Bayesian-Based Indoor Factory Positioning Using AOA, TDOA, and Hybrid Measurements

This work proposes and assesses Bayesian-based localization methods using time difference of arrival, angle of arrival, and hybrid measurements. First, a Bayesian localization model is constructed, and the Markov chain Monte Carlo method approximates the target’s 3D posterior distribution. Then, the model’s performance is evaluated in a 3GPP indoor factory environment using a distributed antenna system with a centralized controller for synchronizing and merging information. The received signal strength is used to select a subset of available anchors, and the estimation accuracy is measured in terms of horizontal and vertical errors. The results show that the Bayesian framework meets the horizontal and vertical errors requirements of the 3GPP for commercial cases with over 100 measurements. The accuracy can be improved by acquiring more measurements or increasing the number of active remote radio heads. However, when information fusion is applied (hybrid model), increasing the number of active anchors decreases the estimation performance.

Abstract-This work proposes and assesses Bayesian-based localization methods using time difference of arrival, angle of arrival, and hybrid measurements.First, a Bayesian localization model is constructed, and the Markov chain Monte Carlo method approximates the target's 3D posterior distribution.Then, the model's performance is evaluated in a 3GPP indoor factory environment using a distributed antenna system with a centralized controller for synchronizing and merging information.The received signal strength is used to select a subset of available anchors, and the estimation accuracy is measured in terms of horizontal and vertical errors.The results show that the Bayesian framework meets the horizontal and vertical errors requirements of the 3GPP for commercial cases with over 100 measurements.The accuracy can be improved by acquiring more measurements or increasing the number of active remote radio heads.However, when information fusion is applied (hybrid model), increasing the number of active anchors decreases the estimation performance.

I. INTRODUCTION
I NDUSTRY 4.0, also known as the 4th industrial revolution, aims to revolutionize traditional manufacturing through automation and the use of modern technology, such as the Industrial Internet of Things (IIoT) [1].The IIoT creates a wireless network of connected machines, people, processes, services, and products [2], [3].A typical IIoT scenario comprises an ultradense network with different machines and connections [4], e.g., an indoor factory environment involves many connected devices and requires reliability in real-time between objects, machines, and workers.High-position accuracy is crucial in these environments to increase production, make timely deliveries, and make critical decisions.However, conventional localization methods are limited or sometimes impractical in indoor factory scenarios due to, e.g., low-power signal devices and obstructed Line-of-Sight (LoS) satellite links.Indoor factory scenarios have an even bigger challenge because environments containing a large density of machinery create a problematic radio condition for the factory, where the metallic machines' surfaces generate several signal reflections, and the production process may create random electromagnetic noises that interfere with the desired signal [5].The 3rd Generation Partnership Project (3GPP) has published in 2019 the Rel-16 [6], which covers the wireless channel model and signal propagation characteristics for different scenarios, including indoor factory environments.This document supports the implementation of channel models and can improve the study of indoor positioning systems [7].
Combining the challenging wireless channel for IIoT scenarios and the required high-positioning accuracy in indoor factory environments, new technologies and approaches must be developed to satisfy the positioning demands.Published in 2021, Rel-17 [8] provides requirements, additional scenarios, evaluations, and technical proposals on positioning enhancements aiming indoor factory scenarios.For commercial use cases, the accuracy for 90% of user equipment is defined as follows.
2) Vertical Position Accuracy: < 3 m.Nevertheless, for IIoT use cases, the accuracy is more stringent and defined as follows.

A. Indoor Positioning Systems Overview
In an indoor positioning system, depending on the radio access interface, propagation features, and the intended accuracy of the localization strategy, it is possible to use distinct measurements to estimate the user equipment location.The received signal strength (RSS) is commonly used in wireless localization systems when the channel model is known in advance.However, this measurement can be impacted by radio channel effects and intensified in indoor factory scenarios with high-signal reflections.Increasing transmission power to improve quality may also shorten the life of some energylimited devices.To this end, there are two common schemes used for wireless positioning technologies: 1) schemes based on the Angle of Arrival (AOA) [9], [10] and 2) the ones based on time difference of arrival (TDOA) [11].
Besides determining the measurement type, selecting an appropriate localization technique is essential.Traditional techniques and algorithms used for indoor positioning system, e.g., nonlinear least squares and maximum likelihood estimators [12], require a large number of measurements or many iterations to reach a suitable estimation accuracy.However, actions, such as acquiring a large data set or applying the same algorithm routines, are limited or even impractical in certain indoor applications.

