# High Resolution Imaging Spectroscopy of a Sigmoidal Mini-filament Eruption

Jiasheng Wang Institute for Space Weather Sciences, New Jersey Institute of
Technology, University Heights, Newark, NJ 07102-1982, USA  Big Bear Solar
Observatory, New Jersey Institute of Technology, 40386 North Shore Lane, Big
Bear City, CA 92314-9672, USA  Center for Solar-Terrestrial Research, New
Jersey Institute of Technology, University Heights, Newark, NJ 07102-1982, USA
Jeongwoo Lee Institute for Space Weather Sciences, New Jersey Institute of
Technology, University Heights, Newark, NJ 07102-1982, USA  Big Bear Solar
Observatory, New Jersey Institute of Technology, 40386 North Shore Lane, Big
Bear City, CA 92314-9672, USA  Center for Solar-Terrestrial Research, New
Jersey Institute of Technology, University Heights, Newark, NJ 07102-1982, USA
Jongchul Chae Department of Physics and Astronomy, Seoul National University,
Gwanak-gu, Seoul 08826, Republic of Korea  Yan Xu  Institute for Space Weather
Sciences, New Jersey Institute of Technology, University Heights, Newark, NJ
07102-1982, USA  Big Bear Solar Observatory, New Jersey Institute of
Technology, 40386 North Shore Lane, Big Bear City, CA 92314-9672, USA  Center
for Solar-Terrestrial Research, New Jersey Institute of Technology, University
Heights, Newark, NJ 07102-1982, USA  Wenda Cao  Institute for Space Weather
Sciences, New Jersey Institute of Technology, University Heights, Newark, NJ
07102-1982, USA  Big Bear Solar Observatory, New Jersey Institute of
Technology, 40386 North Shore Lane, Big Bear City, CA 92314-9672, USA  Center
for Solar-Terrestrial Research, New Jersey Institute of Technology, University
Heights, Newark, NJ 07102-1982, USA  Haimin Wang Institute for Space Weather
Sciences, New Jersey Institute of Technology, University Heights, Newark, NJ
07102-1982, USA  Big Bear Solar Observatory, New Jersey Institute of
Technology, 40386 North Shore Lane, Big Bear City, CA 92314-9672, USA  Center
for Solar-Terrestrial Research, New Jersey Institute of Technology, University
Heights, Newark, NJ 07102-1982, USA

###### Abstract

Minifilament (MF) eruption producing small jets and micro-flares is regarded
as an important source for coronal heating and the solar wind transients
through studies mostly based on coronal observations in the extreme
ultraviolet (EUV) and X-ray wavelengths. In this study, we focus on the
chromospheric plasma diagnostics of a tiny minifilament in quiet Sun located
at (71”, 450”) on 2021–08–09 at 19:11 UT observed as part of the ninth
encounter of the PSP campaign. Main data obtained are the high cadence, high
resolution spectroscopy from the Fast Imaging Solar Spectrograph (FISS) and
high-resolution magnetograms from the Near InfraRed Imaging Spectropolarimeter
(NIRIS) on the 1.6 m Goode Solar Telescope (GST) at Big Bear Solar Observatory
(BBSO). The mini-filament with size $\sim$1”$\times$5” and a micro-flare are
detected in both the H$\alpha$ line center and SDO/AIA 193, 304 Å images. On
the NIRIS magnetogram, we found that the cancellation of a magnetic bipole in
the footpoints of the minifilament triggered its eruption in a sigmoidal
shape. By inversion of the H$\alpha$ and Ca ii spectra under the embedded
cloud model, we found a temperature increase of 3,800 K in the brightening
region, associated with rising speed average of MF increased by 18
$km~{}s^{-1}$. This cool plasma is also found in the EUV images. We estimate
the kinetic energy change of the rising filament as 1.50$\times$$10^{25}$
ergs, and thermal energy accumulation in the MF, 1.44$\times$$10^{25}$ ergs.
From the photospheric magnetograms, we find the magnetic energy change is
1.57$\times$$10^{26}$ ergs across the PIL of converging opposite magnetic
elements, which amounts to the energy release in the chromosphere in this
smallest two-ribbon flare ever observed.

Quiet sun(1322) – Solar chromosphere(1479) –
Solar filament eruptions(1981) – Solar magnetic reconnection(1504)

††journal: ApJ††facilities: GST(FISS and NIRIS), SDO(AIA)

##  1 Introduction

Solar filaments are typically dense, cool plasma formulated in chromosphere
along the polarity inversion line (PIL) during flux emergence (e.g., Bruzek,
1967; Foukal, 1971; Zwaan, 1985; Shibata et al., 1989; Toriumi et al., 2017).
The arch filament enveloped in the emerging magnetic bipole expands as it
rises from the solar surface. The filament erupts as its configuration with
surrounding fields is destabilized by reconnection in different scenarios,
such as breakout, kink instability, and tether-cutting (Bi et al., 2015; Kumar
et al., 2019; Kim & Yurchyshyn, 2022). Solar filament eruptions are often
associated with a wide range of dynamic solar activities from as large as
coronal mass ejections (CMEs) to small-scale spicule activities by which free
energy is transported to the corona from the chromosphere. In active regions,
the sigmoid configurations, which are defined as S-shaped loop arcades of non-
potential magnetic field, have typical length of 100$\arcsec$. Non-
potentiality of the kinked magnetic field flux ropes tend to drive eruption in
the PIL (polarity inversion lines) of the sigmoidal regions by tether-cutting
reconnection and form arcade loops (Liu et al., 2010, 2012). Strong non-
potential bipolar magnetic configuration is also identified in X-ray and EUV
micro-sigmoids (Mandrini et al., 2005; Raouafi et al., 2010; Zheng et al.,
2013).

Mini-filaments are scale-down version of solar filaments, which extend less
than 25 Mm and last for about 50 min in H$\alpha$ images (Wang et al., 2000a).
Similar to large-scale filaments, loop-like topology of mini-filaments usually
lie above the photospheric polarity inversion line (PIL) of small-scale
adjacent opposite-polarity magnetic fields, which are distinguished from
“surge-like macrospicules” that emanating from network field concentrations on
the same size scale. The earliest systematic studies of small-scale filaments
utilizing H$\alpha$ observations from Big Bear Solar Observatory (BBSO) by
Hermans & Martin (1986a) discovered elongated dark features and characterized
that: 1) With occurrence rate of 600 per 24 hour day, the average duration of
eruptive phase was 26 minutes in total lifetime of 70 minutes; 2) Majority of
events are spatially related to cancelling magnetic features; 3) Small-scale
filament share similarities with large scale filaments that often has one or
both ends terminate at cancellation site. The total mass and kinetic energy
are estimated to be 1013 g and 1025 ergs, respectively.

The eruptions of mini-filaments are also found to often produce small scale
coronal mass ejections, coronal jets, and microflares (Innes et al., 2009;
Raouafi et al., 2016), which is thus of interest to the recent topic a small
solar eruption that could contribute correspondingly small transients to the
coronal heating and solar wind acceleration. Observations of the roots of
coronal heating in chromosphere proved that it cannot be driven solely by
energy deposition in the coronal heights; instead, mass and energy supply from
lower atmosphere are required through ubiquitous upflows across the magnetic
configuration in the heating process of corona (De Pontieu et al., 2009; Tian,
2017). The ejected cold material from filament eruptions in chromosphere
called surges have been observed in H$\alpha$ filtergrams (Harrison et al.,
1988; Schmieder et al., 1995; Chen et al., 2022; Schmieder et al., 2022) with
bright/flaring points separating from footpoints of the surge during onset of
eruptions. The brightening primarily observed in soft X-ray (SXR) wavelengths
is located where the H$\alpha$ surges originate, in which magnetic
reconnection between twisted cool loops and open field lines leads to the
formation of surges that coincide with X-ray subflares (Shibata et al., 1992;
Alexander & Fletcher, 1999). Coronal counterparts of the erupted filament,
jets, are observed in extreme ultraviolet filtergrams (EUV, Wang et al., 1998;
Li et al., 2023), as well as at SXR (Innes et al., 2016; Sterling et al.,
2022). Observational facts of the scenario that magnetic reconnection bewteen
confined flux loop and overlying field in different wavelengths suggest the
formation of cool and hot ejecting components from minifilament eruptions
(Shen et al., 2017). Coronal bright points (CBPs) of network field
configuration of opposite polarities are found to be heated to their
brightness temperature maxima with occurrence of minifilament eruptions (Hong
et al., 2014).

A schematic model of MFs eruptions that has characteristics different from
standard jet model (Shibata et al., 1992) was proposed by Moore et al. (2010),
in which the sheared core field carrying cool plasma to begin erupting when
X-ray brightening in the base arch is observed. Such dynamic small-scale
filament eruptions are often observed to initiate blowout jet scenario through
interchange reconnection at the boundary of open fields and closed flux loops
(Shen, 2012; Adams et al., 2014; Li et al., 2015), and show obvious asymmetric
brightening in the loop called jet bright points (JBPs) (Sterling et al.,
2015). The external reconnection is driven by flux cancellation (Panesar et
al., 2017; McGlasson et al., 2019; Sterling & Moore, 2020) at the polarity
inversion line and opens filament field to allow plasma ejection through the
undergoing internal reconnection (Shen, 2012; Adams et al., 2014; Panesar et
al., 2016, 2018). Recently, by reviewing jet producing minifilament events and
flux cancellation model, Sterling & Moore (2020) proposed a possible mechanism
for magnetic switchbacks in the solar wind detected by Parker Solar Probe
(PSP), in which the jetlets carry twist from flux cancellation in the
interplanetary propagation. In-situ observations of solar wind show expansion
of scale and duration small-scale magnetic flux ropes over propagation (Chen
et al., 2020), which inversely, are used to investigate their possible solar
sources (Bale et al., 2021; Lee et al., 2022). The occurrence rate of small-
scale solar eruptions 0.02 per unit area per hour is arguably sufficient to
heat corona and generate Alvénic pulses for switchbacks (Wang et al., 2022;
Raouafi et al., 2023).

