Structure Comparison and Optimal Design of 6-RUS Parallel Manipulator Based on Kinematic and Dynamic Performances

The effects of distinctive parameters such as revolute joint angle or spherical joint location of mobile platform in a 6-DOF 6-RUS parallel manipulators on workspace, kinematic, and dynamic indices are investigated in this study to select proper structure commensurate with performance. Intelligent multi-objective optimization method is used to design the manipulator. Considering distinctive parameters, relevant relations for developing inverse kinematic and Jacobin matrix are obtained. In order to study dynamic properties, mass matrix is obtained from calculating the total kinetic energy of the manipulator. After modifying multi-objective Bees algorithm, it used to optimize the manipulator structure considering all geometrical parameters with proper constraints. In addition of comparison of three well known 6-RUS manipulators’ types, variation diagram of workspace, local and global dynamics and kinematics performance indices have been drawn with respect to structural parameters variation and limitation of these parameters with proper value are determined. Moreover, considering all dimensional parameters, Pareto front line of multi objective optimization of structure is presented based on dynamic and kinematic performance in pre-determined workspace. Based on the results, a fairly comparison among various types of 6-RUS manipulators can be conducted and the most appropriate set of dimensional parameters are selected based on specific demand.

Latin American Journal of Solids and Structures 13 (2016) 2414-2438 are made of mobile platform which is connected to a fixed platform by several parallel arms.Parallel manipulators can be classified based on degree of freedom, number of arms, order of joints in each arm and type of actuator (Merlet, 2006).according to this, various 6-DOF parallel manipulators have been proposed.One of the most important 6-DOF parallel manipulators is 6-RUS manipulators that revolute joint, universal joint, and spherical joint are used in each arm, respectively.
A 6-RUS manipulators has some advantages that the most important one, is low weight of movable parts because of installing motor in the fixed platform.Therefore, bigger and cheaper electrical motors can be used.In addition, thinner connecting rod can be used to the mobile platform that conclude reduction of collision of links to each other (Bonev, 2002).Moreover, these manipulators can be balanced statically (Gosselin and Wang, 2000).Although this type of manipulator has some disadvantages such as bending in connecting rods and complicated mechanical analysis.Also, because of great number of chains, connecting the fixed base with the moving platform, and movment limitations of passive joints, such as spherical and universal, workspace of the 6-RUS manipulators are restericted.To overcome such drawbacks, some researchers demonstrate the tendency for the use less than 6-degree-of-freedom parallel mechanisms (Chablat and Wenger, 2003; Gosselin  et al., 2007; Clavel, 1991) and for 6-dof applications hybrid manipulators are studied (Caro et al., 2015).
The first 6-RUS manipulator was proposed by Hunt(1983).Kinematics and dynamics of Hunt type manipulator was examined by Gil et al.(2004).Moreover, Aginaga et al.(2012), improved static stiffness of this manipulator using inverse singularities.Another type of 6-RUS manipulators is Hexa that proposed by Pierrot(1990) based on 3 DOF delta (Clavel, 1988).This manipulator has same advantages of Delta manipulator, such as high velocity and high acceleration and in addition it has three rotary DOF.The first model of this manipulator was made by Uchiyama (1993) and the direct kinematic of Hexa was developed by Hesselbach et al.(2005).Moreover, calibration of this manipulator was conducted by Dehghani et al.(2014).Another arrangement of 6-RUS manipulator was proposed according to the required performance by various references such as proposed manipulators by Zamanov (Merlet, 2006)), Takeda et al.(1997) and Campos et al.(2013).In addition, various types of this manipulator with industrial application were proposed by Servos & Simulation Inc. (Merlet, 2006).All mentioned types have the same topology but they are different according to the revolute joint axis angle and the connection point to the fixed platform.Generally, few studies have been done on the 6-RUS structures rather than other 6-DOF manipulators such as 6-UPS and 6-PUS (6-DOF manipulators with linear actuators).
In order to compare all structures, some criteria are presented to evaluate manipulators performance.These criteria can be classified to kinematic-static criteria (Cui et al., 2014;Rezaei and Akbarzadeh, 2015) such as isotropy, manipulability, dexterity, workspace and singularity, and dynamic criteria (Nguyen et al., 2015;Zhao, 2013) such as dynamic isotropy, dynamic dexterity and natural frequency.Depending on extensive application of these manipulators, one or more criteria are required to mention.
Optimization of the structural parameters is one of the subject for researchers in the parallel manipulators field (Wang et al., 2015;Wang et al., 2015).Optimizing the parallel manipulators have many difficulties.First, optimizing the parallel manipulators is a multi-objective and multi-criterion problem based on different performance specifications and sometimes the mentioned specifications Latin American Journal of Solids and Structures 13 (2016) 2414-2438 have diverse relationships.Another issue is that there are no direct relationships between the performance specifications and the structural parameters and solving the optimization problems will have several responses (Khan and Angeles, 2006).
Hence, it is seen by studying the available references that in spite of great applications of 6-RUS parallel manipulators and proposed new various types by researches, structural difference between all manipulators and comparing their performances haven't been examined yet and no guidance has been proposed to select the proper dimensional parameters based on desired application.moreover, Although, a simplified type of 6-RUS manipulator in Liu et al.(2002) has been examined based on kinematic dexterity index, no reference was seen to examine this manipulator based on dynamic index.In addition, multi-objective optimization problems of 6-RUS manipulator were not mentioned especially if dynamical performance included.
Three samples of the most well-known 6-RUS group manipulators are classified in this research after determination distinctive parameters.Dynamic and kinematic performance indices are proposed to examine performance and effect of structure variation on performance criteria.In addition, 6-RUS manipulator structure is optimized by using the modified multi objective Bees Algorithm (Pham and Ghanbarzadeh, 2007).

