OPTIMUM DESIGN OF 3R ORTHOGONAL MANIPULATORS CONSIDERING ITS TOPOLOGY

Transcription

1 Copyight by ABCM Page 38 OPTIMUM DESIGN OF 3R ORTHOGONAL MANIPULATORS CONSIDERING ITS TOPOLOGY Giovana Tindade da Silva Oliveia, School of Mechanical Engineeing, Fedeal Univesity of Ubelândia, 6 João Naves de Ávila Av., Campus Santa Mônica, CEP 384-9, Ubelândia, Bazil. Antônio Calos Nogueia, anogueia@ufu.b Sezimáia Fátima Peeia Saamago, saamago@ufu.b College of Mathematics, Fedeal Univesity of Ubelândia, 6 João Naves de Ávila Av., Campus Santa Mônica, CEP 384-9, Ubelândia, Bazil. Abstact. Seveal studies have investigated the popeties of the wokspace open chains obotics with the pupose of emphasizing its geometic and kinematic chaacteistics, and to devise analytical algoithms and pocedues fo its design. The wokspace of a manipulato obot is consideed of geat inteest fom theoetical and pactical viewpoint. The wokspace topology is defined by the numbe of kinematic solutions, the numbe of cusps and nodes that appea on the wokspace bounday. In the classic applications in the industy, manipulatos need to pass though singulaities of the joint space to change thei postue. A 3-evolute (3R) manipulato can execute a non singula change of postue if and only if thee is at least one point in its wokspace which has exactly thee coincident solutions of the invese kinematic model. In this wok, a mulit-objective optimization poblem is fomulated with the aim of obtaining the optimal geometic paametes of obot which must obey the topology specified by the designe. The maximum wokspace volume, the maximum system stiffness and the optimum dexteity ae consideed as the multi-objective functions. In addition, the optimization poblem is subject to penalties that contol the topology, focing it to occupy a cetain potion of the wokspace. A sequential technique and evolutions algoithms ae applied in the solution of the poblem. Some applications ae pesented to show the efficiency of the poposed methodology. Keywods: obotics, manipulatos topology, optimization, wokspace volume, obot stiffness, obot dexteity.. INTRODUCTION In the classic applications in the industy, manipulato obots need to pass though singulaities of the joint space to change thei postue. A 3-evolute (3R) manipulato can execute a non singula change of postue if and only if thee is at least one point in its wokspace which has exactly thee coincident solutions of the invese kinematic model (IKM), esulting in one of the sepaation suface which divide the wokspace in seveal egions that have manipulatos with same popeties (binay o quatenay, egions with the same numbes of cusps and node points). These egions ae called domains. So, to study such manipulatos is essential to know the topology of the singulaity sufaces in the wokspace. These singulaities ae defined as places whee the deteminant of the Jacobian matix of diect kinematic model (DKM) is annulled, defining the othes equations of sufaces which divide the wokspace. Wenge and El Omi (996) showed that fo some choices of the paametes, manipulatos with thee otational joints (3R) may be able to change postue without meeting a singulaity in the joint space. They succeed in chaacteizing such manipulatos (Wenge, 998), but they needed geneal conditions on the design paametes. Covez () found impotant esults about this issue. In 4, Baili ealized eseaches on the popieties of 3R manipulatos with othogonal axes and made a classification in the paametes space. Oliveia et al. (9) showed a sepaation sufaces fomulation by using the algebaic tool Gobne basis to obtain an analytical expession of the sufaces of the paametes space that sepaate the diffeent types of manipulatos. The othe sufaces wee obtained by annulment of the deteminant of Jacobian matix of the diect kinematic model. In this pape, a multi-objective optimization poblem is fomulated with the aim of obtaining the optimal geometic paametes of obot which must obey the topology specified by the designe. The maximum wokspace volume, the maximum system stiffness and the optimum dexteity (expessed with condition numbe of Jacobian matix) ae consideed as multi-objective function. This