Parallel Manipulator

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Liping Wang - One of the best experts on this subject based on the ideXlab platform.

  • optimal design of a 2 dof Parallel Manipulator with actuation redundancy considering kinematics and natural frequency
    Robotics and Computer-integrated Manufacturing, 2013
    Co-Authors: Jun Wu, Tiemin Li, Xiaomeng Chen, Liping Wang
    Abstract:

    In this paper, a planar 2-DOF Parallel Manipulator with actuation redundancy is proposed and the optimal design considering kinematics and natural frequency is presented. The stiffness matrix and mass matrix are derived, and the structural dynamics is modeled. The natural frequency is obtained on the basis of dynamic model. Based on the kinematic performance, the range for link length is given. Then, considering the natural frequency, the geometry is optimized. The natural frequency is simulated and compared with the corresponding non-redundant Parallel Manipulator. The designed redundant Parallel Manipulator has desired kinematic performance and natural frequency and is incorporated into a 4-DOF hybrid machine tool. Highlights? A novel Parallel Manipulator with actuation redundancy is proposed. ? The natural frequency and displacement response are determined. ? Optimal design of the Manipulator based on kinematics and natural frequency is presented. ? The redundant Parallel Manipulator has a higher natural frequency than its counterpart.

  • Research on the inertia matching of the Stewart Parallel Manipulator
    Robotics and Computer-integrated Manufacturing, 2012
    Co-Authors: Zhufeng Shao, Xiaoqiang Tang, Xu Chen, Liping Wang
    Abstract:

    With the development of the Parallel Manipulator, inertia matching as an essential factor to realize good potentials of the Parallel Manipulator is taken serious gradually. However, neither definite inertia index nor inertia matching method has been proposed so far. In this paper, the above issues are discussed by taking the Stewart Parallel Manipulator as a study object. Firstly, adopting limb Jacobian matrices, the concise algebraic expression of the joint-space inertia matrix of the Stewart Parallel Manipulator is deduced, based on the dynamic modeling. Next, on the basis of the coupling analysis of the joint-space inertia matrix, the inertia index of the Parallel Manipulator, the Joint-Reflected Inertia, is proposed. Then, the practical inertia matching principles of the Stewart Parallel Manipulator are concluded on the basis of simulations, considering multiple factors, such as mechanical resonance frequency, acceleration torque and dynamic performance. Finally, the available range of the motor inertia is deduced, and the inertia matching of the Stewart Parallel Manipulator is finished as the case study. The inertia index and inertia matching method suggested in this paper can be further used in other Parallel Manipulators for dynamic analysis and motion system design.

  • Effect Study on Deformation of Links to Error of Parallel Manipulator
    Advanced Materials Research, 2010
    Co-Authors: Li Wen Guan, Liping Wang
    Abstract:

    Aimed at the effect problem of deformation of links on position error of Parallel Manipulator, the 3-TPT Parallel Manipulator is taken as a study object. The deformation error of links of Parallel Manipulator was built by error independent principle based on kinematics; the simulation analysis also is carried out by virtual prototype technique, and the effect laws of deformation of links to error of Parallel Manipulator are given. The results showed that when the deformation length of links is given, the position error can be affected by deformation of each link and the coordinate position of origin in reference frame; At the beginning of Parallel Manipulator moving, the deformation of links are small, and with the time increasing, the deformation of links obviously affect the final position error of Parallel Manipulator. The two method proposed in this paper can be used for pre-estimating the error and reducing harmful effects.

  • dynamic formulation of a planar 3 dof Parallel Manipulator with actuation redundancy
    Robotics and Computer-integrated Manufacturing, 2010
    Co-Authors: Liping Wang, Jun Wu, Jinsong Wang
    Abstract:

    In this paper, based on the conventional Newton-Euler approach, a simplification method is proposed to derive the dynamic formulation of a planar 3-DOF Parallel Manipulator with actuation redundancy. Closed-form solutions are developed for the inverse kinematics. Based on the kinematics, the Newton-Euler approach in simplification form is used to derive the inverse dynamic model of the redundant Parallel Manipulator. Then, the driving force optimization is performed by minimizing an objective function which is the square of the sum of four driving forces. The dynamic simulations are done for the Parallel Manipulator with both the redundant and non-redundant actuations. The result shows that the dynamic characteristics of the Manipulator in the redundant case are better than that in the non-redundancy. The redundantly actuated Parallel Manipulator was incorporated into a 4-DOF hybrid machine tool which includes a feed worktable.

