Mechanical Design and Kinematics Analysis of a Hydraulically Actuated Manipulator

Transcription

1 Send Order for Reprint to The Open Mechanical Engineering Journal Open Acce Mechanical Deign and Kinematic Anali of a Hdraulicall Actuated Manipulator Xuewen Rong Rui Song * Hui Chai and Xiaolin M School of Control Science and Engineering Shandong Univerit Jinan 00 China School of Mechanical and Electrical Engineering Shandong Jianhu Univerit Jinan 00 China Abtract: Thi paper give a mechanim deign of a i DOF hdraulicall actuated manipulator firtl. Then it DH frame and link parameter are given. Net it forward kinematic equation are derived according to homogeneou tranformation method. Fourthl the analtical olution of it invere kinematic are olved b given the poition and poture of the end-effector imultaneoul. The poture of the end-effector i given with three -- Euler angle for the have obviou geometr meaning and are ea to be meaured. In addition the correctne of the invere kinematic equation i verified in Simulink b comparing man et of randoml produced joint variable in workpace and their correponding invere olution. Keword: DH frame DOF Euler angle homogeneou tranformation hdraulic manipulator.. INTRODUCTION The traditional electric powered indutrial manipulator have been ued in all kind of indutrial production. The are alread ver mature in technolog and with high preciion. But the are not uitable for everal tak in evere environment for their electric drive mode and low powerma ratio uch a underwater contruction electric power line maintenance etc. In order to make up for the inadequac of electric powered manipulator ome hdraulicall actuated multi-joint manipulator have been developed in everal countrie. Uninterrupted power uppl ha become indipenable during the maintenance tak of active electric power line a a reult of toda highl information-oriented ociet and increaing demand of electric utilitie. The maintenance tak ha the rik of electric hock and the danger of falling from high place. Therefore it i necear to realie an autonomou robot tem uing electro-hdraulic manipulator becaue hdraulic manipulator have the advantage of electric inulation and high power-ma ratio [ ]. Dunnigan et al. [3-] introduce everal hdraulic manipulator working for underwater foret and contruction tak repectivel. Talor et al. [ 7] introduce two kind of hdraulic manipulator for nuclear indutrie. Wang et al. [8] introduce a hdraulic manipulator developed in China but there i no application cae given. Zhao Y. L. give two application cae of hdraulic manipulator uing in electric power line maintenance [9 0]. Thi paper give the mechanical tructure of a i DOF hdraulic manipulator in ection which i named HdM and developed b Shandong Univerit China. In ection 3 *Addre correpondence to thi author at the School of Control Science and Engineering Shandong Univerit Jinan 00 China; Tel: ; Fa: ; rong@du.edu.cn the forward kinematic equation are given firtl. Then the analtical olution of it invere kinematic are olved b given the poition and poture of the end-effector imultaneoul.. MECHANISM DESIGN OF HdM The mechanical tructure ketch of HdM i hown in Fig. (. HdM ha i rotar degree of freedom (DOF which are all driven b hdraulic actuator. The firt the fourth and the fifth joint are driven b helical rotar actuator. The econd and the third joint are driven b hdraulic linear clinder. The ith joint i driven b a miniature hdraulic motor through a worm reducer o it can turn continuoul. In fact the end-effector i a gripper and it opening and cloing movement i driven b a miniature linear clinder. The movement limit of each joint are lited in Table. The movement of all the i joint and the gripper are controlled b ervo valve which are mounted on a fourth joint third joint hdraulic clinder rotar actuator hdraulic clinder hdraulic motor ith joint end-effector hdraulic clinder econd joint rotar actuator hdraulic manifold firt joint Fig. (. The mechanical tructure ketch of HdM. rotar actuator fifth joint 87-X/ 0 Bentham Open

