# Detection of the actinide Th in an r-process-enhanced star with accretion
origin

Qianfan Xing Key Lab of Optical Astronomy, National Astronomical
Observatories, Chinese Academy of Sciences (CAS)
A20 Datun Road, Chaoyang, Beijing 100101, China  Gang Zhao Key Lab of Optical
Astronomy, National Astronomical Observatories, Chinese Academy of Sciences
(CAS)
A20 Datun Road, Chaoyang, Beijing 100101, China  Schoold of Astronomy and
Space Science, University of Chinese Academy of Sciences
No.19(A) Yuquan Road, Shijingshan District, Beijing, 100049, China  Wako Aoki
National Astronomical Observatory of Japan
2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan  Astronomical Science Program,
Graduate Institute for Advanced Studies, SOKENDAI, 2-21-1 Osawa, Mitaka, Tokyo
181-8588, Japan  Haining Li Key Lab of Optical Astronomy, National
Astronomical Observatories, Chinese Academy of Sciences (CAS)
A20 Datun Road, Chaoyang, Beijing 100101, China  Jingkun Zhao Key Lab of
Optical Astronomy, National Astronomical Observatories, Chinese Academy of
Sciences (CAS)
A20 Datun Road, Chaoyang, Beijing 100101, China  Tadafumi Matsuno National
Astronomical Observatory of Japan
2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan  Astronomical Science Program,
Graduate Institute for Advanced Studies, SOKENDAI, 2-21-1 Osawa, Mitaka, Tokyo
181-8588, Japan  Astronomisches Rechen-Institut, Zentrum für Astronomie der
Universität Heidelberg, Mönchhofstraße 12-14, 69120 Heidelberg, Germany
Takuma Suda Department of Liberal Arts, Tokyo University of Technology, Nishi
Kamata 5-23-22, Ota-ku, Tokyo 144-8535, Japan  Research Center for the Early
Universe, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033,
Japan

###### Abstract

The thorium and six second-peak r-process element ($56\leq$ Z $\leq 72$)
abundances are determined for the $\alpha$-poor star LAMOST J1124+4535 based
on a high-resolution spectrum obtained with the High Dispersion Spectrograph
(HDS) on the Subaru telescope. The age of J1124+4535 is 11.3 $\pm$ 4.4 Gyr
using thorium and other r-process element abundances. J1124+4535 is confirmed
to be a Galactic halo metal-poor ($\mbox{[Fe/H]}={-1.27}$$\pm$0.1) star with
extreme r-process element over-abundance ([Eu/Fe] = 1.13$\pm$0.08) and
$\alpha$ element deficiency ([Mg/Fe] = -0.31$\pm$0.09) by the LAMOST-Subaru
project. Along with the sub-solar $\alpha$ to iron ratios (e.g. [Mg/Fe],
[Si/Fe], [Ca/Fe]), the relatively low abundances of Na, Cr, Ni and Zn in
J1124+4535 show significant departure from the general trends of the Galactic
halo but are in good agreement with those of dwarf galaxies. The chemical
abundances and kinematics of J1124+4535 suggest it was formed in the late
stage of star formation in a dwarf galaxy which has been disrupted by the
Milky Way (MW). The star formation of its progenitor dwarf galaxy lasted more
than 2 Gyr and has been affected by a rare r-process event before the
occurrence of accretion event.

Galaxy: halo - abundances - stars: abundances - stars: atmospheres

††journal: ApJ††facilities: LAMOST, Subaru††software: IRAF (Tody, 1986, 1993),
MOOG (v2017; Sneden 1973), galpy (Bovy, 2015)

##  1 Introduction

The rapid neutron-capture (r-) and slow neutron-capture (s-) processes are
mainly responsible for the nucleosynthesis of the elements heavier than iron.
For decades, the origin of r-process elements and the main astrophysical site
of r-process are still debated (Shen et al., 2015; van de Voort et al., 2015,
2020; Cescutti et al., 2015; Safarzadeh et al., 2019; Haynes & Kobayashi,
2019). The proposed viable site for production of r-process elements include
neutron star mergers (NSMs) and subsets of core-collapse supernovae (CCSNe),
including CCSNe driven by neutrino-driven wind (Wanajo et al., 2018) and
magneto-rotational instability SNe (Winteler et al., 2012). Most notably, the
gravitational wave event GW170817 (Abbott et al., 2017) is confirmed to be
merging of two neutron stars in a binary system. The follow-up observations to
obtain light curve and spectra for this event have identified r-process
elements (Sr and lanthanides) in the ejecta of NSM (Watson et al., 2019;
Chornock et al., 2017; Domoto et al., 2022), providing strong supports for NSM
as the most promising r-process site. However, it is not clear whether NSMs
could occur frequently enough at the early stage of the MW and produce
sufficient r-process yields for the r-process enrichment observed in very
metal-poor stars without other site of r-process (Argast et al., 2004).

The records of the nucleosynthesis of heavy elements via r-process events are
imprinted on the chemical abundances of long-lived metal-poor stars. A great
deal of observational research (e.g. Hansen et al. 2018; Honda et al. 2004;
Zhao & Magain 1991) has been performed for understanding the main
astrophysical site of r-process by analyzing the detailed chemical abundances
of metal-poor stars with extreme r-process enhancement, such as so-called r-II
stars (Beers & Christlieb, 2005) that have Eu/Fe ratios higher than solar
ratios by more than 10 times ([Eu/Fe] $>+1.0$)111 The standard notations
[X/Y]$=\log(N_{\rm X}/N_{\rm Y})-\log(N_{\rm X}/N_{\rm Y})_{\odot}$ and $\log
A({\rm X})=\log(N_{\rm X}/N_{\rm H})+12$ for elements X and Y are adopted in
this work.. With the effort of the R-Process Allicance (RPA; Hansen et al.
2018) project, the number of r-II star has increased to over 60. The studies
of chemical abundances of such stars like CS 22892-052 (Sneden et al. 1994;
[Eu/Fe] = +1.6 and [Fe/H] = -3.1) have revealed that the abundances pattern of
the heavy neutron-capture elements (at least for Ba to Hf) in r-II stars from
the Galactic halo and satellite dwarf galaxies (Aoki et al., 2007; Ji et al.,
2016) agree well with that of the solar system r-process component, suggesting
the r-process pattern is universal and may result from a dominating
astrophysical site.

Several theoretical research aimed to test whether NSM could be the main site
of r-process by comparing the Eu abundance trend in Galactic halo stars with
simulation results from galactic chemical evolution (GCE) models. Early
studies (e.g. Argast et al. 2004) found that the NSM could not well explain
the distribution and scatter of [Eu/Fe] vs. [Fe/H], because the timescale of
mergers of binary neutron stars is longer than that of metal enrichment in the
early Galaxies in these models. However, recent studies have revealed that the
satellite dwarf galaxies around MW are important building blocks for the
formation of the Galactic halo (Helmi, 2020; Zhao & Chen, 2021; Zhao et al.,
2009). The contribution of the dwarf galaxies should be considered in the
theoretical analysis for the Eu abundance trend in the halo (Tsujimoto &
Shigeyama, 2014; Ishimaru et al., 2015). In recent years, dozens of r-I and
r-II stars have been found in stellar streams (e.g. Hansen et al. 2021),
classical dwarf galaxies, and even in ultra faint dwarf galaxies (UFD). Ji et
al. (2016) have discovered 7 r-rich stars with [Fe/H] $<-2$ in Reticulum II
(Ret II) and concluded a rare r-process event such as NSM is responsible for
the r-process enhancement in this UFD. Since most of r-II stars found in the
early researches have metallicities in a relatively narrow region ([Fe/H]
$<-2.5$ ; Sneden et al. 2008), overlapping with the metallicities of UFD
stars, UFDs, such as Ret II, are suggested to be the source of r-rich stars in
the Galactic halo. This scenario is supported by the kinematic analysis of 35
r-rich halo stars by Roederer et al. (2018a), which showed that most of them
do not follow the Galactic rotation and are clustered in the phase space and
metallicity. Furthermore, Roederer et al. (2018b) and Xing et al. (2019) have
found two r-II stars in medium metal-poor ([Fe/H] $\sim-1.4$) halo stars.
Their metallicities are higher than the r-rich stars in UFDs, but in
consistent with r-rich stars in classical dwarf galaxies, such as Ursa Minor
(UMi) and Fornax (Reichert et al., 2021). Notably, the r-II star found by Xing
et al. (2019) exhibit sub-solar [X/Fe] ratios for $\alpha$ elements (e.g. Mg,
Si, Ca, and Ti) at [Fe/H] $<-1$, which are well-known chemical labels for
identifying accreted halo stars (e.g. Helmi 2020). Similar $\alpha$-poor stars
with extreme r-process enhancements have been identified by the following
high-resolution spectroscopic observations of the stars in the Galactic halo
(e.g. Ezzeddine et al. 2020; Sakari et al. 2019) and satellite dwarf galaxies
(e.g. Reichert et al. 2021), suggesting dwarf galaxies are potential source of
r-rich halo stars.

