INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol. X, No. X, pp. X-XX XXXX 201X / 1

DOI: XXX-XXX-XXXX

1. IntroductionMechanisms and machines are developed to support and ease the

ventures of human kind. Mechanisms are a combination of links, in which minimum of one link will the remain fixed and the rest of the links will be in motion to do their task. It is classified into four types based on its structure: (1) Serial, (2) Parallel, (3) Hybrid and (4) Com-plaint Mechanisms. Serial mechanisms have an open chain linkage and parallel mechanisms have a closed chain linkage. Hybrid mecha-nisms are a combination of both open and closed chain linkage. Com-plaint mechanisms are made of single flexible bodies mechanism and it performs the task by its elastic deformation. Before the develop-ment of parallel mechanism, conventional serial type is widely used in the field of automation. Serial mechanism was also used in applica-tions such as pick and place, positioning, machining operations, mate-rial handling, etc., though it necessitaties a large workspace and dex-terous manoeuvrability.The major drawbacks of the serial mechanism are limited precision in positioning and low pay load carrying capa-bilities. In order to avoid the shortcoming parallel mechanism is em-ployed in recent days. Eventhough parallel mechanisms have more advantage than serial mechanisms, it also have some technical diffi-culties such as occurance of many singular configurations in working condition and compact workspace. Study of literature reveals that, singularities in parallel mechanism designs have to be avoided. Be-

cause, when the mechanism moves towards a singular configuration,

its stiffness and accuracy properties quickly deteriorate besides the change in degrees of freedom(DoF). This review focuses on identifi-cation of singularity and methodologies to avoid the same while oper-ating the parallel mechanism.

The physical meaning of the kinematics singularity of a closed loop of a parallel mechanism is changed. Mathematically it is empha-sized by using the fall of rank in inverse kinematics Jacobian. The main characteristics issues of a singularity are torque saturation, un-desirable motion(instantaneous mobility) and breakdown of the sys-tem. These are the main factors that influence the performance of par-allel mechanism. Therefore it is mandatory to identify the avoidable and unavoidable singular configurations. It is the first step to frame

Singularity Identification and Avoidance in Parallel Mech-anisms

Balaji K1 and Sreekumar M1,# 1 IIITD&M Kancheepuram, Off Vandalur-Kelambakkam Road, Melakottaiyur, Chennai - 600127

# Corresponding Author / E-mail: msk@iiitdm.ac.in, TEL: +82-2-123-4567, FAX: +82-2-123-4567

KEYWORDS: Parallel Mechanism, Jacobian analysis, Singularity avoidance, singularity identification, Kinematic configuration index.

In parallel mechanisms, degrees of freedom was altered from its normal behaviour in singularity configuration. The methodology used to identify the singularity is observing the fall in rank of a Jacobian matrix. It can be described by the joint variables with its end effector pose. The occurance of singularities influence the performance of mechanism and it controls it. This scenario consequently leads to unacceptable forces and torques of the links, loss in stiffness are its compliance and a sudden disturbance in its control algorithm. Thus the analysis of singularities place an important role in design and synthesis of parallel mechanisms. Particularly the classification of singularities in the spatial and planar parallel mechanism provides knowledge about the singularity occured in various configurations. Methods to identify the singularity are represented mathematically. Different approaches related to the avoidance of singularity are explained in order to avoid algorithmic singularity. Different singularity free algorithms for various parallel mechanisms and parallel haptic devices are investigated in this review paper.

Manuscript received: August XX, 201X / Accepted: August XX, 201X

NOMENCLATURE AND ABBREVIATIONS

= End-effector velocity vec-tor with ‘m’ dimension

= Actuator velocity vector with ‘n’ dimension

= Jacobian matrix and its inverse

= First and second manipu-lating force

= Joint torque

= Angular and linear length dimension of the mech-anism

= Desired acceleration

= Gain matrices of the con-trol loop

= Positional and orientation error of moving plat-form

= desired angular and linear velocity

LS = Leg SingularityRS = Redundancy SingularitySC = Stationary ConfigurationIIM = Increased Instantaneous

MobilityLS-SC = Leg Singularity leading to

Stationary Configura-tion

RS-SC = Redundancy Singularity leading to Stationary Configuration

LS-IIM = Leg Singularity leading to Increased Instantaneous Mobility

RS-IIM = Redundancy Singularity leading to Increased Instantaneous Mobility

ACS = Active-Constraint SingularityALS = Actuated-Leg SingularityPCS = Passive-constraint singularitySTS = Static singularityDoF = Degree of Freedom