B. Related Works
Ding et al. [13] proposed a passive 3D indoor localization system using WiFi signals and channel state information (CSI).The system divides the 3D space into regions and constructs a spatial radio map using CSI measurements.Features of the mobile target's location are characterized using CSI tensors and learned by a recurrent neural network model for accurate indoor localization.The proposed method has been implemented and evaluated on commodity WiFi devices with good results in terms of accuracy.Kaltiokallio et al. [14] proposed a method applying Bayesian filters for RSS-Based localization and tracking in indoor scenarios, where the filters demonstrated to be efficient and achieved low-estimation error.Madigan et al. [15] proposed a Bayesian network using the Markov chain Monte Carlo to estimate targets' positions.This promising alternative does not require preliminary measurements about the environment, and the only prior knowledge required is the relation between the random variables used to build the probabilistic graphical models that represent the Bayesian network.
Bayesian networks are a promising solution for indoor positioning problems.Alhammadi et al. [16] proposed a Bayesian network-based indoor 3D positioning method using Wi-Fi signal strength measurements and off-the-shelf equipment.In contrast, de Lima et al. [17] and Terças et al. [18] examined the impact of the number of measurements on Bayesian network performance and explore ways to improve the estimation through information fusion.Additionally, Hilleshein et al. [19] suggested an iterative Bayesian network method to minimize estimation error variance.In a nutshell, the utilization of these networks in indoor localization problems offers several advantages, including the representation of the probabilities, associated with different variables and the probability distribution over the location of an object, the efficient combination of information from multiple sources, robust handling of missing or unreliable data by performing probabilistic inference considering the available information, dynamic updating capabilities for tracking moving objects, scalability to handle large complex systems, and simple interpretation through the graphical representation of relationships between variables, all of which make Bayesian networks a suitable solution for indoor localization problems.

C. Problem Formulation and Contribution
In this work, we develop a Bayesian framework to improve positioning accuracy in indoor factory scenarios.Our framework can determine the 3D position of a single target at a time, and we evaluate localization accuracy by examining different measurement types and sensor fusion techniques.Implementing estimators in indoor factory environments faces numerous challenges, including the high density of scatterers and clutter sources, that cause multiple signal reflections on metallic machine surfaces.The probability of LoS in these situations depends on factors like clutter size, clutter density, and height of base station and remote radio heads [6], [7].To assess our Bayesian framework, we assume that most remote radio heads have a clear LoS, considering that clutters are not excessively obstructive for all remote radio heads, increasing the probability of LoS.We use the mean clutter signal measurement value as the estimator input.The framework accommodates a variable number of active anchors, as not all remote radio heads need to be active to acquire measurements.To select the most appropriate set of remote radio heads, we prioritize those with high RSS, considering those with low RSS are located farther from the target's actual position.Our goal is to optimally combine various measurements to minimize the average mean and variance of the target's position estimate while optimizing the number of measurements and active anchors.To our knowledge, no data sets are currently accessible for indoor factory scenarios; therefore, our experiments relied on simulations rather than real measurements.Consequently, we have assessed our results using the 3GPP performance indicators, and we anticipate that our work can serve as valuable benchmarks for future works.To summarize, this article's main contributions are as follows.
1) We formulate a Bayesian framework for determining the 3D position of IIoT devices in the 3GPP indoor factory scenarios [6], [7], by handling either AOA or TDOA.2) To the best of our knowledge, a Bayesian framework by applying information fusion has not been proposed earlier for localizing IIoT devices in the 3GPP indoor factory scenarios.Thus, compared to the prior works in [17] and [18], we propose a hybrid setup, where measurements are intelligently combined utilizing both AOA and TDOA measurements.3) We evaluate how the number of measurements and active anchors impact the performance concerning accuracy.4) We evaluate the proposed solutions using the performance indicators defined by 3GPP for commercial and IIoT factory scenarios.The remainder of this work is organized as follows.Section II presents the Bayesian networks, first introducing the probabilistic graphical models and the Markov chain Monte Carlo algorithms used to estimate the posterior distribution of the target position.In Section III, we discuss the localization model and present the evaluation scenarios.Section IV shows the results for the estimation in terms of the cumulative density function (CDF) and Violin plot of the error position estimation.Finally, Section V concludes this work and provides final remarks.

