Task-based Configuration Optimization of Modular and Reconfigurable Robots using a Multi-solution Inverse Kinematics Solver

Transcription

1 CARV 2007 Task-based Configuraion Opimizaion of Modular and Reconfigurable Robos using a Muli-soluion Inverse Kinemaics Solver S. Tabandeh 1, C. Clark 2, W. Melek 3 Absrac: Modular and Reconfigurable Robos (MRR) are a breed of indusrial robos designed for oday s flexible and versaile producion faciliies. MRRs can be assembled in various ways o achieve numerous disinc Kinemaic configuraions (KC). In MRRs, he problem of finding he mos suiable KC for a specific ask from a predefined se of modules is called Task-based Configuraion Opimizaion (TBCO). To assess a KC in TBCO, a soluion o he Inverse Kinemaics (IK) problem is required. In his paper, a TBCO algorihm based on Geneic Algorihms (GA) is proposed ha uilizes a muli-soluion IK solver. By solving for muliple soluions of he IK, The possibiliy of preferring a sub-opimal KC over an opimal configuraion is decreased. Moreover, in he proposed TBCO algorihm a prioriy based GA Selecion operaor is used o ensure reaching he righ soluion in case of more han one opimizaion crieria. Keywords: Modular and Reconfigurable Robos, Task-Based Configuraion Opimizaion, Geneic Algorihms, Inverse Kinemaics 1 Inroducion A main building block of many auomaed manufacuring plans is he indusrial robo manipulaor. The majoriy of he exising indusrial robos are based on a 1 Universiy of Waerloo, Mechanical Engineering Deparmen 2 California Polyechnic Sae Universiy, Deparmen of Compuer Science 3 Universiy of Waerloo, Mechanical Engineering Deparmen

2 2 S. Tabandeh, C. Clark, W. Melek fixed join and link configuraion and are designed o perform heir duies for a general se of asks. Alhough hese robos can perform well in a se of paricular workspaces, hey have limied adapabiliy owards changes in eiher environmen or he prescribed ask. MRRs, on he oher hand, are assembled from a variey of modular componens and can be physically configured o mee he requiremens of he work space and he ask a hand. The se of modules may consis of joins, links, and end-effecors. Join modules are he acuaors ha provide he degree of freedom of each robo. Link modules of varying lengh connec he joins o each oher. The end-effecor modules consis of he ools required o inerac wih he robo environmen. Figure 1 shows he schemaic diagram of a join module and an MRR ha is assembled ino a hree degree-of-freedom (DOF) robo. As can be seen from he figure, he join module has wo mechanical inpu and wo mechanical oupu pors. Inpu pors should be conneced o he links closer o he base of he robo while he oupu pors should be conneced o he links closer o he end-effecor. The join will produce differen ypes of moion depending on which inpu and oupu pors he links are conneced o. a b Figure 1) a. The schemaic of a Join module wih wo mechanical inpus and wo mechanical oupus b. An MRR assembled ino a 3 DOF KC Differen KCs can be achieved by using differen join, link, and end-effecor modules and by changing heir relaive orienaion and, in he case of he illusraed example, he join pors used. The number of disinc KCs aainable by a se of modules can vary wih respec o he size of he module se from several ens o several housands. Finding he mos suiable configuraion for a specific ask from a predefined se of modules is a highly nonlinear opimizaion problem in a discree search space called TBCO. In he lieraure, differen approaches o solve he have been presened. In (Chen 1994) Chen inroduced a new represenaion called Assembly Inciden Marix (AIM) for Modular Robos. AIM is hen ranslaed o a sring and is used in a Geneic Algorihm, wih reachabiliy and manipulabiliy as he objecive funcions. In (Chen, e al., 1998), he same concep is expanded o make a modified AIM ha includes he por vecors. The muaion and crossover operaors of he Geneic Algorihm are also modified o be direcly applicable o he AIMs. In (Leger 1999) an objec orienaed srucure o represen robos is used wih a GA. In some cases, he algorihm relies on a human exper o redirec he search. In (Kim 1992) Task based configuraion opimizaion problem is ackled wih he

3 CARV addiion of numerous opimizaion crieria. A wo level Geneic Algorihm approach is used in (Chocron, a al., 1997). The upper level GA searches for he mos suiable configuraion and he lower GA solves for a single soluion o he Inverse Kinemaics. Mos of he exising TBCO algorihms use a GA o opimize he KC. A each ieraion of hese GAs, a soluion o he Inverse Kinemaics problem is required. The approaches referenced mosly rely on numerical mehods o solve he IK problem. Unforunaely, numerical IK mehods only converge on he soluion ha is closes o heir saring poin. Therefore, only one soluion of he Inverse Kinemaics will be considered when wo KCs are being compared. In his paper, a TBCO algorihm is proposed ha uses a Geneic Algorihm as he core opimizaion engine (as in previous works). However, in conras o he exising mehods he proposed algorihm uilizes muliple soluions of he IK problem. To faciliae ha, an IK solver capable of calculaing muliple soluions of he IK for a wide range of roboic manipulaors has been developed. In summary he conribuions of his paper can be highlighed as: A novel MRR represenaion called KMR is proposed. A muli-soluion Inverse Kinemaics solver is used for assessing he KCs. A novel prioriy base Tournamen Selecion operaor specifically modified for TBCO is proposed. This operaor enables he algorihm o search for a KC ha can perform a cerain ask beer han he ohers according o an efficiency crierion. TBCO specific Crossover and Muaion operaors have been proposed. By using he proposed operaors, The GA is applied direcly on he KMR wihou he necessiy of coding o a sring. This paper is organized as follows. In Secion 2 he proposed TBCO algorihm is explained in deail. The subsecions include problem formulaion, Module invenory and kinemaic represenaion and he Inverse Kinemaics. In Secion 3 he resuls of applying he proposed TBCO algorihm in finding a 3DOF kinemaic configuraion for a ask is presened. Secion 4 discusses he conclusion and he fuure work of he projec. 2 Task Based Configuraion Opimizaion The goal of TBCO is finding a KC, or an assembly sequence of he modules, capable of performing a cerain ask more efficienly han he oher configuraions. A ask in he Caresian space is defined by a se of ask poins. Each ask poin consiss of a posiion and an orienaion in space. The vecor of he posiion of ask poin wih respec o he origin is represened by P OT and is shown in Eq.(1.1). The orienaion of ask poin is represened by O OT and can be defined as in Eq.(1.2). In his equaion, ( α, β, γ ) are he Euler angles of he ask poin. P OT x = y z (1.1)

4 4 S. Tabandeh, C. Clark, W. Melek O OT α = β γ (1.2) The measure ha represens he abiliy of a KC o accomplish a ask is called reachabiliy. Reachabiliy shows how capable a KC is in reaching he posiion and orienaion of he ask poins. Posiion reachabiliy for ask poin is represened by p f and can be defined as in Eq.(1.3). rch, f = P P (1.3) p rch, OT OE Where POE is he posiion of he end-effecor of KC when i is as close as possible p o P OT. In more specific erms, f rch is he disance of he posiion of he ask poin o ha of he KC when he KC is as close as possible o he ask poin. Eq.(1.4) shows he orienaion reachabiliy definiion. o f = min( ( α, β, γ ) ( α, β, γ ), ( α, β, γ ) ( α + π, β, γ π ) ) (1.4) rch e e e e e e Where ( αe, βe, γ e ) are he orienaion of he end-effecor when i is as close as possible o ha of he ask poin. The oal reachabiliy can be defined as he p o p o weighed sum of f rch and f rch as in Eq.(1.5) Where wrch and wrch are he weighing facors. p p o o f = w. f + w. f (1.5) rch rch rch rch rch A posiion and orienaion reachabiliy of zero for a cerain ask indicaes ha he KC is capable of performing he ask. In pracice, usually more han one KC exis ha can accomplish a cerain ask. Hence, he KC ha can perform he ask more efficienly han he ohers according o an opimizaion crierion (or a se of crieria) should be sough. The TBCO can be formulaed ino an Opimizaion Problem as in Eq.(1.6). min f R Κ Subjec o f op rch ε (1.6) As can be seen from he equaion, he goal is finding a Kinemaic Configuraion R capable of achieveing he ask wihin a reachabiliy less han a olerance value ε while minimizing an opimizaion crieria f op. In his reaseach, as he opimizaion crierion f op, Relaive Manipulabiliy is used. Κ represens he space of all he possible kinemaic configuraions aainable by a se of modules. Figure 2 shows he archiecure of he proposed TBCO algorihm. The inpus o he algorihm are module invenory, opimizaion crieria, and he ask. These inpus mus be coded before enering he algorihm. The way hese parameers are

5 CARV coded grealy depends on he opimizaion algorihm ha is used, and has a dramaic effec on he performance of he algorihm. Figure 2 ) Archiecure of he proposed TBCO algorihm The firs se of inpus, module invenory, is he group of available modules. This se deermines he size and complexiy of he Kinemaic Configuraion space Κ. The opimizaion parameers are anoher group of inpus o he algorihm. These parameers are applicaion dependen and he se of parameers relevan o he asks are highlighed and seleced as he inpus of his sage. The ask is he las inpu o algorihm. The ask usually depends on he produc or objec ha will be manipulaed by he robo. From he ask, a se of he more dominan ask poins are exraced. An opimizaion algorihm is he core of TBCO. This par of he algorihm reads he inpus and finds he opimized kinemaic configuraion. In his research, Geneic Algorihms are used because of heir high efficiency in searching highly nonlinear discree or coninuous spaces. In he GA block, an iniial populaion of he Kinemaic Configuraions is randomly generaed. Each of he individuals of his populaion represens a kinemaic configuraion. According o heir finess value he bes ones are seleced o form a paren pool. To calculae he finess value for each of he KCs, he Inverse Kinemaics and Forward Kinemaics should be solved. Afer undergoing he Crossover and Muaion operaors a new generaion of kinemaic configuraions is formed. This process will coninue unil a erminaion crierion has been me. Finally, a decoding sage will conver he oupu of he opimizaion algorihm o a kinemaic configuraion. The oupu of his sage can include DOF, join and link ypes, assembly sequence, and relaive orienaion of joins. In he following secions, each of he blocks of he proposed TCO algorihm will be explained in more deail. 1.1 Module Invenory Encoding and Kinemaic Configuraion Represenaion Figure 3 shows he module invenory used in he proposed algorihm. As can be seen, all he links and joins are symmerical along he axis of he mechanical pors. Therefore, a join can be conneced o a link in four differen orienaions.

6 6 S. Tabandeh, C. Clark, W. Melek a b c d Figure 3 ) Module Invenory a) Roaional Join Module b) Pivoal Join Module c) Perpendicular-Roaional Join Module d) Link Modules in hree differen sizes To represen a KC assembled from he module invenory, a represenaion called Kinemaic Marix Represenaion (KMR) is proposed. Eq.(1.7) shows he KMR of an n DOF robo. 0 0 L0 φ1 M1 L1 R M L = φ2 2 2 φ M M L n n n (1.7) In KMR, he elemens of he firs column ( φ 1 ) represen he orienaion of each join module relaive o he previous one. This orienaion is measured along he axis perpendicular o each mechanical por. In he second column, he join module ypes ( M i ) are sored. The hird column represens he link module ypes ( L i ). The values ha each of hese variables can sore are shown in Table 1. The KMR marix has 3 column and n+1 rows, where n is he DOF of he robo. The firs row represens he firs link of he robo. This link is perpendicular o he ground and connecs he firs join o he base of he robo. Las elemen of he marix, a row n+1 and column 3, is he las link which can sore a value corresponding o a ool or a spherical wris. Table 1 ) Eligible Values for he variables in KMR Variable Value Meaning Variable Value Module φ i 0 0 Radians L i 0 No Links 1 π 2 Radians 1 Small Link 2 π Radians 2 Medium Link 3 3π 2 Radians 3 Large Link M i 0 Roaional Join 1 Pivoal Join 2 Perpendicular-roaional 1.2 Opimizaion Crieria As he opimizaion crieria Relaive Manipulabiliy ( M r ) is used (Kim 1992). M r is defined as in Eq.(1.8). Where m and n are he number of rows in he Jacobian Marix (he dimension of he ask space) and he number of join respecively. ai and d i are he link lengh and join offse of he ih join. Larger values of Manipulabiliy in robos are preferable.

7 CARV M r = m T de( J. J ) n M = ( i + i ) i= 1 f M f a d (1.8) 1.3 Geneic Algorihm and GA operaors The Algorihm A Geneic Algorihm, hrough Selecion, Cross-over and Muaion operaors, finds he individuals (in our case KCs) ha have he bes finess values and combines hem o produce idividuals ha offer beer finess values han heir parens. This process coninues unil he populaion converges around he individual ha have he bes finess value (Goldberg 1989). In our applicaion, a Microbial GA (Harvey 1994) (Harvey 2001) which has been modified for TBCO was implemened. In he original Microbial GA, Tournamen Selecion is used. In Tournamen Selecion wo individuals are randomly picked from he GA populaion and afer comparison he one wih he beer finess value is seleced as he winner. In our modified microbial GA, o find he wo parens, wo ses of Tournamen Selecions are performed in parallel. The losers of he wo Tournamens will be replaced by he offsprings of he crossover of he winners. To esimae he poenial of individuals, a finess value should be allocaed o hem. In TBCO, he finess value is evaluaed by frch and f op. To fi he TBCO problem ino he GA srucure, he opimizaion problem of Eq.(1.6) can be reformaed as in Eq.(1.9). In he new equaion, he goal is finding an R which is able o minimize boh frch and f op. min f, f R Κ rch op (1.9) In mos of he exising TBCO algorihms, a weighed sum of frch and f op is used as he finess value. Bu i should be noed ha f op becomes of imporance only if a KC can saisfy he reachabiliy requiremens. Hence he necessiy of a selecion scheme which considers he prioriy of Reachabiliy over he opimizaion crieria arises. Figure 4 shows he Pseudo code of he proposed Selecion Scheme. if f 1 2 rch rch if f else 1 2 op op 1 2 elseif ( frch frch ϕ & rand ρ ) ( ϕ κ ) else if f else if f else f f f 1 2 op op f 1 2 rch rch κ Individual 1 is he winner Individual 2 is he winner Individual 1 is he winner Individual 2 is he winner Individual 1 is he winner Individual 2 is he winner Figure 4 ) Pseudo code of he proposed Selecion Scheme

8 8 S. Tabandeh, C. Clark, W. Melek Where κ, ϕ and ρ are consans. According o he scheme, when wo KCs are being compared hree cases could happen. If he reachabiliy of he wo KCs are very close o each oher he one wih he beer fop is he winner. If he reachabiliies are fairly close, he one wih beer f op wih a cerain probabiliy of ρ will be he winner. Finally, if he difference beween he reachabiliies is large he one wih he beer reachabiliy is seleced as he winner. I can be observed ha he scheme considers f op only if he KCs have approximaely he same f rch The GA operaors In he proposed TBCO algorihm, he Cross-over operaor is implemened as follows. One elemen of he kinemaic configuraion marix is seleced randomly. Wihin he wo marices undergoing crossover, all elemens locaed afer he seleced elemen are swapped. To accomplish he Muaion, one of he elemens of he kinemaic configuraion marix is seleced randomly. This elemen is replaced wih anoher randomly generaed number. The muaed number should comply wih he elemen ype i is being sored in. Figure 5 shows an example of applying he operaors on he KMR Ri = R j = R i = R j = R i = R i = a Figure 5 ) The proposed GA operaors: a. Crossover b. Muaion b 1.4 Inverse Kinemaics In order o calculae he reachabiliy and mos of opimizaion crieria a soluion o he IK problem is required. In mos of he exising TBCO algorihms, IK is solved numerically. The numerical IK algorihms converge o he closes soluion o heir iniial value. As a resul, only he finess values ha corresponding o only ha IK soluion will be used in he Tournamen Selecion. In his case, here always is a chance ha a good IK soluion of a less suiable KC will be preferred o a worse soluion of a desirable KC. To solve his problem, a muli-soluion IK solver based on Niching Geneic Algorihms has been developed by he auhors (Tabandeh, e al., 2006). The algorihm is developed based on a Niching GA algorihm. Niching GA has he abiliy o converge on muliple soluion regions. In he menioned algorihm, he oupu is filered and he soluion regions are deeced afer being passed hrough a clusering algorihm. In he presen research, he oupus pass hrough wo more blocks, a numerical IK and a selecion phase. The numerical IK is used o reduce he error even more. In he selecion phase, he bes IK soluion according o he opimizaion crieria is chosen o be included in he TBCO algorihm. Figure 6 shows a block diagram of he IK. The inpus o he algorihm are he robo kinemaic configuraion and he ask in space.