MLG = Mechanism Line of GraphNL-DOP

= Non Linear Dynamic Optimization Problem

CHDET

= Constraint Handling Differential Evo-lution Technique

PM = Parallel Mechanism

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the free path planning(tracking trajectory) and to creat the singular free zones. This review article was organised as follows. The singu-larity representation in mathematical form is discussed in detail in section 3 with the sub divisions of 3.1, 3.2 , 3.3, 3.4, 3.5 respectively. The methodology for identification of singularity is discussed in sec-tion 4 with a sub section based on geometric approach, screw theory, graphical representation, singularity free zones and path planning ap-proaches in 4.1,4.2,4.3,4.4 respectively. Approaches to avoid singu-larity are vividly explained in section 5. The discussion and the con-clusion are highlighted in the sections 6 and 7 respectively

2. Singularity-An OverviewThere are lots of review papers on the basis of its application. The

author reviews about the initial stage of parallel mechanism inventions, due to the drawback of serial one . In 1932, the automated spray painting machine,1 was conceptually idealized by Willard L.V.Pollard and in 1942, his son Willard L.V.Pollard Jr. got a patent for this machine. A universal testing machine, Autohedral hexapod was invented by Dr.Eric Gough in 1947. In 1962, Klaus cappel invented conventional 6-DOF vibration system based on hexapod. In 1965, Stewart demonstrated a 6-DOF motion platform which was used as a flight simulator. Thus the gap between the industry, academia and individual research were exhibited. The classification of planar spherical kinematic bond of many parallel mechanism were published2. It introduced many novel parallel mechanisms.The need for haptic devices and the suitability of parallel mechanism for haptic devices are discussed here.3 The haptic devices are mainly used in medical surgical robotics and in the field of nuclear robotics to perform the unsafe task. CRIGOS, SHaDe, SML haptic devices were the great good examples of parallel haptic devices. The wire actuated parallel mechanisms are also used as haptic devices, as in Falcon, Skycam(Charlotte robots). Parallel mechanism as haptic devices shows its success in large number of applications throughout the world. Low inertia and high stiffness properties of parallel mechanism matches the need of the haptic devices. The low inertia parallel kinematic mechanism has great merits for its nano-micro positioning and precised handling. This survey article focuses on hexapods from a leading supplier named Physik instrumente (PI). Here the main concentration is on the products of hexapod’s for the various application such as scanning microscopy,4 image stabilization for astronomy application,4 medical robot, photonics alignment, satellite dish reflector and feed alignment4. The main advantage of hexapod is 6-DOF compactness, micro to macro alignment and positioning.

(a) (b)

(c) (d )

( e) (f)

(g)

Fig. 1: (A)Hexapod as flight simulator4 ,(b)Satellite dish reflector and feed alignment4 ,(c) Hexapod as medical robot4 ,(d) Shade,(e)Hexapod application in Scanning microscopy4

(f)ABB Flexible5 automation,(g) automated spray painting machine1

(a) (b)

(c) (d)

Fig 2: (A)3-UPU configuration,(B)SNU translation parallel mechanism6, (C)The linear MANTA 7mechanism; (D)The linear

KANUK 7mechanismThe brilliant idea of Clavel et.al., was implementing

parallelogram in parallel mechanism to generate the structural idea of delta robot. Parallelogram was implemented to get the output link w.r.t the orientation of input link. Based on the delta robots,5 36 patents were created all over the world. The salient feature on the

© KSPE and Springer 2011

INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol. X, No. X, pp. X-XX XXXX 201X / 3

DOI: XXX-XXX-XXXX

Delta robot was its fast and quick response for the command. It was applied for the pick and place operation.A translation parallel mechanism(TPM) was built in Seoul national university by Tsai et al.,. The configuration of TPM is 3-UPU, 3 limbs in which each limb connects the base and the moving platform by universal-prismatic-universal joint. The author proves that SNU,6 TPM doesn’t have its architecture singularity. But due to the manufacturing tolerances and flexibility in universal joints it posses a constraint singularity. For the industrial material handling two novel 4-DOF parallel mechanism were designed named as MANTA,7 and KANUK,7. The advantage of this parallel mechanism is its high stiffness through multiple kinematic chain for low mass design. These mechanisms are applied in the field of automated ware house manipulation, media theque manipulation, machine tool changers, and loading and unloading. Moreover the parallel mechanisms applications are achieved in mining machines, walking machines, both terrestrial and space applications including areas such as high speed manipulation, material handling, motion platforms. It is also used in the field of machine tools, medical fields, planetary exploration, satellite antennas,8 haptic devices, vehicle suspensions, variable-geometry trusses, cable-actuated cameras and telescope positioning systems and pointing devices.

