SQP Optimization of 6dof 3x3 UPU Parallel Robotic System for Singularity Free and Maximized Reachable Workspace

Cenk Eryılmaz; V. E. Omurlu

doi:10.1155/2019/3928705

SQP Optimization of 6dof 3x3 UPU Parallel Robotic System for Singularity Free and Maximized Reachable Workspace

Eryılmaz, Cenk;Omurlu, V. E. 2019-02-03 00:00:00 Hindawi Journal of Robotics Volume 2019, Article ID 3928705, 14 pages https://doi.org/10.1155/2019/3928705 Research Article SQP Optimization of 6dof 3x3 UPU Parallel Robotic System for Singularity Free and Maximized Reachable Workspace 1 2 Cenk EryJlmaz andV.E.Omurlu Technology Transfer and Commercialization, TUBITAK BILGEM P.K., 74, Gebze, 41470 Kocaeli, Turkey Department of Mechatronic Engineering, Yildiz Technical University, 34349 Istanbul, Turkey Correspondence should be addressed to Cenk Eryılmaz; cenk.eryilmaz@tubitak.gov.tr Received 3 September 2018; Revised 13 October 2018; Accepted 24 December 2018; Published 3 February 2019 Academic Editor: Yangmin Li Copyright © 2019 Cenk Eryılmaz and V. E. Omurlu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The determination of kinematic parameters for a parallel robotic system (PRS) is an important and a critical phase to maximize reachable workspace while avoiding singular configurations. Stewart Platform (SP) mechanism is one of the widely known PRS and it is used to demonstrate the proposed technique. In the related literature, GCI (Global Condition Index) and LCI (Local Condition Index) are the commonly used performance indexes which give a measure about the dexterity of a mechanism. In this work, Sequential Quadratic Programming (SQP) method is used to optimize kinematic parameters of a 6dof 3x3 UPU SP in order to reach maximum workspace satisfying small condition numbers. eTh radius of mobile and base platforms and the lengths of the legs used in the platform are chosen as kinematic parameters to be optimized in a multiobjective optimization problem. Optimization is performed at different stages and the number of optimized kinematic parameters is increased at each level. In conclusion, optimizing selected kinematic parameters at once by using SQP technique presents the best results for the PRS. 1. Introduction to find alternative parallel structures like redundant PRS [2– 4]. In recent years, many different parallel robotic structures In 1947, Gough [1] established the basic principles of a mech- have been investigated for various applications. anism with a closed-loop kinematic structure that allows The performance of a PRS is mainly affected by its geo- positioning and orientation of a mobile platform to test tire metrical parameters. Determining geometrical parameters wear and tear. In 1965, Stewart oer ff ed a very similar structure is called geometric synthesis. One of the most important to Gough’s mechanism to be used as an aircraft simulator. measures for kinematic performance is called dexterity which Ironically, the Gough platform is most oeft n known as Stewart describes the ability of a mechanism to move and apply platform [1]. In recent years, there have been many researches forces in arbitrary directions as easily as possible. However, about PRS whose principles are originated from Gough. They the most suitable index to define the accuracy and rigidity have numerous advantages compared to the serial robotic performance is the condition number, which LCI (Local systems such as high load carrying capacity and precise Condition Index) and GCI (Global Condition Index) are positioning ability. A PRS has a mobile platform connected related to. LCI is defined as the ratio of the maximum and to a fixed base by parallel legs instead of serial links. minimum singular values of the Jacobian matrix of a PRS Besides their advantages compared to serial manipu- [5]. It varies between 1 and infinity [6, 7]. GCI index is a measure of dexterity and the average of LCI calculated for lators, PRS have a small workspace and require a more complex mechanical design. Controlling the motion of a entire workspace. parallel mechanism is a challenging task because of its Dexterity and workspaceare themost important cri- complex kinematic structure and variable dynamics through teria for optimal designs of parallel manipulators. There its workspace. To overcome this issue, many researchers tried are many researches investigating the effects of geometric 2 Journal of Robotics 6 ０Ｊ： →  →  →  r Ｃ 1 b 2 →  Figure 1: Coordinate systems, {P} and {B},ofSP. parameters on dexterity, workspace, and singularity. Some of described. Section 4 illustrates the SQP and optimization them are about maximizing well-conditioned workspace [8] algorithm. In the last section, we discussed the results of the using different optimization methods. Monte Carlo method SQP optimization procedure. [8, 9], multiobjective/genetic algorithm [10–12], controlled random search (CRS) [13, 14], analytic algorithm [15], and neurogenetic algorithm are mostly used ones [16]. G. Abbas- 2. Mechanism of the PRS nejad offered a practical algorithm to find the singularity A PRS is constructed from parallel connected legs between free cylindrical workspace of parallel manipulators using mobile and base platforms. There are two major classification particle swarm optimization method (PSO) [17, 18]. In the criteria of a PRS which are based on (a) joint types of literature, there are other performance indexes such as GSI connections of legs to the mobile and base platform and (Global Stiffness Index) and VTF (Velocity Transformation (b) the number of legs. The array of the types of joints are Factor) [19]. Therefore, some researchers optimized PRS for universal, prismatic, and universal, respectively. So it is called multiperformance indexes which are called MOO (Multiob- UPU. Specifically, linearly driven six legs which are connected jective Optimization) [20–22]. These optimization methods to a mobile and a base platform cause a full spatial motion are used for investigating the best geometric parameters of 3-axis translational and 3-axis rotational motion. This PRS for structures of parallel manipulators. On the other hand is called Stewart Platform [20, 25, 26] and classiefi d by the some researchers compared convergence performances of number of leg connections on upper and lower platforms. A optimization methods [23]. 3x3 SP includes 3 connections on lower and upper side of the In this research, we used SQP method which is Lagran- legs. Each two of six legs share cocentered joints on each side gian based [24] for the optimization of kinematic parameters of the legs. “Figure 1” illustrates the general structure of SP to reach the maximum workspace for a PRS. This paper is that 3x3 special configuration is used for the current research. organized as follows. In the following section, the mechanism of a PRS and its geometric parameters and Jacobian matrix An SP has 48 design parameters affecting its workspace formulation for 3x3 SP are briefly described. In Section 3, and kinematic behavior [27]. These parameters are spatial our performance indexes and workspace assumption are positions of upper and lower joints (2x18 parameters) and Journal of Robotics 3 upper/lower leg lengths (2x6 parameters). By placing connec- “Eq.” (6) is representing the rotation of mobile coordinate tion points on a circle, these would reduce to upper/lower frame {P} with respect to base frame {B}. circle diameters/radiuses passing through upper/lower joint 𝑇 𝑇 centers, r and r . In this research, optimization parameters (7) 𝑝 ={𝑝 ,𝑝 ,𝑝 } ={𝑟 . cos(𝜆 ),𝑟 . sin(𝜆 ),0} p b 𝑝 𝑖 𝑝 𝑖 are these two radiuses and length of the legs, 𝑙 . “Eq.” (7) describes joint positions P of themobileplatform with respect to {P}. 2.1. Geometric Parameters of PRS. The lengths of the legs (𝑙 ), which change according to the desired position and 𝑖 𝑇 (8) 𝑏 ={𝑏 ,𝑏 ,𝑏 } ={𝑟 . cos(Λ ),𝑟 . sin(Λ ),0} orientation of the mobile platform, can be calculated using 𝑖 𝑏 𝑖 𝑏 𝑖 the inverse kinematics equations, x (x, y, z, 𝛼 , 𝛽 , 𝛾 ). In this “Eq.” (8) describes joint positions B of the base platform with case, there are six unknowns, 𝑙 (i = 1....6).“Figure1” respect to {B}. helps us understand the geometry of SP. Mobile and base C = cos(𝛼),S = sin(𝛼),C = cos(𝛽),S = sin(𝛽), coordinate frames, {P} and {B}, are den fi ed. They each are 𝛼 𝛼 𝛽 𝛽 C = cos(𝛾) ,S = sin(𝛾) linked to the mobile platform and the base. The origin of 𝛾 𝛾 frame {P} is located at the center of a circle passing through C C C S S − S C S S + C S C the center of joints between the mobile platform and the legs. 𝛼 𝛽 𝛼 𝛽 𝛾 𝛼 𝛾 𝛾 𝛼 𝛼 𝛽 𝛾 [ ] The Z axis is pointing upward, and the X axis is the bisecting [ ] p p S C C C + S S S C S S − S C 𝑅 = 𝛼 𝛽 𝛼 𝛾 𝛼 𝛽 𝛾 𝛾 𝛽 𝛼 𝛾 𝛼 𝑃 [ ] line of the angle (O P -O P ). The origin of frame {B} is P 6 P 1 −S C S C C 𝛽 𝛽 𝛾 𝛽 𝛾 located at the center of a circle passing through the center of [ ] (9) joints between the base and the legs. The Z axis is pointing 𝑟 𝑟 𝑟 11 12 13 upward, and the X axis is the bisecting line of the angle [ ] (O B -O B ). 𝑟 𝑟 𝑟 = [ ] b 6 b 1 21 22 23 These two coordinate frames lead us to determine the 𝑟 𝑟 𝑟 [ ] 31 32 33 position of the links attached to the base at (B ...,B )and 1 6 the mobile platform at (P .., P ). where 𝛼 is the rotation of {P} with respect to {B} about Z . 1 6 𝛽 is the rotation of {P} with respect to {B} about Y . 𝛾 is the The angle (O P ,O P ) is denoted as 𝜃 .Angles (O P , p 2 p 3 𝑝 p 1 rotation of {P} with respect to {B} about X . O P ), (O P ,O P ), and (O P ,O P ) are each equal to p 3 p 3 p 5 p 5 p 1 Consequently, 120 . Similarly, angle (O B ,O B ) is denoted by 𝜃 .Angles b 1 b 2 𝑏 (O B ,O B ), (O B ,O B ), and (O B ,O B )are,also, b 1 b 3 b 3 b 5 b 5 b 1 equal to 120 . Now, the length of the legs (𝑙 )can be calculated for 𝐶 𝐶 .𝑟 𝐶(𝜆 )+(𝐶 𝑆 𝑆 −𝑆 𝐶 ).𝑟 .𝑆(𝜆 ) 𝛼 𝛽 𝑝 𝑖 𝛼 𝛽 𝛾 𝛼 𝛾 𝑝 𝑖 (10) [ ] [ ] 𝑖 = 1,2,...,6 (1) = 𝑆 𝐶 .𝑟 𝐶(𝜆 )+(𝑆 𝑆 𝑆 +𝐶 𝐶 ).𝑟 .𝑆(𝜆 ) [ 𝛼 𝛽 𝑝 𝑖 𝛼 𝛽 𝛾 𝛼 𝛾 𝑝 𝑖 ] −𝑆 .𝑟 𝐶(𝜆 )+𝐶 𝑆 .𝑟 .𝑆(𝜆 ) 𝛽 𝑝 𝑖 𝛽 𝛾 𝑝 𝑖 [ ] The angles (X , O P ) and (X , O B ) are denoted as 𝜆 and p p i b b i i Λ , respectively. Therefore, we have The length of the legs can be calculated as follows: 2 2 Λ =60𝑖− , (11) 𝑙 = (𝑆 ) +(𝑆 ) +(𝑆 ) 𝑝 (2) Using (4), we get length of the legs as 𝜆 =60𝑖− , 2 2 2 2 2 2 𝑙 =𝑥 +𝑦 +𝑧 +𝑟 +𝑟 𝑖 𝑝 𝑏 𝑖 = 1,3,5 +2(𝑟 𝑝 +𝑟 𝑝 )(𝑥−𝑏 )+2(𝑟 𝑝 +𝑟 𝑝 ) (12) 11 12 21 22 Λ =Λ +𝜃 , 𝑖 𝑖−1 𝑏 (𝑦 − 𝑏 )+2(𝑟 𝑝 +𝑟 𝑝 ) (𝑧 )−2(𝑏𝑥 +𝑦𝑏 ) 𝜆 =𝜆 +𝜃 (3) 31 32 𝑖 𝑖−1 𝑝 𝑖 = 2,4,6 2.2. Jacobian Matrix. Jacobian matrix of a robotic system represents the linear transformation between the speeds in 𝐵 𝐵 𝐵 𝐵 (4) 𝑆 =− 𝑏 + 𝑡 + 𝑝 𝑖 𝑖 𝑖 ∙ joint space, ,and world space, 𝑥 ̇,of a robotic manipulator. 𝐵 𝑇 (5) 𝑡 ={,𝑥 ,𝑦 }𝑧 Jacobian matrix provides the velocity ratio from the joint space to the space of the end-effector [26]. “Eq.” (5) represents translational position of {P} with respect For a parallel manipulator, the length of the legs (𝑙 )as a to {B}, vector is given as 𝐵 𝐵 𝑃 𝑝 = 𝑅 𝑝 (6) l = l ,..., l (13) ( ) 𝑖 𝑃 1 6 𝑖𝑦 𝑖𝑥 𝑖𝑦 𝑖𝑥 𝑖𝑦 𝑖𝑦 𝑖𝑥 𝑖𝑥 𝑖𝑦 𝑖𝑥 𝑖𝑧 𝑖𝑦 𝑖𝑥 𝑖𝑧 𝑖𝑦 𝑖𝑥 𝑖𝑧 𝑖𝑦 𝑖𝑥 4 Journal of Robotics The position of the end-eeff ctor is If the spectral norm is introduced, the dexterity indexes can be described as (14) x =(,𝑥 ,𝑦 ,𝑧 ,𝛼 ,𝛽 𝛾) 𝑆𝑉 max(𝐽) Cond(J) = (24) 𝑆𝑉 min(𝐽) So, the relation equation between the position of the end- effector ( x) and the length of the legs (𝑙 ) can be represented where SV and SV represent the maximum and min- max(J) min(J) as imum singular values of the Jacobian matrix (J), respectively. The values of Cond (J), which are directly related to the l = g(x) (15) singular values of the Jacobian matrix, are between one and positive infinity. All singular values will be the same in the By differentiating (15), we obtain Jacobian matrix, and the manipulator is isotropic when Cond (J) is equal to 1. l = Jẋ (16) Otherwise, when Cond (J) is potentially equal to positive infinity, then the Jacobian matrix is singular. So, when the condition number is considered, the value of the condition J ≡ (17) 𝑖 number should be minimum for a better dexterity. The condition number isusedasanindex formea- 𝑇 𝑇 ̇ ̇ ̇ (18) [𝑙 ,...,𝑙 ] = J[𝑥, ̇ 𝑦, ̇ 𝑧, ̇ 𝛼, ̇ 𝛽, 𝛾]̇ 1 6 suring the accuracy/dexterity of a robotic system, and also it possesses information of whether a position is close to 𝜕l singularity or not. However, it is not entirely possible to den fi e J = (19) 𝜕x 𝑗 a mathematical proximity to singularity for a PRS whose degree of freedom is a mix of translational and rotational Consequently J is the Jacobian matrix of the mechanism, motion. This is because the use of condition number is just an which is used in calculating the performance indexes of the index and yet it has an advantage as a single number which is PRS, and is utilized in this research as used to describe the overall kinematic behavior of a PRS. Because of the complex definition of the condition num- J = [J , J , J , J , J , J ], 𝑖 = 1,...,6 (20) 𝑖1 𝑖2 𝑖3 𝑖4 𝑖5 𝑖6 ber, we cannot calculate its analytical form as a function of the position parameters except for very simple robots. However, robust linear algebra tools can calculate it numerically for a 3. Performance Indexes and Workspace given spatial position. Parallel manipulators that have both 3.1. Dexterity, GCI, and LCI. Dexterity is an indicator of the translational and rotational dof have a major drawback about ability of a parallel manipulator about how rates of joints condition number. Because the unit of translational motion transform to the rates of the mobile platform and vice versa. is different from the rotational motion, the Jacobian matrix It depends on the position and the structure of the parallel is not homogenous [28]. u Th s, there exist some researches manipulator and can be determined by using the condition about the normalization of Jacobian matrices which are partly number of its Jacobian matrix as rotational and translational. Some of them defined their own methods for normalization and some of them divided Dexterity = Cond(J) (21) Jacobian matrix by the degree of the matrix [29–32]. where Cond (J) is the condition number of Jacobian matrix 3.2. Uniformity. Uniformity is another global performance and is defined as index for PRS. It is an indicator of the uniform dexterous 󵄩 󵄩 workspace. Calculation of uniformity index is shown in −1 󵄩 󵄩 󵄩 󵄩 Cond J = 𝐽 𝐽 (22) ( ) ‖ ‖ 󵄩 󵄩 󵄩 󵄩 equation (25). Uniformity index value changes between 1 and infinity. The index value of the best uniform dexterous ‖.‖ represents the norm of the related vector or matrix. workspace is equal to 1. While calculating the condition number of a Jacobian 𝐼𝐿𝐶 matrix, different types of norms are used in the literature, max 𝑢= (25) and the most used ones are Frobenius and Spectral norms [1]. 