  • dynamics and control of a planar 3 dof Parallel Manipulator with actuation redundancy
    Mechanism and Machine Theory, 2009
    Co-Authors: Jun Wu, Liping Wang, Jinsong Wang, Tiemin Li
    Abstract:

    This paper deals with the dynamics and control of a planar 3-DOF Parallel Manipulator with actuation redundancy. According to the kinematics of the redundant Manipulator, the inverse dynamics is derived by using the virtual work principle, and the driving force is optimized by utilizing the least-square method. Based on the dynamic model, a position and force switching control strategy is proposed for the Parallel Manipulator. In this control strategy, two extendible links are controlled by position and force modes, respectively. Moreover, in the whole workspace, their control modes are switched. The critical angles for switching control modes are planned and the position compensation method is investigated. The control tests show that the contour error of the redundantly actuated Parallel Manipulator is similar to that of its corresponding non-redundant Parallel Manipulator without one extendible link. However, the redundant Parallel Manipulator has a larger orientation workspace and higher stiffness. The redundantly actuated Parallel Manipulator is incorporated into a 4-DOF hybrid machine tool which also includes a feed worktable.

Shuang Cong - One of the best experts on this subject based on the ideXlab platform.

  • Adaptive compensation of dynamics and friction for a planar Parallel Manipulator with redundant actuation
    2010 IEEE International Conference on Robotics and Biomimetics, 2010
    Co-Authors: Weiwei Shang, Shuang Cong
    Abstract:

    In order to improve the actual tracking accuracy of a planar Parallel Manipulator with redundant actuation, a new adaptive compensation of dynamics and friction is developed. Firstly, the dynamic model is established in the task space for the Parallel Manipulator, and a linear parametrization expression with respect to the model parameters is formulated. Then, based on the dynamic model, a new adaptive control law which contains dynamics compensation, friction compensation and tracking error elimination items is designed. The parameter adaptation law is derived by the Lyapunov method, and the convergence of the tracking error and the error rate is proved by the Barbalat's lemma. At last, the proposed controller is implemented in the trajectory tracking experiments of an actual planar Parallel Manipulator with redundant actuation.

  • nonlinear adaptive task space control for a 2 dof redundantly actuated Parallel Manipulator
    Nonlinear Dynamics, 2010
    Co-Authors: Weiwei Shang, Shuang Cong
    Abstract:

    A nonlinear adaptive (NA) controller in the task space is developed for the trajectory tracking of a 2-DOF redundantly actuated Parallel Manipulator. The dynamic model with nonlinear friction is established in the task space for the Parallel Manipulator, and the linear parameterization expression of the dynamic model is formulated. Based on the dynamic model, a new control law including adaptive dynamics compensation, adaptive friction compensation and error elimination items is designed. After defining a quadratic performance index, the parameter update law is derived with the gradient descent algorithm. The stability of the Parallel Manipulator system is proved by the Lyapunov theorem, and the convergence of the tracking error and the error rate is proved by the Barbalat’s lemma. The NA controller is implemented in the trajectory tracking experiments of an actual 2-DOF redundantly actuated Parallel Manipulator, and the experiment results are compared with the APD controller.

  • nonlinear computed torque control for a high speed planar Parallel Manipulator
    Mechatronics, 2009
    Co-Authors: Weiwei Shang, Shuang Cong
    Abstract:

    A new computed torque (CT)-type controller termed nonlinear CT (NCT) controller is developed and applied to a high-speed planar Parallel Manipulator. The NCT controller is designed by replacing the linear PD in the conventional CT controller with the nonlinear PD (NPD) algorithm. The stability of the Parallel Manipulator system with the NCT controller is proven using the Lyapunov theorem, and the proposed controller is further proven to guarantee asymptotic convergence to zero of both tracking error and error rate. The superiority of the proposed NCT controller is verified through the trajectory tracking experiments of an actual high-speed planar Parallel Manipulator, and the experiment results are compared with the CT controller.

