Robust immune particle filter design for terrain‐referenced navigation with interferometric radar altimeter

IET Radar, Sonar & NavigationVolume 14, Issue 9 p. 1402-1409 Research ArticleFree Access Robust immune particle filter design for terrain-referenced navigation with interferometric radar altimeter Suktae Kang, Suktae Kang orcid.org/0000-0002-7034-8518 Weapon Engineering, University of Science and Technology, Agency for defense development, 160 Bugyuseong-dareo 488 beon-gil, Yuseong-gu 34060, Daejeon, Republic of KoreaSearch for more papers by this authorKyung Jun Han, Kyung Jun Han orcid.org/0000-0002-7033-4941 Agency for defense development, 160 Bugyuseong-dareo 488 beon-gil, Yuseong-gu 34060, Daejeon, Republic of KoreaSearch for more papers by this authorMyeong-Jong Yu, Corresponding Author Myeong-Jong Yu mjyu@add.re.kr orcid.org/0000-0001-9239-9028 Agency for defense development, 160 Bugyuseong-dareo 488 beon-gil, Yuseong-gu 34060, Daejeon, Republic of KoreaSearch for more papers by this author Suktae Kang, Suktae Kang orcid.org/0000-0002-7034-8518 Weapon Engineering, University of Science and Technology, Agency for defense development, 160 Bugyuseong-dareo 488 beon-gil, Yuseong-gu 34060, Daejeon, Republic of KoreaSearch for more papers by this authorKyung Jun Han, Kyung Jun Han orcid.org/0000-0002-7033-4941 Agency for defense development, 160 Bugyuseong-dareo 488 beon-gil, Yuseong-gu 34060, Daejeon, Republic of KoreaSearch for more papers by this authorMyeong-Jong Yu, Corresponding Author Myeong-Jong Yu mjyu@add.re.kr orcid.org/0000-0001-9239-9028 Agency for defense development, 160 Bugyuseong-dareo 488 beon-gil, Yuseong-gu 34060, Daejeon, Republic of KoreaSearch for more papers by this author First published: 10 July 2020 https://doi.org/10.1049/iet-rsn.2020.0047Citations: 1AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract Although interferometric radar altimeter (IRA) provides accurate three-dimensional terrain information, it is difficult to eliminate some IRA uncertainties by signal processing. This calls for a filter that possesses strong robustness against outliers. An auxiliary particle filter (APF), which can be used to perform terrain referenced navigation using IRA technology as an alternative to an integrated navigation system using a global navigation satellite system, can mitigate the effect of outliers by introducing auxiliary variables. To improve APF robustness, the authors propose an immune APF, which introduces an artificial immune system to enhance outlier performance by replacing inferior particles with superior particles. The mutation process is applied to obtain particle diversity using a regularised kernel. Monte Carlo simulation results verify the robustness of the proposed filter. 1 Introduction Navigation technology informs a vehicle of its current position, velocity, and altitude. Navigation performance is significantly dependent on three major factors: system, measurements, and filter design. Although navigation precision has highly improved with the development of the global navigation satellite system (GNSS), the GNSS is vulnerable to disturbances like jamming and spoofing. This is a crucial disadvantage, particularly in military application. To overcome this issue, many alternative navigation technologies have been studied [1-3]. One of the major alternative methods is terrain referenced navigation (TAN), which uses terrain information to perform independent navigation and compare the correlation of the vehicle's altitude and digital terrain elevation data. The performance of TAN is dependent on terrain roughness and accuracy of terrain measurement information, which is obtained in this study using interferometric radar altimeter (IRA) technology to provide three-dimensional (3D) terrain information using the slant range and look angle of the closest point on the zero Doppler line. Since IRA technology estimates terrain based on interferometry, it can offer more accurate measurement information with wider operation variation than a conventional radar altimeter, which estimates altitude as the straight vertical distance. As the vehicle goes higher, the footprint of a radar altimeter widens, increasing uncertainty and eventually resulting in a catastrophic impact on navigation performance. On the other hand, an IRA uses at least two antennas, and its measurement is not affected by height, providing a diverse operation range. For this reason, Honeywell applied the TAN system using an IRA sensor to Tomahawk cruise missiles [4]. However, IRA output has many uncertainties caused by environmental and geometric factors and IRA-inherent measurement errors [5]. IRA uncertainty causes a high non-linearity between the measured altitude and estimated vehicle altitude. Since TAN requires filters which can handle high non-linearity, suggested methods of solving this issue [6, 7] include the extended Kalman filter, unscented Kalman filter, point mass filter and particle filter (PF). In this paper, we adopt the PF, which estimates states of the vehicle by implementing Monte Carlo sampling. The idea represents the