#  Robust, High-Precision GNSS Carrier-Phase Positioning
with Visual-Inertial Fusion

Erqun Dong1,2∗, Sheroze Sheriffdeen2, Shichao Yang2, Jing Dong2, Renzo De
Nardi2, Carl Ren2,
Xiao-Wen Chang1, Xue Liu1 and Zijian Wang2 *This work was performed during an
internship at Meta.1School of Computer Science, McGill University, Montreal,
Quebec, Canada<EMAIL_ADDRESS>{chang<EMAIL_ADDRESS>Reality Labs Research, Redmond, WA, USA {sheroze, shichaoy, jingdong, renzo,
carlren<EMAIL_ADDRESS>
###### Abstract

Robust, high-precision global localization is fundamental to a wide range of
outdoor robotics applications. Conventional fusion methods use low-accuracy
pseudorange based GNSS measurements ($>5m$ errors) and can only yield a coarse
registration to the global earth-centered-earth-fixed (ECEF) frame. In this
paper, we leverage high-precision GNSS carrier-phase positioning and aid it
with local visual-inertial odometry (VIO) tracking using an extended Kalman
filter (EKF) framework that better resolves the integer ambiguity concerned
with GNSS carrier-phase. We also propose an algorithm for accurate GNSS-
antenna-to-IMU extrinsics calibration to accurately align VIO to the ECEF
frame. Together, our system achieves robust global positioning demonstrated by
real-world hardware experiments in severely occluded urban canyons, and
outperforms the state-of-the-art RTKLIB by a significant margin in terms of
integer ambiguity solution fix rate and positioning RMSE accuracy.

##  I Introduction

In this paper, we tackle the robustness issue of GNSS carrier-phase-based
global positioning in challenging urban canyons with the aid of VIO.

Robust, centimeter-level global localization is a cornerstone in many outdoor
applications such as mobile robots, self-driving cars, and augmented reality.
Currently, a large family of localization solutions rely on visual-inertial
odometry (VIO) [1, 2, 3] because of the accurate short-distance tracking.
Without pre-built maps that are costly to build and maintain, VIO has limited
usage in large-scale applications that require accurate global positioning.
Inspired by some recent works that fuse GNSS pseudorange positioning (GNSS-PR)
into VIO to mitigate VIO’s drift [4, 5, 6, 7, 8], we fuse VIO into GNSS
carrier-phase positioning (GNSS-CP) towards a low-cost but high-precision
global positioning for outdoor applications.

Figure 1: Demonstration of GNSS pseudorange trajectory (red), GNSS carrier-
phase trajectory (green) and VIO trajectory aligned with GNSS carrier-phase in
the first $50\text{\,}\mathrm{m}$ (blue). The light-green numbers and dashed
lines indicate the direction of four major parts of the trajectory. We can
observe that the pseudorange trajectory is much noisier than the other two
with a positioning error of up to $10\text{\,}\mathrm{m}$. This indicates the
potential of GNSS carrier-phase for more precise global positioning than GNSS
pseudorange.

GNSS-CP has manifested ability for drift-free centimeter-level localization of
reported 2–5cm accuracy under open sky. In contrast, GNSS-PR with $>$5m
accuracy has limited potential for centimeter-level localization. See Fig. 1
for a demonstration. However, GNSS-CP is more susceptible and fragile to line-
of-sight blockage and multi-path disturbance in dense urban areas. Section V-D
shows that the accuracy of GNSS-CP in urban canyons can degrade from
centimeter-level to be even tens of meters. So towards robust GNSS carrier-
phase-based global positioning in challenging urban canyons, we address these
points in this paper: 1) leveraging the local tracking and uncertainty
estimation from VIO to better condition and constrain the search space of
integer least sqaures, and 2) from an engineering perspective, reject
measurements from outlier satellites in urban environments so that the GNSS
carrier-phase resolution are less prone to noisy multi-path signals.

To achieve this, we carefully register the VIO trajectories to the ECEF frame
with a proposed calibration algorithm to solve the translation offset between
the IMU and the center of the GNSS antenna. We loosely couple VIO and GNSS-CP
in an EKF framework, where VIO serves as the motion model that propagates
prior state estimates and covariances, and GNSS-CP serves as the measurement
update to obtain a posterior float solution. We design strict outlier
rejection to tackle signal blockage and multi-path reflection in challenging
environments such as urban canyons. Altogether, the integer ambiguity
resolution can be seeded with good float solutions (where the integer
ambiguities are treated as real numbers) to solve for fixed solutions (where
the integer ambiguities are fixed as integers) with a higher success rate.

We conduct hardware experiments in urban canyons using the Aria glasses [9]
with a rigidly attached, low-cost, multi-band GNSS antenna. Results show that
our calibration renders centimeter-level alignment error, and the overall
positioning and integer ambiguity fixing rate of our method outperforms the
state-of-the-art GNSS-CP named RTKLIB [10]. It is worth mentioning that the
urban canyons in our experiments (Fig. 4, 5) are severer than that in the
experiments of recent papers [11, 12, 13, 7] in terms of the heights and
density of the buildings.

##  II Related Work

The fusion of visual-inertial information with GNSS has been explored since
the success of visual and visual-inertial localization and mapping [1, 14, 2].
Some works [15, 6, 4, 5, 12, 8] localize in the local ENU coordinate system by
a fusion of visual-inertial with GNSS pseudorange positioning. Others can
align the local odometry coordinate frame onto the global ECEF coordinate
frame with GNSS pseudorange measurements [16, 7]. They model either
pseudorange positioning results or pseudorange measurements as Gaussian
noises. However, due to the noisiness of GNSS pseudorange
measurements/positioning, the alignment of VIO and GNSS are subject to large
errors in the range of several meters even under the open sky (refer to Table
II in [7]). Importantly, most work that fuses pseudorange GNSS measurement and
visual inertial sensing (such as [5, 6, 8]) report results against vanilla RTK
or RTKLIB as groundtruth, whereas in this paper we achieve better performance
than the RTK groundtruth as used by others.

There were prior works that attempted global positioning with GNSS carrier-
phase, camera, and IMU. Li et al. [11] and Yoder et al. [13] fused visual-
inertial information as a prior estimate to aid the integer ambiguity
resolution. Shepard et al. [17] used a bundle adjustment method for
estimation. These papers assumed that the IMU-Antenna extrinsics calibration
parameters are known constants beforehand without characterizing their values.
In contrast, we propose a lightweight optimization method to estimate it and
experimentally validate it. Moreover, all three works lack real-world
experiments in severely challenging urban canyons like ours. Moreover, [13]
has the limitation of requiring two GNSS antennas installed on the device.

Research efforts have been made to assist carrier phase positioning in
unfavorable sky visibility condition. Takasu et al. [18] use dead-reckoning by
IMU to take over the navigation. Building heights constructed by Lidar [19]
are used to reduce satellite signal outliers. Thresholding-based outlier
detection methods are also used to mitigate challenging urban canyon effects
[11, 12, 13, 7], but the real-world environments in their experiments are not
as challenging as that in ours. In our paper, we consider VIO’s incremental
motion as a strong prior and reject outliers in both measurement update and
integer ambiguity resolution.

##  III Problem Formulation

Suppose there are $g$ groups of signals received by our GNSS module and the
reference station. Each group are in the same constellation and have the same
frequency, and suppose group $j$ has $m_{j}$ satellites. Then the GNSS
measurement at a certain epoch is
$\boldsymbol{y}=[\boldsymbol{\phi}_{1}^{\top},\dots,\boldsymbol{\phi}_{g}^{\top},\boldsymbol{\rho}_{1}^{\top},\dots,\boldsymbol{\rho}_{g}^{\top}]^{\top}$,
where $\boldsymbol{\phi}_{j}$ is the double-differenced carrier-phase
measurements [20] in the $j$-th group, and $\boldsymbol{\rho}_{j}$ is the
double-differenced pseudorange measurements in the $j$-th group. With one
satellite selected as the reference satellite for double-differencing,
$\boldsymbol{\phi}_{j}$ and $\boldsymbol{\rho}_{j}$ both have $(m_{j}-1)$
dimensions.

