Direct Kinematics Problem of Parallel Robot Based on LS-SVM

Forward kinematics problem of parallel robots is very difficult to solve in comparison to the serial manipulators because of the highly nonlinear relations between joint variables and position and orientation of the end effector. This problem is almost impossible to solve analytically. Numerical methods are one of the common solutions for this problem. But, convergency of these methods is the drawback of using them. In this paper, wavelet based neural network (wave-net) approach is used to solve the forward kinematics problem of the HEXA parallel manipulator. This problem is solved in the typical workspace of this robot. Simulation results show the advantages of employing wavelet neural networks in enhancement of convergency and decreasing modeling errors.

Zeroing dynamics (ZD), as a powerful method proposed for time-varying problems solving, has found successful applications in motion control of serial manipulators. Compared with serial manipulators, parallel manipulators possess some attractive advantages such as high carrying capacity and high operation accuracy. However, the inherent differences between parallel and serial manipulators set great difficulties on the applications of ZD to the control of parallel manipulators. Making progress along this direction, the first ZD based motion controller of parallel manipulators is proposed and the design procedure of the ZD based scheme is presented. As a typical application, the proposed scheme is utilised to control a Stewart platform for tracking a quatrefoil-shaped path, which illustrates the effectiveness of the ZD based scheme.

The type/number of operation modes of a parallel manipulator (PM) may vary with the link parameters of the PM. This paper presents a systematic classification of a 3-RER PM based on the type/number of operation modes. Here, R and E denote revolute joint and planar joint, respectively. The 3-RER PM was proposed as a 4-degree-of-freedom (DOF) 3T1R PM in the literature. Using the proposed method, classification of a PM based on the type/number of operation modes can be carried out in four steps, including formulation of constraint equations of the PM, preliminary classification of the PM using Gröbner cover, operation mode analysis of all the types of PMs using primary decomposition of ideals, and identification of redundant types of PMs. The classification of the 3-RER PM shows that it has 13 types. Besides the two 4-DOF 3T1R operation modes, different types of 3-RER PMs may have up to two more 3-DOF or other types of 4-DOF operation modes. Motion characteristics of the moving platform of 3-RER PMs are also identified using Euler parameter quaternions. This work is the first systematic study on the impact of link parameters on the operation modes of the 3-RER PM and provides a solid foundation for further research on the design and control of 3-RER PMs and other multi-mode (or reconfigurable) PMs.

This paper presents a transformation method to generate the workspace of an n(3RRS) serial–parallel manipulator (S–PM). Firstly, the boundary of the workspace of the 3RRS parallel manipulator (PM) is obtained using the boundary search method and verified by MATLAB/SimMechanics simulation. Secondly, the workspace of the 2(3RRS) S–PM is obtained using the proposed method and verified by MATLAB/SimMechanics simulation. Finally, the workspace of the 3(3RRS) S–PM is obtained using the proposed method. The results indicate that the n(3RRS) S–PM can significantly expand its workspace with the increase in the number of PMs connected in series, and the proposed method can generate the workspaces of n(3RRS) S–PMs efficiently with the advantages of simple operation, high efficiency, and high accuracy. The proposed method is applicable to all S–PMs that are composed of PMs, and it provides a reference for solving the workspaces of other types of redundantly hybrid manipulators.

Parallel manipulators have been proposed to overcome accuracy problem in the end effector positioning, exhibited by serial manipulators(Stewart, 1965)(Reboulet, 1988)(Merlet, 2000). These parallel robots are primarly used in the applications for which the considered processes require a high degree of accuracy, high speeds or accelerations. Aircraft simulator (Stewart, 1965), machining tools (Neugebauer et al., 1998)(Poignet et al., 2002), and various other medical applications (Merlet, 2002)(Leroy et al., 2003)(Plitea et al., 2008) constitute some of the many possible applications of parallel robots. The computation of the inverse dynamic model is essential for an effective robot control. In the field of parallel robots, many approaches have been developed for efficient computation of the inverse dynamics. The formalism of d’Alembert has been used to obtain an analytical expression of the dynamics model (Fichter, 1986)(Nakamura & Ghodoussi, 1989). The principle of virtual works has been applied in (Tsai, 2000) for solving the inverse dynamics of the Gough-Stewart platform and in (Zhu et al., 2005) for a Tau parallel robot. Lagrangian formalism is applied in (Leroy et al., 2003) for the dynamics modeling of a parallel robot used as a haptic interface for a surgical simulator. These various approaches do not seem effective for a robot dynamic control under the real time constraint. A better computational efficiency can be achieved by the development of approaches using recursive schemes, in particular, based on the Newton-Euler formulation. Gosselin (Gosselin, 1996) proposed an approach for the computation of the inverse dynamic model of planar and spatial parallel robots, in which all the masses and inertias are taken into account. This proposed method is difficult to generalize for all the parallel architectures. Dasgupda et al (Dasgupta & Choudhury, 1999) applied this method to several parallel manipulators. Khan (Khan, 2005) has developed a recursive algorithm for the inverse dynamics. This method is applied to a 3R planar parallel robot. Bi et al (Bi & Lang, 2006) use the Newton-Euler recursive scheme for the computation of the articular forces of a tripod system. Khalil et al (Khalil & Guegan, 2004) proposed a general method for the inverse and direct dynamic model computation of parallel robots, which is applied to several parallel manipulators (Khalil & Ibrahim, 2007). Despite the large amount of contributions in this field, there is still a need for improving the computational effeciency of the inverse kinematic and dynamic model clculation for real-time control. In this paper, a parallel robot is considered as a multi robot system with 17

