Kinematics Programming for Cooperating Robotic Systems

Transcription

1 Kinematic Programming for Cooperating Robotic Sytem Critiane P. Tonetto, Carlo R. Rocha, Henrique Sima, Altamir Dia Federal Univerity of Santa Catarina, Mechanical Engineering Department, P.O. Box 476, Univerity Campu, Trindade, Florianópoli-SC, Brazil, {cri.tonetto, ima, Abtract. Thi paper preent the kinematic programming for Cooperative Robotic Sytem (CRS), baed on crew theory approach. It include a ytematic for modeling and programming robotic ytem compoed by any number of robot (not necearily identical), working cooperatively to perform different tak. In order to illutrate the application of the ytematic, an example of CRS including four robot i preented. The kinematic computation of a CRS i made through the crew theory approach and it tool, like the Davie method and Aur virtual chain. Keyword: Cooperative Robotic Sytem, Tak Programming, Kinematic, Modeling Sytematic. Introduction Some indutrial tak need to ue more than one robot to perform ome tak, ince one ingle robot may not complete them alone. So, a Cooperative Robotic Sytem (CRS, in hort) can be applied to a tak or et of tak that preent ome complexitie to be performed by a ingle robot. Some example of cooperative robot tak include load haring, aembling of part and part reorientation while having another operation over it. Since a CRS i compoed by more than one robot, it i neceary to deal with many variable and their interrelationhip for the robot programming. Many reearcher have been tudying the application of more than one robot for tak execution. Lewi [] introduce the relative Jacobian concept baed on the Denavit-Hartenberg convention for two robot. In Tzafeta [2] the ytem i compoed by three identical robot that need to move an object from one poition to another. Dourado [3] tudie the differential invere kinematic baed on the crew theory for CRS. Owen, Croft and Benhabib [4] applied the relative Jacobian developed by Lewi [] in a ytem compoed by two planar robot. The cooperative Jacobian concept i introduced by Ribeiro and Martin [5], developed uing the Denavit-Hartenberg convention or the crew theory aociated with the Davie method. Alo, Ribeiro and Martin [5] propoe the cooperative Jacobian for ytem compoed by an arbitrary number of robot for tak execution. Some pecific cae olution for CRS are found in the literature, but no ytematical and general approache are applied for the kinematic computation and CRS programming.

2 9 C. P. Tonetto et al. In thi paper, an approach for the CRS programming i propoed. It alo preent a general view of the programming proce (which i detailed in [6], [7] and [8]). Thi novel ytematical approach intend to eae the programming proce for cooperating robot while performing indutrial tak. Thi paper i divided in five ection. In Section 2 the main contribution of thi work i highlighted. In ection 3 it i preented a brief urvey of the baic theory for the kinematic computation. In ection 4 the CRS concept i analyzed, decribing the ytematic applied to cooperative robotic. Finally, concluion are ummarized. 2 Contribution to Value Creation Some quetion appeared in the modeling and computing kinematic proce of CRS. In the firt place, how to generalize the addition of an arbitrary number of robot in the ytem. Secondly, how different phyical tructure influence the programming tak and other variable in the ytem, and, alo, how to combine different number and type of joint in order to perform one or more tak. Third, ince the main purpoe of the robot programming proce i to have all tak completed, how to include them in the ytematic the tak pecification(). To program a CRS it i needed to know two main component: robot and tak [8]. Thoe et of data are better divided into three main environment: the robotic tructure, the tak environment and the differential kinematic. Thee environment make poible to plit and evaluate robot data and tak eparately, allowing to know in advance how the programming change when the number of robot and tak increae in the ytem configuration. Let take a baic tructure for compoe a CRS, a a function of the number of robot, and aociated to a way of adding the Aur virtual kinematic chain. It give an initial knowledge over the complexity growth of the kinematic chain reolution involved in the CRS compoition. Such ytematic i expanible and allow to olve general problem of CRS, like colliion avoidance in the workpace of robot and tak a well a to imulate relative diplacement of the robot bae, when neceary in the ytem. The trategy to olve the CRS programming allow to imulate CRS compoed by any number of robot and tak. It give flexibility to the ytem and allow parameter change, uch a the initial configuration, robot poitioning, which robot will perform a tak, among other. All CRS programming logic and trategie propoed in thi paper are baed on the crew theory, graph theory, the Davie method and the Aur virtual chain. Thee tool are briefly decribed in the next ection.