The visual-inertial inputs are processed by VIO providing IMU-centric device
poses $\\{\mathrm{I}\\}$ in the local odometry frame $\\{\mathrm{O}\\}$,
denoted as $\boldsymbol{T}_{\scriptscriptstyle\mathrm{OI}}\in\mathrm{SE}(3)$
where $\mathrm{SE}(3)$ is the three-dimensional Special Euclidean Group. The
VIO also outputs associated uncertainty [1]. Then we interpolate the VIO
trajectory at the timestamps of GNSS carrier-phase measurements for time-
interpolated VIO poses. We take a loosely-coupled fusion approach by feeding
the output of VIO unilaterally to our GNSS-CP algorithm. We assume the
intrinsics and extrinsics of cameras and IMUs are assumed to be calibrated,
and the extrinsics between GNSS antenna and IMU, denoted
$\boldsymbol{t}_{\scriptscriptstyle\mathrm{IA}}\in\mathbb{R}^{3}$, need to be
estimated.

We assume that an initial convergence of carrier-phase positioning (not
necessarily fixed solution) is available before our fusion algorithm starts,
which allows us to initially align the VIO local odometry frame to the ECEF
frame (shown in Fig. 4(b) and Fig. 5(b)) to warm start our algorithm.

The state to estimate is denoted
$\boldsymbol{x}=[{}^{\scriptscriptstyle\mathrm{E}}{\boldsymbol{p}}_{\scriptscriptstyle\mathrm{u}}^{\top},\boldsymbol{d}_{1}^{\top},\dots,\boldsymbol{d}_{g}^{\top}]^{\top}$,
where
${}^{\scriptscriptstyle\mathrm{E}}{\boldsymbol{p}}_{\scriptscriptstyle\mathrm{u}}$
means the current position of the user in the ECEF frame $\\{\mathrm{E}\\}$,
and
$\boldsymbol{d}_{j}=[d_{j}^{(1,2)},d_{j}^{(1,3)},\dots,d_{j}^{(1,m_{j})}]^{\top}$
is the double-differenced integer ambiguities for satellites in group $j$. We
build our fusion system upon a baseline EKF named RTKLIB [10]. The prediction
step of the baseline EKF uses a constant velocity model; whereas we use VIO
for better kinematic modeling, introduced in Section IV-B. For the measurement
update step, the measurement equations are double-differenced carrier-phase
and pseudorange measurements. Specifically, for each group $j$,

$\displaystyle\boldsymbol{\phi}_{j}=\boldsymbol{h}_{\phi,j}(\boldsymbol{x})+\boldsymbol{\epsilon}_{\phi,j},\
\ \ \
\boldsymbol{\rho}_{j}=\boldsymbol{h}_{\rho,j}(\boldsymbol{x})+\boldsymbol{\epsilon}_{\rho,j},$

where $\boldsymbol{\epsilon}_{\phi,j}$ is the Gaussian noise of the double-
differenced carrier-phase measurements and $\boldsymbol{\epsilon}_{\rho,j}$ is
the Gaussian noise of the double-differenced pseudorange measurements (For the
formulation of $\boldsymbol{\epsilon}_{\phi,j}$ and
$\boldsymbol{\epsilon}_{\rho,j}$, readers are referred to [20], Section 7.5).
The indivisual measurement functions in
$\boldsymbol{h}_{\phi,j}(\boldsymbol{x})$ and
$\boldsymbol{h}_{\rho,j}(\boldsymbol{x})$ are

$\scalebox{0.83}{\mbox{$\displaystyle
h_{\phi,j}^{(1,s)}(\boldsymbol{x})=(||{}^{\scriptscriptstyle\mathrm{E}}{\boldsymbol{t}}_{\scriptscriptstyle\mathrm{u}}^{(1)}||_{2}-||{}^{\scriptscriptstyle\mathrm{E}}{\boldsymbol{t}}_{\scriptscriptstyle\mathrm{r}}^{(1)}||_{2})-(||{}^{\scriptscriptstyle\mathrm{E}}{\boldsymbol{t}}_{\scriptscriptstyle\mathrm{u}}^{(s)}||_{2}-||{}^{\scriptscriptstyle\mathrm{E}}{\boldsymbol{t}}_{\scriptscriptstyle\mathrm{r}}^{(s)}||_{2})+\lambda_{j}d_{j}^{(1,s)}$}},$
$\scalebox{0.9}{\mbox{$\displaystyle
h_{\rho,j}^{(1,s)}(\boldsymbol{x})=(||{}^{\scriptscriptstyle\mathrm{E}}{\boldsymbol{t}}_{\scriptscriptstyle\mathrm{u}}^{(1)}||_{2}-||{}^{\scriptscriptstyle\mathrm{E}}{\boldsymbol{t}}_{\scriptscriptstyle\mathrm{r}}^{(1)}||_{2})-(||{}^{\scriptscriptstyle\mathrm{E}}{\boldsymbol{t}}_{\scriptscriptstyle\mathrm{u}}^{(s)}||_{2}-||{}^{\scriptscriptstyle\mathrm{E}}{\boldsymbol{t}}_{\scriptscriptstyle\mathrm{r}}^{(s)}||_{2})$}},$

where subscripts $\mathrm{u}$ and $\mathrm{r}$ mean the user and the reference
station, superscripts $1$ and $s$ are satellite ID’s where without loss of
generality we assume the reference satellite’s ID is $1$, $\lambda_{j}$ means
the wavelength of group $j$’s signal,
$||{}^{\scriptscriptstyle\mathrm{E}}{\boldsymbol{t}}_{\scriptscriptstyle\mathrm{u}}^{(1)}||_{2}$
means the distance between the user $\mathrm{u}$ and satellite $1$ satisfying
$||{}^{\scriptscriptstyle\mathrm{E}}{\boldsymbol{t}}_{\scriptscriptstyle\mathrm{u}}^{(1)}||_{2}=||{}^{\scriptscriptstyle\mathrm{E}}{\boldsymbol{p}}_{\scriptscriptstyle\mathrm{u}}-{}^{\scriptscriptstyle\mathrm{E}}{\boldsymbol{p}}^{(1)}||_{2}$,
and similar meanings apply to
$||{}^{\scriptscriptstyle\mathrm{E}}{\boldsymbol{t}}_{\scriptscriptstyle\mathrm{r}}^{(1)}||_{2}$,
$||{}^{\scriptscriptstyle\mathrm{E}}{\boldsymbol{t}}_{\scriptscriptstyle\mathrm{u}}^{(s)}||_{2}$,
and
$||{}^{\scriptscriptstyle\mathrm{E}}{\boldsymbol{t}}_{\scriptscriptstyle\mathrm{r}}^{(s)}||_{2}$.