posterior density in the form of random particles with associated weights. PF performance is dependent on three major factors: accuracy of system model, reliability of measurement model, and sample impoverishment. The first two factors are determined by the hardware of the system and sensors. Thus, the question of how to solve sample impoverishment plays a key role in PF design. This issue can be solved by either increasing particle numbers, setting decent proposal distribution, or resampling particles. Among these, resampling is the most popular way to solve the impoverishment issue. Various resampling strategies that have been suggested are shown in [8]. Additionally, the recent rapid development of artificial intelligence technology has led to several new strategies. For example, Tian et al. used the artificial fish swarm algorithm shown in [9], while Han et al. presented a hybrid algorithm using the evolutionary algorithm and artificial immune system shown in [10]. Ju et al. also used an artificial immune system in the resampling procedure shown in [11]. Although such filters work efficiently under a general PF scheme, to implement AIS in auxiliary PF (APF), the immune strategy needs to be modified. The major difference between general immune PF and immune APF (IAPF) is resampling frequency. General immune PF resamples when it does not hold enough effective particles [12] but IAPF resamples particles for every iteration, it always contains enough effective particles. Thus, the purpose of immune system in APF is different from that of PF. The purpose of an immune system in an APF is to obtain optimal variance. Previous IAPF has been shown to enhance APF robustness [13] but it has required the presence of some inferior particles in order to maintain particle diversity. This indicates that the immune system has not managed to fully separate superior and inferior particles and due to leftover, computation cost is raised. Besides, previous IAPF tends to diverge on low roughness terrain. Thus, new IAPF is proposed. The proposed IAPF introduces confidence interval analysis and new hyper-mutation process. Confidence interval analysis indicates reliability of the filter performance and helps AIS to separate particles. While previous IAPF requires different size of kernels for each particle, new IAPF only rescales a mutation kernel using tuning parameters. The proposed IAPF is more cost-effective and robust compare to previous IAPF. New IAPF determines which particles are more efficiently engaging in estimation of state based on affinity measure and confidence interval analysis. After particles have been sorted into superior and inferior groups, all inferior particles (omegas) are replaced with randomly chosen superior particles. Hyper-mutation is executed to obtain particle diversity. This study is organised as follows. Section 2 briefly explains the basic principles of TAN and APF, while Section 3 explains AIS. Section 4 describes the proposed immune-boosted PF, whose efficiency is verified via computer simulation in Section 5. The work is concluded in Section 6. 2 Terrain-referenced navigation 2.1 Estimation model The TAN system estimates a vehicle's horizontal position using IRA measurement. The system model for TAN is expressed in (1) (1) where is a function of the system model and is a state vector denoting the vehicle's position composed of latitude, longitude and altitude at time (k), respectively. The measurement model for TAN, is expressed as relative displacement of the closest point and the height of the vehicle (2) (3) where indicate the north, east and down (NED) direction of relative displacements between the closest point and vehicle position in NED frame. is a direction cosine matrix forming inertial navigation system (INS) altitude, and is the misalignment between the INS and IRA, which assumes that the misalignment is known. I is the identity matrix. Fig. 1 shows TAN measurement using an IRA. A barometer is used to estimate barometric height , which offers altitude information for the vehicle. The slant range and look angle of the IRA are expressed as R and , respectively. Once the relative positions are obtained, TAN compares the relative positions with the digital elevation model (DEM) to find the correction value. TAN updates the state of the vehicle by matching the DEM with the relative positions. Thus, a miscalculation of the relative position error has a catastrophic effect on the TAN system. This indicates that IRA uncertainty highly influences TAN. Fig. 1Open in figure viewerPowerPoint TAN measurements using IRA sensor 2.2 Auxiliary PF The basic concept of a PF involves representation of prior and posterior distribution using random particles with associated weights. The knowledge of this random measure features an estimation of the instantaneous filtering posterior distribution as (4) (4) where is the total number of particles, denotes delta function, is the weighting factor, is the sequence of all target states up to t and is the particle estimation in discrete time k. The corresponding sequential importance sampling solution