Dynamic Performance Evaluation of Serial and Parallel RPR Manipulators with Flexible Intermediate Links

Abstract

This paper presents a comprehensive study of the workspace, dynamic characteristics and accuracy of three planar flexible manipulators with 3-RPR, 2-RPR and 1-RPR structures moving at high speed. A geometrical procedure is employed to obtain the workspaces of the manipulators. The flexible intermediate links are modeled as the Euler–Bernoulli beams with fixed-free boundary conditions based on the assumed mode method. Using the Lagrange multipliers, a generalized set of differential algebraic equations of motion is developed for the planar RPR manipulators. Three moving constraints, which are obtained from an inverse kinematics analysis and applied to the actuated base joints, impose the end-effector to follow a high-speed circular motion as the desired trajectory. From this analysis, the dynamic performance of 1-RPR flexible serial manipulator and 2-RPR and 3-RPR flexible parallel manipulators in tracking a desired trajectory is evaluated. Based on the results, it is concluded that, in addition to the specific structure of the manipulator, the accuracy generally depends on the operation conditions. The results contest the general assertion which claims that parallel manipulators have more accuracy and stiffness than serial counterparts.

Appendix

Mass matrix:

$${\mathbf{M}}_{\rho \rho } = m\left[ {\begin{array}{*{20}c} 1 & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & 1 \\ \end{array} } \right] \in R^{n \times n} ,\,\,\,{\mathbf{M}}_{\theta \theta } = \left( {I_{c} + \frac{{ml^{2} }}{3}} \right)\left[ {\begin{array}{*{20}c} 1 & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & 1 \\ \end{array} } \right] + m\left[ {\begin{array}{*{20}c} {l\rho_{1} + \rho_{1}^{2} } & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & {l\rho_{n} + \rho_{n}^{2} } \\ \end{array} } \right] \in R^{n \times n}$$

$${\mathbf{M}}_{{X_{\text{p}} X_{\text{p}} }} = \left[ {\begin{array}{*{20}c} {m_{\text{p}} } & 0 & 0 \\ 0 & {m_{\text{p}} } & 0 \\ 0 & 0 & {I_{\text{p}} } \\ \end{array} } \right], \quad {\mathbf{M}}_{\eta \eta } = m\left[ {\begin{array}{*{20}c} {{\hat{\mathbf{M}}}} & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & {{\hat{\mathbf{M}}}} \\ \end{array} } \right] \in R^{nr \times nr} , \quad {\hat{\mathbf{M}}} = \left[ {\begin{array}{*{20}c} {\int_{0}^{1} {\psi_{1}^{2} {\text{d}}\xi } } & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & {\int_{0}^{1} {\psi_{r}^{2} {\text{d}}\xi } } \\ \end{array} } \right] \in R^{r \times r}$$

$${\mathbf{M}}_{\eta \theta } = ml\left[ {\begin{array}{*{20}c} {{\hat{\mathbf{M}}}_{1} } & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & \ddots & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{0}} & {{\hat{\mathbf{M}}}_{n} } \\ \end{array} } \right] \in R^{nr \times n} , \quad {\hat{\mathbf{M}}}_{i} = \left[ {\begin{array}{*{20}c} {M_{1i} } \\ \vdots \\ {M_{ri} } \\ \end{array} } \right],\,\,M_{ji} = \int_{0}^{1} {\psi_{j} \xi {\text{d}}\xi } + \frac{{\rho_{i} \int_{0}^{1} {\psi_{j} {\text{d}}\xi } }}{l}$$

Stiffness matrix:

$${\mathbf{K}}_{\rho \rho } = - m\left[ {\begin{array}{*{20}c} {\dot{\theta }_{1}^{2} } & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & {\dot{\theta }_{n}^{2} } \\ \end{array} } \right] \in R^{n \times n} ,\quad {\mathbf{K}}_{\theta \rho } = 2m\left[ {\begin{array}{*{20}c} {\dot{\rho }_{1} \dot{\theta }_{1} } & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & {\dot{\rho }_{n} \dot{\theta }_{n} } \\ \end{array} } \right] \in R^{n \times n}$$

$\,\,{\mathbf{K}}_{\eta \eta } = \frac{\text{EI}}{{l^{3} }}\left[ {\begin{array}{*{20}c} {{\hat{\mathbf{K}}}} & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & \ddots & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{0}} & {{\hat{\mathbf{K}}}} \\ \end{array} } \right] \in R^{nr \times nr} ,{\hat{\mathbf{K}}} = \left[ {\begin{array}{*{20}c} {\int_{0}^{1} {\psi_{1}^{{{\prime \prime }2}} {\text{d}}\xi } } & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & {\int_{0}^{1} {\psi_{r}^{{{\prime \prime }2}} {\text{d}}\xi } } \\ \end{array} } \right] \in R^{r \times r} ,$

