Normalization Scheme Research Articles

Concurrent multi-scale design optimization of fiber-reinforced composite material based on an adaptive normal distribution fiber optimization scheme for minimum structural compliance and additive manufacturing

Structural lightweight is a core technical requirement for the structural design of aerospace and new energy power equipment structures. For multi-scale variable stiffness design optimization of discrete fiber-reinforced composite laminates, one of the challenges is how to avoid the explosion of design variable combinations caused by the increase in the number of candidate discrete fiber laying angles. The Normal Distribution Fiber Optimization (NDFO) interpolation scheme has the numerical advantage that the number of design variables does not increase with an increase in the number of candidate discrete fiber laying angles. However, the traditional NDFO interpolation scheme uses uniform penalty parameters across all elements, which means that normalizing the penalty parameters for all the elements ignores the convergence differences of discrete fiber laying angles in different elements within the macro-scale structure topology. This leads to time-consuming and unstable optimization iteration of the macro-scale structural topology and micro-scale discrete fiber laying angle selection. Especially, it easily causes the micro-scale discrete fiber laying angle selection to fall into the local optimum prematurely. Therefore, considering the difficulties and challenges of the traditional NDFO interpolation scheme in the multi-scale variable stiffness design optimization of fiber-reinforced composites. This paper proposes an Adaptive Normal Distribution Fiber Optimization (ANDFO) interpolation scheme, and the feedback mechanism of the convergence rate of the element design variable and the objective function is introduced to achieve the adaptive reduction of the penalty parameters. Based on the proposed ANDFO interpolation scheme, a multi-scale design optimization model of fiber-reinforced composite laminates is established, considering the macro-scale structure topology and micro-scale discrete fiber laying angel selection. The explicit sensitivity of the objective function of minimizing structural compliance to the macro-scale topological design variables and the micro-scale fiber laying angle design variables is derived. Considering the manufacturability of additive manufacturing based on the optimized design results, a multi-scale nonlinear continuous filtering strategy for discrete fiber laying angle is adopted to improve the continuity of the local fiber laying path. Numerical examples systematically present the coupling effects of macro-scale structural topology and micro-scale fiber laying path, multi-scale nonlinear discrete fiber continuous filtering laying path structure, and continuous fiber additive manufacturing multi-scale optimized structure. The proposed ANDFO scheme provides a new theoretical and methodological approach for the lightweight and integrated multi-scale design and manufacturing of fiber-reinforced composite laminates through additive manufacturing technology.

Lithospheric magnetic structure of cratonic regions in Central-Eastern China inferred from aeromagnetic anomalies: Insights into magnetization in the uppermost mantle

Most of the subcontinental lithospheric mantle (SCLM) beneath Proterozoic cratons consists of refertilized Archaean SCLM. Variations in SCLM composition and its physical properties significantly affect the stabilization and preservation of the ancient continents. In this paper, aeromagnetic data are analyzed to reveal the magnetic structure of the lithospheric mantle beneath two major Precambrian blocks in central-eastern China, i.e., the Upper Yangtze Block (UYB) and Ordos Block (OB). After being reduced to the pole, the Fourier power spectrum of the aeromagnetic anomalies is calculated to determine the depth to magnetic sources. Considering the lower spatial resolution of the power spectral analysis in dealing with the long-wavelength aeromagnetic anomalies, we applied the scale-normalized continuous wavelet transform (CWT) on the magnetic data to trace the magnetic sources, with special focus on deeper ones. Synthetical model of a magnetic layer and application to the profile data validate the effectiveness of this scale normalization scheme in improving the wavenumber/spatial resolution in the CWT scalogram.In order to present a detailed magnetic structure, we carried out 2.5D forward modeling work on the magnetic data of a 2280 km-long nearly N-S profile across the UYB and OB. Due to the inherent ambiguity in the modeling results, the CWT-based spectral analysis is successfully adopted to provide source depth constraints for the initial model. The constrained forward modeling results indicate strong inhomogeneities among main tectonic blocks of studied area, like humans have different fingerprints. The magnetization of OB is larger than that of UYB since its Archean to Paleoproterozoic metamorphic basement are widely exposed at the surface, while the Precambrian basement of UYB is mostly overlain by unmetamorphosed Sinian cover and weakly metamorphosed Neoproterozoic strata. The most interesting aspect is that deep-seated magnetic sources might reside in the uppermost mantle of UYB and OB, suggesting vertical layering in the SCLM in cold cratonic regions.

