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AAS 19-529

EXPECTED ACCURACY OF DENSITY RECOVERY USING

SATELLITE SWARM GRAVITY MEASUREMENTS

William Ledbetter∗

, Rohan Sood†

, and Jeffrey Stuart ‡

Asteroids are of particular interest in the modern space industry due to their poten-

tial for advancing knowledge about the origins of the solar system. Identifying and

characterizing asteroids with sustainable resources for future space exploration is

critical. Additionally, limited knowledge of near-Earth asteroids’ physical char-

acteristics such as shape, density, gravity ﬁeld, and composition pose a challenge

to any manned exploration. A stochastic gradient method from the ﬁeld of deep

learning is applied to the problem of asteroid density recovery, with implications

for target selection and mining. The algorithm outperforms the predicted observ-

ability, and is minimally affected by noisy measurements.

INTRODUCTION

Questions about the origins of life on Earth have been the overarching motivation that has driven

the majority of space-faring missions over the last half-century. Scientists and engineers have rec-

ognized that asteroids, the leftover building blocks from the time of planetary accretion, may be the

richest source of information capable of providing a glimpse into the prehistoric solar system. De-

veloping a comprehensive understanding of the composition of asteroids may offer insight into the

early processes of solar system formation, with implications for planetary exploration, exoplanetary

investigations, and, ultimately, the origins of life.

Missions such as NEAR/Shoemaker1and Hayabusa2have been invaluable to the scientiﬁc com-

munity, and successful completion of the OSIRIS-REx and Hayabusa 2 missions will catalyze fur-

ther research. However, the number of cataloged asteroids recently surpassed 780,000, yet, even

with these recent missions, fewer than 20 have been visited by spacecraft. Small-body exploration

missions, such as Lucy,3often utilize ﬂybys to tour multiple bodies with a single spacecraft. Lucy

plans to visit multiple targets in the Jupiter-Trojan system by utilizing sequential ﬂybys. Although

the asteroid-tour-style mission is an elegant solution to the problem of broad exploration, some

trade-offs are made with respect to the traditional single-target mission. In order for the spacecraft

to continue on its tour trajectory, it must maintain signiﬁcant velocity as it passes each body of in-

terest, substantially reducing the time available for close-proximity observations. As an attempt to

augment the quality of data obtained during such short encounters, the authors propose a method of

in-situ gravimetry to reveal internal structures and density variations.

∗Ph.D. Student, Astrodynamics and Space Research Laboratory, Department of Aerospace Engineering and Mechanics,

The University of Alabama, Tuscaloosa, AL, 35487, USA. wgledbetter@crimson.ua.edu

†Assistant Professor, Astrodynamics and Space Research Laboratory, Department of Aerospace Engineering and Mechan-

ics, The University of Alabama, Tuscaloosa, AL, 35487, USA. rsood@eng.ua.edu

‡Research Technologist, Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Dr., Pasadena,

CA 91109, USA, MS 301-121, jeffrey.r.stuart@jpl.nasa.gov

1

Previous work4outlined the principles of density recovery for a heterogeneous body, and revised

the polyhedral gravity model5, 6 to express such a body. Preliminary developments showed promis-

ing results using a simple matrix inversion recovery technique; however, the method had limitations

when complex recoveries were attempted. In this paper, the authors expand the research on two

fronts: gravity model observability, and recovery algorithm development. Analysis of the gravity

model’s observability will help deﬁne bounds on the expected performance of an optimal recov-

ery algorithm, and research into alternative recovery methods will aim to achieve the best expected

performance.

THEORY

Dynamical Model

Recent developments in asteroid gravity modeling have utilized the polyhedral formulation of

Werner and Scheeres.5Updates to this model have investigated techniques for expressing variable

density,6, 7 but still maintain a priori assumptions about the internal geometry, such as core/shell

or hemispherical distributions. The authors’ contribution expands the techniques of Takahashi6to

express fully heterogeneous polyhedra at the same level of resolution as the shape model itself.

The expression of heterogeneous polyhedra relies on vectorization of Werner’s basic formulation,

Equation (1).

