# The Origin of Universality in the Inner Edges of Planetary Systems

Konstantin Batygin Division of Geological and Planetary Sciences, California
Institute of Technology, Pasadena, CA 91125  Fred C. Adams Physics Department,
University of Michigan, Ann Arbor, MI 48109  Astronomy Department, University
of Michigan, Ann Arbor, MI 48109  Juliette Becker Division of Geological and
Planetary Sciences, California Institute of Technology, Pasadena, CA 91125

###### Abstract

The characteristic orbital period of the inner-most objects within the
galactic census of planetary and satellite systems appears to be nearly
universal, with $P$ on the order of a few days. This paper presents a
theoretical framework that provides a simple explanation for this phenomenon.
By considering the interplay between disk accretion, magnetic field generation
by convective dynamos, and Kelvin-Helmholtz contraction, we derive an
expression for the magnetospheric truncation radius in astrophysical disks,
and find that the corresponding orbital frequency is independent of the mass
of the host body. Our analysis demonstrates that this characteristic frequency
corresponds to a period of $P\sim 3$ days, although intrinsic variations in
system parameters are expected to introduce a factor of $\sim 2-3$ spread in
this result. Standard theory of orbital migration further suggests that
planets should stabilize at an orbital period that exceeds disk truncation by
a small margin. Cumulatively, our findings predict that the periods of close-
in bodies should span $P\sim 2-12$ days – a range that is consistent with
observations.

Planet formation (1241), Exoplanet formation (492), Circumstellar disks (235),
Magnetohydrodynamics(1964)

##  1 Introduction

Planetary systems and satellites of gas giants are widely believed to form
within moderately massive circumstellar and circumplanetary disks, which are
composed primarily of Hydrogen and Helium. Although the exact mode of their
growth depends on the local nebular conditions (Safronov, 1972; Lissauer,
1993; Lambrechts & Johansen, 2012; Batygin & Morbidelli, 2023), the accretion
process – if fast enough – unavoidably triggers some degree of orbital
migration (Ward, 1997). After becoming massive enough to raise substantial
wakes within their parent nebulae, planets and satellites begin to exchange
energy and angular momentum with the surrounding gas (e.g., see Paardekooper
et al. 2022 for a recent review). This exchange is irreversible, and typically
results in orbital decay, which drives the forming objects towards the inner
edge of the disk. Near this edge, which is characterized by a sharp decline in
the nebular surface density, the migratory torques reverse sign so that inward
migration comes to a halt (Masset et al., 2006).

The inner cavities of astrophysical disks are carved out by the magnetospheres
of their central objects, such that the truncation radius is determined by a
balance between Lorentz torques and viscous torques, where the latter
facilitate gas accretion (Koenigl, 1991; Bouvier et al., 2007; Batygin, 2018).
As a result, the present-day orbits of short-period planets and satellites
must, to some extent, reflect this primordial equilibrium (e.g., Lee & Chiang
2017). Given that the masses, magnetic moments, and mass-accretion rates of
central objects can vary dramatically from system to system, it comes as no
surprise that the semi-major axes of the inner-most orbits within the galactic
census of planets and moons shows enormous variability. On the other hand, the
corresponding orbital periods display a striking pattern of uniformity, as
shown in Figure 1 (see also Figure A1). Specific examples of this pattern
include the orbital periods of Io, Trappist-1b, and Kepler-11b (as well as
many others), which are all measured to be several days, despite the fact that
the masses of their parent bodies span three orders of magnitude.

Beyond the cursory view of the data provided by Figure 1, the inner edges of
planetary systems have been quantified in a statistically rigorous manner by
Petigura et al. (2018); Mulders et al. (2018). For the Super-Earth population,
Petigura et al. (2018) find a knee in the planet occurrence distribution at a
period of $P\sim 6$ days, while Mulders et al. (2018) derive a distribution
for the inner-most orbital periods within the Kepler sample that exhibits a
flattened peak between 2 and 8 days, and drops off rapidly outside of these
bounds. In the same vein, it is worth noting that the knee in the occurrence
distribution of sub-Neptunes111Following the work of Fulton et al. (2017),
Petigura et al. (2018) define sub-Neptunes as objects with a radius greater
than $1.7\,R_{\oplus}$. ensues at a larger period (i.e., $P\sim 12$ days;
Petigura et al. 2018) than that of the Super-Earths. Though interesting, this
difference likely stems from instability of H/He envelopes to evaporation at
shorter orbital periods (Rogers & Owen, 2021), meaning that the characteristic
value corresponding to the Super-Earth population is likely to be more
representative of the primordial inner edges of planetary systems. At the same
time, we acknowledge that the dataset for planets with radii smaller than
Earth remains highly incomplete. Yet, we expect that the reversal of type-I
migration at the disk cavity would operate similarly for super-Earths and sub-
Terrestrial planets (Masset et al., 2006), and for this reason, we assume that
the orbital distribution of undetected planets is similar to that of the
observed ones.

The case of short-period giant planets is somewhat more complex. While a
power-law occurrence distribution does not provide an adequate match to the
data, it is important to understand that giant planets are expected to
appreciably modify the structure of the disk through gap-opening. Given the
complexity of type-II migration in 3D, it is not clear that Jovian-class
objects must systematically migrate all the way down to the magnetospheric
cavities, as in the case of lower-mass bodies that migrate in the type-I
regime (Lega et al., 2022). Nevertheless, previous work has shown that the
inner boundary of the hot Jupiter population can be fully explained by
magnetic disk truncation, and the characteristic $r\propto m^{2/7}$
relationship of the dividing line emerges self-consistently from consideration
of gas accretion and the appearance of a magnetospheric cavity within the
nebula (Bailey & Batygin, 2018). An additional complication, however, lies in
that an unknown fraction of the hot Jupiter population migrated inward through
the high-eccentricity channel, and the $r\sim m^{1/3}$ relationship expected
within the context of this picture is also consistent with the data (Matsakos
& Königl, 2016; Owen & Lai, 2018).

Finally, within the context of planetary satellites, our analysis is limited
to large natural satellites of giant planets that formed within primordial
circumplanetary disks. Moons that originated from giant impacts, capture, or
from rings are beyond the scope of this work222Our work therefore excludes the
large satellites of Uranus and Neptune — which likely formed through a giant
impact (Ida et al., 2020) and capture (Agnor & Hamilton, 2006), respectively..
Correspondingly, the minuscule Jovian moons such as Metis, Adrastea, Amalthea,
and Thebe represent a separate category of objects when compared to the
massive Galilean moons. Similarly, there is no reason to suspect that the
small moons in the Saturnian system — which likely formed from the rings and
experienced tidal migration outwards (Crida & Charnoz, 2013) — should adhere
to the magnetospheric edge of Saturn’s circumplanetary disk, since they
emerged after the dissipation of the gas. As for Titan, it is worth noting
that while its current orbital period is approximately two weeks, recent work
by Lainey et al. (2020) convincingly demonstrates that Titan has undergone
rapid orbital expansion, with evidence indicating that its primordial period
was on the order of a few days.

The principal goal of this letter is to explain this near-universality of the
characteristic orbital period associated with magnetospheric disk truncation.
Toward this end, we construct an analytic framework that draws upon
conventional results from astrophysical dynamo theory, stellar structure, and
disk accretion. The relevant physical processes are summarized in Figure 2.
Taken together, these considerations reveal how planetary architectures are
assembled within their natal nebulae, and shed light on the fundamental role
played by magnetospheric cavities in shaping the outlines of the inner
boundaries of the resulting planetary systems.

Figure 1: Census of well-characterized extrasolar planets (circles) and moons
of gas (squares) and ice (triangles) giants of the solar system. The mass
ratio of the planet or satellite to the central body ($m/M$) is shown as a
function of the orbital period. The points are color-coded by the spectral
type of the host (with P denoting a planetary central body). The top panel
shows a histogram of the orbital period of the innermost planet or satellite
in each system, similarly color-coded by the spectral type of the host star. A
peak in the $P\sim 2-12$ day range is evident within the distribution. A
complementary view of the orbital distribution is provided in Figure A1.

