Analysis of an adaptive controller for manipulators: Robustness versus flexibility

Systems & Control Letters 12 (1989) 251-258 North-Holland

251

Analysis of an adaptive controller for manipulators: Robustness versus flexibility G. C A M P I O N

* a n d G. B A S T I N **

Laboratoire d'Automatique, Dynamique et Analyse des Systgmes, Unioersitd Catholique de Louvain, Batiment Maxwell, Place du Levant 2, B-1348 Louvain-la-Neuve, Belgium Received 15 February 1988 Revised 8 August 1988

Abstract: The purpose of this paper is to analyse the robustness of an adaptive computed torque control for robotic manipulators, against high-frequency modes arising from flexibility. We present a robot model taking into account the flexibility of the transmission and we show that, for sufficiently small values of this flexibility, the adaptive algorithm guarantees the boundedness of the closed-loop tracking error system.

Keywords: Adaptive control; robotics; manipulators; robustness.

Introduction

A well known approach to improve the control of robotic manipulators is 'the computed torque method' where the control law is designed explicitly, on the basis of a detailed non-linear dynamical model, in order to compensate the robot non-linearities and to guarantee a desired linear closed-loop behaviour (see e.g. [1]). However, this method requires an accurate knowledge of the physical parameters of the manipulators and instability of the control algorithm can occur in case of uncertainty of the parameters. A possible way to handle with parameter uncertainty is to implement adaptive versions of the computed torque method, based on a suitable parametrisation of the robot model. We have proposed an 'adaptive computed torque controller' of this kind in a previous paper (see [2]). Many other adaptive algorithms for manipulators

* Chercheur qualifi6 FNRS. ** Member of GRECO-SARTA (CNRS, France).

can be found in the literature (see, for instance [3,4,5]). Our algorithm was designed for rigid-link manipulators and was proved to be convergent and robust against external input disturbances. Unfortunately, for existing robots, this perfect rigidity assumption is not satisfied and highfrequency parasitic modes can be excited. The robustness against neglected dynamics is critical in adaptive control (see, for instance Ioannou and Kokotovic [6]). Our contribution in the present paper is to analyse the robustness of our algorithm versus these unmodelled high-frequency modes which arise, for instance, from flexibility. Our main result (Theorem 1) shows that sufficiently small flexibility does not cause the unstability of the closed-loop error system. In Section 1 and 2 we introduce a simplified model of the robot with flexible joints. In Section 3 we point out the property of linearity of the robot equations with respect to the inertial parameters allowing to design an adaptive control algorithm (Section 4). In Section 5 we state our robustness result.

1. Modelling of a flexible transmission

Consider a rigid arm driven by an actuator through a flexible joint, for instance a gear mechanism. The flexibility is represented by a linear torsion spring between the rotor and the arm. We consider in addition viscous friction for the relative motion between the rotor and the arm. Define 16, q~a and I b, epb as the inertia moments and the angles of rotation of the rotor and the arm respectively (Ia << Ib). Define U~ and U b as the torques applied respectively to the rotor by the actuator and to the arm by the transmission, and k and c as the stifness of the spring and the viscous friction coefficient.

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

G. Campion, G. Bastin / Adaptive controller for manipulators

252

Without gravity effects the equations of motion are, for the arm: lh~ h = Up = k(O. - Oh) + c(4;. - Oh)

T~(q, c), p) is the n-vector of torques generated by Coriolis and centripetal accelerations and can be expressed as

(1) ~.(q, q, P)

and for the rotor:

d

I ~ + c(~,~ - ~pp) + k(ep~ - Op) = Ua.

- dt [M(q, P ) ] q - ~

(2)

Tg(q, p) is the n-vector of gravity torques.

Defining 7) = k ( 0 , - q'b) we obtain

Tr (q, P) is the n-vector of viscous and Coulomb frictions. It is assumed to be modelled as suggested in [7], i.e. each component of Tf is written

Ip k I , , + I p ~ + k ~ + ~ = I~+It~----~,U" 1

I~Ip

c.

as

and

[a, +a 10,1 10,

c

U p = n + ~,). For I . << Ip these relations reduce to Ia

C

T ~ + ~,~ + ~) = u~

(3)

and

Up=

~ + k, h

(4)

which are a representation of the transfer function between U, and Up, c l+~s

up v~

1 0 ~ q [0TM(q, P ) 0 ] .

