^{2} Faculty of Electrical Engineering, MalekAshtar University of Technology, Tehran 158751774, Iran;
^{3} Faculty of Electrical Engineering, K.N.Toosi University of Technology, Tehran 163151355, Iran
Nowadays, many factors threaten the global positioning system (GPS) receivers installed in the unmanned aerial vehicles (UAVs). GPS spoofing is one of the most important threats which deviates the positioning process in GPS receivers. When the flying UAV is in GPS spoofing threat condition, there will be some mistakes in position calculation in the GPS receivers. Therefore, the navigation data (position data generated by UAV navigation system) which are sent through datalink to the central monitoring station is counterfeited and the UAV will be lost and crashed. So, the significance of estimation in the process of UAV true positioning in the conditions of spoofing attacks is well established.
The basic navigation systems in UAVs are inertial navigation system (INS) and GPS which carry out the positioning process. The INS provides position, velocity and attitude of vehicles with good short term accuracy. The performance of INS is enhanced due to performance enhancement of microelectromechanical systems (MEMS), but it has unbounded long term errors which increase in time during its performance that are called drift error. Unlike the INS, GPS has good long term accuracy with bounded errors in few meters. By considering complementary characteristics of these two systems, their integration eliminates their individual drawbacks which leads to accurate and robust navigation solution. So, GPS eliminates the drift error of INS and INS helps to provide continuous navigation solution^{[1, 2]}.
With respect to the noises of sensors and measuring devices, integration of systems or sensors is suggested in [2–4]. On the other hand, data fusion (DF) algorithms are used in several fields such as system integration^{[5]}, aircraft navigation^{[6]}, autonomous UAV positioning^{[7]}, robust navigational system^{[8]}, wheelchair navigation^{[9]}, denoising INS and GPS data^{[10]}. Therefore, to improve the accuracy, redundancy and reliability of navigation systems in noisy environments, DF algorithms are used in INS/GPS integration system.
INS/GPS integration in navigation systems is based on data fusion. In addition, variable state estimation to find dynamic error in the integration of INS/GPS systems is done through linear and nonlinear DF algorithms. The study and application of methods in state estimation problems for soft sensor development are presented in [11]. In linear INS/GPS integration model with Gaussian noises, Kalman filter (KF) is used^{[1]}. Also, the extended Kalman filter (EKF) is developed due to the nonlinearity of the system and is used in the integration of INS and GPS systems. But, the EKF may diverge when an initial state estimation error is large^{[12]}. Therefore, the unscented Kalman filter (UKF) was proposed to avoid the divergence problem of the EKF^{[12]}. Going from KF to UKF and to particle filter (PF) in linear Gaussian systems will increase computational effort but will not improve estimation accuracy. While going from EKF to UKF and to PF in nonlinear or nonGaussian systems will increase both computational effort and estimation accuracy^{[13]}. By considering the effects of GPS spoofing as source of nonGaussian noise, the problem of spoofing attacks can be solved by using INS/GPS integration model and appropriate estimators. Due to GPS spoofing condition, applying the adapted particle based algorithm for eliminating spoofing effects provides better estimation in UAV positioning process. So, the spoofing attacks will be recognized and the UAV trajectory estimation will be done.
In this article,the GPS spoofing effects are modeled as a cumulative uniform error which is a random noise with uniform distribution to detect and eliminate the impacts of GPS spoofing attacks and estimate the UAV true position with high precision and robustness. Hence, estimating the true position of UAV is done by applying the adapted particle based filters. The UAV true position estimation is done in two steps of spoofing detection and spoofing compensation which are achieved by hypothesis test procedure and adapted particle based filters, respectively. To implement this approach, three GPS spoofing attack scenarios are considered. The particle based filters such as PF, adaptive unscented particle filer (AUPF) and particle swarm optimization filter (PSOF) and the adapted form of them are applied in the estimator part of nonlinear INS/GPS integration model. By studying the basic structure of the particle swarm algorithm, it can be concluded that the PSO can be used as an estimator in the integrated navigation system problems^{[14]}. The process of applying the adapted PSOF to the INS/GPS integration system for estimation of UAV true position is completely described. Unlike other heuristic techniques, the mechanism of PSOF is flexible and wellbalanced to increase the global and local exploration abilities^{[15]}. The algorithm is based on particles which move through multidimensional space. The particles fly through the search spaces to be attracted towards the best solution found by the best experiences of particles and their neighbors. All of the particles have a position vector and velocity vector that keep tracking to their best position values to reach extremum points. In this approach, according to artificial intelligence (AI) of PSOF, it will be concluded that PSOF has better performance in reducing spoofing error effects than PF and AUPF.