The EKF measurement update treats all components of $\boldsymbol{x}$,
including both position variables and integer ambiguities, as real numbers to
solve a float solution [20]. Based on the float solution, we then use the
LAMBDA algorithm [21, 22] to resolve the fixed solution, where a core step is
solving (searching) an integer least squares problem

$\displaystyle\boldsymbol{D}=\operatorname*{argmin}_{\boldsymbol{D}\in\mathbb{Z}^{N}}\
(\boldsymbol{D}-\hat{\boldsymbol{D}})^{\top}\boldsymbol{W}_{\hat{\boldsymbol{D}}}^{-1}(\boldsymbol{D}-\hat{\boldsymbol{D}}),$
(1)

where
$\boldsymbol{D}=[\boldsymbol{d}_{1}^{\top},\dots,\boldsymbol{d}_{g}^{\top}]^{\top}$
is all the double-differenced integer ambiguities in all groups,
$\hat{\boldsymbol{D}}$ is its float solution, $N$ is its dimension, and
$\boldsymbol{W}_{\hat{\boldsymbol{D}}}^{-1}$ is the inverse of the covariance
of $\hat{\boldsymbol{D}}$. After integer ambiguity is searched and validated
with the ratio test [23], we can get a fixed solution of
${}^{\scriptscriptstyle\mathrm{E}}{\boldsymbol{p}}_{\scriptscriptstyle\mathrm{u}}$
that is more accurate than the float solution [21]. In this integer least-
squares problem, the float solution and its covariance form an ellipsoidal
search space for the correct integer, so their quality affects the success
rate of the integer ambiguity search, and equivalently affects the fixed
solution rate.

In a nutshell, this paper calculates the state $\boldsymbol{x}$ at every epoch
with the GNSS measurements $\boldsymbol{y}$ at each epoch and the VIO poses
$\boldsymbol{T}_{OI}$.

##  IV Method

We achieve the robust positioning by improving the quality of the float
solution and covariance in Eq. (1) for better integer ambiguity resolution
with a fusion of VIO into GNSS carrier-phase. Well-aligned VIO can be much
more accurate than RTKLIB’s constant velocity model in terms of both prior
prediction and covariance propagation. Plus, the accuracy of VIO tracking can
also effectively reject outlier satellites so that only the line-of-sight GNSS
signals are used for positioning. We introduce these methods in this section,
and demonstrate the system performance in the experiments Section V-C and V-D.

###  IV-A Alignment of VIO and GNSS with Extrinsics Calibration

We obtain the transformation from the odometry frame $\\{\mathrm{O}\\}$ to the
ECEF frame $\\{\mathrm{E}\\}$,
$\boldsymbol{T}_{\scriptscriptstyle\mathrm{EO}}$, by aligning the VIO
trajectory with the GNSS trajectory with a least squares optimization. For
precise alignment, the extrinsic parameter
$\boldsymbol{t}_{\scriptscriptstyle\mathrm{IA}}$, which represents the
translation between the center of the IMU and the center of the GNSS antenna,
should also be considered. Given a GNSS trajectory consisting of a sequence of
the user’s 3D positions
${}^{\scriptscriptstyle\mathrm{E}}{\boldsymbol{p}}_{\scriptscriptstyle\mathrm{u}_{i}},\
i=1,\dots,k$, and the corresponding time-interpolated VIO trajectory
${\boldsymbol{T}}_{\scriptscriptstyle\mathrm{OI}_{i}},\ i=1,\dots,k$, we
optimize the transformation $\boldsymbol{T}_{\scriptscriptstyle\mathrm{EO}}$
and the extrinsic parameter $\boldsymbol{t}_{\scriptscriptstyle\mathrm{IA}}$
with

$\displaystyle\min_{\begin{subarray}{c}\boldsymbol{T}_{\scriptscriptstyle\mathrm{EO}}\in\mathrm{SE}(3),\
\\\
\boldsymbol{t}_{\scriptscriptstyle\mathrm{IA}}\in\mathbb{R}^{3}\end{subarray}}\sum_{i=1}^{k}\left\lVert\boldsymbol{T}_{\scriptscriptstyle\mathrm{EO}}\cdot{\boldsymbol{T}}_{\scriptscriptstyle\mathrm{OI}_{i}}\cdot\begin{bmatrix}\boldsymbol{t}_{\scriptscriptstyle\mathrm{IA}}\\\
1\end{bmatrix}-{}^{\scriptscriptstyle\mathrm{E}}{\boldsymbol{p}}_{\scriptscriptstyle\mathrm{u}_{i}}\right\rVert^{2}.$
(2)

For the alignment ${\boldsymbol{T}}_{\scriptscriptstyle\mathrm{EO}}$ to be
well constrained on all degrees of freedom, the trajectories needs persistence
of excitation [24] along translation and rotation components. To this end, we
use principal component analysis [25] to decide if the first eigenvalue ratio
(FER) to the sum of all eigenvalues is smaller than a threshold.

We solve the optimization Eq. (2) using a two-pass process. First, we
initialize $\boldsymbol{T}_{\scriptscriptstyle\mathrm{EO}}$ to identity, fix
$\boldsymbol{t}_{\scriptscriptstyle\mathrm{IA}}$ to 0, and optimize Eq. (2)
for a coarse estimate of $\boldsymbol{T}_{\scriptscriptstyle\mathrm{EO}}$.
Second, we initialize $\boldsymbol{T}_{\scriptscriptstyle\mathrm{EO}}$ to the
coarse estimate from the first pass, and optimize Eq. (2) again for
$\boldsymbol{T}_{\scriptscriptstyle\mathrm{EO}}$ and
$\boldsymbol{t}_{\scriptscriptstyle\mathrm{IA}}$ jointly. The two-pass process
is necessary for ensuring that the nonlinear optimization is seeded with a
good initialization to avoid bad local minimum. Section V-B demonstrates the
improvement of alignment accuracy with our proposed calibration method.

Considering VIO’s drifting nature, we need to frequently re-align VIO with the
ECEF frame. All the later alignments optimize
${\boldsymbol{T}}_{\scriptscriptstyle\mathrm{EO}}$ in Eq. 2 with
$\boldsymbol{t}_{\scriptscriptstyle\mathrm{IA}}$ fixed. In real-world urban
canyon environments, GNSS positioning can degrade severely and impede accurate
trajectory alignment. Therefore, we use an root mean square error (RMSE)
threshold of 0.1m and an FER threshold of 97% to decide whether to perform a
new trajectory alignment at a new epoch. The frequency of re-alignment is
annotated in Fig. 4(b) and Fig. 5(b).

###  IV-B Fusion Algorithm

With the VIO registered to the ECEF frame, we then fuse VIO with GNSS in a
loosely-coupled fashion. We use VIO to track the incremental motion in ECEF as
the EKF’s prior prediction and propagates covariance. Then we follow the
routine in RTKLIB [10] to resolve integer ambiguity with the MLAMBDA algorithm
[22]. Major designs of our system are introduced in this section. The
pseudocode for the overall fusion routine is summarized in Algorithm 1.

Algorithm 1 Fusion Algorithm for Each Epoch

1:NewEpochData in epoch $i+1$ captured

2:$\hat{\boldsymbol{T}}_{\scriptscriptstyle\mathrm{EO}}\leftarrow$AlignNewSegmentationOrReuseOldAlignment()

3:${}^{\scriptscriptstyle\mathrm{O}}\boldsymbol{t}_{\scriptscriptstyle\mathrm{I}_{i}\scriptscriptstyle\mathrm{I}_{i+1}}\leftarrow$
VIO(NewEpochData)

4:${}^{\scriptscriptstyle\mathrm{E}}{\boldsymbol{p}}_{\scriptscriptstyle\mathrm{u}_{i+1|i}}\leftarrow$Eq.
(3)

5:$\boldsymbol{P}_{i+1|i}\leftarrow$Eq. (6)

6:$\bar{\boldsymbol{y}}\leftarrow$OutlierSatellitePruning(NewEpochData)

7:Float
solution$\leftarrow$MeasurementUpdate($\bar{\boldsymbol{y}},\boldsymbol{x}$)

8:if Float solution validation passed then

9: Fixed solution$\leftarrow$ MLAMBDA($\bar{\boldsymbol{y}}$)

10: if Fixed solution validation passed then

11: Final solution$\leftarrow$Fixed solution and covariance

12: else

13: $\bar{\boldsymbol{y}}^{\prime}\leftarrow$Further prune satellites that
produce large residuals in integer least squares

