Space Research Articles

The Alpha Distance Formulae

In this study, we define the concept of tangent in two and three-dimensional alpha spaces concerning the alpha circles and the alpha spheres. Then using this concept, we derive the alpha distance formulae between points, a point and a line, between two lines and a point and a plane of the alpha spaces. Finally, we give simple area and volume formulas in the three dimensional space in terms of the alpha distances.

Collision Avoidance of UAV Formation Using Geometric Approach and Relative Velocity Vector in Three-dimensional Space

Collision Avoidance of UAV Formation Using Geometric Approach and Relative Velocity Vector in Three-dimensional Space

Accelerating quad Airy beams-based point response for interferenceless coded aperture correlation holography.

Interferenceless-coded aperture correlation holography (I-COACH) is a promising single-shot 3D imaging method in which a coded phase mask (CPM) is used to encode 3D information about an object into an intensity distribution. However, conventional CPM encoding methods usually lead to intensity dilution, especially in the recording of point spread holograms (PSHs), resulting in low-resolution reconstruction of I-COACH. Here, we propose accelerating quad Airy beams with four mainlobes as a point response to enable weak diffraction propagation and a sharp maximum intensity in the transverse direction. Moreover, the four mainlobes exhibit lateral acceleration in 3D space, so the PSHs in different axial positions show a unique and concentrated intensity distribution on the image sensor, thereby realizing a high-resolution reconstruction of I-COACH. Compared with conventional CPM encoding methods, the proposed accelerating quad Airy-beam-encoding method has superior performance in improving the resolution of I-COACH reconstruction even in the presence of external interference.

Reshaping the future of Design Education: Takeaways from the accounts of the ancient Indian Rishis

Design Education today is being imparted across the globe by varied institutions. We frequently witness new academic approaches, teaching methods, and pedagogies being discussed, and presented at various Design symposiums, and conferences, in the attempt of improving Design Education. While some investigate new strategies via experimentation and imaginative lenses, others investigate past wisdom to identify its relevance in the present context. The paper explores the latter through the teaching methods of ancient Indian teachers (also called Rishis) and seeks to determine its significance in the modern context. Ancient India had a long and illustrious history of learning and education. There are several examples that stand out and have valuable lessons for educators of today and future. The paper incorporates instances from Pt. Sriram Sharma Acharya's ‘Pragya Purana: Devsanskriti Khand (fourth volume)’, which contains references to Indian culture, gurukuls, and the old Indian educational system. The paper discusses three instances of which one presents an intriguing relationship between diet and education as explained by Maharishi Uddalaka. Another instance is Maharishi Uddalaka's description of how he teaches in the absence of any physical infrastructure, while the last instance is from Takshashila, where Acharya Brahaspati employs a novel method of assessing his Ayurvedic pupil before he graduates. The paper goes into additional detail about the instances cited above from the ‘Pragya Purana’, that demonstrate significant takeaways for the Design education.

Structure of Primitive Pythagorean Triples in Generating Trees

A Pythagorean triple is a triple of positive integers $(a,b,c)$ such that $a^2+b^2=c^2$. If $a,b$ are coprime, then it is called a primitive Pythagorean triple. It is known that every primitive Pythagorean triple can be generated from the triple $(3,4,5)$ using multiplication by unique number and order of three specific $3\times3$ matrices, which yields a ternary tree of triplets. Two such trees were described by Berggren and Price, respectively. A different approach is to view the primitive Pythagorean triples as points in the three-dimensional Euclidean space. In this paper, we prove that the triple of descendants of any primitive Pythagorean triple in Berggren's or Price's tree forms a triangle (and therefore defines a plane), and we present our results related to these triangles (and these planes).