  • active joint synchronization control for a 2 dof redundantly actuated Parallel Manipulator
    IEEE Transactions on Control Systems and Technology, 2009
    Co-Authors: Weiwei Shang, Shuang Cong, Yaoxin Zhang, Yanyang Liang
    Abstract:

    This brief applies the synchronization to solve control problem of redundantly actuated Parallel Manipulators. With the synchronization method, a new controller termed active joint-synchronization (AJ-S) controller is developed for a 2-degree-of-freedom (DOF) redundantly actuated Parallel Manipulator. The dynamic model of the Parallel Manipulator is formulated in the active joint space, in which the internal force is calculated by the projection method and the friction is depicted with the Coulomb + viscous friction model. By defining the tracking error, synchronization error, coupled error, and the referenced trajectory vector of the active joints, the AJ-S controller based on the dynamic model is designed. And the AJ-S controller is proven to guarantee asymptotic convergence to zero of both tracking and synchronization errors by the Barbalat's lemma. The AJ-S controller is implemented in the trajectory tracking experiments of an actual 2-DOF redundantly actuated Parallel Manipulator, and the superiority of the AJ-S controller over the well-known tracking controller is studied.

  • augmented nonlinear pd controller for a redundantly actuated Parallel Manipulator
    Advanced Robotics, 2009
    Co-Authors: Weiwei Shang, Shuang Cong, Zexiang Li, Shilong Jiang
    Abstract:

    In order to improve trajectory tracking accuracy for a redundantly actuated Parallel Manipulator, a so-called augmented nonlinear PD (ANPD) controller based on the conventional dynamic controller is proposed in this paper. The ANPD controller is designed by replacing the linear PD in the augmented PD controller with a nonlinear PD algorithm. The stability of the Parallel Manipulator system with the proposed ANPD controller is proven using the Lyapunov stability theorem, and the ANPD controller is further proven to guarantee asymptotic convergence to zero of both the tracking error and error rate. The superiority of the ANPD controller is verified through trajectory tracking experiments of an actual 2-d.o.f. redundantly actuated Parallel Manipulator.

Jinsong Wang - One of the best experts on this subject based on the ideXlab platform.

  • dynamic formulation of a planar 3 dof Parallel Manipulator with actuation redundancy
    Robotics and Computer-integrated Manufacturing, 2010
    Co-Authors: Liping Wang, Jun Wu, Jinsong Wang
    Abstract:

    In this paper, based on the conventional Newton-Euler approach, a simplification method is proposed to derive the dynamic formulation of a planar 3-DOF Parallel Manipulator with actuation redundancy. Closed-form solutions are developed for the inverse kinematics. Based on the kinematics, the Newton-Euler approach in simplification form is used to derive the inverse dynamic model of the redundant Parallel Manipulator. Then, the driving force optimization is performed by minimizing an objective function which is the square of the sum of four driving forces. The dynamic simulations are done for the Parallel Manipulator with both the redundant and non-redundant actuations. The result shows that the dynamic characteristics of the Manipulator in the redundant case are better than that in the non-redundancy. The redundantly actuated Parallel Manipulator was incorporated into a 4-DOF hybrid machine tool which includes a feed worktable.

  • Dimensional Optimization Design of the Four-Cable-Driven Parallel Manipulator in FAST
    IEEE ASME Transactions on Mechatronics, 2010
    Co-Authors: Xiaoqiang Tang, Jinsong Wang, Peng Huang
    Abstract:

    For the design of the five-hundred-meter aperture spherical radio telescope (FAST), a four-cable-driven Parallel Manipulator, which is long in span and heavy in weight, is adopted as the first-level adjustable feed-support system. The purpose of this paper is to optimize dimensions of the four-cable-driven Parallel Manipulator to meet the workspace requirement of constraint condition in terms of cable tension and stiffness. Accordingly, this optimization method adopts catenary simplification in order to set up the cable tension equilibrium equations, preliminarily optimizing three important dimensional parameters. Stiffness of the cable is also taken into consideration because of its effect on work performance of a cable-driven Parallel Manipulator. However, the stiffness value of a cable-driven Parallel Manipulator is not totally credible by traditional theoretical analysis. Therefore, an experimental method for stiffness analysis is presented in this paper. It applies Buckingham π theorem to set up an experimental stiffness similarity model to obtain a more credible stiffness value, which is used in the optimization process. In this way, dimensional optimization is realized with a set of optimized dimensions for building the feed-support system in the FAST. More importantly, the stiffness similarity model can be universally adopted in stiffness analysis of other large cable-driven Parallel Manipulator.