pape also discusses a compaative study of thee diffeent numeical techniques, i.e., sequential quadatic pogamming (SQP), genetic algoithms (GA) and diffeential evolution (DE). The pesence of voids and singulaities and the discontinuous geneation of the envelope geatly incease the complexity of the calculation of the algebaic fomulation fo a coect mathematical model of the obot. Moeove, the objective function pesents seveal local maxima and is extemely nonlinea. These factos geatly incease the difficulties involved in the optimization pocess, justifying the use of diffeent optimization techniques to validate the esults. It is known that conventional methods ae based on a ule of point-to-point and has the dange of falling into local optima. The evolutionay algoithms ae based on the population-to-population ule. These techniques have the advantages of obustness and good convegence popeties. They equie no knowledge o gadient infomation about the optimization poblems, in this case only the objective function and coesponding fitness levels influence the diections

2 Copyight by ABCM Page 39 of seach. The discontinuities pesent on the optimization poblems have little effect on the oveall optimization pefomance. The pobabilistic tansition ules ae used and they pefom well fo lage-scale optimization poblems in the pesence of local optima. In this wok two evolutions algoithms ae applied in the solution of the poblem: Genetic Algoithms (GA) and Diffeential Evolution (DE). GA and DE have been shown efficient to solve linea and nonlinea poblems by exploing all egions of seach space and exponentially exploiting pomising aeas though mutation, cossove, and selection opeations applied to individuals in the population. Theefoe, they ae the suitable fo the optimization poblems studied hee. The manipulatos with thee otational joints with othogonal axes as descibed in Fig.. The study of this type of manipulato is done accoding to the Denavit-Hatenbeg paametes: d, d 3, d 4,, and 3. To educe the numbe of paametes, will be consideed d = and 3 =. The joint vaiables ae θ, θ and θ 3 which epesent the input angles of the actuatos. Fo this type of manipulato, the diect kinematic model is given in Eq. (): ( ) ( ) x = + d + d cθ cθ cθ + d sθ sθ ( ) ( ) y = + d + d cθ cθ sθ + + d sθ cθ ( ) z = d + d cθ sθ () in which c i = cosθ i and s i = sinθ i, fo i =,, 3. Figue. Manipulato with thee otational joints (3R) with othogonal axes A poweful algebaic tool is used: the Gobne basis. With this appoach, it is possible to obtain analytical expessions of the sufaces of the paametes space that sepaate the diffeent types of manipulatos. The annulment of the deteminant of Jacobian matix of the invese kinematic model (IKM) enables to obtain the othe sufaces that sepaate the vaious egions fo diffeent topologies (Oliveia et al, 9). Thus, it is possible obtain: ( ) d + d + C : d4 = d3 + AB 3 3, whee A = ( d 3 + ) + and B ( d 3 ) + = () The Equation () is the suface of sepaation between the manipulatos of domain and domain. The manipulatos that belong to domain ae binay, have a tooidal cavity in its wokspace and do not have cusps and nodes points. The domain epesents the manipulatos that have 4 points of cusp, but do not have the same numbe of nodes. The suface of sepaation C between the domains and 3 is defined by: ( ) C : d = d /( + d ). d + + (3) The domain 3 is composed by manipulatos which pesent cusps points on intenal envelopment. In the case of domain 4, the manipulatos have 4 points of cusp and 4 nodes. The suface C 3, which sepaates the manipulatos of the domains 3 and 4, is given by: ( ) C : d = d /( d ). d +, with d 3 > (4) Finally, the domain 5 coesponds to manipulatos that have no cusp points. Unlike of manipulatos of the type, the intenal envelope is not defined by a tooidal cavity, but by a egion with 4 solutions in IKM. The suface of sepaation C 4 between the domains 3 and 5 is: ( ) C : d = d /( d ). d +, with d 3 < (5) Summaizing, the space of paametes (d 3, d 4 and ) of a 3R othogonal manipulato is divided into 5 domains sepaated by sufaces C, C, C 3 and C 4, defined by Eqs. (), (3), (4) and (5), espectively.

3 Copyight by ABCM Page 4 The Figue a) shows the cuves of sepaation in a plane section (d 3, d 4 ) of the space of paametes, esulting in 5 domains, adopting a fixed value fo =. The Figue b) shows the space of paametes divided accoding to the numbe of cusps points and nodes points. The domains accoding to the numbe of cusps points ae divided into subdomains that contain the same numbe of nodes. Each sub-domain defines a topology of the wokspace denoted WT i (α, β), whee α epesents the numbe of cusp points and β the numbe of nodes points. d Domain 5 C 4 Domain 3 Domain C 3 C.5 C d 3 Domain Domain 4 (a) (b) Figue. Division of paametes space consideing = : (a) Accoding to the suface of sepaation of topologies; ( b) Accoding to the numbe of points of cusps and nodes WT 8 (, ) WT 9 (, ) WT 6 (, 3) WT 5 (, ) WT (, ) WT 7 (4, 4) WT 4 (4, ) WT 3 (4, ) WT (4, ) As explained peviously, the manipulatos of the domain have a tooidal cavity and do not have cusps and nodes points. The manipulato epesented in Fig. 3a) chaacteize the fist type of manipulato, whose topology is known as WT (, ). The manipulatos that belong to domain have 4 points of cusp. This egion can be subdivided into 3 sub-domains though the sufaces E and E. The topology of the wokspace WT (4, ), epesented by Fig. 3b), has 4 cusp points, nodes, a tooidal cavity, two egions with 4 solutions and a egion with solutions in IKM. The topology WT 3 (4, ) contains manipulatos with 4 cusp points, zeo node, without tooidal cavity, a egion with 4 solutions and othe with solutions in IKM, as illustated in Fig. 3c). The tansition between the topologies WT and WT 3 is the bounday between the manipulatos containing a tooidal cavity in its wokspace and those that do not contain. Accoding to Baili (4), the suface of sepaation between these topologies is given by the expession: E : d 4 =.5 (A - B), whee A and B ae given in Eq. (). (6) In domain is still possible to chaacteize the topology epesented in the Fig. 4d), denoted by WT 4 (4, ), containing 4 points of cusp and nodes. These nodes ae diffeent fom nodes of WT since not delimit a tooidal cavity but a egion of 4 solutions in IKM. In this case, the suface of sepaation E, between topology WT 3 and WT 4 is defined by: E : d 4 = d 3 (7) The domain 3 is composed by manipulatos which have cusp points and can be divided into sub-domains though the suface E 3. The manipulatos descibed by WT 5 (, ) have cusps points on intenal envelope, a node point and has the shape of a fish, as shown Fig. 3e). Moeove, the Fig. 5f) pesents a manipulato that belongs to the wokspace WT 6 (, 3), which has points of cusp and 3 nodes. The Eq. (8) defines the sepaation suface between the topology WT 5 and WT 6. Besides, this suface also sepaates the topology of the wokspace WT 8 and WT 9 that ae contained in the domain 5. E 3 : d 4 =.5 (A + B) (8) In domain 4, the manipulatos ae of type 4, epesented by WT 7 (4, 4), have 4 cusp points and 4 nodes, as can be seen in Fig. 3g). The 4 points of cusp ae shaed between the intenal and extenal singulaity sufaces. Finally, the domain 5 coesponds to manipulatos that have no cusp points. Unlike of manipulatos of the type, the intenal envelopment is not defined by a tooidal cavity, but by a egion with 4 solutions in IKM. The domain 5 coesponds to manipulatos of type 5 and do not have cusp points. This egion is divided into subdomains though the suface E 3. In the Fig. 3h), the topology epesented by WT 8 (, ) does not have cusp points and no nodes. As mentioned ealie, its intenal envelope is not defined by a tooidal cavity, but by a egion with 4

4 ABCM Symposium Seies in Mechatonics - Vol. 5 Copyight by ABCM Page 4 solutions in IKM. Finally, Fig. 3i) featues a manipulato which belongs to the topology WT 9 (, ), with cusp points and nodes points obtained by the intesection of intenal and extenal envelopment (a) Type, WT (,) (b) Type, WT (4,) (c) Type, WT 3 (4,) (d) Type, WT 4 (4,) (e) Type 3, WT 5 (,) (f) Type 3, WT 6 (,3) (g) Type 4, WT 7 (4,4) (h) Type 5, WT 8 (,) (i) Type 5, WT 9 (,) Figue 3. Radial section fo 3R othogonal manipulato, showing the 5 types of manipulatos. WORKSPACE OF 3R MANIPULATORS Accoding to Begamaschi et al (6), the wokspace W is the set of all attainable points fo a point P of the endeffecto when the joint vaiables sweep its definition inteval entie. Point P is usually chosen as the cente of the endeffecto, o the tip of a finge, o even the end of the manipulato itself. The fist pocedue to investigate the wokspace is to vay the angles θ, θ and θ 3 in thei inteval of definition and to estimate the coodinates of point P with espect to the manipulato base fame. The wokspace of this obot is a solid of evolution. Thus, it is natual to imagine that the wokspace is the esult of otation aound the z axis of a adial plane section. Figue 4. (a) A scheme fo evaluating the wokspace volume of 3R manipulatos; (b) Discetization of coss section aea by using a ectangula mesh The wokspace of a thee-evolute open chain manipulato can be given in the fom of the adial each and axial each z with espect to the base fame, accoding to Begamaschi et al. (6). Fo this epesentation, is the adial distance of a geneic wokspace point fom the z-axis, and z is the distance of this same point at the XY-plane (see Fig. 4b). Thus, using Eq. (), the paametic equations (of paametes θ and θ 3 ) of the geometical locus descibed by

5 Copyight by ABCM Page 4 point P on a adial plane ae: =x + y and z, whee x, y and z ae given in (). (9) The wokspace volume V can be evaluated by the Pappus-Guldin Theoem, using the following equation (see Fig. 4a): V = π g A, whee A is the coss section aea, which is fomed by the family of cuves given by Eq. (). () This eseach poposes numeical fomulation to appoximate the coss section aea, though its discetization within a ectangula mesh. Initially, the exteme values of vectos and z should be obtained as: min = min {}, max = max{}, z min = min {z} and z max = max{z} () Adopting n and n z as the numbe of intevals chosen fo the discetization along the and z axis, the sizes of the elementay aeas of the mesh can be calculated: = ( max min ) / n and z = ( zmax zmin ) / nz () The n and n z values must be adopted so that the sizes of the elementay aeas ( o z) ae at least % of the total distances consideed in the discetization ( max - min o z max - z min ). Evey point of the family of cuves fom the coss section of the wokspace is calculated by Eq. (9). Using this equation, vaying the values of θ and θ 3 in the inteval [ π, π], it is possible to obtain the family of cuves of the wokspace. Given a cetain point (, z), its position inside the discetization mesh is detemined though the following index contol: i = int[( min ) / ] + and min j = int [( z z ) / z] + (3) whee i and j ae computed as intege numbes. As shown in Fig. 5b), the point of the mesh that belongs to the wokspace is identified by P ij =, othewise P ij =, which means: P =, i f P W ( P) o, if P W ( P) ; whee W(P) indicates wokspace egion. (4) ij ij ij In this way, the total aea is obtained by the sum of evey elementay aeas of the mesh that ae totally o patially contained in the coss section. In Eq. (4), it is obseved that only the points that belong to the wokspace contibute to the calculation of the aea A T. The coodinate g of the cente of the mass is calculated consideing the sum of the cente of the mass of each elementay aea, divided by the total aea, using the following equation: imax jmax imax j A T = ( Pij z) and max g = ( Pij z)( ( i ) + ( / ) + min ) / AT i= j= i= j= (5) Finally, afte the calculation of the coss section aea and the coodinate of the cente of the mass, given by Eqs. (4) and (5), espectively, the wokspace volume of the manipulato can be evaluated by using Eq. (). 3. SYSTEM STIFFNESS Fom the viewpoint of mechanics, the stiffness is the measuement of the ability of a body o stuctue to esist defomation due to the action of extenal foces. The stiffness of a seial mechanism at a given point of its wokspace can be chaacteized by its stiffness matix. This matix elates the foces and toques applied at the gippe link in Catesian space to the coesponding linea and angula Catesian displacements. Two main methods have been used to establish mechanism stiffness models. The fist one is called matix stuctual analysis, which models stuctues as a combination of elements and nodes. The second method elies on the calculation of the seial mechanism's Jacobian matix which is adopted in this wok. Matix J is usually temed Jacobian matix which is descibed in Eq. (6). By consideing the case in that d =, its deteminant is calculated by using the Eq. (7).

6 Copyight by ABCM Page 43 sinθ3 cosθd4 cosθ sinθ3d 4 [ J ] = sinθ3 sinθd4 + sinθ d3 + cosθ3d4 cosθd3 cosθ cosθ3d4 d cosθ3d ( θ ) θ ( θ ( θ θ ) θ ) det( J ) = d d + d cos d sin + d sin + d sin cos cos (6) (7) The stiffness matix of the mechanism in the Catesian space is then given by the Eq. (8), whee K j is the joint stiffness matix of the mechanism, with K j =[k, k, k 3 ]. In this case, each actuatos of the mechanism is modeled as an elastic component. k i is a scala epesenting the joint stiffness of each actuato, which is modeled as linea sping: T K = [ J ] K [ J ] (8) C j Paticulaly, in the case fo which all the actuatos have the same stiffness, i.e., k=k =k =k 3, then Eq. (8) will be educed to: T K = k[ J ] [ J ] (9) C Futhemoe, the diagonal elements of the stiffness matix ae used as the system stiffness value. These elements epesent the pue stiffness in each diection, and they eflect the igidity of machine tools moe clealy and diectly. The objective function fo system stiffness optimization can be witten as Eq. (). In this case, the stiffness index S can be maximized: S = K + K + K33 () 4. DEXTERITY The condition numbe of the Jacobian matix will be used as a measue of dexteity indices fo the 3R manipulato. By using the spectal nom, these indices will be descibed as. Cond( J ) = λ ( J ) / λ ( J ) () max min whee λ max and λ min means the maximum and minimum singula values of Jacobian matix J, espectively. Regading the computing time of optimization pocess, this expession is selected as the objective function fo the optimization of dexteity. The value of Cond(J), which is diectly elated to singula values of Jacobian matix, is between and positive infinity. All the singula values of the Jacobian matix will be the same and the manipulato is isotopic if Cond(J) is equal to. While Cond(J) is pone to be positive infinity it means that the Jacobian matix is singula. Theefoe, fo the optimization of dexteity, the condition numbe must to be minimized. 5. NUMERICAL SIMULATIONS The optimization poblem is fomulated with the objective of obtaining the optimal geometic paametes of the 3R manipulato to maximize the wokspace and the system stiffness and to optimize the dexteity such as the topologies specified by the designe ae obeyed. Since the poblem have seveal objectives, it deals with a multi-objective optimization poblem, it is equied to find all possible tadeoffs among multiple objective functions that ae usually conflicting with each othe. The constaints depend on the topology chosen fo the obot, accoding to Fig. 3. In this wok, the optimization is investigated using a Sequential Quadatic Pogamming (SQP), the Diffeential Evolution (DE) and Genetic Algoithms (AG). The evolutionay algoithms wee developed fo unconstained poblems. So, in the case of constained optimization poblems, it is necessay to intoduce modifications in this method. This wok uses the concept of Penalty Function (Nocedal and Wight, ). In this technique, the poblems with constains ae tansfomed in unconstained poblems adding a penalty function P(x) to the oiginal objective function to limit constaint violations. This new objective function is penalized, accoding to a facto p, evey time that meets an active constaint. The scala p is a multiplie that quantifies the magnitude of the penalty. Fo the efficiency of the evolutionay method, a lage value of the penalty facto p should be used to ensue nea satisfaction of all constaints, in this eseach, was adopted p =. Then, the poblem can be ewite as follows: Maximize F(x) = f(x) + p P(x), whee f(x) = [V, Cond(J), S] and P(x) = max (, g j (x) ) () Subject to: g j (x) ; j=,..,k and x l x i x u, i=,, 3

