AIC ELECTRONICB DIVIglON OlENERAL MOTORS CORlUORATIO_I <_

INTERPLANETARY GUIDANCE SYSTEM REQUIREMENTS STUDY

VOLUME II

COMPUTER PROGRAM DESCRIPTION

PART 4

PERFORMANCE ASSESSMENT FOR

ALL INERTIAL GUIDANCE SYSTEMS

Prepared for

ERC Systems Laboratoriesunder Contract NAS 12-7

AC ELECTRONICS DIVISION

General Motors Corporation

E1 Segundo, California

https://ntrs.nasa.gov/search.jsp?R=19680002376 2020-03-27T07:31:54+00:00Z

AC ELECTRONICB DIVIEIONGENERAL MOTORS CORPORATION _)

PARAGRAPH

1.0

2.0

3.0

4.0

TABLE OF CONTENTS

TITLE

INTRODUCTION AND SUMMARY

MATHEMATICAL MODEL FOR INERTIAL SYSTEMERRORS

2.1 INERTIAL MEASUREMENT UNIT CONFIGURATION 2-1

2.1.1 Gimballed 2-2

2.1.2 Strapped-Down 2-2

2.1.3 Carousel 2-2

2.2 INSTRUMENT ERROR MODELS 2-3

2.2.1 Gyro Error Models 2-3

2.2.2 Accelerometer Error Models 2-5

2.3 VE LOCITY AND POSITION ERROR DETERMINATION 2-6

COMPUTER PROGRAM DESCRIPTION 3-1

3.1 INTRODUCTION 3-1

3.1.1 Schema for Flow Chart Presentation 3-1

3.1.2 Definition of Flow Chart Symbols 3-2

3.1.3 Definition of Mathematical Symbols 3-4

3.2 BASIC ORGANIZATION OF THE PROGRAM 3-13

3.2.1 Coordinate Systems 3-14

3.3 INPUT, GENERAL INITIALIZATION, OUTPUT 3-18

3.3.1 Definition of Flags 3-18

3.3.2 Definition of Input Quantities 3-19

3.3.3 Input - Block A 3-37

3.3.4 General Initialization - Block B 3-39

3.3.5 Output - Block C 3-40

3.4 BASIC COMPUTATIONAL BLOCK - BLOCK IH 3-42

USER'S GUIDE: BLOCK III 4-1

INTRODUCTION AND SUMMARY 4-1

PROCEDURE 4-1

4.2.1 Input Sheets 4-2

4.2.2 System Specification 4-2

4.2.3 Inertial Instrument Selection 4-2

4.2.4 Geometrical Orientations 4-3

4.1

4.2

PAGE

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TABLE OF CONTENTS

PARAGRAPH

5.0

6.0

7.0

TIT LE

4.2.5

4.2.6

4.2.7

4.2.8

4.2.9

Initial Covariance Matrix [P0 ] andError Budgets

Reference Trajectory and State Transistion

Matrix Definition

Program Control

Program Output

Example

REFERENCES

APPEND_ A

A-1 INTRODUCTION

A-2 PROGRAMMER'S OPERATIONALINFORMATION

A-3 PROGRAM LISTING, PROGRAM 117.1

APPENDIX B

PAGE

4-4

4-5

4-5

4-7

4-7

5-1

A-1

A-1

A-2

A-9

B-I

..o

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AC ELECTRONICS DIVISIONGENERAL MOTORS CORPORATION _>

1.0 INTRODUCTION AND SUMMARY

AC Electronics has defined and programmed a digital computer program, (Program

117_ !), which evaluates the performance of an inertial guidance system. The computer

program provides the capability of conducting a performance analysis of an all-inertial

guidance system along the reference trajectory.

The basic output of the program consists of the covariance matrix of the velocity and

position errors due to independent error sources of the inertial guidance system. The

reference trajectory and the state transition matrix are recorded on magnetic tape for

use in repetitive runs of Block III for trade-off studies of inertial guidance systems.

The inertial guidance system error model is capable of simulating an inertial measure-

ment unit in a gimballed, strapped-down, or carousel mode of operation. A total of

three gyro error models and two accelerometer error models are currently program-

med with provision for additional instrument error models included in the program

structure. There are a total of 51 (maximum) independent error sources defined in

the inertial guidance system model.

The program is designed to interface with free-flight programs for interplanetary

guidance and navigation. The covariance matrix of velocity and position errors at the

beginning of a boost trajectory is combined with the covariance matrix of position and

velocity errors due to the inertial guidance system to furnish as output the total

covariance of position and velocity errors at the end of the boost trajectory. Hence, a

complete study capability is provided for trajectories composed of both boost and free-flight phases.

The description of the program given in the following paragraphs will furnish the

engineer and programmer with the necessary information required to use and modify

the program. A mathematical model and the detailed equations of the program are

given in Paragraphs 2.0 and 3.0. A user's guide (Paragraph 4.0) and the input forms

(Appendix B) give details of the actual operation of the program for the engineer. The

programmer will find a program listing and details of computer operation in Appendix A.

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2.0 MATHEMATICAL MODEL FOR INERTIAL SYSTEM ERRORS

The error models representing inertial guidance systems are discussed briefly in

the following paragraphs. Detailed digital computer programs incorporating these

models exist and are described in Paragraph 3.0 and Paragraph 4.0 of this document.

The error models are a mathematical representation of the factors that cause errors

in the guidance system outputs as a function of the mission profile being simulated.

The output errors are in the form of errors in the measured velocity and position and

the error model must thus provide a valid method of calculating the errors in position

and velocity for each mission profile for the guidance system being simulated. The

results are handled statistically in terms of the covariance of the output errors as a

function of the covariance of the guidance system errors in terms of the system errorparameters as defined by the system model.

Inertial guidance systems employ two types of basic inertial sensors, namely,

gyroscopes and accelerometers. Gyroscopes are sensitive to, and hence measure,

angular motion relative to an inertial or non-accelerating coordinate frame. Accelero-

meters respond to or measure the difference between total acceleration and acceleration

due to gravitational forces, again with respect to an inertial coordinate frame. The

system determines its position and velocity by calculating and "remembering" all

changes from its initial position and velocity using the angular and acceleration meas-

urements from the system sensors. The gravitational accelerations are calculated as

functions of position using the known gravitational fields of the appropriate celestialbodies.

The inertial guidance system accuracy is thus dependent upon its accurate determina-

tion of position and velocity changes, or of acceleration. The system error model,therefore, calculates the (vector) error in measured acceleration due to each of the

error parameters incorporated in the model. The resultant position and velocity

errors are obtained by integration of these acceleration errors along the missiontrajectory profile.

The specific models by which system errors are related to acceleration errors and

by which the acceleration errors are integrated into position and velocity errors are

discussed in the following subparagraphs.

2.1 INERTIAL MEASUREMENT UNIT CONFIGURATION

Inertial instruments are subject to errors from a variety of causes, some of which

are mission- and trajectory-independent and others that are functions of the mission-

dependent environment in acceleration and angular rate, the inputs to which the

instruments are sensitive. Thus, the errors will depend not only on the instrument

error parameters but also on the trajectory profile. Furthermore, the errors will

also be dependent upon the angular orientation of each instrument; that is, upon how it

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AC ELECTRONIC_S nlVISION NERALMOTORSCO.PORAT,ON

is oriented with respect to the impressed accelerations and angular rates.

The basic coordinate system in which all errors are calculated in the AC simulations

is a planet-centered, inertially fixed (PC1), coordinate frame with reference axes

denoted by X, Y, and Z. The trajectory data for the mission being simulated is input

with respect to this coordinate frame. The pertinent trajectory data for inertial system

error analyses consists of the components of acceleration, ax(t}, ay(t), and

aT(t ) , and the components of angular rate, cox(t ), coy(t), and-_0_(t}, and/or the vehicle

o_-ientation relative to the PCI coordinates in-t_erms 5f Euler an_les, _](t), _2(t), and__(t). The acceleration error from each error source and the resultant posifion and3velocity errors are calculated in PCI coordinates.

In the current AC Electronics inertial system error model, it is possible to evaluate

three different types of inertial measurement unit. These are the gimballed, strapped-

down, and carousel. From the point of view of inertial component errors, these are

different in the manner in which the instrument orientation varies. These are dis-

cussed briefly as follows.

2.1.1 Gimballed

In a conventional gimballed inertial measurement unit, the instrument package or

platform remains fixed with respect to inertial space. In the error model, the instru-

ment package orientation is thus fixed in PCI coordinates. In the notation used in

Paragraphs 3.0 and 4.0, the orientation of the platform with respect to X, Y, Z

coordinates is specified by means of an orthonormal 3 x 3 matrix [ A4], the orientationof each accelerometer with respect to the platform by an orthonormaI matrix [ J.], andthe orientation of each gyro with respect to the platform by a matrix [ M.]. Thu_,

zfor a gimballed system the orientation of all instruments with respect to the trajectory-

dependent parameters (acceleration) is specified by the 3 x 3 matrices [ Ji], [ Mi], and

[A4].

2.1.2 Strapped-Down

In a strapped-down inertial system, the instrument package remains fixed with respect

to the vehicle, and thus its orientation changes with angular motion of the vehicle. The

initial orientation of the instrument package is specified in the same manner as for the

gimballed system, but the orientation changes as a function of time are given by a

trajectory-dependent attitude time history in terms of Euler angles al(t), a2(t), and

_3(t).

2.1.3 Carousel

A carousel inertial measurement unit is essentially a gimballed unit in which the

instrument package or platform is rotated with respect to an inertial reference accord-

ing to a prespecified time program, usually a constant angular rate. Inertial systems

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AC ELECTRONICS OIVISION GENERAL MOTORS CORPORATION _

of this type are employed in some missions, usually missions involving extended time

periods,because the effect of many component errors, particularly bias errors, is

reduced by virtue of the "geometric averaging" effect. This system is analyzedvery rn,,_h J.llr_,_h e _+..... ,_ _----_ ............... _,_,-_vw,, system except that the time-varying platform attitude

is specified by a non-trajectory-dependent,time-varying program of Euler angles

o l(t), _2 (t), and u3 (t).

2.2 INSTRUMENT ERROR MODELS

The determination of measured acceleration error attributable to the various error

parameters for each instrument type is discussed below_ gyro errors in Paragraph

2.2.1 and accelerometer errors in Paragraph 2.2.2. In addition to instrument errors,

inertial systems are subject to errors due to initial misalignment. These are incorporated

in the model in a manner similar to the treatment of gyro errors.

2.2.1 Gyro Error Models

Gyroscopes serve to measure orientation or direction relative to an inertial frame,

and thus all gyro errors will result in attitude errors that cause the system to measure

the impressed acceleration along the wrong direction. For each gyro error model,

the instantaneous misalignment _(t) due to each error parameter is calculated along

the mission profile. The vector

I _,x(t)1_D(t)= Cpy(t) ,-

z(t)

is computed in PCI coordinates and is based on a small angle representation, each

element representing the misalignment about one of the reference axes, X, Y, Z.

The resultant measured acceleration error, Aa_,is then given by

where a (t) is the acceleration vector in PCI coordinates from the mission trajectory

profile.

There are currently error models for three single-degree-of-freedom gyros included

in AC Electronics simulation program. It should be pointed out that there are a

number of types of gyros, some of unique or exotic design, for which none of the

included error models is exactly correct. However, the models described are

representative of the majority of gyroscope types likely to be employed in spacemissions.

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AC ELECTRONICS DIVISION GENERAL MOTORS CORPORATION_)U_

2.2.1.1 Single-Axis Rate Gyro

A single-axis gyro has a mutually orthogonal axis system consisting of spin, (S), out-

put (O), and input (1), axes. The ideal single-axis rate gyro exactly measures rotation

about its input axis. The error terms that are accounted for in the model include a

random or non-trajectory-dependent drift, acceleration-dependent drifts caused by

mass unbalances within the gyro, second-order acceleration-dependent drifts caused

by anisoelasticity of the gyro structure, errors due to the gyro's sensing axes being

mechanically misaligned, and first- and second-order scale factor errors. In parti-

cular, there are error parameters denoted in the notation of Paragraphs 3.0 and 4.0,

as K1..., K8 that result in drift rates about the gyro's input axis proportional to the2magnitude of the parameters and to 1, ai, as, aias, WO, WS, wi, and wi, where arepresents acceleration, w represents angular rate, and subscripts I, S and O indicate

gyro input, spin, and output axes, respectively.

In addition, the error model incorporates a gyro parameter denoted by K0 that representsthe "stiffness" of the constraint torque about the gyro output axes. The gyro axes tilt

about the output axis an amount proportional to wI' and K0 is the proportionality factor.This effect is incorporated in the model.

The drift rate about the input axis due to each of the error parameters K ,..., K1 8

is resolved into PCI coordinates and integrated to obtain the appropriate misalignment

vector, =_.

2.2.1.2 Single-Axis Torque-Rebalanced Gyro

This is a single-axis rate gyro with a torque feedback servo loop. It is represented

by an error model of the same type asthe single-axis rate gyro above except that

because of the feedback loop the "stiffness" of the constraint torque is essentially

infinite, that is, equivalent to having the parameter K 0 = 0.

