Terminal Sliding Control for a SCARA Robot

Transcription

1 Intenational Confeence on Contol, Instumentation and Mechatonics Engineeing (CIM '7), Joho Bahu, Joho, Malaysia, May 8-9, 7 Teminal Sliding Contol fo a SCARA Robot E. Muñoz, C. Gaviia, A. Vivas Depatment of Electonics, Instumentation and Contol, Univesity of Cauca [elenam, cgaviia, avivas]@unicauca.edu.co Abstact This pape pesents a teminal sliding contol law fo contolling obot manipulatos with paamete uncetainty, applied to a -DOF obot. By exploing some popeties of obot manipulatos, the poposed appoach simplifies significantly the standad scheme of a teminal sliding contol. Majo simulations esults using numeical values of an industial SCARA obot pove the effectiveness of the poposed appoach. The teminal sliding contol and the widely used computed toue contol ae compaed. Tacking pefomance in the task space and obustness (paametic uncetainty and distubance ejection) ae enlightened.. Intoduction The lineaizing and decoupling contol, so called invese dynamic contol [], [], is consideed to be the best theoetical solution fo the contol of obots. The success of this contol law is based on a good knowledge of the dynamic paametes: moe emote they ae, moe the esponse will be unsatisfactoy. In pactice, industial obots have a pioi unknown dynamic chaacteistics (stuctue, paametes) o involve seveal uncetainties such as lage distubances, inaccuacies, unmodeled dynamic, etc. Compensation fo model uncetainties fo obot tacking contol has eceived a consideable amount of attentions in the liteatue, and a detailed suvey can be found in []. In spite of the extensive and successful development of contol techniues such as obust contol [], and adaptive contol [5], [6], [7], the sliding mode contol appoach stays pobably the main choice when one needs to deal with non-paametic uncetainties and unmodeled dynamics. In the past yeas, a teminal sliding mode contol was poposed in [8], [9], []. Simila to the conventional linea sliding mode techniue, stong obustness with espect to lage uncetain dynamics can be achieved. By intoducing a nonlinea tem in the sliding contol and suitably designing the contolle, the ultimate tacking eo and the gain of the teminal sliding mode contolle can be significantly educed in compaison with the conventional linea sliding mode contol. One of the majo difficulties to achieve obustness to paametic uncetainties is the establishment of a pioi uncetainty bounds. In [] a teminal sliding contol law whee the uncetainty bound depends only on the inetia paametes of the obot, which can be easily estimated a pioi, was poposed taking advantage of stuctual popeties of obot manipulato. Accodingly to the last appoach, this pape pesents a tacking teminal sliding contol law applied on a SCARA obot, a fou DOF manipulato widely used in the industy. This study attempts to extend the nonlinea algoithm poposed in [] on an industial obot, with complex machining tajectoies defined in the task space. In ode to evaluate the pefomance of this nonlinea stategy, it will be compaed in tacking pefomance and obustness to the computed toue contol [], widely used in obotic contol. The pape is oganized as follows: in Section, a dynamical model of the obot is intoduced. Design of the sliding contol law is discussed in Section. Contol stategies fo this study ae shown in Section. Simulation esults ae given in Section 5. Finally, conclusions ae pesented in Section 6.. Dynamic Model fo the Robot It is well known that a obot manipulato can be modeled as a set of n igid bodies connected in a seial chain with toues/foces acting at the joints. By applying the Lagange euation of motion [9], the obot dynamic euation is deived as: M( ) C( ) F( ) G( ) τ d = τ () whee is the nx vecto of joint positions, τ is the nx vecto of applied joint foces/toues, M() is the 9