3 Kinematic Programming for Cooperating Robotic Sytem 9 3 Bae Theory for the CRS Programming 3. Screw Theory The methodology preented for kinematic computation of CRS i baed on the crew theory. In thi theory, a crew $ i a geometric element defined by a directed line in pace and by a calar parameter h, that define the crew pitch [9]. One crew can be decompoed in a magnitude q& and it normalized axi $ˆ : i $ = $ˆq& where = oi i + hi $ˆ () and alo, i i an unitary vector with the direction of the axi related to the tranlation and rotation of the crew diplacement. The oi vector define the poition of the i vector related to a fixed coordinate ytem, h i the crew pitch and oi i i the cro product of oi and i vector. Now, the crew can be adapted to each type of body motion. When the movement i a rotation, the crew tep i null (h=), and Equation () lead to $ˆ revolute. Moreover, when the movement i a tranlation, Equation () lead to $ˆ tranlation. i $ˆ revolute = ; tranlation = oi i i $ˆ (2) So, the crew movement decription may be ued to define the differential diplacement between two bodie related to a reference coordinate ytem (baed on Chale theorem and on Mozzi theorem). More detail of the crew theory and it application can be found in the following work: [9], [] and []. 3.2 Succeive Screw Method Thi method can be extended to compute the crew due to the action of a body in another and a well a it coupling among them. The complete approach i preented in the reearch of Tai [2], where he deduce the equation that model the rigid body diplacement by uing the Chale Theorem. Alo, Tai [2] preent the Rodrigue matrix, written to evaluate the diplacement of a rigid body in the pace. So, the Rodrigue matrix i given by:

4 92 C. P. Tonetto et al. co+ co in, = co+ in co+ co in co in co+ in co in co+ By uing the crew diplacement concept for the kinematic analyi, the crew diplacement of n ucceive crew diplacement i jut the premultiplication of the tranformation matrix, given by: (3) A r = A A 2 A 3... A n (4) In thi way, the poition of the manipulator end-effector can be computed, while the robot joint move, by uing the crew evaluation procedure. In other word, the ucceive crew technique can be extended over the axi of the joint in order to calculate the robot effector poe. For the CRS, the crew technique i employed to etablih the influence of each one of the joint over the other, and, thu, to get the and o variable that include uch influence. 3.3 Aur Virtual Chain and Graph Theory Another tool to tudy CRS i the Aur virtual chain. The Aur virtual chain, when added to a CRS, help to analyze the diplacement of a kinematic chain or even to impoe deired movement to a kinematic chain, a decribed before. By definition, a virtual chain i a kinematic chain compoed by virtual link and joint, that can be added to a real kinematic chain without changing the main behavior of a real chain [3]. The virtual chain hould be ued to decribe the relationhip among robot link, tak and part in the cenario of CRS planning. Alo, a CRS can be better repreented by uing graph theory, to viualize and compute interrelation between bae, robot and tak. Such relationhip, repreented by graph, can be ummarized in a circuit matrix B. Thi matrix B i a way to decribe the preence of each edge in the graph mehe. The matrix i aembled o that each row i reerved to one meh of the graph and each column to one edge. Each element of the matrix i defined a:, if the edge i not preent on the meh;, if the edge i on the meh and on the ame direction of the circuit (arbitrary choen, but contant); -, if the edge i on the meh, but in the oppoite direction of the circuit. Together, Aur virtual chain and graph theory are very helpful to build the interrelationhip in a CRS tudy.

5 Kinematic Programming for Cooperating Robotic Sytem Davie Method The Davie method i a way to compute and relate the joint velocitie magnitude of a cloe virtual chain. The method i an adaptation of the Kirchhoff circuit law and tate that the um of the relative peed between kinematic pair throughout any cloed kinematic chain i null []: n i= $ i n = i= $ˆ q& = Nq& = (5) i i in which n i the number of joint of the ytem and N i the network matrix containing the normalized crew. To tudy the kinematic chain behavior it i better to claify the ytem joint a primary N p and econdary N. Thu, the Equation 5 can be written a: q& [ N MN p ] L = Nq& = N pq& p (6) q& p If N i invertible, the magnitude of the econdary joint the following equation: q & p p q& can be computed by = N N q& (7) By ubtituting N, N p and q& on Equation 7, the econdary joint velocitie magnitude turn to be decribed a a function of the primary joint. 4 Repreentation of the CRS The CRS compoition i baed on everal factor related to the tak, and thoe aociated to the available robot to perform them. Alo, other important factor that ha to be conidered i the environment in which the robot are located, like their phyical arrangement around the tak to be performed and the global workpace where the tak will be performed. In thi ection a cooperative tak example will be preented, in which the CRS i compoed by four robot and one compoed tak. The CRS tak programming ha fixed and variable data input. So, the CRS ytem i compoed by the robot: ABB IRB 662, ABB IRB 6, ABB IRB 4 and ABB IRB 2. Their data are preented on the Table and 2. On thee table the value of and o for each joint are decribed. Thi information i ued a input for the robot kinematic programming computation. It i important to notice that the firt joint of the ABB IRB 662 robot i a primatic joint. The tak will be performed by two poitioning robot, one inpection robot, and the lat robot mut follow a trajectory deigned to be painted (write a word - the UFSC

