Nonlinear PID Predictive Control of Robotic Manipulators

Copyright ® IFAC Robot Control, Vienna, Austria, 2000

NONLINEAR PID PREDICTIVE CONTROL OF ROBOTIC MANIPULATORS Wen-Hua Chen

Cent7'e /01' Systems fj Contml, {md Department 0/ Elecimnics fj Elt:ct7'ical Enginee7'ing, U7Iive7'sity of Glasgow, Glasgow G12 8QQ, UK , Email:w.chen@eier;.gla.ac. -uk

Abstract: This paper proposes a new nonlinear PID controller for two-link robotic manipulators, This new PID controller, named nonlinear PID predictive controller, is derived by integration of a nonlinear predictive controller with a non linear disturbance observer , which consists of a nonlinear PID controller and a predictive part. The global stability of the closed-loop system is then established. This controller is implemented and validated ou a two-link robotic manipulator. Experimeutal results show that compared with widely used computed torque method , the developed controller significantly improves performance robustness against uncertainty and friction, Copyright @2000 IFAC

Keywords: PID controller , l"oulinear control, Robotic manipulators , Disturbance rejection , Predictive coutrol

1. INTRODUCTION

However as pointed out in Chen et a/. (2000), almost all the disturbance observers used in robot control are designed based on liuear observer analysis and design techniques , where nonlinear dynamics are ignored and considered as an extemal disturbance , It is obvious that both performance and robustness against 7'eal disturbances and unmodeled dynamics can be improved by including the knowledge of the non linear dynamics of a robot (Oh and Chung, 1999), That is , instead of a lineaT' model , a nonlinear model which is closer to the real robot should be employed in the controller and observer design. Analysis and design of nonlinear controllers and nonlinear observers for robotic manipulators is , however , much lIIore difficult and challenging than analysis and design of linear oues.

Disturbance observer based control (DOBC) has been used in robot control for a long time and for different situatious including independent joint control , friction compensation, and sensorless torque or force control; for example, see (Kim et ul., 1996; Zhang and Furusho , 1998; Komada and Ohuishi , 1990; Murakami et al., 1993), The uuderlyiug idea iu disturbance observer based coutrol is to deduce unknown disturbances possibly includiug friction , unmodeled dyuarnics aud nonlinearity, and unknown load using observer design techniques amI then compensate for them. Recently, the relationship betweeu the passivitybased approaches in Sadegh amI Horowitz (1990) and the disturbance observer based robot control in Bickel and Tomizuka (1995) and Slotine and Li (1987) has been investigated by Bickel aud Tomizuka (1999), It is shown that the disturbance observer based algorithm can be made equivaleut to the passivity-based approach. Stability of the disturbance observer based coutrol is also established ,

This paper proposes a new nonlinear PID controller which is derived by integratiou of a n07llineU7' predictive control with a 7lOnlmeaT' disturbance observer. This new nonlinear controller is referred to as a 1101tii7leaT' P ID pT'ediciive C07ltrol/eT' since in addition to the t raditional PID structure, a predict ive part is illcluded to predict

7S

which is a function of the current system state x( t) and future input in the time period [t, t+TzJ . Then a control profile ·u(t + T)' , T E [0, TzJ is generated by minimising the tracking error performance index (3) . However as in other receding horizon control algorithms , only the control action at time instant t is implemented , i.e. :

output trend by taking into account dynamics information of a system. Stability of the closed loop system under the new nonlinear PID predictive controller is established based on Lyapunov theory. Experimental results demonstrate effectiveness of this new nonlinear controller.

"U(t)

2. PREDICTIVE CONTROLLER

The dynamics of a two-link robotic manipulator can be described by a second order matrix equation, given by

J(B(t»B(t) + G(B(t) , ti(t))

= B(u(t) + d(t»(1)

where B E R2, ·u E R2 , d are the vectors of displacement, the generalised torque and/or force and the unknown disturbance vector. G(B , ti) is the inertia matrix and consists of COl·iolis and centrifugal terms and the gravitational term , etc. The inertial matrix J(B) is always positive definite for all allowable B. When the first order dynamics of DC motors are included in the above model, ·u is the voltage vector imposed on the motors instead of the torque vector. In general the input matrix B E R2 x 2 is of full rank. For the sake of simplicity, all disturbances are equivalent to the disturbance d on the control input "U(t) in this paper.

