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A matrix algorithm based on controlled Lagrangians for stabilizing mechanical systems with underactuation degree one

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Abstract

A matrix control algorithm based on controlled Lagrangian method is presented in this paper to stabilize a class of mechanical systems with underactuation degree one. Firstly, a desired controlled system with Lagrangian structure and desired properties is constructed. By equating the underactuated system with the desired system, the matching condition and controller structure are determined. A sufficient condition for the matching condition to be held is derived, and from this sufficient condition desired kinetic energy, potential energy, gyroscopic forces and dissipative forces of the desired system can be solved explicitly. Compared with the existing matching conditions, for the proposed sufficient condition at most two partial differential equations need to be solved, and the rests are all algebraic equations, which is easier to solve. An algorithm to solve this sufficient condition is given in detail, and with the obtained solution a nonlinear smooth feedback controller can be constructed to stabilize the underactuated systems. Finally, the novel control algorithm is applied to achieve almost global stability of a vertical takeoff and landing aircraft and to locally stabilize a Pendubot with two degrees of freedom at its the highest equilibrium point. Simulation results demonstrate effectiveness of the proposed method.

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Data availability

All data generated or analyzed during this study are included in this published article, and all MATLAB codes for simulations are available on Github at https://github.com/guanjunchen/Stabilization_algorithm.

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Acknowledgements

This work is supported by the National Natural Science Foundation (NNSF) of China under Grant 61673043.

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  1. The Seventh Research Division, Beihang University, Beijing, 100191, China

    Guanjun Chen & Wei Huo

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  1. Guanjun Chen
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  2. Wei Huo
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Correspondence to Wei Huo.

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Chen, G., Huo, W. A matrix algorithm based on controlled Lagrangians for stabilizing mechanical systems with underactuation degree one. Nonlinear Dyn 108, 3623–3642 (2022). https://doi.org/10.1007/s11071-022-07417-3

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