7 Copyight by ABCM Page 44 The geometic paametes ae design vaiables given by x = (d 3, d 4, ) T. The lowe and uppe bounds adopted fo the am length (side constaints) ae:. x i 3., i =,,3. In this simulation, two methods of multi-objective optimization ae utilized: Weighting Objectives Method and Global Citeion Method (L -metic and L 3 -metic) pesented on Oliveia and Saamago (). The weighted sum stategy convets the multi-objective poblem vecto f(x) into a scala optimization poblem by building a weighted sum of all the objectives as Eq. (3). The weighting coefficients w i epesent the elative impotance of each citeion. Thus, Maximize F( x) = wvc wcond ( J ) c + w3sc3 p P( x), whee 3 w i= i = (3) whee the wokspace volume V is given by Eq. (), the stiffness S is calculated using the Eq. () and the condition numbe Cond(J) is epesented in the Eq. (). Objective weighting is obviously the most usual substitute model fo vecto optimization poblems. The touble hee is attaching weighting coefficients to each of the objectives. The weighting coefficients do not necessaily coespond diectly to the elative impotance of the objectives o allow tade-offs between the objectives to be expessed. Fo the numeical methods fo seeking the optimum of (3) so that w i can eflect closely the impotance of objectives, all the functions should be expessed in units of appoximately the same numeical values. The best esults ae usually obtained if c i = /f i o, whee f i o epesents the ideal solution, that indicates the minimum value of each i-th function. To detemine this solution, one must find the minimum attainable fo all the objective functions sepaately. In this case, the vecto f o = [V id,, S id, Cond(J) id ] T is ideal fo a multi-objective optimization poblem. In Global Citeion Method, the multi-objective optimization poblem is tansfomed into a scala optimization poblem by using a global citeion. The function that descibes this global citeion must be defined such as a possible solution close to the ideal solution is found. In this case, the L -metic and L 3 -metic, ae given espectively by: Minimize Minimize Vid V Cond ( J ) id Cond( J ) Sid S F( x) = p P( x) Vid Cond( J ) id S id Vid V Cond ( J ) id Cond( J ) Sid S F( x) = p P( x) Vid Cond( J ) id S id 3 (4) (5) The computational code of the DE was developed in MATLAB by the authos. The paametes used wee: numbe of population individuals Np = 5; geneations, epesentation of individuals by eal vectos using multiplie of the diffeence vecto F =.8 and cossove pobability CR =.5. The Genetic Algoithms Optimization Toolbox (GAOT) pogam developed by Houck et al. (985) has been used to pefom the GA, adopting Np = 8 individuals, geneations, cossove and mutation pobabilities:.6 and.. The Sequential Quadatic Pogamming (SQP) it was pefomed by using the toolbox fmincon of the MATLAB. 5. Example - WT (, ) In Example is consideed an application whee the manipulato must belong to the topology WT (see Fig. 3). In this case, the following constaints ae adopted: Side limits:. < d 3 < 3.;. < d 4 < 3. e. < < 3. [u.l.] and (6) Points below the cuve C, given by Eq. (). The ideal solution calculated using DE is: V id =35.98 [u.v.]; Cond(J) id =.57 and S id =7.9 [u.s.]. It is woth noting that when SQP is applied the optimum depends on the initial estimate povided by the use. Thus, tests wee pefomed fo diffeent initial values esulting in diffeent answes. This behavio clealy indicates the pesence of seveal local minima. Seveal stating points wee tested: the uppe and lowe limits of the seach space, the midpoint of the ange and the optimal solution obtained by DE. In the esults pesented in the Tab. to Tab 4 the stating points ae the solution obtained by DE. The optimal esults obtained though the optimization pocedue Weighting Objectives Method, Eq. (3), ae showed in Tab.. Obseving this table one can be noted that the best solution depends on inteest of the designe because each objective function is conflicting with othe. In this example, when was adopted the weighting coefficients equal to.8 fo the volume (w ) o fo the stiffness (w 3 ) it is obtained simila esults. This is due to the fact that both ae maximized and pesented the simila behavio. But given the weighting coefficients equal to.8 fo the condition numbe (w ) it is obseved that was obtained a diffeent esult and the dexteity was significantly impoved. The esults

