Studies in System Science (SSS) Volume 2, 2014 www.as‐se.org/sss
Variable Structure Control of Spherical Robots with Exponential Reaching Law Tao Yu*1, Hanxu Sun2 1
Faculty of Mechanical Engineering and Automation, Liaoning University of Technology, Jinzhou, Liaoning, China
2
School of Automation, Beijing University of Posts and Telecommunications, Haidian District, Beijing, China
*1
yutaolanjie@163.com; 2hxsun@bupt.edu.cn
Abstract In this study, the dynamics and control aspects of the longitudinal motion of pendulum‐driven spherical mobile robots are investigated. A simplified dynamic model is established for the longitudinal motion of a pendulum‐driven spherical robot by using Lagrangian dynamics. By appropriate definitions the equations of motion for the robotic system are transformed into the state space form. A hierarchical sliding mode controller based on a new exponential reaching law is proposed to achieve set‐point regulation of the longitudinal motion. The asymptotic stability of the whole system is verified through Lyapunov analysis, and the validity of the proposed approach is illustrated through numerical simulations. Keywords Spherical Robot; Longitudinal Motion; Hierarchical Sliding Mode Control; Exponential Reaching Law
Introduction The extensive research on spherical mobile robots has the foundation in the belief that there are certain applications where spherical mobile robots are more advantageous than traditional wheeled mobile robots. Spherical mobile robots are more versatile, less exposed to physical conditions, and they have greater resistance levels towards object collisions. Moreover, the spherical shell of this class of mobile robots provides an efficient cover for the internal driving mechanisms and sensory equipments. Spherical mobile robots can be categorized into different types according to their internal driving mechanisms [1‐9]. Compared with other types of spherical mobile robots [1‐6], a pendulum‐driven spherical mobile robot [7‐9] has a simpler structure, making it easier to be manoeuvred. The schematic diagram of a pendulum‐driven spherical mobile robot with dual inputs is illustrated in Fig. 1. Longitudinal motion is a basic form of locomotion of pendulum‐driven spherical mobile robots, and it is realized by moving a motor‐controlled pendulum forwards or backwards. In this paper, a hierarchical sliding mode control approach based on a novel exponential reaching law is presented for stable control of the longitudinal motion. In the proposed controller, a double layer structure is used to guarantee the stability of the whole system, and the sub‐sliding surfaces are utilized to drive the tracking errors to zero.
Y
O'
m1
B(D), m2
R
l
E, m3
O
X
FIG. 1 STRUCTURE OF A PENDULUM‐DRIVEN SPHERICAL ROBOT FIG. 2 SIMPLIFIED MODEL OF THE LONGITUDINAL MOTION
Dynamic Analysis We start with a simplified planar model, only considering no slip longitudinal motion on flat surfaces. Fig. 2 illustrates the simplified model with a side view of a pendulum‐driven spherical robot. It represents the spherical shell with its center of mass B, the internal mechanism with its center of mass D, which coincides with that of the
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