Wavelet Decomposition for Denoising GPS/INS Outputs in Vehicular Navigation System

Transcription

1 Wavelet Decompoition for Denoiing GPS/INS Output in Vehicular Navigation Syem SALAM ISMAEEL 1 and AYMAN AL-KHAZRAJI 2 Department of Computer Science Ryeron Univerity 35 Victoria St, Toronto, ON M5B 2K3 CANADA alam.imaeel@ryeron.ca 2 Department of Electrical and Computer Engineering Caledonian College of Engineering CPO Seeb 111 OMAN auman.alkhazraji@caledonian.edu.om Abract: - Thi paper aim to bring together Global Poitioning Syem (GPS) and Inertial Navigation Syem (INS) integration approach for fuing uing wavelet analyi. The wavelet i employed to compare the enor output at different reolution level and for everal type of error in INS and GPS a well a to mooth and predict the INS error. Simulation reult how that, the type of filter which are ued in multireolution algorithm have large effect in reducing the INS/GPS error in poition and epecially in velocity component. Furthermore, oft threholding technique i introduced in multi-reolution algorithm, where ix rule are propoed to elect the threhold value. Real GPS and INS meaurement were ued in the Matlab imulation to illurate the effectivene of the propoed method. Key-Word: - Vehicular navigation, Inertial navigation yem, GPS, Wavelet Analyi, Denoiing. 1 Introduction Strapdown Inertial Navigation Syem (SINS) technologie are baed on the principle of integrating pecific force and rate meaured by accelerometer and rate gyro of an Inertial Meaurement Unit (IMU) fixed to the navigating body. Given the initial condition of poition, velocity and attitude, accurate real time integration of IMU output will produce poition and attitude information in ome given navigation coordinate yem [1]-[2]. On the other hand, the GPS relie on the technique of comparing ignal from orbiting atellite to calculate poition (and poibly attitude) at regular time interval [3]. Neverthele, being dependent on the atellite ignal make GPS le reliable than elf-contained INS due to the poibility of drop-out or jamming. Typically, the dynamic error model for a terrerial INS algorithm require three poition error, three velocity error and three attitude error in an INS (i.e. the yem error ate). Thee error are alo augmented by ome enor error ate uch a accelerometer biae and gyrocope drift, which are modeled a ochaic procee. In fact, there are everal random error aociated with each inertial enor. Therefore, it i uually difficult to et a certain ochaic model for each inertial enor capable of working efficiently in all environment and reflecting the long-term behavior of enor error [4]. Hence the introduction of wavelet algorithm to perform the elf-following of the vehicle under all-condition maneuvering will be not only beneficial but alo required. In thi work, the effect of wavelet filter type on the INS/GPS error will be dicued and implemented on all INS and GPS component. Threholding technique with many rule for electing the threhold value and it effect on the performance of INS/GPS error will be alo dicued and examined. Figure 1 i a general block diagram for the wavelet multi-reolution denoiing for GPS/INS output. 2 Propoed Method Thi ection decribe the algorithm of Multi- Reolution Analyi (MRA), the election of wavelet decompoition level a well a wavelet filter and the calculation of threhold value. E-ISSN: Volume 16, 217

