Slider crank mechanism control using adaptive computed torque technique

r-crank mechanism control using adaptive uted torque technique F.-J Lin Y.-S Lin S.-L.Chiu Indexing ternis: Slider-crank mechanism, Adup rive computed torque technique Abstract: The position control of the slider of a slider-crank mechanism, which is driven by a permanent magnet (PM) synchronous motor, using an adaptive computed torque technique, is studied First, the mathematical model of the motor mechanism coupling system is described, where the Hamilton principle and the Lagrange multiplier method are applied to formulate the equation of motion Secondly, assuming that the parameters of the system are well known, according to the computed torque technique, a robust controller is designed to control the slidercrank mechanism Then, considering the existence of the uncertainties of the system, an adaptive computed torque controller is designed based on the Lyapunov stability Moreover, to increase the execution rate of the control algorithms, a digital signal processor (DSP)-based control computer is devised to control the motor mechanism coupling system Introduction Computed torque, or the inverse dynamics technique is a special application of feedback linearisation of nonlinear systems A number of works related to computed torque control of robotic manipulators have been published [I] The computed torque controller is utilised to linearise the nonlinear equation of robot motion by cancellation of some, or all, nonlinear terms [l] However, the objection to the real-time use of such a control scheme is the lack of knowledge of uncertainties, which include parameter variations and external disturbances of the system The adaptive control technique is essential for providing a stable and robust performance for a wide range of applications (e.g robot control, process control, etc), and most of the applications are inherently nonlinear with uncertainties [2-4] Therefore, several adaptive controllers have tried to circumvent using adaptive the problem of uncertainties techniques [5-71 In Su and Leung [SI, a computed torque control approach using the sliding mode tech0 IEE, 1998 IEE Proceedings online no 19982051 Paper first received 15th July 1997 and in revised form 23rd March 1998 The authors are with the Department of Electrical Engineering, Chung Yuan Christian University, Chung Li 32023, Taiwan 364 nique is introduced, and the uncertainty bound is estimated by an adaptive scheme; Imura, Sugie and Yoshikawa [6] described an adaptive robust computed torque control, where a time-varying gain in the proposed robust controller is estimated using an adaptation law; Teshnehlab and Watanabe [7] proposed a self-tuning computed torque controller, where the gains of the computed torque controller are tuned by neural networks The slider-crank mechanism is widely used Examples of its application are found in petrol and diesel engines, where the gas force acts on the slider and the motion is transmitted through the links Steady-state solutions and the elastic stability of the motion of a slider-crank mechanism were obtained in Jasinski et al [8], Zhu and Chen [9] and Badlani et al [lo] In addition, the response of the system has been found to be dependent upon the five parameters: length, mass, damping, external piston force and frequency in Viscomi and Arye [l 11 The transient responses have been investigated on the basis of the ratios, the length of the crank to the length of the connecting rod and the rotating speeds of the crank to the rotating speeds of the rod, etc in Fung [ 121 However, in the previous studies, the applications of electric motors to drive the slider-crank mechanism were not considered and, moreover, no control theory was applied to control the position, velocity, or trajectory of the slider-crank mechanism A slider-crank mechanism system actuated by a fieldorientated control PM synchronous motor drive [13, 141 is investigated in this study The slider-crank mechanism driven by a PM synchronous servo motor has applications in areas where the transfer of the rotation motion to the translation motion is needed, and high precision is required Since the application of the slider-crank mechanism has similar control problems to those of the robotic systems, the computed torque based robust and adaptive controllers are designed to control the motor mechanism coupling system in this study To achieve this objective, a robust controller based on computed torque control is designed to control the coupled mechanism in the nominal condition In addition, an adaptive computed torque controller, in which the gains are tuned using adaptive scheme, is proposed to control the coupled mechanism considering the existence of uncertainties With the great advances in microelectronics and very large-scale integration (VLSI) technology, today, highperformance microprocessors and DSPs can be effectively used to realise advanced control schemes [15] A DSP-based control computer, which is based on a perE E Proc.