In this paper we obtain the abundances of 12 neutron-capture elements for
exploring the abundance pattern of neutron-capture elements for J1124+4535
(Xing et al., 2019), along with measurement of age from Th abundance.
J1124+4535 is selected from the LAMOST survey as candidate $\alpha$-poor
stars. We have performed twice follow-up high-resolution spectroscopic
observations with different resolution and wavelength ranges for J1124+4535 as
part of observation task in the LAMOST-Subaru project (Aoki et al., 2022; Li
et al., 2022). Based on the first spectrum (R $\sim 45,000$) obtained with
Subaru/HDS on 2017 February 27, our previous work (Xing et al., 2019)
confirmed J1124+4535 is a Galactic halo star with extremely enhanced r-process
element abundances and sub-solar [$\alpha$/Fe] ratio. The spectrum with higher
resolution (R $\sim 60,000$) used in this paper was obtained on 2019 June 9,
covering 3500 to 5250 Å wavelength range. The determination of chemical
abundances using those spectra is presented in Section 2. In Section 3 we
discuss the r-process pattern and measurement of age from the Th abundance.
The kinematic analysis based on the updated parallax and proper motions from
Gaia DR3 (Gaia Collaboration et al., 2023) is also included in Section 3.

##  2 Observations and data analysis

LAMOST J1124+4535 was identified as a metal-poor star to have a relatively low
Mg abundance from a sample of candidate $\alpha$-poor stars (Xing et al.,
2015) observed by LAMOST spectroscopic survey (Zhao et al., 2006). All of
these candidates were found to have sub-solar [Mg/Fe] based on spectrum
synthesis analysis of Mg Ib lines from spectra with a resolving power $R\sim
1800$ (Xing & Zhao, 2014).

A high-resolution spectroscopic observation of J1124+4535 took place on 2017
February 27 using HDS on the Subaru Telescope (Noguchi et al., 2002). The
high-resolution spectrum covers the wavelength range 4000-6800 Å with a gap of
5330-5430 Å. The resolving power R $\sim 45,000$ is obtained using a
0${}^{{}^{\prime\prime}}$.8 slit and 2 x 2 CCD pixel binning. The signal-to-
noise ratios (S/N) per pixel at 4300 Å and 5000 Å are 50 and 70, respectively.
Based on this spectrum, Xing et al. (2019) revealed the r-process enhancement
and deficiencies of Na and $\alpha$ elements in J1124+4535 with its
metallicity of [Fe/H] $=-1.27$. The highly enhanced Eu abundance ([Eu/Fe]
$>1$) and sub-solar [Mg/Fe] make this star to be the first known $\alpha$-poor
r-II star.

In order to obtain detailed r-process pattern of J1124+4535, we observed
J1124+4535 again with Subaru/HDS on 2019 June 9 with 90 minutes exposure. The
high-resolution spectrum covers the wavelength range 3500-5250 Å with a
resolving power of $R\sim 60,000$ using a 0${}^{{}^{\prime\prime}}$.6 slit and
2 x 2 CCD pixel binning. The S/N ratios in the continuum range are 80 per
pixel at 4000 Å and 100 per pixel at 4500 Å. The portion of the spectrum of
J1124+4535 covering the absorption line of Th at 4019 Å is presented in the
top panel of Figure 1. The standard data reduction was performed with
Subaru/HDS pipeline222https://www.naoj.org/Observing/Instruments/HDS/ based on
IRAF333IRAF is distributed by the National Optical Astronomy Observatories,
which is operated by the Association of Universities for Research in
Astronomy, Inc. under cooperative agreement with the National Science
Foundation., including overscan subtraction, bias subtraction, linearity
correction, cosmic-ray rejection, scattered light subtraction, spectrum
extraction and wavelength calibration. The radial velocity (RV) of J1124+4535
is measured from Fe I lines that are used for abundance analysis with
equivalent width (EW) measurements. The heliocentric RV derived from the high-
resolution spectra, 54.8$\pm$0.5 km s-1, is in good agreement with that
derived from the LAMOST low-resolution spectrum (57.1$\pm$2.5 km s-1),
indicating there is no evidence of RV variations caused by unknown companion.
The EWs are measured with Splot task in IRAF package by fitting Gaussian or
Voigt line profiles relative to the normalized continuum. We compare the EWs
of more than 50 lines in common between the two high-resolution spectroscopic
observations and find no obvious difference. The mean EW difference is 4.5 mÅ.
We here adopt the stellar parameters (Teff = 5180 K, log $g$ = 2.7, $v_{\rm
t}$ = 1.5 km s-1) and iron abundance ([Fe/H] = -1.27) determined from the
previous high-resolution spectroscopic observation as reported in Xing et al.
(2019). The basic information of J1124+4535 are given in Table 1.

Table 1: Stellar Parameters and kinematics Parameter | Symbol | Value | References
---|---|---|---
RA |  $\alpha$ (J2000) | 11:24:56.61 | Gaia DR3
DEC |  $\delta$ (J2000) | 45:35:31.32 | Gaia DR3
Parallax | $\varpi$ | 0.09 $\pm$ 0.02 mas | Gaia DR3
Distance | $D$ | 7653.22852${}^{+1083.5}_{-838.5}$ pc | Bailer-Jones et al. 2021
Proper motion ($\alpha$) | PMRA | -0.83 $\pm$ 0.01 mas yr-1 | Gaia DR3
Proper motion ($\delta$) | PMDEC | -6.72 $\pm$ 0.01 mas yr-1 | Gaia DR3
Effective temperature | Teff | 5180 $\pm$ 100 K | Xing et al. 2019
Surface gravity | log $g$ | 2.7 $\pm$ 0.3 cgs | Xing et al. 2019
Microturbulent velocity | $v_{\rm t}$ | 1.5 $\pm$ 0.3 km s-1 | Xing et al. 2019
Metallicity | [Fe/H] | -1.27 $\pm$ 0.1 | Xing et al. 2019
Orbital pericentric radius | Rperi | 4.2${}^{+1.42}_{-0.48}$ kpc | this study
Orbital apocentric radius | Rapo | 13.5${}^{+0.94}_{-0.73}$ kpc | this study
Maximum height | Zmax | 13.1${}^{+0.17}_{-1.29}$ kpc | this study
Eccentricity | $e$ | 0.53${}^{+0.02}_{-0.09}$ | this study

Based on the newly obtained high-resolution spectrum covering the wavelength
range from 3500 Å to 5250 Å, we are able to determine abundances for 12 new
elements heavier than Zn, including Mo I, Ru I, Rh I, Ho II, Er II, Tm II, Yb
II, Lu II, Hf II, Os I, Ir I and Th II. The Mo I line at 3864 Å was used to
determine the Mo abundance, and the Ru I line at 3798 Å was identified for
measuring Ru abundance. The Rh abundance was derived from the 3700 and 3958 Å
lines, while Ho abundance was derived from 3810, 3890, and 4045 Å lines. The
three Er II lines at 3692, 3729, and 3830 Å were used to estimate the
abundance of Er. The Tm, Yb, and Lu abundances were determined using the 3701
and 3795 Å Tm II lines, the 3694 Å Yb II line, and the 3507 Å Lu II line,
respectively. The 3719 and 4093 Å Hf II lines, the 4135 Å Os I line, and the
3800 Å Ir I line were also used to determine the abundances. Chemical
abundances for these elements were determined from EW measurements and
spectral synthesis using the 2017 version of MOOG (Sneden, 1973) and the 1D
plane-parallel model atmospheres (Castelli & Kurucz, 2003) under the
assumption of local thermodynamic equilibrium (LTE). The excitation potential
and oscillator strength for lines used for abundance determination are adopted
from Ivans et al. (2006) and Roederer et al. (2018b), while atomic data and
isotopic ratios from Sneden et al. (2008) and Kurucz & Bell (1995) are
employed for spectral synthesis of lines with hyperfine structure and isotopic
splitting.

