ON THE TRAIL OF VUJIČIĊ'S COORDINATES-INDEPENDENT POSITION VECTOR FORM: ROTATIONALLY-INVARIANT/(CLASSICALLY-)COVA-RIANT TRAJECTORIAL COORDINATES SYSTEM FORMULATION AND OTHER REPERCUSIOS FOR THE MECHANICS / DYNAMICS MODELING 1 (original) (raw)

Trajectories in Non-Inertial Frames

In aspecte practice. Modelele şi metoda elaborată permit rezolvarea unui numă mare de probleme de dinamică sistemelor în camp gravitaţional. Some problems of dynamics in non-inertial frames are presented in this paper. Motion of particles in gravitational field are studied. Lagrange equations for relative motion with respect to a reference frame with known trajectory are written and a Binet type equation for relative motion is established. When the motion of a system of bodies which compos a large orbital station is described within reference frames having the origin in the center of the attractive body (Earth), the problem of integration of motion equations presents some difficulties, because some coordinates (like the vector radii) have very great values, and others (like distances between bodies) have very small values. Some difficulties can be avoided if relative motion of the system is studied with respect to a reference frame with a known motion. Relative motion study isn'...

DETC2005-84127 Intrinsic Formulation of Dynamics of Curvilinear Systems

2020

The paper concerns the dynamics of curvilinear systems which are often met in mechanical systems (robots, artificial satellites and so on). We only suppose that each section is rigid. Using Lie group theory, a general curvilinear system is then equivalent to a differentiable distribution of displacements, elements of the Lie group of Euclidean displacements the algebra of which may be identified with the Lie algebra of screws. The kinematics is described by the lagrangian field of deformations and the lagrangian field of velocities elements of the Lie algebra and with standard hypotheses about the distribution of external forces, the intrinsic equations are obtained, the displacements or deformations being small or large. The non linearities (of inertia terms as for internal strenghts ) appear by the adjoint mapping and its derivation : the Lie braket. Last, the elements to automatically obtain scalar equations and to come back to more classical models (beam, cable,) are given.

CURVILINEAR COORDINATES FOR COVARIANCE AND RELATIVE MOTION OPERATIONS

Relative motion studies have traditionally focused on linearized equations, and inserting additional force models into existing formulations to achieve greater fidelity. A simpler approach may be numerically integrating the two satellite positions and then converting to a modified equi-distant cylindrical frame as necessary. Recent works have introduced some approaches for this transformation as it applies to covariance operations, with some approximations. We develop an exact transformation between Cartesian and curvilinear frames and test the results for various orbital classes. The transformation has applicability to covariance operations which we also introduce. Finally, we examine how the transfor-mation affects graphical depiction of the covariance matrix.

Velocity, Acceleration and Equations of Motion in the Elliptical Coordinate System

Archives of Physics Research, 2018

The canonical coordinate systems (rectangular, polar and spherical) are sometimes not the best for studying the trajectories of some forms of motions. For example, motion of objects in an elliptical orbit being described by polar or spherical coordinates may not be accurate. It is due to this that we have derived the position vectors, velocity vectors, acceleration vectors, simple representation of magnitude of the velocity and equations of motion in the elliptical coordinate system. An attempt was also made towards solving the derived equations of motion. The general algorithm for conversion among coordinate systems was also provided.

On Six D.O.F Relative Orbital Motion Parametrization using Rigid Bases of Dual Vectors

AAS/AIAA Astrodynamics Specialist Conference, Hilton Head, South Carolina, August 11-15, 2013, 2013

The relative orbital motion is a six d.o.f motion, generated by the coupling of the relative translational motion with the rotational one. This paper is focused on developing a new relative orbital motion parametrization method using dual rigid bases. Our studies showed that, in the dual tensors free module, the dual rigid bases can completely characterize the relative orbital motion from the Euclidean three dimensional space. The combination between a dual rigid basis and its reciprocal provides a natural computational instrument that can be used to solve many problems in the kinematics, dynamics and control of relative orbital motion setups. * Professor,

