A solution of UAV localization problem using an interacting multiple nonlinear fuzzy adaptive H∞models f ilter algorithm

Elzoghy MOSTAFA , Li FU ,*, Arf IBRAHIM.I., Arif USMAN

a School of Automation Science and Electrical Engineering, Beihang University, Beijing 100191, China

b School of Control and Automation, MTC, Al-Khalifa Al-Maamoon Street Kobry Elkobbah, Cairo 11331, Egypt

KEYWORDS Interacting multiple models;Integrated navigation system;Multi-mode estimation;Nonlinear fuzzy adaptive f ilter;Sensor data fusion;UAV localization

Abstract The purpose of this research is to improve the robustness of the autonomous system in order to improve the position and velocity estimation of an Unmanned Aerial Vehicle (UAV).Therefore, new integrated SINS/GPS navigation scheme based on Interacting Multiple Nonlinear Fuzzy Adaptive H∞Models (IMM-NFA H∞) f iltering technique for UAV is presented. The proposed IMM-NFA H∞strategy switches between two different Nonlinear Fuzzy Adaptive H∞(NFA H∞) f ilters and each NFA H∞f ilter is based on different fuzzy logic inference systems. The newly proposed technique takes into consideration the high order Taylor series terms and adapts the nonlinear H∞f ilter based on different fuzzy inference systems via adaptive f ilter bounds (δi),along with disturbance attenuation parameter γ. Simulation analysis validates the performance of the proposed algorithm,and the comparison with nonlinear H∞(N H∞)f ilter and that with different NFA H∞f ilters demonstrate the effectiveness of UAV localization utilizing IMM-NFA H∞f ilter.

1. Introduction

Recent years have seen an expansion in the deployment of aircraft without a pilot on board which is known as a drone, an Unmanned Aerial Vehicle (UAV) or an Unmanned Aircraft System(UAS).Initially,UAVs of all shapes and sizes were utilized for military missions or dangerous missions for people as well. But recently their utilization quickly extends to civilian missions such as scientif ic,agricultural,commercial,and other applications. The achievement of increasing demands for special autonomous vehicle missions attracts the researchers’interest in realizing the autonomy of the whole vehicle. Due to the wide spreading of UAVs, many requirements should be met to satisfy demands nowadays. A standout amongst the most restrictive requirements is the navigation and control of the autonomous vehicle, which gives the required and expected knowledge to execute a task or move in their environment. The control and navigation systems hugely affect an autonomous vehicle’s execution performance and robustness.At present, autonomous vehicles can just play out the most fundamental operations. It is from these issues and the potential advantages of these vehicles that this research is motivated.

As the demand for position accuracy and the interest of reliability and real-time processing execution expand quickly,stand-alone navigation system cannot meet the navigation necessities step by step.1,2The autonomous vehicles depend on an IMU to detect the vehicle’s angular rate and increasing speed (acceleration), which is utilized to obtain vehicle states,for example, position, speed, and attitude.3,4However, its diverging error nature because of the integration procedure requires absolute sensors as GPS keeping in mind the end goal to constrain the drift.5The innovation of technology of Multi-Sensor Data Fusion(MSDF)is quickly developing.The multinavigation-sensor data fusion system turns into a fundamental improvement of guidance and navigation innovation, specifically; the inertial navigation system based integrated framework is given a lot of consideration in the most recent years.6

There is much simultaneous progressing research to introduce developed new algorithms and calculations of integrated navigation, to enhance the existing algorithms, and to gather and collect these techniques into general engineering design skills to increase the autonomy of vehicles. The autonomous navigation route framework is based on integrated navigation INS/GPS which is progressively utilized, and it depends on navigation f iltering to offer a robust and practical solution for UAV navigation. In this regard, this research has been made to enhance the competence and robustness of algorithm with a specific end goal to give an accurate localization.

Usually the data fusion of SINS and GPS is achieved using different types of Kalman Filter (KF).7But, the performance of KF estimation can be preserved only when the characteristics of process and measurement noises are known.8Consequently, the accuracy of SINS/GPS navigation framework will decrease in case that priori information utilized as a part of KF is not consistent with the real conditions.9Because of that,the interest in f ilters which provide high execution performance without the demand of presumptions regarding the characteristics of measurement and process uncertainties increases day after day.

