ADAPTIVE LIMIT-CHECKING FOR SPACECRAFTUSING RELEVANCE VECTOR AUTOREGRESSIVE MODEL

Ryohei Fujimaki1, Takehisa Yairi2, and Kazuo Machida2

1Dept. of Aero. and Astronautics, Univ. of Tokyo, 4-6-1 Komaba Meguro-ku Tokyo Japan,fujimaki@space.rcast.u-tokyo.ac.jp

2RCAST, Univ. of Tokyo

ABSTRACT

Development of advanced anomaly detection and failurediagnosis technologies for spacecraft is a quite signifi-cant issue in the space industry, because the space envi-ronment is harsh, distant and uncertain. While severalmodern approaches based on qualitative reasoning, ex-pert systems, and probabilistic reasoning have been de-veloped recently for this purpose, any of them has a com-mon difficulty in obtaining accurate and completea pri-ori knowledge on the space systems from human experts.A reasonable alternative to this conventional anomaly de-tection method is to reuse a vast amount oftelemetrydatawhich is multi-dimensional time-series continuously pro-duced from a number of system components in the space-craft.

This paper proposes a novel ”knowledge-free” anomalydetection method for spacecraft based on data-miningtechniques. This method constructs a non-linear proba-bilistic model regarding the behavior of a spacecraft inthe learning phase by applying therelevance vector re-gressionandautoregressive modelto a massive teleme-try data of a spacecraft, and then monitors the on-linetelemetry data using the constructed model and detectsanomalies.

We have confirmed the effectiveness of the proposedanomaly detection method by applying it to the telemetrydata obtained from a simulator of an orbital transfer ve-hicle designed to make a rendezvous maneuver with theInternational Space Station and the real-telemetry data ofEngineering Test Satellite VII(ETS-VII).

Key words: anomaly detection, autoregressive model,relevance vector regression, telemetry data.

1. INTRODUCTION

Anomaly detection is a key issue in the development ofrecent advanced complex spacecraft. The space environ-ment is very harsh for spacecraft due to a variety of fac-tors such as direct radiation, great temperature difference,risk of clash with space debris, and so on. It is practi-cally impossible to completely eliminate the possibilityof anomalies or faults, even if we increase the reliabilityof the system components to the limit. In addition, thespace is so distant from the earth that it is extremely diffi-cult to directly inspect and repair a damaged component.Therefore, early detection of anomalous symptoms in thesystem behavior is significantly important to avoid dis-astrous situations such as loss of control. Although sev-eral anomaly detection / diagnosis methods using mod-ern reasoning techniques such as qualitative reasoning,expert systems and probabilistic reasoning have been de-veloped, they have difficulties in acquiring accurate andcomplete models and knowledge of the spacecraft sys-tems and in monitoring the system behavior exhaustivelyand efficiently.

One major reason for these difficulties is that conven-tional anomaly detection systems are heavily dependenton a priori knowledge on the system behavior for eachspacecraft. For example, the model-based method(4; 7;9; 16) requires a perfect dynamics model of the space-craft, and the expert system(8) demands a set of produc-tion rules. In practice, however, preparing such completeand accuratea priori knowledge of the systems is verydifficult and expensive, partly because of the so-called”bottle neck of knowledge acquisition”, and partly be-cause of the difference between theory, experiment andactual behavior of the system on the orbit. To make mat-ters worse, the reuse of models and knowledge of the pastsystems is also limited, because the spacecraft is one ofthe ultimate custom-made productions in the world, un-like household appliances.

Another difficulty in the anomaly detection and diagnosisfor the spacecraft lies in the complexity of the spacecraft

Proc. of 'The 8th International Symposium on Artifical Intelligence, Robotics and Automation in Space - iSAIRAS’, Munich, Germany.5-8 September 2005, (ESA SP-603, August 2005)

systems. Although the spacecraft subsystems (such asattitude control, power supply, and so on) are designedto minimize mutual interference, actual anomalies oftenoccur across several subsystems, which makes it diffi-cult for human experts and model/knowledge-based de-tection/diagnosis systems to investigate and understandthe phenomena.