2 Intenational Confeence on Contol, Instumentation and Mechatonics Engineeing (CIM '7), Joho Bahu, Joho, Malaysia, May 8-9, 7 nxn symmetic positive-definite manipulato inetia matix, C(, ) is the nx vecto of centipetal and Coiolis toues, F( ) is the nx vecto epesenting toues due to fiction acting at the manipulato joints, G() is the nx vecto of gavitational toues and τ d is the nx vecto of unknown signals due to unmodeled dynamics and extenal distubances. In the absence of fiction o othe distubances, the dynamic model () of an n-link manipulato can be witten as: ( ) ( ) ( ) M C, G = τ () This euation can also be expessed in the following fom: ( ) ( ) ( ) = = ( ) M C, G τ Y,, θ () whee θ is a constant m-dimensional vecto of inetia paametes and Y is an nxm matix called egesso, which is a function of joint position, velocity and acceleation.. Teminal Sliding Contol Thee ae two diffeent appoaches to the chaacteization of dynamic systems: In linea systems theoy, one can assume eithe some stuctue in the signals o some stuctue in the system. Attempts have been made to combine these two appoaches e.g. hamonic identification techniues in the Fouie domain. Key facts on teminal sliding contol can be easily undestood in the single-input single-output case as follows. Conside the usual linea sliding suface [], s=x β x, with x the tacking eo and β R >. Then, on the sliding egime ( s= ), the dynamic eo β t becomes x(t) = x()e, which exponentially asymptotically conveges to zeo when time tends to infinity. Additionally, if the contol law is designed to accomplish with the eaching condition ss = - η s [], the suface is eached at eual speed given by s= ηsign(s) which implies the eaching of the sliding manifold in the finite time, s( ) t = () η It is clea that a lage η will speed up the eaching phase but the cost is an amplified chatteing in the sliding phase. The same stands fo the convegence time t s on the sliding egime. As it was poposed fo [] y [], a sliding suface of the fom: z z s=x β x (5) being z and z odd integes satisfying < z z, yields the dynamic eo on sliding egime: z z-z z z z -z z x(t) = βt x( ) (6) z which implies that x(t) becomes zeo in a finite time (teminal) given by: z-z z x( ) z t = s β(z z ) (7) Thus, once the sliding suface is eached, the convegence time can be impoved without inceasing the value of the η paamete and so educing the chatteing effect. Moeove, the discontinuous sliding contol tem ηsign(s) is eplaced by the continuous nonlinea tem, which in tun also educes the chatteing effect: z z ηs (8) The wok in [] poposes the so called fast teminal sliding mode (FTSM), which allows the impovement of the eaching time. [6] woks with the pevious concept but extended to the multi-input multioutput case fo n-link igid obot manipulatos descibed by the euation (), taking advantage of the stuctual popeties of obot manipulatos epoted by [8]. Conside the vecto sliding vaiable s = Λ p, with = d, being d the desied tajectoy and the joint position, Λ =diag[ λ λ n] >, p p T z n and p= as it was defined z befoe. The poblem is to design a contol law fo () assuming only the knowledge of a nominal paametes

3 Intenational Confeence on Contol, Instumentation and Mechatonics Engineeing (CIM '7), Joho Bahu, Joho, Malaysia, May 8-9, 7 vecto θ, and a bound on paametic uncetainty defined by ρ θo - θ. Imposing the euality s=, the euivalent contol which foces the obot dynamic to stay in s= can be obtained as: ( ) ( ) ( ) M C, G τ = Yθ (9) = - e with = - Λ p d Defining the vaiable s=-, the dynamic of the obot () can be witten as: which satisfies ( ) ( ) s θ M s C, Y = τ () τ = τ when s=s= e. The eade is efeeed to [6] fo the details of the poof that the teminal sliding contol law, which is defined as: ( )( ) τ Y,, θ w Ks () =, - o with defined as p