1. Introduction
Deformation of the machining tool itself, due to cutting force, is one of the major causes of machining error in precision machining.  Such conventional methods of adding rigidity to the machine frame, chuck, and tool for higher precision make the machines larger, heavier and more complex.  The authors propose a different solution to the problem that measures the machining force, and actively compensates the machine deformation.

Fig.1(a) shows general configuration of a machining system.  Machining force is expressed by a curve.   The force flow curve forms a 'C' loop starting at the machining tool and ending at the work.  The machining system deforms along the C loop, and dislocation of the start and end points causes machining error.  Fig.1(b) sketches how we compensate the machining error.  The dislocation is corrected by detecting the machining force and estimating the machine deformation.  The authors previously proposed a force sensor with actively controlled compliance[1].  This sensor provides negative compliance.  The sensor deforms in the direction opposite to the force applied.

This paper reports our analysis of the machining process to compensate machining error using negative compliance.  It further shows a 3-axis negative compliance planar grinding system using a ring-shaped active force sensor we developed to verify our method.  It also evaluates our method of machining error compensation from tests using a planar grinding system, we developed, with a cup grinder and water pressure turbine motor that actually grinds silicon wafers.

2. Machining process analysis
Fig. 2(a) sketches the conventional machining process. Even if the system controls the machining tool to follow a predetermined trajectory, the resulting surface differs from the desired shape. The machining force deforms the tool, the work piece and the structural frame to produce machining error (spring-back).

Fig. 2(b) shows the machining process with negative compliance. To compensate the spring-back, the negative compliance device adds cutting depth. When the initial depth of cut is Z, the following equations (1), (2), and (3) describe machining force F, spring-back S and machining error E, respectively. The suffix 0 indicates initial state. Cm is machining compliance defined as the ratio (cutting depth)/(machining force). Cs is the machining system compliance.