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A geometric formulation of multirotor aerial vehicle dynamics

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A Correction to this article was published on 14 March 2022

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Abstract

A geometric dynamic modeling framework for generic multirotor aerial vehicles (MAV), based on a modern Lie group formulation of classical screw theory, is presented. Our framework allows for a broad range of rotor-wing configurations: any number of rotors can be attached in arbitrary configurations to either the body or wings, with the rotors and wings also tiltable. Our framework takes into account all masses and inertias of the MAV body and rotors, and accounts for both rotor thrust forces and moments as well as external aerodynamic and other forces. Compared to existing methods, our Lie group framework possesses several practical advantages useful for applications ranging from design optimization to model identification and trajectory optimization: (1) the dynamic equations can be easily transformed to coordinates of any reference frame; (2) kinematic and mass–inertial parameters can be easily factored from the dynamic equations; (3) exact, closed-form analytic derivatives of the dynamics with respect to the configuration variables are easily derived. We demonstrate our systematic modeling procedure on examples of fixed-tilt, variable-tilt and hybrid MAVs with wings.

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Data availability statements:

Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

Change history

Notes

  1. Most textbooks contribute this theorem only to the work of M. Chasles in 1830; however, it dates back to G. Mozzi in 1763.

  2. Although one has that \((\mathbb {R}^{6})^*\cong \mathbb {R}^{6}\), we denote in this work the space of wrenches by \((\mathbb {R}^{6})^*\) to stress their covector nature which is crucial to note as wrenches and twists change coordinates differently.

Abbreviations

\(\{{a}\}\) :

Reference frame a

\(\varvec{T}_{a}^{b} \in \text {SE(3)}\) :

Relative configuration of \(\{{a}\}\) w.r.t. \(\{{b}\}\)

\(\varvec{R}_{a}^{b} \in \text {SO(3)}\) :

Relative orientation of \(\{{a}\}\) w.r.t. \(\{{b}\}\)

\(\varvec{p}_{a}^{b} \in \mathbb {R}^{3}\) :

Position of the origin of \(\{{a}\}\) expressed in \(\{{b}\}\)

\({[}\varvec{x}{]} \in \text {so}(3)\) :

Skew-symmetric \(3\times 3\) matrix representation of \(\varvec{x}\in \mathbb {R}^{3}\)

\(\varvec{\mathcal {V}}_{a}^{c,b} \in \mathbb {R}^{6}\) :

Twist (generalized velocity) of \(\{{a}\}\) w.r.t. \(\{{b}\}\) expressed in \(\{{c}\}\)

\(\varvec{\omega }_{a}^{c,b}, \in \mathbb {R}^{3}\) :

The angular part of the twist \(\varvec{\mathcal {V}}_{a}^{c,b}\)

\(\varvec{v}_{a}^{c,b}\in \mathbb {R}^{3}\) :

The linear part of the twist \(\varvec{\mathcal {V}}_{a}^{c,b}\)

\({[}\varvec{\mathcal {V}}{]} \in \text {se}(3)\) :

The \(4\times 4\) matrix representation of the twist \(\varvec{\mathcal {V}}\in \mathbb {R}^{6}\)

\(\varvec{\mathcal {S}}_{a}^{c,b} \in \mathbb {R}^{6}\) :

The screw-vector corresponding to a unit twist between the bodies attached to \(\{{a}\}\) and \(\{{b}\}\), expressed in \(\{{c}\}\)

\(\varvec{\mathcal {W}}_{\text {src}}^{c,b} \in (\mathbb {R}^{6})^*\) :

The applied wrench by the source (src) to the body associated with \(\{{b}\}\), expressed in \(\{{c}\}\)

\(\varvec{m}_{\text {src}}^{c,b} \in (\mathbb {R}^{3})^*\) :

The moment part of the wrench \(\varvec{\mathcal {W}}_{\text {src}}^{c,b}\)

\(\varvec{f}_{\text {src}}^{c,b} \in (\mathbb {R}^{3})^*\) :

The force part of the wrench \(\varvec{\mathcal {W}}_{\text {src}}^{c,b}\)

\(\varvec{\mathcal {G}}_{b}^{c} \in \mathbb {R}^{6\times 6}\) :

The generalized inertia tensor of the body associated with \(\{{b}\}\), expressed in \(\{{c}\}\)

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Acknowledgements

This work was partially supported by the PortWings project funded by the European Research Council [Grant Agreement No. 787675].

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Authors and Affiliations

  1. Department of Mechanical and Aerospace Engineering, Seoul National University, Seoul, Korea

    Youngsuk Hong, Soocheol Noh, Taeyoon Lee & Frank C. Park

  2. Robotics and Mechatronics Group, University of Twente, Enschede, The Netherlands

    Ramy Rashad & Stefano Stramigioli

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  1. Youngsuk Hong
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Correspondence to Youngsuk Hong.

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Hong, Y., Rashad, R., Noh, S. et al. A geometric formulation of multirotor aerial vehicle dynamics. Nonlinear Dyn 107, 495–513 (2022). https://doi.org/10.1007/s11071-021-07042-6

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