A. Probabilistic Graphical Models and Bayesian Networks
Bayesian networks utilize probabilistic objects to model and tackle complex problems.Essentially, they describe the joint probability distribution of a set of random variables through Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
their conditional dependencies, represented by directed acyclic graphs known as probabilistic graphical models with no cyclical relationships.This enables the calculation of the likelihood function, given prior knowledge about the random variables, allowing for event prediction and validation of the model and random variable assumptions against real-world data [20].Thus, Bayesian networks can be applied to localization problems while incorporating prior knowledge about the received samples, such as channel model, measurement type, error distribution, and environment size.This allows us to assess if our initial assumptions and prior knowledge are valid.Bayesian networks are depicted through directed acyclic graphs, the vertices in these models correspond to random variables, and the edges represent their conditional interdependencies [21].The random variables are assumed to be conditionally independent, meaning that each random variable is only influenced by its parent variables while being independent of its nondescendants given its parents.This property enables the factorization of the joint probability density function.For instance, consider a directed acyclic graph represented by G = (V, E), where V are the vertices and E are the edges (corresponding to the random variables and their interdependencies, respectively).According to [22], the joint probability density function, p(V), of the random variables X v , v ∈ V, that represents a Bayesian network model for G, is given by where pa(v) represents the parents of v.The conditional distribution of a random variable v given its parents is given by where child(v) yields all the children of v and \ means containing.In this context, when computing the joint distribution of v, it is necessary to consider only the vertices in V that are children or parents of v.However, finding the final joint distribution that satisfies all model conditions can be challenging depending on the complexity of the model, such as the number of variables and their associated distributions.To simplify the process and perform approximate inference of the Bayesian statistical model, it is necessary to use a sampling method, such as the Markov chain Monte Carlo method [23].This method utilizes Bayes' theorem and prior system knowledge to estimate the posterior distribution.According to [24], the posterior distribution in a Bayesian statistical model is estimated as follows: where H is the hypothesis of the system, and D is the observed data, p(H) is the prior distribution representing the initial hypothesis about the system parameters, p(D|H) is the likelihood function, and p(D) is a normalization factor that represents the probability of all possible values of the parameters in the sampling space.

B. Markov Chain Monte Carlo Sampling Approach
The Markov chain Monte Carlo method combines the concepts of Markov chain and Monte Carlo simulations to approximate a target distribution by generating a sequence of random samples [25].The Markov chain Monte Carlo algorithm explores the sample space by creating a collection of distributions (chains) whose initial value and state are arbitrary, then adjusting the chain state to align with the current random variable distribution and assigning a higher probability to those states.Thus, the resulting distribution represents the Markov chain, which, with a sufficient number of samples, will converge to an equilibrium distribution according to the law of large numbers2 [20].
Distinct algorithms can be utilized to sample the posterior distribution in Bayesian networks, such as Metropolis-Hastings, Hamiltonian Monte Carlo, Gibbs sampling, slice sampling, and no-U-turn sampler [20], [26].In general, Metropolis-Hastings-based algorithms generate samples by making a random proposal for random variable values and then accepting or rejecting the proposal based on certain criteria.
This work employs the no-U-turn sampler algorithm for Bayesian network inference due to its ability to adaptively find good estimates, avoid revisiting local spaces, and significantly reduce simulation time compared to the Hamiltonian Monte Carlo algorithm.Unlike random walk methods, such as the Metropolis-Hastings, no-U-turn sampler utilizes a recursive algorithm to identify a set of probable points in the target distribution.It automatically stops when it begins to retrace its steps.Efficient exploration of the sampling space is crucial, and the no-U-turn sampler achieves this by avoiding revisiting previously explored local spaces [26].It is worth mentioning that the appropriate choice of the prior distribution and input data impacts the convergence speed of the posterior distributions.

