implementation of an intelligent sins navigator based on ... - IEEE Xplore

2009 6th International Multi-Conference on Systems, Signals and Devices

IMPLEMENTATION OF AN INTELLIGENT SINS NAVIGATOR BASED ON ANFIS Karim M. Ahjebory1, Salam A. Ismaeel2, and Ahmed M. Alqaissi 3 1 Al-ISRA University, Amman/Jordan 2 Computer Man College for Computer Studies, Khartoum/Sudan [email protected], [email protected], 3 Faculty of Engineering, University Putra Malaysia email: [email protected]

In this work an intelligent navigator developed to overcome the limitations of existing Strapdown Inertial Navigation Systems (SINS) algorithm. This system is based on Adaptive Neuro-Fuzzy Inference System (ANFIS). As in previous work, which is based on Artificial Neural Network, the window based weight updating strategy was used, and the intelligent navigator evaluated using several SINS hypothetical field tests data. And the results show that the intelligent navigator based on ANFIS more powerful compared with other (traditional and intelligent navigator based on ANN). Index Terms — Inertial navigation systems, adaptive fuzzy system, intelligent navigator

reference [5], the artificial neural network (ANN) was used. On the other hand, the fuzzy-neural network (FNN) can be realized as a neural network (NN) structure, and the parameters of fuzzy rules can be expressed as the connection weights of the neural network. So the backpropagation (BP) training algorithm to update the FNN parameters can be used [6]. In this work, we hope to achieve at least reduce the impact of limitations of the conventional algorithms by proposing an intelligent navigator that uses Adaptive Neuro-Fuzzy Inference System (ANFIS) as the core algorithm for SINS algorithm. Unlike the work described in [5], that uses ANN. Such navigator is expected to provide positioning information (X, Y, Z), velocity information in North, East, and Down (VN, VE, VD) and to be able to overcome the limitations of the SINS algorithm described in [7].

I. INTRODUCTION

II. TERRESTRIAL STRAPDOWN SYSTEM DYNAMIC EQUATION

ABSTRACT

The Strapdown Inertial Navigation System (SINS) performs the navigation and guidance, using the measurements from gyros and accelerometers installed directly on the vehicle body. The key calculations performed in SINS are updating the body frame attitude by integrating the angular rate from gyro, and updating the vehicle velocity by integrating the specific force acceleration from accelerometer. The noncommutativity of finite rotations is one of the major error sources in numerical solutions of the SINS calculation [1]. Strapdown Inertial Navigation System algorithms are the mathematical definition of processes, which convert the measured outputs of inertial sensors that are fixed to a vehicle body axis into quantities, which can be used to control the vehicle [2]. These algorithms suffer from unbounded error due to integration process in it, most articles overcome this problem by integration the INS system with other system like global positioning system (GPS) [3], or by using Kalman filter to overcome such problems [4]. While in

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The differential equation of the relative quaternion between body coordinate and geographic coordinate given by [7]:

u& = 12 Ω bib ⋅ u − 12 Ω bin ⋅ u

(1) Where, the angular velocity skew-symmetric matrix Ωini and Ωbnb are given by: ⎡ 0 ⎢ w Ω bin = ⎢ D ⎢ − wE ⎢ ⎣− wN

− wD

wE

0

− wN

wN − wE

0 − wD

⎡ 0 ⎢− w Y Ω bib = ⎢ ⎢ wP ⎢ ⎢⎣− wR

wY 0

− wP wR

− wR − wP

0 − wy

and

wN ⎤ wE ⎥⎥ wD ⎥ ⎥ 0 ⎦

wR ⎤ wP ⎥⎥ wY ⎥ ⎥ 0 ⎥⎦

(2)

(3)

(

)

⎡ wN ⎤ ⎡ wie + l& cos L ⎤ ⎥ ⎢w ⎥ = ⎢ − L& ⎥ ⎢ E⎥ ⎢ & ⎢⎣ wD ⎥⎦ ⎢⎣− wie + l sin L ⎥⎦

(

(4)

and

[

)

g e = g 0 − 3.0877 × 10 −6 − 0.0044 × 10 −6 sin 2 ( L) ⋅ h + 0.072 × 10

−12

h

]

2

(11)

where [L, l, h]: are geodetic positions (latitude, longitude, and height) wR, wP, wY are the body angular velocities in the body coordinate (roll, pitch, and yaw), respectively.

