| Topic: |
Science > Physics |
| User: |
"mugwomp" |
| Date: |
02 Feb 2005 11:54:40 PM |
| Object: |
Coning Rate - What do they use this for? |
Estimation of Vehicle Dynamic and Static Parameters from Magnetometer Data
John F. Kinkel and Mitchell Thomas
A method of estimating the spin rate, coning rate, and coning half-angle of
a spinning space vehicle using the data from a single-axis magnetometer
mounted transversely to the spin axis of the vehicle is descried. In
addition, estimates are available for the angle between the total angular
momentum vector of the vehicle and the local magnetic field vector. The
ratio of the spin axis to transverse axes moments of inertia may also be
calculated. Estimates for several other parameters, including the local
magnetic field intensity, are available. An example is given for a
magnetometer mounted in a laboratory fixture to provide spin and coning
motion. Finally, the use of the data to obtain absolute pointing information
is illustrated, including an error analysis.
Introduction
For low-budget space experiments we desire to provide as much data as
possible regarding the dynamics of a spinning vehicle while satisfying
severe constraints on the size, weight, power, and cost of the
instrumentation package. As demonstrated in this paper, a single-axis
magnetometer mounted transversely to the spin axis of the vehicle provides a
great deal of information when the data are appropriately processed.
Although the technique has been demonstrated to work for real data (a sample
problem is included here), the existence of a unique solution has not been
proven.
Deduction of Dynamics Parameters
We show how the magnetometer data will provide estimates for several
parameters describing the vehicle dynamics. The basic approach is to
formulate a parameter estimation problem based on a mathematical model (the
estimation model) representing the motion of the vehicle-mounted
magnetometer. The parameter set in the estimation model includes several
parameters of interest in the space experiment, e.g., the vehicle spin rate,
coning rate, coning half-angle, and other parameters to be described.
Loosely speaking, the problem is to determine the model parameter set that
minimizes (in some sense) the difference (distance) between the model
magnetometer data stream and the actual magnetometer data stream. A
nonlinear programming algorithm is used to perform the required
minimization. The details of the problem formulation and solution are
described in the sequel.
Estimation Problem
There is a strong connection between estimation and approximation. For our
present purposes we prefer to use the approximation problem formulation
described by Rice.
After a fright of the magnetometer, we are given a data sequence f(~). We
devise an estimation model(a, incorporating a vector a whose elements
comprise the set of parameters to be estimated. The problem is to find the
optimal estimate for a that minimizes the distance between f(ti) and F(cx,
fi). To make use of this basic problem formulation, it is necessary to do
the following.
1) Define the estimation model F (CY, ti) and associated parameter vector a.
2) Define the distance function p.
3) provide a means of solving the minimization problem.
The next section defines the estimation model. For convenience, we use a
least-squares solution.
We may apply a nonlinear programming algorithm to solve the minimization
problem.
Estimation Model
We assume that the vehicle is symmetrical about the spin axis, which we
designate as the z axis; i.e., we assume that Ix = 1~ and IXY = I*z = Iyz =
0, where the I designates the moments of inertia indicated by the
subscripts.
Using the notation of Hughes we may write the output of the magnetometer as
the dot product of the magnetometer sensitivity vector and the local
magnetic intensity vector.
We will use a standard set of Euler angles to define the geometry. A body
fixed set of axes is used on the spinning object with the z axis being the
axis of symmetry. The magnetometer is mounted so that its sensitive axis is
parallel to the x axis of the body-mounted coordinates. To evaluate Eq. (4)
we transform the magnetometer sensitivity vector (K, 0, 0) in the body fixed
coordinates, to an inertial system. K is the sensitivity coefficient for the
magnetometer. We choose an inertial system with its z axis oriented along
the constant direction of the vehicle's angular momentum, and with the
Earth's magnetic field vector in its X-Z plane. The conversion of the
magnetometer sensitivity vector to the inertial system using the Euler
angles is described by Hughes" and Thomson?
Note that this analysis assumes that the magnetic field is constant over the
trajectory interval examined. If the field cannot be considered constant,
additional variables should appear in Eq. (5). For flight vehicles we have
examined, the approximation appears to be valid, but other investigators are
cautioned to consider its validity for their particular case.
Solving the Minimization Problem
The nonlinear programming algorithm selected here to solve this minimization
problem is the Davidon-Fletcher-Powell (DFP) algorithm. Although no one
nonlinear programming has been developed that will solve all problems, the
DFP algorithm has been found to be very effective in a wide class of
problems and is generally recognized as being among the very best available,
if not the best. The specific algorithmic code used is as described by Press
et al. The minimization algorithm selected will find only a local minimum;
therefore, reasonable initial conditions for the state (parameter) vector
must be provided. If these are not carefully selected, the algorithm may
find some local minimum far from the desired one. The ratio of pitch to roll
moments of inertia is assumed known, at least approximately. The initial
estimates for fs and fp are used in Eq. (19). This leaves only ~2 = B to be
assigned an initial value. Comparing the appearance of plots of F(cx, ti) vs
time will aid the selection of an initial value of a~. Given the initial
values as described, the algorithm will provide an optimal estimate &
corresponding to the local minimum discovered.The value of i found by Eq.