A. Evaluation Scenario
In our investigations, we use the small hall scenario from [8] (see Fig. 1).We denote the length of the scenario by L and its breadth by B. The scenario has a rectangular shape of size 120 × 60 m, with 18 remote radio heads equidistantly spaced by a distance d equal to 20 m and placed at known positions.The remote radio heads are positioned facing down at a height of 8 m.All remote radio heads operate at 28 GHz and are connected to a centralized controller installed on the cloud radio access network.The cloud radio access network is, in turn, responsible for synchronizing the anchors (remote radio heads), merging information, and estimating the position of a target device after acquiring a minimum amount of measurements from the distributed antenna system.The target's position is unknown to the centralized controller and is randomly located inside the network deployment area on a 3D Cartesian coordinate system.
The selection of active anchors collecting measurements is determined based on the highest RSS detected.At the beginning of the process, we ascertain the number of active anchors that will collaborate in the estimation.The distributed antenna system receives a set of measurements from the target device.Subsequently, the cloud radio access network analyzes the received data.It identifies which remote radio heads will most likely aggregate relevant information about the system, guided by the highest detected RSS.The rationale is that if the signal quality is poor, it may not aggregate enough knowledge about the system to estimate the random variables and may decrease the estimation performance, e.g., accuracy and processing time.By selecting a subset of anchors to collect measurements, we aim to assess the optimal number of active remote radio heads that can improve the accuracy of the estimation.The RSS measurements are affected by a log-distance shadowed path-loss model, and the 3GPP standardization body chose the alpha-beta-gamma (ABG) model as the path-loss effect in indoor factory scenarios after studying the models, as seen in [6] and [7].The ABG equation for LoS is given by where f c is the central frequency, d 3D i is the tridimensional Euclidean distance between the target and the ith receiver, X ABG σ ABG is the shadow fading which follows a normal distribution with zero mean and the standard deviation σ ABG = 4.3.

B. Measurement Metrics
We consider two distinct measurement metrics to design the source localization models: 1) the AOA and 2) the TDOA.

1) Angle of Arrival:
The AOA-based source localization method finds the position of the target node by using the AOA measurements collected by distinct nodes.We can collect two measurement angles, namely, the AOA azimuth (φ) and the AOA zenith (θ ) [27].The azimuth and zenith angles, respectively, correspond to the xy-and yz-plane as follows: where (X, Y, Z) represents the 3D target coordinate, (x i , y i , z i ) yields the coordinate of the ith remote radio head, X ASA is the azimuth angle spread of arrival (ASA) which follows a normal distribution with mean μ log 10 ASA and standard deviation σ log 10 ASA , and X ZSA is the zenith ASA (ZSA) which follows a normal distribution with mean μ log 10 ZSA and standard deviation σ log 10 ZSA .Table I shows the angle spread distributions in the indoor factory scenario [6], [7].

2) Time Difference of Arrival:
The TDOA-based source localization method finds the position of the target node by using the difference between the time of flight at the remote radio heads concerning a common reference anchor.This method requires synchronization to calculate the TDOA measurements properly.Assume the target emits a signal at instant 0 and the remote radio head receives it at time t i , the respective time of flight is where c is the speed of light, and X DS is the delay spread (DS), which follows a normal distribution with mean μ log 10 DS and standard deviation σ log 10 DS (see Table I).Assuming the anchor node with index i = 1 is arbitrarily selected as the reference node, the TDOA measurements are given by TDOA j = t i − t 1 (8) where j = i − 1, i ∈ {2, . . ., n}, and n is the number of remote radio heads.

C. Source Localization Procedure
Considering the evaluation scenarios under radio channel degrading effects (as described in Sections III-A and III-B), it is still possible to determine the unknown position of the target node by using the measurements gathered by at least three different receivers, i.e., via multilateration [28], [29].By using this concept, the target position is estimated within the intersecting region of overlapping circles, lines, or hyperbolas, which depends on the source localization method applied.In our model, the variable d 3D is a vector containing the distances between each anchor and the target. 3Therefore, if we take the coordinates' joint distribution that matches these distances, we have a distribution representing the target's position.The challenges in estimating the joint distribution of these coordinates emerge from the number of interdependencies between the random variables, often large and complex.Therefore, tackling this problem requires a technique to reproduce a complex joint distribution by exploring its interdependencies.As previously described in Section II, this can be achieved by using Bayesian inference.By employing the Bayes theorem, we obtain the target distribution based on the available prior knowledge about the joint distribution and new system evidence through a sampling method based on Markov chain Monte Carlo algorithms to generate samples following a given probability distribution and the random variables interdependencies.Next, we present the directed acyclic graphs that represent the Bayesian network for our system.

