# VARIATIONS IN LINE PROFILES OF ATOMIC TRANSITIONS IN RR LYRAE STARS

Christopher Sneden  Department of Astronomy and McDonald Observatory, The
University of Texas, Austin, TX 78712, USA<EMAIL_ADDRESS>George
W. Preston  Carnegie Observatories, 813 Santa Barbara Street, Pasadena, CA
91101, USA<EMAIL_ADDRESS>
###### Abstract

We have investigated the absorption shapes of atomic lines and H$\alpha$ in RR
Lyrae stars. We used the database of high resolution spectra gathered with the
Las Campanas Observatory du Pont Telescope, analyzing a set of about 2700
short exposure spectra of 17 RRab and 5 RRc variables. To increase the signal-
to-noise of the spectra for each star, we first co-added spectra in small
photometric phase bins, and then co-added metallic line profiles in velocity
space. The resulting line absorption shapes vary with photometric phase in a
consistent manner for all RRab stars, while exhibiting no obvious phase-
related variations for the RRc stars. We interpret these line profile
variations in terms of velocity gradients in the photospheric layers that
produce absorption line profiles. The H$\alpha$ profiles are much broader,
indicative of shock temperatures of order 100,000 K.

††facilities: Du Pont (echelle spectrograph)††software: IRAF (Tody 1986, Tody
1993), SPECTRE (Fitzpatrick & Sneden 1987)

##  1 INTRODUCTION

RR Lyrae stars are old low-mass pulsating variable stars located on the HR
Diagram horizontal branch. They are in the post helium flash evolutionary
stage, and are undergoing hydrostatic core helium fusion. The relative
brightness of RR Lyraes absolute V-band magnitudes $<M_{V}>$ of +0.7) has made
them attractive standard candles for distance determinations of Milky Way
field stars, globular clusters and nearby galaxies. RR Lyraes exhibit
photometric and spectroscopic variations with pulsational periods mostly in
the range 0.25$-$0.75 days, but those cyclical variations are essentially
constant over human timescales. Most importantly, the mean absolute V-band
magnitudes of RR Lyrae stars are similar with a well-calibrated dependence on
metallicity.

Our group has concentrated on studies that use high resolution optical
spectroscopy to gain insight into surface activity and envelope properties of
RR Lyrae variables. The goal was to explore the spectroscopic properties of
these stars in detail at many phase points with a single telescope/instrument
configuration, rather than to conduct a large-sample survey using possibly
heterogeneous data sets with limited phase information. We have used these
spectra for many purposes, such as determination of metallicities and
abundance ratios (For et al. 2011a, b, Govea et al. 2014, Chadid et al. 2017,
Sneden et al. 2017), studies of envelope kinematics (Chadid et al., Sneden et
al.), rotation and macroturbulence estimation (Preston et al., 2019), and
description of helium emission/absorption variations (Preston et al., 2022).
Other investigators have used these du Pont echelle spectra as bases for
studies incorporating other data sets to improve our understanding of RR Lyrae
properties such as the Oosterhoff dichotomy (Fabrizio et al., 2019; Braga et
al., 2021a), the Bailey diagram (Bono et al., 2020), the $\Delta$S metallicity
parameter and [$\alpha$/Fe] ratios (Crestani et al., 2021a, b), radial
velocity curves (Braga et al., 2021b).

In our previous studies we have seen evidence of line shape variations in
atomic transitions at various pulsational phases in many RR Lyrae stars. Only
in very limited phase domains are observed line profiles symmetric and
described reasonably well by Gaussian functions. In this paper we present
evidence that line profile distortions are systematic with phase and are
similar in amplitude for all stars. These variations can be interpreted
through velocity changes in photospheric material that is in constant motion
during the pulsational periods of RR Lyrae stars. In section §2 we describe
the spectroscopic database and reduction steps for the present study. Section
§3 shows how we coadded the spectra in two ways in order to produce single
mean profiles for each star at multiple pulsational phase points. We
demonstrate that the shapes of these profiles vary systematically throughout
the pulsational cycles in essentially the same manner in all of our program
stars. Finally In §4 we develop a simple model for velocity gradients that can
account for the RRab line profile variations.

##  2 THE SPECTROSCOPIC DATA SET

From 2006 to 2014 we gathered more than 5400 RR Lyrae spectra with the Las
Campanas Observatory du Pont Telescope Echelle Spectrograph111
https://www.lco.cl/technical-documentation/echelle-spectrograph-users-manual/.
In this work we use the database of several thousand du Pont echelle spectra
of RR Lyrae stars (Sneden et al., 2021). The spectra have resolving power
$R\equiv\lambda/\delta\lambda$ = 27,000 at 5000 Å. Here we have selected a set
of stars that have a large number of spectra over their complete pulsational
cycles. In Table 1 we give the photometric properties of the chosen stars,
along with their metallicities, the total number of du Pont spectra available,
and the number of co-added spectra (discussed below). The $P$ and $T_{0}$
values are taken from Table 1 of Chadid et al. (2017). The metallicities are
means of the [Fe i/H] and [Fe ii/H] values listed in Table 2 of that paper.

The pulsational periods of RR Lyrae stars are mostly in the range 0.25 d $<$ P
$<$ 0.75 d. Spectroscopic exposure times were confined to small fractions of
these periods, never more than 600 s ($\sim$0.01P), usually 400 s or less.
This restriction resulted in relatively poor signal-to-noise ratios, S/N
$\sim$ 15$-$20 near 4300 Å for most spectra.

The reduction procedures for the du Pont data have been described in detail by
For et al. (2011a); see that paper for details of the methods. Standard
IRAF222 https://iraf.noirlab.edu/ (Tody, 1986, 1993; Fitzpatrick et al., 2024)
procedures were employed in these tasks. The resulting fully reduced “raw
spectra” have about 50 echelle orders spanning the approximate wavelength
range 3900$-$8800 Å. Then, for the present work we created continuous
flattened spectra from 4000$-$4600 Å, a spectral region that has the largest
collection of atomic transitions available for study in typical RR Lyrae
stars. The wavelength step size in these raw spectra is $\delta\lambda$ =
0.045 Å/pixel, or velocity displacement $\delta V$ = 3.1 km s-1 at 4300 Å.

Figure 1:  An example of raw and co-added spectra in one spectral region at a
single phase for program star WY Ant. This relatively bright star already with
good $S/N$ has been chosen for display. The identified atomic lines are those
that were measured in this particular co-added spectrum.

The observed line profiles of photospheric transitions in all stars arise from
the combined broadening effects of the spectrograph instrumental profile and
stellar temperature, microturbulence, macroturbulence, and rotational
conditions. RR Lyraes have continuous envelope and atmospheric motions that
are accompanied by shock waves at multiple phase points. These motions have
been well documented, and are the sources of the asymmetries under
investigation here. Preston et al. (2019) attempted to detect rotation in RRab
stars, concluding that its effects cannot be disentangled from those arising
from macroturbulence. The combined effects of macroturbulence and rotation
were estimated as $V_{macrot}$ $\geq$ 5 $\pm$ 1 km s-1; see their Figure 16.

Here we follow the discussion in Preston et al. (2019) to estimate various
broadening components. The instrumental broadening of the du Pont echelle was
explored extensively in that paper, which derives $<$FWHM${}_{inst}>$ = 11.4
$\pm$ 0.5 km s-1. The combined effects of thermal and microturbulent
broadening, expressed as $\sigma_{th,micro}=2\sqrt{2kT/M+V_{micro}^{2}}$ can
be reasonably estimated because of the regularity of RR Lyrae variations.
Figure 6 of Preston et al. suggests that for RRab stars $T_{\rm eff}$ varies
from $\sim$6100 K in the phase range $\phi$ $\sim$0.35$-$0.75, to $\sim$7300 K
near $\phi$ $\sim$ 1.0 . Similarly, microturbulent velocity $V_{micro}$
changes in the range $\sim$2.9$-$4.0 km s-1 in concert with the $T_{\rm eff}$
variations. This yields a small range of thermal and microturbulent broadening
during RRab cycles: 3.2 $\lesssim$ $\sigma_{th,micro}$ $\lesssim$ 4.2 km s-1.
For our purposes, understanding the imprecision of these estimates, we will
hereafter assume $\sigma_{th,micro}$ = 3.7 km s-1.

As a preliminary to our investigation of skew we coadded observations in small
intervals of phase to increase the S/N. Several considerations are involved in
the creation of coadded spectra for a given star, the most important being
brightness of the star, its location in the sky, and the dates of observing
runs when it is observable. Thus, many observations were made of stars in the
constellations Apus and Octans with declinations $<$ -80∘, because such stars
can be observed at very large hour angles up to $\pm$6 h. Conversely, limited
numbers of observations were made of stars with northerly declinations.

