3-PRR Planar Research Articles

Vibration Modeling and Analysis of a Flexible 3-PRR Planar Parallel Manipulator Based on Transfer Matrix Method for Multibody System

This paper presents the vibration model of a 3-prismatic–revolute–revolute (PRR) planar parallel manipulator (PPM) with three flexible intermedia links, utilizing the linear transfer matrix method for multibody systems (MSTMM). The dynamic characteristics of the PRR PPM are also investigated. The dynamic model of the 3-PRR PPM is derived, and the transfer matrix and transfer equation of each component in the system, as well as the overall transfer equation and transfer matrix of the system are obtained. The vibration characteristics of the whole system are determined using the MSTMM and verified through ANSYS simulation. Furthermore, the relationship between the natural frequencies and the flexible PPM configurations is analyzed under a specific circular trajectory. The results demonstrate that the natural frequency of the system changes constantly with the configurations, and the trends of the first six orders are similar. This novel modeling approach does not require global dynamic equations and is both efficient and accurate. Moreover, it can be easily extended to other parallel manipulators with flexible components.

A Novel Semi-Supervised Graph-Guided Approach for Intelligent Health State Diagnosis of a 3-PRR Planar Parallel Manipulator

As one of the most important operation and maintenance approaches, health state diagnosis technology plays a crucial role in ensuring the safety and reliability of mechanical equipment. The planar parallel manipulator, as a typical actuator, is widely employed in the field of precision manufacturing due to its advantages of high stiffness, large load support capability, and high precision. However, compared with common key functional components (such as bearings and gearboxes), planar parallel manipulators have more complicated operating mechanisms and failure behaviors. To satisfy the health state diagnosis demands of planar parallel manipulators in the scenario of insufficient label information, a novel intelligent health state diagnosis approach, termed semi-supervised graph-guided network with perception attention (SGN-PA), is developed for a 3-PRR (P and R represent prismatic and revolute pairs, respectively) planar parallel manipulator. Specifically, an improved multiorder graph perception module is constructed to extract multiscale feature information, and achieve feature fusion by combining perceptual attention mechanism, which enables the proposed SGN-PA model to have adaptation adjustment capabilities. Following that, local and nonlocal feature constraint strategies are employed with pseudo-label technology to reduce intraclass differences and maximize interclass differences, and then to fit the demands of health state diagnosis tasks. Eventually, based on the simulation and experimental scenarios of a 3-PRR planar parallel manipulator, the effectiveness and feasibility of the proposed SGN-PA model is extensively confirmed, and the diagnosis results show that it can significantly relax the constraints of label information while maintaining superior performances.

Multiscale Graph-Guided Convolutional Network With Node Attention for Intelligent Health State Diagnosis of a 3-PRR Planar Parallel Manipulator

The data-driven intelligent health state diagnosis strategy has been successfully applied in modern industrial equipments. However, many of existing research works suffer from two major deficiencies: First, the sample independence assumption is widely adopted, so the influences of the relationship between samples on the overall performances are not explored. Second, most of the abovementioned application objects are typical key functional components (such as bearings, gearboxes, etc.), and research works on the intelligent health state diagnosis of planar parallel manipulator are rarely reported. To address these issues, a novel intelligent health state diagnosis method, termed multiscale graph-guided convolutional network with node attention (MSGCN-NA), is proposed for a 3-PRR (P and R represent prismatic and revolute pairs, respectively) planar parallel manipulator. Specifically, the developed MSGCN-NA model mainly contains the following two parts: first, the one is an unsupervised convolutional autoencoder, which is employed for the extraction of deep representation features, and then combined with Pearson metric to establish the adjacency matrix. Second, The other part is the constructed multiscale graph convolutional network, in which the node attention mechanism is adopted to achieve cross-scale fusion of different neighborhood information. The effectiveness of the proposed MSGCN-NA method is fully verified based on the simulation and experimental scenarios, the results show that MSGCN-NA can achieve superior diagnosis performances.

