^Research Mathematician, Military Products Group Research Deot* I25

TECHNIQUES F O R E R R O R ANALYSIS OF TRAJECTORIES A r n o l d Peske^- and M i c h a e l W a r d ^

M i n n e a p o l i s - H o n e y w e l l R e g u l a t o r C o . , M i n n e a p o l i s , M i n n . A B S T R A C T

A n error analysis technique is presented w h i c h can b e used to predict the p r o p a g a t i o n of errors along a trajectory in terms of the e r r o r v e c t o r at the end of the trajectory. The method, called the conjugate solution method, uses p e r t u r b a - tion theory and assumes that the equations of disturbed motion result in a solution that remains in a region of linear a p - proximation about a given nominal trajectory. The e r r o r a n a l - ysis technique can b e used to predict the effect of random errors as w e l l as non-random e r r o r s . A s an example, the tech- nique is applied to the analysis of error p r o p a g a t i o n along a typical ICBM trajectory. The application of the conjugate so- lution technique and conventional techniques to similar p r o b - lems are evaluated qualitatively through a comparison of com- p u t e r requirements and in the ease of evaluating the d a t a .

INTRODUCTION

A n investigation of the p e r f o r m a n c e of a complex p h y s i c a l system generally involves the solution of a set of equations that describe the system m a t h e m a t i c a l l y . It is usually n e c e s ^ sary to make approximations and assumptions in the mathematical statement of the system to simplify the e q u a t i o n s . F o r e x - ample, in the p r e l i m i n a r y design of a ballistic missile a r e l - atively simple set of d i f f e r e n t i a l equations is derived from physical l a w s . These equations are set up in a suitable coor- dinate system, and solutions are found. Some of the assump- tions a r e :

1 ) System parameters are known or can b e measured exactly.

2) Certain random disturbing forces in the measurement of the system p a r a m e t e r s , e.g., radar n o i s e , are absent.

Presented at ARS G u i d a n c e , C o n t r o l and Navigation C o n f e r e n c e , Stanford, C a l i f . , A u g . 7-9, 1961.

•^-Research E n g i n e e r , M i l i t a r y Products Group Research D e p t .

^Research Mathematician, Military Products Group Research Deot*

I25

3) Gravitational and atmospheric forces can b e computed exactly.

h) Random disturbing forces external to the system, e.g., wind or deviations from a standard atmosphere, are absent.

It is thus necessary to evaluate the system b y determining the effect of these simplifications in the form of an error a n a l y s i s . From a knowledge of the errors introduced in a s y s - tem, component tolerances can be specified, and a system can b e designed that meets a desired p e r f o r m a n c e specification.

A n o t h e r factor that arises in the analysis of error p r o p a - gation along t r a j e c t o r i e s , e.g., during the boost phase of a ballistic m i s s i l e , is that the equations of motion have t i m e - varying coefficients. Thus the conventional frequency t r a n s - form methods are not a p p l i c a b l e , and analysis techniques must be based on time-domain descriptions of the system and the e r r o r s .

M E T H O D S OF ERROR ANALYSIS

There are several methods available for time-domain analysis of systems described b y differential equations w i t h time-

varying c o e f f i c i e n t s . E v a l u a t i o n of these systems requires solution b y digital c o m p u t e r s .

These methods are described in literature dealing w i t h p a r - ticular m i s s i l e s y s t e m s . T h e methods have general applica- tion, although they are described here w i t h application to the boost phase of a ballistic m i s s i l e . M o o r e (l)3 describes one technique that uses the original system equations w i t h a p p l i - cation to the propagation of burnout errors to impact t i m e . Perturbation methods are described b y A n d e r s o n (2) in evalu-

ating inertial guidance s y s t e m s , and b y Rosenberg (3) for com- puting trajectories in the neighborhood of a nominal t r a j e c - tory. T h e methods using adjoint techniques are described b y B l i s s (4) with application to free fall missile b a l l i s t i c s , b y M a r s h a l l (8) w i t h an extension to the evaluation of errors in- troduced b y uncertainties in physical c o n s t a n t s , and b y P f e i f - fer (6) w i t h application to t h e guidance p r o b l e m for a b a l l i s - tic m i s s i l e . Laning and B a t t i n (7) describe application of adjoint techniques to the study of random inputs to linear systems. The purpose of this p a p e r is to discuss the several methods and to evaluate them as an aid in selecting the m e t h o d best suited to a p a r t i c u l a r p r o b l e m .

^Numbers in parentheses indicate References at end of p a p e r .

