- Email: [email protected]
Firefly, Teaching Learning Based Optimization and Kalman Filter Methods for GPS Receiver Position Estimation
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Procedia Computer Science 143 (2018) 892–898
8th International Conference on Advances in Computing and Communication (ICACC-2018) 8th International Conference on Advances in Computing and Communication (ICACC-2018) 8th International Conference on Advances in Computing and Communication (ICACC-2018) Firefly, Teaching Learning Based Optimization and Kalman Filter MethodsLearning for GPS Based Receiver Position Estimation Firefly, Teaching Optimization and Kalman Filter a* Methods for GPS Receiver Position Estimation Lavanya Bagadi , G Sasibhushana Raob, Ashok Kumar Nc a* of ECE,MVGR(A),Vizianagaram,535005,India Department Lavanya Bagadi , G Sasibhushana Raob, Ashok Kumar Nc Department of ECE,AUCE(A),Andhra University,Visakhapatnam,530003,India. a
b,c
b,c
Abstract
a Department of ECE,MVGR(A),Vizianagaram,535005,India Department of ECE,AUCE(A),Andhra University,Visakhapatnam,530003,India.
Global Positioning System (GPS) is an all-weather worldwide navigation system that provides position, navigation and timing Abstract information. A lot of research has been done in improving the position accuracy but still researchers are working towards the betterment of position accuracy using augmentation methods like system WAAS,that EGNOS, andposition, GAGANnavigation etc. because ever Global Positioning System (GPS)without is an all-weather worldwide navigation provides and of timing increasing demand of research user accuracy. In this Firefly Algorithm (FA)accuracy and Teaching Based (TLBO) information. A lot of has been donepaper in improving the position but stillLearning researchers areOptimization working towards the techniques of have been considered for improving the position accuracy the results the existing KalmanofFilter betterment position accuracy without using augmentation methods and like compared WAAS, EGNOS, andwith GAGAN etc. because ever method. considering KF, correctable factors are Algorithm not included. is simple in implementation without(TLBO) tuning increasingWhile demand of user accuracy. In this paper Firefly (FA)TLBO and Teaching Learning Based Optimization parameters have whereas requires control parameters but the results show FA outperforms forexisting GPS precise position techniques beenFA considered for improving the position accuracy and that compared the resultsTLBO with the Kalman Filter accuracy.While FA hasconsidering 4m more accuracy in terms of x andare z coordinates whereas 2mismore accuracy in y-coordinates compared to method. KF, correctable factors not included. TLBO simple in implementation without tuning TLBO. The resultsFA show that with proper selectionbutofthe control parameters methods yield optimal solutions. Numerical parameters whereas requires control parameters results show thatthese FA outperforms TLBO for GPS precise position results clearly verify significance of terms the proposed The whereas computation time taken by FA and TLBO to compared converge to accuracy. FA has 4mthe more accuracy in of x andalgorithms. z coordinates 2m more accuracy in y-coordinates desired result 50.377sec respectively. TLBO. The isresults showand that3.132sec with proper selection of control parameters these methods yield optimal solutions. Numerical results clearly verify the significance of the proposed algorithms. The computation time taken by FA and TLBO to converge to desired result is 50.377sec and 3.132sec respectively. © 2018 The Authors. Published by Elsevier B.V. © 2018 The Authors. by Elsevier B.V. This is an open accessPublished article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection andAuthors. peer-review underbyresponsibility of the scientific committee of the 8th International Conference on Advances in © 2018 The Published Elsevier B.V. Selection and peer-review under responsibility of the scientific committee of the 8th International Conference on Advances in Computing and Communication (ICACC-2018). This is an open article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Computing and access Communication (ICACC-2018). Selection and peer-review under responsibility of the scientific committee of the 8th International Conference on Advances in Keywords: kalman Filter; Position Estimation; TLBO. ComputingFirefly; and Communication (ICACC-2018). Keywords: Firefly; kalman Filter; Position Estimation; TLBO. * Corresponding author. Tel.: +919885512625 E-mail address: [email protected] * Corresponding author. Tel.: +919885512625 E-mail address: [email protected]
1877-0509 © 2018 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection peer-review under responsibility of the scientific 1877-0509and © 2018 The Authors. Published by Elsevier B.V. committee of the 8th International Conference on Advances in Computing and Communication (ICACC-2018). This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection and peer-review under responsibility of the scientific committee of the 8th International Conference on Advances in Computing and Communication (ICACC-2018). 1877-0509 © 2018 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection and peer-review under responsibility of the scientific committee of the 8th International Conference on Advances in Computing and Communication (ICACC-2018). 10.1016/j.procs.2018.10.365