For the above reasons, the necessity for ”knowledge-free”anomaly detection methods which are not dependent ona priori expert knowledge such as dynamics models andproduction rules is being widely recognized. Fortu-nately, most modern spacecraft including artificial satel-lites and orbital transfer vehicles are transmitting multi-dimensional time series calledtelemetry datacontaininga variety of low-level system information to ground sta-tions. While the telemetry data is originally for simplisticlimit-checking and manual analyses by engineers and ex-perts, recent researches have shown that a variety of datamining techniques can be applicable to this data.

In this paper, we propose a new anomaly detectionmethod for spacecraft based on data-mining techniques.This method is based both on the classicalautoregressivemodel(AR model) and therelevance vector regression,which has been one of the most successful nonlinear re-gression learning algorithm in recent years(1; 2; 3; 11;12; 13), and constructs a predictive model for each time-series in the telemetry data in the initial operation phase.After that, it monitors online telemetry data and detectanomalies by checking the probability density of the ob-servation.

The remainder of this paper is organized as follows. First,we briefly explain the conventional approaches to theanomaly detection problem in space systems, such aslimit-checking, simulation, and expert system in section2. Then, we explain therelevance vector autoregressionand propose a new anomaly detection system based onthis method in section 3. In section 4, we show and dis-cuss some results of an experiment. We made use ofthe telemetry data of the unmanned spacecraft based onthe simulation of rendezvous with the ISS (InternationalSpace Station) and real telemetry data of EngineeringTest Satellite VII(ETS-VII) and compared the proposedmethod with the limit-checking. Finally, we concludethis study in section 5.

2. CONVENTIONAL TECHNIQUES OFANOMALY DETECTION / DIAGNOSISFOR SPACE SYSTEMS

Limit-checking has been the most basic and commontechnique of detecting anomalies in spacecraft systemsfor a long time. It constantly monitors some impor-tant time series in the telemetry data and checks whetherthe value is within the pre-defined upper and lower lim-its. Ref.(6) proposed a method of automatically comput-ing limit values using autoregressive model. Though thelimit-checking has an advantage that it is simple enough

to be applied to any types of spacecraft, it lacks flexibilityand expressiveness and suffers from the problem of falsealarms.

In contrast to the limit-checking, the model-based faultdetection and diagnosis method will be the most sophis-ticated approach to the problem, in which system modelsare utilized to simulate the spacecraft behavior and ex-amine the validity of the actual telemetry data. Some re-searchers use qualitative models(4; 16), while others usemathematical (probabilistic) models(7). Ref.(9) does notdirectly detect anomalies but intelligently estimates theimportance of each sensor information using BayesianNetworks and gives a higher priority to the more impor-tant one when it is displayed on a monitor. The model-based approach would provide an ideal performance if aaccurate and complete model and infinite computationalpower were available. In practice, however, both of themare not available, which limits the applicability of thismethod.

Expert systems (knowledge-based systems) also havebeen developed for this purpose, in which knowledge foranomaly detection acquired from human experts is used.The knowledge is generally represented in the form of”if-then” production rules(8). Though the expert systemsare powerful and flexible, it has a difficulty in preparinga set of accurate and complete knowledge on the space-craft. This problem has been well known as the bottle-neck of knowledge acquisition.

In summary, the above methods of limit-checking,model-based and knowledge-based approaches have acommon problem that they are too dependent on theknowledge of human experts.

As mentioned in the previous section, a reasonable ap-proach to this problem is the application of data min-ing / machine learning techniques to the spacecrafttelemetry data. Actually, some researchers have de-veloped anomaly detection methods for spacecraft us-ing the DM/ML techniques such as regression treelearning(14), temporal pattern clustering(17), associationrule mining(18) and support vector machine(5). Most ofthem are based on a common idea that they attempt to de-tect significant unexpected changes in the system behav-ior by automatically constructing some behavior modelsinductively from a set of training data and comparingthem with newly arriving data.

The proposed method based on the AR model and therelevance vector regression learns a probabilistic enve-lope model for each time-series in the telemetry fromthe past data of itself and other relevant series, whichcan be obtained through the initial operation phase whenthe spacecraft is normal, and executes online monitor-ing of the target telemetry series by estimating the properrange of the series. In this sense, it can be called adap-tive limit-checking. It overcomes the limitations of tra-ditional limit-checking method by defining each limit se-quentially and relationally.