14: Fixed solution$\leftarrow$ MLAMBDA($\bar{\boldsymbol{y}}^{\prime}$)

15: if Fixed solution validation passed then

16: Final solution$\leftarrow$Fixed solution and covariance

17: else

18: Final solution$\leftarrow$Float solution and covariance

19: end if

20: end if

21:else

22: Final solution$\leftarrow$the prior
${}^{\scriptscriptstyle\mathrm{E}}{\boldsymbol{p}}_{\scriptscriptstyle\mathrm{u}_{i+1|i}},\boldsymbol{P}_{i+1|i}$

23:end if

24:Return Final solution

####  IV-B1 EKF Prior Prediction

Suppose the state at epoch $i$ is already estimated and we are to estimate the
state at epoch $i+1$. Since VIO has a higher frequency than GNSS, we seek in
the VIO’s sliding window for two poses with the nearest capture times to GNSS
epochs $i$ and $i+1$, denoted
$\boldsymbol{T}_{\scriptscriptstyle\mathrm{OI}_{i}}$ and
$\boldsymbol{T}_{\scriptscriptstyle\mathrm{OI}_{i+1}}$. Then the prior
position estimate,
${}^{\scriptscriptstyle\mathrm{E}}{\boldsymbol{p}}_{\scriptscriptstyle\mathrm{u}_{i+1|i}}$,
can be predicted by the position estimate of the previous epoch,
${}^{\scriptscriptstyle\mathrm{E}}{\boldsymbol{p}}_{\scriptscriptstyle\mathrm{u}_{i|i}}$,
and the incremental translation tracked by VIO in the odometry frame,
${}^{\scriptscriptstyle\mathrm{O}}\boldsymbol{t}_{\scriptscriptstyle\mathrm{I}_{i}\scriptscriptstyle\mathrm{I}_{i+1}}$,
with

$\displaystyle{}^{\scriptscriptstyle\mathrm{E}}{\boldsymbol{p}}_{\scriptscriptstyle\mathrm{u}_{i+1|i}}$
$\displaystyle={}^{\scriptscriptstyle\mathrm{E}}{\boldsymbol{p}}_{\scriptscriptstyle\mathrm{u}_{i|i}}+\boldsymbol{R}_{\scriptscriptstyle\mathrm{EO}}\cdot{}^{\scriptscriptstyle\mathrm{O}}\boldsymbol{t}_{\scriptscriptstyle\mathrm{I}_{i}\scriptscriptstyle\mathrm{I}_{i+1}},$
(3)

where $\boldsymbol{R}_{\scriptscriptstyle\mathrm{EO}}$ is the rotation part of
$\boldsymbol{T}_{\scriptscriptstyle\mathrm{EO}}$.

####  IV-B2 Covariance Propagation

Denote the prior covariance of
${}^{\scriptscriptstyle\mathrm{E}}{\boldsymbol{p}}_{\scriptscriptstyle\mathrm{u}_{i+1}}$
as $\boldsymbol{P}_{i+1|i}$, and the posterior covariance of
${}^{\scriptscriptstyle\mathrm{E}}{\boldsymbol{p}}_{\scriptscriptstyle\mathrm{u}_{i}}$
as $\boldsymbol{P}_{i|i}$. We propagate $\boldsymbol{P}_{i+1|i}$ from
$\boldsymbol{P}_{i|i}$ combining VIO’s covariance. We obtain the $12\times 12$
covariance matrix
$\mathop{\mathrm{cov}}(\begin{bmatrix}\boldsymbol{\xi}_{\scriptscriptstyle\mathrm{OI}_{i}}\\\
\boldsymbol{\xi}_{\scriptscriptstyle\mathrm{OI}_{i+1}}\end{bmatrix})$ from the
covariance of the joint Gaussian distribution maintained by VIO, where
$\boldsymbol{\xi}_{\scriptscriptstyle\mathrm{OI}_{i}}$ and
$\boldsymbol{\xi}_{\scriptscriptstyle\mathrm{OI}_{i+1}}$ are the corresponding
Lie algebra elements [26] for
$\boldsymbol{T}_{\scriptscriptstyle\mathrm{OI}_{i}}$ and
$\boldsymbol{T}_{\scriptscriptstyle\mathrm{OI}_{i+1}}$. Then the covariance of
the incremental translation in the odometry frame is

$\displaystyle\mathop{\mathrm{cov}}({}^{\scriptscriptstyle\mathrm{O}}\boldsymbol{t}_{\scriptscriptstyle\mathrm{I}_{i}\scriptscriptstyle\mathrm{I}_{i+1}})=[\boldsymbol{J}_{\scriptscriptstyle\mathrm{I}_{i}}\boldsymbol{J}_{\scriptscriptstyle\mathrm{I}_{i+1}}]\mathop{\mathrm{cov}}(\begin{bmatrix}\boldsymbol{\xi}_{\scriptscriptstyle\mathrm{OI}_{i}}\\\
\boldsymbol{\xi}_{\scriptscriptstyle\mathrm{OI}_{i+1}}\end{bmatrix})\begin{bmatrix}\boldsymbol{J}_{\scriptscriptstyle\mathrm{I}_{i}}^{\top}\\\
\boldsymbol{J}_{\scriptscriptstyle\mathrm{I}_{i+1}}^{\top}\end{bmatrix},$ (4)

where $\boldsymbol{J}_{\scriptscriptstyle\mathrm{I}_{i}}$ and
$\boldsymbol{J}_{\scriptscriptstyle\mathrm{I}_{i+1}}$ are the partial
derivatives of
${}^{\scriptscriptstyle\mathrm{O}}\boldsymbol{t}_{\scriptscriptstyle\mathrm{I}_{i}\scriptscriptstyle\mathrm{I}_{i+1}}$
w.r.t. the Lie algebra element of
$\boldsymbol{T}_{\scriptscriptstyle\mathrm{OI}_{i}}$ and
$\boldsymbol{T}_{\scriptscriptstyle\mathrm{OI}_{i+1}}$, respectively. In Eq.
(4),
$\mathop{\mathrm{cov}}({}^{\scriptscriptstyle\mathrm{O}}\boldsymbol{t}_{\scriptscriptstyle\mathrm{I}_{i}\scriptscriptstyle\mathrm{I}_{i+1}})$
is $3\times 3$, and $\boldsymbol{J}_{\scriptscriptstyle\mathrm{I}_{i}}$ and
$\boldsymbol{J}_{\scriptscriptstyle\mathrm{I}_{i+1}}$ are both $3\times 6$.
Then in the ECEF frame,

$\displaystyle\mathrm{cov}({}^{\scriptscriptstyle\mathrm{E}}\boldsymbol{t}_{\scriptscriptstyle\mathrm{I}_{i}\scriptscriptstyle\mathrm{I}_{i+1}})=\boldsymbol{R}_{\scriptscriptstyle\mathrm{EO}}\cdot\mathrm{cov}({}^{\scriptscriptstyle\mathrm{O}}\boldsymbol{t}_{\scriptscriptstyle\mathrm{I}_{i}\scriptscriptstyle\mathrm{I}_{i+1}})\cdot\boldsymbol{R}_{\scriptscriptstyle\mathrm{EO}}^{\top}.$
(5)

The final form of the proposed covariance propagation of the GNSS prior
position in EKF prediction step is

$\displaystyle\boldsymbol{P}_{i+1|i}=\boldsymbol{P}_{i|i}+\mathrm{cov}({}^{\scriptscriptstyle\mathrm{E}}\boldsymbol{t}_{\scriptscriptstyle\mathrm{I}_{i}\scriptscriptstyle\mathrm{I}_{i+1}}).$
(6)

####  IV-B3 Outlier Satellite Pruning

In practice, as GNSS signals undergo severe blockage and reflection in urban
canyons, signals traveling along non-line-of-sight (NLoS) paths should not
participate in the positioning. Therefore, we need to detect and rule out such
outlier satellite measurements. We do outlier rejection in both the
measurement update step and the integer ambiguity resolution step.

