Rodrigues-Hamilton Parameters Research Articles

Quaternion methods and models of regular celestial mechanics and astrodynamics

This paper is a review, which focuses on our work, while including an analysis of many works of other researchers in the field of quaternionic regularization. The regular quaternion models of celestial mechanics and astrodynamics in the Kustaanheimo-Stiefel (KS) variables and Euler (Rodrigues-Hamilton) parameters are analyzed. These models are derived by the quaternion methods of mechanics and are based on the differential equations of the perturbed spatial two-body problem and the perturbed spatial central motion of a point particle. This paper also covers some applications of these models. Stiefel and Scheifele are known to have doubted that quaternions and quaternion matrices can be used efficiently to regularize the equations of celestial mechanics. However, the author of this paper and other researchers refuted this point of view and showed that the quaternion approach actually leads to efficient solutions for regularizing the equations of celestial mechanics and astrodynamics.This paper presents convenient geometric and kinematic interpretations of the KS transformation and the KS bilinear relation proposed by the present author. More general (compared with the KS equations) quaternion regular equations of the perturbed spatial two-body problem in the KS variables are presented. These equations are derived with the assumption that the KS bilinear relation was not satisfied. The main stages of the quaternion theory of regularizing the vector differential equation of the perturbed central motion of a point particle are presented, together with regular equations in the KS variables and Euler parameters, derived by the aforementioned theory. We also present the derivation of regular quaternion equations of the perturbed spatial two-body problem in the Levi-Civita variables and the Euler parameters, developed by the ideal rectangular Hansen coordinates and the orientation quaternion of the ideal coordinate frame.This paper also gives new results using quaternionic methods in the perturbed spatial restricted three-body problem.

Математические модели углового движения космических аппаратов и их использование в задачах управления ориентацией

Предложена общая структура кинематических уравнений эволюции ориентации космического аппарата (КА) (системы координат, связанной с КА (ЗСК)) относительно опорной системы координат (ОСК). Предполагается, что начала систем координат совпадают и расположены в произвольной точке КА. Каждая из систем координат оборачивается произвольной абсолютной угловой скоростью (скоростью относительно инерциального пространства), заданной проекциями на их оси. В качестве параметров ориентации могут использоваться углы Эйлера-Крылова, параметры Родрига-Гамильтона, модифицированные параметры Родри-га. Показано, что известные представления уравнений эволюции ориентации ССК относительно ОСК в параметрах Родрига-Гамильтона (компонентах нормируемых кватернионов) могут быть получены из решения задачи Н.П. Ерюгина — отыскание всего множества дифференциальных уравнений, имеющих заданный интеграл движения. Отмечены преимущества и недостатки использования каждого из указанных параметров ориентации. Предложен общий для всех этих уравнений метод синтеза управления ориентацией, основанный на декомпозиции исходной задачи на кинематическую и динамическую задачи и использовании известных обобщений прямого метода Ляпунова для их решения. С помощью компьютерного моделирования проиллюстрировано свойство структурной грубости в смысле А.А. Андронова-Л.С. Понтрягина полученного алгоритма. А именно, на конкретном примере проиллюстрирована свойство даже намеренно структурно упрощенного алгоритма стабилизации заданной постоянной ориентации КА с достаточной точностью отслеживать программу ее изменения. Задача наблюдения типична в управлении стыковкой КА, спуском КА с орбиты, выполнении маршрутных съемок поверхности Земли.

