Algebraic solution to forward kinematics of a 3‐DOF spherical parallel manipulator

A three degree‐of‐freedom (3‐DOF) spherical parallel manipulator consists of two tetrahedrons (pyramids). The base tetrahedron is fixed while the moving tetrahedron is rotating at the joint apex of the two tetrahedrons. This article studies the forward kinematics to a special 3‐DOF spherical parallel manipulator, where the three apical angles of the moving tetrahedron are equal to their counterparts in the base tetrahedron, respectively. The final result of the forward kinematics to this parallel manipulator is a univariate quartic polynomial equation, which has a direct algebraic solution. In addition, a special right‐angle case of the manipulator is investigated and its forward kinematics can be obtained directly. © 2001 John Wiley & Sons, Inc.     

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.     

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.     

A new method for actuating parallel manipulators

This paper presents a new technique of actuating a parallel platform manipulator using shape memory alloy (SMA). This is a type of smart materials that can attain a high strength-to-weight ratio, which makes them ideal for miniature application. The work is mainly to develop a new SMA actuator and then incorporating the actuator in building the parallel manipulator prototype. The SMA used in this study is a commercial NiTi wire. The SMA wire provides an actuating force that produces a large bending and end displacement. A 3-UPU (universal–prismatic–universal) parallel manipulator using linear SMA actuators was developed. The manipulator consists of a fixed platform, a moving platform and three SMA actuators. The manipulator workspace was specified based on the restrictions due to actuator strokes and joint angle limits. System identification techniques were used to model both heating and cooling processes. An ON/OFF control was performed and the results showed closeness in simulation and experimental results. This study showed that shape memory alloy actuated beam can successfully be used to provide linear displacement. The built prototype indicates the feasibility of using SMA actuators in parallel manipulators.     

Formal analysis of the kinematic Jacobian in screw theory

As robotic systems flourish, reliability has become a topic of paramount importance in the human–robot relationship. The Jacobian matrix in screw theory underpins the design and optimization of robotic manipulators. Kernel properties of robotic manipulators, including dexterity and singularity, are characterized with the Jacobian matrix. The accurate specification and the rigorous analysis of the Jacobian matrix are indispensable in guaranteeing correct evaluation of the kinematics performance of manipulators. In this paper, a formal method for analyzing the Jacobian matrix in screw theory is presented using the higher-order logic theorem prover HOL4. Formalizations of twists and the forward kinematics are performed using the product of exponentials formula and the theory of functional matrices. To the best of our knowledge, this work is the first to formally analyze the kinematic Jacobian using theorem proving. The formal modeling and analysis of the Stanford manipulator demonstrate the effectiveness and applicability of the proposed approach to the formal verification of the kinematic properties of robotic manipulators.     

Kinematics analysis of a hybrid manipulator for computer controlled ultra-precision freeform polishing

As one of the final processing steps of precision machining, polishing process is a very key decision for surface quality. This paper presents a novel hybrid manipulator for computer controlled ultra-precision (CCUP) freeform polishing. The hybrid manipulator is composed of a three degree-of-freedom (DOF) parallel module, a two DOF serial module and a turntable providing a redundant DOF. The parallel module gives the workpiece three translations without rotations. The serial module holds the polishing tool and gives it no translations on the polishing contact area due to its particular mechanical design. A detailed kinematics model is established for analyzing the kinematics of the parallel module and the serial module, respectively. For the parallel module, the inverse kinematics, the forward kinematics, the Jacobian matrix, the workspace and the dexterity distribution are analyzed systematically. Workspaces are also investigated for varying structural parameters. For the serial module, the inverse kinematics, the forward kinematics, the workspace and the precession motion analysis are carried out. An example of saddle surface finishing with this manipulator is given and the movement of actuators with respect to this shape is analyzed theoretically. These analysis results illustrate that the proposed hybrid manipulator is a very suitable machine structure for CCUP freeform polishing.     

On the dynamic model and kinematic analysis of a class of Stewart platforms

In this paper, a dynamic model for a class of Stewart platform (six degrees of freedom parallel link robotic manipulators) is derived by using tensor representation. A set of six Lagrange's equations are obtained. The kinematics analysis for a class of Stewart platform is conducted and a sixteenth order polynomial equation corresponding to the forward kinematic solution of the Stewart platform is obtained, which gives all possible global solutions of a manipulator configuration for a given set of six leg lengths.     