Evaluation of state parameter interpretation methods using CPT calibration chamber data

The cone penetration test (CPT) is widely used to determine the in situ state parameter of soils because it provides continuous data and excellent repeatability at a relatively low cost. Accurate interpretation of the state parameter from CPT is the basis for evaluating the strength of granular soils, including assessing liquefaction susceptibility in important structures such as tailings storage facilities. A few interpretation methods are used in practice. They use two different overburden stress normalisation schemes on tip resistance. These methods vary in how much information they utilise to differentiate among soils. This paper evaluates these methods by applying them to an extensive database of calibration chamber tests. Then, the state parameter interpreted by each method is compared with that determined from laboratory data. The database includes manufactured sands, natural sands, and clean sand tailings. The soils were selected such that both calibration chamber testing and triaxial compression data were available from the literature. This evaluation serves as a minimum requirement for applying these methods in engineering projects, especially those dealing with challenging soils such as fines-rich tailings. This study suggests that methods that account for soil properties and in situ horizontal stresses perform better than those that do not.

Estimation of liver standardized uptake value in F18-FDG PET/CT scanning: impact of different malignancies, blood glucose level, body weight normalization, and imaging systems.

The aim of this work was to investigate homogeneity and stability of liver SUV in terms of different malignancies considering different body normalization schemes and blood glucose concentrations as well as PET/CT imaging systems. The study included 207 patients with four different types of cancers namely breast, lymphoma, lung, and bone-metastasis. Data acquisition was performed with GE Discovery IQ, Biograph mCT, uMI 550, and Ingenuity TF64 after a single intravenous injection of 194 ± 67.5 MBq of 18F-FDG. In body weight normalization, SUVmax and SUVmean in bone-mets as well as SUVmean in lung patients were not statistically different among scanners especially for data corrected for glucose levels (p = 0.062, 0.121, and 0.150, respectively). In SUVlbm derived from lung patients, there was no significant differences in Philips in comparison to GE and Siemens (both, p > 0.05) for data corrected and not corrected for glucose levels. In SUVbsa, the only non-significant difference revealed among scanners was in the measurements of SUVmean obtained from lung and bone-mets (p = 0.107 and 0.114) both corrected for glucose levels. In SUVbmi, SUVmean of lung and bone-mets as well as SUVmax of bone-mets showed a non-significant differences among the four different scanning systems (p = 0.303, 0.091, and 0.222, respectively) for data corrected for glucose levels. Liver glucose correction needs further investigations in individual tumors but could be potentially affected by whether measurements are made on SUVmean versus SUVmax, body weight normalization, as well as the imaging system. As such, selection of normalization to body weight method should be carefully selected before clinical adoption and clinically adopted and body surface area would provide the highest correlation. As such, normalization of body weight should be carefully made before clinical adoption. SUVmean proves to be useful and stable metric when liver is corrected for blood glucose levels.

Frequency-domain diffusion adaptation over networks with missing input data

Recently, a modified adapt-then-combine diffusion (mATC) strategy has been developed to handle distributed estimation problem with missing regressions (inputs). However, the mATC algorithm only considers the white input scenario and suffers from high complexity for long model filter lengths. To overcome these shortcomings, this paper proposes novel regularization-based frequency-domain diffusion algorithms for networks with missing input data. First, bias-eliminating cost function based on regularization is established by using the frequency-domain diagonal approximation. Then, with stochastic gradient descent, periodic update, and power normalization schemes, we design the regularization-based frequency-domain least mean square (R-FDLMS) algorithm as well as its normalized variant (R-FDNLMS). The latter converges faster than the former under colored inputs. The stability and steady-state behavior of the R-FDNLMS algorithm are also analyzed. Moreover, two effective power estimation methods are presented for both situations without and with the power ratio between the input signal and perturbation noise, along with a reset mechanism in the first case to enhance tracking performance. Finally, simulations are conducted to illustrate the superiority of the proposed algorithms and the validity of theoretical findings.

