Dynamics of the 6-6 Stewart parallel manipulator

Recursive matrix relations in kinematics and dynamics of the 6-6 Stewart–Gough parallel manipulator having six mobile prismatic actuators are established in this paper. Controlled by six forces, the manipulator prototype is a spatial six-degrees-of-freedom mechanical system with six parallel legs connecting to the moving platform. Knowing the position and the general motion of the platform, we develop first the inverse kinematics problem and determine the position, velocity and acceleration of each manipulator’s link. Further, the inverse dynamics problem is solved using an approach based on the principle of virtual work, but it has been verified the results in the framework of the Lagrange equations with their multipliers. Finally, compact matrix relations and graphs of simulation for the input velocities and accelerations, the input forces and powers are obtained.     

Dynamics of the spherical 3- \mathit{U\underline{P}S/}S parallel mechanism with prismatic actuators

Recursive relations in kinematics and dynamics of the symmetric spherical 3- parallel mechanism having three prismatic actuators are established in this paper. Controlled by three forces, the parallel manipulator is a 3-DOF mechanical system with three parallel legs connecting to the moving platform. Knowing the position and the rotation motion of the platform, we develop first the inverse kinematics problem and determine the position, velocity, and acceleration of each manipulator’s link. Further, the inverse dynamic problem is solved using an approach based on the principle of virtual work, but it has been verified using the results in the framework of the Lagrange equations with their multipliers. Finally, compact matrix relations and graphs of simulation for the input forces and powers are obtained.     

Inverse dynamics of a parallel manipulator

Presented is an analysis of the kinematics and the inverse dynamics of a proposed three degree of freedom (dof) parallel manipulator resembling the Stewart platform in a general form. In the kinematic analysis, the inverse kinematics, velocity, and acceleration analyses are performed, respectively, using vector analysis and general homogeneous transformations. An algorithm to solve the inverse dynamics of the proposed parallel manipulator is then presented using a Lagrangian technique. In this case, it is found that one should introduce and subsequently eliminate Lagrange multipliers to arrive at the governing equations. Numerical examples are finally carried out to examine the validity of the approach and the accuracy of the numerical technique employed. The trajectory of motion of the manipulator is also performed using a cubic spline. © 1994 John Wiley & Sons, Inc.     

Recursive modelling in dynamics of Agile Wrist spherical parallel robot

Recursive matrix relations for kinematics and dynamics of the 3-RRR Agile Wrist spherical parallel robot are established in this paper. The prototype of the robot is a three-degrees-of-freedom mechanism with three identical legs. Controlled by concurrent torques, which are generated by some electric motors, the active elements of the robot have three independent rotations. Knowing the rotation motion of the moving platform, we develop first the inverse kinematical problem and determine the velocities and accelerations. Further, the principle of virtual work is used in the inverse dynamic problem. Matrix equations offer iterative expressions and graphs for the power requirement comparison of each of three actuators in two computational complexities: complete dynamic model and simplified dynamic model.     

Dynamic formulation of a planar 3-DOF parallel manipulator with actuation redundancy

In this paper, based on the conventional Newton–Euler approach, a simplification method is proposed to derive the dynamic formulation of a planar 3-DOF parallel manipulator with actuation redundancy. Closed-form solutions are developed for the inverse kinematics. Based on the kinematics, the Newton–Euler approach in simplification form is used to derive the inverse dynamic model of the redundant parallel manipulator. Then, the driving force optimization is performed by minimizing an objective function which is the square of the sum of four driving forces. The dynamic simulations are done for the parallel manipulator with both the redundant and non-redundant actuations. The result shows that the dynamic characteristics of the manipulator in the redundant case are better than that in the non-redundancy. The redundantly actuated parallel manipulator was incorporated into a 4-DOF hybrid machine tool which includes a feed worktable.     

Workspace and dynamic performance evaluation of the parallel manipulators in a spray-painting equipment

The paper deals with the workspace and dynamic performance evaluation of the PRR–PRR parallel manipulator in spray-painting equipment. Functional workspace of planar fully parallel robots is often limited because of interference among their mechanical components. The proposed 3-DOF planar parallel manipulator with two kinematic chains connecting the moving platform to the base can reduce interference while still maintaining 3 DOFs. Based on the kinematics, four working modes are analyzed and singularity is studied. The workspace is investigated and the inverse dynamics is formulated using the virtual work principle. The dynamic performance evaluation indices are designed on the basis of maximum and minimum magnitude of acceleration vector of the moving platform produced by a unit actuated force. The index not only can evaluate the accelerating performance of a manipulator, but also can reflect the isotropy of accelerating performance. Workspace and dynamic performances of the four working modes are compared and the optimal working mode for the painting of a large object with conical surface is determined.     

