THE WHALE ALGORITHM OPTIMIZED SUPPORT VECTOR MACHINE FOR CHANNEL QUALITY CONTROL OF GNSS VECTOR TRACKING LOOP

THE WHALE ALGORITHM OPTIMIZED SUPPORT VECTOR MACHINE FOR CHANNEL QUALITY CONTROL

OF GNSS VECTOR TRACKING LOOP

Hui Chang Jiang Shuai Chen* Yu Ming Bo Chao Chen Wang Yi Ping Wang Chen Zhao

School of Automation

Nanjing University of Science and Technology Nanjing, China 210094, P.R.C.

Key Words: the whale optimization algorithm, support vector machine, vector tracking loop, local filter, global navigation satellite system (GNSS).

ABSTRACT

For the Global Navigation Satellite System (GNSS) Vector Tracking Loop (VTL), the primary drawback is that the presence of low-quality signals or even a fault in one channel (signal blockage) will affect all channels, and possibly lead to receiver instability or loss of lock on all available satellites. Motivated by this problem, this paper introduced a Whale Algorithm optimized Support Vector Machine (WA-SVM) to monitor the running state of the vector tracking loop channels. The WA was employed to optimize the parameters of the SVM for higher accuracy of classification. In this method, a type of sub-filter was designed for each channel, and the innovative sequences from the sub-filter were employed as the input vector of the WA-SVM. The output was the state of the corresponding channel (negative: faulty and positive: nor- mal). The state variables of each local filter corresponding to the channel were the pseudo-range error and pseudo-range rate error, the measurement information were the code loop discriminator outputs and the frequency discriminator out- puts. A trajectory with random pseudo-range and pseudo-range rate interfer- ence and low-quality signal was generated by a GPS signal simulator to validate the effectiveness of the method. The results demonstrated the WA-SVM me- thod could quickly and effectively detect channel abnormality, which could keep the vector tracking loop working well.

I. INTRODUCTION

The Vector Tracking Loop (VTL) based on a Vector Delay Lock Loop (VDLL) and a Vector Frequency Lock Loop (VFLL) proposed by Spilker [1] is a promising architecture for next generation Global Navigation Satellite System (GNSS) re- ceivers, and many researchers have been studying the VDLL/

VFLL vector tracking loop. A VDLL was first proposed [2]

for the GPS receivers to provide better performance over tra- ditional scalar-based receiver for weak GPS signals. Lashley

implemented a vector delay/frequency lock loop and analyzed the thermal noise performance of the VDLL and VFLL using the rule of thumb tracking thresholds [3]. Lashley also ope- rated a valid comparison of vector and scalar tracking loops, the improvements in signal tracking of VTL were quantified using covariance analysis and Monte Carlo simulations.

The VTL showed an improvement of tracking threshold of 6.2 dB with an eleven satellites and an improvement of 2.4 dB with a five satellites [4]. Furtherly, Lashley analyzed a VTL algorithm based on discriminator and explored the ability

*Corresponding author: Shuai Chen, e-mail: c1492@163.com

of the vector tracking algorithm to track weak GPS signals.

The results showed the VTL operated at a carrier to noise power density ratio of 19dB/Hz through 2 g, 4 g, and 8 g co- ordinates turns [5]. Pany described a VDLL and VFLL GPS receiver and employed the receiver to analyze signal power of GPS C/A code with two different C/N0 estimators [6].

It allowed signal power estimation even below 10 dB/Hz for GPS C/A code signals. JH Won analyzed different signal fad- ing influence on the VTL and defined a Geometry-related Range Accuracy Coefficient; the details of the interaction between the VTL channels were presented [7]. KH Kim also proposed an adaptive VTL for low-quality GPS signals [8]. Jafarmia also assessed the performance of the VTL in detection and mitigation of spoofing attacks [9-10]. SJ Ko assessed the performance of the VTL under radio frequency interference environments [11].

In general, the VTL exploits the inherent relation between signal tacking and navigation state estimation, which combines signal tracking and position/velocity estimation into one al- gorithm and processes received signals in aggregate instead of separately. In this way, the VTL can process signals of lower carrier-to-noise power density ratio (C/N0) and has increased immunity to interference and jamming compared with the Scalar Tracking Loops (STL). Meanwhile, VTL has the ability to bridge signal outages and immediately re- acquire the blocked signals [12]. Under this condition, all the tracking channels are dependent on each other. The low- quality signals or even a fault in a channel may lead to the na- vigation filter to be abnormal and a fault detection of the channel is necessary for improving the robustness of VTL [13].