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Lavanya Bagadi et al. / Procedia Computer Science 143 (2018) 892–898 Lavanya Bagadi/ Procedia Computer Science 00 (2018) 000–000
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1. Introduction Metaheuristic algorithms have gained eminence and form a vital part of global optimization algorithms, soft computing and computational intelligence. The popularity of metaheuristic algorithms [1] is partly due to their ability of dealing with nonlinear global optimization problems. The important feature of optimization algorithm is utilization of population. There are many studies analysing the impact of different population sizes [2] on the performance of algorithms. Indeed, increasing the population size has been a common strategy in tuning an optimization algorithm when it does not perform well as expected for a given problem. He and Yao (2002) [3] theoretically showed that large population size helps in reduction of runtime from exponential to polynomial time. Firefly Algorithm (FA) and Teaching Learning Based Optimization (TLBO) techniques have been considered for improving the position accuracy and compared the results with existing Kalman Filter method. TLBO is simple in implementation without tuning parameters whereas FA requires control parameters but the results show that FA outperforms TLBO for GPS precise position accuracy. 2. Kalman Filter The fitness/objective function considered for evaluation in both the optimization techniques is
O f POR PCR Where
(1)
POR is the observed pseudorange and PCR is calculated pseudorange. The function O f is evaluated until the
difference between observed pseudorange and calculated pseudorange becomes zero or minimum value.
PCR ( x sx xrx )2 ( y sy yry )2 ( z sz zrz )2 Where
(2)
x sx , y sy , z sz is satellite coordinates and xrx , yry , zrz are receiver coordinates to be estimated.
Kalman filter [4] is an efficient filter used for state estimation from a series of incomplete and noisy data set. It consists of two phases in 5 basic steps. In first phase, 3 and 4 steps form the basis for prediction or time update. Steps 5 to 7 form correction or measurement-update for the second phase. The five basic steps used in KF are given below Time update (predict)
ˆ x
t
ˆt 1 wt Ft x T
Pt Ft Pt 1Ft Qt
(3) (4)
Measurement-update (correct) T
T
K t Pt H t ( H t Pt H t R) 1
xˆt xˆt Kt ( z t H t xˆt ) Pt ( I Kt H t ) Pt
Where
xt 1 is the state vector to be estimated = [ xrx , yry , zrz ] Ft Transition matrix to transfer state vector from one state to other w System noise P State covariance matrix Q process noise covariance K Kalman gain Ht Measurement matrix R Measurement noise covariance
z Vector of observed values
(5) (6) (7)
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3. Firefly Algorithm Firefly Algorithm (FA) was first developed by Xin-She Yang in 2008 [1, 5], which was based on flashing behaviour of fireflies. To speed up the convergence as iterations proceed, FA can be tuned to control the randomness of parameters. Step 1: Generate firefly initial population randomly using unifrnd function in matlab for given parameters Step 2: Select current best solution by evaluating the fitness function [6]. Step 3: Update the position of firefly based on movement and attraction using following equations (8) 0ed 2
x
t 1 i
Where
x e t
dij 2
i
( x j xi ) t t t
t
t is step size controlling parameter, while t is a vector of gaussian or other distribution.
Step 4: If stopping criteria is satisfied then terminate.
Fig. 1 presents the above steps [6] in the form of a flowchart as shown below
Start
Generate firefly initial population randomly Select current best solution by evaluating the fitness of all fireflies from the objective function Based on attraction and movement update the position
No
Stopping criteria reached?
Yes Optimal result
Fig.1 Flowchart of FA
(9)
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Lavanya Bagadi/ Procedia Science 00Science (2018) 143 000–000 Lavanya Bagadi et al. /Computer Procedia Computer (2018) 892–898
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Table 1. FA parameter values used for computation Parameter np MIt α β γ αd
Description Number of Fireflies Max Number of iterations Mutation Coefficient Attraction Coefficient Base Value Light Absorption Coefficient Mutation Coefficient Damping Ratio
value 20 350 0.2 2 1 0.98
4. TLBO algorithm TLBO operation can be discussed in two phases [7], Teaching Phase and Learning Phase. In the first phase, students (learners) learn from teachers and in second phase, students learn through interaction or discussing with other students. Step 1: Generate randomly an initial population of size N S , where N indicates the size of population, i.e., number of learners and S indicates the dimension of problem i.e., number of subjects offered. The marks scored by distinctive learners in the kth subject are characterized by the kth column of the initial population. p11 p 21 Initial population, P . . p N 1
p12 p 22
.
pN2
.
p1S p 2 S p NS
.
