AN EVALUATION OF ALIGNMENT PROCEDURES FOR STRAPDOWN INERTIAL SYSTEMS

ABSTRACT

K.P. Schwarz and Ziwen Liu Department of Surveying Engineering

The University of Calgary 2500 University Drive N.W.

Calgary, Alberta, Canada, T2N IN4

The alignment of strap down inertial systems is typically achieved by a real-time estimation process which uses a reduced Kalman filter model for coarse alignment and a more complete model for fine alignment. In surveying applications with strapdown systems, there is usually no need for real-time results and the alignment procedure can be optimized for the complete data set taken during the alignment period. This gives much greater freedom in the choice of the estimation model used for alignment. The paper compares different approaches and evaluates the results with respect to external accurate azimuth information.

1. INTRODUCTION

In navigation applications of inertial systems, alignments are performed in real time. This often goes hand in hand with a requirement to minimize the alignment time. Studies of alignment procedures therefore have the objective of balancing accuracy and time requirements. Usually this amounts to achieving a specified accuracy in as short a time as possible. Highest possible accuracy of the alignment is therefore usually not an issue.

In survey applications of strap down inertial technology, real-time computations are often not needed and high alignment accuracy is more important than short alignment time. Thus, post-mission alignment methods are an alternative in this case. The paper investigates the question whether or not the accuracy of the initial values can be improved by making use of refined analysis techniques. Good initial alignment values are especially important in inertial positioning and inertial gravimetry.

In other applications, especially in a number of the emerging industrial applications, the stability of the alignment is an important requirement. A typical case is machinery alignment where shafts or rollers have to be aligned parallel in space with very little room to do the measurements. The use of an inertial reference unit has been proposed because it allows the transport of a reference direction from one point to the next without awkward forward and backsights. In this case, it is very important that the reference direction remains stable during transport and during the actual measuring periods. Occasional determination of linear gyro drift seems to be feasible by other methods.

Post-mission alignment has a number of advantages compared to real-time alignment. It widens the range of methods that can be used to estimate the alignment parameters. It allows the analysis of residuals and thus makes it possible to identify the steady state measurement phase. Finally, by comparing different types of update measurements for the same data set,

553 K.-P. Schwarz et al. (eds.), Kinematic Systems in Geodesy, Surveying, and Remote Sensing© Springer-Verlag New York Inc. 1991

conclusions can be drawn on the observability of certain states in an operational environment and on optimal data combinations for specific applications.

Post-mission methods that can be applied besides Kalman filtering are spectral methods for data screening and elimination of unwanted peaks, improvement of the initial values by repeated Kalman filtering, optimal smoothing, batch least-squares adjustment of the whole data set, and non-statistical methods of eliminating transient drifts. Since this fIrst series of tests was exploratory in nature and time was limited, not all methods have been used. However, the results presented should give a good indication of their suitability for this task.

Observability of specific states is obviously dependent on the type of update measurements used. Alignment in a stable environment allows to introduce updates for zero velocity, for zero position change, and for zero direction change. A comparison of the effectiveness of these update measurements for the estimation of specific states will therefore be included in the study.

The paper uses four sets of alignment data on the same station to fInd answers to some of the above questions. Each set consists of about one hour of data taken at a 64 Hz rate with the Litton LTN-90-100 unit owned by the University of Calgary. An external astronomically determined azimuth is available as an absolute reference on this station. It has been related to the internal azimuth of the inertial unit by a mirror-collimator system and calibration of the x­axis of the body frame. Although the measurement procedures and mathematical techniques are applied to data of a specific system, the results should be applicable with small modifications to any navigation-grade inertial strapdown unit.

2. ESTIMATION MODELS

All methods used in the following are based on the same mathematical model. It consists of the linearized solution of the first-order system of differential equations

x = Fx + Gu (1)

with initial values

x(to) = c

and update measurements of the form

y = Hx (2)

Assuming a linearized, time-invariant system, the solution of (1) can be written in the standard form

tk X(tk) = <l>(tk,tk-l) X(tk-l) + f <l>(tk,t) G(t) u(t) dt

lk-l (3)

where <l>(tk,tk-l) is the transition matrix between time tk-l and tk. In the following, the state vector will be of the form

x = {E,or,ov,b,d}T (4)

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where

E are attitude errors

Br are position errors

Bv are velocity errors

b are accelerometer biases

d are gyro drifts.

