Efficient Algorithm Research Articles

Application of Deep Reinforcement Learning to Defense and Intrusion Strategies Using Unmanned Aerial Vehicles in a Versus Game

Drones are used in complex scenes in different scenarios. Efficient and effective algorithms are required for drones to track targets of interest and protect allied targets in a versus game. This study used physical models of quadcopters and scene engines to investigate the resulting performance of attacker drones and defensive drones based on deep reinforcement learning. The deep reinforcement learning network soft actor-critic was applied in association with the proposed reward and penalty functions according to the design scenario. AirSim UAV physical modeling and mission scenarios based on Unreal Engine were used to simultaneously train attacking and defending gaming skills for both drones, such that the required combat strategies and flight skills could be improved through a series of competition episodes. After 500 episodes of practice experience, both drones could accelerate, detour, and evade to achieve reasonably good performance with a roughly tie situation. Validation scenarios also demonstrated that the attacker–defender winning ratio also improved from 1:2 to 1.2:1, which is reasonable for drones with equal flight capabilities. Although this showed that the attacker may have an advantage in inexperienced scenarios, it revealed that the strategies generated by deep reinforcement learning networks are robust and feasible.

Long-term outcomes of fistula-tract laser closure for complex perianal fistulizing Crohn's disease.

Fistula-tract laser closure (FiLaC™) has shown promising outcomes in perianal fistulizing Crohn's disease (pfCD). However, most studies assessed a mixed cohort encompassing pfCD and cryptoglandular fistulas during a short follow-up period. This study aimed to evaluate the long-term treatment outcomes of FiLaC™ in patients with complex pfCD. Data from patients with complex pfCD who underwent FiLaC™ during deep remission of Crohn's disease between January 2019 and December 2020 were retrospectively analyzed. Patient demographics, surgery history, and medication strategy were registered before surgery. Follow-ups were scheduled at 1, 2, and 3months after FiLaC™, and at 2-month intervals thereafter. The primary endpoint was clinic healing, while clinic remission/unhealed/recurrence were classified as unhealed. Additionally, adverse events and Wexner fecal incontinence score were documented. Forty-nine patients (40 men and 9 women) with a median age of 26.0 (19.0-35.5) years were included with a median follow-up of 50.0 (39.5-54.0) months. Of these, 31 (63.3%) patients achieved fistula healing, 3 (6.1%) experienced improvement, 3 (6.1%) remained unhealed, and 12 (24.5%) experienced recurrence. Montreal A category was lower in the healed group (P < 0.001). No major complications, such as bleeding or fecal or urinary incontinence, were observed, and pain was transient. The Wexner incontinence score decreased significantly at the last available follow-up, indicating an intact postoperative continence function (P = 0.014). PCDAI scores were significantly higher in the unhealed group (P = 0.041). FiLaC™ is an efficient and safe sphincter-saving procedure for patients with complex pfCD.

Ergonomic Analysis of Small-Scale Palm Sugar Starch Processing Industry in Rancakalong Village, Sumedang Regency

Rancakalong village, Sumedang Regency, has been known for its small-scale palm sugar starch processing industry since 2013. This industry produces starch from palm trees, which is crucial for food and beverages. Generally, the processes involved in processing palm sugar starch include (i) splitting of the palm, (ii) grating, (iii) coarse fiber screening, (iv) fine fiber screening, (v) harvesting, and (vi) drying. Initial evaluations indicated that the workers experienced physical discomfort during their work. Therefore, this study aimed to analyze the ergonomic aspects of processing palm sugar starch, including the working posture, standard time, workload, noise, and vibration. Based on the analysis, the risk level for each work element in processing palm sugar starch fell into the high-risk (score 8-10) and very high-risk (score 11-15) categories. These risks cause discomfort due to inappropriate work posture and duration, necessitating significant changes. The standard time required for each work element was as follows: 212.88±28.43 seconds/kg (palm splitting), 363.45±12.90 seconds/kg (grating), 95.08±9.74 seconds/kg (coarse fiber screening), 192.05±21.27 seconds/kg (fine fiber screening), 35.57±5.78 seconds/kg (harvesting), and 1821.01±41.09 seconds/kg (drying). Regarding workload analysis, processing palm sugar starch activities fell into the moderate category with total energy cost (TEC) values ranging from 92.66±1.50 265.55±3.88 kcal/hour. Regarding noise and vibration analysis, the grating work element was identified as the station with the highest exposure, i.e., 96.00±0.82 dB and 1.6±0.05 m/s2, respectively. The results of this study could be used as a basis for developing more efficient work procedures, maintaining health, and improving safety in the processing of palm sugar starch.