U=Gρ X

eedges

¯re·Ee·¯re·Le−Gρ X

ffaces

¯rf·Ff·¯rf·ωf,(1)

where Gis the gravitational constant, ρis the average density of the body, ¯reis a vector from the

point in space to a point on edge e,Eeis a matrix describing the geometry of edge e, and Leis the

potential of a 1D ‘wire’ expressed in terms of the distances to the edge’s endpoints. In the second

term, ¯rfis a vector from the query point in space to a point on face f,Ffis a matrix describing

the geometry and orientation of face f, and ωfis the signed solid angle subtended by face f, from

the perspective of the point in space. Equation (1) assumes homogeneity, but by distributing the

density term into the summations, each face and edge can be assigned a unique density, ¯ρfand ¯ρe,

respectively, thus enabling variation in the latitude and longitude of the central body. Furthermore,

the potential equation can now be expressed as a vector dot product:

U= ¯u·¯ρ= [¯ue|¯uf]·[¯ρe|¯ρf],(2)

where the elements in the ¯uvectors are given as

ue=GLe(¯re·Ee·¯re)(3)

uf=−Gωf(¯rf·Ff·¯rf).(4)

Radial variation is achieved through layering of multiple polyhedra. Assuming the shape of each

inner layer is a scaled and concentric copy of the outermost layer preserves the Eeand Ffmatricies.

Therefore, only the relative position vectors, ¯reand ¯rf, and the scalars, Leand ωf, vary with

each subsequent layer. The procedure necessary for layering is illustrated in Figure 1, and can be

mathematically expressed by Equation (5).

U= (¯u1−¯u2)¯ρ1+ (¯u2−¯u3)¯ρ2+ ¯u3¯ρ3= ¯u·¯ρ(5)

¯u= [( ¯u1−¯u2)|(¯u2−¯u3)|¯u3](6)

2

Figure 1:Visualization of Polyhedral Layering

¯ρ= [ ¯ρ1|¯ρ2|¯ρ3](7)

In Equations (5) to (7), each color wheel’s size is analogous to ¯uiand the orientation of the color

pattern is analogous to ¯ρi. Takahishi describes a similar technique called the block model, which is

used to simulate zones of homogeneous density.

Observability and Correlation

Park et al., in 2010,8presented an analysis of ﬁnite-cube and ﬁnite-sphere gravity models wherein

observability of internal density was estimated from on-orbit data. The method utilized a batch-

least-squares ﬁlter applied to the range and range-rate measurements: no direct gravity measure-

ments were considered. In recent years, innovations in sensor technology have opened the possi-

bility of measuring gravity gradient (GG) directly from small-scale spacecraft.9The same mathe-

matical technique used by Park et al. can now be applied to predict the effectiveness of this new

gravimetry technique.

Assume that a gravity gradient measurement is taken at a point in space, and the measurement

can be deﬁned by a polyhedral gravity model. The equation that would predict that measurement is

GG =¯

gg ·¯ρ= [ ¯

gge|¯

ggf]·[¯ρe|¯ρf],(8)

where GG is the gravity gradient matrix, ¯

gg is the concatenation of Equations (9) and (10), ¯ρis

the density vector describing the body, and ¯

ggeand ¯

ggfare vectors of matrices. Each matrix in the

edge and face vector is given as,

gge=−GLeEe,(9)

ggf=GωfFf,(10)

respectively. The recovery problem assumes that the position and gravity gradient can be measured.

Position is used to calculate ¯

gg, the gravity measurements are substituted for GG, and the goal of

the problem is to ﬁnd the density vector ¯ρthat most closely satisﬁes Equation (8).

For a given position, the relationship between the density and the gravity gradient is linear. Sub-

sequently, partial derivatives with respect to the density are found simply to be

∂GG

∂¯ρ=¯

gg =G[−LeEe|ωfFf].(11)

Then, following Park et al., deﬁne

Λ=¯

ggT¯

gg,(12)

and to obtain the covariance matrix,

P=Λ−1.(13)

3

The covariance matrix provides the correlations and standard deviations of elements in the density

vector for a given set of measurement data. Analysis of these values informs predictions about the

accuracy of a recovery attempt. For example, a high standard deviation indicates high uncertainty

about the value of a density zone, and high correlation with other zones implies that the gravitational

effects caused by that zone are hard to distinguish from the effects of other zones.