-

.

(5)

c

1 + ~-s + ~ - s

2. The flexible robot model

M(q, p)~l+ T~(q, ~1, P) + Tg(q, p) + Tf(~t, p)

The general dynamical behaviour of an n-degrees of freedom manipulator with n fast parasitic stable modes arising from the flexibility of the joints can therefore be modelled as follows:

i~+ IxClO + IxZc2v = u.

= u.

(9)

We shall often use the following compact notation in equation (6):

f ( q , (t, P) = T~(q, (t, P) + Tg(q, p) + Tf(O. p).

M(q, p)i]+ T~(q, q, P) + Tg(q, p) + Tf(~t, p ) = v + ~ t C , b,

with a 1 and a 2 positive constants. u is the n-vector of torques produced by the actuators. v is the n-vector of internal flexibility state variables. Each component of v is thus of the form ~/= k(q, a - 0b) as described in Section 1. C 1 and C 2 are definite positive n × n matrices describing the dynamics of the parasitic modes. The entries of these matrices are thus of the form c/k and la/k respectively, as described in Section 1. The definite positiveness of C 2 is characteristic of mechanical systems whereas that of C1 results from the assumption of stability of the parasitic modes. The eigenfrequencies of the parasitic modes are of the order of bt 1. The limit when /~ ~ 0 clearly corresponds to the special case of a completely rigid link manipulator described by

(lo) (6) (7)

In these expressions: q is the n-vector of generalized coordinates. p is an N-vector of 'pseudo-physical' constant parameters (to be defined in the next section). M(q, p) is the n × n symmetric definite positive inertia matrix.

3. The parametrisation It is well known that the inertia matrix M(q, p) and the vector f(q, (1, P) are linear with respect to a suitable set of N 'pseudo-physical' parameters p = (Pl, P2 . . . . . PN) T"

G. Campion, G. Bastin /Adaptive controllerfor manipulators

These parameters are on one hand the coefficients of the friction models (8) and on the other hand the barycentric parameters also called 'inertial parameters'. The latter (which were introduced in the early seventies, e.g. [7]) are themselves nonlinear functions of robot mechanical parameters like mass, inertia moment, center of mass, etc. Several examples of such parametrisations can be found in [2] and [4]. This means that M(q, p) and f(q, 4, P) may be written in the following form: N

M(q, p) = Mo(q) + Y'~ p~M~(q),

(11)

i=1 N

f(q, 4, P) = f 0 ( q , q) + ~ P~f~(q, 4),

(12)

i=1

where Mi(q) and fi(q, q) are known functions of q and 4 while the pi's are unknown but constant coefficients. We assume however, in accordance with the usual reality, that physical bounds on the parameters Pi are known by the user: 0
(13)

with

3i &Pi

max

--

Pi min"

(14)

In addition we restrict ourselves to robots with revolute joints for which the inertia matrix M(q, p) is definite positive and bounded for any q and any p satisfying (13).

253

Our adaptive control law (see [2]) is then obtained by using a standard 'computed torque' control law, based on the rigid-link model (9), but with the parameters p, substituted by their on line estimates/3~:

u(q, 4, P)=f(q, 4, P)+ M(q, ~) • [qd

-'[-

Kp(qd--q)

+ K v ( q d -- q)]

(17) The on-line parameter-adaptation law is carried out as follows: ~i=Oi2(pT[Ra(qd--q)+R2(~d--4)],

(18a)

:, (0 pi =

(18b)

if~i=Pimax and~i>~O or if ~ i = P , min and ~i~<0, i

otherwise.

The regressor Oi is the solution of the linear system

M(q, ,3)~,= [Mi(q)t]+f~(q, 0)].

(19)

The coefficients oi are design parameters at the user's disposal and the (n × n) matrices R 1 and R 2 are solutions of the following Lyapunov equations:

gpRl + R1gp = Ol, KvR2 + R2Kv = Q2 + 2R1,

(20a) (20b)

with Q1 and Q2 arbitrary constant symmetric definite positive matrices, or, equivalently (with an obvious definition of R3):

4. The adaptive control law

Suppose that the control purpose is to track a 'desired' linear dynamical behaviour described by a reference model which relates a reference input r to the desired generalized coordinate qd as follows: qd + Kvqd + Kpqd =

r.

