The Neural Control of Joint Torques in Tendon-Driven Limbs Is Underdetermined

Part of the Biosystems & Biorobotics book series (BIOSYSROB, volume 8)


This chapter introduces the mathematical foundations of the classical notion of muscle redundancy. As presented in Chap.  4, a sub-maximal net torque at a joint actuated by tendons can be produced by a variety of combinations of individual forces at each tendon. We see this already in the simplest case of a planar joint with 2 tendons—one on each side of the joint. Of course, each combination of tendon forces will produce different loading at the tendons and joint, and will incur different metabolic or energetic costs, etc. But in principle there are multiple solutions to the problem of achieving a given mechanical output. This underdetermined problem is called the problem of muscle redundancy , and it begs the question of how the nervous system (or a robotic controller) should select a particular solution from among many. This has been called the central problem of motor control and has occupied much of the literature in this field. The main goal of this chapter, however, is to introduce and cast this problem for high-dimensional multi-joint, multi-muscle limbs (it is often only presented in simplified joints). This will serve as the foundation of subsequent chapters where we critically assess this classical notion of muscle redundancy—and challenge its assumptions and conclusions. As mentioned in Chap.  1, however valuable and informative the concept of muscle redundancy has been, it is also paradoxical with respect to the evolutionary process and clinical reality, and should be revised.


Cost Function Muscle Activation Inequality Constraint Muscle Force Unique Optimal Solution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    E.R. Kandel, J.H. Schwartz, T.M. Jessell et al., Principles of Neural Science, vol. 4 (McGraw-Hill, New York, 2000)Google Scholar
  2. 2.
    F.J. Valero-Cuevas, H. Hoffmann, M.U. Kurse, J.J. Kutch, E.A. Theodorou, Computational models for neuromuscular function. IEEE Rev. Biomed. Eng. 2, 110–135 (2009)CrossRefGoogle Scholar
  3. 3.
    F.E. Zajac, Muscle and tendon: properties, models, scaling, and application to biomechanics and motor control. Crit. Rev. Biomed. Eng. 17(4), 359–411 (1989)Google Scholar
  4. 4.
    F.J. Valero-Cuevas, F.E. Zajac, C.G. Burgar, Large index-fingertip forces are produced by subject-independent patterns of muscle excitation. J. Biomech. 31, 693–703 (1998)CrossRefGoogle Scholar
  5. 5.
    V. Chvatal, Linear Programming (W.H. Freeman and Company, New York, 1983)MATHGoogle Scholar
  6. 6.
    G. Strang, Introduction to Linear Algebra (Wellesley Cambridge Press, Wellesley, 2003)Google Scholar
  7. 7.
    P.E. Gill, W. Murray, M.H. Wright, Practical Optimization (Academic Press, New York, 1981)MATHGoogle Scholar
  8. 8.
    G.B. Dantzig, Linear Programming and Extensions (Princeton University Press, Princeton, 1998)MATHGoogle Scholar
  9. 9.
    F.J. Valero-Cuevas, Muscle coordination of the human index finger. Ph.D. thesis, Stanford University, Stanford (1997)Google Scholar
  10. 10.
    E.Y. Chao, K.N. An, Graphical interpretation of the solution to the redundant problem in biomechanics. J. Biomech. Eng. 100, 159–167 (1978)CrossRefGoogle Scholar
  11. 11.
    D.P. Bertsekas, Nonlinear Programming (Athena Scientific, 1999)Google Scholar
  12. 12.
    R. Horst, E.H. Romeijn, Handbook of Global Optimization, vol. 2 (Springer, Berlin, 2002)Google Scholar
  13. 13.
    D.G. Luenberger, Y. Ye, Linear and Nonlinear Programming. International Series in Operations Research and Management Science (2008)Google Scholar
  14. 14.
    E. Todorov, M.I. Jordan, Optimal feedback control as a theory of motor coordination. Nat. Neurosci. 5(11), 1226–1235 (2002)CrossRefGoogle Scholar
  15. 15.
    R. Shadmehr, S. Mussa-Ivaldi, Biological Learning and Control: How the Brain Buildsrepresentations, Predicts Events, and Makes Decisions (MIT Press, Cambridge, 2012)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag London 2016

Authors and Affiliations

  1. 1.Department of Biomedical EngineeringThe University of Southern CaliforniaLos AngelesUSA
  2. 2.Division of Biokinesiology and Physical TherapyThe University of Southern CaliforniaLos AngelesUSA

Personalised recommendations