Equations (1, 6, and 9), represent the mechanization equation for the terrestrial navigation system.

Body fixed coordinate to navigation coordinate (Cbn) can be described in terms of the quaternion parameters:

The main advantage of using a hybrid intelligent system like ANFIS, over other classical filtering algorithms is its ability to deal with noise in the input data in dynamic environments. In this intelligent system not only combine the learning capabilities of a neural network but also incorporate reasoning by using fuzzy inference, there by enhancing the capability of the system for prediction. The goal of ANFIS is to find a model or mapping that will correctly associate the inputs (initial values) with the target (predicted values). The equation which represents a fuzzy logic system with center average defuzzifier, product interface rule, nonsingleton fuzzifier, and bell-shaped membership function is [8]:

⎡u02 + u12 − u 22 − u32 2(u1u 2 − u0 u3 ) 2(u1u3 + u0 u 2 ) ⎤ ⎢ ⎥ 2(u 2 u3 − u0 u1 ) ⎥ C = ⎢ 2(u0 u3 + u1u 2 ) u02 − u12 + u 22 − u32 2 2 2 2⎥ ⎢ 2(u1u3 − u0 u 2 ) 2(u0 u1 + u 2 u3 ) u0 − u1 − u 2 + u3 ⎦ ⎣

(5)

n b

The differential equations of the vehicle position in terms of latitude, longitude, and heading can be arranged in matrix form: ⎡ L& ⎤ ⎡1 /( R N + h) 0 0 ⎤ ⎡V N ⎤ ⎢ &⎥ ⎢ ⎥ ⎢V ⎥ 0 1 / ( ) cos 0 l = + R h L ( ) N ⎢ ⎥ ⎢ ⎥⎢ E ⎥ & ⎢ h ⎥ ⎣⎢ 0 0 1 − ⎦⎥ ⎣⎢V D ⎦⎥ ⎣ ⎦

(6)

where [VN VE VD] = V n: geodetic velocity vector (north, east, and down) RN and RE: are the radii of curvature in the north and east direction and given by: RN =

RE =

(1 − e

re 2

(1 − e

sin 2 ( L ) re 2

(7)

)

1 .5

)

(8)

sin 2 ( L ) and e : eccentricity (= 0.0818)

The differential equations relating the second derivative of the geodetic position and velocities can be derived as: ⎡ ⎤ ⎡ ⎤ VE VV −⎢ + 2wie ⎥VE sin L + N D ⎢ ⎥ (RE + h) cos L RN + h) ( ⎣ ⎦ ⎢ ⎥ & ⎡VN ⎤ ⎢ ⎥ ⎡ ⎤ V V V ⎢& ⎥ n b E E D ⎢VE ⎥ = Cb ⋅ f + ⎢⎢ (R + h) cos L + 2wie ⎥VN sin L + (R + h) + 2wieVD cos L⎥ ⎣ ⎦ E E ⎢ ⎥ ⎢V&D ⎥ ⎣ ⎦ ⎢ ⎥ VE2 VN2 − − − 2w V cos L + ge ⎢ ⎥ (RE + h) (RN + h) ie E ⎢⎣ ⎥⎦

(9)

where fb

:

Specific force outputs in the body coordinate = [fx fy fz]T

ge

:

Gravity force applied on down direction

Gravity force (ge) can be found from initial gravity g0:

[

g 0 = 9.780327 1 + 0.0053024 sin 2 ( L) − 0.0000058 sin 2 (2 L)

]

(10)

