#  TabPert: An Effective Platform for Tabular Perturbation

Nupur Jain
IIT Kanpur
<EMAIL_ADDRESS>
&Vivek Gupta
University of Utah
<EMAIL_ADDRESS>
Anshul Rai
IIT Kanpur
<EMAIL_ADDRESS>
&Gaurav Kumar
IIT Kanpur
<EMAIL_ADDRESS>

###### Abstract

To truly grasp reasoning ability, a Natural Language Inference model should be
evaluated on counterfactual data. TabPert facilitates this by assisting in the
generation of such counterfactual data for assessing model tabular reasoning
issues. TabPert allows a user to update a table, change its associated
hypotheses, change their labels, and highlight rows that are important for
hypothesis classification. TabPert also captures information about the
techniques used to automatically produce the table, as well as the strategies
employed to generate the challenging hypotheses. These counterfactual tables
and hypotheses, as well as the metadata, can then be used to explore an
existing model’s shortcomings methodically and quantitatively.

##  1 Introduction

Given the factual evidence, a crucial part of NLP model reasoning capacity is
if it can evaluate whether a given hypothesis is an entailment (true),
contradiction (false), or neutral (can’t be determined). Current transformers-
based models have been shown to outperform humans on these tasks when the
evidence is presented as simple unstructured text Wang et al. (2018, 2019);
however, when tested with semi-structured evidence Gupta et al. (2020); Chen
et al. (2019), such as tables, as shown in Figure 1, the very same models
struggled to match human accuracy Neeraja et al. (2021); Wang et al. (2021);
Aly et al. (2021).

Furthermore, there can be several reasons for a model’s correct predictions on
a particular example. For example, Poliak et al. (2018); Gururangan et al.
(2018) show that multiple NLI datasets, such as SNLI and MNLI dataset Bowman
et al. (2015); Williams et al. (2018) exhibit hypothesis-bias, i.e., the
hypothesis-only model performs significantly better than the majority label
baseline. In the context of tables, Gupta et al. (2020) shows that the right
prediction doesn’t always imply reasoning: there can be dataset biases in
semi-structured datasets too, such as hypothesis or premise artifacts
(spurious patterns) which can wrongly support a particular label.

Furthermore, a model can also ignore the ground evidence and use its pre-
trained knowledge for making predictions. These models when deployed in the
real world on out-of-domain (different category) or counterfactual (stories
tables) examples fail embarrassingly. One way to avoid this inflated
performance projection is to test models on several challenging sets before
actual deployment. For example, Gupta et al. (2020) evaluate the
RoBERTa(Large) Liu et al. (2019) models on two additional adversarial sets
(hypothesis perturbed and out-of-domain) and observe a significant performance
drop. However, manually creating such challenge sets can be tricky, both in
terms of the annotation cost involved and the actual annotation process,
especially with tabular data of semi-structured nature.

New York Stock Exchange
---
Type |  Stock exchange
Location |  New York City, New York, U.S.
Founded |  May 17, 1792; 226 years ago
Currency |  United States dollar
No. of listings |  2,400
Volume |  US$20.161 trillion (2011)

  1. H1:

NYSE has fewer than 3,000 stocks listed.

  2. H2:

Over 2,500 stocks are listed in the NYSE.

  3. H3:

S&P 500 stock trading volume is over $10 trillion.

Figure 1: A tabular premise example. The hypothesis H1 is entailed by it, H2
is a contradiction and H3 is neutral i.e. neither entailed nor contradictory.

Recently, Ribeiro et al. (2020a) showed that one can deploy simple tricks to
semi-automate this process and develop several adversarial counterfactual
contrast sets by altering existing data to perform behavioral testing of a
model. However, such tricks currently only work for unstructured text and
cannot be directly adopted for semi-structured text such as tables. To fill
this gap, in this work, we present TabPert. TabPert which is an annotation
platform especially designed to work on semi-structured tabular data. TabPert
support semi-automatic creation of tabular counterfactual data. Through
TabPert annotators can modify tables in several ways such as  (a) deleting
information: deleting an attribute-value pair or an existing row completely,
(b) inserting information: inserting an attribute-value pair for an existing
row or creating a fresh row,  (c) modifying information: editing the attribute
or values cells of an existing row, and  (d) modifying hypothesis or label:
modifying an existing hypothesis and its inference label.

TabPert also automatically logs the modification operation for each attribute-
value of the table with respect to the original table. Furthermore, users can
manually log information about the relevant rows and the strategy used for
perturbing a table-hypothesis pair, in addition to the gold label, through
TabPert. Such metadata is very important in assessing annotated data toughness
and can be later utilised to systematically study failure modes of a model.

The contributions of our work can be summarised as below:

  1. 1.

TabPert can delete, modify, and insert information in semi-structured tabular
data for creating counterfactual examples

  2. 2.

TabPert auto-logs table perturbation metadata, and support manual hypothesis
modification and inference labels selection.

  3. 3.

TabPert assists users in logging metadata including hypothesis-related rows
and the perturbation strategy used, which is useful for data and model
quantitative analysis

  4. 4.

We present a case study for TabPert via the generation and evaluation of a
counterfactual InfoTabS dataset and models respectively.

The TabPert source code, the annotated counterfactual InfoTabS dataset, along
with the RoBERTaLarge model, the annotation instructions and examples set, and
all other associated scripts, are available at
https://github.com/utahnlp/tabpert. The instruction video describing TabPert
usage is accessible at https://www.youtube.com/watch?v=sbCH_zD53Kg.

##  2 Tables are Challenging

One might argue that creating a counterfactual dataset for tables is not a
challenging task, and that table modification can be fully automated by merely
_‘shuffling’ or ‘inserting’_ attribute values of one table row into another
table row (with the same attribute) as long as they are from similar
categories, e.g. shuffle ‘producer’ of one film with ‘producer’ of another
film). One can extend this further by shuffling rows with different attributes
in the same as well as different tables (same category) as long as the name-
entity type for values are similar, e.g. shuffle ‘producer’ with the
‘director’ of the same or a different film with each other.

However, this approach does not automate the modification of corresponding
hypotheses and their inference labels. Furthermore, such automatic shuffling
encourages the violation of certain natural common-sense logical constraints,
such as a person’s ‘Birth Date’ must be before their ‘Died Date’, a person’s
‘Marriage Date’ should be after their ‘Birth Date’ and before their ‘Died
Date’, an album’s ‘Released Date’ should be after its ‘Recording date’ and so
on. Without the enforcement of these constraints, the table will be self-
contradictory. While some of these constraints can be automatically satisfied
and hence not violated, a majority still slip through because of their sheer
variety and variation. 111In real data, these constraints are naturally
satisfied.. Furthermore, enforcing these constraints automatically during
perturbation is a challenging job due to its domain-specific nature. However,
such automatic perturbations can be a good initialization for our TabPert
tables, which can then be manually inspected and modified by human annotators
for self-consistency i.e. no natural common sense violation.