  • dynamics and control of a planar 3 dof Parallel Manipulator with actuation redundancy
    Mechanism and Machine Theory, 2009
    Co-Authors: Jun Wu, Liping Wang, Jinsong Wang, Tiemin Li
    Abstract:

    This paper deals with the dynamics and control of a planar 3-DOF Parallel Manipulator with actuation redundancy. According to the kinematics of the redundant Manipulator, the inverse dynamics is derived by using the virtual work principle, and the driving force is optimized by utilizing the least-square method. Based on the dynamic model, a position and force switching control strategy is proposed for the Parallel Manipulator. In this control strategy, two extendible links are controlled by position and force modes, respectively. Moreover, in the whole workspace, their control modes are switched. The critical angles for switching control modes are planned and the position compensation method is investigated. The control tests show that the contour error of the redundantly actuated Parallel Manipulator is similar to that of its corresponding non-redundant Parallel Manipulator without one extendible link. However, the redundant Parallel Manipulator has a larger orientation workspace and higher stiffness. The redundantly actuated Parallel Manipulator is incorporated into a 4-DOF hybrid machine tool which also includes a feed worktable.

  • an experimental study of a redundantly actuated Parallel Manipulator for a 5 dof hybrid machine tool
    IEEE-ASME Transactions on Mechatronics, 2009
    Co-Authors: Liping Wang, Jun Wu, Jinsong Wang
    Abstract:

    This paper deals with the experimental study of a redundantly actuated Parallel Manipulator for a 5-DOF hybrid machine tool. Based on the kinematics of the redundantly actuated Parallel Manipulator, the inverse dynamics is derived by using the virtual work principle. A position and force switching control strategy is used for the two extendible chains of the Parallel Manipulator. Further, the difference predictive control is presented for the extendible chain with force mode. The Parallel Manipulator is incorporated into a 5-DOF hybrid machine tool. Linear and circular contouring experiments of both the hybrid machine tool as well as its corresponding nonredundant counterpart with the redundant limb removed are conducted to evaluate the machine performance. Finally, the machine experiment is used to demonstrate the machine's applicability.

  • on the analysis of a new spatial three degrees of freedom Parallel Manipulator
    International Conference on Robotics and Automation, 2001
    Co-Authors: Jinsong Wang, Liping Wang
    Abstract:

    In this paper, a new spatial three-degrees-of-freedom (two degrees of translational freedom and one degree of rotational freedom) Parallel Manipulator is proposed. The Parallel Manipulator consists of a base plate, a movable platform, and three connecting legs. The inverse and forward kinematics problems are described in closed forms and the velocity equation of the new Parallel Manipulator is given. Three kinds of singularities are also presented. The workspace for the Manipulator is analyzed systematically; in particular, indices to evaluate the mobility (in this paper, mobility means rotational capability) of the moving platform of the Manipulator will be defined and discussed in detail. Finally, a topology architecture of the Manipulator is introduced. The Parallel Manipulator has wide application in the fields of industrial robots, simulators, microManipulators, and Parallel machine tools.

Weiwei Shang - One of the best experts on this subject based on the ideXlab platform.

  • Adaptive compensation of dynamics and friction for a planar Parallel Manipulator with redundant actuation
    2010 IEEE International Conference on Robotics and Biomimetics, 2010
    Co-Authors: Weiwei Shang, Shuang Cong
    Abstract:

    In order to improve the actual tracking accuracy of a planar Parallel Manipulator with redundant actuation, a new adaptive compensation of dynamics and friction is developed. Firstly, the dynamic model is established in the task space for the Parallel Manipulator, and a linear parametrization expression with respect to the model parameters is formulated. Then, based on the dynamic model, a new adaptive control law which contains dynamics compensation, friction compensation and tracking error elimination items is designed. The parameter adaptation law is derived by the Lyapunov method, and the convergence of the tracking error and the error rate is proved by the Barbalat's lemma. At last, the proposed controller is implemented in the trajectory tracking experiments of an actual planar Parallel Manipulator with redundant actuation.