Research on multi-beam survey line system based on dynamic programming algorithm

This paper establishes a calculation model for the coverage width and overlap rate of a multi-beam survey line system [1], and optimizes the multi-beam bathymetric survey plan for marine areas. Through the dynamic programming algorithm, a reasonable survey line plan is designed based on the actual marine area depth. The research content includes: Model Establishment: Construct a calculation model for coverage width and overlap rate, determining the coverage width and overlap rate at each position. Three-Dimensional Space Model: Using the survey vessel as the origin and combining marine area information, solve for the coverage width at different angles. Shortest Survey Line Plan: Using the dynamic programming algorithm, determine the shortest survey line plan, with the dynamic transfer equation determined by the overlap rate index. The total length of the survey line in this plan is calculated to be 114,824 meters. Specific Survey Line Plan Design: Using sea depth data and cubic spline interpolation to complete the discrete point plane, construct approximate width and slope. Under the condition of relaxing the overlap rate to 5%-30%, the shortest survey line length is obtained as 398,059 meters, with a missing survey area ratio of 0%, and the total length of parts with an overlap rate exceeding 20% is 25,759 meters, accounting for 6.4%. This study provides theoretical basis and calculation methods for optimizing multi-beam bathymetric survey plans.

Investigation of the static and low‐velocity impact performance of 3D braided spacer composites

AbstractTo meet the lightweight requirements of advanced composite materials, the ultralightweight three‐dimensional (3D) braided spacer composites were designed and manufactured by ultra‐high molecular weight polyethylene (UHMWPE). The key structural effects of core column distances and core column heights on the static bending, compression, and low‐velocity impact properties of the composites were further investigated. Results indicated that the mechanical properties of 3D braided spacer composites was closely depended on their structural parameters. The flexure and compression of 3D braided spacer composites decreased with increasing core column distance and core column height. W1/H1 specimen showed the optimal bending and compressive performance. The maximum flexural strength and flexural modulus were approximately 54.77 MPa and 4.95 GPa, respectively. The low‐velocity impact properties were further investigated where the stiffness, energy absorption, and impact resistance of the composite decreased with increasing core column distance. The peak load and energy absorption of the W1 specimen were 2.39 kN and 7.99 J, respectively. While the low‐velocity impact properties increased with increasing core column distance. The H3 specimen exhibited the greatest stiffness, impact resistance and energy absorption with peak loads and energy absorption of 2.81 kN and 8.49 J, respectively. In addition, the macro morphology and scanning electron microscopy (SEM)micrographs were examined to investigate the damage morphology and failure mechanism of the composites. The main failure modes were matrix cracking, interface debonding, fiber buckling, and dislocation, as well as tilting, and failure of the core columns. This work provides guidance for structural design and engineering application of 3D lightweight braided spacer composite.

Asymptotic estimates of solution to damped fractional wave equation

It is known that the damped fractional wave equation has the diffusive structure as t→∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$t\\rightarrow \\infty $\\end{document}. Let u(t,x)=e−tcosh(tL)f(x)+e−tsinh(tL)L(f(x)+g(x))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$u(t,x)=e^{-t}\\cosh (t\\sqrt{L})f(x)+e^{-t} \\frac{\\sinh (t\\sqrt{L})}{\\sqrt{L}}(f(x)+g(x))$\\end{document} be the solution of the Cauchy problem for the damped fractional wave equation, where L\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\sqrt{L}$\\end{document} involves the fractional Laplacian (−△)α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(-\ riangle )^{\\alpha}$\\end{document} on the space variable. We can study the decay estimate of the solution u(t,x)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$u(t,x)$\\end{document} over the time t by means of the Cauchy problem for the parabolic equation. In this paper, we consider, for 0<α<1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$0<\\alpha <1$\\end{document}, the Cauchy problem in the two- and three-dimensional spaces for the damped fractional wave equation and the corresponding parabolic equation and obtain the Triebel–Lizorkin space estimate of the difference of solutions. At the same time, we also consider, for α=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\alpha =1$\\end{document}, the case of the Cauchy problem in the four-dimensional space and obtain a Triebel–Lizorkin space estimate.

Spatial Localization of Transformer Inspection Robot Based on Adaptive Denoising and SCOT-β Generalized Cross-Correlation