= CB(t).

t;;/t31 Kz = t3:/t32 t i +j t 2 KJ =

(5) (6) J _

t

i 'd 1

.) - (-i - 1)' (j - 1)! (i

-

1

+ j - 1)

(7)

and

ti = diag{T;, T;},

(2)

-i

= 1, 2

(8)

=

:\'ote that the notation 0' 1 is used here. It is obvious that the gain matrices KJ and Kz depend on the choice of the predictive times Tl and Tz explicitly. By adjusting these two design parameters, the desired system response can be achieved . Chen et al. (1999b) provides the criterion for choosing the design parameters in epe based on overshoot and rising time specifications. Stability of the above nonlinear predictive control is given in Theorem 1 which is modified from the result in Ch en et al. (1999a).

Model based predictive control approach is used in this paper and the generalised predictive control (epC) performance index is adopted (Chen et al., 1999a) , given by

Theurem 1: Suppose that Yd and Yd are defined for all t 2: 0 and bounded. In the absence of disturbances, the closed-loop system under the nonlinear predictive control (4) can asymptotically track the desired reference Yd(t) for all t 2: o.

T

T,

(Y(t + T) - Yd(t + T»)dT

Then the above process is repeated as time goes . When the future output is predicted using Taylor expansion up to second order , Chen et al (1999 a) show that the model predictive controller can be given in a closed form Chen et al. (1999a). For the robotic manipulator (1) and (2) in the absence of disturbances, the Nonlinear epc control is given by

where the feedback gain matrices KJ and K2 are determined by

In the controller design , first it is supposed there are no disturbances. In Section 3, we will discuss how to design a non linear observer to estimate the disturbance d and then compensate for it.

J=~/ (y(t+T)-Yd(t+T»)

=0

+ K2(Y - Yd) - CJ(B)-lG(B,ti) - Yd }(4)

where C E R2 x 2 is a constant matrix of full rank and in many cases , C is an unit matrix.

To

for T

"U(t)* = -(CJ(B)-J B)-J {KdY - Yd) +

Suppose that the controlled output Y E R2 is the combination of the displacements of the robotic manipulator , i.e.,

yet)

= "U(t + T)'

(3)

The structure of the nonlinear epc control in (4) for a two links robotic manipulator is same as the well known computed torque control when the output in the moving receding time horizon is approximately predicted by Taylor expansion up to second order. This establishes the relationship between the predictive control which is widely

where TJ and T2 are the minimum and maximum predictive times respectively. Yd E R2 is the reference trajectory vector. At time illstallt t , the future output y (t + T) , T E [TJ, T 2 J, is predicted using Taylor series expansion ,

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used in process control and the computed torque control (dynamic inversion control) which is well known ill mechallical and electrical engineering and pro\'ides deep insight into these two methods. 3. NO:\LINEAR DISTURBA:\CE OBSERVER Fig. 1. Robotic manipulator controller structure with the nonlinear disturbance observer

In Sectioll 2, it is assumed that there are no disturbances. To compensate for the effect of the unknown disturbance d, a non linear disturbance ouserver is designed to estimate it. The non linear disturuance observer used in this paper is given uy

i = -L(B)z + L(B) (C(B , 8)

-

Bu -

The controller consists of two parts- the nonlinear predictive controller (or the computed torque controller) in Section 2 and the nonlinear disturbance ouserver in Section 3. In this and the following sections, we will investigate the properties of this control system schellle. It will be shown that this composite controller is equivalent to a nonlinear PID controller amI the asymptotic stability of the composite controller will be proved.

p(8)) (9)

and (10)

Following the control system diagram in Figure 1, the control action imposed on the robotic manipulator is given by

where z E R2 and J E R2 are the ouserver state and the estimate of the disturuance d, respectively,

u(i)

= W J(B)-I ,

(15)

- d(t)

where u(i)' and J are given uy the nonlinear GPC (4) and the 1I0nlinear disturbance observer (9) and (10) respectively.

and the non linear 0 userver gain matrix is gi ven uy L(B)

= u(t)"

(12)