In the measurement update step, since VIO provides accurate EKF prediction, we
use 5 times of VIO’s incremental motion as a threshold to filter the
innovation of the satellite measurements. The remaining measurements
$\bar{\boldsymbol{y}}$ is a subset $\boldsymbol{y}$. After this, some elements
of $\bar{\boldsymbol{y}}$ can still be unreliable with some NLoS signals
passing the threshold. Then in the integer ambiguity resolution step, if the
integer ambiguities solved by the MLAMBDA [22] algorithm cannot pass the
validation by the ratio test [23], the system further prunes satellites that
produce largest residuals in the integer least squares problem, and then solve
it again. This way, some more spurious outliers can be ruled out.

(a)

(b)

Figure 2: Device used in the hardware experiments.

##  V Experiments

###  V-A Hardware Implementation

We perform hardware experiments in real-world challenging urban canyons
(Bellevue, Washington, USA) using the Aria glasses [9], which has a suite of
machine perception sensors including two global-shutter monochrome SLAM
cameras, two 6-DOF IMUs, and a built-in GPS pseudorange module. We rigidly
attach an external multi-band GNSS carrier-phase antenna to the Aria glasses
with a metal ground plate, as shown in Fig. 2(a). The GNSS carrier-phase
measurements are received by a u-BLox ZED-F9P GNSS receiver, and the raw data
is further logged by an Arduino into an SD-card for offline processing111Our
algorithm is casual, which is the same as EKF.. We use three constellations in
our experiments: GPS, GLONASS, and Galileo. We set up our own reference
station using the same antenna and receiver, about $20\text{\,}\mathrm{km}$
from the user’s receiver. This baseline length is common in many cities in
north America between a public base station and its nearby urban environment.
The two SLAM cameras operate at $10\text{\,}\mathrm{Hz}$ at $640\times 480$
resolution, and the two IMUs operate at $1000\text{\,}\mathrm{Hz}$ and
$800\text{\,}\mathrm{Hz}$, respectively. The internal GPS pseudorange module
runs at $1\text{\,}\mathrm{Hz}$ while the external GNSS carrier-phase module
at $5\text{\,}\mathrm{Hz}$.

For synchronization between GNSS carrier-phase and VIO, since the built-in GPS
of the Aria glasses and the external GNSS carrier-phase antenna both record
GPS Time, we can simply use the built-in GPS of Aria glasses as a bridge to
synchronize the VIO (system time on Aria glasses) and the external GNSS
carrier-phase signals.

(a) APE under open sky, with 10 satellites blocked, on average 4 satellites
remain. RTKLIB fails some epochs.

(b) APE in lightly-blocked area, with 5 satellites blocked, on average 5
satellites remain visible.

(c) The comparison of the our trajectory, RTKLIB’s trajectory, and the ground
truth in ECEF frame in the lightly-blocked area

Figure 3: Performance comparison with ground truth for simulating signal
blockage in urban canyons. Best viewed zoom in.

###  V-B Analysis of Calibration for Alignment Accuracy

Table I shows alignment experiments with and without the proposed calibration
for three trajectories, with lengths $100\text{\,}\mathrm{m}$,
$50\text{\,}\mathrm{m}$ and $30\text{\,}\mathrm{m}$ respectively, under
unobstructed sky visibility. We can observe that the calibration significantly
reduces the alignment error from decimeter-level to centimeter-level, up to
$12\text{\,}\mathrm{cm}$ in trajectory 3, and thus largely boosts the
alignment accuracy. Another implication is that the alignment with the
calibration has the same magnitude of error as carrier-phase positioning
itself, and thus the accuracy of VIO and carrier-phase positioning are
commensurate to be suitable for complementary sensor fusion.

| Traj 1 | Traj 2 | Traj 3
---|---|---|---
Aligned Length (m) | 100 | 50 | 30
RMSE without Calibration (m) | 0.11 | 0.11 | 0.16
RMSE with Calibration (m) | 0.071 | 0.028 | 0.037

TABLE I: Alignment errors with and without calibration

Moreover, we show with a field experiment under open sky that GNSS pseudorange
measurement is too noisy to provide VIO with a reliable centimeter-level ECEF-
referenced registration. In Fig. 1, the trajectory under good sky visibility
is perceived by VIO, GNSS carrier-phase, and GNSS pseudorange. We can observe
that GNSS-PR is much noisier than the other two modalities, with a positioning
error w.r.t. the other two modalities up to $10\text{\,}\mathrm{m}$. This
suggests that it is not suitable to fuse visual-inertial information with GNSS
pseudorange because VIO cannot have centimeter-level registration to the ECEF
frame by aligning to GNSS pseudorange positioning. This conclusion is also
supported by the ECEF registration error reported in Table II of GVINS [7],
where meter-level registration is observed even in an open-sky sports field.

| Open Sky, 10 sats ex | Lightly-blocked, 5 sats ex
---|---|---
| RTKLIB : Ours | RTKLIB : Ours
RMSE (m) $\downarrow$ | 0.68 : 0.38 | 0.78 : 0.033
Mean (m) $\downarrow$ | 0.55 : 0.34 | 0.45 : 0.025
Median (m) $\downarrow$ | 0.40 : 0.32 | 0.26 : 0.019
Std (m) $\downarrow$ | 0.41 : 0.17 | 0.63 : 0.023
Max (m) $\downarrow$ | 2.97 : 0.88 | 4.90 : 0.43
FSR (%) $\uparrow$ | 10.8 : 20.0 | 48.6 : 92.9

TABLE II: Statistics of groundtruth comparison experiments. “10 sats ex” means
10 satellites are manually excluded. Same applies to “5 sates ex”.

(a) RTKLIB in UC1

(b) Our model in UC1

(c) RTKLIB, UC1 (zoom-in view)

(d) Ours, UC1’s (zoom-in view)

Figure 4: UC1 environment. Green: fixed solution. Yellow: float solution.
White: aligned VIO when there is no valid float solution. Best viewed in color
and zoom in.

(a) RTKLIB in UC2

(b) Our model in UC2

(c) RTKLIB, UC2 (zoom-in view)

(d) Ours, UC2 (zoom-in view)

Figure 5: UC2 environment. Green: fixed solution. Yellow: float solution.
White: aligned VIO when there is no valid float solution. Best viewed in color
and zoom in.

###  V-C Groundtruth Comparison Experiment

To characterize the performance of our method under severe sky blockage in
urban scenarios, we need to be creative about how to get the groundtruth. It
is also inappropriate to compare with methods that fuse low-accuracy
pseudorange GNSS measurement into VIO [5, 6, 7, 8], which lack the ability to
fuse carrier-phase GNSS measurement and instead directly use carrier-phase
positioning as the groundtruth. To this end, we collected two datasets, one
open-sky and one lightly-blocked sky by trees, and manually exclude satellites
to simulate sky blockage, and then use RTKLIB to process full-constellation
data as the groundtruth.

We run our method and RTKLIB to process a subset of satellite measurements
after manual satellite removal (10 satellites excluded for open-sky dataset
and 5 excluded for lightly-blocked dataset). The removed satellites contain
the most frequently observed ones and moderately frequent ones during the
observation period. The absolute position error (APE) over time is shown in
Fig. 3 plotted by the evo toolkit [27]. The RMSE, mean APE, median APE,
standard deviation of APE, max APE, and fixed solution rate (FSR, higher is
better) are listed in Table II. For both RTKLIB and our method, we use the
same standard for the integer solution validation, which is the popular ratio
test [23], the ratio between the second best and the best residuals being 3.
We can observe that our method achieves lower position errors while having
much higher fixed solution rate (FSR) than RTKLIB compared to the groundtruth.