In the detection process of the internal defects of large oil-immersed transformers, due to the huge size of large transformers and metal-enclosed structures, the positional localization of miniature inspection robots inside the transformer faces great difficulties. To address this problem, this paper proposes a three-dimensional positional localization method based on adaptive denoising and the SCOT weighting function with the addition of the exponent β (SCOT-β) generalized cross-correlation for L-type ultrasonic arrays of transformer internal inspection robots. Aiming at the strong noise interference in the field, the original signal is decomposed by an improved Empirical Mode Decomposition (EMD) method, and the optimal center frequency and bandwidth of each mode are adaptively searched. By extracting the modes in the frequency band of the positional localization signal, suppressing the modes in the noise frequency band, and reconstructing the Intrinsic Mode Function (IMF) of the independently selected superior modal components, a signal with a high signal-to-noise ratio is obtained. In addition, for the traditional mutual correlation algorithm with a large delay estimation error at a low signal-to-noise ratio, this paper adopts an improved generalized joint weighting function, SCOT-β, which improves the anti-jamming ability of the generalized mutual correlation method at a low signal-to-noise ratio by adding an exponential function to the denominator term of the SCOT weighting function’s generalized cross-correlation. Finally, the accurate positional localization of the transformer internal inspection robot is realized based on the quadratic L-array and search-based maximum likelihood estimation method. Simulation and experimental results show the following: the improved EMD denoising method better improves the signal-to-noise ratio of the positional localization signal with a lower distortion rate; in the transformer test tank, which is 120 cm in length, 100 cm in width, and 100 cm in height, based on the positional localization method in this paper, the average relative positional localization error of the transformer internal inspection robot in three-dimensional space is 2.27%, and the maximum positional localization error is less than 2 cm, which meets the requirements of engineering positional localization.

MULTI-AGENT SIMULATION MODEL OF INFECTIOUS DISEASE SPREAD

The aim of the research is to develop a multi-agent simulation model for predicting the spread of infectious diseases, particularly COVID-19. In the context of the COVID-19 pandemic, there emerged an urgent need to create tools for forecasting and analyzing the dynamics of epidemics, as well as for evaluating the effectiveness of management decisions. The use of mathematical models in this process allows for an adequate description of the infection spread dynamics, which is essential for making informed decisions. The article discusses traditional approaches to epidemic modeling, such as the predator-prey model and the compartmental SIR (Susceptible-Infectious-Recovered) model. The predator-prey model describes the interaction between two species in an ecosystem using differential equations, which allows for modeling population dynamics. The compartmental SIR model divides the population into three groups: susceptible, infected, and recovered, which enables the analysis of the spread of infectious diseases. However, these models have limitations, particularly due to assumptions about population homogeneity and constant parameters. To more accurately model complex epidemic processes, a multi-agent simulation model was developed. In this model, agents interact within a defined area, mimicking real conditions of infection spread. Agents are divided into three classes: healthy, infected, and recovered. The movement of agents is modeled using random walk in a two-dimensional space, taking into account the possibility of contact between them, which can lead to infection. Infected agents transition to the recovered class after a certain period of illness and can no longer be infected. Modeling results showed that the multi-agent model allows for more accurate prediction of infection spread dynamics. Numerous experiments were conducted, demonstrating the model's adequacy in replicating the infection process, peak infection rates, and recovery periods. The influence of various parameters, such as the duration of illness, on the epidemic dynamics was investigated. The obtained results confirm that considering individual characteristics and behavioral traits of agents improves the accuracy of modeling. This allows the multi-agent simulation model to be used for developing effective control strategies and predicting the spread of infectious diseases, which can be useful for making management decisions in real pandemic conditions.

Spectroscopic Phenological Characterization of Mangrove Communities

Spaceborne spectroscopic imaging offers the potential to improve our understanding of biodiversity and ecosystem services, particularly for challenging and rich environments like mangroves. Understanding the signals present in large volumes of high-dimensional spectroscopic observations of vegetation communities requires the characterization of seasonal phenology and response to environmental conditions. This analysis leverages both spectroscopic and phenological information to characterize vegetation communities in the Sundarban riverine mangrove forest of the Ganges–Brahmaputra delta. Parallel analyses of surface reflectance spectra from NASA’s EMIT imaging spectrometer and MODIS vegetation abundance time series (2000–2022) reveal the spectroscopic and phenological diversity of the Sundarban mangrove communities. A comparison of spectral and temporal feature spaces rendered with low-order principal components and 3D embeddings from Uniform Manifold Approximation and Projection (UMAP) reveals similar structures with multiple spectral and temporal endmembers and multiple internal amplitude continua for both EMIT reflectance and MODIS Enhanced Vegetation Index (EVI) phenology. The spectral and temporal feature spaces of the Sundarban represent independent observations sharing a common structure that is driven by the physical processes controlling tree canopy spectral properties and their temporal evolution. Spectral and phenological endmembers reside at the peripheries of the mangrove forest with multiple outward gradients in amplitude of reflectance and phenology within the forest. Longitudinal gradients of both phenology and reflectance amplitude coincide with LiDAR-derived gradients in tree canopy height and sub-canopy ground elevation, suggesting the influence of surface hydrology and sediment deposition. RGB composite maps of both linear (PC) and nonlinear (UMAP) 3D feature spaces reveal a strong contrast between the phenological and spectroscopic diversity of the eastern Sundarban and the less diverse western Sundarban.

