Decentralised Fault Detection and Diagnosis in Navigation Systems for Unmanned Aerial Vehicles

S.M. Magrabi and P.W. Gibbens Department of Aeronautical Engineering

University of Sydney

Autonomous Unmanned Aerial Vehicles (UA Vs) are a recent technological phenomenon sweeping the world stage. Full autonomy implies that the guidance and navigation system employed must exhibit the highest level of integrity. This paper looks at the parity space Fault Detection and Diagnosis (FDD) methods its applicability in fully autonomous guidance and navigation systems in a decentralised system architecture. Using the existing work as a starting point this paper identifies the effectiveness of these methods when applied to situations where both the hardware and analytical redundancy exist. One of the most important issues in FDD in navigation systems using redundant sensors relates to the integrity of the solution processing architecture. Recently this has motivated the development of multiple FDD solutions running on numerous separate processors in a decentralised computing network. Typical solutions to this problem are based on decentralised or multiple Kalman Filters running in parallel. This paper will address the use and merits of the Information Filter form of the Kalman Filter in a fully decentralised FDD framework.

1. Introduction

Autonomous Unmanned Aerial Vehicles (UAVs) are a recent technological phenomenon sweeping the world stage. The UAV program at the University of Sydney is geared towards the development of a filly autonomous flying platform for potential use in various applications. UAVs have a wide number of uses ranging from agricultural, weather monitoring and other such civil applications to the more specialized military ones. They also provide excellent research platforms for the development of sensors and sensor systems such as the Global Positioning System (GPS), Inertial Measurements Units (IMU), Inertial Navigation System (INS), vision, radar, Terrain Aided Navigation (TAN) etc. Full autonomy implies that the guidance and navigation system employed must exhibit the highest levels of integrity whilst also being robust.

Even though the Global Positioning System (GPS) offers a potential for very high accuracy, in its application on-board a UAV it is plagued with inherent errors which manifest themselves as cycle slips, multipath errors, satellite masking, dynamic influences on the receiver, electromagnetic influences etc. Bom out of this need are integrated navigation systems, incorporating the relative advantages of their constituents apparent at various stages of operation.

These integrated navigation systems coupled together on a single platform like a UAV result in a multisensor system. Sensors allow a system to leam and continually update its model. Hence a multisensor system better reflects the complex model of the system providing a more complete description [l]. Moreover such a system provides a large amount of information, which when fused together in an appropriate architecture, can be used to effectively carry out the task of fault detection and diagnosis.

A fault is defined as an unexpected change that leads to the corruption of the overall performance of the system [2]. Navigation systems are susceptible, in their normal course of operation, to the occurrence of faults - which tend to degrade the navigation solution. As such, the integrity of the navigation solution is questionable. To ensure that the solution is feasible for use in guidance and navigation, the occurrence of the “fault” must be known to the system. Not only that, but the system must be able to isolate i.e. determine the exact location of, the fault and in some cases identify i.e. determine the magnitude of the fault. Having detected, isolated and identified the fault the system must finally be able to re-configure itself so as to overcome the deficiency induced by the fault. Thus the requirement is for Fault Detection Diagnosis (isolation/identification) and Re- configuration (FDDR). It must be noted that in literature this is largely known as Fault Detection Isolation and system Reconfiguration (FDIR), however we have opted to use diagnosis as this better explains the task of fault isolation and identification that are both often required.

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The remainder of the paper is organized as follows: Section 2 will review fault detection theory and briefly survey the methods used. Section 3 will discuss the Parity Space method and its use. A novel system architecture is sought to provide a framework for robust FDD in section 4 and the use and merits of the Information form of the Kalman Filter are addressed. Section 5 presents an example illustrating the implementation of the Information Filter, Kalman Filter and'the Parity Space Method. Section 6 offers conclusions and directions for future work.

2. Review of Fault Detection and Diagnosis Theory

Fault detection and diagnosis is based primarily on the use of redundancy, both hardware or analytical. Analytical redundancy involves the use of a mathematical model of the system and the relationships, between sensor outputs. This is distinct from hardware redundancy whereby two identical sensors measure the same data, and the detection of faults is simply a matter of identifying when a discrepancy between the two occurs.

