Dual tensors based solutions for rigid body motion parameterization

Condurache, Daniel; Burlacu, Adrian

doi:10.1016/j.mechmachtheory.2013.12.016

Cited by 51 publications

(36 citation statements)

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“…The internal structure of any orthogonal dual tensor R ∈ SO 3 is illustrated in a series of results which were detailed in our previous work [17,18,23]. …”

Section: Rigid Body Motion Parameterization Using Dual Lie Algebramentioning

confidence: 99%

“…The next Theorem permits the reconstruction of the rigid body motion knowing in any moment the twist of the rigid body that is equivalent with knowing the dual angular velocity [5,18].…”

Section: Poisson-darboux Equation In Dual Lie Algebramentioning

confidence: 99%

“…(2), a dual number α ¼ α þ εd and a dual unit vector u ¼ uþεu 0 can be computed to have the following Eq. [17,18]:…”

Section: Rigid Body Motion Parameterization Using Dual Lie Algebramentioning

confidence: 99%

“…Orthogonal dual tensorial maps are a powerful instrument in the study of the rigid motion with respect to an inertial and noninertial reference frames. More on dual numbers, dual vectors and dual tensors can be found in [2,[16][17][18][19][20][21][22][23].2.1. Isomorphism between Lie group of the rigid displacements SE 3 and Lie group of the orthogonal dual tensors SO 3…”

mentioning

confidence: 99%

“…More on dual numbers, dual vectors and dual tensors can be found in [2,[16][17][18][19][20][21][22][23].…”

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confidence: 99%

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On Six DOF Relative Orbital Motion of Satellites

Condurache¹

2018

Space Flight

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In this chapter, we reveal a dual-tensor-based procedure to obtain exact expressions for the six degree of freedom (6-DOF) relative orbital law of motion in the specific case of two 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 center. 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.Keywords: relative orbital motion, full body problem, dual algebra, Lie group, Lie algebra, closed form solution IntroductionThe relative motion between the leader and the deputy in the relative motion is a six-degreesof-freedom (6-DOF) motion engendered by the joining of the relative translational motion with the rotational one. Recently, the modeling of the 6-DOF motion of spacecraft gained a special attention [1][2][3][4][5], similar to the controlling the relative pose of satellite formation that became a very important research subject [6][7][8][9][10]. The approach implies to consider the relative translational and rotational dynamics in the case of chief-deputy spacecraft formation to be modeled using vector and tensor formalism.In this chapter we reveal a dual algebra tensor based procedure to obtain exact expressions for the six D.O.F relative orbital law of motion for the case of two Keplerian confocal orbits.© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.Orthogonal dual tensors play a very important role, the representation of the solution being, to the authors' knowledge, the shortest approach for describing the complete onboard solution of the six D.O.F relative orbital motion problem. Because the solution does not depend on the LVLH properties involves that is true in any reference frame of the Leader with the origin in its mass center. 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 local-vertical-local-horizontal (LVLH) frame. For the full body initial value problem, a general representation theorem is given. More, the real and imaginary parts are split and representation theorems for the rotation and translation parts of the relative orbital motion are obtained. Regarding translation, we w...

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“…The internal structure of any orthogonal dual tensor R ∈ SO 3 is illustrated in a series of results which were detailed in our previous work [17,18,23]. …”

Section: Rigid Body Motion Parameterization Using Dual Lie Algebramentioning

confidence: 99%

“…The next Theorem permits the reconstruction of the rigid body motion knowing in any moment the twist of the rigid body that is equivalent with knowing the dual angular velocity [5,18].…”

Section: Poisson-darboux Equation In Dual Lie Algebramentioning

confidence: 99%

“…(2), a dual number α ¼ α þ εd and a dual unit vector u ¼ uþεu 0 can be computed to have the following Eq. [17,18]:…”

Section: Rigid Body Motion Parameterization Using Dual Lie Algebramentioning

confidence: 99%