estimation of the local attitude of orbiting spacecraft

Automatica, Vol. 7, pp. 163-180. Pergamon Press, 1971. Printed in Great Britain.

Estimation of the Local Attitude of Orbiting Spacecraft* Estimation de l'attitude d'un vaisseu spatial sur orbite

Sch/itzung der Lage eines Raumfahrzeuges auf einer Orbitalbahn

OLIeHra nO;lOgeHvia KOCMHqeCI(OFO opa6 a Ha op6Ha'e

A. E. BRYSON, JR.1" and W. K O R T O M ~

An improved gyrocompass filter is designed to estimate spacecraft roll and yaw angles, using measurements from two rate-integrating gyros and an horizon sensor. Improve- ment comes largely from estimating the yaw-gyro drift.

Summary--In orbit an horizon sensor can measure pitch and roll angles but cannot measure yaw angle. Gyroscopic instruments can sense changes in yaw angle, but the measurement error variance steadily increases with time. However, by combining measurements from an horizon sensor and two rate-integrating gyros (in roll and yaw), yaw angle can be estimated with an error variance that is bounded as time increases. Such a device is called an orbital gyrocompass. We show that its logical structure follows directly from Kalman-Bucy filter theory if the measurement uncertainties are modelled as white noise, and only the kinematic equations of motion are used as a system model with the gyro signals considered as known-forcing functions.

However, white noise is not a very good model of gyro drift for long periods of operation. A better model is a constant bias plus exponentially-correlated noise. The Kalman-Bucy filter corresponding to this improved gyro model produces significantly smaller error variances in yaw angle than the orbital gyrocompass, largely through estimation of the yaw-gyro drift. RoU-gyro drift is shown to be unobservable.

1. INTRODUCTION

AS SPACECRAFT technology develops, precise deter- minat ion o f spacecraft attitude becomes of increas- ing interest for aiming telescopes, cameras, and antennas, and as an essential ingredient in active att i tude control systems. A recent symposium was devoted entirely to this subject [1]. In this paper we present a brief survey of existing techniques for

* Received 8 April 1970; revised 24 August 1970. The original version of this paper was presented at the 3rd IFAC Symposium on Automatic Control in Space which was held in Toulouse, France during March 1970. It was recom- mended for publication in revised form by associate editor B. Morgan.

This research was supported by National Aeronatics and Space Administration Grant NASA NGR 05-020-007, by a NASA International University Fellowship extended to Dr. Kortiim at Stanford University, and by the Deutsche Forshungs-und Versuchsanstalt fi~r Luft-und Ramfahrt.

t Professor of Applied Mechanics and of Aeronautics and Astronautics, Stanford University, Stanford, California.

*+ Dipl.-Ing., Ph. D. Deutsche Forschungs-und Versuchs- anstalt fi~r Luft-und Raumfahrt, Oberpfaffenhofen, Ger- many.

163

determining the orientation o f spacecraft and treat in some detail the determination of local attitude angles o f a spacecraft pointed nominally toward the earth.

Attitude sensors Some of the techniques for establishing angular

orientation of spacecraft are:

(i) Gyros. Gimbaled with 1, 2, or 3 degrees o f f reedom; on an inertially stabilized platform or "s t rapped d o w n " to the vehicle.

(ii) Stars. Star telescopes and trackers, star field measurements (star mapping).

(iii) Sun. Sun sensors.

(iv) Earth (or other planets). Horizon sensors, magnetometers, gravity gradient sensors, landmark telescopes and trackers, l andmark field measurements (mapping).

(v) Active reference stations. Interferometric, or other, determination o f the direction o f electromagnetic signals sent f rom a station with known location.

(vi) Atmospheric. Angle-of-attack and sideslip sensors.

InertiaUy stabilized platforms, see e.g. Ref. [2], are used for both inertial navigation and attitude determination, but they are very expensive. In s trap-down systems the platform is replaced by an on-board computer , see e.g. Ref. [3], which con- t inuously computes attitude angles using gyro inputs. The uncertainty o f attitude determination increases with time for both systems; currently this drift can be kept below 0"1 degrees/hr. Over long periods o f time, both systems must be updated by using other measurements.

Star sensors and star mapping techniques are under investigation [4, 5] which promise to deter- mine attitude with an accuracy o f one second of arc

164 A.E. BRYSON, JR. and W. KORTUM

and less. However, the systems require very com- plex, and hence expensive, mechanization.

Current solar sensors and earth magnetometers seem to be limited in accuracy to 1 or 2 degrees [6, 7].

Infrared horizon sensors measure local pitch and roll attitude [8]. The fuzzy nature of the horizon appears to be the main limitation on accuracy [9]. Another means of establishing the local vertical uses gravity gradient measurement devices, which include the vehicle itself [10].

The use of sightings of stars, the sun, planets, landmarks, and reference station signals requires complicated trigonometric computations to estab- lish the attitude angles. In most cases it also requires an accurate knowledge of position.

A ttitude est#nation

Averaging sensor data in some manner over a period of time relaxes the sensor accuracy requirements. Synthesizing appropriate data-processing logic is the domain of estimation theory [11, 12].

Estimation theory is already widely used for determining translational position of spacecraft [13, 14]. Less attention seems to have been paid to its use for attitude determination, particularly local attitude. A paper by KNOLL and EDELS~IN [15] is an exception since it discusses the estimation of the local vertical for an earth satellite using horizon sensor measurements. However, there is no discussion of determining yaw attitude.

POTTER and VANDER VELDE [16] discuss the mixing of gyro and star tracker data. A gyro-stabilized platform provides wide bandwidth information about angular rotations of the vehicle with respect to inertial space but suffers from drift, which they model as a random walk process. Star tracker data have small bandwidths but have no drift. They use only the kinematic equation for a single axis, assuming that the three axes are uncoupled.

JACKSON [17] discusses application of nonlinear estimation using strapdown gyros. He models gyro drift as a white noise process.

FARRELL [18] uses Kalman filtering of magnetic and solar measurements for attitude determination of spinning and non-spinning satellites. Monte Carlo simulation indicated that the estimation scheme is less successful for non-spinning, earth- oriented satellites. Clearly the inertial attitude reference is better adapted to spin-stabilized satellites and solar measurements.

LYONS and SCOTT [19] and TINL1NG and MERRICK [20] discuss local attitude determination and prediction using horizon and sun sensors. Yaw information is deduced from sun sensors and ephemeris data: the yaw accuracy seems to be poor due to sun sensor bias. Crude yaw information can also be obtained for low-altitude satellites by

sensing vehicle motion with respect to the atmo- sphere.

For earth-pointing satellites it is preferable to have sensors that measure local attitude angles, Le roll ~, pitch 0, and yaw i/i, as defined~in Fig. 1.*

ORBIT

FIG. 1. Definition of attitude angles.

Other sensors require computation of rotation transformations and a knowledge of the orbit ephemeris. Pitch and roll information can be obtained from an horizon sensor or a gravity gradient sensor, but neither can provide instan- taneous information on yaw. However, an estimate of yaw angle can be made by a weighted integral of roll angle using knowledge of the kinematic coupling of yaw rate to roll angle and the roll rate to yaw angle; such a device is called an orbital gyrocompass [21]. It is similar to the gyrocompass used on ships [2, 22], where a mass attached to one gimbal puts a gravitational torque on a three- gimbal gyro that, combined with the earth's rotation, keeps it pointing toward geographic north. In the orbital gyrocompass electromagnetic torques are applied to rate-integrating gyros using the horizon sensor measurement of roll angle and a knowledge of the orbital frequency.

Outline of paper

In section 2, the logical structure of the second- order orbital gyrocompass filter [21] is deduced from estimation theory. In section 3, a more realistic model for gyro drift is suggested and used with estimation theory to deduce an improved gyrocompass filter of fourth order. In section 5, a simplified third-order stationary version of the latter filter is shown to be almost as good. In section 6, the "observer" approach to filter design

* We shall assume that these angles are small so that order of rotation is not important.