The rest of this paper is managed as follows. In Section 2, the problem definition, such as mathematical model of INS/GPS integration for a flying UAV and the mathematical model of GPS spoofing attacks in different scenarios is given. In Section 3, the proposed algorithm is completely described comprising of the GPS spoofing detection phase and the trajectory estimation phase. Section 4 includes the simulation results and analysis of applying adapted particle based filters to the integrated INS/GPS navigation system in various GPS spoofing scenarios. Finally, Section 5 concludes the paper.
2 Problem definitionTo solve the GPS spoofing problem, mathematical model of INS/GPS integration and the GPS spoofing impacts on positioning must be specified. The integration system model is divided into INS navigation equations as process model and GPS calculation equations as measurement model. The GPS spoofing is modeled as cumulative uniform error with different signal to noise ratios (SNRs).
2.1 Process model equationIn INS, accelerometers and gyroscopes measure vehicle′s velocity, position and attitude^{[1]}. The problem of UAV position estimation is formulated in a statespace framework. Then, the process model that describes the UAV position is defined by a discretetime process model of motion equation as
${{x}}(k) = {{F}}({{x}}(k  1),{{u}}(k  1),k  1) + {{\omega }}(k  1)$  (1) 
where
$\left[\!\! \begin{array}{l}{{{p}}^n}(k)\\{{{\nu }}^n}(k)\\{{{\varPsi }}^n}(k)\end{array} \!\!\!\right] \!=\! \left[\!\!\! \begin{array}{l}{{{p}}^n}(k  1) + {{{\nu }}^n}(k)\Delta t\\{{{\nu }}^n}(k  1) + [{{C}}_b^n(k  1){{{f}}^b}(k) + {{{g}}^n}]\Delta t\\{{{\varPsi }}^n}(k  1) + {{E}}_b^n(k  1){{{\omega }}^b}(k)\Delta t\end{array}\!\!\! \right]$  (2) 
where
The observation model in the INS/GPS integration system is the output of GPS receiver. According to the GPS positioning calculation^{[1]}, the nonlinear observation function is obtained from measured pseudoranges. The pseudorange between GPS satellite vehicles and the GPS receiver is calculated from [1].
$\rho _{\rm{GPS}}^m = {r^m} + c\delta {t_r} + \tilde \varepsilon _\rho ^m$  (3) 
where
So the nonlinear observation function is as
${{Z}}(k) = {{h}}({{x}}(k),k) + {{{\upsilon }}_k}$  (4) 
where
Spoofing attack is one of the most important threats for UAVs which is known as the source of nonlinear effects in GPS dynamic system. Typically, spoofers send fake GPS satellite signals with false information to the GPS receivers. As soon as spoofed signals hit the receiver, position calculation is done with wrong data so the position and time outputs of GPS receiver will be incorrect. Generally, spoofers are categorized as three basic types such as simplistic spoofers, intermediate spoofers and sophisticated spoofers^{[16]}. The common effect of the mentioned spoofed signals leads to some mistakes in the position calculation in GPS receivers. So, the nonlinear observation function in INS/GPS integration can be rewritten as
${{{Z}}_{\rm {Spoofed}}}(k) = {{h}}({{x}}(k),k) + {{{\upsilon }}_{{k_{\rm Spoofed}}}}$  (5) 
where
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Fig. 1. Impacts of spoofing signals on position calculations in GPS receiver 
The counterfeited signals alter the navigation message which includes the transmission time of each subframe, satellite positions data, satellite clocks correction factors, receiving time delay and preamble bits during position calculations in GPS receiver. The GPS spoofing error is added to allinview satellites. Referring to [17, 18], it can be inferred that all of the extreme effects of the counterfeited signals in GPS receiver impact the calculated pseudoranges and then in position variations as illustrated in Fig. 2.