3 Singularity - Mathematical Modeling

3.1 Jacobian matrix The Jacobian matrix, J or simply Jacobian is defined as an

matrix that transforms the joint rates, in the actuator space (n - dimensional) to the velocity state, in the end- effector space (m-Dimensional). Thus given the joint rates, we can compute the end-effector velocities directly. In a trajectory planning problem, however the end-effector velocities are usually given along a desired path in the end effector space and these velocities must be converted into the joint rates in the joint space.9 This requires a computation of the inverse transformation of .

For a 6 dof spatial manipulator for which m = n = 6, the Jacobian matrix is a 6 x 6 square matrix. For a manipulator with a less than 6 dof, the end effector velocity state may contain just the linear velocity vector, or the angular velocity vector, or a combination of both linear and angular velocity components. In case of planar manipulators (two dimensional space), a three component vector

is sufficient to describe the velocity state of the

end effector. Here the Jacobian matrix is a 3 x 3 matrix. Similarly for

a point positioning device and for a body

orienting mechanism, .

For deriving a general expression for Jacobian matrix, the following procedure can be followed. Let ,

be a set of equations, then

(1)

(2)

Or simply, . Hence .

3.2 Singularity AnalysisAt Certain Manipulator Configurations, The Jacobian May

Lose its full rank or otherwise the determinant is equal to zero and this configuration is called singular configuration. For the parallel manipulator, let the actuated joint variables be denoted by the vector, q and the location of the moving platform be described by a vector, x.Then the kinematic constraint imposed by the limbs can be written in the general form f(x,q) where f is an n- dimensional implicit function of q x and 0 is an n - dimensional zero vector velocity as follows

(3)

where

Due to the existence of two Jacobian matrices and

,a parallel manipulator is said to be asingular configuration when

either or ,or both are singular. Hence the parallel manipulator

possesses three types of singularities namely Direct, inverse and combined singularities.3.3 Classification of Singularities

Direct Kinematic Singularity occurs when the determinant of is equal to zero, Inverse Kinematic Singularity occurs when

the determinant of is equal to zero and Combined Singularity

Occurs when the determinants of and are both zero. In direct

kinematic singularities the moving platform can posses infinitesimal motion in some directions while all the actuators are completely locked. In inverse kinematic singularities the end effector cannot accomplish certain motion. Hence it loses one or more degrees of freedom. In combined singularities, it experiences both the above singularities in combined manner. A hierarchial relation between the singularities were proposed based on the actuation and unactuation of kinematic chain10 which is given in Fig.3.

4 / XXXX 201X INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol. X, No.X

Fig. 3 Hierarchical relationships between the main critical phenomena occurring in the actuated and the unactuated kinematic status

3.4 Condition NumberFor manipulators with only one type of joint and for manipulators

with one type of task like point positioning or body orienting, but not both, the Jacobian matrix characterized by a measure called condition number. It is applicable only when the Jacobian matrix is non singular (i.e., ) Assuming there is a certain velocity error, on the actuated legs and some velocity errors, in the moving platform.

(4)

(5)

According to the matrix theory and using the previous equations, the relationship between the relative error of

and can be expressed as follows

(6)

Where is norm matrix of a vector.

From the above expression , is just the condition

number C(J) of the Jacobian matrix .which can be written as cond(J) or simply C(J).This parameter is an amplifying factor of the kinematic errors of linear actuators to the moving platform. So the condition number of the Jacobian matrix can be used as an optimal design performance index for higher accuracy tracing, for which the condition number should be minimum in the whole work space.

3.5 Point of IsotropyThe Condition number of the Jacobian matrix depends on the link

lengths and the manipulator configuration. As the end effectors move from one location to another location, the condition number will assume different values. The minimum condition number of any matrix is one .Those points in the workspace of a manipulator where the condition number of the Jacobian matrix is equals to 1 is called isotropic points or the point of Isotropy .The condition of isotropy shall be applied to both the matrices separately or to the combined matrix . If the isotropic condition is satisfied then the Jacobian matrix will be proportional to an identity matrix, namely

(7)

(8)

where is an identity matrix and and are Scalar 6.

3.6 Configuration IndexThe Aspect of Singularity and isotropy can be related with a term

called configuration index. The Configuration Index is expressed as

(9)

A matrix with a of 100 is isotropic while that with a of 0% is singular. Hence it can be inferred that a makes a

matrix closer to isotropic condition and a lower makes it closer to singularity11.