All states are defined as corrections to an initial estimate and all subvectors are of dimension 3.

The three methods discussed in the following estimate an optimal x by using the measurements (2) together with the model structure (3). The differences between the methods are due to either a different definition of optimality or different assumptions on the statistical processes involved. In addition, preftltering of the data will be used in certain cases to eliminate unwanted effects. This can either take place in the spectral domain and thus amounts to data cleaning, or can be built into the time-domain filter and then amounts to a change of the optimality criterion.

Kalman filtering

This is the standard real-time method giving an optimal estimate for a specific time tk based on all previous measurements and their accuracy. The unknown alignment parameters at time tk are estimated by using the standard algorithm

x(+) = x(-) + K{y - Hx(-)}

K = P (_)HT {HP(-) HT + R }-l } (5)

where (-) and (+) denote estimates before and after update, K is the Kalman gain matrix, and P and R are the covariance matrices of the system noise and the measurement noise. The time subscript tk has been omitted to simplify notation. For a detailed derivation, see for instance Gelb (1974).

Theoretically, results of this method will agree with other methods only at the end of the alignment period but not at any intermediate points because it process a subset of the data used with the other methods. Practically, convergence of the Kalman filter estimates towards the results of the other methods may be achieved earlier and is a good indicator of having reached the steady-state phase.

Filtering with improved initial values and covariance estimates

This method is in principle a double run of the Kalman filter on the same data set. The idea is simply to improve the initial values and covariance estimates of the alignment which are known to be very poor for the first run. The procedure is quite simple. After processing a specific data set through the Kalman filter, the state vector estimate at the end of the run and its covariance matrix are taken as initial values for another run with the same data set. By starting out with values close to the true ones, nonlinearities in the filter will be minimized and the state variations in the second run should be a good indicator for the system accuracy during the alignment phase.

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Filtering and optimal smoothing

This method improves the Kalman filter results by taking into account the effect of all measurements - before, at and after time tk+ 1. It can therefore only be applied post mission. The Rauch-Tung-Striebel algorithm is often used. It has the form

For a discussion, see again Gelb (1974).

Batch least-squares adjustment

In this case, equation (2) is rewritten as

} (6)

(7)

where Xn is typically the state vector at the end of the alignment period. By rewriting equation (7) as

y = DXn (8)

where y is now the vector of all update measurements and D contains design and transition matrices at all update periods, the least-squares solution for Xn can be written as

(9)

where the xn above a letter denotes an estimate. If the covariance matrices of the system noise and the measurement noise are known, say P and R, then the weighted least-squares solution is obtained from

(10)

Note that the components of R and P are different for different update periods. In formulas (9) and (10), the state vector at the end of the period is estimated from all update measurements. It can then be transferred to intermediate points by using

(11)

The choice of the end point is arbitrary. If instead an intermediate point is chosen, the same procedure applies.

Results from equation (10) are theoretically equivalent to results of filtering and optimal smoothing if the same covariance assumptions are made. The method has, however, some operational advantages, specifically in this application without vehicle dynamics.

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Spectral data cleaning

The data obtained from the inertial system contain a number of large spikes in the spectrum which are most likely due to aliasing caused by the dither frequencies. By transforming the data to the frequency domain, removing the spikes and transforming back to the time domain, clean data are obtained which theoretically should give a better result when used with any of the above methods. The spectral approach is discussed in Czompo (1990) and reference is made to his paper for all details.

3. DATA ANALYSIS AND RESULTS

Experimental setup

The tests were done on the roof of the Engineering Building of the University of Calgary. The choice of this location was mainly determined by the fact that one of the observation pillars installed there has a well-determined astronomical azimuth to a radio beacon a few kilometres away. This azimuth was used as a reference for the internal azimuth of the INS.

To relate the internal azimuth to the reference azimuth, several steps were necessary. First, the inertial system had to be equipped with a mirror in a plane approximately normal to the x-axis of the inertial system. It was then put into a Cardan frame which allowed free rotational motion of the system. The Cardan frame and the mirror arrangement are described in Knickmeyer (1989).

