Mechatronic Servo System Control - M. Nakamura S. Goto and N. Kyura Part 13 pptx

174N omenclature 2.2 Reduced order model of one axis in amechatronic servosystem Sym bo ls Units Meanings G c 1 ( s ) normalized lowspeed 1st order model G c 2 ( s ) normalized middle sp eed 2nd order model K p 1 1/s position loop gain of lowspeed 1st order model K p 2 1/s position loop gain of middle speed 2nd order model K v 2 1/s ve lo cit yl oo pg ain of middle sp eed 2nd order mo del c p 1 1/s position loop gain of normalized lowspeed 1st order model c p 2 1/s position loop gain of normalized middle speed 2nd order model c v 2 1/s velocityloopgain of normalized 2nd order model 2.3 Linear model of the working co ordinates of an artic ulated robot arm Symbols Units Meanings θ 1 rad Angle of 1st axis θ 2 rad angle of 2nd axis p x m position of Xaxis p y m position of Yaxis l 1 m length of 1st axis l 2 m length of 2nd axis ∆T s reference input time interval λ ( t ) s λ ( t )=t +(e K p t − 1)/K p ˆp x m position of Xaxis in working coordinate model ˆp y m position of Yaxis in working linear model v 1 rad/s velocityof1st axis v 2 rad/s velocityof2nd axis v x m/s velocityofXaxis v y m/s velocityofYaxis  x m/s error at the Xdirection of working linear model  y m/s error at the Ydirection of working linear model v m/s objectivevelocity 3. Discrete time interval of amechatronic servosystem 3.1 Sampling time interval Symbols Units Meanings G 1 ( s ) transferfunction of 1st order system G L 1 ( s ) transferfunction of 1st order system with time delay G P 1 ( s ) transferfunction of 1st order system with time delayPade ap- proximation f 0 c Hz Sampling frequency L 1 s sum of required time from statesample of position loop to con- trol input calculation and delaytime in 0th order hold of control input f cP Hz cut-offf requency of transfer function of 1st order system with time delayPade approximation f c 1 Hz cut-offfrequency of transfer function of 1st order system ∆t p s sampling time interval q 1 q 1 = qL 1 /∆t p f s Hz f s =1/∆t p Nomenclature1 75 3.2 Relation between reference input time interval and velocityfluctuation Sym bo ls Units Meanings p s 1 1/s pole of 2nd order model p s 2 1/s pole of 2nd order model e s v maximal constantvelocityfluctuation e t v maximal transientvelocityfluctuation h r 0th order hold in reference input generator u p position command h p 0th order hold in position command part u v velocitycommand r ob jectiv et ra jectory v re f objectivevelocity n pv n pv = K v /K p gain ratio 3.3 Relation be twe en reference input time in terv al and lo cus irregularit y Symbols Units Meanings K px 1/s position loop gain of x axis of 1st order model K py 1/s position loop gain of y axis of 1st order model J Jacobian matrix p z m position of z axis K pz 1/s position loop gain of z axis of 1st order mo del 4. Quantization error of amechatronic servosystem 4.1 Enco der resolution Symbols Units Meanings ∆N rev/min amplitude of velocityfluctuation f r s frequency of velocityfluctuation R E pulse/rev enco der resolution N max pulse/s maximal velocityofservomotor R N R N = ∆N /N max ve lo cit yfl uctuation rate V ref pulse/s command velocity 176N omenclature 4.2 Torque resolution Sym bo ls Units Meanings P ref pulse objectiveposition E s p pulse position decision error ∆t v s sampling time interval of velocityloop T d s timeofangular velocityoutput belowobjectivevelocity T u s timeo fa ngular ve lo cit yo utput ove ro bj ectiv ev elo cit y V d pulse/s velocityofangular velocityoutput belowobjectivevelocity V u pulse/s