Analytical redundancy or model based fault diagnosis methods are increasingly being employed in navigation systems because of the availability of light, high speed computers on flight platforms which can do away with the excess weight and space taken up by redundant sensors. These methods can be categorized into parity space methods, filter based methods, neural networks, fuzzy logic, frequency based methods, expert systems applications, parameter based methods, and the knowledge based approach. For hrther details the reader is referred comprehensive surveys of [2][3][4][5] and the books [6][7][8].

Analytical redundancy can be differentiated into direct and temporal forms. Direct redundancy is when there is an algebraic relationship between sensor outputs, such that a state one sensor measures can be found using the instantaneous output of another or a set of other sensors whereas temporal redundancy is when there is a time related correlation in the histories of sensor outputs and actuator inputs. Fault detection algorithms employing the latter can hence be used for the detection of both sensor and actuator faults [9].

Fault Detection and Isolation consists of two processes: Residual generation and decision making, as shown in figure 1.

Generati on Making

VI

Figure I :Fault Detection Methodology

Residual Generation Residuals are defined as the resulting differences between two algebraically equivalent or analytically redundant quantities in the system model. These are similar to innovations generated by a Kalman Filter, which are the differences between the measured and the estimated outputs. Residuals and innovations, under normal conditions, are small or zero mean. The occurrence of a fault causes the residuals or the innovations to go to a non-zero or large values. Application of the residual generator involves the selection of either a direct implementation through the use of the generic residual form or the parity function or an indirect implementation through the use of a diagnostic observer.

Decision Making This involves assessing the residuals and identifying when and where any abnormalities occur. This is done through threshold testing both static and dynamic, and various statistical tests, where the thresholds are typically based on signal variance. Tests such as the chi-square test, Sequential Probability Ratio Test (SPRT) and modified versions of these are common in literature [lo], [ l l ] , 1121.

3. The Parity Space Method

Parity space methods involve the representation of actuator inputs and sensor outputs, subjected to a linear dynamic transformation. This transformation is not unique and hence can be selected so as to enhance fault isolation and disturbance decoupling properties of the residual. As such it presents a robust framework within which to perform FDD. Early contributions include [ I 13 and [13], that deal with aircraft sensor fault detection and skewed sensor redundancy management respectively. Parity relation concepts were formulated and generalized in [9] and [ 141.

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Given a measurement model, the measurement space can be transformed by virtue of redundancy in the system to a state space and a parity space [ 151.

z(k-s) z(k - S + 1)

Z = z(k-s+2)

- z(k) -

Measurement 1 =, I State Space 1 Space

Parity Space

Figure 2: Relationship between the measurement, state and parity space.

E =

Hence the parity space is spanned by a set of parity vectors.

Consider the discrete state space model of the monitored system

x(k+l) = Ax(k) + Bu(k) ( 1 )

z(k) = Cx(k) + Du(k) (2)

where x is the m-dimensional state vector, U is the n-dimensional input vector and y is the p- dimensional output vector. A, B, C, and D are discrete system matrices of the appropriate dimensions. Combining equations (1) and (2) from time k-s to k, where s specifies the window length, yields the following redundancy relations

C CA

CA2

CA”

CB D K = CAB I D

U =

(3)

u(k-s) u(k-s+l ) ~ ( k - S + 2)

u(k)

Z is of dimension (s+l).p, while E, K, and U are [ (s+ 1 )n] .m, and (s+ 1 ).n respectively. The parity vector (residual) P(k) is given as follows:

[( s+ 1 )p]. [( s+ 1 )n]

P(k) = R(Z - KU) (4)

Note, that P(k) is independent of the state vector x. R is chosen such that its rows span the orthogonal compliment of space of E. R is not unique and can be obtained by subtracting the orthogonal projection onto E from the identity operator i.e. R = I - E ( E ’ E ) - l E ’ ( 5 ) also R E = O (6)

Using this definition it is clear that P(k)=O when the system is functioning correctly but in the presence of the fault P(k) may become non-zero.

With the parity space method, residuals can be generated through the use of the parity function in an open or the closed loop.

