Robot Navigation Using a Pressure Generated Mechanical Stress Field, "The Biharmonic Potential Approach"

Abstract

This paper suggests a new approach for navigation in a known environment. The approach is based on the Bihannonic potential fields which govern mechanical stress fields in homogeneous solids. It is observed that a path generated by such a technique is free of sharp turns that may appear in its counterpart that is generated from an underlying Harmonic potential. This in turns makes a path from the former more dynamically suitable for traversal. Also, the navigation field is extracted from the Biharmonic potential in a manner that bypass the rapidly vanishing field problem which is encountered in the Harmonic potential approach. Development of the Biharmonic approach, simulation results, as well as comparison with the Neumann and Dirichlet setting of the Harmonic approach are provided. I. I W T R O W E T I O ~ In the mid eighties several papers addressing the path planning problem 11.2.31 signaled a major departure from the traditional geometric approach for navigation. The idea of potential fields was suggested. In this approach the flow lines of a vector field that is generated from an underlying potential field (VI is used to safely guide motion to its unique equilibrium point which is situated on the target. The usefulness of such an idea critically hinges on the ability to find a mapping that would accept information about the target and the environment as an input and generate a field that spans the whole workspace of interest. This field has to be locally usable in an a priori specified manner that is independent of the starting point to safely direct motion to the target. The mapping can be seen as a tool for fanning the operator's control over the workspace. With such a mapping a small (seed) space can be used to generate a field that spans a much larger space (both in volume and dimensionality) that is in turn usable for inducing the behavior of interest. Of course, it is possible to construct such a field by properly assigning a vector quantity (to tell a vehicle in which direction to proceed) to sufficient points in the workspace (an interpolation scheme may be used to fill in between the points): however,the effort in doing so is enormous. In the field synthesis process the information about the goal, the environment structure, and the desired behavior are imbedded (encoded) in the fabric of the potential in the form of locally extractable vector feature/s that are designed to operate in a chain-like manner to establish a desired global behavior through a series of local actions. Those vector features and the scheme that utilizes them to lay a path to the target are given the name TRACKING MECHANISM (we also contemplated the name STEERING MECHANISM). The vsctor features are extracted from the potential field by operating on it with a reflexive operator. Usually the Del (V) operator is used to generate the gradient flow of the field (W). In a recent work (4.51 the use of the curl operator(Vx) in addition to V was suggested to generate the navigation field. It ought to be mentioned that the tracking mechanism need not operate in a purely reflexive manner. It can be shown by examples that cognitive aspects can be integrated in the process to efficiently deal with uncertain situations and to enhance the reliability of the process. From the above discussion it can be seen that a potential-based navigation technique can be divided into two interactive stages. A stage to generate the field, and a stage to utilize that field for navigation. Figure-1 shows a block diagram of the suggested structure for such techniques. Gbbal Organization Local Utilization by Opww b y h h l n . 1 Motion Constraints starting point

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