  • nonlinear adaptive task space control for a 2 dof redundantly actuated Parallel Manipulator
    Nonlinear Dynamics, 2010
    Co-Authors: Weiwei Shang, Shuang Cong
    Abstract:

    A nonlinear adaptive (NA) controller in the task space is developed for the trajectory tracking of a 2-DOF redundantly actuated Parallel Manipulator. The dynamic model with nonlinear friction is established in the task space for the Parallel Manipulator, and the linear parameterization expression of the dynamic model is formulated. Based on the dynamic model, a new control law including adaptive dynamics compensation, adaptive friction compensation and error elimination items is designed. After defining a quadratic performance index, the parameter update law is derived with the gradient descent algorithm. The stability of the Parallel Manipulator system is proved by the Lyapunov theorem, and the convergence of the tracking error and the error rate is proved by the Barbalat’s lemma. The NA controller is implemented in the trajectory tracking experiments of an actual 2-DOF redundantly actuated Parallel Manipulator, and the experiment results are compared with the APD controller.

  • nonlinear computed torque control for a high speed planar Parallel Manipulator
    Mechatronics, 2009
    Co-Authors: Weiwei Shang, Shuang Cong
    Abstract:

    A new computed torque (CT)-type controller termed nonlinear CT (NCT) controller is developed and applied to a high-speed planar Parallel Manipulator. The NCT controller is designed by replacing the linear PD in the conventional CT controller with the nonlinear PD (NPD) algorithm. The stability of the Parallel Manipulator system with the NCT controller is proven using the Lyapunov theorem, and the proposed controller is further proven to guarantee asymptotic convergence to zero of both tracking error and error rate. The superiority of the proposed NCT controller is verified through the trajectory tracking experiments of an actual high-speed planar Parallel Manipulator, and the experiment results are compared with the CT controller.

  • active joint synchronization control for a 2 dof redundantly actuated Parallel Manipulator
    IEEE Transactions on Control Systems and Technology, 2009
    Co-Authors: Weiwei Shang, Shuang Cong, Yaoxin Zhang, Yanyang Liang
    Abstract:

    This brief applies the synchronization to solve control problem of redundantly actuated Parallel Manipulators. With the synchronization method, a new controller termed active joint-synchronization (AJ-S) controller is developed for a 2-degree-of-freedom (DOF) redundantly actuated Parallel Manipulator. The dynamic model of the Parallel Manipulator is formulated in the active joint space, in which the internal force is calculated by the projection method and the friction is depicted with the Coulomb + viscous friction model. By defining the tracking error, synchronization error, coupled error, and the referenced trajectory vector of the active joints, the AJ-S controller based on the dynamic model is designed. And the AJ-S controller is proven to guarantee asymptotic convergence to zero of both tracking and synchronization errors by the Barbalat's lemma. The AJ-S controller is implemented in the trajectory tracking experiments of an actual 2-DOF redundantly actuated Parallel Manipulator, and the superiority of the AJ-S controller over the well-known tracking controller is studied.

  • augmented nonlinear pd controller for a redundantly actuated Parallel Manipulator
    Advanced Robotics, 2009
    Co-Authors: Weiwei Shang, Shuang Cong, Zexiang Li, Shilong Jiang
    Abstract:

    In order to improve trajectory tracking accuracy for a redundantly actuated Parallel Manipulator, a so-called augmented nonlinear PD (ANPD) controller based on the conventional dynamic controller is proposed in this paper. The ANPD controller is designed by replacing the linear PD in the augmented PD controller with a nonlinear PD algorithm. The stability of the Parallel Manipulator system with the proposed ANPD controller is proven using the Lyapunov stability theorem, and the ANPD controller is further proven to guarantee asymptotic convergence to zero of both the tracking error and error rate. The superiority of the ANPD controller is verified through trajectory tracking experiments of an actual 2-d.o.f. redundantly actuated Parallel Manipulator.

Jun Wu - One of the best experts on this subject based on the ideXlab platform.

