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A parametrization, given by the Euler angles, of Hermitian matrix generators of even and odd non-degenerate Clifford algebras is constructed by means of the Kronecker product of a parametrized version of Pauli matrices and by the identification of all possible anticommutation sets. The internal parametrization of the matrix generators allows a straightforward interpretation in terms of rotations, and in the absence of a similarity transformation can be reduced to the canonical representations by an appropriate choice of parameters. The parametric matrix generators of second and fourth-order are linearly decomposed in terms of Pauli, Dirac, and fourth-order Gell–Mann matrices establishing a direct correspondence between the different representations and matrix algebra bases. In addition, and with the expectation for further applications in group theory, a linear decomposition of GL(4) matrices on the basis of the parametric fourth-order matrix generators and in terms of four-vector parameters is explored. By establishing unitary conditions, a parametrization of two subgroups of SU(4) is achieved. [ABSTRACT FROM AUTHOR]

This paper reports the successful application of a self-developed, miniaturized, low-power nano-star tracker for precise attitude measurement of a 5-m-long satellite extension boom. Such extension booms are widely used in space science missions to extend and support payloads like magnetometers. The nano-star tracker, based on a CMOS image sensor, weighs 150 g (including the baffle), has a total power consumption of approximately 0.85 W, and achieves a pointing accuracy of about 5 arcseconds. It is paired with a low-cost, commercial lens and utilizes automated calibration techniques for measurement correction of the collected data. This system has been successfully applied to the precise attitude measurement of the 5-m magnetometer boom on the Chinese Advanced Space Technology Demonstration Satellite (SATech-01). Analysis of the in-orbit measurement data shows that within shadowed regions, the extension boom remains stable relative to the satellite, with a standard deviation of 30′′ (1σ). The average Euler angles for the "X-Y-Z" rotation sequence from the extension boom to the satellite are [−89.49°, 0.08°, 90.11°]. In the transition zone from shadow to sunlight, influenced by vibrations and thermal factors during satellite attitude adjustments, the maximum angular fluctuation of the extension boom relative to the satellite is approximately ±2°. These data and the accuracy of the measurements can effectively correct magnetic field vector measurements. [ABSTRACT FROM AUTHOR]

Estimating the location of unmanned aerial vehicles (UAVs) within a global coordinate system can be achieved by correlating known world points with their corresponding image projections captured by the vehicle's camera. Reducing the number of required world points may lower the computational requirements needed for such estimation. This paper introduces a novel method for determining the absolute position of aerial vehicles using only two known coordinate points that reduce the calculation complexity and, therefore, the computation time. The essential parameters for this calculation include the camera's focal length, detector dimensions, and the Euler angles for Pitch and Roll. The Yaw angle is not required, which is beneficial because Yaw is more susceptible to inaccuracies due to environmental factors. The vehicle's position is determined through a sequence of straightforward rigid transformations, eliminating the need for additional points or iterative processes for verification. The proposed method was tested using a Digital Elevation Model (DEM) created via LiDAR and 11 aerial images captured by a UAV. The results were compared against Global Navigation Satellite Systems (GNSSs) data and other common image pose estimation methodologies. While the available data did not permit precise error quantification, the method demonstrated performance comparable to GNSS-based approaches. [ABSTRACT FROM AUTHOR]

In this manuscript, I introduce a convenient method to convert the Euler angles (complete crystal orientations) obtained with EBSD (electron back-scattered diffraction) to the azimuth and inclination angle of crystallographic axes to reconstruct the pole figures. This method, which is subsequently coupled with the rotation of the pole figures, is particularly useful for the analyses of quartz c-axis fabrics in the deformed rocks, where the foliation and/or lineation is not clear or unknown. Although we arbitrarily choose the sample coordinates (Xs-Ys-Zs) in such cases, it is often possible that we can rotate the quartz c-axis pole figures so that they will exhibit a well-defined intrinsic symmetry in the rotated sample frame. The rotated XsYs-plane and Xs-direction can be now called the foliation and lineation, respectively, inversely defined by the quartz c-axis fabrics. On the other hand, the foliation and lineation clearly defined by the shape-preferred orientations (SPOs) of platy or columnar minerals can be oblique to those defined by the quartz c-axis fabrics. In this case, the former foliation and lineation could represent the total strain, while the latter ones could represent the last incremental strain, indicating triclinic deformation symmetry (e.g., triclinic transpression). [ABSTRACT FROM AUTHOR]

The comment disputes some of the inferences in the paper "Helmert Transformation Problem. From Euler Angles Method to Quaternion Algebra", published in this journal. The key points in the dissent are the following: (1) The number of unknown parameters in the reverse transformation problem using dual quaternions. (2) The reliability of both data and the results. (3) There should be no differences between Euler angles and quaternion methods. Our response is summarized as follows: (1) The problem can be solved using either eight or nine unknown parameters. (2) All the data and results are real. (3) There should be differences between methods because of different calculations. [ABSTRACT FROM AUTHOR]

This work systematically investigates the orientation deviation effect on the elastoplastic deformation of a dual‐phase, nickel‐based single‐crystal superalloy through a combined experimental study and crystal plasticity finite element modelling method (CPFEM). Physically based, dual‐phase microstructural model was developed based on scanning electron microscopy (SEM), which was implemented by finite element (FE) modelling using a representative volume element (RVE) with periodic boundary conditions. An extended equivalent yield criterion coupled with CPFEM was adopted to describe the non‐uniform yield behavior induced by octahedral and cubic slip systems. The predicted results have shown that both the bulk behavior and localized stress–strain nature are orientation deviation dependent and that the first Euler angle plays a more important role in elastoplastic behavior than the second Euler angle. This study has thus advanced the basic understanding of the relationship between orientation deviation and the bulk deformation behavior of the dual‐phase nickel‐based single crystal. Highlights: Physically based dual‐phase microstructural model was constructed.The orientation deviation of [001] for nickel‐based single crystal was characterized.An extended equivalent yield criterion coupled with CPFEM was validated.The elastoplastic behavior with orientation deviation was numerically analyzed. [ABSTRACT FROM AUTHOR]

