of Barbalat's lemma [4] shows that, in fact, 'go --+ 0. Since all the signals in the observer have been demonstrated to be bounded,. _o and can be analyzed as.
CDC01-REGI379 A Coupled
Nonlinear
Spacecraft
Attitude
Controller/Observer
Constant
Gyro
J. Deutschmann
Abstract-A with
nonlinear
a nonlinear
gyro
system
is proven
principle
for
excitation,
control bias
to be globally
the
given
resulting
stable,
with
The
in exponential
Sanner
for attitude
for the case
system.
control
of a spacecraft
of constant
zero tracking
nonlinear
gyro
error,
observer
convergence
an Unknown
Bias
and R.M.
scheme
observer
With
bias.
is combined
The
closed
thus proving
loop
a separation
incorporates
persistency
of
of the gyro bias error.
I. Introduction Combined
observer-controller
designs
for
nonlinear
systems
research [1,2,3]. Successful design of such architectures there is, in general, no separation principle for nonlinear systems,
"certainty
converging necessarily In this
equivalence"
observer guarantee
paper
we consider
forcing
the attitude
attitude
using
we propose
utilizing
version
vehicle
from
with
of this
velocity
persistent observer
the nonlinear control law proposed system results in stable closed-loop
a
subject
states
from
problem, track
even
active
an exponentially law does [2,4,5].
in particular
the
a (time-varying)
nonzero from
of
is complicated by the fact that systems. In contrast to linear
state feedback control for the coupled systems
to asymptotically
sensors
an angular
of the
stabilizing operation
a restricted
of a rigid
feedback
fashion with the resulting
substitution
into a nominally stable closed-loop
are
bias
[2,3]
not
task
of
reference
errors.
Specifically,
in a certainty
equivalence
in [6], and demonstrate in this case that operation, with asymptotically perfect
tracking. The
proof
proceeds
the
case
of
demonstrate convergent a similar
in two steps.
constant
gyro
First,
bias,
that the bias estimates to the true bias values. result,
but
under
the
we extend
and
use
a
provided The proof
assumption
the analysis persistency
of the observer of
excitation
in [2,3] to
argument
to
by this observer are in fact exponentially in [2] uses a Lyapunov argument to obtain
that
the biases
themselves
are
exponentially
decaying; this restriction is removed here. Second, we consider the certainty use of these observer estimates in the nonlinear feedback control algorithm
equivalence proposed in
[6]
closed-loop
and
dynamics observer
show
that
the
perturbation
can be represented transients.
introduced
as a bounded
Quantifying
the
impact
function these
by
this
strategy
of the vehicle perturbations
into
the
states have
multiplying on
the
the
Lyapunov
Julie Deutschmann _s with the Flight Dynamics Analysis Branch at the NASA Goddard Space Flight Center, Greenbelt, MD 20771 Robert M. Sanner is with the Department or" Aerospace Engineering at the University of Maryland, College Park, Mac,'land 20742
analysis
given
controller
in [6], we demonstrate
are, in fact,
o f the proposed The paper controller
where
of
a
represents coordinate
the
rotation
system.
Definitions
be
represented
=
COS_
e is the Euler
respectively. from
product
target
attitude
error
in the
controller
actual
used body
of the
our analysis
frame
an
inertial
matrix
for the constant
used gyro
matrix
formed
is represented is defined
_=
that
in the bias is
four
followed
component
quatemion,
r 1 are the
vector
][q[= 1 by definition.
The
system
to
by
axis
the
and
quatemion
spacecraft
from the quatemion
scalar
body
as [6]
+ 2e e r _ 2qS(e)
from
-
the vector
_.
E:y l 0
13z _
quaternion,
as a rotation
according
Similarly, in the observer, the attitude error body frame to the actual body frame as
a
results,
e and
coordinate
by the
=q®qa'=
simulation
as the Euler
can be computed
= (q" - ere)I
and is computed
_=
of the terms
known
axis,
Note
0 - gy
A desired
observer
by
q =
A rotation
S(e) is a cross
II.
vector,
R(q)
where
completing
definitions
V presents
and unit rotation
angle,
of the quaternion,
Section
can
angle
rotation
[I contains
lII the nonlinear
errors.
spacecraft
of a rotation
dp is the
portions
properties
convergence is proven. Section IV presents the nonlinear proof of stability of the closed loop system and the
of the tracking in Section VI,
attitude
Section
In Section
and the exponential design and the
consisting
in the face of the perturbations,
as follows.
and observer.
convergence conclusions
and convergence
methodology.
is organized
developed controller
The
maintained
that the stability
from
q_T = [ ET.,,qd]. the desired
body
The frame
attitude to the
to [7]
_
ta T
is defined
n. JLnJ as the rotation
from
the estimated
E] 7o
q°=
where that
_! represents the
indicates The
spacecraft
,i-'
=q®
[
fiz-s( )
--
_:_
the attitude
state
of the observer.
is aligned
with
the
that the attitude
kinematics
qo
equation
estimate
desired
is aligned
for the quaternion
=
fl
Note
attitude
that and
with the actual is given
=
rl
_'¢ = 0, _¢ = +Z indicates similarly,
qo--+I
attitude.
as
,o='Q,,,
where co is the spacecraft angular velocity. The angular velocity a gyro, which can be corrupted with both systematic and random consider velocity
_'o--0,
only the case of systematic can be written as
errors.
