Systematic Elastostatic Stiffness Model of Over-Constrained Parallel Manipulators Without Additional Constraint Equations

The establishment of an elastostatic stiffness model for over constrained parallel manipulators(PMs),particularly those with over constrained subclosed loops,poses a challenge while ensuring numerical stability.This study addresses this issue by proposing a systematic elastostatic stiffness model based on matrix structural analysis(MSA)and independent displacement coordinates(IDCs)extraction techniques.To begin,the closed-loop PM is transformed into an open-loop PM by eliminating constraints.A subassembly element is then introduced,which considers the flexibility of both rods and joints.This approach helps circumvent the numerical instability typically encountered with traditional constraint equations.The IDCs and analytical constraint equations of nodes constrained by various joints are summarized in the appendix,utilizing multipoint constraint theory and singularity analysis,all unified within a single coordinate frame.Subsequently,the open-loop mechanism is efficiently closed by referencing the constraint equations presented in the appendix,alongside its elastostatic model.The proposed method proves to be both modeling and computationally efficient due to the comprehensive summary of the constraint equations in the Appendix,eliminating the need for additional equations.An example utilizing an over constrained subclosed loops demonstrate the application of the proposed method.In conclusion,the model proposed in this study enriches the theory of elastostatic stiffness modeling of PMs and provides an effective solution for stiffness modeling challenges they present.

Gravity-Based Kinetostatic Modeling of Parallel Manipulators Using Screw Theory

The pose accuracy of parallel manipulators(PMs)is a key index to measure their performance.Establishing the grav-ity-based kinetostatic model of a parallel robot provides an important basis for its error composition and accuracy improvement.In this paper,a kinetostatic modeling approach that takes real gravity distribution into consideration is proposed to analyze the influence of gravity on the infinitesimal twist and actuator forces of PMs.First,the duality of the twist screw and constraint wrenches are used to derive the gravity-attached constraint wrenches independent of the external load and the limb stiffness matrix corresponding to the kinematics-based constraint wrenches.Sec-ond,the gravity model of the mechanism is established based on the screw theory and the principle of virtual work.Finally,the analytical formulas of the infinitesimal twist and the actuator force of PMs are obtained,and the influences of the external load,platform gravity,and rod gravity on the stiffness of the mechanism are decoupled.The non-overconstrained 3RPS and overconstrained 2PRU-UPR PMs are taken as examples to verify the proposed method.This research proposes a methodology to analyze the infinitesimal deformation of the mechanism under the influence of gravity.

Non-Negative Adaptive Mechanism-Based Sliding Mode Control for Parallel Manipulators with Uncertainties

In this paper,a non-negative adaptive mechanism based on an adaptive nonsingular fast terminal sliding mode control strategy is proposed to have finite time and high-speed trajectory tracking for parallel manipulators with the existence of unknown bounded complex uncertainties and external disturbances.The proposed approach is a hybrid scheme of the online non-negative adaptive mechanism,tracking differentiator,and nonsingular fast terminal sliding mode control(NFTSMC).Based on the online non-negative adaptive mechanism,the proposed control can remove the assumption that the uncertainties and disturbances must be bounded for the NFTSMC controllers.The proposed controller has several advantages such as simple structure,easy implementation,rapid response,chattering-free,high precision,robustness,singularity avoidance,and finite-time convergence.Since all control parameters are online updated via tracking differentiator and non-negative adaptive law,the tracking control performance at high-speed motions can be better in real-time requirement and disturbance rejection ability.Finally,simulation results validate the effectiveness of the proposed method.

Topology Search of 3-DOF Translational Parallel Manipulators with Three Identical Limbs for Leg Mechanisms

Three-degree of freedom（3-DOF） translational parallel manipulators（TPMs） have been widely studied both in industry and academia in the past decades. However, most architectures of 3-DOF TPMs are created mainly on designers＇ intuition, empirical knowledge, or associative reasoning and the topology synthesis researches of 3-DOF TPMs are still limited. In order to find out the atlas of designs for 3-DOF TPMs, a topology search is presented for enumeration of 3-DOF TPMs whose limbs can be modeled as 5-DOF serial chains. The proposed topology search of 3-DOF TPMs is aimed to overcome the sensitivities of the design solution of a 3-DOF TPM for a LARM leg mechanism in a biped robot. The topology search, which is based on the concept of generation and specialization in graph theory, is reported as a step-by-step procedure with desired specifications, principle and rules of generalization, design requirements and constraints, and algorithm of number synthesis. In order to obtain new feasible designs for a chosen example and to limit the search domain under general considerations, one topological generalized kinematic chain is chosen to be specialized. An atlas of new feasible designs is obtained and analyzed for a specific solution as leg mechanisms. The proposed methodology provides a topology search for 3-DOF TPMs for leg mechanisms, but it can be also expanded for other applications and tasks.

