Competitive Complexity of Mobile Robot on-Line Motion Planning Problems

Abstract

This paper classifies common mobile robot on-line motion planning problems according to their competitive complexity. The competitiveness of an on-line algorithm measures its worst case performance relative to the optimal off-line solution to the problem. Competitiveness usually means constant relative performance. This paper generalizes competitiveness to any functional relationship between on-line performance and optimal off-line solution. The constants in the functional relationship must be scalable and may depend only upon on-line information. Given an on-line task, its competitive complexity class is a pair of lower and upper bounds on the competitive performance of all on-line algorithms for the task, such that the two bounds satisfy the same functional relationship. The paper classifies the following on-line motion planning problems into competitive classes: area coverage, navigation to a target, and on-line search for an optimal path. In particular, it is shown that navigation to a target whose position is either apriori known or recognized upon arrival belongs to a quadratic competitive complexity class. The hardest on-line problem involves navigation in unknown variable traversibility environments. Under certain restriction on traversibility, this last problem belongs to an exponential competitive complexity class.

Mapping and Pursuit-Evasion Strategies For a Simple Wall-Following Robot

“…Examples include sensorless manipulation [12], [16], [18], [21], [41], bug strategies [27], [28], [35], [42], and gap navigation trees: [33], [45], [62]. On-line exploration strategies make simple motion models and try to reduce the amount of memory or total distance traveled [5], [10], [19], [20], [29], [30], [43], [50], [56].…”

Planning Under Sensing Uncertainty

“…In some contexts, the ratio may still be useful if expressed as a function of the representation. For example, if E is a polygon with n edges, then an O( √ n) competitive ratio means that (12.28) is bounded over all n by c √ n for some c ∈ R. For competitive ratio analysis in the context of bug algorithms, see [65].…”

“…The issue of distinguishability and pebbles arises again in [9,46,47,116,159,185]. For more on competitive ratios and combinatorial approaches to on-line navigation, see [13,38,40,60,65,90,91,119,139,153].…”

Searching and mapping among indistinguishable convex obstacles

“…These paper focuses on developing systematic exploration strategies for robots moving in unknown environments [4], [9], [10]. This is inspired from the lost-cow problem, in which a cow moves along a fence trying to find a gate to access a pasture.…”

“…More importantly, the memory required for the algorithms is constant. Such online exploration strategies make simple motion models and attempt to reduce the amount of memory or total distance traveled [3], [5], [6], [8], [9], [14], [15], [18], [19], [23].…”