to the Bayesian estimation problem is expressed in (5). Posterior distribution becomes (5) where is the proposal importance distribution. is the measurement set through time k, and is also the set of state estimation through time k. A general PF adopts the transition density as the proposal distribution; however, this cannot reflect the current measurement information. Therefore, the proposal density cannot be optimised. To reflect current measurement information, an APF is chosen. The APF uses available measurement information to resample particles at time (k) before the particles are propagated through transition and likelihood distribution. The premise is that a particle representation can approximate the joint posterior at time (k−1). Then, the proposal over the entire path is able to adopt the new data . The basic concept of the APF is to estimate state with minimum variance important distribution by using the auxiliary variable . The auxiliary variable is assigned to the pre-selected particles, which are a 'good fit' to the new data point . These schemes have better outlier performance and posterior tail performance, but at the same time they also induce bias into the estimated posterior. To resolve this issue, the posterior must be corrected by modifying the weight of the remaining particles. However, in practical situations, modifying the remaining particles is not enough to reduce bias. Thus, a roughening process using a regularised kernel is implemented. The APF algorithm is described in Fig. 2. Fig. 2 has been built based on [14, 15]. Especially, resampling algorithm is referred [15]. Fig. 2Open in figure viewerPowerPoint APF algorithm Although the APF possesses better outlier performance than a general PF, it starts to lose its estimation ability rapidly when it does not hold minimum variance or when the auxiliary variables represent improper empirical prediction distribution. Previous study [15] has shown the conditional variance of stratified resampling is always smaller than that of multinomial resampling. For this reason, stratified resampling strategy is adopted to execute resampling procedure. 3 Artificial immune system AIS is a bio-inspired artificial intelligence system which mimics the sophisticated immune systems of vertebrates. The immunology of vertebrates works with T-lymphocytes (T-cell) and B-lymphocytes (B-cell), focusing primarily on their interaction with antigens. An AIS offers various methodologies, from simple heuristic algorithms to machine learning and reinforcement learning [16] based on the utility of memory cells. Memory cells store past antigenic patterns with the interactions between constituent cells. Memory cells store past antigenic patterns with the strength of interactions between constituent cells. This ability allows an immune system to respond to patterns which it has never seen before. For this reason, various AIS methodologies have been suggested [17]. To reduce computational burden, the proposed IAPF mainly consists of two immune processes: clonal selection for particle separation and hyper-mutation for particle diversity. Clonal selection principle is a theory that explains the basic properties of the immune response to an antigenic stimulus. It establishes the idea that cells capable of recognising an antigenic stimulus proliferate, while cells that don't possess such an ability are discarded. The ability to recognise antigens is called affinity. While cells respond to antigens, memory cells reproduce cells with high affinity. Reproduced cells undergo hyper-mutation to obtain diversity. The basic process of clonal selection is shown in Fig. 3. Fig. 3Open in figure viewerPowerPoint Clonal selection scheme Generally, hyper-mutation only works on B-cells and T-cells mainly active secretors to obtain diversity. Hyper-mutation and the selection process allow B-cells to enhance their repertoire diversity and to possess better abilities to recognise selective antigens. In the general evolution of the immune system, the system is expected to encounter an antigenic stimulus. In order to respond to such antigens efficiently, genetic variation and selection have very important roles in immunology. The way an immune system works is similar to that of the resampling process in a PF. Both systems need to discard inferior cells (particles) and reproduce superior cells (particles) while maintaining optimal diversity. 