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Adaptive Probabilistic Planning for the Uncertain and Dynamic Orienteering Problem

Qiuchen Qian, Yanran Wang, David Boyle

TL;DR

The paper introduces the Uncertain and Dynamic Orienteering Problem (UDOP), where edge costs are distributions with unknown, time-varying parameters that also influence prizes and prize-collection costs. It proposes ADAPT, a Bayesian online planning framework that alternates offline path initialization, execution-time observations, and online re-planning using a Normal-Gamma prior and posterior Student-t predictions to derive a safety belief, converting UDOP into a series of deterministic problems solved by an improved Ant Colony System with inheritance. Empirical results in UAV charging scheduling for Wireless Rechargeable Sensor Networks show ADAPT achieving a $100\%$ Mission Success Rate across tested scenarios, with robust solution quality and computation times comparable to baselines; in contrast, competing methods exhibit instability under challenging conditions. The approach enables reliable real-world deployment of autonomous UAV CSPs in uncertain and dynamic environments and offers avenues for extending to prize-collection uncertainty, multi-UAV teams, and broader IoT contexts.

Abstract

The Orienteering Problem (OP) is a well-studied routing problem that has been extended to incorporate uncertainties, reflecting stochastic or dynamic travel costs, prize-collection costs, and prizes. Existing approaches may, however, be inefficient in real-world applications due to insufficient modeling knowledge and initially unknowable parameters in online scenarios. Thus, we propose the Uncertain and Dynamic Orienteering Problem (UDOP), modeling travel costs as distributions with unknown and time-variant parameters. UDOP also associates uncertain travel costs with dynamic prizes and prize-collection costs for its objective and budget constraints. To address UDOP, we develop an ADaptive Approach for Probabilistic paThs - ADAPT, that iteratively performs 'execution' and 'online planning' based on an initial 'offline' solution. The execution phase updates system status and records online cost observations. The online planner employs a Bayesian approach to adaptively estimate power consumption and optimize path sequence based on safety beliefs. We evaluate ADAPT in a practical Unmanned Aerial Vehicle (UAV) charging scheduling problem for Wireless Rechargeable Sensor Networks. The UAV must optimize its path to recharge sensor nodes efficiently while managing its energy under uncertain conditions. ADAPT maintains comparable solution quality and computation time while offering superior robustness. Extensive simulations show that ADAPT achieves a 100% Mission Success Rate (MSR) across all tested scenarios, outperforming comparable heuristic-based and frequentist approaches that fail up to 70% (under challenging conditions) and averaging 67% MSR, respectively. This work advances the field of OP with uncertainties, offering a reliable and efficient approach for real-world applications in uncertain and dynamic environments.

Adaptive Probabilistic Planning for the Uncertain and Dynamic Orienteering Problem

TL;DR

The paper introduces the Uncertain and Dynamic Orienteering Problem (UDOP), where edge costs are distributions with unknown, time-varying parameters that also influence prizes and prize-collection costs. It proposes ADAPT, a Bayesian online planning framework that alternates offline path initialization, execution-time observations, and online re-planning using a Normal-Gamma prior and posterior Student-t predictions to derive a safety belief, converting UDOP into a series of deterministic problems solved by an improved Ant Colony System with inheritance. Empirical results in UAV charging scheduling for Wireless Rechargeable Sensor Networks show ADAPT achieving a Mission Success Rate across tested scenarios, with robust solution quality and computation times comparable to baselines; in contrast, competing methods exhibit instability under challenging conditions. The approach enables reliable real-world deployment of autonomous UAV CSPs in uncertain and dynamic environments and offers avenues for extending to prize-collection uncertainty, multi-UAV teams, and broader IoT contexts.

Abstract

The Orienteering Problem (OP) is a well-studied routing problem that has been extended to incorporate uncertainties, reflecting stochastic or dynamic travel costs, prize-collection costs, and prizes. Existing approaches may, however, be inefficient in real-world applications due to insufficient modeling knowledge and initially unknowable parameters in online scenarios. Thus, we propose the Uncertain and Dynamic Orienteering Problem (UDOP), modeling travel costs as distributions with unknown and time-variant parameters. UDOP also associates uncertain travel costs with dynamic prizes and prize-collection costs for its objective and budget constraints. To address UDOP, we develop an ADaptive Approach for Probabilistic paThs - ADAPT, that iteratively performs 'execution' and 'online planning' based on an initial 'offline' solution. The execution phase updates system status and records online cost observations. The online planner employs a Bayesian approach to adaptively estimate power consumption and optimize path sequence based on safety beliefs. We evaluate ADAPT in a practical Unmanned Aerial Vehicle (UAV) charging scheduling problem for Wireless Rechargeable Sensor Networks. The UAV must optimize its path to recharge sensor nodes efficiently while managing its energy under uncertain conditions. ADAPT maintains comparable solution quality and computation time while offering superior robustness. Extensive simulations show that ADAPT achieves a 100% Mission Success Rate (MSR) across all tested scenarios, outperforming comparable heuristic-based and frequentist approaches that fail up to 70% (under challenging conditions) and averaging 67% MSR, respectively. This work advances the field of OP with uncertainties, offering a reliable and efficient approach for real-world applications in uncertain and dynamic environments.
Paper Structure (24 sections, 2 theorems, 13 equations, 7 figures, 7 tables, 3 algorithms)

This paper contains 24 sections, 2 theorems, 13 equations, 7 figures, 7 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Problem Formulation
  4. UDOP formulation
  5. CSP formulation
  6. System Design
  7. Prior knowledge for edge costs
  8. Offline planning phase
  9. Execution phase
  10. Online planning phase
  11. Experiments, Results and Discussion
  12. Parameter setting
  13. The online planning phase with a Bayesian approach
  14. ADAPT performance analysis of computation time, mission safety and solution quality
  15. Mission Success Rate
  16. ...and 9 more sections

Key Result

Theorem 1

The estimated average power consumption can be modeled as a normal distribution with mean $\mu_{\mathrm{\bar{P}}}=\mu(b_1) \sqrt{\frac{m^3}{\rho}} + \mu(b_0)$ and variance $\sigma^2_{\mathrm{\bar{P}}^*} = \sigma^2(b_1) \frac{m^3}{\rho} + \sigma^2(b_0)$.

Figures (7)

  • Figure 1: Linear regression models, i.e., Reg-Model-R rodrigues2022drone and Reg-Model-A alyassi2022autonomous, for estimating real-time power of DJI M100 dji2016m100.
  • Figure 2: ADAPT framework. The UAV follows an initial path (generated offline), sequentially servicing sensor nodes. During flight, the UAV continuously logs power consumption, which informs subsequent planning triggered upon completing the task at each node. The iterative process of execution and online planning phases continues until the UAV returns to the end depot once all target nodes are recharged, or the residual energy is insufficient to continue.
  • Figure 3: An example of how ADAPT updates posterior distributions using online observations. Nine re-plannings happened during this mission, moving from the most diverged distribution (Post1) to the most centralized one (Post9).
  • Figure 4: An example of ADAPT solving California20 with $\Delta\mu_{\mathrm{\bar{P}}^*}= 20\%$ and $\Delta\sigma_{\mathrm{\bar{P}}^*} = 0\%$. When the UAV recharged sensors 4 and 15, its residual energy (budget) is 283.08 kJ. Here we show four typical candidate paths of ADAPT with safety belief $\Theta\in \{70, 80, 90, 99\}\%$.
  • Figure 5: Mission success rate over 50 executions.
  • ...and 2 more figures

Theorems & Definitions (4)

  • Theorem 1
  • proof
  • Theorem 2
  • proof