3. ADAPTIVE LIMIT-CHECKING USING REL-EVANCE VECTOR AUTOREGRESSIVEMODEL

In this section, we first review multidimensional autore-gressive model and relevance vector regression. Then, wepropose the proposed anomaly detection method,Adap-tive limit-checking.

3.1. Multi-dimensional Autoregressive Model

Multi-dimensional autoregressive model is a most ba-sic modeling technique for time-series data, in which astochastic process is represented as,

τ jARk = Θjxk (1)

where j = {1, · · · , s} represents thejth series of telemetry data, Θj =(Θj

0,Θj1,k−1, . . . , Θ

j1,k−p, . . . , Θ

js,k−1, . . . , Θ

js,k−p)

is a row vector of AR coefficients, xk =(1, τ1

k−1, . . . , τ1k−p, . . . , τ

sk−1, . . . , τ

sk−p) is the data

vector of allp previous time series, and the notation ARrepresents that the target valueτ is based on AR model.Figure 1 shows the concept of multidimensional ARmodel.

Figure 1. Image of multidimensional autoregressivemodel. It sequentially predicts proper envelop of targettelemetry data.

Imagine attitude motion of an artificial satellite. We canrepresent it as the following difference equation,

xt = f(xt−1,u) (2)

wherex andu represent the state vector and control in-put vector, respectively, andf represents the nonlinearfunction ofxt−1 andu. We intend to model this kind ofgoverning equation of the system by learning Eq.(1) fromtraining data set.

We made use of the framework of the relevance vectorlearning for the purpose of extending (1) to nonlinear andprobabilistic model.

3.2. Relevance Vector Regression

The relevance vector regression (RVR) originally pro-posed by Tipping(12) is a state of the art kernel-basednon-linear regression learning method(1; 3; 11; 13). Themost significant advantage of using RVR for the anomalydetection problem rather than other kernel-based learn-ing methods such assupport vector regression(SVR) isthat RVR estimates the probabilistic distribution of a tar-get variable while SVR computes only a point estimate.This is important because we are not interested in predict-ing the most likely values of a telemetry time-series, butin estimating the permissible range of the values. In thisstudy, we used the well-known Gaussian RBF kernel,

k(x, x′) = exp(−‖x− x′‖2

2σ2

). (3)

See ref.(2) for more detail of kernel method.

We write a sample ofN training pairs as{xn, tn}Nn=1 for

the jth telemetry series, correspondingxk andτ jk as de-

scribed above. Hereinafter, we deal with thejth series oftelemetry data and omit the notationj for simplification.

The RVR model which is based on the sparse Bayesianframework assumes that the targets are samples from adistribution model with additive independent zero-meanGaussian noise, with varianceσ2,

t = y + ε = Φw + ε (4)

where t = (t1, . . . , tN ) is the target vector,y is thevector of the predicted value,Φ = (φ1, . . . , φM ) isthe N × M design matrix,w= (w0, . . . , wM )T is theweights,φm(x) = K(x,xm) is the kernel function, andε = (ε1, . . . , εn)T is an error vector.

To achieve the sparsity,M independent hyperparameters,α = (α1, . . . , αM )T overw are introduced. We maxi-mize the following logarithm likelihood,

L(α) = log p(t|α, σ2) (5)

to get the posterior distribution. Through this procedure,the posterior distributions of many of the weights are in-finitely peaked around zero. The training data with theother non-zero weights are called therelevancevectors.

After the optimization of the hyperparameters,αMP andσMP , we obtain a conditional distribution model for anew datumx∗ as

p(t∗|t, αMP , σ2MP ) = N (t∗|y∗, σ2

∗) (6)

wherey∗ is the predict value of the new targett∗ andσ2∗

is the variance of the prediction. See (1; 11; 13) for moredetails of the sparse Bayesian learning.

By combining autoregressive model and relevance vectorregression, we can obtain the posterior density of targettime-series at timek as a nonlinear function of all time-series at previousp time steps.