Quadrotors are expected to play an active role in out-of-sight areas such as equipment inspection and transportation of supplies to disaster areas. However, ensuring stable and highly accurate automatic flights has become a significant challenge. Trajectory tracking control methods that combine adaptive output feedback control based on almost strictly positive real characteristics and backstepping strategy have been proposed for a quadrotor control system with nonlinearities. These methods have simple structures and are robust. However, existing adaptive control methods have the problem of singularities, owing to the use of Euler angles in the modeling of quadrotors. In this paper, we propose an adaptive control method that enables a wider range of attitude changes using quaternions. The effectiveness of the proposed method is validated through numerical simulations. [ABSTRACT FROM AUTHOR]

The aim of this current work is to identify three different gears of cross-country skiing utilizing embedded inertial measurement units and a suitable deep learning model. The cross-country style studied was the skating style during the uphill, which involved three different gears: symmetric gear pushing with poles on both sides (G3) and two asymmetric gears pushing with poles on the right side (G2R) or to the left side (G2L). To monitor the technique, inertial measurement units (IMUs) were affixed to the skis, recording acceleration and Euler angle data during the uphill tests performed by two experienced skiers using the gears under study. The initiation and termination points of the tests were controlled via Bluetooth by a smartphone using a custom application developed with Android Studio. Data were collected on the smartphone and stored on the SD memory cards included in each IMU. Convolutional neural networks combined with long short-term memory were utilized to classify and extract spatio-temporal features. The performance of the model in cross-user evaluations demonstrated an overall accuracy of 90%, and it achieved an accuracy of 98% in the cross-scene evaluations for individual users. These results indicate a promising performance of the developed system in distinguishing between different ski gears within skating styles, providing a valuable tool to enhance ski training and analysis. [ABSTRACT FROM AUTHOR]

For advancing manufacturing, arising AM, with an inverse philosophical approach compared to conventional procedures, has benefits that include intricate fabrication, reduced material waste, flexible design, and more. Regardless of its potential, AM must overcome several challenges due to multi-physical processes with miscellaneous physical stimuli in diverse materials systems and situations, such as anisotropic microstructure and mechanical properties, a restricted choice of materials, defects, and high cost. Unlike conventional experimental work that requires extensive trial and error resources and FEM, which generally consumes substantial computational power, the analytical approach based on physics is an exceptional choice. Understanding the relationship between the microstructure and material properties of the fabricated parts is a crucial focus in AM research. Texture is a vital factor in almost every modern industry. This study first proposed a physics-based model to foreshadow the multi-phase crystallographic orientation distribution in Ti-6Al-4V LPBF while considering the part boundary conditions due to the importance of part geometry in real industry. The thermal distribution obtained from this function operates as the information for the single-phase crystallographic texture model. In this model, we forerun and validate the orientations of single-phase materials utilizing three Euler Angles with the principles of CET and thermodynamics, as well as the intensity of the texture by approximating them with published results. Then, we transform the single-phase texture into a dual-phase texture in Bunge calculation, illustrating visualized by pole figures of both BCC and HCP phases. The tendency and appearances of both BCC and HCP phases in pole figures predicted agree well with the experimental results. This texture evolution model provides a new paradigm for future researchers to model the texture or microstructure evolution semi-analytically and save many computational resources in a real-world perspective. Others have not yet done this work about simulating the multi-phase texture in an analytical approach, so this work bridges the gap in this field. Furthermore, this paper establishes the foundation for future research on materials properties affected by microstructure or texture in academic and industrial environments. The precision and dependability of the results obtained through this method make it a valuable tool for ongoing research and advancement. [ABSTRACT FROM AUTHOR]

For human beings, neck movement will be degraded due to aging, trauma, musculoskeletal disorders, or degenerative diseases. Cervical range of motion (CROM) measurement is one of the popular quantitative neck examinations. Despite radiography is considered as the gold standard, it suffers from invasiveness, radiation exposure, and expensiveness. Recently, vision-based methods have been applied for CROM measurement but achieve large errors and require depth camera. On the other hand, deep neural networks provide good performances on head pose estimation (HPE) from a single image, thus promising for medical CROM measurement. We propose to use CNN networks to extract pyramidal or multi-level image features, which are passed to cross-level attention modules for feature fusion and then to a modified ASPP module and a multi-bin classification/regression module for spatial-channel attention and Euler angle conversion/prediction, respectively. The proposed technique was evaluated on public datasets, such as 300W_LP, AFLW2000, and BIWI, to verify its superior performances (with mean MAE = 3.50°, 3.40°, and 2.31° for different experimental protocols) than state-of-the-art methods. Our pre-trained model was also evaluated with our own collected dataset from hospital for CROM measurement. It also achieved the lowest MAE of 4.58° among other methods and conformed with a medical standard of 5 degrees except the pitch angle (which has a MAE of 5.70°, larger than the standard and the yaw (MAE = 3.60°) and roll angles (MAE = 4.44°)). In general, HPE technique is feasible for CROM measurement and shows its advantages of speed, non-invasiveness, free of anatomical landmark and low cost of operation. [ABSTRACT FROM AUTHOR]

This paper proposes an integrated architecture for navigation and attitude control of low-cost suborbital launch vehicles, propelled by a solid motor. Single-nozzle, two Degrees-of-Freedom (DoF) Thrust Vector Control (TVC) actuation is adopted. For architecture design purposes, a non-linear, unstable, 6 DoF model for the generic thrust-vector-controlled launcher dynamics and kinematics is deduced, and a linear state-space representation is proposed. The navigation system provides full-state estimates resorting to novel complementary kinematic filters, whose design allows to establish an explicit relation with steady-state Kalman filtering. A globally stable estimation solution is obtained, apart from the singularities arising from the use of Euler angles. The attitude control law is derived from the Linear Quadratic Regulator (LQR) using the state-space models for each linearization point of the reference trajectory, with an integral action (LQI) added to improve robustness and to provide null steady state attitude tracking error. A correction method is proposed to allow for pitch and yaw control in the presence of spinning motion, precluding the need of a supplementary roll control system. The control system is implemented through gain scheduling, resorting to an altitude-based linear parametric varying method. The architecture is implemented in a realistic simulation environment, composed by the 6 DoF non-linear model, the Navigation and Control solutions, and the environmental disturbances, to assess its performance through Monte Carlo simulations. The navigation system is able to provide accurate estimates of the state vector, while the control system satisfies attitude tracking performance and robustness to both external disturbances and model parametric uncertainties. • An integrated architecture for navigation and attitude control of suborbital launch vehicles is proposed. • An original time-varying state-space representation is deduced. • Novel complementary kinematic filters are designed to provide accurate full-state estimation. • LQI-based control allows for accurate pitch and yaw reference tracking via thrust vectoring. • The architecture performance is assessed through extensive Monte Carlo simulations. [ABSTRACT FROM AUTHOR]

