This paper considers the problem of rigid spacecraft. A nonlinear control law which uses the feedback of the unit quaternion and the measured angular velocities is proposed and is shown to provide global asymptotic stability. The control law does not require the knowledge of the system parameters, and is therefore robust to modeling errors. The significance of the control law is that it can be used for large-angle maneuvers with guaranteed stability

Joshi, Suresh M.

Kelkar, Atul G.

English

NASA Technical Reports Server

NASA Technical Memorandum 109150
//v -/8
fL/_;
Three-Axis Stabilization of Spacecraft
Using Parameter-Independent Nonlinear
Quaternion Feedback
Suresh M. Joshi
Langley Research Center, Hampton, Virginia
Atul G. Kelkar
NRC Fellow, Langley Research Center, Hampton, Virginia
September 1994
(NASA-TM-I09150) THREE-AXIS
STABILIZATION OF SPACECRAFT USING
PARAMETeR-INDEPENDENT NONLINEAR
QUATERNION FEEDBACK (NASA. Langley
Research Center) I0 p
G3/18
N95-13729
Unclas
0028222
National Aeronautics and
Space Administration
Langley Research Center
Hampton, Virginia 23681-0001
https://ntrs.nasa.gov/search.jsp?R=19950007316 2020-06-16T10:28:28+00:00Z
I' Ii_w,r
Summary
This paper considers the problem of three-axis attitude stabilization of rigid spacecraft.
A nonlinear control law which uses the feedback of the unit quaternion and the measured
angular velocities is proposed and is shown to provide global asymptotic stability. The
control law does not require the knowledge of the system parameters, and is therefore robust
to modeling errors. The significance of the control law is that it can be used for large-angle
maneuvers with guaranteed stability.
Introduction
Attitude control of a free-flying spacecraft has long been known as an important
problem, and has been the subject of many papers since the fifties and sixties [1,2,3]. It
is also a unique problem in dynamics because of the fact that the finite rotation of a rigid
body does not obey the laws of vector addition (in particular, commutativity) and, as
a result, the angular velocity of the body can not be integrated to give the attitude of
the body. The most widely used method of defining the rotation of a body between two
different orientations is an Euler angle description. A 3 × 3 direction cosine matrix (of
Euler rotations) is used to describe the orientation of the body (achieved by three successive
rotations) with respect to some fixed frame of reference. However, there is an inherent
geometric singularity in the Euler representation. This problem can be avoided by using a 4-
parameter description of the orientation [2,3,4], known as 'quaternions', which can be used
to describe all possible orientations. The quaternion approach uses Euler's theorem which
states that any rotation of a rigid body can be described by a single rotation about a fixed
axis. The advantage of using quaternions is that successive rotations result in successive
quaternion multiplications which are commutative. Some early results on the use of
quaternion feedback for attitude error representation and automatic control of the attitude
can be found in [1]. Quaternions were used for the simulations of the rotational motion of
rigid bodies as early as the 1950's [5]. The use of quaternion feedback for controlling robotic
manipulators can be found in [6,7], and for spacecraft control can be found in [8-11].
Various linear and nonlinearquaternion-basedcontrol lawshavebeenrecentlyproposed
[10, I1] for the attitude control of a single-body rigid spacecraft. However, the control laws
proposed in [10] require the knowledge of the system's moments of inertia and also constrain
the choice of the gain matrices. In [11], both model-dependent and model-independent
control laws were presented; however, the control laws used scalar gains.
In this note, a model-independent, nonlinear control law is presented which uses
quaternion feedback and symmetric and positive definite gain matrices. Global asymptotic
