Abstract
In this paper a robotic arm is modelled by a double pendulum excited in its base by a DC motor of limited power via crank mechanism and elastic connector. In the mathematical model, a chaotic motion was identified, for a wide range of parameters. Controlling of the chaotic behaviour of the system, were implemented using, two control techniques, the nonlinear saturation control (NSC) and the optimal linear feedback control (OLFC). The actuator and sensor of the device are allowed in the pivot and joints of the double pendulum. The nonlinear saturation control (NSC) is based in the order second differential equations and its action in the pivot/joint of the robotic arm is through of quadratic nonlinearities feedback signals. The optimal linear feedback control (OLFC) involves the application of two control signals, a nonlinear feedforward control to maintain the controlled system to a desired periodic orbit, and control a feedback control to bring the trajectory of the system to the desired orbit. Simulation results, including of uncertainties show the feasibility of the both methods, for chaos control of the considered system.
Highlights
We define the desired periodic orbits obtained with the nonlinear saturation control Fig. 8a through the use of Fourier series, calculated numerically as:
We consider a robotic arm modeled by a double pendulum excited in its base by a DC motor of limited power via a crank mechanism and a spring
An investigation of the nonlinear dynamics and chaos was carried out based on this model
Summary
Systems with pendulums have important applications. An auto parametric non-ideal system with pendulum was studied by [1], and an auto parametric system with two pendulums harmonically excited was studied by [2]. The first detailed study on non-ideal vibrating systems was done by [3]. The problem of non-ideal vibrating systems has been investigated by a number of authors. A complete review of different theories on non-ideal vibrating systems is to be found in [4]. The effectiveness of the nonlinear saturation control in vibration attenuation for a non-ideal portal frame was investigated by [6]. The Optimal Linear Feedback Control was proposed by [7]. [8] formulated the linear feedback control strategies for nonlinear systems, asymptotic stability of the closed-loop nonlinear system, guaranteeing both stability and optimality.
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