Stellar attitude-control systems and methods with weighted...

Details Stellar attitude-control systems and methods with weighted... Stellar attitude-control systems and methods with weighted...

2000-08-25

2002-03-12

Chin, Gary (Department: 3661)

Data processing: vehicles, navigation, and relative location

Vehicle control, guidance, operation, or indication

Aeronautical vehicle

C701S222000, C701S226000, C244S164000, C244S171000

Reexamination Certificate

active

06356815

ABSTRACT:

BACKGROUND OF THE INVENTION
1. Field of the Invention
The present invention relates generally to spacecraft attitude control systems and methods that employ Kalman filters.
2. Description of the Related Art
Kalman filtering is a statistical technique that combines a knowledge of the statistical nature of system measurement errors with a knowledge of system dynamics, as represented in a state space model, to arrive at an estimate of the state of a system. In general, the system state can include any number of unknowns. Kalman filtering for a spacecraft attitude-control system, for example, is typically configured to have a state matrix that includes attitude and gyroscopic bias but much larger dimensions (e.g., on the order of 60) of state matrices are sometimes used (to enhance generality, the term matrices will be used herein even though some matrices may typically be single column or single row matrices that would otherwise be referred to as vectors).
The Kalman filtering process utilizes a weighting function, called the Kalman gain, which is optimized to produce a minimum estimate variance (i.e., the estimate's accuracy is maximized). In particular, a Kalman filter combines a current measurement y(t
n
) of a parameter x (e.g., an attitude) at a time t
n
with measurement and state predictions y*(t
n
−
) and x*(t
n
−
) of the parameter x that are based on past measurements (and thus apply to a time t
n
−
just before t
n
) to provide a filtered estimate x*(t
n
+
) of x at a time t
n
+
just after the time t
n.
As indicated by the subscript n, the filter successively and recursively combines the measurements and predictions to obtain an estimate with a minimum variance (i.e., maximum accuracy).
This process is succinctly summarized in an estimate update equation
x*
(
t
n
+
)=
x*
(
t
n
−
)+
k
(
t
n
){
y
(
t
n
)−
y*
(
t
n
−
)},  (1)
in which the state prediction x*(t
n
−
) just before the measurement y(t
n
) is updated by a portion k(t
n
) of a residue which is the difference {y(t
n
)−y*(t
n
−
)} between the measurement y(t
n
) and the measurement prediction y*(t
n
−
) to form an estimate x*(t
n
+
) for a time t
n
+
just after the measurement y(t
n
) was made. The portion k(t
n
) is the Kalman gain which is calculated as
k
⁡
(
t
n
)
=
σ
x
*
 
2
⁢
(
t
n
-
)
σ
x
*
 
2
⁢
(
t
n
-
)
-
σ
m
2
(
2
)
in which &sgr;
x
*
2
(t
n
−
) is the estimate variance (i.e., uncertainty of the estimate) just before the measurement y(t
n
) and &sgr;
m
2
is the measurement variance (i.e., uncertainty of the measurement). The measurement variance &sgr;
m
2
is a function of the system under consideration and, more particularly, of the system's measurement hardware (e.g., manufacturers of spacecraft star trackers and gyros typically specify attitude and attitude rate measurement errors).
In contrast, the estimate variance is reduced as the Kalman process continues. In the beginning of the process, the estimate variance is generally much greater than the measurement variance so that the gain k(t
n
) of equation (2) approaches one. As the process continues, the estimate variance is reduced below the measurement variance so that the gain k(t
n
) declines to a value much less than one. It is apparent from equation (1), therefore, that a large portion of the residue {y(t
n
)−y*(t
n
−
)} is initially used to update the state prediction x*(t
n
−
) into the updated estimate x*(t
n
+
) but this portion decreases as the process continues (i.e., the weighting given to new measurements is successively reduced). In particular, the estimate variance is reduced (i.e., updated) at each measurement y(t
n
) in accordance with
&sgr;
x*
2
(
t
n
+
)=(1
−k
(
t
n
))&sgr;
x*
2
(
t
n
−
)  (3)
The updated variance &sgr;
x*
2(t
n
+
) is time delayed so that it becomes the estimate variance &sgr;
x*
2(t
n
−
) that is used in equation (2) for calculating the next gain k(t
n
). It has been shown that the estimate variance can be expressed as &sgr;
m
2

