Flexible Intermediate Links Research Articles

Rigid-flexible coupling dynamic model of a flexible planar parallel robot for modal characteristics research

The increasing applications of flexible parallel robots in industrial production have presented the advantages of light weight and high speed, but at the same time, the elastic vibration problem has emerged. By investigating the modal features of flexible parallel robots so as to suppress the elastic vibration, with pinned-pinned as flexible intermediate links boundary conditions, this article analyzes the rigid-flexible coupling dynamic mathematical model of the 3-RRR (3-Rotate-Rotate-Rotate) flexible planar parallel robot with flexible intermediate links. The effect of the extremity concentrated rotation inertia of flexible intermediate links is considered in the mathematical model. Besides, the effect of inertia and coupling force on the dynamic model and the first three-order vibration responses of flexible intermediate links were discussed based on the established model. The corresponding spectrum characteristics were studied using fast Fourier transform. Comparing the frequency characteristics obtained by theoretical model and modal experiment, it was found that the results obtained by the dynamic mathematical model are quite close to the test results. Less dynamic parameters make it convenient to carry out the control program.

Smooth adaptive sliding mode vibration control of a flexible parallel manipulator with multiple smart linkages in modal space

This paper addresses the dynamic model and active vibration control of a rigid-flexible parallel manipulator with three smart links actuated by three linear ultrasonic motors. To suppress the vibration of three flexible intermediate links under high speed and acceleration, multiple Lead Zirconium Titanate (PZT) sensors and actuators are collocated mounted on each link, forming a smart structure which can achieve self-sensing and self-actuating. The dynamic characteristics and equations of the flexible link incorporated with the PZT sensors and actuator are analyzed and formulated. The smooth adaptive sliding mode based active vibration control is proposed to suppress the vibration of the smart links, and the first and second modes of the three links are targeted to be suppressed in modal space to avoid the spillover phenomenon. Simulations and experiments are implemented to validate the effectiveness of the smart structures and the proposed control laws. Experimental results show that the vibration of the first mode around 92 Hz and the second mode around 240 Hz of the three smart links are reduced respectively by 64.98%, 59.47%, 62.28%, and 45.80%, 36.79%, 33.33%, which further verify the multi-mode vibration control ability of the smooth adaptive sliding mode control law.

Dynamic Model and Vibration Characteristics of Planar 3-RRR Parallel Manipulator with Flexible Intermediate Links considering Exact Boundary Conditions

Due to the complexity of the dynamic model of a planar 3-RRR flexible parallel manipulator (FPM), it is often difficult to achieve active vibration control algorithm based on the system dynamic model. To establish a simple and efficient dynamic model of the planar 3-RRR FPM to study its dynamic characteristics and build a controller conveniently, firstly, considering the effect of rigid-flexible coupling and the moment of inertia at the end of the flexible intermediate link, the modal function is determined with the pinned-free boundary condition. Then, considering the main vibration modes of the system, a high-efficiency coupling dynamic model is established on the basis of guaranteeing the model control accuracy. According to the model, the modal characteristics of the flexible intermediate link are analyzed and compared with the modal test results. The results show that the model can effectively reflect the main vibration modes of the planar 3-RRR FPM; in addition the model can be used to analyze the effects of inertial and coupling forces on the dynamics model and the drive torque of the drive motor. Because this model is of the less dynamic parameters, it is convenient to carry out the control program.

Dynamic Performance Evaluation of Serial and Parallel RPR Manipulators with Flexible Intermediate Links

This paper presents a comprehensive study of the workspace, dynamic characteristics and accuracy of three planar flexible manipulators with 3-RPR, 2-RPR and 1-RPR structures moving at high speed. A geometrical procedure is employed to obtain the workspaces of the manipulators. The flexible intermediate links are modeled as the Euler–Bernoulli beams with fixed-free boundary conditions based on the assumed mode method. Using the Lagrange multipliers, a generalized set of differential algebraic equations of motion is developed for the planar RPR manipulators. Three moving constraints, which are obtained from an inverse kinematics analysis and applied to the actuated base joints, impose the end-effector to follow a high-speed circular motion as the desired trajectory. From this analysis, the dynamic performance of 1-RPR flexible serial manipulator and 2-RPR and 3-RPR flexible parallel manipulators in tracking a desired trajectory is evaluated. Based on the results, it is concluded that, in addition to the specific structure of the manipulator, the accuracy generally depends on the operation conditions. The results contest the general assertion which claims that parallel manipulators have more accuracy and stiffness than serial counterparts.