velocityofangular velocityoutput overobjectivevelocity E d pulse maximal position deviation of angular velocityoutput below objectivevelocity E u pulse maximal position deviation of angular velocityoutput overob- jectivevelocity T f s pe rio do ffl uctuation E r p pulse amplitude of position fluctuation E r v pulse/s amplitude of velocityfluctuation R p pulse/s 2 angular acceleration resolution upper boundary satisfying am- plitude condition of position output error R v pulse/s 2 angular acceleration resolution upper boundary satisfying am- plitude condition of angular velocityoutput error R A pulse/s 2 angular acceleration resolution R T Nm torqueresolution T max Nm maximaltorque B bit bit number corresponding to torque resolution 5. Torque saturation of amechatronic servosystem 5.1 Measurementmethodfor the torque saturation property Symbols Units Meanings a m/s 2 input acceleration t M s momentoftorque taking maximal output sat( x ) saturation curve Nomenclature1 77 5.2 Contour control methodwith avoidance of torque saturation Sym bo ls Units Meanings A max m/s 2 maximal acceleration  m working precision V m/s command tangential velocity r x ( t ) m objectivetrajectory at the direction of x axis r y ( t ) m ob jectiv et ra jectory at the direction of y axis w x ( t ) m input considering working precision at the direction of x axis w y ( t ) m input considering working precision at the direction of y axis u x ( t ) m revised input at the direction of x axis u y ( t ) m revised input at the direction of y axis r min m r min = V 2 /A max possible minimal radius of circular traject ory of movementfor maximal acceleration r m circular radius satisfyingw orking precision  V m m V m = √ A max r velocitysatisfying maximal acceleration A max when drawing radius r a max m/s 2 maximal angular acceleration θ c 1 rad angle with x axis of objectivelocus 1 θ c 2 rad angle with x axis of objectivelocus F ( s ) modificat ion term 6. The modified taughtdata method 6.1 Mo dified taughtdata methodusing amathematical model Symbols Units Meanings r ( t ) m ob jectiv et ra jectory G 1 ( s ) transferfunction of 1st order model F 1 ( s ) mo dification term based on 1st order mo del G 2 ( s ) transferfunction of 2nd order model F 2 ( s ) modificat ion term based on 2nd order model ω c rad/s cut-off frequency γ 1/s pole of pole assignmentregulator by 1st order model K s feedbackgain of pole assignmentregulator φ m rad maximal phase-lead value of modification term ω m rad frequency of maximal phase-lead va lue of mo dification term γ i 1/s pole of pole assignmentregulator by 2nd order model µ 1/s po le of observ er by 2nd order mo del 178N omenclature 6.2 Mo dified taughtdata methodbyusing aGaussian network Sym bo ls Units Meanings φ ( x ) output of Gaussian network w i weightofGaussian network ψ i ( x i ) Gaussian unit M number of Gaussian units x i input to Gaussian net wo rk m i mean of Gaussian unit σ i Variance of Gaussian unit x max linearapproximation region of Gaussian network x p max m constant determining linear appro ximation region of po sition x v max m/s constant determining linear approximation region of velocity x a max m/s 2 constant determining linear approximation region of accelera- tion E rms lose functionoflearning of Gaussian network E l factors of lose function of learning of Gaussian network ( u k , x k ) taughtdata for learning of Gaussian network K number of taughtdata p parameter of Gaussian network p new i modified parameter of Gaussian network p old i parameter of Gaussian network before