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Set-Membership Filter (SMF) has been extensively studied for state estimation in the presence of bounded noises with unknown statistics. Since it was first introduced in the 1960s, the studies on SMF have used the set-based description as its mathematical framework. One important issue that has been overlooked is the optimality of SMF. In this work, we put forward a new mathematical framework for SMF using concepts of uncertain variables. We first establish two basic properties of uncertain variables, namely, the law of total range (a non-stochastic version of the law of total probability) and the equivalent Bayes' rule. This enables us to put forward a general SMFing framework with established optimality. Furthermore, we obtain the optimal SMF under a non-stochastic Markov condition, which is shown to be fundamentally equivalent to the Bayes filter. Note that the classical SMF in the literature is only equivalent to the optimal SMF we obtained under the non-stochastic Markov condition. When this condition is violated, we show that the classical SMF is not optimal and it only gives an outer bound on the optimal estimation., 8 pages, 3 figures

Both routine radiation surveys and incident response--as currently performed by human workers--are time-consuming and involve significant, potentially unanticipated radiation dose (especially for accident response where levels prohibit human presence entirely). Previous efforts addressed routine surveys using robotic systems, but they must also localize and characterize discrete sources when anomalies occur. To be effective, characterization must account for real-world complications, including multiple sources, environmental attenuation, spatial localization, and isotopic identification. Recursive Bayesian estimation (RBE) using grid-based estimators and particle filters has been previously investigated, but we eliminate previous open-space assumptions by including attenuation models and sensor position height in the sequential sampling strategy. The method also incorporates autonomous isotopic identification via gamma spectroscopy, which supports the attenuation modeling and improves the computational efficiency in multisource cases. In addition, survey points are autonomously optimized using the Fisher information. A radioactive decay model is implemented to generalize the method by addressing short-lived nuclides. The developed hardware is evaluated in multiple scenarios where no operator intervention is required, effectively eliminating operator dose uptake from localization/characterization in real-world scenarios. Two different mobile robots are used, demonstrating the portability of the data collection and analysis software. Also, the generalized complexity is documented to support planning in future scenarios.

Bai (2009) proposes recursive estimation for panel data models with interactive effects. We study the behaviours of this recursive estimator. The recursive formula is established that shows the behaviours of recursive estimators depend on the initial estimator, the population structure and the iterative steps. Under some general scenarios, we find that the recursive estimator becomes consistent after the first iteration from any initials. We also obtain the optimal number of iterative steps under some prescribed conditions. The central limit theorem of the recursive estimator is established when the initial estimator is OLS. Various simulations are conducted to support our theoretical findings.

Building occupancy estimation holds great promise for building control systems to save energy and provide a comfortable indoor environment. Existing solutions turn out to be lacking in practice due to their specific hardware requirements and/or poor performances. Recently, an light-emitting diode (LED) light sensor based occupancy estimation system, which is nonintrusive and does not require any additional hardware, has been proposed. However, the performance of the system is limited, especially in a complicated dynamic scenario. In this article, a Bayes filter with neural networks is proposed for the optimal estimation of occupancy based on light sensor data. Specifically, based on the formulation of Bayes filter, the posterior probability of the building occupancy can be decoupled into three components: The prior, likelihood, and evidence. The prior and likelihood are, respectively, estimated from a Markov model and an efficient single-hidden layer feedforward neural network (SLFN). Finally, the evidence can be obtained by the results of prior and likelihood. Real experiments have been conducted to verify the effectiveness of the proposed approach in two complicated scenarios, i.e., dynamic and regular. Results indicate that the proposed Bayes filter outperforms all the benchmark approaches. The impacts of the number of LED sensing units and the number of hidden layers for neural networks are also evaluated. The results manifest that the number of sensing units should be chosen based on the required performance and the SLFN is sufficient for this application.

This work develops a new recursive Bayesian estimation algorithm for gamma-distributed random variables. The time prediction and measurement update steps are developed, and both are shown to have analytic closed formed solutions under certain conditions. Furthermore, the update is shown to be unbiased, this is theoretically guaranteed by the choice of estimator and demonstrated numerically through Monte Carlo methods with a simple example.