𝐼𝐿𝐶 min The difference between these norms is about their calculation methods. If the Frobenius norm is considered, then we can 3.3. Workspace. The number of possible mechanical con- write figurations of serial robotic systems is less than that of parallel robotic system configurations. us, Th comparison √ (23) ‖J‖ = tr(J J) of workspace of serial robotic systems is applicable. For example, one can easily assess that RRR robotic system has In this case, the Frobenius norm is defined as the extracting greater workspace capacity than a cartesian configuration roots of the quadratic sum for each element of the Jacobian because of its angular workspace capacity. However, because matrix. GCI (Global Condition Index) is den fi ed all over the of the vast number of mechanical configuration possibilities workspace as the average of Cond(J) = LCI (Local Condition of parallel robotic systems, no such comparison could be Index). performed easily [24]. 𝑖𝑗 𝜕𝑥 𝜕𝑔 Y Journal of Robotics 5 0.4 0.4 △； =0.04 0.35 0.35 0.3 0.3 0.2 0.2 0.25 −0.2 −0.2 −0.2 −0.1 0 0.1 0.2 y-axis (m) x-axis (m) x-axis (m) 0.2 0.4 0.1 0.35 0.3 −0.1 0.25 −0.2 −0.2 −0.1 0 0.1 0.2 −0.2 −0.1 0 0.1 0.2 y-axis (m) x-axis (m) Figure 2: Volume representation of the workspace to be optimized for the SP and the overall workspace. One can manage to compare 6dof parallel robotic systems for 12000 points in the workspace and then calculated GCI. that are abbreviated as SP in this study. These classes of “Figure 2” represents the volume of the workspace for 3x3 full-motion-capable spatial mechanisms are classified as 3x3, SP and the inner workspace inside of it to be optimized. We 6x3, and6x6 basedon theirnumberof upper and lower used different resolution, “ Δa”, values for each axis of the platform connection points. So, 3x3 has triangular base and workspace as mentioned above. For example, Δa is 0.04 m for mobile platform, while 6x3 configuration has hexagonal base linear motion along y-axis. and triangular mobile platform configuration. Lastly, 6x6 has hexagonal base and mobile platform. Considering the rs fi t, 4. Optimization by SQP the second, and the third configuration of similar dimensions and with the same minimal and maximal leg lengths, the third The primary purpose of the optimization is to increase has the largest workspace, followed by the second and then dexterity and usable workspace of the platform by chang- the rfi st. It must be noted that the workspace volume of the ing the design parameters. Besides various optimization 6x6 structure is always approximately 30% greater than that of techniques, we used SQP method for the optimization of the 6x3 mechanical structure. Similarly the 6x6 congfi uration design parameters 𝑟 , 𝑟 ,and 𝑙 .Min andmax values of the 𝑏 𝑝 𝑖 volume is 70% greater than that of the 3x3 workspace. u Th s objective design parameters are set before the optimization the 6x3 volume is approximately 25% greater than that of the procedure. Another constraint is Z(LCI of workspace point). 3x3 configuration workspace [1]. We calculated all Z (LCI) for each point in the predefined Additionally, the effect of geometrical parameters on workspace shown in “Figure 2.” parallel robotic systems is much greater than that on serial robotic systems. For instance, external stiffness over a given 4.1. Cost Function Assignments. In this research, Matlab workspace of a 6x6 Gough Platform can dieff r by 700% for a Optimization Toolbox (fmincon) is used to perform SQP change of only 10% of the platform radius, 𝑟 [24]. optimization technique for the PRS. The optimization proce- Therefore, in order to extend the reachable workspace of a dure is accomplished in three stages for maximum dexterity 3x3 SP mechanism, objective workspace for optimization and with reachable workspace: performance evaluation is described as follows: (1) One-variable optimization: 𝑟 is intended to be opti- (i) P = Changes from -0.06 to +0.06 m in 0.04 m regular mized intervals (2) Two-variable optimization: 𝑟 and 𝑟 are intended to 𝑝 𝑏 (ii) P = Changes from -0.06 to +0.06 m in 0.04 m regular be optimized intervals (3) Three-variable optimization: 𝑟 ,𝑟 ,𝑙 are optimized all 𝑏 𝑝 𝑖 (iii) P = Changes from +0.3 to +0.4 m in 0.02 m regular together intervals (iv) 𝛼 , 𝛽 , 𝛾 =Changes from -5 to +5 in 2.5 regular For all optimization approaches, a sub cost function is defined intervals as follows: Therefore, total 12000 spatial, 6dof, coordinates need to be (i) Sub cost function for one-variable optimization is satisfied after the optimization performance. First, according 𝑓 𝑥 :𝑟 ⋅ 𝑍−1 to defined workspace, we calculated LCI in Frobenius norm ( ) ( ) (26) 1 𝑝 2,50 z-axis (m) z-axis (m) y-axis (m) z-axis (m) 6 Journal of Robotics r (Radius of moving r (Radius of base p Leg lengths are platform) is defined platform) is defined defined for for min and max as a default min and max values values Reachable points Subcost function is Workspace is created checked selected as f (x) 1 as a rectangular Unreachable points For quick calculation workspace. We used aren't taken into max value of Subcost its corner coordinates account function is defined. layer by layer All condition 7. 9. numbers are For minimum value Cost function is calculated according of r Cost function is minimized according to defined values to min value of r defined as g(x) p except unreachable points Figure 3: Algorithm for one-variable optimization. (ii) Sub cost function for two-variable optimization is 4.2. One-, Two-, and rTh ee-Variable Optimization Algorithm. Figure 3 represents the optimization procedure sequence for 𝑓 (𝑥 ) :𝑟 ⋅𝑟 ⋅ (𝑍−1 ) (27) 2 𝑝 𝑏 one variable in which we have to fix other variables and define their min and max values to constrain them. Secondly, we (iii) Sub cost function for three-variable optimization is limit the leg lengths to check if the points in the workspace 𝑓 𝑥 :𝑟 ⋅𝑟 ⋅ Average of (𝑙 ) 𝑍−1 are reachable or not. Later on, we define the sub cost function ( ) ( ) (28) 3 𝑝 𝑏 𝑖 𝑓 (𝑥) according to one variable or𝑓 (𝑥) for two variables and 1 2 (iv) Cost function for all cases is create a limit value for selected 𝑓 (𝑥).Aeft rthat, we checkif the workspace points are reachable or not, and we eliminate 𝑔 (𝑥 ) :∑Sub cost 𝑓 (𝑥 ) (29) the unreachable ones. Then, cost function 𝑔(𝑥) is defined and condition number 𝑍 for each workspace point is calculated and an optimum value for 𝑟 is investigated which makes the We define cost functions as described above because min- 𝑝 sum of 𝑓(𝑥) (or 𝑔(𝑥) ) minimum. A similar algorithm with a imum g(x) values are investigated, which occur from the little difference is performed for two-variable optimization, 𝑟 sum of minimum sub cost function, f(x), values. It means and 𝑟 . When optimizing for two variables, rfi st step defined that minimum f(x) values are to be found indeed. While 𝑏 in “Figure 3” is skipped and instead of that step, we define searching minimum f(x) values, two criteria are present. One min-max values for 𝑟 as a second stage, and we use sub cost of them is the need for minimum Z and minimum 𝑟 and function 𝑓 (𝑥) set for two variables instead of 𝑓 (𝑥) at the the second criterion is the reachable workspace. Since Z 2 1 fifth step. For three-variable optimization, we constrained all never becomes smaller than 1, the algorithm is constructed variables defining their minimum and maximum values. As accordingly. If the targeted position in the workspace is asub cost function, weused 𝑓 (𝑥). not reachable, large value of Z is assigned not to calculate 3 real Z value in order to eliminate the unreachable point for evaluation. us, Th unnecessary calculations finding minimum 5. Optimization Results f(x) for that corresponding position are avoided. Searching for the minimum of two variables, 𝑟 and 𝑟 , which makes the 3x3 SP mechanism mobile platform diameter is analysed 𝑝 𝑏 sub cost function f(x) minimum, we multiply these variables for constant base diameter utilizing SQP technique. Mobile defining the sub cost function 𝑓 (𝑥). u Th s, while minimizing platform/base platform diameter optimization for better the cost function, we optimize variables. We constrained Z workspace reach and dexterity is performed. Condition (condition number) with a maximum value of 1000 as an number variation for dieff rent mobile platform diame- indicator for dexterity. We limited leg lengths and checked ters in selected range, minimum of condition number for if the coordinates in the workspace are reachable or not. base/mobile platform diameter change, and the number of Searching for the minimum of three variables 𝑟 , 𝑟 , and reachable points for dieff rent base/mobile platform diameter 𝑝 𝑏 𝑙 , which make the sub cost function f(x) minimum, we combinations are discussed. multiply 𝑟 , 𝑟 with an average of 𝑙 and (Z-1) defining the sub In Figure 4, horizontal axis refers to scanned 𝑟 values, 𝑝 𝑏 𝑖 𝑝 cost function 𝑓 (𝑥). We optimized parameters for maximum and vertical axis refers to minimum condition number which reachable dexterous workspace. we get from calculations without homogenization of Jacobian Journal of Robotics 7 Z < 1000 Amount of Reachable Points in Workspace 0.11 4500 0.1 0.09 0.08 30 0.07 0.06 0.05 500 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0 r [m] 0.050 0.070 0.090 0.110 0.130 0.150 Figure 6: Number of the reachable points whose condition numbers r (m) are under 1000. Zmin Figure 4: 𝑍 (min condition number) variations for different 𝑟 𝑚𝑖𝑛 𝑝 Table 1: Optimization of mobile platform diameter for various values while 𝑟 is fixed to 0.175 [m]. range constraints. 0.050 < 𝑟 < 0.180 0.070 < 𝑟 < 0.180 0.080 < 𝑟 < 0.180 𝑝 𝑝 𝑝 Min Z (Condition number in Frobenius norm) 0.15 44 𝑟 = 0.075 𝑟 𝑡𝑝𝑜 = 0.075 𝑟 𝑡𝑝𝑜 = 0.088 𝑝,𝑜𝑝𝑡 𝑝 𝑝 0.090 < 𝑟 < 0.180 0.100 < 𝑟 < 0.180 0.110 < 𝑟 < 0.180 𝑝 𝑝 𝑝 0.14 𝑟 𝑡𝑝𝑜 = 0.090 𝑟 𝑡𝑝𝑜 = 0.100 𝑟 𝑡𝑝𝑜 = 0.110 𝑝 𝑝 𝑝 0.13 0.120 < 𝑟 < 0.180 0.090 < 𝑟 < 0.250 0.050 < 𝑟 < 0.250 𝑝 𝑝 𝑝 𝑟 𝑡𝑝𝑜 = 0.120 𝑟 𝑡𝑝𝑜 = 0.090 𝑟 𝑡𝑝𝑜 = 0.075 𝑝 𝑝 𝑝 0.12 0.11 upper bound of 1000. As indicated in Figure 6, while𝑟 values 28 decrease, 𝑟 values increase, and numbers of reachable points 0.1 elevate. The upper bound of Z value is equal to 1000. us Th ,the numbers of reachable points are high for low mobile platform 0.09 diameters and high base diameters. 0.11 0.12 0.13 0.14 0.15 0.16 0.17 During optimization process of mobile platform diame- r [m] ter, different initial values of 𝑟 with various constraints are Figure 5: 𝑍 with respect to 𝑟 and 𝑟 in predefined reachable 𝑚𝑖𝑛 𝑝 𝑏 used,as shown in Table 1.While 𝑟 = 0.175 [m], minimum workspace. limit of 𝑟 is kept 0.050 [m] and leg lengths are scanned between 0.300 [m] and 0.450 [m]. As a result, the optimum value of 𝑟 is calculated to be 0.075 [m]. When minimum matrix. We get low values of 𝑍 while 𝑟 is increasing. We limit for 𝑟 is 0.080 [m], optimum value is found as 𝑟 = min p 𝑝 𝑝 are searching optimum values for most efficient workspace 0.088 [m]. As mentioned above, different initial values of 𝑟 volume which has reachable points with low condition are used for optimization. Consequently, when the constraint numbers. As a result, high mobile platform diameter refers to value of 𝑟 changes, the optimum value of 𝑟 changes. This is 𝑝 𝑝 calculated minimum condition numbers. because, in dier ff ent workspace regions, we have dieff rent local In Figure 5, horizontal axis refers to scanned 𝑟 values. minimums. Left vertical axis refers to scanned 𝑟 values and right vertical If the optimization procedure in Figure 8 is analyzed axis refers to the mapping of condition numbers which thoroughly, while the lowest constraint of 𝑟 is 0.050 and we calculated in Frobenius norm for predefined reachable 0.070 [m], we selected optimization scanned range between workspace. As seen in Figure 5, minimum condition numbers 0.050 and 0.180 [m]. Then, we obtain an optimum value of 𝑟 occur with high values of 𝑟 and 𝑟 which mean dexterity is 0.075 [m]. When the min constraint value of 𝑟 is elevated to 𝑏 𝑝 𝑝 getting better while they are increasing.However,itis obvious 0.080 [m], the rfi st local minimum of 𝑟 0.075 [m] is passed. from the plot that comparable decrease exists of the mobile Optimized value of 𝑟 shifts from 0.075 [m] to 0.088 [m]. platform diameter with respect to base diameter to obtain Out of nine different optimization combinations, one can minimal condition numbers.Also, the rate of this decrease understand that we have two local minimums which are is getting smaller when the base diameter is a lot higher 0.075 [m] and 0.088 [m]. Then, combinations that include the than mobile platform diameter. The horizontal tendency of initial value of 0.090 [m] and larger will result in an optimized platform diameter is apparent for higher base diameters. value of start point of the scanned range, 0.090 [m], 0.100 [m], In Figure 6, horizontal axis refers to scanned 𝑟 values. 0.110 [m], etc. Left vertical axis refers to different 𝑟 values, and the right Consequently, we found two local minimum values vertical axis refers to the mapping of reachable points in the (0.075 [m] and 0.088 [m]) for 𝑟 having maximum reachable workspace which have condition numbers limited with an points which are limited to upper bound condition number, r [m] Z , min condition number min r [m] p 8 Journal of Robotics Table 2: Optimizing mobile platform diameter, 𝑟 . Cost.F(x) Base Platform Radius, 𝑟 𝑝,𝑜𝑝𝑡[]𝑚 𝑟 0.125 0.130 0.135 0.140 0.145 0.150 0.155 𝑏,𝑜𝑝𝑡[]𝑚 0.070 4.8xE7 4.8xE7 8xE7 4.8xE7 4.8xE7 8xE7 4.8xE7 0.074 0.075 0.082 0.075 0.076 0.090 0.076 0.129 0.130 0.133 0.130 0.125 0.141 0.125 0.075 4.8xE7 8xE7 8xE7 8xE7 8xE7 8xE7 8xE7 0.077 0.075 0.081 0.086 0.087 0.081 0.093 0.125 0.130 0.134 0.137 0.140 0.135 0.145 0.080 6.4xE7 9.6xE7 8xE7 6.4xE7 8xE7 8xE7 8xE7 0.095 0.102 0.084 0.080 0.088 0.093 0.094 0.125 0.128 0.134 0.125 0.142 0.144 0.146 0.085 6.4xE7 6.4xE7 6.4xE7 6.4xE7 6.4xE7 8xE7 8xE7 Mobile Platform Radius, 𝑟 𝑝 0.085 0.085 0.085 0.085 0.085 0.093 0.098 0.125 0.125 0.125 0.125 0.125 0.144 0.148 0.090 6.4xE7 6.4xE7 8xE7 6.4xE7 6.4xE7 8xE7 6.4xE7 0.090 0.090 0.084 0.090 0.090 0.096 0.090 0.125 0.125 0.145 0.125 0.125 0.147 0.125 0.095 6.4xE7 9.6xE7 6.4xE7 6.4xE7 6.4xE7 8xE7 8xE7 0.095 0.095 0.095 0.095 0.095 0.097 0.099 0.125 0.130 0.125 0.125 0.125 0.148 0.150 0.100 9.6xE7 9.6xE7 9.6xE7 9.6xE7 9.6xE7 9.6xE7 8xE7 0.100 0.100 0.109 0.112 0.100 0.112 0.102 0.125 0.130 0.132 0.133 0.125 0.138 0.152 ∗E7= 1x10 Z, of 1000. Optimization results for two variables 𝑟 and 𝑟 Cost Function Values (1x10 ) 𝑏 𝑝 0.1 9.5 and their cost function values with different initial values of the optimization procedure are as shown in Table 2. For 0.095 example, while the initial value of 𝑟 is equal to 0.125 [m] and 8.5 an initial value of 𝑟 is equal to 0.070 [m], cost function value 0.09 8 is equal to 4.8xE7, optimum value of 𝑟 is equal to 0.074 [m], 𝑝 7.5 0.085 and optimum value of𝑟 is equal to 0.12 [m]. Optimum values of diameters 𝑟 and 𝑟 depend on constraints. 𝑝 𝑏 0.08 6.5 While generating Table 2, constraints are r : 0.125 [m] < r < 0.175 [m] and constraints for 𝑟 ; 0.070 [m] <𝑟 < b 𝑝 𝑝 0.075 5.5 0.125 [m]. Low mobile platform diameters with high base diameters seemed to result comparably in small cost function 0.07 5 0.125 0.13 0.135 0.14 0.145 0.15 0.155 values, which means better dexterity in motion. In Figure 7, r [m] minimum cost function value is 4.8 x 10 as indicated in bold areas. In those areas, cost function values are minimum Figure 7: Two-variable ( 𝑟 , 𝑟 ) optimization results. 