and thus, it asymptotically approaches zero as more data (i.e., measurements) is obtained.
FIG. 1
illustrates a block diagram
20
of typical Kalman processes that are expressed in a more general matrix form. The Kalman filter
20
comprises an estimator
22
and a gain calculator
24
that supplies a Kalman gain matrix K(t
n
) to a multiplier
26
of the estimator. To facilitate a description of the filter
20
, an investigation of the gain calculator
24
is preceded by the following description of the estimator.
The estimator
22
receives a measurement matrix Y(t
n
) at an input port
28
and provides an estimate matrix X*(t
n
+
) for a time just after the measurement to an output port
30
. From this estimate matrix, a state prediction matrix X*(t
n
−
) is formed for a time just before the next measurement and this state prediction is provided to a summer
32
. Because the state of the system typically varies dynamically between measurements, the estimate matrix X*(t
n
+
) that corresponds to a time just after the last measurement must be extrapolated over time to form the state prediction matrix X*(t
n
−
).
As shown in
FIG. 1
, this extrapolation is accomplished by passing the estimate matrix X*(t
n
+
) through a delay
34
(to cause it to be time-incident with the next measurement matrix Y(t
n
)) and multiplying it by a state transition matrix
36
which contains extrapolation information in the form of a state transition matrix &PHgr;(t
n
,t
n−1
). For example, if one component of the estimate matrix X*(t
n
+
) is a position x(t
n
+
), the state transition matrix &PHgr;(t
n
,t
n−1
) might be configured to extrapolate the position with a term of Tv(t
n
+
) wherein T is the time between measurements and v(t
n
+
) is the last estimate of velocity.
The state prediction matrix X*(t
n
−
) is also multiplied by a measurement matrix
38
to form a measurement prediction Y*(t
n
−
) which is provided to a differencer
40
where it is differenced with the measurement matrix Y(t
n
). The measurement matrix H(t
n
) conditions the state prediction matrix X*(t
n
−
) so that its elements correspond to those of the measurement matrix Y(t
n
) and can be properly differenced with it.
As a first conditioning example, the measurement may be expressed in one coordinate system (e.g., rectangular) and the estimates tracked in a different coordinate system (e.g., spherical). In this example, the measurement matrix H(t
n
) would be configured to convert the estimates to the coordinate system of the measurements. In a second conditioning example, attitude and gyroscopic bias might be part of the prediction matrix X*(t
n
−
) but only attitude might be present in the measurement matrix Y(t
n
) so that the measurement matrix H(t
n
) would be configured to make the necessary conversion.
The differencer
40
, therefore, generates a residue Y(t
n
)−H(t
n
)Y*(t
n
−
) which is then multiplied in the multiplier
26
to form a correction K(t
n
){Y(t
n
)−H(t
n
)Y(t
n
−
)} that will be used to update the estimate matrix. The updating is performed in the summer
32
where the correction is summed with the state prediction matrix X*(t
n
) to generate the updated estimate matrix
X*
(
t
n
+
)=
X*
(
t
n
−
)+
K
(
t
n
) {
Y
(
t
n
)−
H
(
t
n
)
Y*
(
t
n
−
)  (4)
at the output port
30
.
Attention is now directed to the Kalman gain calculator
24
which performs similar updating and extrapolation processes for the Kalman gain matrix K(t
n
). In the matrix notation of
FIG. 1
, the estimate variance &sgr;
x*
2
(t
n
−
) and the measurement variance &sgr;
m
2
are respectively replaced by an estimate covariance matrix P(t
n
) and a measurement-noise covariance matrix R(t
n
). These matrices ma

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