A novel approach for forward dynamic analysis of 3-PRS parallel manipulator with consideration of friction effect

In this paper, a novel decomposition approach to formulate the dynamic model of a 3-Prismatic, Revolute, Spherical (3-PRS) parallel manipulator is proposed. Since the constraint forces arising from the holonomic constraints would not generate a net force or torque to manipulate the motion of the moving platform, the fact motivates to decompose the reaction forces applied to the connecting joints. The decomposition leads to a transformation matrix which can project the dynamic forces of the moving platform formulated in the task space into the joint space. A sufficient condition is determined to guarantee the existence of the projection matrix. Based on the proposed approach, the inverse of 21×21 augmented matrix formulated by conventional approach in solving forward dynamic problem can be decomposed into that of 6×6 and 15×15 matrices. The computational efficiency can therefore be improved about 23.5%. Besides, since the reaction forces can be calculated simultaneously, each of the actuated legs can be decoupled to calculate the normal forces applied to the sliding planes of the prismatic joints. This feature makes it possible to include the effect of corresponding friction forces into consideration. Since the normal forces applied to the sliding planes vary with the reaction forces, the corresponding friction forces are not only the function of sliding velocities but also the dynamics of the whole mechanism. Computer simulations are performed to verify the proposed approach and analyze the effect of the friction forces on the motion accuracy of the moving platform.     

Modeling and control of micropositioning systems using Stewart platforms

The modeling and control of a 6‐DOF Stewart micropositioning system with each leg actuated by a respective piezoelectric actuator are considered in this paper. The 12 multi‐DOF passive joints are assumed to be well designed and fabricated so that guaranteed guiding precision and lack of backlash can be obtained. The dynamics model of the micropositioning system is derived first, and then a composite control strategy consisting of moving platform model‐based feedback linearization and two sets of simple SISO fuzzy systems is proposed. By considering the internal axial forces of the six legs on the six spherical joints at the moving end as the virtual control inputs of the moving platform, feedback linearization can be easily used to derive the desired control forces for the moving platform. The corresponding desired linear displacement of each piezoelectric actuator can then be computed based on the derived leg model, and each piezoelectric actuator's control voltage can be generated by the first set of independent leg fuzzy controls. The second set of fuzzy controls is suggested for the further enhancement of robustness with respect to uncertainty. Computer simulations are presented to illustrate the effectiveness of the suggested micropositioning control strategy. ©2000 John Wiley & Sons, Inc.     

Dynamics analysis of a novel 5-DoF 3SPU+2SPRR type parallel manipulator

A novel 5-DOF 3SPU+2SPRR type parallel manipulator is proposed. First, the formulae are derived for solving the kinematics parameters of the moving platform. Second, the kinematics of the active legs and connection rod are analyzed, and the formulae for solving velocity and acceleration of the active legs and connection rod are derived. Third, the formulae are derived for solving the dynamic active and constrained forces. Finally, an analytic example is given for solving the dynamics, and the analytic solved results are verified by the mechanism simulation. This paper is aimed at laying a solid theoretical and technical foundation for its prototype manufacture and control.     

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.     

冗余驱动并联机械手的混合位置/力自适应控制

针对冗余驱动并联机构研究一种自适应的混合位置/力控制算法.基于并联机构中约束 子流形的几何性质,将冗余驱动并联机构的逆动力学自然投影到位形空间和约束力空间.基于投 影方程,提出一种统一的具有渐进稳定性的自适应混合位置/力控制算法.采用最小二范数准则 求解冗余解问题,实现了实际驱动关节力矩的优化.仿真结果验证了控制方法的有效性.     

Generation and application of prestress in redundantly full-actuated parallel manipulators

This paper addresses the inverse dynamics of redundantly actuated parallel manipulators. Such manipulators feature advantageous properties, such as a large singularity-free workspace, a high possible acceleration of the moving platform, and higher dexterity and manipulability. Redundant actuation further allows for prestress, i.e., internal forces without generating end-effector wrenches. This prestress can be employed for various goals. It can potentially be used to avoid backlash in the driving units or to generate a desired tangential end-effector stiffness. In this paper, the application of prestress is addressed upon the inverse dynamics solution. A general formulation for the dynamics of redundantly actuated parallel manipulators is given. For the special case of simple redundancy, a closed-form solution is derived in terms of a single prestress parameter. This yields an explicit parametrization of prestress. With this formulation an open-loop prestress control is proposed and applied to the elimination of backlash. Further, the generation of tangential end-effector stiffness is briefly explained. The approach is demonstrated for a planar 4RRR manipulator and a spatial heptapod.     

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.     