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Fig. 2. Loosely INS/GPS integration system in the condition of GPS spoofing attack 
Therefore, the spoofed pseudorange in GPS receiver is modeled as follows:
$\rho _{{\rm{GP}}{{\rm{S}}_{{\rm{Spoofed}}}}}^m = \rho _{{\rm{GPS}}}^m + {\eta _\rho }$  (6) 
where
So, all of the variations in the spoofing signal models, affect in
${{X}}_{{\rm{Spoofed}}}^n(k) = {{X}}_{{\rm{GPS}}}^n(k) + {\zeta _p}$  (7) 
where
To make spoofing scenarios, it is supposed that a GPS spoofer with the model described in [20] is available and the operator wants to deviate the UAV position in desired direction. In [20], the position error during GPS spoofing has uniform distribution when the GPS receiver is in a fixed location.
But, when the UAV is flying, the position of GPS receiver is not constant anymore. So, due to continuous position calculation in GPS receiver during the UAV flight, the spoofing uniform errors in positioning will be added to each step of GPS calculation. Consequently, in this assumption the spoofing error constitutes timeadditive model and acts as a cumulative uniform error when the UAV maneuver is constant. Therefore, due to the random form of the GPS spoofing data which misleads the UAVs in one direction, the effects of the GPS spoofing signals are modeled as a uniform distribution random noise. An example of the GPS spoofing trajectory which shows the direction of misleading a UAV is presented in [22].
As mentioned before, the GPS measurement is the reference point for INS. So, when INS in the INS/GPS integration system adapts itself with the spoofed GPS data, the UAV continues flying in the wrong way and moves away by some distance from the real trajectory. This process happens in all steps of GPS spoofing so the spoofing error acts as cumulative uniform error and the UAV is misled and driven away from the real trajectory.
Indeed, another model for the GPS spoofing in a practical situation with random Gaussian noises is considered. This model is also practical and able to mislead the UAVs in a desired direction. To ensure the performance of the proposed algorithm, it will be tested with random Gaussian noises.
3 Proposed algorithmAccording to effects of GPS spoofing attacks and nonlinear INS/GPS integration model, the proposed algorithm consists of two phases; the spoofing detection phase which is accomplished by hypothesis test and the UAV trajectory estimation phase which is carried out by adapted particle based filter algorithms. The flowchart of UAV trajectory estimation in the proposed algorithm is shown in Fig. 3. In the proposed algorithm, after acquiring data from INS system and GPS receiver, hypothesis test process is done. In the absence of spoofing, the update part in one of the particle based filters is executed with the use of proper GPS measurements. In this case, estimating the real trajectory of UAV in the INS/GPS integration system is done by original particle filters. In case of spoofing existence detected by hypothesis test procedure, generation of particles around the estimation point is done by a specific coefficient and the process of estimating UAV real trajectory is done by these new particles.
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Fig. 3. UAV trajectory estimation flowchart in proposed algorithm 
3.1 Spoofing detection phase
The GPS spoofing recognition is done through hypothesis test procedure. The hypothesis test has two descriptions^{[23]}: the state,
$T(k) = \left {{{X}}_{{\rm{Spoofed}}}^n(k)  {{X}}_{{\rm{INS}}}^n(k)} \right$  (8) 
where
${\rm{Hypothesis}}\;{\rm{Test}} = \left\{ \begin{aligned}& T(k) < \eta, \;\;\;\;{H_0}:{\rm{no}}\;{\rm{spoofing}}\\& T(k) > \eta, \;\;\;\;{H_1} : {\rm{spoofing}}.\end{aligned} \right.$  (9) 
Considering the sensitivity of the GPS spoofing detection and the accuracy of GPS receiver, the value of
As mentioned before, trajectory estimation is done through particle based filters. PF, AUPF and PSOF algorithms are implemented and adapted for the proposed algorithm in the estimation part of the INS/GPS integration system to find out which one acts better in estimating the real trajectory of UAV in the GPS spoofing conditions. PF is based on Monte Carlo technique to solve the state estimation problems^{[24]}. It solves the problem numerically and extracted the Bayesian parameters. According to the type of resampling in PF algorithm, there are two important types of PF; sampling importance resampling particle filter (SIRPF) and sampling importance sampling particle filter (SISPF)^{[25]}. To reduce the computational burden without decreasing the system estimation accuracy, PF algorithm is combined with other filters such as UKF which yields to AUPF. In AUPF algorithm, resampling part is not required which is completely described in [26]. After PF algorithms, PSOF algorithm is implemented. Optimization methods can be classified into two categories: random and derivatives. One of the best random optimization procedures is particle swarm optimization algorithm. Due to having memory, this method has priority above other derivative based methods of optimization. The memory of this method is obtained from its previous experiences of each particle or its neighboring particles which leads to the absolute optimum.