Second, a precise theodolite with a collimator was set up a few meters away with sights to the INS, the radio beacon and the pillar. Directions to these three targets were determined several times and the azimuth of the direction normal to the mirror plane was derived from it A sunshot azimuth was taken independently and the azimuth of the mirror normal was also derived in this way. The two determinations agreed within 10 arc seconds.

Finally, the mirror misalignment, i.e. the directional bias between the direction normal to the mirror plane and the internal x-axis of the INS, was determined by a method described in Knickmeyer (1989). It involves a series of rotations about the systems axes and uses the inertial system output for bias determination. The mirror misalignment was found to be about 35 arc minutes.

By correcting the mirror azimuth by this amount, the azimuth of the x-axis was found to be 355 34' 28". Taking all the error sources into account, the standard deviation of this determination is probably not better than 30 arc seconds.

The INS was kept fixed in this position and five independent alignments were done, each lasting about one hour. All data were recorded at a rate of 64 Hz. Unfortunately one data set was subsequently lost in a computer breakdown, so that only four sets were available for the data analysis described in the following.

Processing characteristics

Since the data sets represent a 'no dynamics' case, processing can be simplified by reducing the number of data points used. This can be done by averaging the raw data output over a certain time period, say 10 seconds. This speeds up the mechanization computations considerably. To test the effectiveness of this method, a comparison was made between the

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standard method using a 64 Hz rate for the mechanization equations and the averaging method using 10 second means of the raw data for the mechanization. Results were similar but in general the use of the high data rate gave slightly better results. It was therefore decided to use the 64 Hz data rate for this test series and to return to the problem of an optimal averaging period later on.

The update rate used in these tests was ten seconds. It gives a detailed picture of the azimuth change as a function of time and this was the main reason for choosing such a high rate. It is not required from an operational point of view but it allows to plot the deviations of the estimated azimuth from the reference azimuth as a continuous curve. Updates were made using the conditions of zero velocity, zero position change and zero direction change. The initial variances of the state vector elements, the spectral densities for the system noise and the variances for the update measurement noise are given in Table 1.

Parameter Initial Variance Spectral Density

EE, EN 40 arcsec. 0.1 arcsec2/sec.

EU 1 degree 0.1 arcsec2/sec.

o<j>,oA.,oh 0.1 m 0

&i>,o~,oh 0.003 m/s 0.25 • 10-6 m2/s3

dx, dy, dz 0.01 deg./h 1.38 • 10-9 deg2/h3

bx, by, bz lOmgal 1.38 • 10-3 mgal2/s

zero velocity update 0.0005 m/s

zero position update 0.01 m

zero angular velocity update 1. • 10-6 deg./h

Table 1: Statistical assumptions

Influence of initial values

All data sets were first processed through the standard Kalman filter using an azimuth error of 1 degree as standard error after the coarse alignment. A typical result of this procedure is displayed as solid line (a) in Figure 1. It shows the deviation of the azimuth estimate from the reference azimuth which is shown as zero line. As expected, this first run shows large variations during the first half hour, even though convergence seems to occur after about 10 minutes or about 60 updates. After half an hour, the CUlve becomes smoother and varies only within 1 arcminute. It appears that any data point from there on gives an estimate of the azimuth which is good to about 1 arc minute. This result does not significantly improve by a longer observation time. It is similar for all other runs.

The estimated state vector at the end of this first run and its covariance matrix were then taken as initial values and the data set was processed again. The result is shown as broken line (b) in Figure 1. This second run is much smoother and does not show the large initial variations. It seems to be quite stable over periods of several minutes but still shows some distinct long term variations. Again, this result is similar for all runs. It indicates that a good approximation is most likely possible with a smaller data set if a number of iterations can be made in post mission.

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3

2

.- 1 C ·s

0 (J

'"' ~ '--'

.c ... -1 =' E ~ -2

-3 85 1085 2085 3085 4085

T ime (sec.)

Figure 1: Effect of initial values (curve (a): poor initial values; curve (b): good initial values)

To investigate this question, only about twelve minutes of the same data set were taken and were processed repeatedly through the standard Kalman filter, improving the initial value for the azimuth and its covariance in each run. The result is shown in Figure 2, where the numbers at the curves indicate their Kalman filter iteration. The result is as expected.

