# Human stick balancing: an intermittent control explanation

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## Abstract

There are two issues in balancing a stick pivoting on a finger tip (or mechanically on a moving cart): maintaining the stick angle near to vertical and maintaining the horizontal position within the bounds of reach or cart track. The (linearised) dynamics of the angle are second order (although driven by pivot acceleration), and so, as in human standing, control of the angle is not, by itself very difficult. However, once the angle is under control, the position dynamics are, in general, fourth order. This makes control quite difficult for humans (and even an engineering control system requires careful design). Recently, three of the authors have experimentally demonstrated that humans control the stick angle in a special way: the closed-loop inverted pendulum behaves as a non-inverted pendulum with a virtual pivot somewhere between the stick centre and tip and with increased gravity. Moreover, they suggest that the virtual pivot lies at the radius of gyration (about the mass centre) above the mass centre. This paper gives a continuous-time control-theoretical interpretation of the virtual-pendulum approach. In particular, by using a novel cascade control structure, it is shown that the horizontal control of the virtual pivot becomes a second-order problem which is much easier to solve than the generic fourth-order problem. Hence, the use of the virtual pivot approach allows the control problem to be perceived by the subject as two separate second-order problems rather than a single fourth-order problem, and the control problem is therefore simplified. The theoretical predictions are verified using the data previously presented by three of the authors and analysed using a standard parameter estimation method. The experimental data indicate that although all subjects adopt the virtual pivot approach, the less expert subjects exhibit larger amplitude angular motion and poorly controlled translational motion. It is known that human control systems are delayed and intermittent, and therefore, the continuous-time strategy cannot be correct. However, the model of intermittent control used in this paper is based on the virtual pivot continuous-time control scheme, handles time delays and moreover masquerades as the underlying continuous-time controller. In addition, the event-driven properties of intermittent control can explain experimentally observed variability.

## Keywords

Inverted Pendulum Psychological Refractory Period Intermittent Control Variable Sample Interval Fixed Sample Interval## Notes

### Acknowledgments

The work reported here is related to the linked EPSRC Grants EP/F068514/1, EP/F069022/1 and EP/F06974X/1 “Intermittent control of man and machine”. Peter Gawthrop was supported by the NICTA Victoria Research Laboratory at the University of Melbourne and is now a Professorial Fellow within the Melbourne School of Engineering; he would also like to acknowledge the many discussions about intermittent control with Ian Loram, Martin Lakie, Henrik Gollee and Liuping Wang. The authors would like to thank the reviewers for their helpful comments on the draft manuscript.

