A new method for optimum design of parallel manipulator based on kinematics and dynamics

This paper presents a new method for the optimum design of parallel manipulators by taking both the kinematics and dynamic characteristics into account. The optimum design of a 3-DOF 4-RRR planar parallel manipulator with actuation redundancy is investigated to demonstrate the method. The kinematic performance indices such as the conditioning index, the velocity index, and workspace area are analyzed. Further, the dynamic dexterity, which is used to evaluate the dynamic characteristics, is investigated. The corresponding atlases are represented graphically in the established design space. Based on these atlases, the geometrical parameters without dimension are determined. Then the optimum dimensional parameters are achieved based on the optimum non-dimensional result. By using the method proposed in this paper, the designer can obtain the optimum result with respect to both kinematic performance indices and dynamic performance indices. Since the dynamic performance is considered in the process of optimum design by using the method proposed in this paper, it is expected to realize the high dynamics of parallel manipulators.     

Design and analysis of a novel 6-DOF redundant actuated parallel robot with compliant hinges for high precision positioning

This paper presents the design and modeling of a new 6-DOF 8-PSS/SPS compliant dual redundant parallel robot with wide-range flexure hinges. This robot can achieve either high accurate positioning or rough positioning as well as a 6-DOF active vibration isolation and excitation to the payload placed on the moving platform. Adopting a kind of wide-range flexure hinge, we establish the kinematics model of the macro parallel mechanism system via the stiffness model and Newton–Raphson method, then we build up the dynamics model using Kane’s method for the micro-motion system. The investigations of this paper will provide suggestions to improve the structure and control algorithm optimization for a novel compliant dual redundant parallel mechanism in order to achieve the feature of larger workspace, higher motion precision and better dynamic characteristics. The results will be helpful in modifying the structure of the prototype platform to enhance its high kinematics and dynamics properties.     

Forward displacement analysis of a quadratic 4-DOF 3T1R parallel manipulator

A quadratic parallel manipulator refers to a parallel manipulator with a quadratic characteristic polynomial. This paper revisits the forward displacement analysis (FDA) and the Type II singularity analysis of a quadratic 4-DOF 3T1R (SCARA) parallel manipulator: the Quadrupteron. It will be proved that there exists a one-to-one correspondence between the two formulas, each producing one solution to the FDA, and the two singularity-free regions. Therefore, a unique solution to the FDA can be obtained in a straightforward way for such a parallel manipulator if the singularity-free region in which it works is specified. The Type II singularity analysis in the joint space will also be investigated in order to identify the conditions on the inputs to keep the Quadrupteron working in the same singularity-free region in its Cartesian workspace.     

Analysis of a 6-DOF redundantly actuated 4-legged parallel mechanism

In this paper, we propose a new 4-legged 6-DOF redundantly actuated UPS parallel mechanism. Kinematics, singularity and dynamics analyses of the proposed mechanism are addressed. Inverse kinematics has a closed-form solution, whereas the forward kinematics has a semi-closed form solution. Also, the Jacobian matrix has been determined using the concept of reciprocal screws. Using the principle of virtual work and minimum norm method, a general formulation for the inverse dynamics of redundantly actuated parallel manipulator is presented. Moreover, the proposed redundant mechanism has been compared with a similar but nonredundant mechanism in three aspects: reachable points, and actuation forces under static and dynamic conditions. We show that the redundancy decreases singular points, and dramatically reduces the actuators’ forces and torques.     

Structural synthesis of fully-isotropic translational parallel robots via theory of linear transformations

The paper presents a new structural synthesis approach of fully-isotropic translational parallel robotic manipulators (TPMs) based on the theory of linear transformations. A TPM is a 3-DOF (degree of freedom) parallel mechanism whose output link, called platform, can achieve three independent orthogonal translational motions with respect to the fixed base. The manipulators presented in this paper have three legs connecting the moving platform and the base (fixed platform). Only one kinematic pair per leg is actuated by a linear motor situated on the fixed base. A one-to-one correspondence exists between the actuated joint space and the operational space of the moving platform. The Jacobian matrix of fully-isotropic TPMs presented in this paper is the identity 3×3 diagonal matrix throughout the entire workspace. The synthesis method proposed in this paper allows us to obtain all structural solutions of fully-isotropic TPMs in a systematic manner. Overconstrained/isostatic solutions with elementary/complex and identical/different legs are obtained. Fully-isotropic TPMs have the advantage of simple command and important energy-saving due to the fact that, for a unidirectional translation, only one motor works as in a serial translational manipulator.     

