Kinematic Calibration of a Six DOF Flexure-based Parallel Manipulator

The absence of friction, hysteresis and backlash makes flexure-based mechanisms well-suited for high precision manipulators. However, the (inverse) kinematic relation between actuators and end-effector is far from trivial due to the non-linear behaviour of the deforming compliant joints. In this paper we consider the kinematic modelling and calibration of a flexure-based parallel manipulator for a six degrees of freedom (DOF) mirror mount. The mount is positioned by six arms, each of which has five joints and is driven by a linear actuator.

Three kinematic models are compared. A simple and computationally fast model that ignores pivot shift is too inaccurate. A flexible multibody model can account for the non-linear deformations of the joints, but is too computationally expensive for real-time applications. Finally, a kinematic model is derived using the Denavit–Hartenberg notation where the pivot shift is described with a polynomial approximation. This model offers nm accuracy with a small
number of terms from a Taylor series and can be evaluated sufficiently fast.

In this way a nominal kinematic model can be derived using the (kinematic) parameters from CAD data. However, the achievable accuracy in an experimental set-up remains inadequate. Hence a geometric calibration procedure has been
developed for the four most critical translations and rotations of the end-effector. The measurement set-up contains two position-sensing detectors to measure these motions. The model is linearized for small errors in the parameters to enable the use of linear regression techniques. With a least squares estimate the errors in the parameters are estimated. The quality of the estimation is checked by combining the singular value decomposition of the (linearised) regression matrix with cross-validation. It was found that the kinematic calibration clearly improves the accuracy of the (inverse) kinematic model.