The maximum number of spectra in coadds is $\sim$10 and a typical number is 5.
In a few instances single observations are included as “coadds” at interesting
phases. Typical phase intervals of coadds is 0.03P. Maximum interval with few
exceptions is 0.1P. In Figure 1 we illustrate the co-addition process. For
this and some subsequent figures we have chosen program star WY Ant to
illustrate the analytical steps. Five raw spectra contributed to the mean
spectrum shown in the figure. Visual inspection will confirm that while the
individual spectra have $<$S/N$>$ $\sim$ 50, the co-added spectrum has
$<$S/N$>$ $\sim$ 100, revealing atomic lines worthy of further analysis while
depressing many noise features.

##  3 LINE COADDITION AND PROFILE MEASUREMENTS

###  3.1 Formation of a Mean Metallic Line

Figure 2:  An example of the transition co-addition process. The 71 individual
lines used for WY Ant at pulsational phase $\phi$ = 0.658 are shown as faint
gray profiles. The mean profile generated by co-addition of all of them is
shown in black. The mean of just the weakest 37 lines is in blue, and that of
the strongest 22 strongest lines is in red.  Figure 3:  An example of the
transition measurement process. The displayed line profile is the co-added
spectrum in velocity space, using all transitions that were previously shown
in Figure 2. The top panel shows a spectral region $\pm$350 km s-1 ($\pm$5.0 Å
at 4300 Å) surrounding the co-added transition. The dotted line indicates the
adopted continuum. The bottom panel has a blowup of just the region $\pm$50 km
s-1 around the transition. The dashed line is placed at line center. The
observed velocity widths to the blue and red at flux levels 0.50 and 0.85 of
the continuum are indicated by arrows of different colors and by labels (e.g.,
B0.50 means the width of the blue part of line profile at flux level 0.50 of
the continuum.  Figure 4:  Statistical measures for WY Ant lines at all co-
added phase points. In the top panel the line widths VB+VR are shown. Open
circles are used for the widths at 0.50 relative line flux level and $\times$
symbols for the 0.85 level, as given in the panel legend. The middle panel
displays differences between blue and red widths VR$-$VB, and the bottom panel
contains the line skew values. Data for this figure are listed in Table 2.

While atmospheric conditions of RR Lyraes are constantly variable, inspection
of Figure 6 in Preston et al. (2019) and other literature studies suggest that
typically for RRab stars away from the shock-wave phases $<$$T_{\rm eff}$$>$
$\sim$ 6200 K, $<$log g$>$ $\sim$ 2.0, and $<$$V_{micro}$$>$ $\sim$ 3.0 km
s-1. Superficially, their spectra look similar to that of the Sun. However,
most RR Lyraes are metal-poor, [Fe/H] $<$ $-$1, and many weak lines in the
solar spectrum disappear into RR Lyrae continuum regions. Additionally, RR
Lyraes have much broader lines than does the Sun, eliminating some otherwise
promising lines that become inaccessible for transition profile analysis. The
relatively low $S/N$ values of our spectra also mangle some profiles of the
remaining lines. Finally, the number of measurable transitions varies
systematically with phase. The sharpest and cleanest line profiles occur in
the $\phi$ $\simeq$ 0.35 $\pm$0.10 domain, while the broadest, most distorted
profiles are often seen when $\phi$ $\simeq$ 0.90 $\pm$0.15.

To systematically determine a candidate line list that might serve for most
program stars at most phases, we first used the solar line compendium of Moore
et al. (1966) in conjunction with one of our higher $S/N$ spectra to identify
promising transitions in the 4000$-$4600 Å that should be examined in all
spectra. This exercise yielded about 100 lines.

For each co-added spectrum, we systematically measured approximate equivalent
widths ($EW$s) in a semi-automatic procedure using the specialized software
code SPECTRE (Fitzpatrick & Sneden, 1987). The purpose was not to determine
accurate $EW$s for detailed study. Rather we wanted to identify which lines
should be included in our line profile analysis of each spectrum, and their
general absorption strengths. Figure 1 identifies lines retained for the
example star/phase in the spectral region displayed in the plot.

For a given spectrum, the spectral region surrounding each identified line was
interpolated in velocity units to small steps (0.3 km s-1). The central depth
of the observed line was found in the interpolated spectrum, and a velocity
array extending $\sim\pm$400 km s-1from this central point was computed. Then
these line arrays in velocity space were co-added to yield the final mean line
profile. The number of individual transitions contributing to the mean
profiles varied greatly from star to star (due to metallicity differences) and
phase to phase (due to changes in line profile sharpness). Additionally, $S/N$
differences for individual stars based on their apparent magnitudes affected
the number of lines available for analysis. In total, the mean line profiles
were composed of between $\simeq$20$-$80 transitions.

When the number of transitions used for a star at a particular phase was
large, $\gtrsim$40, the mean profiles could be constructed with subsets of
lines segregated in line strength, ionization state, and excitation energy. In
Figure 2 we show a typical example of such flexible line averaging, choosing
WY Ant at $\phi$ = 0.658. We have selected this star for display in this and
several more figures because it is typical of our RRab sample, and has spectra
well covering its pulsational phases. At $\phi$ = 0.66 WY Ant has a co-added
mean spectrum with 71 measurable lines having an $EW$ range $\simeq$20$-$280
mÅ. Here we show the 71 line profiles individually as thin black lines and
their mean as a thick black line. The means of the weakest and strongest lines
are shown as blue and red lines, respectively. Inspection of these transition
profiles reveals line asymmetries in all three means, with extended blue wings
in each case. These asymmetries will be quantified in §3.2. In columns 9 and
10 of Table 1 we list the minimum and maximum number of lines that
participated in the line profile calculations at individual phase points for
each star.

###  3.2 Line Statistics

We made 3 sets of line profile measurements: (1) blue and red velocity widths,
labeled $V_{B0.50}$ and $V_{R0.50}$, at fractional line fluxes of 0.50 of the
local continuum (e.g., at the half widths); (2) velocity widths $V_{B0.85}$
and $V_{R0.85}$ at 0.85 of the continuum; and (3) Pearson’s statistical 3rd
moment skew values333 https://en.wikipedia.org/wiki/Skewness at those two flux
levels, $\bar{\mu}_{3,0.50}$ and $\bar{\mu}_{3,0.85}$. In Figure 3 we use WY
Ant at pulsational phase $\phi$ = 0.658 to illustrate our line profile
parameters. The line profile is the one formed from the whole transition set
depicted in Figure 2. The choice of 0.50 central depth is a standard one for
line profile descriptions. The 0.85 level was empirically chosen to measure
the line asymmetries in the wings of the profile, without going so far from
line center that continuum placement and contamination from other spectral
features become major measurement uncertainty sources.

The first task in this process was continuum setting. In Figure 2 the apparent
features that are $\pm$30$-$50 km s-1 away from line center in the coadded
line profile arise from atomic blends in individual transitions that depress
their local continua. Therefore for the coadded line profiles better continuum
placement was accomplished from inspection of a series of high-flux points in
an expanded velocity interval, as we show in the top panel of Figure 3. Then
we measured line quantities defined in the previous paragraph; see the bottom
panel of this figure. Their values in all program stars at all coadded phase
points are listed in Table 2.

In the WY Ant $\phi$ = 0.658 example, our measured quantities were $V_{B0.50}$
= 15.1 km s-1, $V_{R0.50}$ = 12.7 km s-1, $V_{B0.85}$ = 24.9 km s-1,
$V_{R.85}$ = 20.2 km s-1, and skew0.50 = 0.04, and skew0.85 = $-$0.10.. The
blueward asymmetry is well defined at both measurement depths: $\delta
V_{B.50}$ $\equiv$ $V_{B.50}$ \+ $V_{R.50}$ = $-$2.4 km s-1 and $\delta
V_{B.15}$ $\equiv$ $V_{B.15}$ \+ $V_{R.15}$ = $-$4.7 km s-1. The skew values
are $\bar{\mu}_{3,0.50}$ = $-$0.04 and $\bar{\mu}_{3,0.85}$ = $-$0.10. These
line statistics all support the visual impression in Figure 3 of a line with a
blueward distorted profile.

Figure 5:  Variations of metal line skew values for WY Ant for weaker lines
(top panel), strongest lines (middle panel), and the total line set (bottom
panel). The bottom panel is a repeat of the bottom panel in Figure 4.

In Figure 4 we show three descriptors of WY Ant line shapes at the 0.50 and
0.85 relative flux levels for all 21 of its co-added phase points. In the top
panel total line widths $V_{B0.50}$+$V_{R0.50}$ and $V_{B0.85}$+$V_{R0.85}$
are plotted. A relatively smooth variation from minimum width near phase
$\phi$ $\simeq$ 0.35 to maximum near $\phi$ $\simeq$ 0.90 is easy to see and
is not surprising. This trend has been noted in many papers, and has been
explored in detail by For et al. (2011b) and Preston et al. (2019). In the
middle panel we show a simple asymmetry measure of the differences between red
and blue widths VR0.50$-$VB0.50 and VR0.85$-$VB0.85. At phases near line width
minimum, 0.2 $\lesssim$ $\phi$ $\lesssim$ 0.5, WY Ant metal line profiles are
symmetric to our measurement limits. This apparently calm phase interval is
followed by a small but easily detected blueward asymmetry that lasts until
$\phi$ $\simeq$ 0.85, and then by a sharp turn to a much larger redward
asymmetry during the photometric rising-light phase domain. In the bottom
panel of Figure 4 we plot the 3rd moment skew values at the two line flux
points. The skews are consistent with the line width differences and are more
statistically robust. We adopt skew as the primary line asymmetry indicator.

