Vietnam Joumal of Mechanics, VAST, VoL 38, No. 2 (2016), pp. 81 - 88 001:10.15625/0866-7136/38/2/5958
INVERSE KINEMATICS OF A SERIAL-PARALLEL ROBOT USED IN HOT FORGING PROCESS
Chu Anh My
Le Quy Don Technical Universiti/, Hanoi, Vietnam E-mail: [email protected]
Received March 12,2015
Abstract. In hot extrusion forging process, the use of robot arm for transferring heavy billets can reduce downtime, improve productivity, reduce worker fatigue and optimize the use of energy and manpower. To increase tiie stiftiess of the robot and force the end- effector move in directions parallel with the ground surfece, two parallel links are added to a standard serial manipulator. This modification of the structure could make it a bit of challenge in the system modelling and controlling. This paper addresses the inverse kinematics analysis that is the central issue for developing autonomous control modes of the robot application
Keywords: Serial-paralell robot, mverse kinematics, hot extrusion forging.
±. I N T R O D U C T I O N
F o r g i n g is a m a n u f a c h i r i n g process i n v o l v i n g t h e s h a p i n g of m e t a l u s i n g locaUzed compressive forces. F o r g i n g is often classified according to the t e m p e r a t u r e a t w h i c h it is p e r f o r m e d : "cold", "warm", or " h o t " forging. C o n s i d e r a h o t forging process l a y o u t i n Fig 1
Fig.l. Layout of forging station fig. 2. Forging process
© 2016 Wetnam Academy of Science and Technology
A steel bfflet weighteti about 60 kg is heated up to 1100°C ta the healmg fui and it is then h-ansferred to fhe forging machme. The forging machine is a h y d i a u i S press system used for fhe extrusion process. After the billet located ta a die fixed taflii machtae, it is pressed under the force F by the hydraulic press equipment, to product^t ^ forgmg part (see Fig. 2).
Smce bUlets are hot and heavy, two workers are inquired to cany and trai bfflets one by one for the process. This causes instabihty, wtJrker.fct^e alKl dtjsv m the manufachiring. To overcome these disadvantages, an mdustrial'robotics s shown m Fig. 3 is designed to support the workers.
Fig. 3. Robot design for hot forging process
n'"°botc°i;sistsof6jotats,Slmksandonegiipper(theend-effector);andallsy8- tem is actaated by hydrauHc cyltaders. The first jomt connecttag the manipulator wifK the chass^ ts a translation jomt; all the other jotats are prismattc. The hnki connecting jomt 3 and 4, and connecttag jomt 4 and 5 are parallel mechanisms. This structural m o l iftcabon accompanied with the hydrauUc achiators aim to reinfotre the stifiness of the system stj that the robot is capable of carrymg the heavy billet In addition, the orien- tation of the end-effector is restricted as desiied; the end-link moves ta only direcliom parallel wtth the ground surface. This advantage could help to reduce the compMly of the control process Though the hybrid design takes fuB advantages of flexibiUty of fte senal links and makes foU use of the reinforced stiffness of the added paralel mecha- nisms. It possesses complexity ta modelltag and controlltag.
^°' controlltag the robot, the ktaematical modelltag and analysis play a cenlral role ftat needs to be considered specificaUy The kmematic of general manipulator has been
* e fundamental problem [1], ta which the serial chata of links is considered as the stan- dard kmemahcal model. Either Danavit-Hartenberg method or Craig method is usually used for the modelltag. Further studies on the manipulator ktaematic could be found in the hterahire such as the kmemahcs for the redundancy slructares [2], the kinematic of
Inverse tinemallcs if a serM-pamllel robot used In boi fining process a j
the parallel robot [3,4], ktaematic performance analysis [5], kmemahcs of manipulator mounted on a mobile plaleform [6], and the ktaemattcs design of manipulator [7]. How- ever, a few researcher mterests ta modeUng and analyzmg the hybrid serial - parallel robot structures.
This paper addresses the modelltag of which the ktaematical constratat of the par- allel links is taken tato account. A numerical solution to the taverse ktaemalics problem IS proposed. Finally, given the h-ajectory of the end-e£fector, the time history of jotat vari- ables are determtaed that can be used for the conb-olhng process.