stability of the proposed control law is shown by using Lyapunov analysis. The Lyapunov
function used here for proving asymptotic stability does not need a cross term similar to the
one used in [11], and the proof is made much simpler.
Quaternion Feedback Control
The rotational equations of motion of a rigid spacecraft are given by:
&b+wx(Jw)=u (1)
where J is the 3 x 3 inertia matrix; _o is the 3 x 1 angular velocity vector; and u is the 3 x 1
vector of actuator torques. The objective of the control system is to bring the spacecraft to
the desired attitude (orientation) starting from any initial condition.
The orientation of a free-floating body can be minimally represented by a 3-dimensional
orientation vector. However, as stated previously, this representation is not unique. One
minimal representation that is commonly used to represent the attitude is Euler angles.
The 3xl Euler angle vector rl is given by : E(rl)O = w, where E(rl) is a 3x3 transformation
matrix. E(rl) becomes singular for certain values of rl; however, it should be noted that
the limitations imposed on the allowable orientations due to this singularity are purely
mathematical in nature and do not represent physical restrictions. The problem of singularity
in a 3-parameter representation of attitude has been studied in detail in the literature
[2,3,8,10,11]. An effective way of overcoming the singularity problem is to use the quaternion
2
formulation (see[1]-[3]). The unit quaterniona is definedasfollows.
(2)
&3 z •
a = (al, a2, a3) T is the unit vector along the eigen-axis of rotation and 4, is the magnitude of
rotation. The quaternion is also subjected to the norm constraint:
_T_ + a_ = 1 (3)
It can be also shown [12] that the quaternion obeys the following kinematic differential
equations.
• 1
_= _(_ x _+_4_)
1 T--
_4 =-_
The quaternion can be computed [12] using Euler angle measurements (Eq. 4). The
equilibrium solutions of the open-loop system, given by equations (1), (4), and (5), can be
obtained by setting all the derivatives to zero. That is,
(4)
(5)
w x (Jw) = 0 (6)
w X_+Ot4W =0
wT-5 = 0
Taking the dot product of w with both sides of Eq. (7),
(7)
(8)
a4(w'"J) = 0
That is, either a4 = 0, or w = 0, or both are 0. If a4 ¢ 0, then w = 0. If a4 = 0, then from
(7) and (8), w x _ = 0, and w. g = 0, i.e., w = 0, or _ = 0, or both are zero. However, from
(3), a4 = 0 =_ _ ¢ 0; therefore, ,J = 0 when the system is in equilibrium. The system has
multiple equilibrium solutions: (_ss T, c_4ss), where, the subscript 'ss' denotes the constant
steady-state value.
Considerthe control law u, given by:
1 --
u = -_[(a + a4I)Gp + 7(1 - a4)1]-5 - Grw (9)
where Gp and Gr are symmetric positive definite (3 × 3) matrices; 7 is a positive scalar; and
represents the 3 x 3 cross product matrix of the vector _. Equation (9) represents a nonlinear
control law. The following result gives the closed-loop equilibrium solutions.
Lemma 1. Suppose Gp is symmetric and positive definite and 0 < )_M(Gp) < 27, where AM(.)
denotes the largest eigenvalue. Then the closed-loop system given by (1), (4), (5) and (9)
has exactly two equlibrium solutions: [_ = o_= 0, a4 = 1] and [_ = _o= 0, a4 = -1].
Proof.- The closed-loop system is in equilibrium when the derivatives in equations (1), (4),
and (5) are zero. Proceeding as in the open-loop case, the closed-loop equilibrium solution is
given by: _o= 0,_ = -_ss,oc4 -- a4ss. From equation (1), _o = 0 =_ u = 0. From equation (9), we
have
[(_+ a4I)Gp + 7(1 - a4)I]_" = 0 (10)
Pre-multiplying the above equation by _T and noting that the first term vanishes, we have
the following:
u = O =_'h T M'_ = O
where
The eigenvalues of M are given by:
M=a4Gp + 7(1 - a4)I (11)
,_i(M) = a4Ai(Gp) + 7(1 - c_4) = a4(Ai(Gp) - 7) + 3'- M is
singular when )q(M) = 0, i.e., when a4 = L-(-_-_" There are three different subcases that
need to be examined: (a) 0 < ,_i(Gv) < 7, (b) ,_i(Gp) = 7, and (c) 7 < ,_i(Gv) < 27. In subcases
(a) and (c),),i(M) = 0 only if la4] > 1, which is not feasible, since -1 _< a4 <_ 1. That means,