Estimation of the local attitude of orbiting spacecraft 165

is used to deduce a fourth-order stationary filter that has some advantages over the third-order filter. Finally, in section 7, we discuss the desira- bility of a seventh-order filter that would result if we included the dynamic equations in the system model in addition to the kinematic equations.

2. SECOND ORDER GYROCOMPASS-FILTER If gyro drifts and horizon sensor noise are

assumed to be white noise processes, the logical structure of the filter deduced from estimation theory to estimate roll angle and yaw angle is identical to the logical structure of the orbital gyrocompass, provided one also assumes a circular orbit and uses only the linearized kinematic equations as the system model.

The linearized kinematic rotation equations in roll, pitch, and yaw, for a circular orbit with orbital frequency O~o, are

q~ =O9o~ + p (1)

O=coo+ q (2)

I~= -coo~b + r (3)

where q~=roll angle referenced to local vertical, 0 = pitch angle referenced to local vertical, ~ = yaw angle away from orbit plane, and (p, q, r) are com- ponents of vehicle angular velocity resolved in body axes as indicated in Fig. I.

Since the ~b, ~ equations are uncoupled to the 0 equation, we can treat them separately. The 0 filter is quite simple and is not treated here [23].

Let Po, rg be the outputs of rate gyros on the roll axis and yaw axis, respectively. Then

po=p+Dp, ro=r+D r, (4)

where D~, D, are random drift rates in the roll and yaw rate gyros, respectively.

Eliminating p and r in equations (I-4) yields

~=O~o~h +pg- D,, (5)

= - O~oC~ + r , - D,. (6)

Let the horizon sensor measurement be z(t), where

z(t) = c~ + V, (7)

where V is the random error in the measurement. Equations (5-7) are in a form suitable for using

estimation theory if Dp, D,, and V are assumed to be independent white noise processes. The optimum filter for estimating ~b and ~ is then

~=O~o~+Pg+K,(z- ~), ~(0) =0 , (8)

~= -o)o~+rg+Kq,(z-~), ~(0)=0, (9)

where q~, ~b=estimates of ~b, ~b, respectively, and K÷, K , are gains that can be determined in terms of the power spectral densities of the three white noise processes.

Equations (8) and (9) could be implemented by putting the outputs from two rate gyros and an horizon sensor into two electronic integrators as shown in Fig. 2.

HORIZON SENSOR Z = ~.t-V

Pg =p+ Dp ROLL RATE GYRC

r~l"r+D r YAW RATE GYRO

Fio. 2. Roll-yaw filter using electronic integrators.

However, (8) and (9) can also be implemented by introducing appropriate torques into two rate- integrating gyros, (RIGs). The integrations are done by the gyros which eliminates the requirement for electronic integrators.* To see this, recall that a rate-integrating gyro is essentially a floated rate gyro without a restoring torque proportional to t , the angular displacement of the rotor case. The viscous torque, c/9, is so large compared to the

inertia torque, I/~, that the latter is negligible except for very high frequency inputs. Thus,

c l ~ h ~ + h D + T, (10)

where

h =angular momentum of wheel,

hQ = torque due to angular rate fl about sensitive axis of gyro,

hD=random torque, drift,

T=applied, electromagnetic, torque.

Comparing (10) with (8) and (9) we see that we c a n implement (8) and (9) by applying the following

* An even bigger advantage than eliminating the electronic integrators is that current rate-integrating gyros are con- siderably more accurate than current rate gyros.

166 A .E . BRYSON, JR. and W. KORTOM

electromagnetic torques to two RIGs with sensitive axes in roll and yaw:

T. = h[too~. + K~(z - 4,.)], (l i )

where

T,=h[-o~oC~o+K6(z-dp,)], (12)

¢ ~.=~¢o, (13)

C ~g =~ko, (14)

and ~bo, fro = angular displacements of roll and yaw RIG rotor case, respectively.

Substituting (11), (12) into (10) with fl=~b o and ~o respectively yields:

('6)

o r

~g=O~o~Og+pg+ K¢(z-(%), (17)

~g= -O~o~b,+ro+K¢(z-¢.o). (18)

Equations (17) and (18) are identical to equations (8) and (9) if we place q~g-~, ¢ ,g -~ . The imple- mentation of equations (17), (18) is shown in Fig. 3.

I~ hp+hO, I

HORIZON

^

II h r + hD r

FIG. 3. Roll-yaw filter using rate-integrating gyros (orbital gyrocompass).

Referring to equations (l 7) and (18), the torques proportional to (z-~bg) are intended to offset the drift torques, hDp and hDr; the torques proportional to ¢Ooffg and ~Oo~ o are introduced to keep the gyros referenced to the local reference frame, see Fig. l, instead of inertial space. Thus, the RIG outputs are directly proportional to the estimates of the local attitude angles, ~b and ~k.

Tile scheme shown in I-ig. 3 is precis,:ly the scheme arrived at by other considerations iri Ref. [21], and called an "orbital gyrocompass". The use of estimation theory to derive the filter structure shows why the gains in the three feedback loops have the values, to o, -. too, and unity. Furthermore, the selection of the optimal gaines K,~ and K, ca , be made using estimation theory if estimates of the power spectral densities of D m D,., and V are available as indicated in Appendix 1.

However, limited experimental data do not seem to justify the white noise model for gyro drift under- lying this second-order gyrocompass filter. In section 3 we propose a slightly more complicated and, we hope, more realistic gyro drift model and use it in section 4 to design an optimal fourth-order, time-varying gyrocompass tilter, in section 5, we show that a simpler third-order, stationary filter is almost as good as the optimal fourth-order filter.

3. STOCHASTIC MODELS FOR GYRO-DRIFT AND HORIZON SENSOR NOISE

The gyro error, usually called "drift", is caused by disturbance torques acting on the gyro gimbal. These torques arise mainly from friction and un- balances in the presence of gravitational force and acceleration. Eliminating as many of the disturbance torques as possible is the "ar t" of constructing highly accurate gyros, e.g. through low friction bearings and temperature regulation. Nonetheless there will always be some disturbance torques in any gyro.

If we assume that the uncompensated drift is the result of many small and almost independent effects, it is reasonable to approximate it as a Gauss- Markov-process, that is white noise or white noise through a linear shaping filter. The choice of the power spectral density of the white noise and/or the parameters of the shaping filter depends almost entirely on experiments. The open literature contains very little information on such experiments, which are quite expensive. Thus we are faced with choosing a shaping filter, which compromises the desirable features of simplicity, agreement with the limited available data, and physical plausibility, There is also a trade-off between more accurate noise models and the cost and complexity of the corresponding filter.

A model which is quite often used in the literature [ 16, 24-27], for gyro drJ ft is the random walk process, i.e. the drift d(t) is assumed to be the first integral of a white noise process:

d(t) = w(t), (19)

where

E[w(t)w(t')] = Qa6(t - t'), E[a(0) 2] = do 2 . (20)


The mean-squared value of a random walk process grows linearly with time:

E[d(t)2]=do 2 = Qa . t. (21)

The random walk model seems to be justified for short-term operations by available test-data. How- ever, the mean-squared drift cannot increase without bound. Note that this is physically implausible since there are no unbounded uncertainty torques.

Since our main interest here is long-term operation, we have to model the long-term gyro drift behavior. In the absence of an experimentally supported model, most authors [28-31] whose concern is the long-term behavior agree that gyro drift is a low-frequency phenomenon and therefore it can be approximated by an exponentially correlated process with a large correlation time. Such a process is obtained by passing a white noise signal, w(t), through a first order low-pass filter:

d= -ctd+c~w, ~ = I / T d,

Td = correlation time, (22)

where

E[w(t)w(t')] = Q d f ( t - t'). (23)

In this case the time correlation is given by

E[d(t)d(t')] = Ra e x p [ - t'l], where

(24)

Qd= 2RdTd= 2Rd/ot. (25)

The mean-square drift has the value R~ in the steady state and the time correlation is exponential with time-constant Td.

For t ~ Ta the correlated drift model behaves like the random-walk model as indicated in Fig. 4. Thus a short-term random-walk behavior and a long-term bounded behavior can both be modeled by picking the two parameters Td and Qd (or Rd) of a first order exponentially-correlated process.

