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 slider- crank 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 non- linear systems. A number of works related to computed torque control of robotic manipulators have been pub- lished [I]. The computed torque controller is utilised to linearise the nonlinear equation of robot motion by cancellation of some, or all, nonlinear terms [l]. How- ever, the objection to the real-time use of such a con- trol scheme is the lack of knowledge of uncertainties, which include parameter variations and external distur- bances 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,

inherently nonlinear with uncertainties [2-4]. Therefore, several adaptive controllers have tried to circumvent the problem of uncertainties using adaptive techniques [5-71. In Su and Leung [SI, a computed torque control approach using the sliding mode tech-

0 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

process control, etc), and most of the applications are

nique is introduced, and the uncertainty bound is esti- mated by an adaptive scheme; Imura, Sugie and Yoshikawa [6] described an adaptive robust computed torque control, where a time-varying gain in the pro- posed robust controller is estimated using an adapta- tion 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, exter- nal 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 trajec- tory of the slider-crank mechanism.

A slider-crank mechanism system actuated by a field- orientated 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 rota- tion 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 con- trol 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 consider- ing the existence of uncertainties.

With the great advances in microelectronics and very large-scale integration (VLSI) technology, today, high- performance microprocessors and DSPs can be effec- tively used to realise advanced control schemes [15]. A DSP-based control computer, which is based on a per-

E E Proc.-Control Theory Appl., Vol. 145, No. 3, May I998 3 64

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 mecha- nism 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 posi- tion of the coupled mechanism. In practice, the uncer- tainties of the system can not be known exactly. To control the coupled mechanism with robust characteris- tics, an adaptive computed torque controller is designed to control the position of the slider of the coupled mechanism considering the existence of uncer- tainties. The proposed adaptive controller maintains the computed torque structure with a parameter esti- mation scheme. Finally, simulation and experimenta- tion are performed to test the control performance of the proposed robust and adaptive controllers.

2 Dynamic analysis

2. I Field-orientated PM synchronous motor drive A machine model of a PM synchronous motor can be described in a rotor rotating reference frame as follows [16]:

Vq = R s 2 q + PA, + wsxd V d = Rs2d + pxd + W A X q

(1)

(2)

A, = Lq%, ( 3 )

(4)

where

and Ad = L d a d + LmdIfd

In the above equations vd and vq are the d, 4 axis stator voltages, id and i, are the d, 4 axis stator currents, Ld and L, are the d, q axis inductances, Ad and A, are the d, 4 axis stator flux linkages, while R, and U$ are the

er

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:

digital filter

and didt @r

1 -

and the equation for the motor dynamics is:

In eqn. 5 , 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:

The basic prinsciple in controlling a PM synchronous motor drive is based on field orientation. The flux posi- tion in the d-q co-ordinates can be determined by the shaft position sensor because the magnetic flux gener- ated from the rotor permanent magnetic is fixed in relation to the rotor shaft position. In eqns. 4 and 5 , if id = 0, the d-axis flux linkage Ad is fixed since Lmd and Ifd are constant for a surfacemounted PM synchronous motor, and the: electromagnetic torque z, is then pro- portional to iq which is determined by closed-loop coii- trol. 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 q- axis 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-control- led PWM voltage source inverter (VSI), a unit vector (cos 0, + j sin Ox, where e,y is 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, four- pole, 750W 3.9A, 3000rpm type. With the implementa- tion of field-orientated control, the PM synchronous motor drive system can be simplified to a control

;re = r, + B,w, + J mP w T (6)

w, == Pw, ( 7 )

( mechanism )

synchronous motor i encoder

inverter

3-phase

220v rectifier 60HZ

365

system block diagram, as shown in Fig. 2, in which:

I I

I I encoder ( I

I interface I

I and timer I I I _ _ _ _ _ _ - I

-

servo control card

1 J,s + B, =

linear scale interface

~

where iq* the torque current command. A block dia- gram of the DSP-based computer control system for the field-orientated PM synchronous motor drive is shown in Fig. 3. 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 Instru- ments is chosen to realise the robust and adaptive con- troller 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 4 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 fre- quency of 5kHz. To reduce the calculation burden of the CPU, and to increase the accuracy of the three-

phase command current, the co-ordinate transforma- tion in the field-orientated mechanism is implemented by an AD2S100 AC vector processor.