Th abundance is derived from a single line at 4019 Å. This line is blended by
CH, Fe, Ni and Ce. We make use of the spectrum synthesis method and the line
list in Mashonkina et al. (2014) to determine the Th abundance, considering
the affection introduced by these blends. As shown in the bottom panel of
Figure 1, the abundance log $\epsilon$(Th)= $-0.19$ is adopted by the best
fitting synthetic spectrum. The synthetic spectra with abundance variations of
$\pm 0.2$ are also shown for comparison. Another Th II line at 3741 Å is also
detected in the spectrum. The Th abundance derived from this line is 0.05 dex
higher than that from Th II 4019 Å. Taking account of the weakness of the
lines and severe blending with Ti I and Sm II lines, we adopt log
$\epsilon$(Th)= $-0.19\pm 0.09$ from the Th II line at 4019 Å for age
determination. The error in the Th abundance is composed of fitting error and
systematic uncertainties due to changes in stellar parameters. The feature of
U line at 3859 Å is used to derive the upper limit for U (log $\epsilon$(U)
$<-0.85$) based on the spectrum synthesis method. The abundance ratios for
J1124+4535 derived from the Subaru/HDS spectrum are listed in Table 2. The
abundance of Fe I and solar abundances from literature (Asplund et al., 2009)
are adopted for obtaining [X/Fe]. The uncertainties in measurements and
stellar parameters are considered for the abundance uncertainties. The total
error ($\sigma_{total}$) in Table 2 is the quadratic sum of the random error
due to the uncertainty of measurement and systematic errors ($\sigma_{sys}$)
due to the stellar parameter uncertainties ($\delta$$T_{\rm eff}$ = 100 K,
$\delta$log g = 0.3 dex, and $\delta$$v_{\rm t}$ = 0.3 km s-1). The random
error is estimated to be $\sigma$υ$N$${}_{\rm lines}^{-1/2}$, where $\sigma$υ
is the dispersion around the mean abundance obtained from different lines for
each species, and $N$lines is the number of lines used to derive abundance.
The $\sigma$υ of Fe I is adopted for species with $N$lines $<$ 3\. For
elemental abundances measured by the spectrum synthesis method, the errors are
estimated based on comparison with synthetic spectra ($\Delta$$\chi^{2}$
$=1$).

Figure 1: Top panel: Selected region of the spectrum of J1124+4535 covering
the Th line at 4019 Å. Bottom panel: The Eu and Th abundances are determined
by spectrum synthesis method. The isotope ratios from the r-process component
of the Solar System (Arlandini et al., 1999) are adopted for Eu. The observed
spectra are plotted as filled circles. The synthetic spectra with best-fit
abundances are shown in solid lines. The red dashed lines are synthetic
spectra with abundance variations of $\pm 0.2$ dex. Table 2: Chemical
abundances of J1124+4535

Element X | $N_{\rm lines}$ | log $\epsilon$ | $\sigma_{sys}$ | $\sigma_{total}^{a}$ | [X/Fe] | $\sigma_{total}$
---|---|---|---|---|---|---
Co I | 6 | 3.70 | 0.20 | 0.21 |  $-$0.03 | 0.08
Mo I | 1 | 1.01 | 0.16 | 0.16 | 0.15 | 0.12
Ru I | 1 | 1.25 | 0.16 | 0.17 | 0.77 | 0.14
Rh I | 2 | 0.57 | 0.15 | 0.15 | 0.93 | 0.13
Ba II | 5 | 1.24 | 0.15 | 0.16 | 0.24 | 0.12
Eu II | 1 | 0.38 | 0.11 | 0.12 | 1.13 | 0.08
Ho II | 3 | 0.30 | 0.17 | 0.18 | 1.09 | 0.16
Er II | 3 | 0.58 | 0.15 | 0.16 | 0.93 | 0.12
Tm II | 2 | -0.24 | 0.13 | 0.14 | 0.93 | 0.12
Yb II | 1 | 0.37 | 0.19 | 0.20 | 0.80 | 0.16
Lu II | 1 | -0.16 | 0.15 | 0.16 | 1.01 | 0.12
Hf II | 2 | 0.29 | 0.12 | 0.13 | 0.71 | 0.09
Os I | 1 | 1.19 | 0.14 | 0.14 | 1.06 | 0.10
Ir I | 1 | 1.11 | 0.13 | 0.14 | 1.0 | 0.10
Pb I | 1 |  $<$ 1.4 | $\cdots$ | $\cdots$ |  $<$ 0.92 | $\cdots$
Th II | 1 | -0.19 | 0.14 | 0.15 | 1.06 | 0.09
U II | 1 |  $<$ -0.85 | $\cdots$ | $\cdots$ |  $<$ 0.96 | $\cdots$

aafootnotetext: The total error is the quadratic sum of random error and
systematic error ($\sigma_{sys}$) due to the uncertainties on the stellar
parameters ($T_{\rm eff}$, log g, and $v_{\rm t}$). Errors in log $\epsilon$
and [X/Fe] are presented separately.

##  3 Discussion

###  3.1 R-process pattern

Figure 1 illustrates the highly enhanced Eu ([Eu/Fe] = 1.13$\pm 0.1$) in
J1124+4535 with the best fitting synthetic spectrum. Such high enhancements of
Eu with respect to Fe are usually found in very metal-poor stars ([Fe/H]
$<-2$), and are very rare in metal-poor stars with [Fe/H] $>-1.5$, making
J1124+4535 to be one of the two known moderately metal-poor r-II stars in the
MW field halo stars. The other one is HD 222925 ([Eu/Fe] = +1.33, [Fe/H] =
-1.47) discovered by Roederer et al. (2018b). The abundance pattern of
neutron-capture elements measured for J1124+4535 is shown in Figure 2 compared
with the scaled solar r-process pattern. The r-process pattern in solar system
is determined by the abundance residual of solar abundances from which the
s-process abundances are subtracted (Arlandini et al., 1999).

The abundances in J1124+4535 agree well with the scaled solar r-process
pattern (see Figure 2), in consistent with that of other r-II stars. The mean
residual is 0.07 dex with a standard deviation of 0.08 dex, smaller than the
measured abundance uncertainties for r-process elements. This agreement is
prominent for the second-peak elements (56 $\leq$ Z $\leq$ 72) and extends to
the third-peak element (76 $\leq$ Z $\leq$ 88), as well as the actinide
element Th. However, the abundances of light neutron-capture elements Sr, Y,
Zr, Mo, Ru and Rh (first r-process peak elements) in J1124+4535 are
significantly lower than that of solar r-process pattern, in contrast to the
heavier neutron-capture elements. Similar deficiencies of the first-peak
elements have been widely found in other r-process-enhanced stars from the
Galactic halo and dwarf galaxies (see the bottom panel of Figure 2). A
possible explanation is that the light neutron-capture elements, such as Sr, Y
and Zr in the solar system material, are mainly produced by the weak r-process
(Siqueira Mello et al., 2014) rather than the main r-process. The so-called
weak r-process cannot synthesize elements heavier than the first-peak elements
(38 $\leq$ Z $\leq$ 48).

The actinide Th abundance in J1124+4535 ([Fe/H] = -1.27) is in good agreement
with the solar r-process pattern (see Figure 2), similar to the other r-II
star HD 222925 with [Fe/H] = -1.47. The log $\epsilon$(Th/Eu) abundances
(around -0.5) of these two stars are common among the majority of r-II stars.
A noteworthy dispersion among log $\epsilon$(Th/Eu) has been found in very
metal-poor and extremely metal-poor stars. Such dispersion is mainly caused by
“actinide boost” stars with highly enhanced Th (Hill et al., 2002). The origin
of these stars are still unknown and more analysis of Th abundances for metal-
poor stars are required.

Figure 2: Top panel: neutron-capture elemental-abundance pattern for
J1124+4535 in comparison with a scaled solar r-process pattern (Arlandini et
al., 1999), normalized to Ba. Bottom panel: differences between the observed
abundances and the scaled solar r-process pattern for J1124+4535, CS 31082-001
(Hill et al., 2002; Sneden et al., 2008) and RAVE J2038-0023 (Placco et al.,
2017).

###  3.2 Determination of Age

Th has a half-life of 14.05 Gyr and allows us to determine the age of
J1124+4535 through radioactive-decay dating, with assumption that the
r-process material is produced by a single r-process event soon before the
formation of J1124+4535. Based on the log $\epsilon$(Th/r) derived from the
observed spectrum and initial produce ratios (such as log
$\epsilon$(Th/Eu)initial = -0.33 $\pm$ 0.12) given by Schatz et al. (2002),
the age of the r-process event and J1124+4535 is calculated by

$\text{t}=46.67[\log\epsilon(\rm{Th/r})_{\rm initial}-\log\epsilon(Th/r)_{\rm
now}]~{}Gyr~{}~{}~{}~{}~{}~{}(1)$

where $\log\epsilon(\rm{Th/r})_{\rm initial}$ is an initial production ratio,
while $\log\epsilon(\rm{Th/r})_{\rm now}$ is the observed value after the
decay of radioactive Th produced by the r-process event. The resulting age is
very sensitive to the uncertainties of the elemental abundances. So we only
use stable r-process elements with relatively low uncertainties for the
measurement of age. The observed Eu, Hf, Os and Ir abundances are derived by
the spectrum synthesis method and have relatively small errors. We make use of
abundance ratios Th/Eu, Th/Hf, Th/Os and Th/Ir for age estimates, yielding
11.2 $\pm$ 4 Gyr, 13.1 $\pm$ 4.2 Gyr, 10.3 $\pm$ 4.4 Gyr and 10.7 $\pm$ 4.4
Gyr, respectively. The age uncertainties resulting from abundance
uncertainties are listed in Table 3. The average age of 11.3 Gyr is very close
to the age determined from Th/Eu. The uncertainty of the adopted age is 4.4
Gyr including the abundance uncertainty and the dispersion around the mean
age. Finally, we adopt an age of 11.3 $\pm$ 4.4 Gyr for J1124+4535 by
averaging the ages determined from Th/Eu, Th/Hf, Th/Os and Th/Ir ratios. The
uncertainties of the production ratios, such as 0.12 dex for Th/Eu chronometer
in Schatz et al. (2002), are not included on the uncertainties of the age.
They could result in an age uncertainty of 5.6 Gyr.

