Path planning using harmonic functions and probabilistic cell decomposition

Potential-field approach based on harmonic functions have good path planning properties, although the explicit knowledge of the robot’s Configuration Space is required. To overcome this drawback, a combination with a random sampling scheme is proposed. Harmonic functions are computed over computed over a 2d –tree decomposition of a d-dimensional Configuration Space that is obtained with a probabilistic cell decomposition (sampling and classification). Cell sampling is biased towards the more promising regions by using the harmonic function values. Cell classification is performed by evaluating a set of configurations of the cell obtained with a deterministic sampling sequence that provides a good uniform and incremental coverage of the cell. The proposed planning framework open the use of harmonic functions to higher dimensional C-spaces

C-space decomposition using deterministic sampling and distances

Hierarchical cell decompositions of Configuration Space can be of great value for enhancing sampling-based path planners, as well as for other robotic tasks with requirements beyond the planning of free paths. This paper proposes an efficient method to obtain a hierarchical cell decomposition of C-space that is based on: a) the use of a deterministic sampling sequence that allows an uniform and incremental exploration of the space, and b) the use of distance measurements to handle as much information as possible from each sample in order to make the procedure more efficient. The proposed cell decomposition procedure is applied to different path planning methods.Peer Reviewe

Geodesics in Heat

We introduce the heat method for computing the shortest geodesic distance to a specified subset (e.g., point or curve) of a given domain. The heat method is robust, efficient, and simple to implement since it is based on solving a pair of standard linear elliptic problems. The method represents a significant breakthrough in the practical computation of distance on a wide variety of geometric domains, since the resulting linear systems can be prefactored once and subsequently solved in near-linear time. In practice, distance can be updated via the heat method an order of magnitude faster than with state-of-the-art methods while maintaining a comparable level of accuracy. We provide numerical evidence that the method converges to the exact geodesic distance in the limit of refinement; we also explore smoothed approximations of distance suitable for applications where more regularity is required

A novel path planning proposal based on the combination of deterministic sampling and harmonic functions

The sampling-based approach is currently the most successful and yet more promising approach to path planning problems. Sampling-based methods are demonstrated to be probabilistic complete, being their performance reliant on the generation of samples. To obtain a good set of samples, this paper proposes a new sampling paradigm based on deterministic sampling paradigm based on a deterministic sampling sequence guided by an harmonic potential function computed on a hierarchical cell decomposition of C-space. In the proposed method, known as Kautham sampler, samples are not isolated configurations but parts of a whole. As samples are generated they are dynamically grouped into cells that capture the C-space structure. This allows the use of harmonic functions to share information and guide further sampling towards more promising regions of C-space. Finally, using the samples obtained, a roadmap is easily built taking advantage of the known neighbourhood relationships

An adaptative deterministic sequence for sampling-based motion planners

This paper presents a deterministic sequence with good and useful features for sampling-based motion planners, On the one hand, the proposed sequence is able to generate samples over a hierarchical grid structure of the C-space in an incremental low-dispersion manner. On the other hand it allows to locally control the degree of resolution required at each region of the C-space by disabling the generation of mode samples where they are not needed. Therefore, the proposed sequence combines the strength of deterministic sequences (good uniformity coverage), with that of random sequences (adaptive behavior

Block-synchronous Harmonic Control for Scalable Trajectory Planning

ISBN : 978-953-7619-20-6Trajectory planning consists in finding a way to get from a starting position to a goal position while avoiding obstacles within a given environment or navigation space. Harmonic functions may be used as potential fields for trajectory planning. Such functions do not have local extrema, so that control algorithms may reduce to locally descend the potential field until reaching a minimum, when obstacles correspond to maxima of the potential and goals correspond to minima. This chapter presents a parallel hardware implementation of this navigation method on reconfigurable digital circuits. Trajectories are estimated after the iterated computation of the harmonic function, given the goal and obstacle positions of the navigation problem. The proposed massively distributed implementation locally computes the direction to choose to get to the goal position at any point of the environment. Changes in this environment may be immediately taken into account, for example when obstacles are discovered during an on-line exploration. To fit real-world applications, our implementation has been designed to deal with very large navigation environments while optimizing computation time