Experimental Characterization of Operation of a Waist-Trunk System with Parallel Manipulators

An extensive research activity has been focused on the upper and lower limbs of humanoid robots. However, due to mechanical design difficulties and complex control of multi-body system, the torso of humanoid robot is somehow a neglected or simplified design part. In this paper, operation performance of a new waist-trunk system as torso for humanoid robots is presented through results of lab experimental tests. The proposed waist-trunk system is composed of two 3 DOFs （degrees of freedom） parallel manipulators, which are connected in a serial chain architecture. A prototype is built by using two prototypes of CaPaMan （Cassino Parallel Manipulator）, which are convenient stiff architectures with easy-operation characteristics. Experimental tests are carried out with the aims to imitate lateral-bending and transverse-rotation movements of human torso. Operation performances like displacements, accelerations, and actuation torque are measured for a performance evaluation and design characterization of the used manipulator solution imitating human torso. Experimental test results are illustrated and discussed to show the practical operation feasibility of the proposed architecture and the operation characteristics of the built prototype.

A Geometric Approach to the Design of Serial and Parallel Manipulators with Passive Joints

The use of robotic manipulators in remote and sensitive areas calls for more robust solutions when handling joint failure, and the industry demands mathematically robust approaches to handle even the worst case scenarios. For both serial and parallel manipulators torque failure is indeed a worst case scenario. Thus, a systematic analysis of the effects of external forces on manipulators with passive joints is presented. For serial manipulators we find under what conditions the robot is conditionally equilibrated, that is, equilibrated with respect to a specific external force. These conditions are, as expected, very restrictive. The serial, or subchain, case serves as a good platform for analyzing parallel manipulators. In parallel manipulators passive joints can appear as a design choice or as a result of torque failure. In both cases a good understanding of the effects that passive joints have on the mobility and motion of the parallel manipulator is crucial. We first look at the effects that passive joints have on the mobility of the mechanism. Then, if the mobility considering passive joints only is not zero we find a condition similar to the serial case for which the parallel manipulator is conditionally equilibrated with respect to a specific external force.

Geometric Approach for Kinematic Analysis of a Class of 2-DOF Rotational Parallel Manipulators

Euler angles are commonly used as the orientation representation of most two degrees of freedom（2-DOF） rotational parallel mechanisms（RPMs）,as a result,the coupling of two angle parameters leads to complexity of kinematic model of this family of mechanisms.While a simple analytical kinematic model with respect to those parameters representing the geometrical characteristics of the mechanism,is very helpful to improve the performance of RPMs.In this paper,a new geometric kinematic modeling approach based on the concept of instantaneous single-rotation-angle is proposed and used for the 2-DOF RPMs with symmetry in a homo-kinetic plane.To authors＇ knowledge,this is a new contribution to parallel mechanisms.By means of this method,the forwards kinematics of 2-DOF RPMs is derived in a simple way,and three cases i.e.4-4R mechanism（Omni-wrist III）,spherical five-bar one,and 3-RSR1-SS one demonstrate the validity of the proposed geometric method.In addition,a novel 2-DOF RPM architecture with virtual center-of-motion is presented by aid of the same method.The result provides a useful tool for simplifying the model and extending the application of the RPMs.

Workspace of Translation 3-UPU Parallel Manipulators

To determine workspace and relationship between the workspace and geometry of parallel manipulator is important for optimum design of parallel manipulators. In this paper, the workspace and the relationship between the workspace and the geometry of 3-UPU parallel manipulators with pure translation are investigated. Geometric and non-geometric constraints are defined and taken account of in determining the workspace of the translation 3-UPU manipulators. A direct average condition number is used as the global performance index of the workspace. This research shows that there exists an optimal value of the direct average condition number favorable for a good design of parallel mechanisms. The results presented in this paper are useful for the optimum design of 3-UPU parallel manipulators.

Adaptive control of parallel manipulators via fuzzy-neural network algorithm

This paper considers adaptive control of parallel manipulators combined with fuzzy-neural network algorithms （FNNA）. With this algorithm, the robustness is guaranteed by the adaptive control law and the parametric uncertainties are eliminated. FNNA is used to handle model uncertainties and external disturbances. In the proposed control scheme, we consider modifying the weight of fuzzy rules and present these rules to a MIMO system of parallel manipulators with more than three degrees-of-freedom （DoF）. The algorithm has the advantage of not requiring the inverse of the Jacobian matrix especially for the low DoF parallel manipulators. The validity of the control scheme is shown through numerical simulations of a 6-RPS parallel manipulator with three DoF.