DESCRIPTION OF NOTATION FOR 6-RUS
Three 6-RUS manipulators include Hunt type (Hunt, 1983), Hexa (Pierrot, 1990) and a manipulator named Zamanov type by the authors as it was proposed by Zamanov (Merlet, 2006) are shown in Figure 1.Angles between joints on fixed and mobile platforms are B q and P q , respectively.As it is observed, maker vector of 6-RUS manipulator includes 7 parameters: According to Figure 1,with neglecting dimensional differences, the main difference in 6-RUS manipulators has two cases; first is how to place two arms beside each other which is result of difference in revolute angle d q .For example, in Hexa and Hunt type manipulators, arms next to each other move in two parallel platforms but in Zamanov type manipulator, arms move in a plain against each other.The second parameter is the position of spherical joints on mobile platform P q .
Examining other parameters shows that change in parameters size doesn't change in manipulator structure so these two parameters are considered as distinctive parameters.To clarify this issue, structural parameters values for three manipulators are shown in Table 1; therefore, each structure can be introduced by relevant d q and P q .

CRITERIA OF PERFORMANCE EVALUATION
Variation of Geometrical and structural parameters change manipulator performances.Therefore, some criteria should be proposed to compare all types of 6-RUS manipulators to be able to select appropriate structure with proper performances.Applications such as precise, fast or high accelerated displacements.So these criteria include workspace, kinematic and dynamic dexterity are proposed.

Workspace
Workspace is the most important criteria in selecting structure particularly when a manipulator wants to transfer and move objects.Manipulator workspace is collection of all configurations that end-effector can obtain it by proper selection of joints coordinates.Generally, workspace is limited in parallel manipulators, hence this factor is the most important factor in design.Numerical methods (such as discretization (Rezaei et al., 2013), and analytical methods (such as geometrical methods (Bonev, 2002) have been studied.Calculating workspace for 6-RUS fixed manipulator is proposed using geometrical method (Bonev, 2002); however, this method can not be used for reachable workspace.
In this way, discretization method is used to extract workspace.In the mentioned method, space is divided to small distinct spaces then corresponding joint angles to various points are obtained by inverse kinematic.If obtained angles are not in permitted range, that point will belong to workspace.
Workspace of the manipulator may has an irregular shape.However, in some applications, regular workspace shape such as cubic shape workspace is required.In such cases, the biggest volume cubic shape inscribed in the reachable workspace is extracted.In the current study, a procedure described by Hosseini et al. (2011) is used to obtain the maximal inscribed cubic workspace.