Size normalizing planktonic Foraminifera abundance in the water column

AbstractPlanktonic Foraminifera have been collected from the water column with different plankton sampling devices equipped with nets of various mesh sizes, which impedes direct comparison of observed quantifications. Here, we use data on the community size structure of planktonic Foraminifera to assess the impact of mesh size on the measured abundance (ind m−3) of planktonic Foraminifera. We use data from the FORCIS database (Chaabane et al., 2023, Scientific Data 10: 354) on the global ocean at different sampling depths over the past century. We find a global cumulative increase in abundance with size, which is best described using a Michaelis–Menten function. This function yields multiplication factors by which one size fraction can be normalized to any other size fraction equal to or larger than 100 μm. The resulting size normalization model is calibrated over a range of different depth intervals, and validated with an independent dataset from various depth ranges. The comparison to Berger's (1969, Deep. Res. Oceanogr. Abstr. 16: 1–24) equivalent catch approach shows a significant increase in the predictive skill of the model. The new size normalization scheme enables comparison of Foraminifera abundance data sampled with plankton nets of different mesh sizes, such as compiled in the FORCIS database. The correction methodology may be effectively employed for various other plankton groups such as diatoms and dinoflagellates.

Impacts of neutral collisions and finite electron inertia on longitudinal thermal instability of two-component radiative plasma for star formation in interstellar medium (ISM)

The impact of neutral friction, finite electron inertia and radiative heat-loss function on the thermal instability of viscous two-component plasma has been explored, integrating the impacts of thermal conductivity. A general dispersion relation is gained by means of the normal mode analysis scheme with the assist of appropriate linearized perturbation equations of the problem, and an amended circumstance of thermal instability is acquired. For the case of longitudinal propagation, it is obvious that the circumstance of thermal instability is independent of the finite electron inertia, magnetic field strength and viscosity of two components but relies on the radiative heat-loss function, thermal conductivity and neutral particle. From the curves, it is clear that the temperature-dependent heat-loss function, thermal conductivity and viscosity of two components represent stabilizing impact, while finite electron inertia and neutral collision depict destabilizing impact. Results discussed in this problem are helpful for star formation in ISM.

Investigating food production-associated DNA methylation changes in paleogenomes: Lack of consistent signals beyond technical noise.

The Neolithic transition introduced major diet and lifestyle changes to human populations across continents. Beyond well-documented bioarcheological and genetic effects, whether these changes also had molecular-level epigenetic repercussions in past human populations has been an open question. In fact, methylation signatures can be inferred from UDG-treated ancient DNA through postmortem damage patterns, but with low signal-to-noise ratios; it is thus unclear whether published paleogenomes would provide the necessary resolution to discover systematic effects of lifestyle and diet shifts. To address this we compiled UDG-treated shotgun genomes of 13 pre-Neolithic hunter-gatherers (HGs) and 21 Neolithic farmers (NFs) individuals from West and North Eurasia, published by six different laboratories and with coverage c.1×-58× (median = 9×). We used epiPALEOMIX and a Monte Carlo normalization scheme to estimate methylation levels per genome. Our paleomethylome dataset showed expected genome-wide methylation patterns such as CpG island hypomethylation. However, analyzing the data using various approaches did not yield any systematic signals for subsistence type, genetic sex, or tissue effects. Comparing the HG-NF methylation differences in our dataset with methylation differences between hunter-gatherers versus farmers in modern-day Central Africa also did not yield consistent results. Meanwhile, paleomethylome profiles did cluster strongly by their laboratories of origin. Using larger data volumes, minimizing technical noise and/or using alternative protocols may be necessary for capturing subtle environment-related biological signals from paleomethylomes.

Lagrange points and regionally conserved quantities

Lagrange points are the equilibrium points within a restricted three-body system, epitomized by the Trojan asteroids near the L4 and L5 points of the Sun–Jupiter system. They also play a crucial role in some space missions, including the James Webb Space Telescope which is located at the Sun–Earth L2 point. While the existence of five Lagrange points is a well-known feature of the restricted three-body problem, the equations describing the precise location of all five points are not extensively documented. This work presents a derivation of all Lagrange points using polar coordinates and a new normalization scheme that offers a simpler interpretation of solutions compared to prior analyses. A subtle issue concerning the treatment of angular momentum in the potential formulation of this problem is addressed and resolved. The supplementary material to this work contains additional mathematical details and discussion.

Dynamical systems for arithmetic schemes

Motivated by work of Kucharczyk and Scholze, we use sheafified rational Witt vector rings to attach a new ringed space Wrat(X) to every scheme X. We also define R-valued points Wrat(X)(R) of Wrat(X) for every commutative ring R. For normal schemes X of finite type over specZ, using Wrat(X)(ℂ) we construct infinite dimensional R-dynamical systems whose periodic orbits are related to the closed points of X. Various aspects of these topological dynamical systems are studied. We also explain how certain p-adic points of Wrat(X) for X the spectrum of a p-adic local number ring are related to the points of the Fargues–Fontaine curve.

