high-gain observers in the state and parameter estimation of robots having elastic joints

Systems & Control Letters 13 (1989) 331-337 331 North-Holland

High-gain observers in the state and parameter estimation of robots having elastic joints

Salvatore NICOSIA

Dipartimento di Ingegneria Elettronica, H Universith di Roma 'Tot Vergata" Via O. Raimondo, 00173 Roma, Italy

Antonio TORNAMBI~

Fondazione Ugo Bordoni, Via Baldassarre Castiglione 59, 00142 Roma, Italy

Received 6 April 1989 Revised 7 August 1989

Abstract: The paper deals with the state and parameter estimation of robots having elastic joints. The estimation of the unknown parameters of the robot is reduced to the estimation of its state variables by a state space immersion. The Luen- berger observer is used in the state estimation of the extended robot model. The use of high gains is studied in the cancellation of nonlinearities in order to simplify the observer design. The high gain induces a time scale separation between the robot model and the observer and therefore the singular perturbation theory can be used in the stability analysis of the estimate error dynamics. In particular, it is shown that the error dynamics reaches the stable equilibrium in a very fast transient ensuring that the slow dynamics of the observer is just that of the given robot.

Keywords: High gain; observers; elastic robots; parameter estimation; singular perturbation theory.

1. Introduction

Modeling flexible robot manipulators is receiv- ing increasing attention by the researchers acting in the robotics area. A particular class of these manipulators is constituted by the robots with elastic joints which have significance not only as a first approximation of robots with distributed flexibility but also as models of some existing robots [18]. Recently, control techniques suitable for rigid manipulators have been extended to flexible robots; among these we mention: linear techniques [2,4], state space pseudolinearization [16], compensation of nonlinearities via dynamic

state feedback [5], optimum control [21] and singular perturbation techniques [7].

A prerequisite to the successful execution of these sophisticated control strategies is the availa- bility of accurate numerical values of the robot parameters and of the measurements of all the state variables [20]. Frequently, flexible robots have parameters completely unknown and state variables that cannot be directly measured; for instance: elastic constants of the joints, and generalized velocities.

The aim of this paper is deriving dynamic estimates of the unknown parameters and of the state variables by the knowledge of the measured outputs and of the torques applied to the joints. Several methods have been proposed in order to answer the above estimation problem; among the best known we mention the extended optimum filter [9]. The estimation of the unknown parameters is obtained by extending the state vector to the unknown parameters and by using the extended optimum filter to estimate the state vector of the so obtained extended system.

The optimum filter has been shown to have the same structure of a Luenberger observer [1] in the case of linear systems. Some recent results have been given to extend the use of the Luenberger observer in the state estimation of nonlinear systems [11,12,13,17,19,23]. Unfortunately, the observer design is very difficult for nonlinear systems and in particular a systematic procedure has never been performed. So, in general, a particular procedure must be performed for each class of systems; an expert system for the automatic gener- ation of asymptotic observers can be found in [3] in the case of the parameter estimation of linear systems.

The high gain is a classical tool in the feedback design [15,22] and is used in the cancellation of the nonlinearities in order to simplify the control law. In addition, the high gain was used to ap-

0167-6911/89/$3.50 © 1989, Elsevier Science Publishers B.V. (North-Holland)

332 S. Nicosia, A. Tornambb / State and parameter estimation of robots

proximate the output derivatives of linear systems so that the estimate errors are almost independent of disturbances. In this paper we will show how an observer, having the structure of a Luenberger observer, can be used with high gains, say a high- gain observer, to reduce the complexity of the observer design for robotic manipulators. The high-gain observer induces a time scale separation between the considered system and the estimate error dynamics; hence, the singular perturbation techniques can be used in the stability analysis of the error dynamics [10,14].

2. Problem definition

Robotic manipulators are considered with N degrees of freedom. A torque u~, i = 1 . . . . . N, is exerted to the i-th link by a linear torsional spring characterized by an elastic constant E~, i = 1 . . . . . N.

