H∞ Optimal Tuning of the Saturated PID Controller for Robot Arms Using the Passivity and Dissipativity

lb-O? 2

Copyright © 1996 IFAC 13th Triennia] World Congress, San Francisco, USA

Hoo OPTIMAL TUNING OF THE SATURATED PID CONTROLLER FOR ROBOT ARMS USING THE PASSIVITY AND DISSIPATIVITY Takayuki Nakayama and Suguru Arimoto

Fuculty of Engineering, University of Tokyo,

Bunkyo~ku,

119 Japan

Abstract. A modified PID controller wbich feeds a saturated position error feeedback instead of proportional one guarantees the global asymptotic stability for setpoint position control. For this controller, this paper proposes an Hoo tuning method to realize the disturbance attenuation against unknown torque disturbances at each joint. In its stability analysis, tbe following facts are sbown, 1)global asymptotic stability of the closed loop system and disturbance attenuation are feasible easily by taking account of physical properties of robot dynamics 2)disturbance attenuation with arbitrary gain can be established independently of the asymptotic stability. 3)as to robot dynamics, the storage function should be taken as a total internal energy and if the total inter~ nal energy is cbosen as the storage function, tbe robot system bas tbe KYP property which connects the passivity concept and dissipativity concept. Keywords. Robot control, circuit

H~infinity

control, Passivity, Dissipativity, Nonlinear

key role in nonlinear H 00 optimal control. Tbe Isaacs equation (inequality), which gives rise to a dissipativity of the system, guarantees attenuation of a disturbance in the input signal, and the other conditions guarantee the stability of the closed loop system. To find a positive function called storage function satisfying the Isaacs equation (inequality) is a crucial problem for disturbance attenuation. In 1994, along Isidori's metbod, (Astolfi and Lanari, 1994b) and (Astolfi and Lanari, 1994a) showed a conditon under which the I/O gain between input reference position and output real trajectory becomes smaller than arbitrarily given gain 'Y and a PD controller with compensation for the gravity term has robustness against an input torque disturbance by choosing the storage function as a modified Lyapunov function.

1. INTRODUCTION Analyses of nonlinear systems have made considerable progress recen tly and in particular an Hoo control problem for nonlinear systems has been researched rigorously for these several years. In 1991, (Van der Schaft, 1991) proposed an Hoo robust analysis method with state feedback for a class of nonlinear systems. He showed that if a linear approxima· tion of a given nonlinear Hamiltonian system at a hyperbolic equilibrium point contains dom(Ric), ISMcs equation is solvable and the corresponding state feedback has desired stabilizing properties. (Isidori and Astolfi, 1992) showed that conditions of n011linear Hoo controllability can be obtained in a similar form for a linear system and two Hamilton - Jacobi - Isaacs equations play a

217

turbances. In section 6, we show simulation results to confirm theoretical results.

However, their method by direct calculation of the Isaacs equation is not feasible for a class of more dexterous motion control problems in robotics. For a dexterous motion control, it is necessary to involve some feedbacks of integration of state or output which might be nonlineat in themselves. Therefore the state space realization of a robot system which is controlled by a complicated controller to achieve dexterous motion is difficult in general. Further, even if the state space realization has been solved~ the results of direct analysis by the lsaacs inequality are too complicated and does not give a clear form, or not analitically solvable. To overcome these difficulties, we start from assuming internal asmptotic stability. For dexterous motion control, a method based on passivity concept and hyper stable theory is very smart (Arimoto, 1994), (Arimoto, 1995). This concept leads to the fact that an admissible controller should be constructed such that it satisfies passivity between disturbance input and controlled output variables. Assuming that the passivity between the disturbance input and the output is satisfied by a suitable controller, a condition of attaining the disturbance attenuation can be derived easily from a property of robot dynamics called the KYP property (Byrnes et al., 1991), (Hill and Moylan, 1980). By this property, we can associate the dissipativity with the passsivity through simple conditions. From this idea, the conditions for disturbance attenuation are obtained by only appending some additional conditions to those of asymptotic stability. For the saturated PID controller, it is shown that robustness against torque ripples can be adjusted by only P-type feedback gain which is irrelevant to stabilization. This fact is convenient for manipulation of robot dynamics because it is possible to choose the I/O gain arbitrarily no matter how other control parameters for stabilization are set.