Adam Optimizer

The problem of high-resolution polyhedral density reconstruction bears some rudimentary re-

semblance to the problem of training a neural network: a large number of parameters must be

tuned to recreate a set of noisy data. The methods which have shown the best performance in

high-dimensional neural network training can be classiﬁed as stochastic gradient methods. One

such algorithm, Adam, has quickly become one of the most popular methods since its introduction

in 2014.10 By keeping a running average of the second moment of the gradient, Adam is able to

scale the step size of each iteration for empirically better convergence. Certain situations11 result in

slower convergence, and prior work has sought to modify the base algorithm to accommodate such

cases.12 In this investigation, the original formulation based on adaptive moment estimation is im-

plemented. The core loop is reproduced from the original paper as Algorithm 1 for reference. One

Algorithm 1 Adam Algorithm (Reproduced for reference)

Require: α: Step size

Require: β1, β2∈[0,1): Exponential decay rates

Require: imax: Maximum iterations

Require: ∇f(¯x): Gradient function

Require: ¯x0: Initial guess

1: i←0

2: ¯m0←¯

0

3: ¯v0←¯

0

4: while i<imax do

5: i←i+ 1

6: ¯gi← ∇fi(¯xi−1)

7: ¯mi←β1¯mi−1+ (1 −β1)¯gi

8: ¯vi←β2¯vi−1+ (1 −β2)¯g[2]

i(Square bracket exponent indicates element-wise power)

9: αi←α·q1−βi

2/(1 −βi

1)(Correct for initialization bias)

10: ¯xi←¯xi−1−αi¯mi/(√¯vi+ε)(Element-wise square root and division)

return ¯xi

modiﬁcation from the original algorithm is the loop control parameter, which was changed from a

convergence evaluation to a set number of iterations. This change allows the user to terminate the

computation and investigate the algorithm’s initial behavior before investing signiﬁcant time into a

convergence-evaluated test case.

In order to minimize the stochastic gradient function, ∇f, running averages of the mean, ¯m, and

the uncentered variance, ¯v, are used to calculate an appropriate step size. Kingma and Ba draw an

analogy between the ratio ¯mi/√¯viand the signal-to-noise ratio (SNR).10 A smaller SNR (larger ¯v)

indicates greater uncertainty in the direction of the true gradient, and the step size is appropriately

scaled in line 10.

4

The optimization problem in this paper is formulated as a least-squares minimization. Because of

its broad applicability, the framework for solving such problems is written in the same C++ class as

the main optimizer, and is set as the default mode. The objective of the least-squares optimization

is

min(¯y−A¯x)2(14)

where ¯xis the design vector, ¯yis the data vector, and Arelates the two, such that, for the optimal

case, ¯y=A¯x. Adam operates using only the gradient, which is

¯g(¯x) = 2ATA¯x−2AT¯y. (15)

Batching is often utilized when training deep neural networks. In its most common form, batch-

ing consists of choosing a few data points with which the gradient function is calculated, performing

a few iterations, and then randomly re-selecting those data points. This technique can help the op-

timizer avoid getting trapped in non-convex regions of the solution space. For the least-squares

formulation, re-sampling data points amounts to selecting rows of the Amatrix and the correspond-

ing elements in the ¯yvector, and using these truncated values in Equation (15). Two parameters

that are used to control batching are the batch size and the batch frequency. Each is fairly intuitive:

batch size refers to the number of points extracted from Aand ¯yin each sample, and batch fre-

quency determines how often the data is resampled. The effects of each parameter on recovery and

convergence are detailed in the Results section.

PROCEDURE

Test scenarios can be described as occurring in 3 steps: parameter selection, simulation, and

recovery. The most signiﬁcant parameters available for selection, as well as their valid values, are

presented in Table 1. The number of measurement directions parameter refers to the elements of the

gravity gradient matrix (xx,xy,xz,yy,yz,zz).