O n e -method that is g e n e r a l l y used involves the repeated so- lution of the n o n l i n e a r equations of m o t i o n of the system.

The results are used for investigating: l) p r o p a g a t i o n of initial condition errors and guidance sensitivities; 2) e v a l - u a t i o n of instrument errors; and 3) evaluation of C E P .

A n o t h e r method makes the assumption of linear perturbations about a nominal trajectory. A set of p e r t u r b a t i o n equations is derived from the n o n l i n e a r equations of m o t i o n , and m a n y solutions of these equations are found. The results are used for investigation: l) p r o p a g a t i o n o f initial condition errors;

2) evaluation of instrument e r r o r s ; and 3) evaluation of C E P . The conjugate solution method emphasized in this p a p e r uses the linear a p p r o x i m a t i o n of p e r t u r b a t i o n techniques and re- quires the solution to the adjoint equation. T h e result is the c o n j u g a t e , or inverse, fundamental solution m a t r i x to a set of l i n e a r d i f f e r e n t i a l equations w i t h time-varying coef- f i c i e n t s . This solution is basic in solving f o r : l ) p r o p a - gation of initial condition e r r o r s ; 2) a set of "guidance sensitivity" functions; 3) p r o p a g a t i o n of errors due to small forcing functions that affect the system; h) evaluating a CEP for a m i s s i l e system from knowledge of expected accuracy of measurement instruments; and 5) analysis of random inputs that affect the system.

T h e conjugate solution m e t h o d has two m a j o r advantages over either of the o t h e r two m e t h o d s : l) fewer solutions of the equations of m o t i o n o r the p e r t u r b a t i o n equations are needed for the e r r o r p r o p a g a t i o n and guidance sensitivity studies t h a n in e i t h e r of the other methods (reducing the cost of a- nalysis b y cutting d o w n on computation t i m e ) ; and 2) the funda- m e n t a l solution m a t r i x used for the e r r o r p r o p a g a t i o n is also used in the guidance sensitivity studies (giving a smooth transition from the study of e r r o r p r o p a g a t i o n of the compu- tation of s e n s i t i v i t i e s ) . T h e r e is n o need for additional derivations as is necessary in the o t h e r m e t h o d s .

Generalized S y s t e m E q u a t i o n s

A system is considered w h i c h is described b y a set of η first-order, ordinary d i f f e r e n t i a l equations

9 * * *,xn ' ^

[ l a ] dt

dX2

• · · y

127

d xⁿ

= fⁿ (X-l, x², . . . , xⁿ, t ) [lc]

at

The coordinates x^ are choosen so that the rate of change of the coordinates is adequately described by functions of the coordinates and time; i.e., these coordinates are a good rep- resentation of the system parameters of interest. The equa- tions are in general, nonlinear and must be solved on a dig- ital computer. The coordinates x^ in the case of a ballistic missile during injection, for example, could be position co- ordinates, the velocity coordinates, vehicle attitude and at- titude rate, mass and mass rates. The particular coordinates chosen will depend on factors such as the precision of control of the system and the feasibility of measurement.

A system of equations of order higher than first can be re- duced to a first-order system by defining new coordinates.

For example

This can be reduced to a system of first order differential equations by defining

x

l =

dx., χ « _ ±

dt

[3a]

= f1( x1, x2, t ) [3b]

* *² [3c]

= ^fn(^,^0,t) = f(x,tj

dt ^{ά 1}

It is convenient to write a system of equations such as E q s . 1 in a more compact form, using matrix notations

χ = [ χ^χ x² ... x ⁿ ] ^T f = [ f^x f² ... f ⁿ ] ^T Then E q s . 1 can be written in the form

g - f(x,t) [ 4 ] Let E q s . 1 or k represent the motion of a ballistic missile

during injection. For a particular set of initial conditions on the η coordinates, a solution x(t) will be found. The trajectory x(t) may be optimized in some manner, such as re- quiring minimum fuel. By imposing constraints on the coordi- nates, e.g., programmed thrust for a particular attitude pro- file, the trajectory will terminate at some prescribed point

in η space χ(τ). This trajectory represents the performance of the missile within the accuracy limits of the mathematical description of the system.