2. SP-ID CONTROL Dynamics of a robot arm is written as

(Ho

+ H(q))ij + (~H + S(q, q) + Bo)q

+g(q) = v.

(1)

Here, q and q represent respectively joint p0sition and velocities, (Ho + H(q)) the positive definite inertia matrix, (!H + S(q,q)) the coefficient matrix of Coriolis and centrifugal forces, Bo the damping coffiecient matrix, g(q) the gravity term, and v the input torque. We consider a problem of setpoint position control. As for this problem, PID controller is not adequate under the condition that the initial angular velocity is large. We modify a PID controller to satisfy the global asymptotic convergence to the desired posture in such a way as

,,= -Blq -

AITan-I(~q) - COY

t

-Cl

J

ydT

(2)

o where ~q = q - qd' qd is a given desired posture, Tan-I(~q) is the saturated position feedback, y is the linear combination of output signals defined as

and El, Ab Co and Cl are positive definite feedback gain diagonal matrices. This controller is named as SP-ID Control1er. Here, Tan-l(~q) is a vector defined as

(Tan- I (~qd," " Tan -I (~qn))T.

In section 2, a saturated PID controller is constructed and the dynamics is rewritten in a nonlinear state space form. In section 3, asymptotic stability of the closed loop system to the desired state is shown. In section 4, dynamical interpretations of passivity and dissipativity of robot dynamics arc discussed. In section 5, we propose a method for Hoc optimal tuning against torque dis-

Tan-I(~q;) denotes the principal value of arctan(~q;).

Now consider the situation that there exist torque ripples at each joint so that a torque disturbance is added to the SP-ID feedback input torque. Then the total dynamics is written as

218

(Ho

+ H(q))ij+ (~il + S(q,q) + B)q

+g(q)

For the setpoint control problem, the reference qd is constant and hence this can be regarded as a parameter of the system. Thus, the system is represented by only state vari-

,

= -A,Tan-I(Ll.q) - Coy -

c,

J

ables.

ydr

o

+111,

As a. consequence, the robot dynamics is written in a simple llonlincar system form

(4)

where w is the torque disturbance and B Bo + B I •

=

(8)

Attention should be paid to the fact that differentiating the saturated position feedback

signal yields d (T - I A) -d an uqi

t

.

= qi 1 + 1w.q, A 2'

3. PASSIVITY AND ASYMPTOTIC STABILITY OF CLOSED LOOP SYSTEM

(5)

';Ve show that if there exists no torque dis-

and integration of it yeilUs

turbance, the SP-ID controller can control the manipulator to the desired position under any fast initial velocity.

Theorem 1 The total system controlled by SP-ID feedback servo loop has the passivity between torque Jisturbance w and output .z:.

Next, we rewrite this dynamics to the state space form. We define the state valiables as

"1

1

= r

~ C1 g(qd) + I

." = Tan-' (Ll.q),

.,. =

i

ydr,

Proof (7)

Input-output pair (%, w) satisfy the follow· ing inequality

q.

.7 w ~

:t

H'I (r, Ll.q, q)

+ Vi (Ll.q, q, y),(9)

Then the total s}'stem is \VI"itten as where

W,(r,Ll.q, q)

= ~qT(Ho + H(q))q

n

where

+ ~)ali + "bi){Ll.qiTan-I(Ll.q;} i=l

I. = -(Ho + H)-IIC'''I + (AI + CO")"" 1 . + {2H(.",.,,,) + S(.",.,.) + B + Co}"" + 9(",) - 9(qd)),

1 -2Iog(1 + t>qi2 ) } +"Tan-T(Ll.q){Ho

+ H(q)}q

+P(q) - P(qd) - Ll.qT g(qd) 1

+2rTClr,

(10)

VI (Ll.q,q, y) = qT(B - «{Ho + H(q)} +cl1)q + yT Coy +uTan-T(Ll.q){A, - c2I}Tan-'(Ll.q).

(11)

219

Here, P( q) is a potential energy satisfying oP/oq = g(q), Cl is a positive constant such that it satisfies

TheOl'em 2 The closed loop system is asymptotically stable and the SP-ID controller transfers the manipulator to the desired posture in a global asymptotic sense in the absence of torque disturbance.

