Model Based Control versus Classical Control for Parallel Robots

Transcription

1 odel Based Control versus Classical Control for Parallel Robots Květoslav Belda Institute of Inforation heory and Autoation, Acadey of Sciences of the Czech Republic, Pod vodárensou věží 4, Praha 8 - ibeň. fa: and e-ail: belda@utia.cas.cz. Abstract. Earthly civilization is a cople syste based on aount of different relationships. Anybody is not alone. Always, there are soe actions that cause soe reactions. It happens on different levels. In general, a syste can be analyzed fro outside as one object with hoogeneous properties - global level. he opposite way is analysis fro inside - local level. et us tae into account one eaple of such analysis applied to huan wor. Specifically, let us focus on coparison of entioned levels, investigated in control probles of industrial robotics: siple clear local control versus global sophisticated control approach. his paper outlines the ideas and solutions obtained during design of the control for specific parallel robots. It is deonstrated by coparative siulation of classical PID/PSD controllers - local level and discrete odel-based predictive controllers representing global level. Nevertheless, the conclusions of the paper universally show the properties of global and local view on different systes and they are being generalized. Keywords: PID/PSD feedbac controllers, predictive controllers, parallel robotic structures. Introduction Nowadays, the further developent in industrial area is constrained by deficit of powerful achines with adequate dynaics and stiffness (sai 999. Utilization of parallel robots controlled by suitable control algoriths sees to be proising way to iprove dynaics, stiffness, accuracy and productivity of achine tools and their centers, which, together with assebly lines, for the bacbone of ass production. he parallel robots parallel structures (Belda 2002 can be siply understood as ovable truss constructions or as ovable wor platfors supported by several parallel ars. he structures are driven by different drives, in alost cases by electrootors. Siple coparison of serial and parallel structures is shown in Figure. stiffness dynaics adequately actuated dynaics stiffness actuated redundantly enable to solve additional conditions serial structures parallel structures Figure. Coparison of serial and parallel structures. he serial structure has the drives in each joint. he drives in parallel configuration are located in the fied basic frae.

2 he fundaental tas of the control of echanical structures driven by electrootors is to design appropriate control actions, which accoplish required oveent of the ovable platfor. In general, the schee in Figure 2 is solved. electrical quantities OOR echanical oent Gears echanical oent ROBO (parallel structure Result ( position of the structure and otors conversion to electrical quantities echanical oent local level CONROER global level Requireents (required trajectory recalculation of otor coordinates to global Cartesian coordinates (Direct ineatic transforation Figure 2. Conceptual schee of control circuit for local (independent and global (centralized odel-based level. here are two levels of the control (Belda hey are shown in Figure 2: local and global. ocal (decentralized level, represented by continuous PID or discrete PSD controllers, controls each drive independently, without any consideration of utual relations. he global (centralized level, represented by predictive controllers (Belda et al. 2002, uses for control the dynaic odel of the robot structure. he odel can include also drives, which represent also appreciable dynaics. he coplete odel is used not only for control design, but also for siulation of a real object - robot structure and it can be written in different fors. In coparison, the classical PID/PSD controllers design the control action only fro feedbac (Souče 997. hey wor up only output inforation included in feedbac regressively without any assuptions to future. On the other hand, the odel-based approach cobines feedforward and feedbac together (Böh et al he odel represents prior inforation that is used for feedforward foring of control actions. Subsequently, the feedbac fro easurable robot outputs copensates odel inaccuracies and certain bounded disturbance. Such design is ore optial and control action can react faster and ore effective than PID/PSD structures. In this paper, firstly, the standard odel of the robot is defined (Stejsal et al.. hen, in the ain part, the theoretical bacground of the control design is introduced. It includes new possibilities to utilize the properties of the parallel robots. Finally, the designed control is deonstrated by coparative siulation of the PID/PSD controllers and discrete odel-based predictive controllers. Predictive controllers are eplained for echanical and global odel including additionally odel of drives. In this paper, for siplicity DC otors are used. hus, let us start with suitable robot odel definition. 2 odel for siulation and for control design 2. echanical odel he robot structure is ideally considered as a syste of rigid bodies. In such case, the classical equations of otion can be coposed e.g. by utilization of agrange s equations. If the obtained odel includes dependent coordinates, then they ust be reoved via nowledge of the null space of the robot jacobian (Stejsal et al. 996, Belda Finally, the resultant independent coordinate syste (in case of parallel robots, Cartesian syste is the following: R R & R R & & = R g R u, ( where is a ass atri, g is a vector of right sides, is a redistributional atri, R is a Jacobian atri, is a robot output (coordinates of a ovable wor platfor, = [ E, y E, ψ] and u is an input vector.