WY Ant at phase $\phi$ = 0.658 has many useful metallic lines with a large
absorption strength range, 20 $\leq$ EW $\leq$ 280 mÅ (reduced width $-$5.3
$\leq$ log10(EW/$\lambda)$ $\leq$ $-$4.2). Following Figure 2’s line strength
divisions, in Figure 5 we show the variation of skew for weak, strong, and all
lines. Inspection of these figure panels strongly suggest that the skews are
qualitatively the same for lines on the linear to the damping portions of the
curve of growth. This further implies that the metallic line formation
photospheric layer is relatively compact. All atomic absorptions occurs in
similar temperature/pressure conditions.

Figure 6:  Skew values for our sample of RRab stars, plotted as functions of
phase. The three panels contain stars grouped for visual clarity according to
their metallicities, with the most metal-poor in the top panel and the most
metal-rich in the bottom panel. Skew values for individual stars are indicated
with different colors, and their metallicities from Table 1 are in the panel
legends.  Figure 7:  Skew measurements for the RRc stars.

###  3.3 Line Profile Variations for RRab and RRc Stars

Our program sample is composed of 17 RRab and 5 RRc stars. In Figure 6 we
gather the skew values for the RRab stars. Three panels are used separate the
stars into lowest, intermediate, and highest metallicity groups. Previous
papers by our group (e.g., Chadid et al. 2017; Sneden et al. 2017, 2018;
Preston et al. 2019 have divided the RR Lyraes into metal-rich (MR) or metal-
poor (MP) categories with the split estimated to be [Fe/H] = $-$1.0. By that
criterion, the top and middle panels of Figure 6 contain all MP stars and the
bottom panel has only MR stars. Inspection of the figure reveals no obvious
skew differences in different metallicity bins. For MR stars (bottom panel)
the phase regime of positive skew appears to be slightly smaller than those of
the more metal poor stars (middle and top panels), but our MR sample has only
4 stars. Additionally MR stars have stronger line spectra at all phases,
yielding more sharply defined average line profiles than for the MP groups.
Gathering spectra for more MR RRab stars will be needed to resolve this
possible difference between the RRab metallicity groups.

In Figure 7 we show the variation in skew with phase for our sample of RRc
stars. The data in this plot strongly suggest that RRc metal lines have
symmetric profiles. Apart from star AS230659–4354.6, the measured skew values
all are contained within $-$0.1 $\lesssim$ $\bar{\mu}_{3,0.85}$ $\lesssim$
+0.1, and there is no obvious phase dependence of the skew values. We caution
that we did not observe RRc stars as intensively as the RRab stars. We have
fewer coadded phase points for any RRc star (Nspec,max = 10, Table 1) than for
any RRab (Nspec,min = 13, Table 1. Additionally, RRc line strengths are weaker
than those of RRab stars of similar metallicities because RRc stars are
warmer. Finally, the S/N values of our RRc spectra are typically lower than
those of our RRab sample. Therefore we regard the skew range in Figure 7 as
consistent with no line profile red/blue distortions. However, this conclusion
is based only on the 5 RRc stars considered in this paper, and may not apply
to all members of the RRc class. For example, Bono et al. (2000) applied
hydrodynamical modeling to detailed light curves of the RRc variable U Com (P
= 0.29 d), finding photometric features that would suggest that structure in
RV variations might be present. A larger spectroscopic survey of RRc stars
should be undertaken.

##  4 INTERPRETATION OF THE LINE SKEW MEASUREMENTS

Detailed dynamical model atmosphere non-local thermodynamic equilibrium (NLTE)
line transfer computations are beyond the scope of the present observational
spectroscopic contribution. Here we sketch the outlines of a possible
interpretation of the phase-dependent skewed line shape profiles of RRab
stars.

###  4.1 Skew due to Putative Velocity Gradients in Metallic Line-forming
Regions

Figure 8:  Coadded metallic line profiles (black dots) and modeled line
profiles (colored lines) for WY Ant at two pulsational phases. The velocities
$\Delta$RV are with respect to the stellar center of mass velocity. See the
text for description of the empirical model line profiles; their relative
velocities are noted in the figure legend.  Figure 9:  Top panel: radial
velocity variations with phase for metal lines (purple color) and H$\alpha$
(orange) in WY Ant. The ordinate $\Delta$RV values are measured relative to
the WY Ant center-of-mass velocity, 203.7 km s-1. Bottom panel: Photometric
V-band magnitudes for WY Ant from the ASAS database (Pojmanski et al. 2005 and
references therein). In both panels vertical dashed lines denote phases $\phi$
= 0.009 (repeated at $\phi$ = 1.009) and 0.849 that have been chosen for line
profile display in Figure 8.  Figure 10:  Heights ($\Delta$R in km) above
Rosseland $\tau$ = 0.67 in a model from the Kurucz (2011, 2018) grid with
parameters indicated in the legend.

Investigations of skew in solar absorption lines were reviewed in detail by
Dravins (1982 and references therein), who interpreted this skew successfully
in terms of a solar surface covered by myriads of hotter upflowing convection
cells surrounded by cooler downflowing intercellular gas. We propose a
different explanation for the time-dependent skew observed in RR Lyrae stars:
namely, the existence of velocity gradients in their stellar atmospheres first
predicted by Bono et al. (1994).

In the top panel of Figure 8 we show the metallic line skew profiles in metal-
poor WY Ant at pulsational phase $\phi$ = 0.849, near minimum light (maximum
infall velocity). In the bottom panel we show the skewed line profiles at
$\phi$ = 0.009, just after maximum light (near maximum outflow velocity).
Black filled circles denote the observed profiles, and solid lines represent
our fits to the observations, to be explained below. The velocities, labeled
$\Delta$RV, are relative to the center-of-mass velocity, defined here to be
the average of the radial velocities of metal lines over a pulsational cycle.
For WY Ant, we derived center-of-mass RV = 203.7 km s-1, using velocity data
originally presented in Chadid et al. (2017). The $\Delta$RV values for both
metal lines and H$\alpha$ are shown in the top panel of Figure 9, with
vertical lines placed at the phases chosen for Figure 8.

The bottom panel of Figure 9 contains the photometric V-band light curve of WY
Ant444V magnitudes were taken from the All Sky Automated Survey (ASAS,
Pojmanski et al. 2005); and references therein;
https://www.astrouw.edu.pl/asas/. This connects photometric and spectroscopic
quantities for WY Ant. Other RRab stars in our study have essentially the same
variations throughout their pulsational cycles. Thus, phase $\phi$ = 0.849
shows very blue-skewed line profiles (Figure 5) for WY Ant, accompanying
maximum infall velocity ($\Delta$RV $\simeq$ 25 km s-1; Figure 9 top panel)
and minimum brightness ($V$ $\sim$ 11.2; Figure 9 bottom panel). Phase $\phi$
= 0.009 (repeated at $\phi$ = 1.009) corresponds to near-maximum positive
(redward) skew, maximum outflow ($\Delta$RV $\simeq$ $-$32 km s-1), and
maximum brightness ($V$ $\sim$ 10.3).

In both panels of Figure 8 the red curves are Gaussians fitted to the unskewed
wings of the observed profiles. Gaussian $\sigma$ values of 12.8 km s-1 and
12.0 km s-1 were used to match these wings for phases 0.849 and 0.009,
respectively. We assume that these red wings represent the intrinsic profiles
of gas in the metal line-forming regions at these phases. We then empirically
added 4 additional line components with the same $\sigma$ values but weaker
line depths and increasing RV offsets in order to adequately match the skewed
profile wings. The observed profiles are formed in ranges of geometric depth
that begin at the photosphere ($\tau$ $\sim$ 0.6) and continue to very small
values in the case of H$\alpha$. Most metal lines are formed at modest optical
depths 0.1 $\lesssim$ $\tau$ $\lesssim$ 0.8 (e.g., Gray 2008), with the far
wings arising from the deepest layers and cores at higher atmospheric optical
levels. As lines grow stronger their mean formation level advances outward in
stellar atmospheres. In early stellar atmospheres modeling H$\alpha$ formation
was considered as arising in a separate outer atmosphere slab by Kraft et al.
(1964). This notion is supported by the computations of Stellingwerf (2013)
that produced an accumulation of gas high in the atmosphere due to the
succession of a multitude of outward progressing shocks.