-Control Theory Appl., Vol 145, No 3, May I998 sonal computer (PC) and a TMS320C32 DSP, is designed to provide a flexible environment with a high execution rate for the field-orientated mechanism and the control algorithms The field-orientated mechanism and computer interface programs are implemented using the Pentium-PC; the computed torque-based robust and adaptive controllers are implemented using the DSP First, the mathematical model of the motor mechanism coupling system is derived Following that, under the situation that the parameters of the coupled system are well known, a robust controller based on computed torque technique is designed to control the crank position of the coupled mechanism In practice, the uncertainties of the system can not be known exactly To control the coupled mechanism with robust characteristics, an adaptive computed torque controller is designed to control the position of the slider of the coupled mechanism considering the existence of uncertainties The proposed adaptive controller maintains the computed torque structure with a parameter estimation scheme Finally, simulation and experimentation are performed to test the control performance of the proposed robust and adaptive controllers Dynamic analysis I Field-orientated PM synchronous motor drive A machine model of a P M synchronous motor can be described in a rotor rotating reference frame as follows [16]: Vq = R s q PA, wsxd (1) Vd + + = Rs2d + pxd + (2) WAXq where A, = Lq%, and (3) + Ad = L d a d LmdIfd (4) In the above equations vd and vq are the d, axis stator voltages, id and i, are the d, axis stator currents, Ld and L, are the d, q axis inductances, Ad and A, are the d, axis stator flux linkages, while R, and U$ are the stator resistancl: and inverter frequency, respectively In eqn 4, Isd is the equivalent d-axis magnetising current, and L,, is the (Saxis mutual inductance The electric torque is: and the equation for the motor dynamics is: + + ;re = r, B,w, J mPwT (6) In eqn , P is the number of pole pairs, ,z, is the load torque, B, is the damping coefficient, w, is the rotor speed and, J , is the moment of inertia The inverter frequency is related to the rotor speed as: w, == Pw, (7) The basic prinsciple in controlling a PM synchronous motor drive is based on field orientation The flux position in the d-q co-ordinates can be determined by the shaft position sensor because the magnetic flux generated from the rotor permanent magnetic is fixed in relation to the rotor shaft position In eqns and , if id = 0, the d-axis flux linkage Ad is fixed since Lmdand Ifd are constant for a surfacemounted PM synchronous motor, and the: electromagnetic torque z, is then proportional to iq which is determined by closed-loop coiitrol In the field-orientated control of a PM synchronous mlotor, the d-axis rotor flux is provided by the PM mounted on the rotor; therefore, only the qaxis torque current component iq is necessary to be generated by the drive Since the generated motor torque is linearly proportional to the q-axis current as the d-axis rotor flux is constant in eqn 5, the maximum torque per ampere can be achieved The configuration of a general field-orientated PM synchronous motor drive system is shown in Fig 1, which consists of a PM synchronous motor coupled with a mechanism, a ramp comparison current-controlled PWM voltage source inverter (VSI), a unit vector (cos 0, + j sin Ox, where e,yis the position of rotor flux) generator, a co-ordinate 'translator, a speed-control loop and a position control loop The PM synchronous motor used in this drive system is a three-phase, fourpole, 750W 3.9A, 3000rpm type With the implementation of field-orientated control, the P M synchronous motor drive system can be simplified to a control ( mechanism ) 3-phase 220v 60HZ rectifier i synchronous motor inverter encoder er digital filter and didt @r 365 system block diagram, as shown in Fig 2, in which: = + J,s B, where iq* the torque current command A block diagram of the DSP-based computer control system for the field-orientated PM synchronous motor drive is shown in Fig The position of the slider is measured by