## References

- Asai Y, Tasaka Y, Nomura K, Nomura T, Casadio M, Morasso P (2009) A model of postural control in quiet standing: robust compensation of delay-induced instability using intermittent activation of feedback control. PLoS ONE 4(7):e6169Google Scholar
- Astrom KJ, Furuta K (2000) Swinging up a pendulum by energy control. Automatica 36(2):287–295CrossRefGoogle Scholar
- Bottaro A, Casadio M, Morasso PG, Sanguineti V (2005) Body sway during quiet standing: is it the residual chattering of an intermittent stabilization process? Hum Mov Sci 24(4):588–615PubMedCrossRefGoogle Scholar
- Bottaro A, Yasutake Y, Nomura T, Casadio M, Morasso P (2008) Bounded stability of the quiet standing posture: an intermittent control model. Hum Mov Sci 27(3):473–495PubMedCrossRefGoogle Scholar
- Boubaker O (2012) The inverted pendulum: a fundamental benchmark in control theory and robotics. In: Education and e-Learning Innovations (ICEELI), 2012 international conference on, pp 1–6, July 2012Google Scholar
- Cabrera JL, Milton JG (2002) On-off intermittency in a human balancing task. Phys. Rev. Lett. 89(15):158702PubMedCrossRefGoogle Scholar
- Craik KJ (1947) Theory of human operators in control systems: part 1, the operator as an engineering system. British J Psychol 38:56–61Google Scholar
- Dobrowiecki TP, Schoukens J (2001) Practical choices in the FRF measurement in presence of nonlinear distortions. Instrum Meas IEEE Trans 50(1):2–7Google Scholar
- Fantoni I, Lozano R (2002) Stabilization of the Furuta pendulum around its homoclinic orbit. Int J Control 75(6):390–398CrossRefGoogle Scholar
- Fitzpatrick R, McCloskey DI (1994) Proprioceptive, visual and vestibular thresholds for the perception of sway during standing in humans. J Physiol 478(Pt 1):173–186PubMedGoogle Scholar
- Garnier H, Mensler M, Richard A (2003) Continuous-time model identification from sampled data: implementation issues and performance evaluation. Int J Control 76(13):1337–1357CrossRefGoogle Scholar
- Garnier H, Wang L (eds) (2008) Identification of continuous-time models from sampled data. Advances in industrial control. Springer, LondonGoogle Scholar
- Gawthrop P, Loram I, Lakie M (2009a) Predictive feedback in human simulated pendulum balancing. Biol Cybern 101(2):131–146PubMedCrossRefGoogle Scholar
- Gawthrop P, Loram I, Lakie M, Gollee H (2011) Intermittent control: a computational theory of human control. Biol Cybern 104(1–2):31–51PubMedCrossRefGoogle Scholar
- Gawthrop PJ (2009) Frequency domain analysis of intermittent control. Proceedings of the Institution of Mechanical Engineers Pt. I: J Syst Control Eng 223(5):591–603Google Scholar
- Gawthrop PJ, Lakie MD, Loram ID (2008) Predictive feedback control and Fitts’ law. Biol Cybern 98(3):229–238PubMedCrossRefGoogle Scholar
- Gawthrop PJ, Wagg DJ, Neild SA (2009b) Bond graph based control and substructuring. Simul Model Pract Theory 17(1):211–227CrossRefGoogle Scholar
- Gollee H, Mamma A, Loram ID, Gawthrop PJ (2012) Frequency-domain identification of the human controller. Biol Cybern 106:359372CrossRefGoogle Scholar
- Goodwin GC, Graebe SF, Salgado ME (2001) Control system design. Prentice Hall, New JerseyGoogle Scholar
- Insperger T, Milton J, Stepan G (2013) Acceleration feedback improves balancing against reflex delay. J R Soc Interface 10(79)Google Scholar
- Johansson R (1994) Identification of continuous-time models. Signal Process IEEE Trans 42(4):887–897CrossRefGoogle Scholar
- Karniel A (2011) Open questions in computational motor control. J Integr Neurosci 10(3):385–411PubMedCrossRefGoogle Scholar
- Kasdin NJ (1995) Discrete simulation of colored noise and stochastic processes and 1/f alpha; power law noise generation. Proc IEEE 83(5):802–827CrossRefGoogle Scholar
- Kleinman D, Baron S, Levison W (1971) A control theoretic approach to manned-vehicle systems analysis. Autom Control IEEE Trans 16(6):824–832CrossRefGoogle Scholar
- Kleinman DL, Baron S, Levison WH (1970) An optimal control model of human response part I: theory and validation. Automatica 6:357–369CrossRefGoogle Scholar
- Kowalczyk P, Glendinning P, Brown M, Medrano-Cerda G, Dallali H, Jonathan S (2011) Modelling human balance using switched systems with linear feedback control, J R Soc Interface 9(67):234–245Google Scholar
- Kwakernaak H, Sivan R (1972) Linear optimal control systems. Wiley, New YorkGoogle Scholar
- Lee K-Y, ODwyer N, Halaki M, Smith R (2012) A new paradigm for human stick balancing: a suspended not an inverted pendulum. Exp Brain Res 221(3):309–328Google Scholar
