Log of the inverse of the distance transform and fast marching applied to path planning

Abstract-This paper presents a new Path Planning method based in the inverse of the Logarithm of the Distance Transform and in the Fast Marching Method. The Distance Transform of an image gives a grey scale that is darker near the obstacles and walls and more clear far from them and it is calculated via Voronoi Diagram. The Logarithm of the inverse of the Distance Transform imitates the repulsive electric potential from walls and obstacles. This method is very fast and reliable and the trajectories are similar to the human trajectories: smooth and not very close to obstacles and walls

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

Nilpotent Approximations of Sub-Riemannian Distances for Fast Perceptual Grouping of Blood Vessels in 2D and 3D

We propose an efficient approach for the grouping of local orientations (points on vessels) via nilpotent approximations of sub-Riemannian distances in the 2D and 3D roto-translation groups $SE(2)$ and $SE(3)$. In our distance approximations we consider homogeneous norms on nilpotent groups that locally approximate $SE(n)$, and which are obtained via the exponential and logarithmic map on $SE(n)$. In a qualitative validation we show that the norms provide accurate approximations of the true sub-Riemannian distances, and we discuss their relations to the fundamental solution of the sub-Laplacian on $SE(n)$. The quantitative experiments further confirm the accuracy of the approximations. Quantitative results are obtained by evaluating perceptual grouping performance of retinal blood vessels in 2D images and curves in challenging 3D synthetic volumes. The results show that 1) sub-Riemannian geometry is essential in achieving top performance and 2) that grouping via the fast analytic approximations performs almost equally, or better, than data-adaptive fast marching approaches on $\mathbb{R}^n$ and $SE(n)$.Comment: 18 pages, 9 figures, 3 tables, in review at JMI

Robot formation motion planning using Fast Marching

This paper presents the application of the Voronoi Fast Marching (VFM) method to path planning of mobile formation robots. The VFM method uses the propagation of a wave (Fast Marching) operating on the world model to determine a motion plan over a viscosity map (similar to the refraction index in optics) extracted from the updated map model. The computational efficiency of the method allows the planner to operate at high rate sensor frequencies. This method allows us to maintain good response time and smooth and safe planned trajectories. The navigation function can be classified as a type of potential field, but it has no local minima, it is complete (it finds the solution path if it exists) and it has a complexity of order n(O(n)), where n is the number of cells in the environment map. The results presented in this paper show how the proposed method behaves with mobile robot formations and generates trajectories of good quality without problems of local minima when the formation encounters non-convex obstacles.This work has been supported by the CAM Project S2009/DPI-1559/ROBOCITY2030 II, developed by the research team RoboticsLab at the University Carlos III of Madrid.Publicad

Robot Formations Control Using Fast Marching

This paper presents the application of the Voronoi Fast Marching (V FM) method to the Control of Robot Formations. The V FM method uses the propagation of a wave (Fast Marching) operating on the world model to de- termine a motion plan over a viscosity map (similar to the refraction index in optics) extracted from the updated map model. The computational effciency of the method allows the planner to operate at high rate sensor frequencies. This method allows us to maintain good response time and smooth and safe planned trajectories. The navigation function can be classiffed as a type of potential field, but it has no local minima, it is complete (it finds the solu- tion path if it exists) and it has a complexity of order n (O(n)), where n is the number of cells in the environment map. The results presented in this paper show how the proposed method behaves with mobile robot formations and generates trajectories of good quality without problems of local minima when the formation encounters non-convex obstacles

SLAM and exploration using differential evolution and fast marching

The exploration and construction of maps in unknown environments is a challenge for robotics. The proposed method is facing this problem by combining effective techniques for planning, SLAM, and a new exploration approach based on the Voronoi Fast Marching method. The final goal of the exploration task is to build a map of the environment that previously the robot did not know. The exploration is not only to determine where the robot should move, but also to plan the movement, and the process of simultaneous localization and mapping. This work proposes the Voronoi Fast Marching method that uses a Fast Marching technique on the Logarithm of the Extended Voronoi Transform of the environment"s image provided by sensors, to determine a motion plan. The Logarithm of the Extended Voronoi Transform imitates the repulsive electric potential from walls and obstacles, and the Fast Marching Method propagates a wave over that potential map. The trajectory is calculated by the gradient method