The first step of the estimation algorithm is the structural modelling of the robot to be controlled.

2.1. Dynamic modelling of robots having elastic joints

The dynamic model is obtained by using the Lagrangian approach. Let qi, i = 1 . . . . . N, be the i-th rotor displacement which, in the case of rigid robots, coincides with the relative displacement between the adjacent links i and i - 1. In order to describe the elastic effects, let us denote with qN+~, i = 1 . . . . . N, the i-th elastic displacement; the relative displacement between the i-th link and the (i - 1)-th link is qi + qu+~.

Generalized coordinates q~, i = 1 . . . . . 2N, can be collected as follows:

qV = ( q v, q~),

q~ = (ql . . . . . qu) v, qe = (qu+, . . . . . qzu) v"

According to the Lagrangian notation, the motion equations are given by

d 0T OT 0U dt 00~ 0q----~ + ~ = u,

d OT OT ~U - - - [ - - -

dt O~e Oq~ ~qe

where u =

( la)

D ( E ) q ¢ , ( lb )

( U | . . . . . UN)T denotes the vector of the

external torques applied to the joints, and - D ( E ) q e represents the vector of the elastic torques. Given a vector E = (E 1 . . . . . EN), D ( E ) indicates the matrix diag[E 1 . . . . . EN]. Notice that

D( E ) qe = D( q~ ) E = ( qu + l E1 . . . . . q2u Eu ) T.

Kinetic energy T is given by

T( q, (t) = ½dtTB( qr + qe)q

where the inertia matrix B(q~+q~) is positive definite. Let the inverse H(q~ + q~) of the inertia matrix B(q r + qe) be partitioned as follows:

H ( q r + q ~ )

"Hal( qr + qe)(N×N)

H2,( qr + q~)

Hl2( qr + qe) ]

H22(qr + qe) "

Equations (1) can be rewritten as follows:

qr=ePl(qr + qe, qr + ~te) + n l l ( q r + qe) u

- H l z ( q ~ + q~)D( E)qe , (2a)

qe=cP2(qr + qe, (L + fl~) + Hza(qr + qe)U

- H2z( q ~ + qe)D( E ) q e. (2b)

2.2. The estimation algorithm

Let continuous observation vector y ( t ) ~ R q be given by

y ( t ) = h ( q r ( t ), qe( t ) ) , q < a N ,

where h is assumed to be a smooth function of its arguments.

Let U t and I1, be the sequences of the available measurements

Yt = (y(~ ' ) , ~'~ [t0, t ]} ,

U t = (u(~ ') , ~ ' ~ [ t 0, t ]} .

Given the pair (U t, Yt), the state vector estimation problem consists in computing an estimate of the robot state vector

(q r ( t ) , qe(t) , t~r(t), qe( / ) ) T

based on (U,, Yt)- We wish to obtain such an estimate so that the estimate error vanishes as time increases for any observation sequence and for any initial estimate, where both are restricted to suitable domains.

S. Nicosia, A. Tornamb~ / State and parameter estimation of robots 333

An answer to this problem is given by the use of an observer for the state estimation. When some parameters of the robot are not well known the state vector can be extended to the unknown parameters which can be considered as particular state variables with constant dynamics. Hence, the observer can be used in the state estimation of the so obtained extended system.

3. Estimation of the generalized velocities

Let measurements be available for the generalized coordinates qr and G- In this section we will show how an observer having high gains, say a high-gain observer, can be used to obtain dynamic estimates for q~ and q¢.

