Two Efficient Time-Marching Explicit Procedures Considering Spatially/Temporally-Defined Adaptive Time-Integrators

In this paper, two explicit time-marching techniques are discussed for the solution of hyperbolic models, which are based on adaptively computed parameters. In both these techniques, time integrators are locally and automatically evaluated, taking into account the properties of the spatially/temporally discretized model and the evolution of the computed responses. Thus, very versatile solution techniques are enabled, which allows computing highly accurate responses. Here, the so-called adaptive [Formula: see text] method is formulated based on the elements of the adopted spatial discretization (elemental formulation), whereas the so-called adaptive [Formula: see text] method is formulated based on the degrees of freedom of the discretized model (nodal formulation). In this context, each adaptive procedure can be better applied according to the specific features of the focused spatial discretization technique. At the end of the paper, numerical results are presented, illustrating the excellent performance of the discussed adaptive formulations.

The Role of Visibility in Pursuit/Evasion Games

The cops-and-robber (CR) game has been used in mobile robotics as a discretized model (played on a graph G) of pursuit/evasion problems. The “classic” CR version is a perfect information game: the cops’ (pursuer’s) location is always known to the robber (evader) and vice versa. Many variants of the classic game can be defined: the robber can be invisible and also the robber can be either adversarial (tries to avoid capture) or drunk (performs a random walk). Furthermore, the cops and robber can reside in either nodes or edges of G. Several of these variants are relevant as models or robotic pursuit/evasion. In this paper, we first define carefully several of the variants mentioned above and related quantities such as the cop number and the capture time. Then we introduce and study the cost of visibility (COV), a quantitative measure of the increase in difficulty (from the cops’ point of view) when the robber is invisible. In addition to our theoretical results, we present algorithms which can be used to compute capture times and COV of graphs which are analytically intractable. Finally, we present the results of applying these algorithms to the numerical computation of COV.

The Role of Visibility in Pursuit/Evasion Games

The cops-and-robber (CR) game has been used in mobile robotics as a discretized model (played on a graph G) of pursuit/evasion problems. The “classic” CR version is a perfect information game: the cops’ (pursuer’s) location is always known to the robber (evader) and vice versa. Many variants of the classic game can be defined: the robber can be invisible and also the robber can be either adversarial (tries to avoid capture) or drunk (performs a random walk). Furthermore, the cops and robber can reside in either nodes or edges of G. Several of these variants are relevant as models or robotic pursuit/evasion. In this paper, we first define carefully several of the variants mentioned above and related quantities such as the cop number and the capture time. Then we introduce and study the cost of visibility (COV), a quantitative measure of the increase in difficulty (from the cops’ point of view) when the robber is invisible. In addition to our theoretical results, we present algorithms which can be used to compute capture times and COV of graphs which are analytically intractable. Finally, we present the results of applying these algorithms to the numerical computation of COV.

Stability and bifurcations in a discrete-time epidemic model with vaccination and vital dynamics

Background The spread of infectious diseases is so important that changes the demography of the population. Therefore, prevention and intervention measures are essential to control and eliminate the disease. Among the drug and non-drug interventions, vaccination is a powerful strategy to preserve the population from infection. Mathematical models are useful to study the behavior of an infection when it enters a population and to investigate under which conditions it will be wiped out or continued. Results A discrete-time SIS epidemic model is introduced that includes a vaccination program. Some basic properties of this model are obtained; such as the equilibria and the basic reproduction number $$\mathcal {R}_0$$ R 0 . Then the stability of the equilibria is given in terms of $$\mathcal {R}_0$$ R 0 , and the bifurcations of the model are studied. By applying the forward Euler method on the continuous version of the model, a discretized model is obtained and analyzed. Conclusion It is proven that the disease-free equilibrium and endemic equilibrium are stable if $$\mathcal {R}_0<1$$ R 0 < 1 and $$\mathcal {R}_0>1$$ R 0 > 1 , respectively. Also, the disease-free equilibrium is globally stable when $$\mathcal {R}_0\le 1$$ R 0 ≤ 1 . The system has a transcritical bifurcation when $$\mathcal {R}_0=1$$ R 0 = 1 and it might also have period-doubling bifurcation. The sufficient conditions for the stability of equilibria in the discretized model are established. The numerical discussions verify the theoretical results.

