Elsevier

Automatica

Volume 47, Issue 3, March 2011, Pages 515-524
Automatica

A method for guidance and control of an autonomous vehicle in problems of border patrolling and obstacle avoidance

https://doi.org/10.1016/j.automatica.2011.01.024Get rights and content

Abstract

We present a sliding mode based method for a unicycle-like vehicle control and guidance. The proposed guidance-control law is applied to the problems of border patrolling and obstacle avoidance. A mathematically rigorous analysis of this law is provided.

Introduction

Unmanned aerial and ground vehicles (UAV/UGVs) have been used greatly in the recent past for a variety of applications in hazardous and complex environments or plenary surveillance mainly due to their light weight, inexpensive components and low power consumption, see e.g. Ahmadzadeh, Keller, Pappas, Jadbabaie, and Kumar (2006), Girard, Howell, and Hedrick (2004), Quigley, Barber, Griffiths, and Goodrich (2005), and Wang, Steiber, and Surampudi (2008) and references therein. Such applications often involve limitations on communications that require the vehicle to operate autonomously for extended periods of time and distance. In these situations, unmanned vehicles should be equipped with a control system by which they can move autonomously and safely operate in populated environments. This paper addresses the problem of control of an autonomous vehicle moving along a border or an obstacle surface with a predefined margin. Such a problem arises, for example, in border surveillance with a UAV equipped with a fixed camera (Girard et al., 2004) or in bypassing obstacles in close range with a safety margin (Minguez & Montano, 2002). The kinematics of the vehicle under consideration are described by the standard model of the Dubins car type, i.e. a nonholonomic system moving along planar paths of bounded curvature without reversing direction (Dubins, 1957). In the literature, this model is applied to many mechanical systems such as wheeled robots, UAVs, missiles and autonomous underwater vehicles, which in certain circumstances, have similar governing differential equations; see e.g. Ben-Asher and Yaesh (1998), Fossen (1994), Low, Manchester, and Savkin (2007), and Manchester and Savkin (2006) and references therein. The use of UAVs to improve the border security has gained a lot of attention over the past few years (Bolkcom and Nuñez-Neto, 2006, Caveney and Sengupta, 2005, Girard et al., 2004). UAVs can be utilized to greatly improve the border surveillance over remote areas. They have quicker responses and can provide real-time information to the ground station operator. Some kinds of UAVs can fly over 30 h without refueling, which is considerable compared with just over 2 h average flight time for the helicopters (Bolkcom & Nuñez-Neto, 2006). The ability of loitering for such a long time enables UAVs to cover greater and further areas, which was not possible by means of the conventional manned aircraft. Border patrolling applications are referred to those missions in which an aircraft (or a group of aircraft) is supposed to fly along the border of a region and visually monitor it with a mounted camera. The constant visual coverage achieves when the aircraft flies autonomously over a region of interest, which could be on the top of border or close to it with a predefined margin (Lee, Huang, Vaughn, Xiao, & Hedrick, 2003).

In order to operate in a cluttered environment, on the other hand, an autonomous ground vehicle should be able to detect and avoid the obstacles along the trajectory towards the target. Current guidance approaches can be generally classified as global or local path planners (Lapierre, Zapata, & Lepinay, 2007). Global sensor-based planners use the a priori and sensory information to build a complete model of the environment and then try to find the best possible solution (Belkhous et al., 2005, Vlassis et al., 1996). To generate a trajectory, on the other hand, local path planners use onboard sensors to locally observe a small fraction of an unknown environment at each time (Deng et al., 2007, Lapierre et al., 2007, Teimoori and Savkin, 2010). The short calculation time in these strategies allows them to be applied in real-time guidance systems. Another example for local path planners, which is alike in flavor to our approach, is the biologically inspired DistBug algorithm (Kamon & Rivlin, 1997), which is a member of Bugs family approaches (Kamon et al., 1991, Lumelsky and Skewis, 1990, Lumelsky and Stepanov, 1987) motivated by a bug’s behavior on crawling along a wall. Similar to ours, in the DistBug algorithm the vehicle directly travels towards the target and bypasses the enroute obstacles by following their boundaries in close range. In practice, however, one problem is that these strategies are all heuristics based and kinematic equations of the vehicles and their nonholonomic constraints are not considered in these algorithms, which is a severe limitation.