An efficient method for inverse dynamics of manipulators based on the virtual work principle

The computational efficiency of inverse dynamics of a manipulator is important to the real-time control of the system. For serial manipulators, the recursive Newton-Euler method has been proven to be the most efficient. However, for more general manipulators, such as serial manipulators with closed kinematic loops or parallel manipulators, it must be modified accordingly and the resultant computational efficiency is degraded. This article presents a computationally efficient scheme based on the virtual work principle for inverse dynamics of general manipulators. The present method uses a forward recursive scheme to compute velocities and accelerations, the Newton-Euler equation to calculate inertia forces/torque, and the virtual work principle to formulate the dynamic equations of motion. This method is equally effective for serial and parallel manipulators. For serial manipulators, its computational efficiency is comparable to the recursive Newton-Euler method. For parallel manipulators or serial manipulators with closed kinematic loops, it is more efficient than the existing methods. As an example, the computations of inverse dynamics (including inverse kinematics) of a general Stewart platform require only 842 multiplications, 511 additions, and 12 square roots.     

基于SimMechanics的新型并联机构仿真平台

针对一种新型六自由度并联机器人进行了运动学和动力学分析,并借助于MatLab的SimMechanics模块的系统动态建模功能,以六自由度并联机器人仿真模型为对象,设计了PID控制器形成闭环控制,最后构成一个仿真平台。结果表明：在平台上可以进行运动学、动力学、控制方法的研究。同时仿真平台由各模块组成,可以灵活改变各模块的参数和结构,并且避开了Stewart平台的复杂的建模过程,对所有并联机构的研究都有借鉴作用。     

Kinematic analysis of a novel 3-DOF actuation redundant parallel manipulator using artificial intelligence approach

Kinematic analysis is one of the key issues in the research domain of parallel kinematic manipulators. It includes inverse kinematics and forward kinematics. Contrary to a serial manipulator, the inverse kinematics of a parallel manipulator is usually simple and straightforward. However, forward kinematic mapping of a parallel manipulator involves highly coupled nonlinear equations. Therefore, it is more difficult to solve the forward kinematics problem of parallel robots. In this paper, a novel three degrees-of-freedom (DOFs) actuation redundant parallel manipulator is introduced. Different intelligent approaches, which include the Multilayer Perceptron (MLP) neural network, Radial Basis Functions (RBF) neural network, and Support Vector Machine (SVM), are applied to investigate the forward kinematic problem of the robot. Simulation is conducted and the accuracy of the models set up by the different methods is compared in detail. The advantages and the disadvantages of each method are analyzed. It is concluded that ν-SVM with a linear kernel function has the best performance to estimate the forward kinematic mapping of a parallel manipulator.     

OPTIMAL MECHANISM DESIGN AND DYNAMIC ANALYSIS OF A 3‐LEG 6‐DOF LINEAR MOTOR BASED PARALLEL MANIPULATOR

This paper presents the optimal mechanism design and dynamic analysis of a prototype 3‐leg 6‐DOF (degree‐of‐freedom) parallel manipulator. Inverse kinematics, forward kinematics, inverse dynamics and working space characterizing the platform motion are derived. In the presented architecture, the base platform has three linear slideways individually actuated by a synchronous linear servo motor, and each extensible vertical link connecting the upper and base platforms is actuated by an inductive AC servo motor. The linear motors contribute high‐speed movements to the upper platform. This kind of architecture using hybrid (linear and AC) motors yields high level performance of motions, especially in the working space. The novel result of maximal working angles is the significant contribution of this architecture. The Taguchi Experimental Method is applied to design the optimal mechanism of the platform system, and the result is used as the actual data to build this system.     

