Improved Nelder-Mead based KF Method for Accurate Localization

Improved Nelder-Mead based KF Method for Accurate Localization | Authorea try { document.documentElement.classList.add('js'); } catch (e) { } var _gaq = _gaq || []; _gaq.push(['_setAccount', 'G-8VDV14Y67G']); _gaq.push(['_trackPageview']); (function() { var ga = document.createElement('script'); ga.type = 'text/javascript'; ga.async = true; ga.src = ('https:' == document.location.protocol ? 'https://ssl' : 'http://www') + '.google-analytics.com/ga.js'; var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(ga, s); })(); Skip to main content Preprints Collections Wiley Open Research IET Open Research Ecological Society of Japan All Collections About About Authorea FAQs Contact Us Quick Search anywhere Search for preprint articles, keywords, etc. Search Search ADVANCED SEARCH SCROLL This is a preprint and has not been peer reviewed. Data may be preliminary. 28 January 2025 V1 Latest version Share on Improved Nelder-Mead based KF Method for Accurate Localization Authors : Zhangqing Gu 0009-0001-6247-0177 , Shijin Ren [email protected] , Guosheng Hao , Zhenshuai Liu , and Bo Zhang Authors Info & Affiliations https://doi.org/10.22541/au.173808901.14339706/v1 281 views 120 downloads Contents Abstract NM-KF Combined Techniques for Data Filtering Experiment Validation Conclusion Information & Authors Metrics & Citations View Options References Figures Tables Media Share Abstract An improved Kalman filter (KF) method based on Nelder-Mead (NM) optimization is proposed to address the issues of slow convergence speed of KF, unstable state estimation, and challenging commencing parameter selection for model parameters in non-smooth environments. First of all, by employing the NM technique to determine the optimal covariance matrix initially value based on the smoothness indicators and root-mean-square error (RMSE), the convergence speed and stability of the KF are enhanced. Following that, a smoothing gain approach using historical information suppresses the problem of Kalman acquire fluctuation due to catastrophic noise, improving the stability and smoothness of state estimation. Last but not least, in order to boost the robustness, stability, and filtering influence of the KF model, decrease the filter noise sensitivity, and tackle the matrix inversion challenge, the attenuation factor and regularization term have been included to the updating process. The results of the experiments demonstrate that this paper’s method has excellent estimation accuracy and improves the filtering accuracy of straight and curved walking data by 44.6% and 85.5%, respectively, when compared to the existing filtering methods. When the GPS data is missing, the filtering accuracy of straight and curved walking data is improved by 80.6% and 81.1%, respectively. The technique presented in this paper eliminates the challenge of nonlinear models’ performance degradation when working with linear systems by significantly increasing the traditional KF’s convergence speed, stability, and robustness in complex dynamic environments. It additionally obviously improves the accuracy of navigation data for both linear as well as nonlinear systems. Introduction At the moment, accurate objects positioning is widely utilized in logistics (Huang et al., 2023), building (Tamura et al, 2019), manufacturing (Deng et al, 2024), vehicle navigation (Yuan et al, 2024), wearable technology (De Pace et al, 2023) and other fields, which significantly lowers production, safety, and management cost. Inertial Measurement Units (IMUs) and GPS continue to be employed in the majority of positioning systems today. GPS location signals, however, can be interfered with or blocked, which can result in issues including inaccurate placement and missing positioning information (Yang et al, 2023). Long-term positioning is not appropriate due to the issue of error buildup, even if IMU positioning data is immune to outside interference (Ji et al, 2024). Environmental noise also frequently affects the accuracy of the positioning system, so it is necessary to take appropriate steps to reduce noise interference. For combined GPS and IMU localization, Kalman filter (KF) and its deformation techniques are frequently employed (Pang et al, 2024;Nagui et al, 2021;Yang et al, 2019). However, when the localization error is elevated, the KF approach experiences divergence (Ge et al, 2024;Geng et al, 2001). By increasing the focus of the present observation, the attenuation factor enhances the data filtering impact. The filtering process can effectively suppress the filtering dispersion phenomenon by incorporating the fading factor (Xiang, 2019;Wang et al, 2018; Jiang et al, 2017) and attenuation factor (Guo et al, 2024;Li et al, 2022). Some researchers have suggested the Extended Kalman Filter (EKF) (Adrian et al, 2022), Unscented Kalman Filter (UKF) (Cahyadi et al, 2023), Volumetric Kalman Filter (Cubature Kalman Filter, CKF) (Li et al, 2024), and numerous other techniques to enhance the conventional Kalman filter in response to the issue that the KF method is ineffective for nonlinear systems with complex noise distributions. The choice of the initial condition value and the level of the system of nonlinearity have an impact on the EKF algorithm’s performance, despite its superior stability and speed of convergence (Sun et al, 2024). In order to effectively respond to the dynamics of the system, UKF uses a traceless transformation to generate multiple discrete points (including a center point and multiple discrete points determined by the covariance matrix) from the state distribution (Papakonstantinou et al, 2022). This avoids the complicated nature of the linearization process and the accumulation of errors within EKF while retaining the higher-order statistical information of the state distribution. Unfortunately, UKF is not appropriate for systems with high real-time needs since it necessitates nonlinear transformations and weight computations for every discrete point, which greatly raises the computational burden for each state update. Other than that, the expansion parameters have a greater impact on the UKF performance, and improper parameter selections might cause the filtering performance to deteriorate or possibly cause the UKF divergence (Liu et al, 2024). The utilization of deep models to increase the accuracy of location data has become a viable navigation and positioning technique due to deep learning’s strong modeling and feature extraction capabilities.Long Short-Term Memory (LSTM) neural networks were used by Zhang et al. (2019) to capture the dynamic relationship