We can define the following state and output variables:

~a = q~ + G, (3a)

f,2 = O~ + 0¢, (3b)

~21 = q¢, (3c)

~22 = O~, (3d)

Yl = qr + q¢, (3e)

Y2 = G- (3f)

Motion equations (2) can be rewritten in state space form,

gil = ~i2, (4a)

ff~2=ff~(~, u), i = 1 , 2 , (4b)

Yi = ~i, , (4c)

where

o , ( t . u) = + t12) + u

-- (H,2(~,1) + H22(~n))D(~2,)E, dp2(,~, U) = q02(,~l, , ~12) + H2I(~ll)U

- H22( gg1,)D( ~2,)E,

Consider the high-gain model

1 i i , = g i 2 a c T K i l ( f l i - - ~ i l ) ,

i = 1, 2, (5)

where K~j = diag(k~j) are fixed constant matrices such that the spectra of

p , ( X ) = X 2 + X k a + k n , i = 1 , 2 ,

are in the left half plane,

and e is a small parameter. ~(t) denotes the dynamic estimate of ~(t). Aim of this section is to show how (5) constitutes a high-gain observer.

The error equations corresponding to (5) take the form

~5]2 = - 1 K a , f a + qh(~5, u), E 2

i = 1, 2, (6)

where gij = (ij - gij, i, j = 1, 2, and

-W = ~12, "

Let us consider the extended system constituted by the motion equations (4) and by the error equation (6). The extended state vector is (~, (). In order to put the extended system into a singularly perturbed model let us introduce the following notation:

Zil = ~'1, i = 1, 2, (7a)

Zi2 = E~i2,

zTT = ( ~IT1, ~T12, zT1 , Z L ) . (7b)

In the new coordinates the extended system can be rewritten as

il = ~i2 '

g,2 u), L ~

EZil = Zi2 -- gi lZi l , 2, eza = - K S a + t2¢~(~, u),

i = 1, 2. (8)

Let us investigate the behavior of (8) as e becomes smaller and smaller.

Remark 3.1. The high-gain model (5) can be inter- preted as the projection of the extended system (8) through

- e2' (9)


where

E~ = block diag[ Ex, E 2 ],

E , = 0

Notice that there exists an e* such that matrices E i, i = 1, 2, are bounded over [0, e*].

An e-dependent submanifold can be defined:

= n × = 0,

K,2~,] = E 2 ~ i ( ~ , u ) , i = 1 , 2), ( 1 0 )

from which

~-~0-- ((~, ~) E n 4 N × ~ 4 N : ~ = 0 ) . (11)

Since the spectrum of each p~(k), i = 1, 2, is in the left half plane, Tikhonov's theorem [6,8] allows us to approximate, as e tends to zero, the orbits of the extended system (8) through the orbits of two reduced subsystems:

the slow subsystem,

~il = ~i2' i = 1, 2; (12)

u ) ,

the fast subsystem,

d Z~ d r = z i2- K i l ~ ' i l ,

d,~i2 dr Ki2~'il '

i = 1, 2, (13)

where • = t / e is the fast time scale.

Remark 3.2. The approximation of the orbits of the extended system (8) through the orbits of subsystems (12) and (13) is valid on a compact time interval [0, T]. In order to extend the validity of this approximation over any closed time interval of [0, ~ ) , we have to assume the uniform asymptotic stability of the slow subsystem (12) or the existence of a feedback control law, based on the high gain model (5), such that the slow dynamics of the closed loop system is uniform asymptotically stable. Further, the slow subsystem is just given by the motion equations (4) and the fast subsystem constitutes the error dynamics in the fast time scale.

Remark 3.3. I2 0 constitutes the equilibrium mani- fold for the error dynamics in the fast time scale.

Remark 3.4. For any (~, ~) ~ ~2~, we have

1K,1~,1 = e K i l K ~ l d p i ( ~ - t - E~ly, u), (14a) E

!2 Ki2,~/1 = dp/(~+ E~-le, u), (lnb)

i = 1 , 2 .

