Inter national J our nal of Electrical and Computer Engineering (IJECE) V ol. 15, No. 3, June 2025, pp. 2735 ∼ 2747 ISSN: 2088-8708, DOI: 10.11591/ijece.v15i3.pp2735-2747 ❒ 2735 Mar ginalized particle ltering f or r eliable land v ehicle na vigation in global na vigation satellite system-denied en vir onments Abdelkabir Lahr ech 1 , Aziz Soulhi 2 1 Multidisciplinary Laboratory of Research and Inno v ation (LMRI), Sultan Moulay Slimane Uni v ersity , Khouribg a, Morocco 2 Laboratory of Applied Mathematics and Decision Support Computing (LMAID), Superior National School of Mines, Rabat, Morocco Article Inf o Article history: Recei v ed Aug 9, 2024 Re vised Dec 18, 2024 Accepted Jan 16, 2025 K eyw ords: Digital maps GPS na vig ation Mar ginalized ltering Multisensor fusion Non-linear ltering ABSTRA CT Accurate locali zation in land v ehicle na vig ation syst ems is highly dependent on the global na vig ation satellite system (GNSS). Ho we v er , GNS S signal outages are common in urban areas due to obstacles such as tall b ui ldings and tunnels. T o mitig ate these issues, digital road maps and dead reck oning sensors, lik e odometers, are often inte grated to pro vide conti nuous v ehicle localization. This paper presents a rob ust estimation method to solv e the fusion problem of GNSS, odometer , and digital road map measurements in the presence of GNSS out- ages. The proposed solution utili zes a mar ginalized particle lter (MPF), which combines the rob ustness of particle ltering with the ef cienc y of a Kalman lter to handle the linear and non-linear parts of the state and/or measurement equations, respecti v ely . When GNSS signals are una v ailable, the MPF fuses all a v ailable pseudo-range data with odometric and map information to enhance v ehicle positioning. The ef fecti v eness of the proposed method is demonstrated using real-w orld data in an urban transportation scenario, highlighting signi- cant performance impro v ements and real-time application potential. This is an open access article under the CC BY -SA license . Corresponding A uthor: Abdelkabir Lahrech Multidisciplinary Laboratory of Research and Inno v ation (LMRI), Polydisciplinary F aculty , Uni v ersity of Sultan Moulay Slimane Khouribg a, Morocco Email: a.lahrech@usms.ma 1. INTR ODUCTION In recent decades, the use of positioning and na vig ation technol o gi es has signicantly increased, es- pecially in transportation applications. The global na vig ation satellite system (GNSS) is a fundamental com- ponent for man y na vig ation systems due to its long-term stability and global co v erage [1], [2]. Ho we v er , in dense urban en vironments, GNSS performance can be compromise d due to poor satellite visibility and signal blockages caused by tall b uildings and tunnels. These challenges necessitate the use of supplementary sensors to ensure continuous and accurate v ehicle positioning [3], [4] Dead-reck oning (DR) sensors, such as odometers, are commonly used to pro vide continuous na vi- g ation support. Inte grated into antilock braking systems (ABS), odometers generate digital pulses for each wheel re v olution, allo wing for distance estimation [5]. Additionally , digital road maps can enhance positioning accurac y through a process kno wn as map-matching, which aligns the v ehicle’ s estimated position with the road netw ork. Man y methods ha v e been de v eloped to s o l v e the map-matching proble m, emplo ying topologic J ournal homepage: http://ijece .iaescor e .com Evaluation Warning : The document was created with Spire.PDF for Python.
