Nonlinear Analysis, Theory. Methods &Applications, Vol. 30, No. 7, pp.

4485-4496, 1997
Proc. 2nd World Congress of Nonlinear Analysts
Pergamon 0 1997 Elsevier Science Ltd
Printed in Great Britain. All rights reserved
0362-546X/97 $17.00 + 0.00
PII: SO362-546X(!V)OO184-3

A STATE OBSERVER FOR NONLINEAR DYNAMICAL SYSTEMS

MANUELA DALLA MORA, ALFRED0 GERMAN1 and COSTANZO MANES
Dipartimento di Ingegneria Elettrica, Universit;l degli studi dell’Aquila
Monteluco di Roio, 67040 L’Aquila, Italy
Key words and phmses: state observers, nonlinear systems, ff, Riccati equation.

1. INTRODUCTION

The problem of state observation for nonlinear systems is of main importance in automatic control.
In recent years many contributions have been presented in literature that solve the design problem
for classes of nonlinear systems. Most works can be roughly classified into two categories. In one the
geometrical properties of vector fields defining the system are exploited, and therefore mathematical
tools of the differential geometry are extensively employed. In the other one the functional properties
of the input-state and state-output maps which define the system are mainly used. While the
geometrical methods often provide a general solution of the problem for systems that belong to
restricted classes, characterized by geometrical properties [1,2,3,4,5,6,7,13,14,17,18,19,], the second
approach gives solutions for classes of nonlinear systems whose geometrical nature is not investigated
and only macroscopic parameters (norms, gains, Lipschitz constants and so on) are involved in the
design procedure [8,9,11,12,21].
One main problem in state observation for nonlinear systems is that the property of drift-
observability does not imply observability for any input function, as shown in [16,19]. Thus it
is interesting to investigate when the drift-observability property allows the solution of the state
observation problem, finding conditions that restricts the class of input functions. Due to the math-
ematical tools employed, this work may be considered as belonging to both the categories above
mentioned.
In this paper nonlinear systems of the form

k(t) = f(49) -I- T7(4t)b($ (1.1)

y(t) = h(4t)), (1.2)
are considered, where z(t) E X c IR”, u(t) E U C R and y(t) E R, g(z) and f(z) are C”(X)
vector fields and h(z) is a C”(X) function.
The paper is organized as follows. Some definitions are given together with introductory material
in section 2, while in section 3 the main results of the paper are presented. Numerical results on a
robotic application are reported in section 4, showing high performance of the proposed observer.
2. PRELIMINARIES

It is useful to recall some concepts and to give some definitions before to state the observer design
problem and to give the relevant theorems.
First of all the concept of Lie derivative of a function X(z) along a vector field f(z) should be
given
L,X(z) f $f(z). (2.1)
Defining as LofX(z) e X(z) the Lie derivative of order 0, repeated Lie derivative of order s 2 1 is
defined as
LjX(z) 4 L,(L;-‘X(z)). (24

4485
4486 Second World Congress of Nonlinear Analysts

The vector function

(2.3)

defines a square mapping, and denoting with Y, the vector of the first n output derivatives (from 0
ton-l)
T
Y, =
[
y i . . . y(+‘)
1 . (2.4)
it is easy to verify that, for u(t) E 0,
Y, = Q(x). (2.5)
Thus, from a theoretical point of view, if the vector Y, where known at a given time t, the invertibility
of the mapping a(.) would allow exact state reconstruction.
This property justifies the following definition.
DEFINITION 2.1. If the map a(z) is a diffeomorfism from an open set 51 C R” in a(n), the system
(1.1) (1.2) is said R-drift-observable. If 51 E R“ than the system (1.1) (1.2) is said globally drift-
observable or simply drift-observable in W.
An important consequence of this definition is that the Jacobian of the map @p(a)

Q(x) f y (2.6)

is nonsingular in R.
In the state observation problem it is important the following concept, that constitutes a weaker
version of the well-known concept of relative degree (see e.g. [15]).

DEFINITION 2.2. The system (1.1) (1.2) is said to have observation relative degree T in a set 0 E IR”
if
vx E R, L,LTh(x) = 0, s = O,l,. . . ,T - 2,
(2.7)
3x E 0, : L,L;-‘h(x) # 0.