The particle swarm optimization algorithm comprised a swarm of candidate solutions considered as particles. Control of the movement of particles in the same direction in this algorithm was invented by Ebrehat and Kennedy^{[27]} in 1995. Particles fly through the search spaces, which are attracted towards the best solution found by the best experiences of neighboring particles and particles themselves. All of the particles have a position vector and velocity vector that keep track of their best position values to reach extremum points. The velocity and position of each particle is obtained from
$\begin{split}& {{{v}}_{k + 1}} = a{{{v}}_k} + {b_1}\,{r_1}\,({{{p}}_1}  {{{x}}_k}) + {b_2}\,{r_2}\,({{{p}}_2}  {{{x}}_k})\\&{{{x}}_{k + 1}} = {d_1}\,{{{x}}_k} + {d_2}\,{{{v}}_{k + 1}}\end{split}$  (10) 
where
In the absence of spoofing in the proposed algorithm, PSOF is executed normally. PSOF algorithm is made of two basic parts: particle swarm optimization algorithm as an estimator part and INS/GPS integration model as an observer and propagation part. The estimator part consists of four fundamental levels: initialization, update velocities, update positions and checking criterion. The flowchart of PSOF algorithm is shown in Fig. 4.
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Fig. 4. Flowchart of applying PSOF to the INS/GPS integration system to estimate real trajectory 
When the UAV moves in its trajectory, INS and GPS data are retrieved and the objective function can be calculated. The objective function in PSOF algorithm acts as a guide for searching and must be defined as an initialization function. In other words, performance of the PSOF algorithm depends on the objective function which leads to have an appropriate search space^{[14]}. With respect to INS/GPS system model, and the purpose of reducing the measurement error to find the real trajectory by UAV navigation systems, the objective function is defined as
$D(k) = \left {{Y_{est}}(k)  {Y_{\rm{GPS}}}(k)} \right$  (11) 
where
Step 1. Assign the swarm size
Do the following
Step 2. Retrieve the data from the INS dynamic model.
Step 3. Retrieve the measurement data from the GPS receiver model.
Do the following for
Step 4. Check the objective function to find out when criterion is met.
Step 5. Apply the initial particle to the GPS measurement model.
Step 6. Find the own best particle.
Step 7. Find the global best particle.
Step 8. Update the velocity of particles.
Step 9. Update the position of particles.
Step 10. Estimate the best particle positions.
In case of spoofing existence detection in hypothesis test procedure, the position measurements in GPS receiver are being changed. The variations in latitude and longitude in GPS receiver change the amount of
Theorem 1. Consider the amount of
$\gamma = \left\{ \begin{aligned}& {\left(1 + \frac{1}{{T(k)}}\right)^{{e^2}}},\;\;\;\;{\rm{if}}\;\xi \;{\rm{is}}\;{\rm{defined}}\;{\rm{as}}\;{\rm{positive}}\\& {\left(1  \frac{1}{{T(k)}}\right)^{{e^2}}},\;\;\;\;{\rm{if}}\;\zeta \;{\rm{is}}\;{\rm{defined}}\;{\rm{as}}\;{\rm{negative}}\end{aligned} \right.$  (12) 
where
Then, new position of particles in PSOF algorithm is generated as follows:
${{{x}}_{k + 1}} = \gamma {d_1}\,{{{x}}_k} + \gamma {d_2}\,{{{v}}_{k + 1}}.$  (13) 
In this case, the static coefficient,
Proof. Refer to (10), the velocity and position of each particle are obtained. To simplify (10), the following parameters can be used as follows:
$b = \frac{{{b_1} + {b_2}}}{2}\quad\quad\quad\quad\quad\quad\quad$  (14) 
${{p}} = \frac{{{b_1}}}{{{b_1} + {b_2}}}{{{p}}_1} + \frac{{{b_2}}}{{{b_1} + {b_2}}}{{{p}}_2}$  (15) 
where
By substituting (14) and (15) in (10) and using (13), (10) can be rewritten as
$\begin{split}{{{v}}_{k + 1}} = a{{{v}}_k} + b({{p}}  {{{x}}_k})\;\;\;\;\\{{{x}}_{k + 1}} = \gamma {d_1}{{{x}}_k} + \gamma {d_2}\,{{{v}}_{k + 1}}.\end{split}$  (16) 