2736 ❒ ISSN: 2088-8708 and geometric techniques such as point-to-point, point-to-arc, and curv e-to-curv e matching [6], [7]. Ho we v er , these methods are susceptible to errors in dense road netw orks. The e xtended Kalman lter (EKF) is frequently used to inte grate GNSS data with measurements from odometers and lo w-cost inertial na vig ation systems (INS). Ho we v er , the EKF’ s reliance on linearization can introduce signicant estimation errors o v er time, and its assumption of Gaussian noise may not hold in non- linear , non-Gaussian conte xts [8], [9]. I n contrast, the unscented Kalman lter (UKF) uses a sampling technique to handle non-linearities and pro vides a more accurate approximation of the system’ s dynamic model, albeit at a higher computational cost [11], [10]. T o address these limitations, recent adv ancements ha v e focused on sequential Monte Carlo (SMC) methods, including the mar ginalized particle lter (MPF). These methods are capable of handling non-linear models and non-Gaussian statistics without requiring a linearization stage, as the EKF does. The MPF , in particular , is designed to reduce computational cost by applying particle ltering to the non-linear components of the state while using a Kalman lter for linear components [12]. This h ybrid approach not only maintains high accurac y b ut also ensures computational ef cienc y , making it suitable for real-time applications [13]. Recent studies ha v e sho wn the ef fecti v eness of these methods in inte grating GNSS with inertial na vig ation systems (INS) and other sensors to enhance v ehicle localization accurac y in urban en vironments [14], [15]. This paper presents a mar ginalized particle lter (MPF) for land v ehicle na vig ation in urban en viron- ments, where partial or total GNSS outages are common. During par tial GNSS outages, the proposed method uses a v ailable pseudo-range data, e v en when fe wer than four satellit es are visible. In the e v ent of total GNSS outages, the algorithm also incorporates dead-reck oning sensors and a digital road map to pro vide continuous localization. The MPF ef fecti v ely solv es the multi-sensor fusion problem by sequentially inte grating measure- ments from all sensors to estimate the v ehicle’ s 3D position. The proposed approach demonstrates signicant impro v ements in both accurac y and computational ef cienc y , particularly in challenging urban en vironments. 2. VEHICLE SYSTEM AND SENSOR MEASUREMENT MODELING 2.1. V ehicle dynamics modeling T o address this estimation problem, we use a state model. The state v ector X k = ( x k , y k , z k , v x k , v y k , v z k , γ x k , γ y k , γ z k ) T includes the 3-D components of the v ehicle’ s position x, y , z , v elocity v , and acceleration γ . The v ehicle’ s dynamics are described using kinematic equations. F ocusing on the x -component, we can write the equations as (1): γ x k = γ x k − 1 + w γ x k v x k = v x k − 1 + γ x k − 1 ∆ t + w v x k x k = x k − 1 + v x k − 1 ∆ t + 1 2 γ x k − 1 ∆ t 2 + w x k (1) where ( w γ x k , w v x k , w x k ) are additi v e white Gaussian noise terms, and ∆ t represents the sampling interv al. These equations can be e xpressed in matrix form as (2): γ x k v x k x k = 1 0 0 ∆ t 1 0 1 2 ∆ t 2 ∆ t 1 γ x k − 1 v x k − 1 x k − 1 + w γ x k w v x k w x k (2) The same set of equations applies to the y and z components, allo wing us to describe the o v erall v ehicle dynamics with the follo wing equation: X k = F X k − 1 + W k (3) where W k ∼ N (0 , Q k ) is a white Gaussian noise v ector , and F is the dynamic transition matrix of the system, structured as: F = F x 0 3 × 3 0 3 × 3 0 3 × 3 F y 0 3 × 3 0 3 × 3 0 3 × 3 F z Int J Elec & Comp Eng, V ol. 15, No. 3, June 2025: 2735-2747 Evaluation Warning : The document was created with Spire.PDF for Python.
