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.

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.

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.

Research on postprocessing of seven-axis linkage parallel kinematics machine with complicated surfaces

Because of restriction of workspace of parallel kinematics Machine (PKM), 6 DOF PKM can’t finish machining of workpiece with complicated surfaces under only once locating. It is necessary to fit workpiece beyond twice and to lead to low machining precision. Therefore the seven-axis linkage PKM is implemented by fixing a turntable on the worktable of the six-axis linkage PKM. However, the turntable angle decomposing problem from the CL file should be well considered. If the traditional decomposing methods are adopted, the nutation angle usually goes beyond the workspace of the machine. Therefore, according to the relation of the machine coordinate system and the workpiece coordinate system, the turntable angle decomposition algorithmic of the consistent coordinate system and the turntable angle decomposition algorithmic of the non-consistent coordinate system are developed to resolve the problem mentioned above. The turntable angle decomposition of the non-consistent coordinate system processes the decomposition which is based on the consistent coordinate system again. It calculates the initial angle of the locating workpiece, and the decomposed angle of the turntable at the machine coordinate system results in the nutation angle not going beyond workspace of the machine, thereby the decomposition process can be simplified.

The Kinematic Analyses of the 3-DOF Parallel Machine Tools

A 3-degree-of-freedom （3-DOF） parallel machine tool based on a tripod mechanism is developed and studied. The kinematics analysis is performed, the workspace is derived, and an analysis on the number of conditions of the Jacobian matrix and manipulability is carried out. A method for error analysis and manipulability is introduced. Hence, the manipulability analysis of the parallel machine tool is accomplished.

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.

A numerical error analysis method and its application on a 4RRR parallel kinematic machine

To guarantee the accuracy of error analysis and evaluate the manufacturing tolerance s influence,anumerical error analysis method for parallel kinematic machines (PKMs) is presented in this paper.Quasi-Newton method and genetic algorithm are introduced for the forward kinematic solution.Based onthe inverse and forward kinematic solutions,the end-effector s error calculation procedure is developed.To solve the accuracy problem caused by the length and angular parameters' different units,a normalizationmethod is proposed based on the manufacturing tolerance.Comparison between the error analysis resultscalculated by the traditional method and the numerical method for a 4RRR PKM shows that,this numericalerror analysis method is more accurate,simpler,and can evaluate the machine s real error basedon the manufacturing tolerance.

Calibration of parallel kinematics machine using generalized distance error model

This paper focus on the accuracy enhancement of parallel kinematics machine through kinematics calibration. In the calibration processing, well-structured identification Jacobian matrix construction and end-effector position and orientation measurement are two main difficulties. In this paper, the identification Jacobian matrix is constructed easily by numerical calculation utilizing the unit virtual velocity method. The generalized distance errors model is presented for avoiding measuring the position and orientation directly which is difficult to be measured. At last, a measurement tool is given for acquiring the data points in the calibration processing. Experimental studies confirmed the effectiveness of method. It is also shown in the paper that the proposed approach can be applied to other typed parallel manipulators.

A Tentative Study on Dynamics of a New Parallel NC Machine Tool with Long Travel, Large Rotating Angle and High Moving Speed

In view of the structure of traditional five-coord in ate machine tool, the work-piece and machine tool often move along their respec tive guides simultaneously on the whole. In this kind of machine structure, the total mass of moving parts including work-pieces, fixtures, rotating table, wor king table and so on is often very large. Besides, the elastic reform of transmi ssion and the viscous friction force of the guide can not be ignored. As a resul t, the machine tool can not move with high velocity and acceleration, and can no t meet the needs of modern fast and efficient production. The emergence of virtual-axis machine tool has provided a new approach for the solution of the above problems. The kernel of the virtual-axis machine tool is the parallel mechanism. So far, research of parallel mechanism in the world has achieved many results and various applied equipments based on parallel mecha nism have been worked out, but the research generally focuses on the working spa ce and kinematics analysis, dynamics are rarely considered. To meet the requirements of the modern fast and efficient production, reduce the cost and promote the machine tool’s acceleration character, not only should we analyze the kinematics of machine tool, but also we should study its dynamics a nd optimize the structure on the basis of analysis. In this paper, the kinem atics and dynamics of a 5-DOF (degrees of freedom) machine tool with novel para llel mechanism that has three moving DOF and one rotating DOF are studied by Rob ot-Wittenberg method. The dynamics character of the parallel robotic machine is analyzed and used to guide the structure design of machine tool. At last, the c orrectness is verified through a numerical simulation of 5-DOF. Hence, the dyna mics model can generally solve the problems existing in the parallel and hybrid machine tools. The dynamics character of the parallel robotic machine is studied and analyzed in quantity. The dynamics equation of the system can be written as This is a set of differential equations of four DOF system. Theoretically, the c losed solution of the forward and inverse problems can be gained by solving the above equations. The system equations quite suit to program at the computer. Whe n the forces are given, the state variables’ numerical solution can be gain ed through integral; and when the dynamics parameters are given, the forces can also be solved. But the multiple valued phenomena can not be avoided. We have developed simulation software based on the dynamics model presented by t his paper. The different effects of the structure parameters can be given by numerical simulation.