A comparative study of elastic motions in trajectory tracking of flexible RPR planar manipulators moving with high speed

SUMMARYThe study of inertial forces effects at high speeds in flexible parallel manipulators, which generate undesired deviations, is a challenging task due to the coupled and complicated equations of motion. A dynamic model of the Revolute Prismatic Revolute (RPR) planar manipulators (specifically 3-RPR, 2-RPR and 1-RPR) with flexible intermediate links is developed based on the assumed mode method. The flexible intermediate links are modeled as Euler-Bernoulli beams with fixed-free boundary conditions. Using the Lagrange multipliers, a generalized set of differential algebraic equations (DAEs) of motion is developed. In the simulations, the rigid body motion of the end-effector is constrained by some moving constraint equations while the vibrations of the flexible intermediate links cause deviations from the desired trajectory. From this analysis, the dynamic performance of the manipulators when tracking a desired trajectory is evaluated. A comparison of the results indicates that in some cases, adding each extra RPR chain in the n-RPR planar manipulators with flexible intermediate links reduces the stiffness and accuracy due to the inertial forces of the flexible links, which is opposite to what would be expected. The study provides insights to the design, control and suitable selection of the flexible manipulators.

Dynamic characteristics of a 3-RPR planar parallel manipulator with flexible intermediate links

SUMMARYThis paper presents the dynamic modeling of a 3-RPR planar parallel manipulator with three flexible intermediate links in order to investigate the effects of the intermediate links flexibility on the undesired vibrations of the end-effector. For this purpose, the intermediate links are modeled as Euler--Bernoulli beams with two types of fixed-pinned and fixed-free boundary conditions based on the assumed mode method (AMM). The equations of motion of the 3-RPR manipulator are formulated using the augmented Lagrange multipliers method in the form of differential algebraic equations (DAEs) by incorporating the elastic and rigid coordinates in the set of generalized coordinates. After defining the initial conditions and imposing external forces to the manipulator, the equations are then solved numerically using the Modified Extended Backward-Differentiation Formula Implicit (MEBDFI) approach. Comparison of the simulation results for two different boundary conditions shows clearly the effects of flexibility of the intermediate links on the vibration of the end-effector trajectory. Results of this work can be used for the dynamic modeling of other manipulators or to design a controller for reducing the undesired vibrations.

Vibration Analysis of a New Type of Compliant Mechanism with Flexible‐Link, Using Perturbation Theory

Vibration analysis of a new type of compliant parallel mechanism with flexible intermediate links is investigated. The application of the Timoshenko beam theory to the mathematical modeling of the intermediate flexible link is described, and the equations of motion of the flexible links are obtained by using Lagrange’s equation of motion. The equations of motion are obtained in the form of a set of ordinary differential equations by using assumed mode method theory. The governing differential equations of motion are solved using perturbation method. The assumed mode shapes and frequencies are to be obtained based on clamped‐clamped boundary conditions. Comparing perturbation method with Runge‐Kutta‐Fehlberg 4, 5th leads to highly accurate solutions, and the results are performed and discussed.

MULTI-MODE VIBRATION CONTROL AND POSITION ERROR ANALYSIS OF PARALLEL MANIPULATOR WITH MULTIPLE FLEXIBLE LINKS

This paper presents multi-mode vibration control and analysis of moving platform position errors of a planar 3-PRR parallel manipulator with three flexible intermediate links using PZT transducers. The active vibration controller is designed in modal space with modal filters and modal synthesizers determined from the flexible link vibration characteristics. Estimation of the moving platform position error is conducted using measurements of the flexible link deflection from PZT sensors mounted on the flexible intermediate links. An effective strategy for determining the control gains to reduce the vibrations of higher order modes is proposed through modification of the independent modal space control (IMSC) method. The proposed independent modal control strategy is experimentally implemented with first two modes targeted for control on a parallel manipulator with multiple flexible links. The experimental results show that the vibrations of the first two modes are effectively suppressed, and the position error of the moving platform is substantially reduced.

Investigation of axial forces on dynamic properties of a flexible 3-PRR planar parallel manipulator moving with high speed

SUMMARYThe effect of axial forces on the dynamic properties is formulated and investigated for a 3-PRR planar parallel manipulator with three flexible intermediate links. A dynamic model of the manipulator system is developed based on the assumed mode method with the consideration of the effect of longitudinal forces on lateral stiffness is included. The flexible intermediate links are modeled as Euler–Bernoulli beams with pinned-pinned boundary conditions, which are verified by experimental modal tests. Natural frequencies of bending vibration of the intermediate links are derived as the functions of axial force and rigid-body motion of the manipulator. Dynamic behavior including the effect of axial forces on lateral deformation is investigated, and configuration-dependant frequencies are analyzed. Numerical simulations of configuration-dependent frequency properties and axial forces are performed to illustrate the effect of axial forces on the dynamic behaviors of the flexible parallel manipulator. Simulation results of mode amplitudes, deformations, axial forces, inertial, and coupling forces are presented, and further validate the theoretical derivations. These analyses and results provide a new and valuable insight to the design and control of the parallel manipulators with flexible intermediate links.