4 Immune APF Immune systems in general PF neither separate nor discriminate against particles based on their affinity. On the other hand, immune systems in APFs operate both procedures. IAPFs aim to enhance APF robustness by modifying empirical prediction distribution. The distinctive difference between general PF and APF application in an immune system is whether to preserve inferior particles or not. While previous IAPF has retained some inferior particles to maintain particle diversity, the proposed IAPF discards all inferior particles. The particles are split into alpha, neutral and omega based on particle selection probability, which is in turn based on affinity measure and confidence interval analysis. The affinity measure determines the similarity between particles and the true value, while confidence interval analysis informs the reliability of the true value. The entire inferior particles are eliminated and randomly chosen alpha and neutral particles become new omega particles. To bring diversity to particle distribution, hyper-mutation is executed. While previous hyper mutation process in [13] requires different kernel sizes for each particle, new hyper mutation process rescales a mutation kernel using tuning parameters, which significantly reduces computation time. Thus, the proposed IAPF is more robust and computationally effective compared to previous IAPF. 4.1 Affinity measure Affinity in AIS refers to interactions between antibodies or between an antibody and an antigen. These interactions are generally evaluated through distance measure. The more similarity to the true value a particle has, or the bigger the weight it holds, the more superior to other particles it becomes. Based on this background, two kinds of fitness functions are defined using affinity measure. The first fitness function indicates the similarity between particles and the true value. True value is defined as filter estimation . The second fitness function is defined using relative fitness function. Relative fitness function describes how one particle is relatively superior to others. Likewise, as affinity measure determines the basis of immune selection, it has a very important role in AIS. To evaluate the affinity of each particle, a slightly modified cell-line suppression method [18] is used. The multimodal function in cell-suppression assumes the likelihood function in the PF. The modified method measures Euclidean distance between the true value and the particles. Euclidean distance is obtained on the Cartesian coordinate, where particle value is on the x-axis and likelihood of the particle is on the y-axis. The modified method assumes three scenarios. In the first scenario (case 1), the particle is located very close to the true state. In this case, the location of the true state is assumed to be at the peak on the curve. To measure the shortest Euclidean distance, the interpolate point is used. The second scenario (case 2) is the most general, where affinity is measured using particle location. In the third scenario (case 3), the particle is located in tail area and the particle and true value are no longer on the same line. To measure straight line affinity, a new point is created. The details of affinity measure are described in Fig. 4. Fig. 4Open in figure viewerPowerPoint Affinity measure algorithm 4.2 Confidence interval analysis When affinity defines two types of fitness functions, confidence interval analysis determines which fitness function is more reliable. The reliability evaluation is required for the case when the true value does not hold valid credit. Usually, probabilistic density function is assumed to follow Gaussian distribution. Gaussian distribution can be interpreted using its own bound. The dependability of two fitness functions is evaluated using this property. The bound of Gaussian distribution is divided into three intervals as seen in Fig. 5. The bounds are defined as the first, second, and third confidence intervals, respectively. The first confidence interval is considered the most reliable interval; 68% of particles lie in this area. This property is used to test the convergence of the filter. At least 65% of particles should be in this area; otherwise, the filter is not converged enough or the measurement is not reliable, which indicates that the filter output is not credible. In this case, particles need to be spread to maintain particle diversity. Thus, immune system keeps the omega particles. Fig. 5Open in figure viewerPowerPoint Confidence interval analysis If the filter is converged, the estimation performance of particles in the first confidence interval is evaluated as follows: (6) where j is the index of particles inside the first confidence interval and is the total number of particles in the first confidence interval. Particles in the first confidence interval are called confidence particles. Statistically speaking, the estimation value of the confidence interval should be within of the true value. This indicates that error bound decides the reliability of confidence particles. When the estimation error of the confidence particle exceeds the error bound , fitness is not trustworthy. Thus, relative fitness should be a major interest in particle selection probability, and is designated as 0.1. To evaluate the filter performance more precisely, a residual concept is implemented to evaluate filter performance as (7) where is the residual parameter. is designated from 0.2 to 0.8 according to the residual parameter . The minimum and maximum number of can be adjusted based on the system and measurement model. A flow chart of decision is depicted in Fig. 6. Fig. 6Open in figure viewerPowerPoint Flow chart of parameter decision 4.3 Particle selection probability Particle selection probability indicates the probability of a