a linear scale The field-orientated mechanism is implemented using the Pentium CPU, moreover, the TMS320C32 floating point DSP from Texas Instruments is chosen to realise the robust and adaptive controller for the coupled mechanism A servo control card is installed in the control computer, which includes A/D, DIA, PI0 and encoder interface circuits Digital filters and frequency multiplied by circuits are built into the encoder interface circuits to increase the precision of position feedback from encoder and linear scale The current-controlled VSI is implemented by IGBT switching components with a switching frequency of 5kHz To reduce the calculation burden of the CPU, and to increase the accuracy of the three- phase command current, the co-ordinate transformation in the field-orientated mechanism is implemented by an AD2S100 AC vector processor The dynamic modelling technique based on curve-fitting to the step response of the position loop is applied to find the drive model of Fig in the nominal condition (no parameter variations and ,z = 0") The results are: Kt = 0.6732 "/A, n=1 Jm = 6.2 x B, = 1.53 x Nm s2 (11) l o p Nm s/rad The '-' symbol represents the system parameter in the nominal condition Fig shows a PM synchronous motor system including a geared speed reducer with a gear ratio of: a -=7 n = -n = - = - wT nb w Tm Qr (12) Substituting eqns and 11 into eqn 6, the following applied torque can be obtained: r = n (re- J,hr - B,w,) ( = n Kti; - nJ,Q - nB,8 ') (13) where z is the torque applying in the direction of w 'p PM synchronous motor drive system Fig.2 Simpllfed control block diugrum mechanism linear scale J 3-phase 220v IGBT rectifier 60Hz I T _- DSP-based control computer inverter synchronous motor t t t I I I d I pentium memory I I I I I 320C32 I I I encoder interface and timer (I - I I ~ linear scale interface _ _ _ _ _ _ - I servo control card I Fig, 366 DSP-based computer control drive system IEE Proc -Control Theory Appl., Vol 145, No 3, May 1998 PM synchronous motor 2.3 Reduced system (of differential equations of motion gear box The differential algebraic; equation of mechanical motion derived above is suimmarised in the matrix form of eqn 14, anld in the constraint equation of eqn 59 The implicit method musl be employed to solve the equation of tlhe system Eqns 14 and 59 may be reordered and partitioned, according to the decomposition of + = [IO@IT = [vT TIT, which is the same as Wehage and Haug [17] If the constraints are independent, the matrix aq has fill-row rank, and there is always at least one nonsingular submatrix Bq of rank The Gauss-.Jordan reduction of the matrix CD , with double pivoting, defines a partitioning of qj = [vY- uqT, r -I Fig.4 Schematic of motor-gear mechanism v = [e], U = [#] so that 'Pv is the submatrix of CDp whose columns correspond to element v of +, and CDu 1s the submatrix of aV, whose columns correspond to elements U of + The elements of the vectors v, U and matrices a,,, @, are detailed in Appendix 10 Thus, eqns 14 and 5'9 can be rewritten as: 2.2 Mathematical model of the coupled mechanism A slider-crank mechanism driven by a PM synchronous motor is shown in Fig , where m,, m2 and m3 are the masses of the rotating disc, the connecting rod and the slider, respectively; I is the length of the connecting rod and 1' is the distance from point B to the mass centre of the slider; R is the radius of the rotating disc, and r is the distance from to A Moreover, p is the coefficient of the dry friction between the slider and the foundation Hamilton's principle and the Lagrange multiplier are used to derive the differential algebraic equation for the slider-crank mechanism in Appendix M""U M""U + hlwwV+ +TA = B"U + D" N" + h'l""v +A+: = B"U + D" - N u n +bUU + cp,v = GT - (17) &I(V)V + N(V,+) = Q u + D (18) [M""- (19) where M = M""- +: [ T N = N"-+;(+,') + pVl""+il * Nu] - X FE (16) or in the matrix form as: 't t (15) Mu"Qi'] CT (20) B I Fig.5 Slider-crank mechanism driven by PM synchronous motor The result is EL set of differential equations with only one independent generalised co-ordinate v The equation is an initial value pro,blem and can be integrated by using the fourth-order Runge-Kutta method The constraint position, velocity, and acceleration eqns' 59, 6o and 61 must be held By using eqns' 59, 60, 61 and 69, the following equation in the matrix form is obtained: BU+D($)-N(@,II) ff Design of a robust controller The robust control system is shown in Fig 6, where X;, XB,O*, and are the command of slider position, slider position, command angle of crank, and angle of crank, respectively Since X, is the desired control objective and is the state of the motor mechanism (I4) This is a system of differential algebraic equations for which the matrices element can be found in Appendix robust cantroller - _ - I errorfunction PT'i S(t)=e+h,e xi- Fig eqn.83 O* I PM synchronous motor mechanism w Schematic of robust control ofslider-crank mechanism driven by P M synchronous motor IEE Proc.