Suppose that there exists a neighboring solution that is dis- placed by a small amount from the nominal trajectory x ( t ) . This means that x^-x^ are bounded by some small positive con- stant along the trajectory in the interval of interest. Then

ι

Conjugate Solution Method of Error Analysis

The equations of disturbed motion, or the first variational equations, about x(t) can be obtained by expanding Eq. 1 in a T a y l o r¹s series about x(t) and truncating after the first-order terms. This gives the linear vector equation

= A ( t ) A x ( t ) [5]

where

x-x = Δ χ = j j^ x ^ A x ^ . . . A xnJ ^

V i

^

* x 7

E q s . 5 are a set of linear, first-order differential equations with time-varying coefficients. They represent the perturbed motion, in a linear sense, of the vehicle about the nominal trajectory x ( t ) . The coefficients a ^ t ) are evaluated from a knowledge of the parameters of the reference trajectory, such as velocity, position, attitude, and acceleration.

A set of η independent solutions to Eq. 5 > Δ χ ' ^ ( ΐ ) , Δχ(2) ( t ) , . . . , A x (ⁿ) ( t ) , and arranged in a matrix gives

7 T ( t ) = [ A x O) ( t ) A x <²) ( t ) . . . A x (ⁿ) ( t ) ] [6]

For the particular set of boundary conditions 7r(t) = I, the η χ η identity matrix 7 T ( t ) is a fundamental solution matrix to Eq. 5, and 7Γ (t) will satisfy Eq. 5, namely

άΤΓ ^A

,^

^ W

— = A(t) 7Γ (t) 129

and for any boundary condition A x ( t = T )

A x ( t ) = 7T(t)

Δχ(τ)

H e r e , the boundary conditions are set u p at t = T , since a s y s - tems analysis is primarily concerned with the value of the miss v e c t o r at the end o f a trajectory. If 77*(t) is a n o n s i n g u l a r matrix, the inverse matrix

ir'^it)

ir"

(t) A x ( t ) = Δχ(τ) [9]

7r-1(t)7T(t) = 7Τ( t) 7T"^-( t) = I [ ]0] since

the η χ η identity matrix, b y definition o f the inverse matrix, for a!3 t. T h u s , the terminal miss v e c t o r Δχ(τ) can b e evaluated for a n y measured p e r t u r b a t i o n on t h e trajectory, b y the matrix multiplication indicated b y E q . 9 · H o w e v e r , this involves inversion of the fundamental solution matrix 7T(t) f o r all points of interest. T h e inverse, or conjugate, fundamental solution matrix can b e obtained for all time t in t h e interval

[θ,τ] b y the solution of a set of linear differential equa- t i o n s .

Differentiating E q . 3.0 with respect to time gives d_ .-1

dt

\rr{t)Tr-Ht)]- J7-7?\t)+7r(t) §JQ = o [11]

w h e r e 0 is the η χ η null m a t r i x . Post multiply E q . 7 b y

ir-Ht)

§f ^π-Ht)

F r o m E q s . 11 and 12

- 7 T ( t ) * ^ - A ( t ) [13]

Premultiplying E q . 2 b y - 7r"1(t) gives

f f — = -TT-Ht) A(t) [lk]

E q .

lk

7,

TT^'it)

equations, as y e t , unspecified. Postulating a set of l i n e a r differential equations to correspond to E q . lk; let

t r - - *<*>A(t) [15]

H e r e , X ( t ) = [^X^tji λ 2( t ) ·.. Xn( t) J is a row v e c t o r . Now arranging η independent solutions of E q . 15 in a matrix

7Γ •Ht) = p> ( t )x ( H )( t ) . . . x(n( t ) ] ) Τ

w h e r e superscript Τ indicates the transposed v e c t o r it is seen that f o r the p a r t i c u l a r b o u n d a r y conditions 7Γ"^(Τ) = I,

ir~^(t)

7T~l(t)

Thus it is seen that 7T"^-(t) can b e generated as a contin- uous function of t i m e , b y solving E q . 15 for the η b o u n d a r y c o n d i t i o n s , such that 7Γ~1(Τ)=Ι· Polynomials can b e fitted to the tabulated values o f 7 T ~ l ( t ) f o r convenience as desired for later parts of the a n a l y s i s . T h e n , from E q . 9^} the ter- minal miss v e c t o r can b e evaluated b y the indicated matrix multiplication. Inversion of the m a t r i x 7T(t) is not required.

In fact, it is n o t necessary, at this p o i n t , to have the s o - lution matrix 77"(t) to E q . 5· It is only necessary to solve for the η solutions of E q . 15· T h e system of e q u a t i o n s , E q . 15, is the adjoint equation to E q . 5· E q . 9 is a basic for- m u l a for the conjugate solution technique of e r r o r a n a l y s i s .