III. ADAPTIVE NEURO-FUZZY INFERENCE SYSTEM STRUCTURE

⎡

⎡ ⎛ x − m ⎞2 ⎤⎤ i ij ⎟ ⎥⎥ σ ij ⎟⎠ ⎥ ⎥ j =1 i =1 ⎝ ⎦⎦ ⎣ ⎣ f ( x) = ⎡ ⎛ x − m ⎞2 ⎤⎤ M ⎡ n ij ⎢ ∏ exp ⎢ − ⎜ i ⎟ ⎥⎥ ∑ ⎢ ⎜⎝ σ ij ⎟⎠ ⎥ ⎥ j =1 ⎢ i =1 ⎣ ⎦⎦ ⎣ M

n

∑ y ⎢⎢ ∏ exp ⎢⎢ − ⎜⎜ j

(12)

where f (x)

: Fuzzy logic system output, which represent a function to n input variables x

xi

: Input variable in the input universe of discourse

yj

: Center of fuzzy set Fj, which is, a point in the universe of discourse V when μFj (y) achieves its maximum value, and μFj (y) is given by a product interface engine

M

: The number of fuzzy rules

N

: The number of input variables

mi,σi

: The center and width of the bell-shaped function of the ith input variable, respectively.

This equation can be implemented on a Forward Neural Network (FNN). This connectionist model combines the approximate reasoning of fuzzy logic into a five layer neural network structure. Based on the error back propagation algorithm for multiinput single-output (MISO) system, the goal is to

Equations (14), (15), and (16) perform an error back propagation procedure.

determine a fuzzy logic system f( x ) in the form of (12), which minimizes the error function: E(k) =

1 (f ( x (k) ) - d (k) ) 2 2

IV. INTELLIGENT SINS NAVIGATOR BASED ON ANFIS

(13)

Figure (1) shows the main block diagram of the intelligent navigator system proposed in this work. Several tests are applied for ANFIS based SINS algorithm to generate the necessary navigation knowledge for the proposed navigator. After that, the concept of a navigation information database is discussed to provide storage space of the navigation knowledge, and then a window based parameters updating strategy is given as a tool for accumulating the navigation knowledge. The intelligent navigator integrates the data from Inertial Measurement Unit (IMU), i.e. accelerometers and gyros, and mimics the dynamical model of the vehicle to generate navigation knowledge. Thus the latest acquired navigation knowledge can be applied to predict the vehicle’s velocity and position during IMU errors in real time. The resulting ANFIS has the structure depicted in Figure (2). It must be mentioned that each component of velocity and position has its own network, so, six networks have been adopted in this work. The input neurons receive the acceleration at current epoch (AINS(t)), and angular velocity at current epoch (AVINS(t)).

where, dj(k) is the jth is the desired output at time k. According to equation (12), if the number of rules is M, then the problem becomes training the parameters yj, mij, and σij such that E(k) is minimized. And based on the back propagation training algorithm the iterative equations for training the parameters yj, mij, and σij are [8]:

1 zj D

y j ( k + 1) = y j ( k ) − η ( f ( x ( k )) − d ( k ) ) m ij ( k + 1) = m ij ( k ) − 2η

(14)

zj

( f ( x ( k )) − d ( k ) ) D x i2 ( k ) − m ij ⎞ ⎟ ⎟ (σ ij ) 2 ⎠

⎛ ⋅ ( y j ( k ) − f ( x ( k ) )⋅ ⎜ ⎜ ⎝ zj σ ij (k + 1) = σ ij (k ) − 2η ( f ( x(k )) − d (k )) ⋅ ( y j (k ) − f ( x(k )) D ⎛ ( xi2 (k ) − mij ) 2 ⎞ ⎟ ⋅⎜ ⎜ ⎟ (σ ij ) 3 ⎝ ⎠

(15)

(16)

where ⎡ ⎛ x2 − m n i ij z j = ∏ exp ⎢ − ⎜ ⎢ ⎜⎝ σ ij i =1 ⎣

⎞ ⎟ ⎟ ⎠

2

⎤ ⎥ ⎥ ⎦

(17)

The output generates velocity and position in the local level frame at the current epoch. Thus, the navigation knowledge can be learnt, stored and accumulated during the availability of the IMU signals. On the other hand, during IMU signal absence or IMU errors, the latest acquired navigation knowledge can be retrieved from the navigation information database of the intelligent navigator to predict the velocity and position in real time.