In this paper, we propose an integrated guidance-control strategy by which an autonomous vehicle moving with a constant linear velocity can patrol the border of a region in a close range. The proposed control strategy originates from the equiangular navigation and guidance law (Teimoori & Savkin, 2010) and belongs to the class of sliding mode control algorithms; see e.g. Utkin (1992). The only information available to the vehicle is the minimum distance to the border. Furthermore, with a slight modification, we apply the proposed strategy in the problem of safe guidance of an autonomous ground vehicle towards a target while bypassing the enroute obstacles with a predefined safety margin. Unlike many other papers in the area of robotic guidance with obstacle avoidance, mathematically rigorous analysis and justification of the proposed strategy are offered. In particular, we prove that with our guidance-control law, the distance between the vehicle and the border tends to a predetermined value and the safety requirement is always satisfied. Notice that we solve the border patrolling problem for a single vehicle, whereas some other papers in the area use multiple vehicles; see e.g. Girard et al. (2004). The applicability of the proposed integrated guidance-control law in both border patrolling and obstacle avoidance problems is confirmed by extensive computer simulations.

The remainder of the paper is organized as follows. Section 2 presents the statement of the border patrolling problem and the proposed control law. Section 3 offers the basic definitions and discusses the main assumptions. Mathematically rigorous analysis of the control strategy for the border patrolling problem is presented in Section 4. Section 5 introduces an algorithm for guidance towards a target while avoiding the enroute obstacles and presents its mathematically rigorous analysis. Illustrative examples and computer simulations for both border patrolling and obstacle avoidance problems are given in Section 6, whereas Section 7 offers brief conclusions. The proofs of all lemmas and theorems are given in Appendix.

Section snippets

System description and border patrolling problem

We consider a planar vehicle or wheeled mobile robot modeled as a unicycle. It travels with a constant speed and is controlled by the angular velocity limited by a given constant. There is a domain D in the plane. The control objective is to patrol the border D of this domain at the given distance d0 from it and at the given speed v. The vehicle has access to the current distance d(t) to the border and the speed ḋ(t) at which the vehicle approaches the border (ḋ(t)0) or runs away from it (ḋ

Main assumptions

We start with a natural technical requirement.

Assumption 1

The domain D is closed and has a smooth boundary D.

At the same time, we do not assume that D is convex.

Let κ(r) and Rκ(r) stand for the signed curvature of the boundary D at the point rD and the (unsigned) curvature radius Rκ(r)|κ(r)|1. Here κ(r)>0 for convexity points, κ(r)<0 for concavity points, and 01+ (see Fig. 2).

Definition 1

A point rD is said to be multiple if there exist two different minimum-distance points r,rD, i.e., such that distD(r)=

Control for border patrolling

Now we are in a position to state the main result concerning the border patrolling problem.

Theorem 3

Let Assumption 1, Assumption 4 hold and the parameters γ>0 and δ>0 of the control rule (4), (5) be chosen so that the following inequality is true, where λ is the constant related to the interval from Assumption 4 by Definition 2,1+γ2(1λ)2u¯2<μvδγ(=(5)vv).Then the control law (4) asymptotically drives the vehicle at the desired distance to D, i.e., d(t)td0 . During this motion, the safety

Guidance with obstacle avoidance

Now we consider a more general problem of vehicle guidance with obstacle avoidance. There is a steady point target T and several disjoint obstacles D1,,Dk in the plane. The objective is to drive the vehicle to the target through the obstacle-free part of the plane R2{D1Dk}.

Assumption 5

Every obstacle Di is a closed, bounded, and convex set with a smooth boundary Di.