2.2.1.3 Single-Axis Platform

This instrument consists of a single-degree-of-freedom gyro mounted in a single-

axis stabilized platform; that is, the gyro is stabilized or isolated from rotation about

its input axis. The output of this instrument is the measured angular displacement about

its input axis between the outer instrument case and the stabilized gyro element. This

angle, 9, is equal to the integral of the angular rate about the instrument's input axis;

that is,

t

0 = ._ wi dt

o

Instruments of this type are used primarily in strapped-down system applications.

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AC ELECTRONICS DIVISION GENERAL MOTORS CORPORATION _1_)

Since this instrument is based upon a single-axis gyro, its error model will include

some of the same type of error parameters as the previous gyros. In this instrument

the gyro element's spin (S) and output {0) axes do not remain fixed with respect to the

instrument package or platform, but, in fact, rotate about its input axis by the angle

0. Thus, the error model for this instrument must resolve the acceleration and angular

rate inputs into the platform fixed initial gyro axis system and then through the angle 9

into the instantaneous gyro instrument axis system to determine the drift rate

attributable to the various error parameters.

The error parameters incorporated in this model are the same as the parameters,

K1, • • •, K8 in the previous models, except for the omission of K7 and K8. These were

the first- and second-order scale factor errors and this type of gyro is not subject to

these errors because the gyro element is isolated from the angular rate about its

input axis. That is, the wI experienced by the gyro element is zero.

2.2.2 Accelerometer Error Models

There are mathematical error models for two different types of accelerometers

included in AC Electronics simulation programs. One of these is the pendulous inte-

grating gyro accelerometer, or PIGA, and the other model represents a torque feed-

back, pendulum type of accelerometer. The following paragraphs discuss the error

parameters for each of these error models and the resultant acceleration error for

each of them.

2.2.2.1 PIGA-Type Accelerometer (See Reference in Paragraph 5.0)

This accelerometer is actually a single-axis stabilized platform, like the third type

of gyro instrument discussed above, in which the gyro instrument has a deliberate and

calibrated unbalance along its spin axis. That is, it is made to have a large "drift _

sensitivity to acceleration along its input axis. Therefore, the rate at which it drifts,

or rotates about its input axis relative to an inertial frame,is a measure of the

acceleration along its input axis. The measurement output of this instrument is the

rotation of the "stabilized" gyro element relative to the instrument case. If the

instrument is being used in a configuration other than a gimballed IMU, then the

angular output must be corrected for the rotation of the accelerometer case in inertial

space.

The error sources for this accelerometer are to a certain extent related to the error

sources in the single-axis platform gyro instrument. The residual torques that result

in a random or bias drift cause a bias or random acceleration error. In the notation

employed in Paragraphs 3.0 and 4.0, this error parameter is called K9. Errors in

the "drift" sensitivity to acceleration, that is, in the ratio of the gyro angular momentum

to the mass unbalance, result in first-order scale factor errors, denoted by K10. There

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AC ELECTRONICS DIVISION BENERAL MOTORS CORPORATION _.',/!_1_)

are second-order scale factor, or nonlinearity, errors and errors due to misalignment

of the accelerometer input axis. In addition, there will be an acceleration measurement

error proportional to the angular rate about the input axis due to errors in measurIngand/or compensating for this rotation.

In summary, there are in the model error P_2rameters that result in acceleration

measurement errors proportional to 1, aI, a.,l an,_ + K'I r_C' a. +K_ coN, and ¢oi,where subscripts I, C, and N represent the instrument's input a_x_s, cross axis, and

normal axis, respectively. The cross and normal axes are mutually orthogonal

reference axes that are normal to the input axis. The terms a C + K 1' w C and a N + K 1' w N

represent errors due to input axis misatignments about the N and C axes, respectively.

The K_w terms reflect the fact that the instrument actually measures angular rotation.

2.2.2.2 Torque Restrained Pendulum Accelerometer

The torque restrained pendulum accelerometer is a simple pendulum with a mass un-

balance and a feedback torque device to balance the torque due to acceleration. The

instrument has an imput axis, I, an output or normal axis, N, and a "cross" axis, C.

There is an unbalanced mass along the cross axis such that an acceleration along the

input axis causes a torque about the output axis. A torque feedback loop employing an

electrical torque mechanism serves to null the rotation about the output axis by providing

a counter torque. The output of the instrument is an electrical signal proportional

to the feedback torque.

For this type of instrument, the error model provides error parameters,that represent

bias acceleration errors and acceleration errors proportional to a I and a_, these errorsJ.

being due primarily to lack of linearity in the torque electronics. There is provision

for an error parameter yielding an acceleration error proportional to a I a C. This isdue to the fact that the torque feedback loop does not have infininte stiffness and therefore

the input axis will tilt about the normal axis an amount proportional to a r The modelalso includes the effect of input axis misalignments about the C and N axes, these

resulting in acceleration measurement errors proportional to a N and a C, respectively.

2.3 VELOCITY AND POSITION ERROR DETERMINATION

The preceding paragraphs have described how the mathematical error model calculates the

acceleration error z_(t) in PCI coordinates due to each error parameter as a function oftime along the reference trajectory. To describe how these errors result in position and

velocity errors, the equations of motion for the mission trajectory must be considered.

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AC ELECTRONICS DIVISION GENERAL MOTORS CORPORATION _--_

Let X (t) represent the six-element position-velocity vector along the trajectory in PCIcoordinates; that is,

x(t)Y(t)Z (t_

_X (t) = Xlt)

.Y(t)Z(t)

D m

Then the differential equations of motion may be written as

where

a _m

I

#

Z

gx(X, Y, Z)

gy(X, Y, Z)

gz(X, Y, z)

-10

ooax

ay

- az

and gx, gY' and gz are the PCI components of the gravitational acceleration.

Then, if there is an error, &a_ in the measured acceleration, the guidance system

will calculate an erroneous position and velocity, _X ÷ AX, satisfying the equation

where

_+ A__ +g_+z__ +a+Aa.

0

0

0

Aa =_ AaxAay

Aaz

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AC ELECTRONICS DIVISION GENERAL MOTORS CORPORATION _.4_)L/',.3

Thus, the position and velocity error AX satisfies the equation

AX = _ _ + ___) - _ _) + An_

Making first-order approximations, this becomes

,2,k = [5_.(X--)-I,_,X + ,_,a-- _x - -

where

I

L0 (3 x 3)

gX,Y, Z

5X,Y,Z

I (3 x 3)

m

O (3 x 3)

(6x6)

The matrix [ 5g/SX] may be evaluated along the nominal mission trajectory so that the

differential error equations are linear with time-varying coefficients.

in which t obv

The error, A X(t), resulting from a particular acceleration error time history, An, may

then be obtained in terms of the integral expression

AX(t) = 51(t, to) fit 511 (_, to) Aa (T) dT-- t O --

is the trajectory start time and 51 (t, to) is the state transition matrix given

I ]_l(t,t0) = __ . _l(t, to) ' _l(t0,t0)= I(identity).

The simulation model provides an option whereby the state transition matrix _i may

_ (to-t) I (3 × 3)-!i I.L t

tO (3x 3) i I (3x3) ._

be approximated by

-1

¢1 (t,to)=

r I(3x3)

This is quite accurate for relatively short mission times, or for situations where the

gravitational field is very small.

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AC ELECTRONICS DIVISIONGENERAL MOTORS CORPORATION _

For each error parameter simulated, there is a resultant acceleration error vector,

0

0

0

_a it) = Aax

AayAa Z

In the notation used in Paragraphs 3.0 and 4.0, the 6 x 51 matrix G it) denotes the

matrix of Aa error vectors, each column being the error associated with a unit value

of a particular error parameter. It foll(_ s then that the covariance matrix of errors

[ P] resulting from all simulated error sources is given by

[P] = _2(t,t 0) [PI ] _T(t,t ),0

where [ PI] represents the covariance matrix of all of the error parameters being simu-lated and

t _

_2(t't°) = _l(t't0) "_0 _11 (7, to) G(7) d_ .

The simulation model also provides for the incorporation of initial position and velocityerrors at the start of the mission, in which case the resultant error covariance matrix

is given by

T (t,t)T

[P(t)] = _l(t,t )[P ]_ (t,t) + _2(t,t ) [Pi] _2 '0 0 0 0 0

where [P ] is the covariance matrix of initial errors.0

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AC ELECTRONICS DIVISION GENERAL MOTORS CORPORATION o_

COMPUTER PROGRAM DESCRIPTION

3.1 INTRODUCTION

This document contains the definitionof a digitalcomputer program for the performance

assessment of all-inertialguidance systems during boost as they occur in inter-

planetary missions. The definitionconsists of flow charts and corresponding equatiolm

in the details necessary to make the presentation self-contained and ready for coding.

No attempt is made to discuss fly physical meaning of the mathematical model in de-

tail. However, necessary references for further studies are included.

Flow charts provide the basic framework around which the remainder of the discus-

sion is constructed. These diagrams serve to indicate the logical flow connecting dif-

ferent functional blocks. They do not describe literally the operation within the com-

puter program ltselt_ because many of the programming details are of little interest tomost engineers.

The flow charts have been arranged and drawn according to a hierarchical structure.

The "highest w level, designated as Level I, depicts the overall structure of the pro--

gram. Each block appearing in this chart is described by another flow chart. These

charts are designated as Level II. This policy is repeated for each block in every

level until no further logic remains to be described. In almost all cases, three levels

of flow charts suffice to accomplish this objective. The final set of flow charts at the

lowest level are supplemented by the detailed equations which are used in the program

3.1.1 Schema for Flow Chart Presentation

As has already been stated, the flow charts are arranged according to Wlevels. w In

the resulting hierarchy, the Level I flow chart provides the most general description

since it depicts the overall program. Each functional block is further described by

lower level flow charts. These charts indicate the logical flow within the block and

describe the input and output requirements of the block. The equations used to obtain

the desired outputs are presented as a supplement to the lowest level flow chart. The

number of levels that are required depends upon the logical complexity of the functionalblock.

LEVEL h This flow chart is designed to provide a very general description of the en-

tire program. The titles assigned to the functional blocks are intended to be sugges-tive of the nature of the role to be performed within the block. Those functions that

are to be performed in the basic computational cycle are designated by Roman numer-

als. Arabic symbols are used for functions that occur only once or play a passiverole.

LEVEL H" The Level H flow charts provide the first concrete description of the pro-

gram. Only the most important logical flow within each functional block is indicated

on these diagrams. The quantities that are required for all logical and computational

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AC ELECTRONICS DIVISION GENERA[_ I_OTORS CORPORATION ._

operations within thisblock are stated on thischart. These quantities are differenti-

ated as being either INPUT (i.e., values provided initiallyby the engineer) or COM-

PUTED (i.e., values determined in other portions of the program). The quantities

that are required in other parts of the program, either for printout or for computa-

tions, are also indicated on this flow chart. The functional blocks thatappear on these

diagrams are denoted by two symbols (e.g., H. 1 when discussing the "first" block in

the Level Ifflow chart of functional block If)and a name. The names have been selec-

ted to provide some insightinto the nature of the block

LEVEL HI (and below): These diagrams provide additional details of the logical flow

within the functional blocks depicted at Level If. In this program definition,Level HI

provide the description of the most intimate logical details in almost every case so no

purpose was served by proceeding to lower levels. These flow diagrams are augmen-

ted by the equations programmed intothe computer. The input and output require-

ments of these blocks are stated on the diagrams. All of these quantities are summar-

ized on the Level H flow chart.

3.1.2 Definition of Flow Chart Symbols

The following symbols represent the only ones that are used in the flow charts pre-sented below.

Set of operations that is to be described

further by additional flow charts or by

equations

Logical Decision

Operations that are predefined (i. e., insome other docum_n_

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Operations are completely defined by thestatements contained within the box

ELECTRONICS DIVISION GENERAL MOTORS CORPORATION _>

Connector used on Level II flow charts to

indicate entry source and exit destimation

Connector used on Level HI flow charts

Summary of all quantities required in com-

putations of flow charts on which this symbol

appears, or, alternatively, summary of all

quantities computed in this flow chart which

are required in other operations.

This broad arrow appears on Level I and

Level II flow charts. It is used to indicate

information flow from one block to another.

The more important information is stated

within the arrow. This symbol has been

introduced to emphasize that many quantities

are transmitted between the functional blocks

in the higher level charts.