[

0

-Kp

/n ]

-K o "

R=

R1

R2 .

(21b)

(15)

The symmetric matrices K~ and Kp are design parameters at the user's disposal, but chosen such that (15) is exponentially stable and therefore such that the following matrix A 1 is strictly stable: AI =

with

Theoreticalpropertiesfor a rigid-link manipulator The adaptive control law (17)-(20), when it is applied to a rigid-link manipulator, (i.e. ~ = 0) can be shown to be globally stable and to ensure convergence of the tracking errors, i.e.

(16) lim (q - qa) = 0,

t -'-~ OG

lim (q - qd ) = 0.

t - - ' OO

(22)

G. Campion, G. Bastin /Adaptive controllerfor manipulators

254

Furthermore, the same control law remains globally stable in case of external non-measured bounded disturbances. These properties are demonstrated in [2].

the following equivalent closed-loop model is obtained: 2 = A~x + dpp + A3y '

(25a)

i ~ = A2y + btA2~B2ft,

(25b)

tor

(25c)

= -Z+*Rx,

5. R o b u s t n e s s analysis for a f l e x i b l e - l i n k m a n i p u l a or

We now investigate the stability properties of the proposed algorithm, when applied to flexible link manipulators, as modelled in (6)-(7), with /~ > 0. The formulation of our main stability result requires a number of definitions and state transformations which are now presented. Notations

= 0

(see (18))

with ,~ = diag(oi2) and R defined in (21). Definitions (a) Define P as the solution of the following Lyapunov equation: A~P + P A 2 =

X =

[-1

z=

/~O'

- Q3

with

(26)

Q 3 > 0,

and

X2

qd O~1 = ~kmin

Q2

'

0~2 = • rain ( Q 3

)

/~= (QI and Q2 have been defined earlier, see (20)). (b) Define the following definite positive function in R an+N"

PN --PN

State transformation

V ( x , y, fi) ~=1 V l ( x , p) + ½V2(y)

(27a)

With u(q, q, P) given by (17), equations (6) and (7) can be written after some manipulations

with

as

Vl(x , ~) = xTRx + pTs-1/3,

(27b)

V2(y ) =yTpy.

(27c)

2=Alx+~P+A3z+BlU(q,

(t, P),

tz2 =A2z + Bzu(q, q, P),

(23a) (23b)

(c) Define Bp as the following compact set in RN:

where A1 is defined by (16) and

gp & { ~ 1 - (P/max--Pimin)

I ° ° [ ° ] B 1 = _ M - a ( q , p) '

A3 = M - l ( q , p )

A2=

[0

_C~1

1

M - l ( q , P)C1 '

'1

-C21C1 '

~
(28)

Clearly the adaptation law (17)-(18) ensures that

B2=

[°1

Define 0 & maximum value of p T z - l p for p ~ Be.

C2-1 ,

0. +1,

Statement of the main result

with ~ defined in (19). With the state transformation y = z + A~lB2u(q, (1, P)

~i

(24)

Our main robustness result will be given in Theorem 1 where it will be shown that - for a sufficiently smooth desired trajectory qd (see assumption below);

255

G. Campion, G. Bastin / Adaptive controller for manipulators

for any initial position and velocity errors (i.e. for any x(0), y(0)); - for any a priori parameter uncertainty satisfying (13), i.e. for any p(0) ~ Bp; there exists an upperbound for /x, say ~*, such that, for a flexible-links robot characterized by ~t ~<~t*, the closed-loop system (5) is globally stable. This result based on a Lyapunov analysis, is inspired by the singular perturbation method of Kokotovic et al. [8].

Proof. See Appendix A1.

Throughout the paper we make the following assumption: qa, qa, /id and [t'a defined in (15) are uniformly bounded, i.e. there exist four positive constants ~1, X2, 2~3, X4 such that

It must be pointed out that this (2 x 2) matrix is strictly positive definite for all # such that 0
-

Iqdl
IOdI
Iq~l
We introduce a new set of definitions: fl & max(2l R IA3I, 21PIk2, 2 1 P l k 3 } , a

A(#)--a [/~a'B

x(0), y(0), /3(0), Define zx

c = co + ½to,

(30a) (30b)

and the compact set D(c), containing the origin: D ( c ) = ( ( x , y, p ) I V ( x ,

y, / 3 ) < c , / 3 ~ B ~ } . (30c)

(31c)

[1,1

lyl]e(~)lY

g(Ixl, lyl,~)~[Ixl -2/~ka

I e II Y I-

(31d)

Lemma 2. For I~ > 0 and (x, y, /3) ~ D(c), the time derivative of V along the solution of (25) satisfies g [ I x l , lYl, l~].

