Research Papers

Stability of Damped Skateboards Under Human Control

[+] Author and Article Information
Balazs Varszegi

Department of Applied Mechanics,
Budapest University of Technology
and Economics,
MTA-BME Lendulet Human
Balancing Research Group,
Budapest H-1111, Hungary
e-mail: varszegi@mm.bme.hu

Denes Takacs

MTA-BME Research Group on
Dynamics of Machines and Vehicles,
Budapest H-1111, Hungary
e-mail: takacs@mm.bme.hu

Gabor Stepan

Department of Applied Mechanics,
Budapest University of Technology
and Economics,
Budapest H-1111, Hungary
e-mail: stepan@mm.bme.hu

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received September 21, 2016; final manuscript received March 20, 2017; published online May 4, 2017. Assoc. Editor: Bernard Brogliato.

J. Comput. Nonlinear Dynam 12(5), 051014 (May 04, 2017) (7 pages) Paper No: CND-16-1442; doi: 10.1115/1.4036482 History: Received September 21, 2016; Revised March 20, 2017

A simple mechanical model of the skateboard–skater system is analyzed, in which a linear proportional-derivative (PD) controller with delay is included to mimic the effect of human control. The equations of motion of the nonholonomic system are derived with the help of the Gibbs–Appell method. The linear stability analysis of the rectilinear motion is carried out analytically in closed form. It is shown that how the control gains have to be varied with respect to the speed of the skateboard in order to stabilize the uniform motion. The critical reflex delay of the skater is determined as functions of the speed, position of the skater on the board, and damping of the skateboard suspension system. Based on these, an explanation is given for the experimentally observed dynamic behavior of the skateboard–skater system at high speed.

Copyright © 2017 by ASME
Your Session has timed out. Please sign back in to continue.