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AC ELECTRONICS OIVISION GENERAL MOTORS CORPORATION _/;J_)L/.3

3.1.3

allaci aNi

aIiaoi asi

ax (t)ay (t)aZ (t)

Definition of Mathematical Symbols

accelerations in it:_ accelerometer

coordinates

accelerations in the ith gyro coordinates

PCI components of specific force to

mass ratio

[A4] (3,3)

BUDG

it] (3, 3)

matrix relating PCI coordinates to

PIO YAO ROO coordinates

number of budgets with the same IMU

system and trajectory

matrix relating PCI coordinates toIMU reference coordinates

[C'] (3,3)

[Di] (3, 3)

matrix relating initial PIO YAO ROOcoordinates to reference IMU

coordinates

matrix defining orientation of ith gyro

with respect to its initial orientation

on the platform (used in rate gyro and

single-axis platform gyro error models)

(EPIi, EP2i, EP3i) PCI direction cosines of the principal

axis magnitude _ (i = 1,2,3)

(EVli, EV2i, EV3i)

Flag A

PCI direction cosines, of the principal

axis magnitude, _ (i= 1,2, 3)

0 = gimbal system

i = strapdown system

2 = Carousel

Flag B

Flag C

Flag D

2 = two-degree-of-freedom gyros

1 = single-degree-of-freedom gyros

2 = (unspecified)

1 = (unspecified)

3 = single-degree-of-freedom platform

gyros

2 = rate gyro

1 = torque rebalanced gyro

ft/sec 2

ft/sec 2

ft/sec 2

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AC ELECTRONICS DIVISION GENERAL MOTORS CORPORATION _AIE;>L/_,3

Flag E

Flag F

Flag G

Flag 1

Flag 2

Flag 3

Flag 4

Flag 5

Flag 6

Flag 7

3 = PIGA

2 = proof mass accelerometer

0 = state transitionmatrix, _11 (to,t),

set equal tothat corresponding to

boost trajectory in a constant

gravitationalfield-1

I = state transitionmatrix, _1 (to,t)

set equal to the _-1 (to,t)of the

trajectory tape which corresponds

to a boost trajectory in an inverse

square gravitational field

O

1=

1=

O=

1=

O=

1=

O=

1=

O=

1=

O=

1=

O=

1=

O=

print _2 (tf, to)

do not print _2 (tf, to)

print a data dump following eachcalculation of Block HI- 1

do not print such a data dump

print a data dump following eachcalculation of Block HI-2

do not print such a data clump

print a data dump following each

calculation of Block IH-3

do not print such a data dump

print a data dump following eachcalculation of Block tIl-4

do not print such a data dump

print a data dump following eachcalculation of Block III-5

do not print such a data dump

print a data dump following eachcalculation of Block Ill-6

do not print such a data dump

print a data dump following each

calculation of Block ttl-7 at t = tfdo not print such a data dump

3-5

AC ELECTRONICS DIVISION DENERAL MOTORS CORPORATION <,_'AI_)

[G] (6,51) matrix of acceleration errors due to

independent unit error sources

_G21 ], EG22], [G23 ] (3, 8)Submatrices of [G] (6, 51) whichrepresent the acceleration errors due

to gyros No. 1,2, 3, respectively

[G24 ], [G25 _, [G26 ] (3, 8) Submatrices of _G] (6, 51)which

represent the acceleration errors due

to accelerometers No. 1, 2,3,

respectively

[G27 ] (3, 3) Submatrix of [G] (6, 51) which represents

the acceleration errors due to initial

misalignments

(I,C, N) reference accelerometer axes system

refering to input, cross, and normal

axes of accelerometer

ft/sec 2

ft/sec 2

ft/sec 2

_/sec 2

(I, O, S)

_J1 ][J2 I-[J3 ] (3,3)

K 0

K'1

K 1

K 2

reference gyro coordinate system refer-

ing to input, output, and spin axes of

gyro

matrices defining orientation of accel-

erometers No. 1, 2, 3 respectively, with

respect to the IMU reference coordinates

rate gyro parameter

PIGA error model parameter

conversion constant for gyro constantdrift term

conversion constant for gyro accelera-

tion dependent drift term

conversion constant for gyro accelera-

tion dependent drift term

sec

ft/sec

(rad/sec)/

[error budget 1_

unit

(rad/sec)/

[ (ft/sec 2) (error

budget la unit) ]

(rad/sec)/

[ (ft/sec 2) (error

budget 1(_ unit)

3-G

AC EI_ECTRONICB DIVIglON GENERAl- MOTOR_ CORPORATION _)

K 5

K 6

K 8

K 9

K10

Kll

K12

K13

K14

conversion constant for gyro accelera-

tion squared dependent drift term

conversion constant for gyro input axes

misalignment term

conversion constant for gyro input axis

misalignment term

conversion constant for gyro rate scalefactor term

conversion constant for gyro rate

nonlinearity term

conversion constant for accelerometer

bias term

conversion constant for accelerometer

scale factor term

conversion constant for accelerometer

acceleration squared dependent term

conversion constant for accelerometer

acceleration squared dependent term

conversion constant for acceleration

input axis misalignment term

conversion constant for accelerometer

input axis misalignment

(rad/sec)/

[ (ft 2/sec 4) (error

budget la unit) 7

(rad/sec)/

[ (rad/sec) (error

budget la unit)

(rad/sec)/

(rad/sec) (error

budget 1(_ unit)

(rad/sec)/

[ (rad/sec) (error

budget la unit)

(rad/sec)/

[ (rad2/sec 2)

(error budget lff

unit) 7

(ft/sec2)/[error

budget la unit_

(ft/sec2)/[ (ft/sec 2)

(error budget la

unit)

(ft/sec2)/

[ (ft2/sec4) (error

budget la unit)

(ft/sec2)/

[ (ft2/sec 4) (error

budget la unit)

(ft/sec2)/

[ (ft/sec2) (error

budget la unit) 7

(ft/sec2V[ (ft/sec z) (error

budget la unit)

3-7

AC ELECTRONICS DIVISION GENERAL MOTORS CORPORATION _l_J_)

K15

K16

KI7

KIS

K19

LPI, LP2, LP3

LV1, LV2, LV3

[M1], [M2], [M3](3,3)

N

[P] (6,6)

[Pll1, [P12 ] , [P21 ] ,

[P221:

/Pio' YAO' Rod

[PINP ] (6,6)

conversion constant for accelerometer

wheel speed error term

conversion constant for accelerometer

angular rate dependent term

conversion constant for initial misalign-

merit of instrument package

conversion constant for initial misalign-

ment of instrument package

conversion constant for initial misalign-

ment of instrument package

eigenvalues of [Pll] (3,3). The squareroots of these values are the principal

axis magnitudes of position error

ellipsoid

eigenvalues of [P22] (3,3). The square

root of these values are the principal axis

magnitudes of the velocity error ellipsoid

matrices defining orientation of gyros

No. 1, 2, 3 respectively, with respectto IMU reference coordinates

number of trajectory records to be

skipped, every (N+l)th point is used as

a trajectory data point

covariance matrix of position and velocity

error at t = tf

(3,3) submatrices of [P] (6,6)

initial body axes; pitch, yaw, and roll

unit vectors at t = t o

covariance matrix of position and velocity

error at t = tf in a downrange, crossrange,

and altitude coordinate system.

(ft/sec2)/

[ (ft/sec 2) (error

budge l(r unit) ]

(ft/sec 2 )/

[ (rad/sec) (error

budget 1(_ unit) ]

(rad)/[error

budget 1(_ unit]

(rad)/[error

budget 1(_ unit]

(rad)/[error

budget lff unit ]

ft 2

(ft/sec) 2

ft2 (PII)

ft2/sec(P21, PI2)(ft/sec) 2 (P22)

3-8

AC ELECTRONICB DIVIBION GENERAL MOTORB CORPORATION _?>

[Po] (6,6)

Po (ij)

PUNC

[Q] (6, 51)/

TAPE NO

TAPEP

t

tf

t o

[V] (3, 3)

(X,Y, Z)

_l(t), ol2(t), a3(t)

°_ii,°I12,a13

°121, °t22, °t23

c_31, _32, _33

fl ll ' fl12'f113

[Yi] (1,8)

F1,F2,F 3

initial covariance matrix of velocity

and position uncertainties

(1J) _' element of Po

punch card option

1 = punch P matrix

2 = punch ¢2 matrix

matrix of velocity and position errors,

at time t due to independent unit error

sources /

number of prerecorded tape of trajectory

tape write option

0 = don't punch cards

1 = do error analysis only

2 = print tape only

3 -- print tape and do error analysis

time

computation end time

Block III-6 state transistion matrix

parameter

matrix defining orientation of initial

misalignment coordinate system with

respect to PCI coordinates

reference PCI coordinate system

inner, middle, and outer gimbal angles

Euler angles defining [M 1 ] (3, 3) matrix

for orientation of gyro No. 1

Euler angles defining [M 2 ] (3, 3) matrix

for orientation of gyro No. 2

Euler angles defining [M3] (3, 3) matrixfor orientation of gyro No. 3

Euler angles defining a factor of [V] (3, 3)

accelerometer error model with unitycoefficients stated in matrix format

Euler angle defining [C'] (3, 3)

/0^_ "1"_.. "lr").

P21, P22)

3 punch both

see

sec

sec

radians

radians

radians

radians

radians

radians

3-9

AC_ ELECTRONICS DIVISION GENERAL MOTORS CORPORATION <._A(_)::/'_U.3

At

_l(t), _2(t), _3 (t)

8Oi

k1

k 2

k 3

k 4

k 5

_6

k 7

k 8

k 9 -* k16

k17 -* k24

k25

k26

k27

k28

input integration step size

Euler angles defining orientationofIMU

reference axes in Carousel mode:

imputted as tabular data

angular displacement ofthe ith single-axis

platform gyro about its input axis

angular displacement ofthe ithrate gyro

about its output axis

first gyro constant drift (1if)2

firstgyro unbalance along spinaxis (1if)2

firstgyro unbalance along inputaxis (10_2

first gyro anisoelasticity (1(_)2

firstgyro misalignment ofgyro inputaxis

in plane of the input and output axis (1if)2

firstgyro misalignment ofgyro inputaxis

in plane of the input and spin axis (1if)2

firstgyro scale factoruncertainty (1(_)2

firstgyro nonlinearity (1(_)2

(error budget values for second gyro)

(error budget values for third gyro)

first accelerometer bias (lo)2

first aceelerometer scale factor (1(_)2

firstaceelerometer nonlinearity(1(_)2

firstaccelerometer cross axis Non-

linearity (1(_)2

k29 firstaccelerometer sensitive axis (are-sec)2*

misalignment in the plane of sensitive

and normal axis (lff)2

seconds

radians

radians

(meru) 2,

(meru/g) 2.

(meru/g) 2.

(meru/g2) 2.

(arc-sec) 2.

(arc-sec)2*

(1 x 10-6)2*

(1 x 10-6)2*

(i x 10-6g)2.

(i x 10-6)2*

(1 x 10-6/g)2.

(1 x 10-6/g)2.

*Note: The units given are representative units, any appropriate units of the error

sources may be employed, provided that the corresponding conversion constants

K 1 --*K19 are appropriately modified.

3-10

AC ELECTRONICS DIVISION GENERAL MOTORm CORPORATION _

k30 (arc- sec) 2.

(1 x 10 -6)'31 sec/rad) 2,

k32 (1 x 10 -6sec/rad) 2.

k33 _ k40

k41 _ k48

k49 (arc-sec) 2.

k50 (arc-sec) 2.

)'51 (arc-sec)2*

(3, s)

[_11( t, to)]

first accelerometer sensitive axis

misalignment in the plane of sensitive

and cross axis (lo) 2

first accelerometer wheel speed change

(10) 2

angular velocity uncertainty about the

input axes of the first accelerometer (10) 2

(error budget values for second

accelerometer)

(error budget values for third

accelerometer)

initial misalignment of instrument pack-

age about first axis of misalignment

coordinate system (10) 2

initial misalignment of instrument pack-

age about second axis of misalignment

coordinate system (10) 2

initial misalignment of instrument pack-

age about third axis of misalignment

coordinate system (10) 2

matrix of drift angles about PCI axes due

to each independent unit error source of

the i th gyro model (i -- 1, 2, 3)

state transition matrix; linearily relating

position and velocity perturbations at t to

position and velocity perturbations at to

_:jl(t, to) the (ij) th component of _-l(t, to) sec, 1/sec, orn.d. depending

upon (ij)

*Note: The units given are representative units, any appropriate units of the error

sources may be employed, provided that the corresponding conversion constants

K 1 -* K19 are appropriately modified.

3-11

AC ELECTRONICS DIVISION GENERAL MOTORS CORPDRATIDN_,/_'JU_

[_2(tf, to)_ (6, 51)

_bll' _b12' ¢13

_21' ¢22' ¢23

_b3I'_b32' _33

o) 1 , 0)2 , 0)3

¢01(t), cO2 (t), o_3 (t)

COil, ¢OCi, ¢°Ni

¢°Ii' ¢_Oi' °_Si

COpi(t), coyA (t) coRO (t)

matrix of velocity and position errors at

t ---tf due to unit independent error sources

Euler angles defining EJ17 (3, 3) matrix

for orientation of accelerometer No. 1

Euler angles defining [J2 7 (3,3) matrix

for orientation of accelerometer No. 2

Euler angles defining [J3 7 (3,3) matrixfor orientation of accelerometer No. 3

angular rates expressed in the IMU

reference coordinate system

angular rates of the IMU in a Carousel

mode; imputted as tabular data

angular rates in the i th accelerometer

coordinates

angular rates of missile in the i th

gyro coordinates

angular rate about the instantaneous

pitch, yaw, and roll axes

radians

radians

radians

rad/sec

rad/sec

rad/sec

rad/sec

rad/sec

3-12

AC IELEECTRONIC! DIVISIONGENERAL MQTI_III COIqlmORATION _

3.2 BASIC ORGANIZATION OF THE PROGRAM

The basic structure of the program is summarized in the flow chart below. It consti-

tutes, according to preceding definitions, the Level i flow chart and consists of two

different classes of blocks. Those which define the basic computational cycles of the

program (Roman numerals), and those necessary to start the program in a prescribed

way or define the required output (Arabic letters A, B, C).