A dimensional synthesis of parallel manipulators (PMs) consists of determining the values of the geometric parameters that affect the platform motion so that a useful workspace with assigned sizes can be suitably located in a free-from-singularity region of its operational space. The main goal of this preliminary dimensioning is to keep the PM far enough from singularities to avoid high internal loads in the links and guarantee a good positioning precision (i.e., for getting good kinematic performances). This paper presents a novel method for the dimensional synthesis of translational PMs (TPMs) and applies it to a TPM previously proposed by the author. The proposed method, which is based on Jacobians’ properties, exploits the fact that TPM parallel Jacobians are block diagonal matrices to overcome typical drawbacks of indices based on Jacobian properties. The proposed method can be also applied to all the lower-mobility PMs with block diagonal Jacobians that separate platform rotations from platform translations (e.g., parallel wrists).

This paper presents a convex hull-based velocity transmission capability analysis of parallel manipulators (PMs) for more accurate design and performance evaluation of PMs. The velocity transmission capability analysis based on the manipulability ellipsoid does not provide accurate information on how fast a PM can move along an arbitrary direction since the joint velocities are normalized by the two-norm (≤ 1). Therefore, for an accurate velocity transmission capability, a computationally efficient convex hull-based approach is proposed here using minimum infinity-norm solutions. Due to the calculation complexity and the dimensional inconsistency, the velocity transmission capability was analyzed separately for the translational and rotational velocity convex hulls. Moreover, for quantitative measures of the translational and rotational velocities, three performance indices (PI), i.e., local, global, and weighted PI, are proposed to visualize dexterousness of PMs. The proposed method is applied to a general 6 degrees of freedom PM to show the effectiveness of the proposed approach.

A new method, Global Newton-Raphson (GNR), for forward kinematics analysis of six degree-of-freedom (DOF) parallel robots is proposed, in order to obtain direct solutions of parallel robots in real time without divergence. The parallel robot can produce 3-DOF linear motions and 3-DOF angular motions, and feeds back positions of actuators measured by position sensors in real time. The GNR for the robot is built by Taylor series expansion and Newton-Raphson iteration with descent operator. Applying the GNR, the real generalized pose of moving platform is solved with respect to the base for a set of given positions of actuators. Accuracy and real time property of the GNR for the 6-DOF parallel robot are analyzed. The theoretical analysis indicated that the presented method can achieve the numerical convergent solution in real time with high accuracy, even the initial guesses are far from a solution.

The unifying idea for most model-based control approaches for parallel mechanism is to derive a minimal-order dynamics model of the system and then design the corresponding controller. The problem with such a control approach is that the controller needs to change its structure whenever the mechanical system changes its number of degrees-of-freedom. This paper presents a projection-based control scheme for parallel mechanism that works whether the system is overactuated or not; it does not require derivation of the minimal-order dynamics model. Since the dimension of the projection matrix is fixed, the projection-based controller does not need to change its structure whenever the mechanical system changes its number of degrees-of-freedom. The controller also allows to specify lower and upper bounds on the actuator forces/torques, making it suitable not only for the control of parallel manipulators with limited force/torque capability of the actuators but also for backlash-free control of parallel manipulators as well as for control of tendon driven parallel manipulators. The stability of the projection-based controllers is rigourously proved, while the condition for the controllability of parallel manipulators is also derived in detail. Finally, experimental results obtained from a simple parallel mechanism, which changes its degrees-of-freedom, are appended. The results also demonstrate that the maximum actuator torque can be reduced by 20% if the actuator saturation is taken into account by the controller.

