Performance Evaluation and Calibration of Gantry-Tau Parallel Mechanism

Abstract

The Gantry-Tau is a family of parallel manipulators with three linear actuators. This mechanism is of interest for various applications because of the large workspace and its performance in terms of high acceleration, precision, and stiffness characteristics. This paper presents workspace analysis and calibration for a Gantry-Tau mechanism using its forward kinematics. The mathematical model of the systematic errors in the kinematics model of the manipulator is obtained. Analysis of the error is then performed to identify the parameters that have a dominant effect on the kinematics error and the regions of the workspace with a high error due to the calibration. Minimization of the mean square and mean absolute errors is employed for calibration through kinematics parameters. For demonstration purposes, a SimMechanics kinematic model of the mechanism is used and its calibration is performed over many sampled positions within the workspace borders of the robot. The result demonstrates that the kinematic error is significantly reduced after the calibration.

Appendix

The geometric parameters of the Gantry-Tau manipulator:

The profile of the guideways is square with 110 mm length.

The length of the guideways:

$$l_{1k} ,l_{2k} ,l_{3k} = 3020\;{\text{mm}}$$

The profile of the links is circle with 25 mm radius.

The length of the links:

$$\left| {{\mathbf{l}}_{ik} } \right| = \, 2000\;{\text{mm}}$$

The start position of the guideways, ${\mathbf{a}}_{i}^{s}$ (mm):

$${\mathbf{a}}_{1}^{s} = \left[ {\begin{array}{*{20}c} 0 & { - 550} & {550} \\ \end{array} } \right]^{\text{T}} ;\quad {\mathbf{a}}_{2}^{s} = \left[ {\begin{array}{*{20}c} 0 & {550} & {550} \\ \end{array} } \right]^{\text{T}} ;\quad {\mathbf{a}}_{3}^{s} = \left[ {\begin{array}{*{20}c} 0 & { - 550} & { - 550} \\ \end{array} } \right]^{\text{T}}$$

The joint position of the carts, ${\mathbf{c}}_{ik}$ (mm):

$${\mathbf{c}}_{1\,\,1} = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 \\ \end{array} } \right]^{\text{T}} ;\quad {\mathbf{c}}_{2\,\,1} = \left[ {\begin{array}{*{20}c} 0 & {0.143} & 0 \\ \end{array} } \right]^{\text{T}} ;\quad {\mathbf{c}}_{2\,\,2} = \left[ {\begin{array}{*{20}c} { - 0.124} & 0 & 0 \\ \end{array} } \right]^{\text{T}} ;$$

$${\mathbf{c}}_{3\,\,1} = \left[ {\begin{array}{*{20}c} 0 & {0.143} & 0 \\ \end{array} } \right]^{\text{T}} ;\quad {\mathbf{c}}_{3\,\,2} = \left[ {\begin{array}{*{20}c} { - 0.124} & { - 0.072} & 0 \\ \end{array} } \right]^{\text{T}} ;\quad {\mathbf{c}}_{3\,\,3} = \left[ {\begin{array}{*{20}c} {0.124} & { - 0.072} & 0 \\ \end{array} } \right]^{\text{T}}$$

The joint position of the platform, ${\mathbf{d}}_{ik}$ (mm):

$${\mathbf{d}}_{1\,\,1} = \left[ {\begin{array}{*{20}c} 0 & {127.01} & { - 15.19} \\ \end{array} } \right]^{\text{T}} ;\quad {\mathbf{d}}_{2\,\,1} = \left[ {\begin{array}{*{20}c} 0 & { - 124.99} & {59.81} \\ \end{array} } \right]^{\text{T}} ;\quad {\mathbf{d}}_{2\,\,2} = \left[ {\begin{array}{*{20}c} 0 & {127.01} & {59.81} \\ \end{array} } \right]^{\text{T}} ;$$

$${\mathbf{d}}_{3\,\,1} = \left[ {\begin{array}{*{20}c} {143.45} & {1.01} & {99.81} \\ \end{array} } \right]^{\text{T}} ;\quad {\mathbf{d}}_{3\,\,2} = \left[ {\begin{array}{*{20}c} 0 & { - 124.99} & { - 90.19} \\ \end{array} } \right]^{\text{T}} ;\quad {\mathbf{d}}_{3\,\,3} = \left[ {\begin{array}{*{20}c} 0 & {127.01} & { - 90.19} \\ \end{array} } \right]^{\text{T}} .$$