RMS

RW:~V'T"

DRIFT / ~ S UNBOUNOEO _ _ . _ ~ A S Y M P T O T I C VALUE INDEPENDENT OF d o

- - ~ : ~ ,/1- EXP (-¢11 i

t

FIG. 4. RMS--behavior of random-walk modeled drift (RW) compared with exponentially-correlated

drift (EC).

A typical value for the correlation time is 5 hr, see e.g. Ref. [31]. The stationary root-mean-square drift rate (RMS d(oo)= ~/Ra) depends very much on the quality of the gyro to be used; a typical value for reasonably priced autopilot-gyros lies in the range

RMSd(oo)- ~ I - 10°/hr- 0-6.10 - s

- 6 . 1 0 -5 rad.sec - l . (26)*

For the following discussion we shall adopt as a "nominal" value for Td,

Td= 1/~=2.104 see, (27)

whereas we consider different gyro drifts according to equation (26) and corresponding spectral densities via (25) assuming that the correlation time is about the same for all these gyros.

Experience indicates that there is, in addition, a random but constant bias, b, so that the total drift rate is given by

D(t) = b + d(t), (28)

with E[b]=0, E[b2l=b 2. (29)

The magnitude of b is in the same range as RMS d(oo), i.e. b = l - 1 0 ° / h r . If it were much lower the gyro is expensive; if it were much higher it could be measured and compensated. In reality b may be slowly changing, with correlation times from days to weeks/f

For very high precision gyros an even more complete model would be

D(t) = b + d(t) + w(t), (30)

where w(t) may be approximated by a white noise process with spectral densities in the order of 10-1 lrad.2sec- 1.~ For the accuracies we are con- cerned with here, however, the first two terms in (30) seem to be dominating.

Horizon sensor error models are also tentative. Most of the error is due to environmental effects, such as atmospheric variability, cloud cover, as well as geographical and seasonal effects [9], which

* The best gyros presently available have much lower RMS drifts (1-2 orders of magnitude) but they are expensive and it does not make sense to "overdesign" the gyros, i.e. make their error contribution much smaller than the other error sources.

1" From (19, 20, 21) it is clear that a constant bias is a special case of a random walk process (Qa=O, do=b); this will result in similar difficulties with respect to observability as we will see in the next sections. However, an exponentially correlated process (22-25) does not include a constant bias, shown in Fig. 4, so it has to be added (28).

:~ Private correspondence D. B. Jackson, NASA Elec- tronics Research Center and Honeywell Corporation.

168 A . E . BRYSON, JR. and W. KORTOM

cause RMS errors of the order of 10 -2 rad. The instrument errors, assuming an ideal horizon, are at least one order less.

The power spectral density of the total error may be of approximately constant magnitude to as high as 1000-times the orbital frequency. Higher frequencies are usually filtered out. This behavior could be modeled by a white noise passed through a low-pass filter. However, since the correlation time, some seconds, is so low compared with the orbital period, 1.5 hr, one might as well model the horizon sensor noise, v, by white noise

E[ v( t)v( t') ] = R'~( t - t') . (31)

A typical value for R appears to be

shall concentrate on the linearized steady state lilter.

System model

We must augment the state equations (23), (24) by the equations describing the gyro drifts, as follows:

4)=(Do~ +p~-dp-bp, (33)

~= - tOo~+%-dr-br , (34)

dr = - :~rdr + ~rWr, (35)

b r = 0 (36)

R = 10- 3rad.2sec. (32) dp = - %dp + %wp, (37)

There is some indication* that the power spectral density of the horizon sensor error is somewhat lower for very low frequencies, so that a more exact model of the horizon sensor noise would be a white noise, w, passed through a bandpass filter of the form F(s)=Tls/(l +Tls)(1 +T2s) 2. However, T 2 is very small and T1 is very large compared with the orbital period, so that we might as well approximate F(s) by unity. This is made even more reasonable since the knowledge about T1 is very vague. Thus the best we can do at present is to model the horizon sensor error by white noise.

There is also some indication that a constant bias, bns, exists in some horizon sensors, so that V=bns+V. However, its average value is quite small (on the order of 10 -a rad.) and it will be neglected for the design.

bp = o, (38)

where wp, w r are independent white noise processes.* The horizon sensor measurement is

z = q~ + v, (39)

where v is another independent, white noise process.

Unobservability of dp and bp

Neither bp nor dp are observable using the measurement q~. To show this we can make the following changes in state variables:

Y 1 = ~ - -__e '/ _ b_e, (40) (Do (Do

4. THE OPTIMAL FOURTH-ORDER, TIME- VARYING GYROCOMPASS-FILTER

In this section we design the logic for the optimal roll-yaw estimator based on (a) gyro drift modelled as a constant bias error plus an exponentially- correlated error (equation 28) and (b) horizon sensor error modelled as white noise (equation 31). As in section 2, we shall again use only the kinematic equations which means that the gyro measurements pg and rg will be considered as forcing functions, not measurements. The main advantage of this approach is that the estimator does not depend on vehicle and environmental parameters. The only requirements are that the orbit be nearly circular and that we know the orbital frequency, (Do.t

Since we are interested mainly in the long-term behavior for small deviations from q~=O=0 we

Y2 = d r - - -dp . (41) (D O

Substituting equations (40, 41) into (33-38), we have for (33-35)

q~ = (DoYt +P0, (42)

0t 3) 1 = - ( D o ~ - Y2 - br + r 9 - - w p , ( 4 3 )

Ok o

0~ 2

) ) 2 : - - c~Y2 + CtWr----Wp, (44) (D o

with (36-38) unchanged. Equations (37) and (38) are not coupled to equations (42-44) which contain q~, the measured quantity; hence dp and bp cannot

* Private communication D. DeBra, Stanford University. t Some considerations made in Ref. [23] show that slight

variations in COo do not affect the accuracy sitmificantly. * We shall assume a~=ar=ct since there is little basis to

assume otherwise.

Est imat ion of the local att i tude of orbit ing spacecraft 169

be observed with a measurement of (k. Further- more, qJ and dr cannot be observed separately;* only ~b, YD Y2, and br are observable with a measurement of ~b.

Optimal filter for the observable quantities The structure of the opt imal filter is therefore

q~ = too.9~ +Po+ K , ( z - dp), (45)

y ,= -too(O-92-[~,+ro+K~(z-6), (46)

92 = - ~332 + K2 (z - q~), (47)

~ r= Kb(z-- 6 ) . (48)

A block diagram of this filter is shown in Fig. 5.

$

~-"~ ' l K- I ' ' k ' l I~ 'L .~ .,z~-,-j • .. .*." L . . . J b,

FiG. 5. Optimal roll-yaw filter for observable quantities.

The opt imal gains Ke,(t), K~(t), K2(t ), and Kb(t) can be determined by solving a matr ix Riccati equation, shown in Appendix 2, for the variances and covariances of the est imation errors.

Figure 6a, b shows the gain histories for

too= 10-3rad. sec- l ,

R = 10- 3rad.2sec,

ct=0"5 x 10-4sec -1 ,

RMS d p ( ~ ) = R M S bp=0"5 × 10- 5rad. sec -1 ,

R M S d r ( ~ ) = RMS br = 3 x 10- 5rad. sec- 1,

and initial variances

P**(0)=0 .25 x 10-2rad. 2 =P11(0) ,

P22(0) =Pbb(O) = 0"9 x 10- 9rad.2sec -2 .

* dp and dr would both be observable if ~tpv~ctr.

The variance Pbb(t)~O as t ~ o v , i.e. the constant bias br can be est imated exactly after an orbit or two. However , this also means that Kb( t )o0 as t ~ o v , so that the steady state filter is effectively third order; in fact, due to integrator drift, the ~r integrator should be turned off after an orbi t or two. The opt imal est imation of br thus requires a t ime-varying gain history.

