Angular Velocity Vector Research Articles

Determination of Velocities of Thread Axis Points When Interacting with a Direct Large Curvature Taking into Account the Deformation in the Contact Zone

The theoretical study of the process of the movement of the thread, which is deformed in the contact zone with the guide, from the point of view of determining the speed, is of great importance for solving a number of specific applied problems. When determining the speed, it was assumed that the start of the Lagrangian and Euler coordinates coincided. Taking into account that the crumpling of the thread in the contact zone occurs at forces much lower than the forces required to stretch the thread, the thread was considered as non-extensible. To determine the law of the distribution of the speed of the points of the thread, an independent vector, which is always connected to the main triangle, was introduced into consideration. The relationship between the vector of total curvature and the vector of absolute angular velocity is obtained. The obtained values of the speed at an arbitrary point of the axis of the thread as a function of the deformation vector, which determines the degree of twisting of the thread in the contact zone. The obtained results can be used to study various technological processes in the sewing, knitting, and textile industries, where the movement of the thread takes place along the guide surface of great curvature.

An asymmetrical body: Example of analytical solution for the rotation matrix in elementary functions and Dzhanibekov effect

We solved the Poisson equations, obtaining their exact solution in elementary functions for the rotation matrix of a free asymmetrical body with angular velocity vector lying on separatrices. This allows us to discuss the temporal evolution of Dzhanibekov’s nut directly in the laboratory system, where it is observed. The rotation matrix depends on two parameters with clear physical interpretation as a frequency and a damping factor of the solution. Qualitative analysis of the solution shows that it properly describes a single-jump Dzhanibekov effect.

Shaking a container full of perfect liquid; a tractable case, a torus shell, exhibits a virtual wall

Manipulation (‘shaking’) of a rigid container filled with incompressible liquid starting from stationary generally results in some displacement, or mixing, of the liquid within it. If the liquid also has zero viscosity, a ‘perfect’ or ‘Euler’ liquid, Kelvin’s theorems dramatically simplify the flow analysis. Response is instantaneous; stop the container and all liquid motion stops. In fact an arbitrary manipulation can be considered as alternating infinitesimal translations and rotations of the container. Relative to the container, the liquid is stationary during every translation. Infinitesimal rotations (an infinitesimal vector along the rotation axis) resolve into three orthogonal components in the container frame. Each generates its own infinitesimal liquid displacement vector field. Their combined consequences are usually obscure. Rather than a volume flow, a surface flow in 3D is considerably easier, the liquid slipping freely in a shell, sandwiched between two nested closed surfaces with constant infinitesimal gap. The closedness avoids extra boundaries. The two dimensionality admits a scalar streamfunction determined by the container angular velocity vector. Manipulation of the container angular velocity at will, usually leads to an infinitely rich variety of area preserving re-configurations of the liquid. In particular, any chosen point of the liquid can be moved, in the shell container frame, to any other chosen point. However, for a torus (shell) with a small enough hole (diameter < 0.195 torus diameter), there exists a virtual wall, a hypothetical axial cylinder intersecting the torus. No matter how the torus is manipulated, liquid inside the cylinder stays inside; outside stays outside. The analysis, solving for the streamfunction, is based on the relative vorticity, and the conformal mapping of a torus to a (periodic) rectangle, which lead to a fairly simple convolution integral formula for the flow.

On the dynamics and wakes of a freely settling Platonic polyhedron in a quiescent Newtonian fluid