Int J Elec & Comp Eng ISSN: 2088-8708 ❒ 2737 where each block F x , F y , and F z corresponds to the dynamic transition sub-matrices for the x , y , and z components, respecti v ely . 2.2. Measur ement equations without satellite masking When GPS data is a v ailable, the est imator incorporates these measurements to enhance the state es - timation. The GPS measurement v ector Z GPS k = ( x GPS k , y GPS k , z GPS k ) T is related to the state v ector through the measurement function h GPS and can be e xpressed as (4): Z GPS k = h GPS ( X k ) + V GPS k (4) where h GPS is the measurement function and V GPS k is an additi v e white Gaussian noise with zero mean and co v ariance matrix R GPS k . 2.3. Measur ement equations with partial GPS outages When the number of a v ailable satellite signals is insuf cient to compute a complete GPS position (i.e., fe wer than 4 satellites), the lter utilizes the limited pseudorange measurements to compute the GPS position and estimate the v ehicle’ s kinematic characteristics. F or e xample, if only three satellites are visible, the GPS pseudorange equations P r j k for the j -th satellite can be e xpressed as follo ws, ne glecting v arious additional biases related to signal propag ation (ionospheric and tropospheric delays): P r j k = q ( x k − x s,j k ) 2 + ( y k − y s,j k ) 2 + ( z k − z s,j k ) 2 + cδ k (5) where ( x k , y k , z k ) are the coordinates of the recei v er’ s position, ( x s,j k , y s,j k , z s,j k ) are the coordinates of the j -th satellite, δ k is the clock of fset, and c is the speed of light. The recei v er’ s position and clock of fset ( x k , y k , z k , δ k ) are unkno wn parameters that must be esti- mated. T ypically , computing the recei v er’ s location directly w ould require more than four pseudorange mea- surements. Ho we v er , the proposed method le v erages the a v ailable pseudorange measurements to estimate a partial GPS position Z PGPS k using the predicted state v ector components and a non-linear least squares method to solv e the system of equations. Pseudorange measurements are initially computed in the earth-centered, earth-x ed (ECEF) coordi- nate system and then transformed into the Uni v ersal T ransv erse Mercator (UTM) coordinate system to be fused with odometric and map measurements. The partial GPS position can be described as (6): Z PGPS k = h PGPS ( X k ) + V PGPS k (6) where h PGPS is the non-linear measurement function used in the pseudorange equations, and V PGPS k represents Gaussian white noise with zero mean and co v ariance R PGPS k . 2.4. Odometer measur ement equations When GPS data is una v ailable, the dif ferential odometry sensor is used to estimate the v ehicle’ s position [16]. This sensor pro vides measurements Z ODO k of the elementary displacements of the left and right wheels [17]. The measurement v ector is gi v en by (7): Z ODO k =   ∆ D R k ∆ D L k ! = h ODO ( X k ) + V ODO k (7) where (∆ D R k ) and (∆ D L k ) are the elementary displacements of the right and left wheels, respecti v ely , and θ ODO k is the v ehicle’ s orientation relati v e to the horizontal axis. The function h ODO represents the odometry measurement function, and V ODO k is an additi v e white Gaussian noise with zero mean and co v ariance matrix R ODO k . The displacements for the right and left wheels can be e xpressed as (8): ∆ D R k = q (∆ D R ,x k ) 2 + (∆ D R ,y k ) 2 r R + V ODO,R k ∆ D L k = q (∆ D L,x k ) 2 + (∆ D L,y k ) 2 r L + V ODO,L k (8) Mar ginalized particle ltering for r eliable land vehicle navigation in ... (Abdelkabir Lahr ec h) Evaluation Warning : The document was created with Spire.PDF for Python.