-Control Theory Appl Vol 145, No 3, May 1998 361 coupling system, X,* and XB should be transformed to 8* and by using eqn 83 Consider the second-order nonlinear, single-inputsingle-output (SISO) motor mechanism coupling system: + e(t) = f(O, t ) G(8, t)U where + W ( ,t ) where ,=~8! ~,- A,e denotes the reference speed In the nominal condition, (22) f ( , t ) = -M-'N G(8, t ) = M-'Q W ( , t )= M-lD a! + e ( t ) = fn(B, t ) Gn(O,t)U Wn(O,t ) (23) where J,(@t ) is the nominal value off(@ t); Gn(@t ) is the nominal value of G(8, t); W,?(O,t ) is the nominal value of W(8, t ) where the external disturbance FE = If the uncertainties occur (i.e the parameters of the system are deviated from the nominal value and there is an external disturbance added into the system), the dynamic equation of the coupling system can be modified as: e(t) = ( f n ( ,t ) + A f )+ (Gn(8,t ) + (Wn(8,t ) + AW) + AG) U ~ - =d + Af) G,l(8,t)(AGU+Wn(B,t)4Aw) ( t )+ P f n ( B , t ) + y (25) where GL1(8,t ) P = -f?% t)Gil(Q, t ) ( f n ( ,t ) + A f ) Q (26) (27) + = -G;'(O,t)(AGU W n ( Q , t ) AW) + (28) Now, the tracking error e of the system and a error function S(t) are defined as follows: e=Q-Bd (29 + S ( t ) = i: &e (30) where 8,is the desired angle of crank; Ae is a positive real value Substitute eqn 29 into eqn 30, then: S(t)= Fig.7 368 s e,,, - AG=O (33) AW=O (34) = GL'(8,t) (35) (36) y = -G,1(8,t)Wn(0,t) (37) Then, according to eqn 25 and the error function S(t), the robust controller can be designed as: + + U(t) = ct8,,~ / f n ( B , t ) y - K S ( t ) Substitute the above equation into eqn 25, then: al,,,+Bfn(B,t) (38) + y - K S ( t ) =ad+/?fn(Q,t)+ y (39) The above equation can be reduced to: a! (8 - 1+ 8,,f K S ( t )= (40) and a S ( t )+ K S ( t ) = (24) where Af, AG and AW denote the uncertainties Eqn 24 can be rewritten using the computed torque technique as follows: U(t) = GZ1(e,t ) d ( t ) G i l (8, t ) ( f n ( ,t ) (32) /3 = -GZ1(8,t) and U(t) is the control input iq* According to tqn 19, M can be computed to ensure the existence of NI-' Now, assume that the parameters of the system are well known as the nominal condition Rewrite eqn 22 to be: + =o Af (41) Let K and a be the same sign, namely, sgn(Q = sgn(a), then the controller will make the error function S(t) converge to zero exponentially, and the coupling system is stable Design of an adaptive controller To extend the usefulness of the computed torque controller, adaptive control techniques [ ] can be employed considering the existence of uncertainties In this study, the uncertainties contain the variation of the mass of the slider m3 and the external force FE With these uncertainties, an adaptive control law is designed to stabilise the motor mechanism coupling system The proposed adaptive controller based on the computed torque technique for the motor mechanism coupling system is shown in Fig 7, and the adaptive controller is designed as: + U ( t ) ~Ei,,,(t) B f n ( , t )+ T - K S ( t ) (42) where and denote the estimated parameters of /3 and y Substitute eqn 42 into eqn 25, then: a!e,,f+pf,(e,t)+T-ris(t) = a ! + / f n ( , t ) + y (31) (43) Schematic of adapfive control of slider-crank mechunism driven by PM synchronous motor IEE Pi.oc.-Control Theory Appl Vol 145, No 3, May 1998 and (j- a> f,(O,t) + (+ - y) = cl (B - + jref) K S ( t ) (44) Rewrite the above equation: p a s @ )= j f n ( ;t ) +{: - K S ( t ) (45) = p p and = f y Multiply sign(a) to where both sides of eqn 45, then: - - /alS(t)= s i g n ( a ) [/?fn(r9,t ) + ;i, - KS(b)] (46) The above equation can be rewritten as: Assumption: The speed of the variation of the system parameters is much slower than the executing speed of the adaptive algorithm and, during the adaptation, the parameters of the system is considered as constants, i.e - ^ ^ p =p -p =p -S/=?