In numerous application cases, the KF can be helpless for some time because the system dynamics possibly change and the statistical process and measurement uncertainty matrices are hard to implement, and therefore the adaptive techniques have gained much attention in the most recent years.5As an alternative solution to overcome this issue, the algorithm of H∞f ilter turned into another consideration where H∞r(nóng)obust f ilter minimizes worst-case estimation error.10

Later, the Nonlinear H∞(N H∞) f ilter has received much attention. In spite of the fact that N H∞f ilter introduces great execution without need presumptions in connection with the attributes of the extended noise signals(although this information can be used if it is available), it requires extra adjusting factors which need to be determined through experimentation.11N H∞f ilter has been used to f ind a solution to the localization issue for the integrated GPS/SINS navigation system of UAV.12

In many practical applications,the f iltering tuning parameters are still selected by the designer experience,the selection of these parameters is important for the applications. Recently,the fuzzy logic has turned into a powerful technique for improving and optimizing the execution performance of numerous practical applications, particularly in the field of control and estimation.13-20Accordingly, the interest in the fuzzy logic adaptive f ilter has increased. Nourmohammadi and Keighobadi used the fuzzy logic to adapt the low-cost SINS/GPS integration system to solve the orientation estimation problem.17Tseng et al. combined the fuzzy logic with cubature Kalman f ilter to adapt the integrated navigation scheme to adjust the weighting factor of the process noise covariance matrix.19Reshma and Raol used fuzzy logic based adaptive H∞f ilter in target tracking and their results demonstrated that the fuzzy logic based adaptive H∞f ilter performed better than the non-fuzzy logic based H∞f ilter.21Not only the target track has received attention for using fuzzy logic based adaptive f ilter, but also the extensive focus has been given to integrated navigation systems. Liu Jiang et al. proposed the fuzzy logic adaptive H∞technique of the GPS/SINS scheme and they designed the Fuzzy Inference System (FIS) by depending on the performance factor of the f ilter and attenuation level to choose the adaptive disturbance attenuation factor γ suitably and adaptively.8Xu and Cui proposed the Fuzzy Adaptive Interacting Multiple Model (FAIMM) algorithm of the integrated navigation system by combining Fuzzy Adaptive KF (FAKF) algorithm with Adaptive Interacting Multiple Model(AIMM)algorithm,and the simulation results demonstrated that the algorithm of FAIMM achieved a better statistical estimation of noise.9Other authors have proposed the fuzzy adaptive interacting multiple models unscented Kalman f ilter (FUZZY-IMMUKF) for GPS/INS integrated navigation system to enhance the estimation accuracy and tracking capability.22Fariz et al. proposed the fuzzy adaptive nonlinear H∞f ilter to enhance the UAV localization and their model based on two FISs for obtaining suitable choice for the adaptive boundary parameters and adaptive disturbance attenuation parameter to enhance the estimation accuracy of position and velocity for UAV.11In this research, we develop the methodology which has been reported by Fariz et al.11We use the technique of the Interacting Multiple Model (IMM)algorithm to switch between two different NFA H∞f ilters,and each f ilter model is based on two FISs with different fuzzy rules.

In this paper, a new integrated SINS/GPS navigation algorithm called Interacting Multiple Nonlinear Fuzzy Adaptive H∞Models (IMM-NFA H∞) f ilter is proposed for localization issue of UAV. Depending on multiple different FISs strategies, IMM-NFA H∞f ilter persistently modif ies high order terms of Taylor expansion by adapting the disturbance attenuation γ（ ） and f iltering bounds (δi). Simulation outcomes are acquired utilizing IMM-NFA H∞, and compared with navigation f ilter results of N H∞and with two different Nonlinear Fuzzy Adaptive H∞(NFA H∞) models as well. What’s more, these results are approved utilizing a 3D trajectory flight of UAV. The comparison demonstrates the effectiveness of the localization process of UAV utilizing IMMNFA H∞f ilter.

