# Optimality vs. variability: an example of multi-finger redundant tasks

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## Abstract

Two approaches to motor redundancy, optimization and the principle of abundance, seem incompatible. The former predicts a single, optimal solution for each task, while the latter assumes that families of equivalent solutions are used. We explored the two approaches using a four-finger pressing task with the requirement to produce certain combination of total normal force and a linear combination of normal forces that approximated the total moment of force in static conditions. In the first set of trials, many force–moment combinations were used. Principal component (PC) analysis showed that over 90% of finger force variance was accounted for by the first two PCs. The analytical inverse optimization (ANIO) approach was applied to these data resulting in quadratic cost functions with linear terms. Optimal solutions formed a hyperplane (“optimal plane”) in the four-dimensional finger force space. In the second set of trials, only four force–moment combinations were used with multiple repetitions. Finger force variance within each force–moment combination in the second set was analyzed within the uncontrolled manifold (UCM) hypothesis. Most finger force variance was confined to a hyperplane (the UCM) compatible with the required force–moment values. We conclude that there is no absolute optimal behavior, and the ANIO yields the best fit to a family of optimal solutions that differ across trials. The difference in the force-producing capabilities of the fingers and in their moment arms may lead to deviations of the “optimal plane” from the subspace orthogonal to the UCM. We suggest that the ANIO and UCM approaches may be complementary in the analysis of motor variability in redundant systems.

## Keywords

Hand Force Moment of force Uncontrolled manifold hypothesis Inverse optimization ANIO approach## Notes

### Acknowledgments

The study was in part supported by NIH grants AG-018751, NS-035032, and AR-048563. We are grateful to Dr. Alexander Terekhov for his help at the early phases of this project.