between GPS and IMU. The results demonstrate that LSTM can successfully build a combined GPS and IMU combined localization model, which is trained to establish the mapping relationship between the difference between the GPS localization data and the position data obtained by filtering and integrating the IMU data. Given the powerful local feature extraction capabilities of Convolutional Neural Networks (CNN) and the dynamic system modeling advantage of LSTM (Klus et al, 2021), Zhi et al. (2022) suggested a CNN-LSTM combined navigation data processing method that can still achieve high positioning accuracy in the absence of GPS data. The CNN-LSTM model is not appropriate for embedded positioning systems with limited arithmetic power, and it cannot meet the demands of applications with high real-time requirements and low-cost terminal equipment. It also requires a large amount of data for training, which necessitates high data quality and diversity of training data. Additionally, its generalization ability and adaptability to unknown environments remain difficult. The CNN-LSTM model is not appropriate for embedded positioning systems with limited arithmetic power, and it cannot meet the demands of applications with high real-time requirements and low-cost terminal equipment. It also requires an enormous amount of data for training, which necessitates high data quality and diversity of training data. Additionally, its generalization ability and adaptability to unknown circumstances remain difficult. The sensor sampling data frequently contains a significant amount of Gaussian and non-Gaussian noise, even if the actual GPS and IMU localization data of slow-moving objects are smoother. Directly filtering GPS and IMU data frequently yields unsatisfactory results. The multiscale analysis method, which is frequently used to remove noise from complex signals, can break down the signal into subbands of varying frequencies and filter each subband to extract the genuine data from the high-noise data. In order to reduce signal noise, traditional multiscale analysis denoising algorithms typically include thresholding techniques (Subhedar et al, 2023). Nevertheless, the filtered signal produced by this processing technique is very smooth, which may cause it to lose its transitory qualities and fail to faithfully capture its true properties. Signal processing has increasingly begun to use the Compressed Sensing (CS) approach, a novel kind of data filtering technique (Poian et al, 2018;He et al, 2023). It can successfully lower the sample amount of data and preserve the important information by sparsely representing and sampling the signal. Nevertheless, the compressed sensing method cannot successfully eliminate the noise and may even lose the key signal components for high-noise complicated dynamic systems since it is difficult to find a suitable sparse transform. By dynamically adjusting the regularization strength parameter and the corresponding ratio parameter of L1 and L2 regularization through data features, the elastic network, on the other hand, combines the benefits of L1 and L2 regularization and offers a more adaptable and reliable denoising scheme while striking an appropriate balance between sparsity and model complexity (Chen et al, 2018). Furthermore, the performance of Kalman filter models is heavily influenced by parameter settings, and due to deep learning models become more complicated, the right mix of parameter initialization is essential to model performance. In addition to being expensive, traditional manual parameter tuning techniques make it challenging to identify the ideal parameter combination. In light of the aforementioned research, this work suggests a Kalman (Nelder-Mead optimized Kalman Filter, NM-KF) exact positioning technique that is based on Nelder-Mead optimization and multi-scale analysis. The approach initially breaks down the data into subbands of varying frequencies based on the features of GPS and IMU data, respectively, using Tunable Q-factor Wavelet Transform (TQWT) and Wavelet Transform (WT). Elastic Net (EN) is then used to suppress the signal of the subbands at all levels. EN) to reduce the noise of the subband signals at every level. The original localization signals are then reconstructed using the filtered signals.In light of this, the Nelder-Mead optimization KF method is applied to enhance the localization data’s accuracy. The primary enhancements are as follows: (1) To enhance the stability of the filtering model and the smoothing of the filtered data, the regularization term is added to the KF covariance matrix, and the initial values of the KF parameters are determined using the Nelder-Mead optimization approach. (2) To increase the model’s resilience and flexibility, the smoothing gain matrix and attenuation factor are added, and the optimized parameters are utilized as the KF’s starting values. To sum up, this paper’s primary contributions are as follows. 1. A multi-scale analysis and Elastic Net (EN)-based denoising technique for localization data is suggested. The technique applies EN filtering to the wavelet coefficients of TQWT and WT decompositions, combining the benefits of L1 and L2 regularization: Whereas L2 regularization can efficiently smooth the regression coefficients to avoid overfitting and hence increase the model’s stability, L1 regularization can automatically choose features and eliminate redundant features. The elastic network finds the optimal balance between feature selection and feature smoothing when the two are used effectively together. When dealing with high-noise data, EN can efficiently eliminate high-frequency noise and smooth the residual coefficients to maintain signal continuity, prevent important information from being lost, and enhance signal quality and robustness. 2. A filtering approach is suggested that improves KF’s robustness and convergence stability. Through the use of regularization terms and attenuation factors, the technique enhances the filter’s resilience to noise and the stability of the filtering process. In order to guarantee matrix non-singularity and minimize Kalman gain fluctuations, the regularization term is added to the observation error covariance matrix. This accelerates convergence and smoothes the state update. The stability of the state estimation is much increased, the filter’s capacity to adjust to the measurement noise is improved, and the covariance matrix is prevented from convergent too quickly by the addition of the attenuation factor. Furthermore, a Kalman gain matrix optimization technique that incorporates historical data is suggested in order to improve the accuracy of data estimation and the robustness of the filter by suppressing the gain fluctuation in situations involving anomalous observations or high noise. 