Remark 3.5. Since the fast subsystem (13) is asymptotically stable, Tikhonov's theorem guaran- tees us that for any (~, ~) ~ 12~,

l im~ i j=0 , i, j = l , 2 . e ---~ 0

Since matrix E~ -] is not bounded over [0, e*], in order to characterize the behavior of ~ as e becomes smaller and smaller, we need the following result:

Proposition 3.6. For any ( ~, ~) ~ £g,, we have

lim E~-I~ = 0, (15) e--*0

that is

1 l i m ~ - 7 ~ i j = 0 , i, j = l , 2 . e---~ 0

Proof. From the definition of I2~, we have that for any (~, ~) ~ I2~,

= :K3lep, u) , 1

= ~Ki]~.il eKilK~lepi(~, u), E i2 ~ =

i = 1, 2, and therefore, taking the limit for e ~ 0, one can see how limits (15) exist. []

Remark 3.7. Since system (13) is asymptotically stable in £ = 0 for any ~, the error dynamics reaches the equilibrium in a very fast transient. Hence, the slow dynamics of the high-gain model (5) can be obtained by taking the limit for e ~ 0 of (9). Taking (27) into account, we get

l i m ~ ( t ) = ~ ( t ) , (~, ~)~12, . (16) e--*0

The dynamic behavior in the slow time scale of the high-gain model (5) can also be obtained in a different way. Before that, we can benefit from the following:

S. Nicosia, A. Tornambb / State and parameter estimation of robots 335

Proposition 3.8. The following fimits exist for any (~, s) ~ G:

lim 1 K , a~,~ = 0, i = 1, 2, (17a) e~O E

l im~K, zSi,=Oi(~,u), i = 1 , 2. (17b) e--*0

Proof. The proof follows straightforwardly from Remark 3.4 and Proposition 3.6. []

Taking the limit for e---, 0 of the projection of the high-gain model (5) through ~2+ we obtain the slow subsystem induced by map (9),

lim ii~ = lim ~i2 + 1 K ,~--, o +-o e it (Y' - gi') (18a)

lim ii2 = lim 1Ki2(y i - ~il), (18b) e ---, 0 t ---, 0 E

i = t , 2 , (~ ,~)~+. (18c)

Results outlined in Proposition 3.8 allow us to compute the limits:

lilm~,2 + 1 K _ 1Ki , Z,t e i,(Y, ( , , ) = ~i~gi2 +

= 42, (19a)

= ; lira Kn(y, -~i , ) lim K,2zil=q,i(g,u), t-~O e---~ 0

(19b)

i = 1, 2. In view of equations (19), from (18) we obtain that the dynamic behavior in the slow time scale of the high-gain model (5) is just the dynamic behavior of the robot model (4):

lim ~",1 = ~i2, ~--,0 i = 1 , 2 , (~, ~ ) ~ 2 + . (20)

lim ii2 = ~bi (~, u), e--*0

It is noted that in going from the extended system (8) to the slow dynamics (20) of the high-gain observer, the initial condition f(t0) is lost and the values of f( t0) and ~(t0) are in general different; the difference is termed a left-side boundary layer, which corresponds to the fast dynamics (13). Equation (13) constitutes the so called boundary layer correction for ~(t) in the sense that explains the conditions under which ~(t) approximates ~(t),

( ( t ) = ~( t ) - E~-'e(,r) + O(e), (21)

where O(e) is a large-O order of e and is defined as a function whose norm is less than ke, with k being a constant [10]. It is noted that the boundary layer correction is only significant for a short time.

4. Estimation of the elastic constants

As in the previous section let us suppose that measurements for the generalized velocities qr and qe are not available. All the elastic parameters E i, i = 1 . . . . . N, are assumed unknown. Since the matrix H22 is nonsingular we can define the following u-dependent state space immersion from U c R 4N to R5N.

