Anti-swing sliding mode control of three-dimensional double pendulum overhead cranes based on extended state observer

Abstract

When the double pendulum crane works in three-dimensional motion mode, it can significantly improve transportation efficiency. However, controlling the two-stage swing angles in the three-dimensional motion mode is complex and challenging. This paper presents a coordinated control method for the track and trolley of the double pendulum crane to improve the working efficiency of the crane, which realizes the anti-swing control of the double pendulum crane in three-dimensional movement mode. A three-dimensional double pendulum crane model is established, and the model is simplified by the differential flatness theory. A sliding mode control (SMC) method with an extended state observer (ESO) is designed to position and two-stage swing suppression of the three-dimensional double pendulum crane. For the actuator deadband, a transition process is introduced. The stability of the system is analyzed by the Lyapunov method. The proposed method has strong robustness and anti-interference ability. Theoretical and experimental results show that the proposed method can achieve fast and accurate positioning and effectively suppress the two-stage swing. This method is introduced into a nonlinear experimental platform. Compared with other technologies in the literature, the proposed method shortens the transit time, improves work efficiency, and reduces the safety risk.

Appendices

Appendix A : Observation effect of state observer

The overhead crane is a powerful engineering machine, and its working environment has unknown external interference and noise interference, which will affect the operating performance of the crane. In section III, an extended state observer is designed to observe these disturbances. The observations of these disturbances are introduced into the controller to enhance the anti-interference ability and robustness of the crane control system. The accuracy of the observed results directly affects the control effect of the controller. An experiment is performed on the ESO designed to detect the observer’s observation effect in this section. The crane works in a noisy environment and, at the 5th second, suffers an external impact of 20N duration of 0.1s. Fig. 16 shows the observer’s observations of a crane subjected to external disturbances of a size of 20N and duration of 0.1s. Fig. 17 shows the observer’s ambient noise results. The ESO designed in this paper can effectively observe external disturbance and environmental noise from the observer’s calculation speed and estimation accuracy. The introduction of the ESO enhances the stability and robustness of the system.

Fig. 16

Observation of external disturbance

Fig. 17

Observation of ambient noise

Appendix B : Crane System Modeling

A three-dimensional double pendulum crane model is shown in Fig. 1. In the XYZ coordinate system, the trolley is located on the plane of XOY, trolley’s coordinate is $({x_M},{y_M},{z_M})$, hook’s coordinate is $({x_{{m_1}}},{y_{{m_1}}},{z_{{m_1}}})$, and load’s coordinate is $({x_{{m_2}}},{y_{{m_2}}},{z_{{m_2}}})$, where

$$\begin{aligned} \left\{ \begin{array}{l} {x_M} = x\\ {y_M} = y\\ {z_M} = 0\\ {x_{{m_1}}} = x + {l_1}\sin {\theta _1}\cos {\theta _2}\\ {y_{{m_1}}} = y + {l_1}\sin {\theta _2}\\ {z_{{m_1}}} = - {l_1}\cos {\theta _1}\cos {\theta _2}\\ {x_{{m_2}}} = x + {l_1}\sin {\theta _1}\cos {\theta _2} + {l_2}\sin {\theta _3}\cos {\theta _4}\\ {y_{{m_2}}} = y + {l_1}\sin {\theta _2} + {l_2}\sin {\theta _4}\\ {z_{{m_2}}} = - {l_1}\cos {\theta _1}\cos {\theta _2} - {l_2}\cos {\theta _3}\cos {\theta _4}. \end{array} \right. \end{aligned}$$

(55)

From the Eq. (55), the speed of the trolley is $V_M$, the speed of the hook is ${V_{{m_1}}}$, and the speed of the load is ${V_{{m_2}}}$, where

$$\begin{aligned}&V_M = \left[ \begin{array}{l} \frac{\partial }{{\partial t}}\mathrm{{\;}}x\left( t \right) \\ \frac{\partial }{{\partial t}}\mathrm{{\;}}y\left( t \right) \\ 0 \end{array} \right] = \left[ \begin{array}{l} {\dot{x}}\\ {\dot{y}}\\ 0 \end{array} \right] , \end{aligned}$$

