Error-free measurement method for 6-DOF systems via matrix decomposition approach

Abstract

A six-degree-of-freedom measurement system possessing a pure algebraic and error-free calculation algorithm is proposed based on the rotation/translation matrix decomposition approach. Generally, such complicated mathematical models are simplified through linearization. However, this inevitably leads to calculation errors. Accordingly, in the method proposed in this study, the object motion behavior in three-dimensional space is described instead using a pose-change homogeneous coordinate transformation matrix consisting of two separate matrices, namely a rotation matrix which describes the nonlinear component of the object motion and a translation matrix which describes the linear component. In the solution process, the image-orientation-change method is first used to determine the rotation matrix. The translation matrix is then obtained using the singular-value-decomposition least-squares method. Finally, the two matrices are multiplied together to obtain the desired pose-change matrix. The validity of the proposed approach is demonstrated by means of an illustrative numerical example and a simple experimental trial. It is shown that, since the calculation algorithm is a purely analytical and algebraic method and involves no approximations or omissions (i.e., the relevant equations are not simplified through linearization and thus retain their original nonlinear integrity), the derived solutions are error-free if the errors introduced by the measurement technology are ignored.