The following paper is divided into three sections. Sec- tion 2 introduces the mechanism of the designed VTL includ- ing the details of the WA-SVM input vectors setting up.

Section 3 shows the details of the WA-SVM and the channel quality control. Section 4 is the simulation, the results and the analysis.

II. VECTOR TRACKING LOOP BASED ON SUB-FILTERS

Fig. 1 shows the structure of the sub-filters based fault tolerant VTL. Each channel has a new designed sub-filter followed by the WA-SVM detector. The state variables of the local filter are pseudo-range and pseudo-range rate, the observation vector are composed of the outputs of code and carrier discriminators. Then, the state vector of each sub- filter is fed forward to the navigation master filter to compose

Fig. 1 The structure of the sub-filter based fault tolerant VTL

the observation vector of the navigation filter. The presence of low-quality signal or a pseudo-range interference or pseudo- range rate interference in a channel will firstly lead to the abnormality of the corresponding channel’s sub-filter and the following WA-SVM is designed to detect the abnormality.

The WA-SVM will output two types of labels: negative and positive. The negative represents the abnormality and the positive represents the normality. Then, the label is employed as the judgment indicator to help determine whether the chan- nel is isolated or included in the navigation master filter.

1. Details of the Designed Sub-Filter

The sub-filter is a Kalman filter, the state and measure- ment equations are presented as Eqs. (1)-(4).

State equation of the sub-filter:

1 1

0 T 0 1

j j

k k

j j

k k

v

v (1)

where, _k ^j ₁ and _k ^j ₁ is the pseudo-range error and the pseudo-range rate error of channel j at the (k 1) th time epoch. _k ^j and _k ^j is the pseudo-range error and the pseudo-range rate error of channel j at the (k 1) th time epoch. v , v is the model errors of pseudo-range and pseudo-range rate of channel j. The variable T denotes the integration time.

Measurement equation of the sub-filter:

, 1 1

, 1 1

10 01

j j

code k k

j j

carrier k k

z

z (2)

where, z _{code k} ^j _{, 1} is the scaled output of the code discrimi-

nator of the channel j at the (k 1) th time epoch; z _{carrier k} ^j _{, 1}

is the scaled output of the carrier discriminator of the chan-

nel j at the (k 1) th time epoch; , is the measurement

error noise of the pseudo-range and pseudo-range rate of

channel j.

, 1

1 2

j

code k code

z E L

E L (3)

2 2

E E

E I Q (4)

2 2

L L

L I Q (5)

Where, I E and I E represent the in-phase early and late cor- relator outputs during (k 1) th time epoch; and the Q E and Q L represent the quadrature early and late correlator outputs during (k 1) th epoch. code is the length of a C/A code chip.

, 1 2 2 1

2 2

( ) 2 ( )(I Q )

j

carrier k L

P P

z sign

t ⁽⁶⁾

1 2 2 1

= I Q _{P P} I Q _{P P} (7)

where, t is the time internal, I P1 and I P2 represent the in- phase prompt correlator outputs during k th and (k 1) th time epoch respectively; and the Q P1 and Q P2 represent the prompt correlator outputs during (k 1) th epoch. L1 is the wave length of the GPS L1 signals.

2. Kalman Filter and the WA-SVM Input Vector Set Up In this section, the prediction and updating of the filters is detailed illustrated as follows Eqs. (6)-(13), including the input vector of the SVM during the training procedure.

Considering a linear dynamic system:

| 1 1 1

k k k k k

X X W (8)

k k k k

Z H X V (9)

where, X k is the state vector, _{k k | 1} is the transition matrix, Z k is the observation vector, H k is the observation matrix.

1

W k and V k are the noise sequences.

1 0

k k

E W E V (10)

0

T k k i

Q i k E W W

i k (11)

0

T k k i

R i k E V V

i k (12)

In the Eqs. (8)-(12), E W _{k 1} and E V _k denote the expectation of W _{k 1} and V k respectively. Q k and R k repre-

sent the covariance matrix of process noise and observation errors respectively.

