a b s t r a c t As one of the final processing steps of precision machining, polishing process is a very key decision for surface quality. This paper presents a novel hybrid manipulator for computer controlled ultra-precision (CCUP) freeform polishing. The hybrid manipulator is composed of a three degree-of-freedom (DOF) parallel module, a two DOF serial module and a turntable providing a redundant DOF. The parallel module gives the workpiece three translations without rotations. The serial module holds the polishing tool and gives it no translations on the polishing contact area due to its particular mechanical design. A detailed kinematics model is established for analyzing the kinematics of the parallel module and the serial module, respectively. For the parallel module, the inverse kinematics, the forward kinematics, the Jacobian matrix, the workspace and the dexterity distribution are analyzed systematically. Workspaces are also investigated for varying structural parameters. For the serial module, the inverse kinematics, the forward kinematics, the workspace and the precession motion analysis are carried out. An example of saddle surface finishing with this manipulator is given and the movement of actuators with respect to this shape is analyzed theoretically. These analysis results illustrate that the proposed hybrid manip- ulator is a very suitable machine structure for CCUP freeform polishing.

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1. Introduction

Freeform surfaces are widely used in many industries [1–4] for high precision optics, semiconductor applications, precision molds and orthopedic joints. However, no matter how precise the milling or grinding process is, it will inevitably leave cusps and stripes on the workpiece during the manufacturing of freeform surfaces. These marks or specific patterns must be removed by polishing process. Polishing is usually one of the final machining processes in precision machining to remove surface and subsurface damage from a ground part and correct its form[5].

Due to the geometrical complexity of freeform surfaces, pol- ishing these surfaces is more challenging and difficult than pol- ishing flat and spherical surfaces. Nowadays, many freeform sur- faces are still polished manually, which means that the process not only relies heavily on the expertize and experience of the operator, but also requires much attention be given to processingand testing. Toachieve agivenlevel ofprecision withhighefficiency

n Corresponding authors.

E-mail addresses: benny.cheung@polyu.edu.hk (C.-F. Cheung), libing.sgs@hit.edu.cn(B. Li).

and reliability, process automation is clearly the way forward.

In recent years, various approaches to automated polishing have been developed, including computer-controlled polishing (CCP) [3–6], stressed-lap polishing (SLP) [7,8], plasma polishing [9,10], and magnetorheological fluid polishing (MFP) [11,12]. CCP is widely used and is regarded as an efficient polishing method. However, it has significant shortcomings for ultra-precision pol- ishing of freeform surfaces. Due to the small size of the polishing tool, high-frequency errors may be produced. If the size of the polishing tool is increased, the polishing tool and the surface may not fit well, which results in poor quality finishing. SLP technology can overcome this problem. The tool can be actively deformed by on-board actuators to conform to the target surface. However, this results in a very complex tooling, control system and a weak capability to correct errors of form. Plasma polishing is a non- contact polishing method that can overcome the shortcomings of traditional contact polishing methods, but it has low polishing efficiencies and high environmental requirements for processing. MFP is a flexible polishing technology that can achieve high pre- cision, but its cost is high and it is difficult to polish concave sur- face and high steepness surfaces using this technology.