Tracking Controller Design for Satellite Attitude Under Unknown Constant Disturbance Using Stable Embedding
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ORIGINAL ARTICLE
Tracking Controller Design for Satellite Attitude Under Unknown Constant Disturbance Using Stable Embedding Wonshick Ko1 · Karmvir Singh Phogat2 · Nicolas Petit3 · Dong Eui Chang4 Received: 3 June 2020 / Revised: 16 September 2020 / Accepted: 27 November 2020 © The Korean Institute of Electrical Engineers 2020
Abstract We propose a tracking control law for the fully actuated rigid body system in the presence of any unknown constant disturbance by employing quaternions with the stable embedding technique and Lyapunov stability theory. The stable embedding technique extends the attitude dynamics from the set of unit quaternions to the set of quaternions, which is a Euclidean space, such that the set of unit quaternions is an invariant set of the extended dynamics. Such a stable extension of the system dynamics to a Euclidean space allows us to employ well studied Lyapunov techniques in Euclidean spaces such as LaSalle– Yoshizawa’s theorem. A robust tracking control law is proposed for the attitude dynamics subject to unknown constant disturbance and the convergence properties of the tracking control law is rigorously proven. It is demonstrated with the help of numerical simulations that the proposed control law has a remarkable performance even in some challenging situations. Keywords Attitude tracking control · Satellite · Embedding · Lyapunov function · Quaternions
1 Introduction The attitude dynamics and the control of a rigid body encounter the unique challenge that the configuration space of attitudes cannot be globally identified with a Euclidean space [7]. More specifically, the attitude representations such as the three-dimensional special orthogonal group SO(3) that is composed of 3 × 3 orthogonal matrices with the determinant of one [6, 7] or the set of unit quaternions S3 encounter * Dong Eui Chang [email protected] Wonshick Ko [email protected] Karmvir Singh Phogat [email protected] Nicolas Petit nicolas.petit@mines‑paristech.fr 1
Hanon Systems, 95, Sinilseo‑ro, Daedeok‑gu, Daejeon, Korea
2
Department of Mathematical Engineering and Information Physics, University of Tokyo, 7 Chome‑3‑1 Hongo, Bunkyo City, Tokyo 113‑8654, Japan
3
CAS, MINES ParisTech, Paris, France
4
School of Electrical Engineering, Korea Advanced Institute of Science and Technology, 291 Daehak‑ro, Daejeon, Korea
the same challenge that they cannot be globally identified with a Euclidean space. Therefore, the controller design and the stability analysis of systems on manifolds require sophisticated differential geometric tools which are often difficult to comprehend for ordinary engineers. An alternative to such cumbersome approaches is to stably embed the system dynamics on manifolds into Euclidean spaces [2] and then design controllers in these ambient Euclidean spaces. Moreover, the controller designed on the ambient space has a global representation in contrast to a local chartwise representation [1, 2, 5]. In this article, we employ the stable embedding technique to design a robust track
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