17th Mediterranean Conference on Control & Automation Makedonia Palace, Thessaloniki, Greece June 24 - 26, 2009 A Novel Energy Pumping Strategy for Robotic Swinging Evangelos Papadopoulos, Senior Member IEEE and Georgios Papadopoulos Abstract - In this wOI'k we show that an Auobot can be made to behave as a !"Obotic swing. This is achieved by contI'oIling the fil'st joint, pI'ovided that a given condition is satisfied. When this condition is not satisfied, the system undel'goes thl'ough singulal' points. Even when this happens, we al'e again able to make the system behave as a swing by contI'oIling the second joint and employing a new Enel'gy Pumping stmtegy. This stmtegy pl'esents impOl-tant advantages compal'ed to pI'eviously PI'oposed stmtegies, as it is the only one that can stal-t the system f!"Om I'est and dI"ive it to lal'ge heights. MOI'eovel', it is fast and I'equi!'es vel'y small tOI·ques. Index Terms - Robotic swing, enel'gy pumping, gymnast robots , underactuated systems. I. INTRODUCTION T he swing problem has attracted the interest of a number of researchers during the last thirty years. Indeed, the problem is very interesting as it involves increasing the energy of a multibody system using internal (chemical in the case of humans) energy or motions. However, most research focused on dynamic analysis rather than on methods that can result in robotic swinging using controls. Also, none of these studied the effects of singular points or verified if the proposed movements are feasible for a particular under-actuated system. The work up-to-date can be classified in two broad categories. The first deals with swing analysis and reports alternative kinematic strategies without a plan to implement them with active control. They also focus on techniques that allow an increase of the width of oscillation of a system (for Energy Pumping) but do not deal with how to make the system swing with a given amplitude and keep this oscillation constant, see for example [1-7]. One of the earliest works considered the swing model as a simple pendulum with variable length, [1]. Several years later, a strategies for initiation and pumping the swing from a standing position was published following a qualitative approach [2]. Swinging from standing and sitting positions was studied and it was concluded that the swing is best characterized as a forced oscillator, [3], [4]. Two different kinds of swinging were compared in [5]. In another study, the question whether people act as self optimizing machines E. Papadopoulos is with the Department of Mechanical Engineering, National Technical University of Athens, (NTUA) 15 780 Athens (te l: +30- 210-772-1440; fax: +30-210-772-1455; e-mail: egpapado@central.ntua.gr). G. Papadopoulos was with the Department of Mechanical Engineering, NTUA, 15 780 Athens (e-mail: gpapado@mit.edu). Currently, he is a graduate student at the Department of Mechanical Engineering at the Massachusetts Institute of Technology, (MIT). while they swing was investigated, [6]. These studies do not address the issue of robotic swinging, which is dealt with in [7], using a sitting swing strategy but relying only on linear controls based purely on common experience. The second type of work deals with the Acrobat problem in which the goal is to bring the system (an underactuated inverted pendulum) to the up right position, [8-12]. In his pioneering work, M. Spong used partial feedback linearization to bring the Acrobot to the upright position, [9]. Later, researchers tried to achieve the same goal, but most controllers were based on energy methods (e.g. [10], [11]). Other works have used Lyapunov methods and were successful to bring the first Acrobot link to any desired position [12]. Bringing the Acrobot to the up right position with constraints to the second link has been studied, [13]. This kind of motion is close to the motion that gynmasts make on the high bar. The aim of this paper is to study robotic swinging of an Acrobot-type robot using partial model based control. Here, the second link is restricted from making a full revolution. The encountered singular points due to the loss of angular momentum coupling are studied. Their dynamic nature is explained, as well as how they can be avoided using a new swinging strategy. A new energy pumping strategy is proposed that presents important advantages over existing strategies. This strategy can start the system from rest, is fast and requires low torques. II. SYSTEM DYNAMICS To study the robotic swing and the pumping of energy that occurs, (i.e. the transfer of energy from the actuated dof to the unactuated one), an Acrobot-type system is employed. The Acrobot is an under-actuated robotic system with two degrees of freedom, (dot), i.e. the angle of the first link, ql ' and the angle of the second link, q2' see Fig. 1. Of those, only the second dof is actuated. Figure 1. Acrobot system and its parameters. 978-1-4244-4685-8/09/$25.00 ©2009 IEEE 928