Blocks A, B, and C are described in Section 3; Block III in Section 4.

The INPUT block represents a summary of the quantities that an engineer must input.

No computations are contained within this block. In the GENERAL INITIALIZATION

block, computations that must be performed once during a specific simulation run and/

or logical decisions that must be made for proper operation within the basic computa-

tional cycle are accomplished. The OUTPUT block defines the quantities that are to

be available for printout purposes (including storing on magnetic tape) and contains

computations that are not required in the basic computational cycle.

The mathematical techniques applied in this program are based on linear perturbationtheory.

i

InputB General All-inertial III

Initialization _1 Guidance Output

I 1 sysw,m ,

Level I Flow Chart - Performance Assessment of All-inertial

Guidance Systems for Boost

The guidance system employed is specified mathematically in Block III. This block

computes the integrals which determine the errors in position and velocity generated

by the errors in the all-inertial guidance system. These errors are statistically

combined in the covariance matrix [P] which constitutes the major output of the

program.

3-13

AC ELECTRONICS OIVIBION _IENERAL MOTORS CORPORATION _'AI_>-_J-)L/.5

3.2.1 COORDINATE SYSTEMS

The basic coordinate system in Program 117.1 is the Planet Centered Inertial (PCI)

coordinate system (X, Y, Z). The X and Y axes are in the Earth's equatorial plane

and Z is along the direction of the Earth's positive rotation. When starting in Phase

1, the vehicles initial position is in the X-Z plane, specified by inputs of launch

altitude, launch latitude, and planet radius. When starting in phases other than

Phase 1, direct input of PCI coordinates is employed and the vehicle is initialized

at an arbitrary point.

Another basic coordinate system in the program is the reference body axis coordinate

system (PI, YA, RO)" This Cartesian coordinate system is initially oriented byinputs of latitude, longitude and azimuth when starting in Phase 1. These coordinates

rotate with respect to the PCI coordinates with the commanded body rates of the space

booster vehicle. If starting in a phase other than 1, the initial orientation is specified

by inputting three gimbal angles which define body axes orientation with respect to the

initial triad computed in initialization of Phase 1.

The coordinate systems employed in Block III, All Inertial Guidance System, are

referenced to the Planet Centered Inertial, (PCI), coordinate system. The rotational

transformations relating one coordinate system to another is, in general, defined in

terms of three Euler angle rotations. These Euler angle transformations have the

following general format.

I= COS 0 3

-sin 0 3

= [F (el, 02, 03) ]

/

0 / cos 8 2

sin 8 3_ 0cos 8 sin 8 2

The positive sense of the angles Ol, O2,hand convention.

0sn21Io sni 0 si 01 cos 81

0 cos 02 0

o Iu

0 V

i W

and 0 3 is specified by the usual right-

There are ten distinct Cartesian coordinate systems used in Block III. They are:

i. PCI axes The reference planet centered inertial coordinates

2. PIO' YAO' ROO axes The initial orientation of the space boos ter body

axes as specified by Block OO

3-14

AC ELECTRONiCB DiViBIDNt"-//

BENERAL MOTORS CORPORATION _>El-,,3

3. IMU axes The inertial measurement unit reference axes

4. Gyro No. i axes The reference ' .... * -.,-.., __.l _,_ . of +hoLLILJUb , UULLLJI,LV O.ll_t DLJLII. _XCS tu._.,

first gyro

5. Gyro No. 2 axes The reference input, output and spin axes of the

second gyro

6. Gyro No. 3 axes The reference input, output and spin axis of the

third gyro

7. Accelerometer No. 1

axes

The reference input, cross and normal axes of

the first accelerometer

8. Accelerometer No. 2

axes

The reference input, cross and normal axes of

the second accelerometer

9. Accelerometer No. 3

axes

The reference input, cross and normal axes ofthe third accelerometer

10. Initial misalignment

axes

The coordinate axes in which initial misalignments

are specified.

The transformation relating the PCI axes and the PIO, YAO, ROO axes is the matrix

[A4].

X(PIo YAO ROE9 = [A4] X(PCI)

[A4] is defined by input initial latitude, longitude and azimuth in Block B, GeneralInitialization, and is inputted into Block III on the reference trajectory tape. All

other transformations between coordinate systems are in the general Euler angleformat just defined.

The interrelationship of these coordinate systems is shown in the following diagram.

The symbol convention employed is defined by the example.

i PCI Axes ]

i

i

I PIo YAO ROO Axes

Df= [A4]-(PIoYAOROO) X(PCI)

3-15

AC ELECTRONICS DIVISION GENERAL MOTORS CORPORATION<_">U_

f

i

¢o

t

n

.,¢

L_ ,'_

<C I

.,¢

03

q0_ •

;>

J

Ir_

I

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¢d

o3o3

¢4

,-403

v03

c'q *

¢q

=is*

,-,4cq

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,'4

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N

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m i

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•¢ i

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¢J

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o0

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o

p-II--4I-4

oo

o

o¢.Hr_

"10

¢d

rx_

,-_

oo

0

o

r/l

o

3-16

AC ELECTRDNICB DIVIBION GENERAL MOTORS CORI=C]RATION _1_1_

All the transformations except C' (1-'1, r2, r 3) are constant. C'_1, I"2, r3) is timevarying for a strapdown system and a carousel system, and constant for a gimballed

stabilized platform system. It is evident from the diagram that when C' _1, I"2 , r3)is time varying, the IMU axes and all inertial instrument axes are rotating together,

maintaining a fixed orientation to each other, but a time varying orientation withrespect to the PCI axes.

3-17

At. ELECTRONICS DIVISION GENERAL MOTORS CORPORATIDN_

3.3 Input, General Initialization, Output

3.3.1 Definition of Flags

FLAG "A"

FLAG "B"

FLAG "C"

FLAG "D"

FLAG "E"

FLAG "F"

FLAG "G"

PUNC

BUDG

TAPEP

0 = gimbal system

1 = strapdown system2 = Carousel

2 = two-degree-of-freedom gyro

1 = single degree

2 = unspecified

1 = unspecified

3 = single-degree-of-freedom platform

2 = rate gyro

1 = torque rebalanced gyro

3 = PIGA

2 = proof mass accelerometer

1 = unspecified

0 = state transition matrix specified by Block III-6

1 = state transition matrix specified by Block II

0 = no print of _2 (tf, to) matrix

1 = print _2 (tf, to) matrix

1 = punch [P] (tf) on cards

2 = punch _2 (tf,to) matrix on cards3 = punch both on cards

specifies number of budgets used for one IMU

configuration and one trajectory

1 = do error analysis only

2 = print tape only

3 = print tape and do error analysis

3-19

AC ELECTRONICS DIVISION GENERAL MOTORS CORPORATION _

O

g]

e_

Definition of Input Quantities

_._ _

Z

.o

0

6

o_

g_

<

3-19

AC ELECTRONICS DIVISIONGENERAL MOTORS CORPORATION _-7

_ _ _ _ 0 _ "_

II II _ II II II II II II _[ II I] II I_ II

0

U

(0

3-20

AC ELE(._TRONICS OIVIEION GENERAL MOTORS CORPORATION _

O

.2Uu_

u0

c_

go

_ _o _ o

_ _1 0 /_ _ .,-_ 0 _ _ _

II II II II II II II

0

_ __

d

_5 5 5 5 5

I

m_

f r_

U

u_

,-4

M

O,

3-21

AC ELECTRONICS DIVISION QENERAL MOTORS CORPORATION _l_j_)

O

Ou3

00

cd

O

o

u_

o

O

_o_0 _,'_ _"

o

o

,,_ I 0

._,-_ 00

• 0 0 o

_o_' _Oo _._

I I

__ _

•_ _ "_,_

_._ _'__o_

_ _ __ 5 _ _

0

0

0

0

3-22

AC ELECTRONICS DIVISION GENERAL MDTORS CORPORATION _

! !

• i I '_ ""

• .__uu N _ _

_, _,_'_ _, vl__ ._ ._ VI

_ _ __ °_v

._._Cl

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0 •I_ m !

0

_ _I I

3-23

AC ELECTRONICS OIVISION GENERAL MOTORS CORPORATION o,_ I

r_

<D_

r_

> > ;>

;>

_._ _ ._ _z _ ,,.._';_ \ _ _a__ ....

0

0.,..i

3-24

AC ELECTRONICS DIVISION GENERAL MOTORS CORPORATION ._

.o

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2

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• _ _:_d d o N .... d x

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o_ _:_ .... _:_

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I-i

_ _ _ ._ _-,_ 3 .... _'':_3

=.-I0

¢..)

r_

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3-25

AC ELECTRONICS DIVISION GENERAL MOTORS CORPORATION __.,_:.,

0

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o__h _ .o _ o _ .o

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r_

3-26

AI3 ELECTRONICS DIVISION GENERAL MOTORS CORPORATION ._.'_

:!

_ _ o

O

_-, _-, _,_

0

_=_ _ o=

._

_r--n O_ Or-n _-_ _ t-"_

_o_?oo_ oo- oo- o_O°}oOO-o°m_ b_

r_

2

3-27

AC ELECTRONICS DIVISION GENERAL MOTORS CORPORATION _.6_

c_

O

r_

,

0 l-__ '_ ,-_

_ 0_

O_ O_ O_

_._ _._ _.__0 _0

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o__ ._ .__ o_ _ _ o_. _ _. __.

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o o._o.:__ __ _._.

_o__2 _._ _._ _

..... _ _ _ _ _ b

v

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3-28

AC ELECTRONICS DIVISIONGENERAL MOTORS CORPORATION _)

_o

Um

,....4

o_o _ _

._ _ _ o _o_ _

® _

T _3T ? T T _®

_o o

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3-29

AC ELECTRONICS DIVISION GENERAL MOTORS CORPORATION _AI[_

O

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3-30

AC ELECTRONICS DIVISIONGENERAL MOTORS CORPORATION _ _ _

°o_ i_._ _=

U _ O

-,_

•_ _ _ _ ._

3-31

AC ELECTRONICS DIVISION

L ! _

GENERAL MOTORS CORPORATION :_''

_3

°i_ o

= ==N

r_

o o o ¢.J o o o

r_ or} c/_ Qr_ r_ r/} 03 r_ g0 r._ r.R rR of} _3 r/}

O O C, O O O

0'3 "_ LO _ ,--_ C'_ O3 '_ U'3 r..O

O O _ O O O O O O O

r/3

C_

cdo...._

03r_

r.j

v v _ v _ v v v v v

0 0 o 0 0 0 0 0

3-32

AC ELECTRONICS DIVISION GENERAL MOTORS CORPORATION .__

! gO

o o _ _ _

_ _ _°° _ _

gO

_ 0

0 0 0

gO °

_i gO0 0

m gO

0

.¢

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gO 0

3-33

AC ELECTRONICS DIVISION GENERAL MOTORS CORPORATION .;__,

O Q;

"*-' _ b.0

m o_ _

• +_ ¢1 _ r_

i °°M _ ° _

p.-,r_

_ v _ v v

3-34

AC ELECTRONICS OIVISION

I=0

_u

I>,,

i..=<

3-35

AC ELECTRONICS DIVISION GENERAL MOTORS CORPORATION _/l_J_U_

¢=O

o_o_ __

.,,,d

_4°,,_

tl. tl. tl. • tl. tl, '1' 41. tl.

I I I ! I I I I I

_t

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3-36

AC ELECTRONICS DIVISION GENERAL MOTORS CORPORATION _)

3.3.3 Input - Block A

t.i IJ.% . • I_% . , 14.%

wpl_Ll' _yAt_l, _0%_s

_l(t), _2(t), q3(t)

ax(t ), ay(t), az(t )

all' _12' _13

a21, a22, a23

a31, a32' a33

• n' */2, *la

•at' %2' ha

fin' tim' Pla

to

FLAG "A"

FLAG "B"

FLAG "C"

FLAG "D"

FLAG "E", FLAG "F", FLAG "G"

O_l(t). w2(t), w3(t)

_,l(t), _2(t), '_3(t)

)_i i= 1,2 ..... 51

[ Po ] (6_6)

3-37

AC E!_ECTRDNIC5 OIVISlON GENERAL MOTORS CORPORATION ._

-1_1( t, t o)

K 1 through K19

[A 4]

K ,K_o

At

N

Rank

Run number

Tape number

Tape option

Number of budgets

Punch card option

3-38

AC ELECTRONICS DIVISION GENERAL MOTORS CORPORATION_)

3.3.4 General Initialization - Block B

Within this block the matrices that relate the instrument orientation to the instrument

package orientations are computed. Also the matrix that relates the coordinate sys-

tem, in which the initial alignment errors are given, to the Planet Centered Inertia_ coor-

dinate system is computed.