3. In order for the Kalman filter to better adapt to complex noise environments and nonlinear changes in the target operation process, the Nelder-Mead optimization algorithm-based initial parameter optimization method addresses the issues of time-consuming and inadequate accuracy of traditional manual parameter tuning. This effectively improves the accuracy and stability of model fusion filtering. Simultaneously, the approach circumvents the issue of adaptive algorithms’ inability to adjust to fast changes and their propensity for direction deviation or adjustment lag in complicated dynamic environments or high noise levels. 4. The rationality and efficacy of the NM-KF model are validated in this paper through comparison and ablation experiments with currently available techniques on self-collected datasets. These experiments also show that the model has higher stability, robustness, and estimation accuracy in non-stationary dynamic environments. Additionally, it alleviates the issue of traditional Kalman state estimation of nonlinear systems with declining performance and significantly improves navigation data accuracy for both linear and nonlinear systems. Related Work By dynamically modifying the model parameters based on local information during the iteration process, adaptive algorithms improve the model’s adaptability and increase the accuracy of data estimation. This allows the model to perform real-time optimization operations in response to changes in the environment.Through the State-Action-Reward-State-Action (SARSA) technique, Chen et al. (2021) proposed an adaptive EKF algorithm that uses a pruning procedure to eliminate the unsuitable process noise covariance matrix in order to achieve autonomous parameter selection. . Zhang et al. (2018) introduced the Adaptive Interactive Multi-model (AIMM) filtering method in EKF to improve the system’s adaptability to different navigation environments through the soft-switching feature and used the Sage adaptive filtering to adjust the online measurement covariance to improve the algorithm’s robustness. Khalaf et al. (2017) estimated and compensated the mean and covariance of the noise online based on the adaptive filtering principle by tightly coupling the structural processing and the unmodeled residual noise in the measurements. On the other hand, improper initialization of the model parameters may result in a sluggish convergence of the algorithm and instability during the convergence process. In the meantime, the adaptive algorithm’s local adjustment features can restrict the scope of its search and raise the possibility that it will encounter local optimal solutions. Rapid changes in a complex dynamic environment may surpass the algorithm’s adaptive capacity, resulting in adjustment lag or failure. In a high noise or complex dynamic environment, the high noise interference will cause the gradient estimation to deviate, creating a changing adjustment direction. To determine the best KF model parameter solution, offline optimization is used by Bayesian, Gray Wolf, and other algorithms. This lowers the system’s computational complexity in real-time applications and increases system operation efficiency. Simultaneously, it can successfully avoid falling into the local optimal solution thanks to its strong global search capability, improving the model’s resilience and adaptability and getting around the drawbacks of adaptive methods that struggle to make adjustments in complex dynamic environments with high noise levels. By optimizing the noise parameters of the extended Kalman filter using Bayesian Optimization (BO) algorithms, Chen et al. (2019) reduced the problem modeling to a Bayesian optimization task and adopted the Normalized Innovation Squared (NIS) residuals as performance metrics. This allowed them to search the solution space efficiently and prevent local optima by maximizing the probability of improving the current optimal solution. The search path may diverge from the ideal answer, though, because the Bayesian approach is more susceptible to noise and might not accurately capture the objective function in high noise conditions. Because of this, Pang et al. (2024) developed an enhanced Gray Wolf Optimization algorithm to address the slow convergence issue with the original Gray Wolf Optimization (GWO) algorithm. This algorithm uses a nonlinear combinatorial parameter tuning tactics to optimize the Kalman filter’s process noise covariance matrix and observation noise covariance matrix in order to increase the filter’s prediction accuracy and robustness. The Gray Wolf optimization algorithm is sensitive to the initial parameters, though, and if they are not set correctly, the search process may veer off course. In complex problems, it is also simple to linger around the local solutions, which can result in unstable convergence. Conversely, the Nelder-Mead algorithm can efficiently exit the local optimal solution by using the direct search strategy (Hassan et al, 2023), which is progressively modified and converges at the simplex vertex. It is highly efficient and robust, making it ideal for rapidly identifying the best parameter combinations in challenging optimization problems. Furthermore, the accuracy of the model’s state prediction is directly impacted by the quality of the data; if the original signal has more noise, the conventional filtering method will simultaneously eliminate the noise, losing some crucial information. In order to preserve more important information, the multiscale analysis approach provides a mechanism for breaking down the original signal into several subbands of varying frequencies. The noise in each subband can then be efficiently eliminated by filtering it. Consequently, complicated signal processing in the domains of pictures (Alquran et al, 2024), medical signals (Jacobsson et al, 2023), sensor data (Yuan et al, 2022), etc., frequently employs multiscale analysis techniques. Wavelet transform (Yu, 2021), empirical modal decomposition (Bonnet et al, 2014), and other techniques are common multiscale analytic techniques. The GPS positioning data typically shows low-frequency components when the object is moving slowly. The wavelet transform (WT) and discrete wavelet transform (DWT) (Yoo et al, 2015) break down the GPS signals into various frequencies of the wavelet coefficients in order to efficiently filter out the high-frequency noise and extract the important information from the low-frequency