2. KINEMATICS MODELLING
to Fig. 4, we define d,, cj2, q^, qt, qs and qs are jomt variables which determme the configuration of the mechanical system. We denote q = [ til q2 q, qi 05 q^ f Due to the paraUel motion of the two parallel finks, the relationship between q, q. and
?5 can be illustrated ta the followtag Fig. 5.
Fig. 4. Kinematical model of serial parallel robot As shown ta the Fig. 5, the followmg constiam can be write
n o t e d 2 " h : Z i ^ ; ° ^ . ^ " r . tatb'f ' S S S ^ a T * " " t T ^ f ^ ^ ^ " ^ " ^ ^ - respect to all finks Jfiiustrated. Notic^ ^ S l ^ n l t r ^ ^ Z T i j S r to m ^ e It easier m writing homogeneous transformation matrixes in the to^ „f rt, A . Hartenbeig-s formulation. 0„ . (Oo^oyozo) is d r S ' ^ L t f ^ ^ J f r ^ e ^ r i h e
Fig. 5. Geometrical relationship between 173,^4 and ^5 Table 1. Kinematical and geometrical parameters of links Link
V 1"
1 2 3 4 5' 5 6
«l 0 0 0 12
?3 94 15 0
*
dl dt 0
"1 i2 0 0 0 d.
ds Hi 0 0 0
«2 03
m
«5 0 0
oi- 0
-nil
0 7r/2
0 0 - 7 1 / 2
0 0
same way, Bie M o w t a g notations O^.O^ Oj are lhe fink frames, correspondtagly.
The ft-amesO;,OJ,0; are added for the refenctag purpose.
The homogeneous tiansfonnation matrix of the end-effector with respect to fhe reference frame can be denoted as
'tTl- (2)
HOE =
m?St',ff ^ ' ""^i ^i =^':1'^ is tite position vector of ttie end-effector; AoE is the rotation matiix of ttie end-effector witti respect to ttie reference frame.
l o w t a g ^ l S M p " " ^ * ' ' " ' ' " ' " ' ° " ° ' * " ^-d-effeclor can be determmed via tttefol- HoE = H„,, (ti,) H,.,Hi2 fe) H23 (,J3) H34 (u) H45. fe)HysHss (qt), (3)
^ i ^ ^ f 5 ' i L * " " l o ^ e o i e o u s tiansfonnation matiix defined for ttie homogeneous mo- tion of ttie frame mdexed i witti respect to ttie previous frame indBted/.
ImxTse MnEmalics ofa senal-parallel mbat used m hal forging process
COS 01 -^ndicosui sm^isma,- ii,-cos0,- sindi cos 9i cos Iti — cos 0; cos ii:,- fl,sta0,-
0 stall;; cos it, tii 0 0 0 1
(4)
Substihiting all ttie matibces H;, yielded ta accordance with Tab. 1 mto Eq. (3) obtatas [ AoE IE
0 1 Ao6(q) Im (q)
0 1 (5)
Eq. (5) and constiatat Eq. (1) describe ttie forward relationship of flie robot. Given a value of q, ttie position and ttie orientation of ttie end-effector can be calculated analvt-
ically. -' If we denote f (q) = ro6 (q) as ftmction vector of q, for flie position forward ktae-
mahcs, the foUowmg equation can be yielded
= f(q)- 3. INVERSE KINEMA-nCS
(6)
To contiol ttie robot operating as desired, flie taveise ktaematics computation must be taken mto account Generally, given flie pafli of ttie end-effector, all flie histories of ttie jomt vanables must be calculated as
1 = ' " ' (n) • (7) f * ^""i*"" ™"t'''<'«'^ '"'"'«' * ^ orientation of ttie end-effector is regardless because
of ttie fahtre of ttie paralel stiuchire mentioned previously ta addition, as discussed before ttie robot tiansfers flie billet for lorgtag process ta discontinuous sequence. There- fore flie continuity of flie tool patt, is not reqmred. The pafli of flie potat r£ of flie end- effector is lUustiated ta the foUowmg Fig. 6. r £ ta.u
Fig 6. Required path of the end-effector
As seen m Fig. 6, flie pafli can be separated tato 6 arts (Acl,..., Ac6) rroresentine 6 periods of time ttiat the robot hilfills flie required task. To S B t ^ % the formulation and implementation of ttie taverse ktaematics, Eq. (7) is solved acooiArg to the separated 6
peritxls of time as foUows. '. "' . Period 1. The robot moves in ZQ direction only to locate the end-effector aM gets mulyfiyrpiainA
The geometrical parameters are given as: di = 0.S m, aj = O.Il m, 1J2 = Q 25 m,\
02 = 0.1 m, 03 = 0.73 m, 04 = 0.63 m, U5 = 0.18 m, i s = 0.03 m, lis = 0.43 m, s The initial values of jotat variables are given as foUows: dt = 0, ^2 = 0, 03 = I
3-, 94 = ^,96 = 0. The end-effector moves so that j:£ = 0.4, y^ = Q.^ and 2£ = 0.5f, '.