A methodology to achieve the set of operation modes of reconfigurable parallel manipulators

The demand for increasingly more versatile machinery has boosted the development of the so-called reconfigurable mechanisms. In this paper, the authors present a general methodology to assess the multioperational capacity of a 6-DOF parallel manipulator basing on the possible motion patterns having Lie group structure that the manipulator owns. This ability of having different operation modes enables the manipulator to adapt to diverse tasks. To show the potential of the methodology, this approach has been applied to the 6-DOF 3-CPCR which is capable of generating multiple motion patterns. In addition to carrying out the complete theoretical study in which all the different operation modes are obtained, and validating the procedure with GIM software designed for kinematic analysis and design of mechanisms, a demonstration prototype of the 3-CPCR parallel manipulator has been also built.

     

Nonlinear adaptive task space control for a 2-DOF redundantly actuated parallel manipulator

A nonlinear adaptive (NA) controller in the task space is developed for the trajectory tracking of a 2-DOF redundantly actuated parallel manipulator. The dynamic model with nonlinear friction is established in the task space for the parallel manipulator, and the linear parameterization expression of the dynamic model is formulated. Based on the dynamic model, a new control law including adaptive dynamics compensation, adaptive friction compensation and error elimination items is designed. After defining a quadratic performance index, the parameter update law is derived with the gradient descent algorithm. The stability of the parallel manipulator system is proved by the Lyapunov theorem, and the convergence of the tracking error and the error rate is proved by the Barbalat’s lemma. The NA controller is implemented in the trajectory tracking experiments of an actual 2-DOF redundantly actuated parallel manipulator, and the experiment results are compared with the APD controller.     

Dexterity measures and their use in quantitative dexterity comparisons

The majority of proposed dexterity measures rely on the use of the condition number of the manipulator’s Jacobian matrix mapping actuator velocities to end effector velocities. Unfortunately, for the vast majority of manipulators, the conventional Jacobian matrix has inconsistent units making the aforementioned measures dependent on units and scale. Recently, a Jacobian matrix mapping the joint velocities to independent Cartesian velocity components of three points on the end effector was proposed. In most cases, this Jacobian matrix has consistent units and in many cases it is dimensionless. This dimensionally-homogeneous Jacobian matrix yields meaningful dexterity measures that allow quantitative dexterity comparisons between manipulators with different architectures. Although these new measures were originally proposed within the context of parallel manipulator design and analysis, they can also be used for serial architectures. In this paper, the analysis of the Tricept manipulator’s dexterous workspace, as a function of architectural variables, is provided through the use of a quantitative metric for dexterity. Furthermore, the generality of this metric is also demonstrated by first employing it to analyze a serial manipulator and then comparing it to a parallel manipulator having the same degrees-of-freedom. The paper also proposes a strategy to deal with singularities introduced mathematically by the novel Jacobian formulation.     

Kinematic analysis of 5-R<Emphasis Type="Underline">P</Emphasis>UR (3T2R) parallel mechanisms

This paper investigates some kinematic properties of a five-degree-of-freedom parallel mechanism generating the 3T2R motion and comprising five identical limbs of the RPUR type. The general mechanism originates from the type synthesis performed for symmetrical 5-DOF parallel mechanism. In this study, two classes of simplified designs are proposed whose forward kinematic problem have either a univariate or a closed-form solution. The principal contributions of this study are the solution of the forward kinematic problem for some simplified designs—which may have more solutions than the FKP of the general 6-DOF Stewart platform with 40 solutions—and the determination of the constant-orientation workspace which is based on the topology of the vertex space (Bohemian dome) and a geometric constructive approach.     

Workspace and Optimal Design of a Pure Translation Parallel Manipulator

This paper presents a study of a pure translation three degrees of freedom fully-parallel manipulator, known as the Tsai manipulator. Based on the static analysis and on the determination of the singularity loci of the manipulator, criteria for the optimal geometric design of the manipulator for a given workspace free from singularities are proposed.     