REMARK 2.3. Let cr = n - r and let U, be the vector of the first o time-derivatives of input (from 0
to u)
u, 2 [u ic . . . uyT. (2.8)
It can be readily proved that a vector function *(x, Ub) exists such that

yn = @(xl t Q(x, Uo), (2.9)

with the property @(x,0) = 0 Vx E 0. Moreover, when D = 0 (i.e. r = n) then rk I 0. In general
drift-observability of the system (1.1) (1.2) d oes not implies invertibility of (2.9) for z. Invertibility
of (2.9) for z depends on the input, through the vector 17,. However, in the case Q = 0 drift-
observability guarantees state reconstructability for any input.

=[ L.&(x)
The product of the Jacobian Q(z) by the matrix g(z) is

Q(xMx) i 1.
L, L;-‘h(x)
(2.10)
 Second World Congress of Nonlinear Analysts 4487

From the definition of observation relative degree in R, the first r - 1 rows of vector (2.10) are
identically zero in R, so that (2.10) can be rewritten as

Q(ZMz) = F H(z),

L,L;-‘h(z) (2.11)

(Ik is the k x
where

k identity
F fi

matrix).
qr-l)x(wtl)
L-r+1 I '

It is also useful to define the function

H(z) 5

1 1 ;

L,L;-‘h(z)
) (2.12a)

L(2) !! Lyz(z). (2.12b)

Using the previous definitions, direct computations show that an R-observable system (1.1) (1.2)
can be rewritten using the coordinate transformation t = @(a~) as

i = Az + BL(@-l(z)) + FH(@-‘(z))u,
(2.13)
y = c.z,

where A E RnXn, B E LY’ and C E IR” are Brunowsky matrices

(2.14)

The pair A,C is observable, and it is an easy matter to assign eigenvalues to the matrix A - KC,
that has the companion structure

(2.15)

If a n-pla A = (Xl,... ,A,) of eigenvalues has to be assigned, the vector K(X) is the vector that
contains the coefficients of the manic polynomial that has X as roots.
If the assigned eigenvalues are distinct, matrix A - KC can be diagonalized by a Vandermonde
matrix

so that
V(X)(A - K(X)C)V(X)-’ = diag{A} = A. (2.17)

REMARK
computed
2.4. Given a set A of TZ eigenvalues
through the formula

K(X) A?
=-v-‘(A)
i[I,
to be assigned to A - KC, the gain K(X) is readily

(2.18)
x
4488 Second World Congress of Nonlinear Analysts

which is not difficult to check.

It is well known that a Vandermonde matrix V(X) is singular if and only if two (or more) eigen-
values in the set X coincide. It is also well known that the smaller is the minimum difference between
eigenvalues in X, the larger is the norm of V-'(X). F or reasons that will be made clear in the follow-
ing section it is important to choose eigenvalues for matrix A - K(X)C keeping bounded the norm of
the inverse of the Vandermonde matrix V(A). In [I] I‘t is shown that if the n eigenvalues are chosen
as Xi = Xj(w) = -wj, for j = 1,. . . ,n, with w > 0, then

JlmIIVel(X(w))II = 1. (2.19)

REMARK 2.5. In [16,19] it is shown that a system (1.1) (1.2) is observable for any u(t) if and
only if the vector function Q(@-‘(z))g(@-‘(2)) h as a triangular structure in in z-coordinates. This
condition restricts the class of nonlinear systems under investigation. For this reason this property
is not required in the following development, where a suitable condition will exclude pathological
inputs.

3. THE OBSERVER FOR SMOOTH NONLINEAR SYSTEMS

In this section it is proved that a dynamic system of the type

i(t) = f(qt)) + g(i(t)Mt) + 9-‘(w)+(~) - Qw)),

is an exponential observer for the system (1.1) (1.2), p rovided that some conditions are satisfied and
that the constant gain matrix A’ is properly chosen. The required conditions are sufficient to ensure
exponential (or, in some cases, only asymptotic) convergence of observation error to zero.
A first result is given by the following theorem.