If E q . 9 is written in component form, the usefulness of the variable X ( t ) can b e seen. L e t the jth component of the v e c t o r \ ( i ) ( t ) b e donoted b y ^ i j ( t ) . Then [TT-Mt)] ij =

\±j, and expanding E q . 9

η

^ X i j ( t ) AX j( t ) = A X i ( T ) [ l 6 ] j=l

That is A x ( t = T ) is a function of Δχ^ and t, and if the partial derivatives of Δ χ ^ τ ) are taken with respect t o A x j ( t ) for fixed time t, it is seen that

T h u s , the functions ^ i j ( t ) can b e regarded as sensitivity functions.

1 5 1

The boundary conditions are set up at time T, i.e., at the end of the trajectory, and E q . 15 is solved b a c k w a r d s in t i m e . Solution w i t h t h e b o u n d a r y condition,

7Γ"^(Τ)

Δχ^(τ)

Alternative Methods of E r r o r Analysis

For the first of these other m e t h o d s , the missile is assumed to b e d e s c r i b e dAb y the system of equations E q s . 1 · A n o m i - nal trajectory x ( t ) that meets some performance criterion is solved. T h e n using small perturbations of the initial c o n d i - tions Δχ° = xi ( t0) - xi ( t0) j the nonlinear set of equations is again solved. This is done repeatedly for m a n y perturbed initial conditions about the nominal trajectory. A functional form is assumed for the burnout error, for example

Δ χ^ Τ ) = ^ K^±3

Δχ°

j=l

The form of these equations could b e expanded to include second- order terms, b u t for evaluating system accuracy, a l i n e a r set o f equations such as E q s . 19 is usually assumed. T h e data from the solutions to E q . 1 for the various initial conditions is fitted to the E q . 19 w i t h conventional curve fitting tech- n i q u e s , e.g., using weighted least squares t e c h n i q u e s . It should b e noted that the coefficients K^* only hold for the nominal trajectory, and f o r only one p a r t i c u l a r t i m e . To get a set of functions equivalent to the λ ^ j of E q s . 19, this p r o c e s s would have to b e repeated w i t h new initial conditions corresponding to points along the trajectory. Enough points are taken so curves can b e fit through them. Then the coef- ficients can b e fitted with p o l y n o m i a l s .

Alternatively, in the perturbation method described above, E q s . 1 are solved to give t h e nominal trajectory x ( t ) . T h e n , a set of perturbation equations are derived from the nonlinear set. This gives the system of e q u a t i o n s , E q s . 5· F o r each time of interest, η solutions to E q s . 5 are obtained to give a set of linear p r e d i c t o r coefficients as in E q s . 19· E q . 5 is solved repeatedly to obtain the coefficients ICy as a function of time along the trajectory, since for each time ( tQ) only the coefficients K j j for t = tQ are obtained.

The advantages of the conjugate solution m a t r i x at this point a r e :

1) The error p r o p a g a t i o n c o e f f i c i e n t s , the sensitivity functions, are obtained b y one set of η solutions to the adjoint set of differential e q u a t i o n s .

2) L e s s d a t a processing is necessary, because only the reference trajectory and the sensitivity functions need to be curve fitted.

EVALUATION OF ERRORS D U E TO D I S T U R B I N G FORCES A L O N G THE TRAJECTORY

In general, the p e r t u r b a t i o n equations are a set of non- homogeneous equations

The disturbance function u ( t ) is quite general and can repre- sent many things of interest. F o r analysis of the control system, u ( t ) can represent disturbing forces, such as wind shears or m o t o r v i b r a t i o n s , or can represent control forces t h e m s e l v e s . F o r analysis of the guidance system, u ( t ) can represent p o s i t i o n and velocity uncertainties of radar or op- tical measurements were u s e d . F o r inertial m e a s u r e m e n t s , u ( t ) can represent errors due to drift in the reference p l a t f o r m and errors due to b i a s , nonlinearity, e t c . , in the velocity s e n s o r s . T h e s e forces may b e included in the original formu- lation of the system equations E q s . 1 . The 7r(t) m a t r i x is then computed from the homogeneous part of the nonhomogeneous set that results from the truncated T a y l o r¹s series expansion.

The m a t r i x B ( t ) is an η χ η m a t r i x that couples these disturb- ing forces or control forces to the system, as coordinate t r a n s f o r m a t i o n s . B y p r o p e r choice of the coordinates of the system, E a . 20 m a y b e w r i t t e n

T h e effect of these disturbing f o r c e s , or forcing functions, can b e evaluated from the equation (see R e f . 8, p p .

