Three-frequency models for determining the orientation of a solid body taking into account the vibration environment

Two new three-frequency reference models of solid motion taking into account the vibrational environment are proposed. They are based on a four-frequency reference model of rotation [1], which implements rotations according to Krylov angles. For the developed models the analytical dependences for quasi-coordinates, projections of the angular velocity vector and components of the quaternion of orientation corresponding to such rotational motion are obtained. The urgency of taking into account the influence of vibration in traffic modeling on the basis of domestic and foreign literature in the field of navigation, including for the last 10 years. The main sources of vibration are described in detail and what types of oscillations they correspond to - harmonic oscillations occur in moving elements of onboard systems, such as the engine rotor, and in the engine unit and its units there are oscillations that have the character of random broadband noise. Methods of correction of such influence for increase of accuracy of definition of orientation of object are analyzed. The location of the components of the platformless inertial navigation system relative to the vibration sources is considered to be related to the strength of the influence of the vibration environment on the accuracy of the obtained data. Numerical implementations of the models are obtained and the drift error for the third-order orientation algorithm is estimated for several sets of specified parameters in a certain way. The parameters are chosen arbitrarily, but taking into account the existing restrictions on angular motion. The corresponding figures show the result for one of these sets of numerical values (which shows the result of the research in the most detail). The obtained results are compared with the corresponding results for the four-frequency rotation model [1]. The expediency of using new three-frequency models under certain conditions is shown.

SOME PROBLEMS ABOUT THE MOTION OF A RIGID BODY IN A RESISTIVE MEDIUM

The dynamics of rotating rigid bodies is a classical topic of study in mechanics. In the eighteenth and nineteenth centuries, several aspects of a rotating rigid body motion were studied by famous mathematicians as Euler, Jacobi, Poinsot, Lagrange, and Kovalevskya. However, the study of the dynamics of rotating bodies of still important for aplications such as the dynamics of satellite-gyrostat, spacecraft, re-entry vehicles, theory of gyroscopes, modern technology, navigation, space engineering and many other areas. A number of studies are devoted to the dynamics of a rigid body in a resistive medium. The presence of the velocity of proper rotation of the rigid body leads to the apearance of dissipative torques causing the braking of the body rotation. These torques depend on the properties of resistant medium in which the rigid body motions occur, on the body shape, on the properties of the surface of the rigid body and the distribution of mass in the body and on the characters of the rigid body motion. Therefore, the dependence of the resistant torque on the orientation of the rigid body and its angular velocity can de quite complicated and requires consideration of the motion of the medium around the body in the general case. We confine ourselves in this paper to some simple relations that can qualitative describe the resistance to rigid body rotation at small angular velocities and are used in the literature. In setting up the equations of motion of a rigid body moving in viscous medium, we need to consider the nature of the resisting force generated by the motion of the rigid body. The evolution of rotations of a rigid body influenced by dissipative disturbing torques were studied in many papers and books. The problems of motion of a rigid body about fixed point in a resistive medium described by nonlinear dynamic Euler equations. An analytical solution of the problem when the torques of external resistance forces are proportional to the corresponding projections of the angular velocity of the rigid body is obtain in several works. The dependence of the dissipative torque of the resistant forces on the angular velocity vector of rotation of the rigid body is assumed to be linear. We consider dynamics of a rigid body with arbitrary moments of inertia subjected to external torques include small dissipative torques.

The curl of angular velocity and its mechanical significance

The curl of the vector field is widely used in modern field theory, fluid mechanics, mathematics, electromagnetic field, and other fields. In this paper, by introducing an auxiliary vector parameter (We called 𝑷𝑼⃗⃗⃗⃗⃗⃗ ) whose direction satisfies the right-hand thread rule the mathematical expression of angular velocity vector curl (𝛁×𝝎⃗⃗⃗ ) was obtained by analogy with the method of defining velocity vector curl (𝛁×𝒗⃗⃗ ). We also pointed out that the laminar flow of viscous fluid in a circular pipe (Hogen-Poiseuille flow) in nature is a typical real example of angular velocity vector curl (𝛁×𝝎⃗⃗⃗ ). Moreover, a concise mathematical equation (Equation(11)) was given, which could be used to describe some motion characteristics of vortex ring theoretically; Therefore, the motion of a single vortex ring has the dual characteristics of the velocity curl(𝛁×𝒗⃗⃗ ) and the angular velocity curl(𝛁×𝝎⃗⃗⃗ ) at the same time.

Attitude reconstruction from strap-down rate gyros using power series

This paper introduces a power series based method for attitude reconstruction from triad orthogonal strap-down gyros. The method is implemented and validated using quaternions and direction cosine matrix in single and double precision implementation forms. It is supposed that data from gyros are sampled with high frequency and a fitted polynomial is used for an analytical description of the angular velocity vector. The method is compared with the well-known Taylor series approach, and the stability of the coefficients’ norm in higher-order terms for both methods is analysed. It is shown that the norm of quaternions’ derivatives in the Taylor series is bigger than the equivalent terms coefficients in the power series. In the proposed method, more terms can be used in the power series before the saturation of the coefficients and the error of the proposed method is less than that for other methods. The numerical results show that the application of the proposed method with quaternions performs better than other methods. The method is robust with respect to the noise of the sensors and has a low computational load compared with other methods.

