Finding the Maximum Subset with Bounded Convex Curvature

We describe an algorithm for solving an important geometric problem arising in computer-aided manufacturing. When machining a pocket in a solid piece of material such as steel using a rough tool in a milling machine, sharp convex corners of the pocket cannot be done properly, but have to be left for finer tools that are more expensive to use. We want to determine a tool path that maximizes the use of the rough tool. Mathematically, this boils down to the following problem. Given a simply-connected set of points P in the plane such that the boundary of P is a curvilinear polygon consisting of n line segments and circular arcs of arbitrary radii, compute the maximum subset Q of P consisting of simply-connected sets where the boundary of each set is a curve with bounded convex curvature. A closed curve has bounded convex curvature if, when traversed in counterclockwise direction, it turns to the left with curvature at most 1. There is no bound on the curvature where it turns to the right. The difference in the requirement to left- and right-curvature is a natural consequence of different conditions when machining convex and concave areas of the pocket. We devise an algorithm to compute the unique maximum such set Q. The algorithm runs in O(n log n) time and uses O(n) space. For the correctness of our algorithm, we prove a new generalization of the Pestov-Ionin Theorem. This is needed to show that the output Q of our algorithm is indeed maximum in the sense that if Q\u27 is any subset of P with a boundary of bounded convex curvature, then Q\u27 is a subset of Q

Mobile robotic network deployment for intruder detection and tracking

This thesis investigates the problem of intruder detection and tracking using mobile robotic networks. In the first part of the thesis, we consider the problem of seeking an electromagnetic source using a team of robots that measure the local intensity of the emitted signal. We propose a planner for a team of robots based on Particle Swarm Optimization (PSO) which is a population based stochastic optimization technique. An equivalence is established between particles generated in the traditional PSO technique, and the mobile agents in the swarm. Since the positions of the robots are updated using the PSO algorithm, modifications are required to implement the PSO algorithm on real robots to incorporate collision avoidance strategies. The modifications necessary to implement PSO on mobile robots, and strategies to adapt to real environments are presented in this thesis. Our results are also validated on an experimental testbed. In the second part, we present a game theoretic framework for visibility-based target tracking in multi-robot teams. A team of observers (pursuers) and a team of targets (evaders) are present in an environment with obstacles. The objective of the team of observers is to track the team of targets for the maximum possible time. While the objective of the team of targets is to escape (break line-of-sight) in the minimum time. We decompose the problem into two layers. At the upper level, each pursuer is allocated to an evader through a minimum cost allocation strategy based on the risk of each evader, thereby, decomposing the agents into multiple single pursuer-single evader pairs. Two decentralized allocation strategies are proposed and implemented in this thesis. At the lower level, each pursuer computes its strategy based on the results of the single pursuer-single evader target-tracking problem. We initially address this problem in an environment containing a semi-infinite obstacle with one corner. The pursuer\u27s optimal tracking strategy is obtained regardless of the evader\u27s strategy using techniques from optimal control theory and differential games. Next, we extend the result to an environment containing multiple polygonal obstacles. We construct a pursuit field to provide a guiding vector for the pursuer which is a weighted sum of several component vectors. The performance of different combinations of component vectors is investigated. Finally, we extend our work to address the case when the obstacles are not polygonal, and the observers have constraints in motion

An Efficient Spatial-Temporal Trajectory Planner for Autonomous Vehicles in Unstructured Environments

As a core part of autonomous driving systems, motion planning has received extensive attention from academia and industry. However, real-time trajectory planning capable of spatial-temporal joint optimization is challenged by nonholonomic dynamics, particularly in the presence of unstructured environments and dynamic obstacles. To bridge the gap, we propose a real-time trajectory optimization method that can generate a high-quality whole-body trajectory under arbitrary environmental constraints. By leveraging the differential flatness property of car-like robots, we simplify the trajectory representation and analytically formulate the planning problem while maintaining the feasibility of the nonholonomic dynamics. Moreover, we achieve efficient obstacle avoidance with a safe driving corridor for unmodelled obstacles and signed distance approximations for dynamic moving objects. We present comprehensive benchmarks with State-of-the-Art methods, demonstrating the significance of the proposed method in terms of efficiency and trajectory quality. Real-world experiments verify the practicality of our algorithm. We will release our codes for the research communit

Real-Time Obstacle and Collision Avoidance System for Fixed-Wing Unmanned Aerial Systems

