Homing, Calibration and Model-Based Predictive Control for Planar Parallel Robots

Transcription

1 Homing, Calibration and Model-Based Predictive Control for Planar Parallel Robots Květoslav Belda*, Pavel Píša** * Department of Adaptive Sstems, Institte of Information heor and Atomation, Academ of Sciences of the Czech Repblic, Pod Vodárensko věží 4, 8 8 Prage 8 Czech Repblic ( belda@tia.cas.cz). **Department of Control Engineering, Faclt of Electrical Engineering, Czech echnical Universit in Prage, Karlovo nám. 3, 35 Prage, Czech Repblic. Abstract: Parallel robots represent wa to considerabl improve accrac and speed of indstrial machine tools and their centres. his paper deals with the preparator operations: homing and calibration, which precede the start-p of the robot work, i.e. real control process. heir procedres are discssed with respect to planar parallel robots and their control. In this paper, as a sitable control strateg, the model-based predictive control is considered. he predictive control offers operator to continosl inflence the control process. he control isses relating to planar parallel robots are discssed here.. INRODUCION Parallel robots (Merlet, 6; sai, 999), i.e. robots based on parallel kinematic strctres (Fig. ) can be simpl characterized as movable trss constrctions or as movable platforms spported b several links, where platforms serve as a place for a tool or for a gripper. hese strctres represent closed-loop constrctions, flexibilit (dnamics) of which allowing high prodctivit follows from possible small nmber of moving masses. herefore, in comparison with serial tpes (Sciavicco and Siciliano, 996), the ma offer higher stiffness and dexterit. his featre is given b the fixing almost all drives directl on the basic frame withot loading of movable parts of the robot constrction. It contribtes to the decrease of inertial forces. However, a larger nmber of the links and passive joints often reqire more sophisticated procedres in a robot adjstment, start-p and real control process. All procedres in general shold ensre a cooperation of each drive participating in robot motion. he first two procedres are limited b nknown initial robot state, which is detected dring them. Once the robot state is determined at least roghl, then the drive cooperation can be ensred. Since adjstment and start-p procedres (calibration and homing) have no so strict time limitation, the ma be performed in sch a wa not to come to ndesired states. he time conditions have to be considered in the control procedre, which follows from reqirements of real technological process. Conventional local control strategies (e.g. cascade drive control) have no energ optimization. It is a limiting factor of a robot capacit. In case of parallel robots, it ma prodce ndesired antagonistic drive behavior i.e. behavior withot necessar drive cooperation. he antagonism, i.e. drive mtal fighting, is cased b presence of interrelations among individal drives throgh parallel links and appropriate movable platform. he conventional control cannot consider these interrelations and therefore it does not represent economic and safe soltion. o avoid mentioned ndesirable states, some model describing energ decopling in the robot strctre is needfl. However, when some model is available, then global model-based control strategies are more effective. he can provide design of control actions, which are optimized for appropriate robot strctre. Fig.. Principle of parallel kinematical strctres movable platform i th link i th drive robot frame his paper is organized as follows. Firstl, homing and calibration procedres intended for planar parallel robots are investigated. hen the paper deals with control isses relating to parallel strctres. he drawbacks of conventional control are discssed. hereafter, as sitable promising model-based strateg, predictive control is introdced. All procedres being described here are docmented in several representative examples arising from their implementation on one real laborator model of parallel robot.

