Efficient Path Planning in Narrow Passages for Robots With Ellipsoidal Components

Abstract

Path planning has long been one of the major research areas in robotics, with probabilistic roadmap (PRM) and rapidly-exploring random trees (RRT) being two of the most effective classes of planners. Though generally very efficient, these sampling-based planners can become computationally expensive in the important case of "narrow passages." This article develops a path planning paradigm specifically formulated for narrow passage problems. The core is based on planning for rigid-body robots encapsulated by unions of ellipsoids. Each environmental feature is represented geometrically using a strictly convex body with a C 1 boundary (e.g., superquadric). The main benefit of doing this is that configuration-space obstacles can be parameterized explicitly in closed form, thereby allowing prior knowledge to be used to avoid sampling infeasible configurations. Then, by characterizing a tight volume bound for multiple ellipsoids, robot transitions involving rotations are guaranteed to be collision free without needing to perform traditional collision detection. Furthermore, by combining with a stochastic sampling strategy, the proposed planning framework can be extended to solving higher dimensional problems, in which the robot has a moving base and articulated appendages. Benchmark results show that the proposed framework often outperforms the sampling-based planners in terms of computational time and success rate in finding a path through narrow corridors for both single-body robots and those with higher dimensional configuration spaces. Physical experiments using the Manuscript

A Closed-From Parametrization and an Alternative Computational Algorithm for Approximating Slices of Minkowski Sums of Ellipsoids in R³

“…Note that in Minkowski sum formula (3), the ellipsoid E 2 is centered at the origin µ 2 = 0 so that the Minkowski sum also presents the configuration space obstacle of the two ellipsoids. This has many applications in robot motion planning, where the geometries of the robot and each obstacle are approximated by the enclosing ellipsoids and then the configuration space obstacle is calculated as the Minkowski sum of two ellipsoids, when the robot ellipsoid is measured in the body-fixed frame of reference located on the center of the ellipsoid µ = 0 [3].…”

“…however the form of the equation in (3) does not imply the symmetry in formulation, such that x E 1 ⊕E 2 (φ o ) = x E 2 ⊕E 1 (φ o ) for a particular angular parameter φ o . Therefore, Chirikjian et al [11] also reformulated a symmetric closed-form parametrization of the Minkowski sum that is more consistent with the commutative characteristic of the Minkowski sum operator.…”

“…As discussed in previous section, there are several ways to bound (see ( 5),( 7), ( 9), ( 11)) or approximate (see (13) and ( 14)) a Minkowski sum of ellipsoids. Consider an 3 for i ∈ {1, ..., m}, which is centered at µ = µ 1 + µ 2 + ... µ m and is successively described by the explicit and implicit descriptions ∂E = {x(φ) = µ + Cu(φ)} and ∂E = x + µ ∈ R 3 : x T C −2 x = 1 . Recall that u(φ) ∈ S 2 ⊂ R 3 is the unit vector, parametrized by two angles φ = [φ 1 , φ 2 ] and S 2 denotes the unit sphere embedded in R 3 and centered at origin.…”

“…From the second definition (2), one can easily imagine that the Minkowski sum of a geometrical shape (object) and a point is the translated shape by the position vector of that point and similarly the sum of an object and a curve is that object swept along the curve. The concepts of Minkowski sum and slice of Minkowski sum are widely used in vast areas of engineering such as crystallography [1,2], robot motion planing [3,4], computer-aided design [5] and assembly planning [6]. One can show that the Minkowski sum of two convex sets is a convex set.…”

Marching-Primitives: Shape Abstraction from Signed Distance Function

“…Therefore, researchers turn to exploring the possibility of interpreting complex objects and scenes with basic geometric primitives. Taking advantage of the primitive-based representation, many higher-level tasks, such as segmentation [14,16,21,30], scene understanding [29,31,41,47], grasping [33,44,45] and motion planning [35,36], are able to be solved efficiently.…”