Hybrid force and motion control of flexible joint parallel manipulators using inverse dynamics approach

An inverse dynamics control algorithm is developed for hybrid motion and contact force trajectory tracking control of flexible joint parallel manipulators. First, an open-tree structure is considered by the disconnection of adequate number of unactuated joints. The loop closure constraint equations are then included. Elimination of the joint reaction forces and the other intermediate variables yield a fourth-order relation between the actuator torques and the end-effector position and contact force variables, showing that the control torques do not have an instantaneous effect on the end-effector contact forces and accelerations because of the flexibility. The proposed control law provides simultaneous and asymptotically stable control of the end-effector contact forces and the motion along the constraint surfaces by utilizing the feedback of positions and velocities of the actuated joints and rotors. A two degree of freedom planar parallel manipulator is considered as an example to illustrate the effectiveness of the method.     

A Study of Reactional Force Compensation Based on Three-Degree-of Freedom Parallel Platforms

Reaction compensation is necessary for space robot applications because reactional forces/torques will cause undesired movement of the spacecraft. Because the reactional torques can be compensated by the existing torque balancing device in the spacecraft, an additional reaction compensating device is necessary to compensate the reactional forces. In this article we study two types of reactional force compensating devices based on three-degree-of freedom, parallel platforms. The first type has three R-P-S legs while the second type has three R-R-P-S leg and a passive R-R-P leg. A three-degree of freedom serial manipulator is used to generate reactional forces, which are to be compensated by the paralle platforms. The kinematics and dynamics of both platforms are analyzed and closed form inverse kinematics solutions are derived. We then design a reactional force compensating device that satisfies the strict volume constraint in a spacecraft. The first type of parallel platform is found to require very long legs due to large orientational motion at certain positions. The second type has smooth motion in both position and orientation, and therefore its size can be very compact. It is concluded that the second type of parallel platform has great potential to be used as a compact reactional force compensating device. © 1995 John Wiley & Sons, Inc.     

Using CAD geometric variation approach machining complex workpiece by a 3-SPR parallel machine tool

A novel CAD geometric variation approach is proposed for machining the complex workpiece, caving letter on a 3D free-form surface, and milling cam by a 3-SPR parallel machine tool. First, a simulation mechanism of the 3-SPR parallel manipulator is created, and a workspace of the moving platform is constructed by the 3-SPR simulation mechanism. Second, the tool path guiding plane with complex workpiece and the simulation mechanism of the 3-SPR parallel machine tool are combined to form the whole simulation mechanism. Third, the extension of the active legs and the position of the moving platform are solved automatically and visualized dynamically by using simulation mechanism. Finally, the formulae for solving displacement kinematics of the 3-SPR parallel machine tool are derived. The results of the simulation mechanism are verified by kinematic analytic results. The CAD geometric variation approach is straightforward without compiling any program.     

Kinematics and dynamics analyses of a parallel manipulator with three active legs and one passive leg by a virtual serial mechanism

A novel CAD variation geometry approach and a virtual serial mechanism approach are proposed for analyzing the kinematics and dynamics of a parallel manipulator with three SPS-type active legs and one PU-type constrained passive leg. First, a simulation mechanism of this parallel manipulator is created, and some kinematic characteristics are analyzed. Second, the inverse formulae for solving pose and Jocabian matrix are derived, and workspace and singularity are determined. Third, a virtual serial mechanism is created, and the analytic formulae for solving active forces and constrained wrench of these parallel manipulators are derived. The analytic results are verified by using its simulation mechanism.     

Type synthesis of 5‐DOF parallel manipulators based on screw theory

With the introduction of virtual chains to represent the motion patterns of 5‐DOF motions, a classification of 5‐DOF PMs (parallel manipulators) is proposed at first. A general method for the type synthesis of 5‐DOF PMs is then proposed based on screw theory and using the concept of virtual chains. The type synthesis of US‐equivalent PMs is presented in detail to show the application of the proposed approach. US‐equivalent PMs are the parallel counterparts of the 5‐DOF US serial manipulators. For a US‐equivalent PM, the moving platform can rotate arbitrarily about a point moving along a spherical surface. The type synthesis of legs for US‐equivalent PKCs (parallel kinematic chains), the type synthesis of US‐equivalent PKCs, as well as the selection of actuated joints of US‐equivalent PMs are dealt with in sequence. US‐equivalent PKCs with and without inactive joints are synthesized. Several US‐equivalent PMs as well as other classes of 5‐DOF PMs with identical type of legs are obtained. © 2005 Wiley Periodicals, Inc.     

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.     

Dynamics and Coupling Actuation of Elastic Underactuated Manipulators

This paper investigates the constraint and coupling characteristics of underactuated manipulators by proposing an elastic model of the manipulator and examining the second order constraint equation. A dynamic model and a coupling constraint equation are developed from a Jacobian matrix and the Newton‐Euler formulation. The inertia matrix and the Christoffel tensor are analyzed and decomposed into the part concerning actuated joints and the part concerning passive joints. This decomposition is further extended to the dynamic coupling equation and generates an actuation coupling matrix and a dynamic coupling tensor. Two new dynamic coupling indices are hence identified. One is related to an actuation input and the other is related to centrifugal and Coriolis forces. The former reveals the dynamic coupling between the input and the acceleration of passive joints and gives the actuation effect on the passive joints. The latter reveals the dynamic coupling between the centrifugal and Coriolis forces and the acceleration of passive joints and provides the centrifugal and Coriolis effect on the acceleration of passive joints. The study reveals the coupling characteristics of an underactuated manipulator. This is then demonstrated in a three‐link manipulator and extended to a serial manipulator with passive prismatic joint. © 2003 Wiley Periodicals, Inc.     