𝑝 𝑏 with optimized 𝑟 and 𝑟 values as shown and are compatible 𝑏 𝑝 with Table 2. While optimized 𝑟 has minimum values, cost function values are minimum. As mentioned previously, we has large spectrum throughout the workspace. Therefore, are searching for minimum cost function value for better comparing the average condition numbers for dieff rent dexterity. In Table 2, there is not one optimum solution for design parameters with different constraints is meaningless. the minimum cost function value. For example, 𝑟 , 𝑟 𝑝,𝑜𝑝𝑡 𝑏,𝑜𝑝𝑡 As a result, calculating the cost function (sum of condition = 0.074 [m], 0.129 [m] has minimum cost function value numbers multiplied by r and r ) and considering the of (4.8xE7) and also 𝑟 , 𝑟 = 0.077 [m], 0.125 [m] has b p 𝑝,𝑜𝑝𝑡 𝑏,𝑜𝑝𝑡 resulting value is more efficient while searching for the best minimum cost function value of (4.8xE7). optimum design parameters. For better accuracy, we increase the resolution in solution area in Table 3. For the purpose of interpreting the results, we Optimum values are in Table 3. Minimum cost function used cost function values. However, the condition number value is 3.2xE7. Solution set with minimum cost function r [m] p Journal of Robotics 9 Table 3: Optimization of mobile platform diameter, 𝑟 , by narrowing constraint target area. Cost.F(x) Base Platform Radius, 𝑟 𝑝,𝑜𝑝𝑡[]𝑚 𝑟 0.125 0.126 0.127 0.128 0,129 𝑏,𝑜𝑝𝑡[]𝑚 0.060 3.2xE7 3.2xE7 3.2xE7 3.2xE7 3.2xE7 0.068 0.072 0.071 0.071 0.068 0.125 0.127 0.127 0.128 0.129 0.065 3.2xE7 3.2xE7 3.2xE7 3.2xE7 3.2xE7 0.069 0.070 0.070 0.071 0.067 0.129 0.128 0.128 0.128 0.129 0.066 3.2xE7 3.2xE7 3.2xE7 3.2xE7 3.2xE7 0.068 0.070 0.071 0.071 0.068 0.125 0.128 0.127 0.128 0.129 0.067 3.2xE7 3.2xE7 3.2xE7 3.2xE7 3.2xE7 0.069 0.069 0.070 0.070 0.068 0.129 0.128 0.127 0.128 0.129 0.068 3.2xE7 3.2xE7 3.2xE7 3.2xE7 4.8xE7 0.068 0.070 0.070 0.070 0.075 0.125 0.126 0.127 0.128 0.130 0.069 4.8xE7 3.2xE7 3.2xE7 3.2xE7 4.8xE7 0.076 0.070 0.071 0.070 0.074 0.125 0.126 0.127 0.128 0.129 0.070 4.8xE7 4.8xE7 3.2xE7 3.2xE7 4.8xE7 0.074 0.074 0.071 0.070 0.075 0.129 0.129 0.127 0.128 0.129 ∗E7= 1x10 1. 2. r ,r (Radius of Sub cost function is Workspace is created b p selected as f3(x) as a rectangular platforms) and l workspace. We used For quick calculation (lengths of the legs ) its corner coordinates max value of Sub are defined for min cost function is defined. layer by layer and max values Reachable All condition For minimum values points checked numbers are of three variables Cost function is calculated according Unreachable to defined values defined as g(x) points aren't taken except unreachable into account for points optimization 7. 8. 9. Cost function is Results are analyzed By selecting the minimized according to select the optimum values of variables to min value of three values for three optimization variables variables procedure is finished Figure 8: Algorithm sequence for three-variable optimization. value occurs for𝑟 ,𝑟 = 0.072 [m], 0.127 [m];𝑟 ,𝑟 of 𝑟 is equal to 0.070 [m], and initial value of 𝑙 is equal to 𝑝,𝑜𝑝𝑡 𝑏,𝑜𝑝𝑡 𝑝,𝑜𝑝𝑡 𝑏,𝑜𝑝𝑡 𝑝 𝑖 = 0.071 [m], 0.127 [m]; 𝑟 , 𝑟 = 0.071 [m], 0.128 [m]. 0.300 [m], cost function value is equal to 4.8xE7, optimum 𝑝,𝑜𝑝𝑡 𝑏,𝑜𝑝𝑡 Optimization results for three variables 𝑟 , 𝑟 , 𝑙 ,and value of 𝑟 is equal to 0.076 [m], 𝑟 is equal to 0.125 [m], 𝑏 𝑝 𝑖 𝑝 𝑏𝑜𝑝𝑡 their cost function values with different initial values of the and𝑙 is equal to 0.300 [m]. Optimum values of parameters 𝑖𝑜𝑝𝑡 optimization are as shown in Tables 4 and 6. For example, also depend on constraints. While generating Table 4, our while the initial value of 𝑟 is equal to 0.125 [m], initial value constraints are as in Table 5. 𝑏 10 Journal of Robotics Table 4: Optimizing mobile platform diameters, 𝑟 , and optimum leg length, 𝑙 . 𝑝 𝑖 Cost.F(x) Base Platform Radius, 𝑟 𝑝,𝑜𝑝𝑡[]𝑚 𝑟 0.125 0.130 0.135 0.140 0.145 0.150 0.155 𝑏,𝑜𝑝𝑡[]𝑚 Leg Length 𝑙 𝑖,[𝑚] 𝑖,𝑜𝑝𝑡[𝑚] 0.300 0.325 0.350 0.375 0.400 0.425 0.450 0.070 4.8xE7 4.8xE7 8xE7 8xE7 6.4xE7 - 4.8xE7 0.076 0.075 0.070 0.091 0.094 - 0.077 0.125 0.130 0.135 0.143 0.125 - 0.125 0.300 0.325 0.350 0.375 0.375 - 0.388 0.075 4.8xE7 4.8xE7 8xE7 4.8xE7 8xE7 6.4xE7 8xE7 0.076 0.075 0.081 0.077 0.097 0.072 0.076 0.125 0.125 0.134 0.125 0.150 0.172 0.133 0.300 0.358 0.352 0.373 0.375 0.375 0.375 0.080 4.8xE7 8xE7 8xE7 6.4xE7 8xE7 4.8xE7 - 0.070 0.087 0.080 0.080 0.080 0.070 - 0.125 0.142 0.135 0.125 0.133 0.125 - 0.322 0.359 0.350 0.375 0.375 0.322 - 0.085 6.4xE7 6.4xE7 6.4xE7 8xE7 6.4xE7 8xE7 6.4xE7 0.085 0.085 0.085 0.085 0.085 0.096 0.085 Mobile Platform Radius, 𝑟 0.125 0.125 0.125 0.140 0.125 0.147 0.125 0.300 0.300 0.367 0.375 0.300 0.375 0.300 0.090 6.4xE7 6.4xE7 6.4xE7 6.4xE7 6.4xE7 8xE7 6.4xE7 0.090 0.090 0.090 0.090 0.098 0.091 0.090 0.125 0.125 0.127 0.125 0.125 0.150 0.125 0.300 0.358 0.353 0.375 0.375 0.375 0.300 0.095 6.4xE7 9.6xE7 6.4xE7 9.6xE7 8xE7 6.4xE7 8xE7 0.095 0.095 0.095 0.110 0.095 0.095 0.095 0.125 0.130 0.125 0.143 0.150 0.126 0.149 0.300 0.325 0.357 0.375 0.375 0.388 0.375 0.100 9.6xE7 9.6xE7 9.6xE7 9.6xE7 9.6xE7 9.6xE7 8xE7 0.100 0.100 0.100 0.100 0.100 0.118 0.094 0.125 0.130 0.135 0.140 0.145 0.150 0.145 0.300 0.325 0.350 0.375 0.400 0.375 0.386 Table 5: Constraints for three-variable optimization of the PRS. 𝑟 𝑟 𝑙 𝑏[]𝑚 𝑝[]𝑚 𝑖 0.125 [m]<𝑟 < 0.175 0.070 [m]<𝑟 < 0.125 [m] 0.300 [m]<𝑙 < 0.450 [m] 𝑏 𝑝 𝑖 In Table 4, there is not only one optimum solution for optimization results with mathematical calculations and the minimum cost function value. For example, 𝑟 , 𝑟 , compared these values with calculated values. 𝑝,𝑜𝑝𝑡 𝑏,𝑜𝑝𝑡 𝑙 = 0.076 [m], 0.125 [m], 0.300 [m] has minimum cost For three-parameter optimization, we used 𝑓 (𝑥) as a sub 𝑖,𝑝𝑡𝑜 3 function value of (4.8xE7) and also 𝑟 , 𝑟 , 𝑙 = cost function and obtained results in Tables 4 and 6 using the 𝑝,𝑜𝑝𝑡 𝑏,𝑜𝑝𝑡 𝑖,𝑝𝑡𝑜 0.077 [m], 0.125 [m], 0.388 [m] has minimum cost function optimization algorithm as shown in Figure 8. value of (4.8xE7). For better accuracy, we zoom into the For Table 7, we used optimized values from Table 3, and solution area in Table 5. So, we generated Table 6. As shown we generated Table 7 using inverse kinematic equations. in Table 6, minimum cost function value is found as 3.2xE7. Table 7 presents L , L , 𝑢 (uniformity index), and max min According to Table 6, optimum parameters are 𝑟 , 𝑟 , reachable points for optimum radius values of mobile and 𝑝,𝑜𝑝𝑡 𝑏,𝑜𝑝𝑡 𝑙 = 0.071 [m], 0.127 [m], 0.350 [m] and 𝑟 , 𝑟 , 𝑙 = base platforms. According to these calculated values using 𝑖,𝑝𝑡𝑜 𝑝,𝑜𝑝𝑡 𝑏,𝑜𝑝𝑡 𝑖,𝑝𝑡𝑜 0.072 [m], 0.127 [m], 0.374 [m]. Optimization results change optimized parameters from Table 7, GCI (Global Condition according to optimization parameters which are initial val- Index) and LCI (Local Condition Index) values are computed ues and convergence tolerances. Hence, we controlled our for each solution. Figure 9 represents LCI values for the first Journal of Robotics 11 Table 6: rTh ee-variable optimization by narrowing target constraint regions. Cost.F(x) Base Platform Radius 𝑝,𝑜𝑝𝑡[]𝑚 𝑟 0.125 0.126 0.127 0.128 0.129 𝑏,𝑜𝑝𝑡[]𝑚 Leg Length 𝑙 𝑖,[𝑚] 𝑖,𝑜𝑝𝑡[𝑚] 0.300 0.325 0.350 0.375 0.400 0.070 4.8xE7 4.8xE7 3.2xE7 3.2xE7 4.8xE7 0.076 0.075 0.071 0.072 0.076 0.125 0.126 0.129 0.300 0.323 0.349 0.374 0.391 0.071 4.8xE7 4.8xE7 3.2xE7 4.8xE7 4.8xE7 0.075 0.074 0.071 0.074 0.076 0.125 0.126 0.127 0.128 0.129 0.300 0.324 0.350 0.373 0.398 0.072 4.8xE7 4.8xE7 - 4.8xE7 4.8xE7 0.076 0.087 - 0.073 0.077 0.125 0.120 - 0.128 0.125 0.300 0.330 - 0.375 0.385 0.074 4.8xE7 - 6.4xE7 6.4 xE7 4.8xE7 0.076 - 0.091 0.087 0.075 Mobile Platform Radius 0.125 - 0.126 0.128 0.130 0.300 - 0.333 0.375 0.399 0.076 4.8xE7 4.8xE7 6.4xE7 4.8xE7 6.4xE7 0.077 0.076 0.076 0.077 0.084 0.125 0.125 0.127 0.125 0.128 0.300 0.300 0.350 0.374 0.385 0.078 6.4xE7 6.4xE7 6.4xE7 6.4xE7 6.4xE7 0.078 0.078 0.078 0.081 0.094 0.125 0.126 0.125 0.129 0.125 0.300 0.325 0.300 0.375 0.355 0.080 4.8xE7 6.4xE7 6.4xE7 6.4xE7 4.8xE7 0.070 0.080 0.094 0.090 0.076 0.125 0.126 0.126 0.126 0.125 0.322 0.325 0.335 0.375 0.335 Table 7: Leg lengths for optimum parameters of the PRS. Solutions 𝑟 /𝑟 𝐿 𝐿 𝐿 𝑈 Reachable Points 𝑏,𝑜𝑝𝑡 𝑝,𝑜𝑝𝑡 𝑥[𝑚] 𝑚𝑖𝑛[]𝑚 𝑔𝑡ℎ𝑛 []𝑚 (1) 0.127/0.071 0.4550 0.2928 0.3739 1.2505 11592 (2) 0.127/0.072 0.4552 0.2926 0.3739 1.2503 11588 (3) 0.128/0.071 0.4554 0.2929 0.3742 1.2486 11588 optimized variables. As shown in Figure 9, there are nearly When 𝑟 is equal to 0.128 [m], 𝑟 is equal to 𝑏,𝑜𝑝𝑡 𝑝,𝑜𝑝𝑡 400 unreachable points in the workspace. LCI values of 0.071 [m], and 𝑙 is between 0.300 and 0.450 [m], LCI values reachable points vary between nearly 8 and 11. The average are as shown in Figure 11 and GCI is equal to 9.57. of LCI values isdenfi ed asGCI. As shown in Figures 9, 10, and 11, LCI and GCI values When 𝑟 is equal to 0.127 [m], 𝑟 is equal to are very similar, because two parameters 𝑟 and 𝑟 𝑏,𝑜𝑝𝑡 𝑝,𝑜𝑝𝑡 𝑏,𝑜𝑝𝑡 𝑝,𝑜𝑝𝑡 0.071 [m], and 𝑙 is between 0.300 and 0.450 [m], LCI values differ only 0.001 [m]. Distribution of LCI values in graphics are as shown in Figure 9 and GCI is equal to 9.62. is different, but it does not aeff ct GCI noticeably. As shown When 𝑟 is equal to 0.127 [m], 𝑟 is equal to in Table 7, we can change constraints of 𝑙 and reach all 𝑏,𝑜𝑝𝑡 𝑝,𝑜𝑝𝑡 𝑖 0.072 [m], and 𝑙 is between 0.300 and 0.450 [m], LCI values workspace. One can redefine leg lengths of the PRS. If we are as shown in Figure 10 and GCI is equal to 9.49. compare Tables 6 and 7, calculated solutions are dieff rent 𝑙𝑒 𝑚𝑎 12 Journal of Robotics LCI Distiribution-3 GCI :9.57 LCI Distribution -1/ GCI = 9.627 1800 1800 1600 1600 1400 1400 1200 1200 0 2 4 6 8 10 12 0 2468 10 12 LCI Values LCI Values Figure 11: Distribution of LCI values based on un/reachable Figure 9: Distribution of LCI values based on un/reachable workspace points for three-parameter optimization, 𝑟 = workspace points for three-parameter optimization, 𝑟 = 𝑏,𝑜𝑝𝑡 𝑏,𝑜𝑝𝑡 0.128 [m], 𝑟 = 0.071 [m], and 𝑙 = 0.300-0.450 [m]. 0.127 [m], 𝑟 = 0.071 [m], and 𝑙 = 0.300-0.450 [m]. 𝑝,𝑜𝑝𝑡 𝑖 𝑝,𝑜𝑝𝑡 𝑖 LCI Distribution-2 GCI : 9.49 are many researches for the optimization of kinematics of 1800 a PRS. In these works, various optimization methods have been used. In this study, we optimized 3x3 SP mechanism using SQP optimization method for predefined workspace and checked the efficiency of the optimization for three parameters which are diameter of mobile and base platform together with lengths of the legs. It is shown that the optimization of multi-parameters is not easy to formulate 800 concerning cost functions and results must be evaluated by calculating the dexterity indexes and the amount of reachable points. Firstly, we optimized𝑟 and then, we optimized𝑟 and 𝑝 𝑝 𝑟 simultaneously. Lastly, 𝑟 , 𝑟 ,and 𝑙 are optimized at the 𝑏 𝑝 𝑏 𝑖 same time. During optimization, tolerances, procedure steps, and applied/defined cost functions aec ff t the convergence efficiency. Results show that initial values for each parameter 0 2468 10 12 at the beginning of the optimization determine the toler- LCI Values ance of the convergence for the optimization. As a further Figure 10: Distribution of LCI values based on un/reachable research, various optimization methods can be performed workspace points for three-parameter optimization, 𝑟 = 𝑏,𝑜𝑝𝑡 while increasing number of kinematic parameters optimized 0.127 [m], 𝑟 = 0.072 [m], and 𝑙 = 0.300-0.450 [m]. 𝑝,𝑜𝑝𝑡 𝑖 searching different cost functions. Notations but very similar to each other because initial values affect optimization procedure and convergence tolerances of opti- 𝑙 : Leg lengths mization. All LCI in Figures 9, 10, and 11 are calculated x: Dir. along the x-axis of mobile platform from the homogenous Jacobian matrix and these calculated y: Dir. along the y-axis of mobile platform values are divided by the dimension of Jacobian matrix for z: Dir. along the z-axis of mobile platform homogenization. 𝛼 : Pitch angleof mobileplatform In detail, rfi st solution is the best for the amount of the 𝛽 : Roll angle of mobile platform reachable points. Second one is the best solution according 𝛾 : Yaw angle of mobile platform to GCI. And the third one is the best for the uniformity. Z: Condition number of Jacobian matrix J: Jacobian matrix O : Mobile platform center of gravity 6. Conclusion O : Base platform center of gravity Determining the values of geometrical parameters is very 𝜆 : Angle between (X , O P ) 𝑖 p p i important for efficiency of a PRS design. In literature, there Λ : Angle between (X , O B ) 𝑖 b b i Number of Workspace Points Number of Workspace Points Number of Workspace Points Journal of Robotics 13 [11] M. A. Hosseini, H.-R. M. Daniali, and H. D. Taghirad, “Dex- 𝑟 : Radius of a circle passing through center terous workspace shape and size optimization of tricept parallel of joints between base platform and legs manipulator,” International Journal of Robotics, vol. 2, no. 1, 2011. 𝑟 : Radius of a circle passing through center [12] M. A. Hosseini, H.-R. M. Daniali, and H. D. Taghirad, “Dexter- of joints between mobile platform and legs ous workspace optimization of a Tricept parallel manipulator,” 𝜃 : Angle between connection points of legs Advanced Robotics,vol.25,no. 13-14,pp.1697–1712,2011. on base platform [13] Y. Lou, G. Liu, N. Chen, and Z. Li, Optimal Design of Par- 𝜃 : Angle between connection points of legs allel Manipulators for Maximum Effective Regular Workspace , on mobile platform University of Science and Technology, Intelligent Robots and R: Rotational transformation matrix. Systems, Hong Kong, 2005. [14] Y. Lou, G. Liu, and Z. Li, “Randomized optimal design of paral- Data Availability lel manipulators,” IEEE Transactions on Automation Science and Engineering,vol.5,no.2,pp.223–233, 2008. The Matlab Codes data used to support the findings of this [15] Q. Jiang and C. M. Gosselin, “Geometric optimization of the study are available from the corresponding author upon MSSM Gough-Stewart platform,” Journal of Mechanisms and request. Robotics,vol.1,no. 3,pp.1–8, 2009. [16] A. M. Lopes, E. J. S. Pires, and M. R. Barbosa, “Design of a Conflicts of Interest parallel robotic manipulator using evolutionary computing,” International Journal of Advanced Robotic Systems,vol.9,article The authors declare that they have no conflicts of interest. 26, 2012. [17] G.Abbasnejad, H.M.Daniali, and S.M.Kazemi,“A new approach to determine the maximal singularity-free zone of 3- References RPR planar parallel manipulator,” Robotica, vol.30, no.6,pp. 1005–1012, 2012. [1] J.-P. Merlet, Parallel Robots,INRIA, Springer, Amsterdam, Netherlands, 2nd edition, 2006. [18] A. R. Shirazi, M. M. S. Fakhrabadi, and A. Ghanbari, “Optimal design of a 6-DOF parallel manipulator using particle swarm [2] P. C. Srinivas, Y. D. Kumar, P. D. Rao, V. Sreenivasulu, and optimization,” Advanced Robotics, vol.26, no. 13, pp.1419–1441, C. U. 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Publisher: Hindawi Publishing Corporation
Copyright: Copyright © 2019 Cenk Eryılmaz and V. E. Omurlu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ISSN: 1687-9600
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DOI: 10.1155/2019/3928705
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Abstract