By referring to (16) and using recursive functions, the velocity function will be eliminated. Then, (16) can be rewritten as
${{{x}}_{k + 1}} + (\gamma b{d_2}  a  \gamma {d_1}){{{x}}_k} + \gamma a{d_1}{{{x}}_{k  1}} = \gamma b{d_2}{{p}}.$  (17) 
In (17), it is clear that the coefficient
$\mathop {\lim }\limits_{k \to \infty } {{{x}}_k} = {{p}}.$  (18) 
Then,
${{p}} + (\gamma b{d_2}  a  \gamma {d_1}){{p}} + \gamma a{d_1}{{p}} = \gamma b{d_2}{{p}}.$  (19) 
By solving (19), we have
$(a  1)(\gamma {d_1}  1) = 0.$  (20) 
Due to the spoofing attack and extreme position variations in GPS receiver, the value of
$\mathop {\lim }\limits_{T(k) \to \infty } \frac{{{d_1}}}{\gamma } = \left\{ \begin{aligned}& \mathop {\lim }\limits_{T(k) \to \infty } {\left( {{{\left(1 + \frac{1}{{T(k)}}\right)}^{{e^2}}}} \right)^{  1}} = 1\\& \mathop {\lim }\limits_{T(k) \to \infty } {\left( {{{\left(1  \frac{1}{{T(k)}}\right)}^{{e^2}}}} \right)^{  1}} = 1.\end{aligned} \right.$  (21) 
Therefore, the greater the amount of spoofing, the greater the amount of
Finally, the adapted PSOF algorithm for estimating UAV true position in the INS/GPS integration system under current condition of GPS spoofing attacks is as follows:
Step 1. Assign the swarm size
Do the following
Step 2. Retrieve a data from the INS dynamic model.
Step 3. Retrieve the measurement data from the GPS receiver model.
Step 4. Calculate the static coefficient
Do the following for
Step 5. Check the objective function to find out when criterion is met.
Step 6. Apply the initial particle with the static coefficient,
Step 7. Find the own best particle.
Step 8. Find the global best particle.
Step 9. Update the velocity of particles.
Step 10. Update the position of particles with the static coefficient
Step 11. Estimate the best particle positions.
The same procedure for propagating particles with new coefficient is applied to all SIRPF, SISPF and AUPF algorithms. So, the adapted PSOF is compared with the adapted SIRPF, adapted SISPF and adapted AUPF algorithms.
4 Simulation resultsIn order to verify spoofing detection and elimination of the GPS spoofing attack effects from the INS/GPS integration system, the simulation scenario of UAV real trajectory is performed and it is assumed that UAV is operating in cruise mode. The UAV flies with constant speed of 30 m/s for 30 s.
To generate INS and GPS navigation data in the UAV trajectory scenarios, some initial parameters, presented in Table 1, are assumed to simulate these two systems. These parameters are pertinent to low cost inertial navigation system and low cost GPS receiver. The sampling rate of INS is 100 Hz and the GPS measurements are generated at 1 Hz. The clock bias error and the clock drift error of GPS receiver are selected as 0.011 9 s and 0.151 64 ms, respectively. The distance between the sensing points of the sensors, lever arm, is supposed to be zero. Refer to initial data in Table 1, INS and GPS measurements are generated based on the reference trajectory.
The INS and GPS trajectory simulations are illustrated in Fig. 5. Referring to Fig. 5, it is obvious that the INS trajectory earnestly diverges from the true trajectory due to the drift error of inertial sensors.
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Fig. 5. INS and GPS trajectory according to the UAV real trajectory 
Refer to the scenarios of GPS spoofing attacks which are mentioned in Section 2.3, the impacts of the GPS spoofing errors on the UAV position during the flight are simulated and depicted in Fig. 6. There are three scenarios related to the different kinds of GPS spoofing attacks. In Matlab simulations, the GPS spoofing errors are applied in pseudorange calculations. The GPS spoofing errors for three scenarios are bounded between 10 m and 150 m for the first scenario, 650 m and 800 m for the second scenario and between 1 500 m and 2 300 m for the third scenario. The spoofing attack occurs at the 15th second and continues to up to 30 s.
