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Robust Nonlinear Control of Aerial Manipulators

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Abstract

This paper presents single-layer robust nonlinear controllers for the unmanned aerial manipulator (UAM) trajectory tracking problem. The multi-body dynamical modeling of an underactuated UAM is conducted from the perspective of its end-effector using the Euler–Lagrange formalism. Accordingly, two single-layer nonlinear controllers are designed based on the classic \({\mathscr {H}}_{\infty }{}\) and the novel \({\mathscr {W}_{\infty }}{}\) control approaches for robust trajectory tracking of the UAM end-effector. The nonlinear \({\mathscr {H}}_{\infty }{}\) and \({\mathscr {W}_{\infty }}{}\) controllers are implemented in a hardware-in-the-loop framework using a simulator developed on Gazebo and ROS platforms, based on the computer-aided design model of the UAM. The performance of the controllers is evaluated when executing tasks such as object grasping, extension and retraction of the manipulator arm, hovering, and tracking a time-varying trajectory, while the UAM is affected by disturbances as ground effect, environment wind, and parametric and structural uncertainties.

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Notes

  1. The ProVANT simulator software is available for download on https://github.com/Guiraffo/ProVANT-Simulator.

  2. For the sake of simplicity, throughout the manuscript, some function dependencies are omitted.

  3. A video recording of the experiments is available in https://youtu.be/qoQXryey3UM.

  4. MoveIt is a ROS application that provides several motion planning algorithms and other functionalities required by path planners such as collision detection (Coleman et al. 2014).

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Acknowledgements

An early version of this paper (Morais et al. 2020) was presented at the XXIII Congresso Brasileiro de Automática (CBA 2020).

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Authors and Affiliations

  1. Graduate Program in Electrical Engineering, Universidade Federal de Minas Gerais, Belo Horizonte, MG, Brazil

    Júnio E. de Morais, Daniel N. Cardoso & Guilherme V. Raffo

  2. Department of Electronic Engineering, Universidade Federal de Minas Gerais, Belo Horizonte, MG, Brazil

    Guilherme V. Raffo

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  1. Júnio E. de Morais
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  2. Daniel N. Cardoso
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Correspondence to Júnio E. de Morais.

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This work was in part supported by the project INCT (National Institute of Science and Technology) under the Grant CNPq (Brazilian National Research Council) 465755/2014-3, FAPESP (São Paulo Research Foundation), Brazil 2014/50851-0. This work was also supported in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES), Brazil (Finance Code 88887.136349/2017-00, and 001), CNPq, Brazil (Grant numbers 315695/2020-0 and 426392/2016-7), and FAPEMIG, Brazil (Grant number APQ-03090-17)

Nonlinear \({\mathscr {H}}_{\infty }{}\) Control Matrices

Nonlinear \({\mathscr {H}}_{\infty }{}\) Control Matrices

As in Raffo et al. (2011), the matrices \({\varvec{{K}}}_{D}\), \({\varvec{{K}}}_{P}\) and \({\varvec{{K}}}_{I}\) in (31) are given by

$$\begin{aligned} \begin{aligned} {\varvec{{K}}}_{D}&= \begin{bmatrix} {\varvec{{K}}}_{D_{ss}} &{} {\varvec{{K}}}_{D_{sc}}\\ {\varvec{{K}}}_{D_{cs}} &{} {\varvec{{K}}}_{D_{cc}} \end{bmatrix},&{\varvec{{K}}}_{P}&= \begin{bmatrix} {\mathbb {O}}{} &{} {\varvec{{K}}}_{P_{sc}}\\ {\mathbb {O}}{} &{} {\varvec{{K}}}_{P_{cc}} \end{bmatrix}, \\ {\varvec{{K}}}_{I}&= \begin{bmatrix} {\mathbb {O}}{} &{} {\varvec{{K}}}_{I_{sc}}\\ {\mathbb {O}}{} &{} {\varvec{{K}}}_{I_{cc}} \end{bmatrix}, \end{aligned} \end{aligned}$$

