE and SIS22. ATMOS may also read user-supplied files that contain rawindsonde

(weather balloon, rocketsonde, or satellite remote sounding) data and user-supplied earthshine and cloud cover data. Reference 1 (OSC BASIC Option Reference Manual) contains the specifics for using these inputs. Figure 2-1 shows the OSC BASIC program structure.

Trajectory/ranging/aspect information may be initially specified in the BASIC option in one of two ways:

Card Input to BASIC - Program Flow Specification valid when TRAJ = CARDS, DATA, or

DATAS. A table of the minimum number of trajectory/ranging/aspect data to allow for a BASIC execution is sufficient. BASIC processes the minimum data and creates a file written by BALIS3.

An execution of BALIS3 - Program Flow Specification valid when TRAJ = CALC or TAPE.

BALIS3 calculates target/sensor trajectory file to be used by other OSC programs.

Once a trajectory file has been written it may be utilized in the following OSC models:

Thermal Response Models (RVTEMP, EXOHT or EXOHT3) - Ablative thermal response models for calculation of surface temperatures and ablation rates of target (velocity, angle of attack, roll angle, and altitude utilized).

Signature Model (BIDIRC) - Spectral radiant intensity model that utilizes target, sensor, Earth, and aspect information on the trajectory file. BIDIRC rewrites the

2-2 trajectory file adding information generated in BIDIRC and RVTEMP. This file is used by the rest of the OSC BASIC models.

Background Models (ATMRAD, ELIM77, LOW5, LWTRN6, and SPACE) - Generate low and high altitude background radiances for use in the sensor model.

Sensor Models (SENSRM and SISMOD) - Sensor models that use trajectory, target intensities, and background radiances to calculate such quantities as Signal-to-Noise ratio, Noise-equivalent Flux Density and target irradiance at the sensor.

Chaff Model (CHRPRG) - Program supplies the optical signature of chaff deployed exoatmospherically.

Wake Models (WAKEPRO, WFM, and HAW) - Programs that calculate the optical signatures of wakes and discrete fragments.

Post-Processing Using a Saved Trajectory

Figure 2-1: General OSC Program Flow

2.2 BALIS3 Parameterization

The number of trajectory points for BALIS3 is a Fortran variable with the name NUMTIM. The upper limit of NUMTIM is controlled by the OSCPref input file. Users may change this value in the OSC modules file and re-compile the code, or it can be adjusted at run time using the

OSCPref file. The parameter names in the modules file have the same names shown in the

2-3

PARFILE figure. Refer to Volume 1 (BASIC Option Reference Manual) for details of the

OSCPref file. Other array dimension parameters explicitly used in BALIS3 are:

NRGIME = Number of altitude regimes for aerocoefficients (≥2: a high altitude

~exoatmospheric set and an endoatmospheric set are suggested.) The default value is 3.

NVEHIC = Number of vehicles (i.e., targets or sensors) with aerocoefficient input. The default parameter value is 2.

Two variables are used to describe the size of arrays for aerodynamic coefficients:

NELEM1 = Number of elements in linear array or rows in matrix, default = 40.

NELEM2 = Number of columns in matrices for aerocoefficients, default = 25.

NMBETA = Maximum number of altitude-dependent ballistic coefficients available in

BALIS3.

3-1

3. BALIS3 Functional Overview

3.1 General Capabilities/Options

BALIS3 may be executed to provide trajectories for a variety of vehicles. Figure 3-1 shows the nine different kinds of objects that count as a trajectory in BALIS3, as well as the required quantities of the 3 DOF state vector: time (t), position (R), and velocity (V). The state vector times need not be the same and they may precede or follow the first time for which calculations are required. BALIS 3 will fly ballistic targets backwards in time, conserving energy, provided the targets do not impact the earth during the reverse flight.

As shown in Figure 3-1 objects may be deployed by a specification of the full 3-DOF state vector or by impulsive deployment from an "RV" vehicle.

Figure 3-2 shows the required inputs for the deployment of an independently trajectoried object, Figure 3-3 shows the inputs required for an impulsively deployed object.

Objects are additionally divided into targets and sensors; target-sensor pairs provide ranging and aspect information that is output for each target-sensor pair.