2 . HOMING Homing procedre represents set of actions atomaticall leading the all robot elements from initial generall nknown position (i.e. arbitrar position from a robot workspace) to predefined known deterministic position, so-called home position. his position ma be an initial position for a sensor calibration, for real work operations or for a work-tool replacement etc. he real homing procedre is determined b sed robot drives: linear drives rotational drives more precisel, b character of the drive sensors (possible other additional sensors): positional (absolte) sensors incremental sensors denoting the pieces of information on the robot position, generall on the robot state. Frthermore, the procedre depends on tpe of the robot strctre and configration, i.e. if the strctre has serial, parallel or hbrid character.. Homing Procedre o start the procedre, it is necessar to know rogh robot position, appropriate drive positions, respectivel, e.g. known qadrant in the case of rotational drives or interval in the case of linear drives in relation home position. For this prpose, specific mechanical or electromechanical elements (marks) are sitable to be considered (Fig. and Fig. 3). hese elements stand for positional (absolte) homing marks, one part of which representing reference (e.g. robot frame, previos lin is fixed, and the second part is monted on appropriate place moving relative to part one (e.g. slide rest, drive spindle). In case of positional (absolte) sensors, mainl at linear drives, sch orientation element ma be directl sensor itself, bt it need not be a rle. In Fig. and Fig. 3, the elements homing marks represent two possible states (e.g. vales or ), if the mark shade is in the view (vale ) or not (vale ). According to mark vale, representing certain drive position related to the position of movable platform, the sense of rotation or linear motion of robot drives at homing procedre is chosen. It represents limited information on the robot position. herefore, the homing procedre can be realized as a feed-forward or rather weak feed-back decentralized control, where drives are fed with safe energ keeping selected motion sense p to changing mark vales. he real homing procedre can be generall done as follows: he robot is switched on and the control nit tests the vales of the marks. In a correspondence to these vales, the control nit set the inpt vale and its sign for each drive. he robot starts to move slightl towards home position. Dring the process, the positional drive sensors shold register their vales, althogh the vales are not directl connected to phsical meaning. Once some of drivers achieves the change of marks (change or ), the vale of position from drive sensor is stored in memor and the appropriate drive is stabilized in this position. he stabilization can be provided b a simple PI controller. When all drives achieve the mark change, the robot reaches the home position and it is stabilized in this position.. Specification of Homing for Parallel Robots Fig.. Example of homing mark - case of linear drive Fig. 3. Example of homing mark - case of rotational drive In case of the parallel robots, which can be moreover redndantl actated, the homing is limited b kinematical relations throgh chained robot links. In case of serial robots, i.e. open-loop kinematical strctres, the individal drives can be homed independentl; there are no direct kinematical relations. Conversel, the homing of parallel robots has to be proceeded for all drives simltaneosl. he individal homing of each drive is not possible. It can case collisions of the robot links and nsafe robot motion, which can lead to the damage of the robot and its neighborhood. However, b the simltaneos homing, the individal drives reach the home position in different time. When one drive reaches mark change, it is stabilized in mark neighborhood. Nevertheless, its stabilization has to be soft not to restrict other drives to reaching the mark changes. After the change of all marks, it is necessar to watch over the all stabilizing actions, especiall in case of redndant robot actation, where mtal drive fighting ma appear. he problem of the mtal drive fighting with sggestion of sitable soltion will be discssed in the section dealing with control design.

3 3. CALIBRAION he calibration is a procedre of identifing the real geometrical parameters in given kinematical strctre of the robot relative to the position and orientation of links and joints in the robot (Andreff, et al. 4). he calibration is important for the garant of the repeatabilit and for meeting the tolerance reqirements. here are distingished two tpes of the calibration: mechanical calibration software calibration either off-line (one-off or reglarl repeated) prior to real control process or on-line dring the process. 3. Calibration Procedre he calibration procedre starts alread dring the installation of a new robot in a place of the se (in the robot workplace). his phase represents one-off mechanical calibration, where relative positions of individal robot elements are necessar to be adjsted. Except strctral robot elements, the adjstment of the homing elements marks, described in previos section, belongs to this phase. hese elements have to be calibrated together with appropriate drives; i.e. their mtal position has to be fixed and to be corresponding to robot homing position. he elements marks can be sed in second phase software calibration, where the accrate drive sensors are calibrated to robot coordinates. Limited mechanical calibration inclding length measrement of robot links and the like ma be prsed also on-line. he reslt of this phase is set of dimensions served for control design. he second phase, sall exected reglarl e.g. pon robot switching on or possibl pon tool replacement, is software calibration. It can be characterized in simple meaning as mapping accrate drive sensors (incremental conters, measring rles) and others to real phsical coordinates (rotational angles, distances). In other words, dring this phase, all offsets and compensator constants are set p in the program, where the data from sensors are processed. 