Tan-T(t.q){ -~H(q) + S(q, q)}q

~

-Cl

11 0111 2

(12)

for all q and q, and C2 is a positive constant such that it satisfies

Proof

In the absence of torque disturbance, (9) gives rise to

Tan-T(t.q){g(q) - g(qd)} ~

-c,1I Tan-l(t.q)II'.

(13)

Existence of such Cl > 0 is evident from the fact that the left hand side of (12) is

From this inequality, since W 1 is positive definite and f.lVl is negative definite, W l

saturated in ~q a.nd quadratic in q, and C2 > 0 also exists because the gravity term

asymptotically converges and VI converges to O. VI is a quadratic form of Tan- l (t.q), 01

{g(q) - g(qd)} is composed of trigonometric functions of t.q and is 0 at t.q O.

and 1/. Therefore, q, 1/ and Tan- l (t.q) converge to 0 as t __ 00.

=

VI and W 1 are positive definite under the following conditions:

1

" < ,f2 ,

AH

a" > 4

4. DYNAMICAL INTERPRETATION OF PASSIVITY AND DISSIPATIVITY OF ROBOT DYNAMICS

+ a, (14)

A passive system satisfies the following inequality:

where ali and bli are the (i, i) component of diagonal matrices Al and BI respectively, AH denotes the maximum eigenvalue of {Ho+

T d lV V .. w>-d - t .+ "

H(q)}, C3 is the absolute value of the minimum eigenvalue of o'P(q}/oq2, C4 is the maximum magnitude of {g(q) - g(qd)} and a2

lV,

a2

>

max(.C2,2C3),

C4

a2

a2

2ca

Then integrating (11) from time 0 to time t yields

!

~ Wl(t) -

Wl(O)

o

+

O.

(18)

On the other hand, a dissipative system sat-

isfies the following dissipation inequality,

,

,

T w ",dT

~

resenting the input energy, W. the internal energy and V,/!I the energy consumed in the system. The equal sign means the principle of the conservation of energy.

(15)

1

tan-:5 ( - -1)'.

,

0, V.

The left hand side of the first equation is rep-

is the constant which satisfies

{

~

!

Wd(t} - Wd(O) :5

VidT

!(-l 11 wlI'-II"II')dT. o

0

(16)

Wd is called a storage function, represent-

This shows the passivity between torque dis-

ing the storage energy of the system that is positive definite. This inequality means that

turbance w and output ...

stored energy is smaller than supplied energy

from the external world.

Remark

Comparing these two inequalities, we notice

that lVd corresponds to W•• " 11 wll' corresponds to .. T wand 11 "11' corresponds to V•.

The control gains Al is not required to be taken so large.

220

More in detail, V" is divided into the part of 2 output energy 11 .011 and the part of energy consumed at damping elements.

Hence, we instead of it use the property of p .....ivity. Substituting the system dynamics into the Isaacs inequality yields

When the dissipativity of the system is considered, the problem is to find a suitable storage function. As to this problem, it is natural from a physical view point to select the internal energy lV6 as a storage function.

The first term of r.h.s. is f,W. On the other hand, it follows from the passivity that if

Thus two properties of the system, '~passiv­ ity" and "dissipativity", are connected from the physical view point concerning robot dynamics.

wT.o - V - W ,gl W + h;hl 1 T T + 41' W.gIg I W, :'0 0

5. H= OPTIMAL CONTROL

is satisfied, the Isaacs inequality is attained. Calculating the term W ~ yeilds

An admissible control is to achieve the closed loop asymptotic stability with attenuation of the inHuence of exogenous input 10 on output .z:. The first one is already discussed in section 3. In this section we discuss the condition to attenuate the influence of a torque disturbance. To do this, it is necessary to cite Isidori's work.

. TIT T W,,,, - W,glw +h l hi + -, W.9 191 Wo' 41

g1W;

= {Ho + H(q)} -lW" =X3 + nz, = .o.