A Swarm Intelligence Solution for the Multi-Vehicle Profitable Pickup and Delivery Problem

Delivery apps are experiencing significant growth, requiring efficient algorithms to coordinate transportation and generate profits. One problem that considers the goals of delivery apps is the multi-vehicle profitable pickup and delivery problem (MVPPDP). In this paper, we propose eight new metaheuristics to improve the initial solutions for the MVPPDP based on the well-known swarm intelligence algorithm, Artificial Bee Colony (ABC): K-means-GRASP-ABC(C)S1, K-means-GRASP-ABC(C)S2, Modified K-means-GRASP-ABC(C)S1, Modified K-means-GRASP-ABC(C)S2, ACO-GRASP-ABC(C)S1, ACO-GRASP-ABC(C)S2, ABC(S1), and ABC(S2). All methods achieved superior performance in most instances in terms of processing time. For example, for 250 customers, the average times of the algorithms was 75.9, 72.86, 79.17, 73.85, 76.60, 66.29, 177.07, and 196.09, which were faster than those of the state-of-the-art methods that took 300 s. Moreover, all proposed algorithms performed well on small-size instances in terms of profit by achieving thirteen new best solutions and five equal solutions to the best-known solutions. However, the algorithms slightly lag behind in medium- and large-sized instances due to the greedy randomised strategy and GRASP that have been used in the scout bee phase. Moreover, our algorithms prioritise minimal solutions and iterations for rapid processing time in daily m-commerce apps, while reducing iteration counts and population sizes reduces the likelihood of obtaining good solution quality.

A Survey on Manta Ray Foraging Optimization Algorithm of Variants and Applications

This review paper delves into the original MRFO and its variants, focusing on single-objective algorithms including but not limited to hybrid algorithms, learning strategies, multiple populations and dynamic parameter adjustment, highlighting the improvements made to enhance the algorithm's efficiency in global optimization, accelerate convergence rates, and improve its capacity to evade local optima. MRFO has emerged as an effective tool for solving complex optimization problems across various domains, including energy optimization, biomedical field, engineering problems, and others. A comprehensive analysis of applications of MRFO in different fields is provided, emphasizing its adaptability and efficacy. The paper concludes with a discussion on the challenges faced by MRFO and potential future research directions, aiming to consolidate the current research status and guide future investigations.

An adaptive stratification algorithm based on gradient fitting deviation and its application to acoustic ray-tracing algorithm

Accurate underwater positioning is a challenging task for complex marine environments where the study of sound velocity profiles (SVPs) is essential. To overcome the problem that the original SVP reduces the computational efficiency of the ray-tracing algorithm due to great capacity of data, this paper simplifies the SVP. The nature of the SVP and its influence on accurate underwater positioning are discussed. An adaptive stratification algorithm applicable to the ray-tracing algorithm is proposed, based on the gradient fitting deviation and characterized by the fitting treatment of the sound velocity gradient by the Fourier method. Measured SVP data and acoustic positioning data were used to validate this algorithm in the context of acoustic ray-tracing algorithm, to analyze and compare the simplification rates before and after SVP simplification, and the effect on the positioning accuracy. The experimental results in the shallow sea (water depth of 60 m) show that the simplification rate of the SVP is more than 95.56%, and the RMS of the deviation of the ray-tracing algorithm is better than 0.0112 m when determining the optimal number of unfolding levels and threshold; the experimental results in the deep sea (water depth of 1900 m) show that the simplification rate of the SVP is more than 99%, and the RMS of the deviation of the ray-tracing algorithm is better than 0.5 m. This indicates that, the algorithm can reconstruct the original SVP well and implement the automatic hierarchical simplification of the SVP while ensuring the positioning accuracy.