3 he odel ( is a nonlinear syste of ordinary differential equations (ODE, and can be rewritten to different fors according to the requireents of appropriate control approach. Since the robot (ultibody syste represents relatively tie-consuing coputation of eleents of the odel (, the discrete approach for global odel-based strategies is used. It siply respects the tie constraints. he odel in the for ( can be used for siulation of echanical structure on siple local level. hus, let us continue with preparation of the odel for global level - for predictive controllers. As entioned previously, we need transfor the odel to discrete tie description. Firstly, to easy describe the structure, we odify the equation syste (: R R & R R & & = R g h, h = R u (2 a, b and siplify it to the general for of echanical systes & = f (, & g( h (3 hen, the siplification (3, transfored to the state-space forulation, is in a for (4 X& = f ( X g( X h, (4 = C X where the state vector consists of positions and velocities X =, & ] = [, ]. [ 2 (Note: o preserve the traditional control notation in the following tet, the sybol u is used instead of h. Now, the forula (4 can be linearized (Valáše et al. 999 and discretized: for absolute algorith of predictive controller: X ( = A X( B u( ( = C X( or for increental algorith of predictive controller (u( = u( u(: X( = AX( Bu( B u( ( = CX( (5 (6 he basis of all predictive controllers is a prediction of new unnown output values fro actual topical state X in considered horizon of prediction N. he prediction of can be defined for both algoriths in coon for: = f Gu, (7 where is a vector = [ (, ( 2,, ( N] (8 for absolute algorith ū is a vector u = [ u(, u(,, u( N ] (9 CA f = X(, N CA C G = N CA B O and for increental algorith, with u( = u( u(, ū is defined as CA f = CA N B 0 (0 CB u = u = [ u(, u(,, u( N ] ( C B C B 0 ( = X u(, G O (2 N N C( A A B C( A A B CB

4 2.2 Global robot odel o obtain the global odel of the robot, a odel of the drives (electrootors ust be added to the odel ( or to its discrete fors (5 or (6 respectively. As entioned above, the siple brushes DC otors are considered. Such otors, having peranent agnets in stator, are described by ordinary differential equation of second order (Souče 997: R & & 2 u& J =, (3 where and 2 are torque and speed constants, R and is terinal resistance and inductance and J is rotor inertia. Equation (3 corresponds with the schee in Figure 3. DC otor echanical torque on the rotor supply voltage u R s J s angular velocity of the rotor ω internal induced 2 voltage u e Figure 3. Bloc schee of brushes DC otor with peranent agnet in the stator. In equilibriu case (ω = 0, the order of equation (3 is reduced to first order: & R = u (4 In transient process, the necessary supply voltage u is a function of load (required torque u r ( and angular velocity u e (ω of the rotor. he global odel of whole robot can be coposed as follows: description of echanical structure: odel of the otors: & = f (, & g( (5 & R = ur (6 and oreover, the copensation of internal induced voltage u e : u = u ( ( = ( ϕ& r ue ω ur (7 2 In general, the state-space forulations (5 and (6 changed only in details. he syste of equations is only etended by equations of otors and the state vector will include one internal state ( ore X = & = (8 [,, ] [, 2, 3] Finally, the epression for prediction can be written also as equation (7. Now, we can provide fullyvaluable control design.