Table 3-1 lists the differences between the different vehicle types and gives the input code word designation of the vehicle type. Up to 20 trajectories may be flown per case, and any number of cases per run.

In addition to the different types of vehicles and 3-DOF trajectory initialization allowed, the kind of trajectory calculated and the method of calculation can be chosen:

3-DOF Solution

Nose aligned with velocity vector

Nose cones about velocity vector with a fixed angle and rate

6-DOF Solution

Aero-coefficients (one of below)

"Nominal" aerodynamic coefficients

Input tables of aerodynamic coefficients

Vehicle attitude (one of below, chosen one per run)

Direct integration of attitude related angles

Integration of attitude rotation matrix

3-2 deployed RV. Times need not be related each other or to first time of calculation.

Note: SI , SF , and GB sensors look at targets (RV and MI) but not each other.

Up to 20 may be "flown." STs and MIs are deployed from the last previously

Last Previous RV

SI

GB

(B > -0.1)

SF RV

MI

t, R, V t, R, V t, V t, V

X

GB , t, V t, R

Figure 3-1: Deployment of Objects Counting as a Trajectory

R

V Ascent Angle

(Latitude, Longitude) Course Angle

North = 0

East = 90

Altitude

3-3

Figure 3-2: Independent Trajectory Required Inputs

Figure 3-3: Impulse Deployment Parameters

Table 3-1: Vehicle Types

Vehicle Types

Vehicle Type* Description

GB Ground-based sensor. This sensor is stationary with respect to the Earth and may be located with respect to an RV impact.

SI Sensor interceptor. It flies ballistically, just like an RV.

SF Sensor that flies. It is deployed from an RV target as a fly-along sensor.

SP Sensor present. The sensor is stationary before sensor state vector input time and flies ballistically after that time. This is used when the target's thermal history must be established long before the observing sensor is launched.

AF Aircraft Flies. It is not ballistic; its position is given at various times during the flight and may be interpolated as necessary. Position points follow namelist STATE. Number of points set from NUMBAF.

SS Satellite sensor. Input using orbital elements.

RV Reentry vehicle. It may be used as a target or as a bus to deploy MIRVs or fly-along sensors. It can also be used to position a stationary sensor with respect to its impact position.

MI MIRV. It is deployed from an RV target.

ST Satellite target. Input using orbital elements.

TM Telemetry data. Boost points only, no succeeding ballistic trajectory calculations, no namelist STATE required and NOBAL = 1.

EN End of vehicle input

*The code word following the vehicle type must be "-OSC4" with an appropriate sensor and target vehicle designator in order to write data to TRAJ (Unit 7), except cases where MULT = YES or CARDS has been selected. Must be “-TRCK” for active track source and destination.

3-4

Information for only one target-sensor pair is made available to other models in the BASIC option.

Table 3-2 shows the input parameters associated with the selection of these options.

Table 3-2: Trajectory Options

Trajectory Options

IDOF 1 2 3 or 4 3

IDOF6 --- --- 0 1

NVAR

6 3-DOF solution with nose aligned with the velocity

3-DOF solution with nose coning at a fixed angle and rate about the velocity

N/A N/A

12*, 18† N/A N/A 6-DOF solution with nominal aerodynamic coefficients

6-DOF solution with tables of aerodynamic coefficients input. IDOF6 indicates data set number

*Integration of angles related to vehicle attitude †Integration of the rotation matrix that defines vehicle attitude IDOF - Indicates 3-DOF or 6-DOF solution option IDOF6 - Mode of obtaining aerodynamic coefficients NVAR - Number of variables to be integrated

For the 3-DOF options, the only forces allowed are fixed ballistic coefficient () drag and gravity.

For the 6-DOF option with nominal aerocoefficient the forces are:

Gravity

Fixed drag

The allowed torques are:

Aerodynamic torque due to separation of centers of mass and pressure (static margin)

Spin damping

For the 6-DOF option with input tables of aerocoefficients (depending on altitude, Mach number, and angle of attack) the forces and torques are computed in the same way, but are now based on variable aerocoefficients.