Hindawi Journal of Robotics Volume 2019, Article ID 3928705, 14 pages https://doi.org/10.1155/2019/3928705 Research Article SQP Optimization of 6dof 3x3 UPU Parallel Robotic System for Singularity Free and Maximized Reachable Workspace 1 2 Cenk EryJlmaz andV.E.Omurlu Technology Transfer and Commercialization, TUBITAK BILGEM P.K., 74, Gebze, 41470 Kocaeli, Turkey Department of Mechatronic Engineering, Yildiz Technical University, 34349 Istanbul, Turkey Correspondence should be addressed to Cenk Eryılmaz; cenk.eryilmaz@tubitak.gov.tr Received 3 September 2018; Revised 13 October 2018; Accepted 24 December 2018; Published 3 February 2019 Academic Editor: Yangmin Li Copyright © 2019 Cenk Eryılmaz and V. E. Omurlu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The determination of kinematic parameters for a parallel robotic system (PRS) is an important and a critical phase to maximize reachable workspace while avoiding singular configurations. Stewart Platform (SP) mechanism is one of the widely known PRS and it is used to demonstrate the proposed technique. In the related literature, GCI (Global Condition Index) and LCI (Local Condition Index) are the commonly used performance indexes which give a measure about the dexterity of a mechanism. In this work, Sequential Quadratic Programming (SQP) method is used to optimize kinematic parameters of a 6dof 3x3 UPU SP in order to reach maximum workspace satisfying small condition numbers. eTh radius of mobile and base platforms and the lengths of the legs used in the platform are chosen as kinematic parameters to be optimized in a multiobjective optimization problem. Optimization is performed at different stages and the number of optimized kinematic parameters is increased at each level. In conclusion, optimizing selected kinematic parameters at once by using SQP technique presents the best results for the PRS. 1. Introduction to find alternative parallel structures like redundant PRS [2– 4]. In recent years, many different parallel robotic structures In 1947, Gough [1] established the basic principles of a mech- have been investigated for various applications. anism with a closed-loop kinematic structure that allows The performance of a PRS is mainly affected by its geo- positioning and orientation of a mobile platform to test tire metrical parameters. Determining geometrical parameters wear and tear. In 1965, Stewart oer ff ed a very similar structure is called geometric synthesis. One of the most important to Gough’s mechanism to be used as an aircraft simulator. measures for kinematic performance is called dexterity which Ironically, the Gough platform is most oeft n known as Stewart describes the ability of a mechanism to move and apply platform [1]. In recent years, there have been many researches forces in arbitrary directions as easily as possible. However, about PRS whose principles are originated from Gough. They the most suitable index to define the accuracy and rigidity have numerous advantages compared to the serial robotic performance is the condition number, which LCI (Local systems such as high load carrying capacity and precise Condition Index) and GCI (Global Condition Index) are positioning ability. A PRS has a mobile platform connected related to. LCI is defined as the ratio of the maximum and to a fixed base by parallel legs instead of serial links. minimum singular values of the Jacobian matrix of a PRS Besides their advantages compared to serial manipu- [5]. It varies between 1 and infinity [6, 7]. GCI index is a measure of dexterity and the average of LCI calculated for lators, PRS have a small workspace and require a more complex mechanical design. Controlling the motion of a entire workspace. parallel mechanism is a challenging task because of its Dexterity and workspaceare themost important cri- complex kinematic structure and variable dynamics through teria for optimal designs of parallel manipulators. There its workspace. To overcome this issue, many researchers tried are many researches investigating the effects of geometric 2 Journal of Robotics 6 ０Ｊ： →  →  →  r Ｃ 1 b 2 →  Figure 1: Coordinate systems, {P} and {B},ofSP. parameters on dexterity, workspace, and singularity. Some of described. Section 4 illustrates the SQP and optimization them are about maximizing well-conditioned workspace [8] algorithm. In the last section, we discussed the results of the using different optimization methods. Monte Carlo method SQP optimization procedure. [8, 9], multiobjective/genetic algorithm [10–12], controlled random search (CRS) [13, 14], analytic algorithm [15], and neurogenetic algorithm are mostly used ones [16]. G. Abbas- 2. Mechanism of the PRS nejad offered a practical algorithm to find the singularity A PRS is constructed from parallel connected legs between free cylindrical workspace of parallel manipulators using mobile and base platforms. There are two major classification particle swarm optimization method (PSO) [17, 18]. In the criteria of a PRS which are based on (a) joint types of literature, there are other performance indexes such as GSI connections of legs to the mobile and base platform and (Global Stiffness Index) and VTF (Velocity Transformation (b) the number of legs. The array of the types of joints are Factor) [19]. Therefore, some researchers optimized PRS for universal, prismatic, and universal, respectively. So it is called multiperformance indexes which are called MOO (Multiob- UPU. Specifically, linearly driven six legs which are connected jective Optimization) [20–22]. These optimization methods to a mobile and a base platform cause a full spatial motion are used for investigating the best geometric parameters of 3-axis translational and 3-axis rotational motion. This PRS for structures of parallel manipulators. On the other hand is called Stewart Platform [20, 25, 26] and classiefi d by the some researchers compared convergence performances of number of leg connections on upper and lower platforms. A optimization methods [23]. 3x3 SP includes 3 connections on lower and upper side of the In this research, we used SQP method which is Lagran- legs. Each two of six legs share cocentered joints on each side gian based [24] for the optimization of kinematic parameters of the legs. “Figure 1” illustrates the general structure of SP to reach the maximum workspace for a PRS. This paper is that 3x3 special configuration is used for the current research. organized as follows. In the following section, the mechanism of a PRS and its geometric parameters and Jacobian matrix An SP has 48 design parameters affecting its workspace formulation for 3x3 SP are briefly described. In Section 3, and kinematic behavior [27]. These parameters are spatial our performance indexes and workspace assumption are positions of upper and lower joints (2x18 parameters) and Journal of Robotics 3 upper/lower leg lengths (2x6 parameters). By placing connec- “Eq.” (6) is representing the rotation of mobile coordinate tion points on a circle, these would reduce to upper/lower frame {P} with respect to base frame {B}. circle diameters/radiuses passing through upper/lower joint 𝑇 𝑇 centers, r and r . In this research, optimization parameters (7) 𝑝 ={𝑝 ,𝑝 ,𝑝 } ={𝑟 . cos(𝜆 ),𝑟 . sin(𝜆 ),0} p b 𝑝 𝑖 𝑝 𝑖 are these two radiuses and length of the legs, 𝑙 . “Eq.” (7) describes joint positions P of themobileplatform with respect to {P}. 2.1. Geometric Parameters of PRS. The lengths of the legs (𝑙 ), which change according to the desired position and 𝑖 𝑇 (8) 𝑏 ={𝑏 ,𝑏 ,𝑏 } ={𝑟 . cos(Λ ),𝑟 . sin(Λ ),0} orientation of the mobile platform, can be calculated using 𝑖 𝑏 𝑖 𝑏 𝑖 the inverse kinematics equations, x (x, y, z, 𝛼 , 𝛽 , 𝛾 ). In this “Eq.” (8) describes joint positions B of the base platform with case, there are six unknowns, 𝑙 (i = 1....6).“Figure1” respect to {B}. helps us understand the geometry of SP. Mobile and base C = cos(𝛼),S = sin(𝛼),C = cos(𝛽),S = sin(𝛽), coordinate frames, {P} and {B}, are den fi ed. They each are 𝛼 𝛼 𝛽 𝛽 C = cos(𝛾) ,S = sin(𝛾) linked to the mobile platform and the base. The origin of 𝛾 𝛾 frame {P} is located at the center of a circle passing through C C C S S − S C S S + C S C the center of joints between the mobile platform and the legs. 𝛼 𝛽 𝛼 𝛽 𝛾 𝛼 𝛾 𝛾 𝛼 𝛼 𝛽 𝛾 [ ] The Z axis is pointing upward, and the X axis is the bisecting [ ] p p S C C C + S S S C S S − S C 𝑅 = 𝛼 𝛽 𝛼 𝛾 𝛼 𝛽 𝛾 𝛾 𝛽 𝛼 𝛾 𝛼 𝑃 [ ] line of the angle (O P -O P ). The origin of frame {B} is P 6 P 1 −S C S C C 𝛽 𝛽 𝛾 𝛽 𝛾 located at the center of a circle passing through the center of [ ] (9) joints between the base and the legs. The Z axis is pointing 𝑟 𝑟 𝑟 11 12 13 upward, and the X axis is the bisecting line of the angle [ ] (O B -O B ). 𝑟 𝑟 𝑟 = [ ] b 6 b 1 21 22 23 These two coordinate frames lead us to determine the 𝑟 𝑟 𝑟 [ ] 31 32 33 position of the links attached to the base at (B ...,B )and 1 6 the mobile platform at (P .., P ). where 𝛼 is the rotation of {P} with respect to {B} about Z . 1 6 𝛽 is the rotation of {P} with respect to {B} about Y . 𝛾 is the The angle (O P ,O P ) is denoted as 𝜃 .Angles (O P , p 2 p 3 𝑝 p 1 rotation of {P} with respect to {B} about X . O P ), (O P ,O P ), and (O P ,O P ) are each equal to p 3 p 3 p 5 p 5 p 1 Consequently, 120 . Similarly, angle (O B ,O B ) is denoted by 𝜃 .Angles b 1 b 2 𝑏 (O B ,O B ), (O B ,O B ), and (O B ,O B )are,also, b 1 b 3 b 3 b 5 b 5 b 1 equal to 120 . Now, the length of the legs (𝑙 )can be calculated for 𝐶 𝐶 .𝑟 𝐶(𝜆 )+(𝐶 𝑆 𝑆 −𝑆 𝐶 ).𝑟 .𝑆(𝜆 ) 𝛼 𝛽 𝑝 𝑖 𝛼 𝛽 𝛾 𝛼 𝛾 𝑝 𝑖 (10) [ ] [ ] 𝑖 = 1,2,...,6 (1) = 𝑆 𝐶 .𝑟 𝐶(𝜆 )+(𝑆 𝑆 𝑆 +𝐶 𝐶 ).𝑟 .𝑆(𝜆 ) [ 𝛼 𝛽 𝑝 𝑖 𝛼 𝛽 𝛾 𝛼 𝛾 𝑝 𝑖 ] −𝑆 .𝑟 𝐶(𝜆 )+𝐶 𝑆 .𝑟 .𝑆(𝜆 ) 𝛽 𝑝 𝑖 𝛽 𝛾 𝑝 𝑖 [ ] The angles (X , O P ) and (X , O B ) are denoted as 𝜆 and p p i b b i i Λ , respectively. Therefore, we have The length of the legs can be calculated as follows: 2 2 Λ =60𝑖− , (11) 𝑙 = (𝑆 ) +(𝑆 ) +(𝑆 ) 𝑝 (2) Using (4), we get length of the legs as 𝜆 =60𝑖− , 2 2 2 2 2 2 𝑙 =𝑥 +𝑦 +𝑧 +𝑟 +𝑟 𝑖 𝑝 𝑏 𝑖 = 1,3,5 +2(𝑟 𝑝 +𝑟 𝑝 )(𝑥−𝑏 )+2(𝑟 𝑝 +𝑟 𝑝 ) (12) 11 12 21 22 Λ =Λ +𝜃 , 𝑖 𝑖−1 𝑏 (𝑦 − 𝑏 )+2(𝑟 𝑝 +𝑟 𝑝 ) (𝑧 )−2(𝑏𝑥 +𝑦𝑏 ) 𝜆 =𝜆 +𝜃 (3) 31 32 𝑖 𝑖−1 𝑝 𝑖 = 2,4,6 2.2. Jacobian Matrix. Jacobian matrix of a robotic system represents the linear transformation between the speeds in 𝐵 𝐵 𝐵 𝐵 (4) 𝑆 =− 𝑏 + 𝑡 + 𝑝 𝑖 𝑖 𝑖 ∙ joint space, ,and world space, 𝑥 ̇,of a robotic manipulator. 