Figure 11:  H$\alpha$ profile for WY Ant at phase $\phi$ = 0.86. The lines and
symbols are as in Figure 8

To understand our skew profiles during rising light ($\phi$ $\gtrsim$ 0.9) we
consider a simple model in which an outward propagating pressure wave,
conserving momentum ($\rho v$ = constant), moves through an atmosphere of
exponentially decreasing density. Conservation of momentum then puts a
condition on the velocity of the wave, $v$ = cRα in which $c$ and $\alpha$ are
positive constants chosen to match the skew contributions at locations given
by equations 24 and 25 of Edmonds (1969). In the post-maximum light phase
($\phi$ $\gtrsim$ 0.1) the velocity field, produced initially by gravitational
infall, is modified by deceleration due to the pressure support predicted by
Hill (1972).

We have made numerical experiments to estimate atmospheric line formation
heights. We adopted the static model atmospheres of Kurucz (2011, 2018)555
http://kurucz.harvard.edu/linelists.html for these calculations. For phase
$\phi$ = 0.847 our estimate of height above the photosphere for the model that
we adopt for minimum light is shown in Figure 10. Unfortunately a similar plot
for $\phi$ = 0.009 cannot be constructed from the observed line profiles of
H$\alpha$ because of the major interfering contribution of the red-shifted
component of the doubled line profile at this phase (see the H$\alpha$
profiles for our program star metal-poor RV Oct in Figure 18 of Chadid et al.
(2017).

In our model the observed metal line profile (Figure 8 black dots) is created
by contributions from a succession of layers at geometric depths within the
line forming regions ($\Delta$R in Figure 10) that move at different redial
velocities indicated in the legends. We use observed $\Delta$RV values of
Figure 8 as proxies for unknown heights, $\Delta$R, within line forming
regions. Actual contribution functions must be calculated by use of equations
24 and 25 of Edmonds (1969) We make no attempt to define these geometric
heights in this initial reconnaissance.

Our profile in Figure 8 (top) requires a velocity gradient $\sim$20 km sec-2;
the range of $\Delta$R corresponding to the range 0.67 $>$ $\tau$ $>$ 0.1 in
which metal lines are formed is some three times smaller than the total
velocity gradient $\sim$60 km sec-2 which we deem appropriate for H$\alpha$ in
Figure 11. The two observed profiles conform to expectations. The curves with
minima at fluxes 0,2, 0,4, 0.6, and 0.8, fitted to the skewed wings, are
idealized Gaussian contributions to the observed skews at geometric depths
defined by equations 24 and 25 of Edmonds (1969). We make no attempt to define
these geometric depths in this first reconnaissance.

###  4.2 Comparison of Metal and H$\alpha$ gradients

Figure 11 shows the H$\alpha$ profile in this star at phase $\phi$ = 0.86,
almost identical the 0.849 phase chosen for one of the metal line plots in
Figure 8.666 The top panel of Figure 9 contains two sets of RV measurements
for H$\alpha$ in WY Ant in the phase range $\phi$ $\sim$ 0.95$-$1.05. This is
the phase regime with obvious doubling of H$\alpha$ profiles mentioned
earlier, limiting our ability to make detailed statements about this
transition in our RRab stars. Comparison of the FWHMs of metal lines
(12.0$-$12.8 km s-1) and H$\alpha$ immediately shows that temperature, not
turbulence, must be the principal cause of observed line broadening in our
spectra. Were turbulence the cause of broadening, line widths would be
independent of atomic mass. To estimate temperatures we first remove
instrumental broadening FWHMinst = 11.36 km s-1 (Preston et al., 2019) from
the observed values. In the absence of better information about the
temperature structure of the line forming regions we calculate Local
Thermodynamic Equilibrium (LTE) temperatures given in Table 3. We regard the
factor of $\sim$2 disparity between our temperature estimates for the metal
lines at phases 0.849 and 0.009 as an indicator of the uncertainty in our
calculations. We are content to conclude that temperatures of order 100,000 K
in shocked gas are required to explain line widths in our spectra of WY Ant,
an RRab star typical of our metal-poor sample.

##  5 SUMMARY

We have investigated metal and H$\alpha$ line profile shapes in 17 RRab and 5
RRc variable stars. To create high resolution, high signal-to-noise line
profiles for each star we first performed coaddions of individual spectra in
narrow phase intervals. Then for each coadded spectrum we stacked together
many metal lines in velocity space. The resulting mean line profiles exhibit
line profile distortions that vary regularly with phase for all of our RRab
stars. No major differences in the line profiles variations are seen over a
large metallicity range. The RRc stars show no obvious phase-based variations.
We have constructed model line profiles that suggest that velocity gradients
in RR Lyrae photospheres can successfully account for the observed cyclic RV
variations.

We thank Giuseppe Bono, Vittorio Braga, Valentina D’Orazi and our referee for
comments that have improved this paper. As always we appreciate the efforts of
the Carnegie support staff for their efficient, cheerful assistance in our
work at LCO. NOIRLab IRAF is distributed by the Community Science and Data
Center at NSF’s NOIRLab, which is managed by the Association of Universities
for Research in Astronomy (AURA) under a cooperative agreement with the
National Science Foundation. This work has been supported by NSF grant
AST-1616040 (C.S.).

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Table 1: Program Star Data Name | $P$ |  $T_{0}$aa$T_{0}$ = HJB $-$ 2,450,000. | $V$ | $V_{amp}$ | [Fe/H] | Nspec | Nspec | #linesbb#lines(min) and #lines(max) are the smallest and largest number of lines contributing to the mean line profiles at a single phase point for a star. | #linesbb#lines(min) and #lines(max) are the smallest and largest number of lines contributing to the mean line profiles at a single phase point for a star.
---|---|---|---|---|---|---|---|---|---
| (d) | (d) | (mag) | (mag) |  | raw | coadded | min | max
RRab Stars
WY Ant | 0.574344 | 3835.592 | 10.4 | 0.85 |  $-$1.66 | 145 | 21 | 32 | 76
SW Aqr | 0.459303 | 1876.138 | 10.6 | 1.28 |  $-$1.24 | 135 | 18 | 8 | 55
XZ Aps | 0.587266 | 3836.276 | 11.9 | 1.10 |  $-$1.65 | 289 | 26 | 17 | 77
X Ari | 0.651170 | 1890.064 | 9.2 | 0.94 |  $-$2.50 | 195 | 19 | 12 | 49
RR Cet | 0.553029 | 1900.312 | 9.3 | 0.82 |  $-$1.52 | 96 | 20 | 19 | 65
W Crt | 0.412013 | 1871.640 | 10.9 | 1.10 |  $-$0.76 | 170 | 23 | 16 | 65
SX For | 0.605342 | 1870.406 | 10.9 | 0.64 |  $-$1.62 | 86 | 14 | 28 | 62
VX Her | 0.455359 | 2699.996 | 10.1 | 1.25 |  $-$1.60 | 81 | 13 | 7 | 56
V Ind | 0.479602 | 1873.466 | 9.6 | 1.06 |  $-$1.62 | 246 | 24 | 25 | 71
SS Leo | 0.626335 | 1873.056 | 10.5 | 1.00 |  $-$1.88 | 110 | 21 | 17 | 73
ST Leo | 0.477984 | 5322.597 | 11.0 | 1.19 |  $-$1.29 | 65 | 13 | 10 | 64
RV Oct | 0.571170 | 3835.895 | 10.5 | 1.13 |  $-$1.34 | 227 | 26 | 19 | 69
AV Peg | 0.390382 | 5360.677 | 10.0 | 0.96 |  $-$0.14 | 164 | 15 | 13 | 40
HH Pup | 0.390745 | 1869.661 | 10.6 | 1.24 |  $-$0.95 | 175 | 24 | 31 | 74
W Tuc | 0.642243 | 5454.570 | 10.9 | 1.11 |  $-$1.64 | 142 | 23 | 12 | 48
CD Vel | 0.573509 | 3835.914 | 11.7 | 0.87 |  $-$1.71 | 208 | 20 | 14 | 64
UU Vir | 0.475609 | 1886.488 | 10.1 | 1.08 |  $-$0.93 | 130 | 18 | 12 | 68
RRc StarsccNames for RRc stars are from the All Sky Automated Survey (ASAS),
Pojmanski et al. (2005) and references therein.
AS023706–4257.8 | 0.311326 | 1868.717 | 8.8 | 0.50 |  $-$1.88 | 96 | 10 | 35 | 58
AS081933–2358.2 | 0.285667 | 4900.180 | 10.4 | 0.28 |  $-$2.50 | 6 | 3 | 8 | 17
AS085254–0300.3 | 0.266902 | 4900.400 | 12.5 | 0.47 |  $-$1.50 | 6 | 2 | 28 | 38
AS211933–1507.0 | 0.273460 | 1873.307 | 11.1 | 0.50 |  $-$1.50 | 30 | 8 | 46 | 56
AS230659–4354.6 | 0.281130 | 5014.925 | 12.8 | 0.30 |  $-$1.24 | 30 | 7 | 13 | 26
Table 2: Observed Line Broadening Parameters