INPUT:

OUTPUT:

[M i ] =

Ull' u12' u13' u21' u22, u23, u31' u32' US3

_11'_12''13'_21'_22'_23'_$1'*$2'_33

811, 812, 81s iA4J

[M1], [Mz], [Ms], [J1], [J2], [J3], IV]

These matrices are computed according to the following equations.

1 0 0 cos 12

sin ¢_i

cosais cos at3..] _sinai2-sin ai31roo , 1 L_.,oO loo.oil0 cos at2 j 0

t.T_l

[vl

i

= 0

I

0

cos _13

-sin _13

i " 1,2,3

I[:0 cos i2 0 -sin _/ig

sin _/i3 1 0

oo.,,_jl_s,o,,_,o co_,__

u

0 0

cos 813 sin 813

-sin 813 cos 813

t = 1,2,3

ICo -os 0 -sin 812

1 0

sin 812 cos 812"

m

cos $11 sin _/tl

-sin _/il cos _tl

0 0

0

W

m

cos fill sin fill 0

-sin 811 cos 811 0

0 0 1

n m

A 4

The M 1, M 2, M R matrices refer to the first, second, and third gyros respectively.

The Jl' _2' J3 :hatrices refer to the first, second, and third accelerometers respec-tively. xne [V] matrix refers to the initial misalignment coordinate system.

Set the following flags - Flag A through Flag G

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AC ELECTRONICS DIVISION BENERAL MOTORS CORPORATION _AIE;_L/,,3

3.3.5

tf

[_i] , [_2] , [_3] (3,8)

[p] (6,6)

EPll

EPlfi

EP13

_LP_I

EP21

EP22

EP23

EP31

EP32

EP33

(LP3

EVll

EV12

EV13

(LVI

EV21

EV22

EV23

(LV2

EV31

EV32

EV33

_'LV3

_2(t, to)(6, 51)

[ PINP] (6,6)

Output - Block C

final time

gyro drift of each potential error source for 1st, 2nd, and

3rd gyros

covariance matrix of erros

first eigenvector of position ellipsoid (Pll)

principal axis of position ellipsoid (Pll)

second eigenvector of position ellipsoid (Pll)

principal axis of position ellipsoid (Pll)

third eigenvector of position ellipsoid (Pll)

principal axis of position ellipsoid (Pll)

first eigenvector of velocity ellipsoid (P22)

principal axis of velocity ellipsoid (P22)

second eigenvector of velocity ellipsoid (P22)

principal axis of velocity ellipsoid (P22)

third eigenvahe of velocity ellipsoid (P22)

principal axis of velocity ellipsoid (P22)

generalized error integrals

covariance matrix of errors in a downrange,

crossrange, altitude coordinate system.

3-40

AC ELECTRONICS DIVISION GENERAL MOTORS CORPORATION _

EPINP II }

EPINP 12EPINpI3 !

,,fLPINpI

EPINp21 }EPINP22

EPINP 23

_'LPINp2

E PINp31

EPINP 32

E PINP33 !

_/LPINp3

EVINP 11 }

EVINP 12EVINp13 Y

_/LVINP 1

EVINP 21

EVINP22_EVINp231

q-L--VINp2

EVINp31_

EVINp32_EVINp33 !

_LVINP 3

first eigenvector of position ellipsoid (PINPll)

principal axis of position ellipsoid (PINPII)

second eigenvector of position ellipsoid (PINPII)

principal axis of position ellipsoid (PINPll)

third eigenvector of position ellipsoid (PINPII)

principal axis of position ellipsoid (PINPII)

first eigenvector of velocity ellipsoid (PINP22)

principal axis of velocity ellipsoid (PiNP22)

second eigenvector of velocity ellipsoid (PINP22)

principal axis of velocity ellipsoid (PINP22)

third eigenvector of velocity ellipsoid (PINP22)

principal axis of velocity ellipsoid (PINP22)

3-41

AC ELECTRONICS DIVISION GENERAL MOTORS CORPDRATION_)U_

3.4 All-Inertial Guidance System Basic Computational Blocks - Block III

3-42

AC ELECTRONICm OlVINIONOENERAL. MOTORS COR FIORATION _

L_

_jI-

F_

x

U

ii

II ii

3-43

AC ELECTRONICS OIVISION GENERAL MOTORS CORPORATION _/1_)L/_.3

3.4.1.1 Detailed Flow Charts and Equations

3-44

AC ELECTRONICS OIVISION GENERAL MOTORS CORPORATION _>

III-I COORDINATE TRANSFORMATION DEFINITION: TRANSFORMATION FROM

PCI INERTIAL COORDINATES TO IMU REFERENCE COORDINATES

_h= r_ 1 -_,_÷..Jv4= .nmn, lf.r]fn nh+_|.+h. fT'an_nl-m_|nn fT'nm _hA P(_,To.oo_'f]|nAf.A

system to the IMU reference coordinate system. The [C ] matrix is time varying

in the case of the strapdown and carousel systems, and constant for the Inertially

stabilized gimballed system.

INPUT: F I, F2, F 3 -BlocklII-.7, .8, or .9

[A4] (3, 3) - Block I

OUTPUT:

[C'] =

[C] (3,3)

1 0

0 cos F3

0 -sin F 3

sin F3

cos F3

cos F 2 0 - sinF2

0 1 O,

sin F 2 0 COS l"2

COS FI

-sin I"1

0

sin I"1

cos F1

0

q

0

0

1

[C] = [C'][A4]

3-45

o

!i,,,=l

I::l

3-46

AC ELECTRONICS DIVISIONBENERAL MOTORS CORPORATION_

m u

.,-4 i

im

®

I

0

0

I •

L

i-(

I

O

IoO

AC ELECTRONICS DIVISION GENERAL MOTORS CORPORATION _>

III-2.3 GYRO ERRORS FOR SINGLE-AXJ.8 PLATFORM

The inertial drift rates o[ the single axis platform gyro due to unit independent gyro

error sources are integrated to define a 3 x 8 matrix of drift angles, measured about

the reference PCI coordinate axes. The independent error sources are the accelera-

tion insensitive drift rate, acceleration sensitive drift rates due to mass unbalance

along the spin axis and along the input axis, anisoelastictty, and input axis misalign-

ments about the spin and output axes. This form of gyro rotates about the input axis

due to angular rates about its input axis. The matrix D i accounts for such rotation.

The following sequence of computations is performed for each of the three gyros(i = 1,2,3).

INPUT: [C] (3,3) - Block _I-i

[M1] (3,3), [M2] (3,3), [M3] (3,3)- Block B-3

[ax(t), ay(t), az(t)] (3,1)- Block I

[coI, co2, _3] (3,1) - BlockHI-.7, .8, or .9

K 1, K2, ..... K8 - Block A

OUTPUT: [_01](3,8), [q_2](3,8), [_3] (3,8)

The total angle measured by the gyro since launoh is computed by integrating the rateabout the input axis.

ei = _t O_lidt oi(o)

o

= 0

The [Di] matrix defines the changing relationship of the spin and output axis to the

instrument package coordinate system due to rotations about the input axis.

[D i]

L.

1 0 0 -

0 cos ei -sin ei

0 sin ei cos ei

The angular velocities about the instantaneous gyro axes are computed.

3-47

AC ELECTRONICS DIVISION GENERAL MOTORS CORPORATION _/Ig_)L/_.3

¢°ii

¢°0i

°°StiL

[Di][M i] w_ I (3,1)

I

L oJ}

The acceleration along the instantaneous three axes of the gyro is computed,

P

all

aoi

last

]

[Di] [Mi] [C] ay

az

(3,1)

The angular error due to each unit independent error source is computed by inte-

grating the corresponding angular rate in the reference PCI coordinate system.

tcTM.TD T

[_i] (3,8) = _ It o

1 aii -asi aiiasi ¢OOi -WSi 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 K 1

0

0 0 K_

3-48

AC ELECTRONICS OIVISION GENERAL MOTORS CORPORATION _llg_>

HI-2.4 GYRO ERRORS FOR THE RATE GYRO

The inertial drift rates for the rate gyro due to unit independent gyro error sources

are integrated to define a 3 x 8 matrix of drift angles, measured about the reference

PCI coordinate axes. The independent error sources are acceleration insensitive

drift rate, acceleration sensitive drift rates due to mass unbalances along the spin

and input axes, anisoelastictty, input axes misalignment about the spin and output

axes, and angular rate scale factor and nonlinearity. This form of gyro has an

elastic restraint upon angular motion about the output axis. The input parameter

o_iS employed to represent this effective spring stiffness. The following sequencecomputations is performed for each of the three gyros, (i = 1, 2, 3).

INPUT: [C] (3, 3) - Block III-1

[M1], [M2], [M3] (3, 3) - Block B-3

[ax(t ) ay(t) az(t)] (3, 1) - Block I

[001 002 003] (3,1)- Block III-.7, .8, or .9

K 0, K I, K2, .... K 8 - BlockA

OUTPUT: [_i] (3,8), [_2] (3,8), [_3] (3,8)

The angular defection about the gyro output axis is defined by:

Cot

[Di]

= Ko 00Ii ;

cos Cot

= 0

sin Col

Cot (o) = o

0 -sin Ooi

1 0

0 cos Ooi

The angular velocities about the instantaneous input, output, and spin axis are

computed.

00ii

00Oi

00Si

[Di] [Mi] 002 (3, 1)

003

3-49

AC ELECTRONICS DIVISION GENERAL MOTORS CORPORATION _J_)u.3

The accelerations along the instantaneous input, output, and spin axis are computed.

ali ax

aoi = [Dil[Mi][C] ay

as i _ a Z

(3, 1)

The angular error due to each unit independent error source is computed by inte-

grating the corresponding angular rate in the reference PCI coordinate system.

[V i](3,8) = ft cTMTDT.to z

1 aIi ast aliaSi -WOi -¢OSi COil

0 0 0 0 0 0 0

0 0 0 0 0 0 0

2¢°ii

p

K 1e

0

0

K 8

dt

3-50

AC ELECTRONICS DIVISION GENERAL MOTORS P-ORPORATION _(l_J_)

III-2.5 GYRO ERRORS FOR THE TORQUE REBALANCED GYRO

The inertial drift rates of the torque rebalanced gyro due to unit independent gyro

error sources are integrated to define a 3 x 8 matrix of drift angles, measured about

the reference PCI coordinate axes. The independent error sources are the accelera-

tion insensitive drift rate, acceleration sensitive drift rates due to mass unbalance

along the spin and input axis, anisoelasticity, input axes misalignments about the spin

and output axes, angular rate scale factor and nonlinearity. This form of gyro

maintains a fixed orientation with respect to IMU reference axes. The following

sequence of computations is performed for each of the three gyros, (i = 1, 2, 3).

INPUT: [C] - Block III-i

[MI], [M2], [M3] (3,3) - Block B-3

[ax(t) ay(t) az(t)] (3,i)- Block I

[_i 0)2 _3 ] (3,1) - Block HI-.7, .8, or .9

K 1, K 2, ...... K 8 - BlockA

OUTPUT: [_1] (3,8), [_2] (3,8), [_3] (3,8)

The angular velocities about the instantaneous input, output, and spin axis arecomputed.

°Jii

WOi

WSi

w 1

I

[Mi] °°2 I (3,1)

u)3

The accelerations along the instantaneous input, output, and spin axis are computed.

aoi I = [Mi][c]

asi I

" 1

ax.I

Iaz J

(3, 1)

The angular error due to each unit independent error source is computed by inte-

grating the corresponding angular rate in the reference PCI coordinate system.

3-51

AC ELECTRONICS r'tlVISlON BENERAL MOTORS CORPORATION _J_)

[_i] (3,8) =_ cTMTDT.o 1

21 aii -asi aiiasi ¢OOi -COSt ¢oii COil

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

K 1

S

0

K8

dt

3-52

AC ELECTRONICS OIVISION GENERAL MOTORS CORPORATION_

III-2.6 ACCELERATION ERRORS RESULTING FROM GYRO ERROR

INPUT: [_1] , [_o2] , [_o3] (3,8) -BlockIII-2.3, III-2.4, orHI-2.5

[a X, ay, a Z] (3,1) -BlockI

OUTPUT: [G21], [G22], [G231 (3,8)

The vector acceleration error due to a misalignment _0 is calculated by applying the

vector relationship Aa = a x _. The calculation of the PCI acceleration error for

each unit independent gyro error source is done in matrix format for each gyro,(i = 1, 2, 3).

[G2i ] =

0 -a z ay

a z 0 -a X

-ay aX 0

i] (3,8)

3-53

AC ELECTRONICS DIVISION OENERALMOTORSCORPORATION_C_U_

O=;r_

r_Uu

I

1

O

i°1

¢qQ,

]- _D

rJ

I I'

3-54

AC ELECTRONICS DIVISION GENERAL MOTORS CORPORATION _-_>

III.-3.1 PIGA ACCELEROMETER ERRORS

The PiGA accelerometer independent error sources are bias, scale factor uncertainty,

nonlinearity, input axis misalignments about the cross and normal axes, which produce

errors by coupling acceleration and angular rate, wheel speed, and the uncertainty of

the angular rate about the input axis. K_ is the accelerometer scale factor that relates

the angular rate about the input axis to the acceleration along that axis.