Figure 6: The urban canyons of the trajectory of Fig. 5(b) visualized in
Google Earth. The average building height in this canyon is 55m and the
tallest building is 81m.

###  V-D Real-World Urban Canyon Experiments

We conduct real-world experiments in two challenging urban canyon
environments, “UC1” and “UC2” in Fig. 4 and Fig. 5. 222Readers are also
referred to https://youtu.be/PCDymbSTndQ for a video demonstration. A 3D
street view of the urban canyon in our dataset is shown in Fig. 6 to
illustrate how challenging our problem is. As is analyzed in Section V-B,
works that fuse GNSS pseudorange and VIO [5, 6, 7, 8] are not suitable for
comparison in our experiments, since their registration to ECEF have meter-
level offset due to the noisiness of pseudorange positioning. Plus, they do
not tackle carrier-phase positioning problem as we do.

In the trajectories, green means fixed solution, yellow means float solution,
and white means “fill-in” positions from aligned VIO when the validations in
Alg. 1 fails. Visualizations of the results from RTKLIB and our model are
shown in Fig. 4 and Fig. 5. We can observe that RTKLIB suffers from the urban
canyon effects both horizontally (Fig. 4(a), 5(a)) and vertically (Fig. 4(c),
5(c)). All the zig-zags and large jumps of the trajectory by the RTKLIB’s EKF
are eliminated by our model (Fig. 4(b), 5(b)). Moreover, the segments formed
by white spots connect smoothly with the GNSS positioning segments formed by
green and yellow, which is a qualitative indicator that the our fusion method
is coherent.

For quantitative evaluation, we pick three segments respectively from the UC1
and UC2 trajectories, where the urban canyon effects are severe, moderate, and
mild, as marked in Fig. 4(a) and Fig. 5(a). In the selected segments, due to
lack of ground truth, we use our method as a reference and evaluate RTKLIB’s
and the VIO’s deviations from our method.333RTKLIB is commonly used as ground
truth in SLAM research. In this sense, we are improving the ground truth used
by many researches. The VIO trajectory is aligned with the selected segments
by neighboring segments. We show that although VIO has a drift rate of about
$0.5\%$, it is much closer to our trajectory than RTKLIB’s EKF, indicating
that our system is much more robust than the original RTKLIB. The RMSE, mean
APE, median APE, and max APE are listed in the following two tables. For
simplicity, RTKLIB is denoted “RTK” in the tables, and VIO denoted “VIO”.

UC1 Segments | Severe | Moderate | Mild
---|---|---|---
| RTK : VIO | RTK : VIO | RTK : VIO
RMSE (m) $\downarrow$ | 45.39 : 1.61 | 9.59 : 0.99 | 6.58 : 1.11
Mean (m) $\downarrow$ | 32.08 : 1.29 | 4.79 : 0.69 | 4.42 : 1.02
Median (m) $\downarrow$ | 17.00 : 0.96 | 1.32 : 0.31 | 1.14 : 0.94
Max (m) $\downarrow$ | 82.06 : 4.15 | 51.16 : 3.30 | 14.94 : 1.85

UC2 Segments | Severe | Moderate | Mild
---|---|---|---
| RTK : VIO | RTK : VIO | RTK : VIO
RMSE (m) $\downarrow$ | 44.29 : 2.33 | 23.21 : 1.55 | 9.51 : 0.79
Mean (m) $\downarrow$ | 37.51 : 1.88 | 14.69 : 1.11 | 9.38 : 0.63
Median (m) $\downarrow$ | 38.14 : 1.02 | 5.57 : 0.87 | 8.59 : 0.59
Max (m) $\downarrow$ | 112.44 : 4.74 | 59.27 : 4.10 | 12.40 : 1.91

Finally, we compute the fixed solution rate of RTKLIB’s EKF and our method in
the following table. We use the same ratio test standard for integer solution
validation as Section V-C. The results show that our method achieves
significant improvement on fixed solution rate (higher is better), which means
that our method has better robustness in urban canyons.

Environments | UC1 | UC2
---|---|---
| RTK : Ours | RTK : Ours
Fixed Solution Rate (%) $\uparrow$ | 39.4 : 50.8 | 8.1 : 36.5

##  VI Conclusion

This paper tackles the robustness issue of GNSS carrier-phase positioning in
urban canyons with a loosely-coupled EKF to fuse GNSS carrier-phase and VIO.
We propose an optimization-based method to calibrate the extrinsics between
IMU and GNSS antenna to accurately align VIO to the ECEF frame. We use VIO as
the prediction step for the EKF and incorporate its covariance estimate into
the covariance propagation of EKF. The performance of our method is
quantitatively validated with simulation experiments and real-world
experiments in challenging urban canyons. Its robustness, positioning accuracy
and fixed solution rate outperform the state-of-the-art GNSS carrier-phase
positioning RTKLIB [10] by a large margin in challenging urban canyons,
filling the gap towards universally global positioning.

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11institutetext: Los Alamos National Laboratory, Los Alamos, NM, 87545, USA

# Nuclear data activities for medium mass and heavy nuclei at Los Alamos

M. R. Mumpower e-mail<EMAIL_ADDRESS>T. M. Sprouse 11 T. Kawano 11 M. W.
Herman 11 A. E. Lovell 11 G. W. Misch 11 D. Neudecker 11 H. Sasaki 11 I.
Stetcu 11 P. Talou 11

###### Abstract

Nuclear data is critical for many modern applications from stockpile
stewardship to cutting edge scientific research. Central to these pursuits is
a robust pipeline for nuclear modeling as well as data assimilation and
dissemination. We summarize a small portion of the ongoing nuclear data
efforts at Los Alamos for medium mass to heavy nuclei. We begin with an
overview of the NEXUS framework and show how one of its modules can be used
for model parameter optimization using Bayesian techniques. The mathematical
framework affords the combination of different measured data in determining
model parameters and their associated correlations. It also has the advantage
of being able to quantify outliers in data. We exemplify the power of this
procedure by highlighting the recently evaluated 239Pu cross section. We
further showcase the success of our tools and pipeline by covering the insight
gained from incorporating the latest nuclear modeling and data in
astrophysical simulations as part of the Fission In R-process Elements (FIRE)
collaboration.

##  1 Introduction

Nuclear data have a profound impact on modern society Kolos2022 . These data
serve as a foundation for nuclear energy, nuclear security, and the wide
variety of research that is carried out in nuclear astrophysics from the
composition of neutron stars to the impact of kilonova light curves Schatz2016
; Bernstein2019 ; Barnes2021 ; Schatz2022 .

Despite its importance, a robust pipeline for nuclear data assimilation and
dissemination still remains on the horizon. Currently, major evaluated
databases of nuclear reactions, decays and structure can be found in ENDF
ENDF8  and ENSDF ENSDF . Inputs valuable for reaction modeling codes can be
found in the Reference Input Parameter Library (RIPL) Capote2009  and
experimental reaction data can be found in EXFOR Otuka2014 . The Atomic Mass
Evaluation (AME) Wang2017 ; Wang2021  and NuBase Kondev2021  provide targeted
information regarding ground state and decay properties.

The disparate nature of these sources, coupled with the difficulty of
extracting information, complicate the use of these database in modern
applications, particularly with cutting edge science. In this contribution, we
provide a brief overview of the nuclear data activities at Los Alamos National
Laboratory (LANL). We discuss the NEXUS framework which represents a first
step towards providing a pipeline between nuclear data, nuclear modeling
efforts and applications. We showcase the utility of this platform by
highlighting its use in a recent evaluation of 239Pu cross sections as well as
in the scientific Fission In R-process Elements (FIRE) collaboration. We end
with a discussion of recent tools developed at LANL intended to advance the
use of nuclear data in astrophysical applications.