Improved Model-Free Predictive Control of a Three-Phase Inverter

Model predictive control (MPC) performance depends on the accuracy of the system model. Moreover, the optimization algorithm of MPC requires numerous online computations. These inherent limitations of MPC hinder its application in power electronics systems. This paper proposes a two-part solution for these challenges for a three-phase inverter with an output LC filter. The first part of the control scheme is a linear and modified model-free approach based on the auto-regressive structure (ARX) with exogenous input. The second part is the computationally efficient optimization algorithm based on the active set method to solve the optimization problem of the MFPC. The objective of the control scheme is to regulate the output voltages of the inverter in the presence of constraints. The constraints are the maximum admissible filter current and optimal duty cycle to avoid any damage to the system. To validate the performance of the proposed control scheme, simulations and hardware-in-loop (HIL) real-time investigations have been performed, comparing the results of the proposed approach with the model-based predictive control. The results showcase the computational efficiency and effectiveness of the MFPC approach, demonstrating its potential for overcoming the limitations of traditional MPC in power electronics systems.

Sequential optimal experimental design for vapor-liquid equilibrium modeling

We propose a general methodology of sequential locally optimal design of experiments for explicit or implicit nonlinear models, as they abound in chemical engineering and, in particular, in vapor-liquid equilibrium modeling. As a sequential design method, our method iteratively alternates between performing experiments, updating parameter estimates, and computing new experiments. Specifically, our sequential design method computes a whole batch of new experiments in each iteration and this batch of new experiments is designed in a two-stage locally optimal manner. In essence, this means that in every iteration the combined information content of the newly proposed experiments and of the already performed experiments is maximized. In order to solve these two-stage locally optimal design problems, a recent and efficient adaptive discretization algorithm is used. We demonstrate the benefits of the proposed methodology on the example of the parameter estimation for the non-random two-liquid model for narrow azeotropic vapor-liquid equilibria. As it turns out, our sequential optimal design method requires substantially fewer experiments than traditional factorial design to achieve the same model precision and prediction quality. Consequently, our method can contribute to a substantially reduced experimental effort in vapor-liquid equilibrium modeling and beyond.

Revisiting Winnow: A modified online feature selection algorithm for efficient binary classification

AbstractWinnow is an efficient binary classification algorithm that effectively learns from data even in the presence of a large number of irrelevant attributes. It is specifically designed for online learning scenarios. Unlike the Perceptron algorithm, Winnow employs a multiplicative weight update function, which leads to fewer mistakes and faster convergence. However, the original Winnow algorithm has several limitations. They include, it only works on binary data, and the weight updates are constant and do not depend on the input features. In this article, we propose a modified version of the Winnow algorithm that addresses these limitations. The proposed algorithm is capable of handling real‐valued data, updates the learning function based on the input feature vector. To evaluate the performance of our proposed algorithm, we compare it with seven existing variants of the Winnow algorithm on datasets of varying sizes. We employ various evaluation metrics and parameters to assess and compare the performance of the algorithms. The experimental results demonstrate that our proposed algorithm outperforms all the other algorithms used for comparison, highlighting its effectiveness in classification tasks.

Efficient and Robust Adaptive Beamforming Based on Coprime Array Interpolation

Unlike uniform linear arrays (ULAs), coprime arrays require fewer physical sensors yet provide higher degrees of freedom (DOF) and larger array apertures. However, due to the existence of “holes” in the differential co-array, the target detection performance deteriorates, especially in adaptive beamforming. To address these challenges, this paper proposes an efficient and robust adaptive beamforming algorithm leveraging coprime array interpolation. The algorithm eliminates unwanted signals and uses the Gauss–Legendre quadrature method to reconstruct an Interference-plus-Noise Covariance Matrix (INCM), thereby obtaining the beamforming coefficients. Unlike previous techniques, we utilize a virtual interpolated ULA to expand the aperture, enabling the acquisition of a high-dimensional covariance matrix. Additionally, a projection matrix is constructed to eliminate unwanted signals from the received data, greatly enhancing the accuracy of INCM reconstruction. To address the high computational complexity of integral operations used in most INCM reconstruction algorithms, we propose an approximation based on the Gauss–Legendre quadrature, which reduces the computational load while maintaining accuracy. This algorithm avoids the array aperture loss caused by using only the ULA segment in the difference co-array and improves the accuracy of INCM reconstruction. Simulation and experimental results show that the performance of the proposed algorithm is superior to the compared beamformers and is closer to the optimal beamformer in various scenarios.

