Over-constrained Parallel Mechanism Research Articles

Research on High Precision Stiffness Modeling Method of Redundant Over-Constrained Parallel Mechanism.

Traditional stiffness modeling methods do not consider all factors comprehensively, and the modeling methods are not unified, lacking a global stiffness model. Based on screw theory, strain energy and the virtual work principle, a static stiffness modeling method for redundant over-constrained parallel mechanisms (PMs) with clearance was proposed that considers the driving stiffness, branch deformation, redundant driving, joint clearance and joint contact deformation. First, the driving stiffness and branch deformation were considered. According to the strain energy and Castiliano's second theorem, the global stiffness matrix of the ideal joint mechanism was obtained. The offset of the branch was analyzed according to the restraint force of each branch. The mathematical relationship between the joint clearance and joint contact deformation and the end deformation was established. Based on the probability statistical model, the uncertainty of the offset value of the clearance joint and the contact area of the joint caused by the coupling of the branch constraint force was solved. Finally, taking a 2UPR-RR-2RPU redundant PM as an example, a stiffness simulation of the mechanism was carried out using the finite element method. The research results show that the high-precision stiffness modeling method proposed in this paper is correct, and provides an effective method for evaluating the stiffness performance of the PM.

Stiffness modeling of a 2-DoF over-constrained planar parallel mechanism

Stiffness model acquisition of over-constrained parallel mechanisms is relatively difficult since they have more than necessary kinematic loops. In this study, a stiffness modeling solution for over-constrained parallel mechanisms is proposed while considering the computational cost efficiency. Three contributions of the paper are: (1) Presenting the stiffness modeling procedure for serially connected closed-loop structures by using the Virtual Joint Method (2) Considering the effect of dynamic auxiliary forces and dynamic external forces on the mobile platform’s deflection and achieving a direct solution by using superposition principle (3) A model fitting procedure for modifying the stiffness coefficients to comply with the experimental data. A 2 degrees-of-freedom over-constrained parallel mechanism is investigated as a case study. However, the proposed stiffness model is 6-DoF since compliant deflections occur in any direction. A finite element analysis and an experimental study verify the model’s results.

Kinematic, Workspace and Force Analysis of A Five-DOF Hybrid Manipulator R(2RPR)R/SP+RR

In the present study, the over-constrained hybrid manipulator R(2RPR)R/SP+RR is considered as the research objective. In this paper, kinematics of the hybrid manipulator, including the forward and inverse position, are analyzed. Then, the workspace is checked based on the inverse position solution to evaluate whether the workspace of the hybrid manipulator meets the requirements, and the actual workspace of the hybrid robot is analyzed. After that, the force analysis of the over-constrained parallel mechanism is carried out, and an ADAMS-ANSYS rigid-flexible hybrid body model is established to verify the simulation. Based on the obtained results from the force analysis, the manipulator structure is designed. Then, the structure optimization is carried out to improve the robot stiffness. Finally, calibration and workspace verification experiments are performed on the prototype, cutting experiment of an S-shaped aluminum alloy workpiece is completed, and the experiment verifies the machining ability of the prototype. This work conducts kinematics, workspace, force analyses, structural optimization design and experiments on the over-constrained hybrid manipulator R(2RPR)R/SP+RR, providing design basis and technical support for the development of the novel hybrid manipulator in practical engineering.

Dynamic analysis of an over-constrained parallel mechanism with the principle of virtual work

ABSTRACT This research presents the mathematical modelling of kinematic and complete dynamic analysis of a novel over-constrained parallel mechanism, which consists of two universal-prismatic-revolute joint limbs and one revolute-revolute-universal joint limb. The kinematic model is constructed based on the closed-loop vector method and the velocity Jacobian matrix is deduced, velocity-mapping relationships between all moving components and moving platform are also performed. Afterwards, inertia and applied forces are analysed, the complete dynamic equations with the classical Stribeck friction model of the proposed structure is established based on the principle of virtual work. A theoretical numerical example is given to solve kinematics and dynamics solutions, and theoretical forces from developed dynamic formulation are verified by the physic model simulation in Simscape and the rigid-flexible coupling model simulation in Adams. A good agreement between the theoretical results and multi-body software simulation is found.