This paper has been organized as follows: Section 2 describes the framework of integrated navigation system for UAV flight; Section 3 presents the algorithm of the nonlinear H∞f ilter;Section 4 provides the design of the first and second nonlinear fuzzy adaptive H∞f ilters; Section 5 presents the design of IMM-NFA H∞f ilter;Section 6 illustrates the simulation framework, where result analyses are given to present the impact of the proposed approach; Section 7 draws the conclusion.

2. SINS/GPS mathematical model

Our research discusses the SINS/GPS integrated navigation system for UAV based on IMM-NFA H∞taking the higher order of the Taylor development into consideration. The integration between SINS and GPS is employed based on four techniques which are N H∞, NFA H∞-1, NFA H∞-2 and IMM-NFA H∞f ilter. Fig. 1 presents the architecture of the integrated navigation system.The INS represents the main system while the GPS represents the measurement system.

SINS nonlinear model can be expressed as

where process noise w t（ ） and measurement noise v t（ ） are uncorrelated, white and zero-mean, with covariance matrices Q and R,respectively;x is state vector,and consists of position and velocity expressed in navigation frame, attitudes in terms of Euler angles, and gyro and accelerometer biases.

The nonlinear navigation model can be written as11

Fig. 1 INS/GPS system architecture.

3. Nonlinear H∞f ilter

The considered type of disturbances specif ies the form of robust f ilter among various forms,24while the normal characteristic of N H∞f ilter is to guarantee fixed energy disturbance gain to ensure the minimum estimation error.25The nonlinear optimal H∞estimator has been studied and reported by different researchers.25-28The nonlinear discrete form of system and measurement model in Eq. (1) can be expressed by using Taylor series expansion as25,26,29

A GPS receiver and antenna are normally used to give the UAV north-east-down position and speed vector in global coordinates.23The measurement model is

where

?fk（x ）, ?fwk（x ） and ?hk（x ） are the Jacobian at xk-1for f,(f/wk） and h, respectively. Δirepresent high order terms of Taylor series norm, and are bounded as ‖Δi‖≤（i =1,2,3）. The predictor state error is=xk-and the f ilter state error is=xk-.

Compared to EKF which assumes that the high order Taylor terms are zero,N H∞f ilter estimates the nonlinear model in Eq. (1) whilst fulfilling the performance characteristic of H∞for all uncertainties as for all uncertainties Δiand their norm bounds.Then the structure of the system in Eq.(5)is extended as25,31,32

The(δ1,δ2,δ3)are the boundary parameters and γ is the disturbance attenuation parameter.The final N H∞structure formulation is similar as EKF except of covariance Pk/kin,which is given by

Fig. 2 Structure of integrated navigation system using NFA H∞-1.

Fig. 3 Structure of fuzzy inference system.

Fuzzy model is a strategy for portraying the attributes of a system utilizing fuzzy rules. The standard fuzzy model comprises 3 parts: the first part is fuzzif ication, the second part is fuzzy inference (fuzzy reasoning), and the third part is fuzzy defuzzif ication.

where Ck=I.

If δ1=δ2=δ3=0, the N H∞r(nóng)eturns back to extended H∞(E H∞)scheme,the same to EKF in linearization process,however minimizing worst-case estimation error every time instant like linear H∞; furthermore, as γ tends to ∞, E H∞r(nóng)eturns to EKF.12,25Accordingly,the thresholds of the boundary parameters and the disturbance attenuation parameter have been characterized in the simulation by the use of expert knowledge.

4. Design of nonlinear fuzzy adaptive H∞f ilters

4.1. Design of the first nonlinear fuzzy adaptive H∞f ilter(NFA H∞-1)

The structure of the first Nonlinear Fuzzy Adaptive H∞f ilter(NFA H∞-1) used in this paper is the same as that of the f ilter developed by Fariz et al.11The formulations characterized in Eqs.(9)and(10)have parameters δiand γ that can be adapted.The NFA H∞-1 consists of 2 fuzzy controllers called FIS-1 and FIS-2 working independently as shown in Fig. 2, while each FIS has been illustrated in Fig.3(a) and (b).