5. A THIRD-ORDER GYROCOMPASS-FILTER

I f R M S b r is thought to be large, it would certainly be valuable to est imate it. However , since we believe that R M S br and RMS dr have abou t the same size, the loop est imating Y2 might also reduce the effect of b r to an acceptable value. To study this conjecture we considered a filter design based on the assumpt ion br--O:

= tooYl +Po + K~(z- ~), (49)

~l= - too6+r , - -~2+Kt ( z -6 ) , (50)

~2 = -- ~2 + K2(z- alp). (51)

In contrast to the op t imal four th-order filter, this filter is asymptotical ly stable in the steady state since the process is completely observable and controll- able. Thus the following considerat ions can be based on the steady state fo rm of that filter.* To evaluate this third-order filter design, we compute its R M S errors and compare them with the R M S errors of the second-order gyrocompass which are derived in Appendix 3. The error sources for both are taken as bp, d r, b,, bHs, dr, and v.

Effect of the unobserved (br) and unobservable (dp, bp, bos ) quantities

The R M S errors of the filter (49-51) due to a constant bias b r are

RMSq (br) = Ibr[ ]K2/c¢-(too + K,) I

(52)

RMS~(br ) = RMSy,(br ) - - - IK*[ . RMSq~(br). (53) too

These errors are so small compared with the total error caused by the other effects that our conjecture abou t neglecting br in the design seems to be justified.

The contr ibut ion of dp and bp to the est imation error is expressed through equat ion (40):

RMS~(bp, dp) = I[ (RMSdp)Z + (RMSbp)2] ~ . (54) too

* In fact, the steady state form of that filter is identical to the steady state form of the optimal fourth-order filter of section 4.

170 A.E. BRYSON, JR. and W. KORTOM

IO I I t x [ / \ K~(O) - 2.5

K,~IcO) = 0"384.1d z rod sec "~ ~ ÷ ( t ) "2- \

I0 "3 , I ' l ] [ l l ' [ ] 0 0'I 0"2 0-3 04 0,5

t / T o

(a) K÷(t), Kff,)

oJ v

2.10 "~

10"5

i0 "s

0 o K b (~) • 0

0"1 0"2 0"3 04 t/To

(b) K2(t), K~(t).

Fzo. 6. Time-varying gain histories for optimal fourth-order filter.


This is the basic error of the estimator. Note that it is impossible to estimate roU-gyro drift,* and a roll gyro has to be chosen in order that this error is acceptable. The roll-gyro does not influence the roll-angle accuracy, RMSq~(bp)=0. Therefore the total roll error due to constant drifts is very low,t on the order of 2 x 1 0 -4 rad. for the range of b r=10 -6 to 10 -3 rad. sec -~ compared with the values of 10 -4 to 3 x 10 -z rad. for the corresponding drift values in the second-order gyrocompass design.

The yaw error due to constant bias terms is improved only by a factor two due to the inability to estimate bp; the improvement comes from estimating part of b~ through the estimation of),2 [equation (51)].

The estimation errors due to constant bias in the horizon sensor, bt~s are:

RMS~(bns) = ( K z - a K , ) b n s , (55)

RMS~(bns ) = RMS~l(bns )

= K~,{. K2-0~K 1 ~_ l']bos . 6% \0~(¢.0 o + K 1 ) - - K 2 ]

(56)

These values are usually very low, too, compared with the other errors. However, they serve as lower bounds on the errors, which cannot be decreased by improving the gyros.

8.10 -6

N

5.10.6

%-- g

to

10"61

3.1o -~]

2nd ORDER

FILTER

(a) Roll

YAW

2 n d

ROLL , , , , L

'RO- BIAS

' VARYING NOISE

IAS

Effect of time-varying noise In addition to the errors caused by constant gyro

biases, we have postulated slowly changing gyro drifts, introduced by the time varying error terms wp, w r and v.:~ The estimation errors due to these error sources were obtained by computing the steady state covariance matrix for the third order filter from the corresponding matrix-Riccati equation given in Appendix 2.

g

hi

2.1(~ 4

10 -4

F VARYING NOISE

Total errors in estimates and choice of gains Bar graphs comparing the error variances in roll

and yaw of the third- and second-order gyrocompass-filters for typical noise values are given in Fig. 7. An improvement of a factor 2-3 in yaw accuracy is obtained whereas the improvement in roll is only about 50 per cent in this region of drift- values.

(b) Yaw

FIG. 7. Comparison of total error variances in roll and yaw of second- and third-order gyrocompass-filters due to time-varying and constant noise (too=10-3 rad. sec -1, R=10 -3 rad.2 see-l, ct=0.5 × 10-4 sec-~, RMS dp ~RMS bp =-~ RMS dr='~ RMS br=0"5 × 10-5

rad. sec-l).

* In Ref. [21] only by, not dp, is considered. t Recall that this error would be zero if we used the

optimal fourth-order time-varying filter of section 4. Note that part of the error caused by dp is already

expressed by equation (54).

A plot of the total RMS errors as a function of RMS gyro-drift values is given in Fig. 8. Even for very good gyros (10-6rad. sec -1 for yaw-gyro) the RMS second-order filter errors do not drop below

172 A . E . BRYSON, JR. and W. KORTOM

/ / /

~ /

RMS~,\ / /

zo"~- / ~ / / ~ - - ~ 2 n d ORDER FILTER /

3rd ORDER FILTER / / / /

/ / / / / /

o / / , ," /

IO -2 / , _ /~ -.-@--- ~ / /

~ - --O-----

163g.---~

of 10-2rad. is required then Fig. 8 indicates that for the third-order filter:

R M S b ~ = R M S d , ~ 4 " 0 × I 0 5rad. sec ~,

R M S b p = R M S d v ~ O ' 7 x 10- Srad. sec -~ . (55)

The corresponding second-order filter would require gyros with RMS drift values about three times less, as indicated in Fig. 8. Note that for both filters the roll-error is about a factor 4-5 below the yaw-error. I f gyros with

R M S b , = RMSd~= 1"8 x 10- 6rad. sec -~ ,

RMSbp = R M S d v = 0"3 x 10- 6rad. sec- 1 (56)

Io-4L__L ~ ~ 10-6 i0 "5 i0 -4 10-3

RMSd r =RMSbr, rod see "1

FIG. 8. Total steady state RMS--errors in roll, t~, and yaw, ~, for second- and third-order gyrocompass filter (tOo-- 10 -3 rad. sec -1, R = 10 3 rad.2 sec-1, ct =0'5 × 10-4

sec 1, RMS dv--RMS bp=~ RMS dr=~ RMS b~).

io')

IG z

10"3

~ 10"4

~g

' ^ 10 " 6

/ / / .~ ~÷p,+K÷~z-+)

~O I I I I I I I • I I ] I I I I IO " 7 I0" - 6 i0-3 i0"4 i0 "S

RMSdr ,RMSb r tad sec "1

FiG. 9. Optimal gains for third-order filter as a function of gyro-drift.

were available, we could design a third-order filter with a total RMS-error o f 10-3rad. for both roll and yaw, whereas the second-order filter errors do not drop below 7 x 10-3rad. and 5 × 10-3. tad. for yaw and roll, respectively.*

Figure 9 shows then how the gains Ko, K~, K 2 have to be set, once the decision on the choice of the gyros is made.- A typical gain setting for a yaw- gyro with 3 × 1 0 - S r a d . sec -1 drift (roll-gyro: 0"5× 10- 5rad. sec -1) is:

K0=0 .348 x 10 -2 tad. s e c - l ,

K 1 =0.753 x 10 -2 rad. sec- ~, L_f, (57)

K z = - 0 " 8 8 8 x 10 -5 rad. sec -2 , J

which lead to total RMS errors of 0.2 x 10-2rad. in roll and 0.8 x 10-2rad. in yaw.

Transient behavior of the third order filter

The speed of decay of initial errors in the estimates for the steady state third order filter has to be judged f rom its eigenvalues. The corresponding characteristic equation is

•3 q.. (K4~ _{_ ~)22 + (K4, ~ + 6ool + K1 coo),E

+ [c~ago(K 1 + coo)- K2COo] = 0, (58)

and the eigenvalues are

6"5 x 10-3rad. for yaw and 2"5 x 10-3rad. for roll,* whereas the RM S third-order filter errors are below 10-3rad. for both roll and yaw.