THEOREM 3.6. For the system (1.1) (1.2) assume that the following hypotheses hold:
1) The system is drift-observable in RR”, and the map z = (a(z) is uniformly Lipschitz together with
its inverse z = a-*(.2) in E, with constants ye and ye-1 respectively;
2) the functions H(@-l(z)) and L(@-‘(z)), defined in (2.12), are uniformly Lipschitz in R”, with
Lipschitz constants -ye and yz respectively;
3) a constant uM > 0 exists such that Iu(t)l 5 UM Vt > 0;
4) for a given cr > 0 a vector Zi E DX” and a symmetric positive definite matrix P E BnXn exist
that satisfy the following H, Riccati-like inequality

(A- KC)P+P(A- liC)T+BBT+ FFT+2aP+y2P2 5 0, (3.2)

where y* = 7: + &yk.
Then the dynamic system (3.1) is such that

IId4 - i(Qll 5 w-atl140) - 4O>ll (3.3)

PROOF. Consider the coordinate transformation z = a(~), invertible by assumption, both for the
system (1.1) (1.2) and for the observer (3.1)

% = a(z), i = Q(i). (3.4)

 Second World Congress of Nonlinear Analysts 4489

Let e, and e, be the observation errors in z and in t coordinates, i.e. e, k x - 2 and e, k z - i.

From assumption (1) they are such that

II4 5 7oll4~ IMI I 7~-~Ilezll. (3.5)

Recalling (2.13), the system (1.1) (1.2) m

. r-coordinates can be written as

i = AZ t BL(Q-l(z)) t FH(iP-‘(z))u,
y = c.2,

while the observer becomes

i = Ai t BL(ib-‘(2)) t FH(W’(;))u + li’(y - Ci). (3.6)

The dynamics of the observation error in z-coordinates is governed by the equation

1, = (A - KC)e, t B(Z(r) - L(i)) + F(H(z) - H(i))u, (3.7)

where z(.) 6 L(+-‘(.)) and ii(.) 2 H(@-I(.)). Equation (3.7)’ 1s 1’mear with nonlinear perturbations
that can be denoted as
q(z,i) ii L(z) - L(i),
(3.3)
v*(z,i,u) !i (H(z) - H(i))u.
From assumption (2) and (3) the perturbations satisfy the inequalities

Il~lll I YLIML P-9)

Moreover the error dynamics can be written as

6, = (A - KC)e, t Bv1 t Fvz. (3.10)

In order to prove that e,(t) exponentially goes to zero, consider the positive definite function of e,

v(e,) = eTP-‘e,, (3.11)

where the positive definite symmetric matrix P satisfies inequality (3.2). The derivative of v along
the error trajectory is

i, = eT(P-‘(A - KC) t (A - KC)TP-l)e,t

(3.12)
t vyBTP-‘e, t e:P-‘Bvl t vFFTP-‘e, + ezP-‘Fvz.

Since any two vectors a and b satisfy the property

aTb t bTa < aTa t bTb, (3.13)

from (3.12) the following inequality is obtained

ti 5 ez(P-‘(A - ICC) t (A - KC)TP-l)e,t

(3.14)
te~P-‘BBTP-‘e,tv~v~ +e~P-lFFTP-le,tv~vz.
4490 Second World Congress of Nonlinear Analysts

Recalling that -$ = 7; + t&y;, after simple transformations inequality (3.14) can be rewritten as

ti 5 e:P-‘((A - KC)P t P(A - KC)T t EBT t FFT t r2P2)Pm1ez, (3.15)

and from assumption (3.2)

ti 5 -2aeTP-‘e, = -2au. (3.16)
By Gronwall’s inequality
v(t) 5 e-2atv(0), (3.17)
and from this, recalling the definition (3.11) of V, we have

IM~II I AMax(p)
Xmin(P)e-ntllez(0)ll, (3.18)
/----
where XM~~(-) and X,i,(*) denote the maximum and minimum eigenvalue of a matrix (the property
XMax(P-‘)/Xmin(P-‘) = ~Max(p)/~min(p) h as b een used). Given the properties (3.5), inequality
(3.18) becomes
IMt)ll I w-LltlledOIll (3.19)

with p = ~~~~yo-~. This proves the thesis.

COROLLARY 3.7. If all assumptions made in theorem 3.6 hold, with o = 0 in assumption (4) and
inequality (3.2) that holds strictly, then the dynamic system (3.1) is such that

piI Ilz(t) - i(t)11 = 0. (3.20)

PROOF. The proof is conducted with the same passages developed in the proof of 3.6 until inequality
(3.15), from which one has simply
I;(t) < 0. (3.21)
This result, together with the definition (3.11) of v gives

(3.22)

and this, from (3.5), proves the thesis.