The questions raised to be addressed are: How do small-scale filaments evolve
in the quiet Sun? What is the energy distribution in the chromosphere and
corona in the event of MF eruption? How much free energy from underlying
photospheric magnetic fields is released by the coronal heating and solar wind
acceleration? With high-resolution H$\alpha$ and Ca II spectra, this study
focus on the above questions by investigating a newly observed two-ribbon
microflare associated with partial minifilament eruption.

In this paper, we study spectroscopic and dynamic properties of a mini-
filament observed with the H$\alpha$ and Ca ii lines offered by FISS together
with EUV wavelengths (AIA) images. Although significant progress has been made
in erupting minifilaments in X-ray or EUV wavelengths, earlier identification
of small scale eruption in H$\alpha$ wavelength minifilaments (Wang et al.,
2000b) suggests that fact that jets may have both hot and cool components. In
this regard, spectral diagnostics available from the chromospheric lines will
complement the existing results obtained from X-ray/EUV wavelengths. We focus
on the spectral diagnostics of the the chromospheric temperature and density
variation around the time of the minifilament eruption along with underlying
magnetic field detected by the GST/NIRIS to provide insight into small-scale
solar eruption along with jet-formation.

Figure 1:  Minifilament eruption and associated brightenings. H$\alpha$ images
in the line center (1) and wings at $\pm$0.8 Å (2, 3) show the rising MF
(labled F) and associated with brightening (B1 and B2) on the sides of the
filament channel. The high-resolution magnetogram (4) shows the magnetic
bipole associated with the brightening (a dashed circle). SDO/AIA images in
the 171 Å and 304 Å channels (5, 6) show the brightening associated with the
erupting filament. The superimposed contours are HMI magnetogram at $\pm$20,
$\pm$30, $\pm$40, and $\pm$50 G. The red box in (5) denotes the FoV of (1–4).

##  2 Observations and Data Analysis

###  2.1 Quiet Sun data

On 2021 August 7, BBSO/GST provided ground-based support to Parker Solar Probe
(PSP) observations with multi-wavelength instruments. GST observations of
quiet Sun filament evolution were conducted on the target that is located at
(71$\arcsec$, 450$\arcsec$) before the ninth encounter of the PSP campaign
passing through perihelion (2021–08–09 19:11 UT) on the far side. We utilized
304 Å and 1600 Å images taken by the Atmospheric Imaging Assembly (AIA; Lemen
et al., 2012) on board Solar Dynamic Observatory (SDO) to observe the
brightening associated with the MF eruprion in UV/EUV channels.

High-resolution observations were carried out by the 1.6 m Goode Solar
Telescope (GST; Cao et al., 2010) in Big Bear Solar Observatory (BBSO), with
the advantage of a high-order adaptive optics system combining 308 sub-
apertures and reconstruction technique for solar speckle interferometric data
(Wöger et al., 2008). Photospheric optical continuum images are obtained by
Broadband Filter Imager (BFI) with a TiO ﬁlter (705.7 nm; 10 Å band-pass).
Spectroscopic polarization measurement of the Fe I 1565 nm line (0.25 Å band-
pass) is taken by NIRIS with a round FOV of 80$\arcsec$ at 0$\farcs$24
resolution and 42 s cadence.

H$\alpha$ observation is achieved with Fast Imaging Solar Spectrograph (FISS),
which is capable of recording dual-band Echelle spectrograph at H$\alpha$ line
and Ca ii 854.2 nm simultaneously. Only the H$\alpha$ line data are used in
the present work. Each H$\alpha$ spectrogram (slit image) consists of 256
spectral profiles that represent the spatial variation along the slit with a
sampling of 0$\farcs$16, and each spectral profile, of 512 data points that
cover 0.97 nm (9.7 Å) with a sampling of 1.9 pm (19 mÅ). The position of the
slit on the image plane was successively shifted with a step size of
0$\farcs$16, and one raster scan comprising 100 steps covered a 32$\arcsec$
$\times$ 41$\arcsec$ area inside the field of view. The raster scan was
repeated every 22 s for half an hour.

###  2.2 Reduction and model fitting

To process the FISS spectrographs, an average of the observed background solar
region is taken as a reference profile. All the profiles are normalized by the
maximum intensity (a proxy of the continuum intensity) of the reference
profile. The data profiles are then corrected to the reference profile by a
multiplication factor that is close to unity so that contrast in far wings
vanishes while maintaining contrast accuracy in the H$\alpha$ line center. The
stray light effect in the observation is removed by following the calibration
process of Chae et al. (2013b). Thermodynamic properties (i.e., temperature
and line-of-sight velocity) of the filaments are obtained from H$\alpha$
spectrum inversion. In this study, we applied the embedded cloud model
proposed by Chae (2014) to the calibrated data, which assumes low-lying cloud
features (e.g., minifilaments, spicules) are partially embedded in the
chromosphere that displays both emission and absorption, then use regression
method to the free parameters to fit model with observation. According to
Beckers’ cloud model, the absorption profile is uniform inside the feature and
subject to Doppler broadening only, giving the optical depth

$\tau_{\lambda}=\tau_{0}\exp[-(\frac{\lambda-\lambda_{1}}{W})^{2}],$

and radiative transfer function

$I_{\lambda,obs}=I_{\lambda,in}\exp[-\tau_{\lambda}]+S[1-\exp(-\tau_{\lambda})],$

where $I_{\lambda,obs}$ and $I_{\lambda,in}$ represents observed and incident
intensity of light; parameters $\tau_{0}$, $S$, $W$, and $\lambda_{1}$ are
optical depth at peak absorption, source function, Doppler width and peak
absorption wavelength at line-of sight (LOS) velocity $v$. By assuming
incident intensity (Iλ,in) to equals intensity of light (Rλ,in) at the same
height in chromosphere, basic radiative transfer function is reduced to

$R_{\lambda,obs}=R_{\lambda,in}\exp(-t_{\lambda})+s[1-\exp(-t_{\lambda})],$

in which the $R_{\lambda,obs}$, $R_{\lambda,in}$, and $s$ are ensemble-average
quantities of observed reference profile, intensity of light, and source
function, respectively. $t_{\lambda}$ is defined as

$t_{\lambda}=t_{0}\exp[-(\frac{\lambda-\lambda_{2}}{\omega})^{2}],$

with three introduced parameters $t_{0}$, $\lambda_{2}$, and $\omega$. By
resolving the previously defined parameters(also see, Beckers, 1968;
Tziotziou, 2007; Bostanci & Al Erdoğan, 2010), we obtain temperature and LOS
velocity from the equations

$v=\frac{\lambda_{1}-\lambda_{0}}{\lambda_{0}}c,$

and

$W=\frac{\lambda_{0}}{c}\sqrt{\frac{2KT}{M}+\xi^{2}}$

Combining spatial and wavelength information of H$\alpha$ and Ca II spectra
taken by FISS, we are able to derive thermal and dynamic properties of the
chromospheric structures. Further, with multilayer spectral inverison (MLSI)
method developed by Chen et al. (2020), which divides the relatively wide
range of atmosphere into finite number of layers consisting of photosphere,
lower and upper chromosphere, we estimate and compare kinetic properties in
different height during the eruption of MF.

##  3 Results

An event of minifilament eruption in multiple channels (FISS H$\alpha$ lines,
AIA 171Å, 304Å, and NIRIS magnetograms) is presented in Figure 1. The x-axis
is along East-West direction in all panels. Figure 1(1)–(4) display the GST
observations of the MF evolution from spicular bundles between opposite
polarities of the same FOV, which is indicated by the red box in panel 5. The
MF in sigmoid configuration is displayed in H$\alpha$ filtergrams at central
line and $\pm$0.8 Å off at blue wing (see panel 1–3). Two parts of the arcade
loops coincides at [10$\arcsec$, 7.5$\arcsec$], with brightening structures B1
and B2 appearing across two sides of the central region of the sigmoid, and
fan-spined dark features observed with H$\alpha$ at $-0.8$ Å wavelength. The
magnetic bipole at the MF footpoints are circled in panel 4. Upper
chromospheric and Coronal counterparts of the MF at higher temperature is
observed in AIA 304 Å(50000 K) and 193 Å(1.25 MK) (see Figure 1(5) and (6)),
respectively. Manifestation of a miniature version of two-ribbon flare is
observed in a supplementary H$\alpha$ and AIA (193 Å and 304 Å)movie. Contours
of longitudinal magnetic field at $\pm$50 G are superimposed on AIA 304 Å and
193 Å, with positive (negative) field represented in red (blue). The polarity
inversion line lies along the direction of MF, which is between opposite
polarities of strong magnetic elements.

Figure 2: Evolution of minifilament in H$\alpha$ spectrographs, Broadband TiO
filtergrams, and magnetograms (top to bottom) from 16:55 UT to 17:08 UT. (a),
(b) and (c) show snapshots of the MF eruption in H$\alpha$ images at line
center and $\pm$0.6 off center wavelengths, respectively. (d) show granulation
underneath the MF in photosphere in broadband TiO images. (e) show LOS
magnetograms of the MF event.

Detailed eruption process of the MF is shown in Figure 2. Footpoints of the
emerging MF was preoccupied by spicule bundles surrounding the bright point as
seen in H$\alpha$ line-center wavelength at 16:55:16 UT. The MF emerged from
the bright points with one footpoint anchored in place, of which the rising
flux formed two conjugated arcade loop that extend in opposite directions at
17:00:50 UT. The MF upflow is shown in the H$\alpha$$-$0.8 Å at the same time
as appearance of MF sigmoid structure. The eruptive activity of the rising
sigmoid is observed at 17:05:50 UT with bright points seperating from the
footpoints, followed by prominent downflow three minutes later as seen in
H$\alpha$$+$0.8 Å, as the rising MF vanishes.

