In this study, we present a simple and intuitive method for accelerating optimal Reeds-Shepp path computation. Our approach uses geometrical reasoning to analyze the behavior of optimal paths, resulting in a new partitioning of the state space and a further reduction in the minimal set of viable paths. We revisit and reimplement classic methodologies from the literature, which lack contemporary open-source implementations, to serve as benchmarks for evaluating our method. Additionally, we address the under-specified Reeds-Shepp planning problem where the final orientation is unspecified. We perform exhaustive experiments to validate our solutions. Compared to the modern C++ implementation of the original Reeds-Shepp solution in the Open Motion Planning Library, our method demonstrates a 15x speedup, while classic methods achieve a 5.79x speedup. Both approaches exhibit machine-precision differences in path lengths compared to the original solution. We release our proposed C++ implementations for both the accelerated and under-specified Reeds-Shepp problems as open-source code.

In battlefield environments, adversaries frequently disrupt GPS signals, requiring alternative localization and navigation methods. Traditional vision-based approaches like Simultaneous Localization and Mapping (SLAM) and Visual Odometry (VO) involve complex sensor fusion and high computational demand, whereas range-free methods like DV-HOP face accuracy and stability challenges in sparse, dynamic networks. This paper proposes LanBLoc-BMM, a navigation approach using landmark-based localization (LanBLoc) combined with a battlefield-specific motion model (BMM) and Extended Kalman Filter (EKF). Its performance is benchmarked against three state-of-the-art visual localization algorithms integrated with BMM and Bayesian filters, evaluated on synthetic and real-imitated trajectory datasets using metrics including Average Displacement Error (ADE), Final Displacement Error (FDE), and a newly introduced Average Weighted Risk Score (AWRS). LanBLoc-BMM (with EKF) demonstrates superior performance in ADE, FDE, and AWRS on real-imitated datasets. Additionally, two safe navigation methods, SafeNav-CHull and SafeNav-Centroid, are introduced by integrating LanBLoc-BMM(EKF) with a novel Risk-Aware RRT* (RAw-RRT*) algorithm for obstacle avoidance and risk exposure minimization. Simulation results in battlefield scenarios indicate SafeNav-Centroid excels in accuracy, risk exposure, and trajectory efficiency, while SafeNav-CHull provides superior computational speed.

This article addresses time-optimal path planning for a vehicle capable of moving both forward and backward on a unit sphere with a unit maximum speed, and constrained by a maximum absolute turning rate $U_{max}$. The proposed formulation can be utilized for optimal attitude control of underactuated satellites, optimal motion planning for spherical rolling robots, and optimal path planning for mobile robots on spherical surfaces or uneven terrains. By utilizing Pontryagin's Maximum Principle and analyzing phase portraits, it is shown that for $U_{max}\geq1$, the optimal path connecting a given initial configuration to a desired terminal configuration falls within a sufficient list of 23 path types, each comprising at most 6 segments. These segments belong to the set $\{C,G,T\}$, where $C$ represents a tight turn with radius $r=\frac{1}{\sqrt{1+U_{max}^2}}$, $G$ represents a great circular arc, and $T$ represents a turn-in-place motion. Closed-form expressions for the angles of each path in the sufficient list are derived. The source code for solving the time-optimal path problem and visualization is publicly available at https://github.com/sixuli97/Optimal-Spherical-Convexified-Reeds-Shepp-Paths.

Safe Periodic Trochoidal Paths for Fixed-Wing UA Vs in Confined Windy Environments Jaeyoung Lim 1, David Rohr 1, Thomas Stastny 1, Roland Siegwart 1 Abstract -- Due to their energy-efficient flight characteristics, fixed-wing type uncrewed aerial vehicles (UA Vs) are useful robotic tools for long-range and duration flight applications in large-scale environments. However, flying fixed-wing UA V in confined environments, such as mountainous regions, can be challenging due to their limited maneuverability and sensitivity to uncertain wind conditions. In this work, we first analyze periodic trochoidal paths that can be used to define wind-aware terminal loitering states. We then propose a wind-invariant safe set of trochoidal paths along with a switching strategy for selecting the corresponding minimum-extent periodic path type. Finally, we show that planning with this minimum-extent set allows us to safely reach up to 10 times more locations in mountainous terrain compared to planning with a single, conservative loitering maneuver . I. INTRODUCTION Uncrewed aerial vehicles (UA Vs) have become crucial tools for information-gathering applications, such as surveying and inspection [1], search and rescue [2], and environment monitoring [3], [4]. For large-scale coverage or long-range applications, fixed-wing type UA Vs are preferred over rotary-wing type systems due to their high endurance and speed. While the wing-borne aerodynamic lift enables energy-efficient flight, it also poses challenges for operating safely.

