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Fuzzy tuning control approach to perform cooperative object manipulation by a rigid–flexible multibody robot


We study cooperative object manipulation control of rigid–flexible multibody systems in space. During such tasks, flexible members like solar panels may get vibrated. Which in turn may lead to some oscillatory disturbing forces on other subsystems and consequently produce errors in the motion of the end-effectors of the cooperative manipulating arms. Therefore, to design and develop capable model-based controllers for such complicated systems, deriving a dynamics model is required. However, due to practical limitations and real-time implementation, the system dynamics model should require low computations. So, first, to obtain a precise compact dynamics model, the rigid–flexible interactive dynamics modeling (RFIM) approach is briefly introduced. Using this approach, the system is virtually partitioned into two rigid and flexible portions, and a convenient model for control purposes is developed. Next, a fuzzy tuning manipulation control (FTMC) algorithm is developed for a simple conceptual model for cooperative object manipulation. In fact, a suitable setup is designed for practical implementation of this controller. After that, a wheeled mobile robot (WMR) system with flexible appendages is considered as a practical case that necessitates delicate force exertion by several end-effectors to move an object along a desired path. The WMR system contains two cooperative manipulators, appended with two flexible solar panels. To reveal the merits of the developed model-based controller, the maneuver is deliberately planned such that flexible modes of solar panels get stimulated due to arms motion. The obtained results show an effective performance of the proposed approach as will be discussed.

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\(a\) :

Cross sectional of the flexible member

\(\mathbf{C}\) :

Vector of quadratic nonlinear terms of velocity

\(\mathit{EI}\) :

Bending stiffness of flexible member

\(\mathbf{G}\) :

Vector of gravity force affects

\(\underline{\mathbf{G}}\) :

Grasp matrix

\(\mathbf{H}\) :

Positive definite mass matrix of system

\(I\) :

Second moment of area

\(\mathbf{J}_{c}\) :

Jacobean matrix for the manipulators

\(\mathbf{K}\) :

Stiffness matrix of flexible member

\(\underline{\mathbf{K}}_{p},\underline{\mathbf{K}}_{d}\) :

Gain matrix of controller for object

\(\underline{\tilde{\mathbf{K}}}_{p},\underline{\tilde{\mathbf{K}}}_{d}\) :

Gain matrix of controller for system in task space

\(L_{b}\) :

Length of flexible members

\(m^{\{ i \}}\) :

Mass of the \(i\)th flexible body

\(\mathbf{M}_{f}\) :

Positive definition mass matrix of flexible member

\(n_{b}\) :

Number of flexible members

\(\mathbf{q}\) :

Entity vector of generalized coordinate of rigid system

\(\bar{\mathbf{q}}\) :

Entity vector of generalized coordinate of flexible body

\(\bar{\mathbf{q}}_{f}\) :

Vector of elastic generalized coordinate of flexible body

\(\bar{\mathbf{q}}_{r}\) :

Vector of reference or rigid generalized coordinate of flexible body

\(\mathbf{Q}\) :

Vector of generalized forces

\(\mathbf{Q}_{e}\) :

Vector of generalized external forces of the flexible members

\(\tilde{\mathbf{Q}}_{f}\) :

Vector of control forces for object motion

\(\mathbf{Q}_{\mathrm{flex}.}\) :

Vector of generalized forces due to stimulation of the flexible members

\(\tilde{\mathbf{Q}}_{m}\) :

Vector of control forces for end-effector motion

\(\tilde{\mathbf{Q}}_{\mathrm{react}}\) :

Vector of forces in task space that is exerted from object to end-effectors

\(\mathbf{Q}_{v}\) :

Quadratic velocity vector of flexible member

\(\underline{\mathbf{Q}},\underline{\boldsymbol{\rho}}\) :

Symmetric positive definite matrices

\(\mathbf{R}_{\mathrm{C}_{0}},\dot{\mathbf{R}}_{\mathrm{C}_{0}},\ddot{\mathbf{R}}_{\mathrm{C}_{0}}\) :

Vector of position, velocity, and acceleration of robot base’s in inertial frame

\(\mathbf{S}\) :

Shape function of flexible member

\(T\) :

Kinetic energy

\(\bar{\mathbf{u}}_{f}\) :

Displacement field of flexible body

\(V,M\) :

Shear force and bending torque in flexible member

\(\mathbf{x},\dot{\mathbf{x}}\) :

Vector of position and velocity of object

\(\mathbf{X}^{(m)},\dot{\mathbf{X}}^{(m)}\) :

Vector of position and velocity of \(m\)th end-effectors

\(\boldsymbol{\beta}_{0}\) :

Generalized Euler angle variables of the robot base

\(\boldsymbol{\delta}_{0}\) :

Orientation of the body reference

\(\boldsymbol{\theta}\) :

Generalized variables of the robot joints

\(\rho\) :

Mass density of flexible member

\(\boldsymbol{\omega}\) :

Angular velocity

\(\xi\) :

Dimensionless length of the flexible member

\(\{ i \}\) :

Counter of flexible member

\(( i )\) :

Counter of rigid member of manipulators

\(f\) :

Showing flexibility for a part of the system

\(r\) :

Showing rigidity for a part of the system


Index of the base


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Correspondence to P. Zarafshan.

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Zarafshan, P., Moosavian, S.A.A. Fuzzy tuning control approach to perform cooperative object manipulation by a rigid–flexible multibody robot. Multibody Syst Dyn 40, 213–233 (2017).

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  • Space robots
  • Rigid–flexible multibody systems
  • Cooperative object manipulation
  • Fuzzy control