IntroductionAt 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.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.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.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.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 WorkBy 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.