ONBOARD COMPLETE SOLUTION TO THE FULL-BODY RELATIVE ORBITAL MOTION PROBLEM

In this paper, we reveal a dual-tensor-based procedure to obtain exact expressions for the six degree of freedom (6-DOF) relative orbital law of motion between two Keplerian confocal orbits. The solution is obtained by pure analytical methods, and it holds for any leader and deputy motion, without involving any secular terms or singularities. The relative orbital motion is reduced, by an adequate change of variables, into a dual Euler fixed-point problem. Orthogonal dual tensors play a very important role, with the representation of the solution being, to the authors' knowledge, the shortest approach for describing the complete onboard solution of the 6-DOF orbital motion problem. The solution does not depend on the local-vertical–local-horizontal (LVLH) properties involves that is true in any reference frame of the leader with the origin in its mass centre. To obtain this solution, one has to know only the inertial motion of the leader spacecraft and the initial conditions of the deputy satellite in the LVLH frame. A representation theorem is provided for the full-body initial value problem. Furthermore, the representation theorems for rotation part and translation part of the relative motion are obtained. Regarding the translation part, a closed-form free of coordinate solution is revealed, based of generalised trigonometric function in space at constant curvature. They hold for all types of reference trajectories of the leader (elliptic, parabolic, hyperbolic) and deputy (elliptic, parabolic, hyperbolic, rectilinear).

On six DOF relative orbital motion of spacecrafts. A complete onboard solution

Journal of Engineering Sciences and Innovation, 2017

In this paper, we reveal a dual-tensor-based procedure to obtain exact expressions for the six degree of freedom (6-DOF) relative orbital motion problem of two spacecrafts, in the specific case of Keplerian confocal orbits. The result is achieved by pure analytical methods in the general case of any leader and deputy motion, without singularities or implying any secular terms. Orthogonal dual tensors play a very important role, with the representation of the solution being, to the authors' knowledge, the shortest approach for describing the complete onboard solution of the 6-DOF orbital motion problem. The solution does not depend on the local-vertical-local-horizontal (LVLH) properties involves that is true in any reference frame of the leader with the origin in its mass centre. A representation theorem is provided for the full-body initial value problem. Furthermore, the representation theorems for rotation part and translation part of the relative motion are obtained.

The equations of motion for a rigid body using non-redundant unified local velocity coordinates

Multibody System Dynamics

A novel formulation for the description of spatial rigid body motion using six non-redundant, homogeneous local velocity coordinates is presented. In contrast to common practice, the formulation proposed here does not distinguish between a translational and rotational motion in the sense that only translational velocity coordinates are used to describe the spatial motion of a rigid body. We obtain these new velocity coordinates by using the body-fixed translational velocity vectors of six properly selected points on the rigid body. These vectors are projected into six local directions and thus give six scalar velocities. Importantly, the equations of motion are derived without the aid of the rotation matrix or the angular velocity vector. The position coordinates and orientation of the body are obtained using the exponential map on the special Euclidean group SE(3). Furthermore, we introduce the appropriate inverse tangent operator on SE(3) in order to be able to solve the incremental motion vector differential equation. In addition, we present a modified version of a recently introduced a fourth-order Runge-Kutta Lie-group time integration scheme such that it can be used directly in our formulation. To demonstrate the applicability of our approach, we simulate the unstable rotation of a rigid body.

Universal formulation of quasi-Keplerian motion, and its applications

New Astronomy, 2009

We derive the universal solution to the Kepler-Coulomb problem with an additional inverse-square potential, valid for any type of orbit, and describe three prominent applications in astrodynamics: the relativistic precession of the apsides, the numerical integration of perturbed Kepler-Coulomb problems with a generalized leapfrog, and the averaged motion of earth-orbiting satellites with the J 2 perturbation. The modified orbital elements and Delaunay variables are presented as well.