## References

- Ait-Haddou R, Jinha A, Herzog W, Binding P (2004) Analysis of the force-sharing problem using an optimization model. Math Biosci 191:111–122CrossRefPubMedGoogle Scholar
- Bernstein NA (1967) The co-ordination and regulation of movements. Pergamon Press, OxfordGoogle Scholar
- 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 Dyn 16:123–154CrossRefGoogle Scholar
- Feldman AG (1986) Once more on the equilibrium-point hypothesis (lambda model) for motor control. J Mot Behav 18:17–54PubMedGoogle Scholar
- Feldman AG, Levin MF (1995) The origin and use of positional frames of reference in motor control. Behav Brain Sci 18:723–806CrossRefGoogle Scholar
- Freitas SM, Scholz JP (2009) Does hand dominance affect the use of motor abundance when reaching to uncertain targets? Hum Mov Sci 28:169–190CrossRefPubMedGoogle Scholar
- Gelfand IM, Latash ML (1998) On the problem of adequate language in movement science. Mot Control 2:306–313Google Scholar
- Goodman SR, Latash ML (2006) Feed-forward control of a redundant motor system. Biol Cybern 95:271–280CrossRefPubMedGoogle Scholar
- Gorniak SL, Zatsiorsky VM, Latash ML (2007) Hierarchies of synergies: an example of two-hand, multi-finger tasks. Exp Brain Res 179:167–180CrossRefPubMedGoogle Scholar
- Harris CM, Wolpert DM (1998) Signal-dependent noise determines motor planning. Nature 394:780–784CrossRefPubMedGoogle Scholar
- Kaiser HF (1960) The application of electronic computers to factor analysis. Psychol Meas 20:141–151CrossRefGoogle Scholar
- Krishnamoorthy V, Latash ML, Scholz JP, Zatsiorsky VM (2003) Muscle synergies during shifts of the center of pressure by standing persons. Exp Brain Res 152:281–292CrossRefPubMedGoogle Scholar
- Latash ML (2008) Synergy. Oxford University Press, New YorkCrossRefGoogle Scholar
- Latash ML (2010) Motor control: in search of physics of the living systems. J Hum Kinet 24:7–18CrossRefGoogle Scholar
- Latash ML, Scholz JF, Danion F, Schöner G (2001) Structure of motor variability in marginally redundant multifinger force production tasks. Exp Brain Res 141:153–165CrossRefPubMedGoogle Scholar
- Latash ML, Scholz JP, Schöner G (2002) Motor control strategies revealed in the structure of motor variability. Exerc Sport Sci Rev 30:26–31CrossRefPubMedGoogle Scholar
- Latash ML, Shim JK, Smilga AV, Zatsiorsky VM (2005) A central back-coupling hypothesis on the organization of motor synergies: a physical metaphor and a neural model. Biol Cybern 92:186–191CrossRefPubMedGoogle Scholar
- Latash ML, Scholz JP, Schoner G (2007) Toward a new theory of motor synergies. Mot Control 11:276–308Google Scholar
- Latash ML, Friedman J, Kim SW, Feldman AG, Zatsiorsky VM (2010) Prehension synergies and control with referent hand configurations. Exp Brain Res 202:213–229CrossRefPubMedGoogle Scholar
- Li ZM, Latash ML, Zatsiorsky VM (1998) Force sharing among fingers as a model of the redundancy problem. Exp Brain Res 119:276–286CrossRefPubMedGoogle Scholar
- Martin V, Scholz JP, Schöner G (2009) Redundancy, self-motion and motor control. Neural Comput 21:1371–1414CrossRefPubMedGoogle Scholar
- Murray RM, Li Z, Sastry S (1994) A mathematical introduction to robotic manipulation. CRC Press, Roca Baton, FLGoogle Scholar
- Mussa Ivaldi FA, Morasso P, Zaccaria R (1988) Kinematic networks. A distributed model for representing and regularizing motor redundancy. Biol Cybern 60:1–16PubMedGoogle Scholar
- Newell KM, Carlton LG (1988) Force variability in isometric responses. J Exp Psychol Hum Percept Perform 14:37–44CrossRefPubMedGoogle Scholar
- Newell KM, Corcos DM (1993) Variability and motor control. Human Kinetics Publishers Champaign, ILGoogle Scholar
- Oldfield RC (1971) The assessment and analysis of handedness: the Edinburgh inventory. Neuropsychologia 9:97–113CrossRefPubMedGoogle Scholar
- Prilutsky BI, Zatsiorsky VM (2002) Optimization-based models of muscle coordination. Exerc Sport Sci Rev 30:32–38CrossRefPubMedGoogle Scholar
- 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 34:1243–1255CrossRefPubMedGoogle Scholar
- Robert T, Zatsiorsky VM, Latash ML (2008) Multi-muscle synergies in an unusual postural task: quick shear force production. Exp Brain Res 187:237–253CrossRefPubMedGoogle Scholar
- Rosenbaum DA, Meulenbroek RJ, Vaughan J, Jansen C (2001) Posture-based motion planning: applications to grasping. Psychol Rev 108:709–734CrossRefPubMedGoogle Scholar
- Schmidt RA, Zelaznik H, Hawkins B, Frank JS, Quinn JT Jr (1979) Motor-output variability: a theory for the accuracy of rapid motor acts. Psychol Rev 47:415–451CrossRefPubMedGoogle Scholar
- Scholz JP, Schöner G (1999) The uncontrolled manifold concept: identifying control variables for a functional task. Exp Brain Res 126:289–306CrossRefPubMedGoogle Scholar
- Scholz JP, Schöner G, Latash ML (2000) Identifying the control structure of multijoint coordination during pistol shooting. Exp Brain Res 135:382–404CrossRefPubMedGoogle Scholar
- Scholz JP, Danion F, Latash ML, Schöner G (2002) Understanding finger coordination through analysis of the structure of force variability. Biol Cybern 86:29–39CrossRefPubMedGoogle Scholar
- Seif-Naraghi AH, Winters JM (1990) Optimal strategies for scaling goal-directed arm movements. In: Winters JM, Woo SL-Y (eds) Multiple muscle systems: biomechanics and movement organization. Springer, New York, pp 312–334Google Scholar
- Siemienski A (2006) Direct solution of the inverse optimization problem of load sharing between muscles. J Biomech 39(Suppl. 1):45CrossRefGoogle Scholar
- Terekhov AV, Pesin YB, Niu X, Latash ML, Zatsiorsky VM (2010) An analytical approach to the problem of inverse optimization with additive objective functions: an application to human prehension. J Math Biol 61:423–453CrossRefPubMedGoogle Scholar
- Todorov E, Jordan MI (2002) Optimal feedback control as a theory of motor coordination. Nat Neurosci 5:1226–1235CrossRefPubMedGoogle Scholar
- Tsirakos D, Baltzopoulos V, Bartlett R (1997) Inverse optimization: functional and physiological considerations related to the force-sharing problem. Crit Rev Biomed Eng 25:371–407PubMedGoogle Scholar
- Whitney DE (1969) Resolved motion rate control of manipulators and human prostheses. IEEE Trans Man Machine Syst 10:47–53CrossRefGoogle Scholar
- Yang JF, Scholz JP, Latash ML (2007) The role of kinematic redundancy in adaptation of reaching. Exp Brain Res 176:54–69CrossRefPubMedGoogle Scholar
- Zatsiorsky VM, Latash ML (2008) Multifinger prehension: an overview. J Mot Behav 40:446–476CrossRefPubMedGoogle Scholar
- Zatsiorsky VM, Li ZM, Latash ML (1998) Coordinated force production in multi-finger tasks: finger interaction and neural network modeling. Biol Cybern 79:139–150CrossRefPubMedGoogle Scholar
- Zatsiorsky VM, Li ZM, Latash ML (2000) Enslaving effects in multi-finger force production. Exp Brain Res 131:187–195CrossRefPubMedGoogle Scholar