The reduced multibody system transfer matrix method is a completely recursive method utilizing joint coordinates and applicable for evaluating the generalized accelerations of a multibody system at any given moment, provided that the generalized coordinates and velocities are known. For an open-loop multi-rigid-body system, the generalized coordinates of the system are composed of the generalized relative coordinates of the joint elements. Typically, for each joint element, the dimension of the generalized relative coordinates is equal to its relative motion degrees of freedom, leading to minimum dimension of the generalized coordinates of the system, which is equal to the degrees of freedom of the system. However, this may result in singularity for a ball-and-socket joint element when evaluating its generalized accelerations if any triad of Euler angles is utilized as its generalized relative coordinates. The tetrad Euler parameters are an alternative to Euler angles to resolve such a singular problem, which is a common practice in the dynamics approaches using body coordinates as generalized coordinates; nevertheless, it has not been observed in the completely recursive methods with joint coordinates. In this paper, the self-constraint equations of Euler parameters are taken into account to establish the corresponding reduced transfer equations characterized by a symmetric generalized inertial matrix, which are in completely recursive form. Fundamental numerical stability analyses are conducted via condition numbers of corresponding matrices, demonstrating that employing Euler parameters to describe the relative kinematics of a ball-and-socket joint element enhances numerical stability compared to Euler angles. [ABSTRACT FROM AUTHOR]

To gain insight into the organization of the underlying structures in the northern Tunisia region, detailed gravity data in combination with geological information have been investigated. Many techniques have been used, including residual, upward continuations and derivative. Results from the edge detection technique based on the tilt angle map generated from the first vertical gradient were discussed and compared with results obtained by Euler deconvolution. The compilation and comparison of gravity maps and geology maps enable the determination of major structural trends. It validates specific structural elements acquired from outcrops and specifies new ones. According to the gravity data interpretation, the area of study is impacted by several underlying structural trends. The main trend direction is NE–SW, which is closely linked to the J Chehid and Bled Tejra Akseb trends. NW–SE direction is the second significant trend that borders Bled El Ghorfa to the north. Additional determined trends include: the N–S direction, which is associated with the Lakouat fault. A strong correlation was noted between results from tilt angle method and Euler deconvolution, showing that they may both be used to delineate the main structural framework of the region. [ABSTRACT FROM AUTHOR]

Frenet and Darboux trihedrons are the objects of research. At the current point of the direction curve of the Frenet trihedron, three mutually perpendicular unit orthogonal vectors can be uniquely constructed. The orthogonal vector of the tangent is directed along the tangent to the curve at the current point. The orthogonal vector of the main normal is located in the plane, which is formed by three points of the curve on different sides from the current one when they are maximally close to the current point. It is directed to the center of the curvature of the curve. The orthogonal vector of the binormal is perpendicular to the two previous orthogonal vectors and has a direction according to the rule of the right coordinate system. Thus, the movement of the Frenet trihedron along the base curve, as a solid body, is determined. The Darboux trihedron is also a right-hand coordinate system that moves along the base curve lying on the surface. Its orthogonal vector of the tangent is directed identically to the Frenet trihedron, and other orthogonal vectors in pairs form a certain angle ε with the orthogonal vectors of the Frenet trihedron. This is because one of the orthogonal vectors of the Darboux trihedron is normal to the surface and forms a certain angle ε with the binormal. Accordingly, the third orthogonal vector of the Darboux trihedron forms an angle ε with the orthogonal vector of the normal of the Frenet trihedron. This orthogonal vector and orthogonal vector of the tangent form the tangent plane to the surface at the current point of the curve, and the corresponding orthogonal vectors of the tangent and the normal of the Frenet trihedron form the tangent plane of the curve at the same point. Thus, when the Frenet and Darboux trihedrons move along a curve with combined vertices, there is a rotation around the common orthogonal vector point of the tangent at an angle ε between the osculating plane of the Frenet trihedron and the tangent plane to the surface of the Darboux trihedron. These trihedrons coincide in a separate case (for a flat curve) (ε = 0). The connection between Frenet and Darboux trihedrons – finding the expression for the angle ε, is considered in the article. The inverse problem – the determination of the movement of the Darboux trihedron at a given regularity of the change of the angle ε, is also considered. A partial case is considered and it is shown that for a flat base curve at ε = const, the set of positions of the orthogonal vector of normal forms a developable surface of the same angle of inclination of the generators. In addition, the inverse problem of finding the regularity of the change of the angle ε between the corresponding orthogonal vectors of the trihedrons allows constructing a ruled surface for the gravitational descent of the load, conventionally assumed to be a particle. At the same time, the balance of forces in the projections on the orthogonal vectors of the trihedron in the common normal plane of the trajectory is considered. This balance depends on the angle ε [ABSTRACT FROM AUTHOR]

Various rotation operations are required in paleomagnetism. These include transformations to the geographic coordinate system, tilt correction, and finding virtual geomagnetic poles. Different methods have been used for each rotation operation—graphical manipulation on stereonet, matrix calculation, and spherical trigonometry—and computer software has been developed based on these methods. Quaternions, which are commonly used in three-dimensional computer graphics, can handle rotations about arbitrary axes and provide a unified description of the various rotation operations in paleomagnetism. The conversion from a sample coordinate system to the geographic coordinate system depends on orientation methods that vary by sample type and laboratory. Traditionally, coordinate transformations have been computed using rotation matrices of Euler angles based on stereonet manipulation, but quaternions can flexibly accommodate samples oriented by different conventions. Tilt correction can be expressed as a single rotation around the strike direction of the formation. Virtual geomagnetic poles can be obtained by a single quaternion rotation instead of complicated spherical trigonometry. Python functions are provided for all of the rotation operations discussed in this paper, so the readers can incorporate these functions into their own programs to perform rotations using quaternions. [ABSTRACT FROM AUTHOR]