8 Copyight by ABCM Page 45 indicate that this poblem is vey sensitive to the dexteity value. When this function is pioitized, the optimal volume and stiffness ae stongly modified. Table shows the optimal esults obtained by using the Global Citeion Method, Eqs. (4) and (5). In this technique the idea is to minimize the elative eo of functions in elation to ideal values. The solutions obtained epesent a compomise between the thee objective functions. Note that the optimal is simila to the values obtained with pevious method when w =.8 o w 3 =.8 wee chosen. Table. Optimal esults obtained with the Weighting Objectives Method fo Example. Weighting coefficients w i w =.33 w =.33 w 3 =.33 w =.8 w =. w 3 =. w =. w =.8 w 3 =. w =. w =. w 3 =.8 Technique d 3 d 4 Volume Stiffness Time Cond(J) (u.l.) (u.v.) (u.s) (min) DE GA SQP DE GA SQP DE GA SQP DE GA SQP Table. Optimal esults obtained with the Global Citeion Method fo Example. L -metic L 3 -metic Technique d 3 d 4 Volume Stiffness Time Cond(J) (u.l.) (u.v.) (u.s) (min) DE GA SQP DE GA SQP Obseving the Tab. and Tab. it appeas that the sequential and andom techniques obtained simila values, diffeing only in the computational cost. In the case of SQP, the esults wee good because the initial estimate is the optimal obtained by DE. Using sequential pogamming, the computational cost was significantly small, but this technique has some limitations, fo example: if the model is multimodal, it can "get stuck" in some local solutions; it only handles eal vaiables; the objective function and the constaints must both be continuous. Consideing the L -metic, the optimal point obtained by Diffeential Evolution is maked in Fig. 5a). The optimal coss section aea of the wokspace is pesented in Fig. 5b). Compaing the adial section of Fig. 5b) to Fig. 3a), one can obseve that the poject paametes esult in a manipulato with a bigge volume (the void of the wokspace was educed). The optimal manipulato belonging to the topology WT (d =, 3 = ) is epesented in Fig. 5c)..5.5 d d 3 (a) Paametes space with d =, 3 = (b) Coss section aea of the wokspace (c) Scheme fo 3R Robot Figue 5. The optimum design of a 3R Robot, consideing the L -metic by Diffeential Evolution Example

9 Copyight by ABCM Page Example WT 3 (4, ) Now, consideing that designe desies a manipulato that belongs to the topology WT 3, the following constaints ae adopted: Side limits:. < d 3 < 3.;. < d 4 < 3. e. < < 3. [u.l.]; (7) Points above the cuve E, given by Eq. (6) and Points below the cuve E, given by Eq. (7). The ideal solution calculated using DE is: V id = [u.v.]; Cond(J) id =.37 and S id =94. [u.s.]. Fo this case, the optimal esults obtained though the optimization pocedue Weighting Objectives Method ae showed in Tab. 3. The Tab. 4 shows the esults obtained by using Global Citeion Method. As obseved in Example, the best solution depends on inteest of the designe. Table 3. Optimal esults obtained with the Weighting Objectives Method fo Example. Weighting coefficients w i w =.33 w =.33 w 3 =.33 w =.8 w =. w 3 =. w =. w =.8 w 3 =. w =. w =. w 3 =.8 Technique d 3 d 4 Volume Stiffness Time Cond(J) (u.l.) (u.v.) (u.s) (min) DE GA SQP DE GA SQP DE GA SQP DE GA SQP Consideing the L -metic, the optimal point obtained by Diffeential Evolution is maked in Fig. 6a). The scheme of the optimal manipulato belonging to the topology WT 3 is illustated in Fig. 6b). Table 4. Optimal esults obtained with the Global Citeion Method fo Example. L -metic L 3 -metic Technique d 3 d 4 Volume Stiffness Time Cond(J) (u.l.) (u.v.) (u.s) (min) DE GA SQP DE GA SQP d (a)paametes space with d =, 3 = d 3 (b) Coss section aea of the wokspace (c) Scheme fo 3R Robot Figue 6. The optimum design of a 3R Robot, consideing the L -metic by Diffeential Evolution Example

