Asymptotic observer design for constrained robot systems

Asymptotic observer design for constrained robot

systems

H.-P. Huang W.-L. Tseng

Indexing term: Robotics

Abstract: Two types of asymptotic observer are constructed for constrained robot systems in this paper. Since a constrained robot, in general, involves a set of differential equations and a set of algebraic equations, both differential and algebraic variables should be estimated. This gives arise to difficulty in estimating the algebraic variables which are the contact forces. The difficulty is eased by introducing a nonlinear transformation. Although the transformation causes nonlinear coupling on the input, asymptotic observers can be constructed in terms of the transformed system by a special treatment. Since both the contact force and the motion of the robot can be directly estimated, the observers may be useful for the con- troller design of the constrained robot system.

1 Introduction

For many operations of the robot, the robot end effector is constrained by its environment. In that case, the direct control of the contact force between the robot end effec- tor and the constraint surface can greatly expand the task capacity. The mathematical model for the constrained robot, explicitly taking into account the contact force, has been given in References 3, 6 and 8. Several control schemes have also been developed to control directly the contact force and the robot motion based on this model [2, 7, lo]. However, all these control schemes implicitly assume that all state and algebraic variables are avail- able. Unfortunately, this is not always true. Usually, some states are very difficult to measure and some are too expensive to be assumed. Particularly, the contact force variables may be very expensive and inadequate to be measured. Thus, it is required that we design an obser- ver to estimate the contact force and the state variables for the constrained robot systems.

Since a constrained robot system consists of differen- tial equations and a set of algebraic constraint equations, the contact force variables, which are the algebraic vari- ables, may be regarded as state variables without govern- ing differential equations. The overall system is referred to as a nonlinear singular system [6, 81. Hence, tradi- tional design procedures for nonlinear observers, such as the Lie-algebra observer [ 13, the extended linearisation observer [SI, Thau's observer [4] and the variable struc- ture system (VSS) observer [9] cannot be directly applied.

Paper 7732D(C13), received 20th March 1990

The authors are with the Robotics laboratory, Department of Mechani- cal Engineering, National Taiwan University, Taipei, Taiwan 10764, Republic of China

To overcome the above difficulty, the McClamroch and Wang method [7] is used to transform the con- strained system into reduced unconstrained subsystems. Then, the observer is designed in terms of the reduced subsystems. The selection of our observer structure is a combination of the Kuo observer [4] and the VSS obser- ver. We use the concept of the Kuo observer to determine the convergent property of the observer and use the idea of the VSS observer to cancel the effect caused by the nonlinear coupling in the control input. Since a linear output is desired for applying the VSS observer tech- nique, a linear output based on a transformed system will be constructed.

In this paper, we present two types of asymptotic observer design for constrained robot systems. The con- strained system and its reduced form are discussed first. Then a linear output generator is constructed. Finally, the design procedures of the observers are presented and examples are discussed.

2

For a constrained robot, the motion of the robot end effector is constrained by its environment. The Lagrang- ian dynamics of the constrained robot systems, explicitly incorporating the effects of contact forces, can be model- led as [ 6 ]

Constrained robots and problem formulation

M(q)ii

+

F(q,

4)

= U

+

J T ( q ) l (1)

where q E R" is the generalised displacement; M ( q ) is an n x n inertial matrix function; F(q, q ) is an n-dimensional vector function, containing the Coriolis, the centrifugal and the gravitational terms; U E R" is the generalised

control input;

4(q)

is the m-dimensional constraint vector function; J ( q ) = [&$(q)/dq] is an m x n Jacobian matrix; and 1 E R" is the generalised contact force vector associ-

ated with the constraints.

The constraints, given in eqn. 2, are assumed to be holonomic and frictionless. Note that, if

4(q)

is identically satisfied, then also J(q)q = 0. Hence, the motion of the robot end effector is constrained in the constraint mani- fold S c R2", defined by S = ((4,

4):

$(q) = 0, J(q)q = O}.

Suppose the output of the constrained robot system is given by

(3) where y E RP is the output vector and C is a p x 2n con- stant matrix. Our objective is to construct an observer such that the displacement q and velocity q of the robot

and the contact force can be estimated. These estimated values can be used for controller design.

