Experimental Brain Research

, Volume 212, Issue 3, pp 417–427 | Cite as

Pointing with the wrist: a postural model for Donders’ law

  • Domenico CampoloEmail author
  • Ferdinan Widjaja
  • Mohammad Esmaeili
  • Etienne Burdet
Research Article


The central nervous system uses stereotypical combinations of the three wrist/forearm joint angles to point in a given (2D) direction in space. In this paper, we first confirm and analyze this Donders’ law for the wrist as well as the distributions of the joint angles. We find that the quadratic surfaces fitting the experimental wrist configurations during pointing tasks are characterized by a subject-specific Koenderink shape index and by a bias due to the prono-supination angle distribution. We then introduce a simple postural model using only four parameters to explain these characteristics in a pointing task. The model specifies the redundancy of the pointing task by determining the one-dimensional task-equivalent manifold (TEM), parameterized via wrist torsion. For every pointing direction, the torsion is obtained by the concurrent minimization of an extrinsic cost, which guarantees minimal angle rotations (similar to Listing’s law for eye movements) and of an intrinsic cost, which penalizes wrist configurations away from comfortable postures. This allows simulating the sequence of wrist orientations to point at eight peripheral targets, from a central one, passing through intermediate points. The simulation first shows that in contrast to eye movements, which can be predicted by only considering the extrinsic cost (i.e., Listing’s law), both costs are necessary to account for the wrist/forearm experimental data. Second, fitting the synthetic Donders’ law from the simulated task with a quadratic surface yields similar fitting errors compared to experimental data.


Human wrist Kinematic redundancy Donders’ law Hopf fibration Multi-objective optimization 



This study was partially funded by the Academic Research Fund (AcRF) Tier1 (RG 40/09), Ministry of Education, Singapore, and by the EU FP7 VIACTORS project. The authors are grateful to Ms Kelly Savin for proofreading the manuscript.


  1. Andrews JG, Youm Y (1979) A biomechanical investigation of wrist kinematics. J Biomech 12:83–93PubMedCrossRefGoogle Scholar
  2. Blohm G, Crawford JD (2007) Computations for geometrically accurate visually guided reaching in 3-D space. J Vision 7(5):4, 1–22Google Scholar
  3. Campolo D, Formica D, Guglielmelli E, Keller F (2010) Kinematic analysis of the human wrist during pointing tasks. Exp Brain Res 201:561–573PubMedCrossRefGoogle Scholar
  4. Ceylan M, Henriques DYP, Tweed DB, Crawford JD (2000) Task dependent constraints in motor control: pinhole goggles make the head move like an eye. J Neurosci 20:2719–2730PubMedGoogle Scholar
  5. Cruse H, Wischmeyer E, Bruwer M, Brockfeld P, Dress A (1990) On the cost functions for the control of the human arm movement. Biol Cybern 62:519–528PubMedCrossRefGoogle Scholar
  6. Cruse H (1986) Constraints for joint angle control of the human arm. Biol Cybern 54:125–132CrossRefGoogle Scholar
  7. Do Carmo MP (1976) Differential geometry of curves and surfaces. Prentice Hall, New JerseyGoogle Scholar
  8. Donders FC (1847) Beitrag zur Lehre von den Bewegungen des menschlichen Auges. Holland Beitr Anat Physiol Wiss 1:104–145Google Scholar
  9. Engelbrecht SE (2001) Minimum principles in motor control. J Math Psychol 45:497–542PubMedCrossRefGoogle Scholar
  10. Franklin DW, Burdet E, Tee KP, Osu R, Chew CM, Milner TE, Kawato M (2008) CNS learns stable, accurate, and efficient movements using a simple algorithm. J Neurosci 28(44):11165–11173PubMedCrossRefGoogle Scholar
  11. Guigon E, Baraduc P, Desmurget M (2008) Optimality, stochasticity, and variability in motor behavior. J Comput Neurosci 24:57–68PubMedCrossRefGoogle Scholar
  12. Hansard M, Horaud R (2010) Cyclorotation Models for Eyes and Cameras. IEEE Trans Syst Man Cybern B Cybern 40:151–161PubMedCrossRefGoogle Scholar
  13. Hepp K (1995) Theoretical explanations of listing’s law and their implication for binocular vision. Vis Res 35:3237–3241PubMedCrossRefGoogle Scholar
  14. Hepp K (1990) On Listing’s law. Commun Math Phys 132:285–292CrossRefGoogle Scholar
  15. Hoffman DS, Strick PL (1993) Step-tracking movements of the wrist. III. Influence of changes in load on patterns of muscle activity. J Neurosci 13(12):5212–5227PubMedGoogle Scholar
  16. Hore J, Watts S, Vilis T (1992) Constraints on arm position when pointing in three dimensions: donders law and the fick-gimbal strategy. J Neurophysiol 68:374–383PubMedGoogle Scholar
  17. Koenderink JJ, van Droon AJ (1992) Surface shape and curvature scales. Image Vis Comput 10:557–564CrossRefGoogle Scholar
  18. Leonard L, Sirkett D, Mullineux G, Giddins GEB, Miles AW (2005) Development of an in-vivo method of wrist joint motion analysis. Clin Biomech 20:166–171CrossRefGoogle Scholar
  19. Montgomery R (2002) A tour of subriemannian geometries, their geodesics and applications. Mathematical surveys and monographs, 91. American Mathematical Society, Providence, RI, USAGoogle Scholar
  20. Murray RM, Li Z, Sastry SS (1994) A Mathematical introduction to robotic manipulation. CRC Press, Boca RatonGoogle Scholar
  21. Pennestrì E, Stefanelli R, Valentini PP, Vita L (2007) Virtual musculo-skeletal model for the biomechanical analysis of the upper limb. J Biomech 40:1350–1361PubMedCrossRefGoogle Scholar
  22. Shoemake K (1985) Animating rotation with quaternion curves. In: Proceedings of the 12th annual conference on computer graphics and interactive techniques (SIGGRAPH), 16(3):245–254, San Francisco, CA, USA, Jul 22–26Google Scholar
  23. Todorov E (2004) Optimality principles in sensorimotor control. Nat Neurosci 7:907–915PubMedCrossRefGoogle Scholar
  24. Tweed D, Vilis T (1990) Geometric relations of eye position and velocity vectors during saccades. Vis Res 30:111–127PubMedCrossRefGoogle Scholar
  25. Wong AMF (2004) Listing’s law: clinical significance and implications for neural control. Surv Ophthalmol 49:563–575PubMedGoogle Scholar
  26. Wu G et al (2005) ISB recommendation on definitions of joint coordinate systems of various joints for the reporting of human joint motion—part II: shoulder, elbow, wrist and hand. J Biomech 38:981–992PubMedCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • Domenico Campolo
    • 1
    Email author
  • Ferdinan Widjaja
    • 1
  • Mohammad Esmaeili
    • 1
  • Etienne Burdet
    • 2
  1. 1.School of Mechanical and Aerospace EngineeringNanyang Technological UniversityNanyang AvenueSingapore
  2. 2.Department of BioengineeringImperial College LondonLondonUK

Personalised recommendations