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Fig. 6. GPS position deviation subjected to spoofing attacks in three scenarios 
The root mean square error is a scale for measuring errors in the whole UAV trajectory that is calculated as follows:
$RMSE = \sqrt {mean({{({x_n} \!\! x_n')}^2} \!+\! {{({x_e} \! \!x_e')}^2} \!+\! {{(h \!\! {h'})}^2})} $  (22) 
where
With respect to the effects of cumulative uniform error which is generated by GPS spoofing attack, the UAV position deviation from real trajectory increases in time. The numerical information about the root mean square error (RMSE) of three spoofing attacks in east, north and up (ENU) coordinates are presented in Table 2. Referring to Fig. 6, it can be inferred that the GPS positioning deviation in Scenario 1 is smoother than Scenario 3. Also, the RMSE of navigation systems in total trajectory is presented in Table 3.
To provide a Matlab implementation for particle based filters, some initial parameters are needed. Implementation of SIRPF and SISPF algorithms was done by
Furthermore, the values that are assigned to the adapted PSOF algorithm are given by the number of particles
According to 10meter threshold of detection value, the numerical information for PSOF algorithm for position detection error at the start of GPS spoofing in the three scenarios are presented in Table 4.
After applying SISPF, SIRPF, PSOF and AUPF algorithms as estimators to the INS/GPS integration model in the condition of spoofing attack in Scenario 1, the Matlab simulation results are gotten and shown in Figs. 7–10. When the UAV starts flying, particle based filters are activated as estimators in the INS/GPS integration system to estimate the real trajectory. Immediately, by detecting spoofing using the hypothesis test, the start point of spoofing attack is declared then particle based filters act as compensators. In this case, the last true GPS measurement will be a basic value to estimate the next steps of trajectory. The numerical information about RMSE of applying estimation algorithms to the INS/GPS integration system in the condition of GPS receiver spoofing errors in Scenario 1 is presented in Table 5.
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Fig. 7. UAV position compensation during GPS spoofing attack in Scenario 1 with the adapted SISPF algorithm 
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Fig. 8. UAV position compensation during GPS spoofing attack in Scenario 1 with the adapted SIRPF algorithm 
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Fig. 9. UAV position compensation during GPS spoofing attack in Scenario 1 with the adapted PSOF algorithm 
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Fig. 10. UAV position compensation during GPS spoofing attack in Scenario 1 with the adapted AUPF algorithm 
As like as Scenario 1, by applying SISPF, SIRPF, PSOF and AUPF algorithms as filters to the INS/GPS integration model in the condition of spoofing attack in Scenario 2, the Matlab simulation results are obtained and illustrated in Figs. 11–14. The numerical information about RMSE of applying estimation algorithms to the INS/GPS integration system in the condition of GPS receiver spoofing errors in Scenario 2 is presented in Table 6.
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Fig. 11. UAV position compensation during GPS spoofing attack in Scenario 2 with the adapted SISPF algorithm 
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Fig. 12. UAV position compensation during GPS spoofing attack in Scenario 2 with the adapted SIRPF algorithm 
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Fig. 13. UAV position compensation during GPS spoofing attack in Scenario 2 with the adapted PSOF algorithm 
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Fig. 14. UAV position compensation during GPS spoofing attack in Scenario 2 with the adapted AUPF algorithm 
By applying the particle based algorithms in the same way to the INS/GPS integration model in the condition of spoofing attack in Scenario 3, the Matlab simulation results are taken and depicted in Figs. 15–18. The numerical information about RMSE of applying estimation algorithms to the INS/GPS integration system in the condition of GPS receiver spoofing errors in Scenario 3 is given in Table 7.
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Fig. 15. UAV position compensation during GPS spoofing attack in Scenario 3 with the adapted SISPF algorithm 
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Fig. 16. UAV position compensation during GPS spoofing attack in Scenario 3 with the adapted SIRPF algorithm 
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Fig. 17. UAV position compensation during GPS spoofing attack in Scenario 3 with the adapted PSOF algorithm 
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Fig. 18. UAV position compensation during GPS spoofing attack in Scenario 3 with the adapted AUPF algorithm 
The results of state estimation errors of latitude and longitude for PSOF algorithm in three scenarios are illustrated in Figs. 19–21. It is obvious that the errors are bounded between
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Fig. 19. Latitude and longitude state estimation errors for PSOF algorithm in Scenario 1 
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Fig. 20. Latitude and longitude state estimation errors for PSOF algorithm in Scenario 2 
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Fig. 21. Latitude and longitude state estimation errors for PSOF algorithm in Scenario 3 
Before GPS spoofing attacks occur, the INS/GPS integration model produces continuous navigation solution by compensating the drift error of INS and making INS more precise. The performance of particle based filter for solving the GPS spoofing problem is as follows. In the first step, when the spoofing occurs, the amount of deviation between the output of compensated INS and the GPS measurement is calculated. If the obtained value exceeds the considered threshold, the spoofing attack is detected. In the next step by activating particle based filters, at the beginning of the spoofing attack detection process, the position of UAV is estimated by the use of particle based filters in the INS/GPS integration system. In this case, INS output in the first moment of spoofing detection is nearly equal to the GPS measurement before spoofing occurred. The estimated position in this case is very close to the UAV real position in the trajectory.
