Fast Ergodic Search with Kernel Functions

TL;DR

The paper addresses the computational bottlenecks of ergodic search in high-dimensional and non-Euclidean spaces by introducing a kernel-based ergodic metric that generalizes to Lie groups with linear complexity. It derives a kernel ergodic control framework, including a Gateaux derivative and an iLQR-based trajectory optimization, and extends the method to SE(3) on Lie groups. Empirical results show at least two orders of magnitude speedup over state-of-the-art methods in 2–6D spaces and successful SE(3) peg-in-hole tasks driven by a 30-second human demonstration, achieving 100% asymptotic success. The approach enables efficient, learning-free ergodic exploration in complex geometries, with potential extensions to non-stationary kernels and RL integration.

Abstract

Ergodic search enables optimal exploration of an information distribution while guaranteeing the asymptotic coverage of the search space. However, current methods typically have exponential computation complexity in the search space dimension and are restricted to Euclidean space. We introduce a computationally efficient ergodic search method. Our contributions are two-fold. First, we develop a kernel-based ergodic metric and generalize it from Euclidean space to Lie groups. We formally prove the proposed metric is consistent with the standard ergodic metric while guaranteeing linear complexity in the search space dimension. Secondly, we derive the first-order optimality condition of the kernel ergodic metric for nonlinear systems, which enables efficient trajectory optimization. Comprehensive numerical benchmarks show that the proposed method is at least two orders of magnitude faster than the state-of-the-art algorithm. Finally, we demonstrate the proposed algorithm with a peg-in-hole insertion task. We formulate the problem as a coverage task in the space of SE(3) and use a 30-second-long human demonstration as the prior distribution for ergodic coverage. Ergodicity guarantees the asymptotic solution of the peg-in-hole problem so long as the solution resides within the prior information distribution, which is seen in the 100% success rate.

Figures (8)

• Figure 1: Trajectories when optimizing the individual and combined elements of the kernel ergodic metric (\ref{['eq:proposed_metric']}). (a) When only optimizing the maximum likelihood estimation term, the system is driven to a (local) maximum of the probability density; (b) When only optimizing the inner product of the trajectory empirical distribution with itself, the system uniformly covers the search space; (c) The kernel metric is the combination of the two elements, optimizing which drives the system to optimally cover the probability distribution.
• Figure 2: (a) Samples from a target distribution. (b) The kernel parameter selection objective function (\ref{['eq:kernel_parameter_obj']}) with the given samples. In this case, the kernel parameter is the value of the diagonal entry in the covariance. (c) A sub-optimal kernel parameter leads to an "over-uniform" coverage behavior. (d) The optimal kernel parameter generates an ergodic trajectory that allocates the time it spends in each region to be proportional to the integrated probability density of the region. (e) Another sub-optimal kernel parameter leads to an "over-concentrated" coverage behavior.
• Figure 3: Illustration of key concepts in the Lie group ergodic search formula. (a) The exponential map maps a Lie algebra element $\tau\in\mathfrak{g}$ to a Lie group element $g\in\mathcal{G}$; The logarithm map is the inverse of the exponential map; The adjoint transformation maps an element from one tangent space (Lie algebra in this case) to an element in another tangent space $\mathcal{T}_{g^\prime}\mathcal{G}$. (b) The difference between two Lie group elements $g_1^{-1}g_2$ is mapped to the Lie algebra $\log(g_1^{-1}g_2)$ through the logarithm map, which allows us to use the Euclidean space formula to define the quadratic function on the Lie group. (c) The Lie group Gaussian distribution is defined in the tangent space of the mean $\bar{g}$. The probability density function is evaluated as a zero-mean Euclidean Gaussian distribution $\mathcal{N}(\mathbf{0},\Sigma)$ over the Lie group sample $g$ in the tangent space $\log(\bar{g}^{-1}g)$. (d) Dynamics is defined through the left-trivialization in the Lie algebra $\lambda:\mathcal{G}\times\mathbb{R}^n\times\mathbb{R}_0^+\mapsto\mathfrak{g}$, which is mapped back to propagate the Lie group system state through the exponential map $\exp(\Delta t{\cdot}\lambda(t))$. The dynamics is defined as continuous, but the Lie group trajectory is integrated piece-wise.
• Figure 4: Comparison of the scalability of different methods. The proposed method exhibits a linear complexity across 2 to 6-dimensional spaces, while the Fourier metric-based methods, even if accelerated by tensor-train, exhibit a close-to-exponential complexity.
• Figure 5: Example ergodic trajectory generated by the proposed algorithm in 6-dimensional space, with second-order system dynamics. The trajectory overlaps the marginalized target distribution.

Theorems & Definitions (53)

• Definition 1: Inner product
• Definition 2: Dirac delta function
• Remark 1
• Lemma 1
• Definition 3: Trajectory empirical distribution
• Lemma 2
• Lemma 3
• Definition 4: Exact ergodic metric
• Lemma 4: Asymptotic coverage
• Definition 5: Fourier basis function