Compliance Modeling and Analysis of a 3-RPS Parallel Kinematic Machine Module

The compliance modeling and rigidity performance evaluation for the lower mobility parallel manipulators are still to be remained as two overwhelming challenges in the stage of conceptual design due to their geometric complexities. By using the screw theory, this paper explores the compliance modeling and eigencompliance evaluation of a newly patented 1T2R spindle head whose topological architecture is a 3-RPS parallel mechanism. The kinematic definitions and inverse position analysis are briefly addressed in the first place to provide necessary information for compliance modeling. By considering the 3-RPS parallel kinematic machine（PKM） as a typical compliant parallel device, whose three limb assemblages have bending, extending and torsional deflections, an analytical compliance model for the spindle head is established with screw theory and the analytical stiffness matrix of the platform is formulated. Based on the eigenscrew decomposition, the eigencompliance and corresponding eigenscrews are analyzed and the platform＇s compliance properties are physically interpreted as the suspension of six screw springs. The distributions of stiffness constants of the six screw springs throughout the workspace are predicted in a quick manner with a piece-by-piece calculation algorithm. The numerical simulation reveals a strong dependency of platform＇s compliance on its configuration in that they are axially symmetric due to structural features. At the last stage, the effects of some design variables such as structural, configurational and dimensional parameters on system rigidity characteristics are investigated with the purpose of providing useful information for the structural design and performance improvement of the PKM. Compared with previous efforts in compliance analysis of PKMs, the present methodology is more intuitive and universal thus can be easily applied to evaluate the overall rigidity performance of other PKMs with high efficiency.

A Finite and Instantaneous Screw Based Approach for Topology Design and Kinematic Analysis of 5-Axis Parallel Kinematic Machines

Unifying the models for topology design and kinematic analysis has long been a desire for the research of parallel kinematic machines(PKMs). This requires that analytical description, formulation and operation for both finite and instantaneous motions are performed by the same mathematical tool. Based upon finite and instantaneous screw theory, a unified and systematic approach for topology design and kinematic analysis of PKMs is proposed in this paper. Using the derivative mapping between finite and instantaneous screws built in the authors’ previous work, the finite and instantaneous motions of PKMs are analytically described by the simple and non?redundant screws in quasi?vector and vector forms. And topological and parametric models of PKMs are algebraically formulated and related. These related topological and parametric models are ready to do type synthesis and kinematic analysis of PKMs under the unified framework of screw theory. In order to show the validity of the proposed approach, a kind of two?translational and three?rotational(2T3R)5?axis PKMs is taken as example. Numerous new structures of the 2T3R PKMs are synthe?sized as the results of topology design, and their Jacobian matrix is obtained easily for parameter optimization and performance evaluation. Some of the synthesized PKMs have outstanding capabilities in terms of large workspaces and flexible orientations, and have great potential for industrial applications of machining and manufacture. Among them, METROM PKM is a typical example which has attracted a lot of attention from global companies and already been developed as commercial products. The approach is a general and unified approach that can be used in the innovative design of different kinds of PKMs.

Kinetostatic Modeling and Analysis of an Exechon Parallel Kinematic Machine(PKM) Module

As a newly invented parallel kinematic machine（PKM）, Exechon has found its potential application in machining and assembling industries due to high rigidity and high dynamics. To guarantee the overall performance, the loading conditions and deflections of the key components must be revealed to provide basic mechanic data for component design. For this purpose, a kinetostatic model is proposed with substructure synthesis technique. The Exechon is divided into a platform subsystem, a fixed base subsystem and three limb subsystems according to its structure. By modeling the limb assemblage as a spatial beam constrained by two sets of lumped virtual springs representing the compliances of revolute joint, universal joint and spherical joint, the equilibrium equations of limb subsystems are derived with finite element method（FEM）. The equilibrium equations of the platform are derived with Newton＇s 2nd law. By introducing deformation compatibility conditions between the platform and limb, the governing equilibrium equations of the system are derived to formulate an analytical expression for system＇s deflections. The platform＇s elastic displacements and joint reactions caused by the gravity are investigated to show a strong position-dependency and axis-symmetry due to its kinematic and structure features. The proposed kinetostatic model is a trade-off between the accuracy of FEM and concision of analytical method, thus can predict the kinetostatics throughout the workspace in a quick and succinct manner. The proposed modeling methodology and kinetostatic analysis can be further expanded to other PKMs with necessary modifications, providing useful information for kinematic calibration as well as component strength calculations.