Kinematic modeling and dexterity evaluation of a PS-RRS-2RUS parallel manipulator used for controllable pitch propeller

This article investigates the kinematic modeling and dexterity evaluation of a PS-RRS-2RUS parallel manipulator. The design concept and mobility analysis of the manipulator are first addressed utilizing the screw theory. Second, the decoupled and closed-form kinematic solutions with multiple configurations are solved through vector method. Then, the analytical expressions for the velocity relationships are derived into a closed-form Jacobian matrix, which is then used to distinguish the singular postures. Finally, the workspace with fixed orientation is calculated and its dexterity is evaluated. The numerical simulations and validation are conducted in a case study.     

Design of a 6-DOF force device for virtual assembly (FDVA-6) of mechanical parts

Among different interaction modalities, force feedback is one of the key technologies to increase the interactivity and immersion of a virtual assembly process. This paper presents a 6-DOF force device with its forward kinematic analysis, workspace simulation, and gravity compensation. To evaluate the device, a prototype system is developed and a case study is conducted to assemble a mechanical product. The users have given positive feedback on the gravity compensation implemented and the general performance of the device.     

Comparison of 3-RPR planar parallel manipulators with?regard to their kinetostatic performance and sensitivity to geometric uncertainties

This paper deals with the sensitivity analysis of 3-RPR planar parallel manipulators. First, the manipulators under study as well as their degeneracy conditions are presented. Then, an optimization problem is formulated in order to obtain their maximal regular dexterous workspace. Moreover, the sensitivity coefficients of the pose of the manipulator moving platform to variations in the geometric parameters and in the actuated variables are expressed algebraically. Two aggregate sensitivity indices are determined, one related to the orientation of the manipulator moving platform and another one related to its position. Then, we compare two non-degenerate and two degenerate 3-RPR planar parallel manipulators with regard to their dexterity, workspace size and sensitivity. Finally, two actuating modes are compared with regard to their sensitivity.     

漂浮载体上机械臂的工作空间

本文讨论载于航天器的机械臂的工作空间问题，导出机械臂位形计算的普遍公式。对不同载体控制方案的机械臂工作空间进行分析和对比，并提出了为扩大工作空间的机械设计原则，最后讨论了负载对工作空间的影响。     

Complete kinematics of a serial–parallel manipulator formed by two Tricept parallel manipulators connected in serials

Complete kinematic is an essential and a challenging work for series–parallel manipulators (S–PMs). This paper studied the complete kinematic of a 2(3-SPS+UP) series–parallel manipulator. First, a S–PM formed by two well-known Tricept parallel manipulators (PMs) connected in serial is presented. Second, the forward and inverse displacements are studied using sylvester dialytic elimination method. Third, the forward and inverse Jacobian matrices are established based on integrating the constraint and coupling information of the single PMs into the S–PM. Fourth, simple and compact formulae for the forward and inverse acceleration are derived using vector approach. Finally, the workspace of this S–PM is constructed using CAD variation geometry approach. The results show that the 2(3-SPS+UP) S-PM has multiple forward and inverse position solutions. The existence and uniqueness of the forward, inverse Jacobian matrices and the acceleration formula are shown from their explicit form. The workspace analysis shows that this S–PM has large workspace. The research works provided a theoretical basis for the novel 2(3-SPS+UP) S–PM, as well as a feasible approach for establishing the complete kinematics for S–PMs.     

Kinematic design of 3-URU pure rotational parallel mechanism to perform precise motion within a large workspace

We present an optimum design of lower-dof parallel mechanism, a 3-URU pure rotational parallel mechanism that reflects issues of workspace and the position error of the center of rotation of the platform. The uncompensatable error determined by position error of center of rotation was used as an evaluation index for the design. The uncompensatable error index, an index used in the optimum design, was proposed taking into account four sources of errors, representing errors between adjacent joints. Based on the application of the mechanism and the error index, the effect of the redundant platform orientation parameter was numerically investigated and the design flow of the mechanism was proposed. We made a kinematic design of a mechanism with a large workspace subject to minimization of platform’s position error of the center of rotation. A prototype of mechanism with a large inclination angle of the platform up to 1.3 rad was shown, and its characteristics are also discussed.     