(56)

$$\begin{aligned}&\begin{array}{l} {V_{{m_1}}} = \left[ \begin{array}{l} \frac{\partial }{{\partial t}}\mathrm{{\;}}x\left( t \right) {+} {l_1}\,{c_2}\left( t \right) \,\frac{\partial }{{\partial t}}\mathrm{{\;}}{s_1}\left( t \right) {+} {l_1}\,{s_1}\left( t \right) \,\frac{\partial }{{\partial t}}\mathrm{{\;}}{c_2}\left( t \right) \\ {l_1}\,\frac{\partial }{{\partial t}}\mathrm{{\;}}{s_2}\left( t \right) + \frac{\partial }{{\partial t}}\mathrm{{\;}}y\left( t \right) \\ - {l_1}\,{c_1}\left( t \right) \,\frac{\partial }{{\partial t}}\mathrm{{\;}}{c_2}\left( t \right) - {l_1}\,{c_2}\left( t \right) \,\frac{\partial }{{\partial t}}\mathrm{{\;}}{c_1}\left( t \right) \end{array} \right] \\ \begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {}&{}{\begin{array}{*{20}{c}} {}&{}{} \end{array}} \end{array}}&{} = \end{array}\left[ \begin{array}{l} \dot{x} + {l_1}\,{c_2}\,{c_1}\, - {l_1}\,{s_1}\,{s_2}\\ {l_1}\,{c_2} + \mathrm{{\;}}\dot{y}\\ {l_1}\,{c_1}{s_2} + {l_1}\,{c_2}\,{s_1} \end{array} \right] , \end{array} \end{aligned}$$

(57)

$$\begin{aligned}&\begin{array}{l} {V_{{m_2}}} = \left[ \begin{array}{l} \frac{\partial }{{\partial t}}x\left( t \right) + {l_1}{c_2}\left( t \right) \frac{\partial }{{\partial t}}{s_1}\left( t \right) + {l_1}{s_1}\left( t \right) \frac{\partial }{{\partial t}}{c_2}\left( t \right) \\ \quad +{l_2}{c_4}\left( t \right) \frac{\partial }{{\partial t}}{s_3}\left( t \right) + {l_2}{s_3}\left( t \right) \frac{\partial }{{\partial t}}{c_4}\left( t \right) \\ {l_1}\frac{\partial }{{\partial t}}{s_2}\left( t \right) + {l_2}\frac{\partial }{{\partial t}}{s_4}\left( t \right) + \frac{\partial }{{\partial t}}y\left( t \right) \\ - {l_1}{c_1}\left( t \right) \frac{\partial }{{\partial t}}{c_2}\left( t \right) - {l_1}{c_2}\left( t \right) \frac{\partial }{{\partial t}}{c_1}\left( t \right) \\ - {l_2}{c_3}\left( t \right) \frac{\partial }{{\partial t}}{c_4}\left( t \right) - {l_2}{c_4}\left( t \right) \frac{\partial }{{\partial t}}{c_3}\left( t \right) \end{array} \right] \\ \begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {}&{}{}&{}{} \end{array}}&{} = \end{array}\left[ \begin{array}{l} \dot{x} + {l_1}{c_2}{c_1} - {l_1}{s_1}{s_2} + {l_2}{c_4}{c_3} - {l_2}{s_3}{s_4}\\ {l_1}{c_2} + {l_2}{c_4} + \dot{y}\\ {l_1}{c_1}{s_2} + {l_1}{c_2}{s_1} + {l_2}{c_3}{s_4} + {l_2}{c_4}{s_3} \end{array} \right] . \end{array} \end{aligned}$$

(58)