Dynamics analysis for a novel 6-DoF parallel manipulator I with three planar limbs

A novel 6-DoF parallel manipulator I with three planar limbs is proposed and its dynamics is analyzed systematically. First, its characteristics and DoF are analyzed and calculated. Second, the formulae for solving kinematics of the moving platform and the planar limbs are derived. Third, the formulae for solving the inertial wrench applied on the planar limbs and the moving platform are derived, and dynamics formula is derived for solving dynamic active forces applied onto the planar limbs. Fourth, a singularity of the proposed parallel manipulator is determined and analyzed. Fifth, an analytic example is given for solving the kinetostatics and dynamics of the proposed parallel manipulator, and the solved results are analyzed and verified by the simulation mechanism. Finally, a workspace is constructed and analyzed by comparing with an existing 6-DoF parallel manipulator.     

Dynamics analysis of a 3-RRP spherical parallel manipulator using the natural orthogonal complement

In the present research, application of the Natural Orthogonal Complement (NOC) for the dynamic analysis of a spherical parallel manipulator, referred to as SST, is presented. Both inverse and direct dynamics are considered. The NOC and the SST fully parallel robot are explained. To drive the NOC for the SST manipulator, constraints between joint variables are written using the transformation matrices obtained from three different branches of the robot. The Newton–Euler formulation is used to model the dynamics of each individual body, including moving platform and legs of the manipulator. D’Alembert’s principle is applied and Newton–Euler dynamical equations free from non-working generalized constraint forces are obtained. Finally two examples, one for direct and one for inverse dynamics are presented. The correctness and accuracy of the obtained solution are verified by comparing with the solution of the virtual work method as well as commercial multi-body dynamics software.     

New Architecture of a Hybrid Two-Fingered Micro–Nano Manipulator Hand: Optimization and Design

This paper presents the synthesis and design optimization of a compact and yet economical hybrid two-fingered micro–nano manipulator hand. The proposed manipulator hand consists of two series modules, i.e., an upper and lower modules. Each of them consists of a parallel kinematics chain with a glass pipette (1 mm diameter and 3–10 cm length) tapered to a very sharp end as an end-effector. It is driven by three piezo-electric actuated prismatic joints in each of the three legs of the parallel kinematics chain. Each leg of the kinematics chain has the prismatic–revolute–spherical joint structure. As the length of the glass pipette end-effector is decreased, the resolution and accuracy of the micro–nano manipulator hand is increased. For long lengths of the glass pipette end-effector, this manipulator works as a micro manipulator and for short lengths it works as a nano manipulator. A novel closed-form solution for the problem of inverse kinematics is obtained. Based on this solution, a simulation program has been developed to optimally choose the design parameters of each module so that the manipulator will have a maximum workspace volume. A computer-aided design model based on optimal parameters is built and investigated to check its workspace volume. Experimental work has been carried out for the purpose of calibration. Also, the system hardware setup of the hybrid two-fingered micro–nano manipulator hand and its practical Jacobian inverse matrices are presented.     

A comparison study on the dynamics of planar 3-DOF 4-RRR, 3-RRR and 2-RRR parallel manipulators

This paper presents a comparison study on the dynamics of three planar 3-DOF parallel manipulators, one with 3-RRR structure, and the other two with 2-RRR and 4-RRR structures. The inverse kinematics and Jacobian matrix of these mechanisms are analyzed. The dynamic formulations in the linear form are derived and a dynamic performance index is given to compare their dynamic performances. The results show that the 2-RRR parallel manipulator has the worst dynamic performance.     

Extended Jacobian inverse kinematics algorithms for mobile manipulators

We consider the inverse kinematic problem for mobile manipulators consisting of a nonholonomic mobile platform and a holonomic manipulator on board the platform. The kinematics of a mobile manipulator are represented by a driftless control system with outputs together with the associated variational control system. The output reachability map of the driftless control system determines the instantaneous kinematics, while the output reachability map of the variational system plays the role of the analytic Jacobian of the mobile manipulator. Relying on a formal analogy between the kinematics of stationary and mobile manipulators we exploit the extended Jacobian construction in order to design a collection of extended Jacobian inverse kinematics algorithms for mobile manipulators. It has been proved mathematically and confirmed in computer simulations that these algorithms are capable of efficiently solving the inverse kinematic problem. Moreover, a choice of the Jacobian extension may lay down some guidelines for the platform‐manipulator motion coordination. © 2002 Wiley Periodicals, Inc.     