-;i=T The adaptive algorithm for the parameters chosen as follows: p and y i s B = - s v ( a ) S ( t ) f n ( Ot, ) (48) ;i/ = -szgn(a)S(t) (49) Theorem I : The motor mechanism coupling system will be globally stable, if the adaptive controller is obtained by eqn 42 with the adaptive algorithm eqns 48 and 49 Prooj Let the Lyapunov function be: V=-lalS2+- p +y verify the simulation results of the control system, an experimental mechanism is designed in this study For simulation andl experimental tests, a practical slidercrank mechanism is set up and its parameters (nominal value) are: ml = 3.64 kg, ma = 1.18 kg, m3 = 1.8 kg, (56) I = 0.305 m c2 1’ = 0.055 m, R = 0.12 m, T = 0.1 m, p = 0.1 (57) The physical meanings of the parameters can be referred to Seclrion 2.2 No.w, using Matlab package to simulate the motor mechanism coupling system with the robust controller The parameters of the robust controller are chosen as: A, = 10, K = (58) All the gains in the robuist controller are chosen to achieve the best transient performance in both siniulation and experimentation considering the limitation of the control effort and the requirement of stability The control objective is to control the slider to move 0.lm Hence, the initial angle of is 4.712 rad; the desired angle of is equal to 5.760 rad, and the stroke of the slider, AK,, is equal to 0.lm Three simulation cases are addressed here First, the nominal case with external disturbance force FE = is considered The responses of the crank angle, the error function, the slider position, and the control effort are shown in Figs 8-1 Next, the parametric variation case is increasing mass of m3 by a 1.4kg load and with external disturbance force FE = 0, and the responses of the system are shown in Figs 12-15 Finally, Figs 16-19 I I -2> I I Differentiate eqn 50 with respect to time: 5.5 Substitute the eqns 47, 48 and 49 into eqn 51, then: H V ( t )= -/KlS2 (54 Let function P(t) = -P(t) = 149, and integrate function P(t) with respect to time: lil’P(t)dt= V ( )- V ( t ) 5.0 (53) Because V(0) is bounded, and V(t) is descent and bounded, then: time, s Response twjectories of roliust contvol ,system (noniinul case M?th ONt) Fig FE Crank angle (54) Differentiate P(t) with respect to time: P ( t ) = -2lKISS (55) Since K, S and S are bounded, P(t) is uniformly continuous By using Barbalat’s lemma [2, 31, it can be shown that lim,,,P(t) = Therefore, S(t) -+ as t -+ As a result, the coupling system is globally stable Moreover, the tracking error of the system will converge to zero according to S(t) = e + &e = W Simulation and experimental results The dynamic model formulation for the slider-crank mechanism with a PM synchronous servo motor in Section is for general case However, to test the effectiveness of the proposed mechatronics system and to IEE Pror.-Control Tlzeory Appl., Vol 145, No M a y 1998 I I I I I I I I I I I time, s Fig Response trajectories oJ robust control systeni lnomind case with FE = ONt) Error function 369 0.45 0.45 I I I I I I I I I 0.4C C XB 0.35 I I I I time, s Fi 10 Response trajectories of robust control system (nominal case w i z FE = PNt) Slider position I I I I time, s I I I I I I I I Fig 14 Response trajectories of robust control system (parametric variation case with FE = ONt) Slider position I I I I I I I I I I 0.30 0.30 I II I I I I I I I I I I I I I I I I I I I I I I I I I I r I I I I I , I I time s Fi I Response trajectories of robust control system (nominul cuse w i FL-= ONt) Contro? effort $ I I I I I I I time, s Fig 15 Response trajectories of robust control system (parametric variation case with ,FE = ONt) Control effort iq 6.0 6.0 5.5 5.5 I I -1- I I I e I / : I I I I I I I I 5.0 5.0 I Y 4.5 -1- I I I I I I I I t I I I I I I 4 time, s Fig 12 Response trajectories of robust control system (parametric variation case with Ft = ONt) Crank angle Fig 16 Response trajectories of robust control system (parametric variation case with FE = 5Nl) Crank angle 5 0 , g -5 -10 -5 -10 I I I I -1 I I I I I I I -15 time, s Fig 13 Response trajectories of robust control system (parametric variation case with FE = ONt) Error function 370 I I time, s ;;r I I 4.5 time, s Fig 17 Response trajectories of robust control system (parametric variation case with FE = 5Nt) Error function IEE ProccContuol Theory A p p l , Vol 145, No 3, May 1998 I I I I I I I I 0.01 II I/ I I I I I I I I I I I I 0.0 I I time, s ! time, s I Fig 18 Response trajectories of robust control system (parametric variation case with FE = SNt) Fi 22 Response trajectories of udaptive control system (nominal case wii! FE = pNt) Slider position Slider position I- iq 0 I I I I I I I time, s ation case with FE = 5Nt i I I I I I I I I Y I I I I I lb-0 I I I I I I I i I Fi 23 Response trajectories of adaptive control system (nominal case w i z FE = ONtj Fi 20 Response trajectories of adaptive control system (nominal case w&FE = ONt) Crank angle I I I # I I i I I I I I I I I r I I I I I I I I I I I I I I Fig.24 Response trajectories variation case with FF = ONt J I I I I I I 15 * 2 I I I I I I I I I I I I I I I I I I I I I I4 I 10 15 20 time, s Fi 21 Fig.25 Error function Error function Response trajectories of adaptive control system (nominal case wi%FE = O N t ) I I I -15 time, s IEE Proc-Control Theory Appl., Vol 145,No 3, May 1998 of adaptive control system (parametric I I I I I Crank angle I I I t time, s time, s r r time, s 4.5 I/ I I I Control effort iq* Control effort i[...]