𝐵 𝑇 (5) 𝑡 ={,𝑥 ,𝑦 }𝑧 Jacobian matrix provides the velocity ratio from the joint space to the space of the end-effector [26]. “Eq.” (5) represents translational position of {P} with respect For a parallel manipulator, the length of the legs (𝑙 )as a to {B}, vector is given as 𝐵 𝐵 𝑃 𝑝 = 𝑅 𝑝 (6) l = l ,..., l (13) ( ) 𝑖 𝑃 1 6 𝑖𝑦 𝑖𝑥 𝑖𝑦 𝑖𝑥 𝑖𝑦 𝑖𝑦 𝑖𝑥 𝑖𝑥 𝑖𝑦 𝑖𝑥 𝑖𝑧 𝑖𝑦 𝑖𝑥 𝑖𝑧 𝑖𝑦 𝑖𝑥 𝑖𝑧 𝑖𝑦 𝑖𝑥 4 Journal of Robotics The position of the end-eeff ctor is If the spectral norm is introduced, the dexterity indexes can be described as (14) x =(,𝑥 ,𝑦 ,𝑧 ,𝛼 ,𝛽 𝛾) 𝑆𝑉 max(𝐽) Cond(J) = (24) 𝑆𝑉 min(𝐽) So, the relation equation between the position of the end- effector ( x) and the length of the legs (𝑙 ) can be represented where SV and SV represent the maximum and min- max(J) min(J) as imum singular values of the Jacobian matrix (J), respectively. The values of Cond (J), which are directly related to the l = g(x) (15) singular values of the Jacobian matrix, are between one and positive infinity. All singular values will be the same in the By differentiating (15), we obtain Jacobian matrix, and the manipulator is isotropic when Cond (J) is equal to 1. l = Jẋ (16) Otherwise, when Cond (J) is potentially equal to positive infinity, then the Jacobian matrix is singular. So, when the condition number is considered, the value of the condition J ≡ (17) 𝑖 number should be minimum for a better dexterity. The condition number isusedasanindex formea- 𝑇 𝑇 ̇ ̇ ̇ (18) [𝑙 ,...,𝑙 ] = J[𝑥, ̇ 𝑦, ̇ 𝑧, ̇ 𝛼, ̇ 𝛽, 𝛾]̇ 1 6 suring the accuracy/dexterity of a robotic system, and also it possesses information of whether a position is close to 𝜕l singularity or not. However, it is not entirely possible to den fi e J = (19) 𝜕x 𝑗 a mathematical proximity to singularity for a PRS whose degree of freedom is a mix of translational and rotational Consequently J is the Jacobian matrix of the mechanism, motion. This is because the use of condition number is just an which is used in calculating the performance indexes of the index and yet it has an advantage as a single number which is PRS, and is utilized in this research as used to describe the overall kinematic behavior of a PRS. Because of the complex definition of the condition num- J = [J , J , J , J , J , J ], 𝑖 = 1,...,6 (20) 𝑖1 𝑖2 𝑖3 𝑖4 𝑖5 𝑖6 ber, we cannot calculate its analytical form as a function of the position parameters except for very simple robots. However, robust linear algebra tools can calculate it numerically for a 3. Performance Indexes and Workspace given spatial position. Parallel manipulators that have both 3.1. Dexterity, GCI, and LCI. Dexterity is an indicator of the translational and rotational dof have a major drawback about ability of a parallel manipulator about how rates of joints condition number. Because the unit of translational motion transform to the rates of the mobile platform and vice versa. is different from the rotational motion, the Jacobian matrix It depends on the position and the structure of the parallel is not homogenous [28]. u Th s, there exist some researches manipulator and can be determined by using the condition about the normalization of Jacobian matrices which are partly number of its Jacobian matrix as rotational and translational. Some of them defined their own methods for normalization and some of them divided Dexterity = Cond(J) (21) Jacobian matrix by the degree of the matrix [29–32]. where Cond (J) is the condition number of Jacobian matrix 3.2. Uniformity. Uniformity is another global performance and is defined as index for PRS. It is an indicator of the uniform dexterous 󵄩 󵄩 workspace. Calculation of uniformity index is shown in −1 󵄩 󵄩 󵄩 󵄩 Cond J = 𝐽 𝐽 (22) ( ) ‖ ‖ 󵄩 󵄩 󵄩 󵄩 equation (25). Uniformity index value changes between 1 and infinity. The index value of the best uniform dexterous ‖.‖ represents the norm of the related vector or matrix. workspace is equal to 1. While calculating the condition number of a Jacobian 𝐼𝐿𝐶 matrix, different types of norms are used in the literature, max 𝑢= (25) and the most used ones are Frobenius and Spectral norms [1]. 𝐼𝐿𝐶 min The difference between these norms is about their calculation methods. If the Frobenius norm is considered, then we can 3.3. Workspace. The number of possible mechanical con- write figurations of serial robotic systems is less than that of parallel robotic system configurations. us, Th comparison √ (23) ‖J‖ = tr(J J) of workspace of serial robotic systems is applicable. For example, one can easily assess that RRR robotic system has In this case, the Frobenius norm is defined as the extracting greater workspace capacity than a cartesian configuration roots of the quadratic sum for each element of the Jacobian because of its angular workspace capacity. However, because matrix. GCI (Global Condition Index) is den fi ed all over the of the vast number of mechanical configuration possibilities workspace as the average of Cond(J) = LCI (Local Condition of parallel robotic systems, no such comparison could be Index). performed easily [24]. 𝑖𝑗 𝜕𝑥 𝜕𝑔 Y Journal of Robotics 5 0.4 0.4 △； =0.04 0.35 0.35 0.3 0.3 0.2 0.2 0.25 −0.2 −0.2 −0.2 −0.1 0 0.1 0.2 y-axis (m) x-axis (m) x-axis (m) 0.2 0.4 0.1 0.35 0.3 −0.1 0.25 −0.2 −0.2 −0.1 0 0.1 0.2 −0.2 −0.1 0 0.1 0.2 y-axis (m) x-axis (m) Figure 2: Volume representation of the workspace to be optimized for the SP and the overall workspace. One can manage to compare 6dof parallel robotic systems for 12000 points in the workspace and then calculated GCI. that are abbreviated as SP in this study. These classes of “Figure 2” represents the volume of the workspace for 3x3 full-motion-capable spatial mechanisms are classified as 3x3, SP and the inner workspace inside of it to be optimized. We 6x3, and6x6 basedon theirnumberof upper and lower used different resolution, “ Δa”, values for each axis of the platform connection points. So, 3x3 has triangular base and workspace as mentioned above. For example, Δa is 0.04 m for mobile platform, while 6x3 configuration has hexagonal base linear motion along y-axis. and triangular mobile platform configuration. Lastly, 6x6 has hexagonal base and mobile platform. Considering the rs fi t, 4. Optimization by SQP the second, and the third configuration of similar dimensions and with the same minimal and maximal leg lengths, the third The primary purpose of the optimization is to increase has the largest workspace, followed by the second and then dexterity and usable workspace of the platform by chang- the rfi st. It must be noted that the workspace volume of the ing the design parameters. Besides various optimization 6x6 structure is always approximately 30% greater than that of techniques, we used SQP method for the optimization of the 6x3 mechanical structure. Similarly the 6x6 congfi uration design parameters 𝑟 , 𝑟 ,and 𝑙 .Min andmax values of the 𝑏 𝑝 𝑖 volume is 70% greater than that of the 3x3 workspace. u Th s objective design parameters are set before the optimization the 6x3 volume is approximately 25% greater than that of the procedure. Another constraint is Z(LCI of workspace point). 3x3 configuration workspace [1]. We calculated all Z (LCI) for each point in the predefined Additionally, the effect of geometrical parameters on workspace shown in “Figure 2.” parallel robotic systems is much greater than that on serial robotic systems. For instance, external stiffness over a given 4.1. Cost Function Assignments. In this research, Matlab workspace of a 6x6 Gough Platform can dieff r by 700% for a Optimization Toolbox (fmincon) is used to perform SQP change of only 10% of the platform radius, 𝑟 [24]. optimization technique for the PRS. The optimization proce- Therefore, in order to extend the reachable workspace of a dure is accomplished in three stages for maximum dexterity 3x3 SP mechanism, objective workspace for optimization and with reachable workspace: performance evaluation is described as follows: (1) One-variable optimization: 𝑟 is intended to be opti- (i) P = Changes from -0.06 to +0.06 m in 0.04 m regular mized intervals (2) Two-variable optimization: 𝑟 and 𝑟 are intended to 𝑝 𝑏 (ii) P = Changes from -0.06 to +0.06 m in 0.04 m regular be optimized intervals (3) Three-variable optimization: 𝑟 ,𝑟 ,𝑙 are optimized all 𝑏 𝑝 𝑖 (iii) P = Changes from +0.3 to +0.4 m in 0.02 m regular together intervals (iv) 𝛼 , 𝛽 , 𝛾 =Changes from -5 to +5 in 2.5 regular For all optimization approaches, a sub cost function is defined intervals as follows: Therefore, total 12000 spatial, 6dof, coordinates need to be (i) Sub cost function for one-variable optimization is satisfied after the optimization performance. First, according 𝑓 𝑥 :𝑟 ⋅ 𝑍−1 to defined workspace, we calculated LCI in Frobenius norm ( ) ( ) (26) 1 𝑝 2,50 z-axis (m) z-axis (m) y-axis (m) z-axis (m) 6 Journal of Robotics r (Radius of moving r (Radius of base p Leg lengths are platform) is defined platform) is defined defined for for min and max as a default min and max values values Reachable points Subcost function is Workspace is created checked selected as f (x) 1 as a rectangular Unreachable points For quick calculation workspace. We used aren't taken into max value of Subcost its corner coordinates account function is defined. layer by layer All condition 7. 9. numbers are For minimum value Cost function is calculated according of r Cost function is minimized according to defined values to min value of r defined as g(x) p except unreachable points Figure 3: Algorithm for one-variable optimization. (ii) Sub cost function for two-variable optimization is 4.2. One-, Two-, and rTh ee-Variable Optimization Algorithm. Figure 3 represents the optimization procedure sequence for 𝑓 (𝑥 ) :𝑟 ⋅𝑟 ⋅ (𝑍−1 ) (27) 2 𝑝 𝑏 one variable in which we have to fix other variables and define their min and max values to constrain them. Secondly, we (iii) Sub cost function for three-variable optimization is limit the leg lengths to check if the points in the workspace 𝑓 𝑥 :𝑟 ⋅𝑟 ⋅ Average of (𝑙 ) 𝑍−1 are reachable or not. Later on, we define the sub cost function ( ) ( ) (28) 3 𝑝 𝑏 𝑖 𝑓 (𝑥) according to one variable or𝑓 (𝑥) for two variables and 1 2 (iv) Cost function for all cases is create a limit value for selected 𝑓 (𝑥).Aeft rthat, we checkif the workspace points are reachable or not, and we eliminate 𝑔 (𝑥 ) :∑Sub cost 𝑓 (𝑥 ) (29) the unreachable ones. Then, cost function 𝑔(𝑥) is defined and condition number 𝑍 for each workspace point is calculated and an optimum value for 𝑟 is investigated which makes the We define cost functions as described above because min- 𝑝 sum of 𝑓(𝑥) (or 𝑔(𝑥) ) minimum. A similar algorithm with a imum g(x) values are investigated, which occur from the little difference is performed for two-variable optimization, 𝑟 sum of minimum sub cost function, f(x), values. It means and 𝑟 . When optimizing for two variables, rfi st step defined that minimum f(x) values are to be found indeed. While 𝑏 in “Figure 3” is skipped and instead of that step, we define searching minimum f(x) values, two criteria are present. One min-max values for 𝑟 as a second stage, and we use sub cost of them is the need for minimum Z and minimum 𝑟 and function 𝑓 (𝑥) set for two variables instead of 𝑓 (𝑥) at the the second criterion is the reachable workspace. Since Z 2 1 fifth step. For three-variable optimization, we constrained all never becomes smaller than 1, the algorithm is constructed variables defining their minimum and maximum values. As accordingly. If the targeted position in the workspace is asub cost function, weused 𝑓 (𝑥). not reachable, large value of Z is assigned not to calculate 3 real Z value in order to eliminate the unreachable point for evaluation. us, Th unnecessary calculations finding minimum 5. Optimization Results f(x) for that corresponding position are avoided. Searching for the minimum of two variables, 𝑟 and 𝑟 , which makes the 3x3 SP mechanism mobile platform diameter is analysed 𝑝 𝑏 sub cost function f(x) minimum, we multiply these variables for constant base diameter utilizing SQP technique. Mobile defining the sub cost function 𝑓 (𝑥). u Th s, while minimizing platform/base platform diameter optimization for better the cost function, we optimize variables. We constrained Z workspace reach and dexterity is performed. Condition (condition number) with a maximum value of 1000 as an number variation for dieff rent mobile platform diame- indicator for dexterity. We limited leg lengths and checked ters in selected range, minimum of condition number for if the coordinates in the workspace are reachable or not. base/mobile platform diameter change, and the number of Searching for the minimum of three variables 𝑟 , 𝑟 , and reachable points for dieff rent base/mobile platform diameter 𝑝 𝑏 𝑙 , which make the sub cost function f(x) minimum, we combinations are discussed. multiply 𝑟 , 𝑟 with an average of 𝑙 and (Z-1) defining the sub In Figure 4, horizontal axis refers to scanned 𝑟 values, 𝑝 𝑏 𝑖 𝑝 cost function 𝑓 (𝑥). We optimized parameters for maximum and vertical axis refers to minimum condition number which reachable dexterous workspace. we get from calculations without homogenization of Jacobian Journal of Robotics 7 Z < 1000 Amount of Reachable Points in Workspace 0.11 4500 0.1 0.09 0.08 30 0.07 0.06 0.05 500 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0 r [m] 0.050 0.070 0.090 0.110 0.130 0.150 Figure 6: Number of the reachable points whose condition numbers r (m) are under 1000. Zmin Figure 4: 𝑍 (min condition number) variations for different 𝑟 𝑚𝑖𝑛 𝑝 Table 1: Optimization of mobile platform diameter for various values while 𝑟 is fixed to 0.175 [m]. range constraints. 