To ensure an autonomous truck can operate safely in a dump area, it is crucial to detect a berm accurately in advance. However, there are two challenges. First, the berm is not a static terrain but a movable one because of soil dumping. Second, berms are often irregular in shape-they are neither straight lines nor smooth curves. We considered two types of possible existing methods, but only to find they are not accurate and can't provide height information. Therefore, this paper proposes a berm detection algorithm, which includes three steps. First, extract berm candidate 3D LiDAR points based on a 2D height difference grid map. Second, use a binary Bayes filter to build and update 3D dynamic probability grid maps. Last, use a fitting rectangle technique to recognize the berm. We call this algorithm a Probability Grid Berm Detection (PGBD) algorithm. Off-line experimental evaluations on PGBD carried on datasets show good performance, compared with two curb detection algorithms, which are Hough Transformation and Haar Wavelet Transformation. And the good performance of the PGBD algorithm is further verified in the real-time experiment.

This work proposes a marginal distribution multi-target Bayes filter with assignment of measurements to track multiple targets in the presence of an unknown and variable number of targets, clutter, and missed detections. Mathematically, the association of the measurements with either a target or clutter may be established by maximizing the joint likelihood function of the measurement partition, which leads to a two-dimensional assignment problem. By the introduction of detecting label, a handling approach for missed detections is also developed and is applied to the proposed filter. This filter greatly reduces the number of hypothesized targets or Gaussian terms by selecting the predicted probability density of a target or one of its multiple updated probability densities as its state distribution at each time step. Experimental results indicate that the proposed filter requires a less computational load than the existing filters and performs better than the efficient implementations of the δ-generalized labeled multi-Bernoulli filter for multi-target tracking at low and moderate clutter densities.

This paper will present an overview of indoor and outdoor drone localization methods. Outdoor scenarios almost always use a GPS with IMU. Indoor systems are using short-range sensors that are sensitive to the external conditions of the environment. Mostly used methods are optical flow and stereovision, while an ultrasonic transceiver system optimizes and provides high precision and orientation of the drone. An ultrasonic preceptor is integrated into a listener/beacon and can be used with referenced beacons inside a WSN. The Crossbow Cricket hardware platform, which is based on TDOA and RSS principle, is used for simulations and code development. The researched direction is the localization of referent nodes (beacons) concerning the listener which is mounted on a flying drone. For that purpose, a probabilistic approach is used, based on a Bayes filter, where the positions of the beacon can be observed like random variables. Considering that these distributions significantly vary from a Gauss distribution, it is appropriate to use a particle filter.

Moving towards comprehensive digitalization of production facilities, it is critical to know the location of work pieces, charges, or other objects of interest that change location over time during production. For the case of a limited traceability of these objects, we first present a theoretical approach that performs a recursive Bayesian estimation of the object’s location over time based on typical passage measurements in production (e. g. light barriers or RFID systems). The probabilistic method is based on a directed acyclic graph modeling the transfer and sojourn of the objects in the production network. Subsequently, the method is validated on simulated data while varying both size and measurement conditions of the process. The results show the benefit of the proposed method against a single estimation and demonstrate its potential for the application in real time scenarios.

In this paper, we present a frame-level constant bit-rate (CBR) control method using recursive Bayesian estimation (RBE) for Versatile Video Coding (VVC). An R - $\lambda $ model for rate control (RC) has handled the total texture and non-texture bits at a time and has worked reasonably well in High Efficiency Video Coding (HEVC). Nevertheless, if the rate estimation is inaccurately performed, that is, the $R$ and $\lambda $ values for a current frame cannot be linearly modeled with their respective values in the previous frames, the resulting RC performance is degraded. In our work, we adopt the RBE which alternates prediction and update steps not only to precisely estimate the rates, but also to allocate target bits based on the changes in the distortions of the previously coded frames, thus considering the rates and distortions simultaneously. Therefore, an elaborate RC can be performed especially at fluctuating frame complexities. Experimental results show that our RC method outperforms the RC of VVC Test Model (VTM-5.0) in terms of normalized root mean square error (NRMSE) with maximum (average) 34.95% (12.35%) improvement, and maintains higher visual quality consistency in terms of standard deviation of PSNR by 33.07% (22.34%) improvement for All Intra (AI), maximum (average) 44.82% (27.29%) and 22.54% (9.50%) for Low Delay (LD), and maximum (average) 47.35% (39.94%) and 30.35% (18.54%) for Random Access (RA), respectively, compared to the default RC method of the original VTM-5.0.