.
(10)
Teaching Phase Step 2: In this phase, each teacher attempts to enhance the mean result of a class in the subject assigned. A highly qualified person, teacher, trains the learner and chosen as the best learner, i.e. identified as the best solution Pbest . The mean value of each column, i.e. the mean value of marks obtained by different students for each subject is calculated as: (11) Dm D1 D2 . . Dm Step 3: The difference between the mean results in a specific subject and the result of corresponding teacher is given by (12) Ddiff rand (0,1)Pbest Tf Dm Where Tf is teaching factor of value 1 or 2. Step 4: Update the existing population by using the equation: Pn P0 Ddiff
(13)
Step 5: Accept elements Pn , if f ( Pn ) f ( P0 ) , otherwise accept P0 . Learning Phase In this phase students learn among themselves through interaction or by discussing with others. A learner may learn new information if the other learners have more knowledge. ' ' ' Step 6: Select two learners, m and n randomly as Pm Pn . Pm and Pn ' are updated values of Pm and Pn respectively. If Pm '' gives better function value accept it at the end of teacher phase. " ' ' ' ' ' Pm Pm rk ( Pm Pn ), if Pm Pn " m
'
'
'
'
P Pm rk ( Pn Pm ), if Pn Pm
'
(14) (15)
Lavanya Bagadi et al. / Procedia Computer Science 143 (2018) 892–898 Lavanya Bagadi / Procedia Computer Science 00 (2018) 000–000
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Unlike FA, TLBO has only two parameters [4] and values are chosen as MIt=40, Maximum number of iterations and np=50, Population size. 5. Results and Discussion In this paper, GPS receiver position error in 3 dimensions is estimated using FA and TLBO and compared with KF. The input data required for implementing the algorithm is collected on 7th March 2016 from the real time dual frequency GPS receiver located at Department of Electronics and Communication Engineering, Andhra University, Visakhapatnam (Latitude 17.730N/Longitude 83.310E), and receiver position error analysis is carried out over a period of 24Hours. The data captured by the receiver is in Receiver Independent Exchange (RINEX) format and it comprises of two files: i) navigation ii) observation data files. The navigation data file comprises of 22 ephemeris data used to compute current satellite position in three dimensions. The observation data file includes all visible satellites at a particular epoch of time and pseudoranges are obtained after correcting for satellite transit time error [8]. The receiver then computes the relative positions between the satellite and receiver based on the information collected from the two files. In this paper, the initial receiver position coordinates is assumed to be (0,0,0) for KF whereas it is chosen randomly for FA and TLBO. The results show the deviation from original position, which reflects the accuracy provided by these techniques and presented in Table 2 to 4 data. Table 2 Comparison of Position error in x- coordinate Receiver position error in x- coordinate [meters] Time TLBO KF FA [Hours] 02.00 66.24816 47.93834 51.46174 04.00 60.45443 49.81224 51.3508 06.00 58.57798 47.88877 51.04332 08.00 54.39499 52.10526 52.04881 10.00 63.07131 48.89236 52.68931 12.00 80.51836 48.65656 51.52528 14.00 58.11651 52.29074 52.93492 16.00 64.03504 53.21291 51.72462 18.00 67.80831 50.28349 58.36029 20.00 65.60448 52.40445 51.79499 22.00 63.38104 49.94254 52.60067 24.00 57.30394 46.53252 52.73997
Table 3 Comparison of Position error in y- coordinate Receiver position error in y- coordinate [meters] Time TLBO KF FA [Hours] 02.00 176.46099 134.7158 133.3004 04.00 139.79821 130.8506 133.6427 06.00 151.43579 130.9277 133.0361 08.00 151.30843 130.0110 133.7941 10.00 152.84302 135.6849 135.0775 12.00 154.37682 133.4049 134.5536 14.00 154.58141 134.1234 133.9595 16.00 147.57552 133.1206 134.0767 18.00 152.16035 128.5886 133.2437 20.00 148.19832 133.4324 134.2277 22.00 141.21756 128.8685 138.7931 24.00 142.71750 130.2463 130.1112
Table 4 Comparison of Position error in z- coordinate Receiver position error in z- coordinate [meters] Time KF FA TLBO [Hours] 02.00 76.69777 45.36232 45. 05515 04.00 51.47047 46.1647 44.98886 06.00 42.96877 31.47469 45.84917 08.00 54.37816 43.24364 45.81734 10.00 46.53391 42.15711 45.71349 12.00 54.96981 45.97469 49.42942 14.00 50.99792 45.43483 44.9967 16.00 52.9261 38.48994 46.54244 18.00 49.43039 42.13732 44.34558 20.00 50.61133 42.32097 46.53361 22.00 48.76521 41.22457 42.24620 24.00 48.39909 37.59788 47.10674 Fig. 2 Comparison of RX position error in X-coordinate
