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2 Two Cooperating Manipulators with Fractional Controllers X Two Cooperating Manipulators with Fractional Controllers N. M. Fonseca Ferreira Departent of Electrical Engineering, Institute of Engineering of Coibra Rua Pedro Nunes, Coibra, Portugal nunoig@isec.pt J. A. Tenreiro Machado Departent of Electrical Engineering, Institute of Engineering of Porto Rua Dr. António Bernardino de Aleida, Porto, Portugal jt@isep.ipp.pt József K. Tar Budapest Tech, John von Neuann, Faculty of Inforatics Inst. of Intelligent Engineering Systes Bécsiút 96/B, H-134, Budapest, Hungary tar.jozsef@nik.bf.hu Abstract This paper analyzes the dynaic perforance of two cooperative robot anipulators. It is studied the ipleentation of fractional-order algoriths in the position/force control of two cooperating robotic anipulators holding an object. The siulations reveal that fractional algoriths lead to perforances superior to classical integer-order controllers. 1. Introduction Two robots carrying a coon object are a logical alternative for the case in which a single robot is not able to handle the load. The choice of a robotic echanis depends on the task or the type of work to be perfored and, consequently, is deterined by the position of the robots and by their diensions and structure. In general, the selection is done through experience and intuition; nevertheless, it is iportant to easure the anipulation capability of the robotic syste (Y. C. Tsai & A.H Soni., 1981) that can be useful in the robot operation. In this perspective it was proposed the concept of kineatic anipulability easure (T. Yoshikawa, 1985) and its generalization to dynaical anipulability (H. Asada, 1983) or, alternatively, the statistical evaluation of anipulation (J. A. Tenreiro. Machado & A. M. Galhano, 1997). Other related aspects such as the coordination of two robots handling

3 28 Robot Manipulators, New Achieveents objects, collision avoidance and free path planning have been also investigated (Y. Nakaura, K. Nagai, T. Yoshikawa, 1989) but they still require further study. With two cooperative robots the resulting interaction forces have to be accoodated and consequently, in addition to position feedback, force control is also required to accoplish adequate perforances (T. J. Tarn, A. K. Bejczy, P. K., 1996) and (N. M. Fonseca Ferreira, J. A. Tenreiro Machado, 2) and (A. K. Bejczy and T. Jonhg Tarn, 2). There are two basic ethods for force control, naely the hybrid position/force and the ipedance schees. The first ethod (M. H. Raibert and J. J. Craig, 1981) separates the task into two orthogonal sub-spaces corresponding to the force and the position controlled variables. Once established the subspace decoposition two independent controllers are designed. The second ethod (N. Hogan, 1985) requires the definition of the ar echanical ipedance. The ipedance accoodates the interaction forces that can be controlled to obtain an adequate response. Others authors (Kuar, Manish; Garg, Devendra 25, Ahin Yildiri, 25, Jufeng Peng, Srinivas Akella, 25) present advance ethodologies to optiize the control of two cooperating robots using the neural network architecture and learning echanis to train this architecture online. This paper analyzes the anipulation and the payload capability of two ar systes and we study the position/force control of two cooperative anipulators, using fractional-order () algoriths (J. A. Tenreiro Machado, 1997) and (N. M. Fonseca Ferreira & J. A. Tenreiro Machado 23, 24 and 25). Bearing these facts in ind this article is organized as follows. Section two presents the controller architecture for the position/force control of two robotic ars. Based on these concepts, section three develops several siulations for the statistical analysis and the perforance evaluation of and classical PID controllers, for robots having several types of dynaic phenoena at the joints. Finally, section four outlines the ain conclusions. 2. Control of Two Ars The dynaics of a robot with n links interacting with the environent is odelled as: T τ H(q)q C(q,q) G(q) J (q)f (1) where is the n 1 vector of actuator torques, q is the n 1 vector of joint coordinates, H(q) is the n n inertia atrix, C(q, q ) is the n 1 vector of centrifugal/coriolis ters and G(q) is the n 1 vector of gravitational effects. The n atrix J T (q) is the transpose of the Jacobian of the robot and F is the 1 vector of the force that the (-diensional) environent exerts in the gripper. We consider two robots with identical diensions (Fig. 1). The contact of the robot gripper with the load is odelled through a linear syste with a ass M, a daping B and a stiffness K (Fig. 2). The nuerical values adopted for the RR (where R denote rotational joints) robots and the object are 1 = 2 = 1. kg, l 1 = l 2 = l b = l = 1., = deg, B 1 = B 2 = 1 Ns 1 and K 1 = K 2 = 1 4 N -1.