Figure 8 together with Fig. 9 which gives the corresponding optimal, steady state, gain settings for the third-order filter, should be useful for design. For example, if an RMS error in yaw angle

* This is due to the use of non-optimal gains in the orbital gyrocompass. This can be seen from the error equations of an observer using non-optimal gains, as described in Appendix 2.

2 1 ~ - 2 × 10-3sec -1 ,

2 2 , 3 = ( - - 0 95_+1 1"95) x 10-3sec -~ (59)

These eigenvalues show that after 1/5 orbit initial errors are reduced to less than 70 per cent of their original values. This speed seems to be fast enough for many missions. For faster error reduction one has to implement the optimal t ime-varying gains.

* Again assuming the bias in horizon sensor is 10 -3 rad. or less.


A faster error reduction with constant gains can only be achieved with the disadvantage of higher steady state errors; however, a compromise between minimum steady state error and sufficient speed of error reduction might be made.*

6. GYROCOMPASS OBSERVERS

Another approach to filter design is to choose the gains K on the measurement deviations z - / - / 2 so that the error equations have desired eigenvalues. Such filters were suggested by LUENBERGER [33] who called them "observers". They are especially useful when the measurements or the forcing functions contain unknown bias terms.t They are also helpful in determining observability.

Observer including all observable quantities

The structure of the optimal filter for the observable quantities was given in section 4. This filter had time-varying gains which led to an un- desirable zero eigenvalue in steady state operation. For this reason we suggested a third order filter that did not make a separate estimate of br but reduced its effect quite adequately.

However, we can design a fourth order time- invariant observer that removes the effect of br completely, which is not only stable, but has eigenvalues that we can specify. The gains of this observer which has the same structure as the optimal filter, see equations (45-48), are determined by equating the coefficients of the characteristic equation obtained from equations (45-48), which is the following,

2 4 + (K~ + ct)2 s + [(Do(K 1 + COo) + K,~]~. 2

+ [~coo(K1 + too) - K2(Do- Kbcoo]2 - Kb(Do~ = 0, (60)

with the corresponding coefficients of the desired polynomial

24 + a323 + a2 ,~2 + ax2 + ao = 0. (61)

The gains can then be computed successively from the following equations:

K~=a3-a, Kl=-~--(a2-K4,a-(Do2), (D O

1 Kb = -- - - a o ,

(DoO~

K2 = - 1 [ a , + Kbco o - or(Do(K, + too) ] . (Do

i

(62)

* See next section for a more detailed discussion of these and related questions.

t Bias terms are reasonable approximations to slowly changing terms with time constants which are large compared with the other time constants of the system.

For example, let three eigenvalues be identical to the third order steady state optimal filter of Fig. 6, but require in addition 2 4 = - 1 0 - 3 s e c -1. This leads to the gains

K~=0.480 x 10 -2 rad. sec- ~,

K 1=1.115 x 10 -2 rad. s ec - l ,

K 2 =0.169 x 10 -6 rad. sec -2 ,

Kb= --0.186 x 10 .3 rad. sec -2 .

(63)

Compared with the optimal filter, the observer has some desirable features:

(1) It has constant gains instead of the time- varying gains of the optimal filter, and these are easier to implement.

(2) Compared with the steady-state of the optimal time-varying filter, it can be designed to be asymptotically stable instead of having a zero eigenvalue.

(3) Its eigenvalues can be chosen arbitrarily so that it recovers from initial errors arbitrarily fast.*

(4) It removes constant yaw-drift of any size completely, diminishing its effect like an exponential function e x p ( - a t ) with arbitrary ct; therefore it is especially suited for gyros with large bias drifts.

(5) As mentioned earlier bias terms are often not really constant but change slowly. An optimal filter removes the effects of such terms only once, due to the fact that K b ~ 0 , but it has no effect on them if they slowly build up again.

The disadvantage of an observer is that the gains are non-optimal, i.e. the steady state errors will be higher than those of the optimal filter. This disadvantage can be minimized by compromising between speed of response and steady state errors.

It is interesting to note that the observer with gains (63) is identical to an optimal steady state filter where each gyro is assumed to have an exponentially correlated error plus a random-walk error. It may be more straightforward to choose K b to get a desired dynamic response than to guess a spectral density for the random walk process.

7. OBSERVERS BASED ON BOTH KINEMATIC AND DYNAMIC EQUATIONS

The estimators, both observers and optimal filters, discussed so far are based on the kinematic equations of motion and do not make use of the dynamic equations. Consequently they yield

* In the presence of noise, however, it will--at any instant of time--have higher RMS errors than the optimal time- varying filter:

174 A . E . BRYSON, JR. and W. KORT/3M

estimates only of the attiude angles. For active control systems, estimates of attitude rates are also needed, to provide damping terms in the control logic. Since the outputs of the gyros which represent the estimates of the roll and yaw attitude angles are quite smooth, they can be differentiated approx;- mately by lead networks of the form (I +T~s)/ ( 1 + T2s ) where Ta > T 2.*

Since an estimator based on both the kinematic and dynamic equations of motion gives rate information directly, we briefly investigate such an estimator, again using the observer viewpoint.

The kinematic and dynamic equations for a satellite in circular orbit under the influence of gravity can be written in the following lbrm [23]:

= coo~b +p , (64)

~/) = - O~oq~ + r, (65)

t ' ~=ar+b(o+tq + n l , (66)

J: = cp + u2 + n2, (67)

where q~, ~b, p, r are defined above, and

a = --O9oXl, b = - 3 ~ o e Z l , c= -~oK3, (68)

tq = ( l y - 1~)/1~, '~3 = ( Ix- - Iy ) / l : , (69)

u x, 2 = normalized control torques,

T~ Tz u ~ ='7~" ~ , u2 = ' ~ ,

n t ,z=normal ized disturbance torques,

T,x T,z H1 ~ - - ~ /'/2

Ix ='-~--~ '

(70)

panels), fuel consumption, and fucl movement. Thus the coeffcients a, b, c as well as the gains ol such an estirnator have to be ~aried in flight which involves additional logic and introduces addifiomd uncertainty.

(2) Disturbam'e torques n~. ~ due to aerodynamic, solar pressure, magnetic, electric, and gravity forces, micrometeroid impacts act on the vehicle. Most of them are difficult to predict and vary over wide ranges depending on the mission.

(3) The control torques ut .2 must be used as inputs for an estimator designed on the basis of equations (64-67, 71-73). Even though electrical signals generating these torques are available, the actual torques may deviate from these signals, e.g. due to gas-leakage. These uncertainties have to be considered as additional, unknown, disturbance torques,

(4) Use of the linearized kinematic and dynamic equations is an approximation. However, errors introduced by this approximation will generally be small for small deviations from the nominal attitude.*

Many of these modeling problems are not present if we use only the kinematic equations.

Another possibility is to try to estimate the unknown quantities. In the case of the inertia parameters this leads to a non-linear estinaation problem. Reference [35] contains some results on trying to estimate inertia parameters.

L)'timator assuming constant biases

A significant portion of the disturbance torques is constant or slowly changing with time. If we assume only constant disturbance torques n~, 2 = b j, 2 and constant drifts Dp.~= bp,,. we have to add to equations (64-67) the following equations:

i h =0 , /~2 = 0 , hp=0, /~=0 . (74)

Ix, Iy, I~ =principal moments of inertia about x, y, z axis.

The horizon sensor and gyro outputs are observation s:

zl = ¢ + v , (71)

z~ =p~=p + Dp, (72)

z 3 = r g = r + D r . (73)

Some of the problems associated with estimators based on equations (64--67) and (71-73) are:

(1) The moments o f inertia change due to relative motion of parts (e.g. extendable booms and solar

* Another possibility is to use "derived-rate modulators" [34]. They need as inputs only the attitude information and generate the required rate information internally.

