va. 45No. I SCIENCE IN CHINA (Series E) February2002 Nonlinearity for a parallel kinematic machine tool and its application to interpolation accuracy analysis WANG Jinsong (i~)1, WANG Zhonghua (E--,~) 1, HUANG Tian (3~ ~)2 & D. J. Whitehouse 3 1. Department of Precision Instruments & Mechanology, Tsinghua University, Beijing 100084, China; 2. Department of Mechanical Engineering, Tianjin University, Tianjin 300072, China; 3. Department of Engineering, University of Warwick, Coventry CV4 7AL, W Midlands, UK Correspondence should be addressed to Huang Tian Received June 1, 2001 Abstract This paper is concerned with the kinematic nonlinearity measure of parallel kinematic ma- chine tool (PKM), which depends upon differential geometry curvature. The nonlinearity can be de- scribed by the curve of the solution locus and the equal interval input of joints mapping into inequable interval output of the end-effectors. Such curing and inequation can be measured by BW curvature. So the curvature can measure the nonlinearity of PKM indirectly. Then the distribution of BW curvature in the local area and the whole workspace are also discussed. An example of application to the interpola- tion accuracy analysis of PKM is given to illustrate the effectiveness of this approach. Keywords : parallel kinematic machine tool, nonlinearity measure, curvature, interpolation, accuracy analysis. The kinematic relationship between the vectors of joints space and the vectors of workspace is nonlinear for a parallel kinematic machine tool. The characteristic features of PKM on accura- cy, stiffness, velocity and control schemes are quite different from serial ones. The most direct and intensive research on the relationship between the input and output vec- tors is the forward kinematics and inverse kinematics of PKM. The inverse kinematics problem is relatively simple with 1 - 1 analytical form. But the direct kinematics problem is very difficult. It involves the solution of a system of nonlinear coupled algebraic equations in the variables describ- ing the platform orientation and has many solutions, referred to as assembly modes. Among the analytical approaches, according to different passive joints and configurations, Carlo [l] and Fitch- er [2] et al. proved that the PKM has 30, 24, 16 and 12 solutions. Based on the inverse model, Raghavan [3] proved strictly that the general Stewart platform has 40 direct solutions in complex domain. Then it was proved by Wampler [43 using double quaternion method and SOMA coordinate that the conclusion is still true in the real domain. Liang [5'6] followed a different approach and analytically solved the problem for the 6 - 6 Stewart platform with planar base and platform by re- ducting the kinematic equations into a univariate polynomial and established the upper bound of 40 solutions. The numerical approaches that directly resort to nonlinear-equation-solving algo- rithms have computational advantages in most practical situations where only one real solution is required and a good starting point is available in the form of a neighboring configuration, but they are not suitable for a theoretical investigation aimed at determining all the possible solutions. To find all real solutions, some approaches [7's] are employed to simplify the problem by geometrical