3-5

3.2 Program Description

The main program of BALIS3 acts as an executive and keeps track of the different procedures required for the different vehicles and trajectory types during initialization and the calculational loop. The calculational sequence in the main program is as follows:

Read RUN data

Read CASE data

Read VEHICLE data

Initialize

Set defaults

Fly trajectory to appropriate starting point (SUBALL call)

Calculational loop

Output trajectory information

Store OSC information

Advance trajectories (SUBALL)

Determine if trajectory is active

End of Case

Write OSC tape (when tag on sensor and target is “-OSC4”)

Next Case

The main functions of subroutine SUBALL are the following:

Initialization of vehicle attitude and ECI coordinates

Flying the trajectory to requested time by a call to RKYN

Convert new ECI position to rotating Earth coordinates (ECR)

Subroutine RKYN performs the fourth order Runge-Kutta numerical integration of the requested equations of motion. Subroutines DIRIV and DIRIV1 calculate force and torque related quantities used by DIRIV and DIRIV1. Both FORCE6 and FORC18 call GRAV, which supplies the gravitational force including a quadrupole term. Both also use atmospheric profiles from subroutine AIR.

Given altitude, Mach number, and angle of attack, GETARO obtains requested aerodynamic coefficients by interpolation of input table of aerodynamic coefficients. Subroutine ENDO reads these tables and makes this data available to GETARO.

Subroutine VAERO calculates the inertial air velocity with respect to the body.

3-6

3.3 Technical Description

3.3.1 Definition of Forces and Torques

Table 3-3 defines the nomenclature of the forces and torques used by BALIS3 for a trajectory solution and the equations of motion, which are listed in Table 3-4.

Position and velocity are solved for in an inertial coordinate system that coincides with the rotating Earth coordinates at the time when the trajectory state vector is initialized. Thus, only these inertial forces need to be considered:

Gravity

Aerodynamic drag

Table 3-3: Aerodynamic Forces and Torques

Aerodynamic Forces And Torques

Symbol Definitions

Symbol Definition

QB Dynamic pressure: ½ ρAIR VAIR

Fa B Aerodynamic acceleration in body system

I a Moments of inertia (typical) (Inertia Tensor is input with aerocoefficient tables)

L B Torque in the body system

M Vehicle mass

A Base (or Reference) Area; (CDA - drag area)

D Base (or Reference) diameter α Angle of attack β Ballistic coefficient

CA Axial force coefficient

CN Normal force coefficient

CM Damping moment coefficient d Center of aerodynamic pressure in body system with respect to center of mass, i.e., static margin.

Negative values denote xcp is aft of xcg.)

ω Angular velocity

Equations

VAIR

2 = −| VAIR | (cos α, sin α cos , sin α sin )

Fa B= −AQ/M (CA, CN cos , CN sin )

LB = −AQ D2 CM ω/ (2VAIR) + d ×Fa BM = Cω + d ×Fa

BM

Value/Method of Calculation

Variable IDOF6 = 0 (Nominal) IDOF6=0: Special Data Set Input β Input Calculated: Q / | Fa |

M Input Table lookup

A M/β Table lookup

CM D

16 M/β Interpolation (α, Mach number)

CA cos α Interpolation (α, Mach number, altitude)

CN sin α Interpolation (α, Mach number)

3-7

Aerodynamic Forces And Torques

Symbol Definitions d Input Interpolation (α, Mach number)

CDA (A √𝐶𝑁

2 + 𝐶𝐴

2= A = M/ β) (A√𝐶𝑁

2 + 𝐶𝐴

2)

Gravity is calculated from a potential that includes the second zonal harmonic. Several models for Earth shape and gravity are available in the data statements in subroutine

EARTHD. Options include WGS 1972 or WGS 1984 oblate spheroid models with quadrupole moment gravity models. A spherical Earth with Newtonian gravity is also available. Consult the OSC documentation or subroutine EARTHD for a complete list of current options.

Aerodynamic drag is calculated from the velocity of the air relative to the body in the inertial coordinate system. Air is assumed to be at rest in the rotating Earth coordinate system. (A modification of VAERO can change this and introduce winds.) The altitude-dependent density and temperature of the atmosphere are obtained from the atmospheric model chosen in

ATMOS.