Simulation of tomato and corn growth using a modified intercropping model considering radiation interception in two-dimensional space and air temperature stress

Intercropping system has been widely applied in the agricultural production worldwide. However, the difference in agricultural productivity in tomato–corn intercropping system with different spatial arrangements has not been determined. Meanwhile, the existing intercropping models cannot precisely capture inter-species crop growth dynamics. Therefore, a 10-year (2010–2019) experimental field study was conducted in Hetao irrigation district, China, for the parameterization and evaluation of the model. The experiment data from the typical years (2018 and 2019) was adopted to analyze the crop growth dynamics in different hydrological years and different systems [two rows of tomato intercropping two rows of corn (IC2–2), four rows of tomato intercropping two rows of corn (IC4–2), sole corn (SC), and sole tomato (ST)]. Moreover, a modified intercropping model considering radiation interception in two-dimensional space and air temperature stress (MICG) was deduced. The results revealed that the MICG model can precisely simulate inter-species crop growth in tomato–corn intercropping system with different spatial arrangements. The mean relative error (MRE), normalized root mean square error (nRMSE), determination coefficient (R2), and percentage bias (PBIAS) of leaf area index (LAI) simulated by the MICG model in different spatial arrangements were 9.0 %, 7.8 %, 0.97, and –5.3 % for corn and 11.0 %, 7.8 %, 0.95, and 0.1 % for tomato, respectively. An apparent difference in the simulation accuracy was observed among different intercropping models in the rapid and senescence crop growth stages. The simulation accuracy of the MICG model was apparently higher than that of ICGw and ICG models. However, the MICG model still needs to be further improved considering its some limitations, e.g., the soil respiration and soil nitrogen transformation modules need be introduced in this model. Additionally, an apparent difference in crop growth dynamics in different spatial arrangements was observed, particularly in different hydrological years. The largest difference in crop growth dynamics in different spatial arrangements was observed in the middle crop growth stage (days after sowing [DAS] 20–100) in the wet year and in the late crop growth stage (DAS 101–140) in the dry year. In general, the highest aboveground biomass and yields of corn and tomato were observed in the IC2–2 and ST systems, respectively. However, the highest water use efficiency for corn (23.9) and tomato (268.7) was observed in the IC4–2 and ST systems, respectively. The agricultural production benefits in the IC4–2 system were better than in the IC2–2 system. Average land equivalent ratio and water equivalent ratio in both years increased by 0.8 % and 12.5 % in the IC4–2 system compared with those in the IC2–2 system, respectively. Overall, the IC4–2 system could be recommended as the most optimal spatial arrangement for the tomato–corn intercropping system.

Hilbert-curve assisted structure embedding method

MotivationChemical space embedding methods are widely utilized in various research settings for dimensional reduction, clustering and effective visualization. The maps generated by the embedding process can provide valuable insight to medicinal chemists in terms of the relationships between structural, physicochemical and biological properties of compounds. However, these maps are known to be difficult to interpret, and the ‘‘landscape’’ on the map is prone to ‘‘rearrangement’’ when embedding different sets of compounds.ResultsIn this study we present the Hilbert-Curve Assisted Space Embedding (HCASE) method which was designed to create maps by organizing structures according to a logic familiar to medicinal chemists. First, a chemical space is created with the help of a set of ‘‘reference scaffolds’’. These scaffolds are sorted according to the medicinal chemistry inspired Scaffold-Key algorithm found in prior art. Next, the ordered scaffolds are mapped to a line which is folded into a higher dimensional (here: 2D) space. The intricately folded line is referred to as a pseudo-Hilbert-Curve. The embedding of a compound happens by locating its most similar reference scaffold in the pseudo-Hilbert-Curve and assuming the respective position. Through a series of experiments, we demonstrate the properties of the maps generated by the HCASE method. Subjects of embeddings were compounds of the DrugBank and CANVASS libraries, and the chemical spaces were defined by scaffolds extracted from the ChEMBL database.Scientific contributionThe novelty of HCASE method lies in generating robust and intuitive chemical space embeddings that are reflective of a medicinal chemist’s reasoning, and the precedential use of space filling (Hilbert) curve in the process.Availabilityhttps://github.com/ncats/hcaseGraphical