##  3 TabPert Functions, Aspects, and Usability

TabPert is currently supported on common web browsers such as Google Chrome
and can be installed to run locally 222Details:
https://github.com/utahnlp/tabpert. There are three main steps required for
successful annotation, as described below.

###  3.1 Automatic Initialization

First, we initialise TabPert with original tables and automatic counterfactual
tables generated via automatic ‘shuffling’ of table rows or attribute values.
Automatic initialization is beneficial as manual table creation is both time-
consuming and highly error-plausible. This automatic shuffling operation is
stored as a metadata of each attribute-value in the first 4 bits of a 7-bit
string, and can be used later to analyse which shuffling was more effective.
Table 1 shows the meaning of each of these bits 333For same table i.e.
$3^{rd}$ bit $0$, the $1^{st}$ and $2^{nd}$ bits are zero. Counterfactual
hypothesis (and label) initialization is done by copying the information from
the original table exactly.

Bit | Location | Same | Different
---|---|---|---
1 | Dataset | 0 | 1
2 | Category | 0 | 1
3 | Table | 0 | 1
4 | Key | 0 | 1
Table 1: First Four Bits of Table Value Metadata. The $1^{st}$ bit represent
the perturb value table location i.e. whether it is coming from the Train set
(1) or the Test set (0), the $2^{nd}$ bit indicates the different (1) or same
(0) category, the $3^{rd}$ bit represents if the value is from the same (0) or
from different (1) table (for same table, $1^{st}$ and $2^{nd}$ bit is always
zero), and the $4^{th}$ represents same (0) or different (1) attribute.

###  3.2 Modifying Tables

The automatically perturbed tables from initialization can now be manually
modified to create counterfactual examples444Also to avoid self-contradiction
or inconsistency.. All the cells (attributes and values) in the three
counterfactual tables can be edited555Original table is fixed to prevent
inadvertent edits. Table rows can be modified via the dragging and dropping of
a value cell from  (a) same counterfactual table (cut-paste effect),  (b) from
another counterfactual table (cut-paste effect),  (c) from the original table
(copy-paste effect) . To minimise errors during this drag-and-drop operation,
a type validation check runs in the background, which prevents a drag and drop
between different key categories (for example, it is forbidden to drag a
_Person Name_ into _Date of Birth_). To achieve such type-check, keys are
automatically grouped using category (entity type) information during
initialization. 666Keys for which a category information is missing can be
dropped without restriction.

TabPert also supports five additional functions for more challenging edits.
The _‘Add’_ box allows annotators to write text and drag and drop it into a
table to create a new value. For deleting a value cell, simply drag and drop
the value cell to the _’Delete’_ Box. One can also edit the text for an
existing attribute or value via _clicking_ it. Lastly, one can delete any row
with the _‘Edit’_ option, and also insert a new row and its details using the
_’Add Section’_ button. These modification details are also recorded
automatically in the last 3 bits of the 7-bit metadata. The $5^{th}$ bit
represents copy-paste from original, the $6^{th}$ represents a new cell or row
addition, and the last $7^{th}$ bit represents a value update operation.
Figure 2(a) shows the main parts of the TabPert platform for counterfactual
table perturbation.

(a) Table Perturbation :  [fill color=black,inner color=white,]1 Table Title
[fill color=black,inner color=white,]2 Values [fill color=black,inner
color=white,]3 Key (table section name) [fill color=black,inner color=white,]4
Edit Section Name or Delete Section [fill color=black,inner color=white,]5 Add
New Value [fill color=black,inner color=white,]6 Delete Value [fill
color=black,inner color=white,]7 Add New Section

1234567

1234567 (b)  Hypotheses Perturbation:  [fill color=black,inner color=white,]1
Hypotheses Table Labels [fill color=black,inner color=white,]2 Label of an
Original Hypothesis [fill color=black,inner color=white,]3 Label of a
Counterfactual Hypothesis of Table A [fill color=black,inner color=white,]4
Original Hypothesis [fill color=black,inner color=white,]5 Counterfactual
Hypothesis in Table A [fill color=black,inner color=white,]6 Option to open
Modal for Selecting Hypothesis metadata (Figure 2(c)) [fill color=black,inner
color=white,]7 Submit Button to Save Work

123456 (c) Selection of Relevant Rows and Hypothesis Perturbation Strategies :
[fill color=black,inner color=white,]1 Hypothesis [fill color=black,inner
color=white,]2 Table Label [fill color=black,inner color=white,]3 Relevant Row
(checkbox ticked) [fill color=black,inner color=white,]4 Irrelevant
Row(checkbox not ticked) [fill color=black,inner color=white,]5 Strategy Used
(checkbox ticked) [fill color=black,inner color=white,]6 Strategy Not Used
(checkbox not ticked)

###  3.3 Hypothesis Modification and Metadata

The text of a hypothesis of a counterfactual table can be edited directly and
its corresponding label can also be selected from drop-down menu options. In
addition to this, other metadata information is also collected:

  1. 1.

The strategies used by the annotator to modify the hypothesis. The five main
strategies can be selected using the multi-value check-box. The ‘Other’ option
corresponds to hypothesis changes that don’t fall into main strategies.

  2. 2.

All the relevant rows of the table which are necessary for deciding the
inference label.

Figure 2(b) shows the main TabPert view for hypothesis modification, with
hypothesis and inference label. Metadata is inserted by the annotator via
clicking ‘+’ symbol on the left side (below label drop-down) for each
hypothesis, as shown in Figure 2(b). This opens a metadata collection window,
as shown in 2(c)). Here too, we use 6 bits: the initial five for each strategy
(bits order is the same as the order in which the strategies are mentioned in
TabPert as shown Figure 2(c)), and the last one for the ‘Other’ option. We
store the relevant rows ‘attribute keys’ in a list (array) for each hypothesis
along with the final hypothesis text modification.

TabPert Aspects: The TabPert web-app’s core tech stack consists of
ReactJS777https://reactjs.org/ and
Flask888https://flask.palletsprojects.com/en/2.0.x/. Here, Flask is used as
the main back-end Python web framework, and javascript library ReactJS is used
for the front-end. We used Flask because is easy-to-extend, giving us the
ability to easily integrate Python libraries for quick manipulation of _JSON_
and _TSV_ files. We used ReactJS because of the react-beautiful-dnd
library999https://www.npmjs.com/package/react-beautiful-dnd essential for
simulating the drag-and-drop function.

