Fig. 1.Precession polishing process.

the tool and the workpiece. According to the Preston law [26], the influence function is not mathematically well-behaved and is not optimizedfor form control algorithms.

In a precession polishing process [13,14],therotation-axisof the polishing tool is inclined to the surface's localnormal direc- tion. The precession polishing process makes use of a compliant spherical shaped polishing tool and polishes with the side of the tool. As a result, the point with zero velocity is shifted outside the contact area. To obtain a uniform surface texture with no direc- tional properties, the tool axis is then precessed about the local normal direction of the surface. Precession averages the polishing texture, providing a tool influence function that is mathematically well-behaved and close to a Gaussian-like polishing influence function.

3. HybridmanipulatorforCCUPfreeformpolishing

3.1. Structureofthehybrid manipulator

To polish a freeform surface, the polishing machine should have five DOF including three traditional DOF and two rotational DOF. The layout of the mechanical structure, transmission system and driving system all directly affect the performance of the machine. In this study, the proposed polishing machine consists of a parallel module and a serial module using a hybrid manipulator as shown in Fig. 2.

The moving platform of the parallel module is connected to the base by three identical serial chains. Each of the three chains contains one spatial parallelogram, and the vertices of which are four ball joints. The parallelogram is connected to the base by a prismatic joint. The output can be obtained through a combination of the actuation to the three prismatic joints. A fixture that holds the workpiece is mounted on the moving platform. It can be ro- tated toprovidethe rotationmotionof theworkpiecewhen

Fig. 2. Configuration of the proposed machine with hybrid manipulator for CCUP freeformpolishing.

polishing axially symmetric surfaces. To improve the stiffness and workspace performance of the parallel module, the three guide- ways are inclined and intersect at avertex.

Theserialmoduleconsistsofarotating/tiltingtableand a

polishing tool. The table is used to rotate the polishing tool about two orthogonal axes. The rotating axis is vertical and the tilting axis is horizontal to the base. The curvature center of the tool head coincides with the virtual pivot intersected by the two axes.

3.2. Mobility

The first issue to address in the design of the manipulator is its motion capability. The output motion of the serial module is ob- vious when the input joints are actuated. The output motion of the parallel module should be designed to have threetranslational DOF with respect to the base. There are twelve ball joints, three prismatic joints and eleven links, including the base. Applying the Grübler–Kutzbach formula [27]

The inverse kinematics problem involves mapping a known pose of the output platform of the manipulator to a set of input

where n is the number of links, g is the number of joints, and fi

represents the DOF of joint i, yields a mobility of nine. Due to the arrangement of the links and joints, each leg's motion is restricted only by ball joints at both ends as shown in Fig. 2. Each leg is thus free to rotate around its axis and six idle DOF exist within the parallel module. Hence, it leaves the parallel module with three DOF.

3.3. Applications and novelties

The machine tool provides the following motions. (1) X, Y, Z: positioning the workpiece in the required location; (2) A, B: or- ientation of the polishing tool to follow the local normal direction of the surface and impose the precession angle; (3) H-axis: rota- tion of the polishing head to create the tool influence function; (4) C-axis: rotation of the fixture to impose rotational symmetry if required (a redundant DOF). This design provides the capability to polish circular and non-circular surfaces, flat, aspherical, off-axis aspherical and freeform surfaces and any tool paths can be implemented.