##  2 Analytic Theory

As a starting point for our analysis, let us recall the standard expression
for magnetospheric truncation of an astrophysical disk. Though formally
incorrect, this condition is routinely derived by equating the magnetic
pressure, $B^{2}/(2\,\mu_{0})$, to the accretionary ram pressure,
$\rho\,v_{r}^{2}$ (e.g., Mohanty & Shu 2008), with the result

$\displaystyle\frac{B^{2}}{2\,\mu_{0}}\sim\frac{\dot{M}}{4\,\pi\,r^{2}}\sqrt{\frac{2\,\mathcal{G}\,M_{\star}}{r}},$
(1)

where we have written the mass-accretion rate of the disk as
$\dot{M}=4\pi\,r^{2}\,\rho\,v_{r}$, appropriate for spherical infall. Assuming
a disk-aligned dipole field for the star $B=B_{\star}\,(R_{\star}/r)^{3}$,
this expression can be solved for the orbital radius $r$ (e.g., Ghosh & Lamb
1978; Blandford & Payne 1982). In a slight variation on this theme, here we
write the truncation condition not in terms of the orbital radius, but rather
in terms of the orbital frequency,

$\displaystyle\Omega=\xi\,\bigg{(}\frac{\mu_{0}^{3}\,\mathcal{G}^{5}\,M_{\star}^{5}\,\dot{M}^{3}}{2\,\sqrt{2}\,\pi^{3}\,B_{\star}^{6}\,R_{\star}^{18}}\bigg{)}^{1/7},$
(2)

where $\xi$ is a proportionality constant of order unity333The exact value of
this constant depends on the field geometry. As an example, detailed dipole
calculations of Najita & Shu (1994) yield $\xi\approx 1.13$.. In the following
discussion, we examine how each of the physical quantities inside the
parenthesis depend on the mass of the central object (and other properties of
the system).

### Mass-Accretion Rate

At the detailed level, nebular accretion is a complex and episodic process
that is facilitated by turbulent viscosity and magnetized disk winds (e.g.,
see Armitage 2020, 2022 and references therein). Nevertheless, the overall
magnitude of $\dot{M}$ is empirically known to increase with the mass of the
central object and to diminish with time (Hillenbrand et al., 1992; Hartmann
et al., 1998; Herczeg & Hillenbrand, 2008). Moreover, gravitational stability
regulates the ratio $\beta$ of the initial disk mass to the central object
mass to be of order 1/10 (Shu et al., 1990), i.e., comparable to its geometric
aspect ratio so that $\beta\sim 0.1\sim h/r$. The simplest parameterization
that we can adopt for the accretion rate thus takes the form

$\displaystyle\dot{M}\sim\beta\,\frac{M_{\star}}{\tau},$ (3)

where $\tau$ is the age of the system. Although highly approximate, this
prescription is sufficient for our purposes and captures the necessary
features of the accretion process.

### Dynamo Generation

To quantify magnetic field generation in young stars and planets, we adopt the
Christensen et al. (2009) scaling law, which connects the surface field
strength to the intrinsic luminosity of the object. As a physical model, this
relation is appropriate for rapidly rotating, fully convective astrophysical
bodies, and effectively relates the magnetic energy density to the convective
(kinetic) energy density, which in turn determines the heat flux,
$q=\sigma\,T_{\rm{eff}}^{4}$. The result can be expressed in the form

$\displaystyle\frac{\langle
B\rangle^{2}}{2\,\mu_{0}}=c\,f_{\rm{ohm}}\,\langle\rho\rangle^{1/3}\,(\mathcal{F}\,q)^{2/3},$
(4)

where $c\approx 0.63$ is a constant of proportionality, $f_{\rm{ohm}}$ is the
ratio of Ohmic to total dissipation, and $\mathcal{F}$ is a numerical factor
that encapsulates the radial dependence of the flux, density, as well as the
ratio of size of the largest convection cell to the temperature scale-height.

For young stars, Christensen et al. (2009) estimate that $f_{\rm{ohm}}\approx
0.5$ constitutes a reasonable lower bound, and demonstrate that
$\mathcal{F}\sim 1$ for a broad range of bodies, including Jupiter and fully
convective M and K dwarfs444In particular, Christensen et al. (2009) obtain
$\mathcal{F}=0.69-1.22$ for stars with masses in the range of
$M_{\star}=0.25-0.7\,M_{\odot}$ and calculate that $\mathcal{F}=1.19$ for
present-day Jupiter.. In addition, Christensen et al. (2009) find that the
surface field is smaller than the average interior field by a factor of
$\gamma=\langle B\rangle/B_{\star}\approx 3.5$. We adopt these fiducial values
here for the sake of definiteness.

### Gravitational Contraction

Young stars and giant planets contract by radiating away gravitational energy.
This contraction occurs on the Kelvin-Helmholtz timescale, which can be
determined by equating the bolometric luminosity, $4\,\pi\,R_{\star}^{2}\,q$,
to the energy loss rate,
$b\,\dot{R_{\star}}\,\mathcal{G}\,M_{\star}^{2}/R_{\star}^{2}$. Integrating
the resulting relation in time, we obtain the well-known result

$\displaystyle
R_{\star}\approx\bigg{(}\frac{b\,G\,M_{\star}^{2}}{12\,\pi\,q\,\tau}\bigg{)}^{1/3},$
(5)

where we have dropped the surface term (which depends only weakly on initial
conditions). The dimensionless constant, $b$, is determined by the interior
structure of the central body. Here we take $b=3/7$, which corresponds to a
$n=3/2$ (fully convective) polytrope (Chandrasekhar, 1939).

Figure 2: A schematic diagram of the physical processes considered in this
work. A central body of mass $M_{\star}$ is encircled by a disk, which
accretes at a rate $\dot{M}$. Convective energy transport within the body
drives gravitational contraction of its radius, $R_{\star}$, as well as dynamo
action, which generates a field, $B_{\star}$, that is regulated by the
bolometric flux, $q$. Magnetospheric interactions between the central body and
the disk generate a cavity with an edge that corresponds to an orbital
frequency, $\Omega$.

A final specification that remains to be made is the mass-radius relationship
of the central body. Although no exact analytic results apply over the full
range of masses considered here, detailed pre-main sequence calculations of
Baraffe et al. (2015) indicate that the radius scales with mass according to
the approximate relation $R_{\star}\propto M_{\star}^{1/3}$, which corresponds
to constant mean density. This effective mass-radius relation provides an
adequate match to the evolutionary models at early times (as an example, for
time $t=2\,$Myr the corresponding isochrones are best fit by a power-law of
the form $R_{\star}\propto M_{\star}^{0.37}$). In fact, this result is
consistent with expectations: the often-quoted “canonical” radius of a solar-
mass T-Tauri star is $R_{\star}\sim 2\,R_{\odot}$, while core-accretion
calculations give $R_{\star}\sim 2\,R_{\jupiter}$ as an estimate for Jupiter’s
radius during the nebular epoch (e.g., Lissauer et al. 2009; Berardo & Cumming
2017). Both of these values translate to a mean density of
$\langle\rho\rangle\approx 0.15\,$g/cm3, which we adopt as a fiducial
estimate. Combining this mass-radius relation with equation (5), we obtain a
linear scaling of the flux on the mass and inverse dependence on the system
age, i.e., $q\sim(b/9)\,\mathcal{G}\,M_{\star}\,\langle\rho\rangle/\tau$.

### Result

Substituting the relationships derived above into equation (2), we obtain an
expression for the magnetospheric truncation frequency of astrophysical disks,

$\displaystyle\Omega$
$\displaystyle=2\,\xi\,\Bigg{[}\frac{\sqrt{2}}{\big{(}3\,b\,\mathcal{F}\big{)}^{2}}\bigg{(}\frac{\pi\,\beta\,\gamma^{2}}{c\,f_{\rm{ohm}}}\bigg{)}^{3}\frac{\big{(}\mathcal{G}\,\langle\rho\rangle\big{)}^{3}}{\tau}\Bigg{]}^{1/7}$
$\displaystyle\approx 2.4\times
10^{-5}\,\mathrm{s}^{-1}\approx\frac{2\,\pi}{3\,\rm{day}}.$ (6)

This expression provides the key result of the paper. Our analysis shows that
basic considerations of gravitational stability of astrophysical disks,
convective dynamo generation, and Kelvin-Helmholtz contraction translate to a
disk truncation orbital frequency that is largely independent of mass.
Moreover, evaluation of this result at time $\tau\sim 3\,$Myr – corresponding
to the typical disk lifetime – yields a value of the orbital period
$2\pi/\Omega\approx 3$ days.