4 Two Cooperating Manipulators with Fractional Controllers 281 y() Load {x2, y2} Robot A {x1, y1} l () Robot B l 12 l22 Upper Elbow 21 l l12 Lower Elbow x() lb () Fig. 1. Two RR robots working cooperation for the anipulation of an object with length l and orientation. l 2 K2 F2 Robot B {x 2, y 2} K1 1 M B2 Contact with the robot gripper Robot A F1 B1 {x 1, y 1} Load akes contact with grippers Contact with the robot gripper Fig. 2. The contact between the robot gripper and the object. The controller architecture (Fig. 3), is inspired on the ipedance and copliance schees. Therefore, we establish a cascade of force and position algoriths as internal an external feedback loops, respectively, where x d and F d are the payload desired position coordinates and contact forces.

5 282 Robot Manipulators, New Achieveents Position/ Force Trajectory Planning x C P Position Controlle r Position Velocity F C F Force Controlle r Kineatic s Kineatics Force Robots Object Fig. 3. The position/force cascade controller. In the position and force control loops we consider controllers of the type C(s) = K p + K s, 1 < < 1, that are approxiated by 4 th order discrete-tie Pade expressions (a i, b i,, k = 4): C z K K k p 4 4 k We copare the response with the classical PDPI algoriths therefore, in the position and force loops we consider, respectively. a b k k z z 4k 4k (2) C Cs K p K d s 1 z K K (1 z ) C C s z p 1 K p K i s z K p K i z 1 d (3) (4) Both algoriths were tuned by trial and error, having in ind getting a siilar perforance in the two cases (Tables 1 and 2). i K p K K p K (a) Position controller (b) Force controller Table 1. The paraeters of the position and force controllers.

6 Two Cooperating Manipulators with Fractional Controllers 283 i K p K d K p K i (a) Position controller (b) Force controller Table 2. The paraeters of the position and force PDPI controllers. 3. Analysis of the syste perforance In order to study the syste dynaics we apply a sall aplitude rectangular pulse y d at the position reference and we analyze the syste response. The siulations adopt a controller sapling frequency f c = 1 khz, contact forces of the grippers {Fx j, Fy j } {.5, 5} N, a operating point of the center of the object A {x, y} {, 1} and a load orientation of = º. In a first phase we consider robots with ideal transissions at the joints. Figure 4 depicts the tie response of robot A under the action of the and PDPI algoriths. In a second phase (figure 5) we analyze the response of robots with dynaic backlash at the joints. For the i th joint (i = 1, 2), with gear clearance h i, the backlash reveals ipact phenoena between the inertias, which obey the principle of conservation of oentu and the Newton law: q J εj q J 1 ε i ii i i i q i (5) J ii J i q i q i J ε q J εj i 1 i i ii (6) J ii J where 1 is a constant that defines the type of ipact ( inelastic ipact, 1 elastic ipact) and q and q q and q ) are the velocities of the i th joint and otor i i ( i i before (after) the collision, respectively. The paraeter J ii (J i ) stands for the link (otor) inertias of joint i. The nuerical values adopted are h i = rad and i.8 (i = 1, 2). In a third phase (figure 6) we study the RR robot with copliant joints. For this case the dynaic odel corresponds to odel (1) augented by the equations: τ J K i q B q K q q (7) q q Jqq Cq,q Gq (8) where J, B and K are the n n diagonal atrices of the otor and transission inertias, daping and stiffness, respectively. In the siulations we adopt K i = N rad 1 and B i = 1 4 Ns rad 1 (i = 1, 2). The tie response characteristics (Tables 3 and 4), naely the percent overshoot PO%, the steady-state error e ss, the peak tie T p and the settling tie T s reveal that, if we consider siilar perforances for robots with ideal transissions at the joints, the is superior to