##  4 Case Study on InfoTabS

We use TabPert to create counterfactual data for the InfoTabS dataset Gupta et
al. (2020). InfoTabS is a semi-structured natural language inference dataset
which consists of entity tables and human-written hypotheses. For this we
sampled 47 tables with 423 table-hypothesis pairs taken from the $\alpha_{1}$
set of InfoTabS. InfoTabS stores each table as a JSON file with key-value as
attribute (column 1) - value (column 2) of the table. For initialization we
used both the $Train$ and other $\alpha_{1}$ set. Including both set creates
more diversity and variation in automatic initialization101010As discussed
earlier, annotators manually fix the constraint violations during annotation.

Annotation Guidelines: Following a similar line as earlier works by Ribeiro et
al. (2020b); Sakaguchi et al. (2020) for creating challenging adversarial
test-sets, we guided the annotators in annotating three counterfactual table
(A, B, C) for each original $\alpha_{1}$ table. This ensures enough diversity
and coverage in the collected counterfactual data. We advised annotators to
follow the below strategies for each counterfactual table:  (a) For Table A:
change the table in a way that it flips entail and contradict labels while
keeping the hypothesis same,  (b) For Table B: change the hypothesis in a way
that it flips entail and contradict labels; modify the table as needed for the
same, and  (c) For Table C: write a new but related hypotheses with similar
reasoning; one can modify table as needed.

Furthermore, for making the neutrals harder we advise annotators to modify
them by adding extra _‘true’_ information from the same table in the
hypotheses. The above discussed procedure ensures that  (a) the final label is
balanced,  (b) hypothesis bias is destroyed by flipped label Gupta et al.
(2020); Chen et al. (2019), and  (c) ‘neutrals’ are now closer to ‘entails’ in
terms of lexical overlap Glockner et al. (2018).  . Finally, after annotation
we have 109 counterfactual tables with a total of 982 table-hypothesis pairs,
where Table A (141) type having 423, Table B (135) type having 405, and Table
C type having 154 (52) pairs.

#### Experiment and Analysis

To check if the annotated counterfactual data is challenging for existing
model, we use InfoTabS RoBERTaLarge ‘para’ representation and hypothesis-only
model for making predictions on the original and counterfactual annotated
data. Table 2 shows the performance result. The data was represented in ‘para’
form, one with all rows’ sentences (all Rows) and other with just relevant row
sentences (coming from the annotation meta-data).

Performance Analysis: Clearly the same model struggles with the counterfactual
annotated data. Furthermore, better performance with relevant rows for
counterfactual data shows that the model is probably using irrelevant rows
tokens as artifacts for making predictions Neeraja et al. (2021). The
hypothesis-only model performance on counterfactual data is close to majority-
label baselines. Additionally, human find the original and counterfactual data
equally challenging, obtaining performance of $\approx$ 85$\%$ on both sets.
111111No difference in performance between A, B, and C type counterfactual
table-example pairs, see appendix Figure 4.

Model Type | Original | Counterfact
---|---|---
Majority | 33.33 | 33.33
Hypo Only | 64.32 | 44.85
All Rows | 78.91 | 61.26
Relevant Rows | 74.11 | 65.85
Human | 84.8 | 85.8
Table 2: InfoTabS RoBERTaLarge model with original and counterfactual
annotated data.

Perturbation Analysis: We also do an analysis using the hypothesis annotation
metadata to check which strategies of hypothesis modification is more
effective. From Figure 3, it is evident that manual table change (TC) for
Label Flip (LF) is more effective than manual hypothesis change (HC) for label
flip (TF). Furthermore, strategies involving label flip is more effective then
Hypothesis Prompt (Hypo Prompt) and text overlap. We suspect this is because
of the ineffectiveness of hypothesis bias with flipped labels. Surprising, on
new hypotheses, there is marginal performance improvement, indicating simple
new data creation is an ineffective approach. Furthermore, there is no
significant performance drop with any other perturbation approaches.

Figure 3: Performance drop after counterfactual perturbation with various
strategies.

We also did a similar analysis on the table perturbation metadata; refer to
appendix section A for details. We also show some qualitative example of
counterfactual perturbation for each strategy in appendix section C.

##  5 TabPert Utility

#### Main Platform:

TabPert is a tool designed particularly for annotating counterfactual tabular
reasoning data, and as such, it has numerous optimizations, tools, and
features that aid in the creation and collection of huge amounts of data, as
well as annotating it faster and better. It allows for a broader range of
tasks than, perhaps, utilizing spreadsheets or MTurk to modify such data. The
drag-and-drop functionality simplifies annotation, making it simple to
visualize a complicated job. In tabular form, all of the data may be examined
at once. The background type validation reduces mistakes while dragging and
dropping.

#### Other Task Usage:

The initialization source code, as well as the platform, are designed to be
modular, so that new components can be readily added, deleted, and existing
ones may be updated. For example, the ability to reorganize table parts;
copying values across table triplets (in addition to cut-paste); and auto-
saving work with an undo option121212Currently, the Save button must be
pressed manually, however it can be pressed several times at different
timestamp to save even semi-completed work., as well as checkpoints added to
reverse mistakes, may all be readily implemented.

#### Meta-Data:

The metadata collected might be used to generate challenging counterfactual
adverserial test sets. For example, if a hypothesis comprises of shuffled rows
from the train set and the original inference label is inverted, it might be a
good candidate for evaluating NLI model overfitting. Furthermore, it is an
excellent test for hypothesis artifacts if the hypothesis remains the same but
the label is flipped. The hypotheses-specific rows can be used to reduce the
table and explain the inference label reasoning. The marked labels can be
utilized as the gold standard for existing label verification. Counterfactual
tables may also be used to assess pre-trained knowledge overfitting. These are
only a few examples of the numerous conceivable application situations.

We also compare and contrast TabPert with Spreadsheet on effectiveness, visual
benefits, and meta-data collection aspects in appendix section B

##  6 TabPert Limitations and Future

During our pilot study, the platform was run locally by the annotators. This
was not problematic since the number of annotators was small and the tables
were divided between them. We need to host our platform on a central server if
we want multiple annotators to be able to make simultaneous edits for large-
scale deployment. This is something which we plan to do in the near future.
Lastly, the counterfactual data created by making changes had to be manually
saved by pressing a button. This was done so that if the user made some
mistake the original data would not be lost and the user could store the data
after being satisfied with the changes. We wish to add a auto-save feature
along with undo options to cater to both these scenarios.