whereteSRisaparameterandf 6 [0,1J. '. ''.
to this period, only di varies. FtaaUy we obtata di = 05i. • 1 Period 2. The robot transfers the iillet to the heating furnace ' ']
ta this period of time, IE can be represented as
^£(') = -0.124i2 + o.8f-|-0.4 y£(f) = -0.92(2 + 1.24f -\- 0.88 . 2E(() = 0.5
Solvmg Eq (7) yields d^ = 0.5, ,2 = 0, ,5 = 0, and ,3, ?4 and ,5 vary as shown ta Fig. 7. , I
4
\
It
• • ; " ^ < . ;
J
r"
" " , " "
Fig. 7. Values of q,, ,4 and ^5 for ttie period 2 Periods, 'ne robot picks up the billet heated and mcyoes it outside thefum^
to ttus penod of time, r£ can be represented as :>:E = 1.324 - yc = 1.2 2E = 0.5
0.9St (. 2E = 0.5
S l ra,™/!^"^'''• , r °i.-^'* = ' ' ' * = ' ' ' ^ " ^ 93'94 and ,5 a« shown ta Fig. 8.
^ ^ 4 . The robot moo^ the billet up and gets ready to turn tran^ng the billet tothejingmg
Imxrse kinematics ifa serlel-paraM roiwf used In lint forgmg pro,
Fig. 8. Values of qs, q^ and qs for the period 3 to this period of time, rn can be represented as
{ iE= 0.374 { yE = 0.2f-l-1.2 { ZE = 0.5
Solvtag Eq (7) yields tJ, = 0.5, q^ = 0, t,, = 0, and ,3,94 and q, as shown ta Pig. 9.
Fig. 9. Values of q,, „ and q^ for the period 4
.!'?'• * i s period, only 1,2 varies from Oto 180°. After ttial«( varies fom, nto on- All ottier jotat variables keep Uieir values as calculated ta flie last p e r i S
Pmtri 6. Tte rote reocte to (te posttiot, ready to release the billet on the forging machine to Bus penod of time, TE can be represented as ^ macnme
\ !£(() = 0.576(2 - 1.452f _ 0 374
< yE(t) = 0.12(2-0.41+ 1.4 (, ZE(() = 0.5
Solvtag Eq. (7) yields d, = 0.5, ,2 = TT, „ = | , and ,3, ,4 and ,3 as shown m Fig 10.
1 V
Fig. 10. Values of 93,94 and qs for the period 6
For 6 given segments of the required patil, the solution to the taverse position kine- matics is found ttiat is appUcable to contiol the jotat actiialors. The values of the joint variables change ta feasible ranges.
4. CONCLUSION * The ktaematical model of the serial-paraUel robot is formulated. The ktaematits '
equation of ttie considering robot must satisfy ttie constiatat equation because of ttie motionconstiataed of ttie paraUel finks. This is diffeient from ttie typical serial robot.
The end-effector moves always paraUel witti ttie ground surface ttiat could simplify ttie contiol prtjcedure. Moreover, ttus is to make sure fliat ttie griper holds flie bfflet in tixed poshire during ttie tiansferring time period.
The numerical results of ttie mverse equations are suitable ta value and flie curves show Its proper change ta shape and magnitude.
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