Enumeration of axial rotation

  • Yoon, Yong-San (Mechanical Engineering, Korea Advanced Institute of Science & Technology)
  • Received : 2012.09.10
  • Accepted : 2014.05.20
  • Published : 2014.04.25


In this paper, two procedures of enumerating the axial rotation are proposed using the unit sphere of the spherical rotation coordinate system specifying 3D rotation. If the trajectory of the movement is known, the integration of the axial component of the angular velocity plus the geometric effect equal to the enclosed area subtended by the geodesic path on the surface of the unit sphere. If the postures of the initial and final positions are known, the axial rotation is determined by the angular difference from the parallel transport along the geodesic path. The path dependency of the axial rotation of the three dimensional rigid body motion is due to the geometric effect corresponding to the closed loop discontinuity. Firstly, the closed loop discontinuity is examined for the infinitesimal region. The general closed loop discontinuity can be evaluated by the summation of those discontinuities of the infinitesimal regions forming the whole loop. This general loop discontinuity is equal to the surface area enclosed by the closed loop on the surface of the unit sphere. Using this quantification of the closed loop discontinuity of the axial rotation, the geometric effect is determined in enumerating the axial rotation. As an example, the axial rotation of the arm by the Codman's movement is evaluated, which other methods of enumerating the axial rotations failed.



  1. Berry, M.V. (1990), "Anticipations of the Geometric Phase", Phys. Today, 43(12), 34-40.
  2. Chao, E.Y. (1980), "Justification of triaxial goniometer for the measurement of joint rotation", J. Biomech., 13(12), 989-1006.
  3. Cheng, P.L., Nicol, A.C. and Paul, J.P. (2000), "Determination of axial rotation angles of limb segments - a new method", J. Biomech., 33(7), 837-843.
  4. Cheng, P.L. (2000), "A spherical rotation coordinate system for the description of three-dimensional joint rotations", Ann. Biomed. Eng., 28(11), 1381-92.
  5. Cheng, P.L. (2006), "Simulation of Codman's paradox reveals a general law of motion", J. Biomech., 39(7), 1201-1207.
  6. Codman, E.A. (1934), The shoulder: Rupture of the supraspinatus tendon and other lession in or about the subacromial bursa, 2nd Edition, T. Todd Co., Boston.
  7. Crawford, N.R., Yamaguchi, G.T. and Dickman, C.A. (1999), "A new technique for determining 3-D joint angles: the tilt/twist method", Clin. Biomech., 14, 153-165.
  8. Dennis, D.A., Komistek, R.D., Mahfouz, M.R., Walker, S.A. and Tucker, A. (2004), "A multicenter analysis of axial femorotibial rotation after total knee arthroplasty", Clin. Orthop. Relat. Res., 428, 180-9.
  9. Digennaro, V., Zambianchi, F., Marcovigi, A., Mugnai, R., Fiacchi, F. and Catani, F. (2014), "Design and kinematics in total knee arthroplasty", Int. Orthop., 38, 227-233.
  10. Grood, E.S. and Suntay, W.J. (1983), "A joint coordinate system for the clinical description of threedimensional motions: application to the knee", J. Biomech. Eng., 105(2), 136-144.
  11. Ishida, A. (1990), "Definition of axial rotation of anatomical joints", Front. Med. Biol. Eng., 2(1), 65-68.
  12. Kawano, D.T., Novelia, A. and O'Reilly, O.M. (2013), "Codman's paradox", Rotations, Lecture Note,
  13. Kelvin, L. and Tait, P.G. (1912), A Treatise on Natural Philosophy, Part 1, Reprinted 6th Edition, Cambridge University Press, Cambridge.
  14. Mallon, W.J. (2011), "On the hypotheses that determine the definitions of glenohumeral joint motion: with resolution of Codman's pivotal paradox", J. Shoulder. Elbow. Surg., 21, 1-16.
  15. Masuda, T., Ishida, A., Cao, L. and Morita, S. (2008), "A proposal for a new definition of the axial rotation angle of the shoulder joint", J. Electromyogr. Kinesiol., 18(1), 154-159.
  16. Miyazaki, S. and Ishida, A. (1991), "New mathematical definition and calculation of axial rotation of anatomical joints", J. Biomech. Eng., 113(3), 270-275.
  17. Novotny, J.E., Beynnon, B.D. and Nichols, C.E. 3rd. (2001), "A numerical solution to calculate internalexternal rotation at the glenohumeral joint", Clin. Biomech. (Bristol, Avon), 16(5), 395-400.
  18. Politti, J.C., Goroso, G., Valentinuzzi, M.E. and Bravo, O. (1998), "Codman's paradox of the arm rotations is not a paradox: mathematical validation", Med. Eng. Phys., 20, 257-260.
  19. Stepan, V. and Otahal, S. (2006), "Is Codman's paradox really a paradox?", J. Biomech., 39(16), 3080-3082.
  20. Wolf, S.I., Fradet, L. and Rettig, O. (2009), "Conjunct rotation: Codman's paradox revisited", Med. Biol. Eng. Comput., 47(5), 551-556.