Figure 3: Time profile of the MF event in EUV, H$\alpha$ wavelengths, and
magnetiograms. Red and yellow curves in (a) show EUV intensity of the
minifilament core region in 304 Å and 193 Å channels. (b) shows changes of
normalized H$\alpha$ intensity at $-$0.5 Å and $-$0.8 Å from absorption line
center. (c) shows changes of positive (red) and negative (blue) magnetic flux
of the magnetic bipole corresponding the brightening in B1 and B2 in Fig.1.
Black and red dashed lines mark the time of minifilament activation ($t_{1}$)
in H$\alpha$ center line and the start time of minifilament eruption
($t_{2}$). The errorbars in each profile indicate estimation of errors at
3-$\sigma$ deviation.

Figure 3 show time variation of the MF intensity in EUV wavelengths by AIA 193
and 304 Å channels(Fig.3(a)). After subtracting background oscillation, we
found the activation and rising of MF ($t_{1}$) is followed by EUV dimming in
304 Å, and intensity starts to increase when start of the MF eruption
($t_{2}$) is observed in H$\alpha$ images (Fig.2(a-c)). And AIA 193 Å profile
shows similar trend, while the intensity reaches maximum $\sim$2 minutes later
than in 304 Å before dimming between $t_{1}$ and $t_{2}$. The H$\alpha$
profiles (Fig.2(b) in both $-$0.5 and $-$0.8 Å show increase of Doppler blue-
shift indicated by absorption contrast starting from $t_{1}$ and decrease to
background level $\sim$3 minutes after $t_{2}$. Fig.2(c) show that positive
magnetic flux decreased by 6.2$\times$$10^{16}$ Mx through MF eruption while
the negative magnetic flux decreased by 3.0$\times$$10^{16}$ Mx from $t_{1}$
to 3 minutes after $t_{2}$.

Figure 4: Images of MF eruptions at different wavelengths. (a) shows magnetic
bipole at footpoint of the MF sigmoid at (positive) flux peak time. (b) and
(d) show the brightenings at footpoint marked (as B1 and B2) in Fig.1(a) and
the MF in H$\alpha$ at line center and $-$0.6 Å. (c) shows the flaring ribbon
along the the MF in AIA 304 ÅṪhe yellow curves in (a) and (b) denote slit S1
across the MF, and the yellow curve in (c) denotes the slit along the MF
sigmoid. Figure 5: Time-distance diagrams of the minifilament and associated
bright ribbons. (a) and (b) show intensity variation of EUV wavelength at 304
Å from AIA and H$\alpha$ $-$0.5 Å off line center, along the slit S2,
respectively. Blue and green dashed lines during 16:57 UT–17:00 UT and 17:03
UT–17:05 UT indicate ribbon and MF motion, respectively.(c) and (d) show
flaring ribbon motion diverging from the MF and divergence of magnetic bipoles
at the MF footpoint along the slit S1 across the MF. Green dashed line
indicates the motion of bright ribbon front. Red (blue) dashed lines indicate
positive (negative) magnetic flux motions. Black and white dashed lines mark
the time of minifilament activation ($t_{1}$ in fig.3(c)) in H$\alpha$ center
line and the start time of minifilament eruption ($t_{2}$ in fig.3(c)) .

Two slits S1 and S2 are created along the magnetic bipole direction and along
the MF sigmoid trace, respectively (see Figure 4(a) and (c)). Figure 5 show
time-distance diagrams of the MF along S1 (Fig.5 (a) and (b)) and S2(Fig.5 (c)
and (d)). In Fig.5 (a), we found the motion of the bright point and the
following dark feature along the MF sigmoid. The apparent speed of bright
point is 38–54 $km~{}s^{-1}$, and the speed of dark feature seen in Fig.5 (a)
and (b) is 21.5$\pm$1.0 $km~{}s^{-1}$. Fig.5(c) shows the ribbon propagation
from footpoint of the MF sigmoid at a speed of 11$\pm$0.8 $km~{}s^{-1}$.
Transverse motions of the magnetic bipole footpoints in Fig.5(d) show speed of
1.9$\pm$0.5 $km~{}s^{-1}$.

In the magnetic field evolution the negative polarity is stationary in
position while the positive polarities emerge and migrate towards negative
polarity in the rising phase of MF. Field strengths in both positive and
negative polarities decrease by $\sim$30$\%$ (24$\%$) in MF eruption (see
Figure 3(c)). With accurate co-alignment in H$\alpha$ images and AIA images
(193 Å, 304 Å), we distinguished this MF eruption with FISS spectra and the
exact location of its coronal counterparts in AIA EUV images. This enables us
to study thermodynamic properties of the MF eruption from characteristics of
the spectra and their connection with coronal dynamics. We focus our spectral
analysis of H$\alpha$in the following subsection, on the transient part of the
MF which survived $\sim$10 minutes in H$\alpha$ images.

###  3.1 Spectrosopic Analysis of the Minifilament

Figure 6: Panels (a)-(d): FISS spectra of H$\alpha$((a) and (b)) and Ca II
8542Å ((c) and (d)) lines in time sequence from rise of the MF to eruption.
Spectra ((a) and (c)) of the dashed slit position pass through the ML loop and
a bright kernel; Spectra ((b) and (d) of the solid slit position pass through
an extending bright ribbon. Panel (e)-(h): Raster images of H$\alpha$((e) and
(f)) and Ca II 8542 Å ((g) and (h)) at 17:00:50 UT and 17:07:06 UT.

We show the spectroscopic data obtained with 22 s cadence between 16:59:33 UT
and 17:07:06 UT in Figure 6. The raster images of H$\alpha$ and Ca II 8542 Å
are taken at line center(see Figure 6(e-h)), respectively. The H$\alpha$
images display an $\epsilon$-shape MF structure near the center of the FOV,
with two J-shaped arms reaching $\sim$5″ from the junction ($\sim$[8$\farcs$0,
7$\farcs$5]). We select a slit position across the prominent rising arm of MF
structure as well as bright point in vicinity. In the H$\alpha$ spectral
profiles, strong blue-shifted absorption is detected from 17:00:50–17:02:30
UT. The Ca II 8542 Å spectral profiles show narrowed absorption in the line
center, with corresponding brightening fibrils ($\sim$[8$\farcs$0,
7$\farcs$0]) in raster images (Figure 6(g)). Slit 2 is selected across the
bright ribbon along the rising segment of the MF and cut through the J-shaped
arm at $\sim$[10$\farcs$5, 10$\farcs$0]. At 17:07:06 UT, The MF feature in low
opacity broaden in northwest direction, with Ca II 8542 Å bright fibrils
ceding underneath the MF loop.

Figure 7:  Figure 8: MLSI results of (a) Ca II and (b) H$\alpha$ line spectra
in the mini-filament. The layers of the spectral fitting are denoted as 0 (top
layer), 1 (middle layer), and 2 (bottom layer) in the chromospheric heights.
Black curves indicate observed spectra of selected feature point F. Reference
profile is represent in dotted cyan curve, and fitting results are represented
in red, blue and green curves from top to bottom layers, respectively.

We applied the most update MLSI code to H$\alpha$ and Ca II 8542 Å spectra at
different points (B,F, and R) in the slit postions displayed in Figure 6.
Given the source function and opacity are variable along chromosheric height,
MLSI generate a composite of spectral parameters to represent different
layers. In Figure 7 and figure 8, for the spectra inside the MF, the H$\alpha$
profile fitting is based on the 4-layer model fit of the profile. Spectral
wavelengths in the cloud feature range [-1.2 Å 0.30 Å] has been fitted by the
4-layer model, while is fitted by 3-layer model outside the wavelengths of the
cloud regime, in which, the top layer is like a cloud. The MF is characterized
by source function of $\sim$0.08, optical depth of 0.6, Doppler width of 0.33
Å, and Doppler velocity of $-$6.7 $km~{}s^{-1}$. The temperature of the MF is
around 1$\times$$10^{4}$ K. The Doppler width of the H$\alpha$ in combination
with the Ca II 8542 Å Doppler width of 0.25 Å yields the temperature estimate
of 9,400 K at the beginning of the MF rising phase. The above values may vary
depending on the choice of the cloud wavelength range, but not much under the
requirement of good fitting. With above selection of fitting parameters, we
have verified that at onset of the filament rising phase, the MF can be
treated as non-emitting cloud. Therefore, in the following results of spectral
inversion, we focus on the central regime of the spectral wavelengths using
parameters of the cloud features.

Figure 9: Absorption profiles of the cloud feature at start of the MF rising
phase (17:00:50 UT). The blue curves and red curves represent background
profiles and observation data in the absorption line profiles, respectively.
The green curves represent fitting of cloud features of the MF in range $-$1.5
– 1.5 Å. In the contrast profiles, blue and red curves represent data
reduction and fitting of cloud features, respectively.

Figure 9 show inversion of H$\alpha$ and Ca II spectral profiles using
embedded cloud model (ECM) at start time (17:00:50 UT) of the MF eruption. The
strongest upflow motion indicated by the blue shift of H$\alpha$ is observed
at this moment. The Doppler velocity is estimated reaching $\sim$18
$km~{}s^{-1}$ by using lambdameter method (Chae et al., 2013a) on H$\alpha$
profile. In Ca II 8542 Å profile, we find the enhanced intensity with
redshift, indicating bright fibrils associated with the MF rising. We
interpret such relation as photospheric magnetic field cancellation driving
embedded filament upward motion.

Figure 10:  Figure 11: Temperature and Doppler velocity of the MF. In the
middle chromosphere temperature decreased by 2500 K in 4 minutes after
eruption while in the higher chromosphere temperature continuously rise until
reaching 12,600 K in 7 minutes. The solid (dashed) red and blue curves
represent temperature and nonthermal speed in the high (middle) chromosphere,
respectively.