modification η learning rate of Gaussian network 6.3 Mo dified taughtdata methodfor aflexible mechanism Symbols Units Meanings R ( s ) objectivetrajectory U ( s ) taughtdata Z ( s ) position of fulcrum of arm Y ( s ) output G 3 ( s ) overall transfer function of control system F 3 ( s ) modificat ion term 7. Master-slave synchronous positioningcontrol 7.1 The master-sla ve sync hronous po sitioning con trol metho d Symbols Units Meanings k c sloping ratio be twe en two axes of ob jectiv et ra jectory 7.2 Contour control with master-slave synchronous positioning Symbols Units Meanings v x s velocityinput to master axis(x axis) e mm lo cus error ˆ F ( s ) modificat ion term if existing modeling error R y R y =(K py + ∆K py ) /K py co efficientfor modeling error evalua- tion Experimental Equipments The main experimental deviceusing in the experimentofthis book areillus- trated. E.1 DEC-1 DEC-1 (made by Yahata Electric MachineryInc.)using in section 2.1, 2.2,3.2, 3.3,4.1, 5.1,5.2 is shown in Fig.E.1. Its specifications are given in table E.1. DEC-1 is composed of servocontroller,servomotor,coupling as mechanismpartaswell as loadgenerator. This experimental device is made from the DC serv om otor ands erv oc on troller useda ctually in industry. It is equivalenttothe driving part or mechanismpartadoptedineachaxis of (a) Profile (citation from catalogue) P o s i t ion c ontroller PC V eloc i ty c ontroller S e rvo a mplifier M o t o r S oft c o u pling M e c h a nis m p a rt (loa d) S e rvo c ontroller (b) Outline structure Fig. E.1. DEC-1 180E xp erimen tal Equipmen ts Table E.1. Specification of DEC-1 rated outputofmotor kW 0.2 rated torque of motor kgm 0.195 rated velocityofmotor rev/min 1000 inertia momentofmotor axisJ M kgm 2 0.00224 inertia momentofmechanism partJ L kgm 2 0.00653 natural angle frequency of mechanism partω L rad/s 94.2 damping rate of mechanism partζ L 0.002 enco der resolution pulse/rev 2000 gear deceleration ratioN G 1 mechatronic servosystem, suchasindustrialrobot, working machine, etc If the analysis or control problems of mechatronic servosystem usingthis device canbesolved,itispossible to analyze theimprovementofcontrol performance of thegeneral industrial mechatronic servosystem regulated for having similar prop ertiesineachaxis, and concrete its improvementstrategy.Motor of DEC- 1and load generator areconnectedbysoft-coupling.Inthis experimental device, velocitycontrol part, current control part, poweramplifier part in servocontroller arestructured by hardwareanaloguecircuit. Position control part is structured by software in computer. Therefore, velocityloopgain K v is needed to be changed with theregulationofchangeable resistance.Position loop gain K p can be easily changed in software of computer. E.2Motoman The profile of Motoman(made by Yaskawa Inc.) used in section 2.3,6.1 is shown in Fig.E.2 and its specifications are given in table E.2, respectively. Motomanisanindustrialarticulatedrobot arm. Itstranspor table weigh is from 3to150[kg]. It is classified fromK3toK150. Most of industrialrobot arms including Motomanare movedaccording to the designated taughtposition seriesand their velocity. The robot arm usingt eac hingb ox is mo ve db yt augh tp osition andh ence its po sition mu st be memorized. The taughtvelocityisgiven by keyinput in operation panel. After given all position and velocity, robotarm will move when pushingplay ke yo fo pe ration panel. E.4X YT able 181 Fig. E.2. Profile of Motoman (citation from catalogue) Ta ble E.2. Sp ecification of MotomanK 10 (a) Overall