Алгоритм начальной инициализации кватерниона пространственной ориентации в параметрах Родрига-Гамильтона

Introduction. The correction time reduction for the spatial orientation evaluation of a solid under the orientation system initiation is considered. Solid spatial orientation is measured by the integrated values from three orthogonal angular speed sensors. Difference between the actual spatial orientation and the orientation estimated by sensors is adjusted due to the data obtained from other sensors, such as accelerometers and magnetometers. Major existing algorithms deduct data acquired from accelerometers and magnetometers from the evaluation of angular rate through the correction factor thus correcting the spatial orientation assessment error. The higher the solid inclination angle in relation to horizon when the orientation system is switched on, the greater the spatial orientation error is. The algorithm presented herein corrects the spatial orientation evaluation in quaternion components without angular rate sensors which allows minimizing the spatial orientation assessment error within shorter time in comparison with the existing algorithms. Materials and Methods. To implement the correction algorithm, the MPU6050 sensor is used. It is made with microelectromechanical technology, and its body includes three orthogonally located angular velocity sensors and three orthogonally located accelerometers. The output data from MPU6050 sensor is processed by dsPIC33EP256MU806 microchip. The spatial orientation is calculated by the Rodrigues-Hamilton parameters in the quaternion components. The result is input to the Matlab software package which executes the program of visualizing the dependencies on the time of four quaternion components graphically. Research Results . In existing algorithms that use the Rodrigues-Hamilton parameters, at the initial initialization of the orientation system, it is suggested to increase the correction factor, or to use the trigonometric formulas to find the Euler angles and translate them into the RodriguesHamilton parameters. In the first case, the initial initialization time remains sufficiently long, in the second case, due to the use of Euler angles, a phenomenon of “gimbal lock” can be observed. The proposed algorithm performs the initial initialization in a time equivalent to the initialization time in the Euler angles parameters, but it applies only the RodriguesHamilton parameters. Discussion and Conclusions. Using the proposed algorithm will allow a minimum of 5-fold reduction in the initial initialization time of the spatial orientation quaternion. In consequence, the total time required for activating the system will be also reduced due to the fact that the initial initialization is necessary every time the orientation system is switched on. For the correct determination of the spatial orientation according to the proposed algorithm, the necessary condition is the absence of any acceleration on the body other than the gravitational acceleration because the initialization occurs only upon the accelerometer readings.

Determination of the Parameters of the Angular Motion of the Spacecraft from the Information of the Star Sensor Using the Dynamic Equation of the Solid Body Rotation in the Rodrigues–Hamilton Parameters

Determination of the Parameters of the Angular Motion of the Spacecraft from the Information of the Star Sensor Using the Dynamic Equation of the Solid Body Rotation in the Rodrigues–Hamilton Parameters

Synthesis of Time Optimal Spatial Slewing of Spacecraft Using Dynamic Equation of Rigid Body Rotational Motion at Rodrigues-Hamilton Parameters

Synthesis of Time Optimal Spatial Slewing of Spacecraft Using Dynamic Equation of Rigid Body Rotational Motion at Rodrigues-Hamilton Parameters

Mathematical modeling of the dynamics of manipulation system for mobile transport and technological machine with reference to the elasticity of elements

The approaches to modeling the dynamics of mobile manipulators of transport and technological machines with reference to the elastic compliance of elements are shown. The equations of motion of the manipulator units are recorded using the method of Lagrange multipliers. As the main origin for each element were taken four Rodrigues-Hamilton parameters, coordinates of the element position as absolutely rigid body and elastic movements. The equations take into account the impact of external and internal elastic and quadratic forces. Internal elastic forces are calculated on the basis of the stiffness matrix, which formation is made through calculation of the manipulator by finite element method. To set the join couplings the required number of constraint equations is built. The methods of creating of widespread kinematic schemes of manipulators of cylindrical and prismatic joints using set of constraints involving mutual perpendicular vectors, which belong to articulated elements. Using the developed model the dynamics of articulated manipulator crane ACT-4-A was calculated. It was found that the inclusion of elements elasticity reveals oscillatory behavior of the kinematic and power factors of the process and, thus, increase the adequacy of modeling of manipulation system operation during machine use. Elastic elements causes their oscillations with a frequency, which is commensurable with the first lower natural frequency of metal manipulation system itself, that indicates the desirability of its test for the absence of resonance and, if necessary, the development of positive measures to improve the dynamic reliability of the manipulation system. It has been shown that to calculate manipulators for some machines the chassis dynamics should be taken into account.

