Basic Theta Research Articles

Bézier Curves-Based Optimal Trajectory Design for Multirotor UAVs with Any-Angle Pathfinding Algorithms.

Multirotor Unmanned Aerial Vehicles (UAVs) play an imperative role in many real-world applications in a variety of scenarios characterized by a high density of obstacles with different heights. Due to the complicated operation areas of UAVs and complex constraints associated with the assigned mission, there should be a suitable path to fly. Therefore, the most relevant challenge is how to plan a flyable path for a UAV without collisions with obstacles. This paper demonstrates how a flyable and continuous trajectory was constructed by using any-angle pathfinding algorithms, which are Basic Theta*, Lazy Theta*, and Phi* algorithms for a multirotor UAV in a cluttered environment. The three algorithms were modified by adopting a modified cost function during their implementation that considers the elevation of nodes. First, suitable paths are generated by using a modified version of the three algorithms. After that, four Bézier curves-based approaches are proposed to smooth the generated paths to be converted to flyable paths (trajectories). To determine the most suitable approach, particularly when searching for an optimal and collision-free trajectory design, an innovative evaluation process is proposed and applied in a variety of different size environments. The evaluation process results show high success rates of the four approaches; however, the approach with the highest success rate is adopted. Finally, based on the results of the evaluation process, a novel algorithm is proposed to increase the efficiency of the selected approach to the optimality in the construction process of the trajectory.

Development of a Non-hazardous Path Planning Algorithm for Mars Rover in Real Terrain Enviroment

In order to fulfill the corresponding task successfully, a crucial issue should be addressed is the path planning for the exploration of the Mars surface owing to the environmental features of the tough terrain. Traditional path planning algorithms, such as the A* algorithm and the improved A* algorithm — the algorithm D* and the Field D*, which have been successfully implemented on the planetary rover during the expeditions of the Moon and Mars, have the problem of finding the shortest optimal path. One of the more effective algorithms derived from the modified A* refers to the Basic Theta* or the Lazy Theta* algorithms, which are faster any-angle path planning. Additionally, the algorithms can find shorter routes. In this paper, derived from a comprehensive comparison of the existing algorithms (A*, Basic Theta* and Lazy Theta*), a novel modification of the Lazy AT methodology is proposed to reduce the calculation time and obtain a shorter path. Based on the analysis of the surface feature of the Mars topography, the corresponding safety indicator is discussed. The principal hazards of the wheeled vehicles during the exploration on the surface of the Mars are the slopes and the obstacles. According to the requirements for avoiding obstacles as well as the exploration stability of the Mars rover in the period of the exploration, the following topographic coefficients have been chosen to develop the hazard indicator, i.e., the inclination angle of the terrain, the surface roughness and the height difference of the terrain. In addition, to obtain a safe trajectory in algorithm Lazy AT on the Mars surface, the terrain hazard indicator (risk indicator) for the modification of the Risk Lazy AT algorithm is also proposed in this paper. The comparing analysis modeling results of the Risk Lazy AT and Lazy Theta* has shown that our proposed algorithm Risk Lazy AT can guarantee the safety movement of a mobile object during the exploration on the surface of the planet. In light of the real-world surface features of the Mars terrain, the digital map of the planet’s surface has been developed and the spatial routing of the rover has been tested with our novel proposed algorithm, so-called Risk Lazy AT.

Strict Theta*: Shorter Motion Path Planning Using Taut Paths

A common way to represent dynamic 2D open spaces in robotics and video games for any-angle path planning is through the use of a grid with blocked and unblocked cells. The Basic Theta* algorithm is an existing algorithm that produces near-optimal solutions with a running time close to A* on 8-directional grids. However, a disadvantage is that it often finds non-taut paths that make unnecessary turns. In this paper, we demonstrate that by restricting the search space of Theta* to taut paths, the algorithm will, in most cases, find much shorter paths than the original. We describe two novel variants of the Theta* algorithm, which are simple to implement and use, yet produce a remarkable improvement over Theta* in terms of path length, with a very small running time trade-off. Another side benefit is that almost all paths found will be taut, which makes more convincing paths.

