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Nonlinear dynamic modeling and performance analysis of a redundantly actuated parallel manipulator with multiple actuation modes based on FMD theory

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Abstract

A novel redundantly actuated parallel manipulator (PM) with multiple actuation modes was developed recently by the author. In this paper, the further research is implemented and a systematic methodology is proposed to develop the rigid–flexible coupling dynamic model (RFDM) of the novel PM based on the flexible multi-body dynamics theory. Firstly, under the floating frame of reference, an arbitrary flexible link of the novel PM is regarded as an Euler–Bernoulli beam and the finite element approach is employed to discretize the flexible beam. Then, one kind of planar beam element with lumped masses and moments of inertia at both ends is presented and a corresponding dynamic model is deduced based on the Lagrangian formulation. On this basis, the RFDM of system is formulated by virtue of the augmented Lagrangian multipliers approach. Given the stiff characteristic of system dynamic model, the hybrid TR-BDF2 numerical algorithm combined with Baumgarte stabilization approach is employed to address the nonlinear RFDM so as to balance the solution efficiency and precision. Based on the RFDM and its degradation model, i.e., the rigid dynamic model, one dynamic simulation experiment is designed to investigate the dynamic performance of the PM. Numerical results indicate that the practical motion of the PM manifests rigid–flexible coupling characteristic, and the redundant actuation modes can attenuate the effect of link flexibility and improve the trajectory tracking precision of end-effector in comparison with the non-redundant actuation modes. Finally, to validate the presented methodology, the obtained numerical results are compared with a virtual prototype model developed based on SimMechanics. The results will be helpful for structure optimization and efficient controller design of the PM with multiple actuation modes.

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Abbreviations

\(O{-}x{-}y\) :

The global coordinate system

\(O_j {-}\bar{{x}}{-}\bar{{y}}\) :

The relative coordinate system

\({\varvec{r}}_{O_j}\) :

The radius vector of the original point of body-fixed coordinate system

\({\varvec{r}}_{0,j}\) :

The radius vector of arbitrary point P in the body-fixed coordinate system before the deformation of flexible body j

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

The vector of deformation displacements

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

The position vector of point P in the global coordinate system

\({\varvec{R}}(\phi _j)\) :

The rotational transformation matrix

\(\phi _j\) :

The angular displacement of flexible body j

\({\varvec{u}}_{f_i }^j\) :

The array of flexible generalized coordinates of an arbitrary element i in flexible body j

\(v_i^j (\bar{{x}}_i ,t)\) :

The axial elastic deformation of element i in flexible body j

\(w_i^j (\bar{{x}}_i ,t)\) :

The transverse elastic deformation of element i in flexible body j

\({\varvec{N}}_i^j (\bar{{x}}_i )\) :

The shape function matrix of element i in flexible body j

\(l_i^j \) :

The length of element i in flexible body j

\({\varvec{U}}_f^j\) :

The array of global deformation displacements of flexible body j

\({\varvec{r}}_{0,i}^j\) :

The coordinate array of point \(P_i^j \) of element i before the deformation of flexible body j

\({\varvec{B}}_i^j\) :

The Boolean indicated matrix

\({\varvec{D}}^{j}\) :

The transformation matrix of global deformation generalized coordinates

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

The vector of generalized coordinates of flexible body j

\(m_{di}^j\) :

The mass of intermediate continuous section of element i

\(\rho \) :

The mass density of element material

\(V_i^j \) :

The volume of the intermediate continuous section of element i

\(T_{t,i}^j \) :

The kinetic energy of intermediate continuous section of element i

\({\varvec{M}}_{t,i}^j \) :

The mass matrix corresponding to the intermediate continuous section of element i

\(m_{O_{1i} }^j , m_{O_{2i} }^j\) :

The lumped masses located at both ends of element i

\(J_{O_{1i} }^j , J_{O_{2i} }^j\) :

The lumped moments of inertia located at both ends of element i

\(l_{O_{1i} }^j , l_{O_{2i} }^j\) :

The distances from the initial position to the both ends points of element i in the undeformed state, respectively

\(T_{t,c}^j\) :

The translational kinetic energy of lumped masses located at both ends of element i

\({\varvec{M}}_{O_{1i} }^j , {\varvec{M}}_{O_{2i} }^j\) :

The mass matrices of the lumped masses of element i

\(\theta _{O_{1i} }^j , \theta _{O_{2i} }^j\) :

The rotational angles of two end points of element i

\(T_{r,c}^j \) :

The rotational kinetic energy of lumped moments of inertia at both ends of element i

\({\varvec{M}}_{J_{O1i} }^j , {\varvec{M}}_{J_{O2i} }^j\) :

The mass matrices of the lumped moments of inertia of element i

\(T_i^j \) :

The total kinetic energy of element i

\({\varvec{M}}_i^j\) :

The mass matrix of element i in flexible body j

\(U_{p,i}^j\) :

The total potential energy of element i in flexible body j

E :

The Young’s modulus of beam material

I :

