Formulating Jacobian matrices for the dexterity analysis of parallel manipulators

The condition number of the Jacobian matrix has been commonly used in determining the dexterous regions of a manipulator workspace. This has been successful when applied to manipulators having either solely spherical or solely Cartesian degrees of freedom. However, for manipulators having a mix of both rotational and translational degrees of freedom, i.e., complex degree of freedom manipulators, the condition number of the Jacobian matrix may not be used due to dimensional inconsistencies with its elements. This paper furthers earlier work introduced in obtaining a Jacobian matrix which may be used to determine the dexterity of parallel mechanisms regardless of the number and type of degrees of freedom of the mechanism. The result of the method introduced in this paper is a dimensionally homogeneous Jacobian matrix mapping m actuator velocities to n independent end effector velocities. In the typical case where m = n, the Jacobian matrix is also square. As opposed to earlier works, the singular values of the Jacobian matrix obtained here have an evident physical significance. Furthermore, the ratio of the maximum and minimum singular values, i.e., the condition number may be used to measure the dexterity of the manipulator at a given pose. To illustrate the concepts introduced in this paper, the 3-PRS manipulator is analyzed.     

On the use of dual numbers,vectors and matrices in instantaneous,spatial kinematics

This paper gives a self-contained account of the algebra of dual quantities, the differential-geometry of dual curves, and their application to theoretical space kinematics. The scope of the paper is mainly didactical, but its understanding by the reader requires a certain amount of mathematical maturity. Due to the way in which the matter is presented, the paper gives some formulae and facts about dual curves and dual motions which are not generally known.     

Workspace boundaries of serial manipulators using manifold stratification

Workspace boundaries of serial manipulator arms have traditionally been presented as numerical curves traced on a boundary or as analytical formulations for a specific class of manipulators. This paper presents a broadly applicable formulation to identify and visualize the workspace, based on methods adapted from differential geometry and topology. Because the position Jacobian for manipulators with more than three degrees of freedom is not square, manifold stratification of the vector function (a topological vector space) yields nonlinear analytic functions. These functions give rise to analytic and semi-analytic varieties, some of which characterize lower-dimensional manifolds (submanifolds). The process is repeated until all strata are identified. The problem of determining which stratum (or part thereof) is on the boundary is addressed by introducing an acceleration function, which yields a quadratic form whose definiteness properties delineate the exact boundary. The method yields the exact boundary of the workspace in the form of closed-form equations and is capable of identifying voids therein. A complete example is first illustrated, followed by numerous tabulated examples (some of which are compared with reported works in the literature).     

Singularity analysis of parallel manipulators using constraint plane method

One of the most challenging problems in dealing with parallel manipulators is identifying their forward singular configurations. In such configurations these mechanisms become uncontrollable and cannot tolerate any external force. In this article a geometrical method, namely Constraint Plane Method (CPM), is introduced with the use of which one can easily obtain the singular configurations in many parallel manipulators. CPM is a methodical technique based on the famous Ceva plane geometry theorem. It is interesting to note that CPM involves no calculations and yields te result quickly. In addition, some of the previous geometrical methods led to many separate singular configurations; however, CPM yields the answer in just one phrase. In this article we have obtained singular configurations of some parallel manipulators. Moreover, a comparison is made between CPM and the previous methods to show the advantages of CPM.     

Solving inertial wrench of parallel manipulators using CAD variation geometry

It has been a significant and challenging issue to solve the inertial wrench of parallel manipulators (PMs) for their dynamics analysis and control. A CAD variation geometry approach is proposed for solving the inertial wrench of PMs. First, an initial simulation mechanism of PM, and a simulation mechanism of PM with linear/angular velocity and acceleration are created. Second, when modifying the driving dimension of the active legs, the simulation mechanisms are varied correspondingly, the position, linear/angular velocity and acceleration of moving platform and legs, and inertial wrench of moving platform and legs are solved automatically and visualized dynamically. Third, a 3-DoF PM is illustrated, and the displacement, linear/angular velocity and acceleration, and inertial wrenches of the moving platform and legs are solved using CAD variation geometry and are verified by the analytic solutions. Finally, inertial wrenches of the legs are transformed onto the moving platform.     

Derivation of dual forces in robot manipulators

According to the principle of transference, a compact three-dimensional representation of a rigid body kinematics is obtained by substituting dual for real numbers. This representation has recently been applied to robotics where in addition to its compactness, it allows to constitute the Jacobian matrix explicitly from the product of the dual transformation matrices with no additional computation.This paper introduces the generalized Jacobian matrix. This matrix consists of the complete dual transformation matrices as opposed to the regular Jacobian matrix which consists of specific columns only. The generalized Jacobian matrix relates force and moment at the end-effector to force and moment in all directions, at the joints. It is therefore possible to use the dual transformation matrices to derive, with no additional computation, the full force and moment vector at the robot's joints. Furthermore, the generalized Jacobian matrix also relates motion in all directions at the joints to the motion of the end-effector, an essential relation required at the design stage of robot manipulators (in particular, flexible ones). An extension of these kinematics and statics schemes into dynamics is possible by applying the dual inertia operator as is shown by an example of a three degrees-of-freedom robot.     