Kinematic Performance Indices
One of the important criterion to evaluate kinematic performance for parallel manipulators is dexterity.Dexterity is end-effector ability of very small volunteering precise and easy movements around a point in workspace.Therefore, manipulator can perform highly precisely if they have high dexterity and are proper to be used in precise location and simulation.To calculate manipulator dexterity, Jacobin matrix is needed to relate end-effector velocity vector X  and actuated joints velocity vector q  as the relation 2.
Many indices are used to estimate parallel manipulator kinematic dexterity which important ones are condition number and manipulability indices.Condition number shows relation of the existed error in operating joints and end-effector error in Cartesian space.Jacobin matrix condition number is obtained by: Condition number based on Jacobin singular values may be written as relation (4).
In which max ( ) s J and min ( ) s J are maximum and minimum eigenvalues of the Jacobian matrix.
1/k J is considered as the local conditioning index (LCI) (Kucuk and Bingul, 2006)  terity of the manipulator.LCI varies from 0 to 1, which o shows the ill situation and 1 indicates good situation.Of course, the mentioned indices have some problems in dimension and physical interpretation (Khan and Angeles, 2006), because Jacobin matrix elements have similar dimension.Some methods are proposed to solve these problems.One of the most applied method is that Jacobin matrix divides to two translational and rotational coefficients parts and condition number of each one is obtained.
Another applicable method is using of part of Jacobin matrix with length dimension that is divided to coefficient with dimension.This coefficient is stated variedly in different references.For example, in a reference based on one of structural parameters, manipulator is as a platform radius.However, what is stated in this article is characteristic length that is said in Fassi et al.(2005) based on Jacobin matrix properties.In this regard, characteristic length is stated as equation ( 5).
Where R J and T J are translational and rotational parts of Jacobin matrix, matrices size is 3*6.
By using this coefficient, homogenized jacobian matrix are expressed as: The proposed indices for measuring manipulator dexterity is depending on manipulator condition, so global conditioning index (GCI) can be used as equation ( 7).

Dynamic Performance Indices
Dynamics examines relation between velocity and acceleration the end-effector with inserted forces and torques on joints.This relation is stated by a series of differential equations that are movement equations.Dynamic equations dominant on manipulator can be expressed by equation ( 8).
In which ( ) M q is inertia matrix, ( , ) C q q  is Coriolis and centrifuge parts matrices, ( ) G q is mass matrix, and t is inserted torque matrix on joints.Dynamic relations has principal role in some applications such as manipulators with high velocity, high load carrying capacity, high pass band manipulators, and high performance manipulators.
Although, dynamic of the system can be modeled, the made models have non-linear equations which make it difficult the manipulator dynamic performance prediction (Alam and Akhlaq(2015)).Solids and Structures 13 (2016) 2414-2438 Hence, proper criteria should be used to be perceived simply by designer to insert necessary changes to improve system dynamic.Therefore, some indices are used to estimate dynamic performance.

Latin American Journal of
The initial methods to describe dynamic specifications proposed by Asada(1984) which examined the relation between generalized manipulator speed and generalized inertia ellipsoid method (GIE).Khatib(1995) presented belted inertia ellipsoid method (BIE) based on (GIE) and analyzed redundant manipulator dynamically using this method.
Dynamic dexterity is one of the important parameters in manipulators, dynamic dexterity states ability of accelerating of end-effector in each point in any desirable direction in manipulator specific condition (Wu et al. 2008).For example, one of the useful indices to show manipulator dynamic dexterity is Yoshikawa (1985) dynamic manipulability.Yoshikawa investigated the relationship between generalized accelerations and forces and proposed dynamic manipulability ellipsoid (DME) ability.
In this part, dynamic dexterity is used as dynamic performance investigation criterion.Dynamic dexterity index is extracted by equation ( 9).
1/k M is considered as a local dynamic conditioning index (LDCI).The proposed index is de- pending on manipulator condition, so global dynamic conditioning index (GDCI) can be used as follows:

Inverse Kinematic
In inverse kinematic, corresponding joint coordinates are mentioned by condition and direction of mobile platform.To extract inverse kinematic equations, this is noticeable that connecting vector size U i to P i is fixed and equal to 2 l : B i P and B i U are condition vectors of points P i and U i from center of fixed coordinate B and obtained from the following relation: x is condition of mobile platform center than fixed coordinate origin.
Latin American Journal of Solids and Structures 13 (2016) 2414-2438 In addition, according to the definition of rotation matrix as following, mobile platform direction than fixed platform is stated as; a j a j a j a y y j y j y a j y a y j y j a j y y a j y a While a , j and cshow rotation angles of mobile platform around general coordinate origin, y and s shows angle of cosine and sine.P i P is coordinate of P i than P mobile coordinate centers that can be obtain as follows; In which, i P q is the i th element of matrix P θ as follows: 2 2 P P P P P P q q q q q q θ (16) A similar expression may be written for B i B coordinate as: r r q q q q q q q q B θ (17) U is position vector of U i than B i obtained by considering revolute joint angle as: In which, i q is the i th arm angle than fixed platform level as Figure 2. In addition, di q is the i th element of matrix d θ as following: By determining vectors B i P and B i U and replacing them in equation ( 11), equation ( 20) is ob- tained.

cos sin
Coefficients of the mentioned relation are: Latin American Journal of Solids and Structures 13 (2016) 2414-2438 Finally, by solving equation ( 20), actuating joint angles are extracted by relation ( 22).

Jacobian
Jacobian matrix is used to indicate the relation between angular velocity of actuating joint and endeffector velocity in mobile platform.To extract Jacobin matrix, both sides of equation ( 11) are squared.
By replacing equation ( 12) in equation ( 23), we will have: by differentiating equation ( 24), following equation can be obtained: In which, i λ is defined as equation ( 26).
1,2,..., 6 In addition, B i i U  and rotation matrix differentiation B P R  may be written as: B 1 -sin cos = -sin sin , 1,2,..., 6 Latin American Journal of Solids and Structures 13 (2016) 2414-2438 By replacing equations ( 27) and (28) in equation ( 25) following equation is obtained: where qi λ , B P x  andare defined as: P ω Finally, relationship between rotational and linear angular velocity vector of mobile platform X  and velocity vector the actuating revolute joints  θ is obtained in equation ( 31). Where: In addition, X J and q J matrices are obtained from relations (33) and (34).

GENERALIZED MASS MATRIX
In order to obtain manipulator mass matrix, kinetic energy of all its elements including mobile platform, arms, and rods are extracted.In the rest, kinetic energy of each part is obtained.
In addition, rotation and linear velocity vector of mobile platform are stated using Jacobin matrix based on angular velocity vector of actuating joints as following: Finally, by replacing equations ( 37) and ( 38) in equation ( 36), kinetic energy of mobile platform is obtained by equation (39).

Kinetic Energy of the Arms
Kinetic energy of the i th arm which rotates by angular velocity i q  around codirecting vector of revo- lute joint is obtained by the following equation: In which, a m is each arm mass and a I is inertia moment around passing vector from mass cen- ter.It is assumed that mass center of each arm is in its center.Finally, total kinetic energy of manipulator arms in matrix form and based on angular velocity vector of operating joint will be as; Which, matrix a M is defined as relation (42): Latin American Journal of Solids and Structures 13 (2016) 2414-2438

