Weifu Wang

A fast streaming spanner algorithm for incrementally constructing sparse roadmaps

Sampling-based probabilistic roadmap algorithms such as PRM and PRM* have been shown to be effective at solving certain motion planning problems, but the large graphs generated to express the connectivity and a metric on the configuration space may require much storage space and be expensive to search. Recent work by Marble and Bekris [14], [19] applied spanner algorithms to PRM* these algorithms prune some edges in a dense graph, while guaranteeably maintaining an approximation to the metric. In this paper, we apply (and improve) a state-of-the-art streaming spanner algorithm to prune PRM* roadmaps. The algorithm we present has the main advantage of computational speed; when applied to PRM*, the processing time per vertex is independent of the number of sampled vertices, n, as compared to O(nlog2 nloglogn) in [19]. In practice, the algorithm we present prunes a graph with about 20 million edges in less than 20 seconds on a modern desktop computer; compared to the time required for generating such a roadmap, this additional processing time is essentially trivial. In fact, because the combination of this algorithm with PRM* avoids the need for many collision detections, the combination runs several times faster than PRM*alone.

Analytical time-optimal trajectories for an omni-directional vehicle

We present the first analytical solution method for finding a time-optimal trajectory between any given pair of configurations for a three-wheeled omni-directional vehicle in an obstacle-free plane. The mathematical model of the vehicle bounds the velocities of the wheels independently. The timeoptimal trajectories can be divided into two categories: singular and generic. An analytical solution method has previously been presented for singular trajectories; this paper completes the work and presents the solution for generic trajectories. The speed and precision of the algorithm allow dense sampling of the configuration space, to determine how the time and structure of time-optimal trajectories change across configurations. Simulation results show that time-optimal trajectories tend to be ten to twenty percent faster than a simple but practical driving strategy: turn until the fastest translation direction faces the goal, drive to the goal, and turn to the current angle.

Knot-tying with four-piece fixtures

We present a class of fixtures that can be disassembled into four pieces to extract the loosely tied knot. We prove that a fixture can be designed for any particular knot such that the knot can be extracted using only simple pure translations of the four fixture sections. We explore some of the issues raised by our experimental work with these fixtures, which show that simple knots can be tied extremely quickly (less than half a second) and reliably (99% repeatability) using four-piece fixtures.

A fast online spanner for roadmap construction

This paper introduces a fast weighted streaming spanner algorithm (WSS) that trims edges from roadmaps generated by robot motion planning algorithms such as Probabilistic Roadmap (PRM) and variants (e.g. k-PRM*) as the edges are generated, but before collision detection; no route in the resulting graph is more than a constant factor larger than it would have been in the original roadmap. Experiments applying WSS to k-PRM* were conducted, and the results show our algorithm’s capability to filter graphs with up to 1.28 million vertices, discarding about three-quarters of the edges. Due to the fact that many collision detection steps can be avoided, the combination of WSS and k-PRM* is faster than k-PRM* alone. The paper further presents an online directed spanner algorithm that can be used for systems with non-holonomic constraints, with proof of correctness and experimental results.

Towards Arranging and Tightening Knots and Unknots With Fixtures

This paper presents a controlled tying approach for knots using fixtures and simple pulling motions applied to the ends of string. Each fixture is specific to a particular knot; the paper gives a design process that allows a suitable fixture to be designed for an input knot. Knot tying is separated into two phases. In the first phase, a fixture is used to loosely arrange the string around a set of rods, with the required topology of the given knot. In the second phase, the string is pulled taut around the tightening fixtures. Two tightening fixture designs are presented. The first design is a fixture with no moving parts; tilted rods whose cross-sections get closer near the tips, guiding string in a controlled fashion as string slides up the rods during tightening. The second design is a collection of straight rods that can move passively along predefined paths during tightening. Successful tying is shown for three interesting cases: a “cloverleaf knot” design, a “double coin” knot design, and the top of a shoelace knot.

The bench mover's problem: Minimum-time trajectories, with cost for switching between controls

Analytical results describing the optimal trajectories for general classes of robot systems have proven elusive, in part because the optimal trajectories for a complex system may not exist, or may be computed only numerically from differential equations. This paper studies a simpler optimization problem: finding an optimal sequence and optimal durations of motion primitives (simple preprogrammed actions) to reach a goal. By adding a fixed cost for each switch between primitives, we ensure that optimal trajectories exist and are well-behaved. To demonstrate this approach, we prove some general results that geometrically characterize time-optimal trajectories for rigid bodies in the plane with costly switches (allowing comparison with previous analysis of optimal motion using Pontryagins Maximum Principle), and also present a complete analytical solution for a problem of moving a heavy park bench by rotating the bench around each end point in sequence.

An online method for tight-tolerance insertion tasks for string and rope

This paper presents a fast tight-tolerance threading technique for string and rope. Instead of relying on simulations of these deformable objects to plan a path or compute control actions, we control the movement of the string with a virtual magnetic vector field emanating from the narrow openings we wish to thread through. We compute an approximate Jacobian to move the tip of the string through the vector field and propose a method to promote alignment of the head of the string to the opening. We also propose a method for re-grasping the string based on the relationship between the strings configuration, the orientation of the opening, and direction of gravity. This re-grasping method in conjunction with our controller can be used to thread the string through a sequence of openings. We evaluated our method in simulation (with simulated sensor noise) and on the Da Vinci surgical robot. Our results suggest that our method is quite robust to errors in sensing, and is capable of real-world threading tasks with the da Vinci robot, where the diameter of the string (3.5mm) and opening (4.9mm) differ by only 1.4 mm.

Minimum-time trajectories for kinematic mobile robots and other planar rigid bodies with finite control sets

This paper presents first attempts at a method for searching for time-optimal trajectories for a general model of mobile robots that includes Dubins and Reeds-Shepp cars, differential-drive robots, and omnidirectional robots as special cases. The paper takes as a starting point recent results by the authors that describe necessary conditions on the trajectories, based on Pontryagins Maximum Principle. These necessary conditions reduce the problem of finding an optimal trajectory between start and goal to a few one-dimensional search problems. This search is not formally guaranteed to find a near-optimal trajectory if the sampling of the search space is not fine enough, but comparison to existing analytical results for specific systems, and a complete numerical search over trajectories with only a few control switches, demonstrates effectiveness of the method.

Sampling Extremal Trajectories for Planar Rigid Bodies

This paper presents an approach to finding the time-optimal trajectories for a simple rigid-body model of a mobile robot in an obstacle-free plane. Previous work has used Pontryagin’s Principle to find strong necessary conditions on time-optimal trajectories of the rigid body; trajectories satisfying these conditions are called extremal trajectories. The main contribution of this paper is a method for sampling the extremal trajectories sufficiently densely to guarantee that for any pair of start and goal configurations, a trajectory can be found that (provably) approximately reaches the goal, approximately optimally; the quality of the approximation is only limited by the availability of computational resources.

Grasping and folding knots

String stretched tightly along a sequence of fixed grasp points takes the shape of a polygonal arc. In this work, we investigate how many points are necessary and sufficient to grasp and tie arbitrary knots while maintaining tension, so that the string remains polygonal. This approach allows reasoning that is entirely geometric, which does not rely on potentially inaccurate dynamic models of the string or detailed knowledge of physical characteristics of the string. Algorithms are proposed to determine the contact locations, and generate the motions needed to tie arbitrary knots. This work shows that a number of grasp points that is linear in the number of crossings in a knot diagram is sufficient to immobilize string in a polygonal shape with the topology of an arbitrary knot, or to fold or unfold the knot from a straight configuration.