##  7 Conclusions

We proposed TabPert, a powerful platform for visualizing semi-structured
tabular data and generating counterfactual tabular perturbations. This
platform allows researchers to alter tables and hypothesis phrases as well as
collect accompanying metadata in order to statistically and methodically
examine the shortcomings of their systems. We hope that TabPert will be useful
for academics working on semi-structured data, such as tables. TabPert may
also be used in non-academic industrial settings that demand table change,
such as e-commerce product specification tables, financial and tax statements
tables, and so on.

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##  Appendix A Performance vs Perturbation

Figure 4 shows the model performance relative accuracy drop with variable
table perturbation strategy. Analysing average accuracy across the A, B, and C
counterfactual type, shows no significant difference. We also shows the number
of examples for each hypothesis perturbation strategy and the relative
accuracy drop in Figure 5 for each strategy.

Figure 4: Performance drop with various counterfactual perturbation with
various strategies (use the bit tables 1 for the analysis). Performance on A,B
and C type of counterfactual.

Figure 5: #Example of each hypothesis perturbation strategy and relative
accuracy drop for each perturbation strategy.

##  Appendix B TabPert vs Spreadsheet

#### Effectiveness:

When utilizing spreadsheets for annotation, it becomes quite difficult and
time consuming to cut(copy)-paste in numerous rows cells several times.
However, the efficient drag(click)-drop with auto checking restrictions in
TabPert makes it a lot easier and faster procedure. Editing and text
alteration are also easier than in a spreadsheet. Our study found that it
takes around 7 minutes on average to annotate a new table with 9 statements
using TabPert, but the same work in the spreadsheet would take more than 40
minutes.

#### Visualisation Benefit:

Our platform’s table visualisation provides a full data view on a single
screen. Seeing the entire picture (table and sentences) is incredibly helpful
for quality checking of annotations. It also allows the annotator to quickly
follow label and hypothesis changes, which is not feasible with the
spreadsheet’s cell type view.

Furthermore, having a single screen focus view on a single counterfactual
table makes altering hypotheses even easy. Using this focus feature, it is
straightforward to update the labels or add new information to the hypothesis.
This focus view is not viable with a spreadsheet; to make appropriate
alterations, one must search and navigate to each spreadsheet cell.

Finally, the lack of scrolling required while dragging and dropping on our
platform saves annotators time. To discover the relevant cells in a
spreadsheet, one must execute numerous scrolling operations to the up, down,
left, or right.

Furthermore, in TabPert, the cell size is automatically set according on the
underlying information, but in spreadsheet, this must be handled manually.

#### Meta-Data Collection:

TabPert also makes it simple to gather information such as methods used to
change a hypothesis and rows utilized to answer each hypothesis by simply
utilizing a checkbox. In a spreadsheet, this would require 9 columns of check-
boxes for each table or manually writing the meta0data, which is now
automatically done with a single click, thus making the process simpler,
efficient and speedier.

##  Appendix C Qualitative Counterfactual Perturbation Examples

Tables 3, 4, 5, 6, 7 illustrate the five strategies used for counterfactual
table-hypothesis perturbation. In the After rows, the 7-bit meta-data
associated with each value is also listed. The Automatic Initialisation row
explains the meaning of the first four bits of this meta-data, and the Manual
Editing row explains the last three bits.

Table 3: Example using Strategy 1
Strategy: Change table to flip label |  Premise |  Hypothesis |  Label |  Predicted
---|---|---|---|---
Before (T14) |  Box Office  1. $61.3 million Budget  1. $26 million |  Flatliners made over double what it cost to make at the box office. |  E |  E
After (T14A) |  Box Office  1. $ 140.7 million (1010 010) Budget  1. $85 million (0111 010) |  Flatliners made over double what it cost to make at the box office. |  C |  E
Automatic |  |  |  |
Initialisation | 1010: different dataset, same category, different table, same key
| 0111: same dataset, different category, different table, different key
Manual |  |  |  |
Editing | 010: value text edited
Table 4: Example using Strategy 2
Strategy: Change hypothesis to flip label |  Premise |  Hypothesis |  Label |  Predicted
---|---|---|---|---
Before (T14) |  Box Office  1. $61.3 million Budget  1. $26 million |  Flatliners made over double what it cost to make at the box office |  E |  E
After (T14B) |  Box Office  1. $ 13.3 million (1010 010) Budget  1. $5.9 million (0110 010) |  Flatliners made over triple what it cost to make at the box office. |  C |  E
Automatic |  |  |  |
Initialisation | 0110: same dataset, different category, different table, same key
| 1010: different dataset, same category, different table, same key
Manual |  |  |  |
Editing | 010: value text edited
Table 5: Example using Strategy 3
Strategy : Add ‘true’ information from the table to confuse the model, or use text from other places in the table. |  Premise |  Hypothesis |  Label |  Predicted
---|---|---|---|---
Before (T14) |  Produced by  1. Michael Douglas 2. Rick Bieber Directed by  1. Joel Schumacher |  Rick Bieber put more money into Flatliners than Michael Douglas did. |  N |  N
After (T14A) |  Produced by  1. Rick Bieber (0000 100) 2. Michael Douglas (0000 100) Directed by  1. Empress Teimei (1111 000) |  Rick Bieber put more money into Flatliners directed by Empress Teimei, than Michael Douglas did |  N |  N
Automatic |  |  |  |
Initialisation | 0000: same dataset, same category, same table, same key
| 1111: different dataset, different category, different table, different key
Manual |  |  |  |
Editing | 000: no change
| 100: copied from the original table
Table 6: Example using Strategy 4
Strategy: Use the original hypothesis as a prompt to write a new hypothesis |  Premise |  Hypothesis |  Label |  Predicted
---|---|---|---|---
Before (T14) |  Edited by  1. Robert Brown Written by  1. Peter Filardi |  Flatliners was Peter Filardi’s first writing credit. |  N |  N
After (T14B) |  Edited by  1. James Newton Howard (0000 100) 2. Robert Brown (0000 000) Written by  1. Lee Beom-seon (1110 000) |  Flatliners was mostly edited by Robert Brown. |  N |  E.
Automatic |  |  |  |
Initialisation | 0000: same dataset, same category, same table, same key
| 1110: different dataset, different category, different table, same key
Manual |  |  |  |
Editing | 000: no change
| 100: copied from the original table
Table 7: Example using Strategy 5
Strategy: Write a completely new hypothesis |  Premise |  Hypothesis |  Label |  Predicted
---|---|---|---|---
Before (T14) |  Box Office  1. $61.3 million Budget  1. $26 million |  Flatliners made 50 million over it’s budget at the box office. |  C |  E
After (T14C) |  Box Office  1. US$85.4 million (December 2017) (0111 000) Budget  1. $26 million (0000 000) |  Flatliners costed around $25 million in making and was a hit. |  E |  C
Automatic |  |  |  |
Initialisation | 0000: same dataset, same category, same table, same key
| 0111: same dataset, different category, different table, different key
Manual |  |  |  |
Editing | 000 : no change


