{"title":"迹不等式和运动学度量","authors":"Yuwei Wu, Gregory S. Chirikjian","doi":"10.1017/s0263574724000778","DOIUrl":null,"url":null,"abstract":"<p>Kinematics remains one of the cornerstones of robotics, and over the decade, Robotica has been one of the venues in which groundbreaking work in kinematics has always been welcome. A number of works in the kinematics community have addressed metrics for rigid-body motions in multiple different venues. An essential feature of any distance metric is the triangle inequality. Here, relationships between the triangle inequality for kinematic metrics and so-called trace inequalities are established. In particular, we show that the Golden-Thompson inequality (a particular trace inequality from the field of statistical mechanics) which holds for Hermitian matrices remarkably also holds for restricted classes of real skew-symmetric matrices. We then show that this is related to the triangle inequality for <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240911135946881-0491:S0263574724000778:S0263574724000778_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$SO(3)$</span></span></img></span></span> and <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240911135946881-0491:S0263574724000778:S0263574724000778_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$SO(4)$</span></span></img></span></span> metrics.</p>","PeriodicalId":49593,"journal":{"name":"Robotica","volume":null,"pages":null},"PeriodicalIF":1.9000,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Trace inequalities and kinematic metrics\",\"authors\":\"Yuwei Wu, Gregory S. Chirikjian\",\"doi\":\"10.1017/s0263574724000778\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Kinematics remains one of the cornerstones of robotics, and over the decade, Robotica has been one of the venues in which groundbreaking work in kinematics has always been welcome. A number of works in the kinematics community have addressed metrics for rigid-body motions in multiple different venues. An essential feature of any distance metric is the triangle inequality. Here, relationships between the triangle inequality for kinematic metrics and so-called trace inequalities are established. In particular, we show that the Golden-Thompson inequality (a particular trace inequality from the field of statistical mechanics) which holds for Hermitian matrices remarkably also holds for restricted classes of real skew-symmetric matrices. We then show that this is related to the triangle inequality for <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240911135946881-0491:S0263574724000778:S0263574724000778_inline1.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$SO(3)$</span></span></img></span></span> and <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240911135946881-0491:S0263574724000778:S0263574724000778_inline2.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$SO(4)$</span></span></img></span></span> metrics.</p>\",\"PeriodicalId\":49593,\"journal\":{\"name\":\"Robotica\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.9000,\"publicationDate\":\"2024-09-12\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Robotica\",\"FirstCategoryId\":\"94\",\"ListUrlMain\":\"https://doi.org/10.1017/s0263574724000778\",\"RegionNum\":4,\"RegionCategory\":\"计算机科学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"ROBOTICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Robotica","FirstCategoryId":"94","ListUrlMain":"https://doi.org/10.1017/s0263574724000778","RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"ROBOTICS","Score":null,"Total":0}
Kinematics remains one of the cornerstones of robotics, and over the decade, Robotica has been one of the venues in which groundbreaking work in kinematics has always been welcome. A number of works in the kinematics community have addressed metrics for rigid-body motions in multiple different venues. An essential feature of any distance metric is the triangle inequality. Here, relationships between the triangle inequality for kinematic metrics and so-called trace inequalities are established. In particular, we show that the Golden-Thompson inequality (a particular trace inequality from the field of statistical mechanics) which holds for Hermitian matrices remarkably also holds for restricted classes of real skew-symmetric matrices. We then show that this is related to the triangle inequality for $SO(3)$ and $SO(4)$ metrics.
期刊介绍:
Robotica is a forum for the multidisciplinary subject of robotics and encourages developments, applications and research in this important field of automation and robotics with regard to industry, health, education and economic and social aspects of relevance. Coverage includes activities in hostile environments, applications in the service and manufacturing industries, biological robotics, dynamics and kinematics involved in robot design and uses, on-line robots, robot task planning, rehabilitation robotics, sensory perception, software in the widest sense, particularly in respect of programming languages and links with CAD/CAM systems, telerobotics and various other areas. In addition, interest is focused on various Artificial Intelligence topics of theoretical and practical interest.