(22)

(23)

This property is called the KYP property (Kalman . Yacubovitch - Popov) and characterizes the robot dynamics. Substituting this property into (22), finally we obtain the condition for satisfying the Isaacs inequality

Theorem 3 (see (Isidori and Astolfi, 1992» If there exists a storage fuctioll fying the Isaacs inequality

V(~)

(24)

satis-

Substituting (11) into VI yeilds Vd("') + h;(x)h l (",) 1 T T + 412 V,gl(x)gl (xW, :'00

0:'0 qT(B - n{Ho + H(q)) + cd)q

(19)

+nTau-T(ilq){AI - c,I}Tan-l(ilq) T 1 +y {Co - (1 + 4121)}y. (25)

for all t > 0, the dissipation inequality

,

V(t) - V(O) :'0

.lh2 11 wll'-

1I.oII')dr(20)

o

Then, under the conditions

Co

is satisfied and under the condition V(O) = 0 the disturbance attenuation

,

l'

.Ill o

{ B - ,,{Ho + H(q)) Al - c2 1 > 0,

,

wll'dr

~ .lII.oII'dT.

(21)

0

is satisfied.

+ cll > 0,

(26)

the Isaacs inequality is satisfied.

This shows that, under the condition of asymptotic stabilizatioll, if we set the control gain

Co to (1 + We take this storage function by W = W I as discussed in section 4. To obtain a condition under which (20) is satisfied, direct calculation of the Isaacs inequality may be successful, hut it requires complicated calculus.

I = (1 + -, )1, 4")

4~' )1, we can attain the distur-

bance attenuation in s11ch a way that 1 w 1I~II.oIl.

Remark

221

11

,

1)Attention should be paid to the fact that Co is independent of asymptotic stabiliza-

l'

tion of the closed loop system, and hence Co can be used purely for the adjustment of robustness.

J

11

o

,

wll2dr :5

J

11

zll2dr.

(27)

0

In this analysis, it is natural from the physical view point in the case of robot dynamics to take the storage function as the internal energy. This yeilds the KYP property, and, from this property, the conditions of disturbance attenuation are obtained by appending an additional condition on P-gain Co to the conditions on asymptotic stability of the closed loop system.

2)ln this discussion, the disturbance attenuation can be proven under the ini tial con· dition W(O) = O. Therefore, the disturbance attenuation is guaranteed only after a sufficient convergence to the desired posture.

6. SIMULATION RESULTS 8. REFERENCES

We carried out a simulation for a single robot manipulator with three joints to see the pro-. cess of attenuation of torque disturbances. We consider a situation that a square waveform torque disturbance is given after the sufficient convergence to the desired posture, i.e., from. t = 150. vVe take the output signal as Coy to nt the physical unit of the output to the input torque. It corresponds to the P

Arimoto, S. (1994). Fundamental problems of robot control: PartI, innovation in the realm of robot servo-loops; part2, a nonlinear circuit theory towards an understanding of dexterous motions. Robotica. Admoto, S. (1995). Stability analysis of setpoint control for robot dynamics via nOlllinear position-dependent circuits. Dynamics of ContinuoU3, Discrete and Impulsive system.o 1-1, 1-17. Astolft, A. and L. Lanad (1994a). Optimal tuning of pd controllers for rigid robots. Asian Control Conference pp. 141-144. Astolfi, A. and L. Lanari (1994b). H= control of rigid robot. Symposium on Robot Control pp. 199-204. Byrnes, C. I., A. Isidori and J. C. Willems (1991). Passivity, feedback equivalence, and the global stabilization of minimum phase nOlllinear systems. IEEE Transaction on A utomatic Control 3611, 1228-1240. Hill, D. and P. Moylan (1980). Dissipayive dynamical systems: Basic input-output and state properties. J. Franklin lrut. 309, 327-357. Isidori, A. and A. Astolfi (1992). Disturbance attlluatioll and hoo control via measurement feedback ill nonlinear system. IEEE 7'Tansadion on Automatic Control 37-9, 1283-1293. Van der Schaft, A. J. (1991). On a state space approach to nonlinear hoo control. Systems (1 Control Letter. 16, 1-8.

feedback torque. In this case, the theoretical upperbound of gain is (.(f.~I))! and the minimum value of it is unity at Co = 2. Fig.l shows the I/O gain behavior for the case Co = 2 and Co = 10. It is confirmed that the maximum value of gain '1 is smaller than the theoretical upperbound and the gain at Co = 2 is smaller than that at Co = 10 as theoretically indicated.

....

-, _

Co = 2.0

:rz

_

..

--:...,.:.

Co

= 10.0

-

Fig.l I/O gain 7. CONCLUSION

If the closed loop system of a robot with SPID controller is asymptotically stable, set1 ting P gain Co to (1 + -, )1 yeilds the dis41 turba.llce attenua.tion

222