Taking into account the standard results of planetary migration theory, which
predict that the equilibrium planetary orbital radius will exceed the disk
truncation radius by a modest factor of $\sim 1.3$ (e.g., Ataiee & Kley 2021),
and considering our adoption of the lower-bound value of $f_{\rm{ohm}}=0.5$ in
our fiducial analysis, we expect the planetary period to be slightly greater.
As a result, it is reasonable to anticipate that the planetary period will
surpass the disk truncation period by a factor of approximately $1.5-2$.
Furthermore, intrinsic variation in system parameters introduce a spread in
the orbital periods, resulting in a range spanning a factor of approximately 2
to 3, as discussed in the Appendix. This factor of $\sim 3$ variance is
roughly consistent with the observed distributions of inner periods, as shown
in Figure 1 and in Figure A1 (which shows the distributions separately for
each spectral type of the host). Equation (6) thus provides a theoretical
explanation for why the inner-most objects within typical extrasolar planetary
systems as well as systems of giant planet satellites orbit at a period of
approximately $P\sim 2-12$ days, and why this quantity is essentially
independent of the mass of the central body.

Though one could consider distinct parameterizations from those adopted here,
it is important to note that the characteristic orbital period varies
extremely slowly as a function of both time and the physical properties of the
central object. That is, at any given time, the benchmark period scales
according to $\Omega\propto\langle\rho\rangle^{3/7}$, so that moderate
departures from the assumed mass-radius relation ($R_{\star}\propto
M_{\star}^{1/3}$) result in only weak dependence on the host mass. In
addition, as systems age, the mean density is expected to increase, but this
dependence is partially compensated by the increase in time scale $\tau$ (and
the entire factor in square brackets is taken to the 1/7 power). As a result,
while intrinsic variation of parameters is expected to introduce some degree
of variability in the period from system to system, under the assumptions that
the relevant variables are independent from each other, and the fluctuations
in the individual variables themselves are weak functions of the mass of the
central body, the predicted boundary range of the orbital period should remain
robustly mass-independent.

##  3 Discussion

The preservation of dimensionless numbers across a broad range of scales is a
common phenomenon in physical systems (Buckingham, 1914). The near-
preservation of a dimensional number, such as a frequency, is a much rarer
occurrence. The theoretical analysis outlined in this work demonstrates that
the characteristic orbital period associated with magnetospheric disk
truncation - and by extension, planetary orbits - systematically evaluates to
a few days, over a wide range of the host object’s mass. Our result explains
why the innermost orbital periods of planets around FGKM stars, and
potentially that of large natural satellites of Jupiter and Saturn, generally
fall within $\sim 2-12$ days, suggesting nearly universal behavior over a wide
range of host masses. One intriguing implication of this result is that
despite their striking diversity, the terminal assembly of orbital
architectures of planetary and satellites systems is governed by the same
fundamental physical mechanism.

While our analysis provides a theoretical explanation for the characteristic
orbital period of the inner-most objects in planetary and satellite systems,
it is crucial to recognize the limitations of our current sample size.
Presently, our knowledge of planetary systems primarily revolves around host
stars with masses ranging from $M_{\star}\sim 0.1-1M_{\odot}$ (Thompson et
al., 2018), and our assessment of relevant satellite systems is constrained to
Jupiter and Saturn. As the census of exo-moons is expected to expand in the
coming years (Limbach et al., 2023; Ruffio et al., 2023), our calculations
robustly predict a characteristic orbital period for the innermost objects
within these systems. The largely uncharted brown-dwarf territory, which spans
masses in the range of $0.01-0.1M_{\odot}$, presents an important additional
opportunity for further exploration: our model predicts that if planetary
systems that encircle brown dwarves exist, they should exhibit a similar
characteristic orbital period of a few days, mirroring the trends observed in
their lower- and higher-mass counterparts. Future observations of such systems
will provide a valuable test of our predictions and improve our understanding
of the processes that govern the formation and evolution of planetary systems.

As important as the data itself, it is crucial to consider the limitation of
our model’s range of applicability, which manifests primarily as bounds on the
central body’s mass. At the lower-mass end of the spectrum, our model is
unlikely to be applicable to sub-Jovian objects, as large-scale numerical
simulations (e.g. Lambrechts et al., 2019) suggest that Neptune-class planets
do not form gaseous circumplanetary disks during the nebular epoch.
Conversely, at the higher-mass end, our model is expected to falter for stars
with masses exceeding $M_{\star}\gtrsim 1.5-2M_{\odot}$. This limitation
arises because these stars develop radiative cores while the gaseous disk
persists (Baraffe et al., 2015), undermining the assumption of a fully
convective interior that leads to a dipolar magnetic field throughout the
T-Tauri phase555We remark that in principle, Jovian-class objects can also
develop radiative regions within their interiors at early epochs (Berardo &
Cumming, 2017), potentially introducing similar complexity into the planetary
sample.. Toward this end, Gregory et al. (2012) have already shown that the
magnetic fields of $M_{\star}<1.5\ M_{\odot}$ stars remain dipolar for
millions of years, while the fields of $M_{\star}>2\ M_{\odot}$ stars
transition to a more complex multipolar structure over the same timescale,
leading to somewhat smaller truncation periods. Although the current census of
sub-Jovian planets around massive stars is too limited to provide a
comprehensive test of this prediction, we may reasonably expect the derived
period uniformity to vanish for sufficiently massive stars.

We conclude by acknowledging that, as with any analytic theory, our results
are imperfect. As already mentioned above, intrinsic variations in disk
accretion rates, dynamo efficiency, and uncertainties in our description of
the scaling between mass and radius, as well as bolometric flux, inevitably
introduce some degree of scatter into our result. It is further worth noting
that the scenario proposed in our work is only relevant for planets and
satellites with enough mass to undergo significant orbital migration during
the lifetime of their birth nebulae. Additionally, post-nebular processes such
as tidal migration and instabilities in multi-planet systems (see Goldberg &
Batygin 2021, 2022 and the references therein) may slightly modify our
predictions. Nonetheless, our framework offers a qualitative foundation for
unveiling the universal role played by magnetospheric cavities in shaping the
outlines of the galactic planetary census. Future observations and refinements
of our theoretical framework will deepen this understanding.

We are thankful to Erik Petigura, Ravit Helled, and Andrew Howard for
insightful discussions. We thank the anonymous referees for providing careful
and insightful reviews of the manuscript. K.B. is grateful to Caltech, the
David and Lucile Packard Foundation, and the National Science Foundation
(grant number: AST 2109276) for their generous support. F.C.A. is supported in
part by the University of Michigan and the Leinweber Center for Theoretical
Physics. J.C.B. is grateful to Caltech and the Heising-Simons foundation for
their support through the 51 Pegasi b fellowship. This Appendix considers the
distribution of orbital periods for the innermost planets, as set by magnetic
truncation. Equation (6) in the text gives the orbital frequency corresponding
to the truncation radius, and shows that the period is about 3 days for
typical values of the input variables. To estimate the distribution of values,
we note that the expression for the period can be written as the product of
variables. These variables (the field geometry factor $\xi$, the binding
energy factor $b$, and so on) can vary from system to system, sampling some
probability distribution, thereby resulting in a range of inner periods. In
the limit where the number of variables is large, the distribution of the
composite variable (the period) would approach a log-normal form. Even with a
finite number of variables, however, we can estimate the width of the
composite distribution with some degree of certainty (see Adams & Fatuzzo 1996
for an analogous calculation). We start by writing the expression for the
period in the form

$P=P_{0}\xi^{-1}\xi_{m}^{-3/7}\xi_{\rho}^{-3/7}\xi_{b}^{2/7}\,,$ (1)

where $P_{0}$ is the fiducial value of about 3 days, and the variable $\xi$ is
the same as before. We have also defined

$\xi_{\rho}={\langle\rho\rangle\over\langle\rho\rangle_{0}}\qquad{\rm
and}\qquad\xi_{b}={b\over b_{0}}$ (2)

and we have collected all of the magnetic parameters into a single
dimensionless quantity

$\xi_{m}={\beta\over\beta_{0}}{\gamma^{2}\over\gamma_{0}^{2}}\left({c\over
c_{0}}{f_{\rm{ohm}}\over f_{\rm{ohm\,}0}}\right)^{-1}\left({{\cal F}\over{\cal
F}_{0}}\right)^{-2/3}\,.$ (3)

Note that we could in principle include each individual variable. However, as
discussed below, the distributions are not known, so that is it simpler to use
the composite ‘magnetic variable’ $\xi_{m}$ defined here. Moreover, the input
variables that make up this composite variable might not be fully independent.
Taking the logarithm of equation (1) we find

$\log(P/P_{0})=\sum_{k}p_{k}\log\xi_{k}\,,$ (4)

where the index runs from 1 through 4 and the exponents $p_{k}$ are those
given in original equation (1).