The changing from an input crisp value to fuzzy is the responsibility of the fuzzif ication procedure, while the responsibility of fuzzy inference is to convert the fuzzy input to the fuzzy output by utilizing IF-THEN type fuzzy rules, and finally the conversion of fuzzy actions of the inference engine into a crisp action using membership functions (MFs) is the responsibility of the defuzzif ication procedure.8,9,21,22

The input linguistic variables of the first fuzzy inference system are ΔPe, ΔVel, and ΔAttwhich represent the errors of position,velocity, and attitude, respectively. And output variables are the adaptive boundary(δ1,δ2,δ3).However,the second fuzzy inference system inputs are gyro drift and accelerometer bias while the output is the adaptive disturbance attenuation parameter (γ). Gyro drift and accelerometer bias are estimated simultaneously. There are two types of fuzzy inference system and these types are a Takagi-Sugeno (T-S) fuzzy system and Mamdani fuzzy system.In this paper,the first and second fuzzy inference systems depend on the methodology of Takagi-Sugeno fuzzy system.

The subset of all input linguistic variables of the first and second fuzzy inference systems are {P, G}, and that of output variables of the first and second fuzzy inference systems are{P, G, M}. The control rules of FIS-1 are summarized in Table 1, while the control rules of FIS-2 are summarized in Table 2.

All the control rules in Tables 1 and 2 are based on an AND rule.There are many researches on comparing the effects of various kinds of membership functions (MFs)33-37; nevertheless, it appears that the conclusion is highly application dependent, and it is hard to make a decision which type of MF is constantly better.38

In this paper, we have designed MFs based on the trapezoidal shape because it is simpler in representation and analysis38,39; what’s more, it works well in most practical applications.40To the best of the authors’ knowledge and based on the test results, the designed trapezoidal MF shape is the more appropriate type of our particular problem.

The MFs of input linguistic variables ΔPe,ΔVeland ΔAttof the first fuzzy inference system are trapezoidal-shaped MFs as shown in Figs.4(a)-(c).

The MFs of input linguistic variables gyro drift and accelerometer bias of the second fuzzy inference system are trapezoidal-shaped MFs as shown in Fig.5(a) and (b).

4.2. Design of the second nonlinear fuzzy adaptive H∞f ilter(NFAH∞-2)

The NFA H∞-2 comprises 2 fuzzy controllers called FIS-3 and FIS-4 working independently as shown in Fig. 6.

In this section, we have developed the second Nonlinear Fuzzy Adaptive H∞f ilter (NFA H∞-2) using the performance factor (Pf) and attenuation level (γ-) as inputs for the fourth fuzzy inference system (FIS-4) instead of the gyro drift and accelerometer bias as in the previous section to choose the disturbance attenuation parameter suitably and adaptively. We have implemented the third fuzzy inference system (FIS-3)which is the same as the first fuzzy inference system (FIS-1)in the previous section. FIS-4 has been illustrated in Fig. 7.

Table 1 Rules of the first fuzzy inference system (FIS-1).

Table 2 Rules of the second fuzzy inference system (FIS-2).

Fig. 4 MFs of input linguistic variables.

Fig. 5 MFs of input linguistic variable gyro drift and variable accelerometer bias.

Fig. 6 Structure of integrated navigation system using NFA H∞-2.

Filtering state can be detected by a residual error in workable applications8. The equation of the residual error is

Fig. 7 Structure of the fourth fuzzy inference system (FIS-4).

The convergence criterion of f iltering is8

where ρ is a repertory factor whose value is signif icant for f iltering convergence, ρ≥1. The covariance calculation of residual error can be

Then,the convergence criterion of f iltering can be rewritten as

So the performance factor (Pf) may be defined depending on the def inition of ρ as8

The Pfexemplif ies the deviation pattern among the actual estimation error and the theoretical error. The relation between attenuation level (ˉγ) and the adaptive disturbance attenuation parameter (γ) can be characterized as8

where

The membership functions of input linguistic variables Pfandof the fourth fuzzy inference system are trapezoidalshaped MFs as shown in Figs.8(a) and (b). Furthermore,the MFs of output variable adaptive disturbance attenuation of the FIS-4 are trapezoidal-shaped MFs as illustrated in Fig. 9.