4 Singularity Identification Methodologies

The methodology to identify the Singularity identification of parallel mechanism is classified as follows:

Line-geometry method

Singularity Identification

Screw theory

Grassmann-Cayley Algebra

Geometry Graphical

Plucker co-ordinates system

Twist-wrench system Lie –Algebra

system

Fig.4. Classification of singularity identification

4.1 Geometry Based Approach

(a) (b)

( c) (d)

Fig. 5: (a) 3-RPUR12, (b) Five-bar parallel mechanism13, (c)

Selectively Actuated Parallel Mechanism17 (d) CaPaMan Manipulator19

A novel 3-DOF 3-RPUR,12 translation parallel mechanism was proposed and its kinematic problems were analysed. In a five-bar parallel mechanism,13 the singularity was identified and analysed. The non-linear tracking controller was designed for this mechanism. The classification of singularities were enumerated and analysed, based on the Jacobian matrix derived from loop closure equations. Based on the dynamic model, a non-linear tracking controller was designed to eliminate the tracking trajectory error. Comparison of performance between the proposed controller and augmented PD controller was carried out. For the simulation of spinal column motion a 5-DOF 3R2T,14 symmetrical parallel mechanism was used. The kinematic study of this mechanism leads to the singularity identifications. The wrist (or) hip joints of a humanoid robots are modelled as spherical parallel mechanism. A 2 DOF RR-RRR-RRR,15 spherical parallel mechanism’s direct kinematics were analyzed. Based on the loop closure framed equations forward displacement analysis had been presented, for all general parallel mechanism,16 and the singularity are classified based on the Jacobian matrices in to 3 groups. By using various configurations of limb combinations such as PRR,PPR,RRR & RPR, 20 different class of parallel mechanism were constructed. A 3 limb Selectively Actuated Parallel Mechanism,17 configuration was designed. Based on the method of actuation, the movable platform of the mechanism is able to achieve 3-DOF spherical motion or 3-DOF translation, 3-DOF hybrid motion, or complete 6-DOF spatial motion This mechanism architecture decouples translation and rotation of the end-effectors for individual actuator control. Based on geometry, Singularity analysis of the mechanism configuration was presented for all actuation schemes to facilitate the kinematic design of the manipulator.A new spatial rotation 4-SPS-1-S parallel mechanism18’s singularity configurations were analyzed. Inverse kinematics and velocity mapping equations of the mechanism were formulated, by the framed Jacobian matrices(J). These were used to detect the singularities of the mechanism. The transformation between end effector (output) velocity and Joint space (input) velocity was computed. The square root of determinant of JJT matrix was adopted as an actuating performance index. Based on the study about the configuration of singularities, two different methods algebraic formulation and vector analysis approach used in CaPaMan manipulator,19 was presented. The formulation can be given singularity related to the failure of the kinematic model at particular configurations of the mechanism was shown . It was proved that this type of singularity can be avoided by a proper analysis of the problem.

A new spatial three-degrees-of-freedom parallel mechanism,20 was proposed. This parallel mechanism consists of three connecting limbs between base and moving platform. The velocity equation of this parallel mechanism is determined from The inverse and forward kinematics problems they were given in closed forms. The three types of singularities were also presented. Finally, topology architecture was introduced.The translational 3-URC,21

made the platform translates with respect to the base. This mechanism belonged to a set of mechanisms with topologically equal 3-URC

architectures that contains another mechanism behaving as a spherical parallel wrist. This Parallel mechanism (nearby or at the point) experienced different types of singularity, due to counteract external forces in certain directions and so it loses the ability. The singularity determination problem was focused from a geometric perspective for planar and spatial three-legged parallel mechanisms,22. Thus necessary condition for the unstable singularities, the constraints on the passive joint velocities were derived. By using this condition, singularities can be found for certain type of platforms. The study about distribution of actuator singularity, can be applied to analyze end-effector singularity. The redundancy of the parallel mechanism was used for designing Optimal kinematic and dynamic control algorithms,23. The singularity-free fully-isotropic parallel mechanisms (PMs) were presented with four degrees of freedom T2R2-type,24. The theory of linear transformations based method was proposed for structural synthesis of fully-isotropic in this PM. According to force and motion transmission capabilities, this mechanism performs very well with the condition number as the determinant of the Jacobian matrix being equal to one. In this paper various singularity-free algorithms, which are appropriate to parallel haptic system25, are discussed. Furthermore, to overcome singularity problems, a new design including redundant actuation is proposed. The combination of two parallel mechanisms with a central axis, A double parallel mechanism,26 has been designed. The mobility of the mechanism was decoupled by the central axis. So it was constrained. This leads to simplicity in geometric constraints but it needs a novel strategy in the mechanism analysis.

