Double-Loop Control with Hierarchical Sliding Mode and Proportional Integral Loop for 2D Ridable Ballbot

Abstract

In this study, we propose a nonlinear double-loop control system for a ridable ballbot. An improved nonlinear double-loop control technique based on double-loop control scheme and sliding mode technology is used, and the inner-loop consists of proportional–integral feedback plus feedforward control. A sliding mode control enables the ridable ballbot to balance and transfer on the floor. Feedforward compensation has been added for the stability of the ballbot. As a result, the control system can stabilize all state variables of the ballbot system despite the uncertainties of the system parameters such as model, friction, and external disturbances, and relatively large body inertia. Experiments demonstrate the effectiveness of the proposed control system and the performance validation of the nonlinear double-loop control.

Appendices

Appendix A

The components of standard nonlinear state-space dynamics of the 2D ridable ballbot in the y–z plane are the following:

$$\begin{aligned} A_{r3} \left( {{\mathbf{x}}_{r} } \right) & = - \,\frac{{r_{k}^{2} }}{{mau_{r} }}\left( {\begin{array}{*{20}l} {2I_{x} b_{y} r_{w}^{2} \dot{y}_{k} - 3I_{w} b_{rx} r_{k} { \cos }^{2} \alpha \dot{\theta }_{x} + 2b_{y} l^{2} m_{a} r_{w}^{2} \dot{y}_{k} } \hfill \\ {\quad + \,3I_{w} b_{y} r_{k}^{2} { \cos }^{2} \alpha \dot{y}_{k} - gl^{2} m_{a}^{2} r_{w}^{2} { \sin }2\theta_{x} + 2l^{3} m_{a}^{2} r_{w}^{2} \dot{\theta }_{x}^{2} { \sin }\theta_{x} } \hfill \\ {\quad + \,2I_{x} lm_{a} r_{w}^{2} \dot{\theta }_{x}^{2} { \sin }\theta_{x} + 2b_{rx} lm_{a} r_{w}^{2} \dot{\theta }_{x} { \cos }\theta_{x} } \hfill \\ {\quad + \,3I_{w} lm_{a} r_{k}^{2} { \cos }^{2} \alpha \dot{\theta }_{x}^{2} { \sin }\theta_{x} + 3I_{w} glm_{a} r_{k} { \cos }^{2} \alpha { \sin }\theta_{x} } \hfill \\ \end{array} } \right), \\ B_{r3} \left( {{\mathbf{x}}_{r} } \right) & = r_{k}^{2} \left( {2m_{a} r_{w} l^{2} + 2m_{a} r_{k} r_{w} l{ \cos }\theta_{x} + 2I_{x} r_{w} } \right)/mau_{r} , \\ A_{r4} \left( {{\mathbf{x}}_{r} } \right) & = - \,\frac{1}{{mau_{r} }}\left( {\begin{array}{*{20}l} {2I_{kx} b_{rx} r_{w}^{2} \dot{\theta }_{x} + 3I_{w} b_{rx} r_{k}^{2} { \cos }^{2} \alpha \dot{\theta }_{x} - 2glm_{a}^{2} r_{k}^{2} r_{w}^{2} { \sin }\theta_{x} } \hfill \\ {\quad - \,3I_{w} b_{y} r_{k}^{3} \dot{y}_{k} { \cos }^{2} \alpha + 2b_{rx} \left( {m_{a} + m_{k} } \right)r_{k}^{2} r_{w}^{2} \dot{\theta }} \hfill \\ {\quad + \,l^{2} m_{a}^{2} r_{k}^{2} r_{w}^{2} \dot{\theta }_{x}^{2} { \sin }2\theta_{x} - 3I_{w} lm_{a} r_{k}^{3} { \cos }^{2} \alpha \dot{\theta }_{x}^{2} { \sin }\theta_{x} } \hfill \\ {\quad + \,2b_{y} lm_{a} r_{k}^{2} r_{w}^{2} \dot{y}_{k} { \cos }\theta_{x} - 2I_{kx} glm_{a} r_{w}^{2} { \sin }\theta_{x} } \hfill \\ {\quad - \,2glm_{a} m_{k} r_{k}^{2} r_{w}^{2} { \sin }\theta_{x} - 3I_{w} glm_{a} r_{k}^{2} { \cos }^{2} \alpha { \sin }\theta_{x} } \hfill \\ \end{array} } \right), \\ B_{r4} \left( {{\mathbf{x}}_{r} } \right) & = \left( {2m_{a} r_{k}^{3} r_{w} + 2m_{k} r_{k}^{3} r_{w} + 2I_{kx} r_{k} r_{w} + 2lm_{a} r_{k}^{2} r_{w} { \cos }\theta_{x} } \right)/mau_{r} , \\ {\text{and}}\;mau_{r} & = \left( {\begin{array}{*{20}l} {3I_{w} \left( {m_{a} + m_{k} } \right)r_{k}^{4} { \cos }^{2} \alpha + 2\left( {l^{2} m_{a} + I_{x} } \right)\left( {m_{a} + m_{k} } \right)r_{k}^{2} r_{w}^{2} } \hfill \\ {\quad + \,3\left( {I_{kx} + I_{x} } \right)I_{w} r_{k}^{2} { \cos }^{2} \alpha + 2I_{kx} I_{x} r_{w}^{2} + 6I_{w} lm_{a} r_{k}^{3} { \cos }^{2} \alpha \cos \theta_{x} } \hfill \\ {\quad - \,2l^{2} m_{a}^{2} r_{k}^{2} r_{w}^{2} \cos^{2} \theta_{x} + 3I_{w} l^{2} m_{a} r_{k}^{2} { \cos }^{2} \alpha + 2I_{kx} l^{2} m_{a} r_{w}^{2} } \hfill \\ \end{array} } \right). \\ \end{aligned}$$