Proof. See Appendix A2.

co zx ½xT(O)Rx(O) + ½yT(O)Py(O),

].

/-t1•

V < ~ - - ~t

with/3(0) ~ Be.

-#flz 2

(29)

Consider given initial conditions for the closed-loop system (25)

(31a) (31b)

ala2

Iqal
for all t.

[]

[]

Lemma 3. There exists a monotonically increasing function, *if/Q, defined on ]0, /~1[ such that: (a) lim~0~(/~) = 0. (b) I f I x 12 + I Y 12 > ~12(/z) then it follows that g ( I x l , lYl, ~ ) > 0 . Proof. See Appendix A3.

[]

Clearly, (x(O), y(O), /3(0)) ~ D ( c ) .

(300)

We now have the three following preliminary lemmas. I.emma 1. For (x, y, if) ~ D(c): - f, A3, B z are bounded; - there exist 3 positive constants kl, k2, k3, such that

We now define the two following constants 3" and #*: 3 *2 ~

1

#* is the largest value of # such that ~ ( / ~ ) ~< 3 " - e,

#* ~
where Xmax is the largest eigenvalue of

]A21B2ftl <~ka + k21xl +k3 lYl along the solutions of (25).

[C O -- E l i .

~kmax

0

½P

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G. Campion, G. Bastin /Adaptive controllerfor manipulators

and e is arbitrarily small but strictly positive constant such that e < m i n [ c 0, 8 * , / q ] . Our main robustness result can now be stated as follows: Theorem 1. For any initial condition [x(0), y(0), /3(0)] with ~(0) ~ Bp and for t~ ~]0, #*], the solution of [25] belongs to D(c) for all t. Proof. (a) For ~ ~ ]0, ~* ],

(see Lemma 3). Define

Cl(~ ) ~ Xmax,q2(~) + l p _ 81 where q is an arbitrarily small but strictly positive constant, and let Da(/~) be the following compact

set in R4n+N:

Dl(tz) ~=((x, y, ~3)IV(x, y, p)<~cl(IZ) and /3 ~ Bp }. (b) For ~ ~ ]0, /~* ],

CI(~ ) ~ ~kmaxT~2(~,) + lp

=Co-e + ½ p = c - e
v(x, y,/3)

c

it follows that

P~ --

g[lx[, lyl, /~]

(Lemma 2) and that

~krnaxT~2(/~)"~"½p--El-~ ~kmax[IX] 2-~

6. Conclusion (1) We have shown the robustness of the proposed adaptive control algorithm in presence of unmodelled transmission flexibility. More precisely for a given reference trajectory, there exists, for any initial error, an upper bound/~* such that the stability of the error system (23) is ensured for a robot with t~ ~
Appendix l y l z] +½0-

Therefore 712(/,) < Ix [ 2 + [ y [ 2 and, from Lemma 3, since/, < #1, g [ l x l , [Yl, t*] > 0 . Hence there exists a strictly positive constant e 2 such that I)"~ - e 2 < 0 along the solution of (25) f o r / , ~]0, /**] and (x, y, p ) ~ D ( c ) \ D I ( # ) . The solution of (25), with initial condition Ix(0), y(0), /3(0)] on the boundary of D(/*) reaches therefore D1(I~) in a finite time t *, such that

c-cl( ) t*~< ~2

(d) The maximum value of V on D~(~) is reached on the boundary of Dl(/z) and is equal to q(/~). The solution of (25) can only leave D1(/~) at a point (belonging to the boundary) where V(x, y, /5) = c1(~) and where I?~< - e 2 < 0. Since V(x, y, p ) > c1(/~) outside Dl(~t ), the trajectory cannot go outside D~(/~). []

A. 1. Proof of Lemma 1 Consider ( x, y, p) ~ D( c ). (a) We first have that

M(q, ~), Mi(q), f ( q , (t, P), f~(q, (1), OM OMi Of Ofi ( Oq (q' p)' ---~-q (q)' -~q(q' q, P), Oq q , q ) , A 3 and B 1 are bounded. (b) Along the solutions of (25) we have

M(q, /3)2 z = - M(q, ~)[Kpxa + K,,x2] + EPifi(q, Cl) + [ I

C,]y

G. Campion, G. Bastin /Adaptive controllerfor manipulators where

257

lYI'

[-]

X -~" X2

Ymax

qd

-

--7

There exists therefore positive constants c I . . . . . c9 such that

Ix21
Xmax

Vi.