7 284 Robot Manipulators, New Achieveents the PDPI algoriths at the cases of robots with joint dynaic phenoena. I conclusion the have good dynaic response for position and force perturbations. N o C(s) PO% e ss [] T p [s] T s [s] PDPI PDPI PDPI Table 3. Tie response characteristics for a pulse y d at the robot A position reference. N o C(s) PO% e ss [] T p [s] T s [s] PDPI PDPI PDPI Table 4. Tie response characteristics for a pulsef d at the robot A force reference. 5 x dx A () dy A () Tie (s) Tie (s) dfx A (N) dfy A (N) Tie (s) Tie (s)

8 Two Cooperating Manipulators with Fractional Controllers x x 1-3 dx B () dy B () Tie (s) Tie (s) dfx B (N) Tie (s) dfy B (N) Tie (s) Fig. 4. Tie response for robots A and B with ideal joints under the action of the and the PDPI algoriths for a pulse perturbation at the robot A position reference y d = 1 3 and a payload with M = 1 kg, B i = 1 Ns 1 and K i = 1 3 N x 1-3 dy A ().1 dx A () Tie (s) Tie (s) dfx A (N) 1 dfy A (N) Tie (s) Tie (s)

9 286 Robot Manipulators, New Achieveents 5 x x 1-3 dx B () dy B () Tie (s) Tie (s) dfx B (N) 1 dfy B (N) Tie (s) Tie (s) Fig. 5. Tie response for robots A and B with joints having backlash under the action of the and the PDPI algoriths for a pulse perturbation at the robot A position reference y d =.1 and a payload with M = 1 kg, B i = 1 Ns 1 and K i = 1 3 N 1. 5 x dx A () dy A () Tie (s) Tie (s) dfx A (N) 1 dfy A (N) Tie (s) Tie (s)

10 Two Cooperating Manipulators with Fractional Controllers x x 1-3 dx B () dy B () Tie (s) Tie (s) dfx B (N) dfy B (N) Tie (s) Tie (s) Fig. 6. Tie response for robots A and B with joints having flexibility under the action of the and the PDPI algoriths for a pulse perturbation at the robot A position reference y d =.1 and a payload with M = 1 kg, B i = 1 Ns 1 and K i = 1 3 N Conclusion This paper studied the position/force control of two robots working in cooperation using fractional and integer order control algoriths. The syste tie response was analyzed for anipulators having several types of dynaical phenoena at the joints. The transient response of the syste shows the superior perforance of the controller. 5. s Y. C. Tsai & A.H Soni. (1981) Accessible Region and Synthesis of Robot Ars. ASME Journal of Dynaic Systes, Measureent and Control. Vol. 13, pp , T. Yoshikawa, (1985) Manipulability of Robotic Mechaniss. Int. J. Robotics Research. Vol. 4, pp H. Asada, (1983) A Geoetrical Representation of Manipulator Dynaics and its Application to Ar Design. ASME Journal of Dynaic Systes, Measureent and Control. Vol. 5, pp J. A. Tenreiro. Machado & A. M. Galhano, (1997) A Statistical and Haronic Model for Robot Manipulators. In: Proc. of IEEE Int. Conf. on Robotics and Autoation, Albuquerque, New Mexico, USA, pp J. A. Tenreiro Machado, (1997) Analysis and Design of Fractional-Order Digital Control Systes. Journal Systes Analysis, Modelling and Siulation. Vol. 27, No 1, pp