Computational assessment of Amazon forest plots regrowth capacity under strong spatial variability for simulating logging scenarios

In this paper, we assess the regrowth capacity of tropical forest plots by developing an original computational procedure based on statistical model checking methods. We calibrate a new mathematical model of forest dynamics with respect to post-logging data, produced in the Amazon basin. Our mathematical model is determined by a mechanistic system of ordinary differential equations and integrates a new nonlinear aging term, which is necessary to reproduce the complex post-logging dynamics of the tropical forest plots. Our method is based on an efficient algorithmic procedure that explores the parameter space of the model and computes a score for each parameter value, depending on the distance between the trajectory of the model and the data. We distinguish a group of four reference plots, for which the logging was the most intense, from another group of five non-reference plots, which are fitted a posteriori. Our results provide a set trajectories of the new model, which successfully fit the non-monotone post-logging data on each forest plot, in spite of the high level of biological variability identified in the study site. Rather than a unique set of parameters, we return a small cell of parameters, extracted from the parameter space, which contains in its close vicinity several relevant sets of parameters that are equally able to reproduce the regrowth dynamics of distinct tropical forest plots. The size of the cells that should be extracted from the parameter space increases with the level of biological variability. Finally, we show how to use the calibrated mathematical model for simulating logging scenarios, so as to better understand the temporal dynamics of forest regrowth.

Efficient Pareto based approach for IoT task offloading on Fog–Cloud environments

In recent times, a new paradigm has emerged in the field of Cloud computing, namely Fog computing. This paradigm has proven to be highly useful in a wide range of domains where both delay and cost were important metrics. Notably, the Internet of Things (IoT) strongly benefits from this, as small devices can gain access to strong computation power quickly and at a low cost. To achieve this, task offloading is used to decide which task should be executed on which node. The development of an efficient algorithm to address this problem could significantly enhance the sustainability of systems in various industrial, agricultural, autonomous vehicle, and other domains. This paper proposes a new variant of the Niche Pareto Genetic Algorithm (NPGA) called Local search Drafting-NPGA (LD-NPGA) to optimize resource allocation in a Cloud/Fog environment, with the objective of minimizing makespan and cost simultaneously. It generates Pareto solutions allowing the user to make choices closer to its intentions. Thus, it addresses various shortcomings identified in the state of the art, including scalability and aggregation formula. A drafting step is implemented to maintain diversity in the population of solutions, resulting in a more varied Pareto set than basic NPGA. LD-NPGA significantly outperforms state-of-the-art metaheuristics in makespan and cost by 15%. Finally, the scalability of our approach and the variety of solutions generated are confirmed in the different experiments.

Efficient Algorithms for Range Mode Queries in the Big Data Era

The mode is a fundamental descriptive statistic in data analysis, signifying the most frequent element within a dataset. The range mode query (RMQ) problem expands upon this concept by preprocessing an array A containing n natural numbers. This allows for the swift determination of the mode within any subarray A[a..b], thus optimizing the computation of the mode for a multitude of range queries. The efficacy of this process bears considerable importance in data analytics and retrieval across diverse platforms, including but not limited to online shopping experiences and financial auditing systems. This study is dedicated to exploring and benchmarking different algorithms and data structures designed to tackle the RMQ problem. The goal is to not only address the theoretical aspects of RMQ but also to provide practical solutions that can be applied in real-world scenarios, such as the optimization of an online shopping platform’s understanding of customer preferences, enhancing the efficiency and effectiveness of data retrieval in large datasets.

A knowledge graph algorithm enabled deep recommendation system.

Personalized learning resource recommendations may help resolve the difficulties of online education that include learning mazes and information overload. However, existing personalized learning resource recommendation algorithms have shortcomings such as low accuracy and low efficiency. This study proposes a deep recommendation system algorithm based on a knowledge graph (D-KGR) that includes four data processing units. These units are the recommendation unit (RS unit), the knowledge graph feature representation unit (KGE unit), the cross compression unit (CC unit), and the feature extraction unit (FE unit). This model integrates technologies including the knowledge graph, deep learning, neural network, and data mining. It introduces cross compression in the feature learning process of the knowledge graph and predicts user attributes. Multimodal technology is used to optimize the process of project attribute processing; text type attributes, multivalued type attributes, and other type attributes are processed separately to reconstruct the knowledge graph. A convolutional neural network algorithm is introduced in the reconstruction process to optimize the data feature qualities. Experimental analysis was conducted from two aspects of algorithm efficiency and accuracy, and the particle swarm optimization, neural network, and knowledge graph algorithms were compared. Several tests showed that the deep recommendation system algorithm had obvious advantages when the number of learning resources and users exceeded 1,000. It has the ability to integrate systems such as the particle swarm optimization iterative classification, neural network intelligent simulation, and low resource consumption. It can quickly process massive amounts of information data, reduce algorithm complexity and requires less time and had lower costs. Our algorithm also has better efficiency and accuracy.

