An Optimum Design Procedure for Two-finger Grippers: a Case of Study

In this paper we have focused the attention on the dimensional synthesis of the mechanisms in two-finger grippers, that we have named as gripping mechanisms to emphasize on their basic gripping purpose. The design problem has been approached and formulated as an optimization problem by a proposed formulation for the basic characteristics of gripping mechanisms. A case of study has been reported as a numerical example to show the soundness of the proposed optimum synthesis procedure by referring to computational and practical results. INTRODUCTION The two-finger grasp is extensively used both for human and industrial grip, (Pham and Heginbotham, 1986), since it may be considered the simplest efficient grasping configuration. Most of the gripping systems that are installed in industrial automations and robots are mechanical two-finger grippers, (Belforte, 1985; Pham and Heginbotham, 1986). They are used both for manipulation and assembling purposes since most of these tasks can be performed with a two-finger grasp configuration, (Potter, 1985; Taylor and Swarz, 1955). A gripper can be considered as a critical component of automated manipulations since it interacts with the environment and particularly with the piece to be machined or manipulated so that the gripper gives a great contribution to a practical success of using an automated or robotized solution. Therefore, a good design of a gripper may be of fundamental importance. The design of a gripper must take into account several aspects of the components and the system together with the peculiarities of a given application or a multi-task purpose. Strong constraints for the gripping system can be considered lightness, small dimensions, rigidity, multi-task capability, simplicity and lack of maintenance. These design characteristics can be achieved by considering specific end-effectors or grippers. In the last case a two-finger gripper corresponds to the minimum number of fingers and the minimum complexity of an hand. This means that a good design of a two-finger gripper requires to take into account several problems, (Chelpanov and Kolpashikov, 1983; Chen 1982; Tanie, 1985; Ceccarelli 1994), whose understanding and consideration are used in this paper with a designing aim. In this paper we have focused the attention on the dimensional synthesis of the mechanisms in twofinger grippers, that we have named as gripping mechanisms to emphasize on their basic gripping purpose. In particular, some peculiarities of the gripping mechanisms have been considered with the aim to deduce a useful analytical formulation. Then, the synthesis problem has been approached and formulated as an optimization problem by the proposed formulation by using the basic characteristics of gripping mechanisms. A suitable algorithm has been developed for a general optimum synthesis of gripping mechanisms starting from previous experiences reported in (Deibe et al., 1997; Ceccarelli 1997). A case of study has been investigated and reported as a numerical example in the paper to show the soundness of the proposed optimum synthesis procedure by referring to computational and practical results. THE OPTIMUM DESIGN PROBLEM FOR TWO-FINGER GRIPPERS Industrial applications in structured environment may not require sensorial means so that a gripper structure may be simplified as sketched in Fig.1. Therefore the basic components of a two-finger gripper are, Fig.1: shaped fingers; gripping mechanism; connecting transmissions; actuator. The mechanical part of a two-finger gripper can be considered as composed by, Fig.1: fingers and finger tips, which are the elements in contact with a grasped object, so that they perform the mechanics of grasp on the object itself; gripping mechanism, which is the transmission component between the actuator and the fingers; actuator, which is the power source for the grasping action of a gripper. Basic features for a gripper are related to capability for the grasping forces and grasp size. In fact the above-mentioned characteristics are fundamental from a practical viewpoint for the grasping purpose, since they may describe the range of exerting force on the object by the fingers and the size range of the objects which may be grasped. Thus, a dimensional design of gripping mechanisms may have great influence on the capability D of a gripper, Fig.1, and on the grasping force, since the mechanism size may affect the grasp configuration and transmission characteristics of the device. These peculiarities can be considered well known when it is taken into account the great variety of mechanisms which have been used and are still used as Fig.1. Mechanical components and design parameters of a two-finger gripper. gripping mechanisms, (Chen 1982; Belforte 1985). However, we