Table 3 Rules of the fourth fuzzy inference system (FIS-4).

Fig. 8 MFs of input linguistic variable performance factor and variable attenuation level.

Fig. 9 MFs of output adaptive disturbance attenuation parameter.

Fig. 10 Output curved surface of FIS-4.

The final 3D curved surface that represents the mapping from input to output of the FIS-4 is depicted in Fig. 10.

5. Design of IMM-NFA H∞f ilter

A new idea regarding the SINS/GPS integrated navigation system based IMM-NFA H∞f ilter for UAV localization issue is examined and researched. The adaptive technique based augmented different FISs are proposed and consequently adjust the parameters (δ and γ) of the N H∞f ilter. High order terms of Taylor advancement are continually tuned by the IMMNFA H∞f ilter throughout the adaptive boundary parameters(δ) and additionally disturbance attenuationγ. The boundedness error and powerful estimation can be achieved via the online adjustment by using the FIS although linearization errors exist and disturbances are unknown.In this manner,the proposed method comprises adapting and modifying δ1,δ2,δ3and γ of N H∞f ilter utilizing 2 nonlinear fuzzy adaptive H∞f ilters which are combined with interacting multiple model technique. Each f ilter utilizes two FISs.

The first f ilter contains 2 fuzzy controllers called FIS-1 and FIS-2 which work independently as illustrated in Fig. 2, while the second f ilter contains 2 fuzzy controllers called FIS-3 and FIS-4 which operate separately as illustrated in Fig. 6.

The IMM utilize finite possible models for the vehicle motion and switch between these models based on probabilistic switching and are designed with parallel f ilters,while all the f ilters work at each sampling time and exchange some information among each other due to the switching among the different models.41

The algorithm of the IMM-NFA H∞consists of parallel NFA H∞f ilters.These f ilters work independently at each sampling time and they are responsible for weights of model probabilities. Each f ilter estimates the state at each sampling time,while the resultant state estimation is the combination of all the state estimation from every f ilter.

One iteration cycle of the proposed IMM-NFA H∞algorithm is depicted in Fig. 11. The major steps of the proposed IMM-NFA H∞algorithm will be discussed in the following subsections.

5.1. Model interaction and mixing step

In this step,the initial mixed state estimate X0jk｜k（ ）and initial mixed covariance matrix P0jk｜k（ ） are the initial mixed condition of f ilter which corresponds to the mode Mjk+1（ ） and they can be computed as

Fig. 11 Flowchart of IMM-NFA H∞algorithm (one cycle with two NFA H∞sub-f ilters).

where the mixing probabilities μi｜jk｜k（ ） are calculated as

Hence, the predicted mode probability μjk+1｜k（ ） is given as

The switching among different modes is done based on the mode transition probabilities:

5.2. Model individual f iltering

The following step executes two individual different NFA H∞f iltering processes which are done in parallel and combined to give an overall estimate state and covariance.

Also, during this step, the innovation sequence and the innovation covariance can be computed by

Here, we can calculate the associated likelihood function for matched model Mjto update the probabilities of different models where the likelihood function is a Gaussian distribution function of residual υjwith covariance Sjand can be calculated as42

And n is the dimension of residual vector υj.

5.3. Mode probability update

After the measurement update step for each model and calculating the likelihood for each model,43we update mode probability μj（k+1｜k+1） by utilizing the predicted mode probability μjk+1｜k（ ） and mode likelihood function Λjfor Mjk+1（ ） as follows:

where c is normalization parameter which is given by

5.4. State estimates and process covariance combination

In this step, resultant state estimate（k+1｜k+1） and final estimated process covariance matrix（k+1｜k+1） are done via combining the state estimate（k+1｜k+1） and the process covariance matrix（k+1｜k+1）from every f ilter by utilizing mode probability update μj（k+1｜k+1）. And they can be computed as44

