Journal of Mathematical Biology

, Volume 61, Issue 3, pp 423–453

An analytical approach to the problem of inverse optimization with additive objective functions: an application to human prehension

Article

Abstract

We consider the problem of what is being optimized in human actions with respect to various aspects of human movements and different motor tasks. From the mathematical point of view this problem consists of finding an unknown objective function given the values at which it reaches its minimum. This problem is called the inverse optimization problem. Until now the main approach to this problems has been the cut-and-try method, which consists of introducing an objective function and checking how it reflects the experimental data. Using this approach, different objective functions have been proposed for the same motor action. In the current paper we focus on inverse optimization problems with additive objective functions and linear constraints. Such problems are typical in human movement science. The problem of muscle (or finger) force sharing is an example. For such problems we obtain sufficient conditions for uniqueness and propose a method for determining the objective functions. To illustrate our method we analyze the problem of force sharing among the fingers in a grasping task. We estimate the objective function from the experimental data and show that it can predict the force-sharing pattern for a vast range of external forces and torques applied to the grasped object. The resulting objective function is quadratic with essentially non-zero linear terms.

Keywords

Inverse optimization Human prehension Uniqueness theorem Principal component analysis 

List of symbols

x

An independent variable

J

An objective function of an optimization problem

\({\mathcal C}\)

Constraints for an optimization problem

\({\left\langle J,\mathcal C\right\rangle}\)

An optimization problem with the objective function J and the constrains \({\mathcal{C}}\)

fi(·), gi(·)

Scalar functions

C, b

A matrix and a vector of the linear constraints Cxb

ai

A scalar value

\({\mathcal I}\)

A set of indexes

Mathematics Subject Classification (2000)