Coriolis and centrifugal forces:

$$F_{c\rho } = \left[ {\begin{array}{*{20}c} {(0.5\,{\text{ml}})\dot{\theta }_{1}^{2} + \sum\limits_{j = 1}^{r} {m\dot{\eta }_{1j} \dot{\theta }_{1} \int_{0}^{1} {\psi_{j} {\text{d}}\xi } } } \\ \vdots \\ {(0.5\,{\text{ml}})\dot{\theta }_{n}^{2} + \sum\limits_{j = 1}^{r} {m\dot{\eta }_{nj} \dot{\theta }_{n} \int_{0}^{1} {\psi_{j} {\text{d}}\xi } } } \\ \end{array} } \right],\,\quad F_{c\theta } = - m\left[ {\begin{array}{*{20}c} {l\dot{\rho }_{1} \dot{\theta }_{1} + \sum\limits_{j = 1}^{r} {\dot{\rho }_{1} \dot{\eta }_{1j} \int_{0}^{1} {\psi_{j} {\text{d}}\xi } } } \\ \vdots \\ {l\dot{\rho }_{n} \dot{\theta }_{n} + \sum\limits_{j = 1}^{r} {\dot{\rho }_{n} \dot{\eta }_{nj} \int_{0}^{1} {\psi_{j} {\text{d}}\xi } } } \\ \end{array} } \right],$$

$$F_{c\eta } = \left[ {\dot{\rho }_{1} \dot{\theta }_{1} \int_{0}^{1} {\psi_{1} {\text{d}}\xi } \cdots \dot{\rho }_{1} \dot{\theta }_{1} \int_{0}^{1} {\psi_{r} {\text{d}}\xi \cdots \dot{\rho }_{n} \dot{\theta }_{n} \int_{0}^{1} {\psi_{1} {\text{d}}\xi \cdots \dot{\rho }_{n} \dot{\theta }_{n} \int_{0}^{1} {\psi_{r} {\text{d}}\xi } } } } \right]^{\text{T}} \in R^{1 \times nr}$$

Jacobian matrix:

$${\mathbf{J}}_{\varGamma 1}^{T} = \left[ {\begin{array}{*{20}c} {\cos \theta_{1} } & {\sin \theta_{1} } & \cdots & 0 & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \cdots & {\cos \theta_{n} } & {\sin \theta_{n} } \\ \end{array} } \right] \in R^{n \times 2n} , \quad {\mathbf{J}}_{\varGamma 2}^{T} = \left[ {\begin{array}{*{20}c} { - s2_{1} } & {c2_{1} } & \cdots & 0 & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \cdots & { - s2_{n} } & {c2_{n} } \\ \end{array} } \right] \in R^{n \times 2n}$$

where $s2_{i} = \left( {\rho_{i} + l} \right) \sin \theta_{i} + \cos \theta_{i} \sum\nolimits_{j = 1}^{r} {\eta_{ij} \psi_{j} \left( 1 \right)} \,\,{\text{and }}c2_{i} = \left( {\rho_{i} + l} \right) \cos \theta_{i} - \sin \theta_{i} \sum\nolimits_{j = 1}^{r} {\eta_{ij} \psi_{j} \left( 1 \right)} \,\,$

$${\mathbf{J}}_{\varGamma 3}^{T} = \left[ {\begin{array}{*{20}c} { - 1} & 0 & \cdots & { - 1} & 0 \\ 0 & { - 1} & \cdots & 0 & { - 1} \\ {s3_{1} } & { - c3_{1} } & \cdots & {s3_{n} } & { - c3_{n} } \\ \end{array} } \right] \in R^{3 \times 2n} ,\,\,$$

where $s3_{i} = r\sin (\phi_{i} + \phi_{p} ),c3_{i} = r\cos (\phi_{i} + \phi_{p} )$

$${\mathbf{J}}_{\varGamma 4}^{T} = \left[ {\begin{array}{*{20}c} {{\hat{\mathbf{J}}}_{1} } & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & \ddots & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{0}} & {{\hat{\mathbf{J}}}_{n} } \\ \end{array} } \right] \in R^{nr \times 2n} ,{\hat{\mathbf{J}}}_{i} = \left[ {\begin{array}{*{20}c} { - s4_{i1} } & {c4_{i1} } \\ \vdots & \vdots \\ { - s4_{ir} } & {c4_{ir} } \\ \end{array} } \right],\,\,\,\,s4_{ij} = \sin \theta_{i} \cdot \psi_{j} (1),\,\,\,c4_{ij} = \cos \theta_{i} \cdot \psi_{j} (1)$$