Star | phase | # lines | VB0.50 | VR0.50 | VB0.85 | VR0.85 | $\bar{\mu}_{3,0.50}$ | $\bar{\mu}_{3,0.85}$
---|---|---|---|---|---|---|---|---
| $\phi$ |  | km s-1 | km s-1 | km s-1 | km s-1 |  |
WY Ant | 0.009 | 41 | 12.3 | 15.1 | 20.1 | 26.4 | 0.13 | 0.17
WY Ant | 0.057 | 46 | 11.8 | 14.3 | 20.0 | 24.5 | 0.11 | 0.12
WY Ant | 0.085 | 46 | 11.9 | 13.5 | 20.7 | 23.6 | 0.08 | 0.09
WY Ant | 0.159 | 65 | 11.0 | 11.8 | 19.1 | 19.6 | 0.05 | 0.03
WY Ant | 0.261 | 71 | 9.6 | 9.8 | 16.4 | 16.3 | 0.03 | 0.01
WY Ant | 0.346 | 76 | 9.1 | 9.3 | 15.8 | 15.5 | 0.02 |  $-$0.01
WY Ant | 0.450 | 72 | 9.9 | 9.9 | 17.2 | 16.3 | 0.01 |  $-$0.03
WY Ant | 0.547 | 77 | 11.9 | 10.9 | 20.8 | 17.8 |  $-$0.02 |  $-$0.07
WY Ant | 0.658 | 71 | 15.1 | 12.7 | 24.9 | 20.2 |  $-$0.04 |  $-$0.10
WY Ant | 0.749 | 68 | 14.4 | 12.9 | 23.4 | 20.6 |  $-$0.02 |  $-$0.06
WY Ant | 0.849 | 65 | 15.0 | 13.4 | 25.2 | 21.5 |  $-$0.02 |  $-$0.08
WY Ant | 0.881 | 43 | 17.0 | 14.0 | 27.3 | 22.8 |  $-$0.06 |  $-$0.07
WY Ant | 0.905 | 35 | 16.3 | 18.0 | 28.4 | 30.4 | 0.06 | 0.03
WY Ant | 0.904 | 27 | 15.5 | 17.1 | 25.6 | 33.9 | 0.09 | 0.18
WY Ant | 0.915 | 35 | 12.6 | 15.3 | 21.7 | 28.3 | 0.12 | 0.15
WY Ant | 0.927 | 39 | 12.2 | 14.0 | 20.8 | 25.0 | 0.09 | 0.11
WY Ant | 0.939 | 34 | 12.6 | 13.9 | 20.7 | 25.7 | 0.08 | 0.13
WY Ant | 0.953 | 34 | 12.3 | 15.2 | 20.6 | 27.1 | 0.14 | 0.15
WY Ant | 0.964 | 32 | 12.5 | 16.2 | 20.1 | 28.4 | 0.15 | 0.18
WY Ant | 0.973 | 36 | 11.1 | 14.9 | 20.1 | 34.6 | 0.24 | 0.29
WY Ant | 0.987 | 35 | 12.2 | 16.0 | 20.1 | 27.1 | 0.16 | 0.19

Note. — (This table is available in its entirety in machine-readable form.)

Table 3: LTE Temperature Estimates phase $\phi$ | spectrum | FWHMobs | FWHMcorr | TLTE
---|---|---|---|---
|  | (km s-1) | (km s-1) | (K)
0.849 | metals | 12.8 | 5.9 | 118000
0.009 | metals | 12.2 | 4.5 | 67000
0.856 | H$\alpha$ | 36.0 | 34.2 | 70000


# SpaceHopper: A Small-Scale Legged Robot for Exploring Low-Gravity Celestial
Bodies

Alexander Spiridonov, Fabio Buehler, Moriz Berclaz, Valerio Schelbert, Jorit
Geurts, Elena Krasnova,
Emma Steinke, Jonas Toma, Joschua Wuethrich, Recep Polat, Wim Zimmermann,
Philip Arm, Nikita Rudin,
Hendrik Kolvenbach, and Marco Hutter

###### Abstract

We present _SpaceHopper_ , a three-legged, small-scale robot designed for
future mobile exploration of asteroids and moons. The robot weighs
$5.2\text{\,}\mathrm{kg}$ and has a body size of $245\text{\,}\mathrm{mm}$
while using space-qualifiable components. Furthermore, _SpaceHopper_ ’s design
and controls make it well-adapted for investigating dynamic locomotion modes
with extended flight-phases. Instead of gyroscopes or fly-wheels, the system
uses its three legs to reorient the body during flight in preparation for
landing. We control the leg motion for reorientation using Deep Reinforcement
Learning policies. In a simulation of Ceres’ gravity
($0.029\text{\,}\mathrm{g}$), the robot can reliably jump to commanded
positions up to $6\text{\,}\mathrm{m}$ away. Our real-world experiments show
that _SpaceHopper_ can successfully reorient to a safe landing orientation
within $9.7\text{\,}\deg$ inside a rotational gimbal and jump in a
counterweight setup in Earth’s gravity. Overall, we consider _SpaceHopper_ an
important step towards controlled jumping locomotion in low-gravity
environments.

###### Index Terms:

Legged Robots, Space Robotics and Automation, Engineering for Robotic Systems

††This research was supported in part by the Swiss National Science Foundation
through the National Centre of Competence in Digital Fabrication (NCCR dfab).
Corresponding author Alexander Spiridonov. Jonas Toma and Wim Zimmerman are
with the ZHAW, 8401 Winterthur. All other authors are with the Robotic Systems
Lab, ETH Zurich, 8092 Zurich, Switzerland. (e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>mahutter@ethz.ch;). © 2024 IEEE. Personal use of this material is permitted.
Permission from IEEE must be obtained for all other uses, in any current or
future media, including reprinting/republishing this material for advertising
or promotional purposes, creating new collective works, for resale or
redistribution to servers or lists, or reuse of any copyrighted component of
this work in other works.

##  I INTRODUCTION

Low-gravity celestial bodies like asteroids and moons are becoming
increasingly popular targets for space missions. Their composition makes them
valuable both from a scientific and economic perspective[1]. Moreover, the
dawn of a new Space Economy also promises to reduce the cost of visiting these
objects. The launch of small-scale CubeSats has already become affordable for
many enterprises[2]. With this in mind, we aim to develop a robot enabling
scalable mobile asteroid exploration. Traditionally used, wheeled locomotion
systems, such as the Mars Rover Perseverance [3] or the proposed MMX Rover
[4], are not very well suited for asteroids. Low gravity causes loss of wheel
traction, and the rough terrain requires a very versatile system [5]. Other

Figure 1: The small-scale low-gravity robot _SpaceHopper_ fully assembled with
labeled parts and dimensions.

robots, such as the asteroid hopper MASCOT [6], JAXAs Minerva [7] or legged
climbing robots like LEMUR [8] and ReachBot [9] offer alternative approaches
to microgravity mobility. However, their non-articulated design or reliance on
specialized grippers or docking adapters make them less suited for efficient,
wide-ranging exploration of asteroids. Dynamically walking robots such as
ANYmal[10] and Spot[11], on the other hand, have shown impressive capabilities
to traverse unstructured terrain on Earth. Thus, a variant of these robots has
a high potential for mobile exploration of celestial bodies[12]. Some
pathfinders, such as Spot Nebula [13] and SpaceBok[14], have already been
tested on analog sites but are not yet specifically engineered for space [15].
Moreover, in the low-gravity environment of asteroids, gaits with extended
flight phases are increasingly efficient [16]. Dynamic jumping locomotion is
also advantageous compared to other walking gaits, as it can swiftly overcome
obstacles by simply jumping over them. However, it also poses the challenge of
stabilizing the robot’s attitude in low-gravity during the jump phases [17].
Traditionally, attitude control in space and on low-gravity celestial bodies
uses reaction wheels. However, to save weight and eliminate the complexity of
an additional subsystem, one can omit reaction wheels and control the robot’s
attitude using the legs only. SpaceBok, which only has two degrees of freedom
(DOF) per leg, used this idea to perform jumping locomotion in a planar
representation of a microgravity environment [18].

In this paper, we present _SpaceHopper_ , a small-scale, lightweight, and
space-qualifiable robot to investigate controlled locomotion in low-gravity
scenarios. The proposed system uses jumping as the primary form of locomotion.
Moreover, the robot can control its attitude in low-gravity using solely its
legs. Besides demonstrating the mechanical and electrical feasibility of the
system, we demonstrate SpaceHopper’s jumping and reorientation capability in
simulation and partly on the real system. Specifically, we provide the
following contributions:

  * •

We present a three-legged, CubeSat-sized, and lightweight robot specifically
designed for controlled low-gravity locomotion.

  * •

In a simulation, we validate the _SpaceHopper_ design by showing accurate low-
gravity attitude control and jumping to commanded positions.