We investigate systematically the free settling of a single Platonic polyhedron in an unbounded domain filled with an otherwise quiescent Newtonian fluid. We consider a particle–fluid density mimicking a rock in water. Five Platonic polyhedrons of increasing sphericity are studied for a range of Galileo numbers $10 \leqslant \mathcal {G}a \leqslant 300$ . We construct a regime map in the parameter space of Galileo number and particle volume fraction ( $\mathcal {G}a, \phi$ ), highlighting how the angularity of the Platonic polyhedron impacts its settling path and the onset of instabilities. We find that the initial angular position solely affects the transient settling process. All the Platonic polyhedrons maintain a stable settling angular position at low $\mathcal {G}a$ . Higher angularity leads to path unsteadiness at lower $\mathcal {G}a$ . Path instability progresses from steady vertical to unsteady vertical, followed by oblique settling observed for highly spherical particles, but helical settling (HS) for more angular particles. The particle autorotation is found to be the pivotal factor influencing path instability and the regime transition of angular particles. Beginning in the unsteady oblique and helical regimes, particle autorotation becomes more prevalent, escalating further in the chaotic regime as $\mathcal {G}a$ increases. The particle angular velocity vector is shown to be predominantly situated in the horizontal plane. A thorough force balance in the horizontal plane reveals that the Magnus force is the primary driving force of the HS regime. Additionally, we establish two new empirical correlations to predict the particle settling velocity and the disturbed wake length that solely require the physical properties of the system ( $\mathcal {G}a$ and $\phi$ ). Our numerical results suggest that an increase of the density ratio from $2$ to $3$ exerts only a marginal impact on the path instability of the most angular particle, the settling tetrahedron.

Kinematic Batch Estimator for Angular Velocity and Associated Uncertainty

Estimating the relative pose and angular velocity of an uncooperative space object is a critical component for many applications such as on-orbit assembly, inspection, and servicing. This paper presents new mathematical insights that permit a kinematic batch filter formulation for angular velocity estimation using quaternion measurements for relative orientation, thereby eliminating the need to know the moment of inertia of the target. In particular, a novel batch approach to computing the associated uncertainties for the angular velocity magnitude, spin-axis direction, and overall angular velocity vector estimates is presented and compared against previous recursive methodologies. Finally, a realistic use-case scenario is demonstrated in real-time on flightlike hardware.

Движение гиростата вокруг центра инерции в полуевклидовом пространстве

The inertial motion of a gyrostat in a semi-Euclidean space with given index and defect is studied. A gyrostat with a constant gyrostatic moment moves so that its carrier rotates around a fixed center of inertia. Criteria for the existence of regular motions are obtained as a condition for the presence of axial structural-kinetic symmetry of the gyrostat. The properties of nutational, precessional, vibrational-rotational motions are studied and their description in configuration and phase spaces is given. The quadrature dependences of the gyrostat motion parameters in elliptic functions of time are determined. Parametric equations of hodographs of angular velocity and kinetic momentum vectors are found. The study was carried out for the case of the gyrostat angular momentum eigenvector.

Gyroscopically coupled in-plane and transverse vibrations of an annular free-clamped microplate

In this work, we construct and study a model of coupled plane-transverse oscillations of a circular thin plate with a concentric hole under the action of Coriolis and centrifugal inertia forces caused by the rotation of the system along an axis located in the plane of the plate. Equations of vibrations in partial derivatives are obtained using the variational principle of Hamilton - Ostrogradsky. Assuming the smallness of the angular velocity of rotation with respect to the frequency of the working skew-symmetric flexural form of the plate oscillations, an approximate analytical solution is found for both the radial and circumferential, and transverse components of the displacement field in the free oscillation mode. Using the Galerkin projection method, the problem was reduced to a system of two second-order linear differential equations for modal coordinates of mutually orthogonal basic skew-symmetric modes of the plate vibrations. It is found that the regime of initially excited harmonic oscillations in the presence of rotation is transformed into the regime of amplitude-modulated beats. Analytical expressions are found both for the frequency of the slow beat envelope and for the relative depth of their amplitude modulation. The fundamental possibility of determining the modulus of the projection of the angular velocity vector onto the plane of the plate from the measured value of the envelope frequency is shown. The problem of choosing the optimal geometric shape of the resonator from the point of view of maximizing the sensitivity of the system to changes in the value of the angular velocity of rotation is studied. The question of determining the direction of the projection of the angular velocity vector onto the plane of the plate from the measured depth of the amplitude modulation of the beat regime is considered.