D : the denominator of equation (12). η : the learning rate. Equations (14), (15), and (16) perform an error back propagation procedure. 6DOF Simulation

Accelerometer Reading

Gyros Reading

Z net Y net B2

Implement SDINS algorithm INS data

B1

A1

A2

XB2 netB1

A2 B1

B2

∏

∏

∏ N

A1

A2

A1

∏

N

N

N

Traini

A1

A2

B1

∏ ∏

N

A1 A2

B2

∏

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Adjust learning parameters

VD net VE net B1 B2 A2 V N net B1 B2 ∏

∏

X ∏axis

N

∏ N

N

N

N

∑

A1

∏

N

∑ ∑

∑

∑

∑

+ Velocity Position

Save the training parameters Save the training parameters in reference template file Save the training parameters in in reference referencetemplate template file file

ANFIS o/p

Save the training parameters Save the training parameters in reference template file Save the training parameters in in reference referencetemplate template file file

NAViBASE

Fig.1. Main block diagram of the intelligent

NAViBASE

IMU Readi ngs

1 ∏ n ∑

INS Positio n or N/D

1 Ti me

∑ ∏

n

Fig.2. The Architecture of an Adaptive Fuzzy System network

V. WINDOW BASED PARAMETERS UPDATING STRATEGY

In fact, this method combines the advantages of both sequential mode and batch mode of training in order to make the training procedure suitable for real-time processes. In addition, the parameters of each window are then updated via batch training mode. In other words, the parameters of each window are updated sequentially. As depicted in [5], the procedure of the window-based parameters updating method is given below: Parameters initialization: The initial parameters can be obtained using previously stored parameters that are stored in navigation database (NAViBASE) or random initialization. In this work, the initial parameters were obtained using random initialization. After that, the parameters were stored in NAViBASE after each navigation mission and could be applied as the initial parameters for the next mission. Accurate initial parameters may significantly reduce training time. INS signal reception: Within the first INS window (i=l), INS (i), the learned parameters are not updated, thus the stored learned parameters are still the initial synaptic weights P (i-1) (i.e., P (0)). INS signal reception: At the next INS window, INS (i+1), the stored parameters, P(i-1), are updated utilizing the previous available INS information (INS(i)). These learned parameters are stored as P(i) after training is completed. These steps are repeated until INS signal blockage is detected. INS Absence: In case of an INS blockage (after INS (i)), P(i-l) is first applied for real time prediction and then P(i) is utilized to replace P(i-l) and carry on real time prediction during the INS blockage.

The prediction using P (i-1) and the training of P (i) can be operated in parallel. For simplification, the update procedure during INS blockage can be paused thus P(i-

1) is applied to provide prediction during entire INS blockage and it can be updated after the reception of next available INS signal window. Since the ANFIS training procedure takes time, updating the learned parameters immediately at the latest available sample of IMU signal before blockage is difficult. However, the utilization of the proposed method can still provide reasonable prediction accuracy during IMU blockage since it provides the latest updated parameters instead of real time updated parameters for real time prediction. Therefore, failure in providing real time updated learned parameters doesn't mean the intelligent navigator is not able to provide real time prediction. On the contrary, it can utilize the latest acquired and learnt navigation knowledge to provide real time solutions. Combining the latest INS window signals, stored parameters can be adaptively updated to follow the latest motion dynamics thus improving the prediction accuracy during IMU blockage. VI. TRAINING PROCEDURES