Forward kinematics,performance analysis,and multi-objective optimization of a bio-inspired parallel manipulator

In this paper, a bio-inspired parallel manipulator with one translation along z-axis and two rotations along x- and y- axes is developed as the hybrid head mechanism of a groundhog robotic system. Several important issues including forward kinematic modeling, performance mapping, and multi-objective improvement are investigated with specific methods or technologies. Accordingly, the forward kinematics is addressed based on the integration of radial basis function network and inverse kinematics. A novel performance index called dexterous stiffness is defined, derived and mapped. The multi-objective optimization with particle swarm algorithm is conducted to search for the optimal dexterous stiffness and reachable workspace.     

A study of neural network based inverse kinematics solution for a three-joint robot

A neural network based inverse kinematics solution of a robotic manipulator is presented in this paper. Inverse kinematics problem is generally more complex for robotic manipulators. Many traditional solutions such as geometric, iterative and algebraic are inadequate if the joint structure of the manipulator is more complex. In this study, a three-joint robotic manipulator simulation software, developed in our previous studies, is used. Firstly, we have generated many initial and final points in the work volume of the robotic manipulator by using cubic trajectory planning. Then, all of the angles according to the real-world coordinates (x, y, z) are recorded in a file named as training set of neural network. Lastly, we have used a designed neural network to solve the inverse kinematics problem. The designed neural network has given the correct angles according to the given (x, y, z) cartesian coordinates. The online working feature of neural network makes it very successful and popular in this solution.     

Performance Analysis and Application of a Redundantly Actuated Parallel Manipulator for Milling

This paper deals with the performance analysis of a 3-degree-of-freedom (3-DOF) planar parallel manipulator with actuation redundancy. Closed-form solutions are developed for both the inverse and direct kinematics about the redundant parallel manipulator. In performance analysis phase, the dexterity is analyzed, three kinds of singularities are investigated, and the stiffness is estimated. Compared with the corresponding non-redundant parallel manipulator with the redundant link removed, the redundantly actuated one has better dexterity, litter singular configurations and higher stiffness. The redundantly actuated parallel manipulator was applied to the design of a 4-DOF hybrid machine tool which also includes a feed worktable to demonstrate its applicability.     

Certified solving of the forward kinematics problem with an exact algebraic method for the general parallel manipulator

This article introduces an exact method to solve the forward kinematics problem (FKP) specifically applied to spatial parallel manipulators. The majority can modeled by the 6-6 parallel manipulator. This manipulator is a hexapod made up of a fixed base and a mobile platform attached to six kinematics chains with linear (prismatic) actuators located between two ball or Cardan joints. In order to implement algebraic methods, the parallel manipulator kinematics will be formulated as polynomial equations systems where the equation number is equal to the unknown numbers. One position-based kinematics model will be identified to solve the difficult FKP. The selected proven algebraic method implements Gröbner bases and constructs an equivalent univariate polynomial system. The exact resolution of this last system determines the real solution which exactly corresponds to the manipulator postures. The FKP resolution of the general 6-6 parallel manipulator outputs 40 complex solutions. We provide several examples of various hexapod types yielding eight real solutions. This algebraic method is exact and computes certified results.     

Kinematic analysis of a 3-PRS parallel manipulator

Although the current 3-PRS parallel manipulators have different methods on the arrangement of actuators, they may be considered as the same kind of mechanism since they can be treated with the same kinematic algorithm. A 3-PRS parallel manipulator with adjustable layout angle of actuators has been proposed in this paper. The key issues of how the kinematic characteristics in terms of workspace and dexterity vary with differences in the arrangement of actuators are investigated in detail. The mobility of the manipulator is analyzed by resorting to reciprocal screw theory. Then the inverse, forward, and velocity kinematics problems are solved, which can be applied to a 3-PRS parallel manipulator regardless of the arrangement of actuators. The reachable workspace features and dexterity characteristics including kinematic manipulability and global dexterity index are derived by the changing of layout angle of actuators. Simulation results illustrate that different tasks should be taken into consideration when the layout angles of actuators of a 3-PRS parallel manipulator are designed.     

Estimation model for kinematic calibration of manipulators with a parallel structure

This article provides an estimation model for calibrating the kinematics of manipulators with a parallel geometrical structure. Parameter estimation for serial link manipulators is well developed, but fail for most structures with parallel actuators, because the forward kinematics is usually not analytically available for these. We extend parameter estimation to such parallel structures by developing an estimation method where errors in kinematical parameters are linearly related to errors in the tool pose, expressed through the inverse kinematics, which is usually well known. The method is based on the work done to calibrate the MultiCraft robot. This robot has five linear actuators built in parallel around a passive serial arm, thus making up a two-layered parallel-serial manipulator, and the unique MultiCraft construction is reviewed. Due to the passive serial arm, for this robot conventional serial calibration must be combined with estimation of the parameters in the parallel actuator structure. The developed kinematic calibration method is verified through simulations with realistic data and real robot kinematics, taking the MultiCraft manipulator as the case. © 1994 John Wiley & Sons, Inc.     

