# Chapter 5

# Chapter 5: The Neural Control of Joint Torques in Tendon-Driven Limbs Is Underdetermined (*under construction*)

*Last updated Dec. 26 2015 by Francisco Valero-Cuevas*

**Abstract:**

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.

**Forum and commentary:**

*Coming soon!*

**Exercises:**

*Coming soon!*

**Additional references and suggested reading:**

*Coming soon!*

**References in book:**

- E.R.Kandel, J.H.Schwartz, T.M. Jessell et al., Principles of Neural Science, vol. 4 (McGraw- Hill, New York, 2000)
- 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)
- 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)
- 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)
- V. Chvatal, Linear Programming (W.H. Freemanand Company, NewYork, 1983)
- G. Strang, Introduction to Linear Algebra (Wellesley Cambridge Press, Wellesley, 2003)
- P.E. Gill, W.Murray, M.H.Wright, Practical Optimization (Academic Press, New York, 1981)
- G.B. Dantzig, Linear Programming and Extensions (Princeton University Press, Princeton, 1998)
- F.J. Valero-Cuevas, Muscle coordination of the human index finger. Ph.D. thesis, Stanford University, Stanford (1997)
- E.Y. Chao, K.N. An, Graphical interpretation of the solution to the redundant problem in biomechanics. J. Biomech. Eng. 100, 159–167 (1978)
- D.P. Bertsekas, Nonlinear Programming (Athena Scientific, 1999)
- R. Horst, E.H. Romeijn, Handbook o fGlobal Optimization, vol.2 (Springer, Berlin, 2002)
- D.G. Luenberger,Y. Ye, Linear and Nonlinear Programming. International Series in Operations Research and Management Science (2008)
- E. Todorov, M.I. Jordan, Optimal feedback control as a theory of motor coordination. Nat. Neurosci. 5(11), 1226–1235 (2002)
- R. Shadmehr, S. Mussa-Ivaldi, Biological Learning and Control: How the Brain Builds repre- sentations, Predicts Events, and Makes Decisions (MIT Press, Cambridge, 2012)

**Code:**

simple_linear_programming.m: Download

simple_linear_programming.py: Download

© Francisco Valero-Cuevas 2015