By estimating the first point of UAV trajectory in the condition of GPS spoofing, INS data will be compensated by the estimated position in the first timestep after spoofing attacks. Once the INS compensation is done, the position estimation will occur in the next timesteps by particle based filters. Consequently, the unfavorable effects of spoofing attacks will be eliminated by SISPF, SIRPF, PSOF and AUPF with reasonable accuracy.
The comparison of these four algorithms in three scenarios showed that PSOF algorithm presents better performance than others due to minimum RMSE in position in ENU coordinates. This is because of using the experiences of each particle and the experiences of neighboring particles in the search space which cause the PSOF algorithm to have memory. Indeed, the time of simulation in implementing PSOF algorithm is less than others due to its memory based property and is almost one quarter. The numerical information about the RMSE of applying estimation algorithms to the INS/GPS integration system in the condition of GPS spoofing attack scenarios is presented in Table 8. Also, to ensure the performance of the proposed algorithm, it is tested with random Gaussian noises which is called Scenario 4. To compare the effects of different random noises in the GPS spoofing models, the amplitude of the random Gaussian noises is considered the same as the amplitude of the cumulative uniform errors in Scenario 3. The amount of spoofing errors in three directions of
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Fig. 22. Amount of spoofing errors in three directions of

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Fig. 23. UAV position compensation with the adapted PSOF algorithm considering the GPS spoofing attack as random Gaussian noise in Scenario 4 
The values of position error in Table 8 shows that the position error is completely decreased due to the use of particle based filters. So, it can be deduced that the positioning process is robust in the condition of GPS spoofing attacks. Investigating the position errors in PSOF algorithm in three scenarios shows that the more the spoofing effects increase, the more the position errors decrease which can be proved mathematically in the next study. The improvement in accuracy or rate of improvement of positioning process in PSOF algorithm which is defined as difference between position errors of counterfeited GPS and estimated value by the PSOF divided by position error of counterfeited GPS is more than 97% in three scenarios.
The key result of this research is the elimination of undesirable effects of the GPS spoofing attacks in different spoofing scenarios by estimating real trajectory from particle based filters. So, the position deviation in GPS receiver during spoofing attacks will be compensated.
5 ConclusionsIn order to obtain robust positioning of UAVs during maneuver in dangerous scenarios such as GPS spoofing attacks and improve the accuracy of UAV position estimation, we proposed an algorithm as a UAV trajectory estimator in the INS/GPS integration system. The proposed algorithm is composed of the GPS spoofing detection and the spoofing effect compensation which is done by hypothesis test and particle based filters, respectively. By referring to effects of spoofing signals in position calculations in GPS receivers, the spoofing effects are modeled as cumulative uniform error which leads to position deviations in GPS measurements. After detecting the start point of spoofing attacks by hypothesis test, we applied particle based filters such as SISPF, SIRPF, PSOF and AUPF to the INS/GPS integration model. The results showed that the PSOF algorithm has better performance in compensating the spoofing effects due to memory nature of PSOF algorithm and use of the experiences of each particle and the experiences of neighboring particles. The results of applying the PSOF algorithm as an estimator in the INS/GPS integration model showed that this approach can provide reliable and accurate navigation solutions. The accuracy of positioning in the condition of GPS spoofing attacks is improved and is more than 97% in three scenarios. The RMSE test for these algorithms proved that PSOF algorithm eliminates the undesirable effects of the GPS spoofing attacks in different spoofing scenarios.
AcknowledgementsThe authors would like to acknowledge Dr. Saeed Nasrollahi from the Sharif University of Technology in Tehran for providing constructive feedback, careful reading and compassionate guidance to improve the article, Ehya Yavari from the Institute of Robotics and Computer (IRC) in MalekAshtar University of Technology in the field of LORANC national project, for his supporting behavior and Dr. Mahdi Majidi, the faculty member of University of Kashan because of his careful advices to improve this article.
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