3

2

.-d 1 ·s (J

0 I.. ~

, ...... . . -._._._._. 4 __ . -._.-._._0_. ....... yHt>I' • • .-..,~. .. -. .... ... .......... ..... .. ,... ..... _ .... ... ~.

'-" 3 .c ... =' - 1 E ~ .

'N -<

-2

-3 85 185 285 385 485 585 685

Time (sec.)

Figure 2: Azimuth iteration for a short data span

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After a good approximation of the reference azimuth has been reached at the end of the fIrst Kalman fIlter run, the subsequent iterations show a very smooth and stable behaviour. The accuracy of the result is, however, very much dependent on the accuracy of the estimate at the end of the fIrst run. If it has a bias, it is unlikely to be eliminated in subsequent runs. The stability of the results stays the same,however. This means that if a bias is of no concern, i.e. if variation about a given reference direction is the important parameter, these results can be still quite useful.

The following conclusions can therefore be drawn from these experiments. Convergence and accuracy of the alignment depend heavily on the initial azimuth estimate and its covariance. For real-time methods these initial values are usually poor and an improvement is only possible by collecting additional data. For post-mission processing, iterative procedures can be applied, where the end results of one processing step are used as initial values for the next step. In this way, shorter data spans can be used and convergence can be achieved after a few iterations. In general, accuracies of estimates are within a standard deviation of about 30 arc seconds. This accuracy can be reached after about 30 to 40 minutes of real-time Kalman filtering and after about 10 minutes of iterative post-mission processing.

Influence of update measurements

Figure 3 shows the effect of different types of update measurements on the same data set. In all cases the update period is ten seconds and the same initial values have been used for all runs. They are typical values obtained after processing the data through the Kalman fllter once. Again, the deviation of the azimuth estimate from the reference azimuth is shown as a function of time.

3 ~ __________________________________________ -,

2

-1

-2

85 585 1085 1585 2085 2585 3085 3585 Time (sec.)

Figure 3: Effect of different update measurements (1: zero velocity; 2: zero position change; 3: zero direction change)

It is clear from the graph that the results obtained from zero velocity updates and zero position change updates are very close. This is an expected result because the effect of the two

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updates on the azimuth is essentially the same for a stationary system. The third update, enforcing a zero direction change, shows a very different pattern. The resulting curve is smoother than the others but shows a more systematic deviation from the reference azimuth. The same characteristical behaviour occurs in all four runs. It appears that this updating procedure eliminates residual gyro drifts very well but affects the estimation of the misalignments in some way.

A summary of results from all four runs is given in Table 2. It shows that the mean of the four runs is in all cases within 10 to 16 arc seconds of the reference azimuth, well within the standard deviation computed by standard least squares error propagation.

Update Estimated Azimuth Standard Deviation

Zero velocity 3550 34'12" ± 26 arcsec.

Zero position 3550 34'11" ± 26 arcsec.

Zero angular velocity 3550 34'18" ± 30 arcsec.

Reference 3550 34'28" ± 30 arcsec.

Table 2: Mean values from different update procedures

Stability of estimates

It has been mentioned before that stability of the azimuth determination over time is more important in some industrial applications than the accuracy of the azimuth itself. Figure 4 shows results for the four runs using zero velocity updates. In this case, the same initial value

3 ~------------------------------------,

2

-

-2 c

85 585 1085 1585 2085 2585 3085 3585 Time (sec.)

Figure 4: Comparison of four runs

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and covariance matrix have been used for all four runs, assuming that the data have been processed one in the usual way. Variations over ten minute intervals are usually not larger than 10 to 20 arc seconds. Over the total period of one hour, variations can be as large as 1 arc minute.

In some of the emerging applications, the typical operational procedure would require setups of 10 minutes at individual stations, during which time the internal azimuth would be used as a reference, followed by a short transport time to the next station. Table 3 indicates what could be expected under such circumstances. It shows the azimuth difference with respect to the control value for 10 minute means. Statistical information in terms of mean and its standard deviation is given for each run, and for the mean across runs. The table shows clearly that the variations between 10 minute means of individual runs are much smaller than the variations between runs. This indicates that the mean of a run may be biased but that the variations around that bias are small. Thus, if the stability of a reference line is more important than the accuracy of the North direction, then the system tested will reliably maintain such a reference direction for 10 minute means within a standard deviation of 10 to 15 arc seconds.

