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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 estimation 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.
References