Kinetic Energy of the Rods
Kinetic energy of connecting rod to the mobile platform is obtained by the equation ( 43) which is resulted from linear velocity of mass center and angular velocity around mass center.
In which, i i U P ω is angular velocity, ri v is linear velocity of mass center, and B ri I is inertial moment of rod that all are determined regarding to basic coordinate.In addition, m ri is value of rod mass.Velocity of the rod mass center of mobile platform is obtained by equation ( 44).
ω is arm angular velocity vector than reference origin coordinate.This vector is obtained by the following equation: In which, B Bi R is rotational matrix from connecting local coordinate to revolute vector than reference coordinate that is obtained by relation ( 46).
Arm angular velocity as a coefficient of joint velocity vector  θ can be stated as: To obtain rod angular velocity i i U P ω , velocity of point P i is written by two ways using equa- tions ( 48) and ( 49): Latin American Journal of Solids and Structures 13 (2016) 2414-2438 In which, By equating the two sides of equation and exterior multiplying of sides in vector i i U P , the fol- lowing equation is obtained: Consequently, angular velocity vector of rods will be: By replacing equations ( 38) and ( 47) in equation ( 50), rod angular velocity based on operators angular velocities  θ will be as following: (1:3,:) (4:6,:) Finally, rod angular velocity based on operating joints angular velocities is stated by equation ( 54).
In addition, by this assumption that rod inertia moment than connecting coordinate to mass center is ri I , inertial moment to general coordinate B ri I is given by: In which, 2i R is rod rotational matrix than reference coordinate as: In which, i x vector is along with i th rod and xi n is unit vector that is obtained by equation ( 58).
Latin American Journal of Solids and Structures 13 (2016) 2414-2438 In addition, other unit vectors related to connecting coordinate to mass center is obtained by the following equation: Consequently, by replacing equations ( 54), ( 55), and (56) in equation ( 43), kinetic energy related to each rod can be written as: In addition, total kinetic energy of all rods is obtained by equation ( 61).

MULTI-OBJECTIVE OPTIMIZATION
As it was mentioned, beside parameters d q and P q , other geometrical parameters in vector λ are effective on manipulator performance.In addition, having all performance properties may not be possible; therefore, it is needed to collect some solutions which have a collection of properties.
In multi-objective optimization, a collection of solutions is indicated instead of getting one solution.In such a condition, problem usually has more than one optimum solution which is called Pareto optimal solution.An important conception in optimization methods based on Pareto is conception of "dominate".A solution like A dominates on B if it is not worse than B in any objective functions and be better at least in one objective function.A collection of solutions having all conditions Latin American Journal of Solids and Structures 13 (2016) 2414-2438 is called "Pareto front".The aim of multi-objective optimization is finding Pareto solutions by collecting of non-dominate solution of problem.

Objective Function
Optimization problem can be inserted in total reachable or pre-determined workspace like a cube or cylinder.Therefore, the aim is finding a vector of geometrical and structural γ parameters based on relation (1) so to satisfy the following conditions in predetermined cubic workspace :

Geometrical Constraints
In this part, related constraints of optimization problem of 6-RUS parallel manipulator is proposed.since angle of revolute joints axis d q are considered as one of the design parameters, The first con- straint is inserted on revolute joints in a way that two adjacent arms don't contact with each other.According to this, if n th and (n-1) th arms are adjacent, arm n th and (n+1) th won't contact with each other.
Another constraint is the movement range related to installed spherical joint on mobile platform as: Latin American Journal of Solids and Structures 13 (2016) 2414-2438 In which, p n is vertical unit vector on mobile platform and 2i l n is unit vector along connecting rod to mobile platform.In addition, there are similar movement constraints on universal joint movements.
1 2 -1 2 2 =cos ( . ) max( ) 60 In addition, distance of two adjacent vector should be in a way to avoid conflict to each other.For this purpose, the following constraint is added to optimization constraint: In which, d is rod diameter and D is the distance between two rods.

Modified Multi-Objective Bees Algorithm
The Bees Algorithm is a swarm intelligent method to solve optimization problems and was developed in 2005Pham et al.(2006)).This optimization algorithm is based on bee colony food foraging behavior in nature.The specific feature of this method makes it as a reasonable nature imitative and non random simple algorithm.Multi-objective Bees algorithm is based on Pareto front extraction Pham and Ghanbarzadeh(2007): 1. Random construction of initial population include n scout bees 2. Calculating qualification value of population and Pareto collection 3. Calculating m prior population places among Pareto non-dominated solutions 4. Indicating neighborhood radius ngh for m bees 5. Sending nsp bees to neighborhood radius and indicating Pareto collection of each neighborhood 6. Selecting the best bees in neighborhood radius according to its Pareto collection 7. Modifying total Pareto collection 8. Selecting (n-m) residual bees randomly 9. Making new population from scout bees As it mentioned, in the multi-objective Bees Algorithm, just the domination concept is used to select the best solution.However, it reduces the quality and continuity of Pareto solution diagram.It means, obtained solutions may be aggregate in some portion of the Pareto frontier.So crowding distance are used as a secondary criterion for evaluation a solution.The crowding distance shows the density of solutions.Considering Figure 3 it may be obtained from equation (70).