This paper discusses the modeling and trajectory tracking control of a Bi-rotor. A novel class of vertical flight robots that can perform vertical takeoff, landing, and passenger transportation. Bi-rotor aircraft utilize a simplified mechanism compared to helicopters while maintaining the ability to perform complex maneuvers. With six degrees of freedom and four actuators, including two tilt actuators for steering and two propellers for thrust generation, they are classified as underactuated systems. The trajectory tracking controller employs a combination of feedback linearization and backstepping control methods, with an inner loop controlling the Euler angles and an outer loop regulating the Bi-rotor position and calculating desired angles for trajectory tracking. Control algorithms in the limited existing literature often rely on simplified mathematical models, which tend to overlook crucial nonlinear coupling terms. However, neglecting these terms can have significant implications for the dynamic behavior of the system. The dynamic modeling of the Bi-rotor aircraft was validated using the ADAMS software and integrated with the Simulink environment in MATLAB software. The obtained results represent the effectiveness of the proposed algorithm for the control of the Bi-rotor. [ABSTRACT FROM AUTHOR]

This paper presents the problem of transporting a suspended load by a quadrotor. A full model considering the quadrotor and the dynamics of the suspended load, in a three‐dimensional space, is proposed considering as control inputs the torques and the thrust force due to the motors. The solution proposed consists of simple control strategies based on the tangent linearization of the model, which is controllable and therefore flat. A sliding mode control technique is developed for the thrust force and torques associated with the Euler angles in order to track a desired trajectory of the vehicle in a three‐dimensional space with minimum oscillation of the suspended load. The control strategy results to be relatively simple to implement and achieves local stability of the tracking errors, as well as robustness to internal nonlinearities, which are neglected in the linearization process and other external disturbances. The attractiveness of the sliding surfaces and the stability of the tracking errors are formally studied using Lyapunov stability theory. Simulation results are given to show the performance of the proposed control strategy. [ABSTRACT FROM AUTHOR]

Generating reduced-order, synthetic grain structure datasets that accurately represent the measured grain structure of a material is important for reducing the cost and increasing the accuracy of computational crystal plasticity efforts. This study introduces a machine-learning-based approach, termed texture adaptive clustering and sampling (TACS), for generating representative Euler angle datasets that accurately mimic the crystallographic texture. The TACS approach employs K-means clustering and density-based sampling in a closed-loop iteration to create representative Euler angle datasets. Proof-of-principle experiments were performed on rolled and recrystallized low-carbon steel. Validation of the TACS approach was extended to twenty-two datasets, varying lattice structures, and complex crystallographic textures, thereby encompassing a broad range of materials and crystal structures. Kolmogorov-Smirnov (K-S) test comparisons underscore the performance of the TACS approach over traditional electron backscatter diffraction EBSD dataset reduction techniques, with average K-S test scores nearing 0.9, indicating a high-fidelity representation of the original datasets. In contrast, conventional methods display scores below 0.3, indicating less reliability of the structure representation. The independence of the TACS approach from material texture and its capability to autonomously generate datasets with predetermined data points demonstrates its unbiased potential in streamlining dataset preparation for crystallographic analysis. [ABSTRACT FROM AUTHOR]

Investigating aircraft flight dynamics often requires dynamic wind tunnel testing. This paper proposes a non-contact, off-board instrumentation method using vision-based techniques. The method utilises a sequential process of Harris corner detection, Kanade–Lucas–Tomasi tracking, and quaternions to identify the Euler angles from a pair of cameras, one with a side view and the other with a top view. The method validation involves simulating a 3D CAD model for rotational motion with a single degree-of-freedom. The numerical analysis quantifies the results, while the proposed approach is analysed analytically. This approach results in a 45.41% enhancement in accuracy over an earlier direction cosine matrix method. Specifically, the quaternion-based method achieves root mean square errors of 0.0101 rad/s, 0.0361 rad/s, and 0.0036 rad/s for the dynamic measurements of roll rate, pitch rate, and yaw rate, respectively. Notably, the method exhibits a 98.08% accuracy for the pitch rate. These results highlight the performance of quaternion-based attitude estimation in dynamic wind tunnel testing. Furthermore, an extended Kalman filter is applied to integrate the generated on-board instrumentation data (inertial measurement unit, potentiometer gimbal) and the results of the proposed vision-based method. The extended Kalman filter state estimation achieves root mean square errors of 0.0090 rad/s, 0.0262 rad/s, and 0.0034 rad/s for the dynamic measurements of roll rate, pitch rate, and yaw rate, respectively. This method exhibits an improved accuracy of 98.61% for the estimation of pitch rate, indicating its higher efficiency over the standalone implementation of the direction cosine method for dynamic wind tunnel testing. [ABSTRACT FROM AUTHOR]

Hand dysfunction is a common symptom in stroke patients. This paper presents a robotic device which assists the rehabilitation process in order to reduce the need of physical therapy, i.e., a 3UPS/S parallel robotic device is employed for repetitive robot-assisted rehabilitation. Euler angle representation was used to solve the robot's inverse kinematics. The robot's joint space and rotational workspace were determined for two scenarios. In the first scenario, the workspace was obtained considering the actuator's stroke limitations, while in the second scenario, the workspace was determined by adding a second condition, i.e., the range of motion of the spherical joints. Singularity analysis was performed using the geometric algebra approach. The robot was manufactured using additive manufacturing technology. The solution of the inverse kinematic problem was employed to control the robot. The robot can perform a full range of motion during wrist ulnar deviation and radial deviation motions, with the exception of limited wrist flexion and extension motions. The robot has singular configurations within its workspace. Although the spherical joints have roles in reducing the workspace, the primary causes are actuator selection, radii of the base and moving platforms, and the length of the central leg. These factors can be considered to improve the workspace. Singularity can be avoided by carefully selecting the rotation of the moving platform about the Z-axis and avoiding same leg lengths. [ABSTRACT FROM AUTHOR]

