Exponential Stability of Two Timoshenko Arms for Grasping and Manipulating an Object
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ISSN:1598-6446 eISSN:2005-4092 http://www.springer.com/12555
Exponential Stability of Two Timoshenko Arms for Grasping and Manipulating an Object Dongming Wu, Takahiro Endo*, and Fumitoshi Matsuno Abstract: This paper discusses a grasping, orientation, and position control of an object by two flexible arms modeled by the Timoshenko beam. By utilizing our previously proposed controller for the cooperative control, we realize to control the grasping force, orientation, and position of an object as well as to suppress the vibration of the flexible arms. In particular, the closed-loop system is formulated in a Hilbert space, and the exponential stability of the closed-loop system is shown by the frequency domain method. The lightweight flexible arm has intrinsic elastic compliance, and thus the grasping by the flexible arms is safer than that by the rigid arms. In addition, the exponential stability of the closed-loop system contributes to the performance improvement of the grasping and manipulation of an object. Finally, experiments are conducted to confirm the validity of the stability analysis. Keywords: Distributed parameter systems, exponential stability, flexible arm, grasping and manipulation.
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INTRODUCTION
It is known that flexible arms consisting of lightweight links have several advantages over the rigid arms, such as high operational speed, greater payload-to-manipulatorweight ratio, low power consumption, and safer operation. Furthermore, the flexible arms have various application areas by utilizing these desired characteristics [1–4]. However, the lightweight links result in the vibration problem due to its low stiffness, and vibration suppression is one of the critical issues in the control of the flexible arm. In the dynamic model of flexible arms, the EulerBernoulli beam model is widely used. On the other hand, when considering the low-slenderness-ratio model, the Timoshenko beam model shows more accurate dynamics of arm models because of including effects of shear and rotational inertia [5]. Therefore, the Timoshenko beam model has a broader application than the Euler-Bernoulli beam model has. In addition, dynamics of the Timoshenko beam are represented by partial differential equations (PDE), and dynamics of a point mass, including actuators and rigid arm tips, are represented by ordinary differential equations (ODE). Thus, the model of the overall system consisting of the flexible arm, actuators, rigid arm tips, and others, is a PDE-ODE model. In this paper, the flexible arm modeled by the Timoshenko beam is called the Timoshenko arm, and we focus on the control of Timoshenko arm based on the PDE-ODE model. Vibration control is a common topic in the control of
flexible arms, and many studies have discussed the vibration suppression problem of the Timoshenko beam based on the PDE-ODE model [6–14]. If we broaden our focus from the Timoshenko beam to a flexible system modeled by the PDE-ODE model, various types of vibration control have been proposed. As the recent literature, see [15–23] and ref
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