Research Papers

A Set-Point Control for a Two-link Underactuated Robot With a Flexible Elbow Joint

[+] Author and Article Information
Xin Xin

Faculty of Computer Science and Systems Engineering,
Okayama Prefectural University,
Okayama 719-1197, Japan
e-mail: xxin@cse.oka-pu.ac.jp

Yannian Liu

Graduate School of Natural Science and Technology,
Okayama University,
Okayama, Okayama 700-8530, Japan
e-mail: ynliujp@yahoo.co.jp

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received April 18, 2012; final manuscript received May 6, 2013; published online July 9, 2013. Assoc. Editor: Alexander Leonessa.

J. Dyn. Sys., Meas., Control 135(5), 051016 (Jul 09, 2013) (10 pages) Paper No: DS-12-1110; doi: 10.1115/1.4024427 History: Received April 18, 2012; Revised May 06, 2013

This paper concerns a set-point control problem for a two-link underactuated robot moving in the vertical plane with a single actuator at the first joint and a spring between the two links (flexible elbow joint). First, we present two new properties of such a flexible robot about the linear controllability at the upright equilibrium point (UEP, where two links are in the upright position) and about its equilibrium configuration. Second, we show that for the robot with a certain range of spring constant, the proportional derivative (PD) control on the angle of the first joint can globally stabilize the robot at the UEP. Third, for the robot not satisfying the above range of spring constant, we study how to extend the energy-based control approach, which aims to control the total mechanical energy and the angle and angular velocity of the first joint of the robot, to design a swing-up controller. We provide a necessary and sufficient condition for avoiding the singularity in the controller, and we analyze the motion of the robot under the presented controller by studying the convergence of the total mechanical energy and clarifying the structure and stability of the closed-loop equilibrium points. We present the results of numerical investigation, which support our theoretical conclusions.

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Grahic Jump Location
Fig. 1

AF robot: two-link flexible robot with a spring between two links

Grahic Jump Location
Fig. 2

Plots of f1 and f2 with 0 < γ ≤ 0.1 and 0.1 < γ ≤ 1: solid lines for f1 in Eq. (57) and dashed lines for f2 in Eq. (58)

Grahic Jump Location
Fig. 3

Time responses of q1 and q2 of the PD control: kP=5

Grahic Jump Location
Fig. 4

Time responses of V0 and τ of the PD control: kP=5

Grahic Jump Location
Fig. 5

Time responses of q1 and q2 of the PD control: kP=4

Grahic Jump Location
Fig. 6

Time responses of V and E-Er of the swing-up control

Grahic Jump Location
Fig. 7

Time responses of q1 and q2 of the swing-up control

Grahic Jump Location
Fig. 8

Phase portrait of (q2,q·2) in 5≤t≤20 and time response of τ of the swing-up control

Grahic Jump Location
Fig. 9

Time responses of q1 and q2 of the swing-up and stabilizing control



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