An Efficient Algorithm for Computing High-Quality Paths amid Polygonal Obstacles

Abstract

We study a path-planning problem amid a set O of obstacles in R 2 , in which we wish to compute a short path between two points while also maintaining a high clearance from O; the clearance of a point is its distance from a nearest obstacle in O. Specifically, the problem asks for a path minimizing the reciprocal of the clearance integrated over the length of the path. We present the first polynomial-time approximation scheme for this problem. Let n be the total number of obstacle vertices and let ε ∈ (0, 1]. Our algorithm computes in time O (n 2 ε 2 log n ε) a path of total cost at most (1 + ε) times the cost of the optimal path. CCS Concepts: • Theory of computation → Approximation algorithms analysis; Computational geometry; • Computing methodologies → Robotic planning;

Adaptively Informed Trees (AIT*) and Effort Informed Trees (EIT*): Asymmetric bidirectional sampling-based path planning

“…The lower limit on δ ensures numerical stability and that the cost of a path is bounded by a multiple of its total variation as in Section 2.1.2. This optimization objective balances the clearance and length of a path and is similar to the objectives presented by Wein et al (2008) and Agarwal et al (2018).…”

Toward certifiable optimal motion planning for medical steerable needles

“…This cost function is by nature well-behaved given c min = c max = 1, L c = 0.(ii) Cost map: this cost function is informed by a cost map derived from medical images (Fu et al, 2018), where each voxel in the 3D cost map is associated with a cost value that represents tissue damage. We forced c min = 0.01 and then used trilinear interpolation to smooth out the voxelized cost map to make it well-behaved.(iii) Obstacle clearance: Cost function ∫0ℓcl(σ(s)) -1 d s , where cl(·) is the clearance from obstacles, has been widely used (Agarwal et al, 2018; Kuntz et al, 2015; Strub and Gammell, 2021; Wein et al, 2008) since it captures both trajectory length and clearance from obstacles. Here, we modify the point-based cost to be c ( x ) = min{cl( x ) -1 , c max }, forcing the cost not to exceed c max = (0.1 mm) −1 to make it well-behaved.…”

“…(iii) Obstacle clearance: Cost function ∫0ℓcl(σ(s)) -1 d s , where cl(·) is the clearance from obstacles, has been widely used (Agarwal et al, 2018; Kuntz et al, 2015; Strub and Gammell, 2021; Wein et al, 2008) since it captures both trajectory length and clearance from obstacles. Here, we modify the point-based cost to be c ( x ) = min{cl( x ) -1 , c max }, forcing the cost not to exceed c max = (0.1 mm) −1 to make it well-behaved.…”

“…To this end, a motion planner should first guarantee that it will compute a solution, when one exists, in finite time, or notify the user that no solution exists. Moreover, the computed solution should strive to maximize patient safety, which can be quantified using metrics such as minimizing trajectory length ( Favaro et al, 2018 ), maximizing clearance from obstacles ( Agarwal et al, 2018 ; Kuntz et al, 2015 ; Strub and Gammell, 2021 ; Wein et al, 2008 ), and minimizing damage to sensitive tissue ( Bentley et al, 2021 ; Fu et al, 2018 ). Thus, a motion planner for steerable needles should be complete , that is, find a solution plan in a finite number of steps, if one exists, and ideally should be optimal , that is, ensure that the returned plan has a cost (for a given cost metric) that is close to the global optimum.…”