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.

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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

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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.

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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)

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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.

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Fig. 5

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

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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)




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