0.050 < 𝑟 < 0.180 0.070 < 𝑟 < 0.180 0.080 < 𝑟 < 0.180 𝑝 𝑝 𝑝 Min Z (Condition number in Frobenius norm) 0.15 44 𝑟 = 0.075 𝑟 𝑡𝑝𝑜 = 0.075 𝑟 𝑡𝑝𝑜 = 0.088 𝑝,𝑜𝑝𝑡 𝑝 𝑝 0.090 < 𝑟 < 0.180 0.100 < 𝑟 < 0.180 0.110 < 𝑟 < 0.180 𝑝 𝑝 𝑝 0.14 𝑟 𝑡𝑝𝑜 = 0.090 𝑟 𝑡𝑝𝑜 = 0.100 𝑟 𝑡𝑝𝑜 = 0.110 𝑝 𝑝 𝑝 0.13 0.120 < 𝑟 < 0.180 0.090 < 𝑟 < 0.250 0.050 < 𝑟 < 0.250 𝑝 𝑝 𝑝 𝑟 𝑡𝑝𝑜 = 0.120 𝑟 𝑡𝑝𝑜 = 0.090 𝑟 𝑡𝑝𝑜 = 0.075 𝑝 𝑝 𝑝 0.12 0.11 upper bound of 1000. As indicated in Figure 6, while𝑟 values 28 decrease, 𝑟 values increase, and numbers of reachable points 0.1 elevate. The upper bound of Z value is equal to 1000. us Th ,the numbers of reachable points are high for low mobile platform 0.09 diameters and high base diameters. 0.11 0.12 0.13 0.14 0.15 0.16 0.17 During optimization process of mobile platform diame- r [m] ter, different initial values of 𝑟 with various constraints are Figure 5: 𝑍 with respect to 𝑟 and 𝑟 in predefined reachable 𝑚𝑖𝑛 𝑝 𝑏 used,as shown in Table 1.While 𝑟 = 0.175 [m], minimum workspace. limit of 𝑟 is kept 0.050 [m] and leg lengths are scanned between 0.300 [m] and 0.450 [m]. As a result, the optimum value of 𝑟 is calculated to be 0.075 [m]. When minimum matrix. We get low values of 𝑍 while 𝑟 is increasing. We limit for 𝑟 is 0.080 [m], optimum value is found as 𝑟 = min p 𝑝 𝑝 are searching optimum values for most efficient workspace 0.088 [m]. As mentioned above, different initial values of 𝑟 volume which has reachable points with low condition are used for optimization. Consequently, when the constraint numbers. As a result, high mobile platform diameter refers to value of 𝑟 changes, the optimum value of 𝑟 changes. This is 𝑝 𝑝 calculated minimum condition numbers. because, in dier ff ent workspace regions, we have dieff rent local In Figure 5, horizontal axis refers to scanned 𝑟 values. minimums. Left vertical axis refers to scanned 𝑟 values and right vertical If the optimization procedure in Figure 8 is analyzed axis refers to the mapping of condition numbers which thoroughly, while the lowest constraint of 𝑟 is 0.050 and we calculated in Frobenius norm for predefined reachable 0.070 [m], we selected optimization scanned range between workspace. As seen in Figure 5, minimum condition numbers 0.050 and 0.180 [m]. Then, we obtain an optimum value of 𝑟 occur with high values of 𝑟 and 𝑟 which mean dexterity is 0.075 [m]. When the min constraint value of 𝑟 is elevated to 𝑏 𝑝 𝑝 getting better while they are increasing.However,itis obvious 0.080 [m], the rfi st local minimum of 𝑟 0.075 [m] is passed. from the plot that comparable decrease exists of the mobile Optimized value of 𝑟 shifts from 0.075 [m] to 0.088 [m]. platform diameter with respect to base diameter to obtain Out of nine different optimization combinations, one can minimal condition numbers.Also, the rate of this decrease understand that we have two local minimums which are is getting smaller when the base diameter is a lot higher 0.075 [m] and 0.088 [m]. Then, combinations that include the than mobile platform diameter. The horizontal tendency of initial value of 0.090 [m] and larger will result in an optimized platform diameter is apparent for higher base diameters. value of start point of the scanned range, 0.090 [m], 0.100 [m], In Figure 6, horizontal axis refers to scanned 𝑟 values. 0.110 [m], etc. Left vertical axis refers to different 𝑟 values, and the right Consequently, we found two local minimum values vertical axis refers to the mapping of reachable points in the (0.075 [m] and 0.088 [m]) for 𝑟 having maximum reachable workspace which have condition numbers limited with an points which are limited to upper bound condition number, r [m] Z , min condition number min r [m] p 8 Journal of Robotics Table 2: Optimizing mobile platform diameter, 𝑟 . Cost.F(x) Base Platform Radius, 𝑟 𝑝,𝑜𝑝𝑡[]𝑚 𝑟 0.125 0.130 0.135 0.140 0.145 0.150 0.155 𝑏,𝑜𝑝𝑡[]𝑚 0.070 4.8xE7 4.8xE7 8xE7 4.8xE7 4.8xE7 8xE7 4.8xE7 0.074 0.075 0.082 0.075 0.076 0.090 0.076 0.129 0.130 0.133 0.130 0.125 0.141 0.125 0.075 4.8xE7 8xE7 8xE7 8xE7 8xE7 8xE7 8xE7 0.077 0.075 0.081 0.086 0.087 0.081 0.093 0.125 0.130 0.134 0.137 0.140 0.135 0.145 0.080 6.4xE7 9.6xE7 8xE7 6.4xE7 8xE7 8xE7 8xE7 0.095 0.102 0.084 0.080 0.088 0.093 0.094 0.125 0.128 0.134 0.125 0.142 0.144 0.146 0.085 6.4xE7 6.4xE7 6.4xE7 6.4xE7 6.4xE7 8xE7 8xE7 Mobile Platform Radius, 𝑟 𝑝 0.085 0.085 0.085 0.085 0.085 0.093 0.098 0.125 0.125 0.125 0.125 0.125 0.144 0.148 0.090 6.4xE7 6.4xE7 8xE7 6.4xE7 6.4xE7 8xE7 6.4xE7 0.090 0.090 0.084 0.090 0.090 0.096 0.090 0.125 0.125 0.145 0.125 0.125 0.147 0.125 0.095 6.4xE7 9.6xE7 6.4xE7 6.4xE7 6.4xE7 8xE7 8xE7 0.095 0.095 0.095 0.095 0.095 0.097 0.099 0.125 0.130 0.125 0.125 0.125 0.148 0.150 0.100 9.6xE7 9.6xE7 9.6xE7 9.6xE7 9.6xE7 9.6xE7 8xE7 0.100 0.100 0.109 0.112 0.100 0.112 0.102 0.125 0.130 0.132 0.133 0.125 0.138 0.152 ∗E7= 1x10 Z, of 1000. Optimization results for two variables 𝑟 and 𝑟 Cost Function Values (1x10 ) 𝑏 𝑝 0.1 9.5 and their cost function values with different initial values of the optimization procedure are as shown in Table 2. For 0.095 example, while the initial value of 𝑟 is equal to 0.125 [m] and 8.5 an initial value of 𝑟 is equal to 0.070 [m], cost function value 0.09 8 is equal to 4.8xE7, optimum value of 𝑟 is equal to 0.074 [m], 𝑝 7.5 0.085 and optimum value of𝑟 is equal to 0.12 [m]. Optimum values of diameters 𝑟 and 𝑟 depend on constraints. 𝑝 𝑏 0.08 6.5 While generating Table 2, constraints are r : 0.125 [m] < r < 0.175 [m] and constraints for 𝑟 ; 0.070 [m] <𝑟 < b 𝑝 𝑝 0.075 5.5 0.125 [m]. Low mobile platform diameters with high base diameters seemed to result comparably in small cost function 0.07 5 0.125 0.13 0.135 0.14 0.145 0.15 0.155 values, which means better dexterity in motion. In Figure 7, r [m] minimum cost function value is 4.8 x 10 as indicated in bold areas. In those areas, cost function values are minimum Figure 7: Two-variable ( 𝑟 , 𝑟 ) optimization results. 𝑝 𝑏 with optimized 𝑟 and 𝑟 values as shown and are compatible 𝑏 𝑝 with Table 2. While optimized 𝑟 has minimum values, cost function values are minimum. As mentioned previously, we has large spectrum throughout the workspace. Therefore, are searching for minimum cost function value for better comparing the average condition numbers for dieff rent dexterity. In Table 2, there is not one optimum solution for design parameters with different constraints is meaningless. the minimum cost function value. For example, 𝑟 , 𝑟 𝑝,𝑜𝑝𝑡 𝑏,𝑜𝑝𝑡 As a result, calculating the cost function (sum of condition = 0.074 [m], 0.129 [m] has minimum cost function value numbers multiplied by r and r ) and considering the of (4.8xE7) and also 𝑟 , 𝑟 = 0.077 [m], 0.125 [m] has b p 𝑝,𝑜𝑝𝑡 𝑏,𝑜𝑝𝑡 resulting value is more efficient while searching for the best minimum cost function value of (4.8xE7). optimum design parameters. For better accuracy, we increase the resolution in solution area in Table 3. For the purpose of interpreting the results, we Optimum values are in Table 3. Minimum cost function used cost function values. However, the condition number value is 3.2xE7. Solution set with minimum cost function r [m] p Journal of Robotics 9 Table 3: Optimization of mobile platform diameter, 𝑟 , by narrowing constraint target area. Cost.F(x) Base Platform Radius, 𝑟 𝑝,𝑜𝑝𝑡[]𝑚 𝑟 0.125 0.126 0.127 0.128 0,129 𝑏,𝑜𝑝𝑡[]𝑚 0.060 3.2xE7 3.2xE7 3.2xE7 3.2xE7 3.2xE7 0.068 0.072 0.071 0.071 0.068 0.125 0.127 0.127 0.128 0.129 0.065 3.2xE7 3.2xE7 3.2xE7 3.2xE7 3.2xE7 0.069 0.070 0.070 0.071 0.067 0.129 0.128 0.128 0.128 0.129 0.066 3.2xE7 3.2xE7 3.2xE7 3.2xE7 3.2xE7 0.068 0.070 0.071 0.071 0.068 0.125 0.128 0.127 0.128 0.129 0.067 3.2xE7 3.2xE7 3.2xE7 3.2xE7 3.2xE7 0.069 0.069 0.070 0.070 0.068 0.129 0.128 0.127 0.128 0.129 0.068 3.2xE7 3.2xE7 3.2xE7 3.2xE7 4.8xE7 0.068 0.070 0.070 0.070 0.075 0.125 0.126 0.127 0.128 0.130 0.069 4.8xE7 3.2xE7 3.2xE7 3.2xE7 4.8xE7 0.076 0.070 0.071 0.070 0.074 0.125 0.126 0.127 0.128 0.129 0.070 4.8xE7 4.8xE7 3.2xE7 3.2xE7 4.8xE7 0.074 0.074 0.071 0.070 0.075 0.129 0.129 0.127 0.128 0.129 ∗E7= 1x10 1. 2. r ,r (Radius of Sub cost function is Workspace is created b p selected as f3(x) as a rectangular platforms) and l workspace. We used For quick calculation (lengths of the legs ) its corner coordinates max value of Sub are defined for min cost function is defined. layer by layer and max values Reachable All condition For minimum values points checked numbers are of three variables Cost function is calculated according Unreachable to defined values defined as g(x) points aren't taken except unreachable into account for points optimization 7. 8. 9. Cost function is Results are analyzed By selecting the minimized according to select the optimum values of variables to min value of three values for three optimization variables variables procedure is finished Figure 8: Algorithm sequence for three-variable optimization. value occurs for𝑟 ,𝑟 = 0.072 [m], 0.127 [m];𝑟 ,𝑟 of 𝑟 is equal to 0.070 [m], and initial value of 𝑙 is equal to 𝑝,𝑜𝑝𝑡 𝑏,𝑜𝑝𝑡 𝑝,𝑜𝑝𝑡 𝑏,𝑜𝑝𝑡 𝑝 𝑖 = 0.071 [m], 0.127 [m]; 𝑟 , 𝑟 = 0.071 [m], 0.128 [m]. 0.300 [m], cost function value is equal to 4.8xE7, optimum 𝑝,𝑜𝑝𝑡 𝑏,𝑜𝑝𝑡 Optimization results for three variables 𝑟 , 𝑟 , 𝑙 ,and value of 𝑟 is equal to 0.076 [m], 𝑟 is equal to 0.125 [m], 𝑏 𝑝 𝑖 𝑝 𝑏𝑜𝑝𝑡 their cost function values with different initial values of the and𝑙 is equal to 0.300 [m]. Optimum values of parameters 𝑖𝑜𝑝𝑡 optimization are as shown in Tables 4 and 6. For example, also depend on constraints. While generating Table 4, our while the initial value of 𝑟 is equal to 0.125 [m], initial value constraints are as in Table 5. 𝑏 10 Journal of Robotics Table 4: Optimizing mobile platform diameters, 𝑟 , and optimum leg length, 𝑙 . 