From Eq.(56), Eq.(57), and Eq.(58), the system kinetic energy, T, is obtained as follows:

$$\begin{aligned} T= & {} \frac{M}{2}V_M^2 + \frac{{{m_1}}}{2}V_{{m_1}}^2 + \frac{{{m_2}}}{2}V_{{m_2}}^2\nonumber \\= & {} \frac{{{M_x} + {m_1} + {m_2}}}{2}{{\dot{x}}^2} + \frac{{{M_y} + {m_1} + {m_2}}}{2}{{\dot{y}}^2} \nonumber \\&+\frac{{{m_1}}}{2}(l_1^2{{{\dot{\theta }} }_2}^2 + l_1^2c_2^2{\dot{\theta }} _1^2 \nonumber \\&+ 2{c_1}{c_2}{l_1}\dot{x}{{{\dot{\theta }} }_1} - 2{s_1}{s_2}{l_1}\dot{x}{{{\dot{\theta }} }_2} \nonumber \\&+2{c_2}{l_1}\dot{y}{{{\dot{\theta }} }_2}) + \frac{{{m_2}}}{2}(l_1^2{{{\dot{\theta }} }_2}^2 \nonumber \\&+ 2{c_{1 - 3}}{l_1}{l_2}{s_2}{s_4}{{{\dot{\theta }} }_2}{{{\dot{\theta }} }_4} \nonumber \\&+2{s_{3 - 1}}{l_1}{l_2}{s_2}{s_4}{{{\dot{\theta }} }_2}{{{\dot{\theta }} }_3} + l_1^2c_2^2{\dot{\theta }} _1^2 \nonumber \\&+ 2{s_{1 - 3}}{l_1}{l_2}{c_2}{s_4}{{{\dot{\theta }} }_1}{{{\dot{\theta }} }_4} \nonumber \\&+2{c_{1 - 3}}{l_1}{l_2}{c_2}{c_4}{{{\dot{\theta }} }_1}{{{\dot{\theta }} }_3} + 2{c_1}{c_2}{l_1}\dot{x}{{{\dot{\theta }} }_1} \nonumber \\&+ l_2^2{{{\dot{\theta }} }_4}^2 + c_4^2l_2^2{{{\dot{\theta }} }_3}^2 \nonumber \\&+2{c_3}{c_4}{l_2}\dot{x}{{{\dot{\theta }} }_3} - 2{s_1}{s_2}{l_1}\dot{x}{{{\dot{\theta }} }_2} - 2{s_3}{s_4}{l_2}\dot{x}{{{\dot{\theta }} }_4} \nonumber \\&+2{l_1}{l_2}{c_2}{c_4}{{{\dot{\theta }} }_2}{{{\dot{\theta }} }_4} + 2{c_2}{l_1}\dot{y}{{{\dot{\theta }} }_2}\nonumber \\&+ 2{c_4}{l_2}\dot{y}{{{\dot{\theta }} }_4}). \end{aligned}$$

(59)

Taking trolley’s plane as the zero potential energy plane, V which refers to the system’s potential energy is obtained as follows:

$$\begin{aligned} V = - ({m_1} + {m_2})g{l_1}{c_1}{c_2} - {m_2}g{l_2}{c_3}{c_4}. \end{aligned}$$

(60)

 The Lagrange equation is a system of second-order differential equations,

$$\begin{aligned} \left\{ \begin{array}{l} L(q,\dot{q}) = T(q,\dot{q}) - V(q,\dot{q})\\ \frac{d}{{dt}}(\frac{{\partial L}}{{\partial {{\dot{q}}_k}}}) - \frac{{\partial L}}{{\partial {q_k}}} - {Q_k} = 0\text {,} \end{array} \right. \end{aligned}$$

(61)

where L refers to Lagrange function; T represents system kinetic energy; V is system potential energy; q stands for Lagrange variable; $Q_k$ refers to external forces.

Substituting Eq.(59) and Eq.(60) into Eq.(61), the three-dimensional double pendulum crane model (Eq.(1) - Eq.(6) ) can be obtained.