6 94 C. P. Tonetto et al. acronym) over the part. Table. Robot data: ABB IRB 662 and IRB 6 IRB 662 IRB 6 Joint o (mm) o (mm) (,,) (,, ) (,,) (,, ) 2 (,,) (,, 46) (,,) (5,, 486.5) 3 (,,) (,, 37) (,,) (5,, 96.5) 4 (,,) (245,, 565) (,,) (45,, 96.5) 5 (,,) (88,, 565) (,,) (75,, 96.5) 6 (,,) (76,, 565) (,,) (75,, 96.5) Table 2. Robot data: ABB IRB 4 and IRB 2 IRB 4 IRB 2 Joint o (mm) o (mm) (,,) (,, ) (,,) (,, ) 2 (,,) (7,, 352) (,,) (,, 292) 3 (,,) (7,, 72) (,,) (,, 632) 4 (,,) (26,, 72) (,,) (56,, 632) 5 (,,) (45,, 72) (,,) (36,, 632) 6 (,,) (45,, 72) (,,) (374,, 632) In the Fig. i depicted the configuration of the ytem on the left and the graph repreentation with virtual chain addition (CV i ymbol) on the right. In thi example ytem, five PPPS type of virtual chain were added. Thee virtual chain are patial and the name PPPS mean that they are compoed by three primatic and one pherical joint. The CV virtual chain define the part movement, related to the fixed coordinate ytem, and the CV i, with i=, 2, 3, 4, decribe the movement of the end- effector related to the part. Fig.. Cooperative Robotic Sytem and it graph repreentation The deired part movement given by CV virtual chain ha a trajectory defined by

7 Kinematic Programming for Cooperating Robotic Sytem 95 following value: CV =(2, t, 77,,, ) where <t<4.8. The CV virtual chain decribe the movement of the IRB 662 robot end-effector, relative to the part. Thi movement i defined a the et of point, joined into a trajectory, planned o that it can be ued to paint the UFSC acronym over the part. The CV 2 i related to the inpection procedure, and the IRB 6 robot i deignated for thi tak. The inpection i given by the following trajectory: CV 2 =(, 5+5t, 2,,, ) with <t<4.8, relative to the part origin. The CV 3 and CV 4 virtual chain define the movement of the IRB 2 and IRB 4 end-effector. Since thee robot are poitioning robot, the end-effector i fixed on the part, thu leading to null trajectorie for CV 3 and CV 4. On Table 3 the poition of the bae relative to the fixed coordinate ytem and the joint initial configuration for each robot are hown. Table 3. Robot bae poition and joint initial configuration Robot Bae poition (mm) Joint initial configuration (rad) IRB 662 (,-38,) (,.52, -.52,,.57, ) IRB 6 (22,, 4) (.6,, 3.4,, -.52, ) IRB 4 (4, 5, 2) (-,52,.26,,, -.26, ) IRB 2 (4, -, 25) (.52,.26,,, -.26, ) The circuit matrix B i: h = h2 h3 h4 (8) The matrix column are preented in compreed form. For example, the CV column refer to the ix joint of the PPPS virtual chain. The normalized crew matrix D i: = $ $ $ $ $ $ $ $ $ (9) The multiplication of D and B reult in the network matrix N: N=D diag{b i } () where B i i the vector extracted from the i th row of the B matrix. The CRS network matrix N i given by:

8 96 C. P. Tonetto et al. $ $ = $ $ $ $ $ $ $ $ $ $ where $ˆ CVi i a 6 6 ized matrix, including the normalized crew of the virtual chain, with $ i =,,2,3,4; and $ˆ Ri i a 6 6 ized matrix including the normalized crew relative to the robot joint. The network matrix i plit into primary matrix N p and econdary matrix N : () $ $ = $ $ $ $ $ $ (2) $ = $ $ $ (3) The value of q&, q q&, N and N p defined above are replaced on Equation 7 and o, p the velocitie magnitude of the robot joint are computed, a a function of the velocitie magnitude defined for each robot end-effector. Fig. 2 how the robot while performing the deignated tak, in four different moment, and a top view of the ytem. Fig. 2. Robot performing the tak cooperatively A movie of the imulation i available on the youtube channel:

