Automatic Centralized Controller Design for Modular and Reconfigurable Robot Manipulators

Transcription

1 Automatc Centralzed Controller Desgn for Modular and Reconfgurable Robot Manpulators Andrea Gust and Matthas Althoff Abstract We address the problem of controllng modular robot manpulators. The challenge of modular-robot control s that the overall system dynamcs are unknown due to ts flexble composton from gven modules. Most prevous work has faced ths problem by desgnng decentralzed controllers. Smple decentralzed controllers do not guarantee global asymptotc stablty wthout knowledge of the overall system dynamcs and alternatve versons nvolvng communcaton wth neghborng modules result n complcated control concepts. Our approach s completely dfferent: we store parameters regardng the dynamcs and knematcs of each module and a unque dentfcaton number n tself. After fnshng the assembly of the modules, the parameters are gathered n a central controller, whch also detects the confguraton usng the dentfcaton numbers. Our centralzed controller uses ths nformaton to synthesze modelbased control laws on-the-fly as f the full system dynamcs are known beforehand. We ntroduce a novel and compact notaton to automate ths procedure and to generalze the dervaton of the knematc and dynamc model for heterogeneous modules. Fnally, a possble applcaton s shown usng smulatons. I. INTRODUCTION Reconfgurable and modular robot manpulators are mechatronc systems composed of nterchangeable modules. The modular nature of these systems allows them to be adapted for dfferent applcatons, whch s a clear advantage wth respect to fxed-structure robots. Ths s especally useful n ultra-flexble envronments, e.g. search and rescue operatons, space exploratons, servce robots and robots for human-robot cooperaton n manufacturng. In general, modular robots meet the dfferent needs of the users through reconfguratons. Furthermore, modularty s also useful for robot manufacturers to reduce the number of parts as dfferent robots can be assembled from a small set of modules. Ths makes t possble to provde a huge portfolo consstng of e.g. Selectve Complance Assembly Robotc Arm (SCARA) robots and anthropomorphc robots wth a few standard modules. In the last three decades several modular robot manpulators have been developed, for nstance the RMMS [], TOMMS [2], IRIS [3], PolyBot [4] and a sprng asssted reconfgurable modular robot [5]. The hgh versatlty of the modular and reconfgurable robot manpulators leads to several challenges, especally for the desgn of the control system. Consderng arbtrary confguratons of a nonunform set of modules, an enormous number of dfferent dynamc systems can be obtaned, challengng the control desgn. The authors are wth the Department of Computer Scence, Technsche Unverstät München, Garchng, Germany. Correspondng emal: {gust, althoff}@n.tum.de We present a new dea for controllng modular and reconfgurable robot manpulators based on dstrbuted data stored n each module usng a centralzed control approach. After assembly of the modules, the modular nformaton s collected by the central control unt, whch automatcally generates a centralzed and model-based control law wth guaranteed global asymptotc stablty. The control desgn for modular robot manpulators has been a longstandng problem n research. Most work has manly focused on decentralzed control approaches. Adaptve decentralzed control methods for modular robots are presented n e.g. [6], [7]. In the former the dynamc model of each subsystem s approxmated to cancel the couplngs and n the latter the parameters of dstrbuted proportonalntegral-dfferental (PID) controllers are adjusted on-lne. A decentralzed control method for modular and reconfgurable manpulators s developed n [8] based on jont torque measurements for automatc compensaton of the couplng effects. A decentralzed and robust control method s presented n [9] that ncreases the performance wth respect to a PID controller. A method for precson control of modular and reconfgurable manpulators wth guaranteed stablty s presented n [] usng the vrtual decomposton approach. A hybrd archtecture for centralzed and decentralzed operatons s proposed n [], where the authors consder the decentralzed approach for controllng the manpulator dynamcs. Even though satsfyng results have been acheved wth advanced decentralzed schemes, every decentralzed control method can be consdered as a specal case of a centralzed one. In prncple, t s always possble to desgn a centralzed controller wth performance equal to or better than a decentralzed control scheme. Especally when fast trajectory trackng s requred or drect drve actuaton s employed, centralzed control schemes are superor compared to decentralzed controllers snce subsystem couplngs can be compensated nstead of treatng them as dsturbance [2]. Moreover, centralzed archtectures are also benefcal for optmal control, complance and mpedance control [2], dynamc scalng of trajectores [3], and falure detecton [4]. Automatc generaton of complete models from modules has prevously been nvestgated n [5]. In [6] a centralzed control desgn s consdered. In contrast to our work, these approaches assume smlar modules wth symmetrc geometry and a central database storng the nformaton of all modules, whch s mpractcal as dscussed later. A prevous approach that consdered the storage of nformaton n the

