Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 728412, 10 pages

http://dx.doi.org/10.1155/2015/728412

## Robust Tracking and Cruise Control of a Class of Robotic Systems

^{1}Departamento de Electrónica y Telecomunicaciones, CICESE, 22860 Ensenada, BCN, Mexico^{2}Facultad de Ingeniería, Arquitectura y Diseño, UABC, 22860 Ensenada, BCN, Mexico

Received 31 October 2014; Revised 20 January 2015; Accepted 21 January 2015

Academic Editor: Luis Rodolfo Garcia Carrillo

Copyright © 2015 Ricardo Cuesta 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

This paper presents a controller for a class of robotic systems, based on a first-order sliding mode with a particular noninvariant, nonconnected surface. With this control it is possible to regulate the position such that the velocity remains, as long as possible, at a specified value until the system is close to the desired position. The properties inherited from the sliding modes make the control exhibit a high robustness to external perturbations and low sensitivity to system parameter variations. It is shown that the desired speed is reached in a finite time and the system converges exponentially to the desired position. This controller can be applied to systems described by a classical model of a fully actuated, *n*-DOF mechanical system, which could be decoupled via a preliminary decoupling control. To illustrate the theoretical results, the proposed control technique is applied to a Cartesian robot, simulated numerically. Moreover, to show the effectiveness of this strategy, some physical experiments on a rotational (mechanical) device were performed.

#### 1. Introduction

Sliding mode control (SMC) is a nonlinear control technique that, by using a discontinuous control signal, forces the system to move along a previously designed switching surface [1]. SMC has attired a great interest because its implementation is simple and it exhibits high robustness to external perturbations and exhibits low sensitivity to parameter variations of the system. These advantages has made SMC very popular in applications, including electrical, mechanical, chemical, industrial, civil, military, aeronautical, and aerospace engineering [2], among others.

Usually, the sliding surface is represented by a linear function of the state. A typical design makes this surface invariant, containing the control target which is made asymptotically stable with a suitable choice of the controller parameters. This has been enough to regulate, for example, the position of mechanical devices. On the other hand, there exist control algorithms that provide a satisfactory performance to regulate the velocity, for example, keeping a constant velocity in a motor shaft [3, 4].

Combination of both goals, that is, attaining a constant state while keeping a constant velocity in the most part of the displacement stage, is, however, a more complicated goal analyzed less frequently. This double control objective is easy to associate in aeronautics, where the aircraft arrives to a point B from a point A using three stages: takeoff (velocity increases until the aircraft reaches the desired constant velocity), cruise speed (keeping the aircraft in the desired constant velocity), and landing (velocity decreases until the aircraft reaches the final destination or desired position). Some advantages of cruise speed in aircraft are as follows: there is a high decrease in fuel consumption, the aerodynamics levels produce less pressure or less force under the thrust resistance, the motors work in optimal levels of temperature, pressure, and wear, increasing the motor life, because they are not submitted to high efforts [5], and so on. These characteristics are also present in land vehicles, submarine vehicles, and ships.

There are many other applications where it is useful that the vehicle kept a constant velocity before arriving to a fixed desired position. Some examples in aerial transport are helicopters, planes, satellites, spaceships, missiles and, in recent years, unmanned aerial vehicles (UAV). Moreover, there are ships, trains, submarines, telescopes, elevators, robots, mobile robots, conveyors, corrosive liquid transportation, and molten metal transportation that do not admit abrupt changes in speed.

There are many works, like [6–10], where the main control objective is to maintain a desired altitude position in planes, satellites, spaceships, missiles, and UAV, respectively, without imposing a speed behavior. In other works like [11–13], the main control objective is to keep a desired velocity (cruise speed) in ships, trains, and telescopes, respectively, without considering a final position strategy.

In the present paper we propose a control law, based on a first-order sliding mode with a particular noninvariant, nonconnected surface, for a class of robotic systems. The shape of this surface produces indirectly a system behavior which switches from a first- to a second-order sliding mode. First, the control forces the system to attain the noninvariant surface, keeping a constant, desired speed while the system moves on this surface. Later, once the system is close to the desired position, it leaves the sliding surface to converge to this position through a second-order sliding mode. With this proposed control it is possible to solve the combined objectives described previously, that is, position regulation, such that the velocity remains, as long as possible, at a specified value until the system is close to the desired position.

The properties inherited from the sliding modes make the proposed control exhibit a high robustness to external perturbations and low sensitivity to system parameter variations. It is shown that the desired velocity is reached in a finite time and the system converges exponentially to the desired position. This control algorithm can be applied to robotic systems described by a classical model of a fully actuated, -DOF mechanical system, which could be decoupled via a preliminary decoupling control.

To illustrate the theoretical results, the proposed control technique is applied to a Cartesian robot, simulated numerically. Moreover, to show the effectiveness of this strategy, some physical experiments on a rotational (mechanical) device were performed.

The paper is organized as follows. In Section 2, the basic system and controller are described. Section 3 presents the main result and the formal demonstration of stability. Section 4 illustrates the performance of the proposed controller through numerical and experimental implementations. Finally, some conclusions and remarks are presented in Section 5.

#### 2. A Controlled Mechanical System

Let us consider an -DOF mechanical system described bywhere is the symmetric, positive definite inertia matrix for all , is the matrix of centrifugal and Coriolis forces, is the vector of gravitational forces, and is the vector of external forces. Ifwhere , is a new control input, and is a diagonal, positive definite matrix, and then, after the application of this control input, system (1) becomesThis system can be seen as a set of decoupled, second-order systems with the formObviously, a similar form can be obtained when there is no gravity effect; that is, , and .

In a general and simple way, airplanes, projectiles, ships, and terrestrial vehicles can be modeled like a simple mass in movement, using Newton’s second law of motion, with an opposition force caused by the viscous friction, which is proportional to the mass velocity. This is a particular case of (1), which can be easily transformed to (4), for which we can employ the following state equations:where, for simplicity, we have dropped the index , , and .

Equation (5) can describe also the dynamical behavior of other mechanical devices, for example, a motor shaft with viscous friction, where the moment of inertia is used instead of the mass, and the position and velocity are angular variables. This behavior is similar in propellers, turbines, drilling machines, and other similar torsional mechanisms. In summary, system (5) describes a fundamental dynamics of a wide variety of mechanical systems.

In what follows we propose a control law for system (5) and analyze the stability properties of the closed-loop system, and then we apply this control strategy to several systems like (1).

Let us consider the sign function, which is defined in the Filippov sense aswhere . Moreover, define a discontinuous function, denoted by and described byNow, consider a control law, applied to system (5), given bywhere is defined aswhere and and are positive real constants. The closed-loop system, formed by the system (5) and the control law (8), is given by

The state space is split in two disjoint sets, and , characterized by the sign of , described by (see Figure 1) and (denoted as ) are delimited by another region, denoted by , which we call the commutation curve, given bywhere