Temperature variation and Doppler velocity change of the MF before and after
eruption is shown in figure 10 and figure 11 from 16:56:58 UT to 17:12:56 UT.
As shown in the velocity profile, the MF has prominent upflow from the rising
phase (17:00:50 UT) to eruption (17:07:06 UT). Peak value 28 $km~{}s^{-1}$ of
Doppler velocity is reached when the dark feature become visible in H$\alpha$
blue-wing images (see Figure 2 (b2)). Temperature of cloud feature show
continuous increase in 403 s from MF rise to eruption, followed by an abrupt
decrease in the next 160 s. Temperature reaches maximum at 17:06:15 UT when
the eruption occurs and filament structure eject laterally (see Figure 2 (b4)
and (b5)). The acceleration in upflow velocity and the photospheric flux
cancellation show close association, during which the magnetic flux reaches
minimum at the same as the upflow speed reaches maximum. However, time
discrepancy is found between temperature rise of the MF feature and flux
cancellation but is co-temporal with the following magnetic flux increase. We
interpret this relation as continuous emergence of flux bring the MF to upper
chromospheric heights through reconnection driven acceleration.

Figure 12: Dopplergram and Temperature correlation of the MF event. (a), (d),
and (g) show Dopplergrams at MF activation (T1, 16:57:49 UT), start time of
eruption (T2, 17:03:19 UT) and post-eruption time(T3, 17:05:25 UT). Yellow
(blue) contours indicate the brightening features in H$\alpha$line center
($-$0.6Å off central wavelength) (b), (e), and (h) show Temperature maps at
T1, T2, and T3, respectively. (c), (f), and (i) show correspondence of
Temperature and Doppler velocity of the MF event at T1, T2, and T3,
respectively.

At [8”,8”] of the Dopplergrams((a),(d), and (g)) and Temperature maps ((b),
(e), and (h)) in figure 12, prominent upflow is shown along the MF sigmoid
associated with temperature burst near the central region. The scatterplots at
activation time 16:57:49 UT in Fig.12(c) show temperature in the MF region is
$\sim$9,700 K with cool materials of 7,500 K rising at Doppler speed of 10
$km~{}s^{-1}$. Then with enhanced blue shift observed at start of eruption in
the Dopplergram (Fig.12(d)), a segment of high speed area (l2) that over 10
$km~{}s^{-1}$ appear in the scattered distribution, while the temperature does
not show promiment change. At post-eruption time 17:05:25 UT, the scatterplot
(Fig.12(i)) shows 2D Gaussian like distribution centered at 9,800 K. We
observed that with upflow speed over 10 $km~{}s^{-1}$, temperature is 9,500 K,
which is 2,000 K higher than start of the MF eruption.

###  3.2 EUV brightenings associated with the MF eruption

Figure 13: MF eruption in UV/EUV images and magnetograms at 17:01 UT. Panel
(a), (b) and (d) show AIA images in 193, 304, and 1600 Å channels.  Figure 14:
MF eruption in UV/EUV images and magnetograms at 17:07 UT. Panel (a), (b) and
(d) show AIA images in 193, 304, and 1600 Å channels.

Figure 13 and Figure 14 show EUV brightenings in AIA 193 Å (Figure 13 (a))
with corresponding MF at 17:01:04 UT (Figure 13 (b) and two-ribbon flare at
17:07:53 UT (Figure 14 (b)). The EUV brightenings occur as the dark filament
feature appear from bright background in AIA 304 Å, and vanishes when two-
ribbon flaring configuration is observed in chromosphere. Chen et al. (2020)
found that at onset of the MF, it has no corresponding brightening signals and
experiences slow-to-fast acceleration. while in this event, not only the
acceleration is associated with appearance of rising filament, brightening is
also observed at coronal heights before flare ribbon appearance. This
indicates reconnection of a blowout jet model, in which the interchange
reconnection with external field and reconnection of internal flux loop
occurred the same time.

To estimate the energy distribution of the MF event, we applied the cloud
model assumption proposed by Tsiropoula & Schmieder (1997) to determine plasma
density in the chromospheric cloud features. This allows us to calculate
kinetic and thermal energy release of the MF eruption from fitted
thermodynamic properties obtained in Section 3.1. Combining AIA 304 Å and
H$\alpha$ line-center images, we find the eruptive filament of 12 ″length and
maximum width of 2$\farcs$5\. Adopting the mass density estimation of
1.1$\times$$10^{-13}$ $cm~{}s^{-3}$, the kinetic energy change of the rising
filament is 1.50$\times$$10^{25}$ ergs, and thermal energy accumulation in the
MF is 1.44$\times$$10^{25}$ ergs. From the photospheric magnetograms, we find
the magnetic energy change is 1.57$\times$$10^{26}$ ergs across the PIL of
converging opposite magnetic elements. Our previous study of quiet Sun small-
scale eruptions suggest that magnetic flux cancellation in granule scale
provides sufficient free energy to heat the corona (Wang et al., 2022).

##  4 Summary and Discussions

In this paper, we investigate the chromospheric thermodynamics of minifilament
eruption associated with brightening and corresponding photospheric magnetic
cancellation with high-cadence spectral data of H$\alpha$ line from FISS, and
high-resolution magnetograms from NIRIS of GST. With updated MLSI of H$\alpha$
data, temperature evolution and energy transportation of chromespheric
features are quantitatively evaluated at a small-scale brightening region. The
coronal bright points are observed in AIA EUV (193, 304 Å) channel imaging
data as micro-flare in the quiet Sun, which is associated with partial
minifilament eruption. Key structural and dynamic properties of the eruption
minifilament are summarized as follows.

  1. 1.

With high-resolution spectroscopy, we analyzed the dynamic process of this
very small scale sigmoidal mini-filament. Such micro-flare related
minifilament eruption is not associated with jet/jetlet in vicinity, while
coronal brightening is observed along the trajectory direction of the
eruption. This resemblances two-ribbon flare with filament eruption in the
large scale.

  2. 2.

Temperature rise in the flaring regions is co-temporal with magnetic flux
emergence while the brightening emission in corona is associated with
photospheric flux cancellation. Time evolution of the temperature in the
filament sigmoid show maximum value when lateral displacement is observed in
the flux loop. Temperature estimation shows increase from 8,800 K to 12,600 K,
an increase of almost 4,000 K during the eruptive phase, while the maximum
Doppler velocity (18 $km~{}s^{-1}$) of the filament is observed at beginning
of the eruption.

  3. 3.

Thermal and kinetic energy releases are both in the order of $1.5\times
10^{25}$ ergs, consist of 18.7% of total magnetic energy change in the
photosphere from emergence phase to flux cancellation. The released energy is
considered as minimum of minifilament eruption release, which is comparable to
enhanced spicular events.

Minifilament eruptions are known as downscale filament activities in the quiet
Sun. Systematic survey of 32 days of BBSO H$\alpha$ data for small-scale
eruptive filamentary events suggest such ubiquitous presence as many as 600
per 24 hour day (Hermans & Martin, 1986b). Aside from topological similarities
between minifilament events and large scale filaments that often has one or
both ends anchor at cancellation site, small-scale filament also undergo
emergence phase (slow rising phase) and eruption phase, which are often
associated with microflares. We report observations of microflare ribbons in
H$\alpha$ that are often unresolved due to resolution limit of instrument. The
microflare is visible in AIA 304 Å as well, with j-shaped dark feature
overlying along the PIL of the magnetic bipole in the photosphere. In the
eruptive phase, the sigmoid feature experienced heating processing and lateral
displacement. Such chromospheric heating can be interpreted as results of
microflare triggered particle beams (Frogner, L. et al., 2020).

Typically, both cool and hot plasma are both ejected from eruptions. In this
event, we have not observed cool plasma ejection into the corona that are
often associated with (mini)filament eruption and (micro)flare. Sterling et
al. (2022) proposed a small-scale analog of coronal jet, in which the cool
materials of erupting minifilamet fill in the open field jet spire. However,
not all coronal jets are associated with presence of erupting cool
minifilament. Kumar et al. (2019) found 33% of the coronal jets in their study
have no evidence of cool plasma at base of the spire. One argument of the
missing coronal jet is that such small-scale eruptive activity have weak
signal in EUV, especially cool spire can be smeared by background signals. The
observed filament is of typical small-scale filament size, with coronal
brightening counterparts of the magnetic cancellation site. Study of coronal
jet eruptions by Panesar et al. (2020) conclude that magnetic cancellation
triggers the eruption before runaway breakout/tether-cutting reconnection when
speed of the rising minifilament is above 1 $km~{}s^{-1}$. Our results show
that the minifilament experience explosive motion in vertical direction with
speed over 10 $km~{}s^{-1}$ following magnetic flux cancellation in
photosphere, which suggest the runaway reconnection at the internal current
sheet drove minifilament eruption. However, transition from slow rise to fast
rise (eruption) is not observed. Doppler velocity decreases when lateral
expansion of the minifilament is observed in H$\alpha$ images. Coincident of
magnetic flux cancellation and acceleration of minifilament eruption in this
study agree with previous studies of large filament eruptions that tether-
cutting reconnection accelerate eruptions (Moore & Sterling, 2006).

With high cadence H$\alpha$ spectral data observed by FISS, we found that the
minifilament in the chromosphere has close connection with CBP phenomena. In
the event of mini-filament eruption, jet is not associated with the
chromospheric eruptive motions as identified in previous observations.
Instead, the loop shape brightening in corona become visible following the
minifilamentary eruption. Analysis of magnetic properties from NIRIS data
indicates that the cancellation of a magnetic bipole in the footpoints of
minifilament triggered the mini-eruption. Line-of-sight velocity and
temperature from an inversion of H$\alpha$ spectra with embedded cloud model
exhibit a temperature increase of 13,300 K in the brightening region,
associated with average minifilament rising speed increased by 21
$km~{}s^{-1}$ in 10 minutes. Such a study of high-resolution observations
provides further details of connections between photospheric magnetic topology
and its chromospheric counterpart, with which we conclude that energy released
through mini-eruptions could be a source of coronal heating.