RESULTS
In order to investigate structural parameters effect on 6-RUS manipulators performance, inertia and geometrical parameters are considered based on Table 2.The arms length for each axis direction of the revolute joint ( d q ) are selected in a way to prevent from contact of the adjacent arms during the movement.By considering parameters d q and P q that specify Hexa, Hunt type, and Zamanov type manip- ulators, Reachable workspace and maximal inscribed cubic workspace of them are extracted and shown in the Figure 4.By observing workspace volumes based on Table 3, it is seen that the maximum workspace volume is related to Hexa manipulator which is 92% and is 45% more than Hunt type and Zamanov type manipulators, respectively.Other noticeable point is high symmetry of Hexa and Hunt type workspace than Zamanov type workspace.As it is seen in Figure 4, workspace in Zamanov type manipulator has more extension toward connection point of two rods to mobile platform.Furthermore, the volume of the maximal inscribed cubic workspaces are presented in Table 3. according the values, percentage of inscribed cubic workspace to reachable workspace for Hexa, Hunt and Zamanov type manipulators are 33%, 32% and 28% respectively.The reason of why Hexa an Hunt have more percentages, can be described by their symmetry workspaces.In the rest, to investigate the effect of distinctive d q and P q parameters on 6-RUS manipulator performance, some diagrams are proposed in Figures 5 to 9, change rate of d q and P q parameters are considered in a way that covers total range of variation.parameters values indicate Hexa, Hunt type, and Zamanov type manipulators by sign of triangle, circle, and square, respectively.Since global performance properties of a manipulators besides mobile platform position depends on its direction; therefore, constant orientation workspace is considered to investigate global properties., that structural parameters of Hunt type manipulator is there, Workspace for all joint axis angles is small and has the minimum workspace in area B and manipulator will have the minimum workspace by selecting parameters of this area.In order to explore effect of parameters on local kinematic dexterity, it is assumed that the mobile platform is in a certain condition [ , , ]=[0, 0, 0.35]

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x y z .LCI changes diagram is determined based on structural parameters in Figure 6.By observing this diagram, it is concluded that in parameters range that indicate by A, LCI value approaches to zero, it means, by selecting structural parameters in this interval, condition of the manipulator approaches to singular condition.By more precise investigations, it is observed that by selecting parameters in this area, rod, arm, and axis of revolute joint are all in a plane that it is called serial singular condition Bonev(2002).However, for better comparison of various structures, it is good to use global index of dexterity which shows dexterity distribution all over the workspace.Therefore, in Figure 7, GCI changes are brought based on structural parameters changes.By examining diagram, it is observed that changes process of homogeneous GCI is relatively similar to local homogenous LCI.A is a parameters area which is generally too near to singular condition that Hexa manipulator is near to this area and does not have proper condition according to condition number.Manipulators located in B and C zones according to their parameters, which are close to Zamanov and Hunt type manipulators, have the best GCI values.In this diagram, the best GCI is correspond to a manipulator with 48

