The Study of Coupling Dynamics Modeling and Characteristic Analysis for Flexible Robots with Nonlinear and Frictional Joints

Abstract

The current research of robots is mainly developing in lightweight, fast speed and large load. The characteristics of flexible robots are mainly reflected in flexible joints and links. However, flexible joints are often assumed to be simply linear. Since the joint of the planetary reducer also needs to consider the flexible deformation and nonlinearity of the structure, this assumption is not applicable to the assumption of this joint. Therefore, considering coupling properties of the more accurate structure and friction, the joints and the flexible links, a more accurate dynamic model of the flexible robot is established. Furthermore, the accuracy of motion and positioning for the flexible robot under the influence of many factors is analyzed. The research is of great significance for flexible robot control. Firstly, the structure of the two-stage planetary gear reducer is analyzed. Using Newton's Euler and force balance principles, the dynamic model of its joint reducer was constructed. Secondly, taking the Euler–Bernoulli beam as the object, using the Lagrange method and the structural damping formula, the coupling dynamic model of the flexible robot is established. Based on the virtual work and variational principles, the input matrix is further modified by friction. Finally, Using MATLAB and the experiment platform, the validity of the coupling dynamics is verified. And then, taking the flexible deformation, the clearance, and friction of the reducer as the influencing factors, the influence characteristics of the flexible robot are analyzed.

Appendix

$$M = \left[ {\begin{array}{*{20}c} {m_{11} } & {m_{12} } & {m_{13} } & {m_{14} } & {m_{15} } & {m_{16} } \\ {m_{21} } & {m_{22} } & 0 & 0 & 0 & 0 \\ {m_{31} } & 0 & {m_{33} } & 0 & 0 & 0 \\ {m_{41} } & 0 & 0 & {m_{44} } & {m_{45} } & {m_{46} } \\ {m_{51} } & 0 & 0 & {m_{54} } & {m_{55} } & 0 \\ {m_{61} } & 0 & 0 & {m_{64} } & 0 & {m_{66} } \\ \end{array} } \right],$$

$$\begin{gathered} m_{11} = \frac{1}{3}\overline{\rho }L_{1}^{3} + \frac{1}{2}\overline{\rho }_{1} L_{1}^{{}} q_{11}^{2} + \frac{1}{2}\overline{\rho }_{1} L_{1}^{{}} q_{12}^{2} \hfill \\ \, + m_{Tip1} L_{1}^{2} + \overline{\rho }_{2} L_{2} L_{1}^{2} + m_{Tip2} L_{1}^{2} \hfill \\ \end{gathered}$$

$$m_{12} = m_{21} = \overline{\rho }_{1} L_{1}^{2} /\pi.$$

$$m_{13} = m_{31} = \overline{\rho }_{1} L_{1}^{2} /2\pi,$$

$$m_{15} = m_{51} = \frac{2}{\pi }\overline{\rho }_{2} L_{2} L_{1} \cos \left( {\theta_{{1}} { - }\theta_{{2}} } \right),$$

$$m_{22} = m_{33} = \frac{1}{2}\overline{\rho }_{1} L_{1},$$

$$m_{45} = m_{54} = \overline{\rho }L_{2}^{2} /\pi$$

$$\begin{aligned} m_{14} &= m_{41} = \frac{2}{\pi }\overline{\rho }_{2} L_{2} L_{1} q_{21} \sin \left( {\theta_{1} - \theta_{2} } \right) \\ &\quad + \frac{1}{2}\overline{\rho }_{2} L_{1} L_{2}^{2} \cos \left( {\theta_{{1}} { - }\theta_{{2}} } \right) \\ & \quad + m_{Tip2} L_{1} L_{2} \cos \left( {\theta_{{1}} { - }\theta_{{2}} } \right) \end{aligned}$$

$$m_{44} = \frac{1}{2}\overline{\rho }_{2} L_{2} q_{21}^{2} + \frac{1}{2}\overline{\rho }_{2} L_{2} q_{22}^{2} + \frac{1}{3}\overline{\rho }_{2} L_{2}^{2} + m_{Tip2} L_{2}^{2}$$

$$m_{46} = m_{64} = \overline{\rho }L_{2}^{2} /2\pi,$$

$$m_{55} = m_{66} = \frac{{\overline{\rho }_{2} L_{2} }}{2}.$$

$$K = diag(\left[ {\begin{array}{*{20}c} 0 & {k_{22} } & {k_{33} } & 0 & {k_{55} } & {k_{66} } \\ \end{array} } \right]),$$

$$k_{22} = \frac{1}{2}E_{1} I_{1} \frac{{\pi^{4} }}{{L_{1}^{3} }},$$

$$k_{33} = 8E_{1} I_{1} \frac{{\pi^{4} }}{{L_{1}^{3} },}$$

$$k_{55} = \frac{1}{2}E_{2} I_{2} \frac{{\pi^{4} }}{{L_{2}^{3} }},$$

$$k_{66} = 8E_{2} I_{2} \frac{{\pi^{4} }}{{L_{2}^{3} }}.$$