K12 should be set to zero when using the PIGA error model as there is no cross axis

nonlinearity error source in this instrument. The following computations are performedfor each accelerometer (i = 1,2, 3).

INPUT: [C ] - Block III-1

LJII' [J2 ]' [J3 j - Block B-3

Fax(t ) ay(t) az(t ) ] (3, 1) - Block I

[0J 1 a_2 a_3 ] (3, 1) - Block III-. 7,-. 8, or -. 9

KV1 - Block A

OUTPUT: _71], [72], IT3 ] (1,8)

The accelerations along the instantaneous input, cross and normal axes of the PIGAaccelerometer are computed.

aci = [Ji] [c] ay

aNi J aZ

The angular rates about the instantaneous input, cross and normal axes of the PIGAaccelerometer are computed.

ixiI Ell_Ci = LJi] _2°_Ni _3

The matrix [7i] is a statement of the PIGA error model with unit coefficients. In

Block IH-3.4 _7i] is scaled by the constants K9, K10 , ..., K16. This scaled productis then the matrix of acceleration errors for each unit independent error source.

Ti [i: : 2 : : : , : := all aii : aliaCi aci+I_ lo_Ci - aNi- K 1toNi ali °_ii]

3-55

AC ELECTRONICS OIVISIQN QENERAL MOTORS CORPORATION q'/l_J_)_/-,.3

III-3.2 PROOF MASS ACCELEROMETER

The proof mass accelerometer error model has as independent error sources: bias,

scale factor, nonlinearity, cross axis nonlinearity, and input axis misalignment about

the cross and normal axes. The following computations are performed for each

accelerometer (i = 1, 2,3).

INPUT: [C 7 - Block III- 1

[Jl _, [J2 _,

[ax(t) ay(t)

OUTPUT: [TlT, [T27, [T3_ (i,8)

rJ3 _ - Block B-3

a z (t) _ (3, I) - Block I

The accelerations along the instantaneous input, cross and normal axes of the

accelerometer are computed.

I aIi 1aci

aNi

= 7[c ax1aZ

The lx8 matrix [Ti_ is a statement of the proof mass accelerometer error model with

unit coefficients. In Block III-3.4, [Ti_ is scaled by the constants K9, K10 , ..., K16.

This scaled product is then the acceleration error for each unit independent error

source.

: : 2 : : : : :0 O]

[yi] = [i : aii : aii :aIi aci : aci : -aNi : :

3-56

AC ELECTRONICS OIVISION GENERAL MOTORS CORPORATION _>

III-3.4 RESULTING ACCELERATION ERRORS IN THE REFERENCE

PCI COORDINATE SYSTEM

The lx8 matrices [7 i] are scaled by K9, K10, ..., K16 to define the accelerationerrors for each unit error source. The resulting acceleration errors are in instru-

ment coordinates and are then transformed to the reference PCI coordinates. This

computation is performed for each accelerometer (i = 1, 2, 3).

INPUT:

OUT PUT:

_C ] - Block M-1

[J1 ], [J2 _, [J3 ] -BlockB-3

[71], _72 ], [3/3]-SlockIII-3

K 9, K10, ..., K16-BlockA

[G24], [G25], [G26] (3,8)

G2 (3+i) F i3 Koo]0 "K16

(Diagonal Matrix)

3-57

AC ELECTRONICS OIVISlrtN__yS

GENERAL MOTORS CORPORATION _'/M_>

III-4.0 ERROR DUE TO INITIAL MISALIGNMENT OF INSTRUMENT PACKAGE

The initial misalignments are measured in a coordinate system which is related to

the reference PCI coordinate system by the IV] matrix. The PCI acceleration errors

for unit misalignments are computed here.

INPUT: EV ] (3, 3) - Block B. 3

(ax ay az) (3,1)- Block I

K17 , K18 , K19-BlockA

OUTPUT: [G27 ] (3,.3)

[G27] (3, 3) =0 -a Za z 0

ay ax

-a 0

0

0

K18

0 0j0

KI9

3-58

AC ELECTRONICS DIVISION GENERAL MOTORS CORPORATION _AI[_>

III-5.0 ARRANGEMENT OF ACCELEROMETER, GYRO, AND INITIALMISALIGNMENT ERROR MATRICES INTO THE FINAL G MATRIX

The acceleration error due to unit, independent error sources of the three gyros,

three accelerometers and the initial misalignments are assembled into a single 3, 51

matrix, _G]. These acceleration errors are expressed in the reference PCI

coordinate system.

INPUT: [G21], [G22], _G23 ] (3, 8) - Block III-2.6

[G24 ], [G25 ], _G267 (3,8) - Block III-3.4

_G27 ] (3,3) - Block HI-4.0

OUTPUT: _G _ (6, 51)

[G] =0 (3, 51)

I I i I I I

1 "G22 ,'G23 IG24 _25 ,'G26 ',G27(6, 51)

3-59

AI_ ELECTRONICS DIVISIONBENERAL MOTORS CORPORATION <._AI_)

III-6.0 SELECTION OF STATE TRANSITION MATRIX AND

CALCULATION OF VELOCITY AND POSITION ERRORS

A simplified state transition matrix is defined in this block. The value of FLAG F is

used as a criteria for selecting either this simplified model or the more complex

state transition matrix computed in Block II and inputted to Block IT[ on the reference

trajectory tape. The matrix [Q _ (6,51) of position and velocity errors is obtained by

integrating the transformed [G] matrix.

INPUT: [G(t)7 (6,51)- Block III-5

_-1 (t, to) (6, 6) - Block II

t , FLAG F - Block Ao

OUTPUT:

-i

[Q ] (6,5i)

-1

1 (t, to) (6,6)

1 0

0 1

0 0

(t, to)(Block III) =

0 i - (t-to) 0I

0 iI 0 - (t- to)I

1 , 0 0

I0 0 0 1

I

0 0 0 i 0I

_ 0 0 0 0 0

0

0

- (t- to)

Select _11 (t, to) = _11

= _i1t

Q J f - (t, G(t)dt= " _i I t o)

to

(t,to)(Block II) FLAG F = 1

(t,to)(Block III)FLAG F ---0

3-60

AC ELECTRONICg DIVISION GENERAL MOTORm CORPORATION _-_>

III-7 COVARIANCE MATRIX OF VELOCITY AND POSITION ERRORS;

MAGNITUDES AND DIRECTION COSINES OF THE PRINCIPAL

AXES OF POSITION AND VEL_)GI'I'Y _:1_1_)1_ ELLII_bL)I2)D

The calculation of the covariance matrix of position and velocity errors at the final

point of the reference powered trajectory is done in this block. The initial covariance

matrix of position and velocity errors, [P0], is combined with the scaled position and

velocity errors resulting from inertial guidance system error budget. The eigenvalues

and eigen vectors of the submatrices Pll and P22 are computed to define the magnitude

and direction of the principal axes of the position and velocity error ellipsoids. These

values and the [P] matrix are outputted in the reference PCI coordinate system.

INPUT: [Q] (6,51) - Block III-6

ki i =1,2,..., 51- BlockA

[P0] (6,6) - Block A

_71± (tf,to) (6,6) - Block III-6

r - Block A

V - Block A

OUTPUT: [P] (6,6), [_2 (tf,to)] (6,51),[PINP] (6,6)

EPII, EPI2, EPI3 _,_--_ EPINpII ,

EP21, EP22, EP23 \_--_ EPINP21,

EP31, EP32, EP33 _L_P3 EPINP31,

EVll, EVI2, EV13 ",_--_ EVINPll,

EV21, EV22, EV23 _,_-V2 EVINP21,

EV31, EV32, EV33 _-V3 EVINp31,

tf _

_2(tf'to) = _l(tf'to) _ _11

t o

COMPUTE:

EPINp12, EPINp13

EPINp22, EPINP23

EPINp32, EPINP33

EVINp12, EVINp13EVINP22, EVINP23

EVINP32, EVINP33

(t, to) G(t)dt

T[P] (6,6) = _l(tf, to) [Po] #1 (tf, to) + _2 (tf, to)

[P] Pll (3,3) : P12 (3, 3) 1

%/LPINP1

\'LPINP2

_'LPINP3

_:LVINP1_rLVINP2

_"LVINP3

_2 (tf,

k51_j

t o)

3-61

AC ELECTRONICE DIVISION GENERAL MOTORS CORPORATION _>

FIND:

P

r

Uv

rxV

I;x

r

i= [INP]

[?I _ 0

I-I[P]

pT' i 0

I INpTI

I, eigenvectors of [Pll] (3,3)

EPll, EP12, EP13

EP21, EP22, EP23

EP31, EP32, EP33

, each of the eigenvalues associated with the eigenvectors

of [PI1] (3, 3)

LP1, LP2, LP3

3. eigenvectors of [P22] (3,3)

EVIl, EV12, EV13

EV21, EV22, EV23

EV31, EV32, EV33

4. each of the eigenvalues associated with the eigenvectors

of [P22] (3,3)

LV1, LV2, LV3

5. the principal axes of the position error ellipsoid

,

%'_Pl, _,L_--P2, %CLLP3

the principal axes of the velocity error ellipsoid

L*,/LV1, _V2 , _'L--V3-

Same (1 through 6.) for [PINP]

3-G2

AC ELECTRONICS DIVISION GENERAL MOTORS CORPORATION _/l_J[__

III-8 BUDGET SET UP

This block reads in the new budget, [Po] matrix and heading and repeats block iii-7.

3-63

AC ELECTRONICS DIVIBION GENERAL MOTORB CORPORATION _

4.0 USER'S GUIDE: BLOCK III

4. ! INTRODUCTION AND SUMMA_RY

A guide to the use of simulation program 117. i, Block III, is presented. Program

117.1 provides the capability for the error analysis of an all-inertlal guidance system

during a boost trajectory. The boost trajectory data is assumed to be previously

generated by program 118.0 or other programs. The guide gives a detailed explana-

tion of the input required to specify the type of guidance system, inertial instrument

error model, inertial instrument orientation, acceleration and angular rate environ-

ment and error budget.

The program outputs are the position and velocity errors at the end of the boost

trajectory, both in the form of a covarlance matrix and as position and velocity errors

due to each of 51 independent error sources. The covariance matrix is presented in

both the PCI coordinate system and in a downrange, erossrange, altitude coordinate

system. An example case is included to illustrate the use of the program.

The function of Program 117.1 is to calculate a statistical estimate of the velocity and

position errors due to the use of an all-inertial guidance system during a reference

powered trajectory. The existence of a magnetic tape record of the reference powered

trajectory as generated by Program 118.0 will be assumed.

References will be made to the input sheets for Block III, as well as to the block

diagrams and equations of the program.

4.2 PROCEDURE

The program requires that the user select and define by input the following:

1. Type of inertial guidance system,

2. Type of gyros and accelerometers,

3. Geometrical orientation of the space booster, the platform, and the instru-

ments in a reference planet centered inertial coordinate system (PCI).

4. Covariance matrix of initial position and velocity errors,

5. Reference trajectory,

6. Program control parameters.

These quantities are defined by entering the appropriate input on the set of 17 inputsheets.

4-1

AC ELECTRONICS DIVISION GENERAL MOTORS CORPORATION _

It is the purpose of this guide to clarify and emphasize the program requirements in

terms of the actual input quantities in order to assist the program user to achieve the

desired and correct program operation.

4.2.1 Input Sheets

All input data, except for the magnetic tape trajectory record, is entered on the set of

17 input sheets. A complete data deck, as defined by the input sheets, is required for

the first run of a set. Additional runs only require the input of the data that is different

from the input of the preceding run.

The integer in column 1 specifies the data array and the integer in columns 3,4 and 5

specifies the location of each input in the array. Data may be input in floating point

with or without exponential notation in columns 11 through 72 using the existing forms

provided. The data may also be entered in octal form by changing the "DEC" to

"OCT _. Data words are separated but not terminated by commas (,) and an optional

asterisk (*) or column 73 will terminate data on a card. Data words are read into

consecutive locations in an array.

The input sheets contain tabular input for the Euler angles _l(t), _2(t), _3(t) and the

angular rates Wl(t), ¢02(t), and w3(t ). For every system configuration, gimballed,strapdown or carousel, a minimum input is required in these tables. The minimum

input consists of tmin, tmax; and four data points, tmin, X, tmax, Y; where X and Y

are arbitrary values for inertial or strapdown systems. For carousel systems, up to

27 time points can be entered, and they need not be at uniform time intervals. The

last table entry should be selected at the time t = tma x > tf.

4.2.2 System Specification

The selection of the type of inertial guidance system is made by the input Flag A (page17 of input sheets). This flag defines the manner in which the inertial measurement

unit (IMU) is oriented and the angular velocity environment of the IMU. For a

gimballed system, Flag A = 0; for a strapdown system, Flag A = 1; for a carousel

system, Flag A = 2. Flag entries are a decimal integral value and if a value not

specified by the program is entered an error return results and the computer run isterminated.