2738 ❒ ISSN: 2088-8708 where r L and r R are the radii of the left and right wheels, respecti v ely , and (∆ D L,x k , ∆ D L,y k ) and (∆ D R ,x k , ∆ D R ,y k ) are the v ehicle’ s left and right displacements along the x and y components for each wheel. The relationship between these displacements and the components of the state v ector can be e xpressed for both wheels as (9), (10): ∆ D R ,x k = v x k − 1 ∆ k + e 2 v y k ∥ v k ∥ − v y k − 1 ∥ v k − 1 ∥ ∆ D R ,y k = v y k − 1 ∆ k − e 2 v x k ∥ v k ∥ − v x k − 1 ∥ v k − 1 ∥ (9) ∆ D L,x k = v x k − 1 ∆ k − e 2 v y k ∥ v k ∥ − v y k − 1 ∥ v k − 1 ∥ ∆ D L,y k = v y k − 1 ∆ k + e 2 v x k ∥ v k ∥ − v x k − 1 ∥ v k − 1 ∥ (10) where e is the distance between the left and right wheels. The odometry measurements pro vide an estimate of the v ehicle’ s speed and direction, al lo wing for corrections to the v ehicle’ s dynamics. Ho we v er , it is important to note that these sensors can accumulate errors o v er time, leading to potential inaccuracies in the v ehicle’ s estimated location. 2.5. Road map measur ement equations When GPS data is una v ailable, the proposed lter also inte grates mea surements from a digital road map to enhance v ehicle positioning. This road map is part of a geographic information system (GIS) and consists of roads represented by tw o-dimensional arcs composed of piece wise se gments. Each se gment is dened by a nite number of nodes, marking endpoints or transitions between se gments . These nodes pro vide essential location coordinates ( x MAP k , y MAP k ) and orientation θ MAP k , which contrib ute to impro ving the v ehicle’ s estimated location. The inte gration of the digital road map is modeled as a measurement equation, with the cartographic database serving as a set of potential candidates for this equation, thus enhancing positioning accurac y [18]. The road map measurement v ector Z MAP k = ( x MAP k , y MAP k , θ MAP k ) includes the coordinates and direction of a se gment and it is gi v en by (11): Z MAP k = h MAP ( X k ) + V MAP k (11) where h MAP is the map measurement function, V MAP k is an additi v e white Gaussian noise with zero mean and co v ariance matrix R MAP k . The equations for the map measurement function are: x MAP k = x k + V MAP ,x k y MAP k = y k + V MAP ,y k θ MAP k = tan − 1 v y k v x k + V MAP , θ k (12) This modeling allo ws for the direct inte gration of road map errors and uncertainties via the map measurement noise statist ics. Impro v ed map measurements, combined with accurate cartographic data, signicantly enhance v ehicle localization, particularly in urban en vironments where GPS co v erage is poor . 3. MARGIN ALIZED P AR TICLE FIL TERING Mar ginalized particle ltering (MPF), also kno wn as Rao-Blackwellization ltering is an adv anced technique used to address the limitations of standard particle lters, especially in applications in v olving high- dimensional s tate spaces and nonlinear models. By mar ginalizing out the linear components of the state, MPF impro v es computational ef cienc y and estimation accurac y [19]. This section details the MPF approach and its application to v ehicle na vig ation. Int J Elec & Comp Eng, V ol. 15, No. 3, June 2025: 2735-2747 Evaluation Warning : The document was created with Spire.PDF for Python.