for subcases (a) and (e), hi(M) # 0 for any feasible values of a4. In subcase (b), i.e., when
)q(Gp) = 7, hi(M) = 7 > 0. Therefore, M is nonsingular, and _ = 0. Then, from equation (3),
we have: a4 = 1 or -1. (Note that, if )q(Gp) > 27 then there are feasible values of a4 for which
)q(M) = 0). "
(The symbo] ,, denotes the end of the proof.)
From Lemma 1, there appear to be two closed-loop equilibrium points corresponding to
_4 = 1 and a4 = -1 (all other state variables being zero). However, from equation (2),
a4 = 1 _ ¢ = 0, and a4 = -1 _ ¢ = 2r (or more generally 2nr), i.e., there is only one
equilibrium point in the physical space. We shall define the desired equilibrium state as:
5. = w = 0, a4 = 1. In order to make the origin of the state space the desired state, define
= (_4 - 1). Equations (4) and (5) can then be rewritten as:
• 1
5= _(w × 5.+ (_+ 1)w) (12)
1 T--
I)= -_ _ (13)
The system represented by equations (1), (12) and (13) can be expressed in the state-space
form as follows:
= f(x, u) (14)
where x = (a T, _,_T)T. Note that the dimension of x is 7, which is one more than the
dimension of the system. However, one constraint (Eq. 3) is now present. It can be easily
verified from (4) and (5) that the constraint (3) is satisfied for all t > 0 if it is satisfied at
t=0.
If the objective of the control law is to transfer the state of the system from one orientation
(equilibrium) position to another orientation (i.e., a rest-to-rest maneuver), then without
loss of generality, the target orientation can be defined to be zero. The initial orientation,
given by (5.(0),fl(0)) can always be defined in such a way that -1 _<_ _< 0 (i.e., 0 _< a4(0) _< 1),
corresponding to I¢1 _< _r.
Consider the control law given by:
1 -
u = -_[(5. + (f_+ 1)I)Gp - 7/3I]a - G,w (15)
The control law given by (15) was stated in [11]; however, conditions for the existence of the
closed-loop equilibrium solutions were not investigated, and stability proof was not given.
The following t.heoremestablishesthe global asymptotic stability of the physical
equilibrium state (the origin of the state-space)of the system.
Theorem 1. SupposeGp and Gr are symmetric and positive definite, and 0 < ),M(Gp) <
27. Then, the closed-loop system given by equations (1), (12), (13), and (15) is globally
asymptotically stable (g.a.s.).
Proof. Consider the candidate Lyapunov function
V = wTJw + -_TGp-_ + 7/32
V is clearly positive definite and radially unbounded with respect to the state vector
x = {_T,/3, wT}T. Taking the time derivative of V, we have:
(16)
_" = 2wr[--w x ( Jw) + u] + 2-5TGv(w x a + (/3 + 1)w) - ^t/3wT-5 (17)
Noting that the first term in li" is zero, and substituting for u from (15), after simplification,
we get: li" = -wTGr,0, i.e., 9 is negative semidefinite. 9 = 0 only when w = 0. Following the
same procedure as Lemma 1, it can be shown that V = 0 only at the two equilibrium points,
= w = 0,/3 = 0 (corresponding to a4 = 1) and g = w = 0,/3 = -2 (corresponding to a4 = -1).
Consistent with the previous discussion, these values correspond to two equilibrium
points representing the same physical equilibrium state. It can be easily verified, from
equation (16), that any small perturbation e in ¢_from the equilibrium point corresponding
to/3 = -2 will cause a decrease in the value of V (_ has to be > 0 because -2 _</3 _< 0). Thus,
in the mathematical sense,/3 = -2 corresponds to an isolated equilibrium point such that
1) = 0 at that point, and li" < 0 in a neighborhood of that point, i.e., _3= -2 is a 'repeller' and
not an 'attractor'. It has been already shown that V is negative everywhere in the feasible
state space except at the two equilibrium points. That is, if the system's initial condition
lies anywhere in the state space except at the equilibrium point corresponding to/3 = -2,
then the system will asymptotically approach the origin (x = 0); and if the system is at
the equilibrium point corresponding to/3 = -2 at t = 0 then it will stay there for all t > 0.
6