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Lavanya Bagadi/ Procedia Computer Science 00 (2018) 000–000 Lavanya Bagadi et al. / Procedia Computer Science 143 (2018) 892–898
Fig.3. Comparison of RX position error in Y-coordinate
897
Fig.4. Comparison of RX position error in Z-coordinate
Table 2 to 4 results show the position deviation from exact value as error in meters in x, y and z coordinates. From results it is evident that compared to KF both optimization techniques FA and TLBO perform better but FA [9] outperforms all. The computation time taken to converge to optimal solution for FA is 50.377sec and TLBO is 3.132sec. FA has taken more time to compute and also contains reasonable tuning parameters compared to TLBO but the accuracy is more compared to TLBO and KF for GPS precise position applications. Figures 2 to 4 presents the receiver position error analysis in x, y and z coordinates for KF, TLBO and FA techniques. It is clear that TLBO and FA optimization techniques performs better interms of accuracy compared to KF but compared to TLBO even FA is much more better. The deviation in x, y and z co-ordinates for TLBO on an average is 10m,15m and 6m respectively compared to KF whereas FA produces deviation in x, y and z co-ordinates on an average as 14m,17m and 10m respectively compared to KF.
Fig. 5. Firefly algorithm convergence graph
Fig. 6. TLBO algorithm convergence graph
Fig. 5 shows the variation of Best cost with respect to number of iterations for FA and it takes minimum of 350 iterations to satisfy the stopping criteria. Fig. 6 shows the variation of Best cost with respect to number of iterations for TLBO and it takes minimum of 40 iterations to reach convergence. Best cost represents the best fitness function value.
898
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6. Conclusion and Future Scope The experimental results are obtained by implementing these algorithms using Matlab 2016a. This paper shows the performance of optimization techniques to analyse GPS receiver position estimation and compared results with existing KF method and for KF result correctable factors are not taken into consideration. From results it is clear that FA outperforms KF and TLBO interms of accuracy but TLBO is simple in implementation and computation time is 47.245 sec less compared to FA. So, to further enhance the TLBO method a hybrid technique can be proposed which increases accuracy along with computation time. But as the paper concentrates on GPS receiver position error accuracy it is worth in stating that FA has more accuracy compared to TLBO and KF. FA has 4m more accuracy in x and z coordinates and 2m more accuracy in y coordinates as compared to TLBO. In future to enhance the accuracy of results, a hybrid technique can be incorporated by taking combination of any two algorithms. References [1]. Xin-She Yang. (2014) “Nature-Inspired Optimization Algorithms”, Elsevier, 1st edition, 111-127. [2]. Tianshi Chen, Ke Tang, Guoliang Chen, XinYao (2012) “A large population size can be unhelpful in evolutionary algorithms”, Theoretical Computer Science, 436: 54-70. [3]. J. He, X. Yao. (2002) “From an individual to a population: An analysis of the first hitting time of population-based evolutionary algorithms”, IEEE Transactions on Evolutionary Computation, 6(5): 495-511. [4]. R. E. Kalman. (1960) “A new approach to linear filtering and prediction problems,” J. Basic Eng., 82(1): 35–45. [5]. Wang H., Wang W., Sun H., Rahnamayan S. (2016) “Firefly algorithm with random attraction. International Journal of Bio-Inspired Computation. 8(1): 33–41. doi: 10.1504/ijbic.2016.074630, Studies in Computational Intelligence. [6]. Lina Zhang, Liqiang Liu, Xin-She Yang, and Yuntao Dai. (2016) “A Novel Hybrid Firefly Algorithm for Global Optimization” , PLOS One 11(9). [7]. M. Crepinsek, S.H. Liu, L. Mernik. (2012) “ A note on teaching–learning-based optimization algorithm Inform Sci.” 212 : 79-93. [8]. G S Rao. (2010) “Global Navigation Satellite Systems”, McGraw Hill Education Private limited, ISBN (13):978-0-07-070029-1. [9]. Xin-She Yang. (2012) “Multiobjective firefly algorithm for continuous optimization”, Eng. Comput 10.1007/s00366-012-0254-1.
