Realization of 5-axis machine tool with the proposed PSH. 

Context in source publication

Context 1

... realize 5-axis machining, the proposed PSH can be mounted on a serial X - Y translational platform, as shown in Figure 2, where the serial platform is responsi- ble for translational motion of the 5-axis machine tool, especially for compensating the parasitic motion caused by the PSH. 17 In workspace design of a practical manipulator, cone angle limits of spherical joints always greatly influence the volume of the workspace. In general, it is impossible for a common spherical joint to realize a cone angle larger than 90 ° ; thereby, the Hook-type spherical joint is always adopted instead. 18 However, in practice, due to its shortage in stiffness, the Hook-type spherical joint is always fabricated in a large size to meet the stiffness requirements, 19 which will introduce other undesired degradations, such as interferences. In this article, a modified spherical joint, as shown in Figure 3, is proposed and used instead of the common one. The modified spherical joint consists of a bearing stud, two upper ball-sockets, a lower ball-socket, a bearing and a bearing chock. As depicted in Figure 3, a deep groove is cut through the upper and low ball- sockets to endow the ball-end bearing stud with a large tilting capacity. The lower ball-socket is mounted on a bearing chock via a taper roller bearing thereby the spin angle of the modified spherical joint has no limits and still three available DOF. But, the stiffness of this modified spherical joint is still relatively lower than it of common one and there will be an additional bending moment on its bearing, so a precise verification is required in its design. Because of the compact structure, relatively high stiffness comparing with Hook-type spherical joint and, especially, the large tilting angle of 120 ° , the modified spherical joint is still inclined to be utilized, which significantly enlarges of the orientation workspace of the S joint. Points C i and C i ( i = 1, 2, 3) indicate the position of the slider along the sliding rail and the position of the R joint, respectively. For the sake of kinematics analysis, a fixed Cartesian reference coordinate frame O f x 0 , y 0 , z 0 g is set on the fixed base at point O , the central point of triangle D A A A , and x -axis passes through A and z 0 -axis is along its normal vector. Similarly, a moving reference frame P f u , v , w g is attached on the moving platform at point P ( B 4 ), the central point of triangle D B 1 B 2 B 3 , u -axis passes through B 1 and w -axis is along its normal vector. In practice, three PRS legs are symmetrically mounted about the central axis of the proposed PSH. The redundant UPS leg is mounted between O and B 4 , ! ! and let d 4 = OB 4 . Meanwhile, r a = j OA i j denotes the ! distance from O to A i , r b = j PB i j indicates the distance ! between P and B i in the i th PRS leg, and r m = j PB 4 j is distance from point P to B 4 in the UPS leg. Relating to fixed frame O f x 0 , y 0 , z 0 g , some vectors are also defined, including p = OP ! = 1⁄2 p x p y p z T , h i = C 0 i ! C i , l i = C i ! B i , d i j i = 1, 2, 3 = A i C ! 0 i . a i = OA ! i and b i = PB ! i . b 0 i denotes b i expressing in P f u , v , w g . The proposed PSH, which is actually developed on the basis of a 3-PRS PM, always has three parasitic motions: one rotation about z 0 -axis of the fixed frame and two translations along x 0 - and y 0 -axes of the fixed frame. 16 In fact, those parasitic motions are generated synchronously with the other independent motions under the mechanical constraints imposed by the R joints. In Figure 4, the position vectors of A i ( i = 1, 2, 3) and B i ( i = 1, 2, 3) with respect to frames O and P can be expressed ...