Parameter Values

Body Any polyhedral model

Layers >0

Number of probes >0

Noise on/off

Number of meas. directions 1-6

Sample Rate >0

Adam batch size >0

Adam batch frequency ≥1

Table 1:Parameters available for selection

A typical simulation for multiple probe ﬂyby of a sample asteroid is illustrated in Figure 2. In

this case, 5 probes are ejected from a relative position of (50, 0, 0) km from the asteroid, and are

propagated forward for 6 hours. Their initial velocities are calculated to be evenly spaced around

the body, resulting in a more complete ground track coverage.

Before noise is applied to the data, the theoretical observability is analyzed using Park’s method

as detailed above. The standard deviations and the correlation matrix are saved in full precision

5

Figure 2:Sample simulation using an Eros shape model

as text ﬁles, and will be visualized later using MATLAB tools. Noise is sampled from a Gaus-

sian distribution with zero mean, where the level of noise is deﬁned by a standard deviation. A

set is sampled from the deﬁned distribution, and is then added to nominal trajectories and gravity

measurements.

In order for Adam to operate on this problem, it must be cast as a least-squares matrix problem.

Recalling the formulation above, the objective is

min( ¯

GG −gg ¯ρ)2,(16)

where ¯

GG is the measurement vector, ¯ρis the density vector, and gg is the system matrix deﬁned

from Equation (8). The noise in position is considered when calculating gg, and the noise in gravity

measurements is included in ¯

GG.

Optimization is initialized using the default values β1= 0.9and β2= 0.999 given in the original

paper.10 The step size, α= 20, was empirically found to produce good results, but this parameter

is problem-speciﬁc, since Kingma and Ba suggested an αof 0.001. Performance is quantiﬁed by

calculating the difference between the optimized density and the true density at each iteration.

RESULTS

Three test cases of varying complexity were selected that constitute a broad sample of the possible

input parameters. For the simplest case, the base results are presented, and then the neighboring

parameter space is explored by changing certain values. The conclusions from the simple analysis

will determine the settings for more complex cases.

Case 1: 1-Layer Octahedral Body

Base Conﬁguration: The theoretical correlations for the base case are given in Figure 3. For

each axis, the Zone ID refers to the index of that zone in the density vector, ¯ρ. Note that all diagonal

components were set to zero in order for the color scale to be more intuitive. The most evident dif-

ference is the emergence of a visual distinction between edges and faces. Figure 3b shows consistent

6

low-magnitude negative correlations of edges with faces, and mostly small positive correlations be-

tween faces. The outliers within the face-zone, such as (13,18), are likely caused by the symmetry

of the body.

(a) Case 1: 1 Probe correlation matrix (b) Case 1: 10 Probe correlation matrix

Figure 3:Correlation comparison for Case 1

The average standard deviation for the base case is 5.57 ×105g/cm3, an order of magnitude

improvement over the one-probe case’s average of 4.64 ×106g/cm3. However, since the average

density of the body is 2.67 g/cm3, both deviations suggest that a recovery will be imprecise.

In the absence of noise, the Adam algorithm shows consistent convergence in Figure 4, although

the same mechanics that make it robust for noise also make it relatively slow. Nonetheless, the

asymptotic convergence pattern was encouraging given the wide standard deviations from the ob-

servability analysis.

Figure 4:Case 1: Adam Convergence

7

Adding Noise: Previous investigation by the authors suggested that noisy optimization would

be difﬁcult, if not impossible. However, adding up to 10 meters in positional uncertainty resulted

in convergence proﬁles nearly indistinguishable from the clean Figure 4. Zooming in to the level

of Figure 5 is required to discern a difference. It is observed in the 10 cm and 1 m cases that the

optimization actually performed better than the clean case. However, the result may be misleading;

if the optimization were continued, the noisy cases will overshoot the target and settle with a higher

ﬁnal error.

Figure 5:Case 1: Final Iterations of Noise Comparison

Varying Sample Rate: Intuition suggests that increasing the amount of measurements would im-

prove the accuracy of the recovery. With the current formulation, there are two ways to obtain more

measurements: increase the number of probes, or increase the sample rate of the probes. The effects

of increased sample rate are illustrated in Figure 6, with an accuracy improvement of about 7×10−7.

