Damped Least-Square Method Based on Chaos Anti-Control for Solving Forward Displacement of General 6-6-Type Parallel Mechanism

Abstract

Machine tools based on a Stewart platform are considered the machine tools of the 21st century. Difficult problems exist in the design philosophy, of which forward displacement analysis is the most fundamental. Its mathematical model is a kind of strongly nonlinear multivariable equation set with unique characteristics and a high level of difficulty. Different variable numbers and different solving speeds can be obtained through using different methods to establish the model of forward displacement analysis. The damped least-square method based on chaos anti-control for solving displacement analysis of the general 6-6-type parallel mechanism was built up through the rotation transformation matrix R , translation vector P and the constraint conditions of the rod length. The Euler equations describing the rotational dynamics of a rigid body with principle axes at the centre of mass were converted to a chaotic system by using chaos anti-control, and chaotic sequences were produced using the chaos system. Combining the characteristics of the chaotic sequence with the damped least-square method, all real solutions of forward displacement in nonlinear equations were found. A numerical example shows that the new method has some interesting features, such as fast convergence and the capability of acquiring all real solutions, and comparisons with other methods prove its effectiveness and validity.