The 3D rotary movement of an asymmetric rigid body (RB) in space around a fixed point is investigated. A gyrostatic torque (GT), a magnetic field (MF), and a Newtonian force field (NFF) all have an impact on the RB’s ability to rotate. Around its minor axis of inertia, the body is thought to begin moving rapidly. The Krylov–Bogoliubov–Mitropolski (KBM) method is used to solve the governing equations of motion (EOMs) analytically once they are created using Euler–Poisson equations. Additionally, Euler angles approximate analytical solutions (AS) are examined. A graphical simulated viewpoint of the obtained results and the equations of Euler angles, which show how the body is orientated at each given moment, are used to debate the interpretation of the motion. Maintaining control over the body’s rotation and position during motion can be achieved by researching the effects of different values of the acting forces and toques, such as the GT, MF, and NFF. To illustrate how the stability of the RB is affected, phase graphs of various solutions have been created. Periodicity behavior is demonstrated by the closed-phase curves’ symmetry around any of their axes. Together with the beneficial impacts of these forces and torques, the movement behavior of the RB is evaluated and simulated. The results obtained are widely relevant to gyroscopic equipment, especially those that incorporate inertial systems like aircraft and satellites. Technologies that guarantee the motion stability of applications are also included.Supplementary InformationThe online version contains supplementary material available at 10.1038/s41598-025-27177-5.

The definition of a reference system plays a crucial role in quantifying geodetic effects and in surface body cartography. Today, the development and specification of the lunar reference system is stimulated by future space missions, some of which will be manned, and by the needs of lunar space navigation. This paper aims to describe the lunar reference system in use and determine the accuracy of its realisation in space and time. At present, two lunar reference systems are defined: the Principal Axis system (PA system), based on the Moon's principal axes of inertia, and the Mean Earth/Polar Axis (ME system), defined by the Earth's mean position on the Moon surface. A first step towards a relativistic definition of the lunar time scale is also introduced in agreement with that proposed by the International Astronomical Union (IAU) and other recent realisations. We based the realisation of a PA lunar reference system on the choice of an ephemeris, which relied on the coordinates of the laser retro-reflectors on the Moon surface. We related the ME system and frame to the PA system and frame through a rotation transformation. This study provides a new method to determine the transformation procedure between the two systems, based on a series decomposition of librations and pole motion. We used the comparisons of the position of the lunar laser retro-reflectors obtained with different ephemerides to estimate the internal and external uncertainties of the different realisations of the PA and ME systems. It also includes comparisons between Euler angles and the propagation of their uncertainties. This work provides the full expression of the transformation and a new series of libration and polar motion for the lunar motion. We also introduce possible realisations of lunar timescales, applicable depending on their use. The proposed procedure, with opportunities for future improvements, can help set new standards for lunar reference systems and their realisations.

Cardan sequence selection influences subtalar and talonavicular joint kinematics.

Optimization control strategy of wide ZVS range and automatic Euler angle for bi-directional wireless power transfer system by TPS

The quantum asymmetric top is rarely covered in quantum classrooms or textbooks. Yet it is one of the most interesting nontrivial applications of quantum mechanics to spectroscopy. Part of the challenge is the relationship between the lab frame and the body-fixed frame, which is not explained well in currently available treatments. We show how one can solve the quantum dynamics of asymmetric tops in a natural way by working with average and relative degrees of freedom. This allows one to derive all of the operator relationships between the two frames starting from quantum operators rather than relying on results from the classical solution. The derivation also avoids the need to use Euler angles to describe the motion. The solution can then be employed to examine pure rotational spectroscopy, as we illustrate with an example of rotations of the water molecule, which is an asymmetric top.

The main motivation of this research is the semi-analytical exploration of the dynamics of an asteroid that is attacked while approaching a planet (with an inelastic collision of the projectile normally to the surface of the asteroid occurring just before approaching). Namely, the particular case of the spin dynamics of the asteroid that has been struck by a projectile almost perpendicularly to the maximal-inertia principal axis, with further perturbing the dynamics of rotation due to gravitational torques during close approach to the planet, is investigated. The initial surface of the asteroid is assumed to be a rubble pile, but preferably with a quasi-rigid internal structure, with circa constant distances between various parts of the asteroid as a first approximation. As a result of an inelastic collision with the surface of the asteroid, the rubble-pile material should be thrown off the surface into outer space in large amounts; thus, the mass of the asteroid and the moments of inertia along its principal axes should be changed (as well as the regime of angular rotation around its maximal-inertia principal axis). The updated Euler’s equations, stemming from the conservation of angular momentum, have been presented with gravitational torques acting during the approach of the asteroid to the planet (taking into account the impact on the asteroid that occurs just before it enters the zone of close approach). The evolution of the non-linear spin dynamical state is studied, along with kinematical findings for Euler angles via the governing equations, in accordance with two main rotational stages: first, immediately after the impact on the asteroid’s surface; and second, at the regime of asteroid rotation during its close approach to the planet, with perturbations caused by gravitational torques (just after being struck by the projectile).

