Conventional machine tools typically use a linear x-y table for mounting a workpiece and a z-axis for mounting a spindle. The x-y table is usually very heavy, and its operating speed is relatively slow. The machining capability of such a conventional machine tool is often limited to straight lines or simple two-dimensional contours. It is essential that non-conventional machine tools be developed for free-form three-dimensional machining of general shapes.
The Stewart platform has been studied extensively for use as a flight simulator and as a parallel manipulator (Stewart, D. 1965, "A Platform with Six Degrees of Freedom," Proc. Institute of Mechanical Engr., London, England, Vol. 180, pp. 371-386). Other variations of the Stewart platform have also been proposed. Kohli et al. studied several six-DOF parallel manipulators which are driven by base-mounted rotary-linear actuators (Kohli, D., Lee, S. H., Tsai, K. Y., and Sandor, G. N., 1988, "Manipulator Configurations Based on Rotary-Linear (R-L) Actuators and Their Direct and Inverse Kinematics," ASME Journal of Mechanisms, Transmissions, and Automation in Design, Vol. 110, pp. 397-404). Hudgens and Tesar introduced a six-DOF parallel micromanipulator (Hudgens, J. C., and Tesar, D., 1988, "A Fully-Parallel Six Degree-of-Freedom Micromanipulator: Kinematic Analysis and Dynamic Model," Trends and Developments in Mechanisms, Machines, and Robotics, Proc. of the 20th ASME Biennial Mechanisms Conference, DE-Vol. 15-3, pp. 29-37). Pierrot, et al. studied a parallel manipulator using spatial parallelograms (Pierrot, F., Reynaud, and Fournier, A., 1990, "DELTA: A Simple and Efficient Parallel Robot," Robotica, Vol. 8, pp. 105-109). Pierrot, et al. introduced a high-speed six-DOF parallel manipulator (Pierrot, F., Fournier, A., and Dauchez, P., 1991, "Toward a Fully Parallel 6 DOF Robot for High-Speed Applications," Proc. of the 1991 IEEE International Conference on Robotics and Automation, pp. 1288-1293). Most of these six-DOF parallel manipulators consist of six limbs connecting a moving platform to a fixed base by spherical joints. These six-limbed manipulators suffer the following disadvantages:
1. Their direct kinematics are very difficult to solve.
2. Position and orientation of the moving platform are coupled.
3. Their workspace is relatively small.
4. Spherical joint is difficult to manufacture with high precision.
Note that the only six-limbed, six-DOF parallel manipulators for which closed-form direct kinematic solutions have been reported in the literature are special forms of the Stewart platform (Nanua. P., Waldron, K. J., and Murthy, V., 1990, "Direct Kinematic Solution of a Stewart Platform," IEEE Transactions on Robotics and Automation, Vol. 6, pp. 438-444; Grffis, M., and Duffy, J., 1989, "A Forward Displacement Analysis of a Class of Stewart Platforms," Journal of Robotic Systems, Vol. 6, pp. 703-720; Innocenti, C., and Parenti-Castelli, V., 1990, "Direct Position Analysis of the Stewart Platform Mechanism," Mechanism and Machine Theory, Vol. 25, pp. 611-612). In these special forms, pairs of spherical joints are concentric on either just the platform or both the base and the platform. However, as mentioned by Griffs and Duffy, pairs of concentric spherical joints may present design problems. As to the general Stewart platform, researchers have to resort to numerical techniques for the solutions. Innocenti and Parenti-Castelli developed an exhaustive search algorithm to solve the direct kinematics problem of the general Stewart platform (Innocenti, C., and Parenti-Castelli, V., 1993, "Forward Kinematics of the General 6--6 Fully Parallel Mechanism: An Exhaustive Numerical Approach Via a Mono-Dimensional Search Algorithm," ASME Journal of Mechanical Design, Vol. 115, pp. 932-937). Raghavan applied the continuation method and found that the general Stewart platform has 40 direct kinematics solutions (Raghavan, M., 1993, "The Stewart Platform of General Geometry Has 40 Configurations," ASME Journal of Mechanical Design, Vol. 115, pp. 277-282).
Although parallel manipulators have been studied thoroughly, most of the studies have concentrated on their applications as a robot manipulator. Recently, Giddings and Lewis (1995) introduced a machine tool called the "VARIAX Machining Center" utilizing the Stewart platform construction. Six legs connect a moving platform to a base. The upper and lower ends of each leg are connected to the moving platform and the base by gimbals. A spindle is mounted on the moving platform to cut the workpiece. Each of the six legs houses a ball screw. Individual servo motors drive the ball screws. Extending and retracting of the legs controls the position and orientation of the spindle making 6-axis machining feasible. This machine represents a revolutionary design in the machine tool industry. However, since the machine tool utilizes the Stewart platform construction, it suffers all the problems mentioned above. Although the minimanipulator introduced by Tahmasebi and Tsai (Tahmasebi, F., and Tsai, L. W., 1994, U.S. Pat. No. 5,279,176) contains only three limbs, it was designed for manipulating an object in a relatively small workspace.
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