Stability and Bifurcations in a Discrete-Time Epidemic Model with Vaccination and Vital Dynamics

BackgroundThe spread of infectious diseases is such important that changes the demography of the population. Therefore, prevention and intervention measures are essential to control and eliminate the disease. Among the drug and non-drug interventions, vaccination is a powerful strategy to preserve the population from infection. Mathematical models are useful to study the behavior of an infection when it enters a population and investigate under which conditions it will be wiped out or continued.ResultsA discrete-time SIS epidemic model is introduced that includes a vaccination program. Some basic properties of this model are obtained; such as the equilibria and the basic reproduction number $\mathcal{R}_0$ . Then the stability of the equilibria is given in terms of $\mathcal{R}_0$ , and moreover, the bifurcations of the model are studied. By applying the forward Euler method on the continuous version of model, a discretized model is obtained and analyzed. ConclusionIt is proved that the disease-free equilibrium and endemic equilibrium are stable if $\mathcal{R}_0<1$ and $\mathcal{R}_0>1$ , respectively. The system has a transcritical bifurcation when $\mathcal{R}_0=1$ and it might also have period-doubling bifurcation. The sufficient conditions for the stability of equilibria in the discretized model are established. The numerical discussions verify the theoretical results.

TMTDyn: A Matlab package for modeling and control of hybrid rigid–continuum robots based on discretized lumped systems and reduced-order models

A reliable, accurate, and yet simple dynamic model is important to analyzing, designing, and controlling hybrid rigid–continuum robots. Such models should be fast, as simple as possible, and user-friendly to be widely accepted by the ever-growing robotics research community. In this study, we introduce two new modeling methods for continuum manipulators: a general reduced-order model (ROM) and a discretized model with absolute states and Euler–Bernoulli beam segments (EBA). In addition, a new formulation is presented for a recently introduced discretized model based on Euler–Bernoulli beam segments and relative states (EBR). We implement these models in a Matlab software package, named TMTDyn, to develop a modeling tool for hybrid rigid–continuum systems. The package features a new high-level language (HLL) text-based interface, a CAD-file import module, automatic formation of the system equation of motion (EOM) for different modeling and control tasks, implementing Matlab C-mex functionality for improved performance, and modules for static and linear modal analysis of a hybrid system. The underlying theory and software package are validated for modeling experimental results for (i) dynamics of a continuum appendage, and (ii) general deformation of a fabric sleeve worn by a rigid link pendulum. A comparison shows higher simulation accuracy (8–14% normalized error) and numerical robustness of the ROM model for a system with a small number of states, and computational efficiency of the EBA model with near real-time performances that makes it suitable for large systems. The challenges and necessary modules to further automate the design and analysis of hybrid systems with a large number of states are briefly discussed.

Optimal Control for Footbridges’ Vibration Reduction Based on Semiactive Control Through Magnetorheological Dampers

A footbridge is a structure designed for pedestrians or animals to cross roads, water or railways, safely. Modern ones are designed as slender and light structures to be more aesthetic and economic, but may lack enough stiffness and damping that might produce excessive vibrations under service conditions, overpassing comfort limits for users and compromise structural integrity. This work presents the synthesis of a nonlinear optimal control strategy for reducing vibrations in footbridges by means of using magnetorheological dampers. The proposed optimal controller considers both, the footbridge linear dynamics and the damper nonlinear dynamics, as the complete system to be controlled. For analysis purposes, the continuous structure of a footbridge is conveniently idealized as an [Formula: see text]-degrees-of-freedom discretized model, such that it can be handled as an [Formula: see text]-order system. Parameters from an actual footbridge are used to propose a discretized model system of 11 translational degrees of freedom and to analyze the system response as a case study. The dynamical response involves displacement, velocity and acceleration for different number of pedestrians crossing in groups. The investigation rests on comparing the structural response over time for two different conditions: with no control device installed and with one magnetorheological damper installed at the span center. Results obtained with the use of the proposed optimal controller show to be an effective way of reducing the structural motion response.