5 3 Siple local and odel-based predictive control 3. ocal PID/PSD control he siplest control approach considered eans taing the robots and anipulators, powered by group of independent drives /actuators/, separately controlled, as a set of single input - single output systes (setsiso. utual interactions aong all drives, caused by different positions during the robot oveent, are included as disturbances entering each single syste constituting the robot. In that view, the classical PID/PSD feedbac control schee can be used. If it is applied, serious proble of utual conflict of drives ay occur (Belda It is indicated by unpredictable increase of integral/su (I/S channels in a controller, caused by the fact, that ineatic description is never perfect. It eans, that the description does not represent eactly the real ineatics of redundant parallel structure that is given by production and partly by topical technological conditions. oreover, in case of drive redundancy, there eists no unique transforation between coordinates of drives φ and independent coordinates here. he PID/PSD controllers try to achieve zero errors for all dependent drive coordinates φ, but it is soeties ipossible. his fact causes the increase of I/S channels in controllers to saturation. Idea of the solution is the following: local decentralized controllers copute agnitudes of actuators u for drives and then soe operation as a certain projection is applied to these agnitudes (Belda he projection transfors the actuators to independent space (i.e. it coputes so-called general force effects, where the undesirable effects are eliinated, and consecutively the inverse projection recoputes the effects bac. 3.2 odel-based predictive control he derivation of predictive control uses the prediction given by (7. In this paper, the root for is considered. In real coputation, it needs atries with saller diensions and if the penalization λ is nonzero value, it eeps redundant properties (if they eist; see Figure. oreover, it can accoplish soe additional control requireents (e.g. antibaclash condition, soothing of torques etc.. o start derivate Predictive Control in the square - root for, let us define quadratic cost function J = ε {( w ( w u λ λ u} (9 whose iniization leads to solving the syste of algebraic equations: A u b = 0 (20 For solution, the orthogonal triangular decoposition is used (awson et al It reduces atri A and vector b to upper triangular atri R and vector c as follows: A u / = b Q (2 A u b R u c = 0 c z (22 Vector c z is a residuu vector, whose Euclidean nor c z is equal to the square root of cost function (9. =

6 Solving the upper part of the syste (22, we obtain either values of supply voltage u r ( or fictitious general force effects u = h or its increents u = h, respectively. he values of the voltage can be directly applied (Figure 4. In the second case, when the increental algorith is used, the final force effects are given by epression h( = h( u( (23 u( = h( o obtain the real values of actuators u, the equation (2 b ust be solved based on fictitious force effects h for otor torques u (return to initial notation. he solution ay not be unique. In general, it represents deficient ran syste, where pseudo-inverse operation can be applied (awson 974. For coputation, the values of torques, supply voltage, positions etc. are taen into account in basic units of the syste SI ( [N], u [V], = [ [], y [], ψ [rad]]. (Note: In Figure 4 and also in the following schees of control circuit, the odel of the robot includes also oent of inertia of the rotor of DC otor J [g -2 ]. 2 2 inverse ineatics d d && Predictive controller (GPC && = f (, & g ( R & = u r s R odel of DC otor ODE OF HE ROBO && & = f (, & g ( Fig. 4. Schee of control circuit with predictive controller applied to global odel. Now, let us focus on structure of control shown in Figure 4. here are several feedbacs there. Position feedbac (state variable is the ain; current (torque and velocity feedbacs are inherent and auiliary feedbacs respectively. Inherent current feedbac represents channel of internal state variable ( = f(current i [A]. Auiliary velocity feedbac (state variable & serves the coputation of eleents of echanical odel and provides copensation of internal induced voltage u e. All entioned state variables, & and are assued to be available.

7 4 Description of the bloc schee in SIUINK environent he Figure 5 shows parallel siulation of the si different structures of controllers applied to echanical odel of the robot - equations of otion (. trajectory data [,CS(:,:9] Fro Worspace Xd ( Selector U(E U U(E U Selector y psifi -4 y psifi Fi -4 y psifi -4 y psifi -4 Fi -4 Fi -4 2 Fi -4 3 PSD controller PSD IndPar IRF PSD controller PSD IndPar2 IRF SfunGpc Abs GPC Predictive Controller 0 u4 uopt4 y psi Int red o Wsp4 usir u4 uopt4 ypsi upsd upsd o Wsp sir SodRob ROBO odel u» [,d]' o Wsp sir Int red SodRob o Wsp5usiR ROBO odel u» [,d]' o Wsp2 sir2 SodRob o Wsp6usiR2 ROBO odel u» [,d]'3 80/pi coputed 80/pi coputed 80/pi3 coputed3 coputed coputed coputed2 In In2 In3 Out Out SfunGpcD Delta GPC Predictive Controller SfunGpcF Abs GPC forces Predictive Controller SfunGpcDF Delta GPC forces Predictive Controller o Wsp7usiR3 o Wsp9usiR4 o WspusiR5 o Wsp3 sir3 SodRob ROBO odel u» [,d]'4 o Wsp8 sir4 SodRob ROBO odel u» [,d]' o Wsp0 sir5 SodRob ROBO odel u» [,d]'2 80/pi2 coputed2 80/pi4 coputed4 80/pi5 coputed5 coputed3 coputed4 coputed5 In4 Out3 In5 Out4 In6 Subsyste u(-4 for trajectory data loading for trajectory preview for qualitative results Figure 5. SIUINK schee for parallel coparative siulation of PID/PSD controllers and predictive controllers; control circuit PID/PSD without copensation of I/S channels; 2 PID/PSD with copensation of I/S channels; 3 and 4 predictive controllers generating the torques directly; 5 and 6 predictive controllers generating control torques via generalized force effects ( 3 and 5 absolute algoriths; 4 and 6 increental algoriths.

