Mobile Robot Sensory Coverage in 2-D Environments: An Optimization Approach with Efficiency Bounds

TL;DR

This paper addresses planning for a mobile robot to observe multiple targets in 2-D using a fixed-radius sensor. It formulates three related NP-hard MINLP problems and develops polynomial-time two-stage approximation algorithms that first fix a sensing-node visitation order and then optimize sensing-node positions via convex programming, including multi-target views and obstacle occlusion. It derives explicit path-length bounds relating approximate solutions to the optimal path and demonstrates scalable execution times with full MATLAB implementations. The work provides a rigorous foundation for provably efficient coverage planning and discusses extensions to uncertain targets, unknown obstacles, and 3-D sensing.

Abstract

This paper considers three related mobile robot multi-target sensory coverage and inspection planning problems in 2-D environments. In the first problem, a mobile robot must find the shortest path to observe multiple targets with a limited range sensor in an obstacle free environment. In the second problem, the mobile robot must efficiently observe multiple targets while taking advantage of multi-target views in an obstacle free environment. The third problem considers multi-target sensory coverage in the presence of obstacles that obstruct sensor views of the targets. We show how all three problems can be formulated in a MINLP optimization framework. Because exact solutions to these problems are NP-hard, we introduce polynomial time approximation algorithms for each problem. These algorithms combine polynomial-time methods to approximate the optimal target sensing order, combined with efficient convex optimization methods that incorporate the constraints posed by the robot sensor footprint and obstacles in the environment. Importantly, we develop bounds that limit the gap between the exact and approximate solutions. Algorithms for all problems are fully implemented and illustrated with examples. Beyond the utility of our algorithms, the bounds derived in the paper contribute to the theory of optimal coverage planning algorithms.

Figures (22)

• Figure 1: The problem geometry showing circular sensing regions centered at the targets and typical robot sensor footprints with detection range $r$.
• Figure 2: A non-valid sensory coverage path starts at $S_1 \!=\! T_1$ and ends at $S_9 \!\in\! R(T_9)$. The disjoint loop through the sensing nodes $W \!=\! \{ S_5,S_6,S_7,S_8 \}$ does not satisfy the constraint $\sum_{i\in W,j\in W} \xi_{ij}$$\!\leq\! 3$.
• Figure 3: MINLP solution using BONMIN for eight targets that lie at the centers of the red sensing circles. (a) Initial path guess, and (b) the optimal sensory coverage path where each sensing node (blue dots) lies in a target centered sensing circle. (c) Another initial path guess with different visitation order for $T_8$, and (d) the MINLP solution gives the same globally optimal path.
• Figure 4: Optimized sensory coverage path computed by Algorithm \ref{['alg:s_t_heuristic']} for $18$ targets surrounded by sensing circles whose radii equal the robot detection range, $r=0.49$. The sensory coverage path is the black curve while the $18$ blue dots indicate the optimized sensing node locations.
• Figure 5: Geometry of a worst case ratio of path length through the targets, $l_{TSP}$, to the optimal sensory coverage path length $l_{opt}$ (see appendix).