Stein Variational Ergodic Search

TL;DR

The paper tackles the challenge of scalable exploration in continuous spaces by casting coverage as a distribution over trajectories and solving for this distribution via Stein variational gradient descent (SVGD). By coupling ergodic coverage with a probabilistic inference framework, the authors derive an SVGD-based method to generate multiple diverse ergodic trajectories in parallel and to adapt online through model-predictive control. They introduce a discretized trajectory formulation, a posterior that combines an ergodic cost with a prior, and kernel-based repulsion to promote diversity, along with various kernel designs and convergence guarantees. The approach is validated through simulations and real-world drone experiments, showing robust online adaptation, scalability, and a spectrum of complementary exploration strategies. This work enables non-myopic, multimodal exploration in dynamic environments with real-time computational efficiency.

Abstract

Exploration requires that robots reason about numerous ways to cover a space in response to dynamically changing conditions. However, in continuous domains there are potentially infinitely many options for robots to explore which can prove computationally challenging. How then should a robot efficiently optimize and choose exploration strategies to adopt? In this work, we explore this question through the use of variational inference to efficiently solve for distributions of coverage trajectories. Our approach leverages ergodic search methods to optimize coverage trajectories in continuous time and space. In order to reason about distributions of trajectories, we formulate ergodic search as a probabilistic inference problem. We propose to leverage Stein variational methods to approximate a posterior distribution over ergodic trajectories through parallel computation. As a result, it becomes possible to efficiently optimize distributions of feasible coverage trajectories for which robots can adapt exploration. We demonstrate that the proposed Stein variational ergodic search approach facilitates efficient identification of multiple coverage strategies and show online adaptation in a model-predictive control formulation. Simulated and physical experiments demonstrate adaptability and diversity in exploration strategies online.

Figures (12)

• Figure 1: The Stein Variational Ergodic Approach. Robotic exploration is challenging as there are many ways to effectively explore an area that robots need to reason about. Calculating all the possible exploration strategies can be computationally prohibitive, especially when there is no guarantee that optimized solutions will coverage to a diverse set of strategies. We propose to solve this problem by posing coverage and exploration as an inference problem over distributions of trajectories. Our approach leverages ergodic exploration techniques in conjunction with Stein variational methods to efficiently optimize diverse exploration strategies in parallel over continuous domains (see above). Illustrated is a set of $4$ exploration strategies optimized to uniformly explore around the cylinders. The green trajectory indicates the selected best strategy.
• Figure 2: Stein Ergodic Solutions Converge to Similar Ergodicity. (a) Log of mean ergodic losses for $50$ trajectories optimized by the Stein variational ergodic search on a uniform $\mu$. Note the tight bound of 2-standard deviation on the ergodic losses suggests trajectory solutions are close in optimality. (b) Overlap of $50$ ergodic trajectories that provide uniform coverage over the $1m \times 1m$ domain.
• Figure 3: Measured Kernel Ergodic Trajectory Diversity. Illustrated is a comparison of trajectory diversity using the Stein variation ergodic approach and the vanilla ergodic trajectory optimization. (a) Ergodic losses for $20$ trajectories using the proposed Stein variational ergodic method and the vanilla ergodic Trajectories. (b) Trajectory diversity measured using the determinant of the RBF matrix kernel $K_{ij}=k(\mathbf{x}^i, \mathbf{x}^j)$, (where $K_{ii} = 1$ and $k$ is given by Eq. \ref{['eq:rbf']}) with a fixed $h=0.01$. The initial sampled trajectories are drawn from a zero-mean Gaussian distribution with $\sigma^2=0.01$. Values approaching $1$ suggest diversity of trajectories (the Kernel matrix is strongly diagonal with values $1$). Smaller values suggest similar trajectories (the Kernel matrix consists of identical values).
• Figure 4: Effect of Kernel on Ergodic Trajectory Diversity . Trajectories are optimized to provide uniform coverage over domain. All $50$ initial trajectory particles are initialized as an interpolating function between initial and final poses additive zero mean noise with $\sigma^2 = 0.01$. Each (a) Radial basis function (RBF) kernels produce smooth diverse paths. (b) Using Eq. \ref{['eq:markov_kernel']} which relies on the Markov property of trajectories $x(t)$, uniform ergodic trajectories can be arbitrarily diverse. (c) Setting the kernel $k(x_i,x_j) = 1$ reduces the Stein ergodic gradient descent to parallel gradient descent on independent trajectory particles $\mathbf{x}_i$. Note that particles collapse on a single ergodic path.
• Figure 5: The many ways to explore a forest. Here, we demonstrate the many different solutions to uniform exploration of a $100m \times 100m$ forest that our proposed method produces. (a) Trajectories are optimized through the Stein variational ergodic approach starting from the same initial conditions. (b) Each solution is shown to be a locally optimal solution to the ergodic metric that provides uniform coverage which can be used for robust environmental monitoring.

Theorems & Definitions (10)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Theorem 1: Informal
• proof
• Theorem 2
• proof
• Theorem 1
• proof