believe that a proper formulation of the design problem for gripping mechanisms may also help for an optimum use of the gripping mechanisms, since a suitable synthesis can give optimum characteristics with respect to the gripping purpose, (Ceccarelli, 1994). Since the main purpose of a two-finger gripper can be recognized in performing a planar grasp, some fundamental design considerations may be deduced by approaching a two-finger grasp and its mechanics, (Ceccarelli and Nieto, 1993). Referring to Fig.2, a configuration of two-finger grasp can be characterized by possible elementary motions of a grasped object among the fingers as: slipping down along a direction which is orthogonal to a plane of the gripping forces exerted by the fingers as couplers of mechanisms; squeezing which may occur in sliding outward or inward to the gripper itself due to an effect of force components pushing the object (in this case an increase or a decrease of the gripping force by fingers may produce a squeezing motion or a release of the object); rolling about the squeezing line and winding about the slipping line consisting in a motion of revolution of the object among the fingers due to an external torque or a force couple of FA and FB; whirling about the contact line which can be also due to a torque by the object's weight when its mass center does not lay on the contact line yet. Although two contact-points cannot considered enough even for a planar grasp, a two-finger gripper can perform a suitable grasp when force constraints are taken into account so that four conditions can be achieved for a firm grasp. In addition, all the above mentioned elementary motions of the object may be avoided when suitable grasping force FA and FB are exerted by the fingers so that friction forces may arise to completely balance the external action on the object. However, for sake of simplicity and because indeed the difference in position can be very small, the grasping forces can be thought with a common value F. Same observations can be developed for the friction evaluation at the points A and B through the coefficients μA and μB, which can be assumed with a common value μ. The angles φA and φB represent the angles of the friction cones at the contact points A and B, respectively, and they are related to the friction coefficients by μA = tan φA and μB = tan φB. The grasping configuration of the object with respect to the fingers gives the angles ψA and ψB, Fig.2. A mechanical model for a two-finger grasp. which strongly depend on the orientation of the fingers, the position of the contact points, and the shape of the object and the fingers. Although the contact points, which define the contact line as a line joining them, Fig.2, can be located in the same relative position on the fingers, the angles ψA and ψB can differ from each other. However, because of the symmetry of a two-finger gripper, in a design procedure it is convenient to assume them as equal to the most unfavorable value ψ, which refers to the case that gives a squeezing component from the grasping force itself. In Fig.2 a planar grasp by two fingers has been modeled by using the above-mentioned considerations. In addition, r A and r B represent the distances of A and B, respectively, from the squeezing line; r G is the distance of G from the contact line; W is the weight of the object and it is oriented with an angle ΦW with respect to the squeezing line. T is an external torque acting on the object, and it includes the inertial actions due to the manipulator movement, as well as W may include the inertial and external forces, so that the model of Fig.2 can describe all the situations, which may occur to an object grasped by a two-finger gripper. The basics of the static equilibrium of the grasped object can be expressed by using the model of Fig.2 along the directions of the contact and the squeezing lines in term of forces as 0 = Ö sin W + sin F sin F + cos F cos F W B B B A A A B B A A ψ μ ψ μ ψ ψ (1) 0 = Ö cos W + cos F cos F sin F + sin F W B B B A A A B B A A ψ μ ψ μ ψ ψ and in term of torque as ( ) ( ) 0 = cos sin F r + cos sin F r Ö sin W r T B B B B B A A A A A W G ψ μ ψ ψ μ ψ (2) Indeed, the directions of the friction forces, and consequently their signs in the Eqs.(1) and (2), will be determined to be contrary to the relative motion between the object and the finger. Dynamic variations can be modeled by assuming T and W variable as a function of the motion of the gripper and object. The above-mentioned design models and performance considerations address great importance to the gripping mechanism since it transmits the motion and force to the fingers for a suitable grasp. In addition, such considerations may strongly suggest to use a design formulation for gripping mechanisms in the form of an optimization problem as