6. Simulation and analysis

The validation of the proposed approach is performed based on SINS/GPS integrated navigation system of a 3D UAV flight scenario. We exhibit the simulation results to approve the proposed IMM-NFA H∞for UAV localization issue.Four f iltering techniques are employed in the fusion calculation:N H∞, NFA H∞-1, NFA H∞-2 and the proposed IMMNFA H∞f ilter.The results of the proposed technique are compared with the other navigation f iltering strategies. The sampling rate of the Strap-down Inertial Navigation System(SINS)utilized in this study is 100 Hz,while the sampling rate of the GPS sensor is 1 Hz. Table 4 demonstrates the kind of the measured information and the sampling rate utilized in the simulation for every sensor.

Furthermore,the updated rate for each navigation f ilter utilized as a part of this study is 10 Hz. The SINS represents the main system while the GPS represents the measurement system. The role of the GPS receiver sensor is to provide the UAV north-east-down position and the speed vector coordinates. The GPS receiver sensor used in simulation has measurement errors with 3 m standard deviation in position measurement and measurement errors with 0.5 m·s-1standard deviation in velocity measurement.

The following parameters were used in the simulation: the values of the fixed drifts equal to 0.02°/h for the embedded gyroscopes in the SINS, and the value of the fixed bias equal to 120 g for the embedded accelerometers in the SINS. A trajectory of 180 s is taken for flight in the simulation.Fig.12 presents the 3D position flight curve of the true positions.

Table 4 Sampling rate and measured data used in simulation for each sensor.

Fig. 12 3D position of a UAV flight trajectory used in simulation.

The trajectory of the UAV has a different type of motions which are composed of straight, turning and climbing accelerated motions.The initial states used in the simulation are a priori chosen as 80% from the true values. We have fixed all the initial conditions for all the different strategies used in this study. However, each sensor has its own random errors and we have taken into consideration the randomness of the sensors errors. Consequently, our simulation is done based on Monte Carlo runs.The values of the N H∞fixed bound parameters δ1, δ2and δ3have been chosen to be 0.01, 0.75 and 0.02,respectively,while the value of the N H∞fixed attenuation disturbance parameter γ has been chosen to be 20. These values have been chosen to be the same as initial values of the proposed IMM-NFA H∞f ilter which is performed in this study.

The initial value of the mode probability has been chosen as μ= 0.8,0.2[ ]T,while the matrix of mode transition probability has been chosen as

By using the proposed approach of IMM-NFA H∞f ilter and comparing among the single model f ilters of N H∞,NFA H∞-1 and NFA H∞-2,Figs.13-15 provide the navigation results of state estimation of the position of north, east and down, respectively. Figs. 16-18 provide the navigation results of state estimation of the velocity following the axes of north,east and down,respectively.Figs.19-21 provide a comparison of north, east and down position estimation errors, respectively, via all the four approaches: N H∞, NFA H∞-1,NFA H∞-2 and IMM-NFA H∞.Figs.22-24 provide a comparison of error estimation of the velocity following the axes of north, east and down, respectively. From these f igures, we can notice the advantage of utilizing the fuzzy adapting of bounds for the N H∞f ilter.

Fig. 13 State estimation of position in direction of north axis.

Fig. 14 State estimation of position in direction of east axis.

Fig. 15 State estimation of position in direction of down axis.

Fig. 16 State estimation of velocity in direction of north axis.

Fig. 17 State estimation of velocity in direction of east axis.

Fig. 18 State estimation of velocity in direction of down axis.

Fig. 19 Error estimation of position in direction of north axis.

Fig. 20 Error estimation of position in direction of east axis.

Fig. 21 Error estimation of position in direction of down axis.

Fig. 22 Error estimation of velocity in direction of north axis.

The proposed IMM-NFA H∞employs the fuzzy logic adaptive systems(FLAS)for automatically tuning f ilter bound parameters δ1, δ2, δ3and attenuation disturbance parameter γ of the N H∞f ilter. It can be seen that signif icant estimation accuracy enhancement is obtained by utilizing the proposed methodology.

Fig. 23 Error estimation of velocity in direction of east axis.