The state prediction and state covariance prediction pro- cesses are as Eqs. (13)-(17):

|k 1 1

ˆ ˆ

k k k

X X (13)

| 1 1 T | 1 1

k k k k k k k

P P Q (14)

The updating is as following Eqs. (10)-(12):

(H ) 1

T T

k k k k k k k

K P H P H R (15)

ˆ _k ˆ _{k k} (Z _k H X ) _k ˆ _k

X X K (16)

( )

k k K K

P I K H P (17)

where X ^ˆ _k is the predicted state vector; P _k is the variance matrix for the predicted state vector; K k is the gain matrix, which defines the updating weight between the predictions and the new measurements; X ^ˆ _k is the estimated state vec- tor; P k is the variance matrix for the estimated state vector.

The defined innovative sequences are as follows:

k k k ˆ k

r Z H X (18)

The input vector at k th epoch is _{k 1, i} , _{k 2, i} , ,

,

k i , the calculation of k,i is as follows:

2 , ,

2 2

, ,

1

, 1, 2 , 1, 2

( ) /

k i

k i m

k j k i

j

r i m j m

r r m

(19)

where m is the amount of the tracking channels, and variables i, j are indexes of the channels. Specially, the variable means the number of the innovative sequence employed for calculating the input vector in each channel.

The detailed information flow is the setting up of the input vector, which is presented as the following Fig. 2.

III. WHALE OPTIMIZATION

ALGORITHM-SUPPORT VECTOR MACHINE 1. Support Vector Machine

The Support Vector Machines (SVM) algorithm has be-

Fig. 2 Flow of the input vector setting up

come one of the most widely used machine learning algorithm for classification in recent years [11-12]. Here it is employed for the quality control of a channel in the vector tracking loop based receiver. The basic principle of SVM is detailed illus- trated as follows:

First, we consider a binary classification problem. The description is as equation (15):

1 1 2 2

( , ), ( , ), , ( , { 1, 1},

l l

d

i i

T x y x y x y

y x R (20)

where x i is the input data, y i is the corresponding label. l denotes the amount of the elements in the training data set.

The linear SVM finds the optimal separating margin by solv- ing the following optimization task:

Minimize

2 1

1 , 0

2

l

i i

i

w C (21)

Subject to

( ^T ) 1 , 1, 2,...,

i i i

y w x b i l ₍₂₂₎

where z is the penalty value, i are positive slack variables, w is a normal vector, b is a scalar quantity. The minimum problem can be reduced by the using the Lagrangian multi- plier i , which can obtain its optimum according to the Karush- Kuhn-Tucker condition. If i > 0, then the corresponding data x i is called the support vector, therefore, the linear dis- criminate function can be expressed with the optimal hyper- plane parameters w and b in the following equation:

1

( ) sgn( ^l _{i i i} ^T )

i

f x y x b (23)

The equation can be transformed into (18) by its uncon- strained dual form:

Maximize

1 , 1

1 2

l l

i i j i j i j

i i j

y y x x (24)

1

0, 1, , , ^l 0

i i i

i

C i l y (25)

Equation can now be solved using the quadratic program- ming techniques and the stationary Karush-Kuhn-Tucker condition. The resulting solution W can be expressed as a linear combination of the training vectors and the b can be expressed as the average of all support vectors shown in

, 1 l

i i i i j

y x

W = (26)

1

1

^SV

( )

N

i i

SV i

Wx y

= N

b (27)

where N SV is the number of the support vectors. The linear SVM can be expanded into the nonlinear cases by replacing x i with a mapping into the feature space (x i ), in other words, the x x ^T _{i i} can be represented as the form of ( ) ( ) x _i ^T x _i in the feature space. Thus, the nonlinear discrimination func- tion can be expressed as follows:

1

( ) sgn ^l _{i i} ( , ) _i )

i

f x y K x x b (28)

where K(x i , x) = ( (x i ), (x)) and the K(x i , x) is the kernel function. The widely used kernel function is the radial ba- sis function (RBF), which is defined as

( , ) exp( _{i i} ) 2

K x x x x (29)

is the predetermined smoothness parameter that controls the width of the RBF kernel; thus, (4) is rewritten as

Minimize

2

1 , 1

1 exp( )

2

l l

i i j i j i

i i j

y y x x (30)

1

0, 1, , , ^l _{i i} 0.

i

C i l y (31)

The accuracy of forecasting is highly dependent on the

selection of the penalty parameter C and the smoothness

parameter of the kernel function. However, there is no gen-

eral method for choosing the appropriate parameters for

SVM. Thus, the whale optimization algorithm is used in the

SVM to optimize the parameter selection.