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.     

Inverse Kinematics and Workspace Analysis of a 3 DOF Flexible Parallel Humanoid Neck Robot

To mimic the human neck’s three degree-of-freedom (DOF) rotation motion, we present a novel bio-inspired cable driven parallel robot with a flexible spine. Although there exists many parallel robotic platform that can mimic the human neck motion, most of them have only two DOF, with the yaw motion being actuated separately. The presented flexible parallel humanoid neck robot employs a column compression spring as the main body of cervical vertebra and four cables as neck muscles to connect the base and moving platform. The pitch and roll movements of moving platform are realized by the two dimensional lateral bending motion of the flexible spring, and a bearing located at the top of the compression spring and embedded in the moving platform is used to achieve the yaw motion of the moving platform. By combing the force and torque balance equations with the lateral bending statics of the spring, inverse kinematics and optimizing the cable placements to minimize the actuating cable force are investigated. Moreover, the translational workspace corresponding to pitch and roll movements and rotational workspace corresponding to yaw movement are analyzed with positive cable tension constraint. Extensive simulations were performed and demonstrated the feasibility and effectiveness of the proposed inverse kinematics and workspace analysis of the novel 3 DOF flexible parallel humanoid neck robot.     

用6-RUS型并联机构实现海上补给自动对接的关键技术

提出了用6-RUS型并联机构来实时调整对接喇叭口的位姿状态以实现海上补给中的自动对接.首先,对该机构支链的相对位置关系及约束进行了讨论,建立了该机构的逆运动学模型;其次,对该机构支链的受力情况进行了理论分析,基于凯恩方程建立了该机构的逆动力学模型;最后,对该机构在运动过程中电机输出转角及输出扭矩进行了数值仿真.仿真结果...     

Control-orientated dynamic modeling of forging manipulators with multi-closed kinematic chains

The dynamic model, particularly with reference to controller design, is an important issue in mechanical control and design. However, this model is often difficult to achieve in complex multi-closed-loop mechanisms, such as parallel mechanisms or forging manipulators. A new approach on the dynamic modeling of a multi closed-chain mechanism in a forging manipulator, which applies screw theory and the reduced system model, is proposed in this paper. The proposed method not only allows a straightforward calculation of actuator forces but also obtains the dynamic equation of the multi-closed-loop mechanism easily. The structure of dynamic model obtained is similar to that of standard Lagrangian formulations, which can extend vast control strategies developed for serial robots to complex multi-closed-loop mechanisms. A complex multi-closed-loop mechanism on a forging manipulator is decomposed into several serial mechanisms or simpler subsystems. The Lagrangian equations associated with each subsystem are directly derived from the local generalized coordinates of the sub-mechanisms. Jacobian matrices are used to interpret the differential equations of the sub-mechanisms into the generalized coordinate or the actuated pairs according to the D’Alembert principle. Hessian matrices are also applied to form a standard Lagrangian formulation. The screw theory is introduced to overcome the difficulties of solving transformed Jacobian matrices, thereby simplifying the calculation of the matrices. Computation difficulties of transformation matrices may decrease considerably by choosing suitable generalized coordinates instead of direct actuator variables. The full dynamics of the complex multi-closed-loop mechanism in a forging manipulator is presented. Simulations and experiments illustrate the reliability of the proposed method and the correctness of the dynamic model of the forging manipulator.     

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.     

Optimal Path Programming of the Stewart Platform Manipulator Using the Boltzmann–Hamel–d'Alembert Dynamics Formulation Model

In this paper, the dynamics formulation of a general Stewart platform manipulator (SPM) with arbitrary geometry and inertia distribution is addressed. Based on a structured Boltzmann–Hamel–d'Alembert approach, in which the true coordinates are for translations and quasi-coordinates are for rotations, a systematic methodology using the parallelism inherent in the parallel mechanisms is developed to derive the explicit closed-form dynamic equations which are feasible for both forward and inverse dynamics analyses in the task space. Thus, a singularity-free path programming of the SPM for the minimum actuating forces is presented to demonstrate the applications of the developed dynamics model. Using a parametric path representation, the singularity-free path programming problem can be cast to the determination of undetermined control points, and then a particle swarm optimization algorithm is employed to determine the optimal control points and the associated trajectories. Numerical examples are implemented for the moving platform with constant orientations and varied orientations.