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Aggregated Hierarchical Sliding Mode Control for a Spatial Ridable Ballbot

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Abstract

In this study, the dynamical model of a ridable ballbot system is investigated to derive a nonlinear model that can describe the spatial dynamics of the mechatronic system by use of a set of differential equations. An aggregated hierarchical sliding mode control is proposed for a spatial ridable ballbot which is a class of multiple-input and multiple-output underactuated systems funder input coupling case and nonholonomic velocity constraints. The spatial ridable ballbot system is divided into ball and body subsystems. The hierarchical structure of the sliding mode surfaces (SMSs) is designed on the basis of the two subsystems as follows. First, the SMS of every subsystem is defined. Then the SMS of each subsystem is defined as the first layer SMS. The first layer SMS is used to construct the second layer SMS with the SMS of the other subsystem. This process continues until the SMS of all subsystems is included. The total control law is deduced from the hierarchical structure. Asymptotic stability of the control system is theoretically proven by Barbalat’s lemma in the ideal of Lyapunov theory. Simulation and experimental results indicate the validity of the proposed controller.

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Abbreviations

m a :

body mass

m k :

ball mass

I k :

moment of inertia of the ball

I x, I y :

moments of inertia of the body about the x- and y-axes

l :

distance from the mass center of the body and that of the ball

r k :

ball radius

r w :

omnidirectional wheel radius

I w :

moment of inertia of each omnidirectional wheel

α :

zenith angle

x k, y k :

ball position

θ x, θ y :

roll and pitch angles of body

τ x, τ y :

toques of the motors corresponding to the x- and y-axes

b rx, b ry, b x, b y :

friction factors

g :

the gravity acceleration

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Authors and Affiliations

  1. School of Mechanical Engineering, Kyung Hee University, 1732, Deogyeong-daero, Giheung-gu, Yongin-si, Gyeonggi-do, 17104, South Korea

    Dinh Ba Pham & Soon-Geul Lee

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  1. Dinh Ba Pham
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  2. Soon-Geul Lee
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Correspondence to Soon-Geul Lee.

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Pham, D.B., Lee, SG. Aggregated Hierarchical Sliding Mode Control for a Spatial Ridable Ballbot. Int. J. Precis. Eng. Manuf. 19, 1291–1302 (2018). https://doi.org/10.1007/s12541-018-0153-5

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  • DOI: https://doi.org/10.1007/s12541-018-0153-5

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