Momentum Preserving Simulation of Cooperative Multi-Rotors with Flexible-Cable Suspended Payload

In this paper, a momentum-preserving integration scheme is implemented for the simulation of single and cooperative multi-rotors with a flexible-cable suspended payload by employing a Lie group based variational integrator (VI), which provides the preservation of the configuration manifold and geometrical constraints. Due to the desired properties of the implemented VI method, e.g. sypmlecticity, momentum preservation, and the exact fulfillment of the constraints, exponentially long-term numerical stability and good energy behavior are obtained for more accurate simulations of aforementioned systems. The effectiveness of Lie group VI method with the corresponding discrete systems are demonstrated by comparing the simulation results of two example scenarios for the single and cooperative systems in terms of the preserved quantities and constraints, where a conventional fixed-step Runge-Kutta 4 (RK4) and Variable-Step integrators are utilized for the simulation of continuous-time models. It is shown that the implemented VI method successfully performs the simulations with a long-time stable behavior by preserving invariants of the system and the geometrical constraints, whereas the simulation of continuous-time models by RK4 and Variable Step are incapable of satisfying these desired properties, which inherently results in divergent and unstable behavior in simulations.

On an upper bound for the spreading speed

<p style='text-indent:20px;'>In this paper, we use the exponential transform to give a unified formal upper bound for the asymptotic rate of spread of a population propagating in a one dimensional habitat. We show through examples how this upper bound can be obtained directly for discrete and continuous time models. This upper bound has the form <inline-formula><tex-math id="M1">\begin{document}$ \min_{s&gt;0} \ln (\rho(s))/s $\end{document}</tex-math></inline-formula> and coincides with the speeds of several models found in the literature.</p>