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A new look at the uncontrollable linearized quaternion dynamics with implications to LQR design in underactuated systems

Lustosa, Leandro Ribeiro and Cardoso-Ribeiro, Flavio Luiz and Defaÿ, François and Moschetta, Jean-Marc A new look at the uncontrollable linearized quaternion dynamics with implications to LQR design in underactuated systems. (2018) In: 2018 European Control Conference (ECC), 12 June 2018 - 15 June 2018 (Limassol, Cyprus).

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Official URL: https://doi.org/10.23919/ECC.2018.8550415


Quaternion algebra is frequently employed for spacecraft attitude description due to its convenient numerical properties when compared to minimal formulations. In parallel, Linear Quadratic Control (LQR)-based attitude controllers are often applied to underactuated vehicles due to its intuitive tuning process and satisfactory stability robustness properties. However, nonlinear quaternion differential equations of motion linearization yields non-stabilizable systems. Thus, LQR techniques cannot be directly employed since the associated algebraic Riccati equation is ill-posed. The commonplace solution resorts to a reduced quaternion model where only three out of four quaternion coordinates are exploited. The present work shows that such choice exhibits numerically unstable regions that impedes solving the LQR problem for all possible operating points. Additionally, we propose two methods to obtain wellposed LQR problems over all operating points. The first is based on the reduced quaternion model with an appropriate change of coordinates. The second is to append a virtual stabilizing input (VSI) to the nonlinear system to attain controllable linearized systems. The VSI direction should be appropriately chosen to not disturb the controllable modes of the system. Finally, we show that a class of constant angular velocity tracking problems is time-invariant under an appropriate change of variables such that time-invariant LQR techniques are applicable.

Item Type:Conference or Workshop Item (Paper)
Additional Information:Thanks to the IEEE (Institute of Electrical and Electronics Engineers). This paper is available at :https://ieeexplore.ieee.org/document/8550415 “© 2018 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.
HAL Id:hal-01994381
Audience (conference):International conference proceedings
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Institution:Université de Toulouse > Institut Supérieur de l'Aéronautique et de l'Espace - ISAE-SUPAERO (FRANCE)
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Deposited On:25 Jan 2019 13:06

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