Revisiting 3D visual grounding with Context-aware Feature Aggregation

3D visual grounding is the task of accurately locating objects in a three-dimensional scene based on textual descriptions. Current approaches mainly depend on downsampling and extracting the point features for fusion with text features. However, the main challenges of these methods are the poor point feature resolution and limited local context during multi-modal fusion, causing visual-linguistic misalignment, particularly for small objects described in the text. The intuitive solution is to get additional object-related point features, gathering more contextual information to augment the representation capability of multimodal features, thereby promoting the representation capabilities of multimodal features. Based on this, we introduce a novel 3D visual grounding framework named Context-aware Feature Aggregation (CFA). The CFA framework includes two key modules: (1) Point Augmented Aggregation Module (PAM), designed to compensate for downsampling-induced information loss by augmenting sampled points with neighboring context for more discriminative features; and (2) Dual Contextual Grouping Attention Module (DCGAM), which iteratively refines features and geometry coordinates from PAM, capturing more global context. We assess the performance of our CFA framework on two point-based datasets: ScanRefer and Nr3D/Sr3D. The CFA framework exhibits efficiency in 3D visual grounding, surpassing the performance of previous methods by experimental results.

An automated sleep staging tool based on simple statistical features of mice electroencephalography (EEG) and electromyography (EMG) data.

Electroencephalogram (EEG) and electromyogram (EMG) are fundamental tools in sleep research. However, investigations into the statistical properties of rodent EEG/EMG signals in the sleep-wake cycle have been limited. The lack of standard criteria in defining sleep stages forces researchers to rely on human expertise to inspect EEG/EMG. The recent increasing demand for analysing large-scale and long-term data has been overwhelming the capabilities of human experts. In this study, we explored the statistical features of EEG signals in the sleep-wake cycle. We found that the normalized EEG power density profile changes its lower and higher frequency powers to a comparable degree in the opposite direction, pivoting around 20-30 Hz between the NREM sleep and the active brain state. We also found that REM sleep has a normalized EEG power density profile that overlaps with wakefulness and a characteristic reduction in the EMG signal. Based on these observations, we proposed three simple statistical features that could span a 3D space. Each sleep-wake stage formed a separate cluster close to a normal distribution in the 3D space. Notably, the suggested features are a natural extension of the conventional definition, making it useful for experts to intuitively interpret the EEG/EMG signal alterations caused by genetic mutations or experimental treatments. In addition, we developed an unsupervised automatic staging algorithm based on these features. The developed algorithm is a valuable tool for expediting the quantitative evaluation of EEG/EMG signals so that researchers can utilize the recent high-throughput genetic or pharmacological methods for sleep research.

Updated Lagrangian particle hydrodynamics (ULPH) simulations of underwater bubble motions in three-dimensional space

Updated Lagrangian particle hydrodynamics (ULPH) simulations of underwater bubble motions in three-dimensional space

Guide RNA structure design enables combinatorial CRISPRa programs for biosynthetic profiling

Engineering metabolism to efficiently produce chemicals from multi-step pathways requires optimizing multi-gene expression programs to achieve enzyme balance. CRISPR-Cas transcriptional control systems are emerging as important tools for programming multi-gene expression, but poor predictability of guide RNA folding can disrupt expression control. Here, we correlate efficacy of modified guide RNAs (scRNAs) for CRISPR activation (CRISPRa) in E. coli with a computational kinetic parameter describing scRNA folding rate into the active structure (rS = 0.8). This parameter also enables forward design of scRNAs, allowing us to design a system of three synthetic CRISPRa promoters that can orthogonally activate (>35-fold) expression of chosen outputs. Through combinatorial activation tuning, we profile a three-dimensional design space expressing two different biosynthetic pathways, demonstrating variable production of pteridine and human milk oligosaccharide products. This RNA design approach aids combinatorial optimization of metabolic pathways and may accelerate routine design of effective multi-gene regulation programs in bacterial hosts.