# Emulating an Atomic Gyroscope with Multiple Accelerometers

Nathan Shettell  Centre for Quantum Technologies, National University of
Singapore, Singapore  Rainer Dumke  Centre for Quantum Technologies, National
University of Singapore, Singapore  School of Physical and Mathematical
Sciences, Nanyang Technological University, 637371, Singapore

###### Abstract

The main advantage of an atomic accelerometer when compared to a classical
accelerometer is negligible bias drift, allowing for stable long-term
measurements, which opens the potential application in navigation. This
negligible drift arises from the fact that the measurements can be traced back
to natural constants, and the system is intrinsically stable due to the simple
design. In this manuscript, we extend this property of long-term stability to
gyroscopic measurements by considering an array of atomic accelerometers, and
comparing the performance to atomic gyroscopes, which are technologically more
prone to bias drifts. We demonstrate that an array consisting of four three-
axis atomic accelerometers can outperform state of the art atomic gyroscopes
with respect to long-term stability.

## Introduction

Atomic interferometry has undergone rapid technological developments in the
past few decades [1, 2, 3, 4], leading to highly sophisticated technologies
that can be used for a variety of precision measurements, including, but not
limited to, Earth’s gravity [5, 6, 7, 8, 9], the fine structure constant [10,
11] and the Aharonov–Casher effect [12]. Another promising application of
atomic intereferometers is navigation and inertial sensing [13, 14, 15, 16,
17, 18, 19, 20, 21, 22], i.e. as an accelerometer or gyroscope. The main
justification for atomic navigation devices (as opposed to a classical device)
is that the precision of the measurement can rival the best (classical)
inertial sensors [16, 23, 24, 25] while maintaining accuracy for a much longer
period of time.

Bias drifts are a common hindrance to classical sensors, mainly due to
temperature dependencies within the hardware, mechanical wear-and-tear, and
aging, which ultimately degrade the accuracy of long-term measurements. In
comparison, the measurements taken with atomic accelerometers are based on
very well-known and controlled scale factors: the inter-pulse duration, laser
wavelength and transition energies, which allows, in principle, for long-term
absolute measurements [26, 6, 7, 15, 27]. Despite atomic gyroscopes depending
on the same aforementioned factors, the mechanical and optical setup is more
complex, consequently, the atom trajectory drifts over time, limiting the
long-term stability [28, 29, 30]. The bias drifts exhibit in gyroscopes vary
over several orders magnitude, from $~{}10^{-5}-10^{-9}\;$rad$\cdot$s-1/day
[31, 13, 32, 17, 33, 29, 34, 35, 36, 37]. A stability limit of $\sim 7\times
10^{-8}\;$rad$\cdot$s-1/day was hypothesized for atomic gyroscopes, due to
wavefront aberrations and off-resonant Raman transitions [28, 29]. Notably
though, this limit has been surpasses by the gyroscopes demonstrated in Refs.
[36, 37].

In this work, we explore the prospect of measuring rotational motion with
sufficiently many atomic accelerometers [38, 39, 40, 41]. Henceforth, such a
contraption is referred to as an array of atomic accelerometers (AAA). To
compute all three vector components of the angular velocity, one must utilize
both vertical and horizontal atomic accelerometers. Although vertical atomic
accelerometers do exhibit negligible drift (for example, no drifting
properties were witnessed after 24 hours of measurements in Ref. [7]) the same
is not true for horizontal accelerometers [13, 42, 43, 44, 20]. From a
technological viewpoint, horizontal and vertical accelerometers are
effectively equivalent. Thus one can imagine that as horizontal accelerometers
garner more research attention, they could similarly exhibit negligible bias
drifts. In the meantime, this article explores how a bias drift present in a
three-axis accelerometer propagates to the gyroscopic measurements of an AAA.

The working principle of an AAA is to deduce rotational velocities from linear
acceleration gradients. This has been implemented with great success for
navigation [45, 46, 47, 48, 49] because of the high bandwidth of classical
accelerometers. Notably, some of the navigational devices are GPS-aided [46,
47]; long-term stability offered by an AAA is vital to allow for navigation in
GPS denied environments. Other interesting applications of arrays of
accelerometers include bio-mechanics [50, 51], stabilization [52, 53, 54] and
seismology [55, 56, 57, 58], the latter of which can be is a potential
application for an AAA. Evidently, the long-term stability of atomic
accelerometers is a favourable quality when measuring dynamics on longer
timescales imposed by nature, for example, volcanic activity, and underground
geothermal activity and glacier ablation.

It is worthwhile to note that using multiple atomic accelerometers does not
multiplicatively increase the technological overhead, for example, the same
laser system and control electronics can be used for multiple sensor heads.
Such a contraption would provide a complete navigational description (linear
and rotational motion) using a sole type of technology, facilitating
maintenance and troubleshooting. Atomic accelerometers vary drastically in
size and performance, but the working principle is the same: an atomic cloud
with discrete energy states is stimulated by a sequence of laser pulses, the
phase shift accumulated through two different paths of the interferometer is
proportional to the projection of the acceleration of the atom relative to an
inertial reference along the interrogation laser direction [4]; said phase
shift is mapped onto a probability distribution and can be deduced through
statistical modelling. A thorough explanation of the underlying physics can be
found in Ref. [5]. As we are investigating this idea from a theoretical
perspective, the atomic accelerometers are considered to be a blackbox which
outputs an estimate of the linear acceleration with statistical constraints
(precision, sensitivity, resolution, stability, et cetera).

This work is organized as follows. We begin by outlining the pertinent
mathematics to an array of atomic accelerometers. Next, we compare the
performance of an AAA to current atomic gyroscopes [31, 34, 30, 35, 37]. This
is done by considering accelerometers with realistic resolutions [59, 60, 20]
and subjecting the array to different degrees of rotational motion. We
investigate how a bias drift within the atomic accelerometers translates to an
effective bias drift with respect to the angular velocity measurement.
Additionally, we incorporate expected technical challenges, such as size
limitations, positional uncertainties, and misalignment, to determine their
effects on the attainable resolution.