2 Thi operation i equivalent to low pa filtering. Step2: calculate the approximation uing the coefficient obtained in ep1 above a following [5]: Fig. 1. Main block diagram of wavelet decompoition uing GPS/ INS output Multi-Reolution Analyi Algorithm (MRA) In order to determine the INS/GPS error that can be ued to model the INS poition and velocity error, a wavelet multi-level decompoition mu be performed for each component of the INS and GPS output ignal. The following ep decribe the mathematical wavelet decompoition procedure: Step1: For each one of INS and GPS output ignal, calculate the approximation coefficient at Sth reolution level uing [4]: ( /2) Ck, 2 x( n) (2 n k) (1) n Where (n) i the wavelet function (the bai function utilized in the wavelet tranform) and ( 2 n k) are caled and hifted verion of (n) baed on the value of (caling coefficient) and k (hifting coefficient). C,k are the correponding wavelet coefficient. x (n) i the original ignal. x ( n) C ( n) (2), k, k k th Step3: Calculate the detail coefficient at reolution level uing: d, k x( n), k ( n) (3) n Thi operation i equivalent to high pa filtering. Step4: Find the detail uing the reult of ep3 and the following equation: g ( n) d ( n) (4), k, k k Step5: Return to ep one and continue the wavelet decompoition proce until appropriate level of decompoition (LOD) i reached which i different from one IMU to another (the appropriate LOD election will be decribed in the next ub ection). It mu be noted that the next wavelet decompoition proce mu be performed on the approximation obtained from the previou wavelet decompoition proce and o on. Step6: Denoiing the detail of the INS and GPS ignal by applying the threholding technique which i decribed later. Step7: Compare the INS and GPS poition and velocity component at everal reolution level (by ubtracting the wavelet coefficient of each of the GPS output from the correponding wavelet coefficient of each of the INS output) a follow: A A A D D D E 1 GPS1 INS1 E 1 GPS1 INS1 A A A D D D E 2 GPS 2 INS 2 E 2 GPS 2 INS 2 nd nd nd nd nd nd A A A D D D E 3 GPS 3 INS 3 E 3 GPS 3 INS 3 rd rd rd rd rd rd A A A D D D E ( 1) GPS ( 1) INS ( 1) E ( 1) GPS ( 1) INS ( 1) th th th th th th A A A D D D th th th th th th E GPS INS E GPS INS A GPS, A INS are GPS and INS approximation repectively. D GPS, D INS are GPS and INS detail repectively. A E Difference between the GPS and INS approximation for different level. D E Difference between the GPS and INS detail for different level. (5) E-ISSN: Volume 16, 217

3 Step8: Reconruct the INS/GPS error ignal from the wavelet coefficient difference found in ep7 a follow: D 1 E1 D nd nd 2 E 2 1 D rd rd nd 3 E3 2 A th th th E( 1) ( 1) 1 LOD. 2 nd (6) Difference between the GPS and INS detail of the fir Summation of the difference between the GPS and INS detail of the econd LOD with the difference of previou LOD ( 1 ). Summation of difference between the GPS and INS th approximation at th level and the difference between the GPS and INS detail of the previou LOD ( ). S th ( 1) 2.2. Selection of the Appropriate Wavelet Level of Decompoition (LOD) To elect an appropriate LOD in thi cae, everal decompoition level are applied and the Standard Deviation (STD) i computed for each approximation difference component (INS/GPS error) and compared with the real INS error. The proper LOD will be the one having the minimum difference between thee two error. Kinematic inertial (Real Data) denoiing the output of the enor contain both effect of the actual vehicle motion dynamic and the enor noie a well a ome other undeirable effect (e.g. vehicle engine vibration). Therefore, the criterion for the election of the appropriate LOD will be different from the atic cae. Before applying the wavelet multi-reolution analyi on kinematic SINS, it hould be enured that the decompoition or denoiing proce doe not remove any actual motion information [5] Threhold Algorithm Analyi Threholding operation are applied on the coefficient of the wavelet and wavelet packet tranform, and generally can be claified into Hard-threholding and Soft-threholding a decribed in [6]. The choice of threhold i crucial to the quality of the denoiing proce and hould be made carefully. In threholding proce, the coefficient maller than threhold value (Thrv) are judged negligible or noie. In thi work ix method are ued to elect the value of Thrv. Thee method are: - Fir Method: One poibility of electing the threhold by eimating the andard deviation x of the noie at each cale. We take into account that threhold value have to be different on each cale. The threhold in thi cae can be calculated a [7]: Thrv 2 (7) x 2 : Noie Power for noiy ignal. x : deviation for the detail coefficient. - Second Method: here, we elect the threhold value Thrv by eimating the andard deviation x of the noie at each cale. We take into account that the threhold value i different on each cale. In thi cae, the value can be calculated a [8]-[9]: 2 2ln Thrv 2 x N (8) x : Standard deviation for each detail coefficient N : the equence length. - Third Method: Selection uing principle of ein' unbiaed rik eimate (SURE) (MatLab code "rigrure"). - Fourth Method: Selection uing fixed form threhold (MatLab code "qtwolog"). - Fifth Method: Selection uing a mixture of the third and fourth election rule (MatLab code "heurure"). - Sixth Method: Selection uing minimax principle (MatLab code "minimaxi") Selection of the Appropriate Filter The wavelet tranform ha a flexible feature of uing a variety of filter that differ by their coefficient. In thi work, all type of the wavelet filter will be applied to original GPS/INS in order to how the be poible filter for each component of poition and velocity. 3 Reult and analyi In thi ection we preent and dicu the reult obtain uing the propoed method. E-ISSN: Volume 16, 217