CONCLUSION
In the present study, three well-known manipulators were classified after determining the axis angle of revolute joint and the spherical joint placement on the mobile platform as the distinctive parameters of 6-RUS parallel manipulators structure.In addition, the effects of these parameters were investigated on performance criteria including workspace and dynamic and kinematic dexterity.Finally, considering all structural and geometrical parameters and determining proper constraints, optimization problem of the 6-RUS manipulator structure was solved using the modified multiobjective Bees Algorithm.Results of this research show that: 1.There is a specific range of parameters that manipulator workspace gets its maximum value.This range is close to the condition which both adjacent chains are parallel that Hexa manipulator is close to such condition.Selecting specified manipulator in this area is proper for moving objects in big spaces.2. By checking local and global dexterity indices values variation, it is seen that areas which rod, revolute vector, and arm are in the same plain, manipulator approaches to singular areas and selecting structure in this area decreases locating precision of manipulators that Hexa manipulator is close to this area.3.By investigating dynamic dexterity, It is observed; although, local values of indices are good in some configuration, generally base on global dynamic dexterity index, when manipulator arms are aligned toward outside, manipulator is in more proper condition than the time manipulator arms are toward inside.4. It is observed by comparing three 6-RUS manipulators that Hexa manipulator has more workspace but has less proper kinematic dexterity condition than other manipulators and less locating precision condition.Moreover, there is no proper dynamic condition so this manipulator has a proper selection for transferring objects with high acceleration in wide workspace.Although, Hunt type manipulator has limited space, it has proper condition according to condition number that increases its locating precision and has similar dynamic and kinematic conditions to other manipulators.Zamanov type manipulator is in middle of two other manipulators and the only point about this manipulator is that workspace of this manipulator has asymmetrical distribution toward manipulator spherical joints.Due to high symmetry of Hexa and Hunt type workspace than Zamanov type workspace, percentage of inscribed cubic workspace to reachable workspace for Hexa, Hunt are more than Zamanov type. 5.In addition, it is observed that the modified multi-objective Bees Algorithm method is able to find optimum solutions of 6-RUS manipulator multi-objective structure designing.Pareto solutions are compatible with mentioned conclusions.6.According to the distribution of local performance indices in each plain in the prescribed workspace, it is concluded that the maximum values for kinematic and dynamic are obtained in the center of plains.Also, lower plains have the better values of performance indices.Finally, various 6-RUS manipulators can fairly compared with each other using results of this research.In addition, this article proposes a criterion to select the appropriate set of dimensional parameters of the manipulator based on its application.
Geometrical and inertial parameters of the selected manipulator.

Figure 5
Figure5shows manipulator workspace changes based on distinctive parameters variation.By examining this diagram, it is observed that maximum workspace volume occurs in parameters zones which are signed by A and C letters that Hexa manipulator parameters is also there.For example, if a manipulator is asked in proper workspace 0 P   , revolute joint angle d q is considered between 25 and 40.Of course, if revolute vector angle d q in area A shows arms conditions to manipulator outside, corresponding point in area C has 180 degree difference from condition A and shows symmetrical condition inside manipulator.Moreover, it is seen that in 120 P  

Figure 5 :
Figure 5: Effect of structural parameters variation on constant orientation workspace.
considered, manipulator will be in singular condition in this situation.As it is observed in Figure6, among three Hexa, Hunt type, and Zamanov type manipulators, Hexa is closer to the singular area rather and the corresponding LCI value is 0.0278.By considering LCI of Zamanov type manipulator in area B (0.1331) and Hunt type manipulator (0.1067) in area C, it is observed that manipulator with selected parameters in these two ranges have proper local dexterity.

Figure 6 :
Figure 6: Effect of structural parameters variation on local conditioning index.

Figure 7 :
Figure 7: Effect of structural parameters variation on global conditioning index.
parameters change on local dynamic conditioning index in certain condition of mobile platform is shown in Figure 8. Checking this Figures shows that the best values are in parameters range indicate by A. in other words, manipulator has more ability to increase acceleration in all directions by selecting structure parameters in this range.It is clear that, Hunt type manipulator is near to this area.Also in zone B that parameters of Hexa and Zamanov type manipulator are located, dynamic dexterity is better than manipulators parameters located in zone C.

Figure 8 :
Figure 8: Effect of structural parameters variation on local dynamic conditioning index.

Figure 9 :
Figure 9: Effect of structural parameters variation on global dynamic conditioning index.

Figure 10 :
Figure 10: Obtained Pareto-front using modified multi-objective Bees algorithm. d

Figure 11 :
Figure 11: Distribution of local kinematic and dynamic performance indices in the workspace.

to evaluate dex- Latin American Journal of Solids and Structures 13 (2016) 2414-2438
Kinetic Energy of Mobile PlatformKinetic energy of mobile platform is produced by rotation and linear velocity.In this regard, kinetic energy is calculated regarding equation (36).

Table 6 :
Parameters of selected points of Pareto front.