并联机构动力学建模及控制研究

为了系统地研究并联机构动力学特性,分析了4种基本建模方法（牛顿-欧拉方法、直接拉格朗日方法、拉格朗日-达朗贝尔方法和虚功原理方法）,总结了各种方法的优缺点。在此基础上,对动力学的3种控制算法（经典PD控制、机器人控制中常用的计算力矩控制以及数控加工中的先进轮廓控制）在并联机构上的应用进行了分析研究。针对开发的三自由度（3-DoF）纯平动并联机构Orthopod,采用拉格朗日-达朗贝尔方法进行了动力学建模,并对其进行了控制仿真。研究结果显示,轮廓控制在轮廓误差控制方面具有更好的性能,适合于雕刻机的控制应用。     

Solving the forward kinematics problem in parallel manipulators using an iterative artificial neural network strategy

In this paper we address the forward kinematics problem of a parallel manipulator and propose an iterative neural network strategy for its real-time solution to a desired level of accuracy. Parallel manipulators are closed kinematic structures that possess requisite rigidity to yield a high payload to self-weight ratio. Because of this unique feature, they have been employed in manufacturing, flight simulation systems, and medical robotics. However, it is this closed kinematic structure that has led to difficulty in their kinematic control, especially the forward kinematics control. The iterative neural network strategy we propose employs a trained neural network and an error compensation algorithm in the feedback loop. The proposed strategy was tested with data from a real-world flight simulation system. Results show that solutions with a maximum error in the position and orientation parameters of 0.25 mm and 0.01°, respectively, can be achieved in less than five iterations (or about 1 second). Because of the nature of this strategy, it is possible to implement it in a hardware form, which can result in a multi-fold reduction in the solution time for the same accuracy level.     

On the forward and inverse displacement of spatial parallel manipulators

In engineering applications, the singularity, and the forward and inverse displacement of spatial parallel manipulators have received a great deal of attention. This paper presents a new simple but effective methodology to investigate these problems based on the analysis of the degree of freedom (DoF) of the spatial parallel manipulator. Through numerical method, the singularity and the forward displacement problems can be cracked simultaneously. With a numerical example, we point out that the Newton-GMRES algorithm is quite a good redeeming method to solve the forward displacement of spatial parallel manipulators, in which cases the differential coefficient matrix of the Newton?Raphson method is nearly singular.     

Workspace analysis of positioning discontinuities due to clearances in parallel manipulators

Clearances at joints produce a loss of accuracy when positioning a mechanism. The end-effector pose error due to clearances depends on the mechanism configuration, the magnitude of the clearance itself and applied external wrenches. Sudden changes which occur in the actual posture of the mechanism owing to a change of contact mode at joints can be detected in the neighbourhood of some configuration. These sudden changes lead to positioning discontinuities on certain trajectories, or on the workspace. In this paper, a methodology for analysing the location of the discontinuities by means of a dynamic or kinetostatic analysis is presented. The advantages of choosing either the dynamic or the kinetostatic approach are analysed, making use of the 5R planar mechanism.     

Inverse kinematic and dynamic analyses of 6-DOFPUS type parallel manipulators

This paper presents inverse kinematic and dynamic analyses of HexaSlide type six degree-of-freedom parallel manipulators. The HexaSlide type parallel manipulators (HSM) can be characterized as an architecture with constant link lengths that are attached to moving sliders on the ground and to a mobile platform. In the inverse kinematic analyses, the slider and link motion (position, velocity, and acceleration) is computed given the desired mobile platform motion. Based on the inverse kinematic analysis, in order to compute the required actuator forces given the desired platform motion, inverse dynamic equations of motion of a parallel manipulator is derived by the Newton-Euler approach. In this derivation, the joint friction as well as all link inertia are included. Relative importance of the link inertia and joint frictions on the computed torque is investigated by computer simulations. It is expected that the inverse kinematic and dynamic equations can be used in the computed torque control and model-based adaptive control strategies.     

新型4-DOF并联机器人位置和工作空问分析

提出研究新型4 -DOF并联机器人的位置和工作空间的问题,利用计算机辅助几何法构造4-DOF并联机器人模拟机构,基于并联机器人的模拟机构,分析出一种4-DOF并联机器人的位置和工作空间.模拟结果表明,计算机辅助几何法的应用不但大大简化了并联机器入机构分析位置和工作空间难度,而且快捷直观,值得推广.     