The training samples acquired for the window based parameters updating strategy can be arranged through using the following two procedures [5]: One step training procedure: The training samples acquired for each INS window during navigation are the combination of stored training samples and available training samples obtained at the end of each INS window. In other words, as the size of training samples increases during navigation, the size of NAViBASE grows during navigation as well. The advantage of the one step training is that it can provide better generalization of the navigation knowledge by incorporating stored and previous training samples during navigation. The one step training procedure is recommended at the early stage for building the intelligent navigator as the navigation knowledge acquired by the navigator at this moment might not be enough to provide acceptable accuracy during INS blockages. As the size of NAViBASE is quite small, the incorporation of stored training samples doesn't slow down the learning process during each window; actually, it can provide better generalization of the navigation knowledge, but when the size of NAViBASE increases this will slow down learning process. Two steps training procedure: The training samples acquired for each INS window during navigation are obtained at the end of each INS window. After navigation, all the training samples acquired during the navigation are recalled and combined with the stored training samples then fed into the navigator to improve the generalization of navigation knowledge using a conventional off-line batch training method. This procedure is recommended for the regular operational stage for building the intelligent navigator. After several field tests, the navigator might accumulate enough navigation knowledge to provide navigation solutions

during navigation without incorporating stored training samples. In other words, the size of training samples is the same as the INS window. Therefore, the training speed during each window is expected to be faster than the previous procedure. After navigation, all the training samples acquired during the current navigation are recalled and combined with the stored training samples first to remove redundant navigation knowledge, and are then re-trained to improve the generalization of the navigation knowledge for future navigation missions. This will ensure to keep NAViBASE at specific size to avoid slowing down learning process. The key factor that can accelerate the learning is the generalization of navigation knowledge. The perfect solution is to obtain the most generalized navigation knowledge that can then be fed into the navigator in one field test. However, that is not the case for real life applications. Therefore, the navigator must have the ability to evolve during each navigation mission to provide generalized navigation knowledge for future missions. Thus, using the proposed INS architectures, NAViBASE, and window based parameters updating strategy, the intelligent navigator has the ability to generate and accumulate the navigation knowledge. In other words, it can learn and evolve continually to provide updated navigation knowledge and fill the gap between IMU blockages.

Figure (6) shows the error between the desired and actual output of the proposed intelligent navigator. Where the desired output is the output of the SINS algorithm position and velocity in (X, Y, and Z) respectively and the actual output of the ANFIS was the estimated position and velocity. In general, the error in the height position usually follows a very smooth and easy-to-predict pattern compared with the latitude and longitude, where the position errors of these two components usually experience several disturbances. The standard deviation (STD) errors after using the ANFIS are 0.0161, 0.0268, and 0.0383 m for Latitude, Longitude, and height in position and 0.0384, 0.0409, and 0.0449 m/s for North, East, and Down directions in velocity respectively. While the STD errors for the position and velocity after using the SINS algorithm as a stand alone system are 1.1910e+003, 1.3885e+003 ,and 648.0428 m for position and 3.3155, 5.5829, and 3.1636 m/s for velocity in all of the three directions. Implement SDINS algorithm

6DOF Simulatio n Accelerometer Reading

Gyro Reading

Position or velocity component

+

VII. PERFORMANCE ANALYSIS OF THE ANFIS NAVIGATOR

According to figure (3) which illustrates the training phase of the ANFIS the network output is compared to the SINS algorithm output, the error is fed to the ANFIS, which adjust the network learning parameters in a way to minimize the mean square value of error. The parameters (m, y, and σ) are the ANFIS learning parameters are computed during the training phase and they are determine the input /output functionality of the network. Figure (4) shows the mean square error (MSE) for all networks after 10 epochs. The initial values used to obtain these results are listed in table (1). As stated early, these values are obtained by trail-and-error. After the training is completed the network is ready to work in the prediction mode. However the parameters of the networks are modified during the availability of the IMU signals AINS(t) and AVINS(t) (i.e. the training procedure continues) and network is considered working in the update mode. In the case of IMU blockage or during IMU errors, the network will use the latest estimation parameters saved in the NAViBASE to perform the prediction process. Figure (5) shows the operation of the ANFIS in the prediction process during absence of IMU signals. It provides a prediction of INS position and velocity. From the results it was found that the error in the X and Z axes position component is easier to predict than Yaxis while the velocity directions are very easy to predict.

