{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,21]],"date-time":"2026-05-21T18:36:54Z","timestamp":1779388614067,"version":"3.53.1"},"reference-count":15,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2024,1,10]],"date-time":"2024-01-10T00:00:00Z","timestamp":1704844800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"University of Dayton Office for Graduate Academic Affairs"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Robotics"],"abstract":"<jats:p>This paper presents a planar four-bar approximate motion synthesis technique that uses only pole locations. Synthesis for rigid-body guidance determines the linkage dimensions that guide a body in a desired manner. The desired motion is specified with task positions including a location and orientation angle. Approximation motion synthesis is necessary when an exact match to the task positions cannot be obtained. A linkage that achieves the task positions as closely as possible becomes desired. Structural error refers to the deviations between the task positions and the linkage\u2019s generated positions. A challenge in approximate motion synthesis is that structural error involves metrics that include location and orientation. A best-fit solution is not evident because the structural error is based on an objective function that combines the location and orientation. Such solutions lack bi-invariance because a change in reference for the motion changes the values of the metric. This work uses only displacement poles, described solely by their coordinates, as they sufficiently characterize the relative task positions. The optimization seeks to minimize the distance between the poles of the task positions and the poles of the generated positions. The use of poles results in a bi-invariant statement of the problem.<\/jats:p>","DOI":"10.3390\/robotics13010013","type":"journal-article","created":{"date-parts":[[2024,1,10]],"date-time":"2024-01-10T05:47:21Z","timestamp":1704865641000},"page":"13","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["A Bi-Invariant Approach to Approximate Motion Synthesis of Planar Four-Bar Linkage"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0009-0002-9057-9257","authenticated-orcid":false,"given":"Tianze","family":"Xu","sequence":"first","affiliation":[{"name":"Department of Mechanical and Aerospace Engineering, University of Dayton, 300 College Park, Dayton, OH 45469, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2326-9048","authenticated-orcid":false,"given":"David H.","family":"Myszka","sequence":"additional","affiliation":[{"name":"Department of Mechanical and Aerospace Engineering, University of Dayton, 300 College Park, Dayton, OH 45469, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5745-4829","authenticated-orcid":false,"given":"Andrew P.","family":"Murray","sequence":"additional","affiliation":[{"name":"Department of Mechanical and Aerospace Engineering, University of Dayton, 300 College Park, Dayton, OH 45469, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2024,1,10]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"McCarthy, J., and Soh, G. (2010). Geometric Design of Linkages, Springer.","DOI":"10.1007\/978-1-4419-7892-9"},{"key":"ref_2","unstructured":"Suh, C.H., and Radcliffe, C.W. (1978). Kinematics and Mechanism Design, John Wiley and Sons."},{"key":"ref_3","unstructured":"Levitskii, N.I. (,  1950). Design of Plane Mechanisms with Lower Pairs. Proceedings of the AH CCCP Izdatelstvo Akademii Nauk, Leningrad, Moscow."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"503","DOI":"10.1115\/1.3438183","article-title":"Kinematic Geometry Associated with the Least-square Approximation of a Given Motion","volume":"95","author":"Sarkisyan","year":"1973","journal-title":"ASME J. Eng. Ind."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"460","DOI":"10.1115\/1.3267382","article-title":"Motion Synthesis using Kinematic Mappings","volume":"105","author":"Ravani","year":"1983","journal-title":"ASME J. Mech. Transm. Autom. Des."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Larochelle, P.M. 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Des."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"302","DOI":"10.1017\/S0263574723001388","article-title":"Metrics proposed for measuring the distance between two rigid-body poses: Review, comparison, and combination","volume":"42","year":"2024","journal-title":"Robotica"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"368","DOI":"10.1115\/1.2829469","article-title":"Finding All Solutions to Unconstrained Nonlinear Optimization for Approximate Synthesis of Planar Linkages Using Continuation Method","volume":"121","author":"Liu","year":"1999","journal-title":"J. Mech. Des."},{"key":"ref_14","unstructured":"Erdman, A.G., Sandor, G.N., and Kota, S. (2001). Mechanism Design: Analysis and Synthesis, Prentice Hall. [3rd ed.]."},{"key":"ref_15","unstructured":"Sandor, G.N., and Erdman, A.G. (1984). Advanced Mechanism Design: Analysis and Synthesis, Prentice Hall."}],"container-title":["Robotics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2218-6581\/13\/1\/13\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T13:43:39Z","timestamp":1760103819000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2218-6581\/13\/1\/13"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,1,10]]},"references-count":15,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2024,1]]}},"alternative-id":["robotics13010013"],"URL":"https:\/\/doi.org\/10.3390\/robotics13010013","relation":{},"ISSN":["2218-6581"],"issn-type":[{"value":"2218-6581","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,1,10]]}}}