(a) (b)

(c) (d)

(e)

(f) (g )

(h)Fig.6 (a)Spatial Parallel mechanism 20,(b) 3-URC- parallel mechanism21, (c) C5-Parallel Robot29, (d) HALF Parallel manipulator27 (e) 3-RPS Parallel Manipulator 30,(f) spherical parallel manipulator31,(g) Parallel spherical wrist Mechanism 32,(h) double parallel mechanism

Based on the geometric constraints, the inverse/forward kinematics and Jacobian to be implemented in real time position and velocity controls were developed by the formulation. The singularity of the HALF parallel mechanism27 with revolute actuators was analyzed With and without actuation redundancy.The singularities Classification of a two-degree-of-freedom planar parallel mechanism,28 were studied respectively. In this study the analytical relationships between the link lengths of the two mechanism and characteristic shapes and distribution of the singularities are illustrated. The equations of the singularity curves are developed, based on Jacobian matrix of the kinematic transformation. The results have reference value in the singularity-free trajectory planning and control. A new parallel mechanism configuration with six degrees of freedom is proposed29. This device is well adapted to perform force feedback control, and under some conditions, it can be fitted with a center of compliance. This mechanism is designed to obtain a symmetric and compact structure. The damped least-square method is used to solve the singularity problem of resolved-acceleration control schemes. It works by damping accelerations of the end-effector, so that accelerations in the degenerated directions are zero at a singular point. The control using damped velocity, being called the damped-rate resolved-acceleration control scheme,30 (DRRAC) is proposed to overcome this drawback. The main advantage of the DRRAC in the 3RPS parallel manipulator control system is not to plan its path to avoid the singular point, and could improve the workspace.

(a)

\

(b)

(c )

Fig.7 (a) 3-RPRR Mechanism39; (b) 3-URS Mechanism35,,(c)Eclipse –II43

The determinations of singular configurations of a spherical parallel manipulator,31 were focused, in which and denote the actuated and unactuated revolute joints respectively. A new kinematic design of a parallel spherical wrist mechanism,32 with actuator redundancy was proposed . The actuator redundancy not only removes singularities but also increases workspace, on improving dexterity and it was shown using detailed kinematic analysis. The structure optimization has been performed with a global dexterity criterion. Using a conditioning measure, a comparison with a non redundant structure of the same type was performed. It shows that a significant improvement in dexterity has been obtained. A new 4-legged redundantly actuated UPS parallel mechanism,33 was proposed. In that proposed mechanism Kinematic redundancy, Kinematics, Singularity and dynamics analysis were analyzed . The effect of constraint singularity in a spatial parallel mechanism,34 was studied. These mechanism35 comprises two platforms connected by three legs, each being composed of one universal (U), one revolute (R) and one spherical (S) joints, which gives the manipulator six degrees of freedom. Hence, two actuators are required per leg. The differential kinematic relations between actuator joint rates and mobile-platform twist were derived. A new numerical procedure,37 for 3UPU and 3UPS spherical wrists, that could count and identify the disjoint regions was identified the workspace is partitioned by the singularity surface and to assess to which region any two points of the workspace belong. If the two points belong to the same region, a singularity-free path connecting them is found.The force redundancy is studied as the series or parallel dual concept of kinematic redundancy and its implications in kinematics and dynamics of

parallel manipulators38 were described.Numerical studies39 are also presented for two parallel

manipulators to demonstrate the singularity reduction by a single degree of redundancy. singularity avoidance of the 3-RRR mechanism(Fig. 15(a)) using kinematic redundancy was presented. The singularity analysis of the proposed 3-RPRR,39 mechanism was described and a simple and effective redundancy resolution algorithm based on local optimization suitable for real-time control was developed. The principle of the singularity elimination of the Stewart Platform42 was mainly addressed. By adding appropriate redundant actuation, the rank of the Jacobian matrix of the platform is always full, accordingly the singular value of the Jacobian matrix of the platform is non-zero. Then the singular configuration of the platform can be eliminated by adding one redundant actuation and also the performances of kinematics and dynamics of the platform can be greatly perfected by adding appropriate redundant actuation.