This paper presents a unified and efficient method of solving the inverse kinematics for six degree-of-freedom serial connections of two three degree-of-freedom serial and parallel manipulators. Based on the kinematic analysis of three degree-of-freedom serial and parallel manipulators, the forward kinematics are formulated for six degree-of-freedom serial connections, including parallel-parallel, serial-parallel, parallel-serial, and serial-serial types. A unified method of solving the inverse kinematics for the serial connections is established: for a given task velocity, the inverse kinematics of two arms are obtained separately, thereby significantly reducing a computational cost. This is made possible by projecting a given velocity onto the infeasible velocity space of one arm independently. Through computational analysis, the proposed method is shown to be effective for the serial connections having at least one parallel manipulator.

The aim of this paper is to characterize the notion of aspect in the workspace and in the joint space for parallel manipulators. In opposite to the serial manipulators, the parallel manipulators can admit not only multiple inverse kinematic solutions, but also multiple direct kinematic solutions. The notion of aspect introduced for serial manipulators in [Borrel 86], and redefined for parallel manipulators with only one inverse kinematic solution in [Wenger 1997], is redefined for general fully parallel manipulators. Two Jacobian matrices appear in the kinematic relations between the joint-rate and the Cartesian-velocity vectors, which are called the "inverse kinematics" and the "direct kinematics" matrices. The study of these matrices allow to respectively define the parallel and the serial singularities. The notion of working modes is introduced to separate inverse kinematic solutions. Thus, we can find out domains of the workspace and the joint space exempt of singularity. Application of this study is the moveability analysis in the workspace of the manipulator as well as path-planing and control. This study is illustrated in this paper with a RR-RRR planar parallel manipulator.

The workspace of a spatial 6-DOF electro-hydraulic parallel manipulator is strongly coupled, due to its multi-closed-loop kinematic structure and the coupling complicates motion planning and control of the parallel manipulator. This paper clearly analyses the strong dynamic coupling property in the workspace of a spatial 6-DOF parallel manipulator, using modal decoupling theory and a frequency responses characteristics analysis method. The dynamic model of a spatial 6-DOF electro-hydraulic parallel manipulator is expressed with the Kane method and hydromechanics principles. The modal analysis method is used to establish the map between strong coupling workspace and decoupled modal space and the dynamic coupling relationship and coupling strength between workspaces are exactly revealed. The quantitative evaluation index of dynamic coupling is presented. Moreover, the relationship between dynamic coupling effects and input is discussed through applying frequency characteristics analysis. Experimental results show the workspace of the parallel manipulator is strongly coupled and the coupling property is coincident with theoretical results.

The linear control method was proposed to control a parallel manipulator with certain motion trajectory of shift operation. Through optimizing the distribution of mechanism, parallel manipulator was controlled with existing commercial controllers. Motion precision of parallel manipulator was improved to the uttermost by parameter optimization. In addition, the mechanism is affected slightly by changes of structural parameters and is calibration-free.

The parallel manipulators with n<6 spherical joint-prismatic joint-spherical joint (SPS) type active legs and one passive constrained leg have attracted more attention and been used in many practical applications. This kind of parallel manipulator has a larger capability of load bearing and a simple structure of active leg. The paper used 4-SPS/PS four degree-of-freedom (DOF) parallel manipulator for the simulation of ship's heave and swing motion. Mobility property of the 4-SPS/PS parallel manipulator is analyzed by screw theory. The inverse position kinematics model is established and the analytical solution is obtained. It uses extra sensor and Sylvester resultant elimination to derive analytical solutions of forward position kinematics including two real solutions and two imaginary solutions, and uses vector cross product to derive the Jacobian matrix of 4-SPS/PS parallel manipulator. Furthermore, it introduces three performance indicators to analyze the singularity. In order to obtain accurate analysis results of singularity, it calculates the minimum singular value, condition number and manipulability degree of Jacobian matrix for different combination of pose, respectively.

In this paper, the kinetostatic performance and collision-free workspace of a developed 3-DOF parallel robot is investigated. Kinetostatic indices that examine the performance of robot are essential in robot design and control. The popular indices, manipulability, dexterity and kinematic sensitivity are all investigated in order to evaluate the performance of the developed 3-DOF Delta parallel robot. In addition, as parallel robots generally have a limited workspace, obtaining the workspace by considering the mechanical interference is of paramount importance. The mechanical interference, including collision of links, collision of links with obstacles, collision of the end-effector with obstacles, are also examined by using a new geometrical approach. The proposed method is based on the segment-to-segment intersection test and can be expanded to a wide range of robotic mechanical systems, including, among others, cable-driven parallel robots. Moreover, in this paper, a new CFW index is defined which can be used to investigate the workspace with respect to mechanical interference. Furthermore, the aforesaid index provides some insight into obtaining a well-conditioned workspace.