Self-excited vibration of a 3-PRR planar parallel robot

Self-excited vibration of parallel robots can seriously affect the motion performance and damage the mechanical structure. In order to study the self-excited vibration characteristics of a 3-PRR (where P and R represent the prismatic and revolute joints respectively and the underlined letter represents the actuated joint) planar parallel robot, dynamic model is firstly established and the singularity is analyzed theoretically. Then the dynamic characteristics at internal and boundary singularities are both explored experimentally. The motion form and generating mechanism of the self-excited vibration are researched. The influencing factors on vibration frequency are obtained and the self-excited vibration during trajectory tracking motion is analyzed. Finally, a singularity escaping strategy is proposed and tested. Theoretical analysis and experimental results show that the performance of parallel robot deteriorates dramatically at singularities. Nevertheless, the parallel robot can pass through the internal singularity successfully with optimized load and motion speed so that the workspace can be expanded. The 3-PRR planar parallel robot exhibits self-excited vibration at internal singularities, which is mainly caused by the singularity, self-regulation of motors and the closed-chain coupling effect. The vibration frequency is mainly determined by singular configuration and the structural parameters. The parallel robot can maintain self-excited vibration state while carrying out trajectory tracking under a singular attitude angle. Moreover, the proposed singularity escaping strategy is verified to be feasible so that the self-excited vibration can be eliminated effectively.

A Double-Layered Elbow Exoskeleton Interface With 3-PRR Planar Parallel Mechanism for Axis Self-Alignment

Abstract Designing a mechanism for elbow self-axis alignment requires the elimination of undesirable joint motion and tissue elasticity. The novelty of this work lies in proposing a double-layered interface using a 3-PRR planar parallel mechanism as a solution to the axis alignment problem. 3-PRR planar parallel mechanisms are suitable candidates to solve this as they can span the desired workspace in a relatively compact size. In this paper, we present the modeling, design, prototyping, and validation of the double-layered elbow exoskeleton interface for axis self-alignment. The desired workspace for the self-axis alignment mechanism is specified based on the estimated maximum possible misalignment between the exoskeleton joint and the human anatomical elbow joint. Kinematic parameters of the 3-PRR planar mechanism are identified by formulating an optimization problem. The goal is to find the smallest mechanism that can span the specified workspace. The orientation angle of the mechanism’s plane addresses the frontal frustum vertex angle of the elbow’s joint, while the translational motion allows the translational offsets between the user’s elbow and the exoskeleton joint. The designed exoskeleton axis can passively rotate around the frontal plane ±15 deg and translate along the workspace 30 mm in the frontal plane. Experimental results (quantitative and qualitative) confirmed the capability of the proposed exoskeleton in addressing the complex elbow motion, user’s satisfaction, and ergonomics.

Integrating Dynamics into Design and Motion Optimization of a 3-PRR Planar Parallel Manipulator with Discrete Time Transfer Matrix Method

This paper presents a novel method of dynamic modeling and design optimization integrated with dynamics for parallel robot manipulators. Firstly, a computationally efficient modeling method, the discrete time transfer matrix method (DT-TMM), is proposed to establish the dynamic model of a 3-PRR planar parallel manipulator (PPM) for the first time. The numerical simulations are performed with both the proposed DT-TMM dynamic modeling and the ADAMS modeling. The applicability and effectiveness of DT-TMM in parallel manipulators are verified by comparing the numerical results. Secondly, the design parameters of the 3-PRR parallel manipulator are optimized using the kinematic performance indices, such as global workspace conditioning index (GWCI), global condition index (GCI), and global gradient index (GGI). Finally, a dynamic performance index, namely, driving force index (DFI), is proposed based on the established dynamic model. The described motion trajectory of the moving platform is placed into the optimized workspace and the initial position is determined to finalize the end-effector trajectory of the parallel manipulator by the further optimization with the integrated kinematic and dynamic performance indices. The novelty of this work includes (1) developing a new dynamic model method with high computation efficiency for parallel robot manipulators using DT-TMM and (2) proposing a new dynamic performance index and integrating the dynamic index into the motion and design optimization of parallel robot manipulators.

Genetic and hybrid algorithms for optimization of non-singular 3PRR planar parallel kinematics mechanism for machining application

SUMMARYThis paper proposes a special non-symmetric topology of a 3PRR planar parallel kinematics mechanism, which naturally avoids singularity within the workspace and can be utilized for hybrid kinematics machine tools. Subsequently, single-objective and multi-objective optimizations are conducted to improve the performance. The workspace area and minimum eigenvalue, as well as the condition number of the homogenized Cartesian stiffness matrix across the workspace, have been chosen as the objectives in the optimization based on their relevance to the machining application. The single-objective optimization is conducted by using a single-objective genetic algorithm and a hybrid algorithm, whereas the multi-objective optimization is conducted by using a multi-objective genetic algorithm, a weighted sum single-objective genetic algorithm, and a weighted sum hybrid algorithm. It is shown that the single-objective optimization gives superior value in the optimized objective, while sacrificing the other objectives, whereas the multi-objective optimization compromises the improvement of all objectives by providing non-dominated values. In terms of the algorithms, it is shown that a hybrid algorithm can either verify or refine the optimal value obtained by a genetic algorithm.