The upper limit of U abundance of J1124+4535 can be used to determine a lower
limit on the age of this star by

$\text{t}=21.8[\log\epsilon(\rm{U/Th})_{\rm initial}-\log\epsilon(U/Th)_{\rm
now}]~{}Gyr~{}~{}~{}~{}~{}~{}(2)$

where $\log\epsilon(\rm{U/Th})_{\rm initial}$ is an initial production ratio,
while $\log\epsilon(\rm{U/Th})_{\rm now}$ is the observed value. A lower limit
on the age of 9.6 Gyr has been derived using the initial production ratio
($log\epsilon(U/Th)_{initial}$ = -0.22 $\pm$ 0.10) from Schatz et al. (2002).
The derived lower limit age is consistent with the age determined from the
observed Th/r ratios.

The relatively low abundances of Na, Cr, Ni, Zn and $\alpha$ elements in
J1124+4535 exhibit significant departure from the general trends of the
Galactic halo but are in good agreement with abundance trends of UMi dwarf
galaxy (see Figure 3 in Xing et al. 2019 for detail), suggesting J1124+4535
was accreted from a disrupted dwarf galaxy which is similar to the surviving
dwarf galaxy UMi. The r-II star UMi COS 82 (Aoki et al., 2007) has been found
in UMi dwarf galaxy. Its metallicity ([Fe/H] = -1.42) and Eu abundance ([Eu/H]
= -0.18) both are close to those of J1124+4535 ([Fe/H] = -1.27 and [Eu/H] =
-0.14). On the other hand, we could calculate the mass of progenitor stellar
system for J1124+4535 based on its Eu abundance and r-process mass yielded by
a r-process event (Tsujimoto et al., 2017), assuming the r-process material in
J1124+4535 are produced by a single event. It turns out the Eu abundance of
J1124+4535 can be explained by ejecting of 10-4.5±1M⊙ Eu into a progenitor
system with a stellar mass $\sim$ 105M⊙. The progenitor system could be an
independent dwarf galaxy or part of a large one, reaching up to a mass of
$\sim$ 106M⊙. The resulting mass of the progenitor system is similar to the
mass of UMi dwarf galaxy (2.9x105M⊙, McConnachie 2012). In addition, Cohen &
Huang (2010) concluded the star formation of UMi dwarf galaxy lasted about 3
Gyr. The stars formed at the late phase (about 11 Gyr ago) in UMi dwarf galaxy
exhibit sub-solar [$\alpha$/Fe] ratio due to the contribution of SN Ia (Helmi
2020; Xing & Zhao 2015), as shown in Figure 3. This closely matches with the
case of J1124+4535, which has a sub-solar [$\alpha$/Fe] ratio and age of 11.3
$\pm$ 4.4 Gyr.

Table 3: Abundance and Age Uncertainties |  $\Delta$Teff |  $\Delta$log $g$ |  $\Delta$$v_{\rm t}$ | $\sigma_{random}$ | $\sigma_{total}$ | $\sigma_{age}$
---|---|---|---|---|---|---
Th/Eu | 0.01 | 0.03 | 0.04 | 0.07 | 0.0866 | 4.0
Th/Hf | 0.01 | 0.01 | 0.04 | 0.08 | 0.0906 | 4.2
Th/Os | 0.04 | 0.06 | 0.01 | 0.06 | 0.0943 | 4.4
Th/Ir | 0.05 | 0.05 | 0.01 | 0.06 | 0.0933 | 4.4

###  3.3 Kinematics

As the parallax of J1124+4535 provided by Gaia DR2 has an error over 40$\%$,
we could not derive any conclusion from the kinematic analysis in our previous
work (Xing et al., 2019). The Gaia DR3 has been released on 2022 June 13 and
updated the parallax for J1124+4535. The new parallax ($\varpi$ = 0.09 $\pm$
0.02 mas) has an error smaller than 25$\%$, resulting a more accurate distance
of 7653.23${}^{+1083.5}_{-838.5}$ pc (Bailer-Jones et al., 2021). Together
with the proper motions from Gaia DR3, we make use of the newly derived
distance and heliocentric radial velocity to calculate the Galactic velocity
component (U,V,W) and orbital parameters for J1124+4535. The MWPotential2014
from galpy (Bovy, 2015) is adopted for the gravitational potential of the MW
in order to derive the properties of the orbit, including pericentric radius
(Rperi), apocentric radius (Rapo), maximum height above the Galactic plane
(Zmax) and eccentricity of the orbit (see Table 1). The results indicate
J1124+4535 has a very radial orbit with a eccentricity of
0.53${}^{+0.02}_{-0.09}$. The orbit allow J1124+4535 to travel from a
4.2${}^{+1.42}_{-0.48}$ kpc area near the Galactic center to a region
13.5${}^{+0.94}_{-0.73}$ kpc away from the Galactic center. The bottom panel
of Figure 3 shows a Toomre diagram for J1124+4535 and Galactic halo and thick-
disk stars from LAMOST DR3 (Xing & Zhao, 2018). Interestingly, Hattori et al.
(2023) has analyzed the orbital actions of 161 r-II stars and found that
J1124+4535 belongs to a r-II cluster named H22:DTC-17. This cluster is
dynamically associated with the LMS-1/Wukong merger group (Malhan et al.,
2022) and includes an r-II star associated with the Indus stream, which is
suggested to be the remnant of a disrupted dwarf galaxy with mass similar to
that of UMi (Ji et al., 2020). The Indus stream, along with the dwarf-galaxy
stellar stream LMS-1/Wukong (Yuan et al., 2020; Naidu et al., 2020), exhibits
a clumpy distribution in the orbital action and energy space, suggesting the
Indus stream likely originated from the satellite dwarf galaxy of the
progenitor LMS-1/Wukong galaxy (Malhan et al., 2022). Limberg et al. (2023)
determined chemical abundances for 14 stars in LMS-1/Wukong and provided
further evidence for this hypothesis. The trends of [Mg/Fe] and [Eu/Fe] versus
[Fe/H] in LMS-1/Wukong are shown in Figure 3 for comparison. Figure 4 shows
the orbital energy and cylindrical actions of $\sim 1700$ r-process-enhanced
([Eu/Fe] $>+0.3$) stars (Shank et al., 2023) and four known MW substructures
such as Gaia-Sausage-Enceladus (GSE, Helmi et al. 2018), Helmi stream (Helmi &
White, 1999), LMS-1/Wukong and Thamnos (Koppelman et al., 2019). The cluster
associated with J1124+4535 is very close to the members of LMS-1/Wukong merger
group. The eccentricity (see Figure 4) of this cluster (e = 0.493 $\pm$ 0.097)
is also consistent with that of LMS-1/Wukong (e = 0.497 $\pm$ 0.094). The
members of this cluster are highly likely to have been brought into the MW by
the same merger event involving LMS-1/Wukong merger group. Identifying more
stars with a common origin with this cluster will be essential for exploring
the properties of their progenitor dwarf galaxy.

Figure 3: Top and middle panels show the [Mg/Fe] and [Eu/Fe] ratios of
J1124+4535 (red filled circle) and stars in the classical dwarf galaxy UMi
(red open stars) and the stellar stream LMS-1/Wukong (yellow filled circles),
in comparison with those of Galactic stars (gray and blue filled circles). The
bottom panel is the Toomre diagram for J1124+4535 (red filled circle). The
Galactic halo (gray filled circles) and thick-disk stars (black crosses) from
LAMOST DR3 (Xing & Zhao, 2018) are shown for comparison. Figure 4: Comparing
the action, energy, and eccentricities (Shank et al., 2023) of dynamically
tagged cluster (DTC) associated with J1124+4535 with the known MW
substructures such as Gaia-Sausage-Enceladus (GSE, Helmi et al. 2018), Helmi
stream (Helmi & White, 1999), LMS-1/Wukong and Thamnos (Koppelman et al.,
2019).

##  4 Summary

We present a detailed abundance analysis of the first $\alpha$-poor r-II star
discovered from the LAMOST survey, J1124+4535, based on the newly obtained
high-resolution spectrum with Subaru/HDS. The abundances of 12 neutron-capture
elements are determined for the analysis of r-process pattern of J1124+4535,
including six elements in the second r-process peak and actinide Th. This is
the first time we have derived radioactive-decay ages from Th/r abundance
ratios for an $\alpha$-poor halo star. The abundance ratios of Th/Eu, Th/Hf,
Th/Ir and Th/Os are adopted for the age determination, resulting an age of
11.3 $\pm$ 4.4 Gyr for J1124+4535.

