Volumetric Ergodic Control

TL;DR

Volumetric ergodic control (VEC) generalizes ergodic control by incorporating a volumetric state representation $g(x,s)$ of the robot body and sensors, enabling coverage that respects physical volume while retaining the Sobolev-motion ergodic metric structure. By redefining the Fourier coefficients as $c_k^{\text{v}}=\frac{1}{T}\int_0^T f_k^{\text{v}}(s(t))\,dt$ with volumetric basis $f_k^{\text{v}}$, VEC preserves asymptotic coverage guarantees and can be solved with standard control optimizers like $\text{iLQR}$ in a receding-horizon setting. The approach supports a sample-based volumetric representation $g(x,s)=\frac{1}{N}\sum_{i=1}^N \delta(x-h_i(s))$, enabling arbitrary geometries and sensor models to be embedded via differentiable mappings $h_i(s)$. Experimental results across double-integrator, differential-drive, quadcopter, and Franka robotic tasks show VEC improves coverage efficiency by over a factor of two while maintaining 100% task success, with modest real-time computational overhead. These findings demonstrate the practical impact of volumetric reasoning for robust, geometry-aware coverage in manipulation and information-gathering tasks.

Abstract

Ergodic control synthesizes optimal coverage behaviors over spatial distributions for nonlinear systems. However, existing formulations model the robot as a non-volumetric point, but in practice a robot interacts with the environment through its body and sensors with physical volume. In this work, we introduce a new ergodic control formulation that optimizes spatial coverage using a volumetric state representation. Our method preserves the asymptotic coverage guarantees of ergodic control, adds minimal computational overhead for real-time control, and supports arbitrary sample-based volumetric models. We evaluate our method across search and manipulation tasks -- with multiple robot dynamics and end-effector geometries or sensor models -- and show that it improves coverage efficiency by more than a factor of two while maintaining a 100% task completion rate across all experiments, outperforming the standard ergodic control method. Finally, we demonstrate the effectiveness of our method on a robot arm performing mechanical erasing tasks.

Figures (6)

• Figure 1: Erasing results using volumetric ergodic control (VEC). The first column shows the target distributions (airplane, Eiffel Tower, fire, heart, lock) and remaining columns show VEC applied with different end-effector geometries (scissors, star, sword, lightning, trophy), which are modeled as volumetric states with the sample-based representation. VEC reasons over the volumetric states and produces efficient and thorough coverage trajectories.
• Figure 2: Qualitative results for erasing benchmark. VEC effectively leverages the geometry of the end-effector through the volumetric state representation, generating both translational and rotational movements to completely erase the target in less time compared to the baseline ergodic control method.
• Figure 3: Validation of control optimization. VEC is compatible with standard control optimization methods, such as iLQR. iLQR consistently minimizes the ergodic metric over time until convergence, across different robot dynamics, volumetric state representations, and randomized initial states. Note that the values of the converged ergodic metric depend on the specific robot dynamics and volumetric state representation, thus should not be directly compared across platforms.
• Figure 4: Erasing benchmark. (Left) VEC achieves $100\%$ task completion in less than half the time required by the baseline, which only completes $17$ out of the $25$ trials. (Right) VEC succeeds in all $25$ trials under 400 timesteps, while the baseline only completes 9 trials within the same amount of time.
• Figure 7: (Top) Qualitative results for ground search benchmark. VEC leverages the volumetric LiDAR model to adapt the coverage trajectory, enabling faster and more reliable target discovery than the baseline ergodic control. (Bottom) Qualitative results for aerial search benchmark. VEC leverages the volumetric ray-casting camera model, enabling the quadcopter to ascend and widen its field of view, thereby completing the search more efficiently than baseline ergodic control, which remains at a constant height and performs sub-optimally.

Theorems & Definitions (10)

• Definition 1: Dirac delta function
• Definition 2: Trajectory empirical distribution
• Definition 3: Normalized Fourier basis functions
• Definition 4: Standard ergodic metric
• Definition 5: Standard ergodic control
• Definition 6: Volumetric state representation
• Definition 7: Volumetric empirical distribution
• Definition 8: Volumetric Fourier basis
• Definition 9: Volumetric ergodic metric
• Definition 10: Volumetric ergodic control