The dynamic modelling technique based on curve-fit- ting to the step response of the position loop is applied to find the drive model of Fig. 2 in the nominal condi- tion (no parameter variations and z, = 0"). The results are:

Kt = 0.6732 "/A, n = 1

Jm = 6.2 x Nm s2 (11) B, = 1.53 x l o p 2 Nm s/rad

The '-' symbol represents the system parameter in the nominal condition.

Fig. 4 shows a PM synchronous motor system including a geared speed reducer with a gear ratio of:

(12) n a 7 wT Qr n=-=-=-=- n b Tm w 4

Substituting eqns. 8 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

Fig,

366

J 3-phase

IGBT d inverter

220v rectifier 60Hz T

1 I - _ DSP-based control computer t t t

I I I I

pentium

memory

320C32

mechanism

synchronous motor

I

DSP-based computer control drive system

linear scale

IEE Proc -Control Theory Appl., Vol. 145, No. 3, May 1998

PM synchronous

motor

xi- eqn.83

gear box r - - - - - - - I

PM O* synchronous -

motor mechanism

- - - - - - - - Fig. 4 Schematic of motor-gear mechanism

2.2 Mathematical model of the coupled mechanism A slider-crank mechanism driven by a PM synchronous motor is shown in Fig. 5 , 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 0 to A. Moreover, p is the coeffi- cient 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 9.

't

t

FE

B

I

* X

Fig. 5 Slider-crank mechanism driven by PM synchronous motor

2.3 Reduced system (of differential equations of motion 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 decomposi- tion of + = [IO @IT = [vT .TIT, which is the same as Wehage and Haug [17]. If the constraints are independ- ent, the matrix aq has fill-row rank, and there is always at least one nonsingular submatrix Bq of rank 2. The Gauss-.Jordan reduction of the matrix CD , with

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 ele- ments 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:

(15)

(16)

double pivoting, defines a partitioning of qj = [v Y- uqT,

M""U + hlwwV + +TA = B"U + D" - N"

M""U + h'l""v + +:A = B"U + D" -Nu

(17)

(18)

n +bUU + cp,v = GT

or in the matrix form as:

where &I(V)V + N(V, +) = Q u + D

M = M"" -

- +: [M"" - (19) T N= [ N"-+;(+,') Nu]

+ pVl""+il - Mu"Qi'] CT

(20)

The result is EL set of differential equations with only one independent generalised co-ordinate v. The equa-

by using the fourth-order Runge-Kutta method.

The constraint position, velocity, and acceleration

61 and 69, the following equation in the matrix form is obtained:

eqns' 59, 6o and 61 must be held. By using eqns' 59, 60, tion is an initial value pro,blem and can be integrated

3

The robust control system. is shown in Fig. 6, where X;, XB, O*, and 8 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 8 is the state of the motor mechanism

Design of a robust controller 1 (I4)

This is a system of differential algebraic equations for which the matrices element can be found in Appendix 9.

BU+D($)-N(@,II) ff

robust cantroller

I errorfunction _ _ - _ _ _ - - - - -

S(t)=e+h,e I P T ' i

w Fig. 6 Schematic of robust control ofslider-crank mechanism driven by PM 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 8 by using eqn. 83.

Consider the second-order nonlinear, single-input- single-output (SISO) motor mechanism coupling sys- tem:

e( t ) = . f ( O , t ) + G(8, t)U + W(8, t ) (22 ) where

f ( 8 , t ) = -M-'N G(8, t ) = M-'Q W ( 0 , t ) = M-lD

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:

(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 = 0. If the uncertainties occur (i.e. the parameters of the sys- tem 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 modi- fied as:

e(t) = ( f n ( 8 , t ) + A f ) + (Gn(8, t ) + AG) U

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 ( 8 , t ) + Af)

e( t ) = fn(B, t ) + Gn(O,t)U + Wn(O, t )

+ (Wn(8, t ) + AW) (24)

