Kernel-Based Ergodic Search for Scalable Non-Euclidean Coverage
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Solution Overview
Problem
Current ergodic search methods have exponential computation complexity, are restricted to Euclidean space, and are impractical for higher dimensional searches, lacking scalability and generalization to non-Euclidean spaces.
Innovation Solution
A kernel-based ergodic search method that generates a kernel-based ergodic metric and gradient, allowing for efficient trajectory planning in higher dimensions and non-Euclidean spaces, ensuring asymptotic coverage and optimal information maximization.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If standard ergodic search methods are used, then asymptotic coverage of search space is guaranteed, but computation complexity becomes exponential in search space dimension
Solution Approach 1:
The patent transforms the ergodic search metric from its standard form to a kernel-based form by changing the mathematical parameters and representation. This transformation maintains the asymptotic coverage guarantee while reducing computation complexity from exponential to polynomial in the search space dimension, making high-dimensional searches feasible
Solution Approach 2:
The patent replaces the standard ergodic metric computation mechanism with a kernel-based mechanism that uses positive definite kernels. This substitution fundamentally changes how the metric is calculated, avoiding the exponential complexity of the original approach while preserving the theoretical guarantees of ergodic search
2Reliability
If standard ergodic search is applied to higher dimensional spaces, then search coverage is maintained, but computation resources increase exponentially
Solution Approach 1:
By changing the metric formulation to use kernel functions with specific parameters, the patent enables efficient computation in high-dimensional spaces. The kernel-based approach with properly selected parameters maintains search coverage while reducing computational resource requirements from exponential to manageable levels
Solution Approach 2:
The patent effectively handles high-dimensional search spaces by transforming the problem through kernel functions that operate in feature spaces. This dimensional transformation allows the search to proceed efficiently even when the original search space has many dimensions, avoiding the curse of dimensionality
3Reliability
If standard ergodic metric is used, then theoretical coverage is achieved, but planning horizon must be infinitesimally small requiring impractically long exploration period
Solution Approach 1:
The kernel-based ergodic metric incorporates information about the entire trajectory and target distribution in advance, allowing the controller to plan over finite horizons. This preliminary incorporation of distributional information eliminates the need for infinitesimally small planning horizons and impractically long exploration periods while maintaining coverage guarantees
4Reliability
If standard ergodic metric is scaled to higher dimensions, then search space coverage is maintained, but computational cost increases significantly
Solution Approach 1:
The patent changes the fundamental parameters of the ergodic metric by introducing kernel functions. This parameter transformation maintains the coverage property while dramatically improving computational efficiency in higher dimensions, converting an intractable problem into a feasible one
5Reliability
If standard ergodic metric is applied, then Euclidean space search is effective, but generalization to non-Euclidean spaces is not trivial
Solution Approach 1:
The kernel-based ergodic metric is designed to be universal and applicable to both Euclidean and non-Euclidean spaces. By using positive definite kernels that can be defined on various spaces, the metric maintains its effectiveness across different geometric contexts, enabling seamless generalization from Euclidean to non-Euclidean domains
Solution Approach 2:
The patent introduces kernel functions as intermediaries that bridge Euclidean and non-Euclidean spaces. These kernels serve as a universal language that allows the ergodic metric to operate effectively on different types of spaces without requiring space-specific formulations, thus enabling generalization
Data Source
AI summary
According to one aspect, kernel-based ergodic search using a robot may include receiving a target distribution indicative of a desired ergodic search coverage, generating a kernel-based ergodic metric based on the target distribution and a candidate trajectory, generating a kernel-based ergodic gradient based on the kernel-based ergodic metric, generating a trajectory based on the kernel-based ergodic gradient, and implementing the trajectory for the robot.