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Copyright ©1996, American Institute of Aeronautics and Astronautics, Inc.

AIAA Meeting Papers on Disc, July 1996A9635620, AIAA Paper 96-3736

Generalized coning compensation algorithm for strapdown system

Chan G. ParkKwangwoon Univ., Seoul, Republic of Korea

Kwang J. KimKwangwoon Univ., Seoul, Republic of Korea

Dohyoung ChungSeoul National Univ., Republic of Korea

Jang G. LeeSeoul National Univ., Republic of Korea

AIAA, Guidance, Navigation and Control Conference, San Diego, CA, July 29-31, 1996

A generalized coning compensation algorithm for a strapdown system is proposed by minimizing the coning error. Theproposed algorithm is different from the previous algorithms in that it allows the design of the corresponding optimalconing compensation algorithm for all combinations of gyro samples. Using the proposed algorithm, it is found thatmany of the existing coning algorithms can be treated as the special cases of the proposed algorithm. It is clearly shownthat the magnitude of the resulting algorithm errors depends mainly on the total number of gyro samples includingpresent and previous gyro samples rather than on the number of gyro samples in the minor interval. The mainadvantage of using the proposed algorithm lies in its easy expansion to various combinations of gyro samples. Hence, itenables attitude algorithm designers to easily develop the most effective and optimal coning compensation algorithmaccording to their attitude computation specifications. (Author)

AIAA-96-3736

GENERALIZED CONING COMPENSATION ALGORITHM FOR STRAPDOWN SYSTEM

Chan Gook Park* and Kwang Jin KimfDept of Control and Instrumentation Engineering & Institute of New Technology

Kwangwoon University, Seoul 139-701, Korea.and

Dohyoung Chungf and Jang Gyu Lee§School of Electrical Engineering & Automatic Control Research Center

Seoul National University, Seoul 151-742, Korea

In this paper, a generalized coning compensationalgorithm for strapdown system is proposed byminimizing the coning error. The proposed algorithm isdifferent from the previous algorithms in that it allows thedesign of corresponding optimal coning compensationalgorithm for all combinations of gyro samples. Using theproposed algorithm it is derived that many of the existingconing algorithms can be treated as the special cases ofthe proposed algorithm. Furthermore it is clearly shownthat the magnitude of resulting algorithm errors dependsmainly on the total number of gyro samples includingpresent and previous gyro samples rather than on thenumber of gyro samples in the minor interval. The mainadvantage of using the proposed algorithm lies in its easyexpansion to various combinations of gyro samples.Hence it enables the attitude algorithm designers to easilydevelop the most effective and optimal coningcompensation algorithm according to their attitudecomputation specifications.

Introduction

The attitude computation plays a key role instrapdown inertia! navigation system because thecomputed attitude is continuously used for transformingthe vehicle acceleration measured by the accelerometersrigidly attached to the host vehicle.

* Assistant Professorf Graduate Studentf Doctoral Student§ Professor, Senior member

Copyright © 1996 by the American Institute ofAeronautics and Astronautics, Inc. All rights reserved.

Commonly used attitude updating algorithms forstrapdown systems are the Euler method, the directioncosine method, and the quaternion method. Among them,the quaternion method is quite popular due to itsadvantages in nonsingularity, simplicity, andcomputational efficiency.1 Further improvements on thequaternion method have been made by Bortz,2 Jordan,3Miller,4 Lee,5 and Jiangs using the concepts of rotationvector. It is proven that the quaternion updating methodwith the rotation vector can effectively suppress thenoncommutativity error, which is one of the major errorsources in numerical solution of the altitude equation.4"6

Recently Ignagni7"8 has summarized the previousalgorithms in terms of the integral of thenoncommutativny rate vector and also proposed theconcept for improving the computational efficiency.

In this paper, a generalized coning compensationalgorithm for strapdown system is proposed. It is derivedby miromizing the coning error, which generates theworst noncommutativity error during the attitudecomputation. The effectiveness of proposed algorithm isdemonstrated by showing that many of the existingattitude algorithms are the special cases of the proposedalgorithm. Furthermore it is shown that the proposedalgorithm can be easily expanded to the cases of highersample numbers.