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This study introduces a novel approach for the path planning of a 6-degree-of-freedom free-floating space robotic manipulator, focusing on collision and obstacle avoidance through reinforcement learning. It addresses the challenges of dynamic coupling between the spacecraft and the robotic manipulator, which significantly affects control and precision in the space environment. An innovative reward function is introduced in the reinforcement learning framework to ensure accurate alignment of the manipulator's end effector with its target, despite disturbances from the spacecraft and the need for obstacle and collision avoidance. A key feature of this study is the use of quaternions for orientation representation to avoid the singularities associated with conventional Euler angles and enhance the training process' efficiency. Furthermore, the reward function incorporates joint velocity constraints to refine the path planning for the manipulator joints, enabling efficient obstacle and collision avoidance. Another key feature of this study is the inclusion of observation noise in the training process to enhance the robustness of the agent. Results demonstrate that the proposed reward function enables effective exploration of the action space, leading to high precision in achieving the desired objectives. The study provides a solid theoretical foundation for the application of reinforcement learning in complex free-floating space robotic operations and offers insights for future space missions. [ABSTRACT FROM AUTHOR]

The endosomal sorting complexes required for transport (ESCRTs) are responsible for membrane remodeling in many cellular processes, such as multivesicular body biogenesis, viral budding, and cytokinetic abscission. ESCRT-III, the most abundant ESCRT subunit, assembles into flat spirals as the primed state, essential to initiate membrane invagination. However, the three-dimensional architecture of ESCRT-III flat spirals remained vague for decades due to highly curved filaments with a small diameter and a single preferred orientation on the membrane. Here, we unveiled that yeast Snf7, a component of ESCRT-III, forms flat spirals on the lipid monolayers using cryogenic electron microscopy. We developed a geometry-constrained Euler angle--assigned reconstruction strategy and obtained moderate-resolution structures of Snf7 flat spirals with varying curvatures. Our analyses showed that Snf7 subunits recline on the membrane with N-terminal motifs α0 as anchors, adopt an open state with fused α2/3 helices, and bend α2/3 gradually from the outer to inner parts of flat spirals. In all, we provide the orientation and conformations of ESCRT-III flat spirals on the membrane and unveil the underlying assembly mechanism, which will serve as the initial step in understanding how ESCRTs drive membrane abscission. [ABSTRACT FROM AUTHOR]

The aim of this study is the determination of the best fit ellipsoid to given points by quaternions. The problem of the fitting ellipsoid is frequently encountered in image processing, computer games, medicine, engineering and science applications, geodesy, etc. The ellipsoid fitting problem is the process of determining the ellipsoid that best fits a given set of points in 3D. In the fitting process, it is generally done over two models. The first of these is the algebraic method and the second one is orthogonal (geometric) method. In this study, we tried to solve the problem of algebraic and orthogonal ellipsoid fitting based on Euler angles for the first time over quaternions. The superiority of quaternions over Euler rotation angles is well known. In addition, the variance–covariance matrix of the parameters of the fitted ellipsoid will also be calculated. Numerical applications show that the proposed method can be used successfully. [ABSTRACT FROM AUTHOR]

The fully aquatic Japanese giant salamander (Andrias japonicus) is a member of the Cryptobranchidae, and is currently distributed in western Japan, with other members of this group restricted to China and North America. Their feeding behaviour is characterized by a form of suction feeding that includes asymmetric movements of the jaw and hyobranchial apparatus. Previous studies on the North American species, Cryptobranchus alleganiensis, have suggested that this specialized jaw movement is produced by a flexible quadrate‐articular joint combined with a loosely connected lower jaw symphysis including two small fibrocartilaginous pads. However, little is known about this feeding behaviour in the Asian species, nor have the three‐dimensional asymmetric jaw movements been fully investigated in any member of Cryptobranchidae. In this study, we explore the asymmetric jaw movements in A. japonicus using three methods: (1) dissection of musculoskeletal structures; (2) filming of feeding behaviour to understand in which situations asymmetric feeding is used; (3) analysis of 3D movement of jaws and skull. In the third component, fresh (from frozen) specimens of A. japonicus were manipulated to replicate asymmetric and symmetric jaw movements, with the specimens CT scanned after each step to obtain the 3D morphology of the jaws at different positions. These positions were combined and their Euler angles from resting (closed) jaw position were calculated for asymmetric or symmetric jaw positions. Our filming revealed that asymmetric jaw movements are linked to the position of the prey in relation to the snout, with the jaw closest to the prey opening asymmetrically. Moreover, this action allows the salamander to simultaneously grasp prey in one side of the mouth while ejecting water on the other side, if the first suction attempt fails. The asymmetric jaw movements are performed mainly by rotation of the mandible about its long axis, with very limited lateral jaw movements. During asymmetric and symmetric jaw movements, the posterior ends of the maxilla and quadrate move slightly. The asymmetric jaw movements are permitted by a mobile quadrate‐articular joint formed by wide, round cartilages, and by two small fibrocartilage pads within the jaw symphysis that act as cushions during jaw rotation. Some of these soft tissue structures leave traces on the jaws and skull, allowing feeding mode to be reconstructed in fossil taxa. Understanding cryptobranchid asymmetric jaw movement thus requires a comprehensive assessment of not only the symphysial morphology but also that of other cranial and hyobranchial elements. [ABSTRACT FROM AUTHOR]

In this study, a linear quadratic integral differential game approach is applied to regulate and track the Euler angles for a quadrotor experimental platform using two players. One produces commands for each channel of the quadrotor and another generates the worst disturbance based on the mini-maximization of a quadratic criterion with integral action. For this purpose, first, the attitude dynamics of the platform are modeled and its parameters are identified based on the Nonlinear Least Squares Trust-Region Reflective method. The performance of the proposed controller is evaluated for regulation and tracking problems. The ability of the controller is also examined in the disturbance rejection. Moreover, the influence of uncertainty modeling is studied on the obtained results. Then, the performance of the proposed controller is compared with the classic Proportional Integral Derivative, Linear Quadratic Regulator, and Linear Quadratic Integral Regulator. The results demonstrate the effectiveness of the Game Theory on the Linear Quadratic Regulator approach when the input disturbance occurs. [Display omitted] • Implementation of a linear quadratic controller based on differential game theory. • Introducing an action integral to the controller to eliminate steady-state errors. • Proof of optimality of the proposed controller. • Evaluation of the controller's performance on a quadrotor experimental setup. • Comparison of the controller's performance with other robust control methods. [ABSTRACT FROM AUTHOR]