However, this is the same equilibrium point in the physical space; hence, it can be concluded
by LaSalle's theorem that the system is globally asymptotically stable. •
Concluding Remarks
The problem of three-axis attitude stabilization of a spacecraft was considered. A
nonlinear quaternion-based feedback control law was given, and was shown to provide
global asymptotic stability. The control law does not depend on the knowledge of the
system parameters (i.e., moments of inertia), and is therefore robust to modelling errors and
parametric uncertainties.
References
1. Mortesen, R. E.: A Globally Stable Linear Attitude Regulator, International Journal
of Control, Vol. 8, No. 3, 1968, pp. 297-302.
2. Ickes, B. P.: A New Method for Performing Control System Attitude Computation
Using Quaternions, AIAA Journal, Vol. 8, pp. 13-17, January 1970.
3. Margulies, G.: On Real Four-Parameter Representations of Satellite Attitude
Motions, Mathematical Analysis Department, Philco Western Lab., Palo Alto, California,
Report 52, September 1963.
4. Kane, T. R.: Solution of Kinematical Differential Equations for a Rigid Body, Journal
of Applied Mechanics, pp. 109-113, March 1973.
5. Robinson, A. C.: On the Use of Quaternions in Simulation of a Rigid Body Motion,
December 1958, U.S. Air Force, Wright Air Development Center, Wright-Patterson Air
Force Base, Ohio.
6. Gu, Y. L.: Analysis of Orientation Representations by Lie Algebra in Robotics, Proc.
1988 IEEE Int. Conf. Robotics and Automation, Philadelphia, Pennsylvania, April 1988,
pp. 874-879.
7. Wampler, C. W. III: Manipulator Inverse Kinematics Solutions Based on Vector
Formulation and Damped Least-Squares Methods, IEEE Trans. Systems, Man, Cybernetics,
Vol. 15, No. 1, 1986, pp. 93-101.
7
8. Meyer, G.: Design and Global Analysis of Spacecraft Attitude Control Systems,
NASA Tech. Rep. R-361, Mar. 1971.
9. Wie, B. and Barba, P. M.: Quaternion Feedback for Spacecraft Large Angle Maneuvei"s,
Journal of Guidance and Control, Vol. 8, No. 3, 1985, pp. 360-365.
10. Wie, B., Weiss, H., and Arapostathis, A.: Quaternion Feedback for Spacecraft
Eigenaxis Rotations, Journal of Guidance and Control, Vol. 12, No. 3, 1989, pp. 375-380.
11. Wen, J. T. and Kreutz-Delgado, K.: The Attitude Control Problem, IEEE Transactions
on Automatic Control, Vol. 36, No. 10, pp. 1148-1163, 1991.
12. Haug, E. G.: Computer-Aided tt'inematics and Dynamics of Mechanical Systems, Allyn and
Bacon Series in Engineering, 1989.

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Three-Axis Stabilization of Spacecraft Using Parameter-lndependent WU 233-01-01-05
Nonlinear Quaternion Feedback
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Suresh M. Joshi and Atul G. Kelkar
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NASA Langley Research Center
Hampton, VA 23681-0001
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National Aeronautics and Space Administration
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A. G. Kelkar (NRC Fellow, NASA LaRC)
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REPORT NUMBER
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AGENCY REPORT NUMBER
NASA TM-109150
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13. ABSTRACT (Maximum 200 words)
This paper considers the problem of three-axis attitude stabilization of rigid spacecraft. A nonlinear control law
which uses the feedback of the unit quatemion and the measured angular velocities is proposed and is shown to
provide global asymptotic stability. The control law does not require the knowledge of the system parameters,
and is therefore robust to modeling errors. The signficance of the control law is that it can be used for
large-angle maneuvers with guaranteed stability.
14. SUBJECT TERMS
Nonlinear Control, Attitude Control, Quatemion Feedback Control
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Three-axis stabilization of spacecraft using parameter-independent nonlinear quaternion feedback

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