Kinetostatic modelling of a 3-PRR planar compliant parallel manipulator with flexure pivots

Flexure pivots are widely used in long stroke complaint parallel manipulators (CPMs) for their large deformation capacity. However the parasitic motion caused by the flexure pivots cannot be neglected as the rotation angle gets large, which will finally affect the absolute positioning accuracy of the manipulators. In this paper, a modelling approach to calculate the nonlinear kinetostatics of a long stroke planar CPM with flexure pivots is proposed, in which the parasitic motion of the flexure joints are taken into account. The displacement constraint equations and the static equilibrium equations of the CPM are established under the deformed configuration. Finite element analysis (FEA) shows the accuracy and efficiency of the nonlinear kinematic model (NLKM) for both the inverse and forward kinematic analysis. Moreover, the prediction accuracy of the CPM’s kinematic behavior has been improved more than 20 times by employing the NLKM compared with the conventional kinematic model (CKM).

Determination of the maximal singularity-free workspace of 3-DOF parallel mechanisms with a constructive geometric approach

This paper proposes a novel approach to obtain the maximal singularity-free regions of planar parallel mechanisms, which is based on a constructive geometric reasoning. The proposed approach consists of two algorithms. First, the borders of the singularity-free region associated with an arbitrary starting point of the moving platform are obtained. Then, the second algorithm finds the center of the maximal singularity-free circle, which is obtained using the so-called offset curve algorithm. The procedure is applied to a 3-PRR planar parallel mechanism as an example and the obtained results illustrate graphically the effectiveness of the proposed algorithm. The proposed approach can be directly applied to obtain the maximal singularity-free circle of similar parallel mechanisms, which is not the case for other approaches proposed in the literature that are limited to a given parallel mechanism, namely, the 3-RPR planar parallel mechanism. Moreover, as the main feature of the proposed approach, it can be implemented both in a CAD system and in a computer algebra system where non-convex and reentrant curves can be considered.

Precision Positioning of a 3-PRR Planar Parallel Manipulator Driven by Linear Ultrasonic Motors Based on Machine Vision

A 3-PRR planar parallel manipulator actuated by liner ultrasonic motor is set up and investigated. The structure of the planar parallel manipulator(PPM) is presented, and then the kinematic model of the manipulator is developed based on vector chain method. To realize the global closed-loop feedback, charge coupled device camera is adopted to measure the output of the manipulator, namely the 3 degrees of freedom of the manipulator. By using the technologies of image processing, the center of the artificial features fixed on the moving platform is located to realize the pose measurement of the moving platform. A perturbation observer-based joint control is adopted to overcome the disadvantage of the traditional PID control in real time control. Experimental results show that the accuracy of the semi-closed loop control of the PPM is around 500 μm while the target position is(20 mm,20 mm), but after the procedure of the visual based closed loop control, the positioning accuracy is further improved to ±5 μm. Compared with the semi-closed loop control, the visual based positioning control provides a substantial increasing in positioning accuracy of the PPM, which could satisfy the positioning requirements of the PPM.

A Method for Solving Dynamic Equations of a 3-PRR Parallel Robot

In this paper, kinematic relationships for a 3-PRR planar parallel robot are first presented. The robot dynamics equations are formulated using Lagrange equations of first kind. The derived equations are a mixed set of differential and algebraic constraint equations, DAE, which must be satisfied simultaneously. In order to solve the robot dynamic equations, a new method is presented in which the dynamics equation is first partitioned into two parts. The constraint equations and the dependent coordinates are next eliminated. This reduces the dynamic equations to a set of differential equations as a function of three independent coordinates. Finally, a trajectory for the robot end-effector is specified and PD controller which follows the desired trajectory is implemented. The proposed method significantly simplifies the solution of the dynamics equations.