The parity function itself can be used as it represents a Moving Average (MA) process i.e. it is a function of a sliding window of the most recent outputs. If the sensor outputs are corrupted by white noise then the parity function values will be correlated over the length of the window. The advantage of using this method in conjunction with a filter is that it does not depend on the filter estimates.

Open loop residuals are generated through a Auto Regressive Moving Average (ARMA) model. In this case, if outputs are corrupted with white noise then the residuals will be correlated with all preceding and future values. Whereas closed loop residuals use an ARMA model but specifically take noise into account. These residuals are generated by a filter such as the Kalman Filter. This produces an uncorrelated sequence of residuals that are not corrupted by processes such as a random walk in the predictions.

4. A Decentralised Fault Detection and Diagnosis Framework and the

Information Filter

The advantages of a decentralised architecture in FDD are presented by Kerr [ 161, where he uses a decentralised Filtering algorithm employing the standard Kalman Filter formulation with a central fusion center to generate a navigation solution. We aim to investigate decentralisation of FDD in a navigation system by employing an information filter. This paper will address the

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key theoretical issues involved with this implementation within a multisensor system.

- INS MS Filter

GPS -b GPS/MS

JTIDS -b JTIDSA Prefilter .+ NS filter

Prefilter -b Filter

-

A decentralised data fusion system as defined in [17] is a network of sensor nodes, each with its own processing facility so that there is no central fusion or central communication facility. Fusion occurs locally at each node through local observations and information communicated from neighbouring nodes.

- -b

-+

-b

Decentralisation in FDD has been shown by Kerr [I61 to expedite significant reductions in computation memory usage and throughput. Kerr's implementation has a structure as depicted in figure 3 and is essentially Speyer's filter [ 181 with the addition of the Voterhionitor block.

The advantages this method offers as documented by Kerr [ 161 are: . Resolving differing timeperiods between

measurement updates for various participating sub-systems. . It offers the best estimate even in the event of a sub-system failure. Cross-correlations are dealt with routinely by the algorithm. . The transmission of minimal information between the various subsystems rather than the entire raw measurements.

, h. ~ "i.1 Sensor Integrat- Prefi I ter inn Filter i

Figure 3: Simplijied representation of Decentralised implementation [I41

Our implementation aims to take the decentralisation one step further in implementing the information filter. The sensor subsystems exist independently of each other and each computes a current best estimate of the state based on its local knowledge as well as the information appropriated from the other sensors in the form of the information state vector and

the associated information matrix. There is no centralised unification filter or for that matter integration filters. Each filter generates its own solution, hence the system is far more robust to the failure of sensor subsystems. Such a system employing an Inertial Navigation System (INS), Global Positioning System (GPS) and an Air Data System (ADS) is depicted in figure 4.

communications bus

f i processor processor processor

Figure 4: A fully decentralized architecture

Note, that the navigation solution and FDD in the implementation of figure 4, where each sensor-processor pair broadcasts onto a communications bus, are exactly equivalent to those generated by an architecture with a mesh of node to node communication channels i.e. a fully decentralised system with local estimators and local fusion centers. The state estimates are also exactly the same as those given by a centralized Kalman Filter.

The Information Filter The information filter is simply an algebraic equivalent of the Kalman Filter rewritten in terms of the information state vector, y, and the information matrix, Y. Where Y is the inverse covariance matrix and y is the state pre- multiplied by Y.

Given a system model:

and an observation model:

where: x(k) is the state vector A(k) is the state transition matrix w(k) is the process noise which is gaussian and white with an error covariance E[w(i)wT(j)]=6,,Q z(k) is the observation vector C(k) is the observation matrix or model 9(k) is the process noise which is gaussian and white with an error covariance E[S(i)L?T(i)] = 6, R

~ ( k ) = A(k)x(k - 1) + B(k)u(k) + 4 k ) (7)

z ( k ) = C(k)x(k)+ q k ) (8)

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The derivation of the Kalman and the Information filter is detailed in [19]. To understand the information filter formulation consider the equations as compared with the Kalman Filter equations in Table 1.