The abundances of light neutron-capture elements (Sr, Y, Zr, Mo, Ru and Rh) in
J1124+4535 are relatively low in comparison with the solar r-process pattern
while the abundance pattern of heavy elements from Ba to Th closely match the
scaled solar r-process pattern. Since the light neutron-capture elements can
be produced by the weak r-process, the deviations from the scaled r-process
pattern for light neutron-capture elements indicate that the r-process
elements in J1124+4535 is dominated by a single r-process event (such as a
NSM) corresponding to the main r-process site. In that case, we may conclude
J1124+4535 is originated from a dwarf galaxy with a stellar mass $\sim$ 105M⊙,
comparable to the classical dwarf galaxy UMi. Such satellite classical dwarf
galaxies are almost as old as the MW. The $\alpha$-poor stars with sub-solar
[Mg/Fe] ratios, such as J1124+4535 ([Mg/Fe] = -0.31, [Fe/H] = -1.27), are
expected to be formed at the late phase of star formation history of its
progenitor dwarf galaxy. The age of J1124+4535 is consistent with this
scenario that the star formation in the progenitor dwarf galaxy lasted more
than 2 Gyr. More spectroscopic and kinematic analysis of r-process-enhanced
stars will help us to identify more accreted components in the Galactic halo.

The authors thank the referee for helpful comments and suggestions. The
authors also thank Derek Shank for providing the dynamical parameters of
r-process-enhanced stars. This work was supported by the National Natural
Science Foundation of China grant Nos. 11988101, the JSPS-CAS Joint Research
Program, and the science research grants from the China Manned Space Project
with No. CMSCSST-2021-B05. Q.X. and H.L. acknowledge support from the Youth
Innovation Promotion Association of the CAS (id. 2020058 and Y202017) and the
Strategic Priority Research Program of Chinese Academy of Sciences, grant No.
XDB34020205. Q.X. is supported by the Beijing Municipal Natural Science
Foundation grant No. 1242031. This work was also supported by JSPS KAKENHI
Grant Numbers 21H04499, 22K03688, and 23HP8014. This research is based on data
collected at Subaru Telescope, which is operated by the National Astronomical
Observatory of Japan. We are honored and grateful for the opportunity of
observing the Universe from Maunakea, which has the cultural, historical and
natural significance in Hawaii. Guoshoujing Telescope (the Large Sky Area
Multi-Object Fiber Spectroscopic Telescope, LAMOST) is a National Major
Scientific Project built by the Chinese Academy of Sciences. Funding for the
project has been provided by the National Development and Reform Commission.
It is operated and managed by the National Astronomical Observatories, Chinese
Academy of Sciences. This work has made use of data from the European Space
Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the
Gaia Data Processing and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement.

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# Temporal Convolutions for Multi-Step Quadrotor Motion Prediction

Samuel Looper1 and Steven L. Waslander2

1Engineering Science, University of Toronto, Canada,
<EMAIL_ADDRESS>for Aerospace Studies, University of
Toronto, Canada<EMAIL_ADDRESS>
###### Abstract

Model-based control methods for robotic systems such as quadrotors, autonomous
driving vehicles and flexible manipulators require motion models that generate
accurate predictions of complex nonlinear system dynamics over long periods of
time. Temporal Convolutional Networks (TCNs) can be adapted to this challenge
by formulating multi-step prediction as a sequence-to-sequence modeling
problem. We present End2End-TCN: a fully convolutional architecture that
integrates future control inputs to compute multi-step motion predictions in
one forward pass. We demonstrate the approach with a thorough analysis of TCN
performance for the quadrotor modeling task, which includes an investigation
of scaling effects and ablation studies. Ultimately, End2End-TCN provides 55%
error reduction over the state of the art in multi-step prediction on an
aggressive indoor quadrotor flight dataset. The model yields accurate
predictions across 90 timestep horizons over a 900 ms interval.

##  I Introduction

While autonomous robotic systems offer tremendous potential benefits in a wide
range of commercial operations, their safe operation will require highly
accurate localization and control methods for collision avoidance and action
execution. Model-based state estimation and control have demonstrated strong
performance and robustness across the operational domain of remote aircraft
[1, 2, 3], autonomous vehicles [4, 5] and flexible manipulators [6], to name a
few. As such, dynamic system modeling is critical to the effort of developing
safe autonomous robotic systems that can perform precise motions throughout
their operating envelopes.

As a primary motivating example, in this work we focus on multi-step
prediction for quadrotor UAVs. Indeed, developing models of quadrotor flight
solely from first principles has proven to be a challenge. Quadrotors are
underactuated systems whose translational dynamics are tightly coupled with
highly nonlinear rotational dynamics. In real-world environments,
aerodynamics, motor dynamics, and asymmetrical mass distributions can be
significant disturbances, but are often poorly characterized in most physics-
based quadrotor models [7].

Another line of research focuses on developing statistical quadrotor models
from measured flight data. Specifically, discrete time neural network designs
have shown the greatest promise in modeling complex quadrotor dynamics due to
their strong expressive power. A recent work to benchmark neural network
models on quadrotor state prediction performance employ Recurrent Neural
Network (RNN) models to sequentially learn time-correlated features in
quadrotor state telemetry time series data [8]. While these models have
demonstrated state-of-the-art performance, they have several limitations. The
sequential nature of these models leads to longer computation times due to the
lack of parallelization, and can cause unstable gradients at training time
[9]. Furthermore, current models are limited in their ability to learn time-
correlated features over long time horizons [10].

Temporal convolution-based architecture provide a potential solution to the
limits of RNNs for the task of quadrotor state modeling. Temporal
Convolutional Networks (TCNs) have demonstrated the ability to accurately
model time series in a variety of contexts [11],[12],[13] and have the
potential to provide a sparse an efficient model able to learn features over
long time histories. In this work, we apply TCNs to a discrete time multi-step
series forecasting problem, which we adapt to the non-autonomous dynamics of
robotic systems. This allows for TCN models to be trained and evaluated on
indoor quadrotor flight telemetry.

Thus, in this paper, we perform the first in-depth study of convolution-based
architectures for quadrotor modeling. We present End2End-TCN: a novel method
of applying TCNs to robotic system modeling by integrating the control input
into the input state vector. This model surpasses the current state of the art
and several alternative models in prediction accuracy, generating useful
future state predictions over longer periods of time for the purpose of model-
based control and state estimation. We perform a comprehensive series of
experiments to characterize the performance of TCNs with respect to model size
and past state history length. We further provide an analysis of prediction
samples and error distributions to characterize model performance. Most
importantly, we demonstrate that a TCN-based model can provide a memory-
efficient representation of quadrotor dynamics and yield a 55% reduction in
prediction error over a 900 ms interval.

##  II Related Works

Empirical Methods. As a result of the success of model-based quadrotor control
methods, the dynamics of quadrotor flight have been extensively studied in
literature. Previous research that developed quadrotor test bed platforms
[14], developed dynamical system models [15],[16] and characterized
significant aerodynamic effects [17] have laid the foundation for a principled
approach to developing quadrotor models. In these works, simplified models of
quadrotor geometry, rotor thrust, and aerodynamics were used to derive
equations of motion. Such physics-based models have been further refined by
deriving more complex aerodynamic models [7] or by using blade element
momentum theory [18] [19] to better characterize motor thrust. While many such
models obtain parameter values through empirical measurement or offline system
identification, recent works have used online parameter estimation to refine
their physics-based models over time [20] [21] [22].

Neural Networks. Neural networks, on the other hand, provide powerful and
flexible statistical models that can model highly complex time-varying
phenomena. In the field of statistical rotorcraft flight modeling, early work
by Punjabi and Abbeel [23] showed significant success in learning a nonlinear
acceleration model for helicopter flight by training a simple artificial
neural network on past flight telemetry, while others [24] learned a simpler
linearizable model for LQR control. Such models may successfully learn a
latent representation of flight data, but are not designed to specifically
learn time-correlated features, which have been demonstrated to improve
performance in sequence domain tasks. On need look no further than the field
of stock price modeling, where early artificial neural networks [25] were
quickly surpassed by LSTM models [26] and TCNs [27] spcifically due to their
ability to learn time-correlated features.