particle to be selected as a superior particle. Particle selection probability is calculated using and the two fitness functions. As the distance between the particle and the true value gets shorter, the affinity of this particle gets higher. Thus, the fitness function is defined as the inverse of the affinity measure and normalised to express it in probability form. (8) (9) The relative fitness function evaluates superiority among particles using weights and affinity. First, the parameter is defined using fitness and weights and their mean value . Then, is also normalised (10) (11) (12) Fig. 7 shows the particle fitness probability of fitness and relative fitness. The relative fitness probability suppresses and exceeds fitness probability. If a particle is relatively superior to any other particle, it possesses relative fitness probability. On the other hand, if the relative fitness probability is close to zero, this particle would never be chosen as a superior particle. In other words, particles with zero relative fitness are defined as inferior omega particles. Fig. 7Open in figure viewerPowerPoint Fitness probability of each particle The next step is to find the alpha particles. Particle selection probability is calculated based on the two fitness functions as follows: (13) The alpha particles are designated as particles whose is higher than other particles. Neither alpha nor omega particles are defined as neutral particles. Particles are divided into each group based on particle selection probability. After particle designation is carried out, omega particles are eliminated. Then, randomly chosen alpha and neutral particles replace omega particles. To introduce diversity into the particle distribution, these alpha and neutral particles are mutated the hyper-mutation process. The point of the hyper-mutation process is to keep optimal diversity for the filter, which means to maintain optimal variance, e.g. minimum variance of the particle distribution. The hyper-mutation process is described in the next section. 4.4 Hyper-mutation After particle separation is finished, omega particles are replaced and mutated to maintain optimal diversity for particle distribution. The most general method for introducing diversity into the PF is by using a kernel density estimator [19]. The basic idea of kernel density estimation is to match an individual simple model at each target sample. Similarly, the hyper-mutation process uses the kernel density estimator method from the regularised PF (RPF). Even if the PF is propagated and updated with enough , a sample degeneracy phenomenon can occur as the empirical posterior distribution is not continuous. To solve this issue, the RPF transforms discontinuous posterior distribution into absolutely continuous distribution using a kernel . The premise of such transformation is that true posterior distribution has unity covariance; however, RPF does not hold this premise. Thus, ensemble means m and the covariance are required (14) (15) where is factorised using Cholesky decomposition and L is the unity covariance (16) The new scaled kernel using the new covariance is defined in (17), where is the bandwidth of the kernel and is the number of states (17) when the underlying density is Gaussian with the unit covariance matrix, the optimal choice for the bandwidth is defined as follows: (18) This is under the assumption that the density is Gaussian with the unit covariance matrix and that the optimal choice for the bandwidth is the Epanechnikov kernel [19]. Here, for the purpose of computational efficiency, the Epanchnikov kernel is replaced with the Gaussian kernel and the optimal bandwidth becomes (19) Jitter is defined in (20) using a rescaled regularised kernel multiplied by the random normal number generator function randn. denotes the set of particles (20) Since the kernel in the mutation process is the same kernel as in the roughness process, it needs to be rescaled using the tuning parameters 𝒶, 𝒷. The hyper-mutation process is summarised in Table 1. Table 1. Hyper mutation process [Jitter generation] = number of states [Replace omega] 4.5 Proposed IAPF The proposed IAPF aims to enhance outlier performance of the APF. The main idea of the APF is to seize particles which would survive at the next step, e.g. particles that have large weight. Likewise, the IAPF favours superior particles by replacing inferior particles with superior particles. The main immune process is executed at the end of the APF algorithm. The main strategy of the proposed filter is optimisation of the probability distribution by replacing omega particles with alpha and neutral particles. The immune process determines alpha, neutral and omega particles based on particle selection probability. Particles which are likely to be selected as alpha hold a high selection probability and vice versa. Particle selection probability consists of the two fitness functions, which are defined using affinity measure and confidence interval analysis. Affinity measure denotes the similarity between the particle and the true value. Confidence interval denotes which