Banach-Mazur Distance from ℓp3\ell_p^3 to ℓ∞3\ell_\infty^3

The maximum of the Banach-Mazur distance $d_{BM}^M(X,\ell_\infty^n)$, where $X$ ranges over the set of all $n$-dimensional real Banach spaces, is difficult to compute. In fact, it is already not easy to get the maximum of $d_{BM}^M(\ell_p^n,\ell_\infty^n)$ for all $p\in [1,\infty]$. We prove that $d_{BM}^M(\ell_p^3,\ell_\infty^3)\leq 9/5,~\forall p\in[1,\infty]$. As an application, the following result related to Borsuk's partition problem in Banach spaces is obtained: any subset $A$ of $\ell_p^3$ having diameter $1$ is the union of $8$ subsets of $A$ whose diameters are at most $0.9$

Resource-Efficient Cooperative Online Scalar Field Mapping via Distributed Sparse Gaussian Process Regression

Cooperative online scalar field mapping is an important task for multi-robot systems. Gaussian process regression is widely used to construct a map that represents spatial information with confidence intervals. However, it is difficult to handle cooperative online mapping tasks because of its high computation and communication costs. This letter proposes a resource-efficient cooperative online field mapping method via distributed sparse Gaussian process regression. A novel distributed online Gaussian process evaluation method is developed such that robots can cooperatively evaluate and find observations of sufficient global utility to reduce computation. The bounded errors of distributed aggregation results are guaranteed theoretically, and the performances of the proposed algorithms are validated by real online light field mapping experiments

Five-Tiered Route Planner for Multi-AUV Accessing Fixed Nodes in Uncertain Ocean Environments

This article introduces a five-tiered route planner for accessing multiple nodes with multiple autonomous underwater vehicles (AUVs) that enables efficient task completion in stochastic ocean environments. First, the pre-planning tier solves the single-AUV routing problem to find the optimal giant route (GR), estimates the number of required AUVs based on GR segmentation, and allocates nodes for each AUV to access. Second, the route planning tier plans individual routes for each AUV. During navigation, the path planning tier provides each AUV with physical paths between any two points, while the actuation tier is responsible for path tracking and obstacle avoidance. Finally, in the stochastic ocean environment, deviations from the initial plan may occur, thus, an auction-based coordination tier drives online task coordination among AUVs in a distributed manner. Simulation experiments are conducted in multiple different scenarios to test the performance of the proposed planner, and the promising results show that the proposed method reduces AUV usage by 7.5% compared with the existing methods. When using the same number of AUVs, the fleet equipped with the proposed planner achieves a 6.2% improvement in average task completion rate