11 288 Robot Manipulators, New Achieveents Y. Nakaura, K. Nagai, T. Yoshikawa, (1989) Dynaics and Stability in Coordination of Multiple Robotic Mechaniss. International Journal of Robotics Research. Vol. 8, pp T. J. Tarn, A. K. Bejczy, P. K., (1996) Analysis of the Dynaic Ability of Two Robot Ars in Object Handling. Advanced Robotics. Vol. 1, No 1, pp N. M. Fonseca Ferreira, J. A. Tenreiro Machado, (2) Manipulability Analysis of Two- Ar Robotic Systes. In: Proc. of IEEE International Conference on Intelligent Engineering Systes. Vol. 1, No 1, pp A. K. Bejczy and T. Jonhg Tarn, (2) Redundancy in Robotics Connected Robots Ars as Redundant Systes. In: Proc. of IEEE International Conference on Intelligent Engineering Systes, Vol. 1, No 1, pp M. H. Raibert and J. J. Craig, (1981) Hybrid Position/Force Control of Manipulators. ASME Journal of Dynaic Systes, Measureent and Control. Vol. 1, pp N. Hogan, (1985) Ipedance control: An Approach to Manipulation, Parts I-Theory, II- Ipleentation, III- Applications. ASME J. of Dynaic Systes, Measureent and Control. Vol. 17, No 1 pp. 1-24,. Kuar, Manish; Garg, Devendra P.(25), Neuro-fuzzy control applied to ultiple cooperating robots, Industrial Robot: An International Journal, Volue 32, Nuber 3, pp Ahin Yildiri, (25) A Proposed Hybrid Recurrent Neural Control Syste for Two Cooperating Robots, Journal of Intelligent and Robotic Systes, v.42 n.1, p Jufeng Peng, Srinivas Akella, (25) Coordinating Multiple Robots with Kinodynaic Constraints Along Specified Paths The International Journal of Robotics Research Vol. 24, No. 4, pp N. M. Fonseca Ferreira & J. A. Tenreiro Machado, (23) Fractional-Order Hybrid Control of Robotic Manipulator. In: Proc. of 1th IEEE Int. Conf. on Advanced Robotics, Coibra, Portugal. Vol. 1, No 1, pp N. M. Fonseca Ferreira, J. A. Tenreiro Machado, J. Boaventura Cunha, Fractional - Order Position/Force Robot Control. In: Proc. of 2 nd IEEE Int. Conference on Coputational Cybernetics, Vienna, Austria. Vol. 1, No 1, pp , 24. N. M. Fonseca Ferreira & J. A. Tenreiro Machado, (25) Fractional-Order Position/Force Robot Control. Journal of Advanced Coputational Intelligence and Intelligent Inforatics. Vol. 9, No 4, pp

12 Robot Manipulators New Achieveents Edited by Aleksandar Lazinica and Hiroyuki Kawai ISBN Hard cover, 718 pages Publisher InTech Published online 1, April, 21 Published in print edition April, 21 Robot anipulators are developing ore in the direction of industrial robots than of huan workers. Recently, the applications of robot anipulators are spreading their focus, for exaple Da Vinci as a edical robot, ASIMO as a huanoid robot and so on. There are any research topics within the field of robot anipulators, e.g. otion planning, cooperation with a huan, and fusion with external sensors like vision, haptic and force, etc. Moreover, these include both technical probles in the industry and theoretical probles in the acadeic fields. This book is a collection of papers presenting the latest research issues fro around the world. How to reference In order to correctly reference this scholarly work, feel free to copy and paste the following: N. M. Fonseca Ferreira, J. A. Tenreiro Machado and Jozsef K. Tar (21). Two Cooperating Manipulators with Fractional Controllers, Robot Manipulators New Achieveents, Aleksandar Lazinica and Hiroyuki Kawai (Ed.), ISBN: , InTech, Available fro: InTech Europe University Capus STeP Ri Slavka Krautzeka 83/A 51 Rijeka, Croatia Phone: +385 (51) Fax: +385 (51) InTech China Unit 45, Office Block, Hotel Equatorial Shanghai No.65, Yan An Road (West), Shanghai, 24, China Phone: Fax:

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