UN-[formula omitted]: An offline adaptive normalization method for deploying transformers

Transformer has become the de-facto architecture for many natural language and vision tasks thanks to its remarkable performance. As an essential part of transformers, normalization functions struggle to improve the robustness and performance of transformer architecture. In this work, we build up a unified framework to describe and compare the advantages and disadvantages of different normalization methods in transformers. As an online normalization method, Layer Normalization (LN) is initially used to normalize activations inside each token to handle variant-length inputs and improve robustness. However, the strong robustness and performance of online normalization methods is at the cost of inefficient inference. On the contrary, Transformers built with hardware-efficient offline normalization schemes such as Batch Normalization, yields subpar performance sometimes even collapsing during training. Furthermore, we discover that abnormal behaviors in activation statistics, such as significant iteration-to-iteration fluctuations and extreme outliers across layers, make a vital effect on the training process and the final result. Based on the unified framework, we propose Unified Normalization with η (UN-η), achieving low-cost inference and good performance. We theoretically prove the effectiveness of our method. Experimental results on various tasks including language, speech, and vision tasks have demonstrated that UN-η can be a hardware-efficient substitute for LN. Besides, we assess the hardware efficiency of our method on GPU.

Towards semantically continuous unpaired image-to-image translation via margin adaptive contrastive learning and wavelet transform

Unpaired image-to-image translation aims to preserve the semantics of the input image while mimicking the style of target domains without paired data. However, existing methods often suffer from semantic distortions if the source and target domains have large mismatched semantics distributions. To address semantic distortions in translation outputs without paired supervision, we propose a Margin Adaptive Contrastive Learning Network (MACL-Net) that drives contrastive learning as a local semantic descriptor while using a pre-trained Vision Transformer (ViT) as a global semantic descriptor to learn domain-invariant features in the translation process. Specifically, we design a novel margin adaptive contrastive loss to enforce intra-class compactness and inter-class discrepancy. Besides, to better retain the semantic structure of the translated image and improve its fidelity, we use Discrete Wavelet Transform (DWT) to supplement the low-frequency and high-frequency information of the input image into the generator, and effectively fuse the feedforward features and inversed frequency information through a novel normalization scheme, Feature-Frequency Transformation Normalization (FFTN). In terms of experimental results, MACL-Net effectively reduces semantic distortions and generates translation outputs that outperform state-of-the-art techniques both quantitatively and qualitatively.

Dimensionality Reduction for Onboard Modeling of Uncertain Atmospheres

Onboard density models are a key aspect of closed-loop guidance systems for hypersonic flight. Traditional approaches model density as a deterministic function of altitude, but a recent drive toward stochastic guidance approaches motivates onboard uncertainty propagation. Existing solutions for efficient uncertainty propagation generally treat density as an exponential function of altitude, but this approach is limited in its ability to capture relevant dispersions. This work models density as a Gaussian random field that is approximated by a Karhunen–Loève expansion, enabling a relatively high-fidelity, finite-dimensional parametric representation. Alternative models are also developed using a variational autoencoder architecture, resulting in greater dimensionality reduction at the expense of analytical description. Normalization schemes are presented and compared by their efficiency in capturing density variability in a limited number of terms, and normalization by reference dynamic pressure is shown to be the most compact approach. The model alternatives are compared both by their approximations of density itself and by their predictions of peak heat flux for dispersed direct-entry and aerocapture trajectories. An extension of this approach for modeling density as a function of multiple independent variables is also presented and demonstrated. Finally, it is shown that the Karhunen–Loève density model can be sequentially updated according to noisy density observations by formulating the problem as a Kalman measurement function.

From NMR to AI: Designing a Novel Chemical Representation to Enhance Machine Learning Predictions of Physicochemical Properties.

A novel approach to the utilization of nuclear magnetic resonance (NMR) spectroscopy data in the prediction of logD through machine learning algorithms is shown. In the analysis, a data set of 754 chemical compounds, organized into 30 clusters, was evaluated using advanced machine learning models, such as Support Vector Regression (SVR), Gradient Boosting, and AdaBoost, and comprehensive validation and testing methods were employed, including 10-fold cross-validation, bootstrapping, and leave-one-out. The study revealed the superior performance of the Bucket Integration method for dimensionality reduction, consistently yielding the lowest root mean square error (RMSE) across all data sets and normalization schemes. The SVR prediction models demonstrated remarkable computational efficiency and low cost, with the best RMSE value reaching 0.66. Our best model outperformed existing tools like JChem Suite's logD Predictor (0.91) and CplogD (1.27), and a comparison with traditional molecular representations yielded a comparable RMSE (0.50), emphasizing the robustness of our NMR data integration. The widespread availability of NMR data in pharmaceutical and industrial research presents an untapped resource for predictive modeling, highlighting the need for accessible methodologies like ours that complement the analytical toolbox beyond conventional 2D approaches. Our approach, designed to leverage the rich spatial data from NMR spectroscopy, provides additional insights and enriches drug discovery and computational chemistry with a freely accessible tool.