Acceleration and singularity analyses of a parallel manipulator with a particular topology

In this work, two essential steps, the kinematics and the singularity analysis, dealing with the design process of a parallel manipulator are investigated by means of the theory of screws. The proposed mechanism for the analysis is a parallel manipulator with three degrees of freedom. A simple and compact expression is derived here for the computation of the reduced acceleration state of the moving platform, w.r.t. the fixed platform, by taking advantage of the properties of reciprocal screws, via the Klein form of the Lie algebra e(3). To this end, the reduced acceleration state of the moving platform is written in screw form through each one of the three actuator limbs of the manipulator. Afterwards, the acceleration analysis is completed by taking into proper account the decoupled motion of the parallel manipulator. Of course, as an intermediate step this contribution also provides the velocity analysis of the parallel manipulator. The expressions obtained via screw theory are compact and can be easily translated into computer codes. A numerical example is provided to demonstrate the efficacy of screw theory to efficiently analyze the kinematics of the chosen parallel manipulator. Finally, the numerical results from the kinematic analysis are compared with results produced with a commercially available dynamic and kinematic simulation program.     

Dynamic modeling and computational efficiency analysis for a spatial 6-DOF parallel motion system

A common approach for simplification analysis of complex dynamic model is presented, and the simplified dynamic model of a spatial 6-DOF parallel motion system with high computational efficiency is proposed for high real time control. By using Kane method, the full dynamic model of a spatial 6-DOF parallel motion system viewed as 13 rigid bodies is built. With rigid body decomposition, the full dynamic model is separated into several parts firstly, and then some separated parts are further divided into many dynamic components in terms of motion separation and the relationship with acceleration or velocity. The contribution of each dynamic term is analyzed for a specified spatial 6-DOF parallel motion system, and the simplified model is derived. Comparing with full dynamic model, the simplified error is analyzed, and the computational efficiency of the simplified model is discussed in a real-time industrial computer. The simplified strategy is confirmed in simulation. The simplified error is less than 8%, the simplified model can improve the computational efficiency by more than 70%, and the execution time is less than 0.1 ms, which can achieve the requirements of high real time control. The numerical results illustrate that the proposed approach is feasible and effective for simplification analysis of dynamics and the derived simplified dynamic model can be used in real-time control system with small simplified error.     

Inverse dynamics analysis of a general spherical star-triangle parallel manipulator using principle of virtual work

Inverse dynamics of a general model of a spherical star-triangle (SST) parallel manipulator (Enferadi and Akbarzadeh Tootoonchi, Robotica 27:663–676, 2009) is the subject of this paper. This manipulator is of type 3-RRP, has good accuracy and relatively a large workspace which is free of singularities (Enferadi and Akbarzadeh Tootoonchi, Robotica, Revised paper, 2009). First, inverse kinematics utilizing the angle axis representation is solved. Next, velocity and acceleration analysis as well as link Jacobian matrices are obtained in invariant form. Finally, a systematic approach based on the principle of virtual work and the concept of link Jacobian matrices is presented. This method allows elimination of constraint forces and moments at the passive joints from motion equations. It is shown that the dynamics of the manipulator can be reduced to solving a system of three linear equations with three unknowns. Moreover, a computational algorithm for solving the inverse dynamics is developed. Two examples with different trajectories for the moving spherical platform are presented and motor torques are obtained. Results are verified using a commercial dynamics modeling package.     

Design and analysis of a compliant parallel pan-tilt platform

In combination of the advantages of both parallel mechanisms and compliant mechanisms, a compliant parallel mechanism with two rotational degrees of freedom is designed to meet the requirement of a lightweight and compact pan-tilt platform. Firstly, two commonly-used design methods i.e. direct substitution and Freedom and Constraint Topology are applied to design the configuration of the pan-tilt system, and similarities and differences of the two design alternatives are compared. Then inverse kinematic analysis of the candidate mechanism is implemented by using the pseudo-rigid-body model, and the Jacobian related to its differential kinematics is further derived to help designer realize dynamic analysis of the 8R compliant mechanism. In addition, the mechanism’s maximum stress existing within its workspace is tested by finite element analysis. Finally, a method to determine joint damping of the flexure hinge is presented, which aims at exploring the effect of joint damping on actuator selection and real-time control. To the authors’ knowledge, almost no existing literature concerns with this issue.