fulfilling < p, <, K =diag [ k k n] >, and w designed fo compensate the paamete uncetainty epesented by θ = θ o - θ and given by: Desciption of SCARA obot: The simplest kinematics of the manipulato is found in the SCARA design, Figue. This design has thee vetical evolute joint axes, and one vetical pismatic joint at the end. SCARA obots ae mainly used fo pick-and-place opeations. Note that the end-effecto gives to the pat to be manufactued only one oientation, with is sufficient fo the majoity of the industial applications. The dynamic paametes of the link s obot ae composed of the inetial tenso paametes, the fist moment of inetia, the mass and the actuato oto inetia, that ae combined in the vecto θ : θ = [ XX j XYj XZ j YYj YZ j ZZ j () MX MY MZ M Ia ] j j j j j An identification pocedue gives the paametes values shown in Table, [5]. Some of these paametes ae gouped o eliminated fo simplicity [9], and will be consideed as the known paametes duing simulations. Table : Paametes of SCARA obot Paametes Values Paametes Values ZZG.8 MXG. ZZG.6 MY. ZZG. MYG. MXG Ia.5 T T T -ρy s/ Y s if Y s >, w= T if Y s = () Note: The units fo the inetia tenso paametes ae Kg.m, fo the fist moment of inetia Kg.m, and fo the moto inetia Kg.m. achieves unifomly boundedness of the signals and zeo tacking eo in the finite time. The linea sliding case stands fo p==.. Contol Stategies Figue. SCARA obot Computed toue contol: Assuming that only the motion is specified as desied position d, the computed toue contol (CTC), [], [9] computes the euied input contol as follows: ( ) w = K K p d v () whee Kp and Kv ae the contolle gains. The tuning of CTC contolle fo the SCARA obot leads to the gains shown in Table. The esulting scheme is shown in Figue, in which two feedback loops ae epesented; an inne loop based on the manipulato dynamic model and a oute loop opeating on the tacking eo. The function of the inne loop is to obtain a linea and decoupled input/output elationship, wheeas the oute loop is euied to stabilize the oveall system.

4 Intenational Confeence on Contol, Instumentation and Mechatonics Engineeing (CIM '7), Joho Bahu, Joho, Malaysia, May 8-9, 7 Table : K p and K v gains K p 5 5 K v 5 7 M G I Figue. Computed toue contol in task space Sliding contol: Figue shows the scheme of the teminal sliding contol to be applied on a SCARA fou DOF obot. The paametes of the unloaded manipulato ae known and the chosen values of the link paametes ae those shown in Table. Note that this contol stategy acts on a non-linea system while the computed toue contol linealize and decouple the model of the obot. In othe wods, the sliding contol law is applied on a moe complex system. d M G I x d x d d d d ( ~ p pλdiag du/dt p Λu Y s Kp Kv Figue. Sliding contol scheme in the task space The components y ij of Y(,,, ) x Ki Stabilizing linea contol W in E. () fo the SCARA obot ae given by (5). Whee d and d ae the length of the obot ams (.5 m and. m espectively), and e.g. C = cos( ), S = sin( ). In pesence of an unknown load caied pesent in the end-effecto of the obot, paametes of the Table will change and constitute the vecto θ. Fo the compaison of this contol law with the computed toue contol, the same contol paametes, K,Λ,p and ae applied fo the simulation tests. These s Non-linea compensation and decoupling w A ˆ Γ ( ) Robot Y Ku θ x Hˆ (, ) τ R o b o t paametes ae chosen as K=diag([ 5 ]), Λ=diag([ 5 ]), ρ=.5 and p==9/ y = ; y = ; y = ; y = y5 = C d y = d C d C C d S S ; ( ) ( )( ) dc ; y7 = ds( ) ( )( 6 y8 = ds ds C dc S ) ds y9 = y = y = ; y = ; y = ; y = y = ; y5 = Cd y6 = ( dc dcc dss dc) dc dc ; y7 = ds y8 = ( ds dsc dc S ds ) ds ds y9 = ; y = y = ; y = y = ; y = y = ; y5 = ; y6 = ( dc dcc dss dc) dc ; ; y7 = ; y8 = ( ds dsc dcs) ds ; y9 = ; y = y = ; y = y = y = y = y = y = y = y = y = y = ; y = Simulation Results (5) Computed toue contol and teminal sliding contol ae implemented in a simulation envionment (Matlab / Simulink ). Tacking pefomance and obustness ae compaed, unning with msecond sampling peiod. Tacking pefomance: The following paths, including inside the obot wokspace, will be tacked by the obot: a. Cicula path with adius = mm and angula velocity ω = ad/s. That means a small cicle achieved in seconds. b. Linea path with change of diection of 55º and linea velocity of v =. m/s, coveed also in seconds. Figue and Figue 5 show espectively the catesian tacking eo fo the tansient and stationay esponse, fo the computed toue contol. Figue 6 shows the tacking eo when the linea path changes the diection, fo the computed toue contol. Figue 7 and

5 Intenational Confeence on Contol, Instumentation and Mechatonics Engineeing (CIM '7), Joho Bahu, Joho, Malaysia, May 8-9, 7 Figue 8 show espectively the catesian tacking eo fo the tansient and stationay esponse, fo the teminal sliding contol, and Figue 9 shows the tacking eo when the linea path changes its diection, fo the teminal sliding contol x Figue. Tansient esponse fo CTC (cicula path) x Figue 7. Tansient esponse fo teminal sliding contol (cicula path) 7 x x Figue 8. Stationay esponse fo teminal sliding contol (cicula path) Figue 5. Stationay esponse fo CTC (cicula path) 6 x - x Figue 6. Catesian eo fo CTC (linea path) Figue 9. Catesian eo fo teminal sliding contol (linea path) Tansient fo the cicula path shows a catesian eo of x - m fo CTC contol and.9 x - m fo the teminal sliding stategy. Moeove, the time esponse is faste and without oscillations fo the teminal sliding contol. Stationay esponse shows a catesian eo of 8.5 x -7 m fo the CTC vesus 6

6 Intenational Confeence on Contol, Instumentation and Mechatonics Engineeing (CIM '7), Joho Bahu, Joho, Malaysia, May 8-9, 7 x -7 m fo the teminal sliding contol. In the case of a linea path with change of diection, the maximal catesian eo is.5 x - m fo CTC vesus x - m fo the teminal sliding contol. It is clea that the teminal sliding contol stategy poduces a bette behavio with espect to the tacking of the complex tajectoies defined in the task space. Paametic uncetainty: Model mismatch is obtained consideing up to 5% eos fom the known values given in Table. Cicula path is pogammed again as desied tajectoy. Figues and show espectively the catesian tacking eo fo the tansient and stationay esponse, fo the computed toue contol, and Figue and show the same tacking eo but fo the teminal sliding contol. CTC contol and x - m fo the teminal sliding stategy, without the oscillations that yields the CTC esponse. Stationay esponse shows a catesian eo of. x -6 m fo the CTC vesus 8 x -7 m fo the teminal sliding contol. Results show that teminal sliding contol povides best esults with paamete uncetainty, achieving good obustness.. x x Figue. Tansient esponse fo CTC (cicula path) with paamete uncetainty..5 x Figue. Stationay esponse fo CTC (cicula path) with paamete uncetainty. Tansient fo the cicula path with paamete uncetainty shows a catesian eo of. x - m fo Figue. Tansient esponse fo teminal sliding contol (cicula path) with paamete uncetainty. 9 x Figue. Stationay esponse fo teminal sliding contol (cicula path) with paamete uncetainty. Distubance ejection: An extenal distubance is intoduced to the system, which is defined as a joint distubance of.8 adians intoduced at. seconds, tested in egulation mode aound a given steady state position. Figues and 5 show espectively the distubance ejection fo CTC and teminal sliding contol. Maximum eo peak to peak is. adians fo CTC esponse vesus.5 adians fo teminal sliding contol. Teminal sliding contol is moe obust than CTC contol with espect to extenal distubances although the time esponse to these distubances is geate.