Reliability-based approach to the inverse kinematics solution of robots using Elman's networks

The solution of inverse kinematics problem of redundant manipulators is a fundamental problem in robot control. The inverse kinematics problem in robotics is the determination of joint angles for a desired cartesian position of the end effector. For the solution of this problem, many traditional solutions such as geometric, iterative and algebraic are inadequate if the joint structure of the manipulator is more complex. Furthermore, many neural network approaches have been done to this problem. But the neural network-based solutions are not much reliable due to the error at the end of learning. Therefore, a reliability-based neural network inverse kinematics solution approach has been presented, and applied to a six-degrees of freedom (dof) robot manipulator in this paper. The structure of the proposed method is based on using three networks designed parallel to minimize the error of the whole system. Elman network, which has a profound impact on the learning capability and performance of the network, is chosen and designed according to the proposed solution method. At the end of parallel implementation, the results of each network are evaluated using direct kinematics equations to obtain the network with best result.     

Inverse Dynamic Model and a Control Application of a Novel 6-DOF Hybrid Kinematics Manipulator

Kinematics with six degrees of freedom can be of several types. This paper describes the inverse dynamic model of a novel hybrid kinematics manipulator. The so-called Epizactor consists of two planar disk systems that together move a connecting element in 6 DOF. To do so each of the disk systems has a linkage point equipped with a homokinetic joint. Each disk system can be described as a serial 3-link planar manipulator with unlimited angles of rotation. To compensate singularities, a kinematic redundancy is introduced via a fourth link. The kinematic concept leads to several technical advantages for compact 6-DOF-manipulators when compared to established parallel kinematics: The ratio of workspace volume and installation space is beneficial, the number of kinematic elements is smaller, and rotating drives are used exclusively. For a singularity-robust control-approach, the inverse dynamic model is derived using the iterative Newton–Euler-method. Feasibility is shown by the application of the model to an example where excessive actuator velocities and torques are avoided.     

Endogenous configuration space approach to mobile manipulators: A derivation and performance assessment of Jacobian inverse kinematics algorithms

By a mobile manipulator we mean a robotic system composed of a non-holonomic mobile platform and a holonomic manipulator fixed to the platform. A taskspace of the mobile manipulator includes positions and orientations of its end effector relative to an inertial coordinate frame. The kinematics of a mobile manipulator are represented by a driftless control system with outputs. Admissible control functions of the platform along with joint positions of the manipulator constitute the endogenous configuration space. Endogenous configurations have a meaning of controls. A map from the endogenous configuration space into the taskspace is referred to as the instantaneous kinematics of the mobile manipulator. Within this framework, the inverse kinematic problem for a mobile manipulator amounts to defining an endogenous configuration that drives the end effector to a desirable position and orientation in the taskspace. Exploiting the analogy between stationary and mobile manipulators we present in the paper a collection of regular and singular Jacobian inverse kinematics algorithms. Their performance is evaluated on the basis of intense computer simulations.     

Dynamics and control of a novel 3-DOF parallel manipulator with actuation redundancy

This paper deals with the dynamics and control of a novel 3-degrees-of-freedom （DOF） parallel manipulator with actuation redundancy. According to the kinematics of the redundant manipulator, the inverse dynamic equation is formulated in the task space by using the Lagrangian formalism, and the driving force is optimized by utilizing the minimal 2-norm method. Based on the dynamic model, a synchronized sliding mode control scheme based on contour error is proposed to implement accurate motion tracking control. Additionally, an adaptive method is introduced to approximate the lumped uncertainty of the system and provide a chattering-free control. The simulation results indicate the effectiveness of the proposed approaches and demonstrate the satisfactory tracking performance compared to the conventional controller in the presence of the parameter uncertainties and un-modelled dynamics for the motion control of manipulators.     

Kinematics,dynamics and dimensional synthesis of a novel 2-DoF translational manipulator

In the past few years, parallel manipulators have become increasingly popular in industry, especially, in the field of machine tools. In this paper, a novel 2-degree-of-freedom (DoF) parallel manipulator, which has two translational DoFs, is proposed. It is characterized by the fact that the output of the manipulator is two planar DoFs of a rigid body, while its orientation remains constant. The inverse and forward kinematics can be described in closed form. The velocity equation, singularity, and workspace of the manipulator are presented. In addition the inverse dynamics problem of the device is investigated employing the Lagrange multipliers approach. The dimensional synthesis based on the workspace and conditioning indices is presented. The proposed manipulator can be applied to the field of machine tools or used as the mobile base for a spatial manipulator. The results of the paper are very useful for the design and application of the new manipulator.