Planar dual-hoist dynamics of quadcopters carrying slender loads

Abstract

Quadcopters typically suspend large-size loads by dual-hoist mechanisms for material-handling services. The dual-hoist dynamics exhibit complex coupling effects between the quadcopter attitude and distributed-mass loads. Therefore, manipulation is a challenging task due to the dual-hoist dynamics. Some progress has been focused on single-hoist dynamics of quadcopters slung loads. However, little attention has been directed at dual-hoist dynamics. A planar dual-hoist dynamic model in near-hover operation can be noted in this article, and simultaneously used to design an oscillation-control method so as to facilitate flying such aerial cranes. Technological achievements presented in simulations exhibit the dynamic behavior of the analytical model and demonstrate that the control method succeeded in restraining the quadcopter attitude and load oscillations. The theoretical findings in this article might extend to other types of aerial cranes, such as helicopters or tiltrotors slung loads.

Appendix

The nonlinear equations of the motion in Fig. 1 are listed below:

$$ m_{D} \ddot{x} + m_{L} W_{1} + F_{z} \sin \theta = 0, $$

(A1)

$$ m_{D} \ddot{z} + m_{L} W_{2} - m_{L} g - m_{D} g + F_{z} \cos \theta = 0, $$

(A2)

$$ \begin{aligned} I_{yy} \ddot{\theta } & + \frac{{(\ddot{\theta } + \ddot{\delta })m_{L} l_{L}^{2} }}{12} - M_{\theta } - m_{L} g\left[ {0.5l_{L} \cos (\theta + \delta ) - l_{A} \cos \theta - l_{S} \sin (\theta + \beta_{2} )} \right] \\ & + m_{L} W_{1} \left[ \begin{array}{l} - l_{A} \sin \theta \hfill \\ + l_{S} \cos (\theta + \beta_{2} ) \hfill \\ + 0.5l_{L} \sin (\theta + \delta ) \hfill \\ \end{array} \right] + m_{L} W_{2} \left[ \begin{array}{l} - l_{A} \cos \theta \hfill \\ - l_{S} \sin (\theta + \beta_{2} ) \hfill \\ + 0.5l_{L} \cos (\theta + \delta ) \hfill \\ \end{array} \right] = 0, \\ \end{aligned} $$

(A3)

$$ \begin{aligned} & m_{L} gl_{S} \sin (\theta + \beta_{2} )\frac{{\cos (\beta_{1} - \delta )}}{{\cos (\beta_{2} - \delta )}} + 0.5m_{L} gl_{L} \cos (\theta + \delta )\frac{{l_{S} \sin (\beta_{1} - \beta_{2} )}}{{l_{L} \cos (\beta_{2} - \delta )}} \\ & \quad + \frac{{\cos (\beta_{1} - \delta )}}{{\cos (\beta_{2} - \delta )}}\left[ {l_{S} \cos (\theta + \beta_{2} )m_{L} W_{1} - l_{S} \sin (\theta + \beta_{2} )m_{L} W_{2} } \right] \\ & \quad - \frac{{l_{S} \sin (\beta_{1} - \beta_{2} )}}{{l_{L} \cos (\beta_{2} - \delta )}}\left[ \begin{array}{l} \frac{1}{12}(\ddot{\theta } + \ddot{\delta })m_{L} l_{L}^{2} + 0.5l_{L} \sin (\theta + \delta )m_{L} W_{1} \hfill \\ + 0.5l_{L} \cos (\theta + \delta )m_{L} W_{2} \hfill \\ \end{array} \right] = 0. \\ \end{aligned} $$

(A4)

The acceleration constraints of the four-bar linkage are listed below:

$$ \ddot{\beta }_{2} = \tan (\beta_{2} - \delta )\dot{\beta }_{2}^{2} - \frac{{l_{L} \dot{\delta }^{2} }}{{l_{S} \cos (\beta_{2} - \delta )}} - \frac{{\left[ {\sin (\beta_{1} - \delta )\dot{\beta }_{1}^{2} - \cos (\beta_{1} - \delta )\ddot{\beta }_{1} } \right]}}{{\cos (\beta_{2} - \delta )}}, $$

(A5)

$$ \ddot{\delta } = \frac{{l_{S} \left[ {\dot{\beta }_{2}^{2} - \cos (\beta_{1} - \beta_{2} )\dot{\beta }_{1}^{2} - \sin (\beta_{1} - \beta_{2} )\ddot{\beta }_{1} } \right]}}{{l_{L} \cos (\beta_{2} - \delta )}} - \tan (\beta_{2} - \delta )\dot{\delta }^{2} . $$

(A6)