The assumption regarding the uniform Lipschitz property in &” is rather strong but essential
to proof the convergence of the observation error to zero. In the sequel we will show that this
assumption can be replaced by assuming a weak stability property for the state evolution of the
system, namely by assuming that given a bound on the input u(t) all the possible state trajectories,
starting from an initial state z(O) in a compact set !? C R”, are restricted in a compact set R C B”,
obviously d C 0. This is often the case in many applications.
Before to give the relevant theorem, let us give some further definition.

DEFINITION 3.8. Let fi be a compact set in IR”. Given a positive 6 we define as 06 the set

St6 = {z E R” 1 115 - ?/II I 6, v y E n} (3.23)

Of course R C fib1 c Rs, for any &J > 6r > 0.

 Second World Congress of Nonlinear Analysts 4491

Let us observe that the smoothness assumption for a drift-observable system (1.1) (1.2) implies
that the functions 1 and H, are locally Lipschitz in R” (recall that a locally Lipschitz function is
uniformly Lipschitz in any compact set).
In the following we will denote as 7k(Q) the Lipschitz constant of a locally Lipschitz function k(z)
in the compact R. Such constant is such that

7k(%,) > 7k(%,) 2 ‘Yk(fl), (3.24)

for 62 2 S1 2 0. The following theorem can be given (the proof is reported in [22]).

THEOREM 3.9. For the system (1.1) (1.2) assume that the following hypotheses hold:
1) for any input u(t) such that j~(t)l 2 UM W 2 0 the state evolution starting from a compact set
fi c R” is restricted to a compact set R c BP.
2) for a given 6 > 0 the system is R&-drift-observable
3) for a given positive a a vector K E B” and a symmetric positive definite matrix P E Rnx” exist
that satisfy the following H, Riccati-like inequality

(A - KC)P + P(A - KC)T + BET + FFT + 2aP + $P2 5 0, (3.25)

where 7: = yi(Q(%)) t &7;(@(%)).

Then the dynamic system (3.1) is such that

112(t) - f?(t)11 5 eeat6 (3.26)

provided that z(O) E fi and the initial observation error is such that

(3.27)

with p = 4=7+(&)7+-I(@(%)).
Looking at the assumptions of theorems 3.6 and 3.9, it can be recognized that a central point
in the construction of an observer of the form (3.1) is the existence of a pair K, P that solves the
inequalities (3.2) and (3.25). The following theorem can be given.

THEOREM 3.10. For any Q: > 0 a 7* > 0 exists such that the H, Riccati-like inequality (3.2) admits
solution (K, P) for any positive 7 5 y*.
PROOF. It is trivial to verify that if inequality (3.2) a d mi t s solution for a given y* then it admits
solution for any positive 7 5 7’. In particular the same pair K’ and P’ that verify (3.2) with 7 = 7’
verifies the inequality for 7 5 7*. Thus, it remains to proof that a 7* exists.
For, consider the following Lyapunov equation, where 1, denotes the n x n identity matrix,

(A - KC + d,)P + P(A - KC + CXI,,)~+ BBT + FFT + p21n = 0, (3.28)

for a given /3 > 0. For any K that assignystable eigenvalues to the matrix (A - KC + al*), a
unique symmetric positive definite solution P exists (K is easily computed through equation (2.18)
by choosing the set X of n eigenvalues of A - K(X)C such that max(x) < -a).
The following inequality

ml P2<I
-n’ (3.29)
4492 Second World Congress of Nonlinear Analysts

easy to check, substituted in (3.28) gives

This proves that for 7* = p/jlpll inequality (3.2) admits solution.

Theorem 3.10 demonstrates that if the constant 7 associated to a given system of the form (1.1)
(1.2), which admits an observability map, is small enough, the nonlinear observer of the form (3.1)
that gives exponential convergence exists.
An interesting point is that the H, Riccati-like inequality admits solution (K, P) for any a > 0
and 7 > 0 if the term FFT is not present in the expression.
LEMMA 3.11. For any cy > 0 and 7 > 0 the H, Riccati-like inequality

(A-KC)P+ P(A-Ir'C)T+ BBT+2aP+y2P2 5 0, (3.31)

admits solution (K, P) with P symmetric positive definite.