signals. The constant quality factor Q used by WT and DWT, however, makes it challenging to break down complex GPS signals into appropriate multi-scale frequencies. On the other hand, by varying the Q factor, the Tunable Q-factor Wavelet Transform (TQWT) may flexibly change the signal’s frequency resolution and time resolution (Liu et al, 2021;Ramkumar et al, 2024). As a result, TQWT is especially compatible with the frequency characteristics of GPS signals, which can handle complicated non-smooth signals and enhance the smoothness and stability of GPS signals in addition to efficiently filtering out transient high-frequency noise (Zhang et al, 2023). High-frequency noise is produced by IMU data being vulnerable to device jitter and noise interference during the sampling process. IMU data has a bigger volume than GPS data, and while TQWT is more flexible, it is more complicated to implement and takes longer to analyze a lot of IMU data, which makes it challenging to swiftly capture signal features (Li et al, 2020). Empirical Mode Decomposition (EMD) is more sensitive to noise (Mohsen et al, 2021), which may affect the extraction and reconstruction effect of IMFs. It may also be ineffective when dealing with IMU data that contains high-frequency noise, despite the fact that it can adaptively decompose the signal into a series of intrinsic mode functions (IMFs) without the need for wavelet basis functions. To guarantee the accuracy of acceleration and angular velocity signals, Wavelet Transform (WT) can, on the other hand, incorporate higher frequency resolution for high-frequency noise (Gourrame et al, 2023). This allows it to swiftly and efficiently remove high-frequency noise from IMU data while preserving low-frequency motion signals. NM-KF Combined Techniques for Data Filtering An enhanced Kalman parameter initialization tuning step, a Kalman state estimation stage initialized by optimal parameters, and a GPS and IMU data preprocessing stage comprise the NM-KF combined data filtering approach. The original data is broken down into subbands of varying frequencies using TQWT and WT, respectively, based on the GPS and IMU data properties. An elastic network is then used to process each subband in order to preserve the most crucial information while removing noise. To improve the KF, the regularization term is added during the initialization and tuning stage. Then, the Nelder-Mead optimization algorithm is used to choose the right initial values for the Kalman filter parameters based on the characteristics of the sensor data and the application scenarios. Then, the smoothing gain matrix and attenuation factor are added to further improve the KF, and the Kalman filter is set up and initialized using the adjusted parameters. Mercator projection is used to convert the GPS data’s longitude and latitude to X-Y plane space data (Bugayevskiy et al, 1995). The converted 2D plane space data of GPS positioning is then aligned with the inertial guidance unit data using time as the base, assuming that GPS navigation receives the data and at time . The nonlinear transformation then yields the actual 2D plane coordinates of the current work and, which can be expressed as follows: where and are the GPS longitude and latitude data, respectively; and are the longitude and latitude data transformed to the 2D plane one the x and y axis, respectively; is the Mercator projection method. Both the latitude/longitude 2D data and the IMU data are represented by the symbols and, respectively, and are broken down into subbands of various frequencies by TQWT and WT. The TQWT decomposition process can be expressed as follows: where is the input GPS signal, is the frequency-selective Q factor, is the redundancy factor, is the number of decomposition layers, and is the wavelet coefficients of the stage. The WT decomposition process can be expressed as follows: where is the input IMU signal, is the wavelet coefficients of the level, is the wavelet basis function, and is the number of levels of wavelet decomposition. Elastic networks, a denoising technique based on L1 and L2 regularization, can preserve more significant information while filtering out the noise, addressing the issue that the soft threshold denoising method will over-smooth the filtered signal. The following elastic network optimization issue, can be solved to yield the denoised signal, which can be expressed as follows: where is the input feature matrix, is the parameter vector of the model, is the target variable, is the length of the wavelet coefficients of each layer, is the regularization strength, and is the proportion of L1 and L2 regularization weights. The elastic network removes the high frequency noise from the data by processing the wavelet coefficients of each TQWT and WT decomposition layer independently, which can be expressed as follows: where is the unit matrix that is produced using the length of each layer’s wavelet coefficients, is the TQWT decomposed wavelet coefficients following the denoising process, and is the WT decomposed wavelet coefficients following the denoising process. The filtered data from the GPS and IMU decompositions are reconstructed using the processed wavelet coefficient sets and, respectively, which can be expressed as follows: where is the initial signal length, is the GPS signal set with high-frequency noise removed, and is the IMU data set with high-frequency noise removed. The filtered GPS latitude and longitude data can be represented as, and the filtered IMU can be represented as . The GPS and IMU data must be properly physically aligned in order for the Kalman filter to effectively filter the GPS and IMU data fusion. After that, the initial correction amount of GPS positioning data can be expressed as follows: where and are the object’s position data on the x and y axis at the beginning, respectively. The relative displacement is determined by the initial correction amount by (9), when the object begins to move, which can be expressed as follows: where and are actual location data of the object on the and axes at time, respectively, and and are the relative displacement data of the object on the and axes at time, respectively. The GPS positioning dataset within the instant window is designated as, and the IMU dataset is designated as, in accordance with the aforementioned procedures. is the data length in this case. To match the requirements of the Kalman filter input state vector and streamline the data processing flow of the filtering process, the aforementioned GPS and IMU datasets are combined to create the dynamic dataset . The precise placement of position data on a two-dimensional