7Γ ( t^Q) A x ( t^Q) represents the initial conditions for the in- tegration of this equation. E q . 22 is the second basic

d A x

dt = A ( t ) A x ( t ) + B ( t ) u ( t ) [20]

d A x

dt « A ( t ) [ Δ x ( t ) + u(t)] [21]

155

equation for the conjugate solution method of error analysis.

It is the equation that is used to solve for the terminal miss vector due to any deterministic disturbing forces that affect the missile along the trajectory. In particular, for t = T, noting that 7Τ(τ) « I

"T -

Δχ(τ) = ΤΓ'Ητ)λ(τΜτ)άτ

t ^{0 0}

To compute the propagation of errors due to forcing functions u(t) which are functions of time in the interval [ θ , Τ ] , it is necessary to have the solution 77"(t) to E q s . 5 and to use Eq.

22. This requires η computer solutions to E q s . 5 corresponding to the boundary conditions 7Τ(τ) = I. For errors at burnout, t = T, no additional preliminary computations are necessary to solve E q s . 23, since all the functions required for the equation are known from the previous analysis.

For each additional input u ( t ) , a set of equations is de- rived to determine how the function u(t) affects the system.

The errors introduced are assumed small, so that linear ap- proximations can be made. The result is a system of linear differential equations, which are solved to determine the effect of the forcing functions. As an illustration, the effect is considered of accelerometer bias and misorientation of the in- ertial platform.

The basic equation is of the form

At

=

where is the true acceleration acting on the system, A^& is the acceleration sensed by the acceleroraeters, and g is the gravitational acceleration. If the deviation from the true acceleration is denoted by Δ Α^&, then

Δ Α^Τ= Δ Α ^{& + Ag} [25]

The term A ^g enters because an error in sensed acceleration causes an error in velocity and position calculation, and hence, in the computation of the gravitational force. The er- ror introduced by misorientation of the accelerometers can b e denoted by the transformation Δ0ΧΑ&, where Δ 0 is the matrix that transforms the measured acceleration from the erroneous coordinates of the platform to the desired coordinates. The resultant equations are set of linear vector differential equations, which are of the form

fl£2-g(Ax,t) [26]

A d d i t i o n a l terms for each of the e r r o r s o u r c e s , as acceleration and nonacceleration sensitive gyro drift and a c c e l e r o m e t e r er- rors are added to E q s . 26. If the e r r o r sources are considered independent, separate equations can b e derived for the individ- u a l e r r o r s o u r c e s . E q s . 26 are then solved to evaluate errors due to these e r r o r s o u r c e s .

Comparing this method w i t h the conjugate solution method, it is seen that for arbitrary deterministic inputs, additional e- quations must b e solved in b o t h c a s e s . In the conjugate solu- tion method, the equation is E q . 22. F o r the conventional m e t h - o d s , the equation is a l i n e a r differential equation of the form of E q . 26. If the n o n h o m o g e n e o u s equation, in the conjugate solution method, is of the form of E q . 20, the derivation of the transformation m a t r i x B ( t ) is equivalent to the derivations required in the o t h e r methods w h i c h lead to the differential e q u a t i o n s , E q s . 26. H o w e v e r , b y judicious choice o f the system coordinates in the original formulation of the system e q u a t i o n s , E q s . 1 no additional derivations are necessary in the conjugate solution method, since E q . 21 can b e u s e d .

H e r e again, the conjugate solution method has advantages over the conventional m e t h o d . T h e solution of the p e r t u r b a t i o n e- quations for the fundamental and inverse fundamental solution matrices leads directly to a set of v e c t o r e q u a t i o n s , E q s . 22, w h i c h are used to evaluate the errors due to small arbitrary deterministic inputs to the system.

EVALUATION OF THE CIRCULAR P R O B A B L E E R R O R

W h e n certain p a r a m e t e r s of the system h a v e a p r o b a b i l i t y uncertainty, only a statistical knowledge of the errors caused b y these p a r a m e t e r s can b e found. The C E P , circular p r o b a b l e error, is defined as the circle about the terminal point where the p r o b a b i l i t y that the trajectory ends w i t h i n this circle is O.5O. F o r evaluation of C E P , all of the analysis methods d e - scribed above use equivalent t e c h n i q u e s .