PROOF. Choose matrix Ii according to formula (2.18), assigning a set of real eigenvalues X, and
set P = (VTV)-l. Left-multiplying (3.31) by V(X) and right-multiplying it by VT(X) the H,
Riccati-like inequality becomes

2A+VBBTVT+2aIn++~V-'V-T~0. (3.32)

To satisfy the matrix inequality (3.32) it is sufficient to verify the following scalar inequality

2max{X} 2 -llVBBTVTjl -2a--f21jV-'V-TII. (3.33)

The product VB and the norm llVBBTVTjl are easy to compute

VB=[l ... l]T E IR”, * pmTVTIj = n. (3.34)

The choice of eigenvalues used to prove (2.19) can be adopted so to keep the norm of matrix
V-'VTT next to 1 as desired. Assuming w > 1 then max{x} = -w, and inequality (3.33) can be
rewritten as
-w 5 -a - in - ;Y211v-1(w)V(w)-TII, (3.35)

where V(w) has been indicated as function of the scalar parameter w that defines all the eigenvalues.
Thanks to (2.19), inequality (3.35) can be satisfied for w sufficiently large.
This proves the theorem.

For small-input systems the following theorem can be given.

THEOREM 3.12. For the system (1.1) (1.2) assume that the following hypotheses hold:
1) the system is globally drift-observable and the map z = Q(Z) and its inverse z = ‘P-‘(z) are
uniformly Lipschitz in R” with constants 7e and 70-1, respectively;
2) the functions H (a-‘(z)) and L(@-‘(2)) are uniformly Lipschitz in R”, with Lipschitz constants
7~ and 7~ respectively;
Then for any a > 0 there exists a vector It’ E II? such that the dynamic system (3.1), for a suitable
p > 0, gives
b(t) - i( I w-“*b(O) - W)ll (3.36)
 Second World Congress of Nonlinear Analysts 4493

provided that Iu(t)l 5 UM Vt, with UM sufficiently small.

PROOF. Using the coordinate transformation (3.4) consider again the positive definite function v
defined in (3.11) and its time derivative (3.12). Now, consider the following inequality

'T b,
aTb + bTa 5 p2aTa + Fb (3.37)

satisfied by any pair of vectors a and b, and for any real p, as it can be checked expanding the
expression
(pa - ‘b)T(pa - lb) 2 0. (3.38)
P P
Thus, from (3.12) the following inequality can be obtained

ti I eT(P-‘(A - KC) + (A - z~C)TP-‘)e,+

+2 (3.39)
t ezP-‘BBTP-‘e, 2 TP-‘FFTP-le,
+ V;V, + u,e, + 2.
UM

This, from (3.5), after simple transformations can be rewritten as

D 5 e:P-‘((A - KC)P + P(A - ZiC)T + BBT + t&FFT + y2P2)Pd1e,, (3.40)

with y2 = 7: + 7;.
It remains to show that for any positive o a us exists such that the following H, Riccati-like
inequality can be satisfied

(A - KC)P + P(A - ZCC)T + BBT + uZ,FFT + 2cyPt y2P2 < 0, (3.41)

This can be done by considering, for a given p, the inequality

(A - KC)P + P(A - ZiC)T t BBT t 2aP t (r2 t P2)P2 5 0, (3.42)

which admits solution K, P, as proved in lemma 3.11.

Since FFT < Z,, 5 x P2, as it can be easily verified, one has
,h

(3.43)

and thus the solution for (3.41) exists with uM 5 p2&,(P).

This completes the proof.

REMARK 3.13. The sufficient conditions for the existence of an exponential observer given in theorem
3.12 do not include the condition of observability for any input. However, a bound on the input
has to be satisfied. Evidently this smallness condition automatically excludes the presence of inputs
that make indistinguishable some system states.

REMARK 3.14. An automatic choice of K can be adopted by taking K = P2PCT, for a given p.
With this choice inequality (3.2) becomes a true H, Riccati inequality

AP + PAT + BBT + FFT + 2aP + y2P2 - 2P2PCTCP 5 0, (3.44)

4494 Second World Congress of Nonlinear Analysts

in which the matrix P is the only unknown.