An obvious advantage in the use of an information filter formulation is the simplicity of the estimate equations. Moreover as pointed out in [17] estimation in a decentralised system occurs at each sensor node, requiring partitioning and summing of the equations, which is far more simpler in the information form. Even though the predictions are more complex in this form because of the reliance on the propagation coefficient, L(k/k-l), there is no dependence on the observations made, thereby making the system simpler to decouple and hence decentralise. Since this is independent of the observations made, it is easier to decouple and decentralise amongst a network of sensors.

The advantages of using an information filter formulation are [ 171:

It would facilitate the “decentralisation” such that there is no one central processor the computes the navigation solution but rather the solution is a fusion of the best estimate obtained and maintained at each sensing node. Such a system has the inherent characteristics of being scalable, flexible, but more importantly from the fault detection point of view, robust. The Information Filter is better suited for application in multisensor estimation environment because the estimate update equations in the Kalman Filter formulation are more difficult to apply when there are multiple observations. This is because the innovation vectors generated by different sensors at the same time are correlated since they use a common prediction, resulting in the innovations covariance matrix difficult to partition and invert.

Filter Innovations and FDD Filter innovations form the basis for sensor validation and hence FDD. The innovations can be defined as the difference between the estimated and the measured output. For a Kalman Filter, this is defined as:

(9) We can now define the normalised innovations: v( k ) = z( k ) - C( k ) i ( k / k - 1)

y ( k ) = v T ( k ) s - ’ ( k ) v ( k ) (10)

The normalised innovations define an ellipsoidal region of constant probability in the measurement space where each sensor measurement is most likely to be found with a specific high probability.

The information innovation vector can be defined as [ 191: Y ( k ) = CT(k)R- ’ (k )v (k ) (1 1) Transforming this to the information form gives: ~ ( k ) = I ( ~ ) [ I + (k)i(k) - Y’ ( k / k - 1)fik / k - 1) ( 12) where

I + ( k ) = C* (k)[C( k)I(k)C‘( k)]-’ C(k ) (1 3) is the projection of I(k) into the measurement space, inverted and projected back in the state space.

The information innovation variance is expressed as:

B ( k ) = I ( k ) [ I + ( k ) + Y-’ ( k / k - l)]l(k) (1 4)

Similar to (1 1) the psuedo-inverse of this is: B+ ( k ) = C T ( k)[C(k)B(k)CT (k)]-’ C( k ) (1 5)

Thus the normalised information innovation can be then defined as:

r ( k ) = v T ( k ) ~ + ( k ) v ( k ) (16)

By tranforming this into the state space form, it can be proved that (16) is exactly equivalent to (IO) [19]. Hence, the normalised Kalman Filter innovation and the normalised information filter innovation share the same properties, meaning that both will have a x2 distribution with n, degrees of freedom, where n, is the dimension of the z vector. Therefore the Information Filter normalised innovation can be used for the same purposes as the Kalman Filter normalised innovation for FDD and sensor validation in a multi-sensor architecture.

5. Example

To demonstrate the use of an information filter in fault detection, we use a simplified system representing only the forward motion of the vehicle. The one dimensional rectilinear motion is modelled such that the input u ( k ) is the accelerationa(k) as measured by an IMU. The discrete linear time invariant model is as presented in equation (7). The state vector is

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The Kalman Filter The Information Filter(Joseph Form) [I 91

Prediction

Estimate Po< / k) = P(k / k - I ) - W(k)S(k)WT (k) ?(k / k) =[I - W(k)C@)]?(k / k - 1) + W(k)z(k)

where i (i/j) estimate of the state x at time i given information upto and including j P(i/j) the estimate error covariance matrix at time j given information upto and including time j

The Gain Matrix is W(k)=P(k/k-l)CT (k)S' (k)

The Innovation covariance S(k)=C(k)P(k/k-l)C (k)+R(k)

Y(k / k) =Y(k / k -1) +I(k) ?(k / k) = p(k / k - I ) + i(k)

where Y(k / k) = P-'(k / k) i ( k / k) = Y(k / k)?(k / k) 9 (i/j) estimated information state vector at time i given information upto and including j Y(i/j) the information matrix or the inverse covariance matrix

The information state contribution from an observation i(k) = C (k)R-' (k)z(k)

The information matrix contribution from an observation I(k\=C (k\R-' Ik\C(k\

given by x ( k ) = [ p ( k ) o ( k ) l T , where p is the position of the vehicle and o is its velocity.