One of the main methods of the path localization of moving objects is positioning using Global Navigation Satellite Systems (GNSSs) in cooperation with Inertial Navigation Systems (INSs). Its basic task is to provide high availability, in particular in areas with limited access to satellite signals such as forests, tunnels or urban areas. The aim of the article is to carry out the testing and analysis of selected navigation parameters (3D position coordinates (Northing, Easting, and height) and Euler angles (pitch and roll)) of the GNSS/INS system for Unmanned Surface Vehicle (USV) path localization during inland hydrographic surveys. The research used the Ellipse-D GNSS/INS system working in the Real Time Kinematic (RTK) mode in order to determine the position of the "HydroDron" Autonomous Surface Vehicle (ASV). Measurements were conducted on four representative routes with a parallel and spiral arrangement of sounding profiles on Lake Kłodno (Poland). Based on the obtained research results, position accuracy measures of the "HydroDron" USV were determined using the Ellipse-D GNSS/INS system. Additionally, it was determined whether USV path localization using a GNSS/INS system working in the RTK mode meets the positioning requirements for inland hydrographic surveys. Research has shown that the Ellipse-D system operating in the RTK mode can be successfully used to position vessels when carrying out inland hydrographic surveys in all International Hydrographic Organization (IHO) Orders (Exclusive, Special, 1a/1b and 2) even when it does not work 100% correctly, e.g., loss of RTK corrections for an extended period of time. In an area with limited coverage of the mobile network operator (30–40% of the time the receiver operated in the differential mode), the positioning accuracy of the "HydroDron" USV using the Ellipse-D GNSS/INS system working in the RTK mode was from 0.877 m to 0.941 m for the R95(2D) measure, depending on the route travelled. Moreover, research has shown that if the Ellipse-D system performed GNSS/INS measurements using the RTK method, the pitch and roll error values amounted to approx. 0.06°, which is almost identical to that recommended by the device manufacturer. However, when working in the differential mode, the pitch and roll error values increased from 0.06° to just over 0.2°. [ABSTRACT FROM AUTHOR]

People with upper limb disabilities or high quadriplegia have extremely high requirements for the maneuverability and functionality of power wheelchairs. Normal wheelchairs cannot meet travel tasks, while smart customized wheelchairs are expensive and cannot be popularized. Therefore, a novel extension control instrument for power wheelchairs with low cost, strong scalability, and convenient usage is proposed, which can realize the control of the wheelchair by sensing a change of head posture. The device is divided into a head motion sensing unit (HMSU) and a wheelchair assistance control unit (WACU). The mapping relationship between the head attitude and the subject's motion intention is established. The inertial measurement module in the HMSU collects the head attitude data and uses the Kalman filtering method to obtain the accurate Euler angle. The WACU is fixed on the original controller of the wheelchair. The joystick is inserted into the extended control mechanism and controlled, instead of the hand, through a 2-degree-of-freedom servo system combined with the pinion and rack push rod structure, thus controlling the movement of the wheelchair. In proceeding, the system can also detect the distance of objects in the environment in real time through the three-direction (front, left, right) ultrasonic ranging sensors installed on the WACU, with a certain obstacle avoidance function. The prototype experiments prove that the extension control instrument developed in this paper based on the Kalman filter can quickly and accurately identify head motion and accurately control the movement of the wheelchair. It is easy to operate and has strong universality, which presents a new low-cost solution for the travel of patients with disabilities. [ABSTRACT FROM AUTHOR]

Quantum gravity is expected to introduce quantum aspects into the description of reference frames. Here we begin exploring how quantum gravity induced deformations of classical symmetries could modify the transformation laws among reference frames in an effective regime. We invoke the quantum group SU q (2) as a description of deformed spatial rotations and interpret states of a representation of its algebra as describing the relative orientation between two reference frames. This leads to a quantization of one of the Euler angles and to an aspect of agency dependence : space is reconstructed as a collection of fuzzy points, exclusive to each agent, which depends on their choice of reference frame. Each agent can choose only one direction in which points can be sharp, while points in all other directions become fuzzy in a way that depends on this choice. Two agents making different choices will thus observe the same points with different degrees of fuzziness. [ABSTRACT FROM AUTHOR]

Purpose: This work intends to investigate the rigid body's motion around a specific fixed point (analogous to Lagrange's scenario) in the presence of a gyrostatic moment (GM) besides the attraction of a Newtonian force field (NFF). This task is carried out by presuming that the body is quickly rotating about one of the major or minor principal axes of the inertia ellipsoid. Method: The controlling system of six nonlinear differential equations (DEs) along with three first integrals is boiled down to an appropriate system of two DEs in addition to only one integral. Therefore, the analytic solutions of this system are obtained utilizing the approach of Poincaré small parameter (APSP). Results: Euler's angles for the motion under investigation are derived to assess this motion at any instant of time. Additionally, phase plane graphs are displayed using computer codes to depict the stability behavior of the dynamical motion at any time. Conclusion: These achieved outcomes are thought of as a generalization of the ones that were found in some of previous works, in the absence of all applied forces and moments. This work presents a distinctive contribution in several crucial areas, particularly in engineering applications that have used the gyroscopic theory to determine the orientation and maintain the stability of various vehicles, such as spaceships, airplanes, submarines, and racing cars. [ABSTRACT FROM AUTHOR]

The agile eye is classified as a 3-degrees of freedom (DOF) type 3-RRR spherical parallel mechanism (SPM) designed to replicate the movement patterns seen in the human eye. The end organ exhibits a range of motion inside a cone of vision spanning 140°, with a twist tolerance of ±30°. Additionally, the mechanism can achieve angular velocities exceeding 1000 °/s and angular accelerations surpassing 20 °/ $\text{s}^{2}$ . The objective of this research is to conduct a comprehensive examination of the direct and inverse kinematics of the spatial parallel manipulator (SPM) on a manipulator. The purpose of doing a kinematic analysis on the manipulator is to enhance the design optimization process, accurately determine the dimensions of all components, and improve the functionality of the computer-aided design (CAD) system. This study aims to ensure the efficient operation of the SPM and maximize its available workspace. Additionally, this study enables the achievement of a high level of stiffness inside the workspace and the establishment of clearly comprehensible limits for the workspace. Moreover, it facilitates the precise control of the SPM. An evaluation is conducted to assess the efficacy of the existing technique by comparing its outputs with those obtained from a virtual reality simulation using the commercially available software CopeliaSim. The control mechanism described in the present paper demonstrates a surplus in the resolution of precision problems while ensuring competitiveness at a controlled cost. The implementation of the robot revealed the effectiveness of the design approach adopted by recording an error of no more than 1%.