We gratefully acknowledge the use of data from the Goode Solar Telescope (GST)
of the Big Bear Solar Observatory (BBSO). BBSO operation is supported by US
NSF AGS-2309939 and AGS-1821294 grants and New Jersey Institute of Technology.
GST operation is partly supported by the Korea Astronomy and Space Science
Institute and the Seoul National University. This work was supported by NSF
grants, AGS-2114201, AGS-2229064 and AGS-2309939. It is also supported by NASA
grants, 80NSSC19K0257, 80NSSC20K1282 and 80NSSC24K0258.

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# Sailing Through Point Clouds: Safe Navigation
Using Point Cloud Based Control Barrier Functions

Bolun Dai∗1, Rooholla Khorrambakht∗1, Prashanth Krishnamurthy1, Farshad
Khorrami1 ∗ Equal Contribution.This work was supported in part by the NYUAD
Center for Artificial Intelligence and Robotics (CAIR), funded by Tamkeen
under the NYUAD Research Institute Award CG010.1Bolun Dai, Rooholla
Khorrambakht, Prashanth Krishnamurthy, and Farshad Khorrami are with
Control/Robotics Research Laboratory, Electrical & Computer Engineering
Department, Tandon School of Engineering, New York University, Brooklyn, NY
11201 {bolundai, rk4342, prashanth.krishnamurthy<EMAIL_ADDRESS>
###### Abstract

The capability to navigate safely in an unstructured environment is crucial
when deploying robotic systems in real-world scenarios. Recently, control
barrier function (CBF) based approaches have been highly effective in
synthesizing safety-critical controllers. In this work, we propose a novel
CBF-based local planner comprised of two components: Vessel and Mariner. The
Vessel is a novel scaling factor based CBF formulation that synthesizes CBFs
using only point cloud data. The Mariner is a CBF-based preview control
framework that is used to mitigate getting stuck in spurious equilibria during
navigation. To demonstrate the efficacy of our proposed approach, we first
compare the proposed point cloud based CBF formulation with other point cloud
based CBF formulations. Then, we demonstrate the performance of our proposed
approach and its integration with global planners using experimental studies
on the Unitree B1 and Unitree Go2 quadruped robots in various environments.

##  I Introduction

With the increasing capabilities of robotic systems, mobile robots have been
deployed to navigate unstructured environments autonomously for various
operations [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. In all these applications of
autonomous mobile robots, safety is of paramount importance [3, 4, 5, 6].
Recently, control barrier functions (CBFs) [7] have been gaining popularity in
synthesizing safe control actions. One of the main benefits of CBF-based
approaches is that they can transform nonlinear and nonconvex constraints into
linear ones, greatly increasing the computation speed.

For CBF-based methods, a CBF-based quadratic program (CBFQP) is commonly used
to generate safe actions, which can be seen as an MPC with a preview horizon
of one timestep. However, this myopic nature makes CBF-based methods
susceptible to getting stuck in spurious equilibria [8], which makes it
challenging to utilize CBF-based methods for navigation tasks. In [8], a
circulation constraint was introduced to reduce the possibility of getting
stuck in spurious local equilibria and to induce the robot to circulate the
obstacles under such events. There are recent efforts in combining CBFs with
MPC [1, 10]. Although the CBF constraint is a linear constraint for one
timestep, it becomes a nonlinear and nonconvex constraint for longer preview
horizons. This often makes the CBF-MPC a nonlinear MPC (NMPC), which is
computationally expensive to solve, especially for long preview horizons.

Another main challenge in utilizing CBF-based methods for navigation is the
synthesis of CBFs [11, 12]. Traditionally, robots are modeled as spheres to
simplify the synthesis of CBF for robotic systems. However, this can be overly
conservative when operating in a constrained setting. Recently, work [13, 14,
15] has been done using differentiable optimization based growth distance [16,
17] CBFs to model robots and their surroundings using a wider range of
geometric primitives [18]. This approach has been extended in [14] to consider
dynamic obstacles. Although both [13] and [14] are effective in synthesizing
safe controls, they require an extra step of decomposing the environment into
a collection of primitives. For navigation tasks, the main mode of
representation of the environment is often in the form of point clouds
obtained via LiDAR or depth cameras. There are recent papers on synthesizing
CBFs directly from point cloud data, such as [1, 2]. In [1], the CBF
formulation relies on encapsulating the obstacle point cloud with ellipsoids
and the robot as a sphere (possibly causing overbounding). In [2], the
obstacles are modeled directly using point clouds and a CBF formulation is
used based on the smoothed Euclidean distance between a point and a rectangle.
Compared with prior point cloud based CBF approaches, the CBF formulation
proposed in this paper directly models the obstacles using point clouds and
enables the robot to be modeled using more general shapes, i.e., higher-order
ellipsoids, which is a superset of how the robots are modeled in [1, 2].

To address the various challenges outlined above, this paper proposes a novel
point cloud CBF based local planner. The main contributions of this paper are
as follows:

  1. 1.

proposed Vessel: a novel growth distance based point cloud CBF formulation;

  2. 2.

proposed Mariner: a novel CBF-based local preview controller that can mitigate
spurious equilibrium;

  3. 3.

empirically validated the proposed safe local planner (Vessel \+ Mariner) in
simulation and the real world on the Unitree B1 and Unitree Go2 quadruped
robots;

  4. 4.

integrated the proposed safe local planner with a global planner and validated
its performance in indoor environments.

Compared with previous works, the main benefits of our proposed method are:

  1. 1.

the CBF formulation (Vessel) is more general and does not require post-
processing of the point cloud data;

  2. 2.

the local planner (Mariner) can replan at the sensor rate given its
computational efficiency and its ability to directly operate on sensor
readings without relying on a mapping algorithm;

  3. 3.

the combination of Mariner and a global planner (e.g., RRT⋆) makes the
generated motion plan biased towards shorter paths and straight lines.

The remainder of this paper is structured as follows. Section II briefly
reviews CBFs. In Section III, the safe indoor navigation problem is
formulated. In Section IV, we present the Vessel and Mariner formulation. In
Section V, we first compare our proposed method with existing point cloud
based CBF methods for safe navigation; then, we perform ablation studies to
show the effectiveness of each component of our proposed approach.
Additionally, real-world experiments were performed using the Unitree B1 and
Unitree Go2 quadruped robots to demonstrate the effectiveness of our proposed
approach. Finally, in Section VI, we conclude the paper with a discussion on
future directions.

##  II Preliminaries

In this section, we provide a brief introduction to CBFs [7] and CBF-based
safe control synthesis. Consider a control affine system

$\dot{\bm{x}}=\bm{F}(\bm{x})+\bm{G}(\bm{x})\bm{u}$ (1)

where the state is represented as $\bm{x}\in\mathbb{R}^{n}$, the control as
$\bm{u}\in\mathbb{R}^{m}$, the drift as
$\bm{F}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$, and the control matrix as
$\bm{G}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n\times m}$. Following [7],
define two sets $\mathcal{C}$ and $\mathcal{D}$ where the relationship between
the two sets is $\mathcal{C}\subset\mathcal{D}\subset\mathbb{R}^{n}$. Let
$\bm{h}:\mathcal{D}\rightarrow\mathbb{R}$ be a continuously differentiable
function that has $\mathcal{C}$ as its 0-superlevel set, i.e.,

$\mathcal{C}=\\{\bm{x}\mid\bm{h}(\bm{x})\geq 0,\bm{x}\in\mathbb{R}^{n}\\}$ (2)

and $\partial\bm{h}/\partial\bm{x}\neq 0$ for all
$\bm{x}\in\partial\mathcal{C}$, where
$\partial\mathcal{C}\subset\mathbb{R}^{n}$ represents the boundary of
$\mathcal{C}$. Then, if

$\sup_{\bm{u}\in\mathcal{U}}\Big{[}\frac{\partial\bm{h}(\bm{x})}{\partial\bm{x}}\Big{(}\bm{F}(\bm{x})+\bm{G}(\bm{x})\bm{u}\Big{)}\Big{]}\geq-\Gamma(\bm{h}(\bm{x}))$
(3)

holds for all $\bm{x}\in\mathcal{D}$ with
$\Gamma:\mathbb{R}\rightarrow\mathbb{R}$ being an extended class
$\mathcal{K}_{\infty}$ function111Extended class $\mathcal{K}_{\infty}$
functions are strictly increasing with $\Gamma(0)=0$., $\bm{h}$ is a CBF on
$\mathcal{C}$. A common approach to utilize the CBF constraint is to formulate
the safe control problem as a CBFQP [7], which has the form of

$\displaystyle\min_{\bm{u}}\ $ $\displaystyle\
\|\bm{u}_{\text{ref}}-\bm{u}\|_{2}^{2}$ (4) subject to $\displaystyle\
\frac{\partial\bm{h}(\bm{x})}{\partial\bm{x}}\Big{(}\bm{F}(\bm{x})+\bm{G}(\bm{x})\bm{u}\Big{)}\geq-\Gamma(\bm{h}(\bm{x}))$

with $\bm{u}_{\text{ref}}\in\mathbb{R}^{m}$ being a reference control action
obtained using a performance controller [7]. The performance controller can be
obtained using a variety of methods. Common performance controllers include
proportional-derivative (PD) controllers, MPC-based controllers, and control
Lyapunov function (CLF) based controllers.

##  III Problem Formulation

This work addresses the problem of safe navigation for robotic systems. We
make the following assumptions.

  1. 1.

We have access to an initial occupancy map outlining the floor plan structure
of the environment for global path planning and a localization module
providing the position of the robot on this map.

  2. 2.

The robot can obtain information regarding its surroundings in the form of
point cloud measurements in its body frame.

A common scenario that satisfies the first assumption is when we have the
floor plan of a building containing static features but not transient
structures and obstacles like garbage cans, boxes, or others. The focus of
this work is to propose a local planner that, given point cloud data and a
global path, plans an efficient local path while ensuring the robot’s safety.