4 Table 1. Multi-reolution algorithm applied to obtain INS/GPS andard deviation error. Type of Component Direction Eimated INS /GPS error LOD X-axi Poition (m) Y-axi Z-axi Be INS with original GPS Wor INS with original GPS Table 1 illurate the reult of applying MRA algorithm to original GPS meaurement. It reflect alo, two cae of INS (be and wor cae) elected from the eight type of IMU error decribed in [6]. The purpoe of thi i to inveigate and udy variou IMU pecification where the accuracy level of IMU can be categorized a high (rategic grade), medium (navigation grade), and low (tactical grade) at the end of the vehicle' journey. We have concluded that, it i unneceary to increae the order of LOD becaue the feature of the INS/GPS-error will diappear. In other word, it can't be ued to model the INS error becaue the reulting error (INS/GPS error) will not equal the deired INS-error. On the other hand it mu be mentioned that the main GPS error can be denoied by wavelet unlike the INS error where ome of the error can be eliminated by wavelet denoiing (optimal low pa filtering). Such error i called hort-term error while the other part of the INS error i called long-term error. The latter i reduced by GPS/INS integration, which i accomplihed by the multi-reolution algorithm decribed previouly. The output of the multi-reolution for the GPS and INS i ubtracted to obtain the INS/GPS error which can t be eliminated by the denoiing algorithm. After uing all type of the wavelet filter, Table 2 how the be poible filter conidered for each component of poition and velocity uing original GPS meaurement (with be and wor INS ). In thi work, oft threholding technique wa adopted to remove ome of the detail part noie while keeping the original feature of the ignal and improving the ignal to noie ratio (SNR). In Table 3, we illurate the SNR and Root Mean Square Error (RMSE) for the two cae of INS (wor and be) and GPS before applying threholding. To compare with the reult obtained after uing threholding, we conider Table 4. It illurate the SNR and RMSE after applying threholding uing the ix method mentioned previouly for the INS and GPS poition and velocity component where the labeled value in Table II which repreent be reult, ued to determine the performance of threholding election rule. Table 2. Reult of uing different type of wavelet filter for be and wor INS Standard Deviation for 1 LOD of Filter (m/) Poition (m) (m/) Poition (m) North Ea Down X-axi Y-axi Z-axi North Ea Down (m/) X-axi Y-axi Z-axi North Ea Down Be Db4 Db9 Db6 Bior5.5 Bior2.2 Coif2 Wor Db1 Bior2.2 Db4 Bior5.5 Bior2.2 Coif2 E-ISSN: Volume 16, 217

5 The Root Mean Square Error (RMSE) and Signal to Noie Ratio (SNR) can be evaluated uing equation (9) and (1), repectively [5]: - Root Mean Square Error (RMSE). 1 1 N N n 2 RMSE Xˆ X ( n) (9) - Signal to Noie Ratio (SNR) in decibel. SNR( X, Xˆ ) 1Log N 1 n 1 N 1 2 n X( n) 2 Xˆ X ( n) (1) where X ˆ ( n ) : the proceed ignal after removing noie. X (n) : the ignal without any noie. From thee reult it i conclude that: 1. The fourth method of rule election i better for all the type of poition component. 2. The fir or third election rule i very efficient for all type of velocity component. 3. The effect of uing optimum election rule to pecify threhold value i very important for poition and velocity. Moreover, it ha a great impact on velocity denoiing performance. Poition Table 3. Performance comparion before uing threholding technique Type of Be INS Wor INS Original GPS Newton GPS Spline GPS X-axi SNR (db) RMSE (m) e Y-axi SNR (db) RMSE (m) e Z-axi SNR (db) RMSE (m) North SNR (db) RMSE (m/) 4.621e Ea SNR (db) RMSE (m/) Down SNR (db) RMSE (m/) 9.495e D A T A Be cae of INS Wor cae of INS Original of GPS Threhold election Method Table 4. Performance comparion after uing threholding technique Poition X-axi Y-axi Z-axi North Ea Down SNR RMSE SNR RMSE SNR RMSE SNR RMSE SNR RMSE SNR RMSE Fir e e e-13 Second Third e e e-13 Fourth Fifth Sixth Fir e e e-12 Second Third e e e-12 Fourth Fifth Sixth Fir e Second Third Fourth Fifth Sixth E-ISSN: Volume 16, 217