Since the constrained system, given in eqns. 1 and 2, contains a set of algebraic equations, it is not suitable for observer design. McClamroch and Wang [ 7 ] use a non- linear transformation to convert the constrained system into two reduced, unconstrained subsystems in which the constraints are satisfied automatically. Our observer design will be based on these reduced subsystems. The transformation method is briefly summarised in the fol- lowing:

Suppose that there exists an open set V c R"-" and a function

R

: V .+ R", such that

4 ( w ? 2 ) , 42) = 0 for all q 2 E

v

(4) If rank J ( q ) = m, then according to the implicit function theorem, eqn. 4 holds for some V = R"-". Consider the nonlinear transformation

which is differentiable and has a differentiable inverse transformation Q : R" + R", such that

Let the nonsingular Jacobian matrix of the inverse trans- formation be

( 7 ) Then the constrained system, given in eqns. 1 and 2, can be transformed to reduced subsystems

E 1 M ( x 2 ) E ; x 2

+

E 1 F ( x 2 , i , ) = E,TT(X,)U E , M(x2)E;X2

+

E , F ( x , , i,) = E , T T ( x 2 ) u

+

El T T ( X 2 ) J T ( X 2 ) 1 (8) ( 9 ) x1

= o

(10) where M ( x 2 ) = TT(x2)MCQ(x2)1 77x2) F ( x 2 , i 2 ) = T T ( X , ) (1 1) x {FCQ(x2), 7Tx2P21

+

MCQ(x2)l T ( x 2 P 2 ) (12)

Note that the partition of the identity matrix I,, = [ET, E l ] , where E , is an m x n matrix and E , is an (n - m) x n matrix, is used to partition x as x T = [x:, x l ] = [ ( E , x ) ~ , ( E , x ) ~ ] . The relation E , T T ( x 2 ) J T ( x 2 ) = 0 is used in deriving eqn. 9. Furthermore, the constraint eqn. 2 is transformed to eqn. 10. Under this transform- ation, the output y becomes

which is a nonlinear relation. For simplicity, the above equation is rewritten as

Our problem turns out to design observers for the trans- formed system, eqns. 8-10 and 14. Since the differential eqn. 9 completely governs the reduced state vector x z , and the output eqn. 14 is only in terms of x z , the sub- system, eqns. 9 and 14, can be treated as an ordinary unconstrained nonlinear system. The contact force 1 can be determined from eqn. 8. The observability of the con- strained system depends on the observability of the subsystem, eqns. 9 and 14. It can be easily verified that if the n x m matrix E l T T ( x 2 ) J T ( x , ) is nonsingular for all x 2 E R " - " , then the overall system in eqns. 8-10, and 14 is observable if the subsystem of eqns. 9 and 14 is obser- vable. Throughout this paper, we assume that the system is always observable.

As shown in eqn. 14, the output y is now nonlinear vector function of x z

.

To use the VSS observer technique, a linear output is required. The generation of the linear output will be discussed in the next section.

3 Linear output generator

Consider the subsystem in eqns. 9 and 14. Since there is nonlinear coupling in control, the typical nonlinear observer design, such as the Lie-algebra observer [l] and the Kou observer

[4],

cannot be applied. The VSS obser- ver [ 9 ] can handle the nonlinear coupling in control for the system with linear output. Thus, to use the idea of the VSS observer a linear output will be constructed from eqn. 14.

Since C is a p x 2n matrix, if the singular value decomposition is applied, there exist two unitary matrices U and V such that

(15) where

C

= [ D IO] is a p x 2n matrix and D is a p x p diagonal matrix. Let 2n

>

p

>

in. We further partition matrices C and V into

c

= UTCV

where D , , ER""", D , , E R ( p - m ) x ( p - - m ) , V,, E R""", v12 E 7 v31 E

R m x 2 ( n - m )

v

E R ( P - m ) x m ,

v2,

E ~ ( p - m ) x W - m ) R ( Z f l - P ) " m 9 ' I / 3 2

':

R ( 2 n - p ) x ~ ( n - m )

.