Appendix B

The elements of standard nonlinear state-space dynamics of the 2D ridable ballbot in the x–z plane are the following:

$$\begin{aligned} A_{p3} \left( {{\mathbf{x}}_{p} } \right) & = - \frac{{r_{k}^{2} }}{{mau_{p} }}\left( {\begin{array}{*{20}l} {2I_{y} b_{x} r_{w}^{2} \dot{x}_{k} + 3I_{w} b_{ry} r_{k} \dot{\theta }_{y} { \cos }^{2} \alpha + 2b_{x} l^{2} m_{a} r_{w}^{2} \dot{x}_{k} } \hfill \\ { + \,3I_{w} b_{x} r_{k}^{2} { \cos }^{2} \alpha \dot{x}_{k} + gl^{2} m_{a}^{2} r_{w}^{2} { \sin }2\theta_{y} - 2l^{3} m_{a}^{2} r_{w}^{2} \dot{\theta }_{y}^{2} { \sin }\theta_{y} } \hfill \\ { - \,2I_{y} lm_{a} r_{w}^{2} \dot{\theta }_{y}^{2} { \sin }\theta_{y} - 2b_{ry} lm_{a} r_{w}^{2} \dot{\theta }_{y} { \cos }\theta_{y} } \hfill \\ { - \,3I_{w} lm_{a} r_{k}^{2} { \cos }^{2} \alpha \dot{\theta }_{y}^{2} { \sin }\theta_{y} - 3I_{w} glm_{a} r_{k} { \cos }^{2} \alpha { \sin }\theta_{y} } \hfill \\ \end{array} } \right), \\ B_{p3} & = - 2r_{k}^{2} r_{w} \left( {m_{a} l^{2} + m_{a} r_{k} l{ \cos }\theta_{y} + I_{y} } \right)/mau_{p} , \\ A_{p4} & = - \frac{1}{{mau_{p} }}\left( {\begin{array}{*{20}l} {2I_{kx} b_{ry} r_{w}^{2} \dot{\theta }_{y} + 2b_{ry} \left( {m_{a} + m_{k} } \right)r_{k}^{2} r_{w}^{2} \dot{\theta }_{y} + 3I_{w} b_{ry} r_{k}^{2} { \cos }^{2} \alpha \dot{\theta }_{y} } \hfill \\ { + \,3I_{w} b_{x} r_{k}^{3} { \cos }^{2} \alpha \dot{x}_{k} - 2glm_{a}^{2} r_{k}^{2} r_{w}^{2} { \sin }\theta_{y} - 2I_{kx} glm_{a} r_{w}^{2} { \sin }\theta_{y} } \hfill \\ { + \,l^{2} m_{a}^{2} r_{k}^{2} r_{w}^{2} \dot{\theta }_{y}^{2} { \sin }2\theta_{y} - 3I_{w} lm_{a} r_{k}^{3} { \cos }^{2} \alpha \dot{\theta }_{y}^{2} { \sin }\theta_{y} } \hfill \\ { - \,2b_{x} lm_{a} r_{k}^{2} r_{w}^{2} \dot{x}_{k} { \cos }\theta_{y} - 2glm_{a} m_{k} r_{k}^{2} r_{w}^{2} { \sin }\theta_{y} } \hfill \\ { - \,3I_{w} glm_{a} r_{k}^{2} { \cos }^{2} \alpha { \sin }\theta_{y} } \hfill \\ \end{array} } \right), \\ B_{p4} \left( {{\mathbf{x}}_{p} } \right) & = 2r_{k} r_{w} \left( {I_{kx} + m_{a} r_{k}^{2} + m_{k} r_{k}^{2} + lm_{a} r_{k} { \cos }\theta_{y} } \right)/mau_{p} , \\ {\text{and}}\;mau_{p} & = \left( {\begin{array}{*{20}l} {2I_{kx} I_{y} r_{w}^{2} + 3I_{w} \left( {m_{a} + m_{k} } \right)r_{k}^{4} { \cos }^{2} \alpha + 2l^{2} m_{a}^{2} r_{k}^{2} r_{w}^{2} } \hfill \\ { + \,2I_{kx} l^{2} m_{a} r_{w}^{2} + 2l^{2} m_{a} m_{k} r_{k}^{2} r_{w}^{2} + 2I_{y} \left( {m_{a} + m_{k} } \right)r_{k}^{2} r_{w}^{2} } \hfill \\ { + \,3I_{w} l^{2} m_{a} r_{k}^{2} { \cos }^{2} \alpha + 6I_{w} lm_{a} r_{k}^{3} { \cos }^{2} \alpha { \cos }\theta_{y} + 3I_{w} I_{y} r_{k}^{2} { \cos }^{2} \alpha } \hfill \\ { + \,2I_{kx} l^{2} m_{a} r_{w}^{2} - 2l^{2} m_{a}^{2} r_{k}^{2} r_{w}^{2} { \cos }^{2} \theta_{y} + 3I_{kx} I_{w} r_{k}^{2} { \cos }^{2} \alpha } \hfill \\ \end{array} } \right). \\ \end{aligned}$$