3. Specification of Calibration for Parallel Robots he first phase called mechanical calibration is more or less eqivalent for both serial and parallel robots. he second phase software, actall local sensor calibration differs alread again. In comparison with the homing procedre of parallel robots in previos section, that calibration phase proceeds individall for each sensor and appropriate drive b reason of accrac. Contrar to serial robots, where the movement dring the calibration is not limited except for occrrence of link collisions, the movement of each drive is strictl limited to small vicinit of selected representative positions (e.g. home position), since larger movement cases ndefined movement of a movable platform, links and other robot elements. hs, the second phase is discontinos. After each sensor calibration, the homing procedre like has to be exected in order to provide the same initial conditions for other sensors waiting for calibration. Single calibration procedre inclsive initial homing procedre for planar parallel robots e.g. with n drives and their appropriate sensors can be formlated b the following for steps: Step. Let the homing procedre from section. is done. Step. Let the all drives slowl change the marks and stop immediatel after that change on the predefined mark side (i.e. in the niqe mark vale, the same for all cases). Step 3. Let the i th drive is moved back and when the new drive sensor hit is appeared, let the sensor vale is reset to its new origin, which vales correspond to real phsical coordinates for given position. Step 4. Let the procedre goes to Step. and all for steps are repeated for other drive and sensor pairs p to state when all sensors are reset to real vales. he correctness of the procedre can be verified b backward controlled motion with the reference vales recompted from record of sensor data and inversed in their own direction. he robot, which starts in this backward motion in known home position, shold reach accratel the initial nknown position. his proof can be done de to the known starting home position. Moreover, the robot can be alread controlled in both coordinates sstems: drive coordinate sstem or operational sstem. De to the known relation between the sensor vales and real robot position, the direct and inverse kinematical transformations are comptable even for redndantl actated parallel strctres (Böhm, et al., ). he mentioned kinematical transformations enable ser to se the both coordinate sstems. Another possibilit, in case of redndantl actated robots, is on-line calibration (Valášek, et al., ). Since all drives (adeqate and redndant) are eqipped with drive sensor, then dring each motion of the robot, there is a redndant nmber of measrements. he assembled eqations of constraints for certain nmber of robot positions can be solved for both robot dimensions and initial positions. his approach enables the redndant parallel robots to be calibrated on-line dring their operations withot sing an external eqipment or breaking their work. 4. CONROL OF PARALLEL ROBOS he general isse of the robot control is tracking the desired trajector path control. o design sitable control, the following fact has to be considered. he robot strctres represent nonlinear mltivariate sstems, dnamics of which is relativel fast in comparison with comptation time for control actions. For that reason, the discrete methods are considered. he can ensre the finite comptation time of the control. he control can be solved either on local level, b design of independent control of the drives as servos, or b global control design for the whole robot. his approach can be especiall considered, becase sfficientl accrate model of the robot can be composed.

4 4. Model of the robot In general, the parallel robots represent mlti-bod sstems, which can be straightforwardl described in phsical coordinates b Lagrange s eqations of mixed tpe (Stejskal and Valášek, 996). hese eqations lead to the sstem of differential algebraic eqations DAE () M && s Φ = g + s λ f ( s) = where M is a mass matrix, s is a vector of phsical coordinates (their nmber is sall greater than nmber of degrees of freedom DOF), Φ s is a Jacobian, λ is a vector of Lagrange s mltipliers, g is a vector of other internal relations, matrix connects inpts to appropriate differential eqations and algebraic eqations f(s) = represent geometrical constraints. As mentioned, the model () is a DAE sstem and moreover nonlinear. It is not sitable for control design. However, it can be transformed to different form (Stejskal and Valášek, 996), to the sstem of ordinar differential eqations based on independent coordinates, which correspond to DOF: R MR & + R MR & & = R g + R () where matrix R is the basis of the nll space of the overall Jacobian Φ s. he eqation () can be rewritten in condensed notation as follows & = f (, & ) + g( ) (3) where f (, & ) represents robot dnamics and g () is inpt matrix. he model (3) can be transformed to the state-space formla: x& = f ( x) + G( x), = H x x = & which is simpler and more transparent for handling with mlti-inpt mlti-otpt sstems as robots sall are. De to discrete realization of the control, the model (4) is discretized. o se standard discretization via expansion of exponential fnctions, the nonlinear vector f (x) in (4) has to be linearized. It can be provided b decomposition according to Valášek and Steinbaer (999), which leads to the linear form: x& = F( x) x + G( x), = H x x = & he obtained form (5) represents the robot dnamics identicall as model (3) or (4) ; the individal elements of state and inpt matrices F (x) and G (x) has to be recompted on-line for appropriate topical robot state x. Otpt matrix H is rectanglar identit matrix (it is eqal otpt matrix C in (6)). he real se of the decomposition is shown in (Belda, et al., 3). () (4) (5) hen, after discretization of (5), the obtained model has following form: x( k + ) = A( x( + B( (, ( = C x( ( x( = & ( hat form is convenient for control design. 