Inertial Measurement Units (IMUs) represent a valid alternative to standard clinical assessment methods, such as clinical scales, for evaluating upper limb kinematics. A key aspect of utilizing IMUs effectively is ensuring precise sensor-to-segment calibration, which accounts for the relative orientation between the sensor and the attached body segment. This calibration is crucial to obtain accurate results. Although reliable calibration methods are available, their application in clinical and home environments remains challenging due to their complexity. This study aimed to validate a picture-based calibration method feasible for a clinical context and compare it against other standard methods. Ten healthy subjects performed daily activity tasks while upper limb kinematics was recorded using an optoelectronic motion capture system and an IMU-based system. Four calibration methods were compared using error metrics, including root mean square deviation (RMSD) and cross-correlation (XCORR). The results demonstrate that the proposed picture-based method provides highly accurate measurements for the first and second Euler rotation angles of the shoulder, with RMSD < 15° and XCORR > 0.75 across most of the tasks. For the elbow joint, all calibration methods consistently yielded precise results for the first rotation (RMSD < 15° and XCORR > 0.95) across the majority of tasks. The proposed sensor-to-segment calibration method improves the accuracy of upper limb motion data recorded with an IMU-based system compared to traditional methods. Moreover, the calibration approach is easy to use, making it suitable for clinical and home environments.

The aim of a coordinate transformation is to determine the coordinates of a point in a second coordinate system, given their known values in the first system. The subject of coordinate transformations has applications in a wide range of areas. There are different types of coordinate transformation methods: orthogonal, similarity, affine, projective, and polynomial transformations. In this study, we considered a symmetric 12-parameter 3D affine transformation based on the dual quaternion. The performances of different transformation models were compared. We also sought to determine which transformation model to use. A transformation implementation including gimbal lock has been added as an Appendix to clearly demonstrate the advantages of quaternions over Euler angles.

Snapping, driven by stored elastic energy, enables versatile and rapid shape changes in nature; yet replicating such autonomous, reprogrammable morphogenesis in free-standing volumetric structures remains elusive. Here we report a lantern-shaped ribbon-cluster meta-unit that harnesses programmable and reprogrammable elastic energy to achieve over 13 distinct volumetric snapping morphologies from a single unit. Governed by three Euler angles, the meta-unit post-fabrication offers a tunable mechanical design space spanning up to quadrastable states. Unlike single-ribbon or mechanism-based designs, our system autonomously selects snapping pathways via nastic coupling between multiple ribbons, enabling the inverse design of complex snapping morphologies. We harness magnetically actuated bud-to-bloom and tristable morphogenesis to enable fast, non-invasive grasping and remote flow regulation in confined environments. These results establish a general framework for architected materials with programmable shape, stability and function, offering potential applications in soft robotics, deployable devices and mechanical logic.

Calculation of a force effect from muscle action to a quaternion-based musculoskeletal model.

Differences in 3-dimensional Spinal Angles During Thoracic Axial Rotation in the Sitting Position Between Thoracic Kyphosis Groups.