4.2.3 Inertial Instrument Selection

The selection of the instrument error model, corresponding to the type of inertial

instrument specified, is accomplished by setting Flags B through E (page 17 of inputsheets). The values of the flags for different instruments are shown in the Level II

and III flow charts and in the Input Data listing. The instrument error models consist

of drift rate error equations for the gyros and acceleration error equations for the

4-2

AC ELECTRONICB DIVIBIONGENERAL MOTORB CORPORATION _

accelerometers. Flag entries are a decimal integral value and if a value not specified

by the program is entered an error return results and the computer run is terminated.

In the present program formulation, there are no two-degree-of-freedom gyro error

models, but the program control has provision for the inclusion of such models. Two

instrument-dependent parameters, Ko and K' 1, are entered on page 1 of the input

sheets. Ko is a rate gyro parameter related to the effective spring restraint on

angular motion about the output axis. Ko _ 0 corresponds to an infinitely stiff spring

constraint. K' 1 is a PIGA parameter relating acceleration error to that component of

cross and normal axis angular rates sensed by the input axis when an input axis mis-

alignment is present. The value of I_ 1 is the acceleration scale factor of the PIGA,with nominal units of (ft/sec2)/(rad/sec).

The selection of instrument error models appropriate to various trajectories is left

to the engineerts judgment.

4.2.4 Geometrical Orientations

4.2.4.1 Reference Inertial Coordinates to Reference IMU Coordinates

The relationship between the planet-centered inertial coordinates (PCI) and the refer-

ence IMU axes is specified in three different manners, depending upon the value of

Flag A. The reference IMU axes are a right-handed orthogonal triad labeled Epitch,

yaw, roll].

If Flag A -_ 0, a gimballed system is specified and the transformation from PCI

coordinate to IMU reference coordinates is given by the matrix EA4]. EA4] is input to

the 117. i program from the reference trajectory tape. EA4] relates the PCI coordin-

ates to the initial (t = 0) orientation of the pitch, yaw, and roll axes of the space

booster. These orientations are controlled by space booster input. The A4 matrix is

a constant matrix and hence simulates the function of a gyro-stabilized gimballed IMU.

If Flag A = 1, a strapdown system is specified and the relationship between PCI

coordinates and the reference IMU coordinates is time varying. The time-varying

matrix is defined in Block III-1 and is a function of three Euler angles: Fl,I'2, I'3,

and the matrix [A4]. The Euler angles are the gimbal angles of a three-gimbal plat-

form and are input to the program from the reference trajectory tape. The angular

rate environment of the IMU is also input to the program from the reference trajectorytape as angular rates about the reference IMU coordinate axes.

If Flag A = 2, a carousel system is specified and the attitude of the IMU reference

coordinate is defined by the same time-varying matrix as for a strapdown system,

except that the Euler angles are not specified by the program user as time-varying

tabular input of the Euler angles, _1 = I'1; _2 = I'2; and _3 = r3. In addition, the

4-3

AC ELECTRONICS DIVISION GENERAL MOTORS CORPORATION _)

the inertial angular rates about the reference IMU coordinates are also inputted as

tabular functions of time. There is one constraint when using this mode, that the time

history of the Euler angles is consistent with the tabular inputted angular rate.

4.2.4.2 IMU Reference Coordinates and Inertial Instrument Coordinates

The rotational transformations relating individual inertial instrument reference axes

to the reference IMU coordinates are constant transformations. There are threetrans-

formations for the three gyros [Mi] , and three transformations [Ji] for the three

accelerometers. Each transformation is defined by three Euler angles; hence a total

of 18 Euler angles are needed to define the orientation of a set of three gyros and three

accelerometers. Each instrument has a right-handed orthogonal triad associated with

it: the gyro's triad is (input, output, spin); the accelerometer triad is (input, cross,

normal). These transformations are defined in the Initialization Block B. 3.

4.2.4.3 Initial Alignment Coordinate System

In specific system applications, initial alignment errors are specified in a coordinate

system of convenience; e. g., a prism mirror, theodolite system that is not aligned

with any of the preceding coordinate systems. The transformation [V], also defined

in Initial System Block B, relates the PCI system to the coordinate system in which

the misalignment angles are given. The [A4] matrix and three Euler angles are used

to define the IV] transformation.

4.2.5 Initial Covariance Matrix Po and Error Budgets

4.2.5. 1 InitialCovariance Matrix [Po]

The covariance matrix [Po] of initialposition and velocity errors can be inputted

directly on the input sheets. The units of Po elements are designated on the input

sheets and are specified in the PCI coordinate system.

4.2.5.2 Error Budget

The error budget input consists of two parts: the firstis the set of 19 conversion

constants K 1..... KI9 which are used to scale and properly dimension the driftrate

error matrices [_i], [_2], and [_P3];and the acceleration error matrices [YI], [Y2],

and [Y3] • These 19 conversion contants must be consistent with the units of the

error budget proper ,%1..... %51; e.g. (meru's) 2, _g's2 [degrees/hr/g)] 2, [arc-

seconds] 2, etc. The use of conversion constants permits the error budget to be

entered directly in convenient units. This convenience puts an added responsibility

on the program user, but use of the program at AC Electronics has justifiedthis pro-

cedure. For thisreason, the units for K I,..... ,K19 and kI..... %51 are not specified

on the inpat sheets. The error budget values are entered as variances ((y2), no__tas

standard deviations (I(Y). In general, K i is of the form A/B where A provides the cor-rect dimension, either rad/sec or ft/sec 2 to an element of the error matrix; and B is

the scaling into units of the l(y error source.

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AC ELECTRONICS OIVISION QENERAL MOTORS CORPORATION _

4.2.6 Reference Trajectory and State Transition Matrix Definition

4.2.6.1 Reference Trajectory

The interface with the reference trajectory is only concerned with the selection of the

proper reel of tape, which is a function of the computing facility procedure and the

selection of the desired trajectory on that reel of tape, since several reference tra-

jectories can be recorded on the same reel of tape. Such trajectory selection is made

by a run number identification and requires a coordination of information between the

user of the error analysis portion of 117.1 and whoever has made the reference tra-

jectory tape. The run number must be a decimal integral value r, 1. K r < 100. and

is entered on the input sheet on page 1.

4.2.6.2 State Transition Matrix

Flag F is used to select the state transition matrix model used in the calculations:

Flag F = 1, selects a state transition matrix that is input on the reference trajectory

tape. This state transition matrix is formulated for the reference trajectory in an

inverse square, spherically symmetric, planet-centered gravitational field. This

formulation supplies what is termed "gravity feedback _ in the calculation of the co-variance matrix P.

Flag F = 0 selects a simplified format for the state transition matrix, which is calcu-

lated in Block HI-6 of 117.1. This state transition matrix is constant with respect to

position and does not provide a coupling of position errors into gravity errors, as does

the alternate formulation. The input to (sheet 1) is used in this matrix and is set equal

to the time of the first trajectory record, which will, in general, be zero.

The use of Flag F = 1 is recommended as it will always provide the most accurate

result and its use does not increase computer run time.

4.2.7 Pro_ram Control

Input other than data specifying the physical system and its environment is classed as

program control data. Program control type data are the Rank, heading, At, tf, N,

Flag G, Flags 1 through 7, BUDGETS, PUNCH, and TAPE option.

The Rank input controls the amount of printed output and is discussed in Section 3.0

At is the input integration step size and is functionally related to the input N. N is an

integral number input which prescribes the number of reference trajectory data points

which are skipped as input to the program. For example, N = 1 will control the read-

ing of the reference trajectory tape so as to input the first, third, fifth, seventh, etc.

data points on the tape to the program. N = 3 will input the first, fifth, ninth, etc.

reference trajectory data points. The integration routine in the program will select

the minimum (At, time between input data points) as the actual integration step size.

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AC ELECTRONICS DIVISION GENERAL MOTORS CORPORATION _

Therefore, it is usually satisfactory to set At equal to the time interval between nom-

inal input data points of the reference trajectory. If a carousel system is employed,

it may be necessary to decrease At to accurately integrate Over the inputted tabular

values of 71, 72, 73, ¢Ol, 0)2, and ¢o3.

The computer time used by 117.1 is proportional to the actual integration step size

defined by a At and N. Values of At and N, resulting in integration step sizes of 2 to

8 seconds have been found satisfactorily accurate for various space booster trajectories

at AC Electronics. Larger At's are used when the trajectory has acceleration and

angular rates that have small variations in magnitude and direction. The large accel-

eration discontinuities at staging may be skipped if N _ 0. It is recommended that N = 0

for any space booster trajectories that involve staging.

The final time tf is nominally set to the last time on the reference trajectory record.

In certain cases, it is extremely difficult to stop the reference trajectory run at the

terminal condition desired, hence a procedure of letting the run proceed beyond this

condition is used. An examination of the trajectory record determines the time at

which the desired terminal condition is met and this value is used as tf in Block III

input, tf need not be a reference trajectory record time; any time less than or equal

to the final record time is the only criterion for tf.

Flag G controls the output print of _2 (to, tf). This print is not controlled by Rankinput.

Flags 1 to 7 control data dumps at various points in the program cycle. Flags 1 to 6

set equal to 1. will trigger a data dump after each calculation of Blocks III-ItoIII-6,

respectively. These dumps occur at each integration step. (See Level II Flow Chart

in Preliminary Program Description.) Flag 7 = 1 will trigger a data dump after

Block III-7 which occurs once only for each run. Flags 1 to 7 are used primarily for

a programmer's diagnostic purposes, although Flag 7 can also be used to obtain cal-

culated data that is not in the regular output.

BUDG specifies the number of budgets to be used on a single specified system and

a single trajectory. This input is only necessary on the first run of the set. (BUDG

I.) This option allows varying budgets without integratingthrough the entire tra-

jectory each time.

The punch option provides data output on cards of the final P-matrix or the _2 matrix

or both. An input of 0 means no punched cards desired, 1. provides the P-matrix

only, 2. provides the _2 matrix only and 3. provides both. The P-matrix (6x6) is out-

put row wise 3 to a card and the _2 matrix (6x51) is output column wise 3 to a card.

The tape option provides a means for reading the tape only, performing the error

analysis only or both. An input of 1. provides the error analysis only, 2. provides

reading and printing of the tape only and 3. provides both.

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AC ELECTRONICS DIVIBIONGENERAL MOTOR8 CORPORATION _

The second type of input required are the computer control cards.

At the end of each data set is an _END _ card and at the end of final data set a "FIN n

card is also used. The runs can be stacked withthe very first run consisting of a complete

input deck as specified by the input forms. Succeeding input need only be the data

which is different from the preceding run. When the budget option is used, succeeding

data can consist of a heading and changed budget inputs only.

The sequence of runs to be made by stacking input in this fashion must utilize refer-

ence trajectories that are all on one tape.

4.2.8 Program Output

The program printed output is controlled by the Rank input. The printed output carries

a Rank value with its descriptive title. For Rank = 0., the output consists of tf, [cOl](3x8), [_P2] (3x8), [_03] (3x8), P(6x6), the magnitude and direction cosines of the

principle axes for both the position and velocity error ellipsoids, the PINP (6x6) andits associated error ellipsoids.

For Rank = 1., the punched card input is printed in addition to the output for Rank = 0.

For Rank = 2., the [A4] matrix and the actual data points used from the reference

trajectory tape are printed in addition to the output for Rank = 1.

The arrangement of output and the titles are self-explanatory with the possible excep-

tion of the direction cosines of the principle axes. The direction cosines are listed

directly below the corresponding magnitude and are ordered in an X, Y, Z sense.

These cosines are defined with respect to the reference PCI coordinate system for the

[P] (6x6) and with respect to the downrange, crossrange, altitude system (defined by

the final position and velocity vectors) for the [PINP] (6x6).

4.2.9 Example

As an example of the use of program 117.1, two runs, selected from tradeoff studies

of strapdown inertial guidance systems, will be described in detail. The reference

powered trajectory is a deboost maneuver prior to earth atmospheric re-entry. The

trade-off studies varied error budget magnitudes and type of gyros to determine the

relationship to re-entry accuracy. A copy of the program output for this example isgiven in Paragraph 4.2.9.7.

4-7

AC ELECTRONICS DIVISION GENERAL MOTORS CORPORATION <%A_'_

4.2.9.1 Guidance System Specification

A strapdown system is specified by setting Flag A = 0.

4.2.9.2 Inertial Instrument Specification

A proof mass accelerometer model was selected by setting Flag E = 2. Single-degree-

of-freedom gyros were selected by setting Flag B = 1. Two types of gyros were com-

pared in the tradeoff studies. One run for each type is included in the attached

example. The first run uses a single-degree-of-freedom platform gyro (Flag D = 3),

the second run uses a rate gyro (Flag D = 2). The rate gyro parameter K o was set

equal to zero. Flag C is set to 1.0 to avoid an error return as 0 is an illegal entry

for Flag C.