Comparison of Parallel Kinematic Machines with Three Translational Degrees of Freedom and Linear Actuation

The development of new robot structures, in particular of parallel kinematic machines（PKM）, is widely systematized by different structure synthesis methods. Recent research increasingly focuses on PKM with less than six degrees of freedom（DOF）. However, an overall comparison and evaluation of these structures is missing. In order to compare symmetrical PKM with three translational DOF, different evaluation criteria are used. Workspace, maximum actuation forces and velocities, power, actuator stiffness, accuracy and transmission behavior are taken into account to investigate strengths and weaknesses of the PKMs. A selection scheme based on possible configurations of translational PKM including different frame configurations is presented. Moreover, an optimization method based on a genetic algorithm is described to determine the geometric parameters of the selected PKM for an exemplary load case and a prescribed workspace. The values of the mentioned criteria are determined for all considered PKM with respect to certain boundary conditions. The distribution and spreading of these values within the prescribed workspace is presented by using box plots for each criterion. Thereby, the performance characteristics of the different structures can be compared directly. The results show that there is no ＂best＂ PKM. Further inquiries such as dynamic or stiffness analysis are necessary to extend the comparison and to finally select a PKM.

KINEMATIC DESIGN OF A RECONFIGURABLE MINIATURE PARALLEL KINEMATIC MACHINE

The kinematic design of a reconfigurable miniature parallel kinematic machineis dealt with. It shows that the reconfigurability may be realized by packaging a tripod-basedparallel mechanism with fixed length struts into a compact and rigid frame with which the differentconfigurations can be formed. Utilizing a dual parameter model, the influences of the geometricalparameters on the dexterous performance and the workspace/machine volume ratio are investigated. Anovel global performance index for the dimensional synthesis is proposed and optimized, resulting ina set of dimensionless geometrical parameters.

Running Accuracy Analysis of a 3-RRR Parallel Kinematic Machine Considering the Deformations of the Links

Parallel kinematic machines have drawn considerable attention and have been widely used in some special fields.However,high precision is still one of the challenges when they are used for advanced machine tools.One of the main reasons is that the kinematic chains of parallel kinematic machines are composed of elongated links that can easily suffer deformations,especially at high speeds and under heavy loads.A 3-RRR parallel kinematic machine is taken as a study object for investigating its accuracy with the consideration of the deformations of its links during the motion process.Based on the dynamic model constructed by the Newton-Euler method,all the inertia loads and constraint forces of the links are computed and their deformations are derived.Then the kinematic errors of the machine are derived with the consideration of the deformations of the links.Through further derivation,the accuracy of the machine is given in a simple explicit expression,which will be helpful to increase the calculating speed.The accuracy of this machine when following a selected circle path is simulated.The influences of magnitude of the maximum acceleration and external loads on the running accuracy of the machine are investigated.The results show that the external loads will deteriorate the accuracy of the machine tremendously when their direction coincides with the direction of the worst stiffness of the machine.The proposed method provides a solution for predicting the running accuracy of the parallel kinematic machines and can also be used in their design optimization as well as selection of suitable running parameters.

Homing Strategy for a 4RRR Parallel Kinematic Machine

Returning home is the most important process of a parallel kinematic machine （PKM） with incremental encoders.Currently,most corresponding articles focus on the accuracy of homing process,and there lacks the investigation of the operation＇s safety.For a 4RRR PKM,all servoaxes would be independently driven to their zero positions at the same time based on the traditional homing mode,and that can bring serious interfere of the kinematic chains.This paper systemically investigates this 4RRR PKM＇s safety of homing process.A homing strategy usually contains three parts which are the home switches＇ locations,the platform＇s initial moving space,and each links＇ homing direction,and all of them can influence the safety of homing operation.For the purpose of evaluating and describing the safety of the homing strategy,some important parameters are introduced as follows：Safely homing ratio （SHR） is used to evaluate the probability of a machine＇s successfully returning home from an initial moving space;Synchronal rotational angle （SRA） is the four links＇ largest synchronal rotational angle with given directions from a given pose.Whether a machine can safely return home from a given pose can be judged by comparing the SRA with all four home switches＇ mounting angles.By meshing the initial moving space and checking the safeties of returning home from all the initial poses on the nodes,the SHR of this initial moving space can be calculate.For the sake of convenience,the platform＇s initial moving space should be as large as possible,and in this 4RRR PKM,a square zone in the center of the workspace with a giving initial rotation range is selected as the platform＇s initial moving space.The forward direction is selected as each link＇s homing direction according to custom,and the platform＇s initial rotational angle is selected as larger than 0° based on this 4RRR PKM＇s kinematic characteristics.The platform＇s initial moving space can be defined only by the side length of the initial moving square.By setting a probable searching step and calculating the SHR of the initial moving square,an optimal procedure of searching for the largest side length of the platform＇s initial moving square is proposed.The homing strategy proposed is based on a systemic research on the safety of homing process for PKM,and the two new indexes SHR and SRA can clearly describe the safety of homing operation.The homing operation based on this strategy is fast and safe,and the method can also be used in other PKMs with the situation of serious components＇ interference.