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.     

Synthesis on forward kinematics problem algebraic modeling for the planar parallel manipulator: displacement-based equation systems

Based on a proven exact method which solves the forward kinematics problem (FKP) this article investigates the FKP formulation specifically applied to planar parallel manipulators. It focuses on the displacement-based equation systems. The majority of planar tripods can modeled by the 3-RPR parallel manipulator, which is a tripod constituted by a fixed base and a triangular mobile platform attached to three kinematics chains with linear (prismatic) actuators located between two revolute joints. In order to implement the algebraic method, the parallel manipulator kinematics are formulated as polynomial equation systems where the number of equations is equal to or exceeds the number of unknowns. Three geometrical formulations are derived to model the difficult FKP. The selected proven algebraic method uses Gröbner bases from which it constructs an equivalent univariate system. Then, the real roots are isolated using this last system. Each real solution exactly corresponds to one manipulator assembly mode, which is also called a manipulator posture. The FKP resolution of the planar 3-RPR parallel manipulator outputs six complex solutions which become a proven real solution number upper bound. In several typical examples, the resolution performances (computation times and memory usage) are given. It is then possible to compare the models and to reject one. Moreover, a number of real solutions are obtained and the corresponding postures drawn. The algebraic method is exact and produces certified results.     

Inverse Kinematics of Binary Manipulators Using a Continuum Model

Binary actuators have only two discrete states, both of which are stable without feedback. As a result, manipulators with binary actuators have a finite number of states. The major benefits of binary actuation are that extensive feedback control is not required, reliability and task repeatability are very high, and two-state actuators are generally very inexpensive, resulting in low cost robotic mechanisms. These manipulators have great potential for use in both the manufacturing and service sectors, where the cost of high performance robotic manipulators is often difficult to justify. The most difficult challenge with a binary manipulator is to achieve relatively continuous end-effector trajectories given the discrete nature of binary actuation. Since the number of configurations attainable by binary manipulators grows exponentially in the number of actuated degrees of freedom, calculation of inverse kinematics by direct enumeration of joint states and calculation of forward kinematics is not feasible in the highly actuated case. This paper presents an efficient method for performing binary manipulator inverse kinematics and trajectory planning based on having the binary manipulator shape adhere closely to a time-varying curve. In this way the configuration of the arm does not exhibit drastic changes as the end effector follows a discrete trajectory.     

A New Approach to the Architecture Optimization of a General 3-PUU Translational Parallel Manipulator

This paper presents a new approach to the architecture optimization of a general 3-PUU translational parallel manipulator (TPM) based on the performance of a weighted sum of global dexterity index and a new performance index-space utility ratio (SUR). Both the inverse kinematics and forward kinematics solutions are derived in closed form, and the Jacobian matrix is derived analytically. The manipulator workspace is generated by a numerical searching method with the physical constraints taken into consideration. Simulation results illustrate clearly the necessity to introduce a mixed performance index using space utility ratio for architectural optimization of the manipulator, and the optimization procedure is carried out with the goal of reaching a compromise between the two indices. The analytical results are helpful in designing a general 3-PUU TPM, and the proposed design methodology can also be applied to architectural optimization for other types of parallel manipulators.     

Analysis of kinematics/statics and workspace of a 2(SP+SPR+SPU) serial–parallel manipulator

The kinematics/statics and workspace of a 2(SP+SPR+SPU) serial–parallel manipulator are studied systematically. First, a 2(SP+SPR+SPU) manipulator including an upper and a lower SP+SPR+SPU parallel manipulators is constructed, and the inverse/forward displacements, velocity, acceleration and statics of the lower/upper parallel manipulators are studied, respectively. Second, the kinematics and statics of the lower/upper manipulators are combined and the displacement, velocity, acceleration, statics, and workspace of a 2(SP+SPR+SPU) manipulator are analyzed systematically. Third, the analytic solutions’ examples are given and verified by the simulation mechanism. This manipulator has some potential applications for the robot’s arm, leg, and twist, the serial–parallel machine tools, the sensor, the surgical manipulator, the tunnel borer, the barbette of warship, and the satellite surveillance platform.     

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.