Planar Parallel Mechanism Research Articles

Parallel singularity-free design with actuation redundancy: A case study of three different types of 3-degree-of-freedom parallel mechanisms with redundant actuation

Typical parallel mechanisms suffer from parallel singularity due to kinematic coupling of multichains. This paper investigates how to remove parallel singularities by using redundant actuations. First, actuation wrenches and constraint wrenches forming the full direct kinematic Jacobian matrix are derived. After briefly addressing conditions for their constraint singularities, Grassmann–Cayley algebra is employed to identify parallel singularities. Then, employing Grassmann line geometry, the locations and the minimum number of redundant actuators are identified for the parallel mechanisms to have parallel singularity-free workspace. Three different types of 3-degree-of-freedom parallel mechanisms such as planar, spherical, and spatial parallel mechanisms are given as exemplary devices.

On the maximal singularity-free ellipse of planar 3- parallel mechanisms via convex optimization

This paper investigates the maximal singularity-free ellipse of 3-RPR planar parallel mechanisms. The paper aims at finding the optimum ellipse, by taking into account the stroke of actuators, in which the mechanism exhibits no singularity, which is a definite asset in practice. Convex optimization is adopted for the mathematical framework of this paper which requires a matrix representation for the kinematic properties of the mechanism under study in order to solve the latter optimization problem. Based on the nature of the expressions involved in the problem, two situations may arise: dealing with either convex or non-convex expressions. Both situations are treated separately with two very fast and systematic algorithms. For the first situation, an exact method is applied while for the second one, which is a general form of the first situation, convex optimization is accompanied with an iterative procedure. The computational time for the two proposed algorithms are considerably low compared with other methods proposed in the literature which opens an avenue to use the proposed algorithms for real-time purposes.

Singularity analysis of a planar parallel mechanism with revolute joints based on a geometric approach

The singularity is a critical part of a parallel kinematic machine (PKM) in which the degree of freedom decreases or actuator forces conflict with one another to break down the PKM. Therefore, defining and avoiding the region of singularities are essential for safely operating PKMs. This paper proposes a new method for determining the singularity of a planar-type PKM with revolute joints by taking a geometric approach. The main idea of the paper is to use the wrench, not velocity, equation to calculate the Jacobian matrix. In addition, a distinct feature of the proposed method is that it uses perpendicular distances between end-effectors, actuators, and screws to determine singularities based on screw theory. This proposed method can analyze many types of PKMs, including non-redundant as well as redundant ones. The paper considers non-redundant and redundant 5-bar, 6-bar, 3-RRR, and 4-RRR PKMs for case studies. The proposed method is intuitive and simple and thus can be used for analyzing the singularity of various PKMs.

Applicability and generality of the modified Grübler-Kutzbach criterion

A generally applicable criterion for all mechanism mobility has been an active domain in mechanism theory lasting more than 150 years. It is stated that the Modified Grubler-Kutzbach criterion for mobility has been successfully used to solve the mobility of many more kinds of mechanisms, but never before has anyone proven the applicability and generality of the Modified Grubler-Kutzbach criterion in theory. In order to fill the gap, the applicability and generality of the Modified Grubler-Kutzbach Criterion of mechanism mobility is systematically demonstrated. Firstly, the mobility research background and the Modified Grubler-Kutzbach criterion are introduced. Secondly, some new definitions, such as half local freedom, non-common constraint space of a mechanism and common motion space of a mechanism, etc, are given to demonstrate the correctness and broad applicability of the Modified Grubler-Kutzbach criterion. Thirdly, the general applicability of the Modified Grubler-Kutzbach criterion is demonstrated based on screw theory. The mobilities of the classical DELASSUS mechanisms and a modern planar parallel mechanism, are determined through the Modified Grubler-Kutzbach criterion, which are as examples to show the practical application of the Modified Grubler-Kutzbach criterion.