Co-simulation of dynamic systems in parallel and serial model configurations

Recent advancements in simulation software and computation hardware make it realizable to simulate complex dynamic systems comprised of multiple submodels developed in different modeling languages. The so-called co-simulation enables one to study various aspects of a complex dynamic system with heterogeneous submodels in a cost-effective manner. Among several different model configurations for co-simulation, synchronized parallel configuration is regarded to expedite the simulation process by simulating multiple submodels concurrently on a multicore processor. In this paper, computational accuracies as well as computation time are studied for three different co-simulation frameworks: integrated, serial, and parallel. For this purpose, analytical evaluations of the three different methods are made using the explicit Euler method and then they are applied to two-DOF mass-spring systems. The results show that while the parallel simulation configuration produces the same accurate results as the integrated configuration, results of the serial configuration show a slight deviation. It is also shown that the computation time can be reduced by running simulation in the parallel configuration. Therefore, it can be concluded that the synchronized parallel simulation methodology is the best for both simulation accuracy and timeefficiency.     

Speed and force criteria for the proximity to singularities of parallel structure manipulators

The criteria of proximity to singularities of parallel structure mechanisms are considered. The criteria are based on the relative moments of forcing screws acting on the output unit of the mechanism and kinematic screws acting at the input of kinematic pairs.     

并联式混合动力汽车的建模和仿真

描述了并联式混合动力汽车(PHEV)仿真模型的建模思想,利用ADVISOR软件,建立了PHEV的仿真模型,并以SP-PHEV 2000为例,对所建模型进行了仿真.结果表明,所建立的模型和提出的仿真方法是合理有效的.     

Singularity kinematics principle and position-singularity analyses of the 6-3 Stewart-Gough parallel manipulators

This paper presents a new principle and method of kinematics to analyze the singularity of Stewart-Gough parallel manipulators and addresses the property identification of the position-singularity loci of the 6-3 Stewart-Gough manipulators for special orientations. Based on the kinematic relationship of a rigid body, a necessary and sufficient condition that three velocities of three non-collinear points in a moving rigid body can determine a screw motion is addressed and some typical singular configurations of the 6-3 Stewart-Gough parallel manipulators are also addressed in detail. With the above-mentioned condition, a symbolic analytical polynomial expression of degree three in the moving platform position parameters, representing the position-singularity locus of the 6-3 Stewart-Gough manipulators for special orientations, is derived; and the property identification of the position-singularity loci of the 6-3 Stewart-Gough manipulator for these special orientations is investigated at length. It is shown that position-singularity loci of the 6-3 Stewart-Gough parallel manipulator for these special orientations will be a plane and a hyperbolic paraboloid, even three intersecting planes.     

Sub-workspace design of binary manipulators using active and passive joints

This paper is concerned with an efficient continuous variable optimization method for determining an optimal configuration of passive and active joints of a binary manipulator when it operates within a desired sub-workspace. When joints are passive, their states, either stretched or contracted, are also determined by the developed method. Manipulator operations in a sub-workspace can avoid unnecessary actuations, resulting in energy saving. A technique to increase the sub-workspace concentration degree for a given maximum number of allowable active joints is also suggested.     

Algorithms for flexible flow shop problems with unrelated parallel machines, setup times, and dual criteria

In textile industries, production facilities are established as multi-stage production flow shop facilities, where a production stage may be made up of parallel machines. This known as a flexible or hybrid flow shop environment. This paper considers the problem of scheduling n independent jobs in such an environment. In addition, we also consider the general case in which parallel machines at each stage may be unrelated. Each job is processed in ordered operations on a machine at each stage. Its release date and due date are given. The preemption of jobs is not permitted. We consider both sequence- and machine-dependent setup times. The problem is to determine a schedule that minimizes a convex combination of makespan and the number of tardy jobs. A 0–1 mixed integer program of the problem is formulated. Since this problem is NP-hard in the strong sense, we develop heuristic algorithms to solve it approximately. Firstly, several basic dispatching rules and well-known constructive heuristics for flow shop makespan scheduling problems are generalized to the problem under consideration. We sketch how, from a job sequence, a complete schedule for the flexible flow shop problem with unrelated parallel machines can be constructed. To improve the solutions, polynomial heuristic improvement methods based on shift moves of jobs are applied. Then, genetic algorithms are suggested. We discuss the components of these algorithms and test their parameters. The performance of the heuristics is compared relative to each other on a set of test problems with up to 50 jobs and 20 stages.     

并联混合动力台架测试系统设计与应用

为了满足并联混合动力系统的开发与试验需求,搭建了包括混合动力快速原型控制器、测功机控制器和主控计算机系统的集成化混合动力测控系统。基于LabVIEW实时系统,用PXI硬件平台搭建了混合动力快速原型控制器,用PC-104工控机硬件平台搭建了测功机控制器,用LAN通信实现了各个控制器和主控计算机之间的数据传输。测试了混合动力系统的部件特性,设计了测功机的道路负载模拟模式,初步验证了混合动力系统的控制策略。