It is supposed that there are a number of independent error sources characterized b y coefficients that are fixed for a p a r t i c u l a r source, b u t that are statistically distributed for an ensemble of s o u r c e s . It is further supposed that the d i s - tribution o f each of these sources has a zero mean v a l u e . Examples are gyro drift coefficient and accelerometer nonlin-

earity coefficient.

155

F r o m an analysis of the sources acting on the system, the contribution to the terminal error from each source w i l l b e found. If the sources are independent, the total average error from the sources will b e the sura of the contribution of the average of the individual s o u r c e s .

T h e Central Limit T h e o r e m states that for a large number of independent random functions, the p r o b a b i l i t y distribution of the sum of these random functions approaches a normal d i s t r i - bution regardless of the distribution of the individual s o u r c e s . B y the Central Limit Theorem, the terminal error due to all of the previously mentioned sources w i l l b e approximately normally distributed. A l s o , since the sources are independent, the variance of the total error w i l l be the sum of the individual v a r i a n c e s . It should b e noted that the v a r i a n c e , for a random function w i t h zero m e a n v a l u e , equals the mean squared value of the function. W i t h the mean value and the mean squared value of the individual sources, the multidimensional normal distribution for the output can b e determined. T h u s , the CEP can b e computed b y integration of the multivariate normal d i s - tribution function to determine the value for w h i c h the miss probability is 0.50.

STOCHASTIC INPUTS

O n e of the p r i n c i p a l advantages of the conjugate solution method is that the equations can b e extended to stochastic i n p u t s . Consideration is given to E q . 22, for w h i c h u ( t ) is a random function for w h i c h the statistical p r o p e r t i e s , in the form of a correlation m a t r i x , are k n o w n . The following d e r i - vation shows h o w the m e a n squared error at time Τ due to ran- dom inputs can h e evaluated.

The mean or expected value of a function f[x(t)]is defined as

Ε [f(x)] = f °° yO(x,t) f(x)dx [27]

w h e r e p(x,t) is the probability density function of x ( t ) . F o r a function of η variables f ( x1, x2, . . . , xn) , the expected value of the function is defined as

Ε [ f ( x ! , x2, . . . , xn) ] =

η

• · · ,

w h e r e ρ ( x1, t1; x 2 , t2; · · · >χ η> ^η) i s the Joint probability density function of the variables x ^ .

F r o m E q . 23

Δχ(τ) =

w h e r e the initial condition notation is dropped for convenience in the derivation.

One c o m p o n e n tΔ χ^ Τ ) is taken o f the function Δχ(Τ)

Δ χ ^ Τ )

Let

T h e n

T

Σ Σ X ^ C D a (T)u (T)dT

ο j=l k = l 1J k K J

Σ

λ „ ( τ ) * ^ ( τ ) = c

( r )

Δ χ ^ τ ) =

Squaring E q . 32

[ Δ

( τ ) ]

Τ

Σ

( T ) \ ( r ) d r

ο k=l

[31]

[32]

ο k=l

^Τ

ο

i = l

[33]

ds

ο ->

Σ Σ

(

K (s)u (r ) u . ( s ) d T d s

k = Ü = l ^{1 K} Η ^{k χ} T h e expected v a l u e of

^Δχ ^Τ)]

ε Γ Δ χ ^ Τ ) ]

- / / 7 ρ(η

,η

,...,η

), J—OD J-(D J - C D

Σ Σ c (T)c (e)u (Du/ e J d T

k=l i= l 1K

H

*

[35]

ds > du-du .. .du

1 2 η

157

It is necessary to use the following relation for joint prob- ability density functions:

r

^Ρ(\>\)

which is readily expanded to the case of an η-variable distri- bution function.

Using Eq. 36 in Eq. 35

η η

Σ Σ c^{i k}( T ) c (s)u ( T ) u . ( s ) d T d s γ dujdu [37]

k=l

J=l Jl

Interchanging the orders of integration and using the fact that

Γόο Too [38]

0u

u

(t

,t

)

J

is the correlation function of the variables u ^ ( t ^ ) , u²( t ^ )

E[ A^{X i}( T ) ]² =

Σ^Σ^^

(ε)0

(τ,3)

The following cases are of interest:

1) u^ are uncorrelated white noise

Ä y i ^ (

τ

8

y

^[ho]

where §k (Τ-s) is the dirac delta function, then E q . kO becomes *

ε[δχ

(τ)]

= £ h ^cik fr)] ² ^ ^l>l]

k=l

Jo

2) U j . are uncorrelated stationary random functions, i.e., all cross-correlation functions are zero

2 η

Τ Τ

Ε [ ΔΧ ι( Τ ) ] = ^ Γ ΓCik(T) C. (s) φ uku?( r- s ) d T d s [ U 2 ]

k=l

Jq

3) u ^ are random functions with known correlation functions

° Ν

It should b e noted that if u ^ are stationary

Φ

k

(

>

) = Φ ^

(

-

) M

E q . 39 or the equivalent forms, E q s . 4l-^3> as required for the p a r t i c u l a r random functions of interest, is used to e v a l - uate the mean squared output error for the case of random in- puts to the system.