######  Remark 1.

The assumption of having an initial floor plan can be relaxed if the global
planner is allowed to replan at a fixed rate.

##  IV Method

In this section, we present our proposed local planner and how it integrates
into a navigation pipeline.

###  IV-A Vessel: Point Cloud Based Control Barrier Function

Figure 1: Visualization of the proposed point cloud CBF in a 2D scenario for
different versions of higher-order ellipsoids. The blue dots represent the
measured point cloud. The dark orange higher-order ellipse represents the
unscaled primitive defined in (5), and the light orange higher-order ellipse
represents the primitive after it is scaled by the scaling factor. The order
of the ellipse is given in the upper left corner.

Assume the robot, in its body frame, is encapsulated by an ellipsoid
$\mathcal{E}$ defined as

$\mathcal{E}=\Big{\\{}\bm{p}\ \Big{|}\ \bm{p}^{\top}\bm{P}\bm{p}\leq
1\Big{\\}}$ (5)

and the matrix $\bm{P}$ given by

$\bm{P}=\begin{bmatrix}1/a^{2}&0&0\\\ 0&1/b^{2}&0\\\
0&0&1/c^{2}\end{bmatrix}.$ (6)

with $a,b,c\in\mathbb{R}_{+}$ denoting the lengths of its semi-axes.

######  Remark 2.

We call this ellipsoid (including the higher-order ellipsoids as defined in
(10)) a Vessel that contains the robot.

Let the obstacles be given as a point cloud $\mathcal{P}\subset\mathbb{R}^{3}$
in the body frame of $\mathcal{E}$. Then, for each point
$p_{j}\in\mathcal{P}$, we can compute the uniform scaling factor
$\bm{s}_{j}\in\mathbb{R}_{+}$ between the ellipsoids and $\bm{p}_{j}$ as

$\bm{s}_{j}^{2}=\bm{p}_{j}^{\top}\bm{P}\bm{p}_{j}.$ (7)

As seen in Fig. 1, a larger-than-one scaling factor implies that the robot is
not in collision with the point cloud. Thus, we can specify the safe set as
the set of states where $\bm{s}_{j}^{2}\geq 1$. Define $\bm{\alpha}_{j}$ and
$\bm{h}_{j}$ as

$\bm{\alpha}_{j}=\bm{s}_{j}^{2},\quad\bm{h}_{j}=\bm{\alpha}_{j}-\bm{\beta}$
(8)

where $\bm{\beta}\geq 1$. Then, the CBF between point cloud $\mathcal{P}$ and
$\mathcal{E}$ can be computed as

$\bm{h}=\min_{j}{\bm{h}_{j}}.$ (9)

######  Remark 3.

From the derivations in [14], we know that by ensuring safety with respect to
the point with the smallest CBF value, we can ensure safety with respect to
all points in the point cloud.

A visualization of the effect of scaling an ellipsoid using the smallest
scaling factor is given in Fig. 1. This formulation can be extended to higher-
order ellipsoids defined as

$\mathcal{E}^{\bm{d}}=\Big{\\{}\bm{p}\Big{|}\bm{p}_{\bm{d}}(\bm{p},\bm{d})^{\top}\bm{P}^{\bm{d}}\bm{p}_{\bm{d}}(\bm{p},\bm{d})\leq
1\Big{\\}}$ (10)

with $\bm{d}\in\mathbb{N}$ representing the order of the ellipse and
$\bm{p}_{\bm{d}}:\mathbb{R}^{3}\times\mathbb{N}\rightarrow\mathbb{R}^{n}$
defined as

$\bm{p}_{\bm{d}}(\bm{p},\bm{d})=\begin{bmatrix}x^{\bm{d}}\\\ y^{\bm{d}}\\\
z^{\bm{d}}\end{bmatrix}\in\mathbb{R}^{3},\ \mathrm{with}\
\bm{p}=\begin{bmatrix}x\\\ y\\\ z\end{bmatrix}\in\mathbb{R}^{3}.$ (11)

For higher-order ellipsoids, the scaling factor and CBF would be computed as

$\displaystyle\bm{s}_{j}$
$\displaystyle=\sqrt[\bm{d}]{\bm{p}_{\bm{d}}(\bm{p}_{j},\bm{d})^{\top}\bm{P}_{i}^{\bm{d}}\bm{p}_{\bm{d}}(\bm{p}_{j},\bm{d})}$
(12a) $\displaystyle\bm{\alpha}_{j}$ $\displaystyle=\bm{s}_{j}^{\bm{d}}$ (12b)
$\displaystyle\bm{h}_{j}$ $\displaystyle=\bm{\alpha}_{j}-\bm{\beta}.$ (12c)

######  Remark 4.

In this paper, ellipsoids refer to a higher-order ellipsoid of order one. The
order for higher-order ellipsoids with $\bm{d}>1$ will be specified.

A visualization of using different orders of higher-order ellipsoids is also
given in Fig. 1. Given that the minimum function is not differentiable, we
follow the formulation in [8] and adopt a soft version, i.e., the
$\mathrm{softmin}$ function, which is defined as

$\operatorname*{softmin}_{j}\bm{h}_{j}=-\delta\ln\Bigg{(}\frac{1}{N}\sum_{j=1}^{N}{e^{-\bm{h}_{j}/\delta}}\Bigg{)}$
(13)

where $N\in\mathbb{N}$ represents the number of points in the point cloud
$\mathcal{P}$ and $\delta\in\mathbb{R}_{+}$. The value of $\mathrm{softmin}$
approximates the minimum function when $\delta$ is close to zero. The
expression in (13) is susceptible to being subjected to numerical instability,
which we address by adopting the following equivalent form [8]

$\displaystyle\operatorname*{softmin}_{j}\bm{h}_{j}$
$\displaystyle=\min_{j}\bm{h}_{j}+\operatorname*{softmin}_{j}\widehat{\bm{h}}_{j}$
(14a) $\displaystyle\widehat{\bm{h}}_{j}$
$\displaystyle=\bm{h}_{j}-\min_{k}\bm{h}_{k}.$ (14b)

With this, the final form of our CBF formulation becomes

$\bm{h}=\min_{j}\bm{h}_{j}-\delta\ln\Bigg{(}\frac{1}{N}\sum_{j=1}^{N}{e^{-\widehat{\bm{h}}_{j}/\delta}}\Bigg{)}.$
(15)

The partial derivative of the CBF with respect to the position and orientation
of the ellipsoid can be computed as

$\frac{\partial\bm{h}}{\partial(\bm{r},\bm{q})}=\frac{\displaystyle\sum_{j=1}^{N}{\Bigg{[}e^{-\widehat{\bm{h}}_{j}/\delta}\frac{\partial\bm{h}_{j}}{\partial(\bm{r},\bm{q})}}\Bigg{]}}{\displaystyle\sum_{j=1}^{N}{e^{-\widehat{\bm{h}}_{j}/\delta}}}\in\mathbb{R}^{7}$
(16)

where $\bm{r}\in\mathbb{R}^{3}$ and $\bm{q}\in\mathbb{H}$ ($\mathbb{H}$
represent the Hamiltonian) represents the position of and orientation as
quaternions of $\mathcal{E}$ in the world frame, respectively. Next, we
present the theoretical results that show the proposed point cloud CBF
formulation in (15) is a valid CBF.

######  Theorem 1.

For a point cloud $\mathcal{P}$, the CBF proposed in (15) is continuously
differentiable with respect to the position and orientation of its
encapsulating higher-order ellipsoid.

###### Proof.

The proof utilizes the following properties: for two continuously
differentiable functions $\bm{f}_{1}:\mathbb{R}^{n}\rightarrow\mathbb{R}$ and
$\bm{f}_{2}:\mathbb{R}^{n}\rightarrow\mathbb{R}$, we have
$\bm{f}_{1}(\bm{f}_{2}(\bm{x}))$, $\bm{f}_{1}(\bm{x})+\bm{f}_{2}(\bm{x})$, and
$\bm{f}_{1}(\bm{x})\bm{f}_{2}(\bm{x})$ are all continuously differentiable
functions. For a point $\bm{p}_{j}$ measured in the body frame of
$\mathcal{E}$, its corresponding world frame coordinate is

$\bm{p}_{w,j}=\bm{Rp_{j}+r}=\begin{bmatrix}x_{w,j}&y_{w,j}&z_{w,j}\end{bmatrix}^{\top}.$
(17)

where $\bm{R}\in\mathrm{SO}(3)$ represents the orientation of $\mathcal{E}$ in
the world frame as a rotation matrix. Then, expanding the CBF in (12c) for an
order $\bm{d}$ ellipsoid yields

$\displaystyle\bm{h}_{j}=$ $\displaystyle\
\frac{1}{a^{2\bm{d}}}\Big{[}-(\bm{r}_{x}-x_{w,j})(2\bm{q}_{w}^{2}+2\bm{q}_{x}^{2}-1)$
$\displaystyle-\
2(\bm{r}_{y}-y_{w,j})(\bm{q}_{w}\bm{q}_{z}+\bm{q}_{x}\bm{q}_{y})$
$\displaystyle+\
2(\bm{r}_{z}-z_{w,j})(\bm{q}_{w}\bm{q}_{y}-\bm{q}_{x}\bm{q}_{z})\Big{]}^{2\bm{d}}$
$\displaystyle+\
\frac{1}{b^{2\bm{d}}}\Big{[}2(\bm{r}_{x}-x_{w,j})(\bm{q}_{w}\bm{q}_{z}-\bm{q}_{x}\bm{q}_{y})$
$\displaystyle-\ (\bm{r}_{y}-y_{w,j})(2\bm{q}_{w}^{2}+2\bm{q}_{y}^{2}-1)$
$\displaystyle-\
2(\bm{r}_{z}-z_{w,j})(\bm{q}_{w}\bm{q}_{x}+\bm{q}_{y}\bm{q}_{z})\Big{]}^{2\bm{d}}$
$\displaystyle+\
\frac{1}{c^{2\bm{d}}}\Big{[}-2(\bm{r}_{x}-x_{w,j})(\bm{q}_{w}\bm{q}_{y}+\bm{q}_{x}\bm{q}_{z})$
$\displaystyle+\
2(\bm{r}_{y}-y_{w,j})(\bm{q}_{w}\bm{q}_{x}+\bm{q}_{y}\bm{q}_{z})$
$\displaystyle-\
(\bm{r}_{z}-z_{w,j})(2\bm{q}_{w}^{2}+2\bm{q}_{z}^{2}-1)\Big{]}^{2\bm{d}}-\bm{\beta}.$
(18)

Then, based on the aforementioned properties, we can see that $\bm{h}_{j}$ is
continuously differentiable, with respect to
$\bm{r}=[\bm{r}_{x},\bm{r}_{y},\bm{r}_{z}]$ and
$\bm{q}=[\bm{q}_{w},\bm{q}_{x},\bm{q}_{y},\bm{q}_{z}]$. ∎

######  Remark 5.

The proposed point cloud CBF formulation is for velocity control, i.e.,
systems with the dynamics $\dot{\bm{x}}=\bm{u}$. Thus, $\bm{u}=\mathbf{0}$ is
always a feasible solution to the CBFQP within the safe set $\mathcal{C}$.