plane is the primary focus of this paper. To improve computational efficiency and decrease the dimensionality of the data input, we compressed the dynamic dataset, kept only the most important data that significantly affects the positioning results, and updated the dynamic dataset as . KF is a linear dynamic system-based state estimation technique that combines the system’s state transfer model with the observation model to recursively update the state estimates using Bayesian theory (Wang et al, 2021). The more crucial phase in the KF updating stage is the Kalman gain computation, which can be expressed as follows: where is the information gain matrix at moment, is the uncertainty from moment to moment, is the observation matrix at moment, and is the observation error covariance matrix. It is clear from the state update formula and the state covariance matrix formula that a significant fluctuation in will cause the state estimation update value to deviate significantly from the real state while also causing the state covariance matrix to sharply increase. This will raise the uncertainty of the state estimation and decrease the accuracy of the predicted value. Because of this, the smoothing gain matrix is introduced, which minimizes the fluctuation amplitude of the Kalman gain, increases the robustness and estimate accuracy of the model, and successfully integrates the previous knowledge to undermine confidence in the current data (Jin et al, 2024). The matrix of smoothing gains in its iterative form can be expressed as follows: Where is the Kalman gain at moment, is the smoothed gain containing historical information, is the smoothed gain matrix containing historical information at the previous moment, and is the Kalman gain weight, with a value range of . A smaller will make the gain more dependent on the historical information, responding slower to the new observations, and the gain process is smoother, but it will reduce the system’s adaptability to the dynamics of the change. A larger will make the gain more dependent on the Kalman gain at the current moment, can respond faster to the new observation data, and the gain changes faster, but this could result in unstable state estimation. Therefore, the weight of this paper is 0.8. Furthermore, the non-singularity of the observation error covariance matrix must be guaranteed throughout the KF row changing procedure in accordance by (12), which can be expressed as follows: where is the state covariance matrix at moment, is the observation matrix at moment, and is the observation noise covariance matrix at moment . The state covariance matrix falls as the Kalman gain is adjusted through iteratively changing the filter parameters. will tend to zero when the system stochasticity is very tiny, and will also tend to zero when the measurement noise is very small. This will cause to tend to singularity, which will result in strong oscillation or even divergence in the state estimation. The regularization term (Zhang et al, 2023) can be added to the inverse process to guarantee non-singularity in order to prevent singularity and the inability to solve the inverse correctly. Which can be expressed as follows: where the value range of the regularization term is typically . To prevent a significant impact on the Kalman gain calculation process, the system weak noise can select a smaller value; when the system strong noise is high, a bigger value should be selected to keep from approaching the singular matrix (Matin et al, 2021). This paper chooses the medium value from a practical simulation. Additionally, the update phase, in which Kalman gain updates the state covariance matrix and the projected data, which can be expressed as follows: where is the vector of state transfer estimates between moments and, is the amplitude of the observations’ residuals at moment, and is the vector of state estimates at moment . The state covariance matrix is used in the update phase to estimate the smooth system’s actual state. However, in high noise situations, an attenuation factor is used to regulate the update amplitude of throughout the update process (Liu et al, 2019) to prevent the state estimate from becoming unstable and to avoid making the filter overly sensitive to noise. Which can be expressed as follows: where is the attenuation factor, its range of values is typically . A smaller attenuation factor causes the state covariance matrix to change more quickly, which is appropriate for systems with faster dynamic changes; a larger attenuation factor causes the state covariance to change more subtly, which is appropriate for systems with slower changes or more noise (Gabriel et al, 2024). After combining the aforementioned analysis, this publication selects to be 0.93. This study takes into account how the optimal combination of regularization term, smoothing gain, attenuation factor, and filter model parameters affects the filter model’s performance. To improve stability, the regularization term should be added first, followed by the smoothing gain and attenuation factor. This prevents the optimization results from falling into the local optimal solution and improves the state estimation’s smoothness and accuracy. Enhance KF training process status updates in the form can be expressed as follows: where is a vector of state estimates at moment, and is a vector of state transfer estimates from moment to moment . is a matrix of state transfers from moment to moment, is a matrix of control inputs, is a vector of control inputs, is a matrix of state transfer covariances from moment to moment, is a matrix of state covariances at moment, and is a matrix of process noise covariances; The improved KF update phases are shown as follows: where is the observation vector at moment, is the magnitude of the residuals of the observations at moment, is the observation matrix at moment, is the moment to moment state transfer estimation vector, is the moment state estimation vector at moment, is the observation error covariance matrix, is the Kalman gain at the current moment, is the moment-to-moment state transfer covariance matrix, is the moment-to-moment state covariance matrix, is the process noise covariance matrix, is the regularization term, is the Kalman gain weights, and is the decay factor. The accuracy of state estimation depends on the confidence in process and observation noise, which is directly impacted by the initial values of the process noise covariance matrix and observation noise covariance matrix . In this study, the NM method optimizes the enhanced KF to find the best initialization of and (Hu et al, 2023). So this