Sequence Modeling. In recent years, deeper networks with new neural network
architectures have led to major breakthroughs in sequence modeling. Much of
this research has focused on Recurrent Neural Networks (RNNs). Mohajerin et
al. leveraged recurrent architectures towards quadrotor modeling by training
RNNs with Long-Short-Term Memory gated units on an indoor quadrotor dataset,
which greatly improved prediction accuracy for future flight trajectories [8].
This sequential approach mirrors the way discrete dynamical system models are
integrated forward in time. However, the ability of an RNN to model time-
varying phenomena is limited by the size of its hidden state representation
[28], and RNN performance degrades significantly as time horizons extend [10],
both of which limit their usability for quadrotor flight modeling. RNNs also
have limitations that make them ill-suited for online robotics applications.
They are less computationally efficient than convolution-based architectures
that can leverage parallel computation hardware [12] due to the cost of
processing time series in a sequential method. Furthermore, RNNs can be
challenging to train due to backpropagation through time, which can lead to
gradient instability [9].

Temporal Convolutional Networks. While RNNs were the dominant approach for
time series predictive modeling [29],[30],[31], convolution-based approaches
have emerged recently as a viable alternative. Early work by van den Oord et
al. on WaveNet [11] introduced the causal convolution, which modified the
standard discrete convolution operation to maintain the temporal structure of
time series inputs. Dilated convolutions can be employed to make predictions
over large, fixed time horizons and the resulting network can be parallelized
for computational efficiency. This results in sparse networks that learn time-
correlated features in an efficient and deterministic manner, which are called
Temporal Convolutional Networks (TCNs).

Studies have shown that TCNs outperform recurrent networks across a wide range
of sequence modeling tasks [12]. TCNs were further explicitly applied to time
series modeling by Borovykh et al. [32]. More relevant to quadrotor modeling,
TCNs were used in action segmentation tasks[13] and were combined with
Empirical Mode Decomposition (EMD) to predict dynamic climate effects [33].
These prior works demonstrate that TCNs have the ability to learn temporal
patterns in robotic motion over long periods and model highly complex
dynamical systems.

Many applications of deep learning in robotics learn temporal patterns by
simply concatenating images or system state inputs [34]. However, this only
works over short time periods. Recent work by Kaufmann et al. leveraged TCNs
to process sensor input information in an end-to-end learning-based
architecture for quadrotor control [35]. While this study demonstrates the
utility of TCNs in the context of quadrotor state information processing,
there is still a clear lack of research on the ability of TCNs to explicitly
model robotic systems over a long horizon of future state predictions.

##  III Problem Formulation

By treating quadrotor flight dynamics as a time series predictive modeling
problem, we can perform sequence-to-sequence modeling to learn a function that
can predict future states. We first define a parameterization of the quadrotor
state,
$\mathbf{x}(t)=\begin{bmatrix}\mathbf{\eta}(t)&\mathbf{r}(t)&\mathbf{\xi}(t)&\mathbf{v}(t)\end{bmatrix}\in\mathbb{R}^{n}$,
which includes position, $\mathbf{r}(t)\in\mathbb{R}^{3}$, and velocity,
$\mathbf{v}(t)\in\mathbb{R}^{3}$, in a world frame, $\mathcal{F}_{w}$,
orientation, $\mathbf{\eta}(t)\in SO(3)$, represented by Euler rotation angles
from a body frame, $\mathcal{F}_{b}$, about axes XYZ to the world frame,
$\mathcal{F}_{w}$, and rotation rate, $\mathbf{\xi}(t)\in\mathbb{R}^{3}$,
represented by the time derivative of XYZ Euler angles with respect to the
body frame, $\mathcal{F}_{b}$. The diagram below denotes the world frame,
$\mathcal{F}_{w}$, and body frame, $\mathcal{F}_{b}$, with respect to the
quadrotor’s geometry. The way the geometry and reference frames are denoted is
based on a quadrotor X-configuration, where the roll and pitch axes are offset
by 45 degrees from the rotor arms.

Figure 1: Model of a quadrotor as a rigid body from a body and inertial frame.
Figure 2: Full End2End-TCN architecture.

This state represents the quadrotor’s pose with 6 degrees of freedom (given
the orientation representation) and a measure of its first rate of change. The
full system is further characterized by a control input,
$\mathbf{u}(t)=\begin{bmatrix}u_{1}(t)&u_{2}(t)&u_{3}(t)&u_{4}(t)\end{bmatrix}\in\mathbb{R}^{d}$,
representing four motor commands which are generated by the quadrotor’s
controller and linearly map to desired rotor speeds,
$\mathbf{\varpi}=\begin{bmatrix}\varpi_{1}(t)&\varpi_{2}(t)&\varpi_{3}(t)&\varpi_{4}(t)\end{bmatrix}\in\mathbb{R}^{d}$.

In this discrete quadrotor state formulation, we consider a dynamic system
represented by the function $f\colon\mathbb{R}^{n}\to\mathbb{R}^{n}$ that maps
a past state representation, $\mathbf{x_{t_{0}-1}}$, to a future state
representation, $\mathbf{x_{t_{0}}}$, and a function
$g\colon\mathbb{R}^{n}\to\mathbb{R}^{m}$ that maps a state representation,
$\mathbf{x_{t_{0}}}$, to a state observation, $\mathbf{y_{t_{0}}}$. In the
non-autonomous case, the function $f$ maps both the past state,
$\mathbf{x_{t_{0}-1}}$, and a control input,
$\mathbf{u_{t_{0}}}\in\mathbb{R}^{d}$, to the state observation,
$\mathbf{y_{t_{0}}}$.

However, to fully leverage the ability of convolutional neural networks to
compute state predictions in parallel, we extend this formulation to a multi-
step prediction case of length $F$. Note that in the non-autonomous case, past
and future control inputs will be required as inputs to this function, as the
future state, $\mathbf{x_{t_{0}+i}}$, is dependent on the future control
input, $\mathbf{u_{t_{0}+i}}$. Furthermore, given the complexity of dynamic
effects such as aerodynamics on quadrotor motion, the state parameterization
above may not meet the Markov condition. Thus, we theorize that prediction
accuracy will be improved by providing a sequence of $P$ input states. As
such, we seek to model the function $f^{(P,F)}$ mapping a series of $P$ past
states, $P$ past control inputs, and $F$ future control inputs to the series
of $F$ future states. Note that this model assumes access to the full state
representation, which is only possible in the case of weak nonlinear
observability.

Modeling this discrete function can thus be formulated as a sequence-to-
sequence modeling problem. We consider a sequence of prior system states,
$\mathbf{X}_{p}=\begin{bmatrix}\mathbf{x}_{t_{0}-P+1}&\mathbf{x}_{t_{0}-P+2}&...&\mathbf{x}_{t_{0}}\end{bmatrix}\in\mathbb{R}^{n\times
P}$, prior control inputs,
$\mathbf{U}_{p}=\begin{bmatrix}\mathbf{u}_{t_{0}-P+1}&\mathbf{u}_{t_{0}-P+2}&...&\mathbf{u}_{t_{0}}\end{bmatrix}\in\mathbb{R}^{d\times
P}$, and future control inputs,
$\mathbf{U}_{f}=\begin{bmatrix}\mathbf{u}_{t_{0}+1}&\mathbf{u}_{t_{0}+2}&...&\mathbf{u}_{t_{0}+P}\end{bmatrix}\in\mathbb{R}^{d\times
F}$, and seek to estimate future system states,
$\mathbf{X}_{f}=\begin{bmatrix}\mathbf{x}_{t_{0}+1}&\mathbf{x}_{t_{0}+2}&...&\mathbf{x}_{t_{0}+F}\end{bmatrix}\in\mathbb{R}^{n\times
F}$. Thus, given a sequence-to-sequence function
$\hat{f}\colon\mathbb{R}^{(n+d)\times P}\to\mathbb{R}^{n\times F}$ generating
a future system state prediction
$\mathbf{\hat{x}}_{f}=\begin{bmatrix}\mathbf{\hat{x}}(t_{0}+1)&\mathbf{\hat{x}}(t_{0}+2)&...&\mathbf{\hat{x}}(t_{0}+F)\end{bmatrix}$,
we can use statistical methods to minimize a reconstruction loss
$\ell(\mathbf{\hat{x}},\mathbf{x})$ over a set of known future quadrotor
states.

$\mathcal{L}=\frac{1}{F}\sum^{i=t_{0}+F}_{i=t_{0}}\ell(\mathbf{\hat{x}}_{i},\mathbf{x}_{i})$
(1)

##  IV Methodology

Given historical quadrotor state data, neural network model inputs and labels
are generated in a semi-supervised manner. As per the problem formulation,
model inputs include prior quadrotor states $\mathbf{X}_{p}$, control inputs
$\mathbf{U}_{p}$, and future control inputs $\mathbf{U}_{f}$. The sample
labels $\mathbf{Y}_{f}$ correspond to a series of truncated quadrotor states,
$\mathbf{y}_{k}=\begin{bmatrix}\mathbf{\xi}_{k}&\mathbf{v}_{k}\end{bmatrix}$,
which include the translational and rotational velocities from time $t_{0}+1$
to time $t_{0}+F$.