Figure A1: Distribution of innermost orbital periods, sorted by host type.
Unlike in Figure 1, here we do not restrict the data-set to systems with
measured masses, yielding a somewhat distinct view of the galactic planetary
census. The orbital periods of Jupiter’s moon Io and Saturn’s moon Titan are
also shown for comparison. It is important to note that both bodies have
experienced some degree of outward tidal migration. In the case of Titan, the
work of Lainey et al. (2020) suggests that long-range tidal migration might
have occurred. The distributions for M, K, G, and F-type hosts appear quite
similar. For A and B-type hosts, there is a slight preference for lower
periods. However, it’s worth noting the planetary sample for these types of
stars is different: while the fractions of planets with radii smaller than 4
Earth radii are 56%, 66%, 69% and 45% for M, K, G, and F-type stars,
respectively, it is 14% and 0% for A and B-type stars, illustrating the sample
disparity.

Each random variable $\xi$ has the form $\xi=x/x_{0}$. As a result, the mean
of the logarithm of each variable is given by

$\langle\log\xi\rangle=\langle\log x\rangle-\log x_{0}\,,$ (5)

where angular brackets denote taking the average over the distribution of the
variable. If we take the fiducial values $x_{0}$ of the physical variables to
be defined such that $\log x_{0}$ = $\langle\log{x}\rangle$, then the
logarithm of the random variables $\xi$ will have zero mean
$\langle\log\xi\rangle$ = 0. Keep in mind that $x_{0}$ is not the mean value
itself, but rather $x_{0}$ = $\exp[\langle\log{x}\rangle]$
$\neq\langle{x}\rangle$.

If we then assume that the variables are independent, the variance of the
quantity $\log(P/P_{0})$, which is a sum of random variables, is given by the
sum

$\sigma_{T}^{2}=\sum_{k}p_{k}^{2}\sigma_{k}^{2}\,,$ (6)

where $\sigma_{k}^{2}$ is the variance of the variable $\log\xi_{k}$.

To fix ideas, consider the case where each physical variable $x_{k}$ can vary
by a factor of $\Lambda_{k}$ from its fiducial value. In the simplest case,
the corresponding variable $\log\xi_{k}$ will thus have a uniform distribution
over the range $[-\log\Lambda_{k},\log\Lambda_{k}]$. The variance of the
individual distribution is given by

$\sigma_{k}^{2}={2\over 3}(\log\Lambda_{k})^{2}\,.$ (7)

The total variance of the period distribution, the composite variable
$\log(P/P_{0})$ then has the form

$\sigma_{T}^{2}={2\over 3}\sum_{k}p_{k}^{2}(\log\Lambda_{k})^{2}\,.$ (8)

Although we do not know the distributions of the variables in question, they
are expected to vary by modest factors. For the sake of definiteness, let’s
assume that $\Lambda_{k}$ have the same value (denoted now as just $\Lambda$)
for all variables/indices $k$. Our expression becomes

$\sigma_{T}^{2}={2\over 3}(\log\Lambda)^{2}\sum_{k}p_{k}^{2}={142\over
147}(\log\Lambda)^{2}\approx(\log\Lambda)^{2}\,.$ (9)

Recall also that $\sigma_{T}^{2}$ is the variance in $\log(P/P_{0})$. As a
result, if each individual variable $x_{k}$ can vary by a factor of $\Lambda$,
then the period will vary by a factor of $\sim\Lambda$ also. The increase in
variance that arises from the multiple random variables is offset by the small
exponents. Since the individual variables are expected to vary by factors of 2
to 3, the predicted period should lie in a range of about 2 to 12 days.
Although not negligible, this factor of $\sim 3$ is small compared to the
range of masses (a factor of $\sim 1000$) for the central bodies.

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# A review of the security role of
ISP mandated ONUs and ONTs in GPONs

Max Franke  INET
Technische Universität Berlin
Berlin, Germany  Sebastian Neef  SecT
Technische Universität Berlin
Berlin, Germany

###### Abstract

Home fiber connections are largely realized by using passive optical networks,
in their most common form today relying on the GPON standard. Among other
things, this standard specifies how the first node inside of customers’ homes,
the so called ONU or ONT, has to behave, and which security features have to
be supported. Currently, customers in some European countries, including
Germany, have freedom of choice between using terminal equipment provided by
the ISP or a self-selected open market device. We analyze the security
implications resulting from this freedom of choice and whether or not ISP-
mandated hardware would increase the security of the GPON. Our review reveals
that there are no differences between an ISP-mandated ONU/ONT and a standard
conforming subscriber-selected ONU/ONT that would justify the security based
recommendation of an ISP-mandated ONU/ONT.

##  I Introduction

With the ever-increasing demand for higher bandwidth and the consequent
expansion of fiber networks by Internet Service Providers (ISPs), like Fiber-
to-the-Home (FTTH), reaching more and more customers, it becomes crucial to
focus on the technologies and devices that facilitate this progress. The
prevailing standard for such connections is the gigabit passive optical
network (GPON), which was initially standardized by the ITU-T in 2003. The
market penetration of FTTH utilizing GPON can reach as high as 97% in certain
markets, such as the UAE. However, the adoption has been comparatively low in
many European countries, with Germany for instance at around 8% in June
2022[1], but targeted to increase to 50% by 2025 [2].

The GPON standards encompass numerous security measures and considerations for
both the optical line terminal (OLT), responsible for terminating the
connection at the provider’s end, and the optical network unit (ONU, also
”optical network termination”– ONT) at the customer’s end. These standards are
mandatory for all devices, regardless of whether they are provided by ISPs or
third-party equipment vendors. The permissibility of operating third-party
devices depends on the legislation in each country. In Germany, all network
subscribers are granted by law111See e.g.
https://www.bmwk.de/Redaktion/DE/Artikel/Digitale-Welt/freie-routerwahl.html
the freedom to choose their own terminal device. This has been applied to
fixed and mobile access networks in general, including DSL, DOCSIS, PON fiber,
and 4G/5G Fixed Wireless Access.

In most cases, the PON terminal equipment combines the functionalities of both
a router and a modem, resulting e.g. in space and power savings at the users’
premises. When such an integrated device is used as the terminal device, it
serves the function of an ONU/ONT for the subscriber in the fiber network. It
is also possible to split the functionalities and use two separate devices, a
modem and a router. In this case, the modem serves the function of an ONU/ONT.

As such, a vendor of terminal equipment approached us to investigate the
question whether an ISP-mandated ONU/ONT offers any security advantages
compared to a standard conforming open market ONU/ONT. Our review considers
any security differences affecting the subscriber itself, the ISP’s network,
and the security of other subscribers on the same PON.

##  II Background

###  II-A ITU-T Standards

The GPON standard was initially standardized in 2003 as G.984. It consists of
a multitude of documents [3, 4, 5, 6, 7, 8, 9], each covering specific aspects
of GPON. The details for the information exchange between an OLT and an ONU
are described in G.988, the ONU management and control interface (OMCI)
specification [10]. Closely related is ITU-T G.9807.1 (XGS-PON), which
increases the bandwidth of GPONs to up to 10 GBit up- and downstream [11].
However, this work does not consider XGS-PON.

###  II-B Definition of terms: ONU and ONT

The definitions of ONU and ONT in [3] are as follows:

> optical network termination (ONT): A single subscriber device that
> terminates any one of the distributed (leaf) endpoints of an ODN, implements
> a PON protocol, and adapts PON PDUs to subscriber service interfaces. An ONT
> is a special case of an ONU.

and

> optical network unit (ONU): A generic term denoting a device that terminates
> any one of the distributed (leaf) endpoints of an ODN, implements a PON
> protocol, and adapts PON PDUs to subscriber service interfaces. In some
> contexts, an ONU implies a multiple-subscriber device.

[3] makes no further differentiations when it comes to ONU and ONT and often
uses them either interchangeably or both at the same time. This is also true
in the security section:

> Security Due to the multicast nature of the PON, GPON needs a security
> mechanism adapting the following requirements:
>
>   * •
>
> To prevent other users from easily decoding the downstream data.
>
>   * •
>
> To prevent other users from masquerading as another ONU/ONT or user.
>
>   * •
>
> To allow cost-effective implementation.
>
>

As the ONT is just a special case of ONU, none of the other specifications
make any further differentiation when it comes to security considerations. In
fact the specification in which most of the security related aspects are
included, [10], makes no references to ONTs at all.