Алгоритм синтеза терминального управления выставкой космического аппарата в инерционную систему координат

This paper addresses a terminal control scheme for a basic attitude maneuver of a spacecraft, formation of an "inertial attitude" mode in the finite time bounds. The algorithm is based on determination of the angular velocity program values. This determination uses the analytical expressions obtained by means of the boundary solution proposed in the authors' previous works. The solution assumes discrete model parameters' identification by the modal control decomposition method in the observer synthesis. The further stabilization of the attitude and angular velocity parameters is necessary. The angular motion control process is described by the kinematic equations in Rodrigues-Hamilton parameters and Euler's dynamic equations. Analytical solution of the kinematic equations with the constant values of the angular velocity is used to determine the program values of the angular velocity. This allows us to obtain new values of the program angular velocity in every onboard computer cycle. These values ensure forming up of the inertial attitude for the given time. The next step is to calculate an appropriate momentum control values. Linearized Euler's equations are used to get the control values. Linearization is performed at every cycle of the onboard computer. It gives a high degree approximation to the nonlinear model of a spacecraft angular motion. All the synthesized control laws and observer feedback coefficient matrices have simple analytic forms and can be implemented on the onboard digital computer for a real-time execution to form-up the inertial attitude mode. Numerical examples are presented to demonstrate the successful work of the developed control algorithms for a wide variety of the initial conditions (initial attitude, maneuver time) in the inertial coordinate system.

Uniform distribution of points on hypersurfaces: simulation of random equiprobable rotations

The paper describes a universal method for simulation of uniform distributions of points on smooth regular surfaces in Euclidean spaces of various dimensions. The authors give an interpretation of a set of possible values of Rodrigues–Hamilton parameters used to describe a rigid rotation as a set of points of a threedimensional hypersphere in four-dimensional Euclidean space. The relationship between random equiprobable rotations of a rigid body and a uniform distribution of points on the surface of a three-dimensional hypersphere in four-dimensional Euclidean space is established.

The use of Rodrigues–Hamilton parameters in interpreting the motion of a rigid body with a fixed point

The connection between the different methods of interpreting the motion of a body with a fixed point is investigated. A new form of the equations of a stationary angular velocity hodograph is obtained that allows the Poinsot method to be applied using Rodrigues–Hamilton parameters. The properties of the kinematic parameters are studied for the precessing and isoconic motions of the body. The precessing motions in Dokshevich's solution and the isoconic motions in Steklov's solution are considered.

Construction of the semi-analytical and numerical solutions to the problem of rotational motion of the moon

New high-precision, semianalytical and numerical solutions to the problem of the rotational motion of the Moon are obtained, for use in the long 418.9-year time frame. The dynamics of the rotational motion of the Moon is studied numerically using the Rodrigues-Hamilton parameters, relative to the fixed ecliptic for the epoch J2000. The results of the numerical solution to the problem under study are compared with a compiled semianalytical theory of Moon rotation (SMR). The initial conditions for the numerical integration have been taken from the SMR. The comparative discrepancies derived from the comparison between the numerical solutions and the SMR do not exceed 1.5″ on the time-scale of 418.9 yr. The investigation of the comparative discrepancies between the numerical and semianalytical solutions is performed using the least squares and spectral analysis methods in the Newtonian case. All the periodic terms describing the behavior of the comparative discrepancies are interpreted as the corrections to the semianalytical SMR theory. As a result, the series are constructed to describe the rotation of the Moon (MRS2010) in the time interval under study. The numerical solution for the Moon’s rotation has been obtained anew, with new initial conditions calculated using MRS2010. The discrepancies between the new numerical solution and MRS2010 do not exceed 20 arc milliseconds on the time-scale of 418.9 years. The results of the comparison suggest that that the MRS2010 series describe the rotation of the Moon more correctly than the SMR series.