This paper investigates whether the hexagonal structure of grid cells provides any performance benefits or if it merely represents a biologically convenient configuration. Utilizing the Vector-HaSH content addressable memory model as a model of the grid cell -- place cell network of the mammalian brain, we compare the performance of square and hexagonal grid cells in tasks of storing and retrieving spatial memories. Our experiments across different path types, path lengths and grid configurations, reveal that hexagonal grid cells perform similarly to square grid cells with respect to spatial representation and memory recall. Our results show comparable accuracy and robustness across different datasets and noise levels on images to recall. These findings suggest that the brain's use of hexagonal grids may be more a matter of biological convenience and ease of implementation rather than because they provide superior performance over square grid cells (which are easier to implement in silico).

The paper develops a novel motion model, called Generalized Multi-Speed Dubins Motion Model (GMDM), which extends the Dubins model by considering multiple speeds. While the Dubins model produces time-optimal paths under a constant-speed constraint, these paths could be suboptimal if this constraint is relaxed to include multiple speeds. This is because a constant speed results in a large minimum turning radius, thus producing paths with longer maneuvers and larger travel times. In contrast, multi-speed relaxation allows for slower speed sharp turns, thus producing more direct paths with shorter maneuvers and smaller travel times. Furthermore, the inability of the Dubins model to reduce speed could result in fast maneuvers near obstacles, thus producing paths with high collision risks. In this regard, GMDM provides the motion planners the ability to jointly optimize time and risk by allowing the change of speed along the path. GMDM is built upon the six Dubins path types considering the change of speed on path segments. It is theoretically established that GMDM provides full reachability of the configuration space for any speed selections. Furthermore, it is shown that the Dubins model is a specific case of GMDM for constant speeds. The solutions of GMDM are analytical and suitable for real-time applications. The performance of GMDM in terms of solution quality (i.e., time/time-risk cost) and computation time is comparatively evaluated against the existing motion models in obstacle-free as well as obstacle-rich environments via extensive Monte Carlo simulations. The results show that in obstacle-free environments, GMDM produces near time-optimal paths with significantly lower travel times than the Dubins model while having similar computation times. In obstacle-rich environments, GMDM produces time-risk optimized paths with substantially lower collision risks.

Time-optimal path planning in high winds for a turning-rate constrained UAV is a challenging problem to solve and is important for deployment and field operations. Previous works have used trochoidal path segments comprising straight and maximum-rate turn segments, as optimal extremal paths in uniform wind conditions. Current methods iterate over all candidate trochoidal trajectory types and select the one that is time-optimal; however, this exhaustive search can be computationally slow. In this paper, we introduce a method to decrease the computation time. This is achieved by reducing the number of candidate trochoidal trajectory types by framing the problem in the air-relative frame and bounding the solution within a subset of candidate trajectories. Our method reduces overall computation by 37.4% compared to pre-existing methods in Bang-Straight-Bang trajectories, freeing up computation for other onboard processes and can lead to significant total computational reductions when solving many trochoidal paths. When used within the framework of a global path planner, faster state expansions help find solutions faster or compute higher-quality paths. We also release our open-source codebase as a C++ package. The website and demo can be bound at https://bradymoon.com/trochoids, codebase at https://github.com/castacks/trochoids, and video at https://youtu.be/qOU5gI7JshI .

Knowledge bases (KB), both automatically and manually constructed, are often incomplete --- many valid facts can be inferred from the KB by synthesizing existing information. A popular approach to KB completion is to infer new relations by combinatory reasoning over the information found along other paths connecting a pair of entities. Given the enormous size of KBs and the exponential number of paths, previous path-based models have considered only the problem of predicting a missing relation given two entities or evaluating the truth of a proposed triple. Additionally, these methods have traditionally used random paths between fixed entity pairs or more recently learned to pick paths between them. We propose a new algorithm MINERVA, which addresses the much more difficult and practical task of answering questions where the relation is known, but only one entity. Since random walks are impractical in a setting with combinatorially many destinations from a start node, we present a neural reinforcement learning approach which learns how to navigate the graph conditioned on the input query to find predictive paths. Empirically, this approach obtains state-of-the-art results on several datasets, significantly outperforming prior methods.