The Bayes filter is the basis for many state-of-the-art robot localization algorithms. In the literature, its derivation typically requires the robot controls to be chosen independently of all other variables. However, this assumption is not valid for a robotic system that is to act purposefully. The contribution of this letter is twofold: We prove that the Bayes filter is also exact for an autonomous system that chooses the controls depending on any subset of all observable variables. We further show how to augment the filter if a human agent chooses controls on the basis of parts of the state space that are not directly accessible to the robot. In this case, modeling the agent's purpose improves the pose estimate as the control selection provides additional information about the hidden state space. A careful derivation of the Bayes filter then leads to an additional pseudo measurement update step. Simulation and real-world experiments with a teleoperated mobile robot as well as evaluations on the KITTI dataset show that the localization accuracy significantly improves if we augment a particle filter with the proposed pseudo measurement update. Finally, we present an analytical example for an augmented Kalman filter, which leads to a more accurate estimate than the standard Kalman filter.

Modern production systems have numerous sensors that produce large amounts of data. This data can be exploited in many ways, from providing insight into the manufacturing process to facilitating automated decision making. These opportunities are still underexploited in the metal forming industry, due to the complexity of these processes. In this work, a probabilistic framework is proposed for simultaneous model improvement and state estimation in metal forming mass production. Recursive Bayesian estimation is used to simultaneously track the evolution of process state and to estimate the deviation between the physics-based model and the real process. A sheet bending mass production process is used to test the proposed framework. A metamodel of the process is built using proper orthogonal decomposition and radial basis function interpolation. The model is extended with a deviation model in order to account for the difference between model and real process. Particle filtering is used to track the state evolution and to estimate the deviation model parameters simultaneously. The approach is tested and analysed using a large number of simulations, based on pseudo-data obtained from a numerical sheet bending model.

We present a method to estimate two-dimensional, time-invariant oceanic flow fields based on data from both ensemble forecasts and online measurements. Our method produces a realistic estimate in a computationally efficient manner suitable for use in marine robotics for path planning and related applications. We use kernel methods and singular value decomposition to find a compact model of the ensemble data that is represented as a linear combination of basis flow fields and that preserves the spatial correlations present in the data. Online measurements of ocean current, taken for example by marine robots, can then be incorporated using recursive Bayesian estimation. We provide computational analysis, performance comparisons with related methods, and demonstration with real-world ensemble data to show the computational efficiency and validity of our method. Possible applications in addition to path planning include active perception for model improvement through deliberate choice of measurement locations., Comment: 7 pages, 6 figures, accepted to IEEE ICRA 2021

It has been argued that the trust-modelling problem is characterised by uncertainty, and the only coherent way to deal with uncertainty is through probability. Even though some of the trust models introduced for sensor networks employ probabilistic solutions mixed with adhoc approaches, none of them produces a full probabilistic answer to the problem. In this chapter we introduced a theoretically sound Bayesian probabilistic approach for calculating trust and reputation systems in WSNs. We introduced a new Gaussian Trust and Reputation System for Sensor Networks (GTRSSN), which we believe is a breakthrough in modelling trust in WSNs, as previous studies in WSNs focused on the trust associated with the routing

In this work, we present a novel scheme for nonlinear hyperspherical estimation using the von Mises–Fisher distribution. Deterministic sample sets with an isotropic layout are exploited for the efficient and informative representation of the underlying distribution in a geometrically adaptive manner. The proposed deterministic sampling approach allows manually configurable sample sizes, considerably enhancing the filtering performance under strong nonlinearity. Furthermore, the progressive paradigm is applied to the fusing of measurements of non-identity models in conjunction with the isotropic sample sets. We evaluate the proposed filtering scheme in a nonlinear spherical tracking scenario based on simulations. Numerical results show the evidently superior performance of the proposed scheme over state-of-the-art von Mises–Fisher filters and the particle filter.

In this paper we present an alternative approach to spoofing mitigation. We leverage the fact that under most attack modes, both the authentic and spoofed signals are received by the victim. Once an attack is detected by conventional spoofing detection means, the receiver scans the vicinity of each satellite signal’s correlation peak for secondary peaks. In the meantime, the navigation continues in dead reckoning mode, based on other sensors and the user’s dynamics model. A decision about which signals to trust is then cast in the position domain. Once sufficient secondary peaks are detected, several navigation solutions are created from combinations of main and secondary peaks. We show approaches to drastically reduce the number of considered solutions at every epoch. Through recursive Bayesian estimation of the likelihood of each set of peaks we determine unlikely sets that can be ignored in future epochs. We further reduce the computational load by sharing computations among peak combinations. As an example, we apply the procedure to the TEXBAT dataset. We demonstrate a detection of secondary peaks even in the scenario where the spoofer has the largest power advantage and recover the spoofed and authentic navigation solutions in all cases. We finally discuss how to select the authentic solution among the two using the results in the position domain, leveraging measurements from auxiliary sensors such as an IMU.