The comment disputes some of the inferences in the paper “Helmert Transformation Problem. From Euler Angles Method to Quaternion Algebra”, published in this journal. The key points in the dissent are the following: (1) The number of unknown parameters in the reverse transformation problem using dual quaternions. (2) The reliability of both data and the results. (3) There should be no differences between Euler angles and quaternion methods. Our response is summarized as follows: (1) The problem can be solved using either eight or nine unknown parameters. (2) All the data and results are real. (3) There should be differences between methods because of different calculations.

Head Pose Estimation (HPE) has a wide range of applications in computer vision, but still faces challenges: (1) Existing studies commonly use Euler angles or quaternions as pose labels, which may lead to discontinuity problems. (2) HPE does not effectively address regression via rotated matrices. (3) There is a low recognition rate in complex scenes, high computational requirements, etc. This paper presents an improved unconstrained HPE model to address these challenges. First, a rotation matrix form is introduced to solve the problem of unclear rotation labels. Second, a continuous 6D rotation matrix representation is used for efficient and robust direct regression. The RepVGG-A2 lightweight framework is used for feature extraction, and by adding a multi-level feature fusion module and a coordinate attention mechanism with residual connection, to improve the network's ability to perceive contextual information and pay attention to features. The model's accuracy was further improved by replacing the network activation function and improving the loss function. Experiments on the BIWI dataset 7:3 dividing the training and test sets show that the average absolute error of HPE for the proposed network model is 2.41. Trained on the dataset 300W_LP and tested on the AFLW2000 and BIWI datasets, the average absolute errors of HPE of the proposed network model are 4.34 and 3.93. The experimental results demonstrate that the improved network has better HPE performance. [ABSTRACT FROM AUTHOR]

Many problems in computer vision today are solved via deep learning. Tasks like pose estimation from images, pose estimation from point clouds or structure from motion can all be formulated as a regression on rotations. However, there is no unique way of parametrizing rotations mathematically: matrices, quaternions, axis‐angle representation or Euler angles are all commonly used in the field. Some of them, however, present intrinsic limitations, including discontinuities, gimbal lock or antipodal symmetry. These limitations may make the learning of rotations via neural networks a challenging problem, potentially introducing large errors. Following recent literature, we propose three case studies: a sanity check, a pose estimation from 3D point clouds and an inverse kinematic problem. We do so by employing a full geometric algebra (GA) description of rotations. We compare the GA formulation with a 6D continuous representation previously presented in the literature in terms of regression error and reconstruction accuracy. We empirically demonstrate that parametrizing rotations as bivectors outperforms the 6D representation. The GA approach overcomes the continuity issue of representations as the 6D representation does, but it also needs fewer parameters to be learned and offers an enhanced robustness to noise. GA hence provides a broader framework for describing rotations in a simple and compact way that is suitable for regression tasks via deep learning, showing high regression accuracy and good generalizability in realistic high‐noise scenarios. [ABSTRACT FROM AUTHOR]

Consider a rigid body rolling with one point in contact with a fixed surface. Now suppose that the instantaneous point of contact traces out a closed path. As a demonstration of a phenomenon known as holonomy, the body will typically not return to its original orientation. The simplest demonstration of this phenomenon in rigid body dynamics occurs in the motion of a rolling sphere and finds application to path planning and reorientation of spherical robots. Motivated by earlier works of Bryant and Johnson, we establish expressions for the change in orientation of a rolling sphere after completing a rectangular path. We use numerical methods to show that all possible changes in orientation are possible using a single rectangular path. Based on the Euler angle parameterization of a rotation, we develop a more intuitive method to achieve a desired orientation using three rectangular paths. With regards to applications, the paths we discuss can be employed to achieve any desired reorientation of a spherical robot. [ABSTRACT FROM AUTHOR]

This paper addresses the issue of agile attitude maneuver control for low-Earth-orbit satellites during short arc segments for multi-target observations. Specifically, a configuration design for Control Moment Gyroscopes (CMGs) and a hybrid control law are provided. The control law is adept at avoiding singularities and escaping singular planes. Subsequently, an optimal time-based attitude maneuver path-planning method is presented, rooted in the relationship between Euler angles/axis and quaternions. Furthermore, a novel satellite attitude maneuver controller is developed based on a piecewise power-reaching law for variable structure sliding mode control. The paper theoretically demonstrates that the proposed piecewise power reaching law possesses two favorable properties regarding convergence time. On the other hand, the designed reaching law maintains continuity at all stages, theoretically eliminating buffeting. The simulation results demonstrate that the proposed controller achieves an Euler angle control precision of ±0.03° and angular velocity accuracy of ±0.15°/s, fulfilling the demands of multi-objective observational tasks. Compared to conventional power reaching law controllers, the convergence time is reduced by 3 s, and Euler angle accuracy is improved by 70%. This underscores the effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

This research aims to establish the properties of the directional cosines and quaternion parameters to examine the proper scale, inclination, and numerical drift errors in a four-dimensional space. The research uses transformation matrices in the four-dimensional space whose elements are the directional cosines or quaternions. The kinematic equations are derived in terms of both directional cosines and quaternions. Applying the Kalman filter allows us to minimize the measurement error and refine the numerical calculation. The observation of the Euler angle covariance matrix takes on a crucial feature in the design of inertial navigation systems vital for analyzing efficient attitude. [ABSTRACT FROM AUTHOR]

Traditional heatmap regression methods have some problems such as the lower limit of theoretical error and the lack of global constraints, which may lead to the collapse of the results in practical application. In this paper, we develop a facial landmark detection model aided by offset prediction to constrain the global shape. First, the hybrid detection model is used to roughly locate the initial coordinates predicted by the backbone network. At the same time, the head rotation attitude prediction module is added to the backbone network, and the Euler angle is used as the adaptive weight to modify the loss function so that the model has better robustness to the large pose image. Then, we introduce an offset prediction network. It uses the heatmap corresponding to the initial coordinates as an attention mask to fuze with the features, so the network can focus on the area around landmarks. This model shares the global features and regresses the offset relative to the real coordinates based on the initial coordinates to further enhance the continuity. In addition, we also add a multi-scale feature pre-extraction module to preprocess features so that we can increase feature scales and receptive fields. Experiments on several challenging public datasets show that our method gets better performance than the existing detection methods, confirming the effectiveness of our method. [ABSTRACT FROM AUTHOR]