D. Designing Estimators for Localization Graphical Models
Throughout our investigations, we infer the likelihood of the random variables in distinct directed acyclic graph models and, as a result, estimate the target position.In each of the following directed acyclic graph model diagrams, the parameters inside the lozenge -x i , y i and z i -represent anchors' coordinates which are assumed to be known a priori by the centralized controller.The parameters inside the circles correspond to random variables whose distribution is based on prior knowledge.Next, we present the directed acyclic graph models and respective metrics for each case.
1) AOA-Based Bayesian Network: In the AOA-based network, the interdependencies between the random variables are represented by the directed acyclic graph model shown in Fig. 2.This model combines the azimuth and zenith angle measurements.The parameters inside the circles correspond to the AOA random variables, where φ is the azimuth angle, θ is the zenith angle, and σ φ i and σ θ i are the standard deviations of the measurements collected by the ith receiver point, and n is the number of receivers.
The assumptions on the distributions of the random variables constituting the respective AOA-based Bayesian network are given in Table II, where ∼ indicates that a random variable follows a specific distribution.We select the uniform distribution to initialize our prior knowledge of (X, Y, Z)that is, the random variables representing the target location coordinates -because we want the sample space to cover all the deployment scenarios, once the target can be at any place of the environment with equal probability.The upper bounds, L and B, are the length and breadth of our deployment scenario (120 × 60 m), and H is the height of the environment (10 m).Then, e φ i and e θ i are the errors associated to the φ i and θ i measurements, respectively.These are normally distributed with N (0, AOA ), AOA is the covariance matrix of the measurement error, whose main diagonal is a concatenated 3 In this case, the intersection region that represents the target's position estimation is given by overlapping each d 3D i .

TABLE II RANDOM VARIABLES DISTRIBUTION OF THE BAYESIAN NETWORKS
vector (since they are assumed to be independent) of σ 2 φ i and σ 2 θ i , representing the variance of the measurements collected by the ith receiver point.The variance follow a half-normal distribution with a large variances, so we guarantee that the actual variance value is acquired during the sampling interval.
As shown in Fig. 2, the vectors μ φ i and μ θ i depend on the anchor, the target coordinates, and the respective error variances.Based on (1), we define the joint distribution as where V is the set of all random variables of the joint distribution.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.2) TDOA-Based Bayesian Network: For the TDOA-based network, the interdependencies between the random variables are represented by the directed acyclic graph model shown in Fig. 3.
The assumptions on the random variables describing the respective TDOA-based Bayesian network are shown in Table II, where e TDOA j is the error associated with the TDOA j measurement, which are normally distributed with N (0, TDOA ).Note that TDOA is the covariance matrix of the measurement error, whose main diagonal is given by σ 2 TDOA j that is the variance of the jth measurement following a half-normal distribution with large standard deviation.Similar to the previous case, the joint distribution the random variables in the Bayesian network of Fig. 3 is given by where V is the set of all random variables of the joint distribution.
3) Hybrid-Based Bayesian Network: The hybrid network combines at least two different measurement metrics.When using this method, the centralized controller at cloud radio access network is responsible for synchronizing the remote radio heads and processing data fusion between different sources.The main idea is to provide more knowledge about the propagation environment to the sampling algorithm and improve the posterior distribution estimation describing the actual random variables in the graphical model.We combine the networks above to determine the Hybrid-based Bayesian network and its joint distribution.The directed acyclic graph model shown in Fig. 4 represents the random variables interdependencies of both time and angle.
The assumptions on the random variables describing the respective hybrid-based Bayesian network are presented in Table II.The joint distribution of the random variables in the Bayesian network for hybrid measurements is given by where V is the set of all random variables of the joint distribution.

E. Simulation Environment
We implement the model in Python and used the Markov chain Monte Carlo sampler available in the PyMC3 package to carry out approximate inference [30].This package provides distinct Markov chain Monte Carlo algorithms, such as Metropolis-Hastings, Hamiltonian Monte Carlo, slice and no-U-turn sampler [31], our chosen algorithm.The no-U-turn sampler algorithm adaptively finds a good step size, avoiding re-exploring local sampling spaces and ensuring a much shorter simulation time.Using this implementation framework, we can focus on modeling the system and implementing the directed acyclic graph to evaluate the results by applying the no-U-turn sampler.
To sample the distribution and estimate the target's position, we define that the no-U-turn sampler algorithm draws 2500 samples per chain, representing 10 000 samples in each run.We tune the step size to 1500 to approximate the acceptance rate of samples to 80%.The step size tuned means that we burn the first 1500 samples generated by the algorithm.The idea is to reduce the correlation between the approximated distribution and the initial state and ensure that our samples are from the high-probability region.The next chapter presents different results and analyses using our implementation.

