Robotic search and rescue, exploration, and inspection require trajectory planning across a variety of domains. A popular approach to trajectory planning for these types of missions is ergodic search, which biases a trajectory to spend time in parts of the exploration domain that are believed to contain more information. Most prior work on ergodic search has been limited to searching simple surfaces, like a 2D Euclidean plane or a sphere, as they rely on projecting functions defined on the exploration domain onto analytically obtained Fourier basis functions. In this paper, we extend ergodic search to any surface that can be approximated by a triangle mesh. The basis functions are approximated through finite element methods on a triangle mesh of the domain. We formally prove that this approximation converges to the continuous case as the mesh approximation converges to the true domain. We demonstrate that on domains where analytical basis functions are available (plane, sphere), the proposed method obtains equivalent results, and while on other domains (torus, bunny, wind turbine), the approach is versatile enough to still search effectively. Lastly, we also compare with an existing ergodic search technique that can handle complex domains and show that our method results in a higher quality exploration.

In this paper, we address the problem of safe trajectory planning for autonomous search and exploration in constrained, cluttered environments. Guaranteeing safe (collision-free) trajectories is a challenging problem that has garnered significant due to its importance in the successful utilization of robots in search and exploration tasks. This work contributes a method that generates guaranteed safety-critical search trajectories in a cluttered environment. Our approach integrates safety-critical constraints using discrete control barrier functions (DCBFs) with ergodic trajectory optimization to enable safe exploration. Ergodic trajectory optimization plans continuous exploratory trajectories that guarantee complete coverage of a space. We demonstrate through simulated and experimental results on a drone that our approach is able to generate trajectories that enable safe and effective exploration. Furthermore, we show the efficacy of our approach for safe exploration using real-world single- and multi- drone platforms.