Int J Elec & Comp Eng ISSN: 2088-8708 ❒ 2739 3.1. State model and mar ginalization T o address this estimation problem, we use a state model where the state v ector is partitioned into components Λ k (position and v elocity) and Γ k (acceleration). This partitioning allo ws us to treat the linear and non-linear components separately , impro ving the computational ef cienc y of the ltering process. The dynamics and measurement equations can be e xpressed as (13): Λ k = F ΛΛ Λ k − 1 + F ΛΓ Γ k − 1 + W Λ k Γ k = F ΓΓ Γ k − 1 + W Γ k Z k = h (Λ k ) + V k (13) where W Λ k and W Γ k are Gaussian white noises with zero mean and co v ariance matrices Q Λ k and Q Γ k , respec- ti v ely . The measurement noise V k is also a Gaussian white noi se with zero mean and co v ariance matrix R k . The matrices F ΛΛ , F ΛΓ , and F ΓΓ represent the relationships between the state components. 3.2. Pr oposed solution Solving the dynamic state estimat ion requires determining the probability density function (PDF) of the state v ector X k , based on the observ ed measurements: P ( X k | Z k ) = P (Λ k , Γ k | Z k ) (14) where S k denotes the sequence { S 0 , . . . , S k } , with S representing X , Z , Λ , Γ . Using Bayes’ theorem, this PDF can be decomposed into tw o parts, corresponding to the linear and non-linear components of the state v ector: P ( X k | Z k ) = P (Γ k | Λ k , Z k ) P (Λ k | Z k ) = P (Γ k | Λ k ) P (Λ k | Z k ) (15) The probability density function P (Λ k | Z k ) , which addresses the non-linear component of the problem, is estimated using a particle lter . The recursi v e update is go v erned by Bayes’ theorem: P (Λ k | Z k ) = P ( Z k | Λ k ) P (Λ k | Λ k − 1 ) P ( Z k | Z k − 1 ) P (Λ k − 1 | Z k − 1 ) (16) The calculation of the conditional PDF P (Γ k | Λ k ) is performed under the assumption of kno wn pose at each time instant. Let Y k = Λ k − F ΓΓ Λ k − 1 . The dynamics equations for the state model are linear and Gaussian: ( Γ k = F ΓΓ Γ k − 1 + W Γ k , Y k = F ΛΓ Γ k − 1 + W Λ k . (17) Gi v en the linear Gaussian nature of these equations, the optimal solution for acceleration estimation is pro vided by the Kalman lter . The proposed algorithm for mar ginalized particle ltering in v olv es the follo wing steps: a. Filter initialization parameters: The N particles X i 0 are initialized according to the priori distrib ution: Λ i 0 ∼ P (Λ 0 ) , Γ i 0 ∼ P (Γ 0 ) , P 0 | 0 = P 0 (18) b . Prediction: At each time step k , the Kalman lter estimates the 3-D acceleration parameters for each particle using all a v ailable measurements up to k − 1 . Γ i k | k − 1 = F ΓΓ Γ i k − 1 | k − 1 (19) P k | k − 1 = F ΓΓ P k − 1 | k − 1 ( F ΓΓ ) T + Q Γ k (20) c. P articles propag ation: The particles are propag ated through the state space based on the dynamics equation (13), generating N random sequences according to the distrib ution P ( W Λ k ) . Λ i k = F ΛΛ Λ i k − 1 + F ΛΓ Γ i k | k − 1 + W Λ k (21) W Λ k ∼ N 0 , F ΛΓ P k | k − 1 ( F ΛΓ ) T + Q Λ k (22) Mar ginalized particle ltering for r eliable land vehicle navigation in ... (Abdelkabir Lahr ec h) Evaluation Warning : The document was created with Spire.PDF for Python.