- G,l(8,t)(AGU+Wn(B,t)4Aw) = d ( t ) + P f n ( B , t ) + y

(25)

Q GL1 (8, t ) ( 2 6 )

(27)

where

P = -f?% t)Gil(Q, t ) ( f n ( 8 , t ) + A f )

7 = -G;'(O,t)(AGU + Wn(Q, t )+ AW) (28) Now, the tracking error e of the system and a error function S(t) are defined as follows:

e = Q - B d (29

(30) S ( t ) = i: + &e where 8, is the desired angle of crank; Ae is a positive real value. Substitute eqn. 29 into eqn. 30, then:

S ( t ) = s - e,,, (31)

where 1 9 , ~ ~ = 8!,- A,e denotes the reference speed. In the nominal condition,

Af = o (32 )

A G = O (33)

A W = O (34)

a! = GL'(8,t) ( 3 5 )

/3 = -GZ1(8,t) (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:

(38) U(t) = ct8,,~ + / 3 f n ( B , t ) + y - K S ( t ) Substitute the above equation into eqn. 25, then:

al,,,+Bfn(B,t) + y - K S ( t ) =ad+/?fn(Q,t) +y (39)

(40)

(41)

The above equation can be reduced to:

a! ( 8 .. - 8,,f .. 1 + K S ( t ) = 0

and

aS(t) + K S ( t ) = 0

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 sys- tem is stable.

4

To extend the usefulness of the computed torque con- troller, adaptive control techniques [ 2 4 ] 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 ( 8 , t ) + T - K S ( t ) (42) where 4 and denote the estimated parameters of /3 and y. Substitute eqn. 42 into eqn. 25, then:

Design of an adaptive controller

a!e, , f+pf,(e, t)+T-ris(t) = a ! 8 + / 3 f n ( 0 , t ) + y (43)

Fig. 7

368

Schematic of adap five 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)

(45)

Rewrite the above equation:

where p = p - p and 7 = f - y Multiply sign(a) to both sides of eqn. 45, then:

(46)

as@) = j f n ( 8 ; t ) + :{ - K S ( t )

/alS(t) = s i g n ( a ) [/?fn(r9, t ) + ;i, - KS(b)] 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 p and y i s chosen as follows:

B = - s v ( a ) S ( t ) f n ( O , t ) (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:

c2 - 2 >

1 2 2

V = - l a l S 2 + - p +y Differentiate eqn. 50 with respect to time:

Substitute the eqns. 47, 48 and 49 into eqn. 51, then:

V ( t ) = - /KlS2 5 0 (54 Let function P(t) = -P( t ) = 149, and integrate func- tion P(t) with respect to time:

lil’P(t)dt = V ( 0 ) - V ( t ) (53)

Because V(0) is bounded, and V(t) is descent and bounded, then:

(54)

Differentiate P(t) with respect to time:

P ( t ) = -2lKISS (55) Since K, S and S are bounded, P(t) is uniformly con- tinuous. By using Barbalat’s lemma [2, 31, it can be shown that lim,,,P(t) = 0. Therefore, S(t) -+ 0 as t -+ W. As a result, the coupling system is globally stable. Moreover, the tracking error of the system will con- verge to zero according to S(t) = e + &e = 0.

5 Simulation and experimental results

The dynamic model formulation for the slider-crank mechanism with a PM synchronous servo motor in Section 2 is for general case. However, to test the effec- tiveness of the proposed mechatronics system and to

IEE Pror.-Control Tlzeory Appl., Vol. 145, No. 3. M a y 1998

verify the simulation results of the control system, an experimental mechanism is designed in this study. For simulation andl experimental tests, a practical slider- crank mechanism is set up and its parameters (nominal value) are:

ml = 3.64 kg, ma = 1.18 kg, m3 = 1.8 kg, (56)

(57)

I = 0.305 m

1’ = 0.055 m, R = 0.12 m, T = 0.1 m, p = 0.1

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 = 3 (58) All the gains in the robuist controller are chosen to achieve the best transient performance in both siniula- tion 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 8 is 4.712 rad; the desired angle of 8 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 exter- nal disturbance force FE = 0 is considered. The responses of the crank angle, the error function, the slider position, and the control effort are shown in Figs. 8-1 1 . Next, the parametric variation case is increasing mass of m3 by a 1.4kg load and with exter- nal disturbance force FE = 0, and the responses of the system are shown in Figs. 12-15. Finally, Figs. 16-19