Choosing Actuation Scheme for Optimal Performance of 3-DOF Planar Parallel Manipulators

This paper describes how the actuation scheme affects the size and shape of the workspace of 3RPR and 3PRR, which are the two popular planar parallel manipulators. This study helps in achieving the required singularity-free zones for a specific task by adopting a suitable actuation scheme without resorting to changing the architecture of the manipulator. In addition, the performance index GCI for each actuation scheme has been evaluated and compared with each other. Finally, the GCI for each actuation scheme has been optimized subjected to the geometric constraints for the 3PRR planar parallel manipulator.

Inverse dynamics of a 3-prismatic—revolute—revolute planar parallel manipulator using natural orthogonal complement

The performance of robotic systems with parallel kinematics can be evaluated by their kinematic, static, and dynamic properties. These properties are directly used in model-based controllers which potentially offer higher accuracy for robotic systems. Inverse dynamic solution is an essential part of these controllers. In the present work, the inverse dynamics model of a 3-PRR (prismatic—revolute—revolute) planar parallel manipulator based on the natural orthogonal complement (NOC) method is developed. To drive the NOC for the 3-PRR closed-loop systems, the explicit expressions of the loop constraints equations and the associated Jacobian matrices are first obtained. Next the NOC matrix, which is a velocity transformation matrix relating the Cartesian angular/translational velocities of various bodies to the motor joint rates, is calculated. Finally results of the NOC method are compared with simulation of a 3-PRR planar parallel manipulator using two commercial softwares: SimMechanics toolbox of Matlab and COSMOSMotion of SolidWorks. In order to verify the theoretical results, two different configurations for the robot are considered: a horizontal and a vertical. Results of the NOC method as well as the two simulations are compared for the two robot configurations.

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.

Inverse dynamics of the 3-PRR planar parallel robot

Recursive modelling for the kinematics and dynamics of the known 3- PRR planar parallel robot is established in this paper. Three identical planar legs connecting to the moving platform are located in a vertical plane. Knowing the motion of the platform, we develop first the inverse kinematics and determine the positions, velocities and accelerations of the robot. Further, the principle of virtual work is used in the inverse dynamics problem. Several matrix equations offer iterative expressions and graphs for the power requirement comparison of each of three actuators in two different actuation schemes: prismatic actuators and revolute actuators. For the same evolution of the moving platform in the vertical plane, the power distribution upon the three actuators depends on the actuating configuration, but the total power absorbed by the set of three actuators is the same, at any instant, for both driving systems. The study of the dynamics of the parallel mechanisms is done mainly to solve successfully the control of the motion of such robotic systems.

Kinematic analysis and path planning of a new kinematically redundant planar parallel manipulator

SUMMARYIn this work, the 3-RPRR, a new kinematically redundant planar parallel manipulator with six-degrees-of-freedom, is presented. First, the manipulator is introduced and its inverse displacement problem discussed. Then, all types of singularities of the 3-RPRR manipulator are analysed and demonstrated. Thereafter, the dexterous workspace is geometrically obtained and compared with the non-redundant 3-PRR planar parallel manipulator. Finally, based on a geometrical measure of proximity to singular configurations and the condition number of the manipulators' Jacobian matrices, actuation schemes for the manipulators are obtained. Different actuation schemes for a given path are obtained and the quality of their actuation schemes are compared. It is shown that the proposed manipulator is capable of following a path while avoiding the singularities.

FORCE-UNCONSTRAINED POSES OF THE 3-PRR AND 4-PRR PLANAR PARALLEL MANIPULATORS

Force-unconstrained (singular) poses of the 3-PRR planar parallel manipulator (PPM), where the underscore indicates the actuated joint, and the 4-PRR, a redundant PPM with an additional actuated branch, are presented. The solution of these problems is based upon concepts of reciprocal screw quantities and kinematic analysis. In general, non-redundant PPMs such as the 3-PRR are known to have two orders of infinity of force-unconstrained poses, i.e., a three-variable polynomial in terms of the task-space variables (position and orientation of the mobile platform). The inclusion of redundant branches eliminates one order of infinity of force-unconstrained configurations for every actuated branch beyond three. The geometric identification of force-unconstrained poses is carried out by assuming one variable for each order of infinity. In order to simplify the algebraic procedure of these problems, the assumed or “free” variables are considered to be joint displacements. For both manipulators, an effective elimination technique is adopted. For the 3-PRR, the roots of a 6th-order polynomial determine the force-unconstrained poses, i.e., surfaces in a three dimensional space defined by the task-space variables. For the 4-PRR, a 64th-order polynomial determines curves of force-unconstrained poses in the same dimensional space.