Shock and Vibration

Volume 2017, Article ID 3979384, 13 pages

https://doi.org/10.1155/2017/3979384

## An Electromechanical Pendulum Robot Arm in Action: Dynamics and Control

Laboratory of Modelling and Simulation in Engineering, Biomimetics and Prototypes, Department of Physics, Faculty of Science, University of Yaoundé I, P.O. Box 812, Yaoundé, Cameroon

Correspondence should be addressed to Paul Woafo; rf.oohay@1ofaowp

Received 22 June 2017; Revised 18 October 2017; Accepted 8 November 2017; Published 5 December 2017

Academic Editor: Francesco Pellicano

Copyright © 2017 A. Notué Kadjie et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The authors numerically investigate the dynamics and control of an electromechanical robot arm consisting of a pendulum coupled to an electrical circuit via an electromagnetic mechanism. The analysis of the dynamical behavior of the electromechanical device powered by a sinusoidal power source is carried out when the effects of the loads on the arm are neglected. It is found that the device exhibits period-n T oscillations and high amplitude oscillations when the electric current is at its smallest value. The specific case which considers the effects of the impulsive contact force caused by an external load mass pushed by the arm is also studied. It is found that the amplitude of the impulse force generates several behaviors such as jump of amplitude and distortions of the mechanical vibration and electrical signal. For more efficient functioning of the device, both piezoelectric and adaptive backstepping controls are applied on the system. It is found that the control strategies are able to mitigate the signal distortion and restore the dynamical behavior to its normal state or reduce the effects of perturbations such as a short time variation of one component or when the robot system is subject to noises.

#### 1. Introduction

Pendulum motion-driven systems have been intensively studied recently by both industries and research institutes because of their applications in different fields [1–10]. Some of these studies concern the analysis of the dynamical states and the development of control strategies to stabilize the dynamical state to a prescribed state. These pendulum models comprise the downward pendulum [5], horizontal pendulum [6], inverted pendulum [7], spherical pendulum [8], the flexible pendulum [9, 10], the pendulum excited by an RLC circuit based on nonlinear shaker [11], and rotating pendulum [12]. When the pendulum is coupled to an electrical part (electromechanical pendulum), its applications with and without control are more interesting in robotics and other fields of engineering. This is due to some particular dynamical states (periodic, quasiperiodic, and chaotic states) that the electromechanical pendulum can generate because of intrinsic angular nonlinearity or due to natural or imposed nonlinearities in the electrical part [5, 9–12].

The working state of a system with particular dynamics can be modified because of the interaction with its environment or the application of some constraints or control laws. In this line, recent years have seen the development of various control strategies applied on electrical, mechanical, electromechanical, and even biological systems: some examples are the adaptive control [13], active control [14], the classical and active-backstepping controls [15, 16], and the sliding mode control [17]. An interesting contribution dealing with chaos control of a double pendulum arm powered through an RLC circuit is reported in [11] where the authors used the state-dependent Riccati equation control and the nonlinear saturation control techniques to suppress chaos in the dynamics of the double pendulum arm. Due to its importance for engineering and robotic applications, the control of pendulum motion has been intensively studied using various approaches, including passivity-based control [18], nonlinear control [19, 20], sliding mode control [21], motion control of two pendulums [22], and bifurcation control [23].

In this work, the dynamics and the control of an electromechanical pendulum with rigid and constant length are studied. The pendulum is coupled to an electrical part through an electromagnetic link. The dynamics considers the effect of a periodic impulsive force due to the instantaneous shock between the pendulum arm and external load masses arriving periodically. This is described mathematically by a pulse-like excitation added to the initial sinusoidal electrical excitation. In terms of the load mass, the critical electrical signal amplitude leading to the displacement of the mass is evaluated. In view of optimizing the action of the pendulum arm by counterbalancing the collision effects due to the arriving loads, a pulse-like activation signal acts periodically on the pendulum arm. Finally, an adaptive backstepping method, based on the automatic variation of an intrinsic parameter, has been developed either to control the perturbations caused by the collision of the pendulum with the load masses or to counterbalance the disturbances generated by unwanted temporal variations of some parameters of the robot device.

The work is structured as follows. Section 2 describes the electromechanical pendulum robot arm and analyzes the dynamics of the arm when the action of the load is neglected. Section 3 considers the situation where the effect of the periodic actions of the load is taken into account. In order to optimize the working conditions of the robot arm, a control strategy consisting of sending pulse-like signals following the detection of the load arrival is also considered in this section. Because the device can be subject to an unknown time variation (of regular or stochastic nature) of some of its parameters, a backstepping adaptive method is used in Section 4 to reduce the effects of these perturbations on the system. Finally, Section 5 concludes the work.

#### 2. Description, Modelling, and Dynamics of the Electromechanical Pendulum with Negligible Interaction with the Moving Mass

Figure 1 shows the electromechanical robot arm. It is constituted of a pendulum with a rod of length and the spherical proof mass (,), coupled electromagnetically to an electrical circuit. The electrical part of the system is constituted of a coil wired around an iron core rigidly fixed on the pendulum over a length of the pendulum whose rotation axis passes at the point O. is a constant indicating the proportion of the coiled rod length inside the magnetic field.