Table 3-4: Trajectory Equations

Trajectory Equations

NVAR Equation Description

6 dR/dt = V ECI position dV/dt = Fa + g ECI velocity:

Fa ECI drag g gravity

12 *

(IDOF=3)

d/dt = Bω ECI rotation angles that determine the transformation matrix B from body to ECI axes.

ω - the body angular velocity vector in the body system.

dω/dt = I -1

[d × Fa BM – ω× Iω + Cω] BODY angular velocity; I moments of inertia;

d deviation of center of pressure from center of mass Cω spin damping torque Fa

BM aerodynamic force in body system dR/dt = V As above dV/dt = Fa + g As above

18 * dω/dt = I -1

[d × Fa BM – ω× Iω + Cω] As above

(IDOF=3) dR/dt = V As above dV/dt = Fa + g As above dB/dt = (Bω) × B Transformation matrix from body frame to ECI frame. B = (BX, BY, BZ) BX is the ECI unit vector parallel to the vehicle nose.

* Fourth Order Runge-Kutta Solution

For 3-DOF trajectories (IDOF=1, 2), the aerodynamic coefficients of the body are lumped into a fixed ballistic coefficient. For 6-DOF trajectories with IDOF6=0, the ballistic coefficient is also fixed; for IDOF6=1, the aerodynamic coefficients are input and hence the ballistic coefficient is obtained as shown in Table 3-3 and depends on Mach number, angle of attack, and altitude.

3-8

For 6-DOF trajectories (IDOF=3, 4) equations for the body attitude are additionally solved.

The attitude equations are solved in a mixed coordinate system:

Components of the angular velocity in the body system are solved in the body system. The torques are all calculated in the body system and consist of:

Aerodynamic torque

Spin damping torque

(I) torque related to the transformation from inertia to body (rotating) coordinates. Table 3-3 defines these quantities.

The rotation matrix B is the matrix that operates on a vector referred to in body coordinates and converts it to an inertial coordinate description:

V (inertial) = B V (body)

Thus the columns of B are the body axis unit vectors referred to the inertial axes:

B = ( X I B , Y

I B , Z

I B )

The equation of motion for these vectors is

Xd I B / dt =

× X

I B where

= B

For NVAR = 18, these equations of motion are integrated directly. This is equivalent to the procedure proposed in Reference 1 and has the same associated dangers: For long times of integration, the determinant for B may drift away from 1, and affect the stability of the solution.

B is an orthogonal matrix: its transpose is its inverse. To correct for the stability problem associated with NVAR = 18, B is replaced after an integration step by:

1/2 [B + (B-1)T]

For NVAR = 12, an explicitly orthogonal solution to the motion of B is used:

B(t) = exp (x) B(0) where d/dt =

3-9

The interpretation of exp(x) and is that exp(x) is a rotation matrix for a rotation whose axis is parallel to and whose magnitude is the magnitude of . (x) is the antisymmetric matrix that when operating on a vector produces the same result as the vector cross product of with the vector.

Both the NVAR = 12 and 18 options are explicitly free of the singularities associated with Euler angle parameterization of the attitude matrix B.

In order to initialize a trajectories state vector at a later time than the first required calculational time, a modification to the drag related forces is required because of the frictional (dissipative) nature of these forces. A trajectory that has a state vector given at time t1 can be flown backwards in time to to. During this process it has lost energy, which increases the body and local air thermal energies. When it flies back to t1, more energy is lost and it cannot return to its initial state vector at t1. To circumvent this problem, the signs of the frictional drag forces are reversed during fly-back to add energy to the system; during fly-forward this same energy is removed and the body returns to its initial state vector.

3.3.2 Treatment of the Numerical Integrations

A standard Runge Kutta fourth-order integration algorithm is used to obtain position, velocity, and vehicle attitude. The parameter that critically affects the accuracy of the Runge

Kutta procedure is the selection of a time step for a Runge Kutta integration step.

Subroutine RKYN performs integration that flies a trajectory between an initial time and final time. At the initial time the Runge Kutta increment is set according to the minimum of

J = 0.05 df/dt f f = f (r, v, , /B)

The Runge Kutta step size is thus loosely associated with a maximum percentage change in integration variable. This time step is used for all Runge Kutta steps within the requested time range (= input time step).