2 modules s descrbed n [7], where the automatc dervaton of knematcs s consdered for jont modules wth smlar geometry. Our proposed approach s dfferent because we consder modules wth arbtrary knematc and dynamc parameters stored n each module. Our knematc modelng procedure s based on the Denavt-Hartenberg conventon, whch s related to the works n [8] [2]. In these works, t s requred to store homogeneous transformaton matrces n a central database, whch s not a requrement of our approach. We ntroduce a novel notaton and an extenson of the standard Denavt-Hartenberg conventon that smplfes the automatc procedure for obtanng the relatve parameters, especally when consderng prsmatc jonts. Moreover, our approach also consders dynamcs wth the fnal goal of automatcally desgnng model-based control laws. The man noveltes that ths work ntroduces are summarzed as follows: ) none of the cted approaches have consdered dstrbutng nformaton to the modules to enable and/or automate the centralzed and model-based controller desgn after nformaton collecton; ) our automatc controller desgn approach s generalzed and can thus be appled when modules are heterogeneous (e.g. wth dfferent shapes and dynamc parameters); ) we provde a systematc method to characterze modules usng a novel notaton and an extenson of the standard Denavt- Hartenberg conventon; v) our approach does not requre symbolc computaton for the automatc centralzed controller desgn of reconfgurable modular robot manpulators. The paper s organzed as follows. In Sec. II we descrbe the control problem for modular robot manpulators n detal. The proposed method s presented n Sec. III followed by smulaton results n Sec. IV. II. PROBLEM DESCRIPTION We consder a modular and reconfgurable robot manpulator composed of serally connected rgd lnks. Throughout ths paper we also assume that: a) each module ncludes a rgd proxmal part, a rgd jont, and a rgd dstal part (see Fg. 3); b) the connectors are standardzed and allow the assembly of two subsequent modules at only one relatve orentaton; c) the motor nerta effects are not consdered for a succnct presentaton of the materal snce ths does not affect the basc dea of our approach. For a concse presentaton of our method, we further exclude modules that do not ntroduce a new degree of freedom (e.g. lnk modules). The consderaton of such modules s trval, as t becomes evdent later. A reconfgurable modular robot manpulator wth N serally connected modules consttutes an open knematc chan and has the followng dynamc model (see [2]): M(q) q+c(q, q) q+f( q)+g(q)=u, () where q R N s the vector of the generalzed coordnates, M(q) R NxN s the symmetrc and postve defnte mass matrx, C(q, q) q (wth C(q, q) R NxN ) s the vector of the Corols and centrfugal terms, f( q) R N and g(q) R N are respectvely the vectors of frcton and gravty terms, fnally u R N s the vector of the actuaton forces/torques. We consder only vscous frcton wthout loss of generalty of our presented approach. The matrces and vectors of the dynamc model n () are dfferent for varous compostons of the manpulator and type of modules. We face the problem of automatcally generatng a centralzed controller n the jont space that guarantees global asymptotc stablty after a reconfguraton: lm e(t) =, t where e(t) = q r (t) q(t) s the Eucldean norm of the error vector n the jont space and q r (t) s the dfferentable (of class C 2 ) desred trajectory. We consder the development n tme of ths error as a performance ndcator. III. PROPOSED METHOD Our proposed approach s llustrated n Fg. : each module s frst characterzed accordng to our proposed notaton, whch s stored wthn the module. After the selecton of the modules and the manual assembly of the robot, the automatc controller desgn process starts. Frst, nformaton related to the knematcs, dynamcs and a unque dentfcaton number (e.g. the Meda Access Control (MAC) address) are collected from all modules by the central control unt. Next, the centralzed controller s automatcally generated to let the robot operate wth guaranteed moton control performance. Manual Assembly select modules Automatc Controller Desgn system n operaton Fg.. Characterzaton of each module and storage of the nformaton n tself nformaton collecton model-based controller generaton (automatc) Illustraton of the proposed approach. The nformaton of the modules can be stored n two ways: ) usng a centralzed database wth stored nformaton of all the modules that can be used; ) usng dstrbuted memory to store nformaton of a module n the module tself. Obvously, the latter opton s preferred snce the frst one requres database updates when new modules are created and addtonally requres storage of all possble modules (leadng to storage and legacy ssues).