objective function can be expressed as follows: where is a measure of smoothness, is the indicator weight (0.5 in this article), and are a set of initial covariance matrices that were manually established, and RMSE is the weighted root-mean-square error in the X-Y direction. Which can be written as expressed as follows: where the observation data in the and directions are denoted by and, and the prediction data in the and directions by and . The smoothness index is added to evaluate the overall fusion filtering results’ stability and smoothness while evaluating the model fusion filtering’s performance by RMSE. In order to minimize the objective function, the Nelder-Mead algorithm looks for an appropriate set of parameters in the search space. When the objective function’s convergence condition is met, the optimal parameter setting result is returned and saved, which can be expressed as follows: where is the convergence condition and and are the objective function values determined from the corresponding parameter values in the current iteration and the previous iteration, respectively, the optimization process is said to have converged when the difference between the objective function in two consecutive iterations is less than the convergence condition. At that point, the optimization is halted and the current parameter setting values are saved as the optimization results. When taken as a whole, Fig 1 displays the flowchart for implementing the NM-KF algorithm suggested in the present article: Fig 1 | Flowchart of NM-KF algorithm The following are the stages involved in fully implementing the NM-KF algorithm: Step 1: Mercator projection is used to translate GPS latitude and longitude data to X-Y 2D plane space; Step 2: Use TQWT and WT, respectively, to break down the GPS 2D and IMU data into distinct wavelet coefficients; use an elastic network to denoise each layer of wavelet coefficients; remove high-frequency noise to extract low-frequency position and motion data; and compress to create the dynamic data set . Step 3: Equipped the traditional Kalman filter with the regularization term . Step 4: Initializing and . using the Nelder-Mead method to optimize the and parameters of the enhanced KF of Step 3 and storing the best result if the convergence criterion is fulfilled; Step 5: Add a smoothing gain matrix and an attenuation factor based on Step 3 to further improve the Kalman filter; Step 6: Initialize the enhanced Kalman filter in Step 5 using the remaining parameters and the ideal solution found in Step 4; for Determine by equation (19). Determine by equation (20). Determine by equation (21). Determine by equation (22). Determine by equation (23). Determine by equation (24). Determine by equation (25). Determine by equation (26). Estimate output state result . end for Step 7: Visualization using raw GPS positional data and the results of Kalman fusion filtering. Experiment Validation In this experiment, the WTGAHRS3-232 inertial navigation attitude sensor from Witte Intelligence is chosen to collect data with a time step of 0.1s and write it to the TF card via ESP32 in a complex environment with numerous buildings, people, and traffic flows. Strolling along a designated path in order to gather several sets of experimental data for later experimental confirmation. With a total data volume of 7656 records, including time nodes, latitude and longitude, three-axis acceleration, three-axis angular velocity, and other data components, we primarily choose one of the more complicated data groups for testing in this work. Fig 2 displays the GPS raw data after the latitude and longitude data have been aligned with the IMU data and mapped to two dimensions using Mercator projection. Fig 2 | Acquired raw GPS data A. Choice of GPS data filtering technique As illustrated in Fig 3 and Fig 4, which illustrates the filtering effect of the GPS data in the X-Y direction, the GPS signal is divided into multiple subbands using TQWT, and the suitable parameters and are chosen through the elastic network to eliminate the high frequency noise while keeping the low frequency localization signal and reconstructing the GPS localization data. Fig 3 | The X-axis direction of the GPS filter plot Fig 4 | The Y-axis direction of the GPS filter plot According to the preceding figure, WT can effectively denoise, however some important information is filtered out since the filtered data is too smooth.Even if some high-frequency noise is eliminated, the DWT filtering effect is rather weak and is better suited for processing smooth signals. By altering the Q factor, TQWT, on the other hand, may efficiently reduce high-frequency noise while maintaining the low-frequency trend, preserving more signal information and making it better suited for processing GPS data that are not smooth. B. Choice of IMU data filtering technique The wavelet inverse transform is used to reduce the processed wavelet coefficients to the original signals after the IMU data is decomposed into multi-layer wavelet coefficients using WT and the parameters and are modified based on various signal characteristics using an elastic network to efficiently filter out the high-frequency noise. A smaller andshould be chosen because acceleration typically reacts to the object’s overall motion trend, whereas angular velocity changes more dramatically than acceleration and must be effectively suppressed by a relatively large and in order to effectively suppress the high-frequency noise. Consequently, in the experiments presented in this study, the angular velocity selects 0.01, selects 0.1, and the acceleration selects 0.001, selects 0.05. The results of the x, y, and z axis acceleration filtering are shown in Fig 5, 6, and 7, respectively, while the results of the x, y, and z axis angular velocity filtering are shown in Fig 8, 9, and 10. Fig 5 | Plot of the x-axis acceleration filter Fig 6 | Plot of the y-axis acceleration filter Fig 7 | Plot of the z-axis acceleration filter Fig 8 | Plot of the x-axis angular velocity filter Fig 9 | Plot of the x-axis angular velocity filter Fig 10 | Plot of the x-axis angular velocity filter The integration of the elastic network allowed WT to retain some important information in the high-frequency signals while denoising, which improved WT’s fidelity ability and ensured the signal’s smoothness and stability. It is evident that WT successfully smoothed the signal, significantly eliminated the noise spikes, and was able to follow the trend of the original signal better, particularly in the steep descent and recovery part of the performance. On the other hand, even though DWT can eliminate high-frequency