The area moment of inertia of beam cross section

A :

The cross-sectional area of beam

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

The stiffness matrix of element i

\(\mathscr {L}_i^j\) :

The Lagrangian function of element i

\({\varvec{F}}_{e,i}^j\) :

The column matrix of generalized external forces of element i

\({\varvec{C}}_i^j \) :

The centrifugal and Coriolis force matrix of element i in flexible body j

\({\varvec{I}}\) :

The identity matrix

\({\bar{\varvec{M}}}_j \) :

The mass matrix of flexible body j

\({\bar{\varvec{C}}}_j \) :

The centrifugal and Coriolis force matrix of flexible body j

\({\bar{\varvec{K}}}_j \) :

The stiffness matrix of flexible body j

\({\bar{\varvec{F}}}_{e,j} \) :

The column matrix of generalized external forces of flexible body j

\(n_j \) :

The number of elements in flexible body j

\({\varvec{q}}^{(\mathrm{s})}\) :

The vector of generalized coordinates of system

\({\bar{\varvec{M}}}^{(\mathrm{s})}\) :

The mass matrix of system without constraint

\({\bar{\varvec{C}}}^{(\mathrm{s})}\) :

The centrifugal and Coriolis force matrix of system without constraint

\({\bar{\varvec{K}}}^{(\mathrm{s})}\) :

The stiffness matrix of system without constraint

\({\bar{\varvec{F}}}_e^{(\mathrm{s})}\) :

The column matrix of generalized external forces of system

\(\varvec{{\varPhi }}({\varvec{q}}^{(\mathrm{s})},t)\) :

The constraint equations of system

\(\varvec{{\varPhi }}_c ({\varvec{q}}^{(\mathrm{s})},t)\) :

The constraint Jacobian matrix of system

\({\varvec{\gamma }}\) :

The right-hand side of acceleration constraint equations

\({\varvec{\lambda }}\) :

The vector of Lagrangian multipliers

\({\bar{\varvec{\gamma }}}\) :

The modification term of the right-hand side of acceleration constraint equations

\(\alpha , \beta \) :

The stability coefficients of Baumgarte stabilization method

\({\tilde{\bar{\varvec{M}}}}^{\mathrm{(s)}}\) :

The equivalent mass matrix of system

\({\tilde{\bar{\varvec{H}}}}^{(\mathrm{s})}\) :

The quadratic velocity forces of system

\({\tilde{\bar{\varvec{K}}}}^{(\mathrm{s})}\) :

The equivalent stiffness matrix of system

\({\tilde{\bar{\varvec{F}}}}_e^{(\mathrm{s})}\) :

The equivalent column matrix of generalized forces of system

\(a_c \) :

The acceleration of end-effector

\(a_{\max }\) :

The maximum acceleration of end-effector

\(v_c \) :

The velocity of end-effector

\(\bar{{\varDelta }}_{\mathrm{error}} \) :

The mean-square deviation error

\(x_i^\mathrm{d} ,y_i^\mathrm{d}\) :

The position coordinates of the \(i{\mathrm{th}}\) sample point on the desired trajectory

\(x_i ,y_i\) :

The position coordinates of the \(i{\mathrm{th}}\) sample point on the practical trajectory

AMM:

Assumed mode method

ANCF:

Absolute nodal coordinate formulation

DOF(s):

Degree(s) of freedom

FEM:

Finite element method

FFRF:

Floating frame of reference formulation

FMD:

Flexible multi-body dynamics

FSM:

Finite segment method

KED:

Kineto-elasto dynamics

ODE(s):

Ordinary differential equation(s)

PDE(s):

Partial differential equation(s)

PM(s):

Parallel manipulator(s)

PSB(s):

Parallelogram structure branche(s)

RAParM:

Redundantly actuated parallel manipulator

RDM:

Rigid dynamic model

RFDM:

Rigid–flexible coupling dynamic model

VPM:

Virtual prototype model

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Acknowledgements

This research work was supported by the National Natural Science Foundation of China (NSFC) under Grant No. 51475321 and Tianjin Research Program of Application Foundation and Advanced Technology under Grant No. 15JCZDJC38900. These supports are sincerely acknowledged. The authors would also like to express the genuine thanks to the editors and reviewers for their contributions to the publications of high-quality papers.

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  1. Key Laboratory of Mechanism Theory and Equipment Design of Ministry of Education, Tianjin University, Tianjin, 300072, China

    Dong Liang, Yimin Song & Tao Sun

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  1. Dong Liang
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  2. Yimin Song
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  3. Tao Sun
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Correspondence to Tao Sun.

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Liang, D., Song, Y. & Sun, T. Nonlinear dynamic modeling and performance analysis of a redundantly actuated parallel manipulator with multiple actuation modes based on FMD theory. Nonlinear Dyn 89, 391–428 (2017). https://doi.org/10.1007/s11071-017-3461-x

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  • DOI: https://doi.org/10.1007/s11071-017-3461-x

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