3 We propose a tree structure as a sutable topology of the communcaton network for modular robotc manpulators wth an open knematc chan. The tree-structured network supports seral and branch-structured manpulators and s composed of a coordnator assocated wth the central control unt, routers assocated wth the ntermedate modules and end-devces assocated wth the end effectors. In ths structure, each module allows the coordnator to get measurements (e.g. jont poston and velocty), set nput commands to the actuators and set/get data n/from ts local memory database (e.g. dynamc and knematc parameters and type of jonts). Moreover, collectng addtonal nformaton (e.g. the routng tables) makes t possble for the central control unt to detect the robot confguraton. Approprate nformaton from the modules enables the central control unt to automatcally compute the forward knematcs based on an extended Denavt and Hartenberg conventon as s descrbed n III-A and the dynamc model wth the recursve Newton-Euler method as presented n III-B. Fnally, model-based control laws are automatcally syntheszed as dscussed n III-C. A. Forward Knematcs usng Modular Informaton The forward knematcs allows obtanng the pose of the end effector (poston and orentaton) gven the jont postons. For an open knematc chan the forward knematcs s commonly derved n a recursve way, by multplyng the homogeneous transformaton matrces that relate the frame of reference of each lnk to the prevous one. A systematc method for the assgnment of the frame of reference of each lnk s the Denavt-Hartenberg (D-H) conventon [2], whch we brefly recall. Consder that the th lnk of the robot has a body-fxed frame of reference wth axes x, y and z. Accordng to the standard D-H conventon (see Fg. 2), z s algned wth the axs of the jont at the connecton wth lnk +. The x axs s chosen along the common normal between the axs z and z pontng toward the lnk +. The orgn of the frame s set at the ntersecton of the common normal wth z. Fnally, the y axs completes the rghthanded coordnate system. Once the frames of reference are assgned, four parameters are ntroduced to defne the relatve transformaton of coordnates: a, d, α, θ. As shown n Fg. 2, a s the dstance along x between the orgn of the frame and, d s the dstance along z between the orgn of the frame and, α s the angle between the axs z and z around x and θ the angle between x and x around z (the angles are postve counterclockwse). The only varable parameter s d when the jont s prsmatc and θ when t s revolute [2]. Consderng that our proposed approach s automatc and that the standard D-H conventon s not unque, we extend t to hane some partcular cases that need specal consderaton (see Fg. 2): when the z axes ntersect, the x unt vector s obtaned from the cross product between them; Fg. 2. n PJ y y p d z x o o z x lnk a θ y α o n z Representaton of a lnk showng the parameters for knematcs. when the z axes are parallel the x unt vector s set along the common normal between them and the orgn o s set at the jont connecton PJ ; when the z axes are supermposed the x unt vector s algned wth x and the orgn o s set at the jont connecton PJ. In order to automate the dervaton of the four D-H parameters for each lnk (D-H table), we ntroduce two addtonal parameters assocated to each lnk. These are obtaned on the bass of the lnk geometry: p and n, and are the z coordnates of the pont PJ from o and of the pont PJ from o, respectvely (see Fg. 2). Usng these two parameters, d can be recursvely computed as: d = n p (revolute jont), x PJ d = n p + q (prsmatc jont), where q s the jont dsplacement. Addtonally, n order to capture the constant angular offset that can be present between x and x when the jont s n ts zero poston and when t s prsmatc, an addtonal parameter γ s ntroduced. Wth ths addton, the D-H parameter θ s: θ = γ + q (revolute jont), θ = γ (prsmatc jont). In the followng, we ntroduce a notaton to characterze arbtrary modules and the automatc procedure to obtan the parameters of the extended D-H conventon (a, α, p, n, γ ) for each lnk of an assembled manpulator. Let us consder the exemplary module shown n Fg. 3(a). Ths module s composed of an nput connector, a jont and an output connector. To defne the new notaton, we frst have to decompose the module nto a proxmal (pl) and a dstal part (). These parts are shown n Fg. 3(c) and Fg. 3(b), respectvely. In ths descrpton, we consder modules wth only one degree of freedom wthout loss of generalty because more complex modules can be modeled as a seres of those consdered, by settng the relevant parameters to zero. The frst step to characterze the module s to fx a frame of reference for the proxmal part and the dstal part. These two frames are located n the center of the nterfaces of the respectve connectors. The axes x n and x out are postoned