Rapid Optimization of Tension Distribution for Cable-Driven Parallel Manipulators with Redundant Cables

The solution of tension distributions is infinite for cable-driven parallel manipulators（CDPMs） with redundant cables. A rapid optimization method for determining the optimal tension distribution is presented. The new optimization method is primarily based on the geometry properties of a polyhedron and convex analysis. The computational efficiency of the optimization method is improved by the designed projection algorithm, and a fast algorithm is proposed to determine which two of the lines are intersected at the optimal point. Moreover, a method for avoiding the operating point on the lower tension limit is developed. Simulation experiments are implemented on a six degree-of-freedom（6-DOF） CDPM with eight cables, and the results indicate that the new method is one order of magnitude faster than the standard simplex method. The optimal distribution of tension distribution is thus rapidly established on real-time by the proposed method.

Inverse kinematic optimal design of 6-DOF parallel manipulators

To obtain the required articular velocities as lower as possible for the given kinematics of the moving platform, this paper focuses on this kind of articular velocities optimization of 6-DOF parallel manipulators. Based on the inverse kinematic analysis, the H∞ norm of the weighted Jacobian matrix was adopted as the performance index to minimize the articular velocities, and then the optimal design problem was formulated to find a manipulator geometry that minimized the global performance index with the constraints of the workspace and structural parameters limits. Since the optimal design problem is a constrained nonlinear optimization problem without explicit analytical expressions, the genetic algorithm was applied to numerically solve the problem. Simulation results indicate that the articular velocities of the optimal manipulators can be the minimum while the kinematic reauirements of the moving platform are satisfied.

Singularities Analysis of Paradoxical Mechanism and Parallel Manipulators

Parallel manipulator is associated with a set of functions defined by its closure constraints. In this paper, using Lie algebra method, we provide a study on the singularities of parallel manipulators, their relations with the second order of the closure functions, and the tangent space of the configuration space of the manipulator. The transverse condition criterion is applied to analyze the behavior of the singularities. This gives a downright explication why the 6R paradoxical mechanisms work in their singular configurations, and allows to gain insight on configuration space singularities and to choose the adequate design parameters for the parallel manipulator.

Clearance Effects in Parallel Manipulators： Position Error Discontinuities and Inertial Effects Influence

Clearances at joints cause an uncertainty in the actual posture of the end-effector of any mechanism. This uncertainty relays on the clearance dimension and the way these clearances are taken up by the mechanism under the load and the inertial effects at every instant. As a matter of fact, the actual measure of the pose error is often replaced by an uncertainty measure. However, a side effect of the existence of clearances is that they can cause sudden changes in the posture of the mechanism as a motion is performed. Such discontinuities in the position produce task defects and impacts. In this work a tool to determine the pose error due to clearances is presented together with a discontinuity analysis. In addition, effects of mass distribution and inertial effects on such discontinuities are expounded, taking a 3-PRS robot as example.

Design issues for human-machine platform interface in cable-based parallel manipulators for physiotherapy applications

We outline problems and potential solutions for feasible human-machine interfaces using cable-based parallel manipulators for physiotherapy applications.From an engineering perspective,we discuss the design constraints related to acceptance by patients and physiotherapist users.To date,most designs have focused on mobile platforms that are designed to be operated as an end-effector connected to human limbs for direct patient interaction.Some specific examples are illustrated from the authors' experience with prototypes available at Laboratory of Robotics and Mechatronics (LARM),Italy.

Error Analysis and Distribution of 6-SPS and 6-PSS Reconfigurable Parallel Manipulators

The purpose of this paper is to analyze an accuracy design method for reconfigurable parallel manipulators including a 6-SPS and a 6-PSS parallel manipulator.An error analysis method,based on the module error model,was used to express the relationship between the module error and the terminal error in the error transmission equation of the reconfigurable parallel manipulator.In addition,an error distribution method using a manufacturing and assembly difficulty coefficient was used to analyze each error module to determine a maximum terminal error.The error distribution result was then used to set up a reconfigurable parallel manipulator.Error experiments with a reconfigurable parallel manipulator show that the error analysis and distribution method for reconfigurable parallel manipulators are effective and the maximum terminal errors of the reconfigurable parallel manipulators are less than 50 μm.

A Comparative Study on Kinematic Calibration for a 3-DOF Parallel Manipulator Using the Complete-Minimal,Inverse-Kinematic and Geometric-Constraint Error Models

Kinematic calibration is a reliable way to improve the accuracy of parallel manipulators, while the error model dramatically afects the accuracy, reliability, and stability of identifcation results. In this paper, a comparison study on kinematic calibration for a 3-DOF parallel manipulator with three error models is presented to investigate the relative merits of diferent error modeling methods. The study takes into consideration the inverse-kinematic error model, which ignores all passive joint errors, the geometric-constraint error model, which is derived by special geometric constraints of the studied RPR-equivalent parallel manipulator, and the complete-minimal error model, which meets the complete, minimal, and continuous criteria. This comparison focuses on aspects such as modeling complexity, identifcation accuracy, the impact of noise uncertainty, and parameter identifability. To facilitate a more intuitive comparison, simulations are conducted to draw conclusions in certain aspects, including accuracy, the infuence of the S joint, identifcation with noises, and sensitivity indices. The simulations indicate that the complete-minimal error model exhibits the lowest residual values, and all error models demonstrate stability considering noises. Hereafter, an experiment is conducted on a prototype using a laser tracker, providing further insights into the diferences among the three error models. The results show that the residual errors of this machine tool are signifcantly improved according to the identifed parameters, and the complete-minimal error model can approach the measurements by nearly 90% compared to the inverse-kinematic error model. The fndings pertaining to the model process, complexity, and limitations are also instructive for other parallel manipulators.