The following definition is illustrated in Fig. 6.

Definition 4

For C>0, the C-neighborhood of the domain DR2 is the set formed by all points at the

Simulations

To illustrate the performance of the control law (4), we consider autonomous vehicles governed by kinematics equations (1) and moving along planar paths with constant linear velocities. Two scenarios are considered to demonstrate applicability of the proposed law in the border patrolling and obstacle avoidance problems. For simulation and testing, we used Matlab and Mobotsim 1.0 simulator, which is a 2D easy to use graphical mobile robot simulator (M.R.S. Mobotsim 1.0, 2008).

Conclusions

A sliding mode based method for a unicycle-like vehicle control has been presented. The proposed vehicle guidance-control law was applied to the problems of border patrolling and obstacle avoidance. We have presented a mathematically rigorous analysis of this law. The efficiency of the proposed algorithm in border patrolling and obstacle avoidance problems has been illustrated by computer simulations.

Alexey S. Matveev was born in Leningrad, USSR, in 1954. He received the M.S. and Ph.D. degrees in 1976 and 1980, respectively, both from the Leningrad University. Currently, he is a professor of the Department of Mathematics and Mechanics, Saint Petersburg University. His research interests include estimation and control over communication networks, hybrid dynamical systems, and navigation and control of mobile robots.

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    Alexey S. Matveev was born in Leningrad, USSR, in 1954. He received the M.S. and Ph.D. degrees in 1976 and 1980, respectively, both from the Leningrad University. Currently, he is a professor of the Department of Mathematics and Mechanics, Saint Petersburg University. His research interests include estimation and control over communication networks, hybrid dynamical systems, and navigation and control of mobile robots.

    Hamid Teimoori was born in 1978 in Iran. He received his B.Sc. degree (Hons 1) in 2000 from Ferdowsi University, the M.Sc. degree (2002) from Iran University of Science and Technology, Tehran, Iran and the Ph.D. degree (2009) in Electrical Engineering from the University of New South Wales (UNSW), Sydney, Australia. From 2002 to 2005, he worked as an associate lecturer in Imam-Reza University, Iran. He is currently a research associate at the University of New South Wales. His research interests currently center on nonholonomic systems and navigation and guidance of wheeled robots and unmanned aerial vehicles.

    Andrey V. Savkin was born in 1965 in Norilsk, USSR. He received the M.S. and Ph.D. degrees from the Leningrad State University, USSR in 1987 and 1991, respectively. From 1987 to 1992, he was with the Television Research Institute, Leningrad, USSR. From 1992 to 1994, he held a postdoctoral position in the Department of Electrical Engineering, Australian Defence Force Academy, Canberra. From 1994 to 1996, he was a Research Fellow with the Department of Electrical and Electronic Engineering and the Cooperative Research Center for Sensor Signal and Information Processing at the University of Melbourne, Australia. In 1996–2000, he was a Senior Lecturer, and then an Associate Professor with the Department of Electrical and Electronic Engineering at the University of Western Australia, Perth. Since 2000, he has been a Professor with the School of Electrical Engineering and Telecommunications at the University of New South Wales, Sydney, Australia.

    His current research interests include robust control and state estimation, hybrid dynamical systems, guidance, navigation and control of mobile robots, applications of control and signal processing in biomedical engineering and medicine.

    He has authored/co-authored five research monographs and numerous journal and coneference papers on these topics. He has served as an associate editor for several international journals.

    This work was supported by the Australian Research Council, the Russian Foundation for Basic Research, grants 08-01-00775 and 09-08-00803, and the Russian program 16.740.11.0042. The material in this paper was partially presented at the American Control Conference (ACC), June 30–July 2, 2010, Baltimore, Maryland, USA and the 49th IEEE Conference on Decision and Control, December 15–17, 2010, Atlanta, GA, USA. This paper was recommended for publication in revised form by Associate Editor Yoshikazu Hayakawa under the direction of Editor Toshiharu Sugie.

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