###  II-C Related Work

Twenty years after the initial standardization, [12] highlighted security
problems in and proposed enhancements for GPON.

From a practical perspective, there have been several reports of security
issues in GPON setups or capable devices [13, 14, 15].

There have also been works that review security flaws in other
telecommunication standards. E.g., [16] demonstrated an attack on DOCSIS over
ten years ago. With GPON, however, a discussion arose about the point of
network termination. One could argue that the network termination happens at
the fiber in the wall, while others might argue that the GPON modem is part of
the network and thus the network termination follows after it.

Network security can be one of the deciding factors in this debate. It is
therefore necessary to establish whether the use of an ISP-mandated device
instead of a subscriber-selected standard conforming device is crucial to
ensure the security of the PON.

##  III Research Questions and Methodology

We evaluate the security of GPON concerning the two scenarios illustrated in
Figure 1 and Figure 2 .

In scenario 1, the ISP’s network ends before the ONU/ONT, and the ONU/ONT can
be selected by the subscriber.

Figure 1: Scenario 1 with subscriber-selected ONU/ONT. The ISP’s network
includes the fiber, but excludes the subscriber’s ONU/ONT.

In scenario 2, the ISP’s network ends after the ONU/ONT, and the ONU/ONT is
mandated by the ISP.

Figure 2: Scenario 2 with ISP-mandated ONU/ONT. The ISP’s network includes the
fiber as well as the subscriber’s ONU/ONT.

###  III-A Research questions

Our review considers the following research questions:

  1. RQ1

Assuming ITU-T conforming ONUs/ONTs, is the security level of a PON in
scenario 1 (subscriber-selected ONU/ONT) at least as high as in scenario 2
(ISP-mandated ONU/ONT) according to the ITU-T standards?

  2. RQ2

According to the ITU-T standards, can an ISP’s PON network equipment enforce a
minimum level of security, so that an ONU/ONT is not unilaterally able to
reduce this level?

  3. RQ3

Are ONUs/ONTs, and thus the subscriber, assigned the task of maintaining
network security?

  4. RQ4

Does the ONU/ONT have to be part of the ISP’s network to guarantee security?

  5. RQ5

Are all the security features the standard offers required to be used?

###  III-B Threat model

The vendor suggested the following boundary conditions as scope for the
research threat model, matching the strong focus on the ITU-T standards:

  * •

All devices are standard-compliant and properly integrated into the PON.

  * •

Scenarios of malicious attackers with physical access to the fiber or to the
intermediary network equipment, i.e. the passive optical components in the
basement or on the street, are excluded. Such scenarios do not imply any
differences between ISP-mandated and subscriber-selected devices and are
therefore not relevant for our review.

###  III-C Methodology

After a literature review of existing academic and non-academic work, our
methodology consists of going through the relevant parts of the GPON ITU-T
standards with a focus on aspects relevant to our security considerations.
Additionally, the vendor provided us a trace of a GPON connection
initialization between an ONU/ONT and OLT.

##  IV Results

With the given threat model and methodology, our analysis results in the
following answers to our research questions:

###  IV-A RQ1

Yes, the security levels are identical for scenario 1 and scenario 2, assuming
ITU-T conforming devices. Our review does not show any differences in the
standards regarding the two scenarios. The standards do not differentiate
between both scenarios, neither in general nor with respect to security. Even
the term ”Network Termination” (NT) that is used within the standards does not
imply an association of a terminal device to an ISP’s network. The ITU itself
states [17]:

> The NT term is used for generic network termination for various services.
> For some services it could be part of the access network, and for others
> not. The inclusion of the NT in the access network and vice versa does not
> necessarily imply the ownership.
>
> […]
>
> The reference configurations in this clause show abstract functional
> groupings, which may or may not correspond to real devices. Real devices may
> comprise one abstract functional grouping, more than one abstract functional
> grouping or a portion of an abstract functional grouping.

In practice, all devices in a PON will need to support the required security
features and properties, e.g. cryptographic algorithms. These can be
implementation specific and vary from PON to PON. Nonetheless, it does neither
weaken nor impact scenario 1, as conforming subscriber-selected devices will
offer the required functionality and only need to provide a suitable
configuration interface.

Therefore we do not see any justification from the standards to propose a
security advantage arising from ISP-mandated ONUs/ONTs over conforming
subscriber-selected ONUs/ONTs.

###  IV-B RQ2

Yes, an ISP’s PON network equipment can enforce a minimum level of security,
so that an ONU/ONT is not unilaterally able to reduce this level. The desired
security properties are exchanged and negotiated between the OLT and ONU/ONT.
In practice, an ONU/ONT will not be able to reach the state ”O5-Operational”
if the OLT does not approve – e.g., because the OLT’s requests to choose a
session key are not answered properly by the ONU/ONT.

Thus, it is left to the implementation of the OLT to require and enforce
acceptable security properties.

###  IV-C RQ3

In GPON, the downstream is received by all ONUs/ONTs sharing the same fiber to
an OLT. The standards recommend the use of encryption to ensure only the
legitimate ONU/ONT can decipher its own downstream traffic. In this regard,
the ONUs/ONTs are a contributing factor to the security for its own
downstream, but it has no influence on the security of the network or other
subscribers of the PON.

Regardless of the considered scenario, the subscriber has to configure the
ISP-mandated or subscriber-selected ONU/ONT with specific credentials.
However, this is not a new task or role, as the subscriber already needs to
provide the ISP with ONU/ONT-specific information in current GPON deployments,
or needs to configure the router with correct credentials in existing DSL
deployments. Obviously, it is in a subscriber’s best interest to correctly
configure the device, as misconfigured devices might not provide the desired
internet connectivity. No security differences between scenario 1 and 2 can be
deduced here.

###  IV-D RQ4

As the location of the ONU/ONT is inside subscribers’ homes and easily
accessible by them, they can not be required to be part of the ISP’s network
to guarantee security. A malicious actor could simply remove the modem and
attach any device to the fiber, thus breaking security. As such, security has
to be ensured and enforced by the OLT. An ONU/ONT with a non-conforming
implementation or security flaws in the implementation would be far from the
worst-case that could happen in such scenarios.

Thus, the position of a standard conforming ONU/ONT – inside or outside the
ISP’s network – has no effect on the rest of the network. A standard
conforming implementation, including all optional security controls (see RQ5),
provides the same security guarantees to the network, disregarding the origin
of the device.

###  IV-E RQ5

The standard [10] includes optional so called ”Enhanced security control” that
can be negotiated. Among those additional security features is a Pre-Shared
Key (PSK) that would be used in e.g. different hash functions. The example
trace of a GPON connection initialization did not show the use of a PSK. From
discussions with the equipment vendor, these enhanced security features are
typically not activated or used by actual PON network operators.

There are several other optional security features the standard allows for. As
such, to further enhance the security of the GPON, ISPs should work on
implementing and utilizing these additional capabilities – and third-party
vendors support them. Additionally, newer standards like XGS-PON include
features like upstream encryption that further reduce the role the ONU/ONT
takes in guaranteeing security for the network.

###  IV-F Limitations

The defined threat model imposed strong limits on possible attack scenarios
against a PON or its users, which may arise from (physical) access to the
fiber, including non-standard behavior or messages on the protocol level.

In reality, physical access can hardly be prevented. For example, in a FTTH
setup, the fiber terminates at _home_ in a ONU/ONT. For FTTB/C and FTTCab,
reports of optical network equipment present in customer or publicly
accessible places (i.e. basements of multi-story building) exist [13]. As a
consequence, such attack scenarios against PONs need to be considered and
mitigated at the OLT level, as the resulting security risks are not due to a
specific type of ONU/ONT being used.

##  V Conclusion

The analysis of the security role of ONUs/ONTs in GPON shows that the
standards do not differentiate between ISP-mandated and subscriber-selected
devices. Differences between both scenarios with regards to security or the
security capabilitites can not be deduced.

Assuming standard conforming devices, the security of PONs is not influenced
by the use of ISP-mandated or subscriber-selected devices, as long as the ISPs
provide the necessary configuration information (i.e. credentials, PSKs, etc.)
and third-party devices provide a suitable configuration interface. The
minimal requirements for the networks will need to be enforced by the OLT
during the exchange of capabilities. The OLT can even ensure and enforce a
certain security level for the individual subscriber. There is no difference
in security for the network if the ONU/ONT is part of the ISP network or not.