𝑝 𝑖 Cost.F(x) Base Platform Radius, 𝑟 𝑝,𝑜𝑝𝑡[]𝑚 𝑟 0.125 0.130 0.135 0.140 0.145 0.150 0.155 𝑏,𝑜𝑝𝑡[]𝑚 Leg Length 𝑙 𝑖,[𝑚] 𝑖,𝑜𝑝𝑡[𝑚] 0.300 0.325 0.350 0.375 0.400 0.425 0.450 0.070 4.8xE7 4.8xE7 8xE7 8xE7 6.4xE7 - 4.8xE7 0.076 0.075 0.070 0.091 0.094 - 0.077 0.125 0.130 0.135 0.143 0.125 - 0.125 0.300 0.325 0.350 0.375 0.375 - 0.388 0.075 4.8xE7 4.8xE7 8xE7 4.8xE7 8xE7 6.4xE7 8xE7 0.076 0.075 0.081 0.077 0.097 0.072 0.076 0.125 0.125 0.134 0.125 0.150 0.172 0.133 0.300 0.358 0.352 0.373 0.375 0.375 0.375 0.080 4.8xE7 8xE7 8xE7 6.4xE7 8xE7 4.8xE7 - 0.070 0.087 0.080 0.080 0.080 0.070 - 0.125 0.142 0.135 0.125 0.133 0.125 - 0.322 0.359 0.350 0.375 0.375 0.322 - 0.085 6.4xE7 6.4xE7 6.4xE7 8xE7 6.4xE7 8xE7 6.4xE7 0.085 0.085 0.085 0.085 0.085 0.096 0.085 Mobile Platform Radius, 𝑟 0.125 0.125 0.125 0.140 0.125 0.147 0.125 0.300 0.300 0.367 0.375 0.300 0.375 0.300 0.090 6.4xE7 6.4xE7 6.4xE7 6.4xE7 6.4xE7 8xE7 6.4xE7 0.090 0.090 0.090 0.090 0.098 0.091 0.090 0.125 0.125 0.127 0.125 0.125 0.150 0.125 0.300 0.358 0.353 0.375 0.375 0.375 0.300 0.095 6.4xE7 9.6xE7 6.4xE7 9.6xE7 8xE7 6.4xE7 8xE7 0.095 0.095 0.095 0.110 0.095 0.095 0.095 0.125 0.130 0.125 0.143 0.150 0.126 0.149 0.300 0.325 0.357 0.375 0.375 0.388 0.375 0.100 9.6xE7 9.6xE7 9.6xE7 9.6xE7 9.6xE7 9.6xE7 8xE7 0.100 0.100 0.100 0.100 0.100 0.118 0.094 0.125 0.130 0.135 0.140 0.145 0.150 0.145 0.300 0.325 0.350 0.375 0.400 0.375 0.386 Table 5: Constraints for three-variable optimization of the PRS. 𝑟 𝑟 𝑙 𝑏[]𝑚 𝑝[]𝑚 𝑖 0.125 [m]<𝑟 < 0.175 0.070 [m]<𝑟 < 0.125 [m] 0.300 [m]<𝑙 < 0.450 [m] 𝑏 𝑝 𝑖 In Table 4, there is not only one optimum solution for optimization results with mathematical calculations and the minimum cost function value. For example, 𝑟 , 𝑟 , compared these values with calculated values. 𝑝,𝑜𝑝𝑡 𝑏,𝑜𝑝𝑡 𝑙 = 0.076 [m], 0.125 [m], 0.300 [m] has minimum cost For three-parameter optimization, we used 𝑓 (𝑥) as a sub 𝑖,𝑝𝑡𝑜 3 function value of (4.8xE7) and also 𝑟 , 𝑟 , 𝑙 = cost function and obtained results in Tables 4 and 6 using the 𝑝,𝑜𝑝𝑡 𝑏,𝑜𝑝𝑡 𝑖,𝑝𝑡𝑜 0.077 [m], 0.125 [m], 0.388 [m] has minimum cost function optimization algorithm as shown in Figure 8. value of (4.8xE7). For better accuracy, we zoom into the For Table 7, we used optimized values from Table 3, and solution area in Table 5. So, we generated Table 6. As shown we generated Table 7 using inverse kinematic equations. in Table 6, minimum cost function value is found as 3.2xE7. Table 7 presents L , L , 𝑢 (uniformity index), and max min According to Table 6, optimum parameters are 𝑟 , 𝑟 , reachable points for optimum radius values of mobile and 𝑝,𝑜𝑝𝑡 𝑏,𝑜𝑝𝑡 𝑙 = 0.071 [m], 0.127 [m], 0.350 [m] and 𝑟 , 𝑟 , 𝑙 = base platforms. According to these calculated values using 𝑖,𝑝𝑡𝑜 𝑝,𝑜𝑝𝑡 𝑏,𝑜𝑝𝑡 𝑖,𝑝𝑡𝑜 0.072 [m], 0.127 [m], 0.374 [m]. Optimization results change optimized parameters from Table 7, GCI (Global Condition according to optimization parameters which are initial val- Index) and LCI (Local Condition Index) values are computed ues and convergence tolerances. Hence, we controlled our for each solution. Figure 9 represents LCI values for the first Journal of Robotics 11 Table 6: rTh ee-variable optimization by narrowing target constraint regions. Cost.F(x) Base Platform Radius 𝑝,𝑜𝑝𝑡[]𝑚 𝑟 0.125 0.126 0.127 0.128 0.129 𝑏,𝑜𝑝𝑡[]𝑚 Leg Length 𝑙 𝑖,[𝑚] 𝑖,𝑜𝑝𝑡[𝑚] 0.300 0.325 0.350 0.375 0.400 0.070 4.8xE7 4.8xE7 3.2xE7 3.2xE7 4.8xE7 0.076 0.075 0.071 0.072 0.076 0.125 0.126 0.129 0.300 0.323 0.349 0.374 0.391 0.071 4.8xE7 4.8xE7 3.2xE7 4.8xE7 4.8xE7 0.075 0.074 0.071 0.074 0.076 0.125 0.126 0.127 0.128 0.129 0.300 0.324 0.350 0.373 0.398 0.072 4.8xE7 4.8xE7 - 4.8xE7 4.8xE7 0.076 0.087 - 0.073 0.077 0.125 0.120 - 0.128 0.125 0.300 0.330 - 0.375 0.385 0.074 4.8xE7 - 6.4xE7 6.4 xE7 4.8xE7 0.076 - 0.091 0.087 0.075 Mobile Platform Radius 0.125 - 0.126 0.128 0.130 0.300 - 0.333 0.375 0.399 0.076 4.8xE7 4.8xE7 6.4xE7 4.8xE7 6.4xE7 0.077 0.076 0.076 0.077 0.084 0.125 0.125 0.127 0.125 0.128 0.300 0.300 0.350 0.374 0.385 0.078 6.4xE7 6.4xE7 6.4xE7 6.4xE7 6.4xE7 0.078 0.078 0.078 0.081 0.094 0.125 0.126 0.125 0.129 0.125 0.300 0.325 0.300 0.375 0.355 0.080 4.8xE7 6.4xE7 6.4xE7 6.4xE7 4.8xE7 0.070 0.080 0.094 0.090 0.076 0.125 0.126 0.126 0.126 0.125 0.322 0.325 0.335 0.375 0.335 Table 7: Leg lengths for optimum parameters of the PRS. Solutions 𝑟 /𝑟 𝐿 𝐿 𝐿 𝑈 Reachable Points 𝑏,𝑜𝑝𝑡 𝑝,𝑜𝑝𝑡 𝑥[𝑚] 𝑚𝑖𝑛[]𝑚 𝑔𝑡ℎ𝑛 []𝑚 (1) 0.127/0.071 0.4550 0.2928 0.3739 1.2505 11592 (2) 0.127/0.072 0.4552 0.2926 0.3739 1.2503 11588 (3) 0.128/0.071 0.4554 0.2929 0.3742 1.2486 11588 optimized variables. As shown in Figure 9, there are nearly When 𝑟 is equal to 0.128 [m], 𝑟 is equal to 𝑏,𝑜𝑝𝑡 𝑝,𝑜𝑝𝑡 400 unreachable points in the workspace. LCI values of 0.071 [m], and 𝑙 is between 0.300 and 0.450 [m], LCI values reachable points vary between nearly 8 and 11. The average are as shown in Figure 11 and GCI is equal to 9.57. of LCI values isdenfi ed asGCI. As shown in Figures 9, 10, and 11, LCI and GCI values When 𝑟 is equal to 0.127 [m], 𝑟 is equal to are very similar, because two parameters 𝑟 and 𝑟 𝑏,𝑜𝑝𝑡 𝑝,𝑜𝑝𝑡 𝑏,𝑜𝑝𝑡 𝑝,𝑜𝑝𝑡 0.071 [m], and 𝑙 is between 0.300 and 0.450 [m], LCI values differ only 0.001 [m]. Distribution of LCI values in graphics are as shown in Figure 9 and GCI is equal to 9.62. is different, but it does not aeff ct GCI noticeably. As shown When 𝑟 is equal to 0.127 [m], 𝑟 is equal to in Table 7, we can change constraints of 𝑙 and reach all 𝑏,𝑜𝑝𝑡 𝑝,𝑜𝑝𝑡 𝑖 0.072 [m], and 𝑙 is between 0.300 and 0.450 [m], LCI values workspace. One can redefine leg lengths of the PRS. If we are as shown in Figure 10 and GCI is equal to 9.49. compare Tables 6 and 7, calculated solutions are dieff rent 𝑙𝑒 𝑚𝑎 12 Journal of Robotics LCI Distiribution-3 GCI :9.57 LCI Distribution -1/ GCI = 9.627 1800 1800 1600 1600 1400 1400 1200 1200 0 2 4 6 8 10 12 0 2468 10 12 LCI Values LCI Values Figure 11: Distribution of LCI values based on un/reachable Figure 9: Distribution of LCI values based on un/reachable workspace points for three-parameter optimization, 𝑟 = workspace points for three-parameter optimization, 𝑟 = 𝑏,𝑜𝑝𝑡 𝑏,𝑜𝑝𝑡 0.128 [m], 𝑟 = 0.071 [m], and 𝑙 = 0.300-0.450 [m]. 0.127 [m], 𝑟 = 0.071 [m], and 𝑙 = 0.300-0.450 [m]. 𝑝,𝑜𝑝𝑡 𝑖 𝑝,𝑜𝑝𝑡 𝑖 LCI Distribution-2 GCI : 9.49 are many researches for the optimization of kinematics of 1800 a PRS. In these works, various optimization methods have been used. In this study, we optimized 3x3 SP mechanism using SQP optimization method for predefined workspace and checked the efficiency of the optimization for three parameters which are diameter of mobile and base platform together with lengths of the legs. It is shown that the optimization of multi-parameters is not easy to formulate 800 concerning cost functions and results must be evaluated by calculating the dexterity indexes and the amount of reachable points. Firstly, we optimized𝑟 and then, we optimized𝑟 and 𝑝 𝑝 𝑟 simultaneously. Lastly, 𝑟 , 𝑟 ,and 𝑙 are optimized at the 𝑏 𝑝 𝑏 𝑖 same time. During optimization, tolerances, procedure steps, and applied/defined cost functions aec ff t the convergence efficiency. Results show that initial values for each parameter 0 2468 10 12 at the beginning of the optimization determine the toler- LCI Values ance of the convergence for the optimization. As a further Figure 10: Distribution of LCI values based on un/reachable research, various optimization methods can be performed workspace points for three-parameter optimization, 𝑟 = 𝑏,𝑜𝑝𝑡 while increasing number of kinematic parameters optimized 0.127 [m], 𝑟 = 0.072 [m], and 𝑙 = 0.300-0.450 [m]. 𝑝,𝑜𝑝𝑡 𝑖 searching different cost functions. Notations but very similar to each other because initial values affect optimization procedure and convergence tolerances of opti- 𝑙 : Leg lengths mization. All LCI in Figures 9, 10, and 11 are calculated x: Dir. along the x-axis of mobile platform from the homogenous Jacobian matrix and these calculated y: Dir. along the y-axis of mobile platform values are divided by the dimension of Jacobian matrix for z: Dir. along the z-axis of mobile platform homogenization. 𝛼 : Pitch angleof mobileplatform In detail, rfi st solution is the best for the amount of the 𝛽 : Roll angle of mobile platform reachable points. Second one is the best solution according 𝛾 : Yaw angle of mobile platform to GCI. And the third one is the best for the uniformity. Z: Condition number of Jacobian matrix J: Jacobian matrix O : Mobile platform center of gravity 6. Conclusion O : Base platform center of gravity Determining the values of geometrical parameters is very 𝜆 : Angle between (X , O P ) 𝑖 p p i important for efficiency of a PRS design. In literature, there Λ : Angle between (X , O B ) 𝑖 b b i Number of Workspace Points Number of Workspace Points Number of Workspace Points Journal of Robotics 13 [11] M. A. Hosseini, H.-R. M. Daniali, and H. D. Taghirad, “Dex- 𝑟 : Radius of a circle passing through center terous workspace shape and size optimization of tricept parallel of joints between base platform and legs manipulator,” International Journal of Robotics, vol. 2, no. 1, 2011. 𝑟 : Radius of a circle passing through center [12] M. A. Hosseini, H.-R. M. Daniali, and H. D. Taghirad, “Dexter- of joints between mobile platform and legs ous workspace optimization of a Tricept parallel manipulator,” 𝜃 : Angle between connection points of legs Advanced Robotics,vol.25,no. 13-14,pp.1697–1712,2011. on base platform [13] Y. Lou, G. Liu, N. Chen, and Z. Li, Optimal Design of Par- 𝜃 : Angle between connection points of legs allel Manipulators for Maximum Effective Regular Workspace , on mobile platform University of Science and Technology, Intelligent Robots and R: Rotational transformation matrix. Systems, Hong Kong, 2005. [14] Y. Lou, G. Liu, and Z. Li, “Randomized optimal design of paral- Data Availability lel manipulators,” IEEE Transactions on Automation Science and Engineering,vol.5,no.2,pp.223–233, 2008. The Matlab Codes data used to support the findings of this [15] Q. Jiang and C. M. Gosselin, “Geometric optimization of the study are available from the corresponding author upon MSSM Gough-Stewart platform,” Journal of Mechanisms and request. Robotics,vol.1,no. 3,pp.1–8, 2009. [16] A. M. Lopes, E. J. S. Pires, and M. R. Barbosa, “Design of a Conflicts of Interest parallel robotic manipulator using evolutionary computing,” International Journal of Advanced Robotic Systems,vol.9,article The authors declare that they have no conflicts of interest. 26, 2012. [17] G.Abbasnejad, H.M.Daniali, and S.M.Kazemi,“A new approach to determine the maximal singularity-free zone of 3- References RPR planar parallel manipulator,” Robotica, vol.30, no.6,pp. 1005–1012, 2012. [1] J.-P. Merlet, Parallel Robots,INRIA, Springer, Amsterdam, Netherlands, 2nd edition, 2006. [18] A. R. Shirazi, M. M. S. Fakhrabadi, and A. Ghanbari, “Optimal design of a 6-DOF parallel manipulator using particle swarm [2] P. C. Srinivas, Y. D. Kumar, P. D. Rao, V. Sreenivasulu, and optimization,” Advanced Robotics, vol.26, no. 13, pp.1419–1441, C. U. 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Wei, “Global stiffness and well-conditioned formance Of 3-RPS Parallel Manipulator Using Evolutionary optimization analysis of 3UPU-UPU robot based on pareto Algorithms,” 12-CSME-12, E.I.C., Accession 3333 India, March front theory,” Springer International Publishing, pp. 124–133, 2015. ¨ [26] I.Yıldız,V.E. Omurl ¨ u, ¨ Z. Ekiciogl ˘ u, and A. Gune ¨ y, “Forward [10] Z. Gao, D. Zhang, and Y. Ge, “Design optimization of a spatial Kinematic Analysis Methods for 6dof Parallel Mechanism,” in six degree-of-freedom parallel manipulator based on artifi- Proceedings of the Automatic Control Conference by Turkish cial intelligence approaches,” Robotics and Computer-Integrated National Automatic Control Committee-TOK,Istanbul, Turkey, Manufacturing, vol.26, no.2,pp.180–189, 2010. 2010. 14 Journal of Robotics [27] J.-P. 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References