## Array of accelerometers

Within a rigid body, the difference of linear acceleration (in the rotating
reference frame) observed at locations $\bm{r}^{(1)}$ and $\bm{r}^{(2)}$ is
given by

$\bm{a}^{(1)}-\bm{a}^{(2)}=W(\bm{r}^{(1)}-\bm{r}^{(2)}),$ (1)

where

$W=\dot{\Omega}+\Omega^{2},$ (2)

with

$\Omega=\begin{pmatrix}0&&\omega_{z}&&-\omega_{y}\\\
-\omega_{z}&&0&&\omega_{x}\\\ \omega_{y}&&-\omega_{x}&&0\\\ \end{pmatrix},$
(3)

and $\bm{\omega}=(\omega_{x},\omega_{y},\omega_{z})$ is the angular velocity
of the rigid body and $\dot{\Omega}=\frac{\partial\Omega}{\partial t}$. It
follows that, with sufficiently many accelerometers one can measure the linear
acceleration at different locations on the rigid body and reverse engineer
said measurements to determine the angular velocity $\bm{\omega}$ [38, 61, 39,
40, 41, 62].

A minimum of nine single-axis accelerometers are needed to infer the symmetric
terms of $W$ (angular velocity) as well as the anti-symmetric terms (angular
acceleration). Because of the quadratic nature of $\Omega^{2}$, any estimate
of $\bm{\omega}$ has intrinsic sign ambiguity, however this can be eliminated
using dependencies between $\bm{\omega}$ and $\dot{\bm{\omega}}$ [63]. In this
manuscript we focus on an array of four three-axis atomic accelerometers:
where the placement of one of the accelerometers is deemed the origin, and
each one of the remaining accelerometers is fixed at a distance $l$ along a
separate axis, as depicted in Figure 1. This choice is made for mathematical
simplicity, nevertheless, much of the analysis can be straightforwardly
adopted to arrays of nine single-axis accelerometers [61], or of arbitrarily
large arrays.

Figure 1: This figure illustrates the AAA we focus on in this study: the
location of one central accelerometer is denoted the origin and the other
three are each placed along a separate axis at a distance $l$ from the origin.
We assume that the atomic accelerometers are simultaneously sensitive to three
different axis of acceleration. Although such devices exist, the measurements
are sequential and not simultaneous [13, 64, 42, 20, 65], however, assuming
the time delay between measurements is small, than this approximation may
still hold. Alternatively, one can use a different setup with single-axis
atomic accelerometers placed strategically [61] and obtain simultaneous
measurement results.

We use the notation $\hat{\square}$ to denote an estimated quantity, e.g.
$\hat{W}$ is the estimate of $W$. There are numerous methods of formulating
the estimate $\hat{\bm{\omega}}$ from $\hat{W}$ [66, 67], most of them diverge
when one or more components of $\bm{\omega}$ approaches zero [68]. The
methodology we adopt is proposed in Ref. [66], where one considers the
diagonal terms of $W$ in a rotated reference frame, specifically a reference
frame where each component of $\bm{\omega}$ has equal magnitude. This method
attains a high degree of precision and diverges solely when $\bm{\omega}$
tends to exactly zero [66]. An a priori estimate $\tilde{\bm{\omega}}$ is
required to determine the correct reference frame, this can be achieved using
a classical sensor, or alternatively, one can use the angular acceleration
measurement to construct a self-contained estimation strategy

$\tilde{\bm{\omega}}(t_{n})=\hat{\bm{\omega}}(t_{n-1})+\frac{\Delta
t}{2}\big{(}\hat{\dot{\bm{\omega}}}(t_{n-1})+\hat{\dot{\bm{\omega}}}(t_{n})\big{)},$
(4)

where $t_{n}$ is used to denote an estimate of a quantity at the $n$th
measurement and $\Delta t$ is the time between measurements.

Because of the non-linearity of the estimation, it is difficult to determine
exact statistical tendencies. Regardless, a guideline for the error of an
array of accelerometers with $N$ three-axis atomic accelerometers is

$\centering||\hat{\bm{\omega}}-\bm{\omega}||\propto\frac{||\hat{\bm{a}}-\bm{a}||}{\sqrt{M}l_{\text{eff}}||\bm{\omega}||},\@add@centering$
(5)

where $M$ is the number of ways to select four accelerometers which are non-
coplanar, and $l_{\text{eff}}$ is the average separation distance between
pertinent accelerometers. The quantity $M$ is dependent on the arrangement of
the accelerometer within the AAA, in addition, other scaling factors may be
introduced depending on the employed inference strategy. More information on
the statistics of accelerometer arrays can be found in Refs. [69, 39, 70, 71].

Evidently, the capabilities of deducing rotational motion will be bounded by
the effectiveness of the accelerometers, hence the proportionality to
$||\hat{\bm{a}}-\bm{a}||$, and similarly improves with the number of
accelerometers within the array ($N$). Moreover, the precision of the
apparatus increases when the acceleration on different sensors vary
substantially, which is achieved for large values of $||\bm{\omega}||$, or can
be artificially achieved by increasing the distance between sensors ($l$).

## Long-term stability of $\hat{\bm{\omega}}$

In order to properly compare the long-term stability of the angular velocity
measurement made by an AAA, we model the statistical properties of the three-
axis atomic accelerometers from Ref. [20]. Namely a single-shot measurement
resolution of $7.6\times 10^{-5}\;$ms-2, and a bias drift of $5.9\times
10^{-7}\;$ms-2/day. Here, the single-shot measurement resolution concerns the
norm of the acceleration vector. For the sake of simplicity, we model the
resolution of the each vector component (i.e. $\hat{a}_{x}^{(i)}$,
$\hat{a}_{y}^{(i)}$, and $\hat{a}_{z}^{(i)}$) to be identical, thus equal to
$7.6/\sqrt{3}\times 10^{-5}\;$ms-2, which is approximately observed in Ref.
[20]. In addition, we assume the sensitivity of the accelerometers are
independent from $\bm{a}$ and $\bm{\omega}$, this is in to more easily isolate
properties of the effective bias drift with respect to $\hat{\bm{\omega}}$. If
one desired to obtain the bias-drift for a different sensitivity, one could
scale the results according to Eq. (5).

