Abstract
In this paper, a novel feedback control strategy for quadrotor trajectory tracking is designed and experimentally tested with proof of exponential stability, using the Lyapunov transformations theory. The controller is derived from an inner-outer loop control structure, namely by considering the position system coupled through an interconnection term with the attitude system. For the design of the position controller, the considered dynamics are worked on the body frame, which is uncommon in the literature, and its synthesis derives from theories such as Pontryagin’s maximum principle, Lyapunov theory, and Linear Quadratic Regulator (LQR), which ensure Input-to-state stability, steady-state optimality, and global exponential stability. The attitude system is based on an error quaternion parameterization via a nonlinear coordinate transformation matrix followed by a state input feedback, rendering the system linear and time-invariant. Under a correct transformation, LQR theory ensures almost exponential stability and steady-state optimality for the overall interconnected closed-loop systems. Experimental and simulation results illustrate the performance of the tracking system onboard a quadrotor.
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Acknowledgements
This work was supported by the Portuguese Fundação para a Ciência e a Tecnologia (FCT) through the scholarship 2020.06736.BD, and the Institute for Mechanical Engineering (IDMEC), under Associated Laboratory for Energy, Transports and Aeronautics (LAETA) [UID/EMS/50,022/2020] projects. This work has also been supported by the European Union under the Next Generation EU, through a grant of the Portuguese Republic’s Recovery and Resilience Plan (PRR) Partnership Agreement, within the scope of the project PRODUTECH R3 - “Agenda Mobilizadora da Fileira das Tecnologias de Produção para a Reindustrialização”, aiming the mobilization of the production technologies industry towards of the reindustrialization of the manufacturing industrial fabric (Project ref. nr. 60 - C645808870-00000067).
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Open access funding provided by FCT|FCCN (b-on).. This work was supported by the Portuguese Fundação para a Ciência e a Tecnologia (FCT) and the Institute for Mechanical Engineering (IDMEC).
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Author J. Madeiras contributed to designing, implementing the experiments, and writing the document. Authors C. Cardeira and P. Oliveira contributed to the analysis and interpretation of the results, development of the theoretical contributions, and composition of the manuscript.
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Appendices
Appendix A: Proof of Lemma 1
The thrust T is retrieved by setting
Furthermore, and due to the vehicle capabilities, the thrust cannot be negative, which implies
Then, in order to compute \(\varvec{R}_1\) Eq. 28 and \(\varvec{R}_2\) Eq. 30 the norm of \(\varvec{u}_t\) and \(\varvec{\vartheta }\) must greater than zero, i.e.,
which can be achieved by guaranteeing the following condition
Appendix B: Proof of Lemma 2
By using the definition of \([\varvec{\omega } \times ]=\varvec{R}^T\dot{\varvec{R}}\) and taking the time derivative of Eq. 28, the following holds
where \(\omega _{13} = - \varvec{b_1}^T \dot{\varvec{b}}_{\varvec{2}} \), since \(\varvec{R_1} [\varvec{\omega }_{1} \times ] \varvec{e_2} = \frac{d}{dt} \left( \varvec{b_2}\right) \). The computation of \(\varvec{\omega }_{2}\) follows the same approach, i.e.,
where \(\omega _{23} = - \varvec{d_1}^T \dot{\varvec{d}}_{\varvec{2}} \), since \(\varvec{R_2} [\varvec{\omega }_{2} \times ] \varvec{e_2} = \frac{d}{dt} \left( \varvec{d_2}\right) \). The expression for the desired angular acceleration \(\dot{\varvec{\omega }}_{1}\) and \(\dot{\varvec{\omega }}_{2}\) follows directly from derivative in order to time of \(\varvec{\omega }_1\) and \(\varvec{\omega }_2\), respectively.
Then, by taking the derivative of Eq. 34, one can verify that
The expression for the angular acceleration \(\dot{\varvec{\omega }}_d\) is computed by taking the time derivative of \(\varvec{\omega }_d\).
By Assumption 1 and backed by Theorem 1, the variables \(\varvec{\omega }\), \(\dot{\varvec{\omega }}\), \(\varvec{x}_1\), \(\dot{\varvec{x}}_1\), and \(\ddot{\varvec{x}}_1\) are bounded signals. Consequently, the first and second time derivatives of \(\varvec{u}_t\) are bounded functions as well. Furthermore, from the assumptions on the references \(\varvec{p}_d\) and \(\psi _d\) given in Section 2.2 and by considering that Eq. 32 holds true, one may conclude, by inspection of \(\varvec{\omega }_d\) and \(\dot{\varvec{\omega }}_d\), that
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Madeiras, J., Cardeira, C. & Oliveira, P. Position and Attitude Tracking Controllers Using Lyapunov Transformations for Quadrotors. J Intell Robot Syst 110, 9 (2024). https://doi.org/10.1007/s10846-023-02016-9
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DOI: https://doi.org/10.1007/s10846-023-02016-9