IV. PERFORMANCE EVALUATION
We evaluate the proposed Bayesian-based source localization solution described in Section III-D.In our investigations, the target location is unknown by the algorithm and the measurements are generated according to the radio propagation channel model described in ( 4)-( 6) and (8).We assume the target's height to be 1.8 m, simulating the target as an worker with an average height, where the transmitting antenna is located on his helmet.The distributed antenna system collects the measurements and forwards them to a centralized controller installed on the cloud radio access network, which employs the no-U-turn sampler algorithm to sample the posterior distribution of the coordinates representing the target position (X, Y, Z) in space.
For each simulation run, we establish a specific count of active remote radio heads to participate in the estimation process.The cloud radio access network determines the selection anchors collecting AOA and TDOA measurements.This choice is based on evaluating the mean value of the initial five RSS measurements, opting for the set with the highest signal strength.
We assume that the clutter's height remains minimal, thereby enhancing LoS conditions among all devices.In our investigations, we do not directly incorporate multipath components.This decision was made to maintain the simplicity of our model, as including the selection and analysis of those components, would result in a more complex model and increase the complexity of estimations.In our work, the effects of multipath components and the resulting distortions in RSS measurements are represented by errors inherent in the RSS data.The proper analysis of these components can be applied to a different work.As previously stated, indoor factory environments present various challenges, including clutter sources.To overcome this problem and evaluate the performance of Bayesian Networks in indoor factory scenarios, we assume that the received measurements by the cloud radio access network to feed our algorithm is the mean value of the measurements received from the clutter.
To evaluate the performance of the proposed models, we employ the error distance metric given by the difference between the mean value of the estimated distributions and the real coordinates.Our performance evaluation focuses on the horizontal and vertical errors related to the plane xy and the coordinate z, respectively.The error distributions are averaged over 200 simulation runs.The target is in a different location in each iteration, and we feed the localization algorithm with new measurements.It is worth mentioning that for each target position, we run the algorithm four times and collect four samples per run; as a result, the error distributions are averaged over 3200 samples.