######  Remark 6.

Apart from degenerate cases where the point cloud fully surrounds the robot,
there always exists a control input that enables the robot to move away from
the obstacles, which means that $\partial\bm{h}/\partial\bm{x}\neq 0$ for all
$\bm{x}\in\partial\mathcal{C}$.

######  Theorem 2.

For systems with dynamics $\dot{\bm{x}}=\bm{u}$, the CBF formulation in (15)
is a valid CBF.

###### Proof.

The proof directly follows from Theorem 1, Remark 5, and Remark 6. ∎

###  IV-B Mariner: CBF-Based Safe Preview Control

Figure 2: Visualization of the proposed preview control method in a 2D
scenario with higher-order ellipsoids with order four. Figure 3: Illustration
of the complete motion planning pipeline. The values along the orange arrows
are updated at 10 $\mathrm{H}\mathrm{z}$ (LiDAR update frequency), the purple
arrows are updated at 2 $\mathrm{H}\mathrm{z}$, and the blue arrows are
updated only once at the beginning of our experiments. Figure 4: Comparison of
the proposed algorithm’s computation time and GPU usage on different
computers. The GPU usage is computed when running the Mariner at 2
$\mathrm{H}\mathrm{z}$ and the Vessel at 10 $\mathrm{H}\mathrm{z}$.

For CBFQP-based methods, one undesirable scenario is when the solution to (4)
is $\bm{u}=0$. When such a scenario occurs, we say that the robotic system has
fallen into an undesired equilibrium state [8]. Since the proposed CBF in
Section IV-A is for velocity control, $\bm{u}=0$ means that the commanded
velocity is zero and the robot is stuck at its current position. One way to
avoid this issue is to use an MPC-based approach with a sufficiently long
preview horizon. However, incorporating the current CBF formulation with an
MPC-based approach is computationally expensive and generally creates a
nonlinear and nonconvex numerical optimization problem. Therefore, we propose
a novel preview controller that, by fixing the shape of the future path,
transforms the preview motion planning problem into a collision-free inverse
kinematics problem.

The simplification of the motion planning problem by using a fixed set of
paths has been studied in dynamic window based approaches [19]. However, in
[19], the paths are generated using a fixed velocity profile, which is less
flexible than desired. In this work, we propose Mariner, a point cloud CBF
based local planner that represents the paths in the configuration space and
exploits GPU acceleration to check hundreds of paths (which are referred to as
needles) in a few milliseconds. Mathematically, for the $i$-th needle, it is
represented using a higher-order ellipsoid defined as

$\frac{(x-\bar{\bm{s}}_{i}\bar{a}_{i})^{\bar{\bm{d}}}}{\bar{\bm{s}}_{i}^{\bar{\bm{d}}}\bar{a}_{i}^{\bar{\bm{d}}}}+\frac{y^{\bar{\bm{d}}}}{\bar{b}_{i}^{\bar{\bm{d}}}}+\frac{z^{\bar{\bm{d}}}}{\bar{c}_{i}^{\bar{\bm{d}}}}=1$
(19)

where $\bar{a}_{i},\bar{b}_{i},\bar{c}_{i}\in\mathbb{R}$ represents the length
of the semi-axes of the unscaled needle, $\bar{\bm{d}}\in\mathbb{N}$
represents the ellipsoid order used to represent the needle, and
$\bar{\bm{s}}_{i}\in\mathbb{R}$ is the scaling factor of the needle. As shown
in Fig. 2, the configuration space is discretized using
$n_{\text{needle}}\in\mathbb{N}$ needles. Each needle is chosen to point
towards a fixed angle within the robot’s body frame obtained from a
distribution function $\mathrm{dist}:\mathbb{N}\rightarrow\mathbb{R}$ that
maps the needle index to its corresponding angle. For example, in Fig. 2,
$n_{\text{needle}}=19$ and

$\mathrm{dist}(i)=\theta_{\text{needle},i}=\frac{2\pi}{n_{\text{needle}}}i-\pi.$
(20)

Note that, unlike the Vessel, only the length is extended when the needles are
scaled. For each needle, its body frame is located at the robot center with
its $z$ axis pointing upwards and its $x$ axis pointing from the robot center
to the tip of the needle. To compute the scaling factor $\bar{\bm{s}}_{i}$, we
first rotate the point cloud by the needle angle to get the points within the
needle’s body frame

$\bm{p}_{j}=\begin{bmatrix}x_{j}&y_{j}&z_{j}\end{bmatrix}^{\top}=\mathbf{R}(\theta_{\text{needle},i})^{\top}\bm{p}_{w,j}$
(21)

with $\theta_{\text{needle},i}\in\mathbb{R}$ represents the angle the $i$-th
needle is pointing towards,
$\mathbf{R}(\theta_{\text{needle},i})\in\mathrm{SO(3)}$ represents the
rotation matrix corresponding to the $i$-th needle angle, and
$\bm{p}_{j},\bm{p}_{w,j}\in\mathbb{R}^{3}$ representing the position of the
$j$-th point in the needle’s body frame and world frame, respectively, For
$\bm{p}_{j}$, define

$m_{ij}^{\bar{\bm{d}}}=1-\frac{y_{j}^{\bar{\bm{d}}}}{\bar{b}_{i}^{\bar{\bm{d}}}}+\frac{z_{j}^{\bar{\bm{d}}}}{\bar{c}_{i}^{\bar{\bm{d}}}}.$
(22)

Since the needle can only change its length, not its width or height, we only
compute the scale against points with $m_{ij}^{\bar{\bm{d}}}>0$. With this
assumption, we have the scale of the $i$-th needle with respect to the $j$-th
point as

$\bar{\bm{s}}_{ij}=\frac{x_{j}}{(1+m_{ij})\bar{a}_{i}}.$ (23)

For each needle, we compute the scale with respect to all valid points. Then,
we choose the minimum value among all of the scales as the scale of the
needle, and the scale is set to be a predefined $\bar{\bm{s}}_{\max}$ if there
are no valid points or if $\bar{\bm{s}}_{i}>\bar{\bm{s}}_{\max}$, i.e.,

$\bar{\bm{s}}_{i}=\begin{cases}\displaystyle\bar{\bm{s}}_{\max}&\displaystyle\text{if
}\min_{j}{\bar{\bm{s}}_{ij}}>\bar{\bm{s}}_{\max}\text{ or no valid points}\\\
\displaystyle\min_{j}{\bar{\bm{s}}_{ij}}&\displaystyle\text{if
}\min_{j}{\bar{\bm{s}}_{ij}}\leq\bar{\bm{s}}_{\max}.\end{cases}$ (24)

Then, we check the size of the needle scales and only consider needles with
$\bar{\bm{s}}_{i}\geq\bar{\bm{s}}_{\min}$ as valid needles (orange needles
shown in Fig. 2) and treat the rest as invalid needles (red needles shown in
Fig. 2). For all valid needles, we compute the distance between the scaled
needle’s tip (green dots in Fig. 2) and the target
$\bm{p}^{\star}\in\mathbb{R}^{3}$ (red dot in Fig. 2). Finally, we choose the
needle with its scaled tip closest to the target as the chosen needle (blue
needle in Fig. 2) and its scaled tip as the preview control target. The
pseudo-code of this process is given in Algorithm 1.

Figure 5: Illustration of the comparison studies between our proposed method
and DCBF-MPC [1]. The first and third plots from the left show a top-down view
of the generated trajectories. The green dot represents the starting position,
the red dot represents the goal position, the orange block represents the
obstacle, the trajectories of our proposed method are the light purple and
dark green curves (they almost overlap), and the trajectory generated DCBF-MPC
is the blue curve. The corresponding simulation scenarios are shown in the
second and fourth images from the left. Figure 6: Comparison between our
method and DCBF-MPC in a hallway. The left figure shows a top-down view of the
simulated scene. The middle figure shows the positions of the MVEE-generated
ellipsoids relative to the robot. The right figure shows the path generated by
our proposed method. It can be seen that DCBF-MPC fails to find a solution due
to starting in an invalid state, while our approach successfully plans a
collision-free path.

###  IV-C Complete Pipeline

Algorithm 1 Mariner: CBF-based Safe Preview Control

1:Get point cloud data $\mathcal{P}$, number of needles $n_{\text{needle}}$, a
distribution function $\mathrm{dist}$, and current global planner target
$\bm{p}^{\star}$;

2:for $i=0\ \text{to}\ n_{\text{needle}}-1$ do

3: $\theta_{\text{needle},i}=\mathrm{dist}(i)$;

4: Transform and filter $\mathcal{P}$ based on (21) and (22);

5: Compute the scale of this needle $\bar{\bm{s}}_{i}$;

6:Find the scaled needle with its tip closest to $\bm{p}^{\star}$;

7:Set the preview control target to be the tip of the chosen needle from the
last step;

The complete navigation pipeline includes the global planner, the Mariner, and
the Vessel. When given a map of the environment, the global planner creates
waypoints $\\{\bm{p}_{k}^{\star}\\}_{k=1,\cdots,n_{k}}\subset\mathbb{R}^{3}$
between the current position of the robot and the target position, with
$n_{k}\in\mathbb{N}$ being the number of waypoints. As mentioned in Section
III, the map does not need to include transient structures. Then, the Mariner
determines the local target ${}^{\mathcal{B}}\bm{p}_{\text{target}}^{\star}$.
Finally, a performance controller is used to command the robot toward the
local target, and the Vessel filters the performance controller command to
ensure safety. Note that any sampling-based [20] or search-based motion
planning algorithm can be deployed for the global planner. For details
regarding examples of the global planners that could be deployed, please refer
to Section V. The complete pipeline is also illustrated in Fig. 3.

##  V Experiments

In this section, we show the effectiveness of our proposed method by comparing
our proposed point-cloud CBF with existing point cloud based CBF formulations
and validating its performance in real-world settings.

###  V-A Setup

We validate our algorithm using a Unitree B1 and Go2 robot. The simulated
experiments are performed in Pybullet with a Unitree B1. All computations for
real-world experiments are performed onboard using an NVIDIA Jetson Orin NX
for the Unitree Go2 and a Jetson Orin AGX for the Unitree B1. Both robots
obtain the point clouds using a 16-channel 3D LiDAR at 10
$\mathrm{H}\mathrm{z}$. In the remainder of this section, the CBFQP models the
dynamics of the quadruped system as a single integrator with orientation,
i.e.,

$\dot{\bm{x}}=\bm{u}=\begin{bmatrix}\bm{v}^{\top}&\bm{\omega}\end{bmatrix}^{\top}\in\mathbb{R}^{4}$
(25)

with $\bm{v}\in\mathbb{R}^{3}$ and $\bm{\omega}\in\mathbb{R}$ being the linear
and angular velocity, respectively. The performance controller has the form of

$\displaystyle\bm{u}_{\mathrm{ref}}$
$\displaystyle=\begin{bmatrix}\bm{v}^{\top}&\bm{\omega}\end{bmatrix}^{\top}\in\mathbb{R}^{4}$
(26a) $\displaystyle\bm{v}$
$\displaystyle=\bm{K}_{v}({}^{\mathcal{B}}\bm{p}_{\text{target}}^{\star}-{}^{\mathcal{B}}\bm{p}_{\text{robot}})$
(26b) $\displaystyle\bm{\omega}$
$\displaystyle=\bm{K}_{\omega}({}^{\mathcal{B}}\bm{\phi}_{\text{target}}-{}^{\mathcal{B}}\bm{\phi}_{\text{robot}})$
(26c)

with $\bm{K}_{v}\in\mathbb{R}_{\succ 0}^{3\times 3}$ and
$\bm{K}_{\omega}\in\mathbb{R}_{+}$ representing the gain matrices,
${}^{\mathcal{B}}\bm{\phi}_{\text{target}}\in\mathbb{R}$ the desired yaw angle
which points towards the target,
${}^{\mathcal{B}}\bm{p}_{\text{robot}}\in\mathbb{R}^{3}$ the robot position in
its body frame (is constantly zero), and
${}^{\mathcal{B}}\bm{\phi}_{\text{robot}}\in\mathbb{R}$ the robot yaw angle
measured in its body frame (also is constantly zero). For the CBFQP, we choose

$\Gamma(\bm{h}(\bm{x}))=\bar{\gamma}\bm{h}(\bm{x})$ (27)

where $\bar{\gamma}\in\mathbb{R}_{+}$. We use an NMPC [21] controller to
transform the commanded linear and angular velocity to joint torques in the
simulation studies. In the experimental studies we use the onboard proprietary
controller to track the commanded linear and angular velocity. For real-world
experiments, the SLAM toolbox [22] is used for mapping and localization. To
reduce the computation time of the point cloud based CBF, we created custom
CUDA kernels using NVIDIA Warp [23] to parallelize the computations. For the
Vessel, we create a 1D kernel, and a 2D kernel is used for the Mariner. We
show the computation time for an NVIDIA Jetson Orin NX and NVIDIA Jetson Orin
AGX in Fig. 4. In all the cases shown in Fig. 4, we set
$n_{\mathrm{needle}}=100$. Additional experimental results, including the
real-world experiments using Unitree B1, can be found in the accompanying
video: https://youtu.be/lDmayeYcu28.

Figure 7: The experimental setting and ablation study results for showing the
effectiveness of each component of the local planner where “Complete” refers
to having both the Mariner and Vessel, “No Vessel” refers to only using the
Mariner, and “No Mariner” refers to only using the Vessel.

###  V-B Simulation Studies

We compare our approach with a recent point cloud CBF based method introduced
in [1] (DCBF-MPC), where the point cloud is first clustered and then
encapsulated using minimum-volume enclosing ellipsoids (MVEE) [24].
Furthermore, [1] models the robot with a sphere, which leads to more
conservatism. We compare the performance where the task of interest is
navigating to a goal location while avoiding collision with obstacles. The
simulation settings are shown in Fig. 5 and Fig. 6. In all settings, the
Vessel is in the form of higher-order ellipsoids of order one with semi-axes
$(a,b,c)=(0.8,0.4,0.4)~{}$\mathrm{m}$$. The Mariner uses needles that are
uniformly distributed with their semi-axes being
$(\bar{a},\bar{b},\bar{c})=(2,0.8,0.8)~{}$\mathrm{m}$$, $\bar{\bm{d}}=2$,
$n_{\mathrm{needle}}=100$, and uses the same distribution function as in (20).

For our implementation of the approach in [1], we set the integration time
step to be 0.5 $\mathrm{s}$, the preview length to be 30 time steps (15
seconds), the safe distance $d_{\mathrm{safe}}=0.8$, $\gamma=0.3$, and use
DBSCAN [25] for clustering. The NMPC is solved using CasADi [26]. It can be
seen that the approach in [1] can solve the left scenario in Fig. 5 where the
obstacle is smaller. However, for larger obstacles, i.e., the right scenario,
the quadruped gets stuck in front of the obstacle. This is likely caused by
getting stuck in a suboptimal local minimum for the NMPC solver. Moreover, the
NMPC scales poorly with respect to the length of the preview horizon, while
our proposed preview controller scales constantly with respect to the length
of the Mariner needles. The requirement for a point cloud preprocessing
pipeline is another limitation of the DCBF-MPC approach. Specifically, the
usage of MVEEs also makes the CBF formulation for DCBF-MPC much more
conservative than our proposed approach, especially in cluttered scenes, e.g.,
hallways. For the scene shown in Fig. 6, the usage of MVEEs causes the robot
to appear to be in a collision state while being sufficiently far away from
the obstacles. This causes the NMPC to fail, given that the initial state is
already invalid. These issues may be alleviated by adopting better point cloud
preprocessing pipelines and introducing a backup controller, which is not
required by our method. Another major claim of the DCBF-MPC approach is that
it can handle dynamic obstacles. Although, for the simplicity of presentation,
dynamic obstacles are not explicitly considered, given the computational
efficiency of both the Vessel and Mariner, the capability to replan at sensor
frame rate makes the proposed method responsive enough to react to dynamic
obstacles, as shown in the accompanied video.

###  V-C Experimental Deployment

This section presents experimental results of deploying our algorithm on a
Unitree Go2 robot. The goal is to reach a target location while avoiding
collision with obstacles (the boxes in Fig. 7). The semi-axes $(a,b,c)$ for
the Vessels are $(0.5,0.3,0.2)~{}$\mathrm{m}$$. The Needles are modeled using
ellipsoids with semi-axes $(0.8,0.1,0.2)~{}$\mathrm{m}$$, $\bar{\bm{d}}=2$,
$n_{\text{needle}}=100$, and the distribution function in (20). The generated
trajectories are shown in Fig. 7. The robot successfully reaches the target
position on the other side of the box wall when using both the Vessel and
Mariner (denoted as complete in Fig. 7). The robot collides with the boxes if
the Vessel is not activated. If the Mariner is not activated, the robot fails
to escape from the equilibrium of the CBFQP controller. For results of our
proposed pipeline on the Unitree B1 robot in a similar setting, please refer
to the accompanying video. Next, we show the results of integrating our
proposed local planner with a global planner. The RRT⋆ implementation in OMPL
[27] is used to generate a global plan, which is shown in Fig. 3. Then, random
obstacles (four boxes as shown in Fig. 8) are added to the scene. Finally,
following the pipeline shown in Section IV-C, the robot tracks the RRT-
generated waypoints. The Mariner and Vessel settings are identical to the
experiments in Section V-C. As shown in Fig. 8, the local planner generates
collision-free paths for the robot without getting stuck at any point along
the trajectory. Additional experiments are in the accompanying video. For the
additional experiments, the obstacles were also not included when planning the
global path.

Figure 8: Result of our complete pipeline. There are five main obstacles in
the illustrated settings. For each obstacle, four snapshots are provided to
depict the motion generated by the proposed local planner.

##  VI Conclusion

In this work, we proposed Vessel a novel point cloud based CBF formulation and
Mariner a novel local planner that can process point cloud data and move the
CBF-based controller out of local equilibriums. We have shown theoretically
that the proposed CBF formulation is a valid CBF. To show the effectiveness of
our approach, we have tested it on quadrupedal robots in both simulation and
the real world, along with the proposed motion planning pipeline.

##  VII Acknowledgement

We would like to thank Tanishq Bhansali, Naren Devarakonda, Raktim Goswami,
and Pranay Gupta for their assistance with the experiments.

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