4 p pl Input connector o n z n yn x n o pl z pl y δ pl pl x pl (b) a pl α pl PJ y z pl o pl p Jont (a) z x o y pl x pl PJ δ j a n pl Output connector y out z out n α z y (c) o x o out δ Fg. 3. Knematc notaton for module characterzaton. The connectors are ndcated n lght-grey color, (a) s the entre module, (b) the dstal part and (c) the proxmal part. along a unque and standardzed drecton on the connecton plane. The z axes are normal to the respectve connector planes, z n ponts n towards the nput connector and z out ponts outwards from the output one (see Fg. 3). The axes y n and y out are selected to complete the respectve rghthanded frames of reference. Accordng to our notaton we characterze both the proxmal and the dstal part wth a set of parameters. To obtan these parameters, the same approach of the extended D-H conventon descrbed prevously s appled. Accordngly, we obtan four parameters for the proxmal part: a pl, α pl, p pl, n pl, and for the dstal part: a, α, p and n. Regardng the proxmal part llustrated n Fg. 3(c), two auxlary frames are consdered. The frst one has ts orgn at o pl, the ntersecton of the common normal between z n and the jont axs wth z n. The axs x pl s set along the common normal pontng toward the dstal part, z pl s set along z n and y pl completes the rght-handed frame of reference. The second auxlary frame has ts orgn n o pl at the ntersecton of the common normal wth the jont axs, ts axs x pl s set along the common normal and ponts toward the dstal part, z pl s set along the jont axs and fnally y pl completes the rght-handed frame of reference. The four parameters for the proxmal part have the followng meanngs: a pl s the dstance between o pl and o pl along the common normal; x out α pl s the angle between the axs z n and the jont axs around x pl ; p pl s the z coordnate of the nput connecton pont o n from o pl ; n pl s the z coordnate of the jont connecton pont PJ from o pl. For the dstal part llustrated n Fg. 3(b), two auxlary frames are analogously ntroduced. The four parameters for the dstal part have smlar meanngs: a s the dstance between o and o, along the common normal; α s the angle between the jont axs and z out around x ; p s the z coordnate of the jont connecton pont PJ from o ; n s the z coordnate of the output connecton pont o out from o. Fnally, three addtonal parameters are requred: δ pl, δ, δ j. As shown n Fg. 3(c) and (b): δ pl s the angle between x pl and x n, δ s the angle between x and x out, and fnally δ j s the angle between x pl and x when the jont s n ts zero poston (all the angles of ths notaton are postve counterclockwse). Partcular cases of the relatve orentaton between the z axes (e.g. parallel, ntersect or overlap) are haned smlarly to the prevously descrbed extenson of the standard D-H conventon. The connecton ponts to consder are: o n and PJ (proxmal part); PJ and o out (dstal part). We collect the parameters requred to characterze a module for knematcs n Tab.I. TABLE I INFORMATION STORED IN EACH MODULE FOR KINEMATICS. Proxmal a pl α pl p pl n pl δ pl Dstal a α p n δ Jont δ j Jont type It s worth notng that the same approach can be appled to modules wth multple nput and output connectors. In those cases, a set of parameters for each realzable combnaton of the connectors s requred and has to be stored n the module. When the nformaton s forwarded to the central control unt, the set of parameters correspondng to the connectors n use are sent. Usng the proposed notaton, the dervaton of the parameters of the extended D-H conventon for each lnk (a, α, p, n, γ ) can be automated. Let us assume that the th connecton between a module and a module s establshed. In order to obtan the parameters for the lnk, we requre the homogeneous transformaton matrx F of a frame orented as the D-H one located at PJ (see Fg. 2) wth respect to a frame parallel to the frst auxlary frame of the dstal part of the module wth orgn PJ (see Fg. 3(b)). From now on, we use a compact notaton to ndcate operatons wth homogenous transformaton matrces. For nstance: T k ( )/R k ( ) are homogeneous transformatons that represent

