## Abstract

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.

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.

Original language | English |
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Title of host publication | ECCOMAS Thematic Conference on Multibody Dynamics |

Subtitle of host publication | Prague, June 19-22, 2017: conference proceedings |

Editors | Michael Valasek, Zbynek Sika, Tomas Vampola |

Pages | 199-211 |

Number of pages | 13 |

ISBN (Electronic) | 978-80-01-6174-9 |

Publication status | Published - 8 Dec 2017 |

Event | Multibody Dynamics 2017: 8th ECCOMAS Thematic Conference - Czech Technical University, Prague, Czech Republic Duration: 19 Jun 2017 → 22 Jun 2017 Conference number: 8 http://multibody2017.cz/ |

### Conference

Conference | Multibody Dynamics 2017 |
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Country/Territory | Czech Republic |

City | Prague |

Period | 19/06/17 → 22/06/17 |

Internet address |

## Keywords

- Kinematic model
- Geometric calibration
- Flexure-based parallel mechanisms
- Flexible multibody analysis
- Iterative linear parameters estimation