4-1

4. BALIS3 Input Instructions

4.1 BALIS3 Access Commands

BALIS3 is a program that is always accessed by a call from OSCDRIV. In order to convert

BALIS3 to an independent program, the "BASIC" statement must be deleted and be replaced by "BALLIS" and the appropriate inputs. Care should be taken with respect to resetting the logical I/O variables if BALLIS is moved to a standalone program.

In order to use BALIS3 within the OSC structure, the input file is required to request a

BALIS3 execution:

In order to use BALIS3 as an independent program, the line of the input deck contains the word BALLIS in the first 6 columns.

To request a BALIS3 execution within the context of the BASIC option, BASIC is first requested and TRAJ is associated with CALC on the BASIC command line.

Use the same input as the BASIC option; namelist SOLAR precedes BALLIS input.

4.2 BALIS3 Input Data Specification

The data described in this section is independent of the way a BALIS3 execution is requested. The nomenclature is as follows:

RUN data is read only once per BALIS3 execution. RUN termination is requested when the first five characters of a CASE title are "EXIT".

Multiple CASES may be requested. CASE data is ended by requesting a vehicle type END followed by a blank card.

Within a CASE, up to 20 "trajectories" may be "flown." Stationary sensors are counted as a trajectory within the context of the program coding (see Figure 3-1 and Table 3-1). Each trajectory is initialized according to its type (vehicle designator) and mode of calculation (one of the 6-DOF or 3-DOF options). Vehicle designators (tags) are mnemonic in nature:

AF: Airplane flying sensor

GB: Ground Based Sensor - stationary with respect of the Earth; may be deployed relative to an "RV" impact

SI: Sensor Interceptor

SF: Sensor that Flies - deployed from an "RV"

SP: Sensor present. The sensor is stationary before sensor state vector input time and flies ballistically after that time. Often used with IPTEST=0.

4-2

MI: MIRV - deployed from an RV

RV: Reentry Vehicle - may be used as a target or as a bus to deploy MIRVs or flying sensors, as well as to position a stationary sensor with respect to its impact position

TM: Target Telemetry Input Data

SS: Satellite sensor

ST: Satellite target

Trajectory specifications and detailed input specifications are given below. Run and title inputs are described in Table 4-1. Table 4-2 describes namelist TRAJOP and Table 4-3 describes namelist STATE. Table 4-4 describes ZSTATE for each vehicle. Table 4-5 describes variable ballistic coefficients. Table 4-6 describes airplane position input. Table 4-7 describes powered flight positions. Table 4-8 describes the optional endoatmospheric aerodynamic data.

Position and velocity input data follow the conventions of Figure 3-2 and Figure 3-3. All quantities are with respect to the rotating earth. Altitudes and latitudes are geometric and not geodetic.

Attitude initialization is illustrated in Figure 4-1, Figure 4-2, and Figure 4-3, which supplement the input tables (Table 4-1 through Table 4-7). For 3-DOF trajectories, reference should be made to Figure 4-1; for 6-DOF trajectories, Figure 4-2 and Figure 4-3 are applicable. Figure 4.4 contains an example of 6DOF aerocoefficient input. Special problems associated with MIRV deployment and 6-DOF initialization are discussed in Appendix A.

Table 4-1: BALIS3 Run Data and Title

BALIS3 Run Data and Title

Column Name Format/Units Description

Run Data. This card is required and is only read once per run.

DOLD A1 Flag that requests old BALLIS input format if not blank.

Variables added in OSC 18 such as NBOOST, TOEXO and TONDO cannot be input. This option allows OSC 16 and 17 users to avoid a complete change to namelist input for existing trajectories. New input files should use the namelist input.

NENDO I4 Number of endo-data sets defining aerodynamic coefficients (= 0, 1, or 2)

NTAPE I5 Dummy: internally set to standard input device

NVAR0 I5 Defines integration variables for 6-DOF solution.

= 12 (default) integrate angles related to attitude = 18 integrate attitude matrix

Endo-data sets (optional)

See Table 4-5. Read NENDO times (NENDO>0). These data are needed only if 6-DOF trajectories with KIND6 0 are to be calculated. These sets are read only once per run.