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.

WORKPIECE LOCATING AND POST PROCESSING SYSTEMS ON 6-DOF CNC MILLING MACHINE

A conventional non-computerized numerical control （CNC） machine is updated by mounting a six degree-of-free （DOF） parallel mechanism on it, thus obtaining a new CNC one. The structure of this CNC milling machine is introduced, and the workpiece locating system and the post processing system of the cutter location （CL） data file are analyzed. The new machine has advantages of low costs, simple structure, good rigidity, and high precision. It is easy to be transformed and used to process the workpiece with a complex surface.

Kinematic calibration of a class of parallel kinematic machines(PKM) with fewer than six degrees of freedom

By taking a 3-DOF translational milling machine as an example, this paper investigates the kinematic calibration of PKM systems with fewer than 6-DOF. The error mapping function is formulated in such a way that the geometric errors affecting the compensatable and uncompensatable pose errors can be separated. Based upon the previous investigation, a hierarchical approach to the geometric error identification is proposed. The compensation strategy is developed which is particularly suitable for the PKM systems with translational moving capability. The experiment has been carried out to verify the effectiveness of the proposed approach and the results show that the accuracy can be significantly improved.

Step kinematic calibration of a 3-DOF planar parallel kinematic machine tool

This paper presents a novel step kinematic calibration method for a 3 degree-of-freedom(DOF) planar parallel kinematic machine tool,based on the minimal linear combinations(MLCs) of error parameters.The method using mapping of linear combinations of parameters in error transfer multi-parameters coupling system changes the modeling,identification and error compensation of geometric parameters in the general kinematic calibration into those of linear combinations of parameters.By using the four theorems of the MLCs,the sets of the MLCs that are respectively related to the relative precision and absolute precision are determined.All simple and feasible measurement methods in practice are given,and identification analysis of the set of the MLCs for each measurement is carried out.According to the identification analysis results,a step calibration including step measurement,step identification and step error compensation is determined by taking into account both measurement costs and observability.The experiment shows that the proposed method has the following merits:(1) the parameter errors that cannot influence precision are completely avoided;(2) it reflects the mapping of linear combinations of parameters more accurately and enhances the precision of identification;and(3) the method is robust,efficient and effective,so that the errors in position and orientation are kept at the same order of the measurement noise.Due to these merits,the present method is attractive for the 3-DOF planar parallel kinematic machine tool and can be also applied to other parallel kinematic machine tools with weakly nonlinear kinematics.

Error modeling, sensitivity analysis and assembly process of a class of 3-DOF parallel kinematic machines with parallelogram struts

This paper presents an error modeling methodology that enables the tolerance design, assembly and kinematic calibration of a class of 3-DOF parallel kinematic machines with parallelogram struts to be integrated into a unified framework. The error mapping function is formulated to identify the source errors affecting the uncompensable pose error. The sensitivity analysis in the sense of statistics is also carried out to investigate the influences of source errors on the pose accuracy. An assembly process that can effectively minimize the uncompensable pose error is proposed as one of the results of this investigation.

Stiffness estimation of a parallel kinematic machine

This paper presents a simple yet comprehensive approach to quickly estimating the stiff-ness of a tripod-based parallel kinematic machine. This approach can be implemented in two steps. Inthe first step, the machine structure is decomposed into two substructures associated with the machineframe and parallel mechanism. The stiffness models of these two substructures are formulated bymeans of virtual work principle. This is followed by the second step that enables the stiffness model ofthe machine structure as a whole to be achieved by linear superposition. The 3D representations of themachine stiffness within the usable workspace are depicted and the contributions of different componentrigidities to the machine stiffness are discussed. The result is compared with that obtained through finiteelement analysis.

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.