2 8 The Open Mechanical Engineering Journal 0 Volume 8 Rong et al. hdraulic manifold. HdM doe not integrate with a hdraulic power unit o it mut be upplied hdraulic power from an eternal hdraulic power unit through high preure hoe. d 0 θ θ Fig. (. The DH frame of HdM. 3. KINEMATICS ANALYSIS OF HdM 3.. The DH Frame and Link Parameter of HdM According to the DH rule [] the global frame {O 0 } and local frame {O i } fied to each link can be build a hown in Fig. (. Then the link parameter in DH frame of each link can be gained a lited in Table. The open interval in lat column how the motion range of each joint. 3.. Forward Kinematic of HdM The homogeneou tranformation method baed DH rule i the mot commonl ued method to olve the forward kinematic of manipulator or other comple erial mechanim. According to Fig. ( and Table it i ea to get the homogeneou tranformation matri between two adjacent link. The HdM manipulator ha i joint and even link including the mounting bae o there have i homogeneou tranformation matrice. The ucceive product of thee matrice decribe the poition and poture of the end-effector in global frame. And it i alo the forward kinematic equation of the manipulator hown a Eq. (. n o a p n 0 T 0 T T T 3 3 T T T o a p ( n o a p n + n + + n o + + o + o a a + θ 3 a p ( ( + θ a θ d d θ p ( ( p + d + d where i inθ i c i coθ i ij iθ i c ij co(θ i ijk iθ i +θ k c ijk co(θ i +θ k. Table. Link No. i Link parameter of HdM. a i- (mm α i- d i (mm θ i Joint Variable Limit 0 0 d θ θ ( θ θ ( θ 3 θ 3 ( θ θ (-0 70 a -90 d θ θ ( d θ θ ( Invere Kinematic of HdM For manipulator invere kinematic anali i ver important ince it i the bae of trajector planning and realtime control. It i the bet reult to obtain analtical olution of the invere kinematic problem for it can well meet the requirement of real-time control. In the cae of a erial manipulator with i degree of freedom it mut meet one of the following two condition to have analtical olution of it invere kinematic problem. One i that it ha three adjacent parallel joint. The other i that it ha three adjacent joint whoe ae interect at one point. HdM meet the firt condition o we can obtain analtical olution of it invere kinematic. Generall the poition and poture of the end-effector are given with homogeneou tranformation matri hown a Eq. (. And the vector [ ] T give the poition while the 3 3 matri at the upper left corner of 0 T! give the poture of the end-effector in global frame {O 0 }. 0 T! According to the homogeneou tranformation rule the given poition and poture of the end-effector decribed in frame {O } can be hown with Eq. (3. Simultaneoul the ame poition and poture of the end-effector can be hown with Eq. ( due to the forward kinematic. T! 0 T 0 T! ( ( (3

3 Kinematic Anali of a Hdraulicall Actuated Manipulator The Open Mechanical Engineering Journal 0 Volume 8 9 T T T 3 3 T T T + + d c + d The Eq. (3 and ( are all decribe the poition and poture of the end-effector in frame {O } o the correponding element of the two matrice hould be equal to each other. The three element ( ( and (3 in homogeneou matri decide the poition of end-effector while a element ( 3 ( 3 and (3 3 decide the poture of ai of the end-effector in a given frame. But the three element ( 3 ( 3 and (3 3 are not independent o an element in other column hould be elected to decide the poture of - or -ai of the end-effector. The even pair of correponding element ( ( (3 ( 3 ( 3 (3 3 and ( in matrice T and T! are elected and lited a following: p ( d p ( + d p (7 a + a (8 a + a (9 a (0 n + n ( From Eq. ( and (9 we have ( a + a d p ( θ arctan p a (3 p a From Eq. (9 we have θ arcco( a + ac a ( ( Since inθ 0 o Eq. (8 and (0 can be re-written a c + a ( a ( Subtitute Eq. ( and ( into Eq. ( and (7 + a + ( + (7 + a + d + (8 Define m + a + ( + n + a + d + Then Eq. ( and (7 can be re-written a m (9 n (0 We can get an equation onl included b adding the two ide quare of Eq. (9 to thoe of Eq. (0 repectivel. m + n θ 3 arcco m + n ( ( Eq. (9 and (0 can be tranformed to the following two equation due to the trigonometric function algorithm. ( 3 m (3 3 + ( n ( From Eq. (3 and (we can eliminate and get! (a + ( 3 c m(a + a c + na ( θ arcco m( + n 3 ( + ( 3 ( According to the homogeneou tranformation method the poition and poture of the end-effector decribed in frame {O 3 } can be derived a the following matri: 3 T 3 T T T + + (7 a + d d p! T 3 T 0 T 0 p! 3 ( (8

4 0 The Open Mechanical Engineering Journal 0 Volume 8 Rong et al. Eq. (7 how the given poition of the end-effector decribed in frame {O 3 }. It hould be equal to the fourth column of the poition and poture matri deduced from the forward kinematic hown a Eq. (8. Therefore we can get two equation a below: + (a (9 (a + d 3 p p (30 Define u v 3 p p Then Eq. (9 and (30 can be re-written a (a u (3 (a + d v (3 From Eq. (3 and (3 we can eliminate and get (a + d v(a ud (33 Then θ arcin v(a ud (a + d (3 From Eq. ( we can get θ a θ ±arcco n + n c (3 According to the DH rule the plu ign hould be elected for the right ide of Eq. (3 if co( i nonpoitive otherwie the minu ign hould be elected. And co( can be derived from the following equation: co( co( T R 3 T R T R T 3 R 0 T R co( ( + ( + + ( + ( The Given Poition and Poture of the End-Effector (3 The end-effector hould approach to the objective in an optional poture and a erie of planned poture in proceing procedure. So the poition and poture of the end-effector are known and the are generall given with a homogeneou tranformation matri hown a Eq. (. The poition vector [ ] T can be given directl according to the poition of the objective in global frame {O 0 }. The poture of the end-effector can be given with -- Euler angle hown in Fig. (3 []. In fact α and β have obviou geometric meaning and are ea to meaure. α i the included angle between ai 0 and the projection line in 0 0 plane of ai. β i the included angle between ae 0 and. From Fig. (3 it can be concluded that the frame {O } i tranformed from the original poture duplicated with the global frame {O 0 } through three rotation tranformation. The firt one make {O } to rotate α around ai to { O!}. The econd one make { O!} to rotate β around ai! to { O!!}. The lat one make { O!!} to rotate γ around!! to { O!!! }. The poture of { O!!! } i the given poture of the endeffector which i hown a Eq. (37. Fig. (3. -- Euler angle R! Rot(αRot( βrot(γ cαcβcγ αγ cαcβγ αcγ cαβ ( + ( cαβcγ + cαγ αcβγ + cαcγ αβ + ( βcγ βγ cβ * The Correctne Verification of the Invere Solution (37 The block diagram in Simulink for verifing the correctne of the invere olution i hown in Fig. (. Firtl a certain number of et of joint variable are produced randoml in their limit. Then the correponding Fig. (. Block diagram for invere olution verification. 0