3.3. Adaptive Limit-checking

The basic idea of the proposed method is to construct arelevance vector autoregressive model for the importantseries of the spacecraft telemetry data, regarding the dataduring the initial operation phase as the training data.This is based on an assumption that the data in the ini-tial phase is ”normal”, which is considered to be accept-able because a spacecraft in the initial phase is constantlymonitored and checked by a team of human experts anddesigners. After that, it receives the incoming teleme-try data, monitors the target series, compares the valuesto the probability density distribution predicted by thelearned model, and checks whether they are within a cer-tain predefined limit. This threshold is defined as follow-ing equation.

∫ −η

−∞N (t∗|y∗, σ2

∗)dt∗ +∫ +∞

+η

N (t∗|y∗, σ2∗)dt∗ ≤ ε,

(7)whereη is the threshold value.

We have still another important issue to solve, that is, thereduction of false alarm. If we directly apply the rede-fined AR model (Eq.(1)) and the RVR (Eq.(6)), the sys-tem is supposed to give many false alarms. The reasonsare,

1. The redefined AR model completely cannot bemodeled as the complicate spacecraft system eventhough it is learned by the relevance vector learning,

t∗(x∗) = tAR∗(x∗) = tTRUE∗(x∗) + εAR

= y∗(x∗) + ε∗(x∗) (8)

whereεAR is the modeling error of the redefined ARmodel.

2. The RVR based on the sparse Bayesian modeladopts the small set of the training data, in otherwords, therelevancevector, for the prototypes of themodel formed as (4)1.

As the result of 1 and 2, some training data can be seenas anomaly data though all training data are completelynormal because these data has relatively so largeεAR thatthe system regards them as the data which is far from theprototype data.

Therefore, we extended the varianceσ2∗ of the RVR for

the purpose of anomaly detection. Unfortunately, it isimpossible to estimateεAR because we cannot obtainthe true target value,tTRUE∗. Instead, we evaluatedthe variance of the difference between another data set{xi, ti}N2

i=1 and corresponding prediction values,y∗(xi).

σ2AR =

1N2

N2∑

i

(ti − y∗(xi))2 (9)

1This is a great advantage of the RVR for the execution speed ofanomaly detection.

andN2∑

i

ε(xi) −→ 0 (10)

If the AR model completely describes the system behav-ior, (9) must be zero in theory. However, it is supposednot to be zero because of the modeling error of the ARmodel. Therefore, we extended (6) as

p(t∗|t, αMP , σ2MP ) =

∫ ∞

−∞p(t∗|y, σ2

∗)p(y|y∗, σ2AR)dy

= N (y∗, σ2∗ + σ2

AR). (11)

In following section, we confirm that the proposed exten-sion to the AR model with RVR removes false alarms.

4. EXPERIMENT AND DISCUSSION

4.1. Simulation Case Study

We first performed an experiment with pseudo-telemetrydata obtained from a simulator of an orbital transfer vehi-cle which is designed to make a rendezvous and dockingwith the ISS. This telemetry consists of 27 time-seriesvariables in total, 13 of which are fromposition and atti-tude control subsystem, and the rest are frompropulsionsubsystem. In more detail, the former group consists of12 numerical observation time-series variables regardingthe position and attitude of the vehicle and one discrete-valued variable representing a command sequence. Thelatter group consists of 14 discrete-valued time-seriesvariables, each of which indicates the command input toeach of the 14 thruster engines installed in the vehicle.

In this experiment, we assume a scenario where the fourththruster engine which is used for the pitch control hassome trouble at time 250 [sec] and the power falls tozero. If the vehicle system failed to detect this engineanomaly immediately, it would miss a chance of recov-ery and result in a disastrous failre such as a crash withthe station. With this scenario, we performed an exper-iment comparing the proposed method and the conven-tional limit-checking. Ideally, the system should detectanomalies in the time-series representing the pitch angleand rate immediately after the thruster trouble occurs.

The anomaly detection system based on the proposedmethod operates as follows,

1. (Learning) Learn the relevance vector autoregres-sive model using a set of validated normal telemetrydata.

2. (Prediction) Compute the next probable range of thetarget series.

3. (Monitoring) Obtain and check the telemetry data.

4. (Alarming) Give an alarm if the data is out of thepredicted range.