Proportion-based normalizations outperform compositional data transformations in machine learning applications

BackgroundNormalization, as a pre-processing step, can significantly affect the resolution of machine learning analysis for microbiome studies. There are countless options for normalization scheme selection. In this study, we examined compositionally aware algorithms including the additive log ratio (alr), the centered log ratio (clr), and a recent evolution of the isometric log ratio (ilr) in the form of balance trees made with the PhILR R package. We also looked at compositionally naïve transformations such as raw counts tables and several transformations that are based on relative abundance, such as proportions, the Hellinger transformation, and a transformation based on the logarithm of proportions (which we call “lognorm”).ResultsIn our evaluation, we used 65 metadata variables culled from four publicly available datasets at the amplicon sequence variant (ASV) level with a random forest machine learning algorithm. We found that different common pre-processing steps in the creation of the balance trees made very little difference in overall performance. Overall, we found that the compositionally aware data transformations such as alr, clr, and ilr (PhILR) performed generally slightly worse or only as well as compositionally naïve transformations. However, relative abundance-based transformations outperformed most other transformations by a small but reliably statistically significant margin.ConclusionsOur results suggest that minimizing the complexity of transformations while correcting for read depth may be a generally preferable strategy in preparing data for machine learning compared to more sophisticated, but more complex, transformations that attempt to better correct for compositionality.DL26Qs8hz41d9wKrFGqt9FVideo

Breaking Time Invariance: Assorted-Time Normalization for RNNs

Methods such as Layer Normalization (LN) and Batch Normalization have proven to be effective in improving the training of Recurrent Neural Networks (RNNs). However, existing methods normalize using only the instantaneous information at one particular time step, and the result of the normalization is a preactivation state with a time-independent distribution. This implementation fails to account for certain temporal differences inherent in the inputs and the architecture of RNNs. Since these networks share weights across time steps, it may also be desirable to account for the connections between time steps in the normalization scheme. In this paper, we propose a normalization method called Assorted-Time Normalization (ATN), which preserves information from multiple consecutive time steps and normalizes using them. This setup allows us to introduce longer time dependencies into the traditional normalization methods without introducing any new trainable parameters. We present theoretical derivations for the gradient propagation and prove the weight scaling invariance property. Our experiments applying ATN to LN demonstrate consistent improvement on various tasks, such as Adding, Copying, and Denoise Problems and Language Modeling Problems.

Linear and nonlinear stability analyses of micropolar fluid flow in horizontal porous layers

The linear and nonlinear stability analyses of micropolar fluid flow in a horizontal porous layer heated from below in the presence of throughflow is numerically investigated. The Brinkman model is considered to govern the micropolar fluid flow within the porous region. The main purpose of the present study is to investigate the behavior of the subcritical region for micropolar fluid parameters in the presence of throughflow. The energy approach is used to analyze nonlinear stability, whereas the normal mode scheme is used to investigate linear stability. The obtained eigenvalue problems related to linear and nonlinear stability analyses are solved numerically using the bvp4c routine in MATLAB. Finally, the critical thermal Rayleigh number is determined for the given values of the governing parameters. It is observed that the subcritical area decreases as the Darcy number (Da), micropolar parameter (m), and absolute value of throughflow parameter (|Pe|) decrease. Furthermore, there is no subcritical gap in the absence of the throughflow effect for micropolar fluid flow, which is a good agreement for the linear and nonlinear thresholds.

Determining internal coordinate sets for optimal representation of molecular vibration.