5 Kinematic Anali of a Hdraulicall Actuated Manipulator The Open Mechanical Engineering Journal 0 Volume 8 poition and poture of the end-effector are calculated. The net procedure i to obtain the correponding invere olution through the invere kinematic equation. The lat procedure i to compare the invere olution and the random joint variable. If the are equal to each other within a certain preciion it can be concluded that the invere olution are correct. CONCLUSION Thi paper give a mechanim deign of a i DOF hdraulicall actuated manipulator which ha ome obviou advantage compared to traditional electric powered manipulator. Thoe are with high power-ma ratio electric inulation uitable for evere tak etc. So it can be ued in underwater electric power line maintenance foret contruction nuclear and other indutrie. Then the analtical olution of it invere kinematic are olved b given the poition and poture of the end-effector imultaneoul. The poture of the end-effector i given through three -- Euler angle for the have obviou geometr meaning and are ea to be meaured. The olving method of invere kinematic and the poture given method of the end-effector can provide important reference to thoe with imilar configuration to it. CONFLICT OF INTEREST The author confirm that thi article content ha no conflict of interet. ACKNOWLEDGEMENTS Thi work i upported b the High Technolog Reearch and Development Program of China under grant no. 0AA00. REFERENCES [] K. Ahn and S. Yokota Application of dicrete event control to the inertion tak of electric line uing -link electro-hdraulic manipulator with dual arm JSME International Journal Serie C: Mechanical Stem Machine Element and Manufacturing vol. no. pp [] L. Q. Cai Hdraulic manipulator tem for live electric line operation Contruction Machiner for Hdraulic Engineering Power Station vol. no. pp (In Chinee [3] M. W. Dunnigan D. M. Lane A. C. Clegg and I. Edward Hbrid poition/force control of a hdraulic underwater manipulator IEE Proceeding: Control Theor and Application vol. 3 no. pp [] P. M. La Hera U. Mettin S. Weterberg and A. S. Shiriaev Modeling and control of hdraulic rotar actuator ued in foretr crane In: Proceeding of the IEEE International Conference on Robotic and Automation 009 pp [] R. F. Abo-Shanab and N. Sepehri On dnamic tabilit of manipulator mounted on mobile platform Robotica vol. 9 no. pp [] C. J. Talor and D. Roberton State-dependent control of a hdraulicall actuated nuclear decommiioning robot Control Engineering Practice vol. no. pp [7] M. J. Bakari K. M. Zied and D. W. Seward Development of a multi-arm mobile robot for nuclear decommiioning tak International Journal of Advanced Robotic Stem vol. no. pp [8] X. Y. Wang R. Liu and G. Z. Jia On hdraulic manipulator control with generaliation pule code modulation Chinee Journal of Mechanical Engineering vol. 38 (upp. pp (In Chinee [9] Y. L. ZhaoS. Y. LuX. C. LvJ. Liand Z. L. Wang Deign of force feedback hdraulic arm for live working robot Chinee Hdraulic Pneumatic no. 8 pp (In Chinee [0] Y. L. ZhaoQ. ZhangH. QiJ. Liand Y. C. Li Deign of non-force feedback hdraulic arm for live working Machiner Deign Manufacture no. 9 pp (In Chinee [] J. J. Craig Introduction to Robotic: Mechanic and Control (3rd Edition Prentice Hall Pre 00. Received: March 0 Revied: M 0 Accepted: Ma 0 Rong et al.; Licenee Bentham Open. Thi i an open acce article licened under the term of the Creative Common Attribution Non-Commercial Licene ( b-nc/3.0/ which permit unretricted non-commercial ue ditribution and reproduction in an medium provided the work i properl cited.