  * •

We demonstrate 2D attitude control and vertical jumping on the hardware using
a gimbal and a counterweight setup.

##  II SYSTEM DESIGN

###  II-A System Overview

_SpaceHopper_ is a three-legged robot used as a research platform for low-
gravity locomotion in space. We tested _SpaceHopper_ in a lab environment (see
Sec. IV and V) and accordingly estimate it to be at technology readiness level
(TRL) four. The body is triangular and symmetric, with one leg attached to
each corner (Fig. 1). _SpaceHopper_ can fit into a 27U CubeSat [19] with its
legs fully tucked in. At $5.2\text{\,}\mathrm{kg}$, the robot is significantly
lighter than the $54\text{\,}\mathrm{kg}$ CubeSat limit.

Figure 2: Cut section of the hip and knee with annotated transmissions,
movements, and components.

###  II-B Leg Design

Jumping and reorientation is the main locomotion goal of _SpaceHopper_ and a
guiding feature for the leg design. Having three legs is a unique feature of
_SpaceHopper_ and allows for weight saving compared to a classic four-legged
design (one leg weighs $1.17\text{\,}\mathrm{kg}$, including all motors).
Static standing is possible with three legs, which increases stability
compared to a two-legged design. Four legs enable static locomotion, but given
the low-gravity environment, this gait is less efficient [16] and not
favorable. _SpaceHopper’s_ legs each have three active DOFs as this vastly
outperforms legs with only two active DOFs in attitude control [18]. Two of
those active DOFs are in the hip and are realized through a differential drive
(Sec. II-C), which enables hip flexion/extension (HFE) and hip
abduction/adduction (HAA). The other active DOF is placed in the knee,
enabling flexion/extension (KFE). Here, we used a classical right-angled drive
to place the knee motor efficiently in the thigh (see Fig. 2). The chosen
actuators are Maxon EC 45 flat [20] in the hip and ECX Torque 22 M [21] in the
knee. We chose these motors for their high torque-to-weight/size ratio and the
availability of space-graded versions with the same specifications. The
resulting maximum joint torques, accounting for the transmission ratios, are
shown in Table I. As depicted in Fig. 2, we chose a pinion-crown gear
combination instead of classic bevel gears for all right-angled transmissions
because pinion-crown gear combinations allow for looser tolerances in axial
positioning [22]. This freedom is advantageous, considering the large
temperature fluctuations in space.

| Knee | Hip
---|---|---
| Motor | Joint | Motor | Pitch & Yaw | Pitch | Yaw
Max torque $\text{\,}\mathrm{N}\text{\,}\mathrm{m}$ | 0.078 | 0.740 | 0.152 | 1 | 2.8 | 1.5
TABLE I: Maximum achievable torques with chosen motors, as limited by winding
temperature, power, and stall torque.

###  II-C Hip Design

Many modern quadrupeds like ANYmal [10] by ANYbotics or Boston Dynamics’ Spot
[23] use a serially linked drivetrain. This drivetrain has a simple mechanical
design and high efficiency [24]. Nevertheless, for _SpaceHopper’s_ special use
case in low-gravity environments, we chose a parallel linked drivetrain in the
form of a differential drive for the hip actuation based on the following
considerations:

####  II-C1 Combined Torque for Jumping

The differential drive allows both motors to contribute to the pitch torque in
all configurations, in contrast to a serially linked drivetrain. _SpaceHopper_
can benefit significantly from the differential drive, as jumps rely on the
HFE motion for which the motors can work in conjunction and provide double the
torque (see Fig. 2). For combined pitch and yaw motions, the torques are split
up according to Equation 1, which is not critical, as combined motion only
occurs during reorientation when the loads are not significant.

$\begin{pmatrix}\gamma_{h1}\\\
\gamma_{h2}\end{pmatrix}=\begin{pmatrix}1/(2i_{hd}i_{hr}i_{hd})&1/(2i_{hp})\\\
-1/(2i_{hd}i_{hr}i_{hp})&1/(2i_{hp})\end{pmatrix}\cdot\begin{pmatrix}\tau_{p}\\\
\tau_{y}\end{pmatrix}$ (1)

The torques of the two hip motors and the pitch, yaw joint are
$\gamma_{h1},\gamma_{h2},\tau_{p},\tau_{y}$. The gear ratios of the
differential, the right-angled, and the planetary transmission are indicated
by $i_{hd},i_{hr},i_{hp}$, respectively.

####  II-C2 Static Motor Placement

Using a differential drive in the hip allows us to place two motors per leg
inside the body. This placement of the motors is very space-efficient and
eases thermal management, dust protection, and cable management for space
applications.

In order to achieve the same range of motion (ROM) in hip flexion/extension as
offered by a serially linked drivetrain, the differential drive is paired with
a right-angled, second-stage transmission, as shown in Fig. 2. The resulting
ROM is $\pm$85\text{\,}\mathrm{\SIUnitSymbolDegree}$$, which allows
_SpaceHopper_ to develop higher take-off velocities while jumping off. The hip
abduction/adduction has a ROM of
$\pm$70\text{\,}\mathrm{\SIUnitSymbolDegree}$$, while the knee has
$\pm$170\text{\,}\mathrm{\SIUnitSymbolDegree}$$. The second stage transmission
also enables fine-tuning the torque of the respective hip motions
independently. As jumping requires higher torque than reorientation,
_SpaceHopper’s_ hip pitch motion has twice the torque of the hip yaw motion
($i_{hr}=2$).

###  II-D Body Design

The triangular body shape, maximizes the ROM in the yaw DOF while minimizing
the size of the body to save weight. The internal structure is 3D printed,
which allows for high design flexibility. Thanks to this flexibility, we can
tightly integrate the body internals and use vibration dampers to decouple the
electronics stack from the load-bearing structure.

###  II-E Materials

Most of the structural parts in the legs and the main triangles of the body
are manufactured out of aerospace aluminum 7075, which has a good strength-to-
weight ratio and is commonly used in space applications [25]. The shin and the
body’s internal electronics structure consists of 3D printable carbon-
reinforced composites, which strikes a good tradeoff between the cost of
manufacturing and structural robustness. The gears and shafts are manufactured
from hardened steel (16MnCr5 and 30CrNiMo8, respectively) to endure the high
surface pressures and bending forces. Finally, the side structure is water-jet
cut out of $2\text{\,}\mathrm{mm}$ thick, layered carbon fiber for high
tensile strength and low weight.

###  II-F Electrical System

In the center of the electronics stack is a 7S1P lithium-ion battery pack
managed by a Tiny BMS s516 v2.1 battery management system [26]. We use
Lithium-ion cells because of their high power-to-weight ratio. A custom-built
power distribution board (PDB) distributes the power from the battery to the
motors and the onboard processing unit (Nvidia Jetson Nano). An STM32
microcontroller [27] connects to an internal temperature sensor, the BMS, and
the Nvidia Jetson Nano. The STM32 chip shuts down the system in case of
overheating or other failures, which adds to the system’s robustness. We use
the Nvidia Jetson Nano because it allows for efficient inference of the
control policy via its GPU cores. We use three Maxon EPOS4 Compact 24/5
EtherCAT 3-axes motor controllers [28]. They are compact and support
regenerative braking, which increases efficiency and prolongs mission
duration. The robot also has three time-of-flight laser range sensors (LRFs)
[29], each with a maximum range of $4\text{\,}\mathrm{m}$, delivering height
information.

##  III CONTROL SYSTEM OVERVIEW

###  III-A Control Architecture

The control framework of _SpaceHopper_ uses shared memory buffers that
communicate with the peripherals via the Robot Operating System (ROS) [30].
The control framework comprises three parts: a low-level controller, a high-
level controller, and a state estimator. The state estimator receives joint
states, motion capture data, and laser-ranging sensor information and uses it
to estimate the height and attitude of the main body (see Fig. 3). The high-
level controller processes the data received by the state estimator and
computes desired joint positions. These desired joint positions are sent to
the low-level controller, which maps them to desired motor positions and sends
them to the motor controllers through an EtherCAT network.

Figure 3: Dataflow visualization inside _SpaceHopper_ during reorientation
test. Green boxes show software modules running on the onboard computer, and
yellow boxes show peripheral modules.

###  III-B Locomotion Controller

Developing a robust controller for low-gravity jumping locomotion is
challenging. Due to the extended flight phases, small torques on the body
during jump-off compound to large changes in orientation. For a safe landing,
controlling the robot’s attitude during the flight phase is paramount.
However, reorientation using the legs is difficult. The kinematics of a free-
floating space manipulator are nonholonomic, and dynamic singularities can
significantly restrict the controllable workspace [31]. Model-based control
methods exist that compute approximately optimal trajectories using non-linear
optimization. However, they are computationally slow and do not scale well to
multiple legs [32]. Other methods avoid the dynamic singularities by applying
small circular motions with the leg [33]. However, these solutions result in
prolonged reorientation times. On the other hand, model-free Deep
Reinforcement Learning (DRL) has already been used successfully for free-
floating attitude control of the robot SpaceBok [18]. Thus, we build upon this
previous result and use a modified version of the proximal policy optimization
(PPO) algorithm [34, 35] to train locomotion policies for _SpaceHopper_. The
resulting locomotion policies are neural networks that take observations as
inputs and output desired joint positions. The policy’s architecture is a
Multi-Layer Perceptron (MLP) with three hidden layers of size $[512,256,128]$.