Figure 2. Result of anomaly detection by normal RV ARmodel. The upper figure shows that the pitch angle andthe predicted envelop. The lower figure shows that theprobability density of the observation and probabilisticlimit.

5. Repeat steps 2-4

4.1.1. Comparison of Proposed Method with normalRV Autoregressive Model

First, we have compared the proposed method with thenormalrelevance vector autoregressive model.

Fig.2 , Fig.3 show the results of anomaly detection in theseries which represents the pitch angle. The solid line inthe upper figure shows the pitch angle and the dotted lineshows the predicted range, and the solid line in the lowerfigure shows the probability density of the observation.The system gives the alarm when the probability becamelower than the computed limit. We can see that the nor-mal model gives many false alarms. On the other hand,the proposed method correctly gives alarms after a littletime delay. The proposed method also succeeded in de-tecting anomalies in the series representing the pitch rateas shown in Fig.4.

4.1.2. Comparison with Conventional Limit-checking

We compared the proposed method with the conventionallimit-checking. We set the limit on the standard deviationfor the proposed method and on the maximum absolutevalue in the normal phase for the limit-checking. Fig.4shows the result. The conventional limit-checking fails todetect slight anomalies like this case. On the other hand,the proposed method is capable of detecting this anomalybecause it can dynamically estimate proper range of thetarget series.

Figure 3. Result of anomaly detection by the proposedmethod.

Figure 4. Comparison of the proposed method with con-ventional limit-checking. The solid line represents thepitch rate and the dotted lines represent the upper andlower bounds given by each method.

4.2. Real telemetry data

We next performed an experiment with real telemetrydata of Engineer Test Satellite Seven(ETS-VII) whichhave been operated from 1997 to 2002 and succeededorbital rendezvous and docking for the first time in theworld. Unlike simulation data, it is difficult to evaluatewhether it can detect anomalies. Instead, we evaluatedfalse alarm rate.

We used the following five data series which are all re-lated with pitch control (Table 1) and chose S4 (pitch rate)as the target series.

Table 1. Telemetry data

Label NameS1 Pitch angle incrementS2 Wheel rate (pitch)S3 Estimated pitch angleS4 Estimated pitch rateS5 Wheel angular momentum (pitch)

And the operation periods of experiment data is repre-sented as Table 2. These periods are the same operationmode because the model learned by (8) in some operationmode cannot be applied to the telemetry data of other op-eration modes since different operation modes have dif-ferent parameter of system.

Table 2. Operation periods of data

Label Date Time1 17/3/1998 21:04 - 21:322 21/3/1998 20:44 - 21:123 22/3/1998 20:32 - 21:004 24/3/1998 21:59 - 22:27

For estimating the optimal learning parameterσ2 inEq.(3), we appliedCross-Validationto the training datawith changingσ2. Figure 5 shows the result. The hori-zontal axis and the vertical axis representσ and the pre-diction error. From this result, we setσ = 1.

Figure 6 shows the result of predicting the pitch angle(training data: S1, test data: S4). We can confirm that theproposed method properly predict the movement of pitchrate. The average of false alarm rate was 5.3%.

5. CONCLUSION

This paper proposed a new anomaly detection methodbased on therelevance vector regressionandautoregres-sive model.

Figure 5. Result of Cross-Validation. The prediction er-ror is smallest atσ = 1.

Figure 6. Predicting limit for real telemetry data (pitchrate) of ETS-VII.

Compared with the conventional anomaly detectionmethod, this method has mainly two great advantages.First, the proposed method requires little a priori knowl-edge on the spacecraft system, as this method obtains itfrom the telemetry data. Therefore, this method can beapplied to various kinds of spacecrafts at small casts. Sec-ond, the proposed method can automatically consider therelationships among some series.

We performed an experiment with telemetry data ob-tained from a simulator of an orbital transfer vehiclewhich is designed to make a rendezvous and docking withthe ISS and the real-telemetry data of ETS-VIIand con-firmed the efficiency of the proposed method.

ACKNOWLEDGEMENTS

The authors would like to thank Japan Aerospace Ex-ploration Agency (JAXA) for providing the simulationtelemetry data and the telemetry data of ETS-VII.

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