This 6 DoF parallel mechanism43 was developed to eliminate the kinematic complexity of Eclipse –II, redundant parallel mechanism. To achieve the desired motion trajectory, it needs a motion planning control algorithm. For the effective control over motion plan and continuous 360-degree rotational motion, modified redundant parallel mechanism was created. The singularities are classified in to architecture, configuration, formulation singularities. The architecture singularity mainly influence over the spatial parallel mechanism44, it usually spans over the whole workspace or significant part. So it is very difficult to control and apply singularity avoidance strategies. The tripod45 such as UPS,RPS and RPR are modelled in MSC ADAMS and are simulated. The simulation results are obtained by considering the geometrical parameters, Transmission angle of the link and Singularity of the mechanism. A singularities avoidance method46 suitable for the trajectory planning of redundant and non-redundant robot manipulators was proposed. This method was based on establishing proper bounds for the rate of change of the Jacobian matrix of the transformation between the joints speed and end effector Cartesian speed. An analytic expression for the singularities of six-degree of freedom Stewart-platform type parallel manipulator(six-DOF SPM) is obtained using the concept of local structurization method47 (LSM) with extra sensors. By comparing it with previously reported singularities the exactness of the singularity was verified. The actuator relocation effect on singularity, Jacobian and kinematic isotropy performance of parallel Mechanism48 were compared. It is mandatory to study the effect on the Jq and Jx as parallel mechanism consists of several serial legs. It is done when actuator in each leg of the parallel robot is mounted remotely and the actuation joint is driven through a transmission. The effect of both transmission ratios and actuator location on Jacobian, and singularity were analyzed. Finally the effect of actuator relocation on kinematic isotropy performance was illustrated. To optimize the kinematic performance of parallel robots, this actuator relocation can be used.

A global explanatory approach49 is used for the better understanding of non-singular assembly-mode changing motions for 3-RPR planar parallel manipulators. A sequential design method 50is explained, in which two arms are designed first to satisfy workspace requirements, then the third arm is designed to provide a singularity-free workspace.The forward singular configurations in many parallel

manipulators is easily obtained by a geometrical approach, namely Constraint Plane Method51 (CPM). CPM is a methodical technique based on the famous Ceva plane geometry theorem. The new architecture of manipulator52 is addressed with its Geometrical structure and mobility analysis Then the forward and the inverse kinematic problems of the manipulator is analyzed. Based on Grassmann-Cayley algebra53, a different approach for analyzing singularities of parallel mechanism, is based on inspecting the actuators’ line dependencies by use of a matrix called the super bracket, which is similar to the Jacobian matrix and contains Plucker coordinates in its columns is used. Using this method the singular configurations occur precisely, defined by the positions of the joints. A new method 54to define closeness to singularities based on constrained optimization and its resulting general eigenvalue problem was proposed. Stiffness analysis for redundantly actuated parallel mechanism55 (Eclipse-IA) is introduced. The relationship between the new structure and the stiffness properties including determinant and isotropy of the respective stiffness matrix was verified.

4.2Screw theory based approach

(a) (b)

(c)

( e ) ( f )Fig.8 (a) spatial hybrid mechanism 62,(b) 5- DOF Haptic

interface66,(c) 3-RUPU65, (e)3-RPS59(hyper,redundant PM) (f)3-UPS-PPS58,

Based on the theory of screw and reciprocal screws,56 some developed kinematic chain were analysed. The Jacobian analysis of the parallel mechanism was also investigated using this theory. Based on the Jacobian ,the singular configurations were identified. A new analysis40 of wrench singularities was presented for spatial parallel platform manipulators consisting of three legs, with up to two actuators each, and connected to the mobile platform by spherical joints. The wrench singularities were identified using characteristics of tetrahedron. A geometric interpretation of wrench singularities was proposed by a theorem. A 5-DOF decoupled parallel mechanism57 is approached. In this mechanism the construction has 3 peripheral limbs and one central limb. The peripheral limb consists of two prismatic joints, one revolute joint and one spherical joint. The central limb consist of 2 prismatic and a spherical joint. Using Sylvester di alytic elimination method. The forward displacement analysis and dynamic characteristics was derived and analysed, using screw theory. By the use of Klein form of the lie Algebra a SE(3) velocity and reduced acceleration state were derived. The kinematic and dynamic analysis of a modular spherical hyper redundant mechanism58 was solved based on the screw theory and also virtual work is also used. Finally this methodology is applied to 18 DOF type redundant mechanism. Based on index of power, a strategy is implemented for the safe drive of hexa parallel robot,59 in machine supervision. The index of power is derived using power inspired measure using screw theory. The framed strategy is also verified experimentally.

The structural synthesis of 4 DOF and 5 DOF over constrained parallel mechanism60 with identical limbs were developed using a systematic methodology. The geometric condition of these mechanisms are analysed by the help of screw theory and reciprocal of screw theory. A novel 4 DOF novel parallel mechanism61 to achieve the schoenflies motion in its output link was proposed to attain the desired direct and inverse Jacobian. To solve the singularity difficulties of spatial hybrid mechanism62, methodologies were proposed.