Kinematic and dynamic analysis of a novel parallel kinematic Schönflies motion generator

This paper analyses the kinematics and dynamics of an over-constrained parallel mechanism with four degrees of freedom providing Schönflies motion with infinite tool rotation. Despite its limited motion capability in one translational degree of freedom, the proposed mechanism has several potential uses, such as assembly of electronic components or fabric cutting.Firstly, the instantaneous kinematic problem of the investigated mechanism is presented, which includes formulating the inverse kinematics for the position, velocity and acceleration of the mechanism. Then, singular configurations of the manipulator are studied, based on the obtained Jacobin matrices. Secondly, based on the foregoing kinematic relations and the concept of link Jacobian matrices, the dynamic model is formulated by means of the principle of virtual work. Finally, the correctness of the derived models is verified by comparing the results obtained from the formulated models with those obtained using MATLAB Simscape Multibody.

Dynamic Modeling Method of Spatial Passive Over-constrained Parallel Mechanism Based on Newton Euler Method

Dynamic Modeling Method of Spatial Passive Over-constrained Parallel Mechanism Based on Newton Euler Method

Elastostatic Stiffness Analysis of a 2PUR-PSR Overconstrained Parallel Mechanism

The elastostatic stiffness performance of a 2PUR-PSR (S: spherical joint; U: universal joint; R: revolute joint; P: prismatic pair) overconstrained parallel mechanism with two rotational and one translational (2R1T) degrees of freedom was studied based on screw theory and strain energy. First, the actuator and constraint wrenches of the mechanism were obtained based on screw theory. Second, by considering the space composite elastic deformation of the rod (including axial tension, and shear, torsional and bending deformation), compact limb stiffness matrixes were obtained in terms of strain energy and Castigliano’s second theorem. Finally, analytic expressions for the overall stiffness matrix of the mechanism and the amplitudes of the actuator and constrained wrenches were obtained based on the virtual-work principle and the balance equation for a moving platform. The relative error between results from the analytical model and the Workbench finite element model is approximately 1.3%, which validates the effectiveness of the proposed elastostatic stiffness model. Unlike traditional stiffness evaluation index, a regional stiffness index was proposed to measure the mechanism’s stiffness performance at various working heights. Using the index, it is possible to determine the optimum stiffness region of a moving platform with known external wrenches.

Geometric error analysis of an over-constrained parallel tracking mechanism using the screw theory

This paper deals with geometric error modeling and sensitivity analysis of an over-constrained parallel tracking mechanism. The main contribution is the consideration of over-constrained features that are usually ignored in previous research. The reciprocal property between a motion and a force is applied to tackle this problem in the framework of the screw theory. First of all, a nominal kinematic model of the parallel tracking mechanism is formulated. On this basis, the actual twist of the moving platform is computed through the superposition of the joint twist and geometric errors. The actuation and constrained wrenches of each limb are applied to exclude the joint displacement. After eliminating repeated errors brought by the multiplication of wrenches, a geometric error model of the parallel tracking mechanism is built. Furthermore, two sensitivity indices are defined to select essential geometric errors for future kinematic calibration. Finally, the geometric error model with minimum geometric errors is verified by simulation with SolidWorks software. Two typical poses of the parallel tracking mechanism are selected, and the differences between simulation and calculation results are very small. The results confirm the correctness and accuracy of the geometric error modeling method for over-constrained parallel mechanisms.