The individual implementation of the standard on any device is specific to
this device. It is therefore not possible to make any general statements on
security differences between ISP-mandated and subscriber-selected ONU/ONTs.
And even if there are potential security vulnerabilities within a device-
specific implementation, this cannot compromise the security of other
subscribers of the PON.

In reality, malicious attackers usually aim to threaten the security of the
network itself. Security threats against GPONs are therefore far more likely
to arise from rogue, non-standard conforming devices or physical access to the
fiber. Such malicious attack scenarios, which were discussed in previous work,
are agnostic to the chosen device and the association to the ISP’s network.

## References

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# Design of Stickbug: a Six-Armed Precision Pollination Robot
††thanks: 1The authors are with the Department of Mechanical and Aerospace
Engineering, West Virginia University, Morgantown, WV 26505, USA ††thanks:
2Trevor Smith is the corresponding author<EMAIL_ADDRESS>This
study was supported in part by USDA NIFA Award 2022-67021-36124 ”Collaborative
Research: NRI: StickBug - an Effective Co-Robot for Precision Pollination,”
the National Science Foundation Graduate Research Fellowship Award #2136524,
and by the NASA West Virginia Space Grant Consortium, Grant #80NSSC20M0055.

∗Trevor Smith1,2, Madhav Rijal1, Christopher Tatsch1, R. Michael Butts1,
Jared Beard 1, R. Tyler Cook1, Andy Chu1, Jason Gross1, Yu Gu1
West Virginia University
Morgantown, USA

###### Abstract

This work presents the design of Stickbug, a six-armed, multi-agent, precision
pollination robot that combines the accuracy of single-agent systems with
swarm parallelization in greenhouses. Precision pollination robots have often
been proposed to offset the effects of a decreasing population of natural
pollinators, but they frequently lack the required parallelization and
scalability. Stickbug achieves this by allowing each arm and drive base to act
as an individual agent, significantly reducing planning complexity. Stickbug
uses a compact holonomic Kiwi drive to navigate narrow greenhouse rows, a tall
mast to support multiple manipulators and reach plant heights, a detection
model and classifier to identify Bramble flowers, and a felt-tipped end-
effector for contact-based pollination. Initial experimental validation
demonstrates that Stickbug can attempt over 1.5 pollinations per minute with a
50% success rate. Additionally, a Bramble flower perception dataset was
created and is publicly available alongside Stickbug’s software and design
files.

_Index Terms_ -precision agriculture, pollination, multi-agent, autonomous
systems, robot design, multi-armed

##  I Introduction

In recent years, evidence of a decreasing population and diversity of wild
natural pollinators has caused widespread concern for food security [1]. These
concerns have led to exploring alternative pollination methods to address the
growing agricultural demand. One proposed method to meet these pollination
needs is to utilize robotic pollinators. With these devices, farmers can
reduce their dependency on natural pollinators while gaining access to a
multitude of potential benefits. Unlike natural pollinators, which often
exhibit specific environmental preferences [2], robotic pollinators can be
designed to accommodate different environments and tasks. This allows robotic
pollinators to suit various environmental conditions, including those found in
indoor agriculture, such as greenhouses, while being able to accomplish other
tasks such as flower thinning [3] and gathering crop data [4, 5, 6, 7, 8].

Figure 1: Displays Stickbug, a six-armed precision pollination robot tasked
with autonomously navigating, mapping, and pollinating bramble flowers in a
greenhouse. Each arm acts as an independent greedy agent that a referee
oversees to resolve inter-agent conflicts

To accomplish robotic pollination, previous works have utilized non-precise
methods that disperse or spray pollen [9]. These methods have been applied to
both ground [10] and aerial-based robots [11, 12, 13, 14], but often cannot
pollinate specific flower types. To overcome these limitations, precision
pollination techniques have been explored [15], demonstrating pollination in
vanilla [16], kiwifruit [17], watermelon [18], tomato [19], apple [20], and
tall tree [21] agriculture. Our previous research has led to the development
of a single-armed contact-based precision pollination robot named BrambleBee,
designed to autonomously map and pollinate bramble flowers (i.e., blackberry
and raspberry flowers) within greenhouse environments [22, 23]. While
BrambleBee demonstrated the utility of mobile precision-based pollination, the
system is limited to a single manipulator, hindering the productivity and
scalability required to pollinate clusters of bramble flowers. Other works
have explored the potential of small, insect-like flying robots that can be
used for pollination [24, 25, 26, 27]. While these devices inherit the ability
to parallelize the pollination process similar to natural swarms, they often
lack the efficiency, maintainability, ease of control, and suite of high-
accuracy sensors available on ground-based systems. An ideal solution would be
to design a robot capable of leveraging the advantages offered by both
systems.

To achieve this, we introduce the design of Stickbug (shown in Fig. 1), a six-
armed pollination robot that bridges the gap between previous precision
pollination robots and robotic swarms. By incorporating multiple robotic arms
onto a single drive base, we seek to significantly increase flower pollination
throughput. To overcome the motion planning challenges a large state space
presents, each manipulator and the drive-base are designed to operate as
independent agents, analogous to a swarm. However, these agents are overseen
by a ”referee” who intervenes only to resolve conflicts, ensuring smooth,
coordinated operations without constant central control. Altogether, this work
contributes to the literature by:

  1. 1.

Presenting the design of Stickbug, a multi-armed robotic platform tailored for
pollinating bramble flowers in greenhouse environments.

  2. 2.

Demonstrating the core hardware and multi-agent autonomy framework with
initial pollination experiments.

  3. 3.

Sharing the robot design, software, and flower perception dataset.

The paper progresses as follows: Section II outlines the design requirements,
establishing Stickbug’s foundational objectives. Section III details the
hardware design, focusing on Stickbug’s physical components. Section IV
discusses the robot’s software. Section V presents experimental design,
results, and analysis. Finally, Section VI concludes the paper with
reflections on the study and potential future directions.

##  II Design Requirements and Challenges

Individual Bramble plants can grow to more than 2m in height and 50 cm in
radius [28, 29], with flower clusters distributed throughout, often obscured
by surrounding branches. These plants are arranged in rows inside a greenhouse
and collectively produce hundreds of flowers that need individual recognition
and tracking. To enable precision pollination, a manipulator must react to the
dynamics of the flower’s movements, presenting additional challenges to flower
tracking and identification. Furthermore, the short blooming period and the
need to avoid damaging delicate flowers present time and precision
constraints. Therefore, a robotic system with a large workspace, quick and
precise manipulators, and the ability to effectively re-identify moving
flowers is required.

The robotic system must also be able to navigate, localize, and map the
greenhouse environment, which is challenged by tight plant row spacing, GPS
signal obstruction, and the lack of a detailed prior map. To address these
challenges, the system must be designed with a compact footprint for easier
maneuvering and employ alternative localization techniques to GPS for accurate
positioning and mapping.

##  III Hardware Design

Figure 1 depicts Stickbug, a wheeled ground robot featuring six arms, each
outfitted with a pollination end effector. The robot is equipped with multiple
sensors, such as a 3D lidar, an Inertial Measurement Unit (IMU), actuator
encoders, and a depth camera attached to each manipulator’s end-effector. The
three main subsystems, drive base, manipulator, and pollination end-effector,
are individually displayed in Fig 2, and the 3D models are publicly available
[30].

Figure 2: Displays Stickbug’s lightweight 3D-printed planar manipulator on the
vertical mast left rail (a), pollination end-effector featuring a Realsense
D405 depth camera for flower identification and felt tip for pollination (b),
and kiwi drive base with a 3D Velodyne LiDAR for navigation and mapping (c).

###  III-A Drive Base

Navigating the narrow aisles of the greenhouse presents significant challenges
for robotic mobility. This challenge is addressed by utilizing a Kiwi drive
base [31], known for its holonomic movement and compactness. Furthermore, its
strafing ability facilitates continuous pollination of bramble plants while
navigating through rows. This design makes Stickbug well-suited for spatial
constraints and operational demands of greenhouse environments.