Construction of the Numerical and Semi-Analytical Solutions of the Rigid Earth Rotation at a Long Time Intervals

This research is the continuation of our studies of the rigid Earth rotation at a long time intervals (Pashkevich V.V. and Eroshkin G.I., 2005). The main purpose of this investigation is the construction of the new high-precision Rigid Earth Rotation Series 2012 (RERS2012), dynamically adequate to the JPL DE406/LE406 ephemeris (Standish E. M., 1998). The dynamics of the rotational motion of the rigid Earth is studied numerically by using Rodrigues-Hamilton parameters over 2000 and 6000 years. The numerical solution of the rigid Earth rotation is implemented with the quadruple precision of the calculations. The orbital motions of the disturbing celestial bodies are defined by the DE406/LE406 ephemeris. The initial conditions of the numerical integration are taken from SMART97 (Bretagnon P. et al., 1998) and S9000 (Pashkevich V.V. and Eroshkin G.I. 2005). The results of the numerical solutions of the problem are compared with the semi-analytical solutions of the rigid Earth rotation (SMART97 and S9000, respectively) with respect to the fixed ecliptic of epoch J2000. The investigation of these discrepancies is carried out by the least squares and spectral analysis methods for the relativistic (Kinematical) case, in which the geodetic perturbations (the most essential relativistic perturbations) in the Earth rotation

Modeling the motion of an uncompensated gyrostabilized platform in the Rodrigues-Hamilton parameters

We construct a model of triaxial gyrostabilizer platform motion at the force stabilization regime. The Rodrigues-Hamilton parameters have been chosen as kinematic parameters describing the changes in the gyrostabilized platform state vector. The advantages of using the model constructed in the initial attitude control systems of inertial navigation systems as compared with the existing ones are shown.

Construction of the New High-Precision Moon Rotation Series at a Long Time Intervals

Construction of the New High-Precision Moon Rotation Series at a Long Time Intervals The main purposes of this research are the construction of the new high-precision Moon Rotation Series (MRS2011), dynamically adequate to the DE404/LE404 and the DE406/LE406 ephemeris, over long time intervals. The comparison of the new highprecision Moon Rotation solutions of MRS2011 with the solution of MRS2010 (Pashkevich and Eroshkin, 2010), which is dynamically adequate to the DE200/LE200 ephemeris over 418.9 year time interval, is performed. The dynamics of the rotational motion of the Moon is studied numerically by using Rodrigues-Hamilton parameters over 418.9, 2000 and 6000 years. The numerical solution of the Moon rotation is implemented with the quadruple precision of the calculations. The results of the numerical solution of the problem are compared with the composite semi-analytical theory of the Moon rotation (SMR) (Pashkevich and Eroshkin, 2010) with respect to the fixed ecliptic of epoch J2000. The initial conditions of the numerical integration are taken from SMR. The investigation of the discrepancies is carried out by the least squares and spectral analysis methods for the Newtonian case. All the secular, periodic and Poisson terms, representing the behavior of the residuals, are interpreted as corrections to SMR semi-analytical theory. As a result, the Moon Rotation Series (MRS2011) is constructed, which is dynamically adequate to the DE404/LE404 and the DE406/LE406 ephemeris over 418.9, 2000 and 6000 years. A numerical solution for the Moon rotation is obtained anew with the new initial conditions calculated by means of MRS2011. The discrepancies between the new numerical solution and the semi-analytical solution of MRS2011 do not surpass 20 mas over 418.9 year time interval, 64 mas over 2000 year time interval and 8 arc seconds over 6000 year time interval. Thus, the result of the comparison demonstrates a good consistency of MRS2011 series with the DE/LE ephemeris.