Quaternion Update Using the Rotation Vector

The key operation in the attitude algorithm is toproperly update the quaternion and rotation vector.4 Thequaternion update Q(t + K) is obtained by the followingquaternion multiplication:

(1)

where q(h) is an updating quaternion during the timeinterval h and is expressed by the usual quaternionrelation, namely,

1American Institute of Aeronautics and Astronautics

?(/») = c o s — s i n -2 9 . 2

(2)

where (j) represents the rotation vector with magnitude<j>, = (<(> .(j))"1. The rotation vector differential equation canbe approximated as2-9

-». . . / -» . .,« \

(3)- — (j) X (j) X (tj) X (0_)

where ffl^ represents the angular velocity vector. The lasttwo terms in Eq. (3) are referred as the noncoinmutativityrate vector. It must be determined and compensated fromgyro measurement in order to maintain the high accuracyof strapdown attitude algorithms. However the triple-cross-product term is assumed to be quite small and canbe neglected.4'5

hi the development of generalized coningcompensation algorithm it is assumed that each majorattitude interval is divided into a number of minor interval,each in turn being divided into a number of data sampleintervals over which the gyro incremental angle ismeasured, as depicted in Fig.l. This presents the mostgeneral representation of interval, since no restriction isplaced on the number of sensor data intervals contained ina major attitude update interval. It is desirable that therotation vector be updated in every minor interval,whereas the quaternion is updated only once in everymajor interval. To make it possible, an algorithm has to beobtained directly to update the rotation vector instead ofusing a quaternion.

Major interval,/?

Minor interval, T

*0 ^m-l MATT* Tm TMData sample interval ,A^

Fig. 1. Intervals associated with coning compensation

Generalized Coning Compensation Algorithm

In order to develop the generalized coningcompensation algorithm, it is assumed that the body isundergoing pure coning motion, defined by the angularrate vector

co = (4)

where co_ = angular velocity vector with componentsexpressed hi the body frame

a,b = amplitudes of the angular oscillations intwo orthogonal axes of the body

n = frequency associated with the angularoscillations

/, J = unit vectors along the two body axes aboutwhich the oscillations are occurring.

Then the coning angle vector is determined as

J (5)

If the angular rate in Eq. (4) is applied for two axes / andJ_, the vehicle generates the coning motion for ^T-axis.10

When one period of the coning motion is terminated, thedrift error which is the component of thenoncommutativity error is generated in I and J-axes.

Let us define the coning correction over a minorcomputational interval from TMto tm as11

5d> = — f " a(T.x ,) x co <—"• 7 JT: , ' ™"! ~4, m-i

(6)

Applying the coning motion in Eqs. (4) and (5) to Eq. (6),we have

54, =0.

(7)

The result given by Eq. (7) reveals the interesting propertythat the coning correction is constant over all minorintervals, regardless of the absolute time at which theinterval begins. It depends only on the duration of theminor interval T.

For the coning environment defined by Eq. (6), theincremental angular velocity over a data sample intervalof duration AT7 from ttJi to tt is

dt

(8)

where t| = cos A. - 1, v = sin A,, and /„ = DAT

American Institute of Aeronautics and Astronautics

The cross product of two incremental attitude vectorsA0(0 x A6(/) taken over different data sample intervalsbecomes

A6(z) x A0(/) = ab {2sin[(/ -/(9)

Just as the coning correction, the value of the crossproduct of two attitude increments is independent of theabsolute time, but depends only on the duration AT of theintervals and their spacing. Hence Eq. (7) can beapproximated using Eq. (9).

Lee. et al.5 introduced the concept of "distance"between the cross products in Eq. (9). It was found thatcross products with equal "distance" behave exactly thesame for coning inputs. Taking advantage of this propertyallows the generalized form for the coning integralalgorithm to be simplified as

(10)

where n is the number of sample for m-th minor interval;p is the number of sample for (m-l)-th minor interval:A6W(/) is i-th gyro sample for m-th minor interval;AG^j (/) is j-th gyro sample for (m-l)-th minor interval;k and k are the respective constants for distance2/l-J n~t r

In—j and n—i for AS^w);

Substituting Eq. (9) into Eq. (10), we obtain

5$ = &m^_1A9m_1(«-p+l)xA9OT(«)+---

„_! {2 sin(« - - sin(n -;K;. (11)

Expanding each term in Eq. (11) using Taylor series, theconing correction over the minor interval is obtained as

2 + ' ' ' + -«2/-O /%+p-l J

where A is a constant defined by

^(27 + 1)!