I I I I

5.5

H

5.0

4

time, s

Fig. 8

Crank angle

Response twjectories of roliust contvol ,system (noniinul case M?th FE ONt)

I I I I I I I I I I I

0 1 2 3 4

time, s Response trajectories oJ robust control systeni lnomind case Fig. 9

Error function FE = ONt)

with

369

0.45 0.45 I I I I 0.45 I I I I I 1 I I I

0.4C

XB

0.35

C

I I I I I I

I I

I I I I I 1 I I I I I

0.30 I

0.30 I I I

0 1 2 3 4 time, s

Fig. 14 ation case with FE = ONt) Slider position

Response trajectories of robust control system (parametric vari-

0 1 2 3 4 time, s

Fi .10 Response trajectories of robust control system (nominal case w i z FE = PNt) Slider position

8 I I I I I I I I I I I I

1 I I 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 1 I I

I I I I

0 1 2 3 4

time. s 0 1 2 3 4

time, s Fig. 15 ation case with ,FE = ONt) Control effort iq

Response trajectories of robust control system (parametric vari- Fi . I 1 Response trajectories of robust control system (nominul cuse wi3 FL- = ONt) Contro? effort $

6.0

5.5

6

5.0

6.0

5.5

5.0

4.5

I I

-1- I I I I I I I

-1- I

I

e I / : I I

I

Y I I I I I I I I t I I I

I I I 4.5 1

0 1 2 3 4

time, s Response trajectories of robust control system (parametric vari- Fig. 16

ation case with FE = 5Nl) Crank angle

I I I I

0 1 2 3 4 time, s

Fig. 12 ation case with Ft. = ONt) Crank angle

Response trajectories of robust control system (parametric vari-

5

0

;;r -5

-10

-1 5

5

0

,... g -5

-1 0

-1 5

I I I I 8 I I I I I I I

0 1 2 3 4 time, s

Fig. 13 ation case with FE = ONt) Error function

370

Response trajectories of robust control system (parametric vari-

0 1 2 3 4

time, s Response trajectories of robust control system (parametric vari- Fig. 17

ation 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 I

0.01

I I I I I I I I

I/ 0.0

I I I I I I I

I I I 0 1 2 3 4

time, s

0 . 0 3 0 1 ! 0 1 2 3 4

time, s Fig. 18 ation case with FE = SNt) Slider position

Response trajectories of robust control system (parametric vari- Fi .22 Response trajectories of udaptive control system (nominal case

Slider position wii! FE = pNt)

I- 8

4

i q

2

0

-2

I I I I I I I I I r --- - --.-- r - - - - - - - - r ----..--- U i lb-- I I

I

I I I I I I

0 I 0 1 2 3 4

time, s Fig. 19 ation case with FE = 5Nt i Control effort i<,

Response trajectories of' robust control system (parametric vuri-

0 1 2 3 4

time, s Fi .23 Response trajectories of adaptive control system (nominal case

Control effort iq* w i z FE = ONtj

6.0 I I I I I I

Y I I

I I I I r I I I

I t I

I I I 4.5 0 1 2 3 4

time, s Fig. 24 variation case with FF = ONt J Crank angle

Response trajectories of adaptive control system (parametric

0 1 2 3 4

time, s 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 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 I

I I

I I I I I I I I 1 I

-1 5 0 5 10 15 20

time, s

Fig. 25 variation case with FE = ONt) Error function

Response ,trajectories of udaptive control system (parametric

311

I I

I I

I I I I I I I * 1

.15 I 0 1 2 3 4

time, s

Fi 21 Response trajectories of adaptive control system (nominal case

Error function

IEE Proc-Control Theory Appl., Vol. 145, No. 3, May 1998

wi%FE = O N t )