~11 = G + q¢, (22a)

~12 = qr q- qe ' (22b)

f2t = q¢, (22c)

~22 = (]e' (22d)

f23 = q)2(G + qe, q~+q¢) - H 2 2 ( q r + qe)D(qe)E + H2,(G + G)u,

(22e)

where

U = {(qr, qe, qr, qe) ~ RaN: d e t ( D ( G ) ) 4: 0}.

The output variables are defined as follows:

Yt = qr + qe,

YZ = qe"

(23a)

(23b)

Motion equations (2) assume the state space form

~11 =~12, ~t2 = qh(~, u), (24a)

~21 = ~ 2 2 , ~'22 = ~23 , (24b)

~23 = q~2(~, u, ~), (24c)

Yl = ~11, Y2 = ~21' (24d)

where

~ T = ( ~ T , ~lT2, ~T21, ~T2, ~2T3) ,

~,(~, . )= +,(~,,, ~,~)+ +~(~,,, ~,~)

+ (H11 (~11) + H 2 1 ( ~ i l ) ) u

- ( H 1 2 ( ~ I I ) + H22(~ l l ) )D(~21)k ,


, : (~ , ,,, a) 0

- 0~11 ((~2(~11, ~12) + H21(~11)u

-H22(~l,)D(~2,)E)~12

+ 0~12 ¢, (~;, U)

3( H22(~,,)D(f2,) E) - - 0~2! ~22 + g21 (~11)/'i

with

E = ( 11)

12) + H21( ll)U- 23)"

Consider the high-gain model

~11 (12 ~11), = + 1KN(y 1 --

~12w E-~K12(YI--~11 ),

1 i2, = (22 + ~K21 (Y2 - ~21),

~22=~23+ -~K22(Y2-~21), ~23 = E-~K23(Y2--~21 ),

(25a)

(25b)

(25c)

(25d)

(25e)

where matrices Kij = diag(ki) are fixed constant matrices such that the spectra of

pl(~ . ) = ~.2 + ~.kll +k12

a n d

p2(~k) = )k 3 + )k2k21 + ~,k22 + k23

are in the left half plane,

~T (~T, ~T, ~T, ^T ~T) = ~22, 23

and e is a small parameter. As in the previous section we can consider the

extended system constituted by the motion equations (24) and by the error dynamics corresponding to (25). The extended system can be expressed in a singularly perturbed form by which we obtain, as e tends to zero, the slow subsystem (24) and the fast subsystem

d~n d'r - Z12 -- KnzTN, (26a)

dz12 dr K12~11' (26b)

d'~21 (26c) dT

dz22 (26d) d'r

d 523 = dr - K23,~21, (26e)

where T= t/e is the fast time scale and £ is defined as

with

- - = *~22 -- K21z21,

= "~23 -- K22 221,

i 0 0 0 0 ] 0 el 0 0 0 E~= 0 0 I 0 0 .

0 0 el 0 0 0 0 e2I

Analogously to the previous section, the singular perturbation technique can be used in order to characterize the time behavior of the high-gain model (25) in the slow time scale. It is easy to show how the slow dynamics of (25) is just that of motion equations (24). Moreover, the boundary layer correction (26) explains the conditions under which ~(t) approximates ~(t):

g(t) = ~(t) -- E~-1$(T) + O(e).

Also in this case the boundary layer correction is only significant for a short time interval.

Once an estimate of ~ has been computed by means of the high-gain model (25), dynamic estimates of the generalized coordinates, of the generalized velocities and of the elastic constants of the robot can be obtained through the inverse of (22):

0r=G-G, 0o=G,

o=G, g = D-1(~21 )n22' (~1,)(~02 (~11, g12)

+/-/21(&)u- &).

S. Nicosia, A. Tornambb / State and parameter estimation of robots 337

5. Conclusions

For robots having elastic joints a simple observer has been proposed which only requires one observability condition to be designed. This observer exhibits some good properties: it is linear and robust. Moreover, in certain cases, it does not require at all any knowledge of the model.

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high-gain observers in the state and parameter estimation of robots having elastic joints

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