Nonlinear dynamics characteristics of a magnetically actuated dual-spin capsule robot

Abstract

To study the attitude dynamics of a magnetically driven dual-spin spherical capsule robot (DSCR), a fourth-order periodic time-varying nonlinear dynamic equation for the DSCR under the action of complex external torque was established based on Euler dynamics. The nonlinear dynamic characteristic of the DSCR was studied by using the incremental harmonic balance (IHB) method and the Runge–Kutta (RK) method. A semi-analytical periodic solution of the attitude dynamics equation was obtained, and the stability of periodic solutions was analyzed based on Floquet theory. By using the bifurcation diagram, time domain diagram, phase diagram, Poincare map and Lyapunov exponent diagram as analysis methods, the global nonlinear dynamic response of the DSCR was studied. The change rule of the global topological structure of the system and the path of the system to chaos were obtained when the magnetic flux density and the magnetic field frequency changed in a large range. The spatial universal uniform rotating magnetic field (SURMF) experimental platform and the axis orientation measurement device were built to verify the dynamic characteristics of the DSCR. The experimental analysis results were in good agreement with the theoretical calculation data.

Appendices

Appendix 1

$$ \overline{\varvec{M}}\left( \varvec{X} \right) = \left[ {\begin{array}{*{20}c} {J_{e} \cos x_{2} } & 0 \\ 0 & {J_{e} } \\ \end{array} } \right],\;\overline{\varvec{N}}\left( \varvec{X} \right) = \left[ {\begin{array}{*{20}c} {k\cos x_{2} - J_{e} \sin x_{2} } & { - J_{1} \omega } \\ {J_{1} \omega \cos x_{2} } & k \\ \end{array} } \right] $$

$$ {\overline{\varvec{K}}}\left( {{\varvec{X}},{\varvec{X^{\prime}}}} \right) = \left[ {\begin{array}{*{20}c} {(J_{1} + J_{2} - J_{e} )x_{1}^{\prime } x_{2}^{\prime } \sin x_{2} } \\ { - (J_{1} + J_{2} - J_{e} )x_{1}^{\prime 2} \sin x_{2} \cos x_{2} } \\ \end{array} } \right],\;{\overline{\varvec{H}}}\left( {\varvec{X}} \right) = \left[ {\begin{array}{*{20}c} {Gl\sin x_{1} } \\ {Gl\cos x_{1} \sin x_{2} } \\ \end{array} } \right] $$

$$ {\overline{\mathbf{F}}}\left( {{\varvec{X}},\tau } \right) = \left[ {\begin{array}{*{20}c} { - (D\cos \tau + E\sin \tau )\cos (\tau - \delta )} \\ {(D\cos \tau + E\sin \tau )\sin (\tau - \delta )} \\ \end{array} } \right] $$

$$ \begin{gathered} D{\text{ = cos}}\alpha_{1} \sin x_{1} \cos x_{2} - \sin \alpha_{1} \cos x_{1} \cos x_{2} \hfill \\ E = \sin \alpha_{1} \sin \beta_{1} \sin x_{1} \cos x_{2} - \cos \beta_{1} \sin x_{2} + \cos \alpha_{1} \sin \beta_{1} \cos x_{1} \cos x_{2} \hfill \\ \end{gathered} $$

Appendix 2

$$ {\overline{\varvec{K}}}_{0} = \left[ {\begin{array}{*{20}c} 0 & {(J_{1} + J_{2} - J_{e} )\alpha^{\prime}\beta^{\prime}\cos \beta } \\ 0 & { - (J_{1} + J_{2} - J_{e} )\alpha^{{\prime}{2}} \cos 2\beta } \\ \end{array} } \right],\;{\overline{\varvec{F}}}_{0} = \left[ {\begin{array}{*{20}c} { - (\frac{dD}{{d{\varvec{X}}}}\cos \tau + \frac{dE}{{d{\varvec{X}}}}\sin \tau )\cos (\tau - \delta )} \\ {(\frac{dD}{{d{\varvec{X}}}}\cos \tau + \frac{dE}{{d{\varvec{X}}}}\sin \tau )\sin (\tau - \delta )} \\ \end{array} } \right] $$

$$ \begin{gathered} \frac{dD}{{d{\varvec{X}}}} = \left[ {\begin{array}{*{20}l} {a_{11} \cos x_{1} \cos x_{2} - a_{31} \sin x_{1} \cos x_{2} } \hfill & { - a_{11} \sin x_{1} \sin x_{2} - a_{31} \cos x_{1} \sin x_{2} } \hfill \\ \end{array} } \right] \hfill \\ \frac{dE}{{d{\varvec{X}}}} = \left[ {\begin{array}{*{20}l} {a_{12} \cos x_{1} \cos x_{2} - a_{32} \sin x_{1} \cos x_{2} } \hfill & { - a_{12} \sin x_{1} \sin x_{2} - a_{22} \cos x_{2} - a_{32} \cos x_{1} \sin x_{2} } \hfill \\ \end{array} } \right] \hfill \\ \end{gathered} $$

$$ {\overline{\varvec{H}}}_{0} = \left[ {\begin{array}{*{20}c} {Gl\cos x_{1} } & 0 \\ { - Gl\sin x_{1} \sin x_{2} } & {Gl\cos x_{1} \cos x_{2} } \\ \end{array} } \right] $$

Appendix 3

$$ {\varvec{M}} = \int_{0}^{2\pi } {{\varvec{S}}^{T} {\varvec{\overline{M}S^{\prime\prime}}}d\tau } ,\;{\varvec{N}} = \int_{0}^{2\pi } {{\varvec{S}}^{T} {\varvec{\overline{N}S^{\prime}}}d\tau } $$

$$ {\varvec{K}} = \int_{0}^{2\pi } {{\varvec{S}}^{T} {\overline{\varvec{K}}}d\tau } ,\;{\varvec{F}}{ = }\int_{0}^{2\pi } {{\varvec{S}}^{T} {\overline{\varvec{F}}}d\tau } $$