- Ljung L (1999) System identification: theory for the user. Information and systems science, 2nd edn. Prentice-Hall, Upper Saddle River, NJGoogle Scholar
- Loram D, Lakie M (2002) Human balancing of an inverted pendulum: position control by small, ballistic-like, throw and catch movements. J Physiol 540(3):1111–1124PubMedCrossRefGoogle Scholar
- Loram ID, Lakie M, Gawthrop PJ (2009) Visual control of stable and unstable loads: what is the feedback delay and extent of linear time-invariant control? J Physiol 587(6):1343–1365PubMedCrossRefGoogle Scholar
- Loram ID, van de Kamp C, Gollee H, Gawthrop PJ (2012) Identification of intermittent control in man and machine. J R Soc Interface 9(74):2070–2084PubMedCrossRefGoogle Scholar
- Marr D, Vision A (1982) Computational investigation into the human representation and processing of visual information. W.H. Freeman, San Francisco. ISBN 0716712849Google Scholar
- Mehta B, Schaal S (2002) Forward models in visuomotor control. J Neurophysiol 88(2):942–953PubMedGoogle Scholar
- Milton JG, Cabrera JL, Ohira T (2008) Unstable dynamical systems: delays, noise and control. EPL (Europhys Lett) 83(4):48001 Google Scholar
- Milton JG (2011) The delayed and noisy nervous system: implications for neural control. J Neural Eng 8(6):065005PubMedCrossRefGoogle Scholar
- Milton JG, Ohira T, Cabrera JL, Fraiser RM, Gyorffy JB, Ruiz Ferrin K, Strauss MA, Balch EC, Marin PJ, Alexander JL (2009) Balancing with vibration: a prelude for drift and act balance control. PLoS ONE 4(10):e7427Google Scholar
- Navas F, Stark L (1968) Sampling or intermittency in hand control system dynamics. Biophys J 8(2):252–302PubMedCrossRefGoogle Scholar
- Neilson PD, Neilson MD, O’Dwyer NJ (1988) Internal models and intermittency: a theoretical account of human tracking behaviour. Biol Cybern 58:101–112PubMedCrossRefGoogle Scholar
- Nijhawan R, Wu S (2009) Compensating time delays with neural predictions: are predictions sensory or motor? Philos Trans R Soc A Math Phys Eng Sci 367(1891):1063–1078CrossRefGoogle Scholar
- Oytam Y, Neilson PD, O’Dwyer NJ (2005) Degrees of freedom and motor planning in purposive movement. Hum Mov Sci 24(5–6):710–730PubMedCrossRefGoogle Scholar
- Pintelon R, Schoukens J (2001) System identification: a frequency domain approach. IEEE press, New York CityGoogle Scholar
- Rao GP, Unbehauen H (2006) Identification of continuous-time systems. Control Theory Appl 153(2):185–220CrossRefGoogle Scholar
- Scholl E, Hiller G, Hovel P, Dahlem MA (2009) Time-delayed feedback in neurosystems. Philos Trans R Soc A Math Phys Eng Sci 367(1891):1079–1096Google Scholar
- Shadmehr R, Wise SP (2005) Computational neurobiology of reaching and pointing: a foundation for motor learning. MIT Press, Cambridge, MAGoogle Scholar
- Smith MC (March 1967) Theories of the psychological refractory period. Psychol Bull 67(3):202–213Google Scholar
- Stepan G (2009) Delay effects in the human sensory system during balancing. Philos Trans R Soc A Math Phys Eng Sci 367(1891):1195–1212CrossRefGoogle Scholar
- Suzuki Y, Nomura T, Casadio M, Morasso P (2012) Intermittent control with ankle, hip, and mixed strategies during quiet standing: a theoretical proposal based on a double inverted pendulum model. J Theor Biol 310:55–79PubMedCrossRefGoogle Scholar
- Telford CW (1931) The refractory phase of voluntary and associative responses. J Exp Psychol 14(1):1–36CrossRefGoogle Scholar
- Todorov E, Jordan MI (2002) Optimal feedback control as a theory of motor coordination. Nat Neurosci 5(11):1226–1235PubMedCrossRefGoogle Scholar
- Unbehauen H, Rao GP (1990) Continuous-time approaches to system identification a survey. Automatica 26(1):23–35CrossRefGoogle Scholar
- van de Kamp C, Gawthrop PJ, Gollee H, Loram ID (2013) Refractoriness in sustained visuo-manual control: is the refractory duration intrinsic or does it depend on external system properties? PLoS Comput Biol 9(1):e1002843Google Scholar
- Vince MA (1948) The intermittency of control movements and the psychological refractory period. British J Psychol 38:149–157Google Scholar
- Volkinshtein D, Meir R (2011) Delayed feedback control requires an internal forward model. Biol Cybern 105:41–53. ISSN 0340–1200Google Scholar
- Young PC (1965) The determination of the parameters of a dynamic process. Radio Electron Eng 29:345–361CrossRefGoogle Scholar
- Young P (1981) Parameter estimation for continuous-time models: a survey. Automatica 17(1):23–39CrossRefGoogle Scholar
- Young PC (2011) Recursive estimation and time-series analysis: an introduction for the student and practitioner. Springer, BerlinCrossRefGoogle Scholar