5 the translaton/rotaton along/around the k axs. Usng the parameters that characterze subsequent modules, we frst calculate an auxlary matrx F. Ths matrx descrbes the transformaton of a frame wth orgn at PJ, parallel to the second auxlary frame of the proxmal part of module, wth respect to a frame parallel to the frst auxlary frame of the dstal part of module and wth orgn at PJ. It s calculated as: F =T z ( p )T x(a )R x(α )T z(n ) R z (δ δ pl )T z ( p pl )T x (a pl )R x (α pl =[ R U T )T z (n pl ) ]. (2) In order to complete the synthess matrx F, an addtonal possble rotaton φ around z s consdered. In fact, we need to algn the x axs accordng to our extended D-H conventon. Wth the transformaton of (2) we only reached a frame located at PJ wth the orentaton of the second auxlary frame of the proxmal part of module. An addtonal rotaton around z may be requred to brng ths frame parallel to the D-H one. In order to fnd ths angle, we have to consder three possble cases for the jont axes. The detecton of the case that occurs s performed usng U, and the unt vector of the z axs n R, whch contans the nformaton of the relatve orentaton of two subsequent jont axes.. When the axes overlap: φ =.. When the axes are parallel: φ = arctan(v y /v x ), from V= [ ] T v x v y v z = R T U.. When the axes are skew or ntersect: φ = arctan(v y /v x ), from V= [ ] T v x v y v z = R T V and V = z z. The synthess matrx s completed as: F = F R z (φ )= r x s x t x u x r y s y t y u y r z s z t z u z. (3) The same transformaton of F can be expressed n terms of the parameters of a lnk and an auxlary rotaton (φ ) as: L =T z ( p )R z (φ )T x (a )R x (α )T z (n ) C φ C α S φ S α S φ a C φ + n S α S φ = S φ C α C φ S α C φ a S φ n S α C φ S α C α n C α p, (4) where S ε /C ε are the abbrevatons of sne/cosne of an angle ε. Equatng (4) and (3), the parameters of our extended D-H conventon for the th connecton (or lnk) can be nferred: a = u x r x + u y r y, α = arctan(s z /t z ), φ {}}{ γ = arctan(r y /r x )+δ j φ. If s z : If s z = : n = (u x r y u y r x ) s z, p = (u x t z r y u z s z u y t z r x ) s z. n =, p = u z. B. Dynamc Model from Modular Informaton The problem of dervng the dynamc model of a rgd lnk manpulator s usually solved ether wth Lagrangan formulaton [22] or recursve Newton-Euler (N-E) methods [23]. A comparson of the methods for the dervaton of the equatons of moton of a rgd lnk manpulator s n [22]. A computatonally effcent varant of the Newton-Euler method has frst been proposed n [23] wth a complexty that grows lnearly wth the number of the jonts ( O(N) ). Subsequently, enhanced versons of ths algorthm have been presented n e.g. [24], [25]. A revew on methods for robot dynamcs can be found n [26]. The recursve N-E algorthm studes the manpulator lnk by lnk and s composed of two recursons, one for knematcs (forward recurson) and one for forces and torques (backward recurson). It can be used ether wth symbolc computatons to obtan the closed form dynamc model or wth numercal ones for control as descrbed later. We consder the recursve N-E algorthm of [2] wthout motor nerta effects. Smlar to the knematcs, we provde a systematc approach to characterze the modules for dynamcs. The set of requred parameters for each module s presented as well as the procedure to synthesze them for each establshed connecton. For each lnk of the manpulator (see Fg. 4), a body-fxed frame of reference wth orgn n D s consdered. We consder ths frame to have the same orentaton as the D-H one so that the results of the forward knematcs are exploted to nfer relatve orentaton of subsequent lnks. Wth reference to the generc lnk of Fg. 4, the vector from D to D s r D,D and the vector from D to C (the center of mass) s r D,C. In addton to these vectors, the mass m, the nerta tensor I and the vscous dampng coeffcent of the jont FJ are also requred for the algorthm. The superscrpt of all the vectors ndcates n whch frame of reference they are expressed. Smlarly to the knematcs, a synthess procedure s requred for the dynamcs based on modular nformaton. A graphcal representaton of the th connecton nvolvng dynamc parameters s shown n Fg. 5. In our approach, the requred parameters for dynamcs are expressed n the output frame for the dstal part and n the nput frame for the proxmal part. For each module, beyond the mass for both the dstal (m ) and the proxmal part (m pl ), we requre ther nerta tensors: I out, I n pl, and the coordnates of the centers of mass: rc out, rn Cpl, expressed n the output and the nput frame, respectvely.

