Abstract
The operation of launched micro aerial vehicles (MAVs) with coaxial rotors is usually subject to unknown varying external disturbance. In this paper, a robust controller is designed to reject such uncertainties and track both position and orientation trajectories. A complete dynamic model of coaxial-rotor MAV is firstly established. When all system states are available, a nonlinear state-feedback control law is proposed based on feedback linearization and Lyapunov analysis. Further, to overcome the practical challenge that certain states are not measurable, a high gain observer is introduced to estimate unavailable states and an output feedback controller is developed. Rigid theoretical analysis verifies the stability of the entire closed-loop system. Additionally, extensive simulation studies have been conducted to validate the feasibility of the proposed scheme.
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Acknowledgements
This work is supported by National Natural Science Foundation of China (Grant No. 61673347, U1609214).
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Appendix
Appendix
Proof of Theorem 1
Considering a candidate Lyapunov function
with
Recalling Eq. 29, the time derivative of Vpis given by
Further, Eq. 32 can be employed to obtain the derivative of Vψ
Thus, the derivative of V is
where c = min{2cp, 2cψ}. Solving Eq. 52 generates
where \(V(0)=\frac {1}{2}({r_{p}^{T}}(0)r_{p}(0)+r_{\psi }^{T}(0)r_{\psi }(0))\)is the initial value.
Obviously, V → 0 as t →∞. This implies that rp, rψ → 0ast →∞. Subsequently, the tracking error \(\delta _{p_{i}}\)and \(\delta _{\psi _{i}}\)defined in Eqs. 16 and 23 also converge to zero asymptotically by definition. Thus, the closed-loop system can asymptotically track the reference trajectories pdandψd. With the help of Eq. 18 and Assumption 1 − 2,v and Tz are bounded. Rη and Qη are also bounded since \(-\frac {\pi }{2}< \theta , \phi < \frac {\pi }{2}\). Similarly, the signals ωx, ωy and \(\dot T_{z}\) are all bounded according to Eq. 18. Through Eq. 25, one can reach that ωz is bounded, and so is \(\dot Q_{\eta }\). Thus,the two control inputs τ and \(\ddot T_{z}\)are also boundedfrom Eq. 33. □
Proof of Theorem 2
Consider the following Lyapunov function candidate
with
Define the estimated filtered position tracking error as
If enforcing the output feedback controller (45), the filtered position tracking error dynamics (27) can be rewritten as
Thus, the derivative of \(\overline V_{p}\)can be given by
Substituting Eqs. 40 and 42 into Eq. 58 generates
Meanwhile, by resorting to Lemma 1, it is apparent that for the vector\(r_{p} \in \mathbb {R}^{3}\) and a positive constant εp > 0,
Thus, one further has
With the help of Young’s inequality and the fact that | tanh(⋅)|≤ 1, Eq. 61 results in
where ∥⋅∥denotes the L2-norm,and ε, ε1, ε2, ε3, ε4\(\in \mathbb {R}^{+}\).
Similarly, with the controlled designed in Eq. 45, the filtered orientation tracking error dynamics can be rephrased tobe
Hence, the derivative of \(\overline V_{\psi }\) can be given by
Combining Eq. 62 with Eq. 64 delivers
where \(c_{p_{1}}\),\(c_{p_{2}}\), C\(\in \mathbb {R}^{+}\)with
Hence, \(\dot {\overline V}\)will becomes negative as long as
or
or
According to the standard Lyapunov analysis, one has that the filtered tracking errorsrp, rψ, and theobserver error epare uniformly ultimately bounded (UUB). Furthermore, the tracking errors can be arbitrarily reduced by increasing control gainscψ, cp, h1, h2andh3. By following the similar analysisin Proof of Theorem 1, the signals Tz, \(\dot T_{z}\),ω are bounded as well asall the elements of Rη, Qηand\(\dot Q_{\eta }\). Furthermore, theestimation signals \(\hat \delta _{p_{2}}\),\(\hat \delta _{p_{3}}\)and\(\hat r_{p}\)are bounded since the observererror vector epis bounded.Thus, the control signals \(\ddot T_{z}\)and τ are proved to bebounded from Eq. 45. □
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Li, J., Yang, Q. & Sun, Y. Robust State and Output Feedback Control of Launched MAVs with Unknown Varying External Loads. J Intell Robot Syst 92, 671–684 (2018). https://doi.org/10.1007/s10846-018-0774-z
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DOI: https://doi.org/10.1007/s10846-018-0774-z