Investigation on the Effort Transmission in Planar Parallel Manipulators

Abstract In the design of a mechanism, the quality of effort transmission is a key issue. Traditionally, the effort transmissivity of a mechanism is defined as the quantitative measure of the power flowing effectiveness from the input link(s) to the output link(s). Many researchers have focused their work on the study of the effort transmission in mechanisms and diverse indices have been defined. However, the developed indices have exclusively dealt with the studies of the ratio between the input and output powers and they do not seem to have been devoted to the studies of the admissible reactions in passive joints. However, the observations show that it is possible for a mechanism to reach positions in which the transmission indices will have admissible values but the reaction(s) in passive joint(s) can reach excessively high values leading to the breakdown of the mechanism. In the present paper, a method is developed to ensure the admissible values of reactions in passive joints of planar parallel manipulators. It is shown that the increase of reactions in passive joints of a planar parallel manipulator depends not only on the transmission angle but also on the position of the instantaneous center of rotation of the platform. It allows the determination of the maximal reachable workspace of planar parallel manipulators taking into account the admissible reactions in its passive joints. The suggested method is illustrated via a 5R planar parallel mechanism and a planar 3-RPR parallel manipulator.

2A2-K01 平面3自由度冗長駆動パラレルメカニズムの姿勢遷移方法の提案(パラレルロボット・メカニズム)

For the purpose of extending the workspace of a 2-PRRRP redundantly actuated 3-degrees-of-freedom planar parallel mechanism by transition configurations, a method of the transition configurations to specialize redundant actuation is proposed. In this paper, in regard to the reverse bending motion which is transition configurations (assembly mode change) between a numbers of solutions of direct kinematics, a method to avoid a singular configuration in transition configurations is proposed. First, by using mechanical properties of a PRRRP sub mechanism, the solutions of direct kinematics are derived and states of transition configurations are defined. Next, a method of transition configurations to avoid a singular configuration for Reverse bending motion is proposed by extending the singular configuration analysis to redundantly actuated mechanism from non-redundantly actuated mechanism. The results to achieve the proposed method of transition configurations using experimental equipment are shown.

Dimensional Synthesis of a Planar Parallel Mechanism for Wafer Transfer with Kinematics Performance

This paper deals with the dimensional synthesis of a planar parallel mechanism for wafer transfer with kinematics performance. Under the relationship between workspace and parametric, a kinematics performance index, which is composed of the conditioning number of the Jacobian matrix and the motion resolution, is considered as the evaluation criterion. And a new index which is proposed by considering both the mean value and fluctuation of the pre-mentioned index is used to optimally design the dimensional parameters. It is testified that the optimal dimensional parameters can satisfied with the kinematics performance.

Positive Analysis of a New Type of Planar 3-DOF Parallel Mechanisms

The planar 3-DOF parallel mechanism was developed as a part of the pushing device for the wood sawing. Using closed vector polygon method to make positive movement analysis of the mechanism, establish the positive solution mathematical model. It provided the theory bases for the error analysis and control algorithm of the planar 3-DOF parallel mechanism.

특이형상의 위치를 고려한 병렬 기구의 작업공간 최적설계

It is well-known that when singularities are located within the workspace of the parallel mechanism (PM), the usefulness of its workspace is significantly deteriorated. To handle this problem, we suggest an optimal design method which leads to more useful and larger workspace of the PM by taking its singularity locations into consideration in design process. Kinematic models of three selected planar PMs, a 5R type PM, a 3-RPR type planar PM, and a 3-RRR type planar PM, are derived via screw theory and their singularity analyses are conducted. Then workspace optimal designs for those three PMs are conducted to verify that the suggested design method leads more useful and larger workspace in which deterioration by singularity is minimal.