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Procedia Computer Science 143 (2018) 892–898
8th International Conference on Advances in Computing and Communication (ICACC-2018) 8th International Conference on Advances in Computing and Communication (ICACC-2018) 8th International Conference on Advances in Computing and Communication (ICACC-2018) Firefly, Teaching Learning Based Optimization and Kalman Filter MethodsLearning for GPS Based Receiver Position Estimation Firefly, Teaching Optimization and Kalman Filter a* Methods for GPS Receiver Position Estimation Lavanya Bagadi , G Sasibhushana Raob, Ashok Kumar Nc a* of ECE,MVGR(A),Vizianagaram,535005,India Department Lavanya Bagadi , G Sasibhushana Raob, Ashok Kumar Nc Department of ECE,AUCE(A),Andhra University,Visakhapatnam,530003,India. a
b,c
b,c
Abstract
a Department of ECE,MVGR(A),Vizianagaram,535005,India Department of ECE,AUCE(A),Andhra University,Visakhapatnam,530003,India.
Global Positioning System (GPS) is an all-weather worldwide navigation system that provides position, navigation and timing Abstract information. A lot of research has been done in improving the position accuracy but still researchers are working towards the betterment of position accuracy using augmentation methods like system WAAS,that EGNOS, andposition, GAGANnavigation etc. because ever Global Positioning System (GPS)without is an all-weather worldwide navigation provides and of timing increasing demand of research user accuracy. In this Firefly Algorithm (FA)accuracy and Teaching Based (TLBO) information. A lot of has been donepaper in improving the position but stillLearning researchers areOptimization working towards the techniques of have been considered for improving the position accuracy the results the existing KalmanofFilter betterment position accuracy without using augmentation methods and like compared WAAS, EGNOS, andwith GAGAN etc. because ever method. considering KF, correctable factors are Algorithm not included. is simple in implementation without(TLBO) tuning increasingWhile demand of user accuracy. In this paper Firefly (FA)TLBO and Teaching Learning Based Optimization parameters have whereas requires control parameters but the results show FA outperforms forexisting GPS precise position techniques beenFA considered for improving the position accuracy and that compared the resultsTLBO with the Kalman Filter accuracy.While FA hasconsidering 4m more accuracy in terms of x andare z coordinates whereas 2mismore accuracy in y-coordinates compared to method. KF, correctable factors not included. TLBO simple in implementation without tuning TLBO. The resultsFA show that with proper selectionbutofthe control parameters methods yield optimal solutions. Numerical parameters whereas requires control parameters results show thatthese FA outperforms TLBO for GPS precise position results clearly verify significance of terms the proposed The whereas computation time taken by FA and TLBO to compared converge to accuracy. FA has 4mthe more accuracy in of x andalgorithms. z coordinates 2m more accuracy in y-coordinates desired result 50.377sec respectively. TLBO. The isresults showand that3.132sec with proper selection of control parameters these methods yield optimal solutions. Numerical results clearly verify the significance of the proposed algorithms. The computation time taken by FA and TLBO to converge to desired result is 50.377sec and 3.132sec respectively. © 2018 The Authors. Published by Elsevier B.V. © 2018 The Authors. by Elsevier B.V. This is an open accessPublished article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection andAuthors. peer-review underbyresponsibility of the scientific committee of the 8th International Conference on Advances in © 2018 The Published Elsevier B.V. Selection and peer-review under responsibility of the scientific committee of the 8th International Conference on Advances in Computing and Communication (ICACC-2018). This is an open article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Computing and access Communication (ICACC-2018). Selection and peer-review under responsibility of the scientific committee of the 8th International Conference on Advances in Keywords: kalman Filter; Position Estimation; TLBO. ComputingFirefly; and Communication (ICACC-2018). Keywords: Firefly; kalman Filter; Position Estimation; TLBO. * Corresponding author. Tel.: +919885512625 E-mail address: [email protected] * Corresponding author. Tel.: +919885512625 E-mail address: [email protected]
1877-0509 © 2018 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection peer-review under responsibility of the scientific 1877-0509and © 2018 The Authors. Published by Elsevier B.V. committee of the 8th International Conference on Advances in Computing and Communication (ICACC-2018). This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection and peer-review under responsibility of the scientific committee of the 8th International Conference on Advances in Computing and Communication (ICACC-2018). 1877-0509 © 2018 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection and peer-review under responsibility of the scientific committee of the 8th International Conference on Advances in Computing and Communication (ICACC-2018). 10.1016/j.procs.2018.10.365
