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Quasi-time-optimal controller design for a rigid-flexible multibody system via absolute coordinate-based formulation

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Abstract

Previous studies have shown that the absolute coordinate-based formulation is an accurate modeling approach for the rigid-flexible multibody system subject to both large rotation and large deformation. However, it is almost impossible to design the complex controller for a rigid-flexible multibody system via the absolute coordinate-based formulation because of its high dimensions. The primary aim of this study is to design a quasi-optimal controller for a rigid-flexible multibody system via the absolute coordinate-based formulation. The design procedure includes two steps. As the first step, the simplified model for a rigid-flexible multibody system is established via the floating frame of reference formulation first, and then, a time-optimal controller is designed for the simplified model by using the Gauss pseudo-spectral method. In the second step, a quasi-time-optimal closed-loop tracking controller is achieved for the absolute coordinate-based formulation of the rigid-flexible multibody system by superposing a PD controller to track the pre-designed optimal trajectory. The paper presents two case studies for a hub-beam system, where only the motion of the rigid part is observable and driven. The numerical results of the two case studies well verify the effectiveness of the proposed controller.

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Abbreviations

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

Vector of element nodal coordinates

EA:

Tensile stiffness of beam

EI:

Bending stiffness of beam

G :

End-point constraints

h :

General state-control path constraints

J :

Bolza cost function

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

Proportional gain matrix

\({{\varvec{K}}}_F \) :

Stiffness matrix in FFR formulation

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

Differential gain matrix

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

Mass matrix of an element

\({{\varvec{M}}}_A \) :

Mass matrix of rigid-flexible multibody system

\({{\varvec{M}}}_f \) :

Mass matrix of flexible part

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

Mass matrix of rigid part

\({{\varvec{M}}}_F \) :

Mass matrix in FFR formulation

\(p_i \) :

ith modal coordinate of cantilever beam

\({{\varvec{Q}}}_{Ae}\) :

External force vector applied on rigid-flexible multibody system

\({{\varvec{Q}}}_{Ak}\) :

Elastic force vector applied on rigid-flexible multibody system

\({{\varvec{Q}}}_{Ff} \) :

Generalized external force vector in FFR formulation

\({{\varvec{Q}}}_{Fv} \) :

Summation of the centrifugal force vector and the Coriolis force vector in FFR formulation

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

Generalized force vector applied on rigid part

\({{\varvec{Q}}}_e^e \) :

External force vector applied on an element

\({{\varvec{Q}}}_k^e \) :

Elastic force vector applied on an element

\({{\varvec{Q}}}_e^f \) :

External force vector applied on flexible part

\({{\varvec{Q}}}_k^f \) :

Elastic force vector applied on flexible part

\({{\varvec{q}}}_A \) :

Generalized coordinate vector of rigid-flexible multibody system

\({{\varvec{q}}}_F \) :

Generalized coordinate vector in FFR formulation

\({{\varvec{q}}}_f \) :

Generalized coordinate vector of flexible part

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

Generalized coordinate vector of rigid part

r :

Displacement field of an element

S :

Shape function matrix

T :

Kinetic energy of an element

\(T_0 \) :

Moment applied on the hub

\(t_f \) :

Final moment

U :

Strain energy

u :

Control variable vector

\({{\varvec{u}}}_F \) :

Nonzero terms in \({{\varvec{Q}}}_{Ff} \)

\({{\varvec{u}}}_{F\mathrm{ref}} \) :

Designed optimal controller

\({{\varvec{u}}}_\mathrm{PD} \) :

PD controller used to track the designed optimal trajectory

\({{\varvec{u}}}^{L}\) :

Vector of minima of control variable

\({{\varvec{u}}}^{U}\) :

Vector of maxima of control variable

x :

State variable vector

\({{\varvec{x}}}_A \) :

State vector in ACB formulation

\({{\varvec{x}}}_{A1} \) :

Observable state vector in ACB formulation

\({{\varvec{x}}}_{A2} \) :

Unobservable state vector in ACB formulation

\({{\varvec{x}}}_F \) :

State vector in FFR formulation

\({{\varvec{x}}}_{F1} \) :

Observable state vector in FFR formulation

\({{\varvec{x}}}_{F2} \) :

Unobservable state vector in FFR formulation

\({{\varvec{x}}}_{F\mathrm{ref}} \) :

Designed optimal trajectory

\(\varepsilon _1 \) :

A very small value

\(\rho \) :

Density of flexible part

\(\theta \) :

Attitude angle of hub

\(\dot{\theta }\) :

Rotation speed of hub

\({\varvec{\Phi }}_f ({{\varvec{q}}}_f )\) :

Constraint equations of flexible part

\({\varvec{\Phi }}_r ({{\varvec{q}}}_r )\) :

Constraint equations of rigid part

\({\varvec{\Phi }}_{r-f} ({{\varvec{q}}}_r ,{{\varvec{q}}}_f )\) :

Constraint equations between rigid part and flexible part

\({\varvec{\Phi }}_A \) :

Constraint equations of rigid-flexible multibody system

ACB:

Absolute coordinate-based

ACF:

Absolute coordinate formulation

AMM:

Assumed mode method

ANCF:

Absolute nodal coordinate formulation

FEM:

Finite element method

FFR:

Floating frame of reference

LGR:

Legendre-Gauss-Radau

OOC-FFR:

Open-loop optimal controller based on FFR formulation

PD:

Proportional derivative

PID:

Proportional integral derivative

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Acknowledgements

This work was supported in part by the National Natural Science Foundation of China under Grants 11290153, 11402111 and 11372130, and in part by Foundation for the Author of National Excellent Doctoral Dissertation of China under Grant 201233.

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Authors and Affiliations

  1. State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, 29 Yudao Street, Nanjing, 210016, People’s Republic of China

    Ti Chen, Hao Wen, Haiyan Hu & Dongping Jin

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  1. Ti Chen
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  2. Hao Wen
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Correspondence to Haiyan Hu.

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Chen, T., Wen, H., Hu, H. et al. Quasi-time-optimal controller design for a rigid-flexible multibody system via absolute coordinate-based formulation. Nonlinear Dyn 88, 623–633 (2017). https://doi.org/10.1007/s11071-016-3265-4

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