Dynamics and control of a set of dual fingers with soft tips Suguru Arimoto,* Pham Thuc Anh Nguyen,* Hyun-Yong Han* and Zoe Doulgeri† SUMMARY This paper attempts firstly to derive a mathematical model of the dynamics of a set of dual fingers with soft and deformable tips which grasps and manipulates a rigid object with some dexterity. To gain a physical insight into the problem, consideration is restricted to the case that the motion of the whole system is confined to a horizontal plane. Secondly on the basis of the derived model it is shown that the rotation of the object can be indirectly controlled by the change of positions of the center points of both contact areas on the object. Then, each of the center points of contact areas can be positioned by inclining the last link of each corresponding finger against the object. It is further shown that, when both forces of pressing the object becomes almost equal, the equation of motion of the object in terms of rotational angles assumes the form of a harmonic oscillator with a forcing term, which can be regulated coordinately by the relative angle between the two last links contacting with the object. It is also shown that dynamics of this system satisfy passivity. Finally, design problems of control for dynamic stable grasping and enhancing dexterity in manipulating things are discussed on the basis of passivity analysis. KEYWORDS: Dual fingers; Multi-fingered hand; Soft fingers; Manipulation; Stable grasping. 1. INTRODUCTION It was about two decades ago that robot engineers were optimistic about predicting that robots would quickly evolve into a higher form and be able to do any kind of tasks that a human being can do. However, even in the beginning of the new millennium 2000, it is said that robots are too clumsy to be used in ordinary tasks that humans must do in their everyday life. In fact, a variety of multi-fingered robot hands carefully designed and manufactured with very high precision has been brought into practice in manufacturing but mainly used in simpler repetitive tasks. Regardless of such fine hand mechanisms, it is quite a difficult problem to endow them with dexterity and versatility in execution of a variety of ordinary tasks. This paper is motivated by the observation that the prime cause of this difficulty may originate from the lack of our knowledge of physical characterizations of dynamics of such fine and sophisticated mechanisms physically interacting (grasping, handling, or manipulating) with things and environment. In particular, the paper attempts to explore dynamics of a set of dual fingers with soft and deformable finger-tips that are grasping and manipulating a rigid object, because most of the past literature treated only the case of multi-fingered hands contacting rigidly and point-wise with an object or environment 1-3 and proposed to use rolling of finger-tips on the surface of the object in order to enhance dexterity. Differently from the case of rigid finger-tips, it is possible to observe from dynamics of dual soft fingers that each center of contact areas can be changed by coordinately inclining the last links against the object. This means that the posture of the object can be controlled without making rigid and point-wise rolling of the finger-tips on the object surface, which may cause dry fiction and slipping. In this paper it is shown that the derived equation of motion of the system naturally satisfies passivity. Further, it is shown that the special equation of motion of the object in terms of rotation in a horizontal plane is of the form of motion equation of a pendulum with a forcing term composed of two moments, each of which is equal to the reproducing force of a deformed contact area times the length from each corre- sponding contact center to the horizontal center line of the object through the mass center. When two reproducing forces arose from deformed contact areas become almost equal, the rotational angle of the object can be controlled by changes of angles of two last links relative to the object, by which lengths of contact centers to the horizontal center line can be also regulated. Then a feedback control scheme for attaining stable grasping in a dynamic sense is proposed and it is shown that the closed-loop system approaches asymp- totically an equilibrium manifold of still states of grasping. Another feedback scheme is also proposed, that can control the rotation angle of the object in addition to attaining a stable grasping. In final two sections a variety of future research directions necessary to unveil secrets of dexterity and versatility of multi-fingered hands with soft finger-tips will be discussed. It is then claimed that, with the evolution of technology of tactile sensing as well as real-time robot vision, soft hands with multiple fingers will be used in more versatile everyday tasks that need automation. 2. DYNAMICS OF DUAL SOFT FINGERS MANIPULATING AN OBJECT For the sake of simplifying the mathematical argument and gaining a physical insight into the problem why soft fingers can manipulate an object with some dexterity and versatil- ity, we assume that motion of the set of dual fingers is confined to a horizontal plane (see Figure 1) and not affected by the gravity force. Further, we treat the case that the object is rigid with a rectangular shape and shape of * Department of Robotics, Ritsumeikan University, H-1 Nojihgh- shi, Kusatsu, Shiga, 525-8577 (Japan) † Department of Electrical and Computer Engineering, Aristotle University of Thessaloniki, Thessaloniki 54006 (Greece) Robotica (2000) volume 18, pp. 71–80. Printed in the United Kingdom © 2000 Cambridge University Press