Purpose of reseach. Development of a software package for the automatic creation of photorealistic threedimensional models of faces based on a single input image, which will provide a high degree of detail and realism of models, as well as ease of use,Methods. The study uses an integrated approach to create photorealistic three-dimensional models of faces from two-dimensional images based on reverse rendering methods and cascading convolutional neural networks (CNN). The main element is a three-dimensional transformable model (3DMM), which describes the geometry and albedo of a face through linear combinations of principal component bases (PCA). To match the 3D geometry and the 2D image, a weak perspective projection is used, taking into account Euler angles and lighting conditions. Optimization of the objective function using the Gauss-Newton method minimizes the differences between the input and rendered images, and correction of depth and facial details is achieved through the adaptation of 3D graphics. Linear albedo interpolation clarifies the details of the model in key areas, which contributes to the creation of high-quality and realistic 3D models of faces.Results. This article successfully implements a software package capable of generating photorealistic threedimensional models of faces from one-dimensional images using reverse rendering and cascading convolutional neural networks. The experiments have confirmed the algorithm's ability to perceive important facial characteristics and create opportunities for further applications in the fields of computer graphics, animation and virtual interfaces.Conclusion. The results obtained indicate the high efficiency of the developed algorithm for generating photorealistic three-dimensional models of faces from two-dimensional images. In addition, the results confirm that the use of reverse rendering methods in conjunction with cascading convolutional neural networks allows for significant improvements in visualization quality.

HighlightsWhat are the main findings?Processing of orientation signals in the form of Euler angles, which is characterized by discontinuities, by CNNs is disadvantageous. Changing the modality of the trajectory and orientation time series into algebraically produced accelerometer and gyroscope signals has a positive influence on the effectiveness of the biometric system.The choice of neural network architecture has a significant impact on the identification metrics achieved. Sample processing by Multi-Input CNNs provides high classification accuracy. In the architecture mentioned above, acceleration and angular velocity signals are processed by separate branches with a set number of kernels. This makes it possible to determine the number of feature-extraction filters, which consequently enables equal consideration of selected modalities at the decision stage.What is the implication of the main finding?It is possible to significantly increase the efficiency of person recognition by changing only the representation of the data. Sample modification is feasible for available data corpora and does not involve the collection of additional data (and therefore costs).The paper summarizes experiments involving several CNN architectures. On the basis of the compiled summary, the Multi-Input CNN is recommended for the processing of sets of varying modalities. This architecture performs feature extraction through filters separately for data processed within a branch.The rapid evolution of virtual reality systems and the broader metaverse landscape has prompted growing research interest in biometric authentication methods for user verification. These solutions offer an additional layer of access control that surpasses traditional password-based approaches by leveraging unique physiological or behavioral traits. Current literature emphasizes analyzing controller position and orientation data, which presents challenges when using convolutional neural networks (CNNs) with non-continuous Euler angles. The novelty of the presented approach is that it addresses this limitation. We propose a modality transformation approach that generates acceleration and angular velocity signals from trajectory and orientation data. Specifically, our work employs algebraic techniques—including quaternion algebra—to model these dynamic signals. Both the original and transformed data were then used to train various CNN architectures, including Vanilla CNNs, attention-enhanced CNNs, and Multi-Input CNNs. The proposed modification yielded significant performance improvements across all datasets. Specifically, F1-score accuracy increased from 0.80 to 0.82 for the Comos subset, from 0.77 to 0.82 for the Quest subset, and notably from 0.83 to 0.92 for the Vive subset.

The phase detection of abnormal gait and the prediction of lower-limb angles are key challenges in controlling lower-limb exoskeletons. This study simulated three types of abnormal gaits: scissor gait, foot-drop gait, and staggering gait. To enhance the recognition capability for abnormal gait phases, a four-discrete-phase division for a single leg is proposed: pre-swing, swing, swing termination, and stance phases. The four phases of both legs further constitute four stages of walking. Using the Euler angles of the ankle joints as inputs, the capabilities of a Convolutional Neural Network and a Support Vector Machine in recognizing discrete gait phases are verified. Based on these discrete gait phases, a continuous phase estimation is further performed using an adaptive frequency oscillator. For predicting the lower-limb motion angle, this study innovatively proposes an input scheme that integrates three-axis ankle joint angles and continuous gait phases. Comparative experiments confirmed that this information fusion scheme improved the limb angle prediction accuracy, with the Convolutional Neural Network–Long Short-Term Memory network yielding the best prediction results.