Time Data Set

Interval A B C D (Minutes)

0- 10 -39" 0" _76" 42"

10- 20 -26" 0" -50" 19"

20- 30 -39" 5" -37" 19"

30 - 40 -36" _3" -37" 10"

40- 50 _43" 7" -20" 22"

50- 60 -37" 3" _44" 22"

60 -70 -30" 5"

Mean -36" 2" _44" 20"

RMS for ±6" ±4" ± 19 ± 12

RMS about total mean: ±29"

RMS about ref. azimuth: ±32"

Table 3: 10 minutes means for different runs

Spectral data cleaning

A few tests were made with a data set where the spikes in the spectrum had been removed before processing the data through the Kalman filter. This method of spectral data cleaning is described in more detail in Czompo (1990). A typical result is shown in Figure 5 where the solid curve shows the result for the raw data, while the broken curve shows the result for the cleaned data. The two curves display a parallel shift but show otherwise exactly the same behaviour. The explanation of this peculiarity is most likely that the spectral procedure introduces a small difference in the mean value which causes the parallel shift. The spikes that have been removed from the data have sufficiently high frequencies and do therefore not show

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up in the curve. Since factory calibration of the system is usually done without spike removal, the calibrated gyro drift should be changed to reflect the change in the mean value.

5

4

3

..-.. 2 c ·5 1 (.,I

0 s... ~ '-'

.::: - 1 -::s E -2 ·N < -3

-4

-5 85 1085 2085 3085 4085

Time (sec.)

Figure 5: Comparison of filtering results for raw (a) and spectrally cleaned data (b)

4. CONCLUSIONS

Surveying applications of inertial strapdown technology do usually not require real-time processing but place high demands on accurate initial azimuth determination and stability of the internal azimuth reference. The paper explores the consequences of these requirements, i.e. it looks at alignment from a post-mission processing point of view. Such an approach gives more flexibility in the choice of estimation procedures and allows the analysis of residuals. The paper formulates a common framework for the different methods and presents a first comparison of real-time and post-mission techniques on the same data set.

Comparisons were done using Kalman filtering as the real-time method and iterative Kalman filtering as the post-mission method. Results show that the accuracy of the estimated azimuth and its stability is strongly dependent on the quality of the initial values. Since the quality of these values is usually poor in real-time methods, because the initial azimuth is dependent on the coarse alignment, a data span of 30 to 40 minutes is needed to get results at the noise level of the system. In post-mission methods this accuracy can be reached for a data span of only 10 to 15 minutes. The standard error with respect to an external reference azimuth is about 30 arc seconds. The largest part of this error is due to a constant bias which changes randomly from one alignment to the next. Its size is typically 20 to 25 arc seconds.

The stability of the estimated azimuth is therefore much better than indicated by the overall standard deviation. Again, post-mission results are in general more stable than real-time results. Variations about the mean value of the run are small, typically 10 to 15 arc seconds.

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The overall accuracy of the result does not depend very much on the type of update measurement used, however, there are differences when azimuth estimation is viewed as a time function. While results obtained from zero velocity and zero position change updates are virtually identical, this is not true for zero direction change updates. The more systematic behaviour in the latter case requires further study.

The use of spectral methods to eliminate unwanted spikes in the spectrum and thus provide a clean set of data, did not improve the overall accuracy of the results. It introduced a small bias in the data which can be easily accommodated by changing the calibrated constant gyro drift by the same amount.

The results reported here should be considered as preliminary. Further tests are planned and different estimation methods will be used to arrive at optimal operational procedures for different applications.

ACKNOWLEDGEMENTS

Financial support for this research was obtained from an NSERC Cooperative Grant, entitled The Industrial Alignment Project. D. Lapucha, H.E. Martell and J. Czompo are acknowledged for their assistance in setting up the tests and in providing support during the data processing phase.

References

Czompo, S. (1990). Use of spectral methods in strapdown ISS data processing. This volume.

Gelb, A. (ed. 1974). Applied Optimal Estimation. M.LT. Press, Cambridge, Massachussets.

Knickmeyer, E.H. (1989). Calibration, handling and use of a Cardan-frame with the Litton LTN 90-100 inertial reference system. UCSE Report 30011, University of Calgary.

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