Accuracy Analysis of Redundantly Actuated and Overconstrained Parallel Mechanisms With Actuation Errors

A general accuracy analysis method of redundantly actuated and overconstrained parallel mechanisms is proposed. Coupled effects of actuation errors and internal elastic forces arose from the elastic deformation are both considered. The accuracy analysis approach is based on the Lie-group theory and screw theory, and it includes three steps. First, stiffness matrices of serial legs are obtained. Second, the movement of each leg is modeled based on group theory and the elastic forces arose from the deformation are represented using the stiffness matrices, following which the multiclosed-loop structure constraint and self-balanced force constraint are modeled. Finally, the error pose is estimated. The proposed method is illustrated by the accuracy study of a redundantly actuated and overconstrained Stewart platform. The error modeling is easy as the use of stiffness matrix can model the passive joint motions and deformation together. Moreover, the proposed method is computationally cheap as all computations are linear.

Kinematic Calibration of a 2-DoF Over-Constrained Parallel Mechanism Using Real Inverse Kinematics

The kinematic accuracy improvement is a key challenge in the development of over-constrained parallel mechanisms (PMs). Taking a two degree-of-freedom (DoF) over-constrained PM applied in assembly line as an example, this paper addresses the kinematic calibration problem of over-constrained PM to improve accuracy and promote its practical application. Instead of establishing conventional error mapping model, a nonlinear error model is built by inserting geometric errors of parts to the real inverse position analysis. On this basis, a set of nonlinear identification equations are formulated. Unlike other methods that identify the geometric errors by an identification Jacobian matrix and pay extra attention to the robustness of the matrix, these nonlinear identification equations are directly solved by optimization technique. Herein, the hybrid genetic algorithm is adopted in the optimization due to its high robustness, efficiency, and accuracy. Finally, error compensation is implemented by modifying the motor outputs in the controller. Simulations and experiments are then carried out to verify the calibration method, which show that the orientation accuracy of the 2-DoF over-constrained PM improves by 93.96% and 90.38%, respectively. Comparative studies to the conventional regularization method and four other optimization algorithms are also investigated. The results further confirm the high accuracy of the proposed kinematic calibration method for over-constrained PMs.

Methods for Force Analysis of Overconstrained Parallel Mechanisms: A Review

The force analysis of overconstrained PMs is relatively complex and difficult, for which the methods have always been a research hotspot. However, few literatures analyze the characteristics and application scopes of the various methods, which is not convenient for researchers and engineers to master and adopt them properly. A review of the methods for force analysis of both passive and active overconstrained PMs is presented. The existing force analysis methods for these two kinds of overconstrained PMs are classified according to their main ideas. Each category is briefly demonstrated and evaluated from such aspects as the calculation amount, the comprehensiveness of considering limbs’ deformation, and the existence of explicit expressions of the solutions, which provides an important reference for researchers and engineers to quickly find a suitable method. The similarities and differences between the statically indeterminate problem of passive overconstrained PMs and that of active overconstrained PMs are discussed, and a universal method for these two kinds of overconstrained PMs is pointed out. The existing deficiencies and development directions of the force analysis methods for overconstrained systems are indicated based on the overview.

Method for Force Analysis of the Overconstrained Parallel Mechanism Considering the Link’s Spatial Composite Elastic Deformations

Method for Force Analysis of the Overconstrained Parallel Mechanism Considering the Link’s Spatial Composite Elastic Deformations

Simple Methods for Distinguishing Instantaneity of Mobility of the Overconstrained Parallel Mechanism Based on the Geometric Constraint Conditions