An algorithm63 was proposed for tracking the trajectory, to follow a continuous path of a redundant parallel mechanism . In this algorithm the singularity avoidance is assured during its achievement of task. Direct kinematics of the 3-RPS64 was analyzed using Sylvester dialytic elimination method with some numerical examples. The forward position analysis of parallel mechanism with identical limb such as 3 RPS configurations was predicted by the Sylvester-

dialytic elimination method. Then the velocity and the acceleration analysis of the mechanism was carried out. The force analysis and deformation analysis of 3-RUPU65 were presented using screw theory. And based on the stiffness analysis, two global index were referred as maximum and minimum singular values of that compliance matrix which is used to evaluate the compliance of parallel mechanisms. A 5-DOF translating parallel haptic device66 was presented. This device devolpment was analyzed in detail from the development of kinematic synthesis to the static dynamic behaviour based on its screw theory. The twist of movements inside the workspace of parallel mechanism67 was also considered as in singularity. (N-1) twists exist for N-DOF mechanisms.

4.3 Graphical representation based approachTo identify the singularities in planar parallel mechanism52,the

graphical representation is used along with screw theory and virtual work principle methodologies. based on Graphical representation, MLG67 (mechanism line of graph)is used to find the singularities of planar parallel mechanism.

4.4 Singularity Free Path Planning approaches

A numerical technique41 for path planning inside the reachable workspace of parallel manipulators to avoid singularity was proposed. A genetic numerical algorithm was described for generating the reachable workspace of parallel mechanism. The singularity points are determined, grouped into several clusters and modelled as obstacles. A path planning algorithm is used to identify an optimal path avoiding these singular places..In general the main task for the parallel mechanism is to move the end effector(moving platform)without any interference.for tracking the trajectory of moving platform in singularity free zone some methodologies proposed. In 3 RPR parallel mechanism49 singularity surfaces are calculated and a method was proposed to determine the maximal joint space singularity free boxes of this configurations and the planning trajectories are highly focused. For the general planar parallel mechanism,two singularity free path planning algorithm68 are illustrated.In which,the first one is deviated from specified path and its computation time is high. The second one is highly efficient but failed to utilize the available leg forces. The singularity free zones of planar parallel mechanism, using the force transmission factor, 69 was also defined. To indicate the force transmission, the pressure angle is used. To attain the optimal control of the pressure angle of the given trajectory, leg with variable structure was utilized. Using 3RPR configurations this singularity free zone procedure was illustrated with numerical simulations. To construct a continuous path within the work space of the Stewart platform, an algorithm,70 was framed by avoiding singularities and ill conditioning. Using the final and initial end pose of the Stewart platform, the proposed algorithm will plans a smooth continuous path. If both the poses are in different branches, then the algorithm indicates the impossibility of a valid path. This path planning strategy was illustrated using numerical example. The design of 5R (2 DOF) parallel robot71 with its controller was stated as NLDOP.In order to track a desired trajectory with singularity avoidance a (CHDET) was used. The effectiveness of this proposed approach was illustrated through numerical simulations. In this

singularity free proposed algorithm72 ensures the singularity avoidance during the task achievement. This algorithm also change the working mode based on its feasible map configurations. A case study was analyzed to present the effectiveness of the proposed algorithm73. Based on the two search mode in singularity free path planning such as depth search mode and width search mode an algorithm was framed to avoid the obstacle and collision free path plan.

Fig.9. Singularity Free Path Planning algorithm73 5 Singularity Avoidance Approaches

For the above mentioned parallel mechanism configurations, following are the proposed singularity avoidance approaches are practically used to avoid the algorithmic singularities.

• Nakamura algorithm25 • Chiaverini’s Algorithm25 • Choi ‘S algorithm25 • Damped Least Square Method25 & 30

• Kinematic redundancy40

• Damped rate resolved acceleration control method30

The paper deals about the conversion of previously developed Kinematic based task-priority algorithms to force-based task-priority algorithms to cope with the singularity problems of parallel haptic system25.5.1 Nakamura’s Algorithm

Relationship between the manipulating forces and joint torque are as follows:

(10)

(11)

Where and denotes the Jacobian matrix

for the first and secondary task respectively.The General Solution of eqn.10 is expressed as

(12)

Where is the pseudo inverse of and is an arbitrary vector which satisfies some secondary requirement.

Minimizing the secondary task error in

the least square sense, yields:(13

)

Where and is an arbitrary

vector.The general solution of is expressed as

(14)

Since is idempotent.