6 Alo, we have applied many level of decompoition and it wa found that the appropriate LOD varie for each component of poition and velocity. Thi depend on the INS/GPS error, which i nearly equal to the real INS-error. Figure 2 how repectively the ignal of error in X, Y and Z before (left column) and after (right coloumn) threholding, for wor INS and original GPS. The correponding type of filter to the lowe andard deviation of INS/GPS error value i the perfect filter to be ued. It hould be mentioned that all calculation in thi ection we performed for fir level of decompoition. Our objective i to chooe the be filter to be ued for each component of poition and velocity 4 Concluion The following point ummarize the main concluion of thi paper: 1. Wavelet analyi wa beneficial in filtering out ome of the noie component and diurbance that may exi at the INS and GPS output. 2. Wavelet MRA algorithm provide the advantage of comparing the INS and GPS poition and velocity component at different level of reolution Error in X-axe (m) Error in X-axe (m) Time (1/1 ec) Time (1/1 ec) Error in Y-axe (m) Error in Y-axe (m) Time (1/1 ec) Time(1/1 ec) Error in Z-axe (m) Error in Z-axe (m) Time (1/1 ec) Time (1/1 ec) Fig.2. Comparion between INS and poition, repectively, in X-axi, Y-axi, and Z-axi for wor INS and original GPS before (left) and after (right) uing threholding. E-ISSN: Volume 16, 217

7 Reference [1] H. Kunpeng, H. Jitao, and S. Yuping, "A Novel redundant inertial meaurement unit and calibration algorithm," in Optoelectronic and Microelectronic (ICOM), 213 International Conference on, 213, pp [2] Q. Guo, O. Bebek, M. Cavuoglu, C. Marangelo, and D. Young, "A peronal navigation yem uing MEMS-baed highdenity ground reaction enor array and inertial meaurement unit," in Solid-State Senor, Actuator and Microyem (TRANSDUCERS), 215 Tranducer th International Conference on, 215, pp [3] A. Leick, L. Rapoport, and D. Tatarnikov, GPS atellite urveying: John Wiley & Son, 215. [4] Y. Gan, L. Sui, J. Wu, B. Wang, Q. Zhang, and G. Xiao, "An EMD threhold de-noiing method for inertial enor," Meaurement, vol. 49, pp , 214. [5] N. Anari and A. Gupta, "Signal-matched wavelet deign via lifting uing optimization technique," in Digital Signal Proceing (DSP), 215 IEEE International Conference on, 215, pp [6] P. Kihore, A. Sary, A. Kartheek, and S. H. Mahatha, "Block baed threholding in wavelet domain for denoiing ultraound medical image," in Signal Proceing And Communication Engineering Syem (SPACES), 215 International Conference on, 215, pp [7] A. Wu, C. Shen, X. Sun, H. Gong, Y. Wei, J. Feng, et al., "An improved wavelet ignal denoiing method for TWT," in Vacuum Electronic Conference (IVEC), 215 IEEE International, 215, pp [8] R. Kokila, S. Gopinathan, and P. Thangavel, "Wavelet and FFT Baed Image Denoiing Uing Non-linear Filter," International Journal of Electrical and Computer Engineering (IJECE), vol. 5, 215. [9] S. Bhargava and A. Somkuwar, "Evaluation of Noie Excluion of Medical Image Uing Hybridization of Particle Swarm Optimization and Bivariate Shrinkage Method," International Journal of Electrical and Computer Engineering (IJECE), vol. 5, 215. E-ISSN: Volume 16, 217