METHOD FOR DETERMINING THE MISALIGNMENT OF OPTICAL-ELECTRONIC AND INERTIAL ORIENTATIONS HELICOPTER SYSTEMS

A unique method for determining the misalignment of the orientation of the instrumental coordinate systems of different posts of the optical-electronic system of a helicopter with each other and with an inertial navigation system is proposed. The method does not require preflight preparation, and is based on processing video information streams generated by thermal imaging and television channels of the optical-electronic system, and using information from an inertial navigation system. The method involves the helicopter performing a special maneuver, which is a rotation of the helicopter at a low altitude. This maneuver can be automated. When the helicopter rotates, trajectories of characteristic points of the underlying surface and airfield infrastructure are formed on the images. In general, the trajectories of these points are hyperbolas, which are approximated by straight lines. The parameters of these straight lines are determined using the least squares method. The angle of inclination of straight lines in the screen coordinate system determines the position of the angular velocity vector in the instrument coordinate systems. Since all the posts of the optical-electronic system measure the same vector, it is possible to determine their mismatch in roll between themselves and with the inertial navigation system. Preliminary modeling showed high potentialities of the proposed method. The method can be considered as an integral part of a more general method for coordinating coordinate systems in roll, pitch and course based on processing video streams of optical-electronic systems. When the method is used in real conditions, the errors in estimating the angular misalignment of the optical-electronic and inertial systems of a helicopter can be in units of arc minutes.

On New Modifications of Some Perturbation Procedures

In this paper, we present new modifications for some perturbation procedures used in mathematics, physics, astronomy, and engineering. These modifications will help us to solve the previous problems in different sciences under new conditions. As problems, we have, for example, the rotary rigid body problem, the gyroscopic problem, the pendulum motion problem, and other ones. These problems will be solved in a new manner different from the previous treatments. We solve some of the previous problems in the presence of new conditions, new analysis, and new domains. We let complementary conditions of such studied previously. We solve these problems by applying the large parameter technique used by assuming a large parameter which inversely proportional to a small quantity. For example, in rigid body dynamic problems, we take such quantity to be one of the components of the angular velocity vector in the initial instant of the rotary body about a fixed point. The domain of our solutions will be depending on the choice of a large parameter. The problem of slow (weak) oscillations is considered. So, we obtain slow motions of the bodies instead of fast motions and find the solutions of the problem in present new conditions on both of center of gravity, moments of inertia, and the angular velocity vector or one of these parameters of the body. This study is important for aerospace engineering, gyroscopic motions, satellite motion which has the correspondence of inertia moments, antennas, and navigations.

Directional magnetic and electric vortex lines and their geometries in Minkowski space

In this study, we firstly introduce a different type of directional Fermi-Walker transportations along with vortex lines of a non-vanishing vector field in three-dimensional Minkowski space. Then we consider some geometric quantities, which are used to characterize vortex lines, in order to express angular velocity vector (Darboux vector) of the system in terms of these quantities. Later we present timelike directional magnetic vortex lines by computing the Lorentz force. Hence, we reach a remarkable relation between timelike directional magnetic vortex lines and angular velocity vector of vortex lines with a nonrotating frame in Minkowski space. We also determine the timelike directional electric vortex lines by considering the electromagnetic force equation. We finally investigate the conditions of being uniform for magnetic fields of timelike directional magnetic vortex lines and we improve such a remarkable approach to find the electromagnetic curvature which contains many geometrical features belonging to timelike directional magnetic and electric vortex line.

The Slow Spinning Motion of a Rigid Body in Newtonian Field and External Torque

In this paper, the problem of the slow spinning motion of a rigid body about a point O, being fixed in space, in the presence of the Newtonian force field and external torque is considered. We achieve the slow spin by giving the body slow rotation with a sufficiently small angular velocity component r 0 about the moving z-axis. We obtain the periodic solutions in a new domain of the angular velocity vector component r 0 ⟶ 0 , define a large parameter proportional to 1 / r 0 , and use the technique of the large parameter for solving this problem. Geometric interpretations of motions will be illustrated. Comparison of the results with the previous works is considered. A discussion of obtained solutions and results is presented.

Validation of gyroscope sensors for snow sports performance monitoring

Wearable sensors that can be used to measure human performance outcomes are becoming increasingly popular within sport science research. Validation of these sensors is vital to ensure accuracy of extracted data. The aim of this study was to establish the validity and reliability of gyroscope sensors contained within three different inertial measurement units (IMU). Three IMUs (OptimEye, I Measure U and Logger A) were fixed to a mechanical calibration device that rotates through known angular velocities and positions. RMS scores for angular displacement, which were calculated from the integrated angular velocity vectors, were 3.85° ± 2.21° and 4.34° ± 2.57° for the OptimEye and IMesU devices, respectively. The RMS error score for the Logger A was 22.76° ± 23.22°, which was attributed to a large baseline shift of the angular velocity vector. After a baseline correction of all three devices, RMS error scores were all below 3.90°. Test re-test reliability of the three gyroscope sensors were high with coefficient of variation (CV%) scores below 2.5%. Overall, the three tested IMUs are suitable for measuring angular displacement of snow sports manoeuvres after baseline corrections have been made. Future studies should investigate the accuracy and reliability of accelerometer and magnetometer sensors contained in each of the IMUs to be used to identify take-off and landing events and the orientation of the athlete at those events.

The Effect of Instrumentation on Sports Balls During Flight, Exemplified by A Smart AFL Ball: Model and Experiment

Sensors incorporated in a sports ball for data collection can affect the properties of a ball, specifically the spin rate of a ball if the mass distribution (moments of inertia, MOI) is altered. This paper provides a method for assessing the MOIs of a smart ball by means of spin rate data, collected from a gyroscopic sensor. The critical elevation angle of the angular velocity vector defines the separatrix condition, which decides whether the angular velocity vector precesses about the axis with the greatest MOI or with the smallest MOI. The critical elevation angle can be directly determined from the experimental of the angular velocity data, and, together with the ratio of precession speed to angular velocity, applied to calculating the three MOIs. In the smart AFL ball used for the experiments, the critical angle was 13.5°, and the ratio of the two small MOIs was 1.014.