Three-Dimensional Representation of Equilibrium Joint Torques in Two-Joint Movements of the Upper Limb

  • I. V. Vereshchaka
  • W. Pilewska
  • M. Zasada
  • A. I. KostyukovEmail author

In this theoretical study, two-joint equilibrium muscle contractions were simulated to determine the end-point forces created by the hand of the human right upper limb within the horizontal plane. For invariable frontally directed end-point forces, 3D surfaces simulating joint torques (JTs) at the shoulder and elbow joints are reconstructed by defining the characteristic angles (CAs) between the frontal axis and lines from the joint axes to the end-point. The 3D shoulder JTs are presented by planes oriented perpendicularly to the sagittal plane with a downward sagittal skewness; the elbow JT surfaces are essentially nonlinear, showing higher gradients within the left half of the working space. Oppositely directed end-point forces demonstrate the invariance of the JT surfaces that turn about the zero-torque plane while keeping their shapes. Differences between the JTs in the same curvilinear trajectories of the movements (concentric circles) are also analyzed; generation of unvaried (frontally directed) and changed (tangential) end-point forces are compared. Despite the fact that a symmetric pattern in the shoulder JTs is maintained, a transition from the frontal to tangential forces essentially influences the asymmetric pattern of the elbow JTs. The obtained results are discussed with regard to the control of multijoint movements of the limbs in humans.