9 CARV Figure 6 ) Schemaic Diagram of he proposed Muli-soluion IK solver 3 Resuls The proposed TBCO algorihm was used o seek he mos suiable 3DOF configuraion for a single poin ask. Since i was assumed ha spherical wris is used, only posiion reachabiliy was considered. As was menioned earlier, Relaive Manipulabiliy was used as he opimizaion crieria. The desired ask poin is shown in Eq.(1.10). P OT R soluion = = (1.10) (1.11) The TBCO algorihm was se o run for 200 generaions. Figure 7-a shows he progress of he minimum Reachabiliy of each generaion wih respec o he generaion number. As can be seen, he Reachabiliy decreases o 7.5 ( 3% of he P OT ) wih he progree of he algorihm. I should be noed ha i is possible o gain even lower reachabiliies by increasing he maximum generaion number. The KMR of he robo ha could reach he leas reachabiliy in he las generaion is shown in Eq.(1.11). Figure 7-b shows he robo corresponding o his KMR. The black sphere close o he end-effecor of he robo is he desired ask poin. The Relaive Manipulabiliy of he robo in reaching he ask is a b Figure 7 ) a. Progress of he Reachabiliy wih respec o he generaion number b. The oupu of he TBCOalgorihm

10 10 S. Tabandeh, C. Clark, W. Melek 4 Conclusions and Fuure work A Task-based Configuraion Opimizaion based on a Geneic Algorihm was presened. The algorihm uses a Muli-soluion IK solver o calculae he finess value of Kinemaic Configuraions. The proposed TBCO algorihm uses a novel Selecion scheme. Moreover, modified Crossover and Muaion operaors o handle KMRs were explained. The algorihm was used o find a KC capable of performing a cerain ask while achieving a high relaive Manipulabiliy. To fully uilize he capabiliies and deficiencies of he algorihm, implemenaion of a larger se of opimizaion crieria is he nex phase of he research. Moreover, addiion of orienaion reachabiliy and expanding he algorihm o search in he space of n DOF Kinemaics configuraion for a larger se of ask poins could be anoher focus of he TBCO developmen. 5 References Chen IM (1994) Theory and Applicaions of Modular Reconfigurable Roboic Sysems. PhD hesis, Division of Engineering and Applied Science, California Insiue of Technology, Pasadena, CA, USA Chen IM, Yan G (2001) Closed-form inverse kinemaics solver for reconfigurable robos. Proc. of he IEEE In. Conf. on Roboics and Auomaion, pp Chen IM, Yang G (1998) An evoluionary algorihm for he reducion of dofs in Modular Reconfigurable Robos. Proceeding of he ASME Dynamic Sysem and Conrol Division, DSC-64, pp Leger C (1999) Auomaed Synhesis and Opimizaion of Robo Configuraions: An Evoluionary Approach. PhD hesis, The Roboics Insiue, Carnegie Mellon Universiy, Pisburg, Pennsylvania Kim JO (1992) Task based kinemaics of robo manipulaors. PhD hesis, Carnegie Mellon Universiy, Pisburgh, Pennsylvania , Augus Chocron O, Bidaud P (1997) Geneic design of 3d modular manipulaors. Proceedings of he 1997 IEEE In. Conf. on Robo. And Auo., Albuquerque, New Mexico Goldberg D (1989) Geneic Algorihms in search, opimizaion, and machine laearning. Addison-Wesley Harvey I (1994) chaper Evoluionary roboics and SAGA: he case for hill crawling and ournamen selecion, Arificial Life III, Sana Fe Insiue Sudies in he Sciences of Complexiy, Proc. Vol. XVI, pp Addison Wesley Harvey I (2001) Arificial evoluion: A coninuing SAGA. Proc. of 8h Inl. Symposium on Evoluionary Roboics (ER2001) Tabandeh S, Clark C, Melek W (2006) A geneic algorihm approach o solve for muliple soluions of inverse kinemaics using adapive niching and clusering. Proceedings of 2006 IEEE World Congress on Compuaional Inelligence, Vancouver, BC, Canada