The mobility of the parallel mechanism calculated by using the screw theory tool is instantaneous,especially for the overconstrained parallel mechanism,not only the nature of the mobility may be changed,but also movement of some mobility could not realized at all,so whether the mobility is instantaneous or not must be checked.Since there have been very few investigations on the methods for distinguishing mobility's instantaneity of the overconstrained parallel mechanism home and abroad,a simple method for distinguishing the mobility's instantaneity of the overconstrained parallel mechanism is put forward based on the geometric constraint conditions of their own.For this method,the geometric constraint equations are established according to geometric constraint conditions of the different limbs of the overconstained parallel mechanism,and then whether the geometric constraint equations exist nonzero real solution is examined to determine the instantaneity of the mobility,and five overconstrained parallel mechanisms are taken as examples to elaborate the specific steps of the method.Then a simulation analysis method for distinguishing mobility's instantaneity of the overconstrained parallel mechanism is proposed based on the simulation software ADAMS.And for this method,the extra degrees of freedom that are automatically added in the simulation model are firstly detected,then whether the extra degrees of freedom produce displacement is examined to judge the instantaneity of the mobility.Similarly,the above five overconstrained parallel mechanisms are analyzed by using this simulation analysis method.The results obtained by using these two kinds of distinguishing methods are consistent,which effectively verifies the correctness of the two distinguishing methods.

Research on the Dynamic Model of an Over-constrained Parallel Mechanism

Due to their redundant links and redundant restriction forces,it has been a different problem for dynamic modeling and analysis of over-constrained parallel mechanism.A planar 2-DOF over-constrained parallel mechanism is discussed as an example.On the basis of kinematics analysis,the equilibrium equations for the force and moment of this mechanism are established by using Newton Euler method and D'Alembert principle.The model cannot be solved because the number of equations is less than that of unknown variables.So a novel method,which can provide feasible supplementary equations through analyzing the deformation compatibility to establish the relationships between constraint forces and position errors of terminal platform,is put forward.Thus the dynamic model can be solved.This method is applied to the dynamic modeling and analysis of parallel mechanism in a hybrid machine tool.By numerical example,driving forces and platform error are simulated.The result shows that this method is feasible and reasonable.The research provides a new method to the dynamic modeling and analysis of over-constrained parallel mechanism.

Structure Synthesis of Single Closed-loop Multi-degree of Freedom of Over-constrained Mechanism Based on Screw Theory

Based on screw theory, the relationship among the numbers of constraints, joints and degrees of freedom that constitute single closed-loop over-constraint mechanism are analyzed. Also the calculation formula of the degree of freedom for single closed-loop over-constrained mechanism is presented. Moreover, over-constraints are classified into two types of forces and couples which have different orders, in terms of reciprocal screws. All possible forms of characters of independent over-constraints and numbers as well as their combinations are summarized. Under the condition of having a given special combination over-constraints, by using the principle of virtual work and taking reciprocal screw theory as calculation means, the structural synthesis method for the multi-degree of freedom single closed-loop mechanism is obtained. The non-instantaneous conditions of mechanism are obtained by using analytical method, and axis layout, number, arrangement and spacegeometric conditions of joints are presented. The problem for synthesis of multi-DOF single loop over-constraint mechanisms is solved systematically and wholly and all the possible single closed-loop over-constrained parallel mechanisms are enumerated. The structure synthesis procedure, and design examples of some typical mechanisms are given.

Mobility of Overconstrained Parallel Mechanisms

AbstractThe Kutzbach–Grübler mobility criterion calculates the degrees of freedom of a general mechanism. However, the criterion can break down for mechanisms with special geometries, and in particular, the class of so-called overconstrained parallel mechanisms. The problem is that the criterion treats all constraints as active, even redundant constraints, which do not affect the mechanism degrees of freedom. In this paper we reveal a number of screw systems of a parallel mechanism, explore their inter-relationship and develop an original theoretical framework to relate these screw systems to motion and constraints of a parallel mechanism to identify the platform constraints, mechanism constraints and redundant constraints. The screw system characteristics and relationships are investigated for physical properties and a new approach to mobility analysis is proposed based on decompositions of motion and constraint screw systems. New versions of the mobility criterion are thus presented to eliminate the redundant constraints and accurately predict the platform degrees of freedom. Several examples of overconstrained mechanisms from the literature illustrate the results.