A fully convolutional neural network model, dubbed End2End-TCN, is trained on
this time series data to provide quadrotor state predictions over $F$ time
steps. Crucially, in order to make multiple predictions over this non-
autonomous dynamical system, past and future control input must be integrated
into the discrete sequence modeling problem formulation. End2End-TCN
integrates this information into a fixed sequence length input,
$\mathbf{\tilde{X}}=\begin{bmatrix}\mathbf{\tilde{X}}_{p}&\mathbf{\tilde{X}_{f}}\end{bmatrix}\in\mathbb{R}^{(n+d)\times(P+F)}$,
composed of augmented states,
$\mathbf{\tilde{x}_{k}}=\begin{bmatrix}\mathbf{\eta}_{k}&\mathbf{r}_{k}&\mathbf{\xi}_{k}&\mathbf{v}_{k}&\mathbf{u}_{k}\end{bmatrix}\in\mathbb{R}^{(n+d)}$,
for prior states ($k<P$) and,
$\mathbf{\tilde{x}_{k}}=\begin{bmatrix}\mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{u}_{k}\end{bmatrix}\in\mathbb{R}^{(n+d)}$,
for future states ($k\geq P$).

The model is built on a series of causal convolutions, as first developed in
[11], and as implemented in [12]. To achieve the desired effect, a causal
convolution block is composed of a series of causal convolutions with
dilations that increase exponentially at every layer, as depicted in figure 3.

Figure 3: Causal convolutions over a series of layers with exponentially
increasing dilation factor.

Causal convolution blocks are stacked with a nonlinear activation function and
batch normalization, with a residual connection applied for gradient
stability. These blocks are stacked in a layered architecture to form a deep,
overparameterized neural network as in [13] and [36] (see figure 2). End2End-
TCN was designed to output a full time series of predicted states at every
forward pass, allowing for simultaneous multi-step prediction of quadrotor
states.

###  IV-A Physics-based Model

A key part of the study of TCNs for quadrotor modeling is ascertaining whether
prior knowledge of the system’s dynamics is required to improve prediction
accuracy. Consequently, we develop a physics-based model of quadrotor flight
derived for the AsTec Pelican flights in the test set. This model is based on
a simplified wire-frame model of the quadrotor as per figure 1, which is
represented by four arms with a uniform mass and a length $d$. For the
specified platform, the arms form a right angle with one another. Fixed to
each arm is a rotor, which is modeled by a point mass generating a
longitudinal thrust $T_{i}$ and rotational torque $Q_{i}$. The body frame is
defined such that the rotors lie on the XY plane, the x-axis points in the
direction directly between the first and second rotors, and the z-axis points
in the direction of the torques generated by any individual rotor. The diagram
in figure 1 depicts the wireframe quadrotor model and the two corresponding
reference frames (inertial and body). The complex motor and rotor dynamics are
approximated by a quadratic relationship between the rotor angular velocity,
$\varpi_{i}\in\mathbb{R}$, in its discrete representation. This is based on
the rotor dynamic equation in stead state with a freestream velocity of zero
[17], which can be parameterized with respect to a thrust coefficient $c_{T}$,
the density of air $\rho$, the rotor radius $r_{i}$ and the rotor area
$A_{r,i}$.

$T_{i}=c_{T}\,\rho\,A_{r,i}\,r_{i}^{2}\,\varpi_{i}^{2}=C_{T}\,\varpi_{i}^{2}$
(2)

$Q_{i}=C_{Q}\,\varpi_{i}^{2}$ (3)

The total thrusts and torques can thus be calculated from individual rotor
contributions in the vectorized equation below.

$\begin{bmatrix}T_{tot}\\\ \tau_{1}\\\ \tau_{2}\\\
\tau_{3}\end{bmatrix}=C_{T}\begin{bmatrix}1&1&1&1\\\
\frac{d}{\sqrt{2}}&\frac{-d}{\sqrt{2}}&\frac{d}{\sqrt{2}}&\frac{-d}{\sqrt{2}}\\\
\frac{-d}{\sqrt{2}}&\frac{-d}{\sqrt{2}}&\frac{d}{\sqrt{2}}&\frac{d}{\sqrt{2}}\\\
\frac{-C_{Q}}{C_{T}}&\frac{C_{Q}}{C_{T}}&\frac{-C_{Q}}{C_{T}}&\frac{C_{Q}}{C_{T}}\\\
\end{bmatrix}\begin{bmatrix}\varpi_{1}^{2}\\\ \varpi_{2}^{2}\\\
\varpi_{3}^{2}\\\ \varpi_{4}^{2}\end{bmatrix}$ (4)

For state derivatives, we reference a quadrotor state in the form
$\mathbf{x}(t)=\begin{bmatrix}\mathbf{\eta}(t)&\mathbf{r}(t)&\mathbf{\xi}(t)&\mathbf{v}(t)\end{bmatrix}\in\mathbb{R}^{n}$
as per the problem formulation in section 3. The derivative of position is
trivial, $\mathbf{\dot{r}(t)}=\mathbf{v}(t)$. The orientation derivative can
be obtained from the body rotation rates with an additional coordinate
transform $\mathcal{F}_{b}\to\mathcal{F}_{w}$ in matrix form
($\mathbf{R_{b}^{i}}$).

$\mathbf{\dot{\eta}}=\begin{bmatrix}\dot{\theta}\\\ \dot{\phi}\\\
\dot{\psi}\end{bmatrix}=\mathbf{R_{b}^{i}}\mathbf{\omega}=\begin{bmatrix}1&0&-\sin(\theta)\\\
0&\cos(\phi)&\sin(\phi)\cos(\theta)\\\
0&-\sin(\phi)&\cos(\phi)\cos(\theta)\end{bmatrix}\mathbf{\omega}$ (5)

Translational acceleration can be written with respect to the force and torque
from equation 4 using Newton’s 2nd Law. Motor thrust $\mathbf{f_{T}}$ is
transformed from $\mathcal{F}_{b}$ to $\mathcal{F}_{w}$, and additional
inertial accelerations due to gravity ($g$) and translational drag
$k_{t}\mathbf{v}$. Lastly, rotational acceleration can be written from Euler’s
Equations of Rotational Motion, with a body frame rotor torque $\mathbf{\tau}$
and rotational drag $k_{R}\mathbf{\omega}$.

$\mathbf{\dot{v}}=\frac{1}{m}\mathbf{R_{b}^{i}}\mathbf{f_{T}}-\mathbf{g}-k_{t}\mathbf{v}$
(6)

$\mathbf{I}\mathbf{\dot{\omega}}=\mathbf{\tau}-\mathbf{\omega}\times(\mathbf{I}\mathbf{\omega})-k_{R}\mathbf{\omega}$
(7)

To perform motion prediction, the equations of motion are discretized for all
state variables used in motion prediction as per as per section III.
Parameters are either empirically measured or estimated using nonlinear system
identification, as in [8]. Numerical forward integration is then performed
using a real-valued variable-coefficient ODE (VODE) solver. The predicted
state variables after an interval $\Delta t=0.01s$ is compared to learning-
based methods trained on motion prediction for the same discrete time
interval.

###  IV-B Hybrid Models

On the other hand, we can use all or part of this physics-based model as a
component in a hybrid architecture. We develop a series of hybrid models
combining fully convolutional Temporal Convolutional Network component(s) with
similar design parameters as End2End-TCN and the same total number of
parameters. Physics-based components generate forward predictions in a
sequential manner by forward integrating some or all of the dynamic system
equations outlined in section IV-A. This results in three different Hybrid
configurations. Motor-Hybrid uses a TCN component to model the aircraft’s
rotor dynamics, generating motor thrust predictions for a given control input.
AccelError-Hybrid uses a TCN component to model an additive term to the
physics-based state derivative estimates, thus modeling the dynamics that are
not captured by the simplified physics-based model. Lastly, Combined-Hybrid
uses both TCN components of the models above.

Figure 4: Components of Motor-Hybrid (top), AccelError-Hybrid (middle) and
Combined-Hybrid (bottom). Figure 5: Velocity and body rate prediction errors
over time for End2End-TCN (green) and reference models on a log plot.

##  V Experimental Results

###  V-A Experimental Design

We validate this approach and characterize model performance with respect to
its prediction accuracy on real quadrotor flights. We evaluate End2End-TCN and
several alternative predictive models on the WAVE Laboratory AsTec Pelican
Quadrotor Dataset [37], which utilized sensor fusion across inertial, GNSS,
and vision-based systems to collect high-precision quadrotor state estimates.
Data are interpolated to report full quadrotor states at a sample rate of 100
Hz. The dataset is comprised of a series of indoor quadrotor flights, bounded
within a 5 x 5 x 5 m volume. This mostly includes near-hover flight, pseudo-
random rotations and curves in all axes, all within the nominal flight
envelope of the AsTec Pelican quadrotor. In total, the dataset consists of 54
flights, with over 1,388,410 total samples of quadrotor telemetry data, 10% of
which is used in the test set for this experiment.