Kane, T. R. , and Levinson, D. A. , 2005, Dynamics, Theory and Applications, The Internet-First University Press, Ithaca, New York.
Hubbard, M. , 1979, “ Lateral Dynamics and Stability of the Skateboard,” ASME J. Appl. Mech., 46(4), pp. 931–936. [CrossRef]
Wisse, M. , and Schwab, A. L. , 2005, “ Skateboards, Bicycles, and Three-Dimensional Biped Walking Machines: Velocity-Dependent Stability by Means of Lean-to-Yaw Coupling,” Int. J. Rob. Res., 24(6), pp. 417–429. [CrossRef]
Kooijman, J. D. G. , Meijaard, J. P. , Papadopoulos, J. M. , Ruina, A. , and Schwab, A. L. , 2011, “ A Bicycle Can Be Self-Stable Without Gyroscopic or Caster Effects,” Science, 332(6027), pp. 339–342. [CrossRef] [PubMed]
Kremnev, A. , and Kuleshov, A. , 2007, “ Nonlinear Dynamics and Stability of a Simplified Skateboard Model,” The Engineering of Sport 7, Vol. 1, pp. 135–142.
Ispolov, Y. , and Smolnikov, B. , 1996, “ Skateboard Dynamics,” Comput. Methods Appl. Mech. Eng., 131(3–4), pp. 327–333. [CrossRef]
Hubbard, M. , 1980, “ Human Control of the Skateboard,” J. Biomech., 13(9), pp. 745–754. [CrossRef] [PubMed]
Rosatello, M. , Dion, J.-L. , Renaud, F. , and Garibaldi, L. , 2015, “ The Skateboard Speed Wobble,” ASME Paper No. DETC2015-47326.
Varszegi, B. , Takacs, D. , and Stepan, G. , 2015, “ Skateboard: A Human Controlled Non-Holonomic System,” ASME Paper No. DETC2015-47512.
Varszegi, B. , Takacs, D. , Stepan, G. , and Hogan, S. J. , 2016, “ Stabilizing Skateboard Speed-Wobble With Reflex Delay,” J. R. Soc. Interface, 13(121), p. 20160345.
Milton, J. G. , Solodkin, A. , Hluštík, P. , and Small, S. L. , 2007, “ The Mind of Expert Motor Performance Is Cool and Focused,” NeuroImage, 35(2), pp. 804–813. [CrossRef] [PubMed]
Stepan, G. , 2009, “ Delay Effects in the Human Sensory System During Balancing,” Philos. Trans. R. Soc., A, 367(1891), pp. 1195–1212. [CrossRef]
Chagdes, J. R. , Rietdyk, S. , Jeffrey, M. H. , Howard, N. Z. , and Raman, A. , 2013, “ Dynamic Stability of a Human Standing on a Balance Board,” J. Biomech., 46(15), pp. 2593–2602. [CrossRef] [PubMed]
Chagdes, J. R. , Haddad, J. M. , Rietdyk, S. , Zelaznik, H. N. , and Raman, A. , 2015, “ Understanding the Role of Time-Delay on Maintaining Upright Stance on Rotational Balance Boards,” ASME Paper No. DETC2015-47857.
Insperger, T. , and Milton, J. G. , 2014, “ Sensory Uncertainty and Stick Balancing at the Fingertip,” Biol. Cybern., 108(1), pp. 85–101. [CrossRef] [PubMed]
McRuer, D. T. , Weir, D. H. , Jex, H. R. , Magdeleno, R. E. , and Allen, E. W. , 1975, “ Measurement of Driver-Vehicle Multiloop Response Properties With a Single Disturbance Input,” IEEE Trans. Syst., Man, Cybern., SMC-5(5), pp. 490–497. [CrossRef]
Gantmacher, F. , 1975, Lectures in Analytical Mechanics, MIR Publisher, Moscow, Russia.
Peterka, R. J. , 2002, “ Sensorimotor Integration in Human Postural Control,” J. Neurophysiol., 88(3), pp. 1097–1118. [PubMed]
Maurer, C. , and Peterka, R. J. , 2004, “ A New Interpretation of Spontaneous Sway Measures Based on a Simple Model of Human Postural Control,” J. Neurophysiol., 93(1), pp. 189–200. [CrossRef] [PubMed]
Vette, A. H. , Masani, K. , Nakazawa, K. , and Popovic, M. R. , 2010, “ Neural-Mechanical Feedback Control Scheme Generates Physiological Ankle Torque Fluctuation During Quiet Stance,” IEEE Trans. Neural Syst. Rehabil. Eng., 18(1), pp. 86–95. [CrossRef] [PubMed]
Kremnev, A. V. , and Kuleshov, A. S. , 2008, “ Dynamics and Simulation of the Simplest Model of a Skateboard,” European Nonlinear Dynamics Conference 2008 (ENOC-2008), Saint Petersburg, Russia, June 30–July 4.
Varszegi, B. , Takacs, D. , Stepan, G. , and Hogan, S. J. , 2014, “ Balancing of the Skateboard With Reflex Delay,” Eighth EUROMECH Nonlinear Dynamics Conference (ENOC2014), Vienna, Austria, July 6–11.
Stepan, G. , 1989, Retarded Dynamical Systems: Stability and Characteristic Functions, Longman Scientific and Technical Co-Published With Wiley, New York.
Insperger, T. , and Stépán, G. , 2011, Semi-Discretization for Time-Delay Systems: Stability and Engineering Applications, Vol. 178, Springer Science and Business Media, Berlin.


Grahic Jump Location
Fig. 1

Mechanical model of the skateboard–skater system [9]; panel (a) shows the back view of the mechanical model, panel (b) shows the top one and panel (c) shows the skateboard suspension system

Grahic Jump Location
Fig. 2

Qualitative structure of the stability charts in the p–d plane, where the shaded domains indicate the stable regimes. The time delay τ increases from left to right, and the damping coefficient kd increases from top to bottom.

Grahic Jump Location
Fig. 3

Effect of the longitudinal speed on the ultimate time delay; continuous line refers to the fore standing (a = 0.1 m), while the dashed line refers to the back standing (a = −0.1 m)

Grahic Jump Location
Fig. 4

Effect of the longitudinal speed and the damping ratio on the critical time delay; continuous lines stand for stability boundaries, dashed lines do for ultimate critical time delay, and the shaded domain indicates p–d stabilizable domains, while the white domains are unstable. The upper row belongs to the standing behind and the row below belongs to the standing ahead.

Grahic Jump Location
Fig. 5

Effect of the speed and the damping ratio on the required control gains

Grahic Jump Location
Fig. 6

Effect of the speed and the damping ratio on the area of stable domain in p–d plane; panel (a) belongs to the standing ahead case (a = 0.1 m), while panel (b) refers to the standing behind one (a = −0.1 m)



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In