0.45

0.30

I I I I I I I I I I I I

1

I I I I I I I I I I I I

0.40

XB

I I I I I

I I I I I I

I I I I I I I I I I I I

0.30 0 1 2 3 4

time, s Fig. 30 variation case with FE = SNt) Slider position

Response trajectories of adaptive control system (parametric

0 1 2 3 4 time, s

Fig. 26 variation case with FE = ONt) Slider position

Response trajectories of adaptive control system (parametric

8 I I I 1

6

i;l 4

2

I I I I I I I I

I I I I \ , I I I

I

2 t

I I I

0 J 0 1 2 3 4

time, s Rmponse trajectories of adaptive control .system (parametric Fig. 27

variation case with FE = VNt) Control dror t iq*

0 1 2 3 4

time, s Response trajectories of adaptive control system (pavametric Fig. 3 1

variation case with FE = 5Nt) Control effort l q

6.0 I I I I I I I I 1 illustrate the parametric variation case with the exter-

nal disturbance force FE = 5Nt. Furthermore, the adap- tive control system shown in Fig. 7 is also simulated by using Matlab package under the same simulating con- ditions as the robust control system, and the parame- ters of the adaptive controller are chosen to be the same as the robust controller. The simulation results in the nominal case, the parametric variation case and the parametric variation with the external disturbance case are shown in Figs. 20-31. According to the simulation results, due to the robust control characteristics of the robust and adaptive controllers, smooth step-command tracking responses are obtained for both the crank angle and the slider position at both the nominal and parametric variation cases. However, according to the existence of the steady-state error as shown in Figs. 16- 19, the robust controller is inadequate when parameter variation with large external disturbance occurs. On the other hand, the steady-state error is eliminated by the adaptive controller as shown in Figs. 28-31. In the sim- ulation results of the adaptive controller, all the control efforts are larger than the respective simulated control effort of the robust controller owing to the adaptation of the and y parameters.

Some experimental results are provided here to dem- onstrate the effectiveness of the proposed robust and adaptive controllers for step-command tracking. The software flowcharts of the DSP-based computer con- trol system are shown in Figs. 32-35. The control processing is performed by the main program and three subroutines, two in the PC and one in the DSP.

e I I I I I I I I / : I I >

5.1

I I I I I I I I I

~~ ~~

0 1 2 3 4 time, s

Fig. 28 vurialion c a ~ c wilh FE = 5Nl) Crank angle

Response trajectories of adaptive control system (parametric

5 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 I

0

vj -5

-10

I I

-1- I I I I I

I I

-15 L 1 0 1 2 3 4

time, s Fig. 29 variation case with FE = 5Nt) Error function

Response trujectories of adaptive control system (parametric

IEE Proc.-Control Theory Appl., Vol. 145, No. 3, May 1998 312

main (-7 f

parameters initialisation

I a initialisation

interrupt intervals

enable interrupt

monitor

disable interrupt

Fig. 32 Main

Sojiware flowcharts of DSP-based computer control system

get XB from linear scale

interface

synchronise I DSP 1

g e t i i from DPRAM

Fi 33 Software jlowcharts of DSP-based computer control system 1.4 1, T, = 1 ms

interface

calculate rotor flux position

output

to vector processor

W Fig.34 ISR 2, T, = 0.25 ms

Software flowcharts of DSP-based computer control system

Fig. 35 DSP, T, = I ms

Sojiware flowcharts oj DSP-based computer control system

In the main program, parameters and input/output (I/ 0) initialisation are performed first. Next, the interrupt intervals for the two interrupt service routines (ISRI and ISR2) are set. After enabling the interrupts, the main program .is used to monitor the control data. The ISRl with 1"s sampling rate (T,) is used for linear scale interface to get the slider position (X,) and syn- chronising the execution of the DSP, and then gets the value of control effort (i;) from the dual-port RAM (DPRAM) if the execution of DSP ends. The ISR2 with 0.25ms sampling rate is used to implement the field-orientated mechanism and control the vector processor. Subroutine in the DSP first gets the value of X , from the DIPRAM, and transfers X , to the crank angle (e) by using eqn. 25, and then performs the robust or adaptive control algorithm. Two test condi- tions, the nominal case with no external disturbance force and the parametric variation case by adding a 7.4kg load on m3 with no external disturbance force, are tested here. The responses of the crank angle, the slider position, and the control effort iq* with the robust controller for the two test conditions are shown in Figs. 36-38 and 3941. The responses of the crank angle, the slider. position, and the control effort iq* with the adaptive calntroller for the two test conditions are shown in Figs. 4 2 4 4 and 4547. The results show that smooth step-command responses are obtained for the angle of the crank and the position of the slider, more- over, the tracking errors converge to zero within Is. Furthermore, the experimental results correspond to the simulation results.