(21) should be reasonably close to the vehicle design value. A check on the
credibility of the estimate & is the standard deviation for the estimate.
The SNR should be approximately the same as the design value for the
magnetometer system.
An SNR resulting from a large value of 0 indicates with high probability
that an erroneous local minimum has been found and that more accurate
initial values are required for the algorithm to converge to the desired
minimum.
Example
Figure 1 shows a real magnetometer output (dotted values) from a laboratory
fixture, which simulates spin and precession. The fixture was such that the
equivalent total angular momentum vector was approximately vertical and the
equivalent coning half-angle was about ~r/4 rad. A plot of the magnetometer
output is shown in Fig. 1. Examination of the figure indicates that the
values for spin rate and precession rate are about 0.5 and 0.0813 Hz.
Because the coning half-angle y was known to be about ~r/4 rad, the design
value of R is found from Eq. (20) to be 8.7. The dip, or inclination of the
Earth's magnetic field from the horizontal in the Tustin area where the test
were concluded, is about 60 deg downward. Therefore, the angle b between the
total angular momentum vector and the local magnetic intensity vector was
about 150 deg or 2.618 rad
The equations presented in the preceding sections were solved numerically.
The operation of the computer program was checked by creating a fictitious
magnetometer data file based on the model previously described. Without
additive noise the residual total square error z(b) was very small and & was
the same as the fictitious model parameters, indicating convergence of the
algorithm. With additive noise in the magnetometer model data sequence, the
SNR found by Eq. (23) was in good agreement with the known value. Because
the fictitious data stream was generated by the estimation model, the tests
just described check only the operation of the nonlinear programming
algorithm, not the validity of the estimation model. The validity of the est
imation model will be checked using real data from the magnetometer test
fixture previously described.
For determining the initial estimate of the unknown parameters, we used a
spreadsheet program. The magnetometer output, Eq. (7), was calculated for a
priori estimates of the parameters. Direct comparison of the graphs of the
actual data vs time to the calculated data allowed selection of the proper
initial values by inspection. Figure 1 a shows the result of this estimation
for our laboratory model and is typical of the kind of agreement possible.
The coning and spin frequencies are estimated first to get proper
frequencies into the data. Then the phases are adjusted to get the proper
shapes. This method could be automated, but for analysis of flight data
after the fact that is not necessary.
Applying the estimation algorithm to the actual magnetometer data gave the
results shown in Table 1. A plot of the magnetometer estimation model
outputs are shown for the optimal parameter vector & in Fig. lb. Figure 1 b
shows that the optimal parameter vector estimate B produces an estimation
model output that closely matches the real magnetometer output. The validity
of the estimation model is thus verified, at least qualitatively.
Quantitative validation is evident from the SNR shown in Table 1 for the
optimal estimate. The table shows SNR = 28.6 dB as compared to the known
value of about 30 dB for the magnetometer.
Deduction of Absolute Pointing Direction
A three-axis magnetometer has been used to obtain accurate pointing
information for an intercontinental ballistic missile (ICBM) distance launch
on the Inflatable Exoatmospheric Object (LEO) program in 1972. More
recently, highly accurate orientations were obtained on the Firefly sounding
rocket launch in 1989. For Firefly, the uncertainty in orientation inherent
with these devices was eliminated through the use of a sun sensor also,
which provided a positioning pulse once during each spin cycle. As shown
subsequently, similar information can be obtained with a single-axis
magnetometer without the use of any additional inputs, relying on the
variation of the Earth's magnetic field along the fright path. A similar
approach was previously used for IEO data reduction for a three-axis
magnetometer. The details of this earlier approach by Massachusetts
Institute of Technology, Lincoln Laboratory, personnel was never published
in the open literature. The approach used here is similar but more
comprehensively quantifies the resulting error.
As already shown, a single-axis magnetometer can be used to obtain the angle
between the payload's angular momentum vector and the Earth's magnetic field
by modeling the motion over a part of the trajectory. The result is that the
payload's angular momentum vector is known to lie on a cone around the
Earth's magnetic field vector.
At a later point in the trajectory, the direction of the Earth's magnetic
field has changed, and if the payload:s angular momentum vector is cons&r&
the result will be a cone of uncertainty different from that found at the
earlier point. This is shown in Fig. 2. The intersection of the two cones of
uncertainty must contain the actual payload angular momentum vector. Thus,
the cone of uncertainty has been reduced to a pair of possible orientation
vectors. However, if the calculation is repeated for a third point along the
trajectory, a third cone of uncertainty results that must also contain the
payload angular momentum vector. When this cone is used together with the
one from the earlier point a new pair of possible payload orientations is
found. However, only one of this new pair will match one vector of the pair
of vectors found from the first two trajectory points. This matching vector
must be the actual payload angular momentum vector.
