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.

翻译：本文针对无人机搜索与覆盖任务，提出一种基于不确定性地图的轨迹规划算法。该算法采用高斯混合模型对目标区域的先验知识进行建模。轨迹规划问题被表述为最优控制问题，其目标是在指定任务时长内最大化不确定性降低程度。然而，这导致了一个难以处理的最优控制问题，其目标函数无法以闭合形式表达。为此，我们提出一种基于目标函数松弛表述的模型预测控制算法来逼近最优解。该松弛方法通过惩罚无人机沿轨迹飞行时可见区域的重叠，促进对地图的高效探索。该算法能够生成高效平滑的轨迹，并可通过标准非线性规划求解器高效实现，适用于实时规划。与传统方法通常依赖任务空间离散化和复杂混合整数规划不同，本方法计算效率更高且更易实现。该模型预测控制算法首先在MATLAB中进行评估，随后在Gazebo仿真环境和户外实际实验场景中进行测试。结果表明，所提策略能够为搜索与覆盖任务生成高效平滑的轨迹。