where

$$\begin{aligned} {\varvec{{K}}}_{D_{ss}}&= {\varvec{{M}}}_{or}^{-1}\left( {\varvec{{C}}}_{or} + \frac{1}{\omega _{us}^{2}} {\mathbbm {1}}{} \right) ,\\ {\varvec{{K}}}_{D_{sc}}&= {\varvec{{M}}}_{or}^{-1}\left( {\varvec{{C}}}_{oc} - {\varvec{{M}}}_{sc}{\varvec{{M}}}_{cc}^{-1} \frac{1}{\omega _{uc}^{2}} \right) \\&\quad \times \frac{\omega _{uc} \omega _{1c}}{\sqrt{\gamma ^{2} - \omega _{uc}^{2}}} \frac{\sqrt{\gamma ^{2} - \omega _{us}^{2}}}{\omega _{us} \omega _{1s}},\\ {\varvec{{K}}}_{P_{sc}}&= {\varvec{{M}}}_{or}^{-1}\left( {\varvec{{C}}}_{oc} - {\varvec{{M}}}_{sc}{\varvec{{M}}}_{cc}^{-1} \frac{1}{\omega _{uc}^{2}} \right) \\&\quad \times \frac{\omega _{uc} \sqrt{\omega _{2c}^{2} + 2 \omega _{1c}\omega _{3c}}}{\sqrt{\gamma ^{2} - \omega _{uc}^{2}}} \frac{\sqrt{\gamma ^{2} - \omega _{us}^{2}}}{\omega _{us} \omega _{1s}},\\ {\varvec{{K}}}_{I_{sc}}&= {\varvec{{M}}}_{or}^{-1}\left( {\varvec{{C}}}_{oc} - {\varvec{{M}}}_{sc}{\varvec{{M}}}_{cc}^{-1} \frac{1}{\omega _{uc}^{2}} \right) \frac{\omega _{uc} \omega _{3c}}{\sqrt{\gamma ^{2} - \omega _{uc}^{2}}} \frac{\sqrt{\gamma ^{2} - \omega _{us}^{2}}}{\omega _{us} \omega _{1s}},\\ {\varvec{{K}}}_{D_{cs}}&= {\varvec{{M}}}_{ic}^{-1}\left( {\varvec{{C}}}_{ir} - {\varvec{{M}}}_{cs}{\varvec{{M}}}_{ss}^{-1} \frac{1}{\omega _{us}^{2}} \right) \frac{\omega _{us} \omega _{1s}}{\sqrt{\gamma ^{2} - \omega _{us}^{2}}} \frac{\sqrt{\gamma ^{2} - \omega _{uc}^{2}}}{\omega _{uc} \omega _{1c}},\\ {\varvec{{K}}}_{D_{cc}}&= \frac{\sqrt{\omega _{2c}^{2} + 2 \omega _{1c}\omega _{3c}}}{\omega _{1s}} {\mathbbm {1}}{} + {\varvec{{M}}}_{ic}^{-1} \left( {\varvec{{C}}}_{ic} + \frac{1}{\omega _{uc}^{2}}{\mathbbm {1}}{} \right) ,\\ {\varvec{{K}}}_{P_{cc}}&= \frac{\sqrt{\omega _{2c}^{2} + 2 \omega _{1c}\omega _{3c}}}{\omega _{1s}} {\varvec{{M}}}_{ic}^{-1} \left( {\varvec{{C}}}_{ic} + \frac{1}{\omega _{uc}^{2}}{\mathbbm {1}}{} \right) + \frac{\omega _{3c}}{\omega _{1s}}{\mathbbm {1}}{},\\ {\varvec{{K}}}_{I_{cc}}&= {\varvec{{M}}}_{ic}^{-1} \left( {\varvec{{C}}}_{ic} + \frac{1}{\omega _{uc}^{2}}{\mathbbm {1}}{} \right) \frac{\omega _{3c}}{\omega _{1s}}, \end{aligned}$$

in which it is considered the particular case where \({\varvec{{Q}}} = {{\,\mathrm{blkdiag}\,}}(\omega ^{2}_{1s}{\mathbbm {1}}{},\) \(\omega ^{2}_{1c}{\mathbbm {1}}{},\) \(\omega ^{2}_{2c}{\mathbbm {1}}{}\), \(\omega ^{2}_{3c}{\mathbbm {1}}{})\), \({\varvec{{R}}} = {{\,\mathrm{blkdiag}\,}}(\omega ^{2}_{ur}{\mathbbm {1}}{},\) \(\omega ^{2}_{uc}{\mathbbm {1}}{}),\) and \({\varvec{{S}}} = {\mathbb {O}}\).

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de Morais, J.E., Cardoso, D.N. & Raffo, G.V. Robust Nonlinear Control of Aerial Manipulators. J Control Autom Electr Syst 34, 1–17 (2023). https://doi.org/10.1007/s40313-022-00944-9

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