An asymptotically stable collision-avoidance system

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Abstract

Artificial potential fields, which are widely used in robotics for path planning and collision avoidance, are normally beset by difficulties arising from the existence of local minima. This article proposes a solution that involves an asymptotically stable point-mass system governed by differential equations. The system represents a planar point robot moving from its initial position to the desired goal whilst avoiding a static obstacle. Because the system is asymptotically stable, its Lyapunov function, which produces artificial potential fields around the goal and the obstacle, has no local minima other than the goal configuration in the pathwise-connected proper subset of free space which contains the goal configuration. As an application, we consider the point stabilization of a planar mobile car-like robot moving in the presence of a static obstacle.

Introduction

An ongoing research in robotics involves the identification in a two- or three-dimensional space a continuous path that allows a robot, or a part of it, to reach its destination without colliding with obstacles that may exist in the space. Sometimes referred to as the findpath problem, it is essentially a geometric problem.

Having being analyzed over the last two decades by many researchers, it is now possible to gather a majority of the proposed solutions under two categories: (1) those that employ some kind of graph search technique and (2) those that employ some kind of physical analogy.

In a graph search technique, a collision-free path is generated by searching a graph formed out of straight lines that connect the starting position and the destination via the vertices of solid obstacles, or via patches of free space that have been decomposed into geometric primitives such as cones and cylinders. Some of the pioneering work include those in 1983 by Schwartz and Sharir [1], Brooks [2], and Lozano-Pérez [3], and that in 1986 by Herman [4]. Relatively recent applications include those by Lam and Srikanthan [5], and Williams and Jones [6] in 2001, Kruusmaa [7] and Sacks [8] in 2003, Roy [9] in 2005 and Zeghloul [10] in 2006. Theoretically, graph search techniques are elegant. However, they could involve computationally intensive algorithms. Simpler algorithms tend to use physical analogies to establish artificial potential fields with repulsive poles around obstacles and attractive poles around goals. A collision-free path is determined by how much the robot is attracted to or repelled by the poles. Pioneers in this area include Khatib [11] in 1986, Connolly et al. [12] in 1990, and Tarassenko and Blake [13], and Kim and Khosla [14] in 1991. Recent extensions and applications include those by Ge and Cui [15] in 2002, Tanner et al. [16] in 2003 and Lin et al. [17] in 2004. In any artificial potential fields method, it is a challenge, however, to construct potential fields that do not have local minima or points of zero potential and kinetic energy other than the goal configuration. Several studies have successfully considered this problem via the use of special functions. The work of Rimon and Koditschek [18] in 1992 with potential functions, and that of Tanner et al. [16] in 2003 with dipolar inverse Lyapunov functions, are noteworthy. An excellent summary of the various methods associated with artificial potential fields can be found in the work of Lee [19] in 2004.

In this article, our intention is to show the viability of directly using the well-known second method of Lyapunov to construct a Lyapunov function that ensures the asymptotic stability of an obstacle-avoidance system, and hence solves the problem of local minima. The main advantage of this global potential approach [19] is the ease in which it can be used to extract control laws. The proposed technique is based on the 1990 pioneering work of Stonier [20], which was later expanded and improved in 1998 and 2001 by Vanualailai et al. [21] and Ha and Shim [22], respectively. In this paper, we consider a planar obstacle-avoidance system governed by differential equations. The system consists of a point-mass being controlled to its destination or target whilst avoiding a fixed object in two-dimensional space. The proposed Lyapunov function for the system produces artificial potential fields both for obstacle-avoidance and for target attraction. After establishing Lyapunov stability, we then show that it is possible to identify a region of asymptotic stability in which the target is the only minimum point. As an application, we consider the point stabilization of planar mobile robot, which is car-like and non-holonomic.

Section snippets

The Lyapunov method

Here, we briefly recall some of the important Lyapunov stability concepts that we will be using to derive our control laws.

Let Rn be the n-dimensional Euclidean space with the Euclidean norm ·. Let x=(x1,x2,,xn) denote an element of Rn. Consider an autonomous non-linear systemx˙=f(x),x(t0)=x0,t00,where f:ΩRnRn is assumed to be smooth enough to guarantee the existence, uniqueness and continuous dependence of solutions x(t)=x(t;t0,x0) of (1) in Ω, an open set in Rn.

For the purpose of

A globally asymptotically stable point-mass system

Consider a point-mass, defined as the disk of radius rP0, and positioned at (x(t),y(t))R2 at time t0. That is, the point-mass isP={(z1,z2)R2:(z1-x)2+(z2-y)2rP2}.Its instantaneous velocity is (v(t),w(t))(x˙(t),y˙(t)). Our general ODE system is therefore of the formx˙(t)=v(x(t),y(t)),y˙(t)=w(x(t),y(t)),(x0,y0)(x(0),y(0)),and our objective is to steer the point-mass to a goal or target in R2. The target is defined as the disk with center (τ1,τ2) and radius rT, that is, T={(z1,z2)R2:(z1-τ1)2

An asymptotically stable point-mass system with a fixed obstacle

We next consider the situation where there is now a fixed obstacle that the point-mass P has to avoid. Precisely, if (o1,o2) is the center of the disk, and rO is the radius of the disk, then the obstacle can be defined as O={(z1,z2)R2:(z1-o1)2+(z2-o2)2rO2}.Next, we construct an artificial potential field function that guarantees target attraction and collision avoidance.

An application: a planar mobile car-like robot

Consider a planar mobile robot, which is car-like with front wheel steering and engine power applied to the rear wheels (see Fig. 4).

If L is the distance between the two axles and l the length of each axle, then as shown in Pappas and Kyriakopoulos [23], the kinematic model of the car-like robot with respect to its center (x,y)R2 isx˙=vcosθ-L2wsinθ,y˙=vsinθ+L2wcosθ,θ˙=w,where the variable θ gives the robot's orientation with respect to the z1-axis of the z1z2 coordinates, and v and w are the

Conclusion

This article shows how artificial potential fields, with asymptotic stability properties, could be generated by the use of the Lyapunov method. The artificial potential fields also ensure a collision-free trajectory of the system state, which represents a point-mass moving to its target in the presence of a static obstacle. The method proposed was applied successfully to the control of a non-holonomic car-like vehicle.

The article marks a starting point in further developing Stonier's pioneering

Acknowledgments

The authors express their gratitude to the referees for their comments, which improve the original version of this paper. We also wish to thank Jun-Hong Ha of the Korea University of Technology for his useful comments.

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