7 Intenational Confeence on Contol, Instumentation and Mechatonics Engineeing (CIM '7), Joho Bahu, Joho, Malaysia, May 8-9, 7 Distubance ejection (ad) Distubance ejection (ad) Figue.. Distubance ejection fo CTC Figue. 5. Distubance ejection fo teminal sliding contol 6. Conclusion Advanced motion contol in the task space fo a SCARA obot has been consideed in this study. Teminal sliding contol has been analyzed and applied to an industial obot. Contol law is based on popeties of obot manipulatos with a nonlinea tem in the sliding vaiable vecto, as is poposed in [6]. Tacking pefomance on complex tajectoies (cicula path with small diamete and linea path with change of diection) ae simulated fo the obust contol and compaed with a classical model based contolle as the computed toue contol. Also obustness is tested, consideing up to 5% eos in the nominal paametes of the model and intoducing an extenal distubance. Teminal sliding contol stategy shows bette pefomance in tacking eo and obustness on pogammed simulations. Since the CTC techniue is based on the assumption of pefect cancellation of dynamic tems, but the compensation may be impefect fo model uncetainty. Howeve in the teminal sliding contol, when in the sliding, the closed-loop esponse becomes totally insensitive to both intenal paamete uncetainties and distubances. Results show the geat capacities that teminal sliding contol has to solve poblems of contol in industial obotics. Futhe woks will concen test of the global fast teminal sliding mode contol, highe ode sliding modes and implementation on a eal SCARA manipulato. 7. Refeences [] H. Sage, M. De Mathelin,and E. Ostetag. Robust contol of obot manipulatos: a suvey. Intenational Jounal of Contol. Volume 7, No 6, pp: 98-5, 999. [] I. Kalaykov, and E. Iliev. Time-optimal sliding mode contol of obot manipulatos. In Poc. IEEE Confeence on Industial Electonics, Contol and Instumentation, IECON, Nagoya, Japan,. [] Z. Man, A. Paplinski, and H. Wu. A obust MIMO teminal sliding mode contol scheme fo igid obotic manipulatos, IEEE Tansactions on Automatic contol, Volume 9, No, pp: 6-69, 99. [] L. Sciavicco, and B. Siciliano. Modeling and contol of obot manipulatos. McGaw-Hill, New Yok, 996. [5] G. Caafloe, M. Indi and B. Bona. Robot dynamic calibation: Optimal excitation tajectoies and expeimental paamete estimation. Jounal of Robotic Systems. Volum 8, No, pp: 55-68,. [6] Y. Tang. Teminal sliding mode contol fo igid obots. Automatica. Volume, No: 5-56, 998. [7] F. Alonge, F. D Ippolito and F. Raimondi. Globally convegent adaptive and obust contol of obotic manipulatos fo tajectoy tacking. Contol Engineeing Pactice, Volume, No. 9, pp: 9-,. [8] M.W. Spong,. On the obuts contol of obot manipulatos. IEEE Tansactions on Automatic Contol. Volume 7, pp: , 99. [9] W. Khalil, E. Dombe. Modeling identification and contol of obots. Hemes Penton Science,. []. Egeland, and M. Godhavn. A note on Lyapunov stability fo adaptive obot contol. IEEE Tansactions on Automatic Contol. Volume 9, No 8, pp: 67-67, 99. [] J. Slotine, and W. Le. Applied nonlinea contol. Pentice-Hall Inc. New Jesey, 99. [] Yu X., and Z. Man. Fast teminal sliding-mode contol design fo nonlinea dynamical systems. IEEE Tansactions on Cicuits and Systems I: Fundamental Theoy and Applications. Volume 9, pp: 6-6,. [] S. Venkataaman, and S. Gulati. Contol of nonlinea systems using teminal sliding modes. Tansactions of the ASME Jounal of Dynamic Systems, Measuement and Contol. Volume 5, No, pp: 55-56, 99. 5