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Dynamics and trajectory tracking control of cooperative multiple mobile cranes

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Abstract

This paper addresses the dynamics and trajectory tracking control of cooperative multiple mobile cranes. Compared with a single mobile crane, cooperative cable parallel manipulators for multiple mobile cranes (CPMMC) are more complex in configuration, which have the characters of both series and parallel manipulators. Therefore, for the CPMMC, the forward as well as the inverse kinematics and dynamics include the difficulties of both series and parallel manipulators. However, the closed kinematic chain brings about potential benefits, including sufficient accuracy, higher cost performance, better lifting capacity and security. Firstly, the forward and inverse kinematics of the CPMMC with point mass are derived with elimination method, and the complete dynamic model of the CPMMC is established based on Lagrange equation and the complete kinematics. Secondly, considering the repetitive tasks and high security and precision requirement, a robust iterative learning controller is designed for trajectory tracking on the basis of the linearization of the dynamics. Thirdly, taking the engineering practice into consideration, two case studies are simulated with the same expected trajectory but with different weights of the loads. Finally, the designed controller is compared with traditional PD control algorithm via numerical simulation. The results demonstrate the feasibility and superiority of the CPMMC and designed controller, and provide a theoretical basis for the cooperation of multiple mobile cranes.

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Abbreviations

\(L_{i}\) :

The length of boom

f :

The horizontal distance between lower pivot point of boom and the slewing axis

s :

The horizontal distance between lower pivot point of cylinder and the slewing center

h :

The vertical distance between lower pivot point of cylinder and the slewing center

\(l_{si}\) :

The distance between upper pivot point of cylinder and lower pivot point of boom

\({{\phi _i}}\) :

The slewing angle

\(l_{i}\) :

The length of cable

\(l_{ci}\) :

The length of hydraulic cylinder

q :

Generalized coordinates of the CPMMC

D :

The distance between the slewing centers of each two cranes

d :

The distance between the top point of boom and hoisting point of load

m :

The mass of the load

\(K_{m}\) :

The kinetic energy of the load

\(P_{m}\) :

The gravitational potential energy of the load

\({\varvec{\tau }_m} \) :

The general force on the actuators with respect to the payload

\(M_{m}\) :

Inertia matrix with respect to the payload

\(C_{m}\) :

Coriolis matrix with respect to the payload

\({{\theta _i}} \) :

The angle of the boom around the corresponding down pivot

\(K_{L}\) :

The kinetic energy of the boom

\(\bar{{J}}_{{Li}} \) :

The rotary inertia of the boom around the corresponding lower pivot

\(\tilde{{J}}_{{Li}}\) :

The rotary inertia of the boom around the corresponding slewing center

\(m_{L}\) :

The mass of the boom

\(P_{L}\) :

The gravitational potential energy of the boom

\(K_{L}\) :

The kinetic energy of the boom

\({\varvec{\tau }}_L \) :

The general force on the actuators with respect to the boom

\(M_{L}\) :

Inertia matrix with respect to the boom

\(C_{L}\) :

Coriolis matrix with respect to the boom

\(J_{r}\) :

The rotary inertia of the turrets around the corresponding slewing center

\(P_{r}\) :

The gravitational potential energy of the turret

\(K_{r}\) :

The kinetic energy of the turret

\(\varvec{\tau }_r\) :

The general force on the actuators with respect to the turret

\(M_{r}\) :

Inertia matrix with respect to the turret

\(C_{r}\) :

Coriolis matrix with respect to the turret

\({\varvec{\tau }} \) :

The general force on the actuators of the CPMMC

M :

Inertia matrix of the CPMMC

C :

Coriolis matrix of the CPMMC

j :

The iteration times

\({\varvec{\tau }}_a^{j}\, ({{t}})\) :

The unknown disturbance

\({\varvec{\tau }}^{j}\,({{t}})\) :

The input torque

\({\varvec{e}}\) :

The positional tracking error matrix of the actuators

\({\varvec{K}}_p^j \) :

The proportional gain

\({\varvec{K}}_d^j \) :

The differential gain

\(\beta (j)\) :

The regulatory factor of the control gains which acts in the j-th iteration

E :

The robust gain

\({\varvec{K}}_p^j \) :

The proportional gain of the j-th iteration

\({\varvec{K}}_d^j \) :

The differential gain of the j-th iteration

\({\varvec{K}}_p^0 \) :

The initial proportional gain

\({\varvec{K}}_d^0 \) :

The initial differential gain

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Acknowledgments

This work was supported by the National Natural Science Foundation of China (51275515) and the Fundamental Research Funds for the Central Universities (2014HGCH0015). The authors appreciate the comments and valuable suggestions of anonymous referees and editors for improving the quality of the paper.

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Authors and Affiliations

  1. School of Mechanical and Electrical Engineering, China University of Mining and Technology, Xuzhou, 221116, China

    Sen Qian & Bin Zi

  2. School of Mechanical and Automotive Engineering, Hefei University of Technology, Hefei, 230009, China

    Bin Zi

  3. School of Mechanical Engineering and Electronic Information, China University of Geosciences (Wuhan), Wuhan, 430074, China

    Huafeng Ding

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  1. Sen Qian
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Correspondence to Bin Zi.

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Qian, S., Zi, B. & Ding, H. Dynamics and trajectory tracking control of cooperative multiple mobile cranes. Nonlinear Dyn 83, 89–108 (2016). https://doi.org/10.1007/s11071-015-2313-9

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  • DOI: https://doi.org/10.1007/s11071-015-2313-9

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