[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81753-en":3,"doc-seo-81753-105":29,"detail-sidebar-cat-0-en-105":91},{"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":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":11,"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},81753,4398048949847,"Eliana","https://ap-avatar.wpscdn.com/avatar/400002536579ef2da7f?_k=1778318612642679267",8,"Research & Report","Guaranteed Escape for a Bouncing Robot in Pipe Chains","A study of symmetric bouncing of a point robot inside orthogonally joined rectangles, treated as pipes with equal width. The work performs an exhaustive classification of every trajectory pattern within a single rectangular pipe segment, determining the exact conditions under which the robot exits. The analysis is extended from L-shaped pipes to linear chains of k orthogonally connected segments, including curved joints. Exit guarantees are proven for the special angle α = π/4, with conditions tied to initial geometry and segment structure.","Guaranteed Escape for a Bouncing Robot in Pipe Chains  \nYeganeh Bahoo∗ Ahmad Kamaludeen∗ Somnath Kundu∗  \narXiv :2607 .00221v1 [ cs .CG] 30 Jun 2026  \nAbstract  \nWe study the symmetric bouncing of a point robot within orthogonally-joined rectangles with equal width, which we refer to as pipes. We provide an exhaustive case analysis of every trajectory pattern inside a single rectangular pipe segment, identifying the conditions under which the robot exits. We then extend the analysis to L-shaped pipes and, more generally, to linear chains ofk orthogonally connected pipe segments. We prove exit guarantees for the special angle α = π/4 . Furthermore, these results extend to pipes with curved joints.  \n1 Introduction  \nA bouncing robot is a mobile robot that moves in straight lines and reflects off obstacles like a billiard ball. We consider such a robot within an orthogonal pipe: an environment composed of axis-aligned rectangular corridors connected orthogonally, where their openings areplaced end to end (at the corners of the each rectangle in the pipe), so that there exactly two openings in the entire composite polygon; see Figure 3 . The robot has no sensors to detect openings except by physically reaching them with an angle α > 0. When it hits a wall, it bounces symmetrically; if it hits a corner, it stops. We assume there is an opening of length equal to the pipe’s width on the boundary through which the robot can escape. The central problem is to decide how the robot should move so that it is guaranteed to find the exit.  \nThis problem connects with billiards and dynamical systems [4, 16 , 8] . In robotics, bouncing robots have been explored as minimalistic agents for coverage and exploration, relevant to sensor-denied environments such as pipes and ducts [7] . Kundu et al. [10] presented an algorithm guaranteeing escape from a single rectangle under co-prime side lengths. This result is extended to arbitrary side lengths by Kundu et [al. in](al. in) [9]. Escaping orthogonal pipe systems, however, remained open.  \nWe start with an exhaustive case analysis of every trajectory pattern inside the End Block of a rectangular pipe segment (Table 1), which we define as the unit square at the pipes closed end, which one edge of which is the exit, and we identify the exact conditions on the initial angle α  and position y0 under which the  \n∗ Toronto Metropolitan University, Toronto, ON M5B 2K3, Canada,  \n{bahoo,ahmad.kamaludeen,[somnath.kundu}@torontomu.ca](somnath.kundu}@torontomu.ca)  \nWe acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC) .  \nrobot exits. This enumeration is the technical foundation for handling junctions between pipe segments. We apply it to characterize the trajectories in L-shaped pipes (two corridors joined orthogonally) and extend to linear chains of k orthogonally connected segments, proving exit guarantees under various conditions on the angle and length of the segments.  \n2 Background and Related Work  \nDefinition 1 (Symmetric Bounce) When a robot collides with a wall, it undergoes a symmetric bounce. The incident angle is the angle between the incoming trajectory and the wall’s normal; the reflected angle is the angle between the outgoing trajectory and the wall’s normal. A symmetric bounce requires the incident angle to equal the reflected angle, following the classical specular reflection law.  \nRobot model. We model the robot as a point moving at unit speed in a two-dimensional orthogonal pipe environment with no interior obstacles. The robot has no sensors to detect openings or measure distances; its only interaction with the environment is through collisions with the boundary. The robot traverses a straight-line trajectory until it collides with a wall or corner. Upon hitting a corner, the robot terminates its motion. When colliding with a wall, the robot undergoes a symmetric bounce (Definition 1) . Following reflection, the robot resumes straight-li","cbCaidgvsMmf2qfO","https://ap.wps.com/l/cbCaidgvsMmf2qfO","pdf",607723,5,1,"English","en",105,"# Introduction\n# Background and Related Work","[{\"question\":\"What motion and collision rules define the bouncing robot in this work?\",\"answer\":\"The robot moves at unit speed in a 2D orthogonal pipe environment and follows straight-line motion until it hits a wall or corner. Walls produce symmetric specular reflection, while corners stop the robot’s motion.\"},{\"question\":\"How is escape from a pipe segment determined?\",\"answer\":\"The paper classifies trajectory patterns inside a rectangular pipe segment and identifies the precise initial angle α and position conditions that lead the robot to exit through the opening.\"},{\"question\":\"Does the exit guarantee extend beyond a single rectangle?\",\"answer\":\"Yes. Results are extended to L-shaped pipes and to linear chains of k orthogonally connected pipe segments, and further to pipes with curved joints, with proved exit guarantees for α = π/4.\"}]",1784175840,20,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"guaranteed-escape-for-a-bouncing-robot-in-pipe-chains","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/guaranteed-escape-for-a-bouncing-robot-in-pipe-chains/81753/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-26","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What motion and collision rules define the bouncing robot in this work?","Question",{"text":75,"@type":76},"The robot moves at unit speed in a 2D orthogonal pipe environment and follows straight-line motion until it hits a wall or corner. Walls produce symmetric specular reflection, while corners stop the robot’s motion.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How is escape from a pipe segment determined?",{"text":80,"@type":76},"The paper classifies trajectory patterns inside a rectangular pipe segment and identifies the precise initial angle α and position conditions that lead the robot to exit through the opening.",{"name":82,"@type":73,"acceptedAnswer":83},"Does the exit guarantee extend beyond a single rectangle?",{"text":84,"@type":76},"Yes. Results are extended to L-shaped pipes and to linear chains of k orthogonally connected pipe segments, and further to pipes with curved joints, with proved exit guarantees for α = π/4.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,109,114,119,122,126,129,133],{"id":21,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":20,"doc_module":4,"doc_module_name":45,"category_name":106,"show_sort_weight":107,"slug":108},"Comic",60,"comic",{"id":110,"doc_module":4,"doc_module_name":45,"category_name":111,"show_sort_weight":112,"slug":113},6,"Technology",50,"technology",{"id":115,"doc_module":4,"doc_module_name":45,"category_name":116,"show_sort_weight":117,"slug":118},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":45,"category_name":124,"show_sort_weight":28,"slug":125},9,"Religion & Spirituality","religion-spirituality",{"id":28,"doc_module":4,"doc_module_name":45,"category_name":127,"show_sort_weight":28,"slug":128},"World Cup","world-cup",{"id":130,"doc_module":4,"doc_module_name":45,"category_name":131,"show_sort_weight":130,"slug":132},10,"Lifestyle","lifestyle",{"id":134,"doc_module":4,"doc_module_name":45,"category_name":135,"show_sort_weight":20,"slug":136},19,"General","general"]