An aviation accident report data-driven approach to scenario design for a centrifuge-based dynamic flight simulator

The use of flight simulators in investigating an aviation incident or accident related to human errors has been identified as an important part of a strategy to improve safety. This study aimed to replicate a real flight of the MiG-29 aircraft using a centrifuge-based dynamic flight simulator and to determine the simulator’s accuracy in recreating in-flight aircraft performance. A 60-second recording of the real flight of the MiG-29 aircraft, captured by the flight data recorder, was chosen for replication in the HTC-07 human training centrifuge simulator. To evaluate how accurately the simulator replicates the performance of the aircraft, the linear accelerations and angular velocities acting on a pilot during the real flight were compared with those during the replication of that flight in the simulator. The fit of these parameters was assessed using the root mean square percentage error (RMSPE) and the correlation coefficient (r). The highest replication accuracy was achieved for the vertical component of the linear acceleration (RMSPE=2068; r=0.98), while the worst result was obtained for the longitudinal component (RMSPE=14205; r=0.31). Inaccuracies were much more pronounced for the angular velocity. The roll angular velocity had the lowest replication error (RMSPE=12640). However, its correlation with the recorded velocity during the real flight was very weak (r=-0.02). Despite some inaccuracies in replicating other components of the acceleration and angular velocity vectors, the HTC-07 simulator seems valuable for investigating aviation incidents or accidents related to human factors.

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.&#x0D; 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.&#x0D; 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.&#x0D; Keywords: orientation quaternion, Miller's orientation algorithm, SINS, norm error, computational drift, analytical reference model, quasi-coordinates, numerical simulation.

On the Stability of Stationary Solutions of the Equations of Motion of the Goryachev–Sretensky Gyrostat

The equations of motion of the Goryachev–Sretensky gyrostat are studied. It has been established that one component of the angular velocity vector oscillates with zero average value on all solutions outside the set of zero level of the area integral. All stationary solutions are found, including two states of rest and two parametric families of permanent rotations. Using Chetaev’s method of integral connectives, sufficient conditions for the stability of stationary solutions are obtained. Based on the analysis of the roots of the characteristic equation, instability conditions were obtained. The possibility of gyroscopic stabilization due to the moment of circular-gyroscopic forces under certain conditions has been established.

Зависимость геодезической прецессии от выбора системы координат на примере галилеевых спутников Юпитера

The relativistic effect of geodetic precession in the rotation of Jupiter and its Galilean moons (Io, Europa, Ganymede and Callisto) are investigated. For this purpose, the method developed in (Pashkevich, 2016) was modified, which is applicable for studying any bodies of the Solar System that have long-term ephemeris. As a result of its application, the most significant secular terms of geodetic precession were determined. In this study, the values of the angular velocity vectors of the geodetic rotation of the Galilean satellites of Jupiter relative to the barycenter of the Solar System and relative to the barycenter of the system of Jupiter satellites were calculated. Their differences are shown depending on the choice of coordinate reference system due to the asymmetry of the formula for the angular velocity vector of geodetic rotation for any solar system bodies used for these calculations. The obtained analytical values of the geodetic precession of the rotation parameters of the studied objects can be used for the numerical study of their rotation in the relativistic approximation.

Identification, Taxonomy and Performance Assessment of Type 1 and Type 2 Spin Bowling Deliveries with a Smart Cricket Ball.

Spin bowling deliveries in cricket, finger spin and wrist spin, are usually (Type 1, T1) performed with forearm supination and pronation, respectively, but can also be executed with opposite movements (Type 2, T2), specifically forearm pronation and supination, respectively. The aim of this study is to identify the differences between T1 and T2 using an advanced smart cricket ball, as well as to assess the dynamics of T1 and T2. With the hand aligned to the ball's coordinate system, the angular velocity vector, specifically the x-, y- and z-components of its unit vector and its yaw and pitch angles, were used to identify time windows where T1 and T2 deliveries were clearly separated. Such a window was found 0.44 s before the peak torque, and maximum separation was achieved when plotting the y-component against the z-component of the unit vector, or the yaw angle against the pitch angle. In terms of physical performance, T1 deliveries are easier to bowl than T2; in terms of skill performance, wrist spin deliveries are easier to bowl than finger spin. Because the smart ball allows differentiation between T1 and T2 deliveries, it is an ideal tool for talent identification and improving performance through more efficient training.

