A Hamilton-Jacobi Formulation for Time-Optimal Paths of Rectangular Nonholonomic Vehicles

We address the problem of optimal path planning for a simple nonholonomic vehicle in the presence of obstacles. Most current approaches are either split hierarchically into global path planning and local collision avoidance, or neglect some of the ambient geometry by assuming the car is a point mass. We present a Hamilton-Jacobi formulation of the problem that resolves time-optimal paths and considers the geometry of the vehicle

A Rotating-Grid Upwind Fast Sweeping Scheme for a Class of Hamilton-Jacobi Equations

We present a fast sweeping method for a class of Hamilton-Jacobi equations that arise from time-independent problems in optimal control theory. The basic method in two dimensions uses a four point stencil and is extremely simple to implement. We test our basic method against Eikonal equations in different norms, and then suggest a general method for rotating the grid and using additional approximations to the derivatives in different directions in order to more accurately capture characteristic flow. We display the utility of our method by applying it to relevant problems from engineering

Continuous-curvature paths for mobile robots

This paper discusses how to plan continuous-curvature paths for car-like wheeled mobile robots. The task is to generate a trajectory with upper-bounded curvature and curvature derivative. To solve this problem we use three path planning primitives, namely straight line segments, circular segments, and continuous-curvature turns (CC turns) in the path planning. We give a classification of the CC turns and we also describe the motion along different kinds of CC turns. We focus on giving computational effective formulae for real-time usage

Models for Human Navigation and Optimal Path Planning Using Level Set Methods and Hamilton-Jacobi Equations

We present several models for different physical scenarios which are centered around human movement or optimal path planning, and use partial differential equations and concepts from control theory. The first model is a game-theoretic model for environmental crime which tracks criminals' movement using the level set method, and improves upon previous continuous models by removing overly restrictive assumptions of symmetry. Next, we design a method for determining optimal hiking paths in mountainous regions using an anisotropic level set equation. After this, we present a model for optimal human navigation with uncertainty which is rooted in dynamic programming and stochastic optimal control theory. Lastly, we consider optimal path planning for simple, self-driving cars in the Hamilton-Jacobi formulation. We improve upon previous models which simplify the car to a point mass, and present a reasonably general upwind, sweeping scheme to solve the relevant Hamilton-Jacobi equation

Optimal Control of Two-Wheeled Mobile Robots for Patrolling Operations

Optimal Control of Two-Wheeled Mobile Robots for Patrolling Operations Walaaeldin Ahmed Ghadiry, Concordia University, 2015 This work studies the use of the two-wheeled mobile robots in patrolling operations, and provides the most distance-e�cient as well as time-e�cient trajectories to patrol a given area. Novel formulations in the context of constrained optimization are introduced which can be solved using existing software. The main concept of the problem is directly related to the well-known Traveling Salesman Problem (TSP) and its variants, where a salesman starts from a base city and visits a number of other cities with minimum travel distance while satisfying the constraint that each city has to be visited only once. Finally, the salesman returns back to the starting base city after completing the mission. Two di�erent patrolling con�gurations that are related to the TSP and its variants, namely the Single Depot multiple Traveling Salesman Problem (mTSP) and the Multidepot multiple Traveling Salesman Problem (MmTSP) are investigated. Novel algorithms are introduced for the trajectory planning of multiple two-wheeled mobile robots, either with two di�erential motors (which can turn on the spot) or with Dubins-like vehicles. The output trajectories for both types of wheeled robots are investigated by using a model predictive control scheme to ensure their kinematic feasibility for the best monitoring performance. The proposed formulations and algorithms are veri�ed by a series of simulations using e�cient programming and optimization software as well as experimental tests in the lab environment