Note that there is minimal beneﬁt of a 1-second sample rate over the 5-second sample rate. These

cases are not identical, due to random batch selection, but they appear to twist around each other,

with neither displaying consistent gains over the other.

Varying Number of Probes: The other option to gather more measurements is to increase the

number of probes. Additional probes are sent on unique trajectories, thereby sampling in previously

unexplored space. The effects shown in Figure 7 are of a similar order of magnitude to the sample

rate analysis above. This similarity indicates that the variation could be a product of the stochas-

ticity of the algorithm itself, via noise sampling and batch sampling, rather than an effect of the

sample rate or the number of probes. Also, the ﬁnal errors after 50,000 iterations do not support

the hypothesis that more probes implies better recoveries. Therefore, other factors are affecting the

system.

Varying Batch Size: The concept of batching, as discussed in the Adam Optimizer section is

implemented in the simulations, and two variables are introduced that control the process: batch

size, and batch frequency. In the previous two analyses, a large number of samples were available for

8

Figure 6:Case 1: Final Iterations of Sample Rate Comparison

Figure 7:Case 1: Final Iterations of Probe Number Comparison

9

use in the optimization, but since the batch size was constant throughout, each iteration only utilized

800 samples. By increasing the batch size, more samples are used in each iteration. The effects of

varying the batch size are illustrated in Figure 8. The drastic improvement in convergence speed

Figure 8:Case 1: Batch Size Comparison

from larger batch size can be explained by Kingma and Ba’s SNR analogy: a more comprehensive

sample of the data results in a more consistent gradient vector. Then, if the gradient is more stable,

the algorithm will take larger steps. However, there are some trade-offs with respect to the runtime.

Increasing the batch size is the same as increasing the height of the Amatrix in Equation (15); as

such, the linear algebra operations take longer to compute.

Considering the standard deviation predictions from above, the result for the 8000 batch size is

particularly surprising. Despite the standard deviation being on the order of 105g/cm3, the densities

are recovered to an accuracy of about 6×10−4g/cm3.

Behavior Near Answer: All the tests, thus far, have investigated the behavior of the optimization

as it leaves the initial guess and moves towards the correct answer. However, it is also of interest to

know how the algorithm behaves in the vicinity of the correct answer. This zone is tested by passing

the true density distribution as the initial guess, and then observing whether the algorithm stays in

the neighborhood of the optimum. Given the results of the cases above, all scenarios in Figure 9

utilize the maximum possible batch size, as well as noise in position and gravity measurements.

The location to which the algorithm converges is a function of the random noise sample. The three

cases shown in Figure 9 use different samples; thus, they converge to a different location. However,

the behavior can still be analyzed. The 5 probe scenario converges to its ﬁnal solution noticeably

slower than the 10 and 15 probe cases.

Case 2: 3-Layer Random Sphere

The polyhedral body for case 2 is a sphere with noise applied to the radius of each vertex. The

body consists of 560 unique density zones, in comparison to 20 density zones of the octahedral

body. Furthermore, two inner layers are added, resulting in a total of 1680 density zones.

The results from case 1 can assist in making an informed choice of parameters for case 2. A

10

Figure 9:Case 1: Behavior of Algorithm Near Optimum

batch size of 2000 is arbitrarily selected as a balance between the predicted runtime and conver-

gence speed. Note, the batch size is greater than the number of density zones, resulting in an over-

constrained Amatrix at each iteration. As a result, it is less probable to select a batch permutation

that does not produce a reasonable gradient direction. In this case, ﬁfteen probes were simulated,

and are assumed to take a measurement every 10 seconds.

Base Conﬁguration: Analysis of theoretical observability, using the same technique as case 1,

produced a less intuitive output. The standard deviation for many zones was returned as NaN,

implying these areas cannot be observed at all. For the areas that returned a number, the average

value was 2.18×1014 g/cm3; however, the results of case 1 suggest that a reasonably accurate answer

can still be found. The NaN zones are illustrated by dark lines in Figure 10. Each layer of the

polyhedron is represented by a third of the x and y axes, with the outermost layer shown in the top

left of Figure 10. The concentration of dark lines in the bottom right implies that the inner layers

are less discernible than the outer layer.

