3D path planning based on nonlinear geodesic equation

Abstract

A lot of methods have been proposed for 2D path planning of mobile robot, which could be a mobile platform or a wheelchair, in planar maps. This paper addresses a concept of the shortest path planning for a mobile robot to traverse a 3D surface, which is a parametrized regular surface that models the non-flat terrain on which the mobile robot traverses. Geodesic curve linking a given start to a given target that is locally shortest on non-flat terrain is used as path. Nonlinear geodesic equations are computed by a gradient descent method with energy function of geodesic, which is shown to converge to the geodesic path in a neighborhood of target position in which a certain Lipschitz condition holds. We present numerical simulations to illustrate the geodesic path planning on non-flat terrains. I. INTRODUCTION The path planning problem is generally stated as: given a start and a goal and a description or representation of an environment, plan a path linking the start and target locations subject to some criteria of safety, mobility and optimality. The path planning for robots is a complex problem in robotics that has been studied for decades, and remain challenging in real-time robot motion in dynamic environment consisting of static or moving obstacles,such as UAV (8)-(11) and underwater robots (11,12). Researchers and engineers have been interested in two-dimensional path planning for mobile robot. Many methods have been proposed for path planning, such as graph search, randomization methods, potential fields, soft computing (e.g., fuzzy logic, neural networks, evolutionary computations) based methods. Depending on whether the environment model is completely known a priori or not, path planning is mainly classified into two categories: The first is called path planning based on the environment model or global path planning as the mobile robot knows all the information about the environment. The path could be planned offline without considering the resources of planning. The second is called path planning based on sensors or local path planning where the information of environment is provided by sensors of the robot or the environment.The robot is required to real-time plan or replan a path to account for the new information of environment gathered by the sensors and planning resources such as computing power and allowed planning time. For the first kind of path planning, we mention harmonic function (10), artifical potential field