Soft subdivision motion planning for complex planar robots

Abstract

The design and implementation of theoretically-sound robot motion planning algorithms is challenging. Within the framework of resolution-exact algorithms, it is possible to exploit soft predicates for collision detection. The design of soft predicates is a balancing act between easily implementable predicates and their accuracy/effectivity. In this paper, we focus on the class of planar polygonal rigid robots with arbitrarily complex geometry. We exploit the remarkable decomposability property of soft collision-detection predicates of such robots. We introduce a general technique to produce such a decomposition. If the robot is an m-gon, the complexity of this approach scales linearly in m. This contrasts with the O(m 3) complexity known for exact planners. It follows that we can now routinely produce soft predicates for any rigid polygonal robot. This results in resolution-exact planners for such robots within the general Soft Subdivision Search (SSS) framework. This is a significant advancement in the theory of sound and complete planners for planar robots. We implemented such decomposed predicates in our open-source Core Library. The experiments show that our algorithms are effective, perform in real time on non-trivial environments, and can outperform many sampling-based methods.

Rods and Rings: Soft Subdivision Planner for R^3 x S^2

“…Since resolution-exactness delivers stronger guarantees than probabilistic-completeness, we expect a performance hit compared to sampling methods. But for simple planar robots (up to XX:15 4 DOFs) [31,19,34,38] we observed no such trade-offs because we outperform state-of-the-art sampling methods (such as OMPL [30]) often by two orders of magnitude. But in the 5-DOF robots of this paper, we see that our performance is competitive with sampling methods.…”

“…We say Fp(B) is "nice" if there are intersection algorithms that are easy to implement (desideratum G2) and practically efficient (desideratum G3). We now formalize and generalize some "niceness" properties of Fp(B) that were implicit in our previous work ([31, 19, 34], especially [38]).…”

“…We are able to do this because of the twin foundations of resolution-exactness and soft-predicates. Although we had already used this foundation to implement a variety of planar robots [31,19,34,38] that can match or surpass state-of-the-art sampling methods, it was by no means assured that we can extend this success to 3D robots. Indeed, the present work required a series of technical innovations: (I) One major technical difference from our previous work on planar robots is that we had to give up the notion of "forbidden orientations" (which seems 'forbidding' for 3D robots).…”

“…Our definitions of Fp(B) in [31,19,34] are all Π 1 -sets. But in [38], we make a further extension: define a Σ 2 -set to be a finite union of the Π 1 -sets, i.e., each Σ 2 -set S has the form…”

On Heaviside Binary Logic: a Mathematical Characterization and its Applications

“…Considering(19), introduce the variable z = α n (σ α + δ α )(−f n ) − ξ and consider that any term d k d k z, k ≥ 2 is zero, therefore it will not be written. As the same time, it can be reasonably assumed that the perturbation is zero at the border and as a consequence terms which are borderrelated are zero too.…”