The conjugate solution method is readily extended to the case of inputs that have known, correlation f u n c t i o n s . A p - plications are calculation of the mean squared error at b u r n -

out conditions or p r o p a g a t i o n of the m e a n squared error due to vibration environments, atmospheric turbulence and rocket motor n o i s e , and random variations in the measurement in- struments .

The conventional methods of error analysis do not have an analogous method for computing the mean squared error due to random functions acting along the trajectory. Instead, in the analysis of the propagation of random functions for p a r t i c u l a r systems, it has b e e n shown that the standard deviation p r o p a - gates according to some functional form. F o r example, for a gaussian random p r o c e s s , in a time-invariant system, or for a gaussian random p r o c e s s through a single integration, the standard deviation propagates in p r o p o r t i o n to t1/*. For a gaussian random p r o c e s s through a double integration, the standard deviation propagates in p r o p o r t i o n to t3/2. These functional forms for random p r o c e s s e s , in conventional analy- sis, are frequently introduced as deterministic inputs, and the analysis is carried out as described previously for d e - terministic inputs.

APPLICATION A N D REPRESENTATIVE RESULTS

The conjugate solution method of error analysis was applied to the study of the p l a n a r m o t i o n of an ICBM during injection, over a flat, nonrotating E a r t h . Results w e r e obtained for error sensitivities, propagation of errors, and error analysis for particular error sources in the measurement instruments.

F i g . 1 shows the reference frame used in the study, and the results are presented in F i g s . 2 and 3·

159

The coordinates w e r e chosen as follows

xl X range

X2 ζ altitude

X3

β

TLX mass

x5 ^vx x-component of velocity

x6 = _vz ζ-component of velocity x?

β

Xg = m mass rate

*9

τ

•|>5]

System time w a s included to enable extension of the study t o include variations in system time f o r synthesis of a controller.

F o r the error analysis, the relation Γ = t w a s assumed.

Using E q . k5, a set of equations w a s derived to correspond to E q s . 1 .

They are dx-^ = d x

" d t d t

d x2 dz

d t " d t

d X r - _ 2

dv X

d t d t

d x⁶

dvz

d t ~ d t

d x ^

aß

d t * " d t

d x³

Λβ

Vx dt dt

= β

d x ^ dm

" d t " dt m

f5 (z> vx >

V ß

ß

t)

f6 (Ζ> Vx '

V ^ > & >

> *>

dXg dm dt dt ^S °

[k6]

From E q s . 4 6 , a set of linear perturbation equations corres- ponding to E q s . 5 was derived. The coefficients were (using the notation

a1 5 - ^a2 6 - ^a2 7 ^{= _ a}2 8 ^{= X}

a5 2 - -[§ S <^cd cosr ⁺ c^Ls i n r ) ]

a

62 = S (-c

sinr

c

cosr)

ÜJii

(r + ζ )³ e 72 \ I ρ / V dz /

„ - [ f ^ ^S i n 0 ⁺

C o s r

^ S i n r ) l

sJ 0\ / *^Cn a

7 3

(— )(—)

a 5^

&

6k

Γ-ϋ,

oos

L ^m2 D L ^m

I & η [ (c^D sin Χ - C^L cos y ) - ^ - p - sin ß\

m

q

s/

ml

a5 5 - - f [^{2 ( c}d^{C o s r +} c^Ls i n y ) ( ! ^ )^{+ (}^ i c o s r

ba

xc

δ L x 1 S i n / ) S i n /

öa

1 4 1

a

65

S i n / l

a 75

I ν

ν

c

a

5 6

a67 = - S [2 ( cd S i n- C/ r + S^ + — i n

cos

r)

I

q s>C r ² C ν η J

*76 ⁼

a5 8 . £ s p C o SQ /

a68 - ^ ^Sni ß

A l l other a^j not listed are zero.