Exploiting higher kinematic performance using a 4-legged redundant pm rather than gough-stewart platforms,

M. H. Abedinnasab, Y.-J. Yoon, and H. Zohoo
A review of redundant parallel kinematic mechanisms,

M. Luces, J. K. Mills, and B. Benhabib
Multiple-goal kinematic optimization of a parallel spherical mechanism with actuator redundancy,

R. Kurtz and V. Hayward
Some investigations into the optimal dimensional synthesis of parallel robots,

R. Kelaiaia, A. Zaatri, O. Company, and L. Chikh
Optimal Design of Parallel Manipulators,

A. Fattah and S. H. Jazi
Global stiffness and well-conditioned optimization analysis of 3UPU-UPU robot based on pareto front theory,

D. Zhang and B. Wei
Design optimization of a spatial six degree-of-freedom parallel manipulator based on artificial intelligence approaches,

Z. Gao, D. Zhang, and Y. Ge
Dexterous workspace shape and size optimization of tricept parallel manipulator,

M. A. Hosseini, H.-R. M. Daniali, and H. D. Taghirad
Dexterous workspace optimization of a Tricept parallel manipulator,

M. A. Hosseini, H.-R. M. Daniali, and H. D. Taghirad
Randomized optimal design of parallel manipulators,

Y. Lou, G. Liu, and Z. Li
Geometric optimization of the MSSM Gough-Stewart platform,

Q. Jiang and C. M. Gosselin
Design of a parallel robotic manipulator using evolutionary computing,

A. M. Lopes, E. J. S. Pires, and M. R. Barbosa
A new approach to determine the maximal singularity-free zone of 3-RPR planar parallel manipulator,

G. Abbasnejad, H. M. Daniali, and S. M. Kazemi
Optimal design of a 6-DOF parallel manipulator using particle swarm optimization,

A. R. Shirazi, M. M. S. Fakhrabadi, and A. Ghanbari
Kinematic optimal design of 6-UPS parallel manipulator,

Y. Zhang and Y. Yao
Multi-objective design of parallel manipulator using global indices,

F. A. Lara-Molina, J. M. Rosario, and D. Dumur
Optimal custom design of both symmetric and unsymmetrical hexapod robots for aeronautics applications,

A. Cirillo, P. Cirillo, G. De Maria, A. Marino, C. Natale, and S. Pirozzi
Multiobjective optimization of 6-dof UPS parallel manipulators,

R. Kelaiaia, A. Zaatri, and O. Company
Optimization algorithms for kinematically optimal design of parallel manipulators,

Y. Lou, Y. Zhang, R. Huang, X. Chen, and Z. Li
Optimization approaches applied in dimensional synthesis of parallel mechanisms,

D. Modungwa, N. Tlale, and B. Twala
Design For Optimal Performance Of 3-RPS Parallel Manipulator Using Evolutionary Algorithms,

S. R. Babu, V. R. Raju, and K. Ramji
Forward Kinematic Analysis Methods for 6dof Parallel Mechanism,

I. Yıldız, V. E. Ömürlü, Z. Ekicioğlu, and A. Güney
Workspace-oriented methodology for designing a parallel manipulator,

J.-P. Merlet
Jacobian, Manipulability, Condition Number and Accuracy of Parallel Robots,

J. P. Merlet
Formulating Jacobian matrices for the dexterity analysis of parallel manipulators,

G. Pond and J. A. Carretero
Kinematic-sensitivity indices for dimensionally nonhomogeneous jacobian matrices,

P. Cardou, S. Bouchard, and C. Gosselin
Optimal design of a 3-leg 6-DOF parallel manipulator for a specific workspace,

J. Fu and F. Gao

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SQP Optimization of 6dof 3x3 UPU Parallel Robotic System for Singularity Free and Maximized Reachable Workspace

SQP Optimization of 6dof 3x3 UPU Parallel Robotic System for Singularity Free and Maximized Reachable Workspace

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SQP Optimization of 6dof 3x3 UPU Parallel Robotic System for Singularity Free and Maximized Reachable Workspace

SQP Optimization of 6dof 3x3 UPU Parallel Robotic System for Singularity Free and Maximized Reachable Workspace

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