15A06 15A09 49K99 62P10 92C10 

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References

  1. Ahuja RK, Orlin JB (2001) Inverse optimization. Oper Res 49(5): 771–783MATHCrossRefMathSciNetGoogle Scholar
  2. Ait-Haddou R, Binding P, Herzog W (2000) Theoretical considerations on cocontraction of sets of agonistic and antagonistic muscles. J Biomech 33(9): 1105–1111CrossRefGoogle Scholar
  3. Anderson FC, Pandy MG (1999) A dynamic optimization solution for vertical jumping in three dimensions. Comput Methods Biomech Biomed Eng 2(3): 201–231CrossRefGoogle Scholar
  4. Bernstein NA (1967) The coordination and regulation of movements. Pergamon, OxfordGoogle Scholar
  5. Biess A, Liebermann DG, Flash T (2007) A computational model for redundant human three-dimensional pointing movements: integration of independent spatial and temporal motor plans simplifies movement dynamics. J Neurosci 27(48): 13045–13064CrossRefGoogle Scholar
  6. Bottasso CL, Prilutsky BI, Croce A, Imberti E, Sartirana S (2006) A numerical procedure for inferring from experimental data the optimization cost functions using a multibody model of the neuro-musculoskeletal system. Multibody Syst Dynam 16: 123–154MATHCrossRefMathSciNetGoogle Scholar
  7. Cole KJ, Johansson RS (1993) Friction at the digit-object interface scales the sensorimotor transformation for grip responses to pulling loads. Exp Brain Res 95(3): 523–532CrossRefGoogle Scholar
  8. Collins JJ (1995) The redundant nature of locomotor optimization laws. J Biomech 28(3): 251–267CrossRefGoogle Scholar
  9. Crowninshield RD, Brand RA (1981) A physiologically based criterion of muscle force prediction in locomotion. J Biomech 14(11): 793–801CrossRefGoogle Scholar
  10. Cruse H, Wischmeyer E, Brwer M, Brockfeld P, Dress A (1990) On the cost functions for the control of the human arm movement. Biol Cybern 62(6): 519–528CrossRefGoogle Scholar
  11. Edelman S, Flash T (1987) A model of handwriting. Biol Cybern 57(1–2): 25–36CrossRefGoogle Scholar
  12. Engelbrecht S (2001) Minimum principles in motor control. J Math Psychol 45(3): 497–542MATHCrossRefMathSciNetGoogle Scholar
  13. Erdemir A, McLean S, Herzog W, van den Bogert AJ (2007) Model-based estimation of muscle forces exerted during movements. Clin Biomech (Bristol, Avon) 22(2): 131–154CrossRefGoogle Scholar
  14. Flash T, Hogan N (1985) The coordination of arm movements: an experimentally confirmed mathematical model. J Neurosci 5(7): 1688–1703Google Scholar
  15. Herzog W (1992) Sensitivity of muscle force estimations to changes in muscle input parameters using nonlinear optimization approaches. J Biomech Eng 114(2): 267–268CrossRefGoogle Scholar
  16. Herzog W, Binding P (1992) Predictions of antagonistic muscular activity using nonlinear optimization. Math Biosci 111(2): 217–229MATHCrossRefMathSciNetGoogle Scholar
  17. Johansson RS, Westling G (1984) Roles of glabrous skin receptors and sensorimotor memory in automatic control of precision grip when lifting rougher or more slippery objects. Exp Brain Res 56(3): 550–564CrossRefGoogle Scholar
  18. Kuo AD, Zajac FE (1993) Human standing posture: multi-joint movement strategies based on biomechanical constraints. Prog Brain Res 97: 349–358CrossRefGoogle Scholar
  19. Kuzelicki J, Zefran M, Burger H, Bajd T (2005) Synthesis of standing-up trajectories using dynamic optimization. Gait Posture 21(1): 1–11CrossRefGoogle Scholar
  20. Niu X, Latash ML, Zatsiorsky VM (2007) Prehension synergies in the grasps with complex friction patterns: local versus synergic effects and the template control. J Neurophysiol 98(1): 16–28CrossRefGoogle Scholar
  21. Pataky TC, Latash ML, Zatsiorsky VM (2004) Prehension synergies during nonvertical grasping, ii: modeling and optimization. Biol Cybern 91(4): 231–242MATHCrossRefGoogle Scholar
  22. Pham QC, Hicheur H, Arechavaleta G, Laumond JP, Berthoz A (2007) The formation of trajectories during goal-oriented locomotion in humans. ii. a maximum smoothness model. Eur J Neurosci 26(8): 2391–2403CrossRefGoogle Scholar
  23. Plamondon R, Alimi AM, Yergeau P, Leclerc F (1993) Modelling velocity profiles of rapid movements: a comparative study. Biol Cybern 69(2): 119–128CrossRefGoogle Scholar
  24. Prilutsky BI (2000) Coordination of two- and one-joint muscles: functional consequences and implications for motor control. Motor Control 4(1): 1–44Google Scholar
  25. Prilutsky BI, Gregory RJ (2000) Analysis of muscle coordination strategies in cycling. IEEE Trans Rehabil Eng 8(3): 362–370CrossRefGoogle Scholar
  26. Prilutsky BI, Zatsiorsky VM (2002) Optimization-based models of muscle coordination. Exerc Sport Sci Rev 30(1): 32–38CrossRefGoogle Scholar
  27. Raikova RT, Prilutsky BI (2001) Sensitivity of predicted muscle forces to parameters of the optimization-based human leg model revealed by analytical and numerical analyses. J Biomech 34(10): 1243–1255CrossRefGoogle Scholar
  28. Redl C, Gfoehler M, Pandy MG (2007) Sensitivity of muscle force estimates to variations in muscle-tendon properties. Hum Mov Sci 26(2): 306–319CrossRefGoogle Scholar
  29. Shim JK, Latash ML, Zatsiorsky VM (2003) Prehension synergies: trial-to-trial variability and hierarchical organization of stable performance. Exp Brain Res 152(2): 173–184CrossRefGoogle Scholar
  30. Siemienski A (2006) Direct solution of the inverse optimization problem of load sharing between muscles. J Biomech 39: S45CrossRefGoogle Scholar
  31. Tsirakos D, Baltzopoulos V, Bartlett R (1997) Inverse optimization: functional and physiological considerations related to the force-sharing problem. Crit Rev Biomed Eng 25(4–5): 371–407Google Scholar
  32. Westling G, Johansson RS (1984) Factors influencing the force control during precision grip. Exp Brain Res 53(2): 277–284CrossRefGoogle Scholar
  33. Zatsiorsky VM, Latash ML (2008) Multifinger prehension: an overview. J Mot Behav 40(5): 446–476CrossRefGoogle Scholar
  34. Zatsiorsky VM, Gregory RW, Latash ML (2002) Force and torque production in static multifinger prehension: biomechanics and control. ii. control. Biol Cybern 87(1): 40–49MATHCrossRefGoogle Scholar
  35. Zatsiorsky VM, Gao F, Latash ML (2003) Prehension synergies: effects of object geometry and prescribed torques. Exp Brain Res 148(1): 77–87CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Department of KinesiologyThe Pennsylvania State UniversityUniversity ParkUSA
  2. 2.Institut des Systèmes Intelligents et de RobotiqueCNRS-UPMC, Pyramide ISIRParisFrance
  3. 3.Department of MathematicsThe Pennsylvania State UniversityUniversity ParkUSA

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