fitness is more trustworthy. After particle selection is carried out, omega particles are eliminated. Alpha and neutral particles fill the deleted omega particle space. These alpha and neutral particles undergo hyper-mutation. Since the APF always has enough particles, it would be dangerous to eliminate too many particles. Thus, the empirical portion of omega particles should not exceed 1% of the entire population. The number of alpha particles is the same as that of the omega particles. The rest of the particles are designated as neutral particles. Memory cells remember the omega particles and keep them for the case in which the filter is not sufficiently converged. Otherwise, the omega particles in the memory cells are discarded and then replaced with randomly chosen alpha and neutral particles. Since the key point of the IAPF is to favour superior particles, the number of alphas is 1.5 times higher than that of the neutral particles. Hyper-mutation of the particles is carried out using the kernel method. The same kernel is used in the hyper-mutation process to reduce computational burden. However, since this regularised kernel was originally scaled for the entirety of the particles, it is necessary to rescale the kernel. The tuning parameters 𝒶, 𝒷 rescale the kernel and control the variance of the particles. The entire scheme of the proposed method is summarised in Fig. 8. Fig. 8Open in figure viewerPowerPoint Proposed IAPF algorithm 5 Simulation TAN simulations with an IRA sensor are conducted to verify the robustness of the proposed filter. Digital terrain elevation data level 1 is used for the terrain information data, which has 90 m of resolution. 100 mil inertial measurement unit (IMU) is implanted in the TAN system. Two trajectories are used for the TAN simulation, with each trajectory aircraft travelling for 200 seconds (SIM 1) and 400 seconds (SIM 2). The previous IAPF [13] was originally applied in 1 mil IMU, but it tends to diverge on low roughness terrain with 100 mil IMU system. Thus, new IAPF is proposed to improve robustness on low roughness terrain. The IRA range profile is generated through the stretch processing in a region of interest [20]. For each distance gate, received pulse signals are then processed using fast Fourier transform (FFT). The results of FFT are compressed along the heads of the vehicle and mapped along the zero Doppler line. The closest terrain point is obtained through a fixed constant false alarm rate method. When the IRA converts the closest terrain point to 3D coordinates, the wind is not considered. The impact of winds affects the Euler angles, which results in errors in IRA output. Other major factors causing IRA errors are velocity error and flat or repetitive terrain characteristics. Velocity error causes the wrong Doppler zero line. Flat or repetitive terrain characteristics cause uncertainty in closest terrain point detection. For these reasons, only meaningful IRA information should be selected for TAN application. Thus, IRA validity check logic is inserted based on IRA sensor specification. IRA validity check conditions contain terrain roughness, IRA range error and IRA look angle error. The elevation of terrain measured by the IRA is stored with the latitude and longitude of the INS. The embedded DEM contains altitude information with respect to the latitude and longitude of the trajectory. The navigation errors are compensated for by finding the matching reference elevation set. Reference elevation is obtained by comparing the measured terrain elevation set with the DEM. The performance of TAN is highly dependent on terrain roughness. If the terrain is flat or repetitive, e.g. the terrain does not hold valid roughness, this can cause an error in position estimation. Even though the IRA can measure precise terrain information, if the terrain is not rough enough, this causes uncertainty of the closest points, which contaminates IRA output. In this case, IRA validity logic does not work efficiently. Additionally, the position value of the terrain data base can be biased by incorrect position estimation value, resulting in miscalculation of the matching reference algorithm. This eventually causes catastrophic results for the TAN system. Thus, terrain roughness is a key factor in TAN. This study focuses on the effect of terrain characteristics in filter performance. To verify the robustness of the filter, two simulation trajectories are chosen. One (SIM 1) is a reference trajectory to show APF performance in general situations, and the other (SIM 2) is chosen to show the robustness of the proposed filter when the terrain does not hold valid roughness. The terrain figures of the simulation trajectory are plotted in Figs. 9a–d. Figs. 9a and c show the orthographic projection of the trajectories. Figs. 9b and d show the side view of the trajectories. The vehicle travels in the diagonal direction of the terrain in SIM 1.The vehicle travels in a longitudinal direction with fixed latitude position in SIM 2. The Monte Carlo simulation is conducted 100 times for the validity of the simulation. The initial position