The motivation for the research presented in this dissertation is to provide a two-fold solution to the problem of non-cooperative reactive mid-air threat avoidance for fixed-wing unmanned aerial systems. The first phase is an offline UAS trajectory planning designed for an altitude-specific mission. The second phase leans on the results produced during the first phase to provide intelligent, real-time, reactive mid-air threat avoidance logic. That real-time operating logic provides a given fixed-wing UAS with local threat awareness so it can get a feel for the danger represented by a potential threat before using results produced during the first phase to require aircraft rerouting. The first original contribution of this research is the Advanced Mapping and Waypoint Generator (AMWG), a piece of software which processes publicly available elevation data in order to only retain the information necessary for a given altitude-specific flight mission. The AMWG is what makes systematic offline trajectory possible. The AMWG first creates altitude groups in order to discard elevations points which are not relevant to a specific mission because of the altitude flown at. Those groups referred to as altitude layers can in turn be reused if the original layer becomes unsafe for the altitude range in use, and the other layers are used for altitude re-scheduling in order to update the current altitude layer to a safer layer. Each layer is bounded by a lower and higher altitude, within which terrain contours are considered constant according to a conservative approach involving the principle of natural erosion. The AMWG then proceeds to obstacle contours extraction using threshold and edge detection vision algorithms. A simplification of those obstacle contours and their corresponding free space zones counterparts is performed using a fixed -tolerance Douglas-Peucker algorithm. This simplification allows free space zones to be described by vectors instead of point clouds, which enables UAS point location. The resulting geometry is then processed through a vertical trapezoidal decomposition where for each vertex defining a contour a vertical line is drawn, and the results of this decomposition is a set of trapezoidal cells. The cells corresponding to obstacle contours are then removed from the original trapezoidal decomposition in order to solely retain the obstacle-free trapezoidal cells. After decomposition, cells sharing part of a common edge are considered from a graph theory perspective so it becomes possible to list all acyclic paths between two cells by applying a depth first search (DFS) algorithm. The final product of the AWMG is a network of connected free space trapezoidal cells with embedded connectivity information referred to as the Synthetic Terrain Avoidance (STA network). The walls of the trapezoidal cells are then extruded as the AWMG essentially approximates a three-dimensional world by considering it as a stratification of two-dimensional layers, but the real-time phase needs 3D support. Using the graph conceptual view and the depth first search algorithm, all the connected cell sequences joining the departure to the arrival cell can be listed, a capability which is used during aircraft rerouting. By connecting two adjacent cells' centroids to their common midpoint located on the shared edge, the resulting flying legs remain within the two cells. The next step for paths between two cells is to be converted into flyable paths, and the conversion uses main and fallback methods to achieve that. The preferred method is the closed-form Dubins paths method involving the design of sequences of arc circle-straight line-arc circle (CLC) in order to account for the minimum radius turn constrain of the UAS. An additional geometric transformation is developed and applied to the initial waypoints used in the Dubins method so the flying leg directions are respected which is not possible by using the Dubins method alone. When consecutive waypoints are too close from one another, a condition called the Dubins condition cannot be respected, and the UAS trajectory design switches to the numerical integration of a system of ordinary differential equations accounting for the minimum turning constraint. Using the Dubins method and the ODE method makes it possible for the AWMG to design flyable offline trajectories accounting for the lateral dynamic of the fixed-wing UAS. The second original contribution of this research is the development and demonstration of the Double Dispersion reduction RRT (DDRRT), an algorithm which employs two new developed logic schemes respectively referred to as Punctual Dispersion Reduction (PDR), and Spatial Dispersion Reduction exploration (SDR). The DDRRT is employed during the real-time in-flight phase where it initially assumes a perfect terrain and no unpredictable threat, consequently following a 100% adaptive goal biasing toward the next waypoint in its list. When a threat such as an unpredicted obstacle is detected, the (PDR) acknowledges the fact that the DDRRT tree branches have met an obstacle and the its goal-biasing toward the next waypoint is decreased. If the PDR keeps decreasing, the DDRRT develops awareness of its surrounding obstacles by relaxing its PDR and switching to SDR which has the effect of increasing the dispersion of its branches, but keeping their extension bounded by the cell containing the last good position of the UAS, Csafe. If a number of branches reach a limit proportional to the Csafe and its relative area, then the STA network is queried for alternative rerouting. The two phases provide real-time reactive mid - air threat avoidance scenarios with the ability for a UAS to develop local and realistic threat awareness before considering intelligent rerouting. Either the local exploration of the DDRRT is successful before reaching a maximum number of points, or the STA Network is required to find another route

Safe open-loop strategies for handling intermittent communications in multi-robot systems

The objective of this thesis is to develop a strategy that allows robots to safely execute open-loop motion patterns for pre-computed time durations when facing interruptions in communication. By computing the time horizon in which collisions with other robots are impossible, this method allows the robots to move safely despite having no updated information about the environment. As the complexity of multi-robot systems increase, communication failures in the form of packet losses, saturated network channels and hardware failures are inevitable. This thesis is motivated by the need to increase the robustness of operation in the face of such failures. The advantage of this strategy is that it prevents the jerky and unpredictable motion behaviour which often plague robotic systems experiencing communication issues. To compute the safe time horizon, the first step involves constructing reachable sets around the robots to determine the set of all positions that can be reached by the robot in a given amount of time. In order to avoid complications arising from the non-convexity of these reachable sets, analytical expressions for minimum area ellipses enclosing the reachable sets are obtained. By using a fast gradient descent based technique, intersections are computed between a robot’s trajectory and the reachable sets of other robots. This information is then used to compute the safe time horizon for each robot in real time. To this end, provable safety guarantees are formulated to ensure collision avoidance. This strategy has been verified in simulation as well as on a team of two-wheeled differential drive robots on a multi-robot testbed.M.S