9 Kinematic Programming for Cooperating Robotic Sytem 97 The joint poition for each one of the robot are preented in the Fig. 3, with the time a the horizontal axi (<t<4.8). For the IRB 662, only four joint are ued, while the 4th and 6th joint value remain null, howing that for thi tak and initial configuration they weren t neceary. The requirement of joint according to tak and initial configuration i an intereting topic for further reearche. Fig. 3. The joint poition for each robot 5 Concluion Planning tak in cooperative robotic ytem require more knowledge about the tak and how to compoe the robotic environment where it will be performed. The propoed methodology for the CRS kinematic computation allow to combine different configuration between robot and part to be teted. In thi paper one example of application wa preented. If one or more robot are added or removed from the ytem, the programming proce i the ame a preented, changing only the tak pecification and the N matrix, ince it wa repreented by the graph and by the normalized crew matrix, it compoition i eaily changed. The network matrix will have different number of row and column according to the number of robot in the ytem. The off-line imulation reult are important for CRS programming, epecially when everal robot are included, leading to a great number of parameter to be analyzed. Becaue of thee characteritic, it i very hard to implement a real tak without previou imulation. The analytical evaluation could not reult in practical reult, due to the great number of variable and it inter-dependence. By uing imulation, it i poible to have a complete view of the ytem behavior, to detect

10 98 C. P. Tonetto et al. and analyze ingularitie, colliion and to enhance the performance of the tak when it i finally performing the tak. Furthermore, the theory preented in thi paper can be applied to CRS compoed by other kind of robot, uch a: mobile robot, parallel robot and manipulator. Beide that, it i poible to imulate the ytem looking for colliion avoidance of defined joint to obtacle of the environment, while the CRS i performing tak. A deeper performance analyi of the ytem i an intereting reearch field, evaluating which of the available parameter can be adjuted in order to enhance the tak performance. Reference. Lewi, C.L.: Trajectory generation for two robot cooperating to perform a tak. Proc. IEEE International Conference on Robotic and Automation. 2, (996) 2. Tzafeta C.S., Prokopiou P.A, Tzafeta S.G.: Path Planning and Control of a Cooperative Three-Robot Sytem Manipulating Large Object. Journal of Intelligent and Robotic Sytem. 22, 99-6 (998) 3. Dourado A.O.: Cinemática de Robô Cooperativo. UFSC. Florianópoli-SC, Brazil (25) 4. Owen W.S., Croft E.A., Benhabib B.: A multi-arm robotic ytem for optimal culpting. Robotic and Computer-Integrated Manufacturing , Elevier Ltd (25) 5. Ribeiro L.P., Martin D.: Screw-baed relative Jacobian for manipulator cooperating in a tak uing Aur virtual chain. In: Proceeding of 2th International Congre of Mechanical Engineering. ABCM, Brazil (29) 6. Tonetto C.P., Dia A.: Trajectory Planning with redundant cooperative robotic ytem. In: Proceeding of 2th International Congre of Mechanical Engineering. ABCM, Brazil (29) 7. Tonetto C.P., Dia A.: The crew theory applied to compute the kinematic of multi-robot ytem in cooperative tak.in: Proceeding of the 2th International Conference on Flexible Automation and Intelligent Manufacturing. California State Univerity, Eat Bay, (2) 8. Tonetto C.P., Rocha C.R., Dia A.: Simulation of Multi-robot Cooperative Sytem programming baed on a three environment definition In: Proceeding of the 2th Mechatronic Forum Biennial International Conference. Swi Federal Intitute of Technology, Zurich, Switzerland (2) 9. Hunt K.K.: Don t cro-thread the crew. A Sympoium Commemorating the Legacy, Work, and Life of Sir Robert Stawell Ball Upon the th Anniverary of A Treatie on the Theory of Screw (2). Davie T.H.: Kirchhoff circulation law applied to multi-loop kinematic chain. Mechanim and Machine Theory, vol. 6, pp (98). Rocha C.R., Tonetto C.P., Dia A.: A comparion between the Denavit-Hartenberg and the crew-baed method ued in kinematic modeling of robot manipulator. Robotic and Computer-Integrated Manufacturing. In: Conference paper of Flexible Automation and Intelligent Manufacturing. vol. 27, pp (2) 2. Tai Lung-Wen,: Robot Analyi: The Mechanic of Serial and Parallel Manipulator. Jonh Wiley & Son, INC (999) 3.Campo A., Guenther R., Martin D.: Differential Kinematic of Serial Manipulator Uing Virtual Chain. Brazilian Society of Mechanical Science and Engineering. ABCM. vol. XXVII, pp (25)