ANFIS

NAViBASE

Adjust learning Parameters Real INS data (True)

Fig.3. Block diagram of ANFIS during training stage VIII. CONCLUSIONS

The conclusions drawn from the results presented in this paper are: 1) The process of selecting the initial values of the parameters (m, y, and σ), number of rules, and value of the learning rate is done through a trial-and-error procedure and determining the appropriate settings for one trajectory may need several attempts; therefore, handling several trajectories separately can be a very long process whereas when these trajectories are handled together, one after the other, the process of selecting the appropriate initial values is done only one time. 2) If a group of trajectories is used in the training process of the ANFIS network, then these trajectories must have somehow similar features.

[4] S. A. Ismaeel, “Intelligent Navigating System Based on SINS and ANN,” Journal of Computer, Communication and Control Engineering, 2006. [5] S. A. Ismaeel and A. M. Hassan, “GPS/INS System Integration Based on Neuro-Wavelet Techniques,” The 2006 International Conference on Artificial Intelligence (ICAI’06 Las Vegas, USA), June 26-29, 2006,. [6] S. A. Ismaeel and K. M. Al-Jebory, “Adaptive Fuzzy System Modeling,” Eng. Technology, vol. 20, no. 4, pp. 201-212, 2001. [7] S. A. Ismaeel and S. A. Aziez, “Development of Six-Degree of Freedom Strapdown Terrestrial INS Algorithm,” Journal of Um-Salama for Science, vol. 2(1), pp. 155-165. 2004. [8] Y. Hao, Z. Xiong, W. Gao and L. Li “Study of Strapdown Inertial Navigation Integration Algorithms,” Proceedings of the 2004 International Conference on intelligent Yechatronics and Automation Chengdu,Chin, August 2004.

3) The results presented in this work strongly indicate the potential of including the intelligent navigator as the core navigation algorithm for the next generation navigation system. X. REFERENCES [1] D. Jwo and H. Huang, “GPS Navigation Using Fuzzy Neural Network Aided Adaptive Extended Kalman Filter,” Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference 2005 Seville, Spain, December 12-15, 2005. [2] J. Farrell and M. Barth, The Global Positioning System and Inertial Navigation, McGraw-Hill Companies, Inc. 1999. [3] Kai-Wei, C., Yun, H., “An intelligent navigator for seamless INS/GPS integrated land vehicle navigation applications,” Applied Soft Computing 8, 722-733, 2008.

Position in Y-axiz

Position in Z-axis

6

Position in X-axis

8

4.5 4

7

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3 2.5 2 1.5

4 M ean S quare E rror (m )

M e a n S q u a re E rro r (m )

M ean S quare E rror (m )

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Fig.4. The relation between the number of epochs and the mean square error for the networks of position and velocity.

Table 1. Initial values for the six networks; position and velocity

m Initial values

M ean S quare E rror (m /s ec )

3.5

Mean Square Error (m/sec)

4

σ y M Learning rate No. of epoch

X-axis [0, 5]

Position Y-axis [-5, 5]

Z-axis [-1, 1]

North [1, 2]

Velocity East [-1, 1]

Down [-2, 1]

[0, 5] [0, 5] 5 0.13 10

[-5, 5] [-1, 2] 5 0.13 10

[-3, 3] [-1, 1] 5 0.13 10

[1, 2] [1, 2] 5 0.13 10

[-1, 1] [-1, 1] 5 0.13 10

[-2, 2] [-2, 2] 5 0.13 10

6DOF Simulation Accelerometer Reading

Use latest learning parameters

Gyro Reading

ANFIS

NAViBASE

Real INS data (True)

Fig.5. Block diagram of ANFIS during prediction stage.

0.4

0.25

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0.3 0.2

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Fig.6. Error between the desired and actual output of the conceptual intelligent navigator along all components for position and velocity.