###  V-B Comparative Study

To validate the performance of End2End-TCN, we compare its performance in
terms of velocity and body rate prediction accuracy with alternative models.
This includes the current state-of-the-art result on this dataset, which was
achieved by Mohajerin in [8] with an LSTM Recurrent Neural Network Hybrid
model to multi-step quadrotor prediction. The model is also compared to a
physics-based model, and a series of hybrid models with both TCN and physics-
based components, as outlined in section 4.

TABLE I: Summary of multi-step prediction results across 90 time steps (900
ms).

Model MSE Error (t=0.01s) MSE Error (t=0.45s) MSE Error (t=0.90s) Velocity
Body Rate Velocity Body Rate Velocity Body Rate Physics-based 0.00003 0.000572
0.0892 0.0981 0.938 1.08 LSTM Hybrid 0.00441 0.616 0.0217 2.30 0.0384 3.01
Motor-Hybrid 0.0100 0.00543 0.115 0.269 0.115 0.632 AccelError-Hybrid 0.0153
0.00356 0.200 0.187 0.205 0.625 Combined-Hybrid 0.0124 0.0126 0.178 0.535
0.192 1.02 End2End-TCN 0.000735 0.00197 0.00881 0.0352 0.0357 0.0464

We find that End2End-TCN outperforms the current state of the art and all
alternative models across nearly the entire 90 step sample (corresponding to
900 ms). The most significant performance improvements are in rotation rates,
where the fundamental kinematics rely on current and past quadrotor states.
This may indicate that dilated convolutions are better suited for this type of
long-term sequence modeling. We find that hybrid models perform significantly
worse than the fully convolutional approach. This can mostly be attributed to
the difficulty of integrating TCNs with numerically integrated dynamical
system equations, which are sequential in nature. Hybrid models that have
multiple TCN components, each with a fraction of a single large End2End-TCN,
likely suffer due to a fundamental lack of expressive power.

Lastly, we see that most TCN-based models represent a 2-10x improvement with
respect to prediction accuracy when compared to the physics-based model over a
longer time horizon, which indicates that these models learn generalizable
unmodeled dynamics that have significant temporal effects. We find that TCN
model errors typically plateau over time. While a constant acceleration error
due to unmodeled disturbances may cause errors growing quadratically over
time, End2End-TCN optimizes for accuracy across the flight sample over longer
time periods where transient effects may not be statistically relevant.

###  V-C Analyzing Flight Samples

While End2End-TCN makes extremely accurate predictions for a majority of
samples, overall accuracy is limited by a long tail in the error distribution
as depicted for body rotation rate error in figure 6.

Figure 6: Distribution of End2End-TCN body rate errors over time (Box-whisker
plot: median (red line), 2nd and 3rd quartiles (blue box) and range)

These uncommon but large errors occur at the extremes of the quadrotor’s
flight envelope. While using an L1-Norm loss function reduces prediction error
overall, it constrains the model to learn the simple hover point dynamics,
that are more frequent in the training and evaluation datasets. As such,
flight samples in more aggressive maneuvers yield predictions that
significantly diverges from the ground truth, as in figure 6. We find that
samples with errors in the 90th percentile have significantly higher rates of
change of position and motor command (i.e. faster and sharper turns). We also
find an increase in the variance of pitch and roll angles, indicating that
samples taken farther from the hover point of the quadrotor.

Figure 7: Flight predictions with respect to ground truth for a selection of
test samples including low error (frequent) cases and high error (infrequent)
cases

It is hypothesized that this behavior is largely data-driven. The current
dataset, comprised of stable, indoor flight, has few samples in the extreme
ranges of the quadrotor’s flight envelope. However, in comparison, hybrid
models appear to be more robust to these outlier samples. These models have
significantly worse mean errors over time but a smaller standard deviation,
which indicates that building models with a prior on the system’s dynamics may
be an effective way to address a lack of data in certain flight modes.

###  V-D Scaling Effects

One of the main potential benefits of a fully convolutional architecture for
quadrotor predictive modeling was its computational efficiency and memory
footprint. Thus, we investigate the impact of model size of its predictive
modeling performance. Table II shows the validation set accuracy results of
End2End-TCN when varying the number of depth layers. Forward pass frequency
was calculated on a test set running on a Nvidia GeForce RTX 2080 Ti Graphic
Processing Unit (GPU).

Overall, we find that End2End-TCN retains a significant amount of its
predictive ability as the size of the network decreases, particularly for
translational velocity. On the other hand, we see significant reductions in
body rotation rate prediction accuracy, likely due to the nonlinear nature of
these dynamics and their higher sensitivity to disturbances. Similarly, we
find that reducing the observation window does not significantly degrade the
performance of End2End-TCN.

One hypothesis for this behavior is that the current model is fundamentally
limited by the size of the dataset rather than the size of the model. As
demonstrated in language models and other sequence learning tasks [38],
performance improvements from increasing model size is fundamentally capped if
the size of the dataset does not increase accordingly. There may be additional
factors about time-correlated data that make it less susceptible to
performance increases from model scale. This view of a data-centric approach
for further model scaling is supported by error distributions and the sparsity
of data in certain flight modes.

TABLE II: Prediction error with respect to model size.

# of layers # of param. / fps (hz) MSE (t=0.45s) MSE (t=0.90s) Vel. Ang. Vel.
Vel. Ang. Vel. 5 298,346 / 492.6 0.0102 0.0387 0.0423 0.0634 8 1,166,794 /
383.7 0.0088 0.0352 0.0357 0.0464 10 4,640,266 / 302.4 0.0087 0.0403 0.0353
0.0663 12 18,517,706 / 243.7 0.0148 0.0398 0.0412 0.0654

###  V-E Ablation Studies

A series of ablation studies is performed on End2End-TCN to validate the
model’s detailed design. We first compare a series of alternative
architectures. This includes models with varying amounts of regularization
layers (Batch Normalization and Dropout) and varying training loss functions
(Euclidean, Manhattan, and Weighted Euclidean). The results of the study are
summarized in table III for Batch Normalization (BN), Dropout (Drop),
Shortened gradient path architecture (SG), Weighted L2-Norm loss function
(WL2), and L1-Norm loss (L1). A crucial element of the design of End2End-TCN
is the integration of future control inputs for the multi-step prediction of
non-autonomous dynamical systems. In our ablation study, we consider two
methods to achieve this. In the baseline model, past quadrotor states, past
control inputs, and future control inputs are concatenated into a single model
input sequence. We compare this approach to an architecture where only past
quadrotor states and control inputs are fed to the first layer, while future
control inputs are fed to an intermediate layer for the purposes of shortening
their gradient path.

Figure 8: Architecture with shortened gradient paths to future control inputs.
TABLE III: Architecture Ablation Study

BN Drop SG WL2 L1 MSE Error Velocity Body Rate 0.0198 0.0715 ✓ 0.0172 0.0401 ✓
✓ 0.0217 0.0433 ✓ ✓ 0.0329 0.0440 ✓ ✓ 0.0317 0.0700 ✓ ✓ 0.0158 0.0396

Firstly, we see that the alternative architecture performance significantly
worse with respect to body rate error when compared to the final model. While
this architecture was hypothesized to increase performance by shortening the
gradient path to the most important features, namely the last quadrotor state
and the control inputs, we see that the number of layers between these
features and the output are too few to properly capture the nonlinear rotation
dynamics of the quadrotor. Furthermore, reducing or eliminating batch
normalization in End2End-TCN decreases performance, as does adding dropout to
the model. These results mirror similar conclusions in literature [13]. We
also find that the L1-Norm loss function, which is more robust to outlier
state errors, leads to better generalization to the test set than do L2 or
weighted L2 loss functions.

##  VI Conclusion

This paper presents a detailed study of the use of Temporal Convolutional
Networks for quadrotor state modeling and motion prediction. While classical
modeling techniques characterize such robotic systems using prior knowledge of
the system’s non-autonomous dynamics, we formulate this as a sequence modeling
problem by performing discrete multi-step prediction. We segment quadrotor
telemetry to train a fully convolutional neural network, End2End-TCN, in a
semi-supervised fashion. End2End-TCN outperforms the previous state of the art
by 55% and proves to be more effective than hybrid models and fully physics-
based models. We demonstrate that End2End-TCN retains over 95% of its
performance over shorter time intervals when the model is compressed by a
factor of 3, and we further characterize model performance with an ablation
study and an analysis of predicted flight samples.

This fully convolutional approach to quadrotor modeling is currently limited
by the scale and distribution of training data, which is a bottleneck shared
by many sequence to sequence models. Collecting data on aggressive quadrotor
flight would reduce the model’s bias towards hover point dynamics and
potentially reduce infrequent low-accuracy prediction samples. Further work is
required to ascertain whether this method will generalize to outdoor
environments with wind disturbances. Finally, End2End-TCN will be applied in
model-based quadrotor control methods to further contextualize its accuracy
and computational efficiency.

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