The resulting system together with the observations

z~ =qS+ v, (75)

z 2 = p , = p + b p , (76)

z 3 = rg = r + br, (77)

is not completely observable. To show this let us rewrite the equations using the following defini- tions:

bp, t78) 030

y2Ab2 - cb r . (79)

* See Ref. [36] for a discussion of non-linear coupling effects.


1 his gives a set of equations for the states ~b, Yl, pg, rg, bl, Y2, b,, bp:

=cooYl +Pg, (80)

If we define

Yt ¢- b2 cooC '

(89)

p l = - c o o ~ b + r g - b , , (81)

[J ~=ar~ + bd; + bl - a b , + ul , (82)

f g = cpg + Yz + us, (83)

b~ - 0 , (84)

=0, (85)

b,=o, (86)

bp=O. (87)

With the observations

z l = ~ + v , zz=pg, za=ra. (88)

Since bp is neither observed nor coupled to any of the equations (80-86), bp is unobservable. This means that ~, b2 and p are also unobservable; only linear combinations of them Yl = ~b - (bp/co°), Yz = b z - cbp, Pc =P + bp are observable. The fact that equations (80-86) and (88) constitute a completely observable system may be seen directly.

Equations (79) and (83) show that an estimate of bp could be obtained if b2 ~. cbp, i.e. Y2 -~ cbr On the other hand, if cbt,~.b2, i.e. y2~-b2 the uncertainty in the estimates of both disturbance torques b~.2 would be small. This latter ease is more likely to occur: e.g. if

I= = 2000 slug-ft 2 ,

T~ = 10- 4Nm,

c=0-9 x 10-3rad. see -1,

coo= 10-arad. see -1 ,

bp= 10- 5rad. see- 1,

then

cbp'~ 10- Srad. see -2 ,~b 2 " 0 . 3 x 10- 6rad. see -2 ,

Thus for this case we could estimate b t and b z fairly accurately. However, there is no need to do so if we use the kinematic-estimator.

y2Ap+ b2, C

(90)

then we obtain the system:

q~---- t°oYl +Ya, (91)

)~t = -coo~b + r , (92)

P2=ar +bdp+bt +u 1, (93)

i" = c Y 2 "lt- U2 ,

bl =0 ,

b~--0,

(94)

(95)

(96)

with the observation

z = ~b + noise. (97)

It is obvious that b2 is not observable, and that ~k and p are only determined up to a constant. Since b2[co.c is much bigger than the unknown constant bffcoo in the previous filter, by a factor of 30 for the above numerical values, the uncertainty in ¢, becomes much bigger if one omits both gyros.*

One advantage of an estimator using the kinematic and dynamic equations is that it is possible to give some, less accurate, estimate of ff and ~b if any one or any two of the observations q~, p, r, are available.t It is impossible then to estimate certain constant bias terms, but a kinematic estimator fails to operate if the horizon sensor or either gyro fails.

8. C O N C L U S I O N S

The logical structure of an orbital gyrocompass follows directly from filter theory if the measurement uncertainties are approximated as white noise, only the kinematic equations of motion are used as a system model, and rate gyro signals are considered as known forcing functions. It is shown that gyrocompass filters can be implemented using rate- integrating gyros instead of the less accurate rate gyros, which also eliminates the requirement for two electronic integrators.

Estimator using only the horizon sensor

Since for both types of estimators, the roll gyro drift is unobservable, one might be tempted to omit the gyros and use only the horizon sensor.

* The same is t rue if one omits only the roll gyro, with its unobservable b~, and uses an hor izon sensor and a yaw gyro.

t I t is easy to see that in any o f these cases we have a completely observable system for the states ~ , ~ , p, r ff one assumes no bias terms, a fact which is sortmtimes overlooked.

t76 A . E . BRYSON, JR. and W. KORTUM

Noise For long periods, white noise is not a very

accurate approximat ion for gyro drift and it is

suggested that exponentially-correlated noise plus a constant bias is a better approximation. On the

other hand, white noise is a reasonable approxima- t ion for horizon sensor noise.

Roll-gyro drifts Roll-gyro dr i f t s - -bo th exponential ly correlated,

dp, and constant , bp--are shown to be unobservable, which means that their cont r ibut ion to the total

error cannot be reduced effectively by any filter design. However, both types of drift cause only

bounded filter errors.

Yaw gyro drifts Exponential ly correlated yaw-gyro drifts, dr,

cannot be observed separately, but only as a l inear

combina t ion of dr and dp (for ~/(no=0.05 .v2 =d~ - 0 . 0 5 dp). Since we also chose d p = l d , the signi-

ficant term in this l inear combina t ion is d, so that,

practically, dr can be treated as an observable quanti ty. Yaw-gyro bias b, is observable, i.e. its

contr ibut ions to the errors can be removed completely.

Optimal fourth-order f i l ter Because the two roll gyro drifts are unobservable,

the opt imal filter is fourth-order instead of sixth-

order. Fur thermore, to observe yaw-gyro bias requires a t ime-varying gain which tends toward

zero after a few orbits. Thus the steady state of the optimal filter is only neutral ly stable with one zero

eigenvalue.

Steady state third-order f i l ter

Without much loss in accuracy, the est imation of yaw gyro bias b~ can be omitted, since the loop

est imating dr estimates most of b, as well. The resulting third-order filter reduces the RMS

est imation error by a factor of 2 to 3 compared to orbital gyrocompassing, essentially a second-order

filter. Using the observer concept for gyrocompass

designs demonstrates again why roll gyro bias is not observable and shows how to construct a stable

est imator for all observable quantit ies having constant gains.

Using the observer concept also indicates why it may not be worthwhile to construct an estimator based on the kinematic and dynamic equations. The roll drift is still unobservable and there are addi t ional uncertainties introduced through the dynamics. However, crude yaw-est imation is now possible with an horizon sensor only, without gyros.

Acknowledgement--The authors are indebted to Dr. Daniel B. DeBra, Stanford University, for many helpful discussions and suggestions concerning orbital gyrocompassing and the related modeling problems,

REFERENCES

[I] Spacecraft attitude determination symposium. (Fhe Aerospace Corporation, El Segundo, California. 30 September-2 October 19691.

[2] G. R. PITMAN, JR., Ed. : Inertial Guidance..Iohn Wiley, New York (19621.

[3] J. L. FARI~tELL: Performance of strapdown inertial attitude reference systems. AIAA Journal Spacecra/) Rockets 3, No. 9 (1966).

[4] M. GORSTEIN and J. HALLOCK : Two approaches to the starmapping problem for space vehicle attitude determination, paper presented at [I].

[5] J. E. CARROLL : Sub-arc second star sensors, design and fabrication, paper presented at [I].

[6] J. C. KOLCODGE' and D. A. Koso: Solar attitude reference sensors, paper presented at [1].

[7] H. D. BLACK: A passive system for determining the attitude of a satellite. AIAA Jaurnal 2, 1350-1351 (1964).

[8] R. W. ASTHEIMER: Characteristics of IR horizon sensor designs, paper presented at [1]. Also High precision attitude determination by sensing the earth and lunar horizon in the infra red. Automatica January 1971.

[9] J. A. DODGEN and H. J. CURFMAN: Accuracy of IR horizon definition as affected by atmospheric considerations, paper presented at [1].

[10] J. W. D~ESEL: A new approach to gravitational gradient determination of the vertical. AIAA Journal 2, 1189- 1196, 1964.

[Ill R. E. KALMAN: A new approach to linear filtering and prediction problems. Trans. ASME, Series D, J. Bas. Engng 82, 35~-5 (1960).

[12] R. E. KALMAN and R. S. Buoy: New results in linear filtering and prediction theory. Trans. ASME, Series D, J. Bas. Engng 83, 95-108 (19611.

[13] R. H. BATTIN: Astronautical Guidance. McGraw-Hill, New York (1964).

[14] R. S. Buoy and P. D. JOSEPH: Filtering for Stochastic Processes with Applications to Guidance. Interscience, New York (19681.