On the Kinematic Description of the Motion of a Rigid Body

A system of ordinary differential equations is derived for a vector of finite rotation corresponding to Euler’s theorem: the vector of finite rotation is directed along the axis of finite rotation of a solid and its length is equal to the angle of plane rotation around this axis. The system of equations is explicitly resolved with respect to the time derivative of the components of the rotation vector. The right part of the system depends on the rotation vector and the angular velocity vector in the main axes. The equivalence of the obtained system of equations to the system of equations for quaternions is shown. The coordinates of the orts of the main axes of a rigid body in fixed axes are expressed in terms of the angles of final rotation and the components of angular velocity according to simple analytical formulas.

A Method of Star Spot Center-of-Mass Localization Algorithm for Star Sensor Under Highly Dynamic Conditions

The starlight guidance system calculates the vehicle attitude information by observing the stars position through the star sensor. When the star sensor operates under high dynamic conditions, the motion blur phenomenon exists in the images taken by it, which seriously affects the computational accuracy of attitude solution. In this paper, based on the motion characteristics of the vehicle under high dynamic conditions, we study and derive the imaging model of the blurred star spot under high dynamic conditions. On this basis, we use the least squares method combined with simulated annealing algorithm to fit the trajectory of the corresponding star spot. The trajectory fitting is based on the information of multiple star streaks in the star field image. And finally, we calculate the center-of-mass coordinates of the star spot. Simulations and experiments demonstrate the feasibility and effectiveness of the method. The fitting error is less than 0.3 pixels for the angular velocity vector [10°/s,0°/s,10°/s] and exposure time 200ms, which is an improvement of at least 0.1 pixel compared to previous studies.

Ultrahigh Precision Angular Velocity Measurement using Frequency Shift of Partially Coherent Beams

AbstractIn the recent decade, the rotational Doppler effect has garnered considerable attention for stimulating the development of applications such as rotational Doppler velocity and topological charge measurements. Previous studies performed measurements under sources with one or multiple amplitude, phase, and polarization modulations. However, the applicability of these schemes is limited by the crucial factor of alignment between the source and object, especially if the magnitude of the source is greater than the object size. Therefore, this study proposes a partially coherent angular velocity measurement model that allows the rotational axes of targets to deviate from the source center and is even less susceptible to external jitters. Accordingly, a proof‐of‐principle experiment to determine the angular velocity under arbitrary alignment conditions is conducted. Tracing the rotational motion by rotating the coherent structure of the source results in a frequency shift—red shift for the same rotation and blue shift for a reverse rotation. The angular velocity vectors (both magnitude and direction) of two anisotropic sub‐Rayleigh objects are successfully measured with ultrahigh precision. The lowest angular velocity is 0.001 r s−1. The average relative error is less than 0.05% with sufficient sampling. Thus, the present findings can be applied to velocity metrology and micromanipulation.

Deriving Euler’s Equation for Rigid-Body Rotation via Lagrangian Dynamics with Generalized Coordinates

Euler’s equation relates the change in angular momentum of a rigid body to the applied torque. This paper uses Lagrangian dynamics to derive Euler’s equation in terms of generalized coordinates. This is done by parameterizing the angular velocity vector in terms of 3-2-1 and 3-1-3 Euler angles as well as Euler parameters, that is, quaternions. This paper fills a gap in the literature by using generalized coordinates to parameterize the angular velocity vector and thereby transform the dynamics obtained from Lagrangian dynamics into Euler’s equation for rigid-body rotation.

Использование топологического анализа данных для визуализации выходной информации приборов

The article discusses Takens embedding for 2D and 3D visualization of 1D time series data. The paper considers the use of topological data analysis in conjunction with Takens embedding to analyze the output information of a dynamic system - a rigid body. 3D images of curves constructed from three components of the angular velocity vector of a rigid body for various values of the main moments of inertia are constructed. Three-dimensional images of the curves constructed on the basis of the Takens embedding for the component of the angular velocity of a rigid body for various values of the principal moments of inertia are constructed. Distances between 3D images of curves are determined by constructing persistent landscape functions. Using the method of topological data analysis in conjunction with Takens embedding for image comparison allows you to classify and identify images and output information of the instrumental composition.