The data is modeled using

$\hat{\bm{a}}^{(i)}=\bm{a}^{(i)}+\bm{\eta}^{(i)}+\beta^{(i)}\bm{u}^{(i)},$ (6)

where $\bm{\eta}^{(i)}$ is a random variable to emulate white noise,
$\beta^{(i)}$ is the bias of the $i$th sensor and $\bm{u}^{(i)}$ is a unit
vector signifying the direction in which the accelerometer drifts towards. To
simulate the dynamical properties of the drift, the bias at the $n$th time
step, $\beta^{(i)}(t_{n})$, is sampled from a normal distribution centered
around $\beta^{(i)}(t_{n-1})+\dot{\beta}\Delta t$, where
$\dot{\beta}=5.9\times 10^{-7}$ ms-2/day is the drift rate.

For (most) classical inertial sensors, bias drifts are gauged by performing
long-term static measurements; thus, the bias drifts are modeled to be
independent of the degree of motion. This is not true for an AAA because of
the non-linearity present in the method in which $\hat{\omega}$ is
constructed. There is an inverse dependency on the angular velocity,
$||\bm{\omega}||$. Hence, the simulation is repeated for different values of
$\omega$. For the sake of symmetry, the AAA is subjected to constant motion
$\bm{\omega}=\omega(1,1,1)/\sqrt{3}$, in which
$\omega\in\\{0.05,0.50,5.00\\}\;$rad$\cdot$s-1. Data is sampled for a total
time of $T=2\times 10^{6}\;$s, with a sampling time of $\Delta t=0.25\;$s.
This value of $\Delta t$ is comparable to the interrogation of time of most
modern atomic accelerometers [19], and would provide data readout at an
acceptable rate for a variety of applications, whether that be navigation or
seismology. Typically, the time difference between data readout is a few
seconds (e.g. $\Delta t\approx 1.65\;$s in Ref. [20]) because of dead time,
however, techniques have been developed which effectively eliminate dead times
[36]. Note though, that the value of $\Delta t$ has very little impact on the
performance of the AAA. The exception of course is the construction of the a
priori $\tilde{\bm{\omega}}$ in the self-contained strategy, Eq. (4), however,
the error added by a marginally worse choice of $\tilde{\bm{\omega}}$ is
greatly overshadowed by white noise; in the event that Eq. (4) is problematic,
one could instead use a classical sensor as a substitute for
$\tilde{\bm{\omega}}$.

Figure 2: (a) Long-term stability analysis for the angular velocity
measurement, in which the accelerometers are subjected to a bias with a drift
rate of $6\times 10^{-7}$ ms-2/day. The accelerometers are subjected to a
constant rotation of $\bm{\omega}=\omega(1,1,1)/\sqrt{3}$ and are separated by
a distance of $l=1\;$m. The total Allan deviation
$\sigma=(\sigma_{x}^{2}+\sigma_{y}^{2}+\sigma_{z}^{2})^{1/2}$ is plotted,
where $\sigma_{x},\sigma_{y},\sigma_{z}$ are the Allan deviations
corresponding to their respective vector component of $\hat{\bm{\omega}}$. (b)
Error analysis of $\hat{\omega}_{x}$, for $l=1\;$m and
$||\bm{\omega}||=1\;$rad$\cdot$s-1. The white noise terms, $\bm{\eta}^{(i)}$,
are omitted to highlight the effects of the bias drift. When the array
undergoes constant rotation, the drift behaves similarly to a traditional
inertial sensor, however, when the array undergoes random motion, the
effective drift behaves as white noise which increases in severity with time.

As a consequence of the subtraction in Eq. (1), the severity of the bias
ultimately depends $\bm{u}^{(i)}-\bm{u}^{(j)}$. Hence, if two accelerometers
drift in the same direction, the quantity
$\hat{\bm{a}}^{(i)}-\hat{\bm{a}}^{(j)}$ will not induce any drift on the
gyroscopic measurements. On the other hand, the effective bias drift will be
largest when $\bm{u}^{(i)}=-\bm{u}^{(j)}$. For the particular layout we are
simulating, see Figure 1, the worst case scenario occurs when the bias of each
axial sensor is opposite to the central sensor, which is what we consider in
the simulation, Figure 2a.

Regardless of the values of $\omega$, we find that the bias becomes the
limiting factor in accuracy after approximately $T\approx 3\times 10^{4}\;$s
of interrogation. The efficacy of Eq. (5) is showcased in the simulation
results of Figure 2a, as increasing $\bm{\omega}$ by a factor of ten,
similarly improves the effective bias drift by a factor of ten. The most-
promising result concerning the simulation when $\omega=5.00\;$ rad$\cdot$s-1,
in which the corresponding bias drift exhibited is approximately $1.2\times
10^{-7}\;$ rad$\cdot$s-1/day. This value surpasses the stability limits of the
majority of modern atomic gyroscopes [34, 30, 35, 31] and is suitable for the
navigation of maritime vehicles [24]. For the chosen configuration of atomic
accelerometers with $l=1\;$m, the long-term stability of the AAA is not
suitable for analyzing rotational processes with small $|\bm{\omega}|$ over a
long period of time, such as seismology. Nevertheless, the effective long-term
stability of the AAA can be enhanced by increasing $l$ or by introducing
additional sensors. Moreover, the simulation performed assumed the worst case
scenario, in which the bias drifts of the radial sensors occur in the exact
opposite direction to the central sensor; on average one expects an
improvement by a factor of $\sqrt{2}$.

Evidently, when $||\bm{\omega}||$ is increased, then the effective bias drift
is reduced. One technique which could ensure this to be the case, is to
subject the AAA to a fixed rotation with angular velocity $\bm{\omega}_{0}$
such that $||\bm{\omega}_{0}||^{2}\gg||\hat{\bm{a}}^{(i)}-\bm{a}^{(i)}||/l$
for all $i$. If the unknown angular velocity is negligible in comparison, i.e.
$||\bm{\omega}_{0}+\bm{\omega}||\approx||\bm{\omega}_{0}||$, then
$||\bm{\omega}_{0}||$ will dictate the bias drift present within the
measurements. Evidently, one must be able to correctly set $\bm{\omega}_{0}$
with a a higher degree of precision compared to the desired precision of
$\hat{\omega}$, otherwise the sensitivity of $\hat{\omega}$ will retrogress.
Additionally, one must be mindful of the dynamic range of the accelerometers
[25], otherwise, the atomic accelerometers may become unusable due to complete
loss of contrast [72]. Hence, the choice of $\bm{\omega}_{0}$ will greatly
dependent on the technological specifications of the hardware and intended use
case of the AAA.