Due to the increased size of the Amatrix, computations for case 2 are much slower than those for

case 1. Therefore, only 1000 iterations are shown in Figure 11. Comparing the graph to Figure 4,

it is clear that case 2 converges relatively slower, and is less consistent. Although the Amatrix is

overconstrained, as previously mentioned, the ratio of the batch size to the total number of mea-

surements is approximately 1:16. The low ratio implies higher turnover in each batch, resulting in a

noisier gradient and a smaller step size.

Behavior Near Answer: In Figure 12, 1000 iterations are not enough to conclusively observe a

point of convergence. The divergence from the true answer is slower than the cases in Figure 9, due

to the high batch turnover and small step size. The ﬂuctuations after the 900th iteration, marked by

the red circle, may indicate that the algorithm has settled near a point and will continue to oscillate,

but a longer simulation would be necessary to conﬁrm this possibility.

11

Figure 10:Case 2: Correlations (Dark Blue = Unobservable)

Figure 11:Case 2: Initial Behavior

12

Figure 12:Case 2: Final Behavior

Case 3: 2-Layer Ellipsoid

Case 3 consists of a 2-layer ellipsoidal body with a total of 3600 unique density zones. The

simulation uses 20 probes and a sample rate of 15 seconds.

Base Conﬁguration: As in case 2, many areas of the body appear unobservable from Park et

al.’s8method. No clear patterns emerge between the outer and inner layers in Figure 13. The

average deviation for the observable zones is 1.52×1015 g/cm3, only one order of magnitude larger

than case 2, leading to the expectation of similar trends in the recovery.

Figure 13:Case 3: Correlations (Dark Blue = Unobservable)

13

The behavior using the average density as the initial guess is illustrated in Figure 14. Although

the initial error is higher than case 2, the reduction in error after 1000 iterations is greater, and

the minimization is relatively more consistent. The consistency is a direct result of the higher

initial error. Closer to the answer, stochastic effects dominate, however, in the initial iterations, the

minimizing direction is clearly evident. Differences between the 20- and 25-probe simulations are

due to the inherent randomness of the procedure, such as noise sampling and batching.

Figure 14:Case 3: Initial Behavior

Behavior Near Answer: When the algorithm is close to the true answer, the noise in the system

is more apparent. No convergence is observed in Figure 15, and the erratic path of both cases

indicates that the gradient is inconsistent. The large jump in the 20 probe case between 600 and 700

Figure 15:Case 3: Final Behavior

iterations (Figure 15) is due to batching effects. In this regime, multiple batches were sequentially

selected that strongly pointed away from the true answer. If the scenario were recalculated, the

increase may take place more gently, or may not be present at all. Gradient inconsistencies are

14

partially caused by the ratio of batch size to the number of density zones in the body. The selected

batch size of 2000 is insufﬁcient to fully constrain a body with 3600 parameters, but any tangible

increase in batch size comes at the expense of the runtime. In future work, it may be beneﬁcial to

dedicate computing time towards obtaining accurate, high-resolution recoveries, but the aim of this

paper is to analyze trends of a prototype algorithm in simple cases.

CONCLUSION

Asteroids are of particular interest in the modern space industry due to their potential for ad-

vancing knowledge about the origins of the solar system. The techniques investigated in this paper

can support ﬂyby missions and target selection for asteroid mining. In the authors’ previous work,

using a different optimization algorithm, a clear trend emerged that suggested more probes would

give a more accurate result. The results from this paper suggest that the choice and tuning of the

optimization technique has a more signiﬁcant effect, especially in noisy conditions, than simply

increasing the number of probes. The most signiﬁcant conclusion from the above research is the

apparent disagreement between the theoretical observability of a body, and the performance of the

Adam algorithm. Even with noise applied to the dataset, Adam was able to consistently move from

a decent initial guess to a more accurate solution. Future work will investigate this disconnect and

attempt to develop techniques for more effective accuracy prediction.

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