##  2 NEXUS

The Los Alamos Python package, NEXUS, is a data-agnostic framework intended to
furnish access to nuclear properties. The phrase ‘data agnostic’ means that it
does not matter how the data was generated or parsed. The output format of the
data is also irrelevant to the framework’s function. Instead the focus of the
code is on a consistent object-oriented representation of various physical
quantities and the relationships between them. This approach is in stark
contrast to the major databases which revolve around the transmission format
of data.

As an example of an object in NEXUS, in its simplest form a reaction cross
section may be reasonably represented by two arrays: the incident particle
energy and the cross section at each energy. Additional information may be
warranted, in which case the base object may be extended. There may be
uncertainties reported on each energy point as well as uncertainties in the
cross section values. For particular applications associated metadata may also
need to be affiliated with attributes, for example, providing the physical
units of each of the arrays. The focal point is on the representation of the
data in code, not on how the information will be transmitted.

Information transmission of data is supported however in NEXUS. This revolves
around the concept of marshalling which is a more general concept of object
serialization (which deals only with string representations). Each object in
NEXUS has a corresponding object, a marshaller, which handles its
transmutation to various representations. A marshaller may translate one
object type into another, or it may convert an object from its code
representation to a form that can be saved on a computer hard disk, e.g. ASCII
or JSON. The power of this approach means that complex derived objects may be
constructed and converted to the desired output format for particular
applications. A concert example is the use of atomic binding energies and
their appropriate parsing into suitable reaction network codes for use in
astrophysical Li2022  or Machine Learning applications Lovell2022 ;
Mumpower2022 .

The NEXUS framework provides a base set of objects for nuclear properties that
are commonly found in the aforementioned nuclear databases. In addition,
access to a host of theoretical outputs from nuclear model codes are also
available. Among other data access methods, this list includes, level
densities, $\gamma$-strength functions, optical model parameters, ground state
binding energies, and reaction cross sections. Information regarding nuclear
isomers has recently been utilized to study the impact of long-lived excited
states in astrophysics Misch2020 ; Misch2021 .

The data-agnostic approach NEXUS framework also provides a set of application
programming interfaces (APIs) to nuclear data libraries, nuclear model codes
and application codes. The transition between these three areas enabled by
this platform is depicted pictorially in Figure 1.

Figure 1: The Los Alamos Python package, NEXUS, provides a series of
application programming interfaces (APIs) to efficiently go from nuclear
theory or data to applications and back again.

##  3 Hierarchical Approach to Nuclear Data Evaluations

We now highlight the power of the NEXUS framework by showcasing a result from
a recent evaluation of 239Pu which uses the Bayesian optimization module. The
evaluation is based on a hierarchical approach where data is deemed most
important followed by model calculation to ‘fill in the gaps’ when empirical
data is lacking. This evaluation combines experimental cross section data from
EXFOR along with output from the Los Alamos statistical model code, CoH
Kawano2016 ; Kawano2019 .

In order to maintain consistency throughout each evaluated cross section
channel, a global set of model parameters must be fit to available data. Model
parameter fitting is performed incrementally channel by channel starting with
the total cross section. Because most statistical model cross sections perform
very well with respect to available data, it is possible to use a Bayesian
approach called hyperparameter optimization to optimize the model with respect
to independent datasets Hobson2002 .

To optimize the calculated total cross section with respect to available data,
$\sigma_{\textrm{tot}}$, a Metropolis random-walk algorithm is used to probe
the optical model parameter space. The Soukhovitskii potential is used as the
basis for the optical model Soukhovitskii2005 ; Capote2005 . The model space
consists of six parameters, including the potential depths, diffuseness and
radii. All other model parameters are held fixed during this optimization as
they do not influence the total cross section. It was determined that holding
the optical model deformation fixed, rather than letting it vary during the
optimization was ideal for approximating a robust minimum. In this procedure
the chi-square goodness of fit is minimized with respect to data which is
weighted based on its uncertainty.

The resultant parameter sensitivities of the fitting procedure are shown in
Figure 2. The optical model parameters are standardized to unity to ensure
that there is no difference in the scale of individual parameters. All of the
model parameters are found to exhibit a relatively strongly peaked
distribution around the optimal value. The optimal values are all within 10%
from the baseline parameters suggested by Soukhovitskii with diffuseness
parameters changing the most.

Figure 2: The determination of optical model parameters using Bayesian
hyperparameter optimization. Shaded regions highlight areas of minimal
$\chi^{2}$.

The cross section fit with the optimal optical model parameters are shown in
Fig. 3. The procedure performs nearly identically to the previous evaluation
of ENDF/B-VIII.0. Slight modifications are seen relative to ENDF/B-VIII.0
below 30 keV where the reported uncertainties of the datasets pull down the
fit to the total cross section. A similar modification, albeit to a much
smaller effect, can be found between 1 and 2 MeV, near the local minimum of
$\sigma_{\textrm{tot}}$.

Figure 3: The evaluation of the total cross section of neutrons on 239Pu using
the Los Alamos code CoH for cross section modeling.

##  4 FIRE Collaboration

The NEXUS framework also serves as the basis for the nuclear data pipeline
used in the highly successful Fission In R-process Elements (FIRE) topical
collaboration. While the name centers around fission, many relevant nuclear
properties for this nucleosynthesis process have been studied by FIRE
researchers Orford2018 ; Vilen2018 ; Vilen2020 ; Wang2020a ; Wang2020b ;
Cote2021 ; Zhu2022 . Nuclear data appropriate for studies of nuclear
astrophysics is assimilated and disseminated via a tailored repository which
transform data into the appropriate format for use in the Los Alamos Portable
Routines for Integrated nucleoSynthesis Modeling (PRISM) reaction network code
Sprouse2021 . This state-of-the-art nuclear reaction network has been
frequently used in contemporary studies from the study of universality among
metal poor halo stars Roederer2022  to the first kilonova actinide opacity
results Fontes2022 . We now cover several recent FIRE studies that highlight
the crucial role accurate nuclear data can play in understanding the formation
of the heaviest elements in the cosmos.

The synthesis of the heaviest elements in nature is thought to occur via the
rapid neutron capture ($r$-) process Kajino2019 . While the astrophysical
event(s) responsible for this process are still under debate Thielemann2011 ,
it is with certainty that nuclear properties (most influentially: masses,
half-lives, reaction rates, and fission properties) are critical for
understanding this phenomenon Mumpower2016 . The central problem to this
endeavor is that accurate data is limited to a small portion of the chart of
nuclides and therefore simulations of the $r$ process must rely on theoretical
extrapolations for thousands of nuclei when this data is not available
Sprouse2020 . This implies that to understand $r$-process nucleosynthesis, it
is of paramount importance to simultaneously include the most accurate nuclear
data and modern theoretical predictions.

Once the $r$ process creates heavy, radioactive elements, they must decay back
to more stable isotopes. In doing so, the radioactive decay of these freshly
synthesized elements power a distinct light curve in compact object merger
environments (neutron star mergers and black hole neutron star mergers) called
a kilonova. The importance of $\beta$-decay half-lives and branching ratios on
the impact of kilonova light curves has been studied in the recent work of
Lund et al. (2022) Lund2022 . Lund and colleagues showed that different
theoretical models of $\beta$-decay properties profoundly impact the
production of elements that undergo $\alpha$-decay and spontaneous fission.
The influence of $\alpha$ and spontaneous fission emitters in turn strongly
impacts nuclear heating that powers the light curves. Neutron-rich isotopes of
nobelium ($Z=102$), lawrencium ($Z=103$), rutherfordium ($Z=104$), and dubnium
($Z=105$) were found to be particularly influential on observational
timescales. This lead the researchers to conclude that the $\beta$-decay
feeders of these nuclei represent a prime target for future precision
experimental campaigns. More comprehensive studies of the impact of nuclear
data and model uncertainties on kilonova were performed in Refs. Zhu2021 ;
Barnes2021 .