specification degree of freedom 6 precision of repeated PTP control mm ± 0.1 powercapacity kVA 8 transportable weigh kg 10 bodymass kg 300 (b) Specification of eachaxis 1a xis 2a xis 3a xis 4a xis 5a xis 6a xis length of arm mm 200 600 770 - 100 - maximal ve lo cit y rad/s 2.09 2.09 2.09 4.59 4.59 6.98 E.3 Pe rformer MK3S The profile of PerformerMK3s (made by Yahata Electric MachineryInc.) usedinsection 5.2 is shown in Fig.E.3 and its specification is given in table E.3. In order to be able to construct controller freely in PerformerMK3S, in the authors’ laboratory,velocityloopisconstructed by hardwareinservo controller. Nevertheless, theposition loop is rebuilt to be able to construct in computer. Therefore, position loop gain can be set freely in computer. E.4 XY Ta ble XY table (made by Yaskawa Inc.) used in section 6.2,7.1, 7.2 is shown in Fig. E.4 and its specification is given in table E.4. XY table is the device used for transferring knives of working machine because of its independentmove- mentof x axisand y axisaccordingtothe ball spring installed in two servo motors, respectively.For makingsimilar of XY table with PerformerMK3S, velocityloopisconstructed by hardwareinservocontroller andposition loop is constructed in computer. Therefore, position loop gain can be set freely in computer. 182E xp erimen tal Equipmen ts Fig. E.3. Profile of Pe rformer MK3S (citation from catalogue) Table E.3. Specification of Performer MK3S (a) Ov erall sp ecification degree of freedom 5 driving properties V/pulse 5.0[V]/(2048[pulse/rev]× 3000[rpm]/60[s]) detection properties V/pulse 0.5[V]/(2048[pulse/rev]× 1000[rpm]/60[s]) enco der resolution pulse/rev 8192 transp ortable mass kg 2(maximal velocity), 3(lowvelocity) bodybrief mass kg 32 (b) Specification of eachaxis 1axis 2axis 3axis 4axis 5axis length of arm mm 135 250 215 100 - output W 80 80 80 30 30 rated torque Nm 0.319 0.319 0.159 0.095 0.095 rated rotation number rpm 2400 2400 3000 3000 3000 rated vo ltage V 100 100 100 100 100 rated curren t A 2.2 2.2 0.9 0.63 0.63 deceleration ratio 1/120 1/160 1/160 1/120 1/88 inertia momentofmotor axis × 10 − 7 Nms 2 4.0 4.0 2.7 2.1 2.1 Table E.4. Specification of XY table rated outputofmotor kW 0.2 rated torque of motor kgm 0.065 rated ve lo cit yo fm otor rpm 3000 spring pitch mm 1.4 E.4X YT able 183 (a) Profile PC S e rvo c ontroller S e rvomo t o r Xax i s Yax i s S e rvomo t o r (b) Outline structure Fig. E.4. XY table [...]... Engineering, vol 64, no 8, pp 115 8-1 164, 1998 (in Japanese) [2] S Goto, M Nakamura and N Kyura: Reduced Order Model Configuration of Industrial Mechatronic Servo Systems and Its Significance, Proceedings of the 17th SICE Kyushu Branch Annual Conference, 1998 (in Japanese) [3] S Goto, M Nakamura and N Kyura: Propriety of Linear Model of Servo System for Industrial Articulated Robot Arms and Evaluation of Its Linearization... 10 3-1 12, 1994 (in Japanese) [4] Y Fujino and N Kyura: Motion Control, Sangyo Tosho, pp 8 5-1 17, 1996 (in Japanese) [5] K Kuki, T Murakami and K Ohnishi: Vibration Control of a 2 Mass Resonant System by the Resonance Ratio Control, Trans of the Institute of Electrical Engineers of Japan, vol 11 3- D, no 10, pp 116 2-1 169, 1993 (in Japanese) [6] Y Hori: Introduction to Control of Torsional Vibration System, ... The papers included in this book and the literature referred by these papers are summarized in terms of each chapter The papers being base of each chapter are indicated in parentheses (in Japanese) Chapter 2 [1][2][3] [1] J Zou, M Nakamura, S Goto and N Kyura: Model Construction and Servo Parameter Determination of Industrial Mechatronic Servo Systems Based on Contour Control Performance, Journal of... S 1 2-1 , 1994 (in Japanese) [7] S Arimoto: Dynamics and Control of Robot, Asakura Shoten, 1990 (in Japanese) [8] Edited by Yaskawa Electric Co.: Basic of Servo Technology for Mechatronics, pp 1 2-2 5, Nikkan Kogyo Shimbun, 1986 (in Japanese) [9] S Takagi: Introduction of Analysis Revision 3, Iwanami Shoten, pp 6 2-6 3, 1973 (in Japanese) Chapter 3 [10][11][12] [13] [14] [10] M Nakamura, H Koda and N Kyura: ... (A.15) the control input is selected as u(t) = −f x(t) (A.16) and the control purpose x(t) → 0 (t → ∞) can be implemented with any initial state x(0) = x0 From equation (A.15) and (A.16), the regulator for setting poles (eigenvalue of A − bf ) of closed-loop system is called as pole assignment regulator [42] 1 Eigen-equation of A is |sI − A| = sn + an−1 sn−1 + · · · + a1 s + a0 (A.17) and conversion... putting these basic inputs into (A.5) and calculating Y (s), and finally can be obtained by inverse Laplace transform In this book, 1st order system G(s) = Kp s + Kp (A.7) is always adopted and its impulse response, step response and ramp response respectively are calculated as g(t) = e−Kp t f (t) = 1 − e −Kp t 1 (1 − e−Kp t ) h(t) = t − Kp (A.8) (A.9) (A.10) For 2nd order system with two different real roots... Shoten, pp 6 2-6 3, 1973 (in Japanese) Chapter 3 [10][11][12] [13] [14] [10] M Nakamura, H Koda and N Kyura: Determination of Required Sampling Rate for Sampling Control of Continuous Contour Control by Servo System, Trans of SICE, vol 28, no 5, pp 64 9-6 51, 1992 (in Japanese) ... value y(t) and control input u(t) is called as observer [43] The observer is implemented in d!(t) ˆ ˆ = A! + Ky(t) + Bu(t) dt ˆ x(t) = D!(t) + Hy(t) (A.22) (A.23) by 1 select proper matrix W with (S = 0) in S= 2 Calculate SAS −1 = C W A11 A12 , SB = A21 A22 (A.24) B1 B2 (A.25) 3 Consider the design parameter ˆ A = A22 − LA12 (A.26) the eigenvalues of matrix L are γ1 , γ2 , · · · , γn−l , and the L... impulse response, step response and ramp response respectively are calculated as A.3 Pole Assignment Regulator s1 s2 (es1 t − es2 t ) s1 − s2 s1 s2 es1 t + es2 t f (t) = 1 + s1 − s2 s2 − s1 1 s1 s2 h(t) = t − + es1 t + es2 t Kp s1 (s1 − s2 ) s2 (s2 − s1 ) g(t) = s1 = s2 = −Kv + (A.12) (A .13) (A.14) 2 Kv − 4Kp Kv 2 −Kv − 187 2 Kv − 4Kp Kv 2 A.3 Pole Assignment Regulator To control object expressed by state...Appendix A.1 Laplace Transform and Inverse Laplace Transform If there is a function f (t) on time t ∞ f (t)e−st dt = L[f (t)] = F (s) (A.1) 0 it is called as Laplace transform of f (t)[36] The inverse transform of equation (A.1) f (t) = L−1 {F (s)} (A.2) is called as inverse Laplace transform In s domain, the inverse Laplace transform of rational function F (s) is transformed by partial fraction factorization . ) modificat ion term 7. Master-slave synchronous positioningcontrol 7.1 The master-sla ve sync hronous po sitioning con trol metho d Symbols Units Meanings k c sloping ratio be twe en two axes of ob jectiv et ra jectory 7.2. Gaussian units x i input to Gaussian net wo rk m i mean of Gaussian unit σ i Variance of Gaussian unit x max linearapproximation region of Gaussian network x p max m constant determining linear appro ximation region of po sition x v max m/ s. Contour control with master-slave synchronous positioning Symbols Units Meanings v x s velocityinput to master axis(x axis) e mm lo cus error ˆ F ( s ) modificat ion term if existing modeling