• A form-finding method considering main cable's flexural and torsional stiffnesses is proposed based on Cosserat rod. • The rotation vector's expression of the principal axis of inertia is derived via Euler angles. • Differential equations of cable forces and Euler angles are derived via the rotation vector. • A semi-analytical algorithm considering inclined hanger forces is constructed via the transfer matrix method. • The bidirectional bending under self-weight and non-eccentric hanger force causes the rotation of main cable. During the configuration transformation, the 3D curved main cable's torsion increased the construction control's complexity in the suspension bridge with the spatial cable system. The calculation of the torsional angle of the main cable must consider the cable's flexural and torsional stiffnesses. However, most form-finding methods adopted the assumption of the ideal flexible cable, failing to obtain the torsional angle of the main cable. Hence, this study proposed a semi-analytical form-finding method for the 3D curved cable that considered the main cable's flexural and torsional stiffnesses. This method was based on the Kirchhoff-Love rod theory, which provided a feasible framework for analyzing the torsion in the main cable. Firstly, the Euler angles were incorporated to facilitate the parameterization of finite rotation and the transformation of static equilibrium differential equations. Then the governing equations of the target configuration were established using the geometric compatibility conditions and the static equilibrium conditions. Moreover, the equations were numerically solved through the finite difference method and the Levenberg-Marquardt method. Finally, four examples were employed to verify the feasibility and accuracy of the proposed method, including two cantilever beams and two suspension bridges with curved 3D cable and 2D cable, respectively. The results of the case study demonstrated that the torsion in the main cable was primarily caused by bidirectional bending rather than internal torque. This torsion was observed under the influence of the cable's self-weight and non-eccentric hanger forces. Furthermore, it was observed that the maximum torsion did not occur at the mid-span point. The position of the inflection point was found to be highly sensitive to the orientation constraints imposed at the fixed end of the cable. [ABSTRACT FROM AUTHOR]

This paper investigates the inner-loop attitude and velocity control problem for air-breathing hypersonic vehicles subject to practically constrained actuators. Traditional nonlinear dynamic inverse (NDI) is enhanced by an intelligent control allocation (ICA) to realise the low-complexity attitude decoupling. The ICA addresses various characteristics of aerodynamic control surfaces while providing an efficient detecting mechanism for allocation errors. Within the ICA-NDI frame, a finite-time disturbance observer-based terminal sliding mode (FTDO-based TSM) attitude controller regulates Euler angles in the fast manner. This attitude controller employs an improved integral sliding surface containing necessary information from both ICA detection and FTDO estimation, which, therefore, is capable of accommodating constrained control surfaces and rejecting mismatched disturbances. Meanwhile, an adaptive governor adjusts the velocity reference according to the current saturation level of the scramjet fuel-to-air equivalency ratio. The governor is further integrated to a baseline adaptive velocity controller to synthetically handle scramjet input saturations and uncertain factors. [ABSTRACT FROM AUTHOR]

Artificial Intelligence (AI) and the constant paradigm shift in road traffic have led to a need for significant improvement in road safety to minimize traffic accidents. LiFi helps minimize accidents by transmitting data between multiple vehicles (i.e. Vehicle-to-Vehicle (V2V)) and between vehicles and infrastructure (i.e. Vehicle-to-Infrastructure (V2I)) without interference. LiFi uses light to transmit data between devices or vehicles, which ensures efficient data transmission speed and is therefore considered a safe technology. A method called Deep Jacobian Regression and Tate Bryant Euler Recommendation (DJR-TBER) is proposed in this paper based on V2V and V2I autonomous vehicle communication. The proposed method DJR-TBER consists of an input layer, four hidden layers and finally an output layer. Sensors are first used to obtain the information. A linear regression-based speed evaluation model is developed and followed by a Jacobi matrix-based distance evaluation model in the hidden layer. The third hidden layer by developing a distance evaluation model. The use of Laplacian function ensures secure V2I communication for the autonomous vehicle. Finally, a Tate-Bryant-Euler angle-based model for emergency handling is proposed in the hidden layer to optimally consider the aspect of braking in emergency situations and thus increase driving safety. [ABSTRACT FROM AUTHOR]

The construction of bounding boxes is an effective method to reduce the complexity of collision detection in virtual assembly. In response to the shortcomings of creating AABB bounding boxes by default for Box components in the X3D environment, a method for creating OBB bounding boxes in this environment is studied. By calculating Euler angles and combining with Transform components, the construction of OBB bounding boxes in this environment is implemented. In order to improve the efficiency of constructing bounding boxes in X3D environment, a hybrid adaptive bounding box generation algorithm based on AABB and OBB is proposed, which adds judgment on the degree of inclination of objects and enables it to adaptively select AABB or OBB methods to construct bounding boxes based on the geometric features of the 3D model. The research results show that when the included angle threshold is set to 15°, this algorithm reduces the total generation time of bounding box for the entire teapot by 5.61% compared to the pure OBB method, and the total volume by 1.53% compared to the pure AABB method. This algorithm combines the fast generation speed of AABB bounding boxes and the good tightness of OBB bounding boxes, making it an effective bounding box construction algorithm. [ABSTRACT FROM AUTHOR]

In the field of modern healthcare, technology plays a crucial role in improving patient care and ensuring their safety. One area where advancements can still be made is in alert systems, which provide timely notifications to hospital staff about critical events involving patients. These early warning systems allow for swift responses and appropriate interventions when needed. A commonly used patient alert technology is nurse call systems, which empower patients to request assistance using bedside devices. Over time, these systems have evolved to include features such as call prioritization, integration with staff communication tools, and links to patient monitoring setups that can generate alerts based on vital signs. There is currently a shortage of smart systems that use sensors to inform healthcare workers about the activity levels of patients who are confined to their beds. Current systems mainly focus on alerting staff when patients become disconnected from monitoring machines. In this technical note, we discuss the potential of utilizing cost-effective sensors to monitor and evaluate typical movements made by hospitalized bed-bound patients. To improve the care provided to unaware patients further, healthcare professionals could benefit from implementing trigger alert systems that are based on detecting patient movements. Such systems would promptly notify mobile devices or nursing stations whenever a patient displays restlessness or leaves their bed urgently and requires medical attention.