Research on wireless sensors represents a continuously evolving technological domain thanks to their high flexibility and scalability, fast and economical deployment, pervasiveness in industrial, civil and domestic contexts. However, the maintenance costs and the sensors reliability are strongly affected by the battery lifetime, which may limit their use. In this paper we consider a wireless smart camera, equipped with a low-energy radio receiver, and used to visually detect a moving radio-emitting target. To preserve the camera lifetime without sacrificing the detection capabilities, we design a probabilistic energy-aware controller to switch on/off the camera. The radio signal strength is used to predict the target detectability, via self-supervised Gaussian Process Regression combined with Recursive Bayesian Estimation. The automatic training process minimizes the human intervention, while the controller guarantees high detection accuracy and low energy consumption, as numerical and experimental results show., Comment: 6 pages, 4 figures, 2 tables, accepted at MED 2021

Autonomous mobile robots, an important research topic today, are often developed for smart industrial environments where they interact with humans. For autonomous movement of a mobile robot in an unknown environment, mobile robots must solve three main problems; localization, mapping and path planning. Robust path planning depends on successful localization and mapping. Both problems can be overcome with Simultaneous Localization and Mapping (SLAM) techniques. Since sequential sensor information is required for SLAM, eliminating these sensor noises is crucial for the next measurement and prediction. Recursive Bayesian filter is a statistical method used for sequential state prediction. Therefore, it is an essential method for the autonomous mobile robots and SLAM techniques. This study deals with the relationship between SLAM and Bayes methods for autonomous robots. Additionally, keyframe Bundle Adjustment (BA) based SLAM, which includes state-of-art methods, is also investigated. SLAM is an active research area and new algorithms are constantly being developed to increase accuracy rates, so new researchers need to understand this issue with ease. This study is a detailed and easily understandable resource for new SLAM researchers. ROS (Robot Operating System)-based SLAM applications are also given for better understanding. In this way, the reader obtains the theoretical basis and application experience to develop alternative methods related to SLAM.

Autonomous system or self-driving car needs to localize itself very frequently or sometimes continuously to determine its proper location that is essential to perform its navigation functionality. The probabilistic models are among the best methods for providing a real-time solution to the localization problem. Current techniques still face some issues connected to the type of representation used for the probability densities. In this paper, we attempt to localize the self-driving car using particle filter with low variance resampling. Particle filter is a recursive Bayes filter, non-parametric approach, which models the distribution by samples [1, 2, 3]. A specially modified Monte Carlo localization method is used for extracting the local features as the virtual poles [4, 5]. Simulations results demonstrate the robustness of the approach, including kidnapping of the robot’s field of view [6]. It is faster, more accurate, and less memory-intensive than earlier grid-based methods.

The probability hypothesis density (PHD) filter is a multi-target tracking (MTT) method based on random finite set (RFS), providing an approximation implementation of the multi-target Bayes filter by propagating the first order moment of multi-target posterior. The PHD filter not only takes into account the accuracy of target state estimation, but also has good real-time performance, which is very suitable for radar MTT applications. In order to apply the PHD filter to scenarios with phased array radar working in track-and-search (TAS) mode, this paper proposes a time-matching recursive PHD filter. In order to handle the MTT in a TAS scenario where adjacent directions of the surveillance area are not continuously scanned, the proposed filter propagate PHD of each sector independent of others. It also introduces the time-matching filtering framework and multi-sensor multi-target recursive filtering framework to update PHD multiple times for each single sector at each time step, which can handle the MTT in scenarios where sectors have different sampling rates. The proposed method is simulated using the Gaussian mixture model, and the results, compared with those of standard PHD filter, demonstrate the effectiveness of it.

In this work, we proposed an indoor location system that makes use of a Raspberry Pi embedded computer and WiFi signals to guide a person inside a region of the faculty of Electrical and Electronic at Universidad Nacional de Ingenieria, Peru. The main advantage with similar indoor location systems like beacons or RFID technology is that the presented system does not require additional hardware since it makes use of the pre-installed WiFi routers. The experimental tests show promising results, achieving a location accuracy of 92.31%.