Experimental Characterization of a Binary Actuated Parallel Manipulator

This paper describes the BAPAMAN（Binary Actuated Parallel MANipulator） series of parallel manipulators that has been conceived at Laboratory of Robotics and Mechatronics（LARM）. Basic common characteristics of BAPAMAN series are described. In particular, it is outlined the use of a reduced number of active degrees of freedom, the use of design solutions with flexural joints and Shape Memory Alloy（SMA） actuators for achieving miniaturization, cost reduction and easy operation features. Given the peculiarities of BAPAMAN architecture, specific experimental tests have been proposed and carried out with the aim to validate the proposed design and to evaluate the practical operation performance and the characteristics of a built prototype, in particular, in terms of operation and workspace characteristics.

Kinematics and Dynamics Hessian Matrices of Manipulators Based on Screw Theory

The complexity of the kinematics and dynamics of a manipulator makes it necessary to simplify the modeling process.However,the traditional representations cannot achieve this because of the absence of coordinate invariance.Therefore,the coordinate invariant method is an important research issue.First,the rigid-body acceleration,the time derivative of the twist,is proved to be a screw,and its physical meaning is explained.Based on the twist and the rigid-body acceleration,the acceleration of the end-effector is expressed as a linear-bilinear form,and the kinematics Hessian matrix of the manipulator（represented by Lie bracket）is deduced.Further,Newton-Euler＇s equation is rewritten as a linear-bilinear form,from which the dynamics Hessian matrix of a rigid body is obtained.The formulae and the dynamics Hessian matrix are proved to be coordinate invariant.Referring to the principle of virtual work,the dynamics Hessian matrix of the parallel manipulator is gotten and the detailed dynamic model is derived.An index of dynamical coupling based on dynamics Hessian matrix is presented.In the end,a foldable parallel manipulator is taken as an example to validate the deduced kinematics and dynamics formulae.The screw theory based method can simplify the kinematics and dynamics of a manipulator,also the corresponding dynamics Hessian matrix can be used to evaluate the dynamical coupling of a manipulator.

Forward kinematics analysis of parallel manipulator using modified global Newton-Raphson method

In order to obtain direct solutions of parallel manipulator without divergence in real time,a modified global Newton-Raphson(MGNR) algorithm was proposed for forward kinematics analysis of six-degree-of-freedom(DOF) parallel manipulator.Based on geometrical frame of parallel manipulator,the highly nonlinear equations of kinematics were derived using analytical approach.The MGNR algorithm was developed for the nonlinear equations based on Tailor expansion and Newton-Raphson iteration.The procedure of MGNR algorithm was programmed in Matlab/Simulink and compiled to a real-time computer with Microsoft visual studio.NET for implementation.The performance of the MGNR algorithms for 6-DOF parallel manipulator was analyzed and confirmed.Applying the MGNR algorithm,the real generalized pose of moving platform is solved by using the set of given positions of actuators.The theoretical analysis and numerical results indicate that the presented method can achieve the numerical convergent solution in less than 1 ms with high accuracy(1×10-9 m in linear motion and 1×10-9 rad in angular motion),even the initial guess value is far from the root.

Singularity Analysis of a 3-RPS Parallel Manipulator Using Geometric Algebra

Singular configurations must be avoided in path planning and control of a parallel manipulator. However, most studies rarely focus on an overall singularity loci distribution of lower-mobility parallel mechanisms. Geometric algebra is employed in analysis of singularity of a 3-RPS parallel manipulator. Twist and wrench in screw theory are represented in geometric algebra. Linear dependency of twists and wrenches are described by outer product in geometric algebra. Reciprocity between twists and constraint wrenches are reflected by duality. To compute the positions of the three spherical joints of the 3-RPS parallel manipulator, Tilt-and-Torsion angles are used to describe the orientation of the moving platform. The outer product of twists and constraint wrenches is used as an index for closeness to singularity（ICS） of the 3-RPS parallel manipulator. An overall and thorough perspective of the singularity loci distribution of the 3-RPS parallel manipulator is disclosed, which is helpful to design, trajectory planning and control of this kind of parallel manipulator.