Extended Kalman Filter Design to Estimate the Attitude of the Nanosatellite System Using Magnetometer

The problem of estimating nonlinear systems is an important issue in many engineering applications. One of these applications is in the nanosatellite for Vehicle-to-everything (V2X) communications, which is used to replace sensors in the event of failure or error. Therefore, the accuracy of pre-filter estimation is of great importance. To control the orientation of a satellite, it is important to estimate the attitude accurately. Time-series estimation is especially important in micro and nanosatellites, whose sensors are usually low-cost and have higher noise levels than high-end sensors. Also, the algorithms should be able to run on systems with very restricted computer power. In this paper, an overview of the algorithms used to determine the attitude of nanosatellites, and especially using only the magnetometer sensor, will be discussed. This paper aims to simulate a nanosatellite system to estimate the magnetic field derivative vector using only the magnetometer data with the desired accuracy. The estimated vectors in the pre-filter, along with the magnetometer sensor data, are used to estimate the attitude of the satellite. The estimation of the state vector consisting of the vector part of quaternion and the angular velocity vector is used for the control calculation. To evaluate the accuracy of this pre-filter, a comparison of the magnetometer vector estimator with the body sensor data has been used.

AXOIDS OF HOOKE’S JOINT

The analysis of literary sources devoted to the research and construction of cardan transmissions using the Hooke's joint showed that the peculiarities of the spherical movement of the cardan joint crosspiece are insufficiently covered. This is explained by the relatively small weight of the crosspiece, due to which it almost does not affect the dynamics of the cardan transmission. The need to take into account the mass-geometric parameters of the crosspiece and to analyze comprehensively its movement may arise during accurate calculations of high-speed cardan transmissions. In this case, it is convenient to consider the movement of the crosspiece as rolling without sliding of the moving axoid on the stationary axoid. This approach makes it possible to more fully investigate the peculiarities of the spherical movement of the crosspiece and gives a visual representation of the orientation of its instantaneous axis of rotation in fixed and moving coordinate systems. Based on this, this work is devoted to the construction of a kinematic model of the Hooke's joint crosspiece, the movement of which is considered as rolling without sliding of a moving axoid on a stationary one.&#x0D; As a result of the kinematic analysis of the Hooke's joint, the equations of the hodographs of the vector of the absolute angular velocity of the crosspiece in the fixed and moving coordinate systems were obtained, and the parameters of the fixed and moving axoids of the crosspiece of the cardan joint were determined. It is proved that the stationary axoid of the crosspiece of the Hooke's joint is an oblique elliptic cone with the apex in the center of the crosspiece, and its opposite generatrices, crossing the major axis of the guide ellipse, coincide with the axes of the drive and driven shafts of the joint. The moving axoid of the crosspiece is a closed conical surface with a variable opening angle, the maximum value of which is equal to twice the value of the interaxial transmission angle. With small values of the interaxial angle, which occur in the practice of using a single Hooke’s joint, a moving axoid can be approximately considered as a straight circular cone with a vertex in the center of the crosspiece and an opening angle equal to twice the value of the interaxial transmission angle. The error of this assumption increases rapidly with the increase of the interaxial transmission angle. The correctness of the obtained mathematical model was confirmed by computer modeling in the Autodesk Inventor environment.&#x0D; The obtained results can be used in the study of dynamic processes that occur in cardan transmissions using Hooke's joint.

Postflight Recovery of the Rotational Motion of a Small Space Vehicle from Solar Sensor Information

The use of the developed technology for solving inverse problems of parametric identification of the rotational motion of a spacecraft (SC) based on the accumulated sample of measurements of various compositions is expanded. On the example of processing telemetry data (measuring the direction vector to the Sun) of the BugSat-1 small spacecraft (SSC), the possibility of determining not only the characteristics of rotational motion but also its inertia coefficient is demonstrated. The rotational motion is determined based on the data obtained for five time intervals from January to May 2022, during which BugSat-1 made an uncontrolled movement around the direction vector toward the Sun. This type of motion does not allow applying traditional approaches to determine the dynamics of its rotation. In the process of data processing, a simplified model of the rotation of a SSC is used. At the same time, the initial conditions of rotational motion are estimated—the angular velocity vector and orientation angles, which made it possible to draw a conclusion about the precessional motion of a SSC. Using the analytical model of a regular precession, it is possible to estimate the inertia coefficient of a SSC. The results of flight information processing are presented and the errors of the obtained results are estimated.