The case of a payload with a constant angular momentum over large portions
of its trajectory is a common and usual situation. Furthermore, the analysis
shows that the deduction of #$ the angle between the payload's angular
momentum vector and the Earth's magnetic field, is feasible by matching
trajectory parameters over a los portion of the flight. Because suborbital
trajectories of interest range from about 500 s in duration for short
sounding rocket flights, to 2400 s for ICBM flights, the opportunity exists
to apply the cone-of uncertainty analysis earlier highlighted to many points
along the trajectory. Although only three points are theoretically needed,
as already shown, many points will be useful to mitigate the effect of
measurement errors on the analysis.
The purpose of this part of the paper is to show that the analysis
highlighted can be used for realistic trajectories in the presence of
measurement errors of the parameter beta. We show that there are conditions
where the method will be of marginal usage, but many cases exist where
absolute pointing data can be found from analysis of the single-axis
magnetometer.
Analysis Approach
We assume that the trajectory parameters (altitude vs time) and the
direction of the Earth's magnetic field are known accurately. The effect we
are investigating is the impact of the measurement error in p.
The model of the Earth's magnetic field is that of a dipole, whose axis of
symmetry is tilted from that of the Earth by the spherical polar angles & =
11.5 deg and &?I = -69 deg. Defining A as the complement of @ (the polar
angle in the spherical coordinate system of the Earth's field), it is the
latitude of the magnetosphere.
This simple model allows the magnetic field direction along typical
trajectories of interest to be easily estimated. The Appendix shows the
derivation of the pointing direction given the cones of uncertainty for two
different points along a trajectory. When this procedure is applied to data
where the value of the cones is known accurately, the actual payload
pointing direction is obtained from the first three points along any
trajectory. For the analysis reported, the values of /I were assigned random
errors along the trajectory, and the ability to again deduce the pointing
direction of the payload was studied.
To aid in deduction of pointing, we had to smooth the resulting inaccurate B
values by curve fitting with a curve of order 1,2, or 3 (in general, order 2
worked best). This has the effect of averaging out much of the error
introduced by the random errors in p. For example, Fig. 3 shows a typical
trajectory run during this analysis. Figure 4 shows the resulting B values
for a particular pointing vector of the payload (direction cosines all
equal). The figure also shows a resulting set of /? assigned a random 1%
error, and the second-order cume fit to these data.
Parametric Analysis
The following paragraphs present the results of the study.
Effect of Number of Intervals Used in the Calculations
Table 2 shows the effect of increasing the number of points calculated along
a trajectory from 10 to SO. See the Appendix for definition of&Z. The
improvement in accuracy results from the additional averaging obtained by
using more data points. For short trajectories,
Effect of Trajectory
One might expect that with a longer trajectory, the magnetic field direction
will vary more, and the analysis will be less sensitive to errors in B (p
will vary more strongly the more the magnetic field varies). Our
calculations show that indeed this is correct. Table 3 presents deduced
orientations as a function of the total time of the trajectory (from launch
to impact), and the trend toward increasing accuracy with trajectory time is
apparent, although for long times an asymptote is approached.
For a long trajectory, it appears that the error in the deduced pointing
direction is of the same order as the error in /S. This is shown in Table 4,
which shows the variation in u from 1 to 10% for an ICBM trajectory. For
shorter trajectories, or those using fewer points, the trend is the same,
but the errors are greater than CL
Effect of Vehicle Orientation
Table 5 shows that the analysis gives similar results if the angular
momentum vector lies in either the xz or yz planes. If the orientation is in
the my plane, however, the effect of error in #J can cause a switch in the
deduced pointing vectors from positive to negative for some trajectory
points. Thus, there remains an uncertainty for cases where the payload
direction has no z component. However, if a small z component is introduced
at a level about 10% of the x or y components, this ambiguity is removed.
Conclusions
What is believed to be a novel technique for obtaining dynamic data from a
spinning spacecraft using only very simple on-board equipment has been
demonstrated. Using only a relatively low-accuracy single-axis magnetometer
we obtain reasonably accurate estimates for the vehicle spin rate,coning
rate, coning half-angle, ratio of pitch to roll moments of inertia, angle
between total angular momentum vector and local magnetic field intensity,
and the magnitude of the local magnetic field intensity. The estimates are
achieved using a nonlinear programming solution of an optimal estimation
problem formulation.
By processing the magnetometer data in successive blocks of a few hundred
data points, it should be possible to obtain a sequence of estimates for
time varying parameters. It should be necessary to manually initialize the
estimation process only for the first data block because the remaining
blocks can utilize the optimal estimate from the preceding block as the
initial parameter vector.
Deduction of the absolute orientation of a spinning payload by using a
single axis magnetometer is feasible for a large variety of trajectories and
orientations. To obtain reasonable accuracy 50 or more points along the
trajectory should be analyzed. The longer the trajectory, the more accurate
the magnetometer supplied data becomes.
The two possible solutions of EZq. (A8) produce the two lines of
intersections between two cones. One of these values of r is equal to the
angular momentum vector M.
Ideally, a new set of points will give a different solution since f will
change. However, one of the new solutions should equal one from the previous
set of solutions because the solutions again contain M, which is assumed to
be constant.
In the presence of measurement errors, the new solution will not precisely
match a previous one, but the set of solutions that from point to point is
least variable is most likely to be, on average, M.
.
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