+ - }K,(12)

(13)

In order to find kd(d = l,2,- -.n + p-l) in Eq. (12), thetrue coning correction in Eq. (7) is expanded using Taylorseries, and it becomes

2 [ 3!

= ab

5!

3!x2 5!x23 _ T 5 . T 5

K

(14)

where c is defined by

(2:1-1)

(2z +l)!x 2(15)

Suppose that the number of total sample N is p+n, thenthe number of unknown parameter kd is N—l. UsingEqs. (12) and (14) the simultaneous equation forunknown parameter kd can be expressed as follows.

AT-1 Af-1

/=! J=y=]

N-l(16)

Once n and p are selected, the correspondingoptimal coning compensation algorithm can be designedby the following procedures. First, the constants A andct are calculated from Eqs.(13) and (15), respectively.Secondly, the unknown coefficient kd is solved by Eq.(16). Finally, the optimal coning compensation algorithmis obtained by inserting kd into Eq. (10). Since thissimple procedure enables us to design optimal coningcompensation algorithms for various combinations of nand p, it is referred as the generalized coningcompensation algorithm.


Furthermore kd in Eq. (16) can be simply expressed in. amatrix form. For the case when the number of totalsample is less than or equal to 5, the optimal set ofcoefficients satisfies the following matrix equation:

1 2 3 4

1 3 19- - — 114 2 41 23 289 283

40 60 120 3017 311 827 12,071

_12,096 6,048 1,344 3,024 _

*,k,k,

L 4-

• «' •12

240n'

10,080n9

_725,760_(17

Hence kd can be obtained by matrix inversion andmultiplication. After kd's are determined, the optimalconing algorithm can be found using Eq. (10).

The 8(b computed from Eq. (10) is later used for—.771

updating the rotation vector <j> over the m-th minor—m

interval such as,

-Lm -Lm-1

1-Lm-1 1=1

(18)

Next the algorithm error is derived to analyze theperformance for the generalized coning compensationalgorithm. The basic relationships used in deriving theoptimal algorithm coefficients may also be employed inestablishing the accuracy associated with each algorithmin a pure coning environment. The error s is denned asthe error contained in approximation as follows.

s = 84 - 5<i> .—— * m l m

(19)

Substituting Eqs. (12) and (14) into (19), it is found thatthe most dominant error hi z becomes

(rad).

(20)

Note that the algorithm error mainly depends on the totalnumber of samples.

Algorithms

It is well known that the^estimation error in updatingthe rotation vector depends mainly on the number of gyrosamples and many coning compensation algorithms havebeen previously developed. In this section the coning errorcompensation algorithm and the resulting errors forvarious sample combinations are derived using theproposed algorithm.

Each algorithm is obtained accordingly from Eqs. (13),(15), (16) and (10). The corresponding error is alsoobtained from Eq. (20).

Algorithm 1 (two samples and one previous sample):When two present samples and one previous sample perupdate are used, then « = 2, p = l, and N = 3. Hencethe algorithm can be determined by computing &, and k-,fromEq. (17) and using Eq. (10) as follows.

=-l.A9 ,(2)xAe (2)+— A9 (l)xA0 (2). (21)"3A —ff^~i^ ' — ̂ ^ ' ic — M^ ^ — "*N ^

The algorithm error can be obtained from Eq. (20) and itbecomes

_ ab 7 _ ab•~140 17920

(22)

The algorithm 1 using our proposed algorithm is identicalto the one proposed by Jiang.6

Algorithm 2 (three samples'): three-sample algorithm ispreviously proposed by Miller.4 In this case n = 3, p = 0,and 7V=3. /^ and k-, can be computed from Eq. (17).Then the algorithm becomes

(23)

and the error becomes

3ab 7 ab280 ~ 204120

(24)

The resulting error of Eq. (24) has the same power of £2Tas that of Eq. (22) has. Hence it can be seen that themagnitude of error depends mainly on the total number ofgyro samples rather than on the number of samples in theminor interval.


Algorithm 3 Cthree samples and one previous sample):When using three present samples and one previoussample, the algorithm can be obtained in the similarmanner as in Algorithm 2, and it becomes

(25)393——A9m(2)xA0m(3)f\Qs\ —'ft* ^ s —.« \- /


Algorithm 6 (six samples): Six-sample algorithm has notbeen previously introduced. However using the proposedmethod it can be easily obtained. In this case n = 6,p = Q, and N=6, and kd can be calculated from Eq.(17) as follows.

15797 3917 _ 608 2279 4631 ~ 4650 ' z ~ 2310'̂ ~ 385 ' 4 " 2310' 5 ~ 924 "

(31)

Hence the algorithm becomes

420 8.266.860(26)

This result is also identical to the Jiang's result6 It impliesthat using one previous sample we can expectconsiderable reduction on the error compared to that ofthree-sample algorithm in Algorithm 2.

Algorithm 4 (four samples'): In this case n = 4, p = 0,and N=4, and the algorithm becomes

(27)214——A9 m(3)xA9 m(4),1 /•> ̂ —TO \ f —FT} V /


ab 9 ab-A = • (28)T S 315 82,575,360

This result is identical to the one proposed by Lee et al.5

Algorithm 5 (five samples'): Five-sample algorithm hasbeen recently introduced by Ignagni.8 In this case n = 5,p = Q, and N=5. Then the algorithm becomes

+

125m

325252 A

25+u1375

and its error becomes

ab5544 54,140,625,000

IQ7Y1

(29)

(30)

A3 A6m(3) (4) x A6ro

(32)

Its error can also be calculated from Eq. (20) and itbecomes

4004 52,295,018,840,064(nr>13. (33)

The error for six-sample algorithm has the 13th power ofQT, and it concurs with the previous statement that themagnitude of error depends mainly on the total number ofsamples.

Conclusion

In this paper, a generalized coning compensationalgorithm for strapdown system is proposed bymininuzing the coning error, which is one of the mostcrucial error sources. It is shown by examples that manyof the existing attitude algorithms can be regarded asspecial cases of the proposed algorithm. Furthermore it isclearly demonstrated that the magnitude of resultingerrors depends mainly on the total number of samplesrather than on the number of gyro samples in the minorinterval. The main advantage of using the proposedalgorithm lies in its easy applications to variouscombinations of sample numbers. Thus it enables theattitude algorithm designers to choose the most effectiveconing compensation algorithm for their attitudecomputation specifications with ease.

References

1- Wilcox, J. C., "A New Algorithm for StrapdownInertial Navigation," IEEE Transactions on Aerospaceand Electronic Systems, Vol. AES-3, No. 5, 1967, pp.796-802


2. Bortz, J. E., "A New Mathematical Formulation forStrapdown Inertia! Navigation." IEEE Transactions onAerospace and Electronic Systems, Vol.7, NO. 1, 1971 ,pp. 61-66

3. Jordan, J. W., "An accurate Strapdown directioncosine algorithm," Report TN D-5384, NASA,Washington, DC, Sept. 1969.

4. Miller, R. B., "A New Strapdown AttitudeAlgorithm," Journal of Guidance, Control, and Dynamics,Vol. 6, No. 4,1983, pp. 287-291

5. Lee, J. G. et al., "Extension of Strapdown AttitudeAlgorithm for High-Frequency Base Motion," Journal ofGuidance, Control, and Dynamics, Vol. 13, NO. 4, 1990,pp. 738-743

6. Jiang, Y. F. and Lin, Y. P, "Improved StrapdownConing Algorithms," IEEE Transactions on Aerospaceand electronic Systems, Vol. 28, No. 2,1992, pp. 484490

7. Ignagni, M. B., "Optimal Strapdown AttitudeIntegration Algorithms," Journal of Guidance, Control,and Dynamics, Vol. 13, No. 4, 1990, pp. 738-743

8. Ignagni, M. B., "Efficient Class of Optimal ConingCompensation Algorithms," Journal of Guidance,Control, and Dynamics, Vol. 19, No. 2, 1996, pp. 424-429

9. Jiang, Y. F. and Lin, Y. P, "On the rotation vectordifferential equation," IEEE Transactions on Aerospaceand Electronic Systems, AES-27, Jan. 1992, pp. 181-183

10. Lee, J. G. et al., Attitude Algorithm for StrapdownInertial Navigation System, ADD Technical Report,1988.4.

11. Savage, P. G., "Strapdown System Algorithms,"AGARD-LS-133,1984.


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