Abstract
The control problem of a multi-copter swarm, mechanically coupled through a modular lattice structure of connecting rods, is considered in this article. The system’s structural elasticity is considered in deriving the system’s dynamics. The devised controller is robust against the induced flexibilities, while an inherent adaptation scheme allows for the control of asymmetrical configurations and the transportation of unknown payloads. Certain optimization metrics are introduced for solving the individual agent thrust allocation problem while achieving maximum system flight time, resulting in a platform-independent control implementation. Experimental studies are offered to illustrate the efficiency of the suggested controller under typical flight conditions, increased rod elasticities and payload transportation.
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Acknowledgements
This research was in part performed by using NYUAD’s Core Technology Platform Kinesis lab motion capture system. The authors thank Mr. Nikolaos Giakoumidis and Dr. Oraib Al Ketan for their technical support and insights.
Funding
The authors declare that no funds, grants, or other support were received during the preparation of this manuscript. This work is supported in part by the NYUAD Center for Artificial Intelligence and Robotics (CAIR), funded by Tamkeen under the NYUAD Research Institute Award CG010.
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All authors contributed to the study conception. The primary design was made by D. Chaikalis (DC), followed by modifications made by DC and A. Tzes (AT). The experiments were performed by DC and N. Evangeliou. The first draft of the manuscript was written by DC and edited by AT. All authors read and approved the final manuscript.
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Appendices
Appendix A
Let \(z=z^{\circ }+\Delta z\), and \(T=T^{\circ }+\sum _{i=0}^{n-1}\Delta T_{i}\), where \(z^{\circ }\) and \(T^{\circ }\) are constants, then \(\dot{z}=\Delta \dot{z}\). According to the backstepping control principle, the controller design relies on computing a virtual control signal to stabilize each sub-state sequentially. Similar to [36], assume the positive definite Lyapunov function \(V_{1} = \frac{{e_{z}^{2}}}{2}\), where \(e_{z} = z - z^{d}\) with \(z^{d}\) corresponding to the desired altitude. Then, \( \dot{V}_{1} = e_{z} \dot{e}_{z} = e_{z} \left( \Delta \dot{z} - \dot{z}^{d} \right) .\) Let the virtual input \(\dot{z}^{*}\) capable of stabilizing the altitude error sub-state \(\dot{z}^{*} = \dot{z}^{d} - K_{z1}e_{z}\), and the velocity error term \( s_{z} = \Delta \dot{z} - \dot{z}^{*}. \) Then \(\dot{V}_{1}\) can be rewritten as \( \dot{V}_{1} = e_{z} \left( s_{z} - K_{z1}e_{z} \right) . \) Let the augmented Lyapunov function \( V_{z} = V_{1} + \frac{1}{2}{s_{z}^{2}}, \) then
where it was assumed for simplicity that \(\ddot{z}^{d} = 0\). Given the altitude dynamics Eq. 13 and the altitude controller (15) applied to Eq. 29 results in \( \dot{V}_{z} = -K_{z1}{e_{z}^{2}} - K_{z2}{s_{z}^{2}} \le 0. \) Hence \(e_{z} \rightarrow 0 \), and \(s_{z} \rightarrow 0\), implying that \(z \rightarrow z^{d}\), \(\dot{z} \rightarrow \dot{z}^{d}\), since the system is asymptotically stable under the proposed controller.
Likewise, asymptotic stability of x and y can be proven.
Appendix B
In the case when there is external payload of unknown mass, the controller needs to be augmented with an online adaptation term for this mass. Assume in Eq. 15, the term m is replaced by its estimate \(\hat{m}\), then the Lyapunov function derivative is
Let the positive quantity
and the adaptation evolution rule
Then the augmented Lyapunov function \(V_{z}+V_{m}\) has derivative \( \dot{V}_{z}+\dot{V}_{m} = -K_{z1}{e_{z}^{2}} -K_{z2}{s_{z}^{2}}, \le 0. \) We should note that there is no guarantee that \(\hat{m} \rightarrow m\) but simply that \(e_{z}\) and \(s_{z}\) converge to zero.
Appendix C
Given the system’s attitude dynamics Eq. 19, the attitude control input \(\varvec{\tau }^{c}\) Eq. 22 is computed using adaptive control principles, in order to guarantee the stability of the vehicle’s attitude.
Given desired roll, pitch and yaw angles \((\phi ^{d}, \theta ^{d}, \psi ^{d})\), let the: a) attitude error vector \( \textbf{e}_{\phi } = \left[ \phi , \theta , \psi \right] ^{T}\! - \left[ \phi ^{d}, \theta ^{d}, \psi ^{d}\right] ^{T}\!, \) b) ideal angular velocity as \( \Omega ^{*} = \Omega ^{d} - K_{\phi } \textbf{e}_{\phi }, \) where \(K_{\phi }\) is a diagonal positive gain matrix, c) velocity error vector \(\textbf{z}_{\phi } = \Omega - \Omega ^{*}\), d) adaptation estimates \(\hat{\textbf{J}}, \hat{\tau }^{s}\) for inertia matrix and asymmetric torques acting on the structure, and e) error matrix \( \tilde{E} = I_{3} - \textbf{J}^{-1} \hat{\textbf{J}}, \) where \(I_{3}\) is the \(3\times 3\) identity matrix.
Based on the backstepping principle, a composite Lyapunov function is defined, incorporating attitude errors and errors in unknown estimates
where \(\Lambda \) is a diagonal positive gain matrix and \(\sigma \) is a positive constant.
Substituting Eqs. 22 to 32 and using the adaptation evolutions
then the derivative of the Lyapunov composite function is
In this derivation the symmetry and positive definiteness of the inertia matrices is used. The inertia matrix estimate \(\hat{\textbf{J}}(0)\) is computed using the application of the parallel axis theorem on the agent masses and \(\hat{\tau }^s(0)=0\). Similarly, the same initial estimate for the system inertia matrix is used for the feedforward component of (22).
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Chaikalis, D., Evangeliou, N., Tzes, A. et al. Modular Multi-Copter Structure Control for Cooperative Aerial Cargo Transportation. J Intell Robot Syst 108, 31 (2023). https://doi.org/10.1007/s10846-023-01842-1
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DOI: https://doi.org/10.1007/s10846-023-01842-1