Flexible linkage structural vibration control on a 3-PRR planar parallel manipulator: Experimental results

Piezoelectric actuators and sensors have been used for vibration control of flexible manipulators and mechanisms since the 1990s. However, very few attempts have been made toward experimental investigations compared with numerical simulations, especially for mechanisms and manipulators with multiple flexible links. This paper presents an experimental study on active vibration control of a moving 3-PRR parallel manipulator with three flexible intermediate links with bonded lead zirconate titanate (PZT) actuators and sensors. First, experimental modal tests are conducted with an impulse force hammer and an accelerometer to identify structural vibration mode shapes and natural frequencies of the flexible manipulator. These modal test results are used to provide guidance for the determination of the location of PZT transducers and the design of an active vibration controller. Then, a strain rate feedback (SRF) controller is designed in modal space. A state-space model is formulated with the control input voltage applied to PZT actuators, and output generated voltage from PZT sensors. The design of an optimal active vibration controller is addressed based on SRF. Furthermore, to suppress the noise introduced through the differentiation of the strain signal, a second-order auxiliary compensator is introduced to the SRF controller. The design of the SRF controller and the noise attenuation compensator is addressed utilizing vibration theories. Finally, active vibration control experiments are implemented to demonstrate that the proposed active vibration control strategy is effective. Power spectral density plots of vibrations illustrate that the structural vibration of flexible links is suppressed effectively when the proposed SRF vibration control strategy is employed.

Coupling characteristics of rigid body motion and elastic deformation of a 3-PRR parallel manipulator with flexible links

Modeling of multibody dynamics with flexible links is a challenging task, which not only involves the effect of rigid body motion on elastic deformations, but also includes the influence of elastic deformations on rigid body motion. This paper presents coupling characteristics of rigid body motions and elastic motions of a 3-PRR parallel manipulator with three flexible intermediate links. The intermediate links are modeled as Euler–Bernoulli beams with pinned-pinned boundary conditions based on the assumed mode method (AMM). Using Lagrange multipliers, the fully coupled equations of motions of the flexible parallel manipulator are developed by incorporating the rigid body motions with elastic motions. The mutual dependence of elastic deformations and rigid body motions are investigated from the analysis of the derived equations of motion. Open-loop simulation without joint motion controls and closed-loop simulation with joint motion controls are performed to illustrate the effect of elastic motion on rigid body motions and the coupling effect amongst flexible links. These analyses and results provide valuable insight to the design and control of the parallel manipulator with flexible intermediate links.

Vibration control of elastodynamic response of a 3-PRR flexible parallel manipulator using PZT transducers

SUMMARYThis paper addresses the dynamic simulation and control of structural vibrations of a 3-PRR parallel manipulator with three flexible intermediate links, to which are bonded lead zirconate titanate (PZT) actuators and sensors. Flexible intermediate links are modelled as Euler–Bernoulli beams with pinned-pinned boundary conditions. A PZT actuator controller is designed based on strain rate feedback (SRF) control. Control moments from PZT actuators are transformed to force vectors in modal space and are incorporated in the dynamic model of the manipulator. The dynamic equations are developed based on the assumed mode method for the flexible parallel manipulator with multiple PZT actuator and sensor patches. Numerical simulation is performed and the results indicate that the proposed active vibration control strategy is effective. Spectral analyses of structural vibrations further illustrate that deformations from structural vibration of flexible links are suppressed to a significant extent when the proposed vibration control strategy is employed, while the deflections caused by inertial and coupling forces are not reduced.

Dynamic Modeling and Experimental Validation of a 3-PRR Parallel Manipulator with Flexible Intermediate Links

This paper presents the development of structural dynamic equations of motion for a 3-PRR parallel manipulator with three flexible intermediate links, based on the assumed mode method. Lagrange's equation is used to derive the dynamic model of the manipulator system. Flexible intermediate links are modeled as Euler---Bernoulli beams with pinned---pinned boundary conditions. Dynamic equations of motion of a 3-PRR parallel manipulator with three flexible links are developed by adopting the assumed mode method. The effect of concentrated rotational inertia at both ends of intermediate links is included in this model. Numerical simulations of vibration responses, coupling forces and inertial forces are presented. The corresponding frequency spectra analysis is performed using the Fast Fourier Transform (FFT). Experimental modal tests are performed to validate the theoretical model through comparison and analysis of modal characteristics of the flexible manipulator system.