Screw theory

Screw theory is the algebraic calculation of pairs of vectors, also known as dual vectors[1] – such as angular and linear velocity, or forces and moments – that arise in the kinematics and dynamics of rigid bodies.[2][3]

Screw theory provides a mathematical formulation for the geometry of lines which is central to rigid body dynamics, where lines form the screw axes of spatial movement and the 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 numbers and addition of vectors.[4]

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;[1] 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,[5][6][7][8] mechanical design, computational geometry and multibody dynamics. This is in part because of the relationship between screws and dual quaternions which have been used to interpolate rigid-body motions.[9] Based on screw theory, an efficient approach has also been developed for the type synthesis of parallel mechanisms (parallel manipulators or parallel robots).[10]

  1. ^ a b McCarthy, J. Michael; Soh, Gim Song (2010). Geometric Design of Linkages. Springer. ISBN 978-1-4419-7892-9.
  2. ^ Dimentberg, F. M. (1965) The Screw Calculus and Its Applications in Mechanics, Foreign Technology Division translation FTD-HT-23-1632-67
  3. ^ Yang, A.T. (1974) "Calculus of Screws" in Basic Questions of Design Theory, William R. Spillers (ed.), Elsevier, pp. 266–281.
  4. ^ Ball, R. S. (1876). The theory of screws: A study in the dynamics of a rigid body. Hodges, Foster.
  5. ^ Featherstone, Roy (1987). Robot Dynamics Algorithms. Kluwer Academic Pub. ISBN 978-0-89838-230-3.
  6. ^ Featherstone, Roy (2008). Robot Dynamics Algorithms. Springer. ISBN 978-0-387-74315-8.
  7. ^ Murray, Richard M.; Li, Zexiang; Sastry, S. Shankar; Sastry, S. Shankara (1994-03-22). A Mathematical Introduction to Robotic Manipulation. CRC Press. ISBN 978-0-8493-7981-9.
  8. ^ Lynch, Kevin M.; Park, Frank C. (2017-05-25). Modern Robotics. Cambridge University Press. ISBN 978-1-107-15630-2.
  9. ^ 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.
  10. ^ Kong, Xianwen; Gosselin, Clément (2007). Type Synthesis of Parallel Mechanisms. Springer. ISBN 978-3-540-71990-8.