error of the TAN system is 30 m, and the number of particles in the filters is 1000. Fig. 9Open in figure viewerPowerPoint 3D trajectory of simulation (a) Orthographic projection of SIM 1, (b) Side view of SIM 1, (c) Orthographic projection of SIM 2, (d) Side view of SIM 2 As seen in Figs. 9a and b, SIM 1 holds valid roughness in the overall trajectory. On the other hand, Figs. 9c and d show that SIM 2 is relatively flat. SIM 2 seems to hold valid roughness in the latitudinal direction, but in the longitudinal direction, flatter terrain areas exist. SIM 2 is divided into three sections: Area #1, Area #2 and Area #3. SIM 2 has valid roughness at the start point of the trajectory, but in Area #1 (150–200 s), the first flat area shows up. In Area #2 (200–240 s), a gulch appears, and the roughness of the terrain is recovered here. However, the terrain starts to lose validity of roughness again in Area #3 (250–270 s). Comparing Area #1 and Area #3, the terrain roughness of Area #3 is less valid than that of Area #1, so Area #3 is expected to cause severe degradation of the TAN system. After the vehicle escapes Area #3, the terrain starts to recover its roughness again. The simulation results of SIM 1 expect that both the APF and IAPF will work well. There is no significant difference between the two filters. On the other hand, the results of SIM 2 show significant difference between them. Both filters tend to diverge around Area #1, the first flat area. However, in Area #2, terrain recovers its roughness, and both filters tend to converge. Finally, in Area #3, the roughness of the terrain loses its validity rapidly. Both filters would diverge again in this area, but the divergence of the proposed IAPF is expected to be less than that of the APF. The robustness of the IAPF would be demonstrated in Area #1 and Area #3, and the impact of improvement would be bigger in Area #3. Simulation results are depicted in Fig. 10. Figs. 10a and b are the latitudinal and longitudinal error of SIM 1. Figs. 10c and d show the latitudinal and longitudinal error of SIM 2. There is no significant difference between the APF and IAPF in SIM 1. However, the IAPF shows significantly improved performance in SIM 2. The biggest performance difference is 15.25 m on the longitudinal axis, around 345 s. Fig. 10Open in figure viewerPowerPoint TAN navigation error plots (a) Latitude error of SIM 1, (b) Longitude error of SIM 1, (c) Latitude error of SIM 2, (d) Longitude error of SIM 2 Figs. 10c and d show interesting results for the filter performance: the longitudinal direction seems to be highly dependent on terrain roughness, while the performance of the latitudinal direction seems to hold stable compared to the longitudinal direction. It is assumed that since the vehicle is travelling in the longitudinal direction, the change of relative position on the longitudinal axis is bigger than the relative position on the latitudinal side, which makes the relative position on the longitudinal axis harder to estimate. Moreover, IRA signal processing is done in the vehicle's head direction, which means that in SIM 2, filter performance on the longitudinal axis is more dependent on terrain roughness. In Fig. 10d, both filters tend to diverge in Area #1, but after the vehicle escapes Area #1, the IAPF recovers its estimation ability faster than the APF. In Area #3, the IAPF shows its distinctive characteristics more clearly. When the vehicle flies over Area #3, both filters tend to diverge again, but the divergence of the IAPF is smaller than that of the APF. Furthermore, even though terrain roughness is slowly recovered as the vehicle escapes Area #3, the APF cannot recover its estimation ability, while recovery of IAPF estimation ability rapidly shows stabilised sustainability. While the APF diverges up to 44 m, the divergence of the IAPF is only 36 m. The total CEP of the IAPF is improved by 3.5 m in SIM 2. The average computation time in each run is measured using MATLAB with Intel® Xenon CPU e5-2620 v.4 @2.1 GHz, 64bit, 64 RAM. In this condition, APF takes 8 ms, previous IAPF [13] takes 30 ms and proposed IAPF takes 12 ms, respectively. The simulation results conclude that the proposed IAPF improves robustness of APF and benefits from computation cost. 6 Conclusion This study proposes an immune APF to improve robustness of APF. The proposed filter shows major improvement on the terrain where the roughness of the terrain is not valid. In this case, the closest point of the terrain is hard to detect, and this terrain characteristic contaminates IRA output. Furthermore, this uncertainty cannot be separated by signal processing. APF resampling of particles which are likely to survive the next iteration holds better outlier performance than that of a general PF. Similarly, the immune process duplicates superior particles and eliminates inferior particles to obtain optimal variance using a hyper-mutation process. This enhances outlier performance of the APF and also possesses better robustness against IRA uncertainty compared to the APF. 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