An asymptotically stable collision-avoidance system
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 be the n-dimensional Euclidean space with the Euclidean norm . Let denote an element of . Consider an autonomous non-linear systemwhere is assumed to be smooth enough to guarantee the existence, uniqueness and continuous dependence of solutions of (1) in , an open set in .
For the purpose of
A globally asymptotically stable point-mass system
Consider a point-mass, defined as the disk of radius , and positioned at at time . That is, the point-mass isIts instantaneous velocity is . Our general ODE system is therefore of the formand our objective is to steer the point-mass to a goal or target in . The target is defined as the disk with center and radius , that is,
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 is the center of the disk, and is the radius of the disk, then the obstacle can be defined as 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 iswhere the variable gives the robot's orientation with respect to the -axis of the – coordinates, and and 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.
References (28)
Use of Lyapunov techniques for collision-avoidance of robot arms
- et al.
On the “Piano Movers”, Problem I, the case of a two-dimensional polynomial body moving amidst polygonal barriers
Commun. Pure Appl. Math.
(1983) Solving the find-path problem by good representation of free-space
IEEE Trans. Syst. Man Cybern.
(1983)Spatial planning: a configuration space approach
IEEE Trans. Comput.
(1983)Fast, three-dimensional collision-free motion planning
- et al.
High-speed environment representation scheme for dynamic path planning
J. Intell. Robotic Syst.
(2001) - et al.
A rapid method for planning paths in three dimensions for a small aerial robot
Robotica
(2001) Global level path planning for mobile robots in dynamic environments
J. Intell. Robotic Syst.
(2003)Path planning for planar articulated robots using configuration spaces and compliant motion
IEEE Trans. Robotics Autom.
(2003)Study on the configuration space based algorithmic path planning of industrial robots in an unstructured congested three-dimensional space: an approach using visibility map
J. Intell. Robotic Syst.
(2005)
A local-based method for manipulators path planning, using sub-goals resulting from a local graph
Robotica
Real-time obstacle avoidance for manipulators and mobile robots
Int. J. Robotics Res.
Path planning using Laplace's equation
Analogue computation of collision-free paths
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