By invoking the GPC controller and the plant model into the observer dynamic equations, it can be shown that when the initial disturbance estimate is chosen as

(13) The convergence rate of this ouserver can ue adjusted uy the choice of the constants WI and W2·

the disturbance estimation is given by

In order to develop the stauility result, it is assumed that the disturuance varies much more slowly than the system state. However, as shown in Chen et a/. (2000) and experimental results in this paper, this assumption call be removed ill some cases. Stauility of the above non linear disturuance ouserver is stated in Theorem 2.

d(t)

=

-B- 1 WC-

I

(ell) +

! t

K2e(t)

+ KI

e(T)dT)

o ( 17) where e is the tracking error , defined by

The01"1;7T1 2: For the two-link robotic manipulator

(1) and (2), the ouserver (9) and (10) is gloually exponentially staule, if W in (13) satisfies

e

(18)

Y

Then substituting the disturuance estimate (17)and the nonlinear GPC (4) into the control law (15) yields

(14)

where 82m denotes the maximum velocity of the second link and X is an inertial parameter depellding on the tip and second link masses and the lengths of the first and second links. 4.

= Yd -

'U

~ONLINE.\R

= u(i)* - d(t) =(CJ(B)-I B)-I (Kle + K 2 c + Yd) + B-IC(B , B) + +B-IWC- I

PID PREDICTIVE CONTROLLER

(ell) +

! t

K2e(l)

+ KI

e(T)dT) (19)

o

The control systelll diagram for robotic manipulators proposed in this paper is shown in Figure 1.

This composite controller can ue further written in the PID controller structure , given uy

77

Suppose that disturbance varies much more slowly than the state, It follows frol1l the observer (9), (10) and the system model (1) that

.' ""hnur 1"10 [·oulrvllu

El

=d - d

=_B- 1L(B)B(d =

d)

_B- 1 L(B)Bel

(26)

Fig , 2, Nonlinear PID predictive controller

J

Furthermore substituting the control law (15) into the manipulator dynamics yields

t

U

= P(B)e(t) + D(B)e(t) + I

e(T)dT

+ N(B,8)

J(B)B

o

(20)

= B- 1 (J(B)C- I KI

D(B)

+ H'C- I K 2) (21)

= B-I(J(B)C- I K2 + WC-I)

= B(u*

- ed

(27)

Invoking (4) into (27) and combining with (26) yields the the closed-loop error dynamics of the robotic manipulator under the composite controller, given by

where P(8)

+ G(B,8)

(22) {

(23)

and

J(B)C-1 (e(t)

+ Kze(t) + K1e(t)) + Bel (t)

ej =

_B- 1L(B)Bel (t)

Theorem 3: The two-link robotic manipulator (1) and (2) with a constant disturbance under the composite controller consisting of the nonlinear epc (4) and the nonlinear disturbance observer (9) , (10) as in Figure 1 , i.e" the non linear PID predictive controller (20), is exponentially stable if the condition (14) is satisfied,

This controller is referred to as a 'IlOnlinear' PID pl'edzctive contlvlleT' as shown in Figure 2 where x denotes the state vector of the robotic manipulator, i.e. , x = [B; 8] , The proportional and differential coefficients are non linear functions of the displacements of the links B, In addition to the traditional PID structure, a prediction part N(x) is included in this controller. It consists of two terms. The first term B-IJ(B)C-IYd takes into account the control input requirement for future output using the second order derivative of the reference signal (note that the first derivative of the reference is employed by the PID part ,) The latter term B- I G(B,8) is to make up the influence of the current system 's dynamics on future output. Hence N (x) takes into account the influence of the current system 's dynamics on future output and the input requirement for tracking future reference, This can be explained from the fact that this controller is derived from the predictive control method in Section 2,

6. EXPERIMENTAL RESULTS 6.1 Expel'intent setting

The proposed nonlinear PID predictive control is implemented on a two link robotc manipulator in the laboratory, In this experiment, a direct drive motor is attached to each joint and potentiometers and tachometer are mounted at the end of each link to measure the position and the velocity of the links, Since the outputs of the tachometers are quite noisy, the signals from the tachometers are filtered by digital filters before used to calculate the control action . The motor dynamics are approximately represented by a first order model. All the calculation in the nonlinear controller and the nonlinear disturbance observer is performed by dSPACE, The physical data and parameters of this system is given in the Table of Appendix. Two controllers are implemented and compared, One is the computer torque control method and the other is the nonlinear PID predictive controller proposed in this paper.

