{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T00:50:02Z","timestamp":1760057402878,"version":"build-2065373602"},"reference-count":26,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2025,2,1]],"date-time":"2025-02-01T00:00:00Z","timestamp":1738368000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Heilongjiang Provincial Natural Science Foundation of China","award":["LH2022A001","19ZR01"],"award-info":[{"award-number":["LH2022A001","19ZR01"]}]},{"name":"Daqing Normal University Natural Science Foundation","award":["LH2022A001","19ZR01"],"award-info":[{"award-number":["LH2022A001","19ZR01"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The rapid advancement of robotic technologies has demonstrated the significant potential of Multi-Robot Systems (MRS) for application across various fields, particularly in automation, manufacturing, and rescue operations. However, enhancing the reliability of Multi-Robot Systems, particularly in critical applications, has emerged as a primary focus of research. A mathematical model of a five-robot system, equipped with early warning capabilities, is developed using Markov process theory and the supplementary variable method in this paper. A model of an abstract Cauchy problem system is developed, employing semigroup theory to investigate the well-posedness of solutions for this five-robot system. The stability of the system is verified using analytical methods, confinal correlation theory, and modern functional analysis techniques. Several key reliability indicators are presented using the eigenvector method. Numerical simulations and comparative methods effectively demonstrate the efficacy of the proposed eigenvector method. Firstly, the innovation of this paper lies in the combination of qualitative and quantitative analyses to improve and enrich the theory and methods of repairable systems. Secondly, mathematical analysis methods and the mathematical software are employed to provide both analytical and numerical solutions for the system.<\/jats:p>","DOI":"10.3390\/axioms14020113","type":"journal-article","created":{"date-parts":[[2025,2,3]],"date-time":"2025-02-03T04:36:51Z","timestamp":1738557411000},"page":"113","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Reliability Analysis and Numerical Simulation of the Five-Robot System with Early Warning Function"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8917-8176","authenticated-orcid":false,"given":"Xing","family":"Qiao","sequence":"first","affiliation":[{"name":"School of Mathematical Science, Daqing Normal University, Daqing 163712, China"}]},{"given":"Dan","family":"Ma","sequence":"additional","affiliation":[{"name":"School of Mathematical Science, Daqing Normal University, Daqing 163712, China"}]},{"given":"Shuang","family":"Guo","sequence":"additional","affiliation":[{"name":"School of Mathematical Science, Daqing Normal University, Daqing 163712, China"}]}],"member":"1968","published-online":{"date-parts":[[2025,2,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1709","DOI":"10.1016\/0026-2714(94)90125-2","article-title":"Optimization of a repairable and redundant system","volume":"34","author":"Dhillon","year":"1994","journal-title":"Microelectron. Reliab."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1037","DOI":"10.1016\/0026-2714(95)90862-K","article-title":"Stochastic analysis of a general standby system with constant human error and arbitrarily system repair rates","volume":"35","author":"Yang","year":"1995","journal-title":"Microelectron. Reliab."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"238","DOI":"10.1016\/j.promfg.2018.10.042","article-title":"A Context-Aware Safety System for Human-Robot Collaboration","volume":"17","author":"Liu","year":"2018","journal-title":"Procedia Manuf."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1016\/j.ejcon.2019.06.009","article-title":"Intelligent fuzzy sliding mode control for complex robot system with disturbances","volume":"51","author":"Zheng","year":"2020","journal-title":"Eur. J. Control"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"957","DOI":"10.1007\/s10514-017-9677-2","article-title":"Progress and prospects of the human-robot collaboration","volume":"42","author":"Ajoudani","year":"2018","journal-title":"Auton. Robot."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Leite, A., Pinto, A., and Matos, A. (2018). A safety monitoring model for a faulty mobile robot. Robotics, 7.","DOI":"10.3390\/robotics7030032"},{"key":"ref_7","first-page":"2225","article-title":"Multi-robot self-organizing cooperative pursuit method based on probabilistic graphical model","volume":"40","author":"Huang","year":"2023","journal-title":"Control. Theory Appl."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1360","DOI":"10.1007\/s11424-020-8186-0","article-title":"Robust Control of Robotic Manipulators in the Task-Space Using an Adaptive Observer Based on Chebyshev Polynomials","volume":"33","author":"Reza","year":"2020","journal-title":"J. Syst. Sci. Complex."