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Lavanya Bagadi et al. / Procedia Computer Science 143 (2018) 892–898 Lavanya Bagadi/ Procedia Computer Science 00 (2018) 000–000
893
1. Introduction Metaheuristic algorithms have gained eminence and form a vital part of global optimization algorithms, soft computing and computational intelligence. The popularity of metaheuristic algorithms [1] is partly due to their ability of dealing with nonlinear global optimization problems. The important feature of optimization algorithm is utilization of population. There are many studies analysing the impact of different population sizes [2] on the performance of algorithms. Indeed, increasing the population size has been a common strategy in tuning an optimization algorithm when it does not perform well as expected for a given problem. He and Yao (2002) [3] theoretically showed that large population size helps in reduction of runtime from exponential to polynomial time. Firefly Algorithm (FA) and Teaching Learning Based Optimization (TLBO) techniques have been considered for improving the position accuracy and compared the results with existing Kalman Filter method. TLBO is simple in implementation without tuning parameters whereas FA requires control parameters but the results show that FA outperforms TLBO for GPS precise position accuracy. 2. Kalman Filter The fitness/objective function considered for evaluation in both the optimization techniques is
O f POR PCR Where
(1)
POR is the observed pseudorange and PCR is calculated pseudorange. The function O f is evaluated until the
difference between observed pseudorange and calculated pseudorange becomes zero or minimum value.
PCR ( x sx xrx )2 ( y sy yry )2 ( z sz zrz )2 Where
(2)
x sx , y sy , z sz is satellite coordinates and xrx , yry , zrz are receiver coordinates to be estimated.
Kalman filter [4] is an efficient filter used for state estimation from a series of incomplete and noisy data set. It consists of two phases in 5 basic steps. In first phase, 3 and 4 steps form the basis for prediction or time update. Steps 5 to 7 form correction or measurement-update for the second phase. The five basic steps used in KF are given below Time update (predict)
ˆ x
t
ˆt 1 wt Ft x T
Pt Ft Pt 1Ft Qt
(3) (4)
Measurement-update (correct) T
T
K t Pt H t ( H t Pt H t R) 1
xˆt xˆt Kt ( z t H t xˆt ) Pt ( I Kt H t ) Pt
Where
xt 1 is the state vector to be estimated = [ xrx , yry , zrz ] Ft Transition matrix to transfer state vector from one state to other w System noise P State covariance matrix Q process noise covariance K Kalman gain Ht Measurement matrix R Measurement noise covariance
z Vector of observed values
(5) (6) (7)
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3
3. Firefly Algorithm Firefly Algorithm (FA) was first developed by Xin-She Yang in 2008 [1, 5], which was based on flashing behaviour of fireflies. To speed up the convergence as iterations proceed, FA can be tuned to control the randomness of parameters. Step 1: Generate firefly initial population randomly using unifrnd function in matlab for given parameters Step 2: Select current best solution by evaluating the fitness function [6]. Step 3: Update the position of firefly based on movement and attraction using following equations (8) 0ed 2
x
t 1 i
Where
x e t
dij 2
i
( x j xi ) t t t
t
t is step size controlling parameter, while t is a vector of gaussian or other distribution.
Step 4: If stopping criteria is satisfied then terminate.
Fig. 1 presents the above steps [6] in the form of a flowchart as shown below
Start
Generate firefly initial population randomly Select current best solution by evaluating the fitness of all fireflies from the objective function Based on attraction and movement update the position
No
Stopping criteria reached?
Yes Optimal result
Fig.1 Flowchart of FA
(9)
4
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Table 1. FA parameter values used for computation Parameter np MIt α β γ αd
Description Number of Fireflies Max Number of iterations Mutation Coefficient Attraction Coefficient Base Value Light Absorption Coefficient Mutation Coefficient Damping Ratio
value 20 350 0.2 2 1 0.98
4. TLBO algorithm TLBO operation can be discussed in two phases [7], Teaching Phase and Learning Phase. In the first phase, students (learners) learn from teachers and in second phase, students learn through interaction or discussing with other students. Step 1: Generate randomly an initial population of size N S , where N indicates the size of population, i.e., number of learners and S indicates the dimension of problem i.e., number of subjects offered. The marks scored by distinctive learners in the kth subject are characterized by the kth column of the initial population. p11 p 21 Initial population, P . . p N 1
p12 p 22
.
pN2
.
p1S p 2 S p NS
.