A. Vertical and Horizontal Error Distribution
In this simulation set, we compare the error distribution statistics (mean and standard deviation) using the Violin plot 4 among distinct models by varying the number of 4 A violin plot combines a box plot and a kernel density plot.It is used for visualizing multiple distributions at once for comparison.It summarizes the active anchors and the number of measurements.Anchors independently acquire AOA and TDOA measurements, and the no-U-turn sampler algorithm is used to estimate the tridimensional position.Figs. 5 and 6 show the vertical and horizontal error results, respectively.Our Violin plot shows the error position estimation, which is the difference between the mean value of the estimated random variables' distribution and the real coordinates.We can observe that obtaining a limited number of measurements leads to significant uncertainty in the estimates.In both AOA and Hybrid cases, collecting 35 measurements results in high-error variance.However, by increasing the number of measurements, the error distribution narrowed, the standard deviation decreased, and the mean distribution approached zero, indicating that the accuracy of coordinate estimation improved with more measurements.The same is valid for the TDOA-based model with 8 and 18 active anchors, even if the estimation is imprecise due to higher standard deviation values.
When considering four active anchors collecting TDOA measurements in Fig. 5, the mean value of the distribution is around 2.5 m, representing that the estimated z coordinate is around 5 m -the middle of the height scenarios -indicating inaccurate estimation.Similarly, by employing four active anchors collecting TDOA measurements in Fig. 6, the mean is closer to zero; however, the standard deviation is high, indicating high-estimation errors.This happens because, in TDOA-based models, one of the anchors is used as a reference to calculate the time difference between the remote radio heads.Therefore, three TDOA measurements are used to estimate the 3D location; however, this amount of measurements is insufficient to provide enough degree of freedom in the system.To solve it, we need to increase the number of active anchors.
By increasing the number of active anchors, we can improve the diversity of measurements and gather more location information, leading to a better posterior distribution estimation.As a result, the mean and standard deviation of the error distribution decrease.In the case of AOA-based models, activating 18 anchors instead of just four and collecting 35 measurements per remote radio head has been shown to improve the vertical error by 0.15 m and horizontal error by 0.20 m.This demonstrates that the improvement can be substantial even with fewer measurements if the measurement diversity is increased.Furthermore, suppose the number of measurements increases to 150 through new acquisitions or historical data (depending on the system's dynamics).In that case, the improvement can be even further, with vertical and horizontal errors improving by an additional 0.10 m.
In TDOA scenarios, the benefits of increasing the number of active anchors are related to the accuracy and precision of the estimation, as it helps reduce the mean error distribution toward zero and narrows the error distribution.Fig. 5 shows that when the number of active anchors increases, the mean value of the error distribution approaches zero, which indicates data set using five indicators: 1) the minimum (beginning of the grey line); 2) the first quartile (beginning of the gray box); 3) the median (white dot); 4) the third quartile (end of the gray box); and 5) the maximum (end of the gray line).
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.that the estimation accuracy has improved as the final position estimation is closer to the real target's location.In Fig. 6, we can observe that, by increasing the number of active anchors, the estimation precision is increased, and the standard deviation of the error drastically decreases, representing that the final estimation has a high probability in a specific location.
The Hybrid-based model combines features of both aforementioned scenarios.The best result regarding vertical estimation is associated with AOA scenarios, and the best result concerning horizontal estimation relates to TDOA measurements.However, by combining these two approaches, it is possible to acquire satisfactory results by activating fewer anchors, which can be beneficial from energy saving and computational effort perspectives.In Figs. 5 and 6, we can observe that, by using four anchors, the distribution of the vertical error slightly decreases because the AOA measurements cannot compensate for the lack of remote radio head to satisfy the degree of freedom related to TDOA measurements.Results show that the activation of 8 anchors has the potential to either preserve or improve the distribution of the vertical and horizontal errors without causing any deterioration.However, using 18 anchors in this scenario leads to a slight compromise in the final distributions, as the mean value and standard deviation are higher than using just one measurement type.This is because approaching information fusion leads to an increase in the number of interdependencies in the model, enhancing the understanding of the system and reducing uncertainty about the event.However, if the number of active remote radio heads is also high, the complexity of the model increases as well.Hence, one must match multiple parameters in the sampling stage, which increases the number of iterations for an accurate estimate since the model has more random variables and interdependences to solve.

B. Cumulative Density Function of the Estimation Error
The 3GPP standardization body provides the vertical and horizontal accuracy requirements for commercial and indoor factory IIoT in the technical report [8].Figs.7 and 8 show the CDF of the proposed models, varying the number of measurements and active anchors.The markers indicate the 90 th percentile of the CDF and represent the achieved model accuracy.
We observe that the AOA and hybrid localization models met the vertical error requirements.However, the TDOA localization model only achieved the desired commercial Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.performance when both the number of active anchors and samples were high.Results related to the TDOA-based model indicate that the estimation is inadequate when the number of active anchors is low.Even increasing the number of samples, the algorithm cannot estimate the coordinate z correctly because, with few active remote radio head, we have insufficient measurement diversity representing this random variable.However, this method has the best result for the plane xy when activating at least 8 anchors.
Fig. 8 shows that the requirements for IIoT cases were not achieved, but reaching the desired commercial accuracy for a few configurations is possible.The best method to achieve the bounds is increasing the number of samples, although this procedure increases computational cost and the time to collect measurements.By collecting more data, the posterior distribution is sampled more effectively, resulting in a more precise estimation, as indicated by the reduction of positioning error.The cumulative distribution functions for the three cases display a similar trend, with the horizontal error decreasing as the number of samples increases and showing a slight improvement when the number of active anchors increases.
Tables III   50 measurements.On the other hand, the TDOA-based model fails to achieve the desired accuracy.The only configuration where the desired accuracy is met is when 18 active anchors and 150 samples are collected by the remote radio heads.However, the error still exceeds 2 m.
Regarding HE, we can observe that the TDOA-based model performs best when we activate over 8 anchors, the opposite of the VE, where this approach performs worst.The AOAbased and Hybrid-based models exhibit good horizontal error performance, with error values below 1 m when more than 100 measurements are collected.We can observe that when the number of anchors is smaller than 8, the information fusion slightly degrades -about 2 cm -or outperforms the other approaches.However, if the number of anchors increases, the Hybrid-based model efficacy declines, and the TDOA-based model outperforms it.It is important to note that although the TDOA-based model has the best horizontal error performance, it does not accurately estimate the z coordinate; we cannot ignore the VE.
Therefore, it is observed that the increase in the number of active remote radio heads may not necessarily lead to improved estimation performance.In some instances, implementing information fusion can achieve results similar to or even better than cases with more active anchors.For example, activating 8 anchors to collect 150 hybrid samples can produce better results than activating 18 anchors to collect the same amount of AOA measurements.This highlights the importance of choosing the appropriate number of active anchors when collecting samples, as the wrong choice can negatively impact the estimation.Alternatively, resources, such as energy and computational processing, can be saved by collecting hybrid measurements with fewer active anchors.