Classifying the Boundaries of the Wrench-Closure Workspace of Planar Parallel Cable-Driven Mechanisms by Visual Inspection

The wrench-closure workspace of parallel cable-driven mechanisms is the set of poses for which any wrench can be produced at the end-effector by a set of non-negative cable tensions. It is already known that the boundary of the constant-orientation wrench-closure workspace of a planar parallel cable-driven mechanism is composed of segments of conic sections. However, the relationship between the geometry of the mechanism and the types of these conic sections is unknown. This technical report proposes a graphical method for determining the types of these conic sections from the mechanism geometry. It is also shown that the proposed method can be applied to find the constant-orientation singularities of a 3-RPR planar parallel robot, since these conic sections correspond to the boundary segment of the analogous three-cable driven planar parallel mechanism.

Dynamic Performance Evaluation of a 2-DoF Planar Parallel Mechanism

The acceleration performance function and dynamic performance evaluation combining the acceleration, velocity, gravity and external force of a 2-DoF planar parallel mechanism are presented in this paper. By means of the principle of virtual work, the inverse dynamic model and acceleration performance function of the planar parallel mechanism are setup. Based on the factors in the acceleration performance function, the effect on the acceleration performance of parallel mechanisms is investigated. Then a new method considering the acceleration factor, velocity factor, gravity factor and external force factor for dynamic performance evaluation of the parallel mechanism is introduced, which can evaluate the dynamic performance of high-speed parallel mechanisms more exactly. Based on this method, the evaluation indices used in dynamic optimization are introduced. These indices overcome the limitation of commonly used indices, and lay the foundation for dynamic optimization of parallel mechanisms.

GEOMETRIC ANALYSIS OF THE KINEMATIC SENSITIVITY OF PLANAR PARALLEL MECHANISMS

The kinematic sensitivity is a unit-consistent measure that has been recently proposed as a mechanism performance index to compare robot architectures. This paper presents a robust geometric approach for computing this index for the case of planar parallel mechanisms. The physical meaning of the kinematic sensitivity is investigated through different combinations of the Euclidean and infinity norms and by means of several illustrative examples. Finally, this paper opens some avenues to the dimensional synthesis of parallel mechanisms by exploring the meaning of the global kinematic sensitivity index.

Determination of Singularity-Free Zones in the Workspace of Planar Parallel Mechanisms with Revolute Actuators

The singularity-free workspace of parallel mechanisms is important in the trajectory planning and the kinematic design of the mechanisms. The planar parallel mechanism with revolute actuators, which the expression for the singularity locus and the algorithms for determining the singularity-free zones involve both the Cartesian and the joint variables, has been studied in the paper. Algorithms are proposed here to simplify the determination of the singularity-free zones in Cartesian space. The methodology and procedures for determining the singularity-free zones of planar parallel mechanisms with revolute actuators are introduced. The numerical results show that the simplified algorithms and the techniques are correct and efficient in determining singularity-free zones in the workspace of planar parallel mechanisms.

Optimal Control of Error Performance of Planar 3-DOF Parallel Mechanism

Abstract. The optimal control strategy is introduced to improve the motion accuracy of planar 3-DOF parallel mechanism. The optimal control of error performance of mechanism is achieved by structuring and quantizing the quadratic form index of the error performance, and taking into account both the system response and energy control in state space, so we study the problem which can be boil down to solve two-point boundary value problem of differential equation. The simulation results show that the actual motion of mechanism keep the better tracking accuracy with the global motion while dynamics parameters is changing, and the controller is ease to accomplishment in the industrial due to the simply structure and the stable change of control energy.

Dynamic Simulation of Hybrid-Driven Planar Five-Bar Parallel Mechanism Based on SimMechanics and Tracking Control

This paper investigates dynamic simulation and trajectory tracking control of hybrid-driven planar five-bar parallel mechanism (HPPM). To begin with, a simulation model of dynamics based on MATLAB/SimMechanics is established. Then, traditional PD control and closed loop PD-type iterative learning control of the HPPM are designed. At the end, the simulation based on SimMechanics is carried out, which acquires angular, angular velocity, angular acceleration of two driving links and constraint reaction of kinematic pairs at any time. In addition, the performance of the closed loop PD-type iterative learning control is compared with that of the traditional PD controller through simulations of the HPPM in the presence of the model external disturbances. The simulation results indicate that a perfect trajectory tracking of end-effector of the HPPM is achieved by the closed loop PD-type iterative learning controller.