noise, the filtered signal exhibits a clear step-like structure, particularly in the signal mutation section, and the curve is excessively smooth, which causes many of the original signal features to be lost. On the other hand, TQWT performs well when working with rapidly changing signals since it has a greater fidelity ability to efficiently eliminate high-frequency noise while maintaining more original signal information. The overall smoothness and utility of the filtered signal are impacted by this feature, too, as more noise is mistaken for signal features and kept. For the parameters and, the Nelder-Mead algorithm’s improved KF reconstruction results are displayed in Table 1 and Fig 11 The diagonal elements of the and parameters are listed in Table 1. The first five elements are the diagonal elements of the reconstruction, and the final two are the diagonal elements of the reconstruction. In Fig 11, the KF reconstruction result using the Nelder-Mead algorithm is shown in blue, and the KF reconstruction result using the regularization improvement is shown in orange. Table 1 | Nelder-Mead optimization results Optimization Models Parameter reconstruction results Kalman filter [0.00010005,0.00114119,0.00100191,0.00143104,0.00122434,0.01235189,0.06991046] Tik Kalman Filter [0.00010004,0.00114401,0.00101233,0.00146912,0.00109551,0.01257446,0.07046595] The results of the parameter reconstruction are shown to be improved by the addition of regularization, which also helps to mitigate the overfitting issue for this parameter. Fig 11 | Comparison graph showing how regularization affects optimization outcomes The figure illustrates how the KF with regularization lowers the objective function value and function evaluations in comparison to the original KF, reducing the number of iterations from 302 to 272, and increasing optimization speed by 9.93%. Regularization expedites the optimization process and facilitates the model’s quicker convergence to a better solution. As a result, incorporating regularization into the optimization process can boost its effectiveness and, to a certain degree, enhance its outcomes. The filtering impact of the modified KF is somewhat improved. The filtering results of straight lines and turns under normal circumstances are shown in Fig 12 and Fig 13, and the filtering results of straight lines and turns when the GPS signal is lost are shown in Fig 14 and Fig 15 These include the following: the red line represents the filtering result of the KF enhanced by NM optimization, the blue line represents the filtering result of the original KF, the green line represents the filtering result of the EKF, the black line represents the actual path, and the cyan dots represent the original positioning data with noise. Fig 12 | Comparison of filtering effect at NM-KF linear travel under normal conditions Fig 13 | Comparison of filtering effect at NM-KF turning travel under normal conditions In contrast, the method of this paper, NM-KF state estimation, is very close to the real path and shows strong tracking ability in all places, and the localization data curve is more smooth and free of obvious deviation. It is evident that on the straight-line trajectory, KF can effectively reduce the noise and the overall smoothness is higher, but there is still some deviation in some areas. EKF can also perform effective state estimation, but the overall effect is slightly inferior to that of KF, and there is even obvious deviation in some places. In the turning nonlinear trajectory, KF performs worse, especially at the bottom of the obvious deviation; EKF is more accurate at tracking path changes and is closer to the real path than KF, especially at the bottom area. In contrast, the NM-KF state estimation results of this paper’s method are the closest to the real path, and the filtering curves are smoother and better capture the path changes. In conclusion, it is demonstrated that, in everyday situations, KF works better in linear systems but worse in nonlinear ones, while EKF performs better in nonlinear systems but less well than KF in linear systems. The NM-KF method described in this study, on the other hand, substantially lowers deviation and jitter, improves stability and robustness, and maintains high estimation accuracy in nonlinear systems in addition to demonstrating higher estimation accuracy and noise immunity in linear systems. Fig 14 | Comparison of filtering effect at NM-KF linear travel when GPS signal disappears Fig 15 | Comparison of filtering effect at NM-KF turning travel when GPS signal disappears The trajectory data clearly exhibits unpredictability when the GPS signal is lost, as can be seen in Fig 14; the EKF filtering result has a significant divergence, but it clearly performs better than the KF. While the NM-KF clearly has a better divergence from the real trajectory than the EKF, the filtered data has a superior smoothing degree and the filtering output is extremely similar to the genuine trajectory data after numerous optimization rounds. The comparison of nonlinear trajectory filtering in Fig 15 shows that EKF has a higher filtering error than slightly growing NM-KF at the turn, and KF is not appropriate for nonlinear data filtering. The NM-KF exhibits a small deviation at the extreme change, but overall performance is better than that of the EKF. This is because the introduction of regularization limits the information of the filter to the current observation, causing a lag in the system’s response.In summary. The NM-KF has superior robustness in complex settings, efficiently suppresses noise, lowers deviations and fluctuations, and exhibits high filtering accuracy for both linear and nonlinear dynamic system filtering when compared to the KF and EKF. To provide a better illustration of the performance of different filtering algorithms, Table 2 compares the RMSE performance indexes of different filtering methods under normal conditions; Table 3 compares the RMSE performance indexes of different filtering methods when linear systems’ GPS signals are absent; Table 4 compares the RMSE performance indexes of NM-KF and other models when nonlinear systems’ GPS signals are lost; and Table 5 displays the NM-KF model’s ablation experiments under normal conditions. Comparison of performance indices. Table 2 | Performance comparison of various filtering algorithms under normal conditions Models RMSEx RMSEy KF straight track 1.65 1.95 turning track 5.99 8.83 EKF straight track 1.97 2.34 turning track 1.85 1.14 NM-KF straight track 0.87 1.02 turning track 0.93 1.38 According to the above table, NM-KF improves 26.1% and 44.6% in the X-Y