Calibration of a Parallel Kinematic Machine Tool

A calibration method is presented to enhance the static accuracy of a parallel kinematic machine tool by using a coordinate measuring machine and a laser tracker. According to the established calibration model and the calibration experiment, the factual 42 kinematic parameters of BKX-I parallel kinematic machine toot are obtained. By circular tests the comparison is made between the calibrated and the uncalihrated parameters and shows that there is 80 % improvement in accuracy of this machine tool.

Robust decoupling control of a parallel kinematic machine using the time-delay estimation technique

This paper proposes a robust decoupling control scheme using a time-delay estimation technique for a parallel kinematic machine to enhance its trajectory tracking performance.The dynamic model of a parallel kinematic machine(PKM)is a multivariable nonlinear strongly coupled system that is always affected by uncertainties and external disturbances.The proposed controller employs the time-delay estimation(TDE)technique to estimate the dynamic model of a PKM with uncertainties and disturbances,thus obtaining a simple model structure.The TDE technique involves estimating the unknown system dynamics by intentionally using a time-delayed signal,which will inevitably lead to estimation errors.Hence,the proposed controller effectively reduces the unfavourable TDE error by combining fast and robust integral terminal sliding mode control with TDE(TDE-ITSMC).In turn,the TDE technique can reduce the upper bound on the switching gain in the sliding mode control(SMC)scheme,which reduces damage to the robot.Finally,comparative experimental studies with other controllers confirm that TDEITSMC offers excellent trajectory tracking accuracy and is a practical robust control scheme for PKMs.

Error analysis and optimization of a 3-degree of freedom translational Parallel Kinematic Machine

In this paper, error modeling and analysis of a typical 3-degree of freedom translational Parallel Kine- matic Machine is presented. This mechanism provides translational motion along the Cartesian X-, Y- and Z- axes. It consists of three limbs each having an arm and forearm with prismatic-revolute-revolute-revolute joints. The moving or tool platform maintains same orientation in the entire workspace due to its joint arrangement. From inverse kinematics, the joint angles for a given position of tool platform necessary for the error modeling and analysis are obtained. Error modeling is done based on the differentiation of the inverse kinematic equations. Variation of pose errors along X, Y and Z directions for a set of dimensions of the parallel kinematic machine is presented. A non-dimensional performance index, namely, global error transformation index is used to study the influence of dimensions and its corresponding global maximum pose error is reported. An attempt is made to find the optimal dimensions of the Parallel Kinematic Machine using Genetic Algorithms in MATLAB. The methodology presented and the results obtained are useful for predicting the performance capability of the Parallel Kinematic Machine under study.

Separation of Comprehensive Geometrical Errors of a 3-DOF Parallel Manipulator Based on Jacobian Matrix and Its Sensitivity Analysis with Monte-Carlo Method

Parallel kinematic machines （PKMs） have the advantages of a compact structure,high stiffness,a low moving inertia,and a high load/weight ratio.PKMs have been intensively studied since the 1980s,and are still attracting much attention.Compared with extensive researches focus on their type/dimensional synthesis,kinematic/dynamic analyses,the error modeling and separation issues in PKMs are not studied adequately,which is one of the most important obstacles in its commercial applications widely.Taking a 3-PRS parallel manipulator as an example,this paper presents a separation method of source errors for 3-DOF parallel manipulator into the compensable and non-compensable errors effectively.The kinematic analysis of 3-PRS parallel manipulator leads to its six-dimension Jacobian matrix,which can be mapped into the Jacobian matrix of actuations and constraints,and then the compensable and non-compensable errors can be separated accordingly.The compensable errors can be compensated by the kinematic calibration,while the non-compensable errors may be adjusted by the manufacturing and assembling process.Followed by the influence of the latter,i.e.,the non-compensable errors,on the pose error of the moving platform through the sensitivity analysis with the aid of the Monte-Carlo method,meanwhile,the configurations of the manipulator are sought as the pose errors of the moving platform approaching their maximum.The compensable and non-compensable errors in limited-DOF parallel manipulators can be separated effectively by means of the Jacobian matrix of actuations and constraints,providing designers with an informative guideline to taking proper measures for enhancing the pose accuracy via component tolerancing and/or kinematic calibration,which can lay the foundation for the error distinguishment and compensation.

