Balance-bot

Research on inverted pendulum has gained momentum over the last decade on a number of robotic laboratories over the world; due to its unstable proprieties is a good example for control engineers to verify a control theory. To verify that the pendulum can balance we can make some simulations using a closed-loop controller method such as the linear quadratic regulator or the proportional–integral–derivative method. Also the idea of robotic teleoperation is gaining ground. Controlling a robot at a distance and doing that precisely. However, designing the tool to takes the best benefit of the human skills while keeping the error minimal is interesting, and due to the fact that the inverted pendulum is an unstable system it makes a compelling test case for exploring dynamic teleoperation. Therefore this thesis focuses on the construction of a two-wheel inverted pendulum robot, which sensor we can use to do that, how they must be integrated in the system and how we can use a human to control an inverted pendulum. The inverted pendulum robot developed employs technology like sensors, actuators and controllers. This Master thesis starts by presenting an introduction to inverted pendulums and some information about related areas such as control theory. It continues by describing related work in this area. Then we describe the mathematical model of a two-wheel inverted pendulum and a simulation made in Matlab. We also focus in the construction of this type of robot and its working theory. Because this is a mobile robot we address the theme of the teleoperation and finally this thesis finishes with a general conclusion and ideas of future work.

Nonautonomous difference equations with applications

This work is divided in two parts. In the first part we develop the theory of discrete nonautonomous dynamical systems. In particular, we investigate skew-product dynamical system, periodicity, stability, center manifold, and bifurcation. In the second part we present some concrete models that are used in ecology/biology and economics. In addition to developing the mathematical theory of these models, we use simulations to construct graphs that illustrate and describe the dynamics of the models. One of the main contributions of this dissertation is the study of the stability of some concrete nonlinear maps using the center manifold theory. Moreover, the second contribution is the study of bifurcation, and in particular the construction of bifurcation diagrams in the parameter space of the autonomous Ricker competition model. Since the dynamics of the Ricker competition model is similar to the logistic competition model, we believe that there exists a certain class of two-dimensional maps with which we can generalize our results. Finally, using the Brouwer’s fixed point theorem and the construction of a compact invariant and convex subset of the space, we present a proof of the existence of a positive periodic solution of the nonautonomous Ricker competition model.