Clarification of the transverse orbital angular momentum of spatiotemporal optical vortices

Advances in the generation and the application of spatiotemporal optical vortices (STOV) are proceeding fast, but fundamental aspects of their nature remain obscure. Bliokh (2023 Phys. Rev. A 107 L031501) (PRA) and Porras (2023 Prog. Electromagn. Res. 177 95) (PIER) provide contradictory results on the transverse orbital angular momentum (OAM) carried by STOVs. We show that the results by Porras in PIER and by Bliokh in PRA refer to different STOVs and are all correct. In PIER, STOVs are elliptical at given cross section and time, or in space-time, but not in three-dimensional space. In PRA, STOVs are elliptical in space but not in space-time. This is evidenced from two dual, equivalent theories on the transverse OAM where a wave packet is seen in space-time evolving with propagation distance or in space evolving in time, that account for all values of the total, intrinsic and extrinsic OAM in PIERS and PRA. However, the intrinsic OAM with respect to the photon wave function center in PRA is not generally conserved, which advocates for the energy center in PIER as the STOV center. We argue that STOVs are generated in experiments to purportedly have elliptical symmetry in space-time. The values provided in PIER should then be taken as the reference for elliptical STOVs, and the theory therein to evaluate the transverse OAM of other wave packets. Hancock et al (2021 Phys. Rev. Lett. 127 193901; 2024 Phys. Rev. X 14 011031) erroneously attribute the transverse OAM of elliptical STOVs in space to the elliptical STOVs in space-time that they consider theoretically and can generate in their experiments.

Three-dimensional line-of-sight-angle-constrained leader-following cooperative interception guidance law with prespecified impact time

To address the problem of multi-missile cooperative interception against maneuvering targets at a prespecified impact time and desired Line-of-Sight (LOS) angles in Three-Dimensional (3D) space, this paper proposes a 3D leader-following cooperative interception guidance law. First, in the LOS direction of the leader, an impact time-controlled guidance law is derived based on the fixed-time stability theory, which enables the leader to complete the interception task at a prespecified impact time. Next, in the LOS direction of the followers, by introducing a time consensus tracking error function, a fixed-time consensus tracking guidance law is investigated to guarantee the consensus tracking convergence of the time-to-go. Then, in the direction normal to the LOS, by combining the designed global integral sliding mode surface and the second-order Sliding Mode Control (SMC) theory, an innovative 3D LOS-angle-constrained interception guidance law is developed, which eliminates the reaching phase in the traditional sliding mode guidance laws and effectively saves energy consumption. Moreover, it effectively suppresses the chattering phenomenon while avoiding the singularity issue, and compensates for unknown interference caused by target maneuvering online, making it convenient for practical engineering applications. Finally, theoretical proof analysis and multiple sets of numerical simulation results verify the effectiveness, superiority, and robustness of the investigated guidance law.

Uncertain random variables and laws of large numbers under U-C chance space

Uncertainty theory was founded by Baoding Liu for modeling belief degrees about human uncertainty. And chance theory was pioneered by Yuhan Liu based on probability theory and uncertainty theory. In many cases, imprecise risk situation in subjective or objective setting and human uncertainty simultaneously appear in a complex system. To measure uncertain random events in this system, this paper combines uncertainty space with convex non-additive probability space into a new kind of two-dimensional chance space called U-C chance space. Moreover, a new framework for uncertain random variables combining uncertainty theory with convex non-additive probability theory is provided. For applications of this new framework, it can be applied to characterize some kinds of phenomena which possess the characteristics of both imprecise risk situations and belief degrees about human uncertainty in situations of great events, such as financial crisis, major natural disaster, etc. The main contribution of this paper is to derive the types of Kolmogorov and Marcinkiewicz-Zygmund LLNs for uncertain random variables satisfying some conditions under U-C chance space, where the convex non-additive probability is totally monotone. Furthermore, several examples are stated and explained.