4.2.9.3 Orientation Geometry

In the strapdown system, the orientation of the IMU reference coordinate system is

done automatically by the program as it selects the gimbal angles from the trajectory

tape as values for the Euler angles rl, F2, and F3 which define [C' ] (3,3). The engin-

eer must define the orientation of the six inertial instruments with respect to the IMU

reference coordinates by inputting six sets of Euler angles; Oql , _i2, °Q3 (i = 1,2,3)

for the three gyros and _il, ¢i2, _i3 (i = 1, 2, 3) for the three gyros. In these example

runs, instrument orientations along the reference body axes were defined by choosing

the values given in Table 1. The units of angle are radians. These values were

GYROS ACCELEROMETERS

°_il °_i2 °_i3 _il ¢i2 _i3

0. -1. 5707963 0. 0 -1. 5707963 0

0. 0. 1. 5707963 0 0 1. 5707963

1. 5707963 0. 1. 5707963 1.5707963 0 1. 5707963

Table 1

determined by aligning the I, O, S axes for gyros or the I, C, N axes for accelerom-

eters along the PIO, YAO, ROO axes, respectively, and defining a set of Euler angle

rotations which result in the desired orientation with respect to the P_ , YAkO" R ?hoaxes. The orientations defined in Table 1 puts the input axis of gyro 1 aIong

roll axis, gyro No. 2 input axis along the pitch axis, and gyro No. 3 input axis along

the yaw axis. The accelerometers have the same orientation.

4-8

AC ELECTRONICS DIVISION GENERAL MOTORS CORPORATION <._'>

The initial misalignment axes were selected to be the PIO, YAO, ROO axes by setting

fill = ill2 = ill3 = 0. A minimum input of 4 entries is required in the tables for _l(t),

_n_ (t_. _n It_. co (t). £o_ (t_. and co _(t_ to avoid error return on table reading ]ngin.'Z .... 3 ..... 1 .... Z ..... 3' " ........................

4.2.9.4 Initial Covariance

The initial errors in position and velocity were defined to be zero by setting each

element of [Po 7 equal to zero in the input sheets.

4.2.9.5 Error Budget and Conversion Constants

The specification of values for the error budget variances )`1, • • • ,)`51, and the

conversion constants is dependent upon the units selected for the error budget. The

basic unit of gyro drift error budget was selected to be the meru [1/1,000 of earth

angular rate, equivalent to 0.015 degrees/houri; the basic unit of accelerometer

error was selected to be the ug [(1/106)(32. 174 ft/sec2)]; and the arc-second is the

unit of angular misalignment. With these as basic units, the error budget units are:

GYRO

}'1 -* (meru)2

}'2 -* (meru/g)2

)`3 -' (meru/g)2

)`4 -* (meru/g2)2

2)`5-* (arc-sec)

)`6-_ (arc-sec)2

)`7 -" (10-6)2

)`8 -_ F10- 6/(rad/sec) ] 2

)`9 -* )`16

)`17 -* )'24

ACCELEROMETER

)`25 -_ (_ag)2

)`26-"(ug/g)2

bias drift rate

mass unbalance

mass unbalance

anisoelasticity

input axis misalignment

input axis misalignment

rate gyro scale factor uncertainty

rate gyro nonlinearity

second gyro

third gyro

bias acceleration error

scale factor uncertainty

4-9

AC ELECTRONICS DIVISION_j'7

GENERAL MOTORS CORPORATION _A(_)

}`27-_(ug/g2)2

}`28-'(ug/g2)2

2

}`29-'(arc-sec)

2

}`30--' (arc-sec)

input axis nonlinearity

cross axis nonlinearity

input axis misalignment

input axis misalignment

for proof mass accelerometer

for proof mass accelerometer

}`33 _ }`40second accelerometer

}`41 _ }`48third accelerometer

INITLA L MISALIGNMENTS

}`49'}`50'}`51

2-. (arc-sec)

With this choice of units, the values of K 1

-57.2921149 x 10

K1 = 1000

-57. 2921149 x 10

K2 = (1000)(32. 174)

-_ K19 are:

-8= 7. 2921149 x 10 (rad/sec)/meru

= 2. 2646320 x 10 -9 (rad/sec)/(meru/g)

K3 = K 2 = 2. 2646320 x 10 -9 (rad/sec)/(meru/g)-5

K4 = 7. 2921149 x 10 = 7. 0330184 x 10 -11 (rad/sec)/(meru/g 2)(1000)(32. 174) 2

-6K 5 = K 6 = 4. 8481368 x 10 (rad/sec)/arc-sec

K 7 = 10 -6 (rad/sec)/(10 6 rad/sec)

K 8 = 10 -6 (rad/sec)/(rad/sec) 2 10-6/(rad/sec)

K9

_ 32. 174 _ 3. 2174 x 10 -5 (ft/sec2)/Ug106

-6 (ft/sec2)/(ft/sec2)(ug/g)K10 = 10

1 -8= = 3. 1080997 x 10

Kll = K12 (32. 174) 106

(ft/sec2)/(ft2/sec 2) (ug/g 2)

4-10

AC ELECTRONICS DIVISION GENERAL MOTORS CORPORATION _.41tl_)

KI3 = K14 = 4.8481368 x 10-6 (ft/sec2)/(ft/sec2)(arc-sec)

--15' --16 _........... = .................... I

-6K17 = K18 = K19 = 4. 8481368 x 10 (rad)/(arc-sec)

The error budget magnitudes that are entered on the input sheet were selected by theengineer, and are entered as variances (la) 2.

4.2.9.6 Program Control

The reference trajectory is characterized by %mooth n behavior of acceleration and

angular rates. The data points on the reference trajectory tape are at 2-second

intervals. Therefore, a large effective At may be used to obtain accurate results.

Input values of At ---8. and N = 3. were selected to produce an actual machine At of 8

seconds as the time interval between input data points is also 8 seconds, to -- 0. and

tf = 418. were values corresponding to data points on the reference trajectory tape.

A run number corresponding to the run number of the reference tape is input as 6.

The input data deck was assembled in the following sequence, since two successive

runs were desired.

Data for run 7D-SDF-10-30

END

Data for run 7E-RATE-3-10

END

FIN

4.2.9.7 Output

The actual output for the example just described is reproduced in this paragraph.

The amount of output is controlled by the Rank input which was set to 1.0. This value

gave all the output except the print of the reference trajectory data which had previously

been recorded. All punched card input is printed and the computed output of [_i 7 (3x8)(i - 1, 2,3) and [P_ (6, 6) with the principal axes of position and velocity error ellipsoids

and their orientations is given; e. g., in the first run the largest position semi-axes is

505.23602 feet, and has direction cosines -0. 22361882, 0. 83163428, and -0. 5083090991

in an X Y Z ordering in the PCI coordinate system.

Flag G was set at 1.0 to obtain an output of _2(tf, to) (6, 51). This output is actually

printed in (51, 6) order to conform to the dimension of machine paper. This matrix

shows the position and velocity errors due to the 51 independent error sources.

4-11

AC ELECTRONICS DIVISIONGENERAL MOTORS CORPORATION _>

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4-27

ELECTRONICS OIVISlON GENERAL MOTORm CORPORATION _

5.0 REFERENCES

. BSD TR 64-347; "Effect of Rotation on Velocity Error in a Pulse Inte-

grating Gyro Accelerometer [PIGA]. Mobile Mid Range Ballistic

Missile Guidance and Control Subsystem n, Systems Division, General

Precision Aerospace. GPI Document 823.077-015, ContractAF 04(694)-55830 June 1964

5-1

APPENDIX A

COMPUTER PROGRAM 117.1

A. 1 INTRODUCTION

The material in this appendix is designed to furnish the data required by a program-

mer for a detailed understanding of Program 117.1. A complete program listing is

given, supplemented by descriptions of utility subroutines, tape format, and opera-

tional guide/or an IBM 7094 Mod II. This data will facilitate the operational applica-tion of Program 117.1 and aid the programmer in possible future modifications oradditions.

A-1

A. 2 PROGRAMMER'S OPERATIONAL INFORMATION

A. 2.1 SYSTEM CONFIGURATION (IBSYS VERSION 13)

FUNCTION SYMBOL PHYSICA L

Library 1 SYSLB1 A1

Library 2 SYSLB2 Unassigned

Library 3 SYSLB3 Unassigned

Library 4 SYSLB4 UnassignedCard Reader SYSCRD RDA

On-Line Printer SYSPRT PRA

Card Punch SYSPCH A0

Output SYSOU1 A3

Alternate Output SYSOU2 A3

Input SYSIN1 B3

Alternate Input SYSIN2 B3

Peripheral Punch SYSPP1 B4

Alt. Peripheral Punch SYSPP2 B2Check Point SYSCK1 B5

Alternate Check Point SYSCK2 B5

Utility 1 SYSUT1 A4

Utility 2 SYSUT2 B1

Utility 3 SYSUT3 A2

Utility 4 SYSUT4 B2

Utility 5 SYSUT5 Unassigned

Utility 6 SYSUT6 Unassigned

Utility 7 SYSUT7 Unassigned

Utility 8 SYSUT8 Unassigned

Utility 9 SYSUT9 Unassigned

ATTACHED UNITS NOT ASSIGNED OR RESERVED

A5 B6

A6 B7

A7 B8

A8 B9

A9 B0

INTER SYSTEM RESERVE UNITS

NONE

LOGI CA L

FORTRAN II FORTRAN IV

1

6 6

5 5

7 7

4 1

8 2

2 3

3 4

A. 2.2 PROGRAMMER:S _un.Jr.,........_ur_ 117. i

Error Conditions

At Error conditions result in program dump in floating point form with

XR4 being key to location where error occurred.

B. Hints on Error Conditions

1)

2)

3)

4)

5)

Make very sure input data is correct

XR4 should tell where error occurred

Make sure end cards END and FIN are present and used

correctly

Double check deck makeup

Table input data is frequent error on engineer's part.

A-3

A. 2.3 OPERATOR'SGUIDE FOR 117.1 ON IBM 7094MOD II

I Machine Configuration {System Requirements)

A. Channel A

1) A1 = IBSYS VERSION 13

2) A2 = Utility tape

3) A3 = List tape (output print tape)

4) A4 = Utility tape

B. Channel B

1) B1, B2, and B5 = Utility tape

2) B4 = Punch tape (Card output)

3) B3 = Card-to-tape (Input)

C. Core Storage

1) 32K

II Deck Setup

A. Control Cards at Beginning of Deck

1) _DATE

2) _IBSYS

3) _RESTORE

4) _OB

5) Sm

6) _EXECUTE

7) IBJOB

A-4

II Deck

Bo

Setup (con't)

Program Deck

1) NINPUT

2) NL_k_IC

3) I93Z

4) MTAPEW

5) CPUNCH

6) NEVAL

7) NINTL

S) NPRCON

9) 79NSUB

10) 31NSUB

1.1) 32NSUB

12) 33NSUB

13) 34NSUB

14) 36NSUB

15) 37NSUB

16) 38NSUB

17) NTAPER

18) N_UTPT

19) ONOUT

20) IN(_UT

21) EIGEN

22) INVERT

23) ADM.AT

24) SBMAT

25) TRMAT

26) YMATMY

27) AMATMY

A-5

II Deck Setup (con't}

B. Program Deck (con't)

C,

H,

Io

28) INTP

29) NINTG

Control Cards at End of Program Deck

1) _ENTRY

2) 7/8

3) Data Cards (Not Control Cards)

4) 7//8

5) IBSYS

6) _ENDFILE SYSOUI

7) _ENDFILE SYSPPI

8) _ST(_P

Devices Used by Program

1) Fortran Logical 2 tape unit = B1

Built-in Pauses

1) Pause 77777 with 77777 right justified in AC means to make

sure either a special tape or a blank tape is to be mounted on B1, then

hit start.

2) Pause 66666 with 66666 right-justified in AC means the problem is

completed and save tape from B1 before going to next job; then

hit start.

A-6

A.2.4 TAPE FORMAT FOR 117. i (58WORDRECORD)

Record No: 1 - Heading Record (All Floating or BCI)

i) -3.0

2) Run No.

3) Phase No.

4-13) Heading

14-54) 0

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I)2)3)

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8)

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Run No.

Phase No.

PIX o 1

PIYo

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YAY o

YAZ o

ROXo

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ROZ °

0

4 BCI

A 4 matrix in row form

Blank Words

A-7

Record No. 3 and All Others Except Special End Records

i) TIME

2) Run No.

3) Phase No.

4-6) ¢_PI, _YA, wRO

7-12) X, Y, Z, X, Y, 7.

13-18) ax, ay, az, Oil, _2' (x3

* 19-54) -1 -I -1 _P41'-1-1 _61-1toll' _021' _031' _051'

)¢P12'

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In Error Analysis Program

Read in as Transpose

Record N (Record Separating Cases) Special End Record

1)

2)3)4)

Time = 1. E20

Run No.

Phase No.

Remaining words are garbage thru word 58

Record (N + 1)

I)

2)

If end of all runs duplicate of record N

If another case same format as preceding case on tape.

* Note:-I -1 -i

(to-t)is in ¢P41'_°52'_63This option is built in error analysis portion

of program

A-S

AC ELECTRONICE DIVISION GENERAL MOTORS CORPORATION _

A. 3 PROGRAM LISTING, PROGRAM 117.1

The original of the compilation listing has been supplied with the program decks.

A-9

AC ELECTRONIC8 DIVI§ION QENERAL MOTORS CORPORATION _

APPENDIX B

Input Forms for Program 117.1

B-1

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