Premultiplying both sides

of eqn. 14 by U gives Dl 1

=I

D22

+

+

' D22

v2,[

3

212 L 2

Note that if rank (C) > m, then D,,Vl, is nonsingular. It is convenient to use the partition of the identity matrix I , = [ E T , E:], where E , is an m x p matrix and E , is a (p - m) x p matrix, to simplify eqn. 16 as

Since

D,,

V,, is nonsingular, {iT[aQ(x,)/ax,]'

i

QT(xZ)}' can be solved from eqn. 18 as

Substituting eqn. 20 into eqn. 19 gives

y =

c

[3

where

Y

=

CE4

-

D,,

V,lV,-,'D;,'E,lUY = CD,, V,,

- D,,

~ 2 1 V ~ , ' V 1 2 1 (22) (23) The new output j is linear in state variables i,and x 2

.

It will be used in our observer design. However, the obser- vability for the subsystem eqns. 9 and 21 must be rechecked.

4 Observer design

In this section, two types of asymptotic observers will be designed for the dynamic system eqns. 8, 9 and 21. The design concept of the first observer resembles the VSS observer, and the second one is a modification of the first one.

To carry out the subsequent developments of the first observer, we make the following assumption:

Assumption 1 :

P-'

=[;;I

p , E ~ ( n - m ) x 2 ( n - m ) , P , E R ( " - m ) x 2 ( n - m ) , P is a 2(n

-

m) x 2(n - m) positive matrix, H is a positive number, and

P ,

CTC

= 0.

Define

x ,

=

[

i 2 ] ,

8 ,

=

[

and e = X , -

8 ,

x2 X 2

Then we select the observer structure as follows

iz

=

i 2

+

G : C ( X , -

8,)

+

P , R ( X , ,

8,,

U ) (24) E 1 M ( 2 , ) E T i , = -E1F(2,,

i,)

+

E,TT(R,)u

+

E , T'(2,)JT(x,)2

+

ElM(2,)ETG2 C ( X 2 -

8,)

I E E P R O C E E D I N G S - D , Vol. 138, N O . 3, M A Y 1991 - E l M ( k 2 ) E ; ( E 2 M ( 2 , ) E ; ) - , E , T'(2,)U

+

E , M ( 2 , ) E ~ P 1 R ( X , ,

8 , ,

U) (25) E , M ( 2 , ) E ; i 2 = - E ,

F(%,,

i2)

+

E , A?(?,)ETGq c ( X , -

8,)

+

E , M ( 2 , ) E T P , R ( X 2 ,

8 , ,

U) (26) where

[

0 , otherwise

and CY, G: and G: are constant matrices. The structure of the observer is shown in Fig. 1.

U

We have the following results

Theorem I: Consider the system in eqns 8, 9 and 21. If the following conditions are satisfied

(a) \ , Assumption 1 holds

(b) P V F ( x , , i,) - P

is uniformly negative definite for all x , , i,E R " - " and some E

>

0, where

1

- ( E , M ( x , ) E 3 - , E , F ( x , , i,)

[

i 2 F ( x , , 2 2 ) = and

( c ) E , T'(x2)JT(x,) and E , TT(2,)J'(2,) are nonsingular for all x 2 , 2, E Rn-", then the observer defined by eqns. 24-27 is an asymptotic observer in the sense that

Ilx(t) - 2(t)ll Q Ke-''('-'O) Vt 3 to (29) where r] = (2eeTe/eTPe) and K depends on x ( t o ) and ?(to). In addition, the estimated contact force I converges to

A

at the same rate as %it) to x(t).

Proof: eqns. 9 and 26 can be conveniently expressed in matrix form as

and

Choose a Lyapunov function V as 1

V(e) = - eTPe 2 Then

x U - e T P [ i:]R(X2,

2,

, U) (33) When IICell # 0, the last two terms in the right hand side can be further simplified as follows

e T P P

CT

h(x,)u

-

eTR(X,,

8,,

U)

Thus, eqn. 33 becomes

V(e) d eT [ P b F ( x 2 , i,) -

[

; E ] V [ ::])(ws) ds e where w, = s["]

+

(1 - s)[f2] and 0

<

s

< 1

x2 X 2 Since V F ( x , , i , ) -

[

;E]v[

;;]

is uniformly negative definite for some E > 0, we have

P(e)

<

-EIIeI12 (34)

Then,

V ( e )

<

V [ e ( ~ , , ) ] e - " ( ~ - ' o ) And the result is followed.

Next, we consider the estimated contact force vector

A.