2. Whale Optimization Algorithm

Inspired by the special hunting behavior of the hump- back whale, a new meta-heuristic algorithm called Whale Optimization Algorithm (WA) was proposed [13-14]. The hunting is named as bubble-net hunting strategy. The whales follow the typical bubbles causing the creation of circular while encircling prey during hunting. Simply bubble-net feeding/

hunting behavior could understand such that humpback whale went down in water approximate 10-15 meter and then after the start to produce bubbles in a spiral shape to encircle prey and then follows the bubbles and moves upward the surface.

The mathematic model for Whale Optimization Algorithm (WA) is given as follows:

i. Ncircling Prey

The humpback whales can recognize the location of prey and then encircle them. In order to get the optimum, the target prey is defined as the current best candidate solution which is close to the optimum. And the other search agents try to update their positions according to the current best search agent. The calculation details of the updating are as Eq. (32):

*

*

( ) ( 1) ( )

D E X X t

X t X t A D

(32)

where, t means the current iteration number, vectors A and E are then the coefficient vectors, X ^* is the position vector of the best solution obtained so far. X is the po- sition vector, is the absolute value, and is the element- by-element multiplication. X ^* will be updated in each iteration if there is a better solution. The vectors A and

E are calculated as follows:

2 , 2

A a r a E r (33)

where a is linearly decreased from 2 to 0 over the itera- tions, r is a random vector and the value range is [0, 1].

ii. Bubble-Net Attacking Method (Exploitation Phase) The following two approaches are designed to mathe- matically model the bubble-net behavior of the humpback whales, the details are as follows:

(i) Shrinking encircling mechanism: This behavior is achieved by decreasing the value of a in the Eq.

(30). Note that the fluctuation range of A is also decreased by a . In other words, A is a random value in the interval a a , where the a decreased

from 2 to 0 over the course of iterations. Setting random values for A in 1,1 , the new position of a search agent (X, Y) and the positing of the current best agent (X*, Y*).

(ii) Spiral updating position: This approach first calcu- lated the distance between the whale located at (X, Y) and the prey (X*, Y*). A spiral equation is then cre- ated between the position of whale and prey to mimic the helix-shaped movement of humpback whales as follows:

' *

( 1) ^bl cos(2 ) ( )

X t D e l X t (34)

where D ^' X t ^* ( ) X t ( ) and indicates the distance of the i th whale to the prey (the best solution of obtained so far), b is a constant for defining the shape of the logarithmic spiral, l is a random number in [-1, 1].

Assuming that there is a 50-50% probability that whale earth follows the shrinking encircling or logarithmic path du- ring optimization. The mathematical model is as follows:

*

' *

( ) 0.5

( 1)

cos(2 ) ( ) 0.5

bl

X t A D if p

X t D e l X t if p (35)

where p is a random number ranging [0, 1].

iii. Search for Prey (Exploration Phase)

The same approach based on the variation of A vector can be utilized to search for prey (exploration). In fact, hump- back whales search randomly according to the position of each other. Therefore, we can use A with the random va- lues greater than 1 or less than -1 to force search agent to move far from a reference whale. In contrast to the ex- ploration phase according to a randomly chosen search agent instead of the best search agent found so far. This mecha- nism and A 1 emphasize exploration and allow the WA optimization algorithm to perform a global search. The ma- thematical model is as follows:

D C X rand X (36)

( 1) _rand

X t X A D (37)

where vector X _rand is a random position vector (a random value) chosen from the current population.

iv. Whale Algorithm Optimized Support Vector Machine

The SVM parameters optimization is operated using

the WA optimization method. The training of the nonlinear SVM is essentially a constrained optimization problem. The constrained optimization usually first decides the fitness function in the algorithm and the range of each parameter.