Fig. 24 Error estimation of velocity in direction of down axis.

It has been verif ied that, by monitoring the linearization errors which have been obtained by the subtraction between the linearized and non-linearized nonlinear state navigation models and represented in the position, velocity and attitude errors, and the bounds δ1, δ2, δ3can be suitably tuned via the first and third fuzzy inference systems. Furthermore, the changes in the gyro drifts and accelerometer biases which influence the estimation environment and navigation accuracy have been monitored throughout the second fuzzy inference system to adjust the attenuation disturbance parameter γ. What’s more,γ can be tuned adaptively in accordance with the fourth fuzzy inference system by monitoring the attenuation level alongside the innovation information and its covariance throughout the performance factor index.

The adapted δ1,δ2,δ3and γ have been used to scale the process and measurement noises and consequently the IMMNFA H∞has good capability to adjust the process and measurement noise covariance matrices to achieve a signif icant navigation accuracy and prevent the divergence.

By comparing the different f iltering techniques as shown in Figs.13-24,we can notice that the accuracy of the NFA H∞-1 and NFA H∞-2 f iltering techniques has almost no difference;however the accuracy of their algorithms is better than the accuracy of the N H∞f ilter.It is because the disturbance noise covariance matrices of the NFA H∞-1 and NFA H∞-2 f iltering techniques are provided with noise covariance adaptability.

Table 5 Comparison of estimated errors among N H∞, NFA H∞-1, NFA H∞-2 and IMM-NFA H∞.

The accuracy of the IMM-NFA H∞algorithm is better than that of the N H∞, NFA H∞-1 and NFA H∞-2 f iltering techniques. It is because the IMM-NFA H∞algorithm is precisely provided with noise covariance adaptability based on the changes of sub-model probabilities of NFA H∞-1 and NFA H∞-2.

The simulation has demonstrated that the IMM-NFA H∞algorithm can give better coverage of variable disturbance noise characteristics adaptability. Consequently, the IMMNFA H∞algorithm has improved accuracy compared with N H∞,NFA H∞-1 or NFA H∞-2 algorithm when the measurement and process conditions change.

From the equations of N H∞f ilter, the equation of process covariance matrix can be influenced by the attenuation disturbance parameter γ and consequently the convergence of the process covariance matrix depends on choosing a suitable value of the attenuation disturbance parameter γ by using the IMM-NFA H∞.From the simulation,we have noticed that the error covariances of position and velocity which are acquired via IMM-NFA H∞are smaller than those obtained via N H∞algorithm. we have noticed from the simulation results that the large changes in the dynamics motion affected the position and velocity estimation accuracy because inaccurate measurements are obtained from accelerometers during the high dynamic turns during flight and this is shown in the previous f igures, where the estimation errors of position in the north, east and vertical channels in case of using the N H∞f ilter increase during the high dynamic flight motion and reach more than 15 m, while these errors have been reduced in case of using the proposed technique of IMMNFA H∞f ilter. As well, the estimation errors of velocity in the vertical channel in case of using the N H∞f ilter increase during the 3D coordinated turn maneuver and reach more than 3 m·s-1while in case of using IMM-NFA H∞f ilter they reach 2 m·s-1.Also,it is evident from the previous f igures that the estimation errors can be more stable within fixed range during the low dynamic flight motions.

As shown from the previous f igures of the position and velocity estimation errors in different directions,it is clear that more stability and good estimation performance have been achieved with respect to the position and velocity estimation errors of the IMM-NFA H∞f ilter than that with predefined fixed and non-adaptive parameters in case of the nonlinear H∞f ilter.It is obvious from Table 5 that the estimation errors based on the N H∞f ilter have been reduced almost by 40%when we used the proposed approach of IMM-NFA H∞f ilter.

From the table,it is evident that the proposed methodology fulfills a slightly accurate estimation velocity and position than the first and second nonlinear fuzzy adaptive H∞f ilters as well.Additionally, the simulation results demonstrate that the proposed approach is well suited to handle with the class of nonlinear integrated SINS/GPS navigation system of UAV localization issue.