2740 ❒ ISSN: 2088-8708 d. Updated estimation: The N Kalman lters are updated with the measurement using Λ i k and Λ i k − 1 : Y i k = Λ i k − F ΓΓ Λ i k − 1 (23) K k = P k | k − 1 ( F ΛΓ ) T F ΛΓ P k | k − 1 ( F ΛΓ ) T + Q Λ k − 1 (24) Γ i k | k = Γ i k | k − 1 + K k Y i k − F ΛΓ Γ i k | k − 1 (25) P k | k = P k | k − 1 − K k F ΛΓ P k | k − 1 (26) e. W eighting: The weighting step e v aluates the probability associated with each particle using Bayes’ rule. This step emplo ys the a v ailable measurement Z k at time k to compute the weight of each particle, p i k , according to (27): p i k = exp − 1 2 ∥Z k − h (Λ i k ) ∥ 2 R k P N j =1 exp − 1 2 ∥Z k − h (Λ j k ) ∥ 2 R k p i k − 1 (27) where ∥ · ∥ 2 R = ( · ) T R − 1 ( · ) . R is the measurement noise co v ariance matrix. This step calculates the weight of each particle by le v eraging the a v ailable observ ation Z k at time k . Depending on the GPS data a v ailability , three main scenarios are considered: − GPS measurements a v ailable: in this scenario, the lter utilizes GPS measurements e xclusi v ely to determine the v ehicle’ s parameters, such as acceleration, v elocity , and position. GPS data is considered a v ailable when pseudo-range measurements from at least four satellites are accessible. In such cases, the weights of the particles are determined based on the a v ailable GPS data to update the particle states accurately . p i k = exp − 1 2 ∥Z GPS k − h GPS (Λ i k ) ∥ 2 R GPS P N j =1 exp − 1 2 ∥Z GPS k − h GPS (Λ j k ) ∥ 2 R GPS p i k − 1 (28) − Limited GPS a v ailability: in the e v ent of partial GPS outages, the a v ailable GPS pseudorange measures are used e v en if fe wer than four satellites are in vie w , as the y still contain positioning information. T o a v oid indetermination, the predicted localization parameters of the state v ector are used to compute a GPS measurement using the lter . F or instance, if only three GPS satellites are a v ailable, the pseudo-range equation can be e xpressed as (29): P r j k = q ( x s,j k − ˆ x k | k − 1 ) 2 + ( y s,j k − y k ) 2 + ( z s,j k − z k ) 2 + cδ t (29) where j = { 1 , 2 , 3 } and ˆ x k | k − 1 represents the prediction of x k calculated by the lter . T o compute a partial GPS position Z PGPS k , the predicted position ˆ x k | k − 1 from the proposed lter , along with the least squares method, can be used when at least three pseudo-range measurements are a v ailable. When only tw o satellites are visible, the method requires using tw o components of the predicted state v ector , ˆ y k | k − 1 and ˆ z k | k − 1 , to solv e the GPS na vig ation problem. This approach ensures that e v en with limited satellite visibility , the system can still pro vide accurate positioning by ef fecti v ely le v eraging the a v ailable data. During periods of limited GPS a v ailability , the lter also inte grates odometer measurements along with data from digital road maps to estimate displacement. The primary challenge is aligning the pre viously predicted state with the map data. This in v olv es matching the predicted location of the v ehicle with the road netw ork on the map. There are se v eral solutions to address the multitar get multisensor tracking problem. These solutions include probabilistic data association and multiple h ypothesis tracking techniques [20], similitude funct ion mini- mization, as well as approaches that utilize fuzzy logic and belief theory [21]. Moreo v er , Mark o v decision processes with reinforcement learning of fer rob ustness ag ainst noisy data [22]. Int J Elec & Comp Eng, V ol. 15, No. 3, June 2025: 2735-2747 Evaluation Warning : The document was created with Spire.PDF for Python.
