Stephen R. Lindemann

On the Relationship between Classical Grid Search and Probabilistic Roadmaps

We present, implement, and analyze a spectrum of closely-related planners, designed to gain insight into the relationship between classical grid search and probabilistic roadmaps (PRMs). Building on quasi-Monte Carlo sampling literature, we have developed deterministic variants of the PRM that use low-discrepancy and low-dispersion samples, including lattices. Classical grid search is extended using subsampling for collision detection and also the optimal-dispersion Sukharev grid, which can be considered as a kind of lattice-based roadmap to complete the spectrum. Our experimental results show that the deterministic variants of the PRM offer performance advantages in comparison to the original PRM and the recent Lazy PRM. This even includes searching using a grid with subsampled collision checking. Our theoretical analysis shows that all of our deterministic PRM variants are resolution complete and achieve the best possible asymptotic convergence rate, which is shown superior to that obtained by random sampling. Thus, in surprising contrast to recent trends, there is both experimental and theoretical evidence that some forms of grid search are superior to the original PRM.

Current Issues in Sampling-Based Motion Planning

In this paper, we discuss the field of sampling-based motion planning. In contrast to methods that construct boundary representations of configuration space obstacles, samplingbased methods use only information from a collision detector as they search the configuration space. The simplicity of this approach, along with increases in computation power and the development of efficient collision detection algorithms, has resulted in the introduction of a number of powerful motion planning algorithms, capable of solving challenging problems with many degrees of freedom. First, we trace how sampling-based motion planning has developed. We then discuss a variety of important issues for sampling-based motion planning, including uniform and regular sampling, topological issues, and search philosophies. Finally, we address important issues regarding the role of randomization in sampling-based motion planning.

Incrementally reducing dispersion by increasing Voronoi bias in RRTs

We discuss theoretical and practical issues related to using Rapidly-Exploring Random Trees (RRTs) to incrementally reduce dispersion in the configuration space. The original RRT planners use randomization to create Voronoi bias, which causes the search trees to rapidly explore the state space. We introduce RRT-like planners based on exact Voronoi diagram computation, as well as sampling-based algorithms which approximate their behavior. We give experimental results illustrating how the new algorithms explore the configuration space and how they compare with existing RRT algorithms. Initial results show that our algorithms are advantageous compared to existing RRTs, especially with respect to the number of collision checks and nodes in the search tree.

Incremental low-discrepancy lattice methods for motion planning

We present deterministic sequences for use in sampling-based approaches to motion planning. They simultaneously combine the qualities found in many other sequences: i) the incremental and self-avoiding tendencies of pseudo-random sequences, ii) the lattice structure provided by multiresolution grids, and iii) low-discrepancy and low-dispersion measures of uniformity provided by quasi-random sequences. The resulting sequences can be considered as multiresolution grids in which points may be added one at a time, while satisfying the sampling qualities at each iteration. An efficient, recursive algorithm for generating the sequences is presented and implemented. Early experiments show promising performance by using the samples in search algorithms to solve motion planning problems.

Real Time Feedback Control for Nonholonomic Mobile Robots With Obstacles

We introduce a method for constructing smooth feedback laws for a nonholonomic robot in a 2-dimensional polygonal workspace. First, we compute a smooth feedback law in the workspace without taking the nonholonomic constraints into account. We then give a general technique for using this to construct a new smooth feedback law over the entire 3-dimensional configuration space (consisting of position and orientation). The trajectories of the resulting feedback law will be smooth and will stabilize the position of the robot in the plane, neglecting the orientation. Our method is suitable for real time implementation and can be applied to dynamic environments

Smoothly Blending Vector Fields for Global Robot Navigation

We introduce a new algorithm for constructing smooth vector fields for global robot navigation. Given a ddimensional cell complex with each cell a convex polygon, our algorithm defines a number of local vector fields: one for each cell, and one for each face connecting two cells together. We smoothly blend these component vector fields together using bump functions; the precomputation of the component vector field and all queries can be done in linear time. The integral curves of the resulting globally-defined vector field are guaranteed to arrive at a neighborhood of the goal state in finite time. Except for a set of measure zero, the vector field is smooth. The resulting vector field can be used directly to control kinematic systems or can be used to develop dynamic control policies. We prove convergence for the integral curves of the vector fields produced by our algorithm and give examples illustrating the practical advantages of our technique.

Smooth Feedback for Car-Like Vehicles in Polygonal Environments

We introduce a method for constructing provably safe smooth feedback laws for car-like robots in obstacle-cluttered polygonal environments. The robot is taken to be a point with motion that must satisfy bounded path curvature constraints. We construct a global feedback plan (or control policy) by partitioning the environment into convex cells, computing a discrete plan on the resulting cell complex, and generating local control laws on the state space that are safe, consistent with the high level plan, and satisfy smoothness conditions. The trajectories of the resulting global feedback plan are smooth and stabilize the position of the robot in the plane, neglecting the orientation.

Computing smooth feedback plans over cylindrical algebraic decompositions

In this paper, we construct smooth feedback plans over cylindrical algebraic decompositions. Given a cylindrical algebraic decomposition on R, a goal state xg , and a connectivity graph of cells reachable from the goal cell, we construct a vector field that is smooth everywhere except on a set of measure zero and the integral curves of which are smooth (i.e., C∞) and arrive at a neighborhood of the goal state in finite time. We call a vector field with these properties a smooth feedback plan. The smoothness of the integral curves guarantees that they can be followed by a system with finite acceleration inputs: ẍ = u. We accomplish this by defining vector fields for each cylindrical cell and face and smoothly interpolating between them. Schwartz and Sharir showed that cylindrical algebraic decompositions can be used to solve the generalized piano movers’ problem, in which multiple (possibly linked) robots described as semi-algebraic sets must travel from their initial to goal configurations without intersecting each other or a set of semi-algebraic obstacles. Since we build a vector field over the decomposition, this implies that we can obtain smooth feedback plans for the generalized piano movers’ problem.

Multiresolution approach for motion planning under differential constraints

In this paper, we present an incremental, multiresolution motion planning algorithm designed for systems with differential constraints. Planning for these systems is more difficult than ordinary path planning due to the presence of momentum (drift) or nonholonomic velocity constraints. Given a motion planning problem for such a system and that a solution to the problem exists, then a finite reachability graph containing a solution trajectory is guaranteed to exist, under very reasonable conditions. In general, this graph can be generated using sufficiently dense input space sampling, sufficiently small time step, and sufficiently large tree depth. We show how to find and search such a tree in an incremental, multiresolution way. We prove the completeness of our algorithm, discuss related practical concerns, and show experimental results for several systems

Iteratively Locating Voronoi Vertices for Dispersion Estimation

We present a new sampling-based algorithm for iteratively locating Voronoi vertices of a point set in the unit cube Id= [0, 1]d. The algorithm takes an input sample and executes a series of transformations, each of which projects the sample to a new face of the Voronoi cell in which it is located. After d such transformations, the sample has been transformed into a Voronoi vertex. Locating Voronoi vertices has many potential applications for motion planning, such as estimating dispersion for coverage and verification applications, and providing information useful for Voronoi-biased or multiple-tree planning. We prove theoretical results regarding our algorithm, and give experimental results comparing it to naive sampling for the problem of dispersion estimation.