. . . . ................................................. . . . . . . . . . . .

. . . . . . . . . . . . . . . .

Fig. 36 Crank angle

Experimental results of robust control system (nominal case)

/ ......... j.. ............................... ................................. . . . . . : : : ' I , .

Fig. 37 Slider position

Experime,ntal results of robust control system (nominal case)

IEE Proc.-Control Theory Appl., Vol. 145, No. 3, May 1998 313

Fig. 38 Control effort iq

Expprimental results of robust control system (nominal case

0 443m i

, : : : : : : : : . . : . : . + : - - 0 5s

t i

Fig. 44 Control effort tq

Expprimental results of adaptive control system (nominal case)

0 443m

0 343m Y

t, 0 5s i

............................................. . . . . . . . . . . .

Fig. 39 tion case) variation casej' Crank angle Crank angle

Experimental results of robust control system (parametric varia- Fig. 45 Ex erimentul results of adaptive control system (parametric

Fig. 40 tion case) variation case) Slider position Slider position

Experimental results of robust control system (parametric varia- Fig, 46 Experimental results of adaptive control system (parametric

Fig. 41 tion case) Control effort i',*

Experimental results of robust control system (parametric varia-

...................

Fig. 42 Crank angle

Experimental results of adaptive control system (nominal case)

Fig. 43 Slider position

Experimental results of adaptive control system (nominal case)

Fig. 47 variation case) Control effort i,'

Experimental results of adaptive control system (parametric

6 Conclusions

This study has successfully demonstrated the applica- tions 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 motor both in the nominal case and in the case where uncertainties 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 to parametric variations and external distur- bances. Furthermore, the proposed adaptive controller is more suitable to be applied when the uncertainties occur.

314 IEE Proc -Control Theory Appl , Vol 145, No 3, May 1998

Thus, the following equation is obtained:

6wA = r6B + (FB - F E ) ~ x B

= r6O + FBE (-r60 sin 8 - IS+ sin q5) (64) where z is given in eqn. 13, FBE 3 FB - FE and:

FE = -pm3g sgn (VB) (65)

sgn(VB) = O if VB = O (66)

1 if VB > 0 i -1 if VB < O and VB is the velocity of slider B. Rewriting eqn. 64 in terms of the geineralised co-ordinate I@, we obtain:

6WA = -6$TQA (67)

7 Acknowledgment

The authors would like to acknowledge the financial support of Chung Yuan Christian University, Taiwan, through grant CY85-RG-004.

8 References

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) SLOTINE. J.-J.E., and LI. W.: ‘Aoolied nonlinear control’ (Pren- 3 _ _ tice-Hall, 1991)

4 JOHANSSON, R.: ‘Adaptive control of robot manipulator motion’, IEEE Trans., 1990, RA-6, pp, 483490

5 SU, C.Y., and LEUNG, T.P.: ‘A sliding 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 com- puted torque gains by using neural networks with flexible struc- tures’, IEE Proc. D, Control Theory Appl., 1994, 141, (4), pp. 235-242

8 JASINSKI, P.W., LEE, H.C., and SANDOR, G.N.: ‘Vibrations of elastic connecting rod of a high-speed slider-crank mechanism’, ASME 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 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 dynam-

ics of AC drives’ (Oxford Universitv. 1996) 15 LE-HUY, H.: ‘Microprocessors ani’digital IC’s for motion con-

16 KRAUSE, P.C.: ‘Analysis of electric machinery’ (McGraw-Hill, trol’, ZEEE Proc., 1994, 82, pp. 1140-1163

1986) 17 WEHAGE, R.A., and HAUG, E.J.: ‘Generalized coordinate par-

titioning for dimension reduction in analysis of constrained dynamic systems’, ASME J. Mech. Des., 1982, 104, pp. 247-255