10 Copyight by ABCM Page 47 The optimal coss section aea of the wokspace is pesented in Fig. 6b). Compaing the adial section of Fig. 6b) to Fig. 3c), one can obseve that the wokspace is inceased. Futhemoe, the manipulatos of this type of topology emain with 4 cusp points, a egion with 4 solutions and othe with solutions in IKM. Is impotant to note that the applied methodologies wee effective to obtain an optimum which obeys the topology constaints. 5.3 Input and Output Window Pogam A input data window and a output optimal esults window was developed to facilitate the use of computational code, as shown in Fig. 7a) and 7b), espectively. In the input window the designe can choose the multi-objective optimization method, the type of topology and define the side constaints. In the output window can be seen the volume, stiffness and dexteity optimal values. Moeove, it pesents the optimal dimensions and the coss section aea of the wokspace. (a) (b) Figue 7. (a) Input data window; (b) Output optimal esults window fo the topology WT 5 (,), Fig.3(e)

11 Copyight by ABCM Page CONCLUSIONS In this wok, a suitable fomulation of the optimal design of manipulatos with thee othogonal otational joints was used. The aim is to obtain the optimal dimensions of the manipulatos so that the maximum volume of the wokspace, the maximum stiffness of the mechanism and the optimization of dexteity ae consideed simultaneously. In addition, wee imposed constaints accoding to the type of wokspace topology by using appopiate equations witten accoding to the sepaation sufaces of diffeent domains. The solutions wee obtained by means of two evolutionay techniques and one sequential. The authos developed a computational code in MATLAB, easy to be used by the designe, allowing the optimum design of manipulations can be calculated consideing the most appopiate topology fo the tasks. The main contibutions of this wok wee: to veify that the dexteity has a geat influence on the optimal dimensions of the manipulatos; enable the designe to choose one type of topology to obtain the best design that matches the desied application. In the futue wok othe examples with diffeent topology constaints will be studied and the geneal case fo the 3R manipulato, adopting the paamete 3, will be consideed. 7. ACKNOWLEDGEMENTS The authos acknowledge the Fundação de Ampao a Pesquisa do Estado de Minas Geais (FAPEMIG) by the financial suppot. 8. REFERENCES Baili, M. and Wenge Ph., Chablat D., 4, Analise et Classification de Manipulateu 3R à axes Othogonaux, Thèse de Doctoat - Univesity of Nantes, Fance. Begamaschi, P.R., Nogueia, A.C. and Saamago, S.F.P., 6, Design and optimization of 3R manipulatos using the wokspace featues, Applied Mathematics and Computation, Elsevie, Vol. 7. No.., pp Covez, S. and Rouillie, F.,, Using compute algeba tools to classify seial manipulatos, Poceeding Fouth Intenational Wokshop on Automated Deduction in Geomety, Lins. Houck, C.R.; Joinez, J.A. and Kay, M.G., 985, A Genetic Algoithms fo Function Optimization: a Matlab Implementation. NCSO-IE Technical Repot, Univesity of Noth Caoline, USA. Oliveia, G.T.S., Nogueia, A.C., Saamago, S.F.P., 9, Use of the Gobne Basis in the Study of Manipulatos Topology, Poceedings of the th Intenational Congess of Mechanical Engineeing, Gamado, th COBEM, 9. v.. Bazil. Oliveia, L.S. and Saamago, S.F.P.,, Multiobjective Optimization Techniques Applied to Engineeing Poblems, Jounal of the Bazilian Society of Mechanical Sciences and Engineeing, Vol.XXXII, pp.94-4,. Nocedal, J., Wight, S.J.,, Numeical Optimization, Spinge Seies in Opeations Reseach. 9. RESPONSIBILITY NOTICE The authos ae the only esponsible fo the pinted mateial included in this pape.