Table 5 demonstrates the comparison of the standard deviations of the estimated velocity and position errors in the X,Y and Z axes among N H∞, NFA H∞-1, NFA H∞-2 and IMMNFA H∞f ilters. From the table, it is apparent that the proposed technique gives further precise positions with no presupposition of process and measurement noise characteristics or N H∞bounds. Moreover, mode probability of the proposed IMM-NFA H∞is portrayed in Fig. 25.

The change of model probability is associated with the change of measurement information. At the point when the information of SINS/GPS data and the process noise change,the residual error and its covariance of each model-matched f ilter will change. And accordingly, the associated likelihood function of every model-matched f ilter will be calculated for updating the probabilities of different models.The mode probability of each mode is updated with the residual error and its covariance by the model-matched f ilter.

Fig. 25 Mode probability of IMM-NFA H∞.

Fig. 26 Comparison of 3D flight trajectory of a UAV among N H∞, NFA H∞-1, NFA H∞-2 and IMM-NFA H∞.

Fig.25 shows the mode probability of IMM-NFA H∞algorithm based on the changes of sub-model probabilities of NFA H∞-1 and NFA H∞-2. This f igure shows that the probabilities of NFA H∞-1 and the probabilities of NFA H∞-2 increase close to one and decrease close to zero in different time periods. The probability of the sub-model is close to one, when the noise covariances of the related sub-model are near enough to true noise covariances in these periods. However, the probability of the sub-model is almost zero at some intervals of time during the period when the noise variance increases greatly.

In contrast, the IMM-NFA H∞f ilter provides more accurate positioning and performs signif icantly better than N H∞.This outcome is aff irmed by the 3D position of a UAV flight profile trajectory shown in Fig. 26, and different trajectories are compared with the true trajectory.

Fig. 26 presents a comparison of 3D UAV flight trajectory during navigation utilizing INS/GPS data fusion. We have concluded from the validation of our methodology with respect to the integrated SINS/GPS kinematic navigation applications that the conventional N H∞with non-adaptive and fixed parameters cannot guarantee the high estimation accuracy and error stability when the characteristics of the disturbances change. The UAV position and velocity estimated by the IMM-NFA H∞f ilter clearly show its robustness against the different issues which are associated with the modeling errors, nonlinearities errors and noise uncertainties.

7. Conclusions

(1) A new algorithm depends on IMM-NFA H∞f ilter is proposed for UAV localization issue utilizing SINS/GPS sensor data fusion scheme and its execution performance is evaluated.

(2) The general evaluation can be obtained by merging the state estimated from parallel processing f ilters depending on separate models which match system modes.

(3) IMM-NFA H∞f ilter algorithm utilizes model probabilities to weight input-output information of the bank of parallel NFA H∞f ilters at every sampling time.

(4) The technique which was initiated considers an arrangement of models to describe the behavior patterns of the system.

(5) The procedure of every one cycle comprises four essential stages: model interaction and mixing step, model individual f iltering, mode probability update, and state estimate and process covariance combination.

(6) The major contribution is designing a nonlinear fuzzy adaptive robust f ilter taking into account high order Taylor expansion terms through adaptive disturbance attenuation, along with the adaptive bounds and therefore averting the linearization errors which is a problem in classical EKF.

(7) FLAS is utilized to f ind the boundary of system noise and attenuation disturbance parameter through various FISs.

(8) Our methodology is employed for the autonomous navigation system to accomplish more robustness and better accuracy and to enhance the localization of UAV, and can be considered as an alternative technique for the localization problem of the autonomous navigation vehicle.

(9) The outcome sensor fusion procedure can effectively manage and handle the nonlinear issue in navigation vehicle.

(10) IMM-NFA H∞f ilter provides signif icant enhancement in navigation estimation accuracy in comparison with the N H∞approach with fixed parameters.

(11) The UAV position and velocity estimated by the IMMNFA H∞f ilter clearly show its robustness against the different issues which are associated with the modeling errors, nonlinearity errors and noise uncertainties.

Acknowledgement

This study is supported by a grant from the National Natural Science Foundation of China (No. 61375082).