The state transition matrix A is time invariant and G is the input matrix given by:

Observations are made of the position and velocity by a linear observation model of equation (S), where the observation model is defined by:

c=[: :] Using the system defined above, a Kalman Filter and an information filter were formulated to estimate the position and the velocity. The observation model (8) was implemented in a multi-sensor form for both filters, such that the innovations of two filters measuring the same observation were subjected to different noise levels. The information filter has been formulated in a fully decentralised manner. For this example two position sensors and two velocity sensors are used to emulate observations

- from a GPS type measurement device. A constant bias fault was introduced midway through the simulation on the first of the two position observations and the normalised innovations were generated for both the Kalman and the Information Filter.

As the estimates and innovations of the two filters are equivalent, the results are identical. Both detect the bias introduced in the position observation. Figure 5 shows the normalised innovations from the Information Filter. The advantage of the Information Filter over the Kalman Filter is its applicability and ease of implementation in a multisensor environment.

The Parity Space Method presented in section 3 has been implemented concurrently within each filter in order to identify the introduced fault. The pertinent relationships are given by:

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K =

Implementing the Parity Space Method we generate the parity vector which is a zero mean distribution. The occurrence of the fault results in parity fault indications also shown in figure 5, and correlate with the fault indications from the filter innovations. Note that the fault indication only persists for the duration determined by the window length.

The benefit of analytical FDD is elucidated here in that isolation of the faulty position observation would not be possible without the aid of the acceleration and velocity observations.

Equivalent fault indications are induced by the introduction of a bias fault on the velocity sensor, identifying the fault correctly.

Figure 5 : Residuals for a constant bias fault for position sensor 1. Top Left: Normalised Information Filter

Top Right: Normalised Information Filter Residuals for unfailed sensor 2

Second Row: Parity Vectors - Failed sensor 1 Third Row: Parity Vectors - Unfailed sensor 2

Residuals for failed sensor 1

As a second demonstration consider a situation where a bias fault occurs on the acceleration input to the system. The results of the FDD process are shown in figure 6. In this case, the

indication of a fault by both innovation and parity space methodologies in all sensors indicates that the fault is not on the position and velocity observations, but on the acceleration input measurement.

Again, this indicates the benefit of analytical redundancy.

Figure 6: Residuals for a constant bias fault for acceleration input. Top Row: Parity Vectors for acceleration fault Second Row: Normalised Information Filter

Third Row: Normalised Information Filter Residuals for position sensors 1 & 2

Residuals for velocity sensors 1 & 2

6. Conclusions

The relative merits in implementing the information filter innovations and parity space FDD methodologies in a decentralised autonomous navigation system have been discussed.

The information filter demonstrates features which make it more attractive as a data fusion methodology in a decentralised framework than a Kalman Filter. These stem from the way in which observation information from distributed sensors can be simply added at each local node to produce state estimates. In addition, only incremental changes in observations need be

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communicated leading to lower communication loads. Given this, the fault detection may be achieved by checking either the information filter innovations and/or the parity vectors arising from a parity space methodology.

Information filter innovations method relies on correct identifications. Observations are only assimilated into the estimation process if they are judged to be non-faulty. The main disadvantage of the innovations based method is that missed detections result in corruption of the state estimates, and can hence ensuing trigger false alarms or hr ther missed detections.

Parity space fault identifications are based on direct temporal relationships between input measurements and output observations. In the linear case they therefore have the advantage of being independent of the state prediction. For the full nonlinear navigation equations relating to autonomous vehicles such as UAV’s, this characteristic is partially lost due to the state dependence of the observation matrix. Nevertheless the robust identification characteristics of the parity space methodology will be retained. Combination of the information filter with parity space FDD therefore presents a desirable state estimation and sensor integrity monitoring approach in decentralised system architectures.

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