.
(10)
Teaching Phase Step 2: In this phase, each teacher attempts to enhance the mean result of a class in the subject assigned. A highly qualified person, teacher, trains the learner and chosen as the best learner, i.e. identified as the best solution Pbest . The mean value of each column, i.e. the mean value of marks obtained by different students for each subject is calculated as: (11) Dm D1 D2 . . Dm Step 3: The difference between the mean results in a specific subject and the result of corresponding teacher is given by (12) Ddiff rand (0,1)Pbest Tf Dm Where Tf is teaching factor of value 1 or 2. Step 4: Update the existing population by using the equation: Pn P0 Ddiff
(13)
Step 5: Accept elements Pn , if f ( Pn ) f ( P0 ) , otherwise accept P0 . Learning Phase In this phase students learn among themselves through interaction or by discussing with others. A learner may learn new information if the other learners have more knowledge. ' ' ' Step 6: Select two learners, m and n randomly as Pm Pn . Pm and Pn ' are updated values of Pm and Pn respectively. If Pm '' gives better function value accept it at the end of teacher phase. " ' ' ' ' ' Pm Pm rk ( Pm Pn ), if Pm Pn " m
'
'
'
'
P Pm rk ( Pn Pm ), if Pn Pm
'
(14) (15)
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Unlike FA, TLBO has only two parameters [4] and values are chosen as MIt=40, Maximum number of iterations and np=50, Population size. 5. Results and Discussion In this paper, GPS receiver position error in 3 dimensions is estimated using FA and TLBO and compared with KF. The input data required for implementing the algorithm is collected on 7th March 2016 from the real time dual frequency GPS receiver located at Department of Electronics and Communication Engineering, Andhra University, Visakhapatnam (Latitude 17.730N/Longitude 83.310E), and receiver position error analysis is carried out over a period of 24Hours. The data captured by the receiver is in Receiver Independent Exchange (RINEX) format and it comprises of two files: i) navigation ii) observation data files. The navigation data file comprises of 22 ephemeris data used to compute current satellite position in three dimensions. The observation data file includes all visible satellites at a particular epoch of time and pseudoranges are obtained after correcting for satellite transit time error [8]. The receiver then computes the relative positions between the satellite and receiver based on the information collected from the two files. In this paper, the initial receiver position coordinates is assumed to be (0,0,0) for KF whereas it is chosen randomly for FA and TLBO. The results show the deviation from original position, which reflects the accuracy provided by these techniques and presented in Table 2 to 4 data. Table 2 Comparison of Position error in x- coordinate Receiver position error in x- coordinate [meters] Time TLBO KF FA [Hours] 02.00 66.24816 47.93834 51.46174 04.00 60.45443 49.81224 51.3508 06.00 58.57798 47.88877 51.04332 08.00 54.39499 52.10526 52.04881 10.00 63.07131 48.89236 52.68931 12.00 80.51836 48.65656 51.52528 14.00 58.11651 52.29074 52.93492 16.00 64.03504 53.21291 51.72462 18.00 67.80831 50.28349 58.36029 20.00 65.60448 52.40445 51.79499 22.00 63.38104 49.94254 52.60067 24.00 57.30394 46.53252 52.73997
Table 3 Comparison of Position error in y- coordinate Receiver position error in y- coordinate [meters] Time TLBO KF FA [Hours] 02.00 176.46099 134.7158 133.3004 04.00 139.79821 130.8506 133.6427 06.00 151.43579 130.9277 133.0361 08.00 151.30843 130.0110 133.7941 10.00 152.84302 135.6849 135.0775 12.00 154.37682 133.4049 134.5536 14.00 154.58141 134.1234 133.9595 16.00 147.57552 133.1206 134.0767 18.00 152.16035 128.5886 133.2437 20.00 148.19832 133.4324 134.2277 22.00 141.21756 128.8685 138.7931 24.00 142.71750 130.2463 130.1112
Table 4 Comparison of Position error in z- coordinate Receiver position error in z- coordinate [meters] Time KF FA TLBO [Hours] 02.00 76.69777 45.36232 45. 05515 04.00 51.47047 46.1647 44.98886 06.00 42.96877 31.47469 45.84917 08.00 54.37816 43.24364 45.81734 10.00 46.53391 42.15711 45.71349 12.00 54.96981 45.97469 49.42942 14.00 50.99792 45.43483 44.9967 16.00 52.9261 38.48994 46.54244 18.00 49.43039 42.13732 44.34558 20.00 50.61133 42.32097 46.53361 22.00 48.76521 41.22457 42.24620 24.00 48.39909 37.59788 47.10674 Fig. 2 Comparison of RX position error in X-coordinate