Dynamic Modeling and Eigenvalue Evaluation of a 3-DOF PKM Module

Due to the structural complexity, the dynamic modeling and quick performance evaluation for the parallel kinematic machines （PKMs） are still to be remained as two challenges in the stage of conceptual design. By using the finite element method and substructure synthesis, this paper mainly deals with the dynamic modeling and eigenvalue evaluation of a novel 3-DOF spindle head named the A3 head. The topological architecture behind the proposed A3 head is a 3-RPS parallel mechanism, which possesses one translational and two rotational capabilities. The mechanical features of the A3 head are briefly addressed in the first place followed by inverse position analysis. In the dynamic modeling, the platform is treated as a rigid body, the RPS limbs as the continuous uniform beams and the joints as lumped virtual springs. With the combination of substructure synthesis and finite element method, an analytical approach is then proposed to formulate the governing equations of motion of system using the compatibility conditions at interface between the limbs and the platform. Consequently, by solving the eigenvalue problem of the governing equations of motion, the distribution of lower natural frequencies of the A3 head throughout the entire workspace can be predicted in a quick manner. Modal analysis for the A3 head reveals that the distributions of lower natural frequencies are strongly related to the mechanism configuration and are axially symmetric due to system kinematic and structural features. The sensitivity analysis of the system indicates that the dimensional parameters of the 3-RPS mechanism have a slight effect on system lower natural frequencies while the joint compliances affect the distributions of lower natural frequencies significantly. The proposed dynamic modeling method can also be applied to other PKMs and can effectively evaluate the PKM＇s dynamic performance throughout the entire workspace.

Parallel Mechanisms with Two or Three Degrees of Freedom

Parallel manipulators for the machine tool industry have been studied extensively for various industrial applications. However, limited useful workspace areas, the poor mobility, and design difficulties of more complex parallel manipulators have led to more interest in parallel manipulators with less than six degrees of freedom (DoFs). Several parallel mechanisms with various numbers and types of degrees of freedom are described in this paper, which can be used in parallel kinematics machines, motion simulators, and industrial robots.

Dynamic modeling and analysis of the 3-PRS power head based on the screw theory and rigid multipoint constraints

This study presents a dynamic modeling and analysis methodology for the 3-PRS parallel mechanism.First,an improved reduced dynamic model of component substructures is proposed using the dynamic condensation technique and the rigid multipoint constraints at the joint/interface level,leading to a minimum set of generalized coordinates for external nodes.Next,the mapping between interface constraint stiffness and global stiffness is illustrated,resulting in an analytical stiffness model of joint substructures.Subsequently,the derived component and joint substructures are synthesized into the entire mechanism based on the Lagrange equation.Finally,a case study illustrates that the lower-order dynamic performances predicted within the proposed approach have the same trend as those obtained from a complete-order finite element model.The root mean square discrepancy of the lower-order natural frequencies between the two models is less than 5.92%,indicating the accuracy and effectiveness of the proposed model.The developed approach can highly and efficiently predict the dynamic performance distributions across the entire workspace and guide the optimal functional design under the virtual machine framework.

Stifness modeling and analysis of a novel 4-DOF PKM for manufacturing large components

Faster response to orientation varying is one of the outstanding abilities of a parallel kinematic machine（PKM）.It enables such a system to act as a reconfgurable module employed to machine large components effciently.The stiffness formulation and analysis are the beforehand key tasks for its parameters design.A novel PKM with four degrees of freedom（DOFs）is proposed in this paper.The topology behind it is 2PUS-2PRS parallel mechanism.Its semianalytical stiffness model is frstly obtained,where the generalized Jacobian matrix of 2PUS-2PRS is formulated with the help of the screw theory and the stiffness coeffcients of complicated components are estimated by integrating fnite element analysis and numerical ftting.Under the help of the model,it is predicted that the property of system stiffness distributes within the given workspace,which features symmetry about a certain plane and is also verifed by performing fnite element analysis of the virtual prototype.Furthermore,key parameters affecting the system stiffness are identifed through sensitivity analysis.These provide insights for further optimization design of this PKM.