If E , T T ( x 2 ) J T ( x 2 ) and E , TT(2&JT(2,) are nonsingular for all x 2 , 2, E Rn-", then

I

and

A

can be solved from eqns. 8

and 25. From eqn. 8,1 is determined as Vt > to

A

= S l ( X 2 , i 2 )

+

S A X , , &)U S l b , , i 2 ) = C E l T T ( X 2 ) J T ( X 2 ) l - 1 ( - E l ~ ( x , , ~ T (35) where x [ E , A?(x,)E;]-'E,

+

E , } F ( x , , i , ) S 2 ( ~ 2 , i 2 ) = [ E i T T ( ~ , ) J T ( ~ 2 ) l - 1 ( E 1 M ( ~ 2 ) E T x [ E , M(x,)ET] ' E , - E,} T T ( x 2 ) (36) Similarly, we have

X

= S,(R,, jZ,)

+

S 2 ( 1 , ,

2 , ) ~

+

[ E , TT(?,)JT(?,)] - x [E,A?(%,)ET][G; - G : ] C ( X ,

-

8,)

(37) Since 1, converges to x 2 , and

i,

converges to i, , as time approaches to infinity, functions S , ( 1 , , i,) and S,(?, , 2,) 214

will converge to S , ( x 2 , i,) and S , ( x , , i,), respectively; moreover, C ( X , - X,) conv_erges to zero. Thus, the esti- mated contact force vector I converges to I at the same rate as 1, to x 2

.

Note that the G: term is used to acceler- ate the convergence rate.

The above observer design, the term R(X,,

%,,

U), may be discontinuous as Ce = 0. Hence, there may exist chattering phenomena in the steady state. To improve the observer performance, the observer is slightly modi- fied as follows.

Take the following assumption: Assymption 2

(ii) V R , ( x , ) = Li(x,)C, IILix2)II

<

Hi, i = 1,

...,

n where

P - ' =

[;:I

p , E ~ ( n - m ) x 2 ( n - m ) , p , E ~ 0 - m ) x W - m ) , P is a 2(n - m) x 2(n - m) positive matrix, Hi is a positive number, and P ,

CTC

= 0.

Then, the observer structure is chosen as follows:

[k2

x2 - -

',I

x2 .~ 2 , = i ,

+

GYC E , M(?,)ET jz, = - E ,

F(?,,

j z , )

+

E , M(2,)ETGX

c[

j C 2 - x2 - x2 - x2 n x HiIIuiII

+

E 2 T T ( 2 2 ) ~ (40) i = 1

where CY, G:, and Gg are constant matrices. The struc- ture of this improved observer is similar to Fig. 1.

We have the following results.

Theorem 2: Consider the system in eqns. 8, 9 and 21. If the following conditions are satisfied

(i) Assumption 2 holds

(ii) P V Q X , , i,) - P

[

3[

;;I

is uniformly negative definite for all x , , i , E Rn-" and some E > 0, where

1

- ( E , A?(x,)ET)- ' E ,

F(x,,

i 2 ) I 2 F ( x 2 , i,) = and

(iii) E , T T ( x 2 ) J T ( x 2 ) and ElTT(?,)JT(jZ2) are non- singular for all x , , ?, ER"-", then the observer defined by eqns. 3 8 4 0 is an asymptotic observer in the sense IEE PROCEEDINGS-D, Vol. 138, N o . 3, M A Y 1991

In addition, the estimated contact force

2

converges to

1.

Proof: The proof is similar to Theorem 1. Hence, it is omitted here.

Remark: In the construction of the observer, the states of the transformed system rather than the states of the original system are estimated. This indicates that it is not necessary to estimate all states for the constrained system. The complete information of the original states can be obtained by the following transformation

(43) and

Another remarkable feature of the observer is that the contact force 1 can be directly estimated rather than obtained by expensive force sensors.

5 Example

In this section, an example is given to illustrate the above developments. Both asymptotic observers are constructed in terms of the same example. Their simulation results are also given for the purpose of justification. Suppose a constrained dynamic system is given by

q 1 = U 1 (45)

(100 - 4q:)42

+

4:

+

1004, - 44,

4:

= U, (46)

4 4 2 1 , 4 2 ) = 4: - q 2 (47)

The constraint and output equations are

where q l , q , , U, u2 E R and y E R 2

'.