4. Conventional Control he conventional control strateg PID/PSD feedback control considers the robots maniplators as a set of singleinpt / single-otpt sbsstems, which represent individal drives. he mtal interactions are taken into accont as distrbance entering each drive sbsstem. However, in the case of the parallel robots, mainl redndantl actated, the nprodctive part of Integral/Sm channels mst be solved. It does not occr at serial open-loop strctres. Undesirable nprodctive part of I/S channels is cased b inaccracies in mechanism. It means that the drive coordinates designed according to inverse kinematical transformation from independent coordinates in some cases cannot be attainable. his cases npredictable increase of I/S channels, which does not contribte to motion and moreover leads to instabilit of the whole robot sstem. o damp this ndesired propert, the specific redction projection is sed. From mathematical point of view the following minimization problem is solved: A = b min where matrix A is R, generall horizontal rectanglar redistribtion matrix, which transforms drive torqes to the generalized force effects, and are control actions reqired on drives, and b is a vector of generalized force effects. o obtain redcing projection in view of, the qadratic criterion is sed: ( )! J = + λ A b = min (8) minimization of which gives the projection formla: = A ( AA ) A (9) red with assmption that A ( AA ) A I. 4.3 Model-Based Predictive Control he model-based control in general ses the knowledge of the dnamic model (e.g. () or its state-space form (6)) and that wa it globall optimises control actions in the view of the whole robot. De to the model, the ndesired properties of conventional control discssed in previos sbsection do not occr. (6) (7)

5 One model-based representative strateg is predictive control. It represents a mlti-step control based on eqations of predictions and the local repetitive minimization of qadratic criterion (Ords and Clarke, 993). he eqations of predictions serve for the expression of feedforward within horizon of predictions N. On their basis, the dominant part of control actions is determined. Using discrete state-space form (6) the eqations have following form: ˆ = f + G, ˆ = [ˆ ˆ k + k+ N ] CA where f = M x, (k ) N CA L () C G = M CA N BL O M BLCB Vector f represents free responds from instant k, i.e. predicted sstem otpts for =. he prodct G compensates differences of these otpts from desired vales within considered horizon of the prediction N. he control is compted from the qadratic criterion () J k = N j = { (ˆ Q w ) Q + } k+ j k+ j k+ j () where Q and Q are penalizations; and w ( k + j ) is a vector of desired vales. Obtained vector represents the control actions for whole horizon N. However, onl the first appropriate actions are reall applied to the robot. his process is repeated in ever time step for appropriatel pdated model (6). o provide effective and stable algorithm for the comptation of control actions, the minimization of the criterion can be sitabl transformed to the minimization in the specific matrix sqare-root form (Belda, 5): Q ( ˆ w) Q G Q ( f w)! J = = + = Q Q () he minimization of sch form leads to the soltion of simple sstem of algebraic eqations relative to nknown vector of control actions. he predictive control, de to mlti-step character, generates more sitable control actions for drives, which fit the desired robot motion withot addition of nprodctive parts. It represents economic soltion in the view of energ consmption in comparison to conventional approaches. 5. APPLICAION EXAMPLES In this section, the application of explained theoretical reslts will be shown on one specific planar parallel robot, which is being developed for top milling machine. Main spindle of the machine, which serves for milling ctters, is located in centre of the movable platform of the robot strctre. he control is focsed on coordinates in plane. Vertical axis (coordinate) is set independentl and is fixed dring motion. 5. Description of Considered Parallel Robot For experiments, the robot called Moving Slide was sed. It is illstrated in Fig. 4. his robot represents horizontal planar parallel configration, which has 4 Rotational + Prismatic + Rotational joints. Moreover, it is redndantl actated, becase there are for drives for onl three degreases of freedom here. his featre frthermore improves robot stiffness and dnamics Fig RPR parallel strctre Moving Slide 5. Robot Homing and Calibration According to sections. and 3., the robot is led to home position, which is, for this case, selected in the centre of the robot workspace. he robot has rotational drives (see Fig. 5). Fig. 5. Homing mark - case of rotational drives Fig. 6. XY graph of real robot motion dring homing In Fig. 6, the lines represent homing trajectories of for different nknown initial robot positions. As mentioned in section 3., their coordinates were reconstrcted b backward robot motion sing direct kinematical transformation (Böhm, et al., ). It is possible de to known coordinates of the home robot position as initial conditions for the direct transformation.

