Convex Tours of Bounded Curvature

We consider the motion planning problem for a point constrained to move along a smooth closed convex path of bounded curvature. The workspace of the moving point is bounded by a convex polygon with m vertices, containing an obstacle in a form of a simple polygon with $n$ vertices. We present an O(m+n) time algorithm finding the path, going around the obstacle, whose curvature is the smallest possible.Comment: 11 pages, 5 figures, abstract presented at European Symposium on Algorithms 199

Curvature-Constrained Shortest Paths in a Convex Polygon

Let B be a point robot moving in the plane, whose path is constrained to have curvature at most 1, and let poly be a convex polygon with n vertices. We study the collision-free, optimal path planning problem for B moving between two configurations inside poly (a configuration specifies both a location and a direction of travel). We present an runtime- time algorithm for determining whether a collision-free path exists for B between xstwo given configurations. If such a path exists, the algorithm returns a shortest one. We provide a detailed classification of curvature-constrained shortest paths inside a convex polygon and prove several properties of them, which are interesting in their own right. For example, we prove that any such shortest path is comprised of at most eight segments, each of which is a circular arc of unit radius or a straight line segment. Some of the properties are quite general and shed some light on curvature-constrained shortest paths amid obstacles

Coverage and Time-optimal Motion Planning for Autonomous Vehicles

Autonomous vehicles are rapidly advancing with a variety of applications, such as area surveillance, environment mapping, and intelligent transportation. These applications require coverage and/or time-optimal motion planning, where the major challenges include uncertainties in the environment, motion constraints of vehicles, limited energy resources and potential failures. While dealing with these challenges in various capacities, this dissertation addresses three fundamental motion planning problems: (1) single-robot complete coverage in unknown environment, (2) multi-robot resilient and efficient coverage in unknown environment, and (3) time-optimal risk-aware motion planning for curvature-constrained vehicles. First, the ε* algorithm is developed for online coverage path planning in unknown environment using a single autonomous vehicle. It is computationally efficient, and can generate the desired back-and-forth path with less turns and overlappings. ε* prevents the local extrema problem, thus can guarantee complete coverage. Second, the CARE algorithm is developed which extends ε* for multi-robot resilient and efficient coverage in unknown environment. In case of failures, CARE guarantees complete coverage via dynamic task reallocations of other vehicles, hence provides resilience. Moreover, it reallocates idling vehicles to support others in their tasks, hence improves efficiency. Finally, the T* algorithm is developed to find the time-optimal risk-aware path for curvature-constrained vehicles. We present a novel risk function based on the concept of collision time, and integrate it with the time cost for optimization. The above-mentioned algorithms have been validated via simulations in complex scenarios and/or real experiments, and the results have shown clear advantages over existing popular approaches

Path Planning for Underactuated Dubins Micro-robots Using Switching Control

In this paper, we develop an optimal path planning strategy for under-actuated Dubins micro-robots. Such robots are non-holonomic robots constrained to move along circular paths of fixed curvature clockwise or counter-clockwise. Our objective is to investigate the coverage and optimal path problems, as well as multi-robot cooperation, for a switching control scheme. Our methods are based on elementary geometry and optimal control techniques. The results in this paper show that the trajectories of micro-robots can cover the entire two-dimensional plane, and that the proposed switching control scheme allows multiple robots to cooperate. In addition, we deduce the minimum-time path under the switching control scheme by converting the robot model into the traditional Dubins vehicle model

Bounded-Curvature Shortest Paths through a Sequence of Points

We consider the problem of computing shortest paths having curvature at most one almost everywhere and visiting a sequence of $n$ points in the plane in a given order. This problem arises naturally in path planning for point car-like robots in the presence of polygonal obstacles, and is also a sub-problem of the Dubins Traveling Salesman Problem. This problem reduces to minimizing the function $F:\R^n\rightarrow\R$ that maps $(\theta_1,\ldots,\theta_n)$ to the length of a shortest curvature-constrained path that visits the points $p_1, \ldots, p_n$ in order and whose tangent in $p_i$ makes an angle $\theta_i$ with the $x$-axis. We show that when consecutive points are distance at least $4$ apart, all minima of $F$ are realized over at most $2^k$ disjoint convex polyhedra over which $F$ is strictly convex; each polyhedron is defined by $4n-1$ linear inequalities and $k$ denotes, informally, the number of $p_i$ such that the angle $\angle(p_{i-1},p_i,p_{i+1})$ is small. A curvature-constrained shortest path visiting a sequence points can therefore be approximated by standard convex optimization methods, which presents an interesting alternative to the known polynomial-time algorithms that can only compute a multiplicative constant factor approximation. Our technique also opens new perspectives for bounded-curvature path planning among polygonal obstacles. In particular, we show that, under certain conditions, if the sequence of points where a shortest path touches the obstacles is known then ``connecting the dots'' reduces to a family of convex optimization problems

Path planning for simple wheeled robots : sub-Riemannian and elastic curves on SE(2)

This paper presents a motion planning method for a simple wheeled robot in two cases: (i) where translational and rotational speeds are arbitrary and (ii) where the robot is constrained to move forwards at unit speed. The motions are generated by formulating a constrained optimal control problem on the Special Euclidean group SE(2). An application of Pontryagin’s maximum principle for arbitrary speeds yields an optimal Hamiltonian which is completely integrable in terms of Jacobi elliptic functions. In the unit speed case, the rotational velocity is described in terms of elliptic integrals and the expression for the position reduced to quadratures. Reachable sets are defined in the arbitrary speed case and a numerical plot of the time-limited reachable sets presented for the unit speed case. The resulting analytical functions for the position and orientation of the robot can be parametrically optimised to match prescribed target states within the reachable sets. The method is shown to be easily adapted to obstacle avoidance for static obstacles in a known environment