Int J Elec & Comp Eng ISSN: 2088-8708 ❒ 2741 In our solution, we ha v e opted for a probabilistic approach that emplo ys the Mahalanobis distance as a coherence metric [23], [24]. This metric is calculated based on the 2-D predicted map position: ˆ Z MAP k | k = h MAP ˆ X i k | k and each a v ailable map attrib ute, which includes the coordinates of nodes and the directions of se gments: Z MAP m , m = { 1 , ..., m max } . These normalized distances d m = ( Z MAP m , ˆ Z MAP k | k ) are computed by using e v ery node and considering the follo wing tw o possible directions of the road: d 1 ,m = min m h ( Z MAP ,1 m − b Z MAP k /k ) T ( e P MAP k /k ) − 1 Z MAP ,1 m − b Z MAP k /k ) i (30) d 2 ,m = min m h ( Z MAP ,2 m − b Z MAP k /k ) T ( e P MAP k /k ) − 1 ( Z MAP ,2 m − b Z MAP k /k ) i (31) where Z MAP ,1 m = ( x MAP m , y MAP m , θ MAP m ) and Z MAP ,2 m = ( x MAP m , y MAP m , θ MAP m + π ) e P MAP k /k represents the error co v ariance of the tw o-dimensional predicted map measurement. The most appropriate map data is identied by minimizing a distance criterion based on a predened thresh- old, i.e., Z MAP ∗ k = ar g min Z MAP [ d 1 m , d 2 m ] ≤ S Assuming that each sensor’ s measurement noise is independent, the log-lik elihood function of equation 27 can be reformulated as follo ws: ∥Z PGPS k − h PGPS (Λ i k ) ∥ 2 R PGPS + ∥Z ODO k − h ODO (Λ i k ) ∥ 2 R ODO + ∥Z MAP ∗ k − h MAP (Λ i k ) ∥ 2 R MAP f. Estimation: The global estimated state is traditionally pro vided by aggre g ating the weighted contrib utions of all particles. This can be mathematically e xpressed as (32): ˆ X k | k = N X i =1 p i k Λ i k T , Γ i k | k T T = N X i =1 p i k X i k (32) g. Resampling and redistrib ution: T o address the de generac y problem in the mar ginalized particle lter , a resampling procedure is emplo yed, duplicating particles with high weights and discarding those with lo w weights. W e use multinomial resampling, where particles are selected based on their wei g ht s compared to a uniform distrib ution [0 , 1] [25]. The number of ef fecti v e particl es N ef f is k ept belo w a threshold N thresh , calculated as (33): N ef f = 1 P N i =1 p i k 2 < N thresh (33) After resampling, the weights are normalized to p i k = 1 N . 4. EXPERIMENT AL RESUL TS In this section, we present e xperimental results to quantify the adv antages of the proposed mar gina l- ized particle lter (MPF) method. The e xperiments were conducted using a v ehicle dri v en in an urban area in Calais, France. The test v ehicle w as equipped with a No v atel GPS recei v er , which calculates the GPS posi- tioning. The measurement campaign w as conducted in an urban en vironment o v er a duration of 323 seconds. The v ehicle reached a maximum speed of 16 . 5 meters/second (approximately 60 kilometers/hour). The speed v aried signicantly due to frequent accelerations and de celerations, as well as stop-and-go situations caused by traf c lights and pedestrians. There were tw o periods where GPS positioning w as una v ailable due to an insuf- cient number of visible satellites: the rst period lasted 40 seconds and the second lasted 30 seconds. During these periods, the estimator used the a v ailable GPS pseudorange measurements, odometric measurements, and the road netw ork database as sho wn in Figure 1. The sensors used in the e xperiments ha v e the follo wing parameters: The GPS pro vides t he v ehi- cle’ s 3-D location at a frequenc y of 1 Hz when a v ailable, with an error co v ariance matrix set to R GPS = Mar ginalized particle ltering for r eliable land vehicle navigation in ... (Abdelkabir Lahr ec h) Evaluation Warning : The document was created with Spire.PDF for Python.