6 he trajector incldes two trn points between segments 7 and 8 [x, ] = [-., ], and segments 8 and 9 [x, ] = [., ]. In them, the robot is stopped. Dring the whole motion, the robot has three acceleration stages (start and two trn points) and three braking stages (two trn and end points). he designed controller sing predictive strateg was exected with horizon of prediction N = and sampling period s =. s. 6. CONCLUSIONS Fig. 7. ime histor of real homing and calibration he Fig. 7 represents time histor for one homing trajector from Fig. 6. After homing of all drives, the robot is stabilized in the home position. It is perceptible ndesirable increase of I/S channels, which cannot be still compensated before finishing the calibration. 5.3 Robot Control he example illstrated in Fig. 8 demonstrates real control. axis [m] axis x [m] (b) Fig. 8. rajector (a) and time histor of control action (b) he strctre (Fig. 4) tracks individal trajector segments in order indicated in Fig. 8 (a) b nmbers - 9. Initial and final point is in the workspace centre [x, ] = [, ]. (a) his paper focses on the homing, calibration and control of planar parallel robots. Homing and calibration procedres were discssed initiall for general robot strctres, then the particlarities of the procedres applied to parallel robots were explained. he individal procedres were demonstrated and verified on laborator model of 4 RPR parallel strctre Moving Slide. In regards to control, predictive strateg proved to be a sitable approach for robot control. ACKNOWLEDGEMENS he athors appreciate the kind spport of the Grant Agenc of the Czech Repblic b the grant (/6/P75, 6/8) Model-based Control of Mechatronic Sstems for Robotics. REFERENCES Andreff, N., Renad, P., Martinet, P. and Pierrot, F. (4). Vision-based kinematic calibration of an H4 parallel mechanism: practical accracies. he Indstrial Robot. Vol. 3, No. 3, Belda, K. (5). Control of Parallel Robotic Strctres Driven b Electromotors. Dissertation. CU in Prage. Belda, K., Böhm, J. and Valášek, M. (3). State-space generalized predictive control for redndant parallel robots. Mechanics Based Design of Strctres and Machines, Vol. 3, No. 3, Böhm, J., Belda, K. and Valášek, M.: (). he Direct kinematics for path control of redndant parallel robots. Advances in Sstems Science: Measrement, Circits and Control. WSEAS Press, Rethmno beach, Greece, Merlet, J.-P. (6). Parallel Robots, Springer, Dordrecht. Ords, A. and Clarke, D. (993). A state - space description for GPC controllers. Int. J. Sstems SCI., Vol. 4, No. 9, Sciavicco, L. and Siciliano, B. (996). Modeling and Control of Robot Maniplators, he Mc-Graw Hill Companies, Inc., New York. Stejskal, V. and Valášek, M. (996). Kinematics and Dnamics of Machiner, Marcel Dekker, Inc. sai, L.-W. (999). Robot analsis, he Mechanics of Serial and Parallel Maniplators, John Wile & sons, Inc., New York. Valášek, M., Belda, K. and Florian, M. (). Control and calibration of redndantl actated parallel robots. Reports from the IWU 6, Franhofer IWU, Chemnitz, German, Valášek, M. and Steinbaer, P. (999). Nonlinear control of mltibod sstems, Eromech 99, Lisabon,