6 Fg. 4. PJ z D r D,D C r D,C y z x D Representaton of a lnk for dynamc related parameters. r out C C th connecton o n x n z n z out y out y n o out x out r n Cpl Cpl PJ S(U)= u z u y u z u x u y u x, wth U= [ u x u y u z ] T. The last parameter that needs to be syntheszed for each consttuted lnk s r D,D. Also for ths vector we can use homogeneous transformatons wth the syntheszed knematc parameters: A A and fnally = T z ( p )R z (γ + q )T x (a )R x (α )T z (n ) (rev. jont), = T z ( p + q )R z (γ )T x (a )R x (α )T z (n ) (prsm. jont), [ A =[ A ] R = U T ], r D,D = U. We store the addtonal parameters n each module for the dynamcs as collected n Tab. II. TABLE II INFORMATION TO STORE IN THE MODULES FOR DYNAMICS. Fg. 5. Representaton of a connecton nvolvng parameters for dynamcs. Connectors are ndcated n lght-grey color. Usng ths notaton, when the th connecton s realzed, the output frame of module and the nput frame of module match. We denote the matched frame as o. The synthess of the total mass of the lnk m, the coordnates of ts center of mass rc o and the nerta tensor I o can thus be automated: m = m + mpl, I o = I out + I n pl, rc o = m rout C + m pl rcpl n. m Our proposed automatc procedure transforms these quanttes to be expressed n a frame parallel to the D-H one and located at D for each realzed lnk (see Fg. 4). Snce the knematc parameters are known, these transformatons of coordnates are performed usng homogeneous transformatons and Stener s theorem [2] for the nerta tensor. The coordnates of the center of mass are computed as: where A o [ r D,C ] = [ [ A o ] r o C = R z ( δ pl )T z ( p pl )T x (a pl )R x (α pl [ R o = U o ] T. ], )T z (n pl )R z (φ ) The nerta tensor n computed as: ( ) I = R ot I o m S T (rc o )S(rC o ) R o +m S T (r D,C )S(r D,C ), where S( ) s an ant-symmetrc matrx: Proxmal m pl I n pl rcpl n Dstal m I out rc out Jont FJ C. Centralzed Controller Desgn The procedures descrbed for automatc dervaton of knematcs and dynamcs enable the automatc generaton of model-based control laws usng the modular nformaton n Tab. I and II. Our proposed centralzed controller for modular and reconfgurable robot manpulators takes as nput the poston vector (q), the velocty vector ( q), the reference trajectory (q r, q r, q r ) and fnally the array of structures contanng the nformaton of the modules: ModRob. The array of structures ModRob contans the nformaton of Tab. I and II for each module. The output s the closed loop force/torque command, whch guarantees global asymptotc stablty. In order to realze the model-based control laws, we have mplemented a functon u=ne mod (q, q, q,modrob), that performs the recursve N-E algorthm usng ModRob as descrbed n Sec. III-A and III-B and returns the force/torque commands. We have mplemented the automatc generaton of two model-based control laws: proportonal and dervatve (PD) control wth gravty compensaton and computed torque control. Computed torque control s one of the most effectve centralzed and model-based control methods for robotc manpulators wth rgd lnks. Assumng perfect knowledge of the system dynamcs, computed torque lnearzes the model of () by compensatng the nonlnear and couplng terms through feedback. The control law s completed by assgnng an nput (υ) to the lnearzed system that guarantees asymptotcally stable dynamcs of the error n jont space [27]. The PD control wth gravty compensaton [2] s sutable for pont to pont moton and s desgned to compensate the gravtatonal terms of the dynamc model and to nclude

7 x n z n z out x out z out z out x z n out z n x n a b c x out x n q:4(t) (rad) ).8 Module.8 Module Module 3.4 Module Module tme (s) q5(t) (m) Fg. 6. Representaton of the modules for llustratve purpose used for smulatons. Connectors are ndcated n lght-grey color. TABLE III INFORMATION OF THE MODULES USED FOR SIMULATION. Parameter a b c a pl (m) α pl (rad) π/2 p pl (m)... n pl (m) δ pl (rad) a (m) α (rad) π/2 p (m)..5 n (m). δ (rad) δ j (rad) π Jont Type Rev. Rev. Prsm. rcpl n (m) [.5] T [.5] T [.5] T m (kg) I n ( pl kgm 2 ) (dagonal) [2 2.65] [2 2.65] [2 2.65] rc out (m) [.5] T [.5] T [.25] T m (kg) I out ( kgm 2 ) (dagonal) [2 2.65] [2 2.65] [ ] FJ (Nms) ( t f n e(t)/q r tme (s) Fg. 7. Smulaton usng PD wth gravty compensaton controller. The dashed lnes represent the desred trajectores. q:4(t) (rad) ) ( t