Dual numbers representation of rigid body dynamics

Brodsky, V.; Shoham, Moshe

doi:10.1016/s0094-114x(98)00049-4

Cited by 121 publications

(63 citation statements)

References 23 publications

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“…4 The result is rather remarkable because it shows that the skew-symmetry of the local stiffness matrix is only dependent on the external wrench; therefore, the stiffness matrix is symmetric if and only if the system is at an unloaded equilibrium. For the gravity load case, the local stiffness K l always has a skew-symmetric part no matter which point on the body is selected.…”

Section: Stiffnessmentioning

confidence: 99%

Time derivatives of screws with applications to dynamics and stiffness

Lipkin

2005

Mechanism and Machine Theory

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International audienceScrew quantities provide geometric insight into three-dimensional mechanics modeled by rigid bodies and lumped parameters. Four distinct cases of time differentiation are examined by combining fixed and moving body derivatives (fundamental to rigid body mechanics) with material and local derivatives (fundamental to continuum mechanics). Three combinations always yield another screw quantity while the most common, the material derivative with respect to the fixed body, does not. Two fundamental formulations are examined with this last derivative, Euler's Laws and the gravitational loading of an elastic system. By coincidence, the formulations appear screw-like when they are expressed at the center-of-mass but, in contrast to actual screw formulations, they do not retain invariant forms when expressed at arbitrary points

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Section: Stiffnessmentioning

confidence: 99%

Time derivatives of screws with applications to dynamics and stiffness

Lipkin

2005

Mechanism and Machine Theory

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“…Considering ω B D being the dual angular velocity vector of the rigid body D in the body frame, the dual form of the Euler equation given in [30] results that:…”

Section: Rigid Body Motion In Arbitrary Non-inertial Frame Revisedmentioning

confidence: 99%

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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“…Brodsky and Shoham 15 proposed a general dual quaternion-based dynamics for the rigid body, and Wang and Sun 29 proposed a dual quaternion-based dynamics for a space target. However, both of these two dual quaternion-based dynamics used a dual operator Second, using this operator makes the mapping between the dual number and the real number much more difficult.…”

Section: Model Dynamics Based On Dual Vector Quaternionsmentioning

confidence: 99%

Adaptive pose and inertial parameters estimation of free-floating tumbling space objects using dual vector quaternions

Hou

Wang

et al. 2017

Advances in Mechanical Engineering

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The robust parameter estimation of unknown space objects is essential to the on-orbit servicing missions. Based on the adaptive filtering techniques along with the dual quaternions modeling methods for pose estimation, this article proposes a dual vector quaternions-based extended Kalman filter and a dual vector quaternions-based adaptive fading factors extended Kalman filter to estimate the parameters of a free-floating tumbling space target. Using the dual vector quaternions to model the kinematics and dynamics of the system, the representation of the model is concise and compact. Also, the translational and rotational coupled effects are considered. In addition, the estimation algorithm is designed by the innovation-based multiple adaptive fading factors. As a result, the dual vector quaternions-based adaptive fading factors extended Kalman filter is robust against the faulty measurements which may lead to divergence of the traditional extended Kalman filter. As far as the authors know, both the designed filters are the first pose and inertial parameters estimation algorithms based on dual vector quaternions, and the dual vector quaternions-based adaptive fading factors extended Kalman filter is the first robust dual vector quaternions-based parameters estimating method. Finally, the proposed dual vector quaternions-based extended Kalman filter and dual vector quaternions-based adaptive fading factors extended Kalman filter are validated by mathematical simulations, and the dual vector quaternions-based adaptive fading factors extended Kalman filter is compared with the dual vector quaternions-based extended Kalman filter to show its robust performances.

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Dual numbers representation of rigid body dynamics

Cited by 121 publications

References 23 publications

Time derivatives of screws with applications to dynamics and stiffness

Time derivatives of screws with applications to dynamics and stiffness

On Six DOF Relative Orbital Motion of Satellites

Adaptive pose and inertial parameters estimation of free-floating tumbling space objects using dual vector quaternions

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