4-3

BALIS3 Run Data and Title

Column Name Format/Units Description

Trajectory Case Title

1-25 TIT(J), J=1,5 5A5

Case title card. This card is required at the beginning of each case. "EXIT" terminates BALIS3, and namelist TRAJOP not read.

Table 4-2: Namelist TRAJOP Input for BALIS3

Namelist TRAJOP Input for BALIS3

Name Units Description

Transition Criteria

*Required when HEATR = XONDO or HEATR = XONDOD or ASCNTXD or ASCNTXP.

**Cannot be used when HEATR = XONDO or XONDOD or ASCNTXD or ASCNTXP.

TOEXO* seconds Transition time on trajectory from DIPREF to EXOHEAT for ASCENT runs. (HEATR =

ASCNT??)

TONDO* seconds Transition time on trajectory from EXOHEAT to RVTEMP/DIPREF for ASCENT runs.

(HEATR = ASCNT??).

TRALT* kilofeet Altitude at which thermal calculation switches from EXOHEAT to RVTEMP. This altitude should correspond to TONDO in ASCENT runs.

Powered Flight Specifications

NOBAL Ballistic trajectory flag.

= 0 (default) ballistic trajectory flown after boost trajectory.

= 1 no ballistic trajectory required after boost trajectory. Include some ballistic state vector input anyway because BALLIS will expect to initialize one.

NBDOF Degrees of freedom allowed for target boost flight.

= 1 (default) 3-DOF, assumes nose aligned with velocity = 3 6-DOF, nose is not assumed to be aligned with velocity but positioned using BALLIS

Euler angles orientation, uses two read statements (input on 2 different lines).

= 9 6-DOF input but done with one read statement (one line)

NBECI

kilofeet feet/second degrees meters meters/second

Polar or Cartesian coordinate flag = 0 (default) boost state vector ZB given in altitude, latitude, longitude, velocity ascent angle, and azimuth = 1 boost state vector ZB is given in ECI Cartesian Coordinates followed by Euler angles:

X, Y, Z Vx, Vy, Vz α, β, γ = 2 boost state vector ZB is given in ECI followed by ECI direction cosines (ECICOS):

X, Y, Z Vx, Vy, Vz

This input matches what is output on ECITARG when METFIL = 2. NBDOF must also be = 9.

NBOOST Number of boost or powered flight points to be input before target state vector flies ballistically which is assumed to begin at TO = TSTATE if NOBAL = 0.

NBSKIP Use every NBSKIP boost points. Default = 1 - use every boost point.

TREF seconds Reference time for target type TM when NBECI = 1.

BSTFIL 80 characters Name of file containing boost points. File name must not be “Z####”.

Ballistic Flight Specifications

N Number of calculation steps. Min = 2.

NPRINT Printing interval (NPTRAJ maximum printed). H*NPRINT is t on trajectory tape TRAJ except if IPTEST 1. If IPTEST is set greater than or equal to 1, NPRINT must be set greater than 1.

4-4

Namelist TRAJOP Input for BALIS3

Name Units Description

NPRIN1 Printing interval changed from = NPRINT*H to NPRIN1*H after TSTART if IPTEST > 0.

NPRIN2 Printing interval changed from = NPRINT*H to NPRIN2*H after TSTAR2 if IPTEST = 1 or 3.

TO seconds Start time for trajectory calculations. Seconds after launch is recommended but not required.

H seconds Runge Kutta step time (internal to BALIS3).

IPTEST Trajectory modification flag = 0 no modification occurs = 1 NPRINT is changed from input value to NPRIN1 at TSTART, and to NPRIN2 at

TSTAR2; NPRINT value should be > 1. Recommended for HEATR- XONDO = 2 signature calculations begin at TSTART = 3 combines both options 1 and 2 = 4** signature calculations begin at TSTART with NPRINT spacing, but thermal calculations are made for each trajectory point.

TSTART seconds Time at which trajectory transfer flag IPTEST operates

TSTAR2 seconds Time at which NPRINT is changed to NPRIN2 if IPTEST = 1 or 3.