Robust optimization approaches in inventory management: Part A—the survey

This work, the first part (Part A) of a comprehensive study, presents a survey on Robust Optimization (RO) in inventory management, highlighting its role in addressing uncertainties. This survey reviews various modeling frameworks, types of uncertainties, decision-making criteria, and decision rules essential to RO. The subsequent part (Part B) offers a comparative study, analyzing robust inventory models and highlighting key analytical and numerical contributions. This survey critically evaluates the effectiveness of RO in managing model uncertainty, enhancing decision-making processes, deriving structural insights, and boosting computational efficiency. Additionally, it discusses the limitations and challenges of applying RO to inventory management. While acknowledging the foundational role of traditional inventory literature in establishing essential theories, optimal policies, and efficient algorithms, this article addresses a significant gap by focusing on inventory management through the lens of RO methodology. Together with the comparative study, these works encapsulate the current state of robust inventory management, shedding light on future research directions and ongoing challenges in this evolving field.

Time-Series Feature Extraction by Return Map Analysis and Its Application to Bearing-Fault Detection

Developing new features for time-series characterization is a current challenge in data science and machine learning. In this paper, we propose a new metric based on a simple and efficient algorithm, namely, the return map. The return map analysis is well established in the field of non-linear dynamics, in particular, for fitting the parameters of a chaotic system from a waveform, or to attack a chaotic communication channel. We show that our metrics work well for both non-linear dynamics and time-series feature extraction problems in the field of machine learning. In an experiment aiming to classify vibration signals of normal and damaged bearings, we compare our method with two other methods that reported to have excellent accuracy, based on entropy and statistical feature distribution, respectively. We show that our method achieves higher accuracy with almost the lowest time costs, which was confirmed in experiments with two different datasets containing three main classes of bearings: normal, with inner race faults, and with outer race faults, having different damage origins and recorded in various conditions. In particular, for the dataset supplied by Case Western Reserve University, our method reached an accuracy of 100% at signals of 5000 sample points length, with a total time of 0.4 s required for feature estimation, while the entropy-based method reached an accuracy of 95% with a time of 100 s, and a statistical feature distribution method reached an accuracy of 93% with a total time of 1.9 s. Results show that the developed method is better suited to real-time bearing condition monitoring applications than most of the methods reported to date.

A cooperative learning-aware dynamic hierarchical hyper-heuristic for distributed heterogeneous mixed no-wait flow-shop scheduling

The distributed heterogeneous mixed scheduling mode in the manufacturing systems emphasizes the cooperation between factories for the entire production cycle, which poses enormous challenges to the processing and assignment of jobs. Discrepancies in the processing environment and types of machines of each factory during various production stages cause diverse processing paths and scheduling. The distributed heterogeneous mixed no-wait flow-shop scheduling problem with sequence-dependent setup time (DHMNWFSP-SDST), abstracted from the industrial scenarios, is addressed in this paper. The mathematical model of DHMNWFSP-SDST is established. A cooperative learning-aware dynamic hierarchical hyper-heuristic (CLDHH) is proposed to solve the DHMNWFSP-SDST. In CLDHH, a cooperative initialization method is developed to promote diversity and quality of solutions. A hierarchical hyper-heuristic framework with reinforcement learning (RL) is designed to select the algorithm component automatically. Estimation of Distribution Algorithm (EDA) guides the upper-layer RL to select four neighborhood structures. A dynamic adaptive neighborhood switching constructs the lower-layer RL to accelerate exploitation with the dominant sub-neighborhoods. An elite-guided hybrid path relinking achieves local enhancement. The experimental results of CLDHH and six state-of-the-art algorithms on instances indicate that the proposed CLDHH is superior to the state-of-the-art algorithms in solution quality, robustness, and efficiency.