In this paper, we propose a nonparametric Bayesian approach for Lindsey and penalized Gaussian mixtures methods. We compare these methods with the Dirichlet process mixture model. Our approach is a Bayesian nonparametric method not based solely on a parametric family of probability distributions. Thus, the fitted models are more robust to model misspecification. Also, with the Bayesian approach, we have the entire posterior distribution of our parameter of interest; it can be summarized through credible intervals, mean, median, standard deviation, quantiles, etc. The Lindsey, penalized Gaussian mixtures, and Dirichlet process mixture methods are reviewed. The estimations are performed via Markov chain Monte Carlo (MCMC) methods. The penalized Gaussian mixtures method is implemented via Hamiltonian Monte Carlo (HMC). We show that under certain regularity conditions, and as n increases, the posterior distribution of the weights converges to a Normal distribution. Simulation results and data analysis are reported.

Localizing and tracking the pose of robotic grippers are necessary skills for manipulation tasks. However, the manipulators with imprecise kinematic models (e.g. low-cost arms) or manipulators with unknown world coordinates (e.g. poor camera-arm calibration) cannot locate the gripper with respect to the world. In these circumstances, we can leverage tactile feedback between the gripper and the environment. In this paper, we present learnable Bayes filter models that can localize robotic grippers using tactile feedback. We propose a novel observation model that conditions the tactile feedback on visual maps of the environment along with a motion model to recursively estimate the gripper's location. Our models are trained in simulation with self-supervision and transferred to the real world. Our method is evaluated on a tabletop localization task in which the gripper interacts with objects. We report results in simulation and on a real robot, generalizing over different sizes, shapes, and configurations of the objects., Comment: Accepted in IROS2020

Due to the harsh and unknown underwater environment, the question of how autonomous underwater vehicles (AUVs) should navigate and maneuver, especially in a dynamic environment with changing flow patterns, is still largely open. This paper presents a systematic background flow sensing framework, which plays an important role in improving the navigation/control intelligence of AUVs. This flow sensing framework utilizes distributed pressure measurements of AUVs to estimate surrounding flow fields. The proposed method first determines the flow pattern/model around AUVs based on fast Fourier transform (FFT) spectrum analysis and then uses recursive Bayesian estimation and dynamic mode decomposition (DMD)-based modeling to identify model parameters. This method is capable of sensing background flow fields even in flow pattern changing environments, e.g., open waters in real-world scenarios, thus dramatically expanding the application scope of the existing flow sensing methods. Simulation results are provided to demonstrate the effectiveness of the proposed flow sensing method.

The Bernoulli filter is a Bayes filter for joint detection and tracking of a target in the presence of false and miss detections. This paper presents a mathematical formulation of the Bernoulli filter in the framework of possibility theory, where uncertainty is represented using {\em possibility} functions, rather than {\em probability} distributions. Possibility functions model the uncertainty in a non-additive manner, and have the capacity to deal with partial (incomplete) problem specification. Thus, the main advantage of the possibilistic Bernoulli filter, derived in this paper, is that it can operate even in the absence of precise measurement and/or dynamic model parameters. This feature of the proposed filter is demonstrated in the context of target tracking using multi-static Doppler shifts as measurements., Comment: 23 pages

Indoor positioning systems for smartphones are often based on Pedestrian Dead Reckoning, which computes the current position from the previously estimated location. Noisy sensor measurements, inaccurate step length estimations, faulty direction detections, and a demand on the real-time calculation introduce the error which is suppressed using a map model and a Bayesian filtering. The main focus of this paper is on grid-based implementations of Bayes filters as an alternative to commonly used Kalman and particle filters. Our previous work regarding grid-based filters is elaborated and enriched with convolution mask calculations. More advanced implementations, the centroid grid filter, and the advanced point-mass filter are introduced. These implementations are analyzed and compared using different configurations on the same raw sensor recordings. The evaluation is performed on three sets of experiments: a custom simple path in faculty building in Slovakia, and on datasets from IPIN competitions from a shopping mall in France, 2018 and a research institute in Italy, 2019. Evaluation results suggests that proposed methods are qualified alternatives to the particle filter. Advantages, drawbacks and proper configurations of these filters are discussed in this paper.