Notably, drifting accelerometers has a vastly different effect on
$\hat{\bm{\omega}}$ when the motion is not constant, as portrayed in Figure
2b. When $\bm{\omega}$ is kept fixed, the effective drift behaves as expected,
however, for random motion (see Figure 3) the effective drift behaves as white
noise which becomes progressively more severe as time increases.

## Resolution analysis

In this section we gauge how different technical challenges inherit to an AAA
influence its resolution, namely size limitations, positional uncertainties of
the individual sensors, and misalignment of the sensor heads. In all of the
simulations, the AAA is subjected to random motion, shown in Figure 3, where
$||\bm{\omega}||$ is kept fixed.

When simulating the effects of positional uncertainties and the misalignment,
we are considering these to be unknown errors which are static in time. In
reality, these values may drift over time leading to another source of bias
drift and ultimately effecting the long-term stability. However, if one
periodically measures these values, then any change can be properly accounted
for when calculating $\bm{\omega}$ (up to the precision of the measurement
device, which is what we deem as positional uncertainty and misalignment).

Figure 3: The simulated random values used for $\omega_{x}$ (red),
$\omega_{y}$ (blue) and $\omega_{z}$ (green), scaled such that
$||\bm{\omega}||=1$.

Figure 4: Resolution analysis with respect to different technological
challenges inherit to an AAA. In all three plots, the values are obtained by
averaging over 25 simulations, this is to consider different values of
$\bm{\eta}$, and different directions for the positional uncertainty in (b)
and axis of rotation in (c). In each of the 25 simulations, the average error
$||\hat{\bm{\omega}}-\bm{\omega}||$ is computed, where the total interrogation
time is $T=2\times 10^{3}\;$s and $\Delta t=0.25\;$s. Note that, the error
fluctuates minimally in time due to the method in which $\hat{\bm{\omega}}$ is
constructed. (a) Resolution analysis for different sizes of an AAA. (b)
Resolution analysis for different degrees of positional uncertainty (with
$l=1.0\;$m). (c) Resolution analysis for different degrees of misalignment
(with $l=1.0\;$m). The trends in (b) and (c) are similar because a positional
uncertainty will induce (with high probability) an angular misalignment; the
semblance in numerical values is due to the choice $l=1.0\;$m.

### Size Limitations

The first aspect we explore is how the size of the AAA affects the attainable
resolution. Here, the data is simply simulated using

$\hat{\bm{a}}^{(i)}=\bm{a}^{(i)}+\bm{\eta}^{(i)},$ (7)

in which different values of $l$ is considered.

Figure 4a clearly exhibits the hypothesized tendencies for the resolution of
the array of atomic accelerometers, Eq. (5), in which the error
$||\hat{\bm{\omega}}-\bm{\omega}||$ is computed for a variety of values for
$l$ and $||\bm{\omega}||$. Even for very large arrays with $l=2.0\;$m, the
attainable single-shot resolution of $\sim 10^{-5}\;$rad$\cdot$s-1 is greatly
outperformed by state of the atomic gyroscopes, whose single-shot resolution
is on the order of $\sim 10^{-8}-10^{-9}\;$rad$\cdot$s-1 [37]. The single-shot
resolution of the AAA can be improved upon by introducing additional
accelerometers, however, the most practical way of rivaling atomic
accelerometers is by hybridizing the array of accelerometers with classical
sensors [15, 73, 42, 74]; note that this does not defeat the purpose of an
AAA, as their most impactful contribution is long-term stability.

### Positional Uncertainty

The second aspect we explore is how positional uncertainties in the placement
of the individual atomic accelerometers affects the attainable resolution.
This positional uncertainty could be caused by fluctuations in the locations
of the atomic cloud within the atomic accelerometers. Here, the data is again
simulated using Eq. (7), with the addition that the position of the atomic
accelerometers, $\bm{r}^{(i)}$, are shifted by a distance $\delta r$ in the
direction of a random unit vector $\bm{u}^{(i)}$, hence
$\bm{r}^{(i)}\rightarrow\bm{r}^{(i)}+\delta r\bm{u}^{(i)}$.

We can deduce from Figure 4b that there is some threshold value for $\delta
r$, which when surpassed causes the resolution to deviate from the form
proposed in Eq. (5). For $l=1.0\;$m, this threshold is between $\delta
r=10^{-4}\;$m and $\delta r=10^{-5}\;$m. Moreover, the deviation is
exacerbated for larger values of $||\bm{\omega}||$. This threshold can be
overcome by utilizing high-precision calibration techniques, such as laser
interferometry, which can be used to measure distances on a $10^{-5}$ m scale.

### Misalignment

The third aspect we explore is how misalignment of the individual atomic
accelerometers affects the attainable resolution. The misalignment could be
caused by fluctuations in the angle of the retro-reflective mirrors present in
the atomic accelerometers, or simply mechanical error when positioning each
atomic accelerometer. Here, the data is simulated using

$\hat{\bm{a}}^{(i)}=M_{\theta}^{(i)}(\bm{a}^{(i)}+\bm{\eta}^{(i)}),$ (8)

where $M_{\theta}^{(i)}$ is a matrix which induces a rotation of an angle
$\theta$ about a random axis.

Similar to the positional uncertainty, $\delta r$, we can deduce from Figure
4c that there is some threshold value for $\theta$, which when surpassed
causes the resolution to deviate from the form proposed in Eq. (5). For
$l=1.0\;$m, this threshold is between $\theta=10^{-4}\;$rad and
$\theta=10^{-5}\;$rad. Fortunately, mechanical alignment can be measured with
extremely high-precision, and similarly retro-reflective mirrors are
calibrated on the order of $\sim 10^{-6}\;$rad [5] and can be stabilized with
extremely precise tilt calibration [75]. Hence, the cumulative misalignment
for each atomic accelerometer is not expected to surpass the aforementioned
threshold.

## Discussion

The long-term stability of atomic accelerometers has greatly improved in the
past few decades [18, 19] and currently outclass classical accelerometers.
Even though atomic gyroscopes have similarly improved, their long-term
stability has not yet surpassed that of classical gyroscopes [18, 16, 19]. In
this manuscript, we have demonstrated that an array consisting of four three-
axis atomic accelerometers can be an effective substitute for an atomic
gyroscope whose long-term stability surpasses the hypothesized stability limit
imposed on atomic gyroscopes [28, 29]. The long-term stability was simulated
using parameters describing a recently developed three-axis gyroscope [20],
which exhibits a non-negligible drift. However, as research and development
progresses, the long-term stability of these devices might improