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Parallel Mechanisms with Two or Three Degrees of Freedom Christiaan J.J. Paredis, H. Benjamin Brown, Pradeep K. Khosla Abstract: Parallel manipulators for the machine tool Industry have been studied extensively for various industrial applications. However, limited useful workspace areas, the poor mobility, and design difficulties of more complex parallel manipulators have led to mare interest in parallel manipulators with less than six degrees of freedom (DoFs). Several parallel mechanisms with various numbers and types of degrees of freedom are described in this paper, which can be used in parallel kinematics machines, motion simulators, and industrial robots. Key words: parallel manipulator; parallel kinematic machine; degree of freedom; robot Introduction Mechanical systems that allow a rigid body to move with respect to a fixed base play a very important role in numerous applications. A rigid body can move in various translational or rotational directions which are called degrees of freedom (DoFs). The total number of degrees of freedom for a rigid body cannot exceed six, for example, three axes, A robot includes a system to control several degrees of freedom of an end – effector. The last few years have witnessed important developments in the use of industrial robots, mainly due to their flexibility. However, the mechanical architecture of the most common robots is not well adapted to certain tasks. Other types of architectures have, therefore, recently been developed for industrial use, including parallel manipulators. A parallel manipulator, which is a closed-loop mechanism, typically consists of a moving platform that is connected to a fixed base by several limbs or legs. Typically, the number of limbs is equal to the number of degrees of freedom such that every limb is controlled by one actuator and all the actuators can be mounted at or near the fixed base. For this reason, parallel manipulators. Because the external load can be shared by the actuators, parallel manipulators tend to have a large load-carrying capacity. Parallel manipulators are always presented as having very good performance in terms of accuracy, rigidity and the ability to manipulate large loads. They have been used in a large number of applications ranging from astronomy to flight simulators, and are becoming increasingly popular in the machine-tool industry, The conceptual design of parallel manipulators can be dated back to 1947, when Gough established the basic principles of a mechanism with a closed-loop kinematic structure to control the position and orientation of a moving platform to test tire wear and damage. He built a prototype in 1955 (Fig, 1a) where the moving element was a hexagonal platform whose vertices were all connected to links by ball-and-socket joints. The other end of the link was attached to the base by a universal joint. Six linear actuators modified the total link length. Stewart designed a platform manipulator as an aircraft simulator in 1965 (Fig, 1b), in which the moving element was a triangular platform whose vertices were all connected by ball-and-socket joints to support mechanisms each constituting of two jacks, also placed in a triangle. In 1978, Hunt made a systematic study of kinematic structure of parallel manipulators, with the planar three-RPS parallel manipulator as typical example. Since then, parallel manipulators have been studied extensively by numerous researchers. Most of the six-DoFs parallel manipulators studied to date have included six extendible limbs. These parallel manipulators possess the advantages of high stiffness, low inertia, and large payload capacity. However, they suffer the problems of relatively small useful workspace and design difficulties. Furthermore, their direct kinematics is very difficult to analyze. Therefore, parallel manipulators with less than six-DoFs have increasingly attracted attention for industry applications. This paper introduces parallel manipulators and the classification of parallel manipulators. Three types of new parallel manipulators are introduced: a spatial three-DoFs parallel manipulator, a two- DoFs parallel manipulator, and a planar three- DoFs serial-parallel manipulator. 1 Definition of parallel Manipulator A parallel manipulator is made of an end-effector with n degrees of freedom with a fixed base linked together by at least two independent kinematic linkages. Actuation takes place through n-simple actuators. These mechanisms have the following characteristics. · At least two linkages support the end-,effector. Each of those linkages contains at least one simple actuator. · The number of actuators is the same as the number of degrees of freedom of the end-effecto. ·The mobility of the manipulator is zero when the actuators are locked. Parallel mechanisms are of interest for the following reasons: ·The load can be distributed on the multiple linkages. ·Few actuators are needed. ·When the actuators are locked, the manipulator remains in position, which is an important safety concern for certain applications. Parallel manipulators for which the number of linkages is strictly equal to the number of degrees of freedom of the end-effector are called fully parallel manipulators. 2 Degrees of Freedom of a Mechanism The degrees of freedom of a mechanism are the number of independent parameters or inputs needed to completely specify the configuration of the mechanism. However, a general mobility criterion cannot be easily defined for closed-loop kinematic linkages，as Hunt and Lerbet already noted. Classical mobility formulae can indeed neglect some degrees of freedom. Grubler's formulae is nevertheless generally used, which may be written as M=d(n-g-1)+ (1) where M is system mobility (degrees of freedom); d is screw system order (d=3 for planar and spherical motion, d=6 for spatial motion); n is number of links including the frame; g is number of joints; and關(guān)are degrees of freedom associated with the i-th joint. 3 Classification of Parallel Manipulators The total number of degrees of freedom of a rigid body cannot exceed 6; therefore，the number of DoFs of a parallel manipulator will be between 2 and 6. Since the first parallel mechanism design， many mechanical designs have been proposed for parallel manipulators with 2 to 6 DoFs. A survey of 87 actuators proposed in the literature showed that 40% has six DoFs, 3.5% five DoFs, 6% four DoFs，40%three DoFs，and the remaining two DoFs. 3. 1 Two-DoFs parallel manipulators Most existing two-DoFs parallel manipulators are planar manipulators with two-translational DoFs. Such designs use only prismatic and revolute joints. McCloy showed that there are 20 different combinations. This number is reduced to 6 as shown in Fig. 2 if the actuators are assumed to be attached to the ground. There is no passive prismatic joint and no actuator is supporting the weight of another actuator. 3. 2 Three-DoFs parallel manipulators There are many three-I?oFs parallel manipulators, so only the classical designs will be presented here. One example is the planar three-RRR (R stands for revolving joint) parallel manipulator as shown in Fig. 3a. The moving platform has three planar DoFs，which are two translations along the x and y axes and one rotation around the axis perpendicular to the O-xy plane. Another example is the spherical three-RRR parallel manipulator as shown in Fig. 3b，in which all the joint axes intersect at a common vertex. The motion of any point in the mechanism is rotation about the vertex. The moving platform has only rotational DoFs with respect to the base. Hunt presented the three-RPS parallel manipulator shown in Fig. 3c, which has complex DoFs，which cannot be strictly defined. The most famous robot with three translations is the DELTA (Fig.3d)，proposed by Clavel and marketed by the Demaurex Company and ABB under the name IRB 340 FpexPicker. DELTA has been widely used in industry. Another type of three-DoFs parallel manipulator has the moving platform connected to the base through four legs，where the fourth leg is passive and is also the leading leg，which means that the leg determines the motion of the moving platform, for example，in the spherical coordinate parallel manipulator shown in Fig. 3e.This parallel manipulator is used for the machine tool design by IFW of the University of Hannover. 3. 3 Four-DoFs parallel manipulators. A four-DoFs fully parallel manipulator has d-=6, n=10, g=I2, and M=4.Substituting these coefficients into Eq. (1) gets F=22/4 which is the degrees of freedom for each leg. Therefore, there axe， actually, no four-DoFs fully parallel manipulators. The early mechanisms with four-DoFs were not fully parallel manipulators, i, e.， manipulators with two actuators per linkage or with passive constraints. 3. 4 Five-DoFs parallel manipulators Five-DoFs fully parallel manipulators must have F=29/5，so there are no five-DoFs fully parallel manipulators. A five-DoFs parallel manipulator proposed by Austad consists of two parallel manipulators. 3. 