Kinematics and Statics Analysis of a 2-DOF Planar Parallel Mechanism with over Constraint

This paper presents a deep research on 2-DOF planar parallel mechanism during developing a new type of hybrid machine tool. The composition of this planar parallel mechanism is introduced, its direct and inverse kinematics solution equations as well as the solution equations for the speed and the acceleration are deduced, meanwhile, the jacobian matrix is obtained. The statics problems of the parallel mechanism are modeled and analyzed based on principle of virtual work, and the balancing driving force solution equation of the driving part is given. Through the solving process we can see that the direct and inverse solution equations of this mechanism have explicit expressions and convenient to realize real-time control. It forms a solid basis for the design and development of this mechanism.

2A1-J05 バック転パラレルリンク機構に関する研究 : 機構解析とプロトタイプの試作(パラレルロボット・メカニズム)

A redundantly actuated 3 degree of freedom (xyθ) planar parallel link mechanism (PLM) with back-flip motion (BF PLM) is proposed. The back-flip means that the end plate of the PLM has working area of 180 degrees or more. It aims at the table mechanism of multi-axes machine tools. In this paper, singular configuration is analyzed by using the kinematics of the PLM. By applying numerical analyses, it has been revealed that the back-flip motion can be achieved stably (not singularity) over the working area excluding the upper limit. Prototype of BF PLM and control system are produced. Linear motor of Parallel Drive is implemented to the prototype of BF PLM.

Optimal Design for Planar Cable-driven Parallel Mechanism with Respect to Maximizing Workspace

Workspace is an essential index of parallel mechanism.Compared to parallel mechanisms actuated by rigid limbs,cable-driven parallel mechanisms have larger workspace as an outstanding merit.The relationship between the workspace and the position where the cables are anchored on the moving platform is discussed.The static model of cable-driven parallel mechanism is set up and the constant workspace of 3-DOF planar mechanism driven by 4 cables is analyzed.After that,the parameters of the cables anchoring on the moving platform are modeled,and the parameters are optimized to realize the biggest constant orientation workspace.The optimized parameters of the planar mechanism have larger workspace than the conventional plane configuration.

Singularity Analysis of Planar Cable-driven Parallel Mechanisms with Grassmann Geometry

Cable-driven parallel mechanism is singular when some wrenches acting on the cables are deficient.Based on the definition,the cables are defined as line vectors.The linear correlation among cables is analyzed with Grassmann line geometry,and the singularity of the mechanism could be easily testified.The geometry condition of singular free planar mechanism is presented.The singularities of some mechanisms are analyzed by the line geometry method,and the singular free mechanisms are designed based on the geometry condition.According to the geometry condition,the singular mechanisms are improved through changing the dimensions or add some cables to avoid the singularity.At last,the numerical method with Jacobian matrix is used to analyze the singularity,and the result demonstrates that the improved mechanisms avoid the singularity which original mechanisms have.The result also shows that Grassmann line geometry is applicable to the analysis of cable-driven parallel mechanisms.

Research on 2-Dof Parallel Robot and its Motion Characteristics

A novel robot, derived from a planar parallel mechanism, is presented. With two sliders driving on the same side of parallel guide rails, this simple but practical mechanism is capable of realizing a large workspace. The direct and inverse kinematical solutions are given. The simulation demonstrates that all kinds of straight lines and circles can be realized by the end actuator of the robot; the corresponding motion disciplines and characteristics driven by the two sliders are analyzed; the results are verified on the prototype. By allowing the end actuator to move in Z-direction or to rotate around A or B-axis, three-, four-, five-axis manipulator would be composed and surface welding and space cutting would be realized. This paper lays the foundation for the real-time control and industrial application of this novel robot.