The designed fitness function is expressed in the following equation:

2

1 ,

MAX ( , , )

1 exp( )

2

i

N N

i i j i j i

i i j

L C

y y x x (38)

The constraints of the solution string are:

(1) 0 _i C i , 1, , , and N ^N _{i 1 i i} y 0 (2) 15 log ₂ C 15,

(3) 5 log ₂ 5

From the above description of the whale optimization algorithm, it is clear that the WA starts with a set of popu- lation (candidate solutions) in the feature space. The whale population of m initial solutions are generated with n 2 dimensions denoted by S:

1 2 3

[ , S S S , , , S _m ] S

1 ⁱ , 2 ⁱ , 3 ⁱ , , log 2 ⁱ , log 2 ⁱ

S i C

where 0 _k ⁱ C , ^N _{k 1 k i} ⁱ y 0 , 15 log ₂ C ⁱ 15 , -5 log 2 ⁱ 5 . The _k ⁱ is the multiplier of k th training data in the i th candidate solution; log ₂ C ⁱ , log ₂ ⁱ are the penalty parameter and smooth parameter of the SVM, respectively, constructed by the solution string S i . The following is the pseudo codes for the training procedure for the one SVM.

All the channels are the same and the training are effective for all.

Initialized the whales population S i (i = 1, 2, , n), n is the population amount

Calculate the fitness of each search agent S* = best search agent

While (t < maximum number of iterations) for each search agent

Update a, A, C, l, and p If1 (p < 0.5)

If 2 A <1

Update the position of the current search agent by the Eq. (29)

else if2 A 1

Select a random search agent

Update the position of the current agent by the Eq. (34)

Fig. 3 The flow chart of the signal collecting

Fig. 4 The 3D plot of the designed trajectory

else if1 (p 0.5)

Update the position of the current search by the Eq. (31) end if1

end for

Check if any search agent goes beyond the search space and amend it

Calculate the fitness of each search agent Update S* if there is a better solution

1 t t end while return S*

IV. EXPERIMENTS AND RESULTS

1. Details and Description of the Set Up

In order to evaluate the performance of the proposed method, the method is implemented in a software defined vector tracking receiver (SDR-VTL). At the first step, a trajectory is generated by a hardware signal generator which is capable of Compass and GPS. The details of the signal generation are showed in Fig. 3. Considering the compu- tation load, the time length of the trajectory is totally 360 s, and the 3D plot is shown as Fig. 4.

2. Performance Evaluation of the WA-SVM

Classification accuracy comparison results of three se-

lected channel between SVM and WA-SVM are presented

in Fig. 5. The X axis represents the sequence length and the

Fig. 5 Classification accuracy rate of the selected 3 channels

Fig. 6 Position errors

Fig. 7 The position errors

Y axis represents the classification accuracy. Classification accuracy increases with the increasing of the sequences length.

WA-SVM has higher classification accuracy than the SVM with the same length of sequence. The results demonstrate the effectiveness of the WA method in optimizing the para- meters of SVM. In addition, the sequence length means the length of the innovative sequences that is calculated as Eq.

(14). The calculation of the input vector is as Eq. (15).

Two scenes are employed to evaluate the performance of the WA-SVM based fault tolerant VTL. Fig. 6 shows the position errors of the Scene 1. Blue line denotes the VTL

position errors and red line represents the SVM VTL posi- ton errors. From the results, it can be seen that when the pseudo range jump occurs, the positon error of VTL is large than that of the WA-SVM VTL. The same results can be observed from the Scene 2 results in Fig. 7.

Scene 1: Radom pseudo-range jumps is added to channel 1 and channel 5, the details are as following Eq. (39), where D means the jump errors for the pseudo-ranges.

0 m

30 m (20 30 , 120 130 )

others

D s t s s t s (39)

Scene 2: A pseudo-range jump is added to channel 1, channel 3, and channel 5. The details are as following Eq. (40), where the D means the jump errors for the pseudo-ranges.

0 m

20 m (20 30 ,120 130 )

others

D s t s s t s (40)

V. CONCLUSION

In this paper, a WA-SVM was employed to improve the robustness of the VTL. First, an experiment was conducted to validate the performance of the WA-SVM. The results showed that WA could help find better parameters for higher classification accuracy, Second, two experiments were con- ducted to evaluate the performance of the WA-SVM with the random pseudo-range jump happening. The results de- monstrate that WA-SVM aided VTL provides stable naviga- tion solutions with the random pseudo-range jump happening.

Future work will include further field tests of the WA- SVM VTL in cars.

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Manuscript Received: Feb. 14, 2017

First Revision Received: Nov. 15, 2017

Second Revision Received: Mar. 09, 2018

and Accepted: Jun. 05, 2018