f n e(t)/qr.8 Module.8 Module Module 3.4 Module Module tme (s) tme (s) Fg. 8. Smulaton usng computed torque controller. The dashed lnes represent the desred trajectores. q5(t) (m) a lnear feedback control law for postons and veloctes. Assumng perfect knowledge of the gravtatonal term, PD control wth gravty compensaton guarantees global and asymptotc stablty for any desred constant equlbrum posture [2]. We obtan the compensatng commands for the control laws numercally by varyng approprate numercal arguments to the functon NE mod. For example, the numercal vector of the gravtatonal terms used for PD control wth gravty compensaton s computed as g(q) = NE mod (q,,,modrob) and the closed loop command of the computed torque control s computed numercally as u cl = NE mod (q, q,υ,modrob) where q, q are the current numercal vectors of postons and veloctes. IV. SIMULATION RESULTS A pont to pont moton n the jont space s smulated for a confguraton usng a gven set of modules. We show and compare results of the automatcally generated control laws: PD control wth gravty compensaton and computed torque control. Gaussan measurement nose and nput dsturbance are ncluded. We consder these dstrbutons wth zero mean and standard devaton of σ n =.3 for measurements of jont postons (rad and m) and veloctes (rad/s and m/s) and σ u =.3 for the nput forces/torques (N and Nm). We perform the smulatons usng three knd of modules shown n Fg. 6: a, b (revolute jonts) and c (prsmatc jont). The parameters of the modules used for these examples are collected n Tab. III. Usng these modules we buld a confguraton composed of fve modules: [b-a-b-a-c]. The trajectory of each jont s desgned wth a quntc polynomal. The coeffcents have been selected to guarantee zero ntal/fnal velocty and zero ntal/fnal acceleraton. The fnal tme of the planned moton s t f n = s wth a startng tme t n = s. The jonts start from zero to reach the fnal postons: q(t f n )=[.2,.4,.6,.8,.] T. The smulatons are performed usng the followng gan matrces [2] of the controllers: K p = 25I and K d = 3I, where I s the dentty matrx of proper dmensons. As shown n Fg. 7 and Fg. 8, computed torque control has better performance observng the development n tme of the Eucldean norm of the error. In these fgures, we show the error vector as a fracton of the respectve fnal postons of the trajectores for normalzed presentaton n one plot. V. CONCLUSIONS We present a systematc and effectve approach for centralzed control of modular and reconfgurable robot manp-

8 ulators. To the best knowledge of the authors, none of the approaches presented n the lterature so far have consdered dstrbutng nformaton to the modules to automatcally generate a pure model-based centralzed controller, especally when consderng heterogeneous modules. Ths dea paves the way for future modular and reconfgurable robotc systems wth performance and capabltes that approach those of fxed structure ones from a control perspectve. Havng the possblty to auto-generate a model-based centralzed controller wth guaranteed asymptotc stablty for a non-unform set of modules, the central control unt does not requre a large database and successve updates. Fast reconfguraton wth guaranteed stable operatons and moton control performance ensures hyper-flexblty of the robotc system. A drawback of ths approach compared to conventonal methods for controllng modular robot manpulators s the necessary storage of nformaton n the modules that may ncrease manufacturng costs. Addtonally, f self-detecton of the confguraton were requred, network solutons that support a tree-structured topology would be needed. A communcaton bus (e.g. CAN bus) could also be sutable but addtonal communcaton lnes have to be employed as descrbed n [7]. However, we beleve that the cost savngs from standardzaton of modular robots would outwegh such cost ncreases. Furthermore, the automated generaton of the controller wth guaranteed moton control performance would reduce the tme and costs requred for commssonng the robot after assemblng or reconfguraton. An mportant contnuaton of ths work s the ntroducton of robustness for uncertan modular nformaton, elastcty n the jonts and measurement nose, especally because n future works ths control approach wll be mplemented on a real robot. 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