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"769","DOI":"10.1007\/s12204-020-2217-9","article-title":"Stability and Numerical Analysis of a Standby System","volume":"25","author":"Qiao","year":"2020","journal-title":"J. Shanghai Jiaotong Univ. Sci."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1279","DOI":"10.1007\/s11424-021-0036-1","article-title":"A Nonlinear Optimal Control Approach for Tracked Mobile Robots","volume":"34","author":"Gerasimos","year":"2021","journal-title":"J. Syst. Sci. Complex."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"3149","DOI":"10.2478\/amns.2023.1.00027","article-title":"Dynamic Optimization of Robot Automatic Control System Based on Differential Algebraic Equations","volume":"8","author":"Wang","year":"2023","journal-title":"Appl. Math. Nonlinear Sci."},{"key":"ref_12","first-page":"1","article-title":"Application of industrial robot based on infrared sensor technology in monitoring of electrical equipment","volume":"9","author":"Jiang","year":"2024","journal-title":"Appl. Math. Nonlinear Sci."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1560","DOI":"10.1007\/s11424-023-1332-8","article-title":"Exponential Stability of Impulsive Neutral Stochastic Functional Differential Equations with Markovian Switching","volume":"36","author":"Ke","year":"2023","journal-title":"J. Syst. Sci. Complex."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1015","DOI":"10.1016\/j.apm.2004.03.009","article-title":"Bilevel control of degraded machining system with warm standbys, setup and vacation","volume":"28","author":"Jain","year":"2004","journal-title":"Appl. Math. Model."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"880","DOI":"10.1016\/j.apm.2006.02.009","article-title":"Vacation policies for machine repair problem with two type spares","volume":"31","author":"Ke","year":"2007","journal-title":"Appl. Math. Model."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"30","DOI":"10.1109\/TR.1963.5218202","article-title":"Time to failure and availability of paralleled systems with repair","volume":"12","author":"Gaver","year":"1963","journal-title":"IEEE Trans. Reliab."},{"key":"ref_17","unstructured":"Pazy, A. (2012). Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer Science Business Media."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"281","DOI":"10.1007\/BF00250793","article-title":"Non-linear age-dependent population dynamics","volume":"54","author":"Gurtin","year":"1974","journal-title":"Arch. Ration. Mech. Anal."},{"key":"ref_19","unstructured":"Dunford, N., and Schwartz, J.T. (1958). Linear Operators Part I: General Theory, Interscience Publishers."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"321","DOI":"10.1112\/plms\/s3-54.2.321","article-title":"Resolvent positive operators","volume":"3","author":"Arendt","year":"1987","journal-title":"Proc. Lond. Math. Soc."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1015","DOI":"10.1007\/s11424-015-4081-5","article-title":"Reliability analysis of a cold standby repairable system with repairman extra work","volume":"28","author":"Zhang","year":"2015","journal-title":"J. Syst. Sci. Complex."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Wang, Y., Qiao, X., Zhan, B., Ma, D., and Zhao, G. (2014, January 26\u201327). The Stability and Reliability Analysis of a System Containing Two Redundant Robots. Proceedings of the 2014 Sixth International Conference on Intelligent Human-Machine Systems and Cybernetics, Hangzhou, China.","DOI":"10.1109\/IHMSC.2014.77"},{"key":"ref_23","first-page":"125","article-title":"Spectral analysis of a two unit deteriorating standby system with repair","volume":"10","author":"Yuan","year":"2011","journal-title":"WSEAS Trans. Math."},{"key":"ref_24","unstructured":"Taylor, A.E., and Lay, D.C. (1958). Introduction to Functional Analysis, Wiley."},{"key":"ref_25","unstructured":"Arendt, W., Grabosch, A., Greiner, G., Moustakas, U., Nagel, R., Schlotterbeck, U., Groh, U., Lotz, H.P., and Neubrander, F. (2006). One-Parameter Semigroups of Positive Operators, Springer."},{"key":"ref_26","unstructured":"(2019, March 12). MATLAB. 9.6.0.1.1072779. MathWorks. March 2019. Available online: https:\/\/ww2.mathworks.cn\/products\/matlab.html."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/2\/113\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T16:25:27Z","timestamp":1760027127000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/2\/113"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,2,1]]},"references-count":26,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2025,2]]}},"alternative-id":["axioms14020113"],"URL":"https:\/\/doi.org\/10.3390\/axioms14020113","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,2,1]]}}}