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Model-Predictive Trajectory Generation for Aerial Search and Coverage

Hugo Matias, Daniel Silvestre

TL;DR

The paper tackles UAV trajectory planning for search and coverage by leveraging an uncertainty map modeled as a Gaussian Mixture Model and formulating the goal as maximizing uncertainty reduction under mission time. It introduces a Model Predictive Control framework with a relaxed objective that combines horizon-based information gain and a smooth exponential penalty to discourage overlaps of sensing footprints, enabling real-time optimization with standard nonlinear solvers. Key contributions include the relaxed MPC formulation, a scalable penalty design that avoids mixed-integer programming, and comprehensive validation in MATLAB, Gazebo, and outdoor experiments. The approach yields efficient, smooth trajectories suitable for real-time planning in practical search and coverage tasks, with potential extensions to time-varying maps and adaptive weighting strategies.

Abstract

This paper introduces a trajectory planning algorithm for search and coverage missions with an Unmanned Aerial Vehicle (UAV) based on an uncertainty map that represents prior knowledge of the target region, modeled by a Gaussian Mixture Model (GMM). The trajectory planning problem is formulated as an Optimal Control Problem (OCP), which aims to maximize the uncertainty reduction within a specified mission duration. However, this results in an intractable OCP whose objective functional cannot be expressed in closed form. To address this, we propose a Model Predictive Control (MPC) algorithm based on a relaxed formulation of the objective function to approximate the optimal solutions. This relaxation promotes efficient map exploration by penalizing overlaps in the UAV's visibility regions along the trajectory. The algorithm can produce efficient and smooth trajectories, and it can be efficiently implemented using standard Nonlinear Programming solvers, being suitable for real-time planning. Unlike traditional methods, which often rely on discretizing the mission space and using complex mixed-integer formulations, our approach is computationally efficient and easier to implement. The MPC algorithm is initially assessed in MATLAB, followed by Gazebo simulations and actual experimental tests conducted in an outdoor environment. The results demonstrate that the proposed strategy can generate efficient and smooth trajectories for search and coverage missions.

Model-Predictive Trajectory Generation for Aerial Search and Coverage

TL;DR

The paper tackles UAV trajectory planning for search and coverage by leveraging an uncertainty map modeled as a Gaussian Mixture Model and formulating the goal as maximizing uncertainty reduction under mission time. It introduces a Model Predictive Control framework with a relaxed objective that combines horizon-based information gain and a smooth exponential penalty to discourage overlaps of sensing footprints, enabling real-time optimization with standard nonlinear solvers. Key contributions include the relaxed MPC formulation, a scalable penalty design that avoids mixed-integer programming, and comprehensive validation in MATLAB, Gazebo, and outdoor experiments. The approach yields efficient, smooth trajectories suitable for real-time planning in practical search and coverage tasks, with potential extensions to time-varying maps and adaptive weighting strategies.

Abstract

This paper introduces a trajectory planning algorithm for search and coverage missions with an Unmanned Aerial Vehicle (UAV) based on an uncertainty map that represents prior knowledge of the target region, modeled by a Gaussian Mixture Model (GMM). The trajectory planning problem is formulated as an Optimal Control Problem (OCP), which aims to maximize the uncertainty reduction within a specified mission duration. However, this results in an intractable OCP whose objective functional cannot be expressed in closed form. To address this, we propose a Model Predictive Control (MPC) algorithm based on a relaxed formulation of the objective function to approximate the optimal solutions. This relaxation promotes efficient map exploration by penalizing overlaps in the UAV's visibility regions along the trajectory. The algorithm can produce efficient and smooth trajectories, and it can be efficiently implemented using standard Nonlinear Programming solvers, being suitable for real-time planning. Unlike traditional methods, which often rely on discretizing the mission space and using complex mixed-integer formulations, our approach is computationally efficient and easier to implement. The MPC algorithm is initially assessed in MATLAB, followed by Gazebo simulations and actual experimental tests conducted in an outdoor environment. The results demonstrate that the proposed strategy can generate efficient and smooth trajectories for search and coverage missions.
Paper Structure (24 sections, 23 equations, 18 figures)

This paper contains 24 sections, 23 equations, 18 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Paper Overview
  4. Notation
  5. Preliminaries and Problem Formulation
  6. Uncertainty Map
  7. Sensing Model
  8. Optimal Control Problem
  9. MPC Approach with Relaxed Formulation
  10. Objective Function
  11. Penalty Function
  12. Computational Complexity
  13. Evaluation Metric
  14. Quadrotor Motion Control
  15. Full Dynamics Model
  16. ...and 9 more sections

Figures (18)

  • Figure 1: Example of an uncertainty map.
  • Figure 2: Sensor and visibility region.
  • Figure 3: Illustration of the set $\mathcal{C}_r(\bm{\gamma})$.
  • Figure 4: Illustration of the observation regions at the discrete-time instant $k = 3$ for a horizon length $N = 4$ (grey - previously covered circles; white - predicted observation circles).
  • Figure 5: Evolution of the penalty function with the distance between the two circles for some values of $\alpha$ with $r = 0.5$ m.
  • ...and 13 more figures