Aggregated Hierarchical Sliding Mode Control for Vibration Suppression of an Excavator on an Elastic Foundation

Abstract

Crawler excavators are important, utilitarian machines in construction industries. Special features of the chain and chassis allow them to operate and move on unstable ground. The shock motions of the links, which coincide with the changes of the forces impacting the foundation, create vibrations within the entire system. The highest level of vibrations appears when the boom moves, as it holds the robotic excavator's entire actuator with large mass and moment of inertia. These undesired fluctuations cause system instability, driver discomfort, reduced operating efficiency, and increased energy consumption. In this study, a hierarchical sliding mode control focusing on boom movement is designed based on the system dynamic model. This controller allows the boom to move precisely with a slight fluctuation angle of the main body, even while the other links are moving using a parameter estimator. Maintaining stable boom movements significantly simplifies arm and bucket control. The effectiveness of the entire work is investigated by numerical simulation and implementation results on a small-scale hydraulic excavator.

Appendix

The Jacobian matrix for translation of the ith link is given as follows:

$${\mathbf{J}}_{T1} = \left[ {\begin{array}{*{20}c} {J_{T1(11)} } & 0 & 0 & 0 \\ {J_{T1(21)} } & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} } \right],\,\,{\mathbf{J}}_{T4} = \left[ {\begin{array}{*{20}c} {J_{T4(11)} } & {J_{T4(12)} } & {J_{T4(13)} } & {J_{T4(14)} } \\ {J_{T4(21)} } & {J_{T4(22)} } & {J_{T4(23)} } & {J_{T4(24)} } \\ 0 & 0 & 0 & 0 \\ \end{array} } \right],$$

$${\mathbf{J}}_{T2} = \left[ {\begin{array}{*{20}c} {J_{T2(11)} } & 0 & 0 & {J_{T2(14)} } \\ {J_{T2(21)} } & 0 & 0 & {J_{T2(24)} } \\ 0 & 0 & 0 & 0 \\ \end{array} } \right],{\mathbf{J}}_{T3} = \left[ {\begin{array}{*{20}c} {J_{T3(11)} } & {J_{T3(12)} } & 0 & {J_{T3(14)} } \\ {J_{T3(21)} } & {J_{T3(22)} } & 0 & {J_{T3(24)} } \\ 0 & 0 & 0 & 0 \\ \end{array} } \right],\,$$