... Experimental results of adaptive control system (nominal case) Crank angle 0 443m 0 343m Y t , i 0 5s Fig 43 Experimental results of adaptive control system (nominal case) Slider position 314 Conclusions This study has successfully demonstrated the applications of computed torque based robust and adaptive controllers to the position control of the slider of a slider- crank mechanism driven by a PM synchronous... control system (parametric varia- tion case) Slider position Fig, 46 Experimental results of adaptive control system (parametric variation case) Slider position Fig.41 Experimental results of robust control system (parametric variation case) Control effort i',* Fig.47 Experimental results of adaptive control system (parametric variation case) Control effort i,' 6 Fig 42 Experimental results of adaptive. ..Fig.38 Expprimental results of robust control system (nominal case Control effort iq Fig 44 Expprimental results of adaptive control system (nominal case) Control effort tq Fig.39 Experimental results of robust control system (parametric varia- Fig 45 tion case) Ex erimentul results of adaptive control system (parametric variation casej' Crank angle Crank angle i 0 443m - , : : : : : :... In this Appendix, Hamilton’s principle and the Lagrange multiplier are used to derive the differential algebraic equation for the slider- crank mechanism The motor mechanism coupled system is shown in Fig 5 The slider- crank mechanism consists of three parts: crank, rod and slider The holomonic constraint equation [18] is: @(G) = r s i n 0 - I s i n 4 = 0 +*?j =0 (60) %4 = - (+,:.4),:.4 (611 @,:.lj= rocos0... mode controller with bound estimation for robot manipulators’, ZEEE Trans., 1993, RA-9, (2), pp, 208-214 6 IMURA J SUGIE T and YOSHIKAWA T.: Adaptive robust control of robot manipulators: theory and experiment’, ZEEE Trans., 1994, RA-10, ( 5 ) , pp 705-710 7 TESHNEHLAB, M., and WATANABE, K.: ‘Self tuning of computed torque gains by using neural networks with flexible structures’, IEE Proc D, Control. .. high-speed slider- crank mechanism , A S M E J Eng Znd., 1971, pp 636-644 9 ZHU, Z.G., and CHEN, Y.: ‘The stability of the motion of a connecting rod’, J Mechanisms, Transmissions, Automation Des., 1983, 105, pp 637-640 10 BADLANI, M., and KLEINHENZ, W.: ‘Dynamic stability of elastic mechanisms’, J Mechanism Des., 1979, 101, pp 149-153 11 VISCOMI, B.V., and ARYE, R.S.: ‘Nonlinear dynamic response of elastic slider- crank. .. ‘Nonlinear dynamic response of elastic slider- crank mechanism , ASME J Eng I d , 1971, 93, pp 251-262 12 FUNG, R.F.: ‘Dynamic analysis of the flexible connecting rod of a slider- crank mechanism , A S M E J Vibration Acoust., 1996, 118, pp 687-689 13 LEONHARD, W.: Control of electrical drives’ (Springer, 1996) 14 NOVOTNY, D.W., and LIPO, T.A.: ‘Vector control and dynamics of AC drives’ (Oxford Universitv... (VB) 1 ORTEGA R and SPONG M.W.: ‘Adaotive motion control of rigid robots: a tutorial’ IEEE Proceeding of the 27th conference on Decision and control, 1988, pp 1575-1584 2 ASTROM, K., and WITTENMARK, B.: Adaptive control (Addison-Wesley, 1995) 3 SLOTINE J.-J.E., and LI W.: ‘Aoolied nonlinear control (Prentice-Hall, 1991) 4 JOHANSSON, R.: Adaptive control of robot manipulator motion’, IEEE Trans.,... are considered Moreover, a DSP-based control computer, which is a coprocessor structure, was successfully devised to implement the field-orientated mechanism and the proposed controllers The design procedure of the proposed controllers have been described in detail Simulation and experimental results show that the dynamic behaviours of the proposed controller-motor mechanism system are robust with regard... behaviours of the proposed controller-motor mechanism system are robust with regard to parametric variations and external disturbances Furthermore, the proposed adaptive controller is more suitable to be applied when the uncertainties occur IEE Proc -Control Theory A p p l , Vol 145, No 3, May 1998 7 Thus, the following equation is obtained: Acknowledgment The authors would like to acknowledge the financial