Miscellaneous

IGEOD Geodetic calculation flag. < 0 signifies no Earth rotation.

= 0 spherical earth (radius = 6371 km) ; <= -4, Round Earth, no rotation = 1 WGS 72 oblate earth; = -1, WGS-72, no Earth rotation = 2 WGS 84 oblate earth (Recommended) ; = -2, WGS-84, no Earth rotation = 3 WGS 66 oblate earth; = -3, WGS-66, no Earth rotation

MSKIP Number of trajectory points that are skipped a few seconds before intercept to speed thermal calculations

TMSKIP

METFIL

seconds Time at which trajectory interval skipping begins, usually near intercept for endgame analysis

Trajectory write flag = 0 no file written = 1 METRICS file written = 2 ECITARG and ECISENS written = 3 NBOOST.DAT written (One line output is available by setting NBDOF=9)

= 4 ATAPTRAJ

= 5 ECRTARG, ECRSENS written. (If active track used, ECRBMEND1, ECRBMBEG1 written.)

Note: Only one type of file is written for each option; i.e., METFIL is not written if ECITARG and ESCISENS are written.

Table 4-3: Namelist STATE Input for BALIS3

Namelist STATE Input for BALIS3

Name Units Description

WT pounds Vehicle weight

IDOF Trajectory kind indicator:

= 1 3-DOF, nose along velocity. EULERS, OMEGAS, BODMOM and CENTP NOT USED.

= 2 3-DOF, nose coning around velocity. EULERS and OMEGAS (1) required. (Figure

4-1)

= 3 6-DOF, short input with user specified orientation and motion. EULERS and OMEGAS required. (Figure 4-2). BODMOM and CENTP or aero coefficient data are also required.

= 4 6-DOF, faster calculation, input same as IDOF = 3. Not recommended for endoatmospheric trajectories. Use IDOF = 3.

4-5

Namelist STATE Input for BALIS3

Name Units Description

IDOF6 6-DOF option:

= 0 nominal aero coefficient based on β, WT. BODMOM and CENTP are required.

= 1 IDOF6 ISET. Endo data set number. Omit BODMOM and CENTP. State vector -rotating Earth coordinate altitudes and latitudes are geometric.

INECI = 0 body position and velocity given in altitude, latitude, longitude, velocity ascent angle, and azimuth

= 1 body position and velocity given in Cartesian Coordinates X, Y, Z, Vx, Vy, Vz.

TREF seconds Used only when INECI = 1. Reference time for coordinate system for body position and velocity as defined by the user. If TREF TSTATE the coordinate system is rotated by the angle the earth rotates in time TSTATE-TREF.

MCARD = 0 no card input for this target = 1 card input to specify the orientation and position of this target relative to target 1

(MULT = CARDS)

TSTATE seconds Time of state vector definition (Seconds TALO is recommended)

BETA pounds/feet Ballistic coefficient

EPOCH Julian days Year, day, and fraction of day at epoch for satellite trajectories (SS and ST tags). Of form yyddd.ff. Jan 1 = day 1. If year is given in 2 digits, 1900 will be added if >56, 2000 if <57.

DRAG revolution/day Half of the derivative of mean motion. Used for satellite trajectories. (SS and ST)

Default = 0.

TUTCZ seconds UTC seconds past midnight at which user's time system is at zero. Used with satellite trajectories. (SS and ST)

NUMBAF Number of position points required to describe airplane's flight path, position and altitude.

1 < NUMBAF NUMTIM

Maximum of 5 airplanes allowed.

AFFIL 80 characters Name of file containing airplane’s flight path. File name must not be “Z####”.

ZSTATE State vector defining initial position and velocity depends on TG1, BETA, and INECI. See Table 4-4.

DEPLOY Logical Deploy a Vehicle Using Several MI or SF Objects.

= .FALSE. (default) not activated; Impulses set from previous RV’s trajectory.

= .TRUE. activated; Impulses set from previous object instead of previous RV. This allows a series of impulses to be used to determine a target’s trajectory.