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Lavanya Bagadi/ Procedia Computer Science 00 (2018) 000–000 Lavanya Bagadi et al. / Procedia Computer Science 143 (2018) 892–898
Fig.3. Comparison of RX position error in Y-coordinate
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Fig.4. Comparison of RX position error in Z-coordinate
Table 2 to 4 results show the position deviation from exact value as error in meters in x, y and z coordinates. From results it is evident that compared to KF both optimization techniques FA and TLBO perform better but FA [9] outperforms all. The computation time taken to converge to optimal solution for FA is 50.377sec and TLBO is 3.132sec. FA has taken more time to compute and also contains reasonable tuning parameters compared to TLBO but the accuracy is more compared to TLBO and KF for GPS precise position applications. Figures 2 to 4 presents the receiver position error analysis in x, y and z coordinates for KF, TLBO and FA techniques. It is clear that TLBO and FA optimization techniques performs better interms of accuracy compared to KF but compared to TLBO even FA is much more better. The deviation in x, y and z co-ordinates for TLBO on an average is 10m,15m and 6m respectively compared to KF whereas FA produces deviation in x, y and z co-ordinates on an average as 14m,17m and 10m respectively compared to KF.
Fig. 5. Firefly algorithm convergence graph
Fig. 6. TLBO algorithm convergence graph
Fig. 5 shows the variation of Best cost with respect to number of iterations for FA and it takes minimum of 350 iterations to satisfy the stopping criteria. Fig. 6 shows the variation of Best cost with respect to number of iterations for TLBO and it takes minimum of 40 iterations to reach convergence. Best cost represents the best fitness function value.
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Lavanya Bagadi et al. / Procedia Computer Science 143 (2018) 892–898 Lavanya Bagadi / Procedia Computer Science 00 (2018) 000–000
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6. Conclusion and Future Scope The experimental results are obtained by implementing these algorithms using Matlab 2016a. This paper shows the performance of optimization techniques to analyse GPS receiver position estimation and compared results with existing KF method and for KF result correctable factors are not taken into consideration. From results it is clear that FA outperforms KF and TLBO interms of accuracy but TLBO is simple in implementation and computation time is 47.245 sec less compared to FA. So, to further enhance the TLBO method a hybrid technique can be proposed which increases accuracy along with computation time. But as the paper concentrates on GPS receiver position error accuracy it is worth in stating that FA has more accuracy compared to TLBO and KF. FA has 4m more accuracy in x and z coordinates and 2m more accuracy in y coordinates as compared to TLBO. In future to enhance the accuracy of results, a hybrid technique can be incorporated by taking combination of any two algorithms. References [1]. Xin-She Yang. (2014) “Nature-Inspired Optimization Algorithms”, Elsevier, 1st edition, 111-127. [2]. Tianshi Chen, Ke Tang, Guoliang Chen, XinYao (2012) “A large population size can be unhelpful in evolutionary algorithms”, Theoretical Computer Science, 436: 54-70. [3]. J. He, X. Yao. (2002) “From an individual to a population: An analysis of the first hitting time of population-based evolutionary algorithms”, IEEE Transactions on Evolutionary Computation, 6(5): 495-511. [4]. R. E. Kalman. (1960) “A new approach to linear filtering and prediction problems,” J. Basic Eng., 82(1): 35–45. [5]. Wang H., Wang W., Sun H., Rahnamayan S. (2016) “Firefly algorithm with random attraction. International Journal of Bio-Inspired Computation. 8(1): 33–41. doi: 10.1504/ijbic.2016.074630, Studies in Computational Intelligence. [6]. Lina Zhang, Liqiang Liu, Xin-She Yang, and Yuntao Dai. (2016) “A Novel Hybrid Firefly Algorithm for Global Optimization” , PLOS One 11(9). [7]. M. Crepinsek, S.H. Liu, L. Mernik. (2012) “ A note on teaching–learning-based optimization algorithm Inform Sci.” 212 : 79-93. [8]. G S Rao. (2010) “Global Navigation Satellite Systems”, McGraw Hill Education Private limited, ISBN (13):978-0-07-070029-1. [9]. Xin-She Yang. (2012) “Multiobjective firefly algorithm for continuous optimization”, Eng. Comput 10.1007/s00366-012-0254-1.