Path Planning with Modified a Star Algorithm for a Mobile Robot

This article deals with path planning of a mobile robot based on a grid map. Essential assumption for path planning is a mobile robot with functional and reliable reactive navigation and SLAM. Therefore, such issues are not addressed in this article. The main body of the article introduces several modifications (Basic Theta*, Phi*) and improvements (RSR, JPS) of A star algorithm. These modifications are focused primarily on computational time and the path optimality. Individual modifications were evaluated in several scenarios, which varied in the complexity of environment. On the basis of these evaluations, it is possible to choose path planning method suitable for individual scenario.

Partition identities from third and sixth order mock theta functions

In a recent paper, the first author gave a connection between bilateral basic hypergeometric series and mock theta functions, which led to many new identities involving mock theta functions. This paper is a sequel, and our goal is to provide partition theoretic properties of new identities involving third or sixth order mock theta functions.

Path Planning Strategies for UAVS in 3D Environments

The graph-search algorithms developed between 60s and 80s were widely used in many fields, from robotics to video games. The A* algorithm shall be mentioned between some of the most important solutions explicitly oriented to motion-robotics, improving the logic of graph search with heuristic principles inside the loop. Nevertheless, one of the most important drawbacks of the A* algorithm resides in the heading constraints connected with the grid characteristics. Different solutions were developed in the last years to cope with this problem, based on post-processing algorithms or on improvements of the graph-search algorithm itself. A very important one is Theta* that refines the graph search allowing to obtain paths with any heading. In the last two years, the Flight Mechanics Research Group of Politecnico di Torino studied and implemented different path planning algorithms. A Matlab based planning tool was developed, collecting four separate approaches: geometric predefined trajectories, manual waypoint definition, automatic waypoint distribution (i.e. optimizing camera payload capabilities) and a comprehensive A*-based algorithm used to generate paths, minimizing risk of collision with orographic obstacles. The tool named PCube exploits Digital Elevation Maps (DEMs) to assess the risk maps and it can be used to generate waypoint sequences for UAVs autopilots. In order to improve the A*-based algorithm, the solution is extended to tri-dimensional environments implementing a more effective graph search (based on Theta*). In this paper the application of basic Theta* to tri-dimensional path planning will be presented. Particularly, the algorithm is applied to orographic obstacles and in urban environments, to evaluate the solution for different kinds of obstacles. Finally, a comparison with the A* algorithm will be introduced as a metric of the algorithm performances.

Theta*: Any-Angle Path Planning on Grids

Grids with blocked and unblocked cells are often used to represent terrain in robotics and video games. However, paths formed by grid edges can be longer than true shortest paths in the terrain since their headings are artificially constrained. We present two new correct and complete any-angle path-planning algorithms that avoid this shortcoming. Basic Theta* and Angle-Propagation Theta* are both variants of A* that propagate information along grid edges without constraining paths to grid edges. Basic Theta* is simple to understand and implement, fast and finds short paths. However, it is not guaranteed to find true shortest paths. Angle-Propagation Theta* achieves a better worst-case complexity per vertex expansion than Basic Theta* by propagating angle ranges when it expands vertices, but is more complex, not as fast and finds slightly longer paths. We refer to Basic Theta* and Angle-Propagation Theta* collectively as Theta*. Theta* has unique properties, which we analyze in detail. We show experimentally that it finds shorter paths than both A* with post-smoothed paths and Field D* (the only other version of A* we know of that propagates information along grid edges without constraining paths to grid edges) with a runtime comparable to that of A* on grids. Finally, we extend Theta* to grids that contain unblocked cells with non-uniform traversal costs and introduce variants of Theta* which provide different tradeoffs between path length and runtime.