EULERS = α, β, γ Degrees Orients body axes in space. Array of 3 elements corresponding to α, β, γ in Appendix B

OMEGAS(1) =ωx radians/second Angular velocity about axis of symmetry = body x IDOF = 2 precession rate of nose about velocity IDOF = 3 angular velocity about body nose (XB) IDOF = 4 same as 3, but ωx must not be zero.

OMEGAS(2) =ωy radians/second Angular velocity about body Y

OMEGAS(3) = ωz radians/second Angular velocity about body axis Z. ωy, ωz dummy for IDOF = 2

Required Only If IDOF = 3 or 4, IDOF6 = 0:

CENTP Feet Center of aerodynamic pressure with respect to center of mass in body axes. Array of 3 elements.

BODMOM slug feet Body moments of inertia (Ixx, Iyy, Izz). Array of 3 elements.

OPTISIG Style Deployment Input and/or Coning Angle Divergence

KPRCSS Flag to access OPTISIG style deployment input and/or coning angle divergence.

= 0 (default) Do not use = 1 OPTISIG style deployment using IREF, XMEAN, XSIGMA, and IDIST. Calculates

EULERS and OMEGAS for IDOF=3 or 4.

= 2 modeling coning angle divergence using XMEAN(5), XSIGMA(5), and IDOF=4 for exoatmospheric runs.

= 3 combine options 1 and 2 above in same run

4-6

Namelist STATE Input for BALIS3

Name Units Description

IREF

XMEAN(1:5)

degrees degrees degrees/second radians/second

Reference axis used for vehicle alignment.

= 1 zero angle of attack at reentry (at 120 km, 400kft) = 2 range insensitive axis

Mean value for each of five values: , , α, ωp, θ̇

1st Value = Angular momentum polar displacement angle from the reference axis ()

2nd Value = Rotation angle of the angular momentum of the ECI trajectory plane () 3rd Value = Coning half angle (α) 4th Value = Precession rate (ωp) 5th Value = Rate of coning angle divergence (θ̇)

XSIGMA(1:5) degrees or degrees/second

Standard deviation for each XMEAN value. If using Monte Carlo Option (VARYSIG = YES) and KPRCSS > 0, only XSIGMA(I) > 0 is needed.

IDIST(1:5) Random distribution type used for each XMEAN value:

= 1 uniform = 2 uniform cosine = 3 normal = 4 log tan normal = 5 log normal

Table 4-4: ZSTATE Definition by Vehicle Designation

ZSTATE Definition by Vehicle Designation

Designation Units Definition of ZSTATE at TSTATE

For INECI = 0

RV, SI Kilofeet ZSTATE(1) = Altitude

Degrees ZSTATE(2) = Latitude

Degrees ZSTATE(3) = Longitude feet/second ZSTATE(4) = Velocity

Degrees ZSTATE(5) = Velocity elevation angle

Degrees ZSTATE(6) = Course angle (referenced from north, 90 degrees = east)

GB Kilofeet ZSTATE(1) = Altitude with BETA > -0.1 Degrees ZSTATE(2) = Latitude

Degrees ZSTATE(3) = Longitude

ZSTATE(4), (5), and (6) are not used

GB Kilofeet ZSTATE(1) = Altitude with BETA < -0.1 Degrees ZSTATE(2) = Relative latitude with respect to calculated impact point

Degrees ZSTATE(3) = Relative longitude with respect to calculated impact point

ZSTATE(4), (5), and (6) are not used

Actual state vector for GB sensor is equal to IMPACT + ZSTATE

SP ZSTATE defined as for SI, but sensor does not move before TSTATE

For INECI = 1

RV, SI Kilofeet ZSTATE(1) = X position

Kilofeet ZSTATE(2) = Y position

Kilofeet ZSTATE(3) = Z position feet/second ZSTATE(4) = X velocity feet/second ZSTATE(5) = Y velocity feet/second ZSTATE(6) = Z velocity

4-7

ZSTATE Definition by Vehicle Designation

Designation Units Definition of ZSTATE at TSTATE where Earth-centered Cartesian coordinate system at time TREF with:

X axis is the Greenwich meridian, 0° latitude Y axis is 90° meridian, 0° latitude Z axis is through the North Pole

SP ZSTATE defined as for SI, but sensor does not move before TSTATE

Format Not Tied to…