2742 ❒ ISSN: 2088-8708 [10 0 0; 0 10 0; 0 0 10] . The tropospheric delay is corrected using the Goodman and Goad model and the ionospheric delay is adjusted with the Klob uchar model [26]. The odometer measurements are tak en at a frequenc y of 1 Hz with an associated error co v ariance ma trix gi v en by R ODO = [0 . 5 0; 0 0 . 5] . This matrix accounts for primary sources of error , such as wheel skids and v ariations in wheel pressure. The road map is discretized into a 5 meter grid, with an error co v ariance matrix of R MAP = [25 0 0; 0 25 0; 0 0 0 . 1 ( rad )] . This discretization accounts for quantization noise and f acilitates matching within a wider neighborhood. The algorithm utilizes a total of N = 1000 particles with a resampling threshold set at N thresh = 2 N 3 . Figure 1. Reference trajectory of the v ehicle 4.1. V ehicle trajectory estimation This section, present the results of the proposed MPF applied to e xperimental data from a real urban transport scenario, focusing particularly on periods of GPS masking. During these periods, t he lter utilizes the a v ailable pseudo-range data, odometric measurements, and the digital road map database to estimate the v ehicle’ s trajectory . T o e v aluate the ef fecti v eness of our MPF algorithm, we calculated the trajectory error using data collected from the dif ferential GPS (DGPS) sensor . Figures 2 and 3 sho w the v elocity and acceleration errors in 3D for the v ehicle, respecti v ely . The GPS signal is partially una v ailable, lasting 40 seconds, between t = 104 and t = 144 . W e observ e that the joint use of the Dead Reck oning sensor and the digital road map database ef fecti v ely bounds the v elocity errors and accurately corrects the v ehicle’ s dynamic characteri stics, e v en when GPS data is una v ailable. The MPF approach ef fecti v ely constrains the kinematic errors when odometric data, partial GPS measurements, and digital road map data are used together . 4.2. Estimation of v ehicle positioning The mar ginalized particle lter sho ws enhanced accurac y in estimating the v ehicle’ s 3-D positioning error , as illustrated in Figure 4. During the rst period of partial GPS masking, which lasts 40 s from t = 104 s to t = 144 s, and the second period of partial GPS m asking, lasting 30 s from t = 249 s to t = 279 s, the combination of partial GPS measurements, the DR sensor , and the digital road map data pro v es ef fecti v e. This combination helps to bound the positioning errors and accurately corrects the v ehicle’ s dynamic characteristics e v en in the absence of full or partial GPS data. Int J Elec & Comp Eng, V ol. 15, No. 3, June 2025: 2735-2747 Evaluation Warning : The document was created with Spire.PDF for Python.
Int J Elec & Comp Eng ISSN: 2088-8708 ❒ 2743 Figure 2. V ehicle’ s 3-D acceleration error across time Figure 3. V ehicle’ s 3-D v elocity error across time Figure 5 illustrates the e v olution of the global dilution of precision (GDOP), a k e y performance crit e- rion for assessing GPS measurement quality , throughout the e xperiment. GDOP pro vides a measure of satellite geometry’ s impact on the accurac y of GPS positioning. During GPS outages, indicated by masking, neither GPS positions nor GDOP v alues can be calculated. The plot highlights periods of higher GDOP , notably from t = 150 to t = 228 and t = 255 to t = 273 , which correspond to the de g r aded quality of the 3-D positioning estimations. This ele v ated GDOP indicates poor satellite geometry , resulting in less accurate GPS localiza- tion during these interv als. The gure also presents the visible satellite count during the sequence, pro viding additional conte xt for interpreting the v ariations in GDOP . Figure 4. V ehicle’ s 3-D position error across time During GPS outage periods, we also plotted the time e v olution of the Mahalanobis dist ance, as sho wn in Figure 6. The v alues corr espond to the minimum distances calculated from potential candidates. The map- matching threshold w as e xperimentally set at S = 7 . Figure 7 displays the road netw ork o v erlaid with v arious estimated trajectories of the land v ehicle on a 2-D map. The estimated paths from dif ferent methods are plotted for comparison with the reference trajectory . The MPF estimated path is sho wn with red stars, representing t he trajectory calculated using the MPF . The reference path is depicted in green, representing the v ehicle’ s actual trajectory . The path es timated using dead reck oning (DR) methods is sho wn as an orange dotted line. Mar ginalized particle ltering for r eliable land vehicle navigation in ... (Abdelkabir Lahr ec h) Evaluation Warning : The document was created with Spire.PDF for Python.
2744 ❒ ISSN: 2088-8708 Figure 5. Ev olution of GDOP and satellite visibility across time Figure 6. Map-matching criterion during the GPS outage Figure 7. Comparison of true and estimated v ehicle trajectories Int J Elec & Comp Eng, V ol. 15, No. 3, June 2025: 2735-2747 Evaluation Warning : The document was created with Spire.PDF for Python.