5 Six-DoFs parallel manipulators Six-DoFs parallel manipulators are the most popular manipulators so they have been studied extensively. The architecture shown in Fig. 4a is a classical six-DoFs parallel manipulator. Most six- DoFs parallel manipulators have six extendible limbs. These parallel manipulators possess the advantages of high stiffness, low inertia, and large payload capacity. However, they suffer the problems of relatively small useful workspace and design difficulties. Furthermore，analysis of their direct kinematics is very difficult. There are also Some exotic chain manipulators in which the manipulator is actuated by a planar mechanism, such as a four-bar mechanism, or a five-bar mechanism, or which have two actuators per leg and which usually have three legs 4 Evolution of Parallel Manipulators After Gough established the basic principles of mechanisms with closed-loop kinematic structures in 1947，as shown in Fig. 1a, many other parallel manipulators with a specified number and type of degrees of freedom have also been proposed. The architecture designed by Stewart in 1965 is shown in Fig, 1b. As shown in Fig. 4a, theoretically speaking, the six legs can be arranged at will to design various six-DoFs parallel manipulators, such as the manipulator shown in Fig. 46, where the legs are arranged in a 3-2-1 style which is a very compact structure that can be used in microsystem. The arrangement of the six legs shown in Fig. 4c makes the manipulator move freely along a specified direction，which is very useful for the industrial applications. A six-DoFs parallel manipulator similar to that proposed by Pierrot has each pair of legs in the manipulator shown in Fig. 4a parallel to each other. The number of DoFs of the manipulator will be different if the inputs to the two legs in each pair are the same. The equivalent manipulator architecture is shown in Fig.5. The manipulator output will be three translations，which is probably the origin of the DELTA robot. The actuated links can be arranged as the well-known, fast robot DELTA shown in Fig. 3d. DELTA has been made in several versions，such as the Pollard mechanism Tsars manipulator, Fig.3f, is also among three translational parallel manipulators. Although Tsars manipulator has translations identical with that of DELTA，it is not exactly a version of DELTA. Their design concepts are different and Tsars manipulator is the first design to deal with the problem of a UPU chain. Another three-translational DoFs parallel manipulator, Star, was design by Herve based on the group theory. Although these design concepts provide ideas to design a new manipulator, additional work is needed to design a robot combining translational and rotational DoFs with less than six DoFs. For example, there are few spatial three-DoFs parallel manipulators combining two spatial translations and one rotation, as will be presented in the following section. 5 New Spatial Three-DoFs 1 Manipulator 5. 1 Manipulator structure The spatial three-DoFs parallel manipulator shown in Fig.6a consists of a base plate, a movable platform，and three legs that connect the two plates. Each connecting leg has four degrees of freedom. Two of the three legs have identical chains with a two-DoFs joint (or two 1-DoF joints) and two 1-DoF joints.The third leg consists of a planar four-bar parallelogram and three 1-DoF joints. One 1-DoF joint in each leg is actuated. The moving platform is an isosceles triangle. The vertices of the platform are connected to a fixed-base plate through legs (1)，(8) and (12). Legs (1) and (12) have identical chains with a constant link connected to a universal joint (or two revolute joints) (15) or (13) at the bottom end and a passive revolute joint (3) or (11) at the other end. The revolute joint is then attached to an active slider (4) or (10)，which is mounted on the guide way (2) or (9) The third leg (8) has a constant link, a planar four-bar parallelogram, which is connected to a revolute joint (16) at the bottom end and a passive revolute joint (5) at the other end. The revolute joint is attached to an active slider (6) which is mounted on the guide way (7).The movement of the moving platform is accomplished by sliding the three sliders on the guide ways. 5. 2 Manipulator capability The proposed manipulator is a general manipulation device that must have three degrees of freedom when the input elements are active. In the arrangement of the links and joints shown in Fig. 6, legs (1) and (12) provide two constraints on the rotation of the moving platform about the z axis and the translation along the x axis. The revolute joints (5) and (16) for the third leg (8) have parallel axes as shown in Fig.6a.The third leg can provide one constraint on the rotation of the moving platform about the axis. Hence，the combination of the three legs constrains the rotation of the moving platform with respect to the z and z axes and the translation along the axis. Therefore, the mechanism has two translational degrees in the O-yz plane and one rotational degree of freedom about the y axis. 5.3 Novelties and applications The mechanical design is interesting because of the third actuating leg mechanism which uses a planar four-bar parallelogram, as used in other parallel mechanisms, such as Star Like robot, the Tsai manipulator, CaPaMan. This unique spatial three -DoFs parallel manipulator (a) has only revolute joints，

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