Typically, DRL-based locomotion controllers are trained in simulation and then
transferred to the real world using sim-to-real techniques [36]. We train the
policies in the IsaacGym simulation environment [37] for fast training, thanks
to its GPU acceleration. To transfer the policies to the real robot, we employ
domain randomization [38] of the robot’s link masses, the center of mass, DOF
frictions, and P&D-Gains.

Policy | Observations
---|---
Attitude Control Simulation |  $\mathbf{q}$, $\mathbf{\dot{q}}$, $\mathbf{q}_{b}$, $\bm{\omega}_{b}$, $\mathbf{a}_{t-1}$
Attitude Control Gimbal |  $\mathbf{q}$, $\mathbf{q}_{b}$, $\bm{\omega}_{b}$, $\mathbf{a}_{t-1}$
Jumping Simulation |  $\mathbf{q}$, $\mathbf{\dot{q}}$, $\mathbf{r}_{b}$, $\mathbf{r}^{*}_{b}$, $\mathbf{v}_{b}$, $\mathbf{q}_{b}$, $\bm{\omega}_{b}$, $\mathbf{a}_{t-1}$

orientation_3d: | $\|\textit{rot\\_vec($\mathbf{q_{b}}$)}\|_{2}$ | orientation_2d: | $7-(|\theta|+|\varphi|)$
---|---|---|---
action_rate: | $\|\mathbf{a}_{t}-\mathbf{a}_{t-1}\|_{2}$ | torques: | $\|\bm{\tau}\|_{2}^{2}$
dof_vel: | $\|\mathbf{\dot{q}}\|_{2}$ | dof_acc: | $\|\mathbf{\ddot{q}}\|_{2}$

collision: |  $\sum_{j}\mathbbm{1}\left(\|\mathbf{f}_{j}\|_{2}>0.2\right)$ height: $|r_{b,3}|$
---|---
pos_cmd: | $1-\|\mathbf{r}_{b}-\mathbf{r}_{b}^{*}\|_{2}/\|\mathbf{r}_{b}^{*}\|_{2}$
dof_limits: | $\|\text{ReLU}(\mathbf{q}_{l}-\mathbf{q})\|_{1}+\|\text{ReLU}(\mathbf{q}-\mathbf{q}_{u})\|_{1}$

$\mathbf{q}$: | Joint Positions |  $\mathbf{\dot{q}},\mathbf{\ddot{q}}$: | Joint Velocities and Accelerations
---|---|---|---
$\mathbf{r}_{b}$: | Body Position |  $\mathbf{v}_{b}$: | Body Velocity
$\mathbf{q}_{b}$: | Body Quaternions |  $\bm{\omega}_{b}$: | Body Angular Velocity
$\mathbf{a}_{t}$: | Actions |  $\mathbf{r}^{*}_{b}$: | Commanded Body Position
$\mathbf{q}_{l}$: | Lower DOF Limit |  $\mathbf{q}_{u}$: | Upper DOF Limit
$\bm{\tau}$: | Joint Torques |  $\mathbf{f}_{j}$: | Contact Forces of Body $j$

TABLE II: The policy observations and reward components.

##  IV ATTITUDE CONTROL

Since low gravity attitude control is a critical capability, we investigate it
separately. In a first step, we train a policy using DRL to reorient while
freely floating in zero gravity. We then validate the reorientation capability
on the real system. Creating an accurate low gravity reorientation test in
Earth’s gravity for a system with 3DOF per leg is challenging. To this end, we
developed a custom gimbal test rig that offloads the gravity from the body
while allowing it to rotate freely. We also train a DRL policy to reorient in
this test setup. While this gravity offload system substantially alters the
robot’s dynamics, it allows us to test the mechanical, electrical and control
systems during attitude control maneuvers. Moreover, it helps us assess the
sim to real performance.

###  IV-A Simulation Study

####  IV-A1 Experiment Setup

Since the robot effectively experiences weightlessness between jump-off and
landing, we simulate the robot floating in zero gravity. The robot starts in a
random initial orientation and then reorients to an upright landing
configuration.

####  IV-A2 Controller

We use a DRL policy. All policy observations and reward components are listed
in Table II. The total reward used during training is:

$\begin{split}r=c_{1}\cdot r_{\text{orientation\\_3d}}+c_{2}\cdot
r_{\text{action\\_rate}}\\\ +c_{3}\cdot r_{\text{torques}}+c_{4}\cdot
r_{\text{dof\\_limits}}\end{split}$ (2)

Where $(c_{1},c_{2},c_{3},c_{4})=(-1,-0.04,-0.15,-3)$. Here,
$r_{\text{orientation\\_3d}}$ penalizes non upright attitudes.
$r_{\text{action\\_rate}}$ prevents high-frequency oscillations in the joints,
$r_{\text{torques}}$ encourages the policy to find a low-torque solution and
$r_{\text{dof\\_limits}}$ penalizes joint positions close to the ROM limit.

####  IV-A3 Results

After training, the robot can reorient itself to an upright orientation
precisely and rapidly. Fig. 4 shows the robot changing its attitude from a
random initial orientation to upright. The policy reaches an upright
orientation within $1\text{\,}\mathrm{s}$ (see left panel of Fig. 5).
Moreover, the attitude control is robust, with a mean smallest angle
orientation error of $0.501\text{\,}\deg$ for twenty random initial
orientations, as seen in the right panel of Fig. 5. Overall, considering that
a jump with a distance of $6\text{\,}\mathrm{m}$ in Ceres gravity
($0.029\text{\,}\mathrm{g}$) takes around $8\text{\,}\mathrm{s}$ (see Fig. 8),
the reorientation is fast enough to reach an upright landing configuration.

Figure 4: _SpaceHopper_ reorienting itself to the upright orientation, which
is displayed as a black line. Figure 5: _SpaceHopper’s_ body orientation
(Euler angles $x,y,z$) in simulation during reorientation from random initial
attitude to upright (left panel), and a box plot of final orientation errors
for twenty random initial orientations (right panel).

###  IV-B Testing on Hardware

####  IV-B1 Experiment Setup

We test the capabilities of _SpaceHopper_ using a custom gimbal, allowing the
robot to rotate freely in two degrees of freedom. The inner ring can rotate
with the angle $\varphi$ and the outer ring with angle $\theta$ (see Fig. 6).
The robot starts in a random initial configuration of $\varphi$, $\theta$ and
has to reach an upright attitude. We track the body orientation and angular
velocity using a Vicon motion capture system.

####  IV-B2 Controller

We use a DRL policy. The total reward is:

$\begin{split}r=c_{1}\cdot r_{\text{orientation\\_2d}}+c_{2}\cdot
r_{\text{action\\_rate}}+c_{3}\cdot r_{\text{torques}}&\\\ +c_{4}\cdot
r_{\text{dof\\_limits}}+c_{5}\cdot r_{\text{dof\\_acc}}+c_{6}\cdot
r_{\text{dof\\_vel}}&\end{split}$ (3)

Where $(c_{1},c_{2},c_{3},c_{4},c_{5},c_{6})=(0.15,-0.06,-0.01,-1,-4\cdot
10^{-6},-0.01)$. $r_{\text{orientation\\_2d}}$ encourages the robot to be
upright. We use $r_{\text{dof\\_vel}}$ and $r_{\text{dof\\_acc}}$ to avoid
sim-to-real problems. The environments terminate when the robot collides with
itself or with a part of the gimbal.

####  IV-B3 Results

We analyze the performance of _SpaceHopper’s_ reorientation capability by
releasing the robot at twenty random initial configurations of the gimbal
angles $\varphi$ and $\theta$. _SpaceHopper_ reaches an upright attitude from
all configurations. The left panel of Fig. 7 shows the change of attitude for
one random initial configuration. It takes the robot $5\text{\,}\mathrm{s}$ to
reach the upright orientation. The mean final orientation error is
$9.7\text{\,}\deg$ over twenty random initial configurations (see right panel
of Fig. 7). Despite the significant dynamics mismatches, which we further
elaborate on in the Discussion VI-C, the promising results of this experiment
serve as a positive indicator for _SpaceHopper’s_ capabilities in future more
realistic micro-gravity experiments.

Figure 6: SpaceHopper mounted in the gimbal test stand. The inner ring angle
is $\varphi$ in orange, and the outer ring angle $\theta$ in blue. Figure 7:
_SpaceHopper’s_ body orientation (Euler angles $x,y,z$) in the gimbal during
reorientation from random initial attitude to upright (left panel), and a box
plot of final orientation errors for twenty random initial orientations (right
panel).