5. STABILITY Stability is essential for a control system, It is important to investigate stability of the composite controller consisting of the non linear predictive control (4) and the non linear disturbance observer (9) and (10). Define the observer error as el

= d - d,

The reference signal for each link is generated by the output of a stable transfer function Gr(s)

(25)

78

=0

~I

Reference

00 •

•

-

Computed torQue control Nonlinear predICtIVe PlO

j

"

, I

.

\ \ \ \

..

I

,,

,

\ \

\

,

20

. -"

':~ ,~);;.

-

-

-, ."

"

'0

20

T ,~ ~ I6C )

"

I

o

Fig. 4. Nonlinear PID predictive controller versus computed torque control: Second Link

trol method, in particular, in steady state. However since the nonlinear PID predictive controller is derived from integration of the nonlinear predictive controller and the non linear disturbance observer , the observer considers the friction as an external disturbance and estimates and then compensates for it. The tracking performance in Figures 3 and 4 shows that the nonlinear PID predictive controller works well against friction. The tracking error in state steady in removed.

+ 1.8s + 1

for the both links in the experiment . It is obvious that the reference signal generated by the above model driven l>y a pulse generator is smooth and differentiable up to second order. The position and velocity gains of the computed torque controller are chosen as 'i

= 1, 2.

In modeling the two-link robotic manipulator, the effects of the sensors, connection , wire, etc, are ignored. The controller is directly generated based on the dynamic model of the robotic manipulator in Chen et al. (1999a) and the physical parameters in the Tal>le of Appendix C. Due to the mismatch between the model and the real robotic manipulator , the coupling effect l>etween the two links cannot be completely removed by the computed torque method and this is evident by the fact in Figures 3 and 4. At the beginning of the experiment , the first link moves to track the reference signal. It is required that the second link mailltaina the relative degree between the first link and the second link to be zero during the period 0-5 seconds. The second link has significant tracking error for the computed torque control method. However the remaining coupling effect due to the unmodeled dynamics is considered as an unknown disturbance in the proposed nonlinear PID predictive controller, where the built-in disturbance observer estimates and then compensates for it. The similar phenomenon occurs at 5 second when the second link starts to track its reference trajectory. As shown in Figure 3, compared with the computed torque control method , the non linear PID predictive controller greatly reduces the tracking error of the first link caused by the coupling effect. Thus the non linear PID controller exhibits quite good performance rol>ustness .

(28)

In the experiment of the proposed nonlinear PID predictive control , the same parameters as that. in the computer torque control, i.e ., (28) , which are with respect to TJ = 0 and Tz = 1/1.2 sec in (3), are used in (4) and the observer gains are selected as Wj

= 1 and

W2

Reference

Tome{sec )

1 GT(s) = - : : - - - -

=3

:"~~ : - .1 -'--------\....:." .,,,"_... . ___ ~_:.~_-:. _:..-_-!

-~L--~====~,O====~"====~20---'~'--~,

,.

driven l>y a pulse generator with the amplitude 90 degree. The transfer function G T (s) can be considered as a desired model that the robotic manipulator should follow. It represents the tracking performance specifications and is chosen as

kZi

•

Computed torque control Nonhnear predictive PlO

Fig. 3. Nonlinear PID predictive controller versus colllputed torque control: first link

S2

I

= 2

The tracking performances of the proposed nonlinear PID predictive control and the computed torque control are further compared in Figures 3 and -I . The nonlinear PID controller significantly improves the tracking performance. There are two important factors degrading the performance of the computed torque control method in this experiment. One is friction. The other is the mismatch between the model used for the controller design and the real robotic manipulator. Friction widely exists in engineering systems and is quite difficult to model and cOlllpensate for. Friction is significant in the two link rol>otic manipulator for the experiment . As shown in Figures 3 and 4, the friction greatly deteriorates the tracking performance of the computed torque con-

