A Singularity Analysis Method for Stewart Parallel Mechanism with Planar Platforms

Singularity analysis is an important problem in the field of parallel mechanism, how to obtain a concise and analytical expression of the singularity locus has been the focus of the study for a long time. This paper presents a new method for the singularity analysis of the Stewart platform parallel mechanism. Firstly, the rotation matrix is described by quaternion; utilizing length constraint equations of the extensible limbs and properties of the quaternion, seven equivalent equations are obtained, and the variables of position and orientation are decoupled preliminarily. Secondly, by taking the derivative of seven the equivalent equations with respect to time, a new kind of Jacobian matrix is obtained, which reflects the mapping relationship between the change rate of the lengths of extensible limbs and the change rate of variables of position and orientation of the moving platform. Finally, the analytical expression of the singularity locus is derived from calculating the determinant of the new Jacobian matrix; when the quaternion are transformed into Rodriguez parameters, there are only 258 items in the fully expanded analytical expression of the singularity locus. Not only can this method be used to study the singularity problem of Stewart platform parallel mechanism, but it can also be used to study the singularity free workspace of the mechanism.

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