  • optimal design of a 2 dof Parallel Manipulator with actuation redundancy considering kinematics and natural frequency
    Robotics and Computer-integrated Manufacturing, 2013
    Co-Authors: Jun Wu, Tiemin Li, Xiaomeng Chen, Liping Wang
    Abstract:

    In this paper, a planar 2-DOF Parallel Manipulator with actuation redundancy is proposed and the optimal design considering kinematics and natural frequency is presented. The stiffness matrix and mass matrix are derived, and the structural dynamics is modeled. The natural frequency is obtained on the basis of dynamic model. Based on the kinematic performance, the range for link length is given. Then, considering the natural frequency, the geometry is optimized. The natural frequency is simulated and compared with the corresponding non-redundant Parallel Manipulator. The designed redundant Parallel Manipulator has desired kinematic performance and natural frequency and is incorporated into a 4-DOF hybrid machine tool. Highlights? A novel Parallel Manipulator with actuation redundancy is proposed. ? The natural frequency and displacement response are determined. ? Optimal design of the Manipulator based on kinematics and natural frequency is presented. ? The redundant Parallel Manipulator has a higher natural frequency than its counterpart.

  • EMEIT - Parameter design and manufacture for the 4RRR Parallel Manipulator
    Proceedings of 2011 International Conference on Electronic & Mechanical Engineering and Information Technology, 2011
    Co-Authors: Xian Huang, Tiemin Li, Boqiang Xu, Jun Wu
    Abstract:

    In order to overcome the drawbacks of 3RRR non-redundant Parallel Manipulators, a redundantly actuated 4RRR Parallel Manipulator is designed and manufactured. In this paper, the relationship of mechanism parameters, workspace and dexterity is examined. A procedure to design a 4RRR Parallel Manipulator is presented in detail and a prototype of the machine is manufactured, preparing for the further studies on kinematic and dynamic performance of 4RRR Parallel Manipulator.

  • Parameter design and manufacture for the 4RRR Parallel Manipulator
    Proceedings of 2011 International Conference on Electronic & Mechanical Engineering and Information Technology, 2011
    Co-Authors: Xian Huang, Tiemin Li, Boqiang Xu, Jun Wu
    Abstract:

    In order to overcome the drawbacks of 3RRR non-redundant Parallel Manipulators, a redundantly actuated 4RRR Parallel Manipulator is designed and manufactured. In this paper, the relationship of mechanism parameters, workspace and dexterity is examined. A procedure to design a 4RRR Parallel Manipulator is presented in detail and a prototype of the machine is manufactured, preparing for the further studies on kinematic and dynamic performance of 4RRR Parallel Manipulator.

  • dynamic formulation of a planar 3 dof Parallel Manipulator with actuation redundancy
    Robotics and Computer-integrated Manufacturing, 2010
    Co-Authors: Liping Wang, Jun Wu, Jinsong Wang
    Abstract:

    In this paper, based on the conventional Newton-Euler approach, a simplification method is proposed to derive the dynamic formulation of a planar 3-DOF Parallel Manipulator with actuation redundancy. Closed-form solutions are developed for the inverse kinematics. Based on the kinematics, the Newton-Euler approach in simplification form is used to derive the inverse dynamic model of the redundant Parallel Manipulator. Then, the driving force optimization is performed by minimizing an objective function which is the square of the sum of four driving forces. The dynamic simulations are done for the Parallel Manipulator with both the redundant and non-redundant actuations. The result shows that the dynamic characteristics of the Manipulator in the redundant case are better than that in the non-redundancy. The redundantly actuated Parallel Manipulator was incorporated into a 4-DOF hybrid machine tool which includes a feed worktable.

  • dynamics and control of a planar 3 dof Parallel Manipulator with actuation redundancy
    Mechanism and Machine Theory, 2009
    Co-Authors: Jun Wu, Liping Wang, Jinsong Wang, Tiemin Li
    Abstract:

    This paper deals with the dynamics and control of a planar 3-DOF Parallel Manipulator with actuation redundancy. According to the kinematics of the redundant Manipulator, the inverse dynamics is derived by using the virtual work principle, and the driving force is optimized by utilizing the least-square method. Based on the dynamic model, a position and force switching control strategy is proposed for the Parallel Manipulator. In this control strategy, two extendible links are controlled by position and force modes, respectively. Moreover, in the whole workspace, their control modes are switched. The critical angles for switching control modes are planned and the position compensation method is investigated. The control tests show that the contour error of the redundantly actuated Parallel Manipulator is similar to that of its corresponding non-redundant Parallel Manipulator without one extendible link. However, the redundant Parallel Manipulator has a larger orientation workspace and higher stiffness. The redundantly actuated Parallel Manipulator is incorporated into a 4-DOF hybrid machine tool which also includes a feed worktable.

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