##  V JUMPING LOCOMOTION

Demonstrating controlled low-gravity jumping locomotion is the overarching
goal of _SpaceHopper_. Integration of jumping, attitude control, and landing
is necessary for successful jumping locomotion. We first demonstrate
controlled jumping locomotion in a simulation of Ceres
($0.029\text{\,}\mathrm{g}$). Since validation of this controller on the real
hardware in Earth’s gravity requires extensive testing facilities [39], we
restrict ourselves to showing the real robot’s jumping capability only for
vertical jumping in a counterweight setup.

###  V-A Simulation Study

####  V-A1 Experiment Setup

The simulation environment consists of a flat ground plane with the gravity of
Ceres. The robot starts on the ground and is commanded to jump to a goal body
position $\mathbf{r}_{b}^{*}$ with radius $6\text{\,}\mathrm{m}$ around the
initial position.

####  V-A2 Controller

We propose an end-to-end approach for jumping, reorientation, and landing. A
DRL policy trained simultaneously for all three tasks is not limited by
heuristic transitions between controllers. The total reward is:

$\begin{split}r=c_{1}\cdot r_{\text{pos\\_cmd}}+c_{2}\cdot
r_{\text{orientation\\_3d}}+c_{3}\cdot r_{\text{action\\_rate}}&\\\
+c_{4}\cdot r_{\text{torques}}+c_{5}\cdot r_{\text{collision}}+c_{6}\cdot
r_{\text{height}}&\end{split}$ (4)

The reward component weights have the following values:
$(c_{1},c_{2},c_{3},c_{4},c_{5},c_{6})=(1.5,-0.4,-0.05,-0.4,-15,0.19)$. Here,
$r_{\text{pos\\_cmd}}$ encourages reducing the distance to the goal position,
$r_{\text{collision}}$ penalizes collisions with the ground,
$r_{\text{orientation\\_3d}}$ enforces an upright body attitude and
$r_{\text{height}}$ incentivizes the exploration of solutions involving
jumping.

####  V-A3 Results

As shown in Fig. 9, this policy can handle jumping, reorientation, and landing
end-to-end. The robot reaches a maximum height of $2.55\text{\,}\mathrm{m}$,
stabilizes its attitude, and manages to land on its feet within
$9\text{\,}\mathrm{s}$. For 100 random position commands at a distance of
$6\text{\,}\mathrm{m}$, the policy achieves an average position error of
$0.316\text{\,}\mathrm{m}$. The worst case position error is
$0.843\text{\,}\mathrm{m}$. Most importantly, the robot’s body, thighs, and
shins never collide with the ground. Figure 8 confirms our assumption that
jumping requires higher pitch than yaw torques.

Figure 8: The yaw, pitch, and knee joint torques during jumps to 100 random
positions in simulation. The thick lines are the mean and the shaded areas are
the standard deviations. We average the joint torques from different legs. The
robots leave the ground around the $0.4\text{\,}\mathrm{s}$ mark. Figure 9:
_SpaceHopper_ in Ceres gravity, jumping to a $6\text{\,}\mathrm{m}$ far
commanded position (white line).

###  V-B Testing on Hardware

####  V-B1 Experiment Setup

_SpaceHopper’s_ drivetrain is not powerful enough to allow jumping on Earth.
We use a simple pulley-counterweight setup that attaches to the top of the
robot’s main body to offload the gravity from the body. Using a counterweight
of $3.9\text{\,}\mathrm{kg}$, we simulate a gravity of
$2.5\text{\,}\mathrm{m}\mathrm{/}\mathrm{s}^{2}$ on the body’s vertical axis.
We chose this value so that the robot does not exceed the maximum height of
the test stand while jumping.

####  V-B2 Controller

Due to the limitations of the test setup (see Subsec. VI-C), we only validate
the jumping capability of the mechanical and electrical systems. Accordingly,
we omit a full DRL-based control approach and use a simple controller that
tracks a hand-crafted joint-space trajectory with the height estimation from
the laser-ranging sensors. The trajectory consists of a leg extension to
initiate the jump and a contraction during the flight phase to prepare for the
next jump. This trajectory is the same for all three legs and gets triggered
when the robot is below a certain height.

####  V-B3 Results

Fig. 10 shows _SpaceHopper_ jumping to a height of $1.2\text{\,}\mathrm{m}$,
landing and immediately jumping off again. In this continuous jumping mode,
_SpaceHopper_ achieves at least four and up to 15 consecutive jumps before a
human has to intervene. The failure mode is a result of the open-loop
controller design. Over time, minor disturbances accumulate, which cause the
robot and rope of the counterweight to swing. After a human stabilizes the
swinging motion, the robot can resume jumping for up to $80\text{\,}\min$.

Figure 10: _SpaceHopper_ jumping continuously with a counterweight of
$3.9\text{\,}\mathrm{kg}$ being attached to the top of the body over a pulley
system.

##  VI DISCUSSION

###  VI-A Space-qualification of components

As mentioned in various sections, we designed _SpaceHopper_ towards operation
in space by using appropriate materials (aluminum 7075 and carbon fiber),
space-qualifiable motors, a differential drive design with crown gears, and
lithium-ion batteries. However, the system is not fully space-graded yet. For
example, the electronics are not radiation-shielded, and adequate thermal
insulation is vital to survive in space. A first feasibility study regarding a
thermal assessment for a lunar mission has already been done [40].
Furthermore, the system relies on an external motion capture system during
testing. Therefore, an internal state estimation is needed that can estimate
the robot’s position and velocity relative to an inertial frame. The current
sensor layout with three LRFs is inadequate to perform this task. Fusing an
IMU (accelerometer and gyroscope), a camera, a LRF, and performing Range-
Visual-Inertial Odometry (range-VIO) could be a possible solution [41, 42].

###  VI-B Safety and Robustness of DRL for Space Exploration

The cost of space missions is significant, making safe and robust controllers
a critical component of the system. Standard model-free RL has no theoretical
safety guarantees. However, as seen in our experiments and prior work [43],
DRL controllers, in practice, display excellent robustness even in highly
complex environments. Thanks to modern training techniques such as domain
randomization, resulting policies can generalize to unseen target
environments. Furthermore, safe RL is an active field of research [44]. In the
future, such findings may be integrated into control approaches for additional
theoretical guarantees.

###  VI-C Simulating low-gravity scenarios

While the gimbal and the counterweight setup allow us to test the
reorientation and jumping of SpaceHopper at Earth’s gravity, they lack in
accurately representing the robot’s dynamics in low-gravity environments. The
Earth’s gravity still acts on the legs for both gravity offload systems. The
inertia added through the gimbal test setup changes the system dynamics
further. Additionally, imperfections in balancing the gimbal lead to constant
torques acting on the robot, which increases the attitude error and leads to
longer stabilization times. Testing only vertical jumping also falls short of
validating pronking locomotion on real hardware. Moreover, the fast jumping
dynamics cause slack in the counterweight rope, which introduces additional
inaccuracies and makes sim to real transfer of DRL policies difficult. In
future work, we plan a parabolic flight testing campaign that will overcome
these limitations and further allow us to test the main locomotion goal and
increase SpaceHopper’s TRL.

###  VI-D Adaptation to Irregular Terrain

The irregular terrain of asteroids poses significant challenges for non-
articulated hopping robots like MINERVA [7]. Unpredictable bouncing after
landing severely limits precise locomotion. We hypothesize that a limbed
system can adapt its legs for a soft landing, which allows for more controlled
interactions with the surface. Future work will need to validate our control
approach in this setting, especially on soft and granular media that may
require different foot designs.

##  VII CONCLUSION AND FUTURE WORK

In conclusion, _SpaceHopper_ is a research platform to investigate highly
dynamic legged locomotion for exploring low-gravity celestial bodies, such as
asteroids and moons. Unique features of the design, such as three legs,
lightweight construction, small size, and a differential drive train, make the
system well-adapted to jumping locomotion in low-gravity. In a zero gravity
simulation, _SpaceHopper_ can change its attitude in $1\text{\,}\mathrm{s}$.
In a simulation of Ceres’ gravity, it can also jump to commanded positions at
a distance of $6\text{\,}\mathrm{m}$ with an average position error of
$0.316\text{\,}\mathrm{m}$. The real robot reaches an upright attitude within
$5\text{\,}\mathrm{s}$ with a mean orientation error of $9.7\text{\,}\deg$
inside a gimbal test setup. The mechanical and electrical systems allow
repeated vertical jumps in a counterweight setup. Due to the low-gravity
optimized design of _SpaceHopper_ , creating a fitting test on Earth is a
significant challenge. A natural follow-up of this work is to test
_SpaceHoppers’_ capabilities in the microgravity environment of a parabolic
flight. Further work is also necessary to show jumping on granular media, as
often found on asteroids and moons. Finally, the state estimation must be
extended and validated to work in the target environment.

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