where

$$\begin{aligned} J_{T1(11)} & = - a_{c1} \sin \left( {\alpha_{0} + \delta_{\alpha } + \delta_{\alpha 0} + \varphi_{1} } \right),J_{T1(21)} = a_{c1} \cos \left( {\alpha_{0} + \delta_{\alpha } + \delta_{\alpha 0} + \varphi_{1} } \right), \\ J_{T2(11)} & = - a_{c2} \sin \left( {\alpha_{0} + \delta_{\alpha } + \delta_{\alpha 0} + \beta - \varphi_{2} } \right), \\ J_{T2(21)} & = a_{c2} \cos \left( {\alpha_{0} + \delta_{\alpha } + \delta_{\alpha 0} + \beta - \varphi_{2} } \right), \\ J_{T2(14)} & = - a_{c2} \sin \left( {\alpha_{0} + \delta_{\alpha } + \delta_{\alpha 0} + \beta - \varphi_{2} } \right) - a_{1} \sin \left( {\alpha_{0} + \delta_{\alpha } + \delta_{\alpha 0} } \right), \\ J_{T2(24)} & = a_{c2} \cos \left( {\alpha_{0} + \delta_{\alpha } + \delta_{\alpha 0} + \beta - \varphi_{2} } \right) - a_{1} \cos \left( {\alpha_{0} + \delta_{\alpha } + \delta_{\alpha 0} } \right), \\ J_{T3(11)} & = - a_{c3} \sin \left( {\alpha_{0} + \delta_{\alpha } + \delta_{\alpha 0} + \beta + \gamma - \varphi_{3} } \right) - a_{2} \sin \left( {\alpha_{0} + \delta_{\alpha } + \delta_{\alpha 0} + \beta } \right), \\ J_{T3(12)} & = - a_{c3} \sin \left( {\alpha_{0} + \delta_{\alpha } + \delta_{\alpha 0} + \beta + \gamma - \varphi_{3} } \right), \\ J_{T3(14)} & = - a_{c3} \sin \left( {\alpha_{0} + \delta_{\alpha } + \delta_{\alpha 0} + \beta + \gamma - \varphi_{3} } \right) \\ & \quad - a_{2} \sin \left( {\alpha_{0} + \delta_{\alpha } + \delta_{\alpha 0} + \beta } \right) - a_{1} \sin \left( {\alpha_{0} + \delta_{\alpha } + \delta_{\alpha 0} } \right), \\ J_{T3(21)} & = a_{c2} \cos \left( {\alpha_{0} + \delta_{\alpha } + \delta_{\alpha 0} + \beta + \gamma - \varphi_{3} } \right) + a_{2} \cos \left( {\alpha_{0} + \delta_{\alpha } + \delta_{\alpha 0} + \beta } \right), \\ J_{T3(22)} & = a_{c2} \cos \left( {\alpha_{0} + \delta_{\alpha } + \delta_{\alpha 0} + \beta + \gamma - \varphi_{3} } \right), \\ J_{T3(24)} & = a_{c2} \cos \left( {\alpha_{0} + \delta_{\alpha } + \delta_{\alpha 0} + \beta + \gamma - \varphi_{3} } \right) \\ & \quad + a_{2} \cos \left( {\alpha_{0} + \delta_{\alpha } + \delta_{\alpha 0} + \beta } \right) - a_{1} \cos \left( {\alpha_{0} + \delta_{\alpha } + \delta_{\alpha 0} } \right), \\ J_{T4(11)} & = - a_{c4} \sin \left( {\alpha_{0} + \delta_{\alpha } + \delta_{\alpha 0} + \beta + \gamma + \theta - \varphi_{4} } \right) \\ & \quad - a_{2} \sin \left( {\alpha_{0} + \delta_{\alpha } + \delta_{\alpha 0} + \beta } \right) - a_{3} \sin \left( {\alpha_{0} + \delta_{\alpha } + \delta_{\alpha 0} + \beta + \gamma } \right), \\ J_{T4(12)} & = - a_{c4} \sin \left( {\alpha_{0} + \delta_{\alpha } + \delta_{\alpha 0} + \beta + \gamma + \theta - \varphi_{4} } \right) \\ & \quad - a_{3} \sin \left( {\alpha_{0} + \delta_{\alpha } + \delta_{\alpha 0} + \beta + \gamma } \right), \\ J_{T4(22)} & = + a_{c4} \cos \left( {\alpha_{0} + \delta_{\alpha } + \delta_{\alpha 0} + \beta + \gamma + \theta - \varphi_{4} } \right) \\ & \quad + a_{3} \cos \left( {\alpha_{0} + \delta_{\alpha } + \delta_{\alpha 0} + \beta + \gamma } \right), \\ J_{T4(23)} & = + a_{c4} \cos \left( {\alpha_{0} + \delta_{\alpha } + \delta_{\alpha 0} + \beta + \gamma + \theta - \varphi_{4} } \right), \\ J_{T4(24)} & = + a_{3} \cos \left( {\alpha_{0} + \delta_{\alpha } + \delta_{\alpha 0} + \beta + \gamma } \right) + a_{2} \cos \left( {\alpha_{0} + \delta_{\alpha } + \delta_{\alpha 0} + \beta } \right) \\ & \quad + a_{1} \cos \left( {\alpha_{0} + \delta_{\alpha } + \delta_{\alpha 0} } \right) + a_{c4} \cos \left( {\alpha_{0} + \delta_{\alpha } + \delta_{\alpha 0} + \beta + \gamma + \theta - \varphi_{4} } \right). \\ \end{aligned}$$