Ergodic Trajectory Optimization on Generalized Domains Using Maximum Mean Discrepancy

TL;DR

This work addresses the limitation of domain-specific ergodic trajectory optimization by introducing a maximum mean discrepancy (MMD) based ergodic metric that operates on samples from any domain. By embedding distributions into a reproducing kernel Hilbert space and using a domain-aware kernel, the method defines an ergodic objective that can be optimized with standard solvers while accommodating Lie groups and differential kinematic constraints. The approach demonstrates competitive ergodicity across diverse domains (SE(3) objects, bunnies, turbines) without requiring explicit utility maps or basis function computation, albeit with quadratic scaling in the number of domain samples. Overall, the method broadens the applicability of ergodic coverage to general objects and environments and offers a practical trade-off between generality and computational cost for robotic exploration tasks.

Abstract

We present a novel formulation of ergodic trajectory optimization that can be specified over general domains using kernel maximum mean discrepancy. Ergodic trajectory optimization is an effective approach that generates coverage paths for problems related to robotic inspection, information gathering problems, and search and rescue. These optimization schemes compel the robot to spend time in a region proportional to the expected utility of visiting that region. Current methods for ergodic trajectory optimization rely on domain-specific knowledge, e.g., a defined utility map, and well-defined spatial basis functions to produce ergodic trajectories. Here, we present a generalization of ergodic trajectory optimization based on maximum mean discrepancy that requires only samples from the search domain. We demonstrate the ability of our approach to produce coverage trajectories on a variety of problem domains including robotic inspection of objects with differential kinematics constraints and on Lie groups without having access to domain specific knowledge. Furthermore, we show favorable computational scaling compared to existing state-of-the-art methods for ergodic trajectory optimization with a trade-off between domain specific knowledge and computational scaling, thus extending the versatility of ergodic coverage on a wider application domain.

Figures (8)

• Figure 1: Ergodic trajectory optimization via maximum mean discrepancy enables robotic search over arbitrary objects. Illustration of our approach for generating ergodic trajectories for inspection of the bunny (red areas are of more importance) while respecting differential kinematic constraints of the Franka panda robot. Note only samples from the bunny surface are needed to compute trajectories.
• Figure 2: Ergodic trajectory optimization via MMD procedure. (a) A typical robotic inspection setting where the goal is to scan a object (box) while respecting robot constraints. (b) Generated samples (e.g., from a depth camera) with inspection areas of higher importance shown in red. (c) Ergodic trajectories optimized using the ergodic MMD metric \ref{['eq:mmd_empirical']} from samples with differential kinematics constraints (in joint space). Samples are shown offset the box used to implicitly induce collision avoidance constraints. (d) Executed ergodic trajectories inspecting the box uniformly across the two-faces.
• Figure 3: Comparison of ergodic search methods. (a) Ergodic trajectories optimized using the derived kernel via the $L^2$-norm sun2024fast. (b) Ergodic trajectories optimized using MMD (ours) directly on domain samples. Grey areas indicate high-information; both approaches generated comparable levels of ergodic coverage $\mathcal{E}_a=-5.78$, $\mathcal{E}_b-6.23$ (lower is better) according to MMD metric; however our approach did not require gradient information of the underlying distribution. Note discrepancy in ergodicity due to kernel approximation of metric integral over samples. (c-d) Ergodic control methods based on one-step control optimization mathew2011ivic2022. These approaches require more time to achieve similar levels of ergodicity due to their formulation. (d) Requires computing a partial differential equation at each step on a mesh on the search domain.
• Figure 4: Computational scale analysis. (Left) Computation time with respect to increasing time horizon. (Right) Computation time with respect to increasing state dimension. Comparisons are done with respect to fast ergodic kernel method sun2024fast. Both short comparable performance where our approach is approximately $2\times$ more computationally expensive (due to the kernel approximation of expectations) with respect to state-dimensionality, though the scale order is polynomial for both methods.
• Figure 5: Uniform coverage over bunny. (a) Uniformly sampled points over bunny mesh. (b) Uniform coverage Traveling Salesperson solution over the sampled points. (c) Uniform ergodic trajectories optimized using MMD (ours) directly on mesh samples. Note that our approach is able to generate uniform coverage trajectories that respect path smoothness constraints globally.

Theorems & Definitions (3)

• Definition 1
• Definition 2
• Theorem 1: Weak Convergence, simon2023, Theorem 7