motor control two-joint movements upper limb joint torques muscle synergy central commands 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    J. M. Hollerbach, “Computers, brains, and the control of movement,” Trends Neurosci., 5, 189-192 (1982).Google Scholar
  2. 2.
    D. M. Wolpertand and Z. Ghahramani, “Computational principles of movement neuroscience,” Nat. Neurosci., 3, 1212-1217 (2000).Google Scholar
  3. 3.
    D. M. Wolpert and M. Kawato, “Multiple paired forward and inverse models for motor control,” Neural Netw., 11, 1317-1329 (1998).Google Scholar
  4. 4.
    M. Kawato, “Internal models for motor control and trajectory planning,” Cur. Opin. Neurobiol., 9, No. 6, 718-727 (1999).Google Scholar
  5. 5.
    E. Bizzi, N. Hogan, F. A. Mussa-Ivaldi, and S. Giszter, “Does the nervous system use equilibrium-point control to guide single and multiple joint movements?” Behav. Brain Sci., 15, No. 4, 603-613 (1992).Google Scholar
  6. 6.
    A. G. Feldman, “Once more for the equilibrium point hypothesis (λ model),” J. Mot. Behav., 18, 17-54 (1986).Google Scholar
  7. 7.
    A. G. Feldman, “Space and time in the context of equilibrium-point theory,” Wiley Interdiscip. Rev. Cogn. Sci., 2, No. 3, 287-304 (2011).Google Scholar
  8. 8.
    A. G. Feldman, “The relationship between postural and movement stability. Progress in motor control,” Adv. Exp. Med. Biol., 957, 105-120 (2016).Google Scholar
  9. 9.
    A. I. Kostyukov, “Muscle hysteresis and movement control: a theoretical study,” Neuroscience, 83, No. 1, 303-320 (1998).Google Scholar
  10. 10.
    T. Tomiak, T. I. Abramovych, A. V. Gorkovenko, et al., “The movement- and load-dependent differences in the EMG patterns of the human arm muscles during two-joint movements (a preliminary study),” Front. Physiol., 7, No. 218 (2016); doi:
  11. 11.
    A. I. Kostyukov, “Theoretical analysis of the force and position synergies in two-joint movements,” Neurophysiology, 48, No. 4, 287-296 (2016).Google Scholar
  12. 12.
    T. Tomiak, A. V. Gorkovenko, A. N. Tal’nov, et al., “The averaged EMGs recorded from the arm muscles during bimanual “rowing” movements,” Front. Physiol., 6, No. 349 (2015); doi:
  13. 13.
    A. I. Kostyukov and T. Tomiak, “The force generation in a two-joint arm model: analysis of the joint torques in the working space,” Front. Neurorobot., 12, No. 77 (2018); doi:
  14. 14.
    O. V. Lehedza, A. V. Gorkovenko, I. V. Vereshchaka, et al., “Comparative analysis of electromyographic muscle activity of the human hand during cyclic turns of isometric effort vector of wrist in opposite directions,” Int. J. Physiol. Pathophysiol., 61, No. 2, 3-14 (2016).Google Scholar
  15. 15.
    O. V. Lehedza, “Manifestations of hysteresis in EMG activity of muscles of the human upper limb in generation of cyclic isometric efforts,” Neurophysiology, 49, No. 3, 220-225 (2017).Google Scholar
  16. 16.
    N. Dounskaia, “The internal model and the leading joint hypothesis: implications for control of multi-joint movements,” Exp. Brain Res., 166, No. 1, 1-16 (2005).Google Scholar
  17. 17.
    N. Dounskaia and J. Goble, “The role of vision, speed and attention in overcoming directional biases during arm movements,” Exp. Brain Res., 209, No. 2, 299-309 (2011).Google Scholar
  18. 18.
    N. Dounskaia, J. Goble, and W. Wang, “The role of intrinsic factors in control of arm movement direction: implications from directional preferences,” J. Neurophysiol., 105, No. 3, 999-1010 (2011).Google Scholar
  19. 19.
    N. Dounskaia and W. Wang, “A preferred pattern of joint coordination during arm movements with redundant degrees of freedom,” J. Neurophysiol., 112, No. 5, 1040-1053 (2014).Google Scholar
  20. 20.
    A. I. Kostyukov, “Muscle dynamics: dependence of muscle length on changes in external load,” Biol. Cybern., 56, Nos. 5/6, 375-387 (1987).Google Scholar
  21. 21.
    A. I. Kostyukov and O. E. Korchak, “Length changes of the cat soleus muscle under frequency-modulated distributed stimulation of efferents in isotony,” Neuroscience, 82, No. 3, 943-955 (1997).Google Scholar
  22. 22.
    A. V. Gorkovenko, S. Sawczyn, N. V. Bulgakova, et al., “Muscle agonist-antagonist interactions in an experimental joint model,” Exp. Brain Res., 222, 399-414 (2012).Google Scholar
  23. 23.
    T. E. Milner and C. Cloutier, “Damping of the wrist joint during voluntary movement,” Exp. Brain Res., 122, No. 3, 309-317 (1998).Google Scholar
  24. 24.
    E. Burde, R. Osu, D. W. Franklin, et al., “The central nervous system stabilizes unstable dynamics by learning optimal impedance,” Nature, 414, No. 6862, 446-449 (2001).Google Scholar
  25. 25.
    P. L. Gribble and D. J. Ostry, “Independent coactivation of shoulder and elbow muscles,” Exp. Brain Res., 123, No. 3, 335-360 (1998).Google Scholar
  26. 26.
    A. M. Hill, A. M. J. Bull, A. L. Wallace, and G. R. Johnson, “Qualitative and quantitative descriptions of glenohumeral motion,” Gait Posture, 27, No. 2, 177-188 (2008).Google Scholar
  27. 27.
    C. D. Bryce and A. D. Armstrong, “Anatomy and biomechanics of the elbow,” Orthop. Clin. North Am., 39, No. 2, 141-154 (2008).Google Scholar
  28. 28.
    B. M. Van Bolhuis, C. C. Gielen, and G. J. van Ingen Schenau, “Activation patterns of mono- and bi-articular arm muscles as a function of force and movement direction of the wrist in humans,” J. Physiol., 508, Part 1, 313-324 (1998).Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • I. V. Vereshchaka
    • 1
  • W. Pilewska
    • 2
  • M. Zasada
    • 2
  • A. I. Kostyukov
    • 3
    Email author
  1. 1.Department of Physical EducationGdansk University of Physical Education and SportGdanskPoland
  2. 2.Faculty of Physical Education, Health, and Tourism, Institute of Physical CultureKazimierz Wielki UniversityBydgoszczPoland
  3. 3.Department of Movement Physiology, Bogomolets Institute of PhysiologyNational Academy of Sciences of UkraineKyivUkraine

Personalised recommendations