# кватернион

## Approximation of the orientation equations of the orbital coordinate system by the weighted residuals method

In the quaternion formulation, the problem of mathematical modeling of the spacecraft movement in an elliptical orbit was considered. Control is an acceleration vector from jet thrust. Control modulus is constant. The control is directed orthogonally to the plane of the spacecraft orbit. The quaternion differential equation of an orbital coordinate system orientation was used to describe spacecraft movement.

## About a problem of spacecraft's orbit optimal reorientation

The problem of optimal reorientation of the spacecraft's orbit is solved with the help of the Pontryagin maximum principle and quaternion equations. Control (thrust vector, orthogonal to the orbital plane) is limited in magnitude. Functional, which determines a quality of control process is weighted sum of time and module (or square) of control. We have formulated a differential boundary problems of reorientation of spacecraft's orbit.

## Dual Matrix and Biquaternion Methods of Solving Direct and Inverse Kinematics Problems of Manipulators, for Example Stanford Robot Arm. I

The methology of solving the direct kinematics problem of manipulators by using screw mechanics methods (dual direction cosine matrices, Clifford biquaternions) is shown on the example of Stanford robot arm. Kinematic equations of motion of the manipulator are found. These equations will be used for solving the inverce kinematics problem with the help of biquaternion theory of kinematic control.

## Dual matrix and biquaternion methods of solving direct and inverse kinematics problems of manipulators for example Stanford robot arm. II

The methology of solving the inverce kinematics problem of manipulators by using biquaternion theory of kinematics control is shown on the example of Stanford robot arm. Solving of the inverce kinematics problem of Stanford robot arm is performed using the simplest control law. The analysis of numerical solution results is made. The efficacy of applying the theory of kinematics control for solving the inverce kinematics problem of manipulators is proved.

## Solution of a Problem of Spacecraft’s Orbit Optimal Reorientation Using Quaternion Equations of Orbital System of Coordinates Orientation

The problemof optimal reorientation of the spacecraft’s orbit is solved with the help of the Pontryagin maximum principle and quaternion equations. Control (thrust vector, orthogonal to the orbital plane) is limited inmagnitude. Functional, which determines a quality of control process, is weighted sum of time and impulse (or square) of control. We have formulated a differential boundary problems of reorientation of spacecraft’s orbit.

## Analytical Solution of Equations of Near-circular Spacecraft’s Orbit Orientation

The problem of optimal reorientation of spacecraft’s orbit with a limited control, orthogonal to the plane of spacecraft’s orbit, is considered. An approximate analytical solution of differential equations of near-circular spacecraft’s orbit orientation by control, that is permanent on adjacent parts of the active spacecraft’s motion, is obtained.

## Investigation of the Problem of Optimal Correction of Angular Elements of the Spacecraft Orbit Using Quaternion Differential Equation of Orbit Orientation

In this paper we consider the problem of optimal correction of angular elements of the spacecraft orbit. Control (jet thrust vector orthogonal to the plane of the orbit) is limited by absolute value. The combined quality functional characterizes the amount of time and energy consumption. With the help of the Pontryagin maximum principle and quaternion differential equation of the spacecraft orbit orientation, we have formulated differential boundary value problem of correction of the angular elements of the spacecraft orbit.

## Analytical Solution of Differential Equations of Circular Spacecraft Orbit Orientation

The problem of optimal reorientation of spacecraft’s orbit with a limited control, orthogonal to the plane of spacecraft orbit is being investigated. We have found an analytical solution of differential equations of circular spacecraft orbit orientation by control that is permanent on adjacent parts of the active spacecraft’s motion.

## Quaternion Models and Algorithms for Solving the General Problem of Optimal Reorientation of Spacecraft Orbit

The problem of optimal reorientation of the spacecraft orbit is considered in quaternion formulation. Control (vector of the acceleration of the jet thrust) is limited in magnitude. To solve the problem it is required to determine the optimal orientation of this vector in space. It is necessary to minimize the duration of the process of reorientation of the spacecraft orbit. To describe the motion of the center of mass of the spacecraft we used quaternion differential equation of the orientation of the spacecraft orbit.