4. SIMULATION RESULTS

The nonlinear observer has been successfully applied in several applications. Among them, it has
been used to observe the rotor flux in induction motors, in which only measurements of stator
currents and of rotor speed are available. For this application both numerical [23] and experimental
[24] results have been obtained.
In this section the performance of the observer is shown in a robotic application. Consider a rigid
robot with 2 rotational joints that moves in the vertical plane. As it is well known its dynamics is
described by 4 state variables: 2 joint positions, angles (ql, qz), and 2 joint angular velocities (Qr ,422).
The input consists of the motor torques on the two joints. We assume that only the position q1
of the first joint is measured, so that the observation problem consists in the reconstruction of the
position of the second joint together with both joint velocities. Defining the vectors ~1 = [ql q21T
and xz = [il &IT, and the state vector z = [XT XT]‘, the robot dynamics in the form (1.1) (1.2) is

k=[I6 =[-M-‘;%l)n(x)]
+[M-:(x1)]
i2 + (4.1)
y=[l 0 0 012,

where matrix M(xr) E Rzx2 is the symmetric and positive definite inertia matrix, that depends
only on joint positions, and the vector n(x) E IR2 contains centrifugal, Coriolis and gravitational
interaction forces between joints. The inertia matrix can be written as a function of joint positions
(41,42) as

M(q1,qz) =
[
j1 + 5, + m21:
JZ t m2llL2
+ 2m2ltL2
cosq2
c0sq2 j2 t m2$lc2
J2
c-42
I- (4.2)
The term n(x), written as a function of (ql, qz, il,c&t) is

-m2111,2(q~ t 2qlq2) sin92 t gmlld cosql t gmzLz(cosm t co:(ql t 42))

m2111c2q~ sinq2 t gm21c2 cos(q1 t 42)
(4.3)
[
The robot parameters are reported in table 1.
The system has relative degree T = 2, and is not globally drift-observable. However it is drift-
observable in suitable open sets of R4, and thus, according to the theory, convergence for suitably
small initial errors is guaranteed.
The references for joint angles are such that position q1 = 0 corresponds to first arm in the
horizontal plane, for q1 = 7r/2 the first arm is directed upward and for q1 = -x/2 downward. q2 = 0
corresponds to the arm fully stretched.
The numerical simulation here reported is performed over a time interval of 0.5 s, with an inte-
gration step of 0.5 ms. The true initial state is x(0) = [-4 $ 0 OIT while the observed initial state
is f(0) = [-z 0 0 OIT. No error has been assumed at 1 = 0 on the observation of joint coordinate
91, because it is available through measurement. Moreover zero initial velocity has been assumed to
reproduce the practical situation of the robot and observation processes switching on at the same
time.
The gain K used is such to assign eigenvalues X = -20 . [l p p* p”] with p = 1.2. The input
torques applied are
u](t) = 5 cos(2&), uz(t) = 5sin(27rt). (4.4)
Note that the input is two-dimensional, so that the application does not correspond exactly to the
theory reported in the paper, where single-input systems are explicitly considered. However, the
extension of the theory to multiple input systems is straightforward.
 Second World Congress of Nonlinear Analysts 4495

The plots of true and observed position and velocity of joint 2, and of true and observed velocity
of joint 1 are reported (fig.% l-3). True and observed position of joint 1 are not reported, because
the difference can not be perceived at the scale used.
It can be noticed that the transient lasts about 0.15 s. A thumb-rule computation suggests that
the time constant is about 0.05 s. This is quite in agreement with the slowest eigenvalue assigned
to the linear part of the observer, that is Xr = -20 s-l.

5. CONCLUSIONS

In this work the problem of state observation for nonlinear systems affine in the control is inves-
tigated. An observer that gives exponential convergence of the observed state to the real one is
proposed. The class of systems under analysis is not restricted to systems that are observable for
any input. The drift-observability property (i.e. observability for zero input) together with the sat-
isfaction of an H, Riccati-like inequality are proved to be sufficient conditions for the existence of
an exponential observer. The second condition automatically excludes the presence of inputs that
make indistinguishable some system states. The observer feedback gain is chosen solving a simple
eigenvalues placement problem. Both global and local convergence results are presented. Simulation
results show a good performance of the observer.

0.00 0,lO 0,20 0,30 0,40 lsl 0,50 0,oo 0,lO 0,20 0,30 0,40 b10,50
Figure 1. Observed and real position ofjoint 2. Figure 3. Observed and real velocity of joint 1.

ml=5 Kg m2=5 Kg link weights

11=0.5 m l,=O.S m link lengthe
1,,=0.25 m 1,2=0.25m mass center positions
Jl=O.5 Kg m2 J2=0.5 Kg m2 moments of inertia
g=9.8 ms-1 gravity acceleration

O,M 0,lO 0,20 0,30 0,40 lslo,50

Figure 2. Observed and real velocity of joint 2. Table 1, Robot parameters
4496 Second World Congress of Nonlinear Analysts

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