18 HAUG, E.J.: ‘Intermediate dynamics’ (Prentice-Hall, 1992)

9 Appendix

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 (59) where qj = [e # I T is the vector of the generalised co- ordinates. The kinematic velocity and acceleration equations are obtained by taking the first and second derivatives of eqn. 59, respectively, as:

and +*?j = 0 (60)

(61 1

(62)

( 6 3 )

%4 = - (+,:.4),:.4 where

@,:.lj = rocos0 - 1jIcosQ

+,:.G = re2 sin o - 14‘ sin 4 The virtual work 6 WA includes the applied torque z with the virtual angle SO, the friction force FB, and the external force FE with the virtual displacement SLYB.

IEE Proc.-Control Theory Appl., Vol. 145, No. 3, A4ay 1998

where QA is the generalised force and given as:

The generalised constraint reaction force can be obtained in terrns of the Lagrange multiplier [18] as:

Qc = +;A (69)

+$ = [rcosO - I c o s ~ ] (70)

where

Thus, the virtual work by all constraint reaction forces is:

6Wc = 6$ Q (71) T C

The general form of Hamilton’s principle is:

0 = llt2 [6L + 6WA + 6Wc] d t

d L d d L = lltz 6gT [ - dt (%) - QA + Qc] d t

(72) where L is Lagrangian and is the total kinetic energy minus the potential energy [12]:

1 1 1 4 6 2

L = -m1~9282 + -vL,I~$’ + -m2r2d2 sin2 o 1 1 2 2

+ -mZrZ84 sin 0 sin 4 + -m,r2Q2 sin2 o + m:,rl&i sin 0 sin 4 + -m312qi2 sin2 4

- -1n2glsin4

1 2

(73 ) 1 2

The variation eqn. 73 must hold for all values of SI) and SI)(^,) = tiy(t,> = 0. Thus, the Euler-Lagrange equations of motion, accounting for both applied and constraint forces, are:

M($)4 + N(G, 4) - BU - D($) + +$A = 0 (74) where

M(1L) =

1 1 - 4 - 7 1 R 2 - ( m 2 +in3)r2 .,n2 8 - n 2 J,,, ( 3 m 2 - ,n 1 7 1 b i n 0 bin 4

315

(77) The transform formulation from 8 to XB is obtained by:

XB = Tcos8 + 1 cos (b + 1'

= r cos 8 -+ (12 - r2 sin2 8) 3 + 1' (78)

The transform formulation from X, to 8 is obtained by:

[(x, - 1 ' ) - r cos 81 = ( 1 2 - r2 sin' 8) ' [(X, - I ' ) - r cos el2 = l 2 - r2 sin2 T9

(xB - 1'12 - 2r (xB - 1'1 cos B + r2 cos29

(79)

(80)

= 1' - r2 sin26' (81)

(82) -27- (XB - I ' ) cos8 = l 2 - r2 - (XB - 1 ' )2

r2 + (X, - q2 - l 2 2r (X, - 1 ' )

8 = c0s-l

10 Appendix

Choose v = [e], U = [$I, aV = [ r cos 81, and $1. The entries of the matrices in eqns. 15 and 16 are:

- (m2 + m3) r2 sin2 Q - n 2 J ,

= [-I cos

(84)

M""= [ - ( a m 2 - m3 ) r l s i n ~ s i r i $ ] (55)

Mu" = [ - (:ria, - m,) rl sin 19 sin q5] (86)

- (m2 + m3) r2$'sin 6' cos 8

1 - (:m2 +ms) r1~2~ in8cosq5-n2B,9

(88)

1 1 2

- m312d2 sin 4 cos Cp - -mag1 cos 4 (89)

B" = [-nKt], U = [ i ; ] , D" = [ F ~ ~ r s i n O ] (90)

B" = [O], D" = [ F ~ ~ I s i n ( b ] (91)

376 IEE Proc.-Control Theory Appl.. Vol. 145, No. 3, May 1998