Localization of mobile agents is to estimate the locations of the mobile agents based on the distance and bearing measurements from neighbors or from surrounding environment, which has attracted tremendous attentions in the past several decades. Most of the traditional localization methods are passive, that the agents are passively localized based their obtained measurements. For the noises and sparsity of measurements, locating ambiguity remains a challenging problem in passive localization methods. Recent works propose active localization, which combines robot location perception and active navigation. It enables an agent to actively adjust its action based on its newest location and the newest observations, so that the agent can act properly to seek better localization, e.g. move properly to avoid ambiguity. This paper presents a survey for the state-of-the-art in filtering based active localization. We also discuss the limitations of recent active localization methods and the future research directions.

Parameter calibration is a burdensome yet essential part of robotics development which traditionally was done through manual calibration routines. Over the recent years, many authors have proposed to automatically calibrate parameters directly from the input data, during the operation of the robot. While the majority of methods are discussed within the context of a specific application and benefit from the particularity of the problem, some generic approaches are proposed that are applicable to a wide spectrum of problems. However in practice, they require re-implementation and customization every time applied to a different domain, due to coupling of formulations with the model, e.g. linearization steps, matrix decomposition, closed form solving, etc. In this paper, we exploit the expressiveness of general purpose probabilistic programming languages (PPLs) to build a generic online calibration framework that can estimate the parameters of arbitrary robotic systems during operation. The proposed approach, based on Bayes filter and Monte Carlo methods, only requires model specification and works as a black-box otherwise. Hence, it spans the generality to the implementation aspect of the calibration problem which facilitates a range of new applications, e.g. fast prototyping of arbitrary robots. We show a short PPL program is capable of calibrating kinematic, extrinsic, and noise parameters of a classic SLAM dataset with minimum knowledge about the system and the parameters.

Accurately locating fire is crucial in autonomous fire suppression systems. In order to minimize secondary damages during fire suppression, it would be idealistic to find the exact location of the fire and suppress it accordingly. However, due to low accuracy and high costs of sensor systems, the work of locating fire is still incomplete. This paper presents a probabilistic fire location estimation in indoor fire environments by fusing two rotary ultraviolet sensors. For this, two ultraviolet sensors are designed to have a narrow range of field of view and are scanned by servomotors to acquire ultraviolet signals for each angle. In order to solve uncertainty of the signals in fire environments, Bayesian estimation is applied to probabilistically estimate the fire location. For its validation, nine actual fire experiments were conducted in different locations. Errors between original and estimated fire locations were measured to analyze its performance.

Autonomous underwater vehicles (AUVs) have received increasing attention among scientific and engineering societies for their superior performances and promising future in marine environments. A key challenge to marine autonomy is the sensing of the background flow environment, which is important for navigation and motion control of AUVs. While existing flow estimation studies typically focus on (quasi-)steady flow fields for AUVs of well-defined shapes, this paper proposes a novel flow sensing algorithm that applies to any dynamic flows for arbitrary AUV shapes. The proposed flow sensing method assimilates distributed pressure measurements through coalescing recursive Bayesian estimation and dynamic mode decomposition (DMD)-based model reduction. To demonstrate the effectiveness of the proposed distributed flow sensing algorithm, both simulation and experimental results are presented on three testing prototypes of different shapes including a Joukowski foil, a triangle, and a rectangle.

Cooperative positioning (CP) is considered as a promising positioning method for multiple autonomous underwater vehicles (multi-AUVs), because CP is characterized by low cost and high precision. In this research, a novel autonomous underwater vehicle (AUV) CP algorithm is proposed to enhance the global localization accuracy of the follower AUV. However, in traditional CP algorithm, the positioning error is large under the condition that the outlier data exists in the observation, which happens commonly. So in this research, a novel CP algorithm based on the factor graph and maximum correntropy(FGMC) for AUV is proposed to enhance the global localization accuracy of the AUV. Different from the traditional algorithms, this presented FGMC-based CP algorithm implements mathematically the Bayes filter by converting the global function estimation problem into the local one. And furthermore, the maximum correntropy is used as the cost function in the factor graph to estimation problem, this can reduce the influence of outliers on positioning accuracy. FGMC based cooperative positioning algorithm is established to mathematically implement the Bayes filter by converting the global function estimation problem into local function estimation problem. Furthermore, the maximum correntropy is used as the cost function in the factor graph to estimate the variables. To demonstrate and verify the proposed algorithm, simulation and real tests in different scenarios are performed in this research. Compared with the traditional CP algorithms, the positioning error of the proposed FGMC cooperative positioning algorithm is obviously smaller than that of the other algorithms.