8 he following Figure 6 deonstrates predictive controller, which considers the global robot odel (5-(7, i.e. it taes into account the odel of the drives (DC otors. [,CS(:,:9] Fro Worspace Xd ( for trajectory data loading for trajectory preview for qualitative results /K2 /K2 /K2 /K2 K.sR ransfer Fcn K.sR o Wsp2 ROBO odel u» [,d]'3 sir Fi -4 dfi -4 Fi -4 y psi d dy dpsi dfi -4 fi-4 coputed 80/pi2 e e SfunGpcotory Abs GPC e u [V] i error Selector2 error w - y e(,y, ^2y^2,psi e usi o Wsp4 Deu u(-4 [V] ransfer Fcn2 K.sR ransfer Fcn3 K.sR ransfer Fcn4 e SodRob u(-4 [N] o Wsp3 usir e e e coputed dfi-4 coputed 80/pi3 coputed Figure 6. Control circuit with global robot odel. he siple ipleentation in SIUINK environent is provided by classically structured S-functions (predictive controllers - SfunGpc*., odel of the robot - SodRob.. 5 Coparative eaple fro siulation Predictive Control (drive - 4 PID/PSD (drive Figure 7. ie histories of four inputs (four torques during parallel siulation of the bloc schee shown in Figure 4. he tie histories (Figure 7 were recorded during the otion along S - shaped trajectory. Figure 7 illustrates all predictive control approaches, with both echanical and global odel. he results of predictive controllers loo siilar, but in real ipleentation, are different.

9 Fro practical point of view, the setting of paraeters of PID/PSD is not siple tas. he Figure 7 illustrates the properties of PID/PSD control deducing the actions only fro the feedbac, which can not be energetically effective. At the sae tie, the Figure 7 shows the advantages of the cobination of feedforward action odulation with feedbac inaccuracy and disturbance copensation ore effective control way with lower energetic deands. 6 Conclusion he paper deals with a set of possible ways of control and their coparison. It shows that the classical approaches are energetically consuing. hey should be, in a future, replaced by such strategies e.g. strategies based on predictive controllers, which consider the ost of the available inforation on given controlled object, in this paper, robotic structure driven by DC otors. Fro general point of view the paper shows that the different levels of analysis, both local and global, are iportant. However, it is necessary to deterine the adequate ratio of both levels. In view of paper subject, if we control individual otor used as only one individual drive of the syste, the local analysis is eaningful. In case of ore nuber of otors as several drives of one syste or syste centers, it is necessary to consider both levels eaple of parallel robotic structures. Global level provides the optial distribution of input energy. hen, all drives can cooperate effectively without utual fighting. Finally, local level perfors just their own oveent change of position of individual drive. his change is transfored through the echanical eleents ineatic structure to resultant oveent global result. he siilar relation can find in huan civilization. Good eaple is a situation of individual countries in gradually integrating Europe, which, as one cople partnership, tries to copete with United States and countries of eastern Asia. Acnowledgeent his research is supported by GA ČR (0/03/0620, Redundant drives and easureent for hybrid achine tools, GA ČR (02/02/0204, Design of adaptive intelligent controllers and GA ČR (02/0/0608, New directions of linear control. References Belda, K. (2002, Control of Redundant Parallel Structures of Robotic Systes, dissertations, CU in Prague. Belda, K., Böh, J. Valáše,. (2002 State-Space Generalized Predictive Control for Redundant Parallel Robots, NAO ASI Worshop, Praha 2002, pp Böh, J., Belda, K., Valáše,. (200 Study of Control of Planar Redundant Parallel Robot. Proceedings of the IASED Int. conference IC 200, Innsbruc 200, pp awson, Ch. J., Hanson, R. J. (974, Solving east Square Probles, Prentice - Hall, Inc. Online anuals (in PDF: Eternal Interfaces; Writing S-Functions. he athwors, Inc Souče, P. (997, Drives of Production Appliances, script of CU in Prague, (in Czech. Stejsal, V., Valáše,. (996, Kineatics and Dynaics of achinery, arcel Deer, Inc. say, ung-wen (999, Robot analysis, he echanics of Serial and Parallel anipulators, John Wiley & sons, Inc. Valáše,., Steinbauer, P. (999 Nonlinear Control of ultibody Systes, In: Euroech 99, isbon, pp