[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84524-en":3,"doc-seo-84524-105":29,"detail-sidebar-cat-0-en-105":90},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},84524,962075006959,"Anda","https://ap-avatar.wpscdn.com/avatar/e0002397efbe92a78e?_k=1776741047341049297",8,"Research & Report","Optimal Any-Angle Path Planning in Static and Dynamic Environments","Any-angle path planning extends graph-based routing by allowing movement between any pair of vertices, producing straighter, shorter trajectories in continuous space while avoiding collisions. Many algorithms exist, yet only a few ensure optimal solutions, particularly with dynamic obstacles. This work presents optimal any-angle planning on grids and two computation-accelerating techniques that preserve optimality: elliptical forward expansion and field of view to accelerate visibility checks. Unified scanning yields Zeta* for static settings and Zeta*-SIPP for dynamic settings, with major speedups over TO-AA-SIPP.","arXiv :2607 .00065v1 [ cs .RO] 30 Jun 2026  \nOptimal any-angle path planning in static and dynamic environments  \nYiyuan Zou∗, Clark Borst  \nControl and Simulation, Faculty of Aerospace Engineering, Delft University of Technology, Delft, The Netherlands  \nAbstract  \nAny-angle path planning extends traditional graph-based path planning by allowing movement between any pair of vertices, rather than being restricted by predefined edges. It can find straighter and shorter paths in continuous space with graphs, making it particularly suitable for navigation in open areas such as airspaces, warehouses, and oceans. Many any-angle path-planning algorithms have been proposed, but only a few can guarantee optimal solutions, especially in the presence of dynamic obstacles. To address this challenge, this article focuses on optimal any-angle path planning on grids and introduces two general techniques that accelerate computation while preserving optimality in both static and dynamic environments: 1) elliptical forward expansion, which leverages ellipse-based neighborhoods to restrict the search space, and 2) field of view, which replaces traditional line-of-sight methods to speed up visibility checks. To integrate these two techniques, inverted and forward scanning are introduced. Inverted scanning establishes visual connections from open nodes, whereas forward scanning initiates scans from closed nodes. Building on the proposed techniques, Zeta* and Zeta*-SIPP are developed for static and dynamic environments respectively. Zeta*, when combined with forward scanning, is similar to the state-of-the-art algorithm Anya and attains comparable performance. Unlike Anya, Zeta* can be readily extended to other settings, such as dynamic environments (e.g., Zeta*-SIPP) . Zeta*-SIPP, with either scanning method, is more than 20 times faster than the corresponding stateof-the-art optimal planner TO-AA-SIPP. Overall, this research identifies the key requirements for achieving optimal any-angle path planning and introduces a unified approach suitable for different environments.  \nKeywords: Any-angle path planning, Static obstacles, Dynamic obstacles, Heuristic search, Optimality  \n1. Introduction  \nPath planning is a fundamental problem across various domains, including aerospace, transportation, robotics, and computer games. It typically involves finding an optimal path between two points in a given space, while avoiding collisions with static obstacles and, if present, dynamic obstacles. Over the years, numerous algorithms have been developed to address this problem under a wide range of real-world conditions. Among them, A* (Hart et al., 1968) stands out as one of the most classic and widely recognized algorithms. However, as a graph-based algorithm,  \n∗ Corresponding author  \n[Email addresses:](Email addresses: y.zou@tudelft.nl)[ y.zou@tudelft.nl](Email addresses: y.zou@tudelft.nl) (Yiyuan Zou ), [c.borst@tudelft.nl](c.borst@tudelft.nl) (Clark Borst) Preprint submitted to Elsevier  \n(a) A* (b) Theta* (c) Anya  \nFigure 1: Search trees (yellow lines) and explored nodes (blue grids or regions) of A*, Theta* and Anya1. The start point is marked by the pink vehicle icon, while the target point is shown in green. The dark gray cells represent static obstacles.  \ntraditional A* is highly affected by the structure of the predefined graph. On regular square grids, A* generally considers only the eight adjacent neighbors of the current node during the search, restricting its movement to 45-degree increments at each step, as shown in Figure 1a.  \nTo address this issue, some post-hoc smoothing techniques can be applied to straighten the final generated paths (Botea et al., 2004) . However, they do not guarantee the discovery of true shortest paths and may be difficult to yield effective results in complex environments. Any-angle path planning has thus been proposed (Nash and Koenig, 2013), which allows any-angle turns at graph vertices rather than being limite","cbCaibwD2ttw4gGG","https://ap.wps.com/l/cbCaibwD2ttw4gGG","pdf",1416818,1,33,"English","en",105,"# Introduction\n## Motivation and limitations of classic graph search\n## Any-angle planning and Theta*\n## From non-optimal any-angle methods to optimal planning","[{\"question\":\"What distinguishes any-angle path planning from traditional graph-based planning?\",\"answer\":\"Any-angle planning allows movement between any pair of vertices rather than restricting motion to predefined edges, enabling straighter and shorter paths in continuous space.\"},{\"question\":\"Why is achieving optimality difficult in the presence of dynamic obstacles?\",\"answer\":\"Many existing algorithms improve path realism but do not guarantee true shortest paths, and dynamic obstacles further complicate maintaining optimal solutions.\"},{\"question\":\"What techniques does the paper introduce to accelerate optimal any-angle planning while preserving optimality?\",\"answer\":\"It introduces elliptical forward expansion to restrict the search space via ellipse-based neighborhoods and a field-of-view mechanism to speed visibility checks, integrated through inverted and forward scanning in the resulting 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distinguishes any-angle path planning from traditional graph-based planning?","Question",{"text":74,"@type":75},"Any-angle planning allows movement between any pair of vertices rather than restricting motion to predefined edges, enabling straighter and shorter paths in continuous space.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"Why is achieving optimality difficult in the presence of dynamic obstacles?",{"text":79,"@type":75},"Many existing algorithms improve path realism but do not guarantee true shortest paths, and dynamic obstacles further complicate maintaining optimal solutions.",{"name":81,"@type":72,"acceptedAnswer":82},"What techniques does the paper introduce to accelerate optimal any-angle planning while preserving optimality?",{"text":83,"@type":75},"It introduces elliptical forward expansion to restrict the search space via ellipse-based neighborhoods and a field-of-view mechanism to speed visibility checks, integrated through inverted and forward scanning in 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