Application of the Spectral Analysis for Modeling the Rotations of the Moon

Application of the Spectral Analysis for Modeling the Rotations of the Moon The main purposes of this research are the development of the optimal spectral analysis schemes for the investigation of the rotational motion of the Moon and then the comparison between the result of the optimal spectral analysis of the rotational motions of the Earth and the Moon. Dynamics of the rotational motion of the Moon is studied numerically by using Rodrigues-Hamilton parameters over 418.9 year time interval. The results of the numerical solution of the problem are compared with the composite semi-analytical theory of the Moon rotation (SMR) represented by Cassini relations and the semi-analytical solutions of the lunar physical libration problem (Eckhardt, 1981), (Moons, 1982), (Moons, 1984), (Pešek, 1982). The initial conditions of the numerical integration are taken from SMR. The investigation of the discrepancies is carried out by the optimal spectral analysis methods for the Newtonian case. All the periodic terms representing the behavior of the residuals are interpreted as corrections to SMR semi-analytical theory. As a result, the Moon Rotation Series (MRS2010) is constructed, which is dynamically adequate to the DE200/LE200 ephemeris over 418.9 year time interval. A numerical solution for the Moon rotation is obtained anew with the new initial conditions calculated by means of MRS2010. The discrepancies between the new numerical solution and MRS2010 do not surpass 20 mas over 418.9 year time interval. The result of the comparison demonstrates that MRS2010 series represent more accurately the Moon rotation than SMR series.

Kinetic energy of an asymmetric rigid body moving around a fixed point: invariant representation in terms of quaternion matrices

The similarity transform and the properties of quaternion matrices are used to derive invariant expressions for the inertia matrix and kinetic energy of an asymmetric rigid body undergoing spatial motion around a fixed point in body-fixed and inertial frames of reference. The elements of the quaternion matrixes are Rodrigues–Hamilton parameters and quasi-velocities

Algorithms of motion planning for a six-legged walking machine

This paper investigates algorithms of motion planning for a six-legged walking machine over complex terrain. Application of the Rodrigues-Hamilton parameters for describing the orientation of the body enables one to develop algorithms of motion planning for the body and legs in an absolute coordinate frame with automatic adaptation to the surface. Experiments with a laboratory scale walking machine have demonstrated the efficiency of proposed algorithms.

On the nonlinear dynamics of elastically interacting asymmetric rigid bodies

A symmetric mathematical model is developed to describe the spatial motion of a system of space vehicles whose structure is represented by regular geometrical figures (Platonic bodies). The model is symmetrized by using the Euler-Lagrange equations of motion, the Rodrigues-Hamilton parameters, and quaternion matrix mathematics. The results obtained enable us to model a wide range of dynamic, control, stabilization, and orientation problems for complex systems and to solve various problems of dynamic design for such systems, including estimation of dynamic loading on the basic structure during maneuvers in space

Analytical Mechanics. Foundations of Engineering Mechanics Series

Analytical Mechanics. Foundations of Engineering Mechanics Series

Dynamics and Control of Machines

Dynamics and Control of Machines

Some aspects of the mathematical modeling of the earth rotation

A numerical theory of the rigid Earth's rotation taking into account the most essential relativistic perturbations (geodetic perturbations) is developed using improved initial conditions of the numerical integration. The differential equations of the rigid Earth's rotation are solved using the Rodrigues-Hamilton parameters, which describe the directions of the principal axes of inertia of the Earth with respect to the fixed ecliptic plane and equinox J2000.0. The rigid Earth rotation is a result of the gravitational interaction of the Earth's body with the point mass disturbing bodies (the Sun, Moon, and major planets). The geocentric motions of the disturbing bodies are provided by the ephemeris DE403/LE403 (Standish et al.,1995). The numerical integration of the rotational motion equations of the rigid Earth is carried out over a 200 year time interval from January 1, 2000. The results of the numerical solution of the problem are compared with the semi-analytical theory of the rigid Earth's rotation SMART97 (Bretagnon et al., 1998). The possible reason of the discrepancies up to ±300μas between the numerical integration results and the kinematical solution SMART97 (Pashkevich, 2000) are studied. In the result of the present investigation the discrepancies over the whole time interval of comparison are diminished to ±7μas.