Abstract Active suspensions composed of self-propelled colloidal particles are considered. The propulsion of these micrometric particles is generated by chemical reactions occurring by heterogeneous catalytic at their surface and by diffusiophoresis coupling the concentration gradients of the reacting molecular species to the fluid velocity. By this mechanism, chemical free energy is transduced into mechanical motion. The non-equilibrium thermodynamics of such active suspensions is developed by explicitly taking into account the internal degrees of freedom of the active particles, which are the Eulerian angles specifying their orientation. Accordingly, the distribution function of the colloidal particles is defined in the six-dimensional configuration space of their position and their orientation, which fully characterises the polar, nematic, and higher orientational orders in the active system. The local Gibbs and Euler thermodynamic relations are expressed in terms of the colloidal distribution function, the dynamics of which is ruled by a six-dimensional local conservation equation. All the processes contributing to the entropy production rate are derived from the local conservation and kinetic equations for the colloids, the molecular species, mass, linear momentum, and energy, identifying their thermodynamic forces – also called affinities – and their dissipative current densities. The non-equilibrium constitutive relations between them are obtained using the Curie symmetry principle and the Onsager-Casimir reciprocal relations based on microreversibility. In this way, all the coefficients of mechanochemical coupling are completely determined for isothermal, incompressible, dilute suspensions composed of spherical Janus particles on the basis of the interfacial properties between the fluid solution and the solid particles and chemohydrodynamics. The complete expression of the entropy production rate is thus established for such active systems.&amp;#xD;

Medical application of deep-learning-based head pose estimation from RGB image sequence.

This work focuses on time and frequency domain analyses of IMU-based orientation estimation algorithms, including indirect Kalman (IKF), Madgwick (MF), and complementary (CF) filters. Euler angles and quaternions are used for orientation representation. A 6-DoF IMU is attached to a 6-joint UR5e robotic arm, with the robot's orientation serving as the reference. Robotic arm data is obtained via an RTDE interface and IMU data via a CAN bus. Test signals include pose sequences, which are big-amplitude, slowly changing signals used to evaluate stationary and low-dynamics responses in the time domain, and small-amplitude, fast-changing generalized binary noise (GBN) signals used to evaluate dynamic responses in the frequency domain. To prevent poor filters' performance, their parameters are tuned. In the time domain, RMSE and MaxAE are calculated for roll and pitch. In the frequency domain, composite frequency response and coherence are calculated using the Ockier method. RMSEs are computed for response magnitude and coherence, and averaged equivalent time delay (AETD) is derived from the response phase. In the time domain, MF and CF show the best overall performance. In the frequency domain, they again perform similarly well. IKF consistently performs the worst in both domains but achieves the lowest AETD.

Shoulder joint angles in supine and upright imaging of the preoperative reverse total shoulder arthroplasty patient

Ball joints are commonly used in graphics and robotics in the simulation of articulated rigid body systems. The ball joint orientation can be decomposed, non-uniquely, into swing and twist components. In modeling with ball joints, it can be advantageous to include limit constraints on the swing, twist, or both. While many physics engines include such limit constraints, we have found that twist limit constraints, when present, are lacking in robustness, typically due to unhandled singularities in commonly employed swing-twist decompositions, where the twist state is undefined. In this paper, we introduce two novel, robust models of the ball joint twist limit constraint. The first utilizes the Euler angle decomposition of the orientation, which has two singularity points. To robustly handle the twist limit constraint, we propose a novel treatment at singularities based on the physical behavior of a three-hinge system. Thus, our model significantly reduces the dramatic motion that can occur around the singularities, eliminating the need to directly avoid them. Second, we introduce an alternative incremental model that avoids singularities altogether during simulations. In this model, the twist is quantified by integrating the instantaneous twist velocity in time, ensuring that the singularity shifts from one time step to the next and remains far from the current state. We investigate the properties of both proposed models. We demonstrate that both models offer robust and practical solutions to common issues encountered in constraining the motion of ball joints.

This study investigates the 3D motion of a rigid body (RB) rotating around a fixed point, focusing on Lagrange's case while considering the effects of a gyrostatic moment (GM) and a Newtonian force field (NFF). It is notably that the center of mass is displaced very much from the principal dynamic symmetry axis. Utilizing the fundamental principle of angular momentum, the equations of motion (EOM) are formulated and solved applying the large parameter approach (LPA) to determine approximate solutions (AS) for the irrational frequencies' case. Euler's angles, which define the body's orientation at any moment, are explicitly calculated. Additionally, to assess the impact of applied moments on motion stability, we utilize advanced computational tools to generate graphical representations of the achieved solutions and the related Euler's angles. This work enhances the understanding of RB dynamics in complex motion scenarios, emphasizing the interaction between external forces and GMs in shaping stability and behavior. The study is poised to greatly influence the aerospace industry by advancing our understanding of rotational motion and the dynamics of celestial bodies, with direct applications in the design and operation of spaceships, spacecraft, and satellites.