Editing Screw theory (section)

{{Short description|Mathematical formulation of vector pairs used in physics (rigid body dynamics)}}

'''Screw theory''' is the algebraic calculation of pairs of [[Vector (mathematics and physics)|vectors]], also known as '''''dual vectors'''''<ref name="McCarthy"/> – such as [[Angular velocity|angular]] and [[linear velocity]], or [[force]]s and [[Moment (physics)|moments]] – that arise in the [[kinematics]] and [[Dynamics (mechanics)|dynamics]] of [[Rigid body|rigid bodies]].<ref>Dimentberg, F. M. (1965) [https://web.archive.org/web/20140407084703/http://oai.dtic.mil/oai/oai?verb=getRecord&metadataPrefix=html&identifier=AD0680993 ''The Screw Calculus and Its Applications in Mechanics''], Foreign Technology Division translation FTD-HT-23-1632-67</ref><ref>Yang, A.T. (1974) "Calculus of Screws" in ''Basic Questions of Design Theory'', William R. Spillers (ed.), Elsevier, pp. 266–281.</ref>  

Screw theory provides a [[mathematical]] [[formulation]] for the [[geometry]] of lines which is central to [[rigid body dynamics]], where lines form the [[screw axis|screw axes]] of spatial movement and the [[Line of action|lines of action]] of forces.  The pair of vectors that form the [[Plücker coordinates]] of a line define a unit screw, and general screws are obtained by multiplication by a pair of [[real number]]s and [[Vector addition|addition of vectors]].<ref name=b1>{{cite book|title=The theory of screws: A study in the dynamics of a rigid body|url=https://archive.org/details/theoryscrewsast00ballgoog|author=Ball, R. S.|publisher=Hodges, Foster|year=1876}}</ref>

Important theorems of screw theory include: the ''transfer principle'' proves that geometric calculations for points using vectors have parallel geometric calculations for lines obtained by replacing vectors with screws;<ref name="McCarthy">{{cite book|author1=McCarthy, J. Michael |author2=Soh, Gim Song |title=Geometric Design of Linkages|url=https://books.google.com/books?id=jv9mQyjRIw4C|date=2010|publisher=Springer|isbn=978-1-4419-7892-9}}</ref>    
[[Chasles' theorem (kinematics)|''Chasles' theorem'']] proves that any change between two rigid object poses can be performed by a single screw;  ''[[Poinsot's theorem]]'' proves that rotations about a rigid object's major and minor – but not intermediate – axes are stable.

Screw theory is an important tool in robot mechanics,<ref>{{cite book|author=Featherstone, Roy |title=Robot Dynamics Algorithms|url=https://books.google.com/books?id=c6yz7f_jpqsC|year=1987|publisher=Kluwer Academic Pub|isbn=978-0-89838-230-3}}</ref><ref>{{cite book|author=Featherstone, Roy |title=Robot Dynamics Algorithms|url=https://books.google.com/books?id=UjWbvqWaf6gC|year=2008|publisher=Springer|isbn=978-0-387-74315-8}}</ref><ref>{{Cite book |last1=Murray |first1=Richard M. |url=https://books.google.com/books?id=D_PqGKRo7oIC&q=murray+li+sastry |title=A Mathematical Introduction to Robotic Manipulation |last2=Li |first2=Zexiang |last3=Sastry |first3=S. Shankar |last4=Sastry |first4=S. Shankara |date=1994-03-22 |publisher=CRC Press |isbn=978-0-8493-7981-9 |language=en}}</ref><ref>{{Cite book |last1=Lynch |first1=Kevin M. |url=https://books.google.com/books?id=5NzFDgAAQBAJ |title=Modern Robotics |last2=Park |first2=Frank C. |date=2017-05-25 |publisher=Cambridge University Press |isbn=978-1-107-15630-2 |language=en}}</ref> mechanical design, [[computational geometry]] and [[multibody dynamics]]. 
This is in part because of the relationship between screws and [[dual quaternion]]s which have been used to interpolate [[rigid-body motion]]s.<ref>Selig, J. M. (2011) "Rational Interpolation of Rigid Body Motions," Advances in the Theory of Control, Signals and Systems with Physical Modeling, Lecture Notes in Control and Information Sciences, Volume 407/2011  213–224, {{doi|10.1007/978-3-642-16135-3_18}} Springer.</ref> Based on screw theory, an efficient approach has also been developed for the type synthesis of parallel mechanisms (parallel manipulators or parallel robots).<ref>{{cite book|author1=Kong, Xianwen |author2=Gosselin, Clément |title=Type Synthesis of Parallel Mechanisms|url=https://books.google.com/books?id=II9_FbbpkL0C|date=2007|publisher=Springer|isbn=978-3-540-71990-8}}</ref>