[15] A. L. KNOLL and M. M. EDELSTEIN: Estimation of local vertical and orbital parameters for an earth satellite using horizon sensor measurements. AL4A Journal 3, 338-345 (1965).

[16] J. E. POTTER and W. E. VANDER VELDE: Optimum mixing of gyroscope and star tracker data. AIAA Journal Spacecraft Rockets 5, 536-540 (][968).

[17] D. B. JACKSON: Applications of nonlinear estimation theory to spacecraft attitude determination systems, paper presented at [1].

[18] J. L. FARRELL: Attitude determination by Kalman filtering, presented at 2nd IFAC Symposium on Auto- matic Control in Space, Vienna, Austria, September 4-8 (1967). Automatica 6, 4][9-430 (19701.

[19] M. G. LYONS and E. D. SCOTT: Attitude determination and prediction for a class of gravity-stabilized satellites, paper presented at [1].

[20] B. E. T1NHNG and V. K. MERRICK: Estimation and prediction of the attitude of a passive gravity stabilized satellite from solar aspect information, Paper presented at [1].

[21] J. L. BOWERS, J. J. ROe)DEN, E. D. SCOTT and D. B. DEBRA: Orbital gyrocompassing heading reference. AIAA Journal Spacecraft Rockets 5, 903-910 (][968).

[22] R. H. CANNON: Alignment of inertial guidance system by gyrocompassing-linear theory. J. Aerospace Sci. 28, 885-895 (][961).

[23] W. KORTOM: Attitude estimation and control of earth- pointing satellites. Ph.D. Thesis, Dept. of Applied Mechanics, Stanford (1970).

[24] J. R. COOPER: A statistical analysis of gyro drift test data. S. M. Thesis TE-13, Dept. of Aeronautics and Astronautics, M.I.T., September 1965.

[25] A. DUSHMAN: On gyro drift models and their evalua- tion. IRE Trans. Aerospace Navigation Electronics ANE-7, December 1962.

Estimation o f the local attitude of orbiting spacecraft 177

[26] J. HENDRICKSON: Guidance systems component correlation testing and studies. M.I.T. Experimental Astronomy Lab., RN-13, October 1965.

[27] W. S. WIDNALL: A comparison of some vertical- indicating-systems, including an optimum time-varying system. Subj. 16.60, Advanced Aeronautical Problems, M.I.T., April 1965.

[28] W. T. MCDONALD: Gyrocompassing spacecraft navi- gator. Ph.D. Thesis, Dept. of Aeronautics and Astro- nautics, M.I.T., June (1968).

[29] L. D. BROCK: Application of statistical estimation to navigation systems. Ph.D. Thesis T-414, Dept. of Aeronautics and Astronautics, M.I.T., June (1965).

[30] S. L. FAGIN: A unified approach to the error analysis of augmented dynamically exact inertial navigation systems. IEEE Trans. Aerospace Navigation Elec- tronics, December (1964).

[31] H. ERZBERGER: Application of Kalman filtering to error correction of inertial navigators. NASA Tech- nical Note, NASA TN D-3874, February 1967.

[32] A. E. BRYSON, JR. and Y. C. Ho: Applied Optimal Control. Blaisdell, New York (1969).

[33] D. G. LUENBERGER: Observers for multivariable systems. 1EEE Trans. Aut. Control AC-II, 190-197 (1966).

[34] J. C. NICKLAS and H. C. VIVIAN: Derived-rate incre- ment stabilization: Its application to the attitude control problem. Trans. ASME, Series D, J. Bas. Engng 54-60 (1962).

[35] G. D. NELSON and G. R. ARNESON: On the develop- ment and performance of an attitude determination data reduction and analysis system. Paper presented at [1].

[36] T. R. KANE: Attitude stability of earth-pointing satellites. A1AA Journal 3, 726-731 (1965).

APPENDIX 1

Roll-yaw fi l ter using measurements o f roll rate, yaw rate, and roll angle containing white noise

The system model is

q~ 0 o°]r¢ ' " " ' L~J Lr,,J LD,I (A.2)

, [:] - - [ , 0 ] + V ( 1 . 3 )

where

E[D,(t)Dp(t ')] = Q p a ( t - t') (A.4)

E[D,(t)D,(t ' )] = Q , a ( t - t') (A.5)

E[ V(t) V(t')] = R 6 ( t - t') (A.6)

Thus the equations for the variances and the covariance o f error in the opt imal estimates of ~b and ~b ale:

{,oo,,4 F o d t L P ~ P ¢ ¢ 3 = L - 0% LP**P¢¢j

o vo] LP~¢P¢¢_I (1 .7 )

o l_,_P'oo,'o.l Q,I R L P # P ~ ¢ I t _P#P ,¢ 1

or

• 1 p 2 (A.8)

p # = Ogo( p ~,~ ' _ p¢4,) _ I p ¢,~p ¢~ (A.9)

l 2 P~,¢ = - 2woP,~ + Q,.- ~ P ~ , . (A.10)

In the statistical steady state P c ~ , = P ~ =P~¢ =0 , so that equations (A.8-A.10) become simultaneous quadratic equations for Poe, P¢~, and P ~ . They are easily solved to yield:

.0,-oo<jl+ 0 ,) coER

(A.11)

Q.l', ,,,.,=, t~ER (o2RA

P q, q,'-* P ¢,¢,X/1 + Q, O92oR •

(A.13)

Thus the steady state gains are

K¢=PRC'~Ogo[2 /I+ Q--~-2+~q ½ °9oR] (1.14)

h+ _d /~ L ¥ ~o2R .J" (A.15)

APPENDIX B

Computation o f error covariances

From filter theory for cont inuous systems, see e.g. Ref. [12], the optimal filter for the process

Yc = F( t )x + G(t)w(t), (A. 16)

with the measurements

z = H x + v , (A.17)

where

x = n x 1, w = m x 1, z, v = p x 1 vectors,

F = n x n, G = n x m, H = p x n matrices,

has the form

= Ffc + K(t)(z - H~),~(0) = 0, (A. 18)

with

K = p H r R -1 (optimal gains), (A.19)

where P is the optimal error covariance matrix, i.e.

P (t) = E [~(t)~ r(t)], ~ = x - ~. (A.20)

178 A.E . BRYSON, JR. and W. KORTOM

P may be computed from the matrix Riccali equation

[ ' = F P + P F r + G Q G r - p H ' r R - J HP, (A.21)

where Q and R are the spectral densities of the independent white noise processes w and v.

To integrate (A.21) on a digital computer one has to discretize it, e.g. by using a Runge-Kutta type of integration scheme. Another way which we employed, is to use the equivalent discrete version of (A.16) to (A.21), see e.g. Ref. [32] section 12.5, and then solve the discrete filter problem [11]. A program which does the con- tinuous-to-discrete conversion and the discrete filter synthesis was written and made available to us by courtesy of Mr. David S. Spain, Electrical Engineering Dept., Stanford University.

In the case where the gains K are chosen or given, the error covariance, denoted by S(t), may be computed from the linear matrix differential equation

= ( F - KH)S + S ( F - KH)T_~_ G QG T+ KRK r. (A.22)

and v, their steady state contribution, denoted by q~(b) and ~(b), can be computed separately from

equations (A.23-A. 26) withq3 = ~ = 0, d e - dr =v = 0:

0 = 0 9 o ~ - - b e - K~b - K~bns, (A.27)

0 = --~oo~b-br-Kjp--Kg, bns, (A.28)

as

RMSq~(b)=~( b, , ]2+{Kc, b,s ,~=l ~ (A.29) J '

L\mo/ \COo(too + K , ) /

+ ( K,bns (A.30) + K,/_1

Note the roll-bias bp has no influence on the roll- error but does have an influence on the yaw-error. The yaw-gyro bias and the horizon sensor bias affect both errors.

If we adopt the following values for the constant bias errors

The steady state errors (if they exist) may be obtained by solving the algebraic equation resulting from letting S = 0 in (A.22).

bp=br=lO-Srad, see -x , bns=lO-3rad., (A.31)

and

APPENDIX 3

Error analysis of the second-order gyrocompass-filter

The error analysis of the second-order gyrocompass-filter given in Ref. [21] is based on constant gyro drifts and constant horizon sensor errors, the bias errors. In the analysis here we use the sensor error models developed in section 3, which include exponentially-correlated noise as well as bias in the gyro drifts, Dr(t) and Dr(t), and white noise as well as bias in the horizon sensor error V(t):

Dr,,(t)=bp, r+dp. r(t), V(t)=bns+ v(t). (A.23)

The measured roll-rate, pg, and yaw rate, ro, are therefore

pg =p + Dp(O, rg = r + Dr(t). (A.24)

K0= 10-Zrad. sec -1 , K~, = 3 x 10- 2rad. sec- l , (A.32)

for the gains as suggested in Ref. [21], we obtain

RMSq~(b) ~ 10- 3rad., RMS q~(b) ~ 10- 2rad. (A.33)

The RMSq3(b) value is due largely to horizon sensor bias whereas the main contribution to RMS~(b) is roll-gyro drift.

The same estimation accuracy can be achieved using a less expensive yaw-gyro with a higher drift and a roll-gyro with a slightly lower drift. For example,

bp~0.5 x 10- Srad. sec -~ ,

br_-__3 x 10- Srad. sec - I . (A.34)

The differential equations for the errors $ A 6 - ff = are

= COo~ -- Dr(t ) - K ~ - K , V (t), (A.25)

= - COo~ - D,(t) - K¢~ - Kq, V(t). (A.26)

Errors due to constant bias terms

If we assume that the constant bias terms bp, ,, ns are uncorrelated with the time-varying terms dr, ,

Since this choice of gyros seems to be reasonable, we will base all further comparisons on the assump- tion that the roll/yaw-gyro drift relation is

Dr/D , = 1/6. (A.35)

Note that it does not make sense to improve the gyros so that they lead to RMS-errors below 10 - 3 rad., since this is a lower bound on the bias error in the horizon sensor.


Errors due to time-varying noise

The comple te set o f e r ror equat ions for the t ime- varying noise sources of the gyros and hor izon

sensor is:

~)a = - K ~ a + ~o~d-- dp(t)- K#v(t), (A.36)

where

~a = - (K~ + eOo)~d-- d,(t)-- K,v( t ) , (A. 37)

d p : -- %dp + OZpW p , (A.38)

d r = -- otrd r + arWr, (A.39)

E[w p( t)w p( t') ] = Qpf( t - t') , - ]

e[wr(t)w,(t')] = Q , f ( t - t'),

E[v( t)v( t')] = R6( t - t') .

(A.40)

The mean-square errors of roll and yaw can be c o m p u t e d f rom a tenth order set o f l inear a lgebraic equat ions governing the covar iance mat r ix in the s teady state [32], p. 334, equa t ion (11.4.32), as

then we obta in

RMSq~a= x / X l l ---2"72 x 10-3rad . . (A.47)

R M S ~ d = x /X22~ 1"28 × 10-2rad . , (A.48)

Total RMS-errors o f the second-order gyrocompass- .filter

The to ta l R M S errors o f the orb i ta l gy rocompass design suggested in [21] using the er ror models o f equat ion (A.23), are then

RMSq~ t o t a l s 2 . 8 9 × 10-3 rad . , (A.49)

R M S ~ t o t a l ~ 1.62 x 10-2rad . . (A.50)

The constant bias errors (A.29, A.39) make no significant contribution to the total RMS errors.

The R M S yaw-er ror is a lmos t one order o f magni tude bigger than the R M S rol l -error . The uncer ta in ty in rol l is caused mainly by hor izon sensor errors, whereas uncer ta in ty in yaw is caused by hor izon sensor and gyro errors, and the con- t r ibut ions of each are abou t the same size.

XI~ = ,~lim E[(%(t) 2 ] = 1((OoX12 - X13 + ½K4,2R),

(A.41)

X22= t~lim E[~ld(t)2]=loI(K~ +cOo)Xal + K~,X12

+ X1,,-Koq-°9°Xa3% - K e K c , R I , (A.42)

where

1 [ ot,Q r

- L[ o( o + + + K,

+ K , Z R 1 ,

O~pQp

X 13 : - - ½K~ + % + (K~, + COo)COo/% '

(A.43)

(A.44)

(-O o O~ r Q r X 1 4 ~ - -½

(K, ¢ ~,)~, ¢ COo(K~, + coo) (A.45)

I f we choose for the R M S values of the exponen- t ial ly cor re la ted drif ts dp, r the same values as for the bias terms bp, , [see equa t ion (A.34)],

Qp= I 0 - 6rad.2sec - 1,

Qr = 36.10- 6rad.2see- 1,

R = 10-arad .2sec , (A.46)

R6sum6----Sur orbite un capteur d'horizon peut m6surer les angles de tangage et de roulis mais ne peut pas m6surer l'angle de d6viation du cap. Des appareils gyroscopiques peuvent detecter des variations de l'angle de deviation du cap, mais la variation de l'erreur de m6sure augmente graduellement avec le temps. Toutefois, en combinant des m6sures a partir d'un d6tecteur d'horizon et de deux gyroscopes int6grateurs de vitesse (en roulis et en deviation du cap), l'angle de d6via- tion du cap peut 6tre estim6 avec une variation d'erreur qui est limit6e lorsque le temps s'6coule. Untel dispositifporte le nom de gyroboussole orbitale. L'article montre que sa structure logique d6coule directement de la theorie de filtrage de Kalman-Bucy si les incertitudes de m6sure sont simul6es par un bruit blanc et seules les equations cin6matiques du mouvement sont utilis6es comme module du syst~me avec les signaux gyroscopiques consid6r6s comme des fonctions d'entr6e connues.

Toutefois, le bruit blanc n'est pas un tr~s bon module de la d~rive d'un gyroscope sur une longue p6riode de fonctionnement. Un meilleur module est constitu6 par un 6cart constant plus un bruit /t correlation exponentielle. Le filtre de Kalman-Bucy correspondant ~ c e module am61ior6 du gyroscope donne lieu ~t des variations de l'erreur sensiblement inf6rieures en angle de deviation du cap que la gyroboussole orbitale en grande partie en raison de l'estimation de la d6rive du gyroscope de deviation du cap. I! est montr6 que la d6rive du gyroscope de roulis n'est pas observable.

Zusammenfassung--Auf der Orbital-Kreisbahn kann ein Horizont-FiJhler Nick- und Rollwinkel, aber keine Gier- winkel messen. Kreiselgerhte k6nnen .~nderungen im Gierwinkel messen, abet die MeBfehlerstreuung whchst mit der Zeit best~indig an. Werden jedoch die Messungen eines Hodzont-Fiihlers und die von zwei Integrierkreiseln (f'tir Rollen und Gieren) kombiniert, so kann der Gierwinkel mit einer in der Zeit begrenzten Fehlerstreuung gesch~itzt werden. Eine solche Einrichtung wird Orbitalkompass genannt. Gezeigt wird, dab seine logische Struktur direkt aus der Filtertheorie von Kalman-Bucy folgt, wenn die Unsicher- heiten in der Messung als weiBes Rauschen modelliert sind und nur die kinematischen Bewegungsgleichungen als

180 A . E . BRYSON, JR. and W. KORTUM

Systemmodell benutzt werden, wobei die Kreiselsignale als bekannte Funktionen der Zwangskraft betrachtet werden. Weil3es Rauschen ist jedoch kein gutes Modell der Kreiselab- wanderung fiber lange Operationszeiten. Ein besseres Modell ist ein konstantes und ein exponentiell-korreliertes Rauschen. Das Kalman-Bucy-Fi l ter , das diesem verbesserten Kreisel- gerfit entspricht, erzeugt bedeutend engere Fehlerstreuungen im Gierwinkel, als der OrbitalkreiselkompaB, zum groBen Teil durch die Schg_tzung der Gierkreiseldrift. Die Roll- kreiseldrift ist, wie gezeigt, nicht beobachtbar.

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estimation of the local attitude of orbiting spacecraft

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