NOMENCLATURE

C^d, C^, C ^ = drag, lift, and moment coefficients I = inertia

I^Sp = specific impulse

g = gravitational specific force m = mass

q = dynamic pressure r^e = radius of Earth x,z = inertial coordinates ν = velocity

α = angle of attack

β = attitude

7 » ß

- α

ρ = atmospheric density

E q . 46 were solved u n d e r the constraint of a p a r t i c u l a r thrust profile to determine a reference trajectory that reached a desired set of b u r n o u t c o n d i t i o n s . This trajectory was then taken as the n o m i n a l , or ideal, flight. The p e r t u r b - ed motion of the vehicle was assumed to remain in a region of linear approximation about this nominal trajectory. The m o - tion of the perturbed flight is described b y the first v a r i - ational e q u a t i o n s , E q s . 5> w h i c h w e r e derived from E q s . 46.

D i g i t a l computer solutions to E q s . 5 were found to give the inverse fundamental solution matrix. 77"" ( t ) . A representa- tive set of solutions is shown in F i g . 2, w h i c h shows the a l - titude sensitivities to errors in the system coordinates.

T h o s e sensitivities not shown are zero. Since the p e r t u r b a - tion equations are linear, the resultant e r r o r in range from a measured deviation Δχ.^ from the nominal trajectory is found b y multiplying Δ χ^± b y its corresponding sensitivity X g i i t ) =

In addition, the error due to g y r o drift rate, gyro m a s s unbalance torque, accelerometer b i a s error, and accelerometer linearity e r r o r w a s computed. P r e v i o u s in-house studies have shown these to b e the p r e d o m i n a n t e r r o r sources from the m e a s u r e m e n t instruments.

The resultant p o s i t i o n and v e l o c i t y errors, are shown in Fig 3· H e r e , the e r r o r sources w e r e assumed to b e independent, and the propagated error is indicated as the square root o f the sum of the squares of the individual e r r o r s .

CONCLUSIONS

T h e methods of e r r o r analysis w e r e presented and explained.

The advantages of the conjugate solution m e t h o d w e r e shown to b e most significant in the guidance sensitivity and error p r o p p a g a t i o n studies and in the study of random i n p u t s . The methods are approximately equivalent in the study of d e t e r - ministic inputs to the system and for evaluating the c i r c u l a r probable error, C E P . Computer requirements are significantly less in determining the error p r o p a g a t i o n and guidance sen- sitivity functions, using the conjugate solution method. The

1*5

assumption of linearity about a nominal solution which is a basic assumption of accuracy analyses allows the computation to b e done b y a normalized (unit) error, so that the output errors can b e scaled linearly according to the magnitude of the input e r r o r s .

REFERENCES

1 M o o r e , R.A., "Determination of missile accuracy," in A n Introduction to Ballistic M i s s i l e s (Space Technology L a b o r a - t o r i e s , Inc., Cojioga P a r k , Calif.,; i960 V o l . 4, p p . 111-227.

2 A n d e r s o n , J.E., "Analysis of errors in inertial guidance systems," M - H A e r o Document U-ED6118, Minneapolis-Honeywell Regulator C o . , M i n n . , 1959·

3 Rosenberg, R.M., "On flight trajectories in the neighbor- hood of a known trajectory," J. Franklin Institute 266, 1958·

4 B l i s s , G.Α., Mathematics F o r E x t e r i o r Ballistics (John W i l e y and S o n s , Inc., N e w Y o r k , 1944).

5 M a r s h a l l , W . C . , "The adjoint method and some applications to a s t r o n a u t i c s , " M - H M P G Document U - R D 6 l7l , M i n n e a p o l i s -

H o n e y w e l l Regulator C o . , M i n n . , i960.

6 P f e i f f e r , C G . , "Guidance for space missions/'Jet P r o p u l - sion L a b . E x t . P u b . 656, C a l i f . Inst. Technology, 1959·

7 L a n i n g , J.H. and B a t t i n , R . H . , Random Processes in

Automatic Control (McGraw-Hill B o o k C o . , I n c . , N e w Y o r k , 1956).

8 B e l l m a n , R., Introduction to M a t r i x A n a l y s i s (McGraw-Hill B o o k C o . , I n c . , N e w Y o r k , i960).

mg

• x

F i g . 1 Reference coordinate frame for a p l a n a r rocket flight

F i g . 2 Sensitivity curves for altitude coordinate errors due to errors in X- and Z-components of velocity

1A5

F i g . 3 Sensitivity curves for altitude coordinate errors due t o errors in altitude, attitude, mass and mass rate

F i g . 4 P r o p a g a t i o n of total error due to deterministic errors