DMT-OMPA: Innovative applications of an efficient adversarial Marine Predators Algorithm based on dynamic matrix transformation in engineering design optimization

This paper introduces an innovative variant of the Marine Predators Algorithm (MPA), termed the Dynamic Matrix Transformation-based Oppositional Marine Predators Algorithm (DMT-OMPA), aimed at enhancing the efficiency of engineering optimization strategies. Traditional MPAs have several shortcomings, including insufficient solution diversity and coverage in the initialization phase, a tendency to become trapped in local optima, and inadequate search capabilities in the later stages of iteration, all of which negatively impact the algorithm’s efficiency and effectiveness. To address these issues, the DMT-OMPA incorporates oppositional learning mechanisms and dynamic matrix transformation strategies, significantly enhancing global search capabilities and accelerating convergence speed, particularly in handling complex multidimensional optimization problems.Experimental results on the CEC2013 and CEC2017 test suites demonstrate that DMT-OMPA outperforms other recent MPA variants, various classical algorithm variants, and newly proposed algorithms, verifying its advantages in precision and reliability. Furthermore, the application of this algorithm to various real-world engineering problems substantiates its broad applicability and high efficiency. The study’s findings not only deepen our understanding of swarm intelligence optimization algorithms but also provide a new efficient tool for solving complex engineering problems. The results indicate a promising potential for wider application in diverse fields, suggesting that the DMT-OMPA algorithm could become an effective tool for tackling complex optimization problems in the future.

Evaluation of biochemical algorithms to screen dysbetalipoproteinemia in ε2ε2 and rare APOE variants carriers.

Dysbetalipoproteinemia (DBL) is a combined dyslipidemia associated with an increased risk of atherosclerotic cardiovascular diseases mostly occurring in ε2ε2 subjects and infrequently in subjects with rare APOE variants. Several algorithms have been proposed to screen DBL. In this work, we compared the diagnostic performances of nine algorithms including a new one. Patients were divided into 3 groups according to their APOE genotype: ε2ε2 ("ε2ε2", n=49), carriers of rare variants ("APOEmut", n=20) and non-carriers of ε2ε2 nor APOE rare variant ("controls", n=115). The algorithms compared were those from Fredrickson, Sniderman, Boot, Paquette, De Graaf, Sampson, eSampson, Bea and ours, the "Hospices Civils de Lyon (HCL) algorithm". Our gold standard was the presence of a ε2ε2 genotype or of a rare variant associated with triglycerides (TG) >1.7 mmol/L. A replication in the UK Biobank and a robustness analysis were performed by considering only subjects with both TGand low-density lipoprotein-cholesterol (LDLc) >90th percentile. Total cholesterol (TC)/ApoB and NHDLC/ApoB are the best ratios to suspect DBL. In ε2ε2, according to their likelihood ratios (LR), the most clinically efficient algorithms were the HCL, Sniderman and De Graaf's. In APOEmut, Sniderman's algorithm exhibited the lowest negative LR (0.07) whereas the HCL's exhibited the highest positive LR (29). In both cohorts, the HCL algorithm had the best LR. We proposed a powerful algorithm based onApoB concentration and the routine lipid profile, whichperforms remarkably well in detecting ε2ε2 or APOE variant-related DBL. Additional studies are needed to further evaluate algorithms performances in DBL carriers of infrequent APOE variants.

VE-LIOM: A Versatile and Efficient LiDAR-Inertial Odometry and Mapping System

LiDAR has emerged as one of the most pivotal sensors in the field of navigation, owing to its expansive measurement range, high resolution, and adeptness in capturing intricate scene details. This significance is particularly pronounced in challenging navigation scenarios where GNSS signals encounter interference, such as within urban canyons and indoor environments. However, the copious volume of point cloud data poses a challenge, rendering traditional iterative closest point (ICP) methods inadequate in meeting real-time odometry requirements. Consequently, many algorithms have turned to feature extraction approaches. Nonetheless, with the advent of diverse scanning mode LiDARs, there arises a necessity to devise unique methods tailored to these sensors to facilitate algorithm migration. To address this challenge, we propose a weighted point-to-plane matching strategy that focuses on local details without relying on feature extraction. This improved approach mitigates the impact of imperfect plane fitting on localization accuracy. Moreover, we present a classification optimization method based on the normal vectors of planes to further refine algorithmic efficiency. Finally, we devise a tightly coupled LiDAR-inertial odometry system founded upon optimization schemes. Notably, we pioneer the derivation of an online gravity estimation method from the perspective of S2 manifold optimization, effectively minimizing the influence of gravity estimation errors introduced during the initialization phase on localization accuracy. The efficacy of the proposed method was validated through experimentation employing various LiDAR sensors. The outcomes of indoor and outdoor experiments substantiate its capability to furnish real-time and precise localization and mapping results.