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Consensus tracking control and vibration suppression for nonlinear mobile flexible manipulator multi-agent systems based on PDE model

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Abstract

This paper evaluates the consensus tracking control of nonlinear multi-agent systems (MASs) described by partial differential equations (PDEs). Each agent of the MASs is a mobile flexible manipulator composed of a mobile carrier, a flexible manipulator, and an end load. By utilizing Hamilton’s principle, the characteristics of each agent are described by nonlinear fourth-order PDEs with the coupling of displacement, angle, and vibration. A consensus tracking control strategy is proposed for the MASs based on the obtained PDE model. Under the designed controller, both the angle of the flexible manipulator and the displacement of the mobile carrier of all agents can achieve consensus via mutual communication and track the desired positions and speeds. In addition, the elastic deformation and deformation speed of each agent’s flexible manipulator can be suppressed. The closed-loop system is proven asymptotically stable by constructing the Lyapunov function and applying the extended LaSalle’s invariance principle. A MAS example is numerically simulated to verify the effectiveness of the proposed control strategy.

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Data availability

The datasets generated and analyzed during the current study are available from the corresponding author on reasonable request.

Abbreviations

\({\mathbb {R}}\) :

Set of real numbers

\({\mathbb {Z}}^+\) :

Set of positive integers

\({\mathbb {R}}^N\) :

Set of \(N\times 1\)-dimensional real vectors, \(N\in {\mathbb {Z}}^+\)

\({\mathbb {R}}^{N\times N}\) :

Set of \(N\times N\)-dimensional real matrices, \(N\in {\mathbb {Z}}^+\)

\((*)_i\) :

States or parameters related to the ith agent, \(i=1,\ldots ,N\)

\(EI_i\) :

Bending stiffness of the flexible manipulator

\(L_i\) :

Length of the flexible manipulator

\(I_{Fi}\) :

Moment of inertia of the joint

\(\rho _i\) :

Mass per unit length of the flexible manipulator

\(M_i\) :

Mass of the mobile carrier

\(m_i\) :

Mass of the end load

\(\phi _i(t)\) :

Angle of the flexible manipulator

\(r_i(t)\) :

Horizontal displacement of the mobile carrier

\(\tau _i(t)\) :

Control input acting on the joint of the flexible manipulator

\(u_{Mi}(t)\) :

Control input acting on the mobile carrier

\(u_{Li}(t)\) :

Control input acting on the end of the flexible manipulator

\(F_i(x_i,t)\) :

Distributed control input acting on the flexible manipulator

\(y_i(x_i,t)\) :

Bending deformation of the flexible manipulator

\({\mathbf {{ P}}}_i(x_i,t)\) :

Position vector from the origin of the inertial frame to point \(x_i\) of the flexible manipulator

g :

Gravitational acceleration

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Number 61873296).

Funding

Funding was provided by National Natural Science Foundation of China (Grant Number 61873296).

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Authors and Affiliations

  1. School of Automation Science and Electrical Engineering, Beihang University, Xueyuan Road No.37, Haidian District, 100191, Beijing, China

    Le Li & Jinkun Liu

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Contributions

All authors contributed to the study conception and design. Material preparation, investigation, control design, analysis and validation were performed by LL and JL. The original draft of the manuscript was written by LL and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

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Correspondence to Jinkun Liu.

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Li, L., Liu, J. Consensus tracking control and vibration suppression for nonlinear mobile flexible manipulator multi-agent systems based on PDE model. Nonlinear Dyn 111, 3345–3359 (2023). https://doi.org/10.1007/s11071-022-07980-9

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