79

7. CONCLUSIOl\S

Kim, B. K., W. K. Chung and Y. Youm (1996). Robust learning control for robot manipulator based on disturbance observers. In : P7'Uceedings 'Of IEEE /ndust7'ial Elect7'U71ics GU7!fer'C7!ce. pp . 1276-1282. Koruada, S. and K. Ohnishi (1990). Force feedback control of robot manipulator by the acceleration tracing orientation method. IEEE T1'Unsactiu7!s '011 IndustTial Elect7'Unics 37(1) , 6-12. Murakami, T., F. Yu and K. OllIlishi (1993) . Torque sensoriess control in IIlultidegree-offreedom manipulator. IEEE T1'Unsactions un /ndustT'iai Elect7'U1Iics 40(2) , 259-265. Oh, Y. and W.K. Chung (1999). Disturbanceobserver- based motion control of redundant manipulators using inertially decoupled dynamics . IEEE/ ASME TT'U1!SactioTls 011 Mechat1'01Iics 4(2) , 133- 145. Sadegh , 1'\. and R . Horowitz (1990). Stability and robustness analysis of an adaptive controller for robotic manipulators. Inte17wtiunai JOUT'nal 'Of Rubotics Researd! 9(3), 74- 92. Slotine, J. J. E. and W. Li (1987). On the adaptive control of robot manipulators. Intematio7!ai JO'a1'7lal of Rubutics ReselJTd! 6(3), 44- 59. Zhang, G . and J. Furusho (1998). Control of robot arms using joint torque sensors. IEEE GontT'ul Systems Magazine 18(1), 48-55 .

A new kind of nOn linear controllers - the nonlinear PID predictive controller for a two-links robotic manipulator has been developed in this paper based on the concept of disturbance observer based control. In theoretical aspect , it has been shown that the integration of a nonlinear predictive control and a non linear disturbance observer leads to a nonlinear PID controller and stability of the composite controller has been established . It should ue noted that since an integrator arises naturally ill the cOIIIPosite controller, it removes all tracking error in steady state if disturbances and unmodeled dynamics do not destroy stability of the closed-loop system. The experimental results show that the integration of a non linear controller and a nonlinear disturbance observer can work well. The composite controller exhibits good performance robustness against friction and ullInodeled uynamics in the experilJlents.

ACKNOWLEDGEMENTS This work was supported by the UK Engineering and Physical Science Research Council under the grants No. GR/N31580 and GR/L62665. The experiment facility provided by Dr. Donald J. Ballance is greatly appreciated.

APPENDIX: PHYSICAL PARAMETERS FOR EXPERIME;'I1TS Parameters First and Second link lengths Second motor mass Tip mass in the end point First and Second link masses First motor torque constant Second motor torque constant First motor voltage constant Second motor voltage constant Armature resistance of Motor 1 Armature resistance of Motor 2

REFERENCES Bickel , R. and M. Tomizuka (1995). Disturbance observer based hybrid impedance control. In: P1'Oceedings of Amer'ican Gont7'01 Gunfer·e1lce. Seattle, WA. pp. 729-733. Bickel, R. and M. Tomizuka (1999). Passivilitybased versus disturbance observer based robot control: equivalence and stability. ASME JO'U17lal 'Of Dynamic Systems, MeaS'U7'emeut and GontmI121(1) , 41-47. Chen , Wen-Hua , D. J. Ballance and P. J. Gawthrop (1999a). Nonlinear generalised predictive control and optimal dynamic inversion control. In: PT'oceedings of 14th IFAG Wu1'ld Gungl'ess, Beijing, Ghi7!a , 1999. Vol. E. pp . 415-420. Chen . Wen-Hua, D. J. Ballance and P. J. Gawthrop (1999b) . Optimal control of nonlinear systems: A predictive control approach. Autumatica. Submitted to Automatica. Chen, \\'en-Hua , D. J. Ballance, P. J. Gawthrop and J. O 'Reilly (2000). A nonlinear disturbance observer for two-link robotic manipulators. IEEE T1'Unsctio71S on /ndustT'ical Elect7'Unics.

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Values 0.38 m 0.44 kg 0.1 kg 0.361 kg 0.23 Nm / A 0.044 Nm/A 0.29 V / rad/sec 0.047 V / rad / sec 3.4

5n

n