5.2 Chiaverini’s AlgorithmIn general, there are two kinds of singularities Kinematic

singularity and algorithmic singularity, when solving inverse kinematics. The kinematic singularity occurs when

(15)

Whereas the algorithmic singularity occurs in either (16

)Where R(*) and N(*)represent the range space and null

space of matrix *, respectively. The general solution of TA eqn.14. is not acceptable near singularities. So, to eliminate the algorithmic singularities in Nakamura’s method, Chiaverini modified eqn.14 as follows:

(17)

Nakamura’s algorithm does not have any task error in the normal case while Chiaverini’s algorithm has the secondary task error

with the expectation of the case when . Thus, there exists

trade-offs between the two algorithms.5.3 Choi’s Algorithm

Choi proposed an algorithm to reduce the secondary task error found in Eqn. (17) . Choi modified Eqn. (17) as follows:

(18)

Where .

Eqn. (18) will have no algorithmic singularities if the weight matrix is chosen to be a positive definite matrix as follows:

(19)

Although Eq. 18 tend to contaminate the performance of the secondary task with a small positive number .

5.4 Damped least square methodThe pseudo inverse of any matrix can be given by,

(20)

Where and are orthogonal

matrices.The matrix has the block matrix and

The damped least-

square method, given by(21

Put the first position in the

queue

The Queue is empty?

Get the head of Queue H

H is close to the goal

position?

There is no path

stop in the algorithm

The path is found stop the

algorithm

Depth mode: put N in the head of

the Queue

yesno

yesno

no

no

Get N the nearest position to the goal in the vicinity of H

Position N is

collision or a

singular?

Width mode: put non singular and non collision points that are in the vicinity of H and near a collision

point in the tail of Queue

yes

)

Where and

.If is much less than the smallest non-zero singular value of ,

then is approximately the minimum norm solution. As a singular

value approaches zero, reaches maximum when and

then decreases rapidly to zero and increases when decreases monotonically. This fact can be promoted to find solutions subject to joint torque limits.

5.5 Kinematic Redundancy Resolution AlgorithmThe singularity avoidance approach is simple and effective

redundancy. Mainly in this approach, the end effector singularity can be avoided. By using this method the singularity free workspace can be highly utilized. For the repeatable and accurate trajectory path generation of the end effector this algorithm approach is used.

Procedure for Kinematic Redundancy Algorithm: • Compute Initial pose(P) of the End effector, Calculate

its Jacobian matrix [A] using its given initial joint space variable. • Assume the desired end pose(Pk) of the end effector and

then calculate its Jacobian matrix (AK ) using its given initial joint space variable.

• Compare the Jacobian matrix [AK] with least constant value (δ). If it is greater than δ, then check with joint space variable else if it is less than δ determine the feasible joint space variable with

its constraint ( ).

• Using the above feasible joint space variable calculate the Jacobian matrix.

• Optimize the joint space variable based on the determinant of Jacobian matrix.

• Compute the angular values for end position. • Repeat step 1 to 5 by assuming the current end position

as starting point end effector.

5.6 DRRAC Control MethodThis is one of the singularity avoidance approach in 3-RPS

parallel mechanism. By using this singular avoidance approach it could improve the workspace of the mechanism, DRRAC30 is asymptotically stable. This method is simple schematic, high quality solution and better computation efficiency.

Fig.10. Damped rate resolved acceleration control method 30

6 DiscussionBased on this review work,the parallel mechanisms which posses

singularities are focused.The hierarchical relationships and the classification of singularities based on the actuation over kinematic link chain are tabulated. Among the singularity identification methods,mathematically represented method such as Jacobian analysis, point of isotropy,conditioning number,and configuration index are discussed. The singularity identifications of different parallel mechanisms configurations are highlighted, based on the geometric approach, screw theory, graphical approach which is used to identify. The singularity avoidance approaches such as Nakamura Algorithm, Chiaverini’s Algorithm, Choi’s algorithm, Damped Least Square Method, Kinematic redundancy and Damped rate resolved acceleration control method are explained and implemented to avoid the algorthmic singularities .

7 Conclusions In this work, the parallel mechanism are focused and critical

aspects such as singularity identification and avoidance, and also about the singularity free path planning approaches have been elaborately discussed. The performance of the singularity identification and avoidance system mostly depends upon the type of controller. Literature confirms that the controller plays a vital role in singularity identification and in avoidance. Though the time consumed by the system is not explicitly mentioned in the literature, method utilizing Kinematic configuration index, Point of isotropy definitely consumed less time compared to other methods.

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