V. CONCLUSION
In this study, we conducted a Bayesian-based localization analysis using distinct measurement types to approximate the tridimensional target node's location coordinates in an indoor factory scenario.Our investigation involved collecting measurements from multiple anchors operating at 28 GHz, positioned within the indoor factory scenario, with known positions.Our results indicate that the estimation accuracy can be improved by collecting additional measurements or activating more remote radio heads.However, combining different approaches makes it possible to achieve satisfactory results with fewer active anchors, which offers benefits from energy savings and computational cost perspective.Our analysis revealed that collecting hybrid measurements may slightly decrease estimation performance when the number of active anchors increases.The proposed methods met the vertical error requirements defined by 3GPP for commercial and IIoT cases when the anchors collected AOA or hybrid measurements.For commercial cases, horizontal error was achieved when more than 100 measurements were collected, but the same was not observed in IIoT cases.
Future work aims to improve horizontal error in IIoT cases.It will involve analyzing the impact of increased measurements and/or antennas per anchor and incorporating different measurement types to increase environmental knowledge and reduce estimation error.Research may also include implementing an iterative routine in which the model continually learns and improves accuracy with each estimation.We envision this research as a foundational step toward enhancing estimation outcomes in indoor factory scenarios.While our current study relied on simulations rather than real measurements, we see opportunities for experimental analysis in specific scenarios where signal data is accessible, thereby enhancing the depth and applicability of our findings.Furthermore, efficient strategies for selecting active remote radio head units should be explored, given that RSS measurements are highly distorted due to environmental factors.Additionally, when positioned directly above the remote radio heads, certain angle measurements cannot be accurately obtained, leading to increased estimation errors.

Fig. 5 .
Fig. 5. Violin plot of the vertical error for the proposed source localization by using different numbers of measurements, number of active anchors, and number of measurements.Additionally, each violin plot shows the error distribution's mean (μ) and standard deviation (σ ).

Fig. 6 .
Fig. 6.Violin plot of the horizontal error for the proposed source localization by using different measurements, number of active anchors, and number of measurements.Additionally, each violin plot shows the error distribution's mean (μ) and standard deviation (σ ).

Fig. 7 .
Fig. 7. CDF of the vertical error for the proposed source localization by using different measurements, number of active anchors, and number of measurements.

Fig. 8 .
Fig. 8. CDF of the horizontal error for the proposed source localization by using different measurements, number of active anchors, and number of measurements.
and IV show the 90% marker of the CDF for coordinate z and plane xy, respectively.The arrows' direction indicates the improvement of the results when we modify the simulation parameters.The vertical arrows show the error reduction of the number of acquired measurements, and the horizontal arrows show the reduction in the number of active antennas.The bold numbers in the Tables show the best results for each model.Results indicate that the AOA-based model demonstrates the most optimal performance for the coordinate z, with a marginal difference of less than 5 cm when the information fusion (Hybrid-based model) is applied to over Bayesian-Based Indoor Factory Positioning Using AOA, TDOA, and Hybrid Measurements Leonardo Terças , Student Member, IEEE, Hirley Alves , Member, IEEE, Carlos H. M. de Lima, Member, IEEE, and Markku Juntti , Fellow, IEEE

TABLE I FAST
FADING PARAMETERS MEASUREMENTS OF LOS INDOOR FACTORY SCENARIOS

TABLE III VERTICAL
ERROR (M): 90% MARKER OF THE CDF FOR COORDINATE Z