( c ) From the expression of reevaluated as

ti can be

u,

Therefore for (x, y, p ) ~ D(c) according to the definitions (30),

dfo ~ dfi = -~-7-(q, q) + Y'./~J/(q, q) + LP,--~-

1Il ~ - c q B l x l 2 + 2B21xll y I

[ dM0 ~ ^ dMi ] + [ dt + E ~ i M i ( q ) + L P ' - - - ~ - I

+(M o + E~iMi)'(ild--KSc2--Kpx2) dfo dt

0fo.

q)(qd + 22)'

aM o aq (qd + X2)' aM,

at

aq (qd + X2).

Then, using assumption (29), there exist three positive constants kl, k2, k 3 such that

Ia21B=f*l<~kl + k = l x l + k 3 [ Y l

1;'2~ - azfl l Y l 2 + f l l y l 2 + 2kl IPIlYl #

.

[]

A.3. Proof of Lemma 3

0f~ af~ aq (q' q)(qd"}- X2) q- ~'-~(q' (1)(qd'q- X2)'

dMo dt aM,.

,

and the result follows immediately.

aq (q' O)(gld+x2) + - ~ ( q '

dr. dt

and

afo dl + afo .. aq --~q

afo

Ixl

Fig. 1.

[]

A.2. Proof of Lemma 2

For ~<~1,

A(/,) is definite positive and g [ I x l , lY[, / * ] = 0 defines an ellipse in the ( I x 1, l Y I)-plane (we are only interested in the portion satisfying Ix [ > 0, I Y I > 0). g is negative for (Ix I, l Yl) in the interior of the ellipse and positive outside. Define Xm~ and Ym~x as the maximum values of I xl and I Y[ on the ellipse (see Figure 1). It can easily be checked that Xmax and Ymax tend to zero for/~ -o 0. Define n2(

) _

2 -- Xmax + Ymax"

Then lim,_~07/(/,)=0 and for I x l 2 + l y l 2 > ~2(/,). (x. y) is outside the ellipse and g is positive. []

Along the solutions of (25),

[? _xTol

oj

(the strict inequality is valid for/3 on the boundary of Bp; see (18)) and 1

T

References

[1] C.P. Neuman and V.D. Tourassis, Robust nonlinear feedback control for robotic manipulators, 1EEE Proc. on Control Theory and Application, Part D 132 (4) (July 1985). [2] G. Campion and G. Bastin, Lyapunovdesign of an adaptive external linearization feedback control for manipulators, in: A. Bensoussan and J. Lions, Ed., Analysis and Optimization of Systems (Springer-Verlag, Berlin-New York, 1986) 172-187.

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G. Campion, G, Bastin /Adapti~:e controller for manipulators

[3] R. Middleton and G.C. Goodwin. Adaptive Computed torque control for rigid link manipulators, IEEE 25th CDC, Athens (Dec. 1986) 68-73. [4] JJ. Craig, Ping Hsu, S.S. Sastry, Adaptive control of mechanical manipulators, 1EEE Conference in Robotics and Automation, San Francisco, CA (1986) 190-195. [5] J.J. Slotine and W. Li, Adaptive robot control: a new perspective, 26th IEEE CDC. Los Angeles, CA (Dec. 1987) 192-198. [6] P.A. loannou and P.V. Kokotovic, Instability analysis and

improvement of robustness of adaptive control, Automatica 20 (5) (1984) 583-594 [7] C. Canudas, C. de Wit, K.J. Astrom and K. Braun, Adaptive friction compensation in D.C. motor drives, IEEE Conference on Robotics and Automation, San Francisco, CA (1986). [8] P. Kokotovic, H. Khalil and J. O'Reilly, Singular Perturbation Methods on Control: Analysis and Design (Academic Press, London. 1986).