In the
case
is typically measured errors. In this work
of a gyro
bias,
the
by we
angular
co = c% +b
where
cog is the angular
velocity
angular
velocity
is given
as
between
the true and estimated
from
the gyro
_ = cog + I).
and b is the gyro bias.
The
bias
error
is defined
An estimate as the
of the
difference
bias
G=b-E, Finally, a measure of the discrepancy the controller is computed as [8]
between
_¢
which
is defined
such that
III. Following
the actual
= co - R(_¢)e)
and
desired
angular
velocity
d
in
(3)
qc = ½Q(q_)o_.
Nonlinear
the development
(2)
Observer
for Constant
of [2] a state observer
Gyro
for the bias can be defined
= I(_RT(_o)(O.),+ 6 + k_'osgn(_o)) =1
Bias
7 :o sgn( o)
as
(4)
(5)
1,
The
gain,
terms
k, is chosen
in the observer
Computing
as a positive
constant.
The
R r (qo)resolves
the angular
velocity
frame.
the derivatives
of _o in (1) and
b in (2) results
in the following
differential
error equations
qo
=
_o
L -
"='7
I
k_
O
_v
b = -7'-eo s_(_o)
The equilibrium states are (0,0,0, Lyapunov function is chosen as
Vo=1-r7b
Taking
the derivative
+ 1,0,0,0).
, b+ 7
of Vo, and noting
Again,
following
(qo-1)'+eo L (_o+l)2+eo
that
_T_
=
of [2], a
q_>O qo_k,I
tbr (7) if Ql(t) that,
and
for all t >_t,_
Q_(t)are
bounded,
cL,[ > _,,f,.r: Q,(.c)Qzr(_:)dz>_c_x,[ The
upper
satisfied,
bound
is satisfied
rewrite
the matrix
Q,(t)Q_r(t)
Since
since
To determine
is bounded.
if the lower
= (_o(t)I
+ S('go(t))(_o(t)I-
for any 8>0, there
any 8TI,
zr [f$:ra Q, (T)Q
S('go(t))
exists
(9) implies
r (z)dz]z
= I-
a Tl(8)>0
Co(t)_:o(t)
such
that
= (1 - 8)(T= - T,)II-II
the required
the observer,
"go and
zero exponentially
establishes
that
IV. attitude
dynamics
Nonlinear
b approach
Controller
for a rigid spacecraft Ho
r
(9)
]'go < 8 for all
=
This demonstrates
The complete
is
that
for any z in R 6 and thus (8) is satisfied. and hence
bound
Q, (t)Q_r(t)as
.go --+ 0 asymptotically,
t > t o + T_. Taking
Ql(t)
(s)
property
for
fast.
Design
are given
- S(Hco)o
UCO
as
= u
= _-Q(q)o
where
matrix
and u is the
applied
example, from attached rocket thrusters. The attitude q(t) to asymptotically track a (generally)
H is a constant,
goal of the time-varying
controller desired
angular
by
velocity
wd(t) is bounded
The passivity
wd(t),
symmetric
related
for consistency
and differentiable
based
controller
with
of
S
Where H results
from
(3),
inertia
co = R(_)co,,
_c
-b _.ge
the composite
----- O'1 --
Taking
-kgc.
is for attitude
torque,
for
the actual qd(t) and
It is assumed
that
dh (t) also bounded.
[6] utilizes
=
dla = _Q(q_)cod.
external
error metric (10)
(l')r
the derivative
of (10)
and
multiplying
by
in Hg = Hd_ - Hor
= u + S(Hco)co
- Hat
where a_ = o,
= R(_,)o
a -S(_)R(_,)coa
- ;,.QI(_)G_
(11)
and
Q, (_)
= _I
+ S(_)
as defined
above.
u = -KDs for any stability In the
+ H%
application,
the
measurements of the angular approach is employed using
control
O_r,
= _-
_c
the control
law
law
(12)
KD as shown
(12)
cannot
velocity co are not available. the estimates _ from above, u = -KD]
_ = 6-
these definitions,
- S(Hc0)o_
symmetric, positive definite matrix and asymptotic tracking properties. current
where
With
in [6] produces
the
be implemented Instead resulting
desired
because
a certainty in
exact
equivalence
+ Hfi r - S(H_)t%
R(qc)¢Od,
(13)
and
a, = R(_o)_ + S(R(_0),-%),_c- zq,@)(oc. Substituting
(13)
g, = [S(R(_c)o_
into
(11),
along
a) -ZQ_(_¢)]b,
with
(10),
and
noting
and _c - _oc = co - ¢.b =
that
produces
"g=s-g=b, the
closed-loop
dynamics Hs - S(Hco)s
The
terms
by definition
on the right and
+ KDs = [-S(m,)H
hand
side of (14)
I1'%11