By introducing the Lagrange multiplier and applying the McClamroch and Wang nonlinear transformation, eqns. 4 5 4 8 can be transformed into

2i:

+

2x,x, = U 1

+

1

₍₄₉₎ x, = -0.01i; - x,

+

0.02x2 U 1

+

O.OlU, (50) x1

= o

(51)

['

01[:'1 i 2 y = 0 1 1

By singular value decomposition, a linear output can be constructed from eqn. 52 as

j = [O 2]y = 2x,

+

21, (53)

I E E PROCEEDINGS-D, Vol. 138, No. 3, M A Y 1991

From eqns. 24-27, the first observer structure is selected as follows: 2 i 2

+

22,

jc,

= U 1

+

x

+

2R, G!(x,

+

i , - R, -

k,)

- 2i,(0.02f,u1

+

O.OlU,)

+

C L

OIRb, 9 i , 2 f, 2 E 2 9 4(2R,) (54) (55) (56)

K,

= -0.01i; - f,

+

2G9(x2

+

I,

- 2, -

i,)

i,

=

k,

+

2G7(x2

+

i , - 2, -

i,)

+ C L

OIRb, 9

I,

9 2 2 3

K,

2 U)

+

c

- 1, lIR(x2, i , , 2 2 ,

k,,

U) where 0 otherwise

Select H = 1 and the observer gains as = - 1.5, G; = 5.5, G! = 0.4, then according to Theorem 1, the observer defined by eqns. 54-57 is an asymptotic observer for the system defined by eqns. 49, 50 and 53. The simulation results are shown in Figs. 2 4 . It is obvious that there are

0 . 5 1

0 0.67 1.33 2.00 2.67 3.33 4.00 time, s

Fig. 2

x 2 (to) = 1.0

Displacement error (x, - a,) ofobserver I

i2 (to) = 0.5 -0.1 2 -0.50 0 0.67 1.33 2.00 2.67 3.33 4.00 time, s Fig. 3 k2 (to) = 1.0

some chattering phenomena as the estimated variables approach the true variables.

From eqns. 3 8 4 0 , the second observer can be selected as follows :

Velocity error ( i , -

k,)

ofobserver I i 2 (to) = 0.5

232,

+

2?,32, = U 1

+

x

+

2R2 G!(x,

+

i, - f 2 -

E,)

₍₅₈₎ 21s

0 0.67 1.33 2.00 2.67 3.33 L.00 time, s

Fig. 4 1 (to) = 2.00

i ( t o ) = 4.94

Contact force error (A - 2) of observer I

-1.00 0 0.5 1.0 1.5 2.0 2.5 3.0 time, s Fig. 5 P, ( t o ) = 0.5 x 2 ( t o ) = 1.0

Displacement error (x, - ?,) ofobserver 2

_J

0 0.5 1.0 1.5 2.0 2.5 3.0

time, s Fig. 6

i 2 ( t o ) = 1.0

Velocity error (k, - i,) ofobserver 2 iZ ( t o ) = 0.5

. 1

A,

= i z

+

2G3x2

+

1, -

22

-

H,)

Select H = 0.15 and the observer gains as Gy = -7, G: = 5.5 and GZ = 0.05, then according to Theorem 2, the observer defined by eqns. 54-56 is an asymptotic observer for the system defined by eqns. 49,50 and 53.

The simulation results are shown in Figs. 5-7. From those* figures, we know that the estimated states i2, j c ,

and

A

will converge to the system states x,, j c , and

A,

respectively. The result is better than the first observer’s.

c

-15.01 I I I I I I I ‘ I I

0 0.67 1.33 2.00 2.67 3.33 L.00

time, s Fig. 7 Contact force error (i - 1) of observer 2 1 ( t o ) = 2.00

6 Conclusion

Two asymptotic observers are constructed for the con- strained robot system. It has been shown that the con- verging properties can be determined by the selection of the observer gain matrices GY, G: an G : . The dificillty caused by the nonlinear coupling in the control has been overcome by introducing the VSS observer idea at the expense of the requirement of linear output. Although the estimates of the states are based on the transformed reduced subsystems, the estimates of the original states can be recovered by applying the inverse transformation. Since the contact force, which is usually not directly available in the constrained robot system, can be esti- mated directly, the observer may be very useful for the controller design of a constrained robot system. The con- troller may design on the basis of i,, i, or ij,

4 ;

however, the stabilisation problem of the overall system should be carefully investigated. This result will be reported in a forthcoming paper.

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