The inclusion of measured $\beta$-decay properties has also been studied in
the context of this pipeline. Kiss et al. measured 28 new $\beta$-decay
neutron emission probabilities and half-lives for important lanthanide species
Kiss2022 . This study not only informed the impact of measurements in
astrophysical environments, it was able to rank the importance of the
measurements through novel Bayesian analysis. Using a new measure to gauge the
impact of each measurement, the final abundances were shown to be most
sensitive to the half-life of 168Sm followed by several Gd isotopes. This work
is in agreement with another experimental effort supported by FIRE scientists
studying $\beta$-delayed neutron emitters that were found to be influential
around the second or $A\sim 130$ $r$-process peak Hall2021 .

FIRE researchers have also teamed up with nuclear experimentalists to study
nuclear structure measurements Orford2022  including the first exploration of
neutron shell structure to the ‘south east’ of lead Tang2020 . The production
of 207Hg led the team to conclude that current data suggests a strong shell
closure at neutron number $N=126$, in agreement with modern theoretical model
predictions. The $N=126$ is the final confirmed closed shell in the chart of
nuclides. The strength of this shell closure acts as a ‘gatekeeper’ for the
production of actinides and is thus extremely important in their production as
well as for how much fission may ensue in an $r$-process event.

In one of the major results of the FIRE collaboration, fission fragment mass
yields were calculated for all fissioning systems using the Finite-Range
Liquid-Drop Model (FRLDM) Mumpower2020 . The statistics of each fissioning
system was carefully compiled using millions of computational hours on the Los
Alamos high-performance computing network. One of the fundamental observations
made by the team was that the spread of the fission yield distribution
increases as a function of neutron excess. As a consequence of this result,
the heaviest neutron-rich nuclei that undergo fission in the $r$ process would
spread their fragments over a much larger mass range than previous studies
anticipated. The impact of these yields were subsequently tested in the
astrophysical $r$ process by Vassh et al. (2020). Vassh and colleagues found
that the wide fission yields of FRLDM drastically improve the fit to the solar
isotopic residuals relative to the commonly used symmetric split technique.
Wide fission yields impact the production of lanthanides as well as the
production of lighter precious metals. The possibility of co-production among
these elements is in agreement with observations of metal poor halo stars
which are thought to be evident of a single $r$-process event.

##  5 URSA

With the dramatic and ongoing acceleration in the technical capabilities
offered by recent and upcoming nuclear experimental facilities and detectors,
such as FRIB Horowitz2019 , N=126 Factory Savard2020 , and storage rings as
GSI, IMP, and RIKEN Zhang2016 , there will be a rapid transformation in state-
of-the-art nuclear data made available to astrophysical applications, e.g., in
astrophysical nucleosynthesis Rauscher2002 ; Kajino2019 ; Cowan2021 , stellar
structure and evolution Paxton2011 , and galactic chemical evolution Cote2018
, among other possibilities. Over the past 50 years B2FH , the intimate
relationship between nuclear physics, astrophysics, and astronomy has become
increasingly interconnected. As such, we are at a critical point where the
need to bridge the connections between the nuclear physics and astrophysics
communities has never been more apparent Meynet2008 ; Arnould2020 .

The rapid neutron capture process, or $r$ process, is an especially prominent
example of the connection between astrophysics and nuclear physics: the $r$
process proceeds through some of the most neutron-rich nuclei thought to exist
in the universe, which establishes this unique category of astrophysical
environments as an excellent probe of the nuclear physics of nuclei far away
from the valley of stability and approaching the neutron dripline Quark2Cosmos
. At the same time, this makes the $r$ process extraordinarily sensitive to
nuclear physics uncertainties which currently plague these same neutron-rich
nuclei Mumpower2016 , highlighting the need within the computational
astrophysics community to incorporate even individual points of new
experimental nuclear data from the moment it becomes available. As an example
of this effect, we refer to Fig. 4, where a single mass measurement is seen to
adjust the $r$ process abundance pattern by a relatively significant amount
Li2022 ; with the advent of many ($\approx$ 100s) of similarly impactful
experiments expected to occur in the immediate- and near-future Surman2018 ,
it is reasonable to expect that the astrophysics community is facing a
wonderful opportunity to capitalize on the ongoing developments in nuclear
physics and nuclear data communities.

Figure 4: The impact on the final $r$-process abundances from the single
measurement of the mass of 123Pd (red curve) relative to the baseline
simulation Li2022 . The $r$-process solar isotopic residuals are shown as
black dots. This result marked the first time a superposition of trajectories
were used to study the impact of a new measurement, in contrast to present day
sensitivity studies which use only single trajectories.

In anticipation of these advances, we have begun development of Unified
Reaction Structures in Astrophysics (URSA), which aims to expedite the
transmission of nuclear data from first experiment to computational
simulations in astrophysics. The overall motivation of URSA is centered around
the complete decoupling of computational applications of nuclear data from
specific details regarding the data formats and/or data sources themselves. In
practical terms, this manifests as an underlying data format that is
sufficiently expressive to capture the many types of nuclear data needs in
astrophysical and related applications, which variously include radiation
decay spectra, reaction cross sections, decay half-lives and branching ratios,
fission properties, among numerous other conceivable possibilities. At the
same time, URSA intends to provide, alongside the data format itself,
numerical libraries that access, process, and interpret data in any such ways
that numerical simulation codes might require. In this way, a computational
astrophysicist need not rely on other researchers or evaluators to import any
particular piece of experimental or evaluated data into one data format or
another; rather, URSA provides the resources to update all aspects of
computational simulations’ nuclear data requirements, as necessary, to
incorporate new nuclear data in real time as it becomes available. The ideal
data-to-application pipeline suggested by this model is demonstrated
schematically by Fig. 5, where we outline the efficiency with which URSA can
enable new and innovative nuclear data to be implemented in astrophysical
applications. The overall effect is to enable an immediate reaction time to
developments in nuclear data across a broad swath of astrophysical nuclear
data applications. Two factors that will be critical as the URSA project
evolves will be community uptake and user feedback, without either of which
the full potential of its impacts will surely not be realized.

Figure 5: An example of the Unified Reaction Structures for Astrophysics
(URSA) data-to-application pipeline. From the moment that even a single point
of nuclear data is made available, URSA provides a complete suite of tools to
update data in existing files within its prescribed format without any need to
modify existing simulation codebases.

##  6 Summary

Access and incorporation of accurate nuclear data is influential to producing
reliable nuclear evaluations and empowers scientific discovery. We have
reviewed a handful of cases where the Los Alamos NEXUS framework along with
theoretical model codes have been used in this regard. With a stream of new
nuclear data expected to be produced by radioactive beam facilities such as at
the Facility for Radioactive Isotope Beams (FRIB), adoption of transmission
protocols such as the Unified Reaction Structures for Astrophysics (URSA) will
be crucial for researchers in order to keep pace with rapidly evolving
scientific advancement.

## Acknowledgements

M.R.M. would like to acknowledge valuable discussions with Mark Chadwick, Kay
Kolos, Filip Kondev, Michael Smith and Patrick Talou. The authors were
supported by the US Department of Energy through the Los Alamos National
Laboratory (LANL). LANL is operated by Triad National Security, LLC, for the
National Nuclear Security Administration of U.S. Department of Energy
(Contract No. 89233218CNA000001). This work is intended for unlimited release
with identification number: LA-UR-22-30993.

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