Arising from the harmonic approximation in solving the vibrational Schrödinger equation, normal modes dissect molecular vibrations into distinct degrees of freedom. Normal modes are widely used as they give rise to descriptive vibrational notations and are convenient for expanding anharmonic potential energy surfaces as an alternative to higher-order Taylor series representations. Usually, normal modes are expressed in Cartesian coordinates, which bears drawbacks that can be overcome by switching to internal coordinates. Considering vibrational notations, normal modes with delocalized characters are difficult to denote, but internal coordinates offer a route to clearer notations. Based on the Hessian, normal mode decomposition schemes for a given set of internal coordinates can describe a normal mode by its contributions from internal coordinates. However, choosing a set of internal coordinates is not straightforward. While the Hessian provides unique sets of normal modes, various internal coordinate sets are possible for a given system. In the present work, we employ a normal mode decomposition scheme to choose an optimal set. Therefore, we screen reasonable sets based on topology and symmetry considerations and rely on a metric that minimizes coupling between internal coordinates. Ultimately, the Nomodeco toolkit presented here generates internal coordinate sets to find an optimal set for representing molecular vibrations. The resulting contribution tables can be used to clarify vibrational notations. We test our scheme on small to mid-sized molecules, showing how the space of definable internal coordinate sets can significantly be reduced.

МЕТОД ТА ЗАСІБ МОНІТОРИНГУ БЕЗПЕКИ В КОМП'ЮТЕРНІЙ МЕРЕЖІ ЗАСОБАМИ SIEM

This work focuses on researching, analyzing, and enhancing methods and tools for security monitoring in computer networks. The study develops security monitoring tools and methods based on SIEM agents, improving the data normalization process from security logs. The research explores SIEM's role in the SIEM-EDR-NDR triad perspective to accelerate responses to network security threats. The investigation is grounded in the experiences of foreign companies and domestic banking networks. The interaction of SIEM-EDR-NDR components, forming a SOC triad, is examined. SIEM is utilized for centralized data analysis, including EDR and NDR, providing a comprehensive security overview. EDR detects and responds to threats on endpoints, complemented by NDR, extending SIEM analysis. This combination ensures effective response to cyberattacks, reducing "dwell time" until detection. The formulation of tasks for EDR components in the SIEM-EDR-NDR triad is discussed. Emphasis is placed on the importance of protecting endpoints at all stages of an attack, and effective strategies, such as traffic analysis, application control, and centralized cybersecurity management, are identified. Integration of EDR with existing security tools to create a comprehensive system is highlighted. Within the SIEM context, data processing stages, from log collection and normalization to event classification and correlation, are illuminated. The role of correlation in incident formation and investigation is underscored. An enhanced normalization scheme with an expanded agent deployment and key data processing stages within the SIEM system is proposed. The work addresses the improvement of event log processing in SIEM for effective network security monitoring and timely threat mitigation. The achieved goal accelerates threat response processes through SIEM agent integration, facilitating the organization and classification of information flows for prompt threat mitigation.

Orientation Quaternion Norm Correction in SINS Algorithms: Calculation Normalization Schemes and Their Efficiency

The problem of correcting the norm of the computed orientation quaternion in the algorithms for the operation of strapdown inertial navigation systems is considered. Two existing approaches to the correction process are considered, the first approach is to normalize the rotation quaternion at the computation cycle, the second approach is to normalize the resulting quaternion. 5 well-known calculation schemes for norm correction are given. To simulate the test motion, we used an analytical quaternion kinematic rotation model based on a sequence of three rotations corresponding to the Krylov angles. The case of linear dependence of elementary rotation angles on time is considered. The model provides analytical representation of the projections of the angular velocity vector of the rigid body on the associated axes and the corresponding quasi-coordinates on the calculation cycle. The results of numerical simulation of the reference motion for a given set of frequencies are presented as dependences of the projections of the angular velocity vector of the rigid body on time and the constructed trajectories in the configuration space of the orientation parameters.
 To determine the rotation quaternion on a cycle, the Miller algorithm was used, which makes it possible to obtain an increase in the orientation vector based on ideal information from the angular velocity sensors in the form of quasi-coordinates. The transformation into a rotation quaternion occurs with the help of the corresponding expansions of the trigonometric functions of the true rotation angle (modulus of the orientation vector) in a series.
 Based on a numerical experiment, it is shown that the best result of correcting the norm of the calculated quaternion in the sense of the minimum error of the norm is given by one of the finite normalization schemes, for which there is no division operation and ensures the stability of the norm correction in time. The results of numerical simulation of the model rotational motion of a rigid body and the development of schemes for correcting the norm of the calculated orientation quaternion are presented.
 Keywords: orientation quaternion, Miller's orientation algorithm, SINS, norm error, computational drift, analytical reference model, quasi-coordinates, numerical simulation.