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.

References

References
1.
Kane
,
T. R.
, and
Levinson
,
D. A.
,
2005
,
Dynamics, Theory and Applications
,
The Internet-First University Press
,
Ithaca, New York
.
2.
Hubbard
,
M.
,
1979
, “
Lateral Dynamics and Stability of the Skateboard
,”
ASME J. Appl. Mech.
,
46
(
4
), pp.
931
936
.
3.
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
.
4.
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
.
5.
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
.
6.
Ispolov
,
Y.
, and
Smolnikov
,
B.
,
1996
, “
Skateboard Dynamics
,”
Comput. Methods Appl. Mech. Eng.
,
131
(
3–4
), pp.
327
333
.
7.
Hubbard
,
M.
,
1980
, “
Human Control of the Skateboard
,”
J. Biomech.
,
13
(
9
), pp.
745
754
.
8.
Rosatello
,
M.
,
Dion
,
J.-L.
,
Renaud
,
F.
, and
Garibaldi
,
L.
,
2015
, “
The Skateboard Speed Wobble
,”
ASME
Paper No. DETC2015-47326.
9.
Varszegi
,
B.
,
Takacs
,
D.
, and
Stepan
,
G.
,
2015
, “
Skateboard: A Human Controlled Non-Holonomic System
,”
ASME
Paper No. DETC2015-47512.
10.
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.
11.
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
.
12.
Stepan
,
G.
,
2009
, “
Delay Effects in the Human Sensory System During Balancing
,”
Philos. Trans. R. Soc., A
,
367
(
1891
), pp.
1195
1212
.
13.
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
.
14.
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.
15.
Insperger
,
T.
, and
Milton
,
J. G.
,
2014
, “
Sensory Uncertainty and Stick Balancing at the Fingertip
,”
Biol. Cybern.
,
108
(
1
), pp.
85
101
.
16.
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
.
17.
Gantmacher
,
F.
,
1975
,
Lectures in Analytical Mechanics
,
MIR Publisher
,
Moscow, Russia
.
18.
Peterka
,
R. J.
,
2002
, “
Sensorimotor Integration in Human Postural Control
,”
J. Neurophysiol.
,
88
(
3
), pp.
1097
1118
.
19.
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
.
20.
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
.
21.
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.
22.
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.
23.
Stepan
,
G.
,
1989
,
Retarded Dynamical Systems: Stability and Characteristic Functions
,
Longman Scientific and Technical Co-Published With Wiley
,
New York
.
24.
Insperger
,
T.
, and
Stépán
,
G.
,
2011
,
Semi-Discretization for Time-Delay Systems: Stability and Engineering Applications
, Vol.
178
,
Springer Science and Business Media
,
Berlin
.
You do not currently have access to this content.