direction when compared to KF and 38.1% and 53.8% when compared to EKF when on the straight line trajectory; on the turning nonlinear trajectory, NM-KF improves 85.5% and 82.1% in the X-Y direction when compared to KF, and improves 53.0% in the X axis direction and 38.6% lower in the Y axis direction. Table 3 | Performance comparison of various filtering algorithms when GPS signal lost Models RMSEx RMSEy KF straight track 2.10 2.51 turning track 5.99 8.83 EKF straight track 3.65 4.34 turning track 1.85 1.14 NM-KF straight track 1.32 1.45 turning track 1.24 1.73 According to the above table, in the more normal case, when the GPS signal was lost in the linear system, it had no effect on the nonlinear system except NM-KF. Despite this, NM-KF improved the X-Y direction error by 42.5% and 8.2% in comparison to the normal case, but it still improved the X-Y direction error by 79.3% and 80.6% in comparison to KF, improved the X direction error by 33.0% in comparison to EKF, and reduced the Y direction error by 51.8%. The NM-KF improved 37.1% and 42.2% over the KF in the X-Y direction, and 63.8% and 66.6% over the EKF in the X-Y direction, respectively, even though all of the linear system’s mistakes resulted in some degree of growth. Table 4 | Comparison of model performance indexes in nonlinear system when GPS signal lost Models RMSEx RMSEy KF straight track 1.59 1.89 turning track 6.08 9.19 EKF straight track 2.01 2.39 turning track 2.14 3.47 NM-KF straight track 1.32 1.17 turning track 2.89 1.74 According to the above table, the linear system’s model performance slightly declines in comparison to the typical scenario when the GPS signal is lost in the nonlinear system. While all of the nonlinear system’s faults rise to some degree, NM-KF outperforms KF by 68.9% and 81.1% in the X-Y direction, while EKF outperforms EKF by 35.1% in the X direction and drops by 49.9% in the Y direction. In conclusion, NM-KF exhibits high accuracy and filtering effect when the GPS signal is normal. It is significantly better than KF and EKF in the X-Y direction in linear trajectory, and it is significantly better than KF in nonlinear trajectory. While it is not as good as the performance of EKF, it is gradually catching up to filtering effect of EKF. The filtering performance may also be kept at a high level in linear and nonlinear systems where GPS signals are lost. This is a major benefit over KF and EKF, demonstrating superior robustness, stability, and noise immunity. Table 5 | Comparison of performance metrics for normal condition NM-KF model ablation experiments Models RMSEx RMSEy KF straight track 1.65 1.95 turning track 5.99 8.83 Improved KF (smoothing gain, attenuation factor) straight track 2.53 1.85 turning track 3.01 6.77 NM optimizes improved KF (Tikhonov Regularization) straight track 1.39 1.25 turning track 1.69 2.16 NM optimizes improved KF (Tikhonov Regularization, smoothing gain) straight track 1.35 1.08 turning track 1.63 1.82 NM optimizes improved KF (Tikhonov Regularization, smoothing gain, attenuation factor) straight track 0.87 1.02 turning track 0.93 1.38 Naturally, the high-frequency noise in the time series is effectively smoothed out and the error is greatly decreased, particularly in the cornering nonlinear trajectory, after the smoothing gain and attenuation factor are introduced. The NM algorithm optimizes the KF, which can effectively capture the changes in the data and determine the best initialization settings based on it. On the other hand, the attenuation factor enhances the filter’s adaptability to the rapidly changing environment and lessens its sensitivity to noise, and the error is greatly decreased, particularly in the curved nonlinear trajectory. In the nonlinear trajectory, the error is much decreased, increasing the system’s resilience and, to a greater degree, improving the estimation accuracy and stability. In conclusion, the NM-KF approach shown in this study outperforms KF in high-noise complicated dynamic situations. It can also adjust to the uncertainty in dynamic environments more effectively, which improves the accuracy of localization. Conclusion In order to solve the issues of slow convergence, unstable state estimation, and challenging parameter tuning of the Kalman filter in a high-noise, non-stationary environment, a solution based on the Nelder-Mead algorithm is given in this study. Before anything else the data preprocessing step breaks down GPS and IMU data using a multi-scale analysis method and substitutes an elastic network for the conventional threshold filtering method, which successfully removes high-frequency noise while keeping more crucial information and enhances the input data’s quality; Consequently, the regularization term greatly enhances the numerical stability of the filtering process by successfully mitigating the phenomena of filter dispersion and estimation jitter. Afterward, the historical details are presented using the Second, the smoothing gain is used to introduce the history information, which effectively suppresses the dramatic fluctuation of the Kalman gain in the high-noise environment and improves the smoothness of the state estimation. at the same time, the attenuation factor is introduced to smooth the response of the filtering process to the noise and prevent the quick convergence of the state covariance matrix, which would make the filter overly sensitive to the noise. Ultimately, the NM algorithm optimizes the starting values of the parameters and, further enhancing the filter’s performance and estimation accuracy. accuracy of estimation. 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Keywords algorithm digital signal processing optimal estimation sensing solution methods and algorithms Authors Affiliations Zhangqing Gu 0009-0001-6247-0177 Jiangsu Normal University School of Computer Science and Technology View all articles by this author Shijin Ren [email protected] Jiangsu Normal University School of Computer Science and Technology View all articles by this author Guosheng Hao Jiangsu Normal University School of Computer Science and Technology View all articles by this author Zhenshuai Liu Jiangsu Normal University School of Computer Science and Technology View all articles by this author Bo Zhang Jiangsu Normal University School of Computer Science and Technology View all articles by this author Metrics & Citations Metrics Article Usage 281 views 120 downloads .FvxKWukQNSOunydq8rnd { width: 100px; } Citations Download citation Zhangqing Gu, Shijin Ren, Guosheng Hao, et al. Improved Nelder-Mead based KF Method for Accurate Localization. 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