Facilitating navigation and mapping, the drive base, shown in Fig. 2, is
equipped with a ZED2i for inertial measurements, 3D Velodyne LiDAR, and a
Realsense D455 camera. The Realsense camera is placed atop the back of the
mast, pointing down to improve immediate obstacle perception and view of areas
occluded to the LiDAR. This dense point cloud and inertial data, along with
the drive base software, are processed by an Intel NUC Enthusiast equipped
with an Nvidia GPU to accelerate computations. Next, to ensure an adequate
driving speed, 23cm omni-wheels are driven by 44rpm DC motors with a three-
channel Roboteq motor controller, granting a traversal speed of 1m/s. To
ensure a low center of gravity, the drive base houses Stickbug’s power system
consisting of three 12v 42Ahr LiFe batteries. Moreover, it includes an onboard
WiFi router for communicating with manipulators.

###  III-B Manipulators

Stickbug’s six-manipulator design aims to boost pollination efficiency through
parallelization but also introduces several design complexities. These
complexities include manipulator arrangement, collision mitigation, and
cooperation. Moreover, the dense plant row arrangement and tall, bramble
plants limit space for multiple large robotic arms. Therefore, Stickbug
utilizes a shared vertical mast design to accommodate access to height
extremes and six small arms. However, due to the shared mast, collisions
become likely, prompting fixed segmented arm workspaces. Yet, the uneven
flower distribution demands cooperative manipulator efforts for efficient
pollination. To reconcile the need for segmentation and cooperation, Stickbug
employs left and right rails on the mast, each supporting three planar
manipulators. This arrangement splits the workspace into dynamic lower,
middle, and upper regions per rail, simplifying collision avoidance and
enabling cross-rail cooperation by creating a shared center workspace.
Furthermore, to ensure successful pollination, the manipulator’s end effector
must be aligned with the flower; thus, a spherical wrist is utilized, omitting
the roll joint due to the flowers’ radial symmetry. By combining a vertical
mast, dual-rail system, small planar arms, and spherical wrist, Stickbug is
well-suited to utilize multiple manipulators to pollinate bramble plants in
complex greenhouse environments.

To facilitate flower identification and tracking, the manipulators, shown in
Fig. 2 utilize a Realsense D405 depth camera on the end-effector. An onboard
Nvidia Jetson then processes the generated color and depth images, manipulator
autonomy, and actuator commands. Next, to adequately reach the flowers,
Stickbug’s mast spans 0.3-1.9m, and each arm has a reach of 50cm from the
mast. This creates a 240-degree fanned cylindrical workspace with 120 degrees
of overlap centered between the right and left rails. In addition,
manipulators employ helix-linear non-captive stepper motors paired with
Leadshine closed-loop stepper drivers to enable rail sharing, ensuring precise
vertical motion. Furthermore, to minimize dynamic mass, thereby decreasing
power needs and enhancing safety, the planar arms and wrist are made from
lightweight 3D printed linkages and powered by Dynamixel servos for precise
control and integrated wiring. Moreover, power is provided to the manipulators
through daisy-chaining from the drive base.

###  III-C Pollination End-Effector

Drawing from our previous work [22, 23], Stickbug addresses the delicate task
of flower pollination with a 3 DoF half-Stewart platform end effector, shown
in Fig. 2, enabling precise pitch, yaw, and forward movements for careful
flower brushing. This end effector is constructed from a custom 3D printed
design and includes a felt surface for efficient pollen transfer. It is
actuated by three Dynamixel XL320 servos, adjusted from the arm’s 12V power to
9V via an inline DC-DC voltage regulator, and features integrated color LEDs
for autonomy state indication. This configuration ensures Stickbug’s gentle
interaction with flowers and easy integration with the manipulators.

##  IV Robot Software and Autonomy

Figure 3: Stickbug’s software architecture is distributed across three main
agents: the drive base for navigation and mapping, the manipulators for flower
identification and pollination, and the referee for inter-agent conflict
detection and resolution.

Centrally operating a complex 51 Degree of Freedom (DoF) system presents a
formidable challenge due to the computational demands of coordinating multiple
manipulators alongside a mobile drive base in the dynamic greenhouse
environment. In addition to the practical difficulties of calibrating all
manipulators and cameras. To address this, Stickbug’s software architecture,
displayed in Fig. 3, adopts a distributed approach, segmenting computation and
tasks across the six manipulators and the drive base. This division reduces
the system’s computational load but introduces the potential for conflicts,
such as manipulators competing for the same flowers. Conflicts are addressed
by introducing an additional referee agent to maintain some centralized
oversight while keeping the computational load manageable. Consequently,
Stickbug’s architecture is spread across these specialized agents—drive base,
manipulator, and referee—with inter-agent communication facilitated by the
Robot Operating System (ROS). All software is publicly available [32]

###  IV-A Drive Base

The drive base agent is tasked with mapping and localizing within the
greenhouse where GPS signal is unreliable and navigating the tight aisles to
ensure the manipulators can access unpollinated flowers. To overcome GPS
obstruction for localization and mapping, LIO-SAM [33], a Lidar-IMU-based
Simultaneous Localization and Mapping (SLAM) algorithm, is utilized. Next, to
navigate within the greenhouse, the waypoint manager utilizes a list of
waypoints to visit all plants based on a rough prior greenhouse map. The
current waypoint is passed to the FALCO path planner[34], which generates a
path using precalculated motion primitives and height-based analysis for local
obstacle avoidance. To safely follow the planned trajectory in the narrow
greenhouse aisles, the path follower leverages the Kiwi drive’s holonomic
capabilities and a state machine switching between three driving modes. When
the waypoint deviates by over 60 degrees, the robot rotates in place to ensure
visibility when traversing due to the mast limiting the LiDAR’s Field of View
(FoV). For heading differences below 60 degrees with obstacles nearby,
translational velocity control ($V_{x}$ and $V_{y}$) is used to avoid
collisions and maintain a constant heading. In all other scenarios twist
control ($V_{x}$, $V_{y}$, $\dot{\theta}$) is utilized.

The drive base’s integration of LIO-SAM, the Waypoint Manager, FALCO, and the
Path Follower enables effective navigation and mapping within the greenhouse.
Figure 4 shows the robot navigating to predetermined waypoints (marked as
small spheres, with the current target enlarged in purple), differentiating
traversable (green) from non-traversable areas (red) and regions beyond local
analysis (grey). The robot’s path, indicated by a dark blue line, demonstrates
its adeptness in navigating the constrained and complex greenhouse
environment.

Figure 4: Shows Stickbug navigating greenhouse rows, leveraging LIO-SAM [33]
for localization and mapping, and Falco [34] for planning. Green areas are
traversable; red indicates untraversable space, and grey areas are outside the
scope of local planning. Small spheres mark waypoints, with the current target
enlarged in purple. The dark blue line displays Stickbug’s path.

###  IV-B Manipulators

Each of the manipulator agents are tasked with identifying and pollinating
flowers while coordinating with other manipulators. Together, these tasks
present significant challenges as presented in Section II, such as detecting
and locating individual flowers within dense foliage, determining their
orientation for adequate pollination, and planning pollination tasks with
dynamically moving manipulators whose interactions with plants can increase
flower sway, exacerbating the difficulty of identifying and reaching flowers.

To identify and position individual flowers, the manipulator agents utilize
depth cameras and a custom-trained YOLOv8 [35] model, which extracts bounding
boxes from color images to locate each flower’s 2D position as shown in Fig.
5.a. The depth image is then utilized to estimate their 3D coordinates in the
identifying manipulator’s base frame. Next, to determine if the camera’s
current orientation is suitable for pollinating a detected flower, the
segmented bounding boxes for each flower are passed to a custom binary
classifier that determines if the center of the flower is visible, as shown in
Fig. 5.a. The flower manager then updates and tracks these flower poses and
pollination statuses based on a distance threshold, a method prone to false
positives due to clustered flower movements exceeding this threshold. To
counteract this, an expiration time is assigned to each flower, extended with
each re-identification, effectively filtering out transient detections. This
strategy enhances the system’s efficiency by focusing on consistently visible
flowers while reacting to the dynamic floral environment.

Figure 5: Stickbug employs YOLOv8 [35] for flower detection, illustrated by
bounding boxes, and uses a custom binary classifier to assess the visibility
of flower centers for pollination (a). Part (b) presents the confusion matrix
of the Flower Center Visibility classifier, with part (c) detailing the
classifier’s performance metrics.

Next, the manipulator arrangement on the mast grants each arm a unique
perspective of the bramble plant, identifying distinct flower sets and
establishing an inherent division of labor that enhances coordination.
However, this setup also introduces the potential for conflicts in overlapping
areas. To mitigate these conflicts, manipulators share their positions and
successfully pollinated flower locations, thus avoiding collisions and
reducing redundant efforts. However, they refrain from sharing unpollinated
flower data, which helps maintain spatial distribution and prevents
calibration errors from misleading the coordination efforts.

After identifying flowers, planning the order of pollination in the dynamic
environment of moving manipulators and flowers presents a significant
challenge. Therefore, a greedy strategy is employed, prioritizing the nearest
unpollinated flower, enabling the manipulators to react to environmental
changes. Once a flower is selected, visual servoing further aids maneuvering
by tracking and compensating for flower movements [36]. Upon reaching the
flower, the end-effector executes gentle brushing motions for adequate
pollination, showcasing a reactive approach to overcome the complexities of
the dynamic environment.

RGB images of natural and artificial flowers were collected to train the
flower perception system with YOLOv8 and the flower center visibility binary
classifier. To enhance the diversity of the dataset and simulate real-world
conditions, several augmentations were applied: scaling ($\pm 25\%$),
flipping, cropping (0-30%), rotating ($\pm 20$ degrees), and adjusting
saturation ($\pm 30\%$), brightness ($\pm 30\%$), and noise (0-5% of pixels).
These modifications were implemented using Roboflow software [37] along with
all image annotations. Then YOLOv8 and the binary classifier were trained
using this dataset, beginning with weights pre-trained on the COCO dataset
[38] for YOLOv8 and random initialization for the binary classifier. The
flower data set and trained weights for flower detection and center visibility
classification are publicly available [39].

After training, the YOLOv8 model achieved a 92% mean average precision for
flower detection. For discerning flower center visibility, the binary
classifier demonstrated robust performance on a 32-image test set, as
reflected in the confusion matrix (Fig. 5.b-c) metrics: accuracy at 90%,
precision at 89%, recall at 94%, F1 score at 91%, and specificity at 86%.
These results underscore the models’ adeptness in identifying and localizing
the visible centers of bramble flowers, which is crucial for accurate
pollination.

###  IV-C Referee

Due to the manipulators’ naive decision-making, there will be conflict
scenarios in which they interfere with each other’s objectives. Thus, the
referee agent is tasked with identifying and providing simple resolutions to
these conflicts as they arise, thereby maintaining high pollination
performance with minimal computational expense. The referee monitors
performance and state data from all agents, such as pollination rate, looking
for drops in performance that indicate a conflict has occurred and analyzing
the agent’s state to determine the type of conflict. Once a conflict is
identified, the involved manipulator(s) are assigned new poses to resolve it.
Currently, the referee focuses on manipulator-manipulator conflicts and
manipulator anomalies.

Manipulator-manipulator conflicts occur when two or more arms target positions
too close to each other, such as attempting to pollinate the same or adjacent
flowers. This leads to a cycle where the manipulators approach, recognize a
potential collision, retreat, and then try to approach again. The referee
identifies these conflicts by noting when two or more nearby manipulators fail
to pollinate within a specific time frame. To resolve this, the referee
randomly selects one manipulator to continue while directing others back to
their home positions. This approach allows the chosen agent to pollinate the
problematic flower, removing it from the unpollinated flower set. The other
agent(s) are sent home to redistribute the agents spatially and to broaden
their view of the plant, possibly exposing new flowers or encouraging
different ones to be pollinated, thus reducing future instances of
manipulator-manipulator conflicts.

The referee also addresses manipulator anomalies, such as repetitive
unfinished pollination attempts induced by flowers moving in and out of reach
or view. These are detected when a manipulator is far from other manipulators
and hasn’t successfully pollinated within a set time frame. They are then
resolved by sending the manipulator back to its home position, similar to
manipulator-manipulator conflicts.

##  V Experiments, Results, and Discussion

Experiments were performed to validate Stickbug’s design and software,
focusing on coordinating multiple manipulators and evaluating Stickbug’s
current pollination performance.

Figure 6: Displays the experimental setup for evaluating Stickbug’s design
within a greenhouse, featuring an artificial bramble plant with ample flowers
for pollination. With the six-armed configuration displayed, this experiment
asses manipulator coordination and cooperation across configurations of 1, 2,
4, and 6 arms.

In this experiment, Stickbug was parked in front of an artificial bramble
plant, as shown in Fig. 6, due to the unavailability of natural flowers in
this season and was tasked with pollinating as many artificial flowers as
possible in five minutes; a human observer recorded the number and time of
pollination attempts and successful pollinations. A pollination attempt is
defined as the manipulator identifying a flower, moving towards it, and
executing the pollination brushing. Given that artificial plants preclude
fruit growth as a means to verify pollination [40], the definition of
successful pollination is limited to the end-effector tip making contact with
the flower during a pollination attempt. The experiment tested configurations
with 1, 2, 4, and 6 manipulators to ensure workspace symmetry beyond the
single-arm setup. Each configuration was repeated five times, totaling twenty
experiments. The average cumulative number of pollination attempts over the
experiment trial and the average percentage of successful pollinations are
shown in Fig 7.a. The pollination attempt rate over time and average
pollination attempt rate are shown in Fig. 7.b.

Figure 7: Part (a) shows the cumulative number of pollinated flowers over the
experimental window for each set of manipulators. In addition, the pollination
rate for each set of manipulators is shown in part (b). As the number of
manipulators increased, the total pollination attempts and pollination attempt
rate increased.

Fig. 7 reveals that Stickbug, with six manipulators, manages over 1.5
pollination attempts per minute but only achieves a 50% success rate. This is
attributed to manipulators’ movements causing flowers to exit the end-effector
camera’s view, primarily when the camera is close to a flower and its
displacement tolerance is limited. Further analysis, as shown in Fig.7.a,
indicates a direct correlation between an increase in the number of
manipulators, therefore increasing disturbances, and a decline in success
rates. To address this, refining flower tracking techniques and coordinating
manipulator actions to minimize disturbances are crucial for improving overall
pollination success.

Fig. 7 also displays that more arms result in higher rates and total numbers
of pollination attempts, indicating efficient spatial coordination despite
each manipulator’s greedy behavior. This improvement in pollination
performance supports the system design of spatially distributing manipulators
to inherently allocate tasks and minimize overlap. The lack of manipulator-
manipulator conflicts, called by the referee agent, further evidences minimal
competition for flowers, reinforcing the effectiveness of this approach in
preventing conflicts.

The absence of manipulator-manipulator conflicts among the greedy agents
suggests limited cooperation, primarily due to not sharing unpollinated flower
locations. This is evidenced in Fig 7, where configurations with four and six
arms persist in pollination efforts beyond three minutes—revealing an uneven
task distribution with certain manipulators addressing more flowers
sequentially rather than all arms pollinating simultaneously. This indicates a
design overemphasis on workspace segmentation that hinders manipulator
cooperation and highlights the necessity for a more integrated approach to
sharing and distributing pollination tasks.

Fig 7.b illustrates a consistent decline in pollination attempts over time,
with manipulators ceasing activity after addressing visible flowers. The
referee identified this performance drop as a manipulator anomaly and sent
them home to reset. However, this strategy, intended to correct failures in
pollination attempts, was ineffective in finding new flowers. This underscores
the need for search strategies to ensure thorough pollination by detecting and
reaching occluded flowers.

##  VI Conclusions and Future Work

This paper introduced the unique design of Stickbug, an autonomous six-armed
and multi-agent precision pollination robotic system tasked with pollinating
bramble flowers in a greenhouse environment. In addition, we presented the
initial demonstration of Stickbug’s complex hardware and autonomy stack and
achieved over 1.5 pollination attempts per minute with approximately 50%
success rate of pollination on artificial plants in a greenhouse setting.
Furthermore, we have created and shared the bramble flower dataset used to
train the flower perception, the 3D design files, and the software for
Stickbug.

The primary limitation of this study was the lack of access to real flowers
for experimentation due to the short pollination season, preventing
verification of actual pollination. Secondly, this experimentation revealed
the system’s limitations in re-identifying and tracking dynamic flower
movements, pollination load balancing, and the need for a flower search
function to find occluded flowers. Future work will provide subsequent
experiments on live plants during the flowering season and improve flower
memory and re-identification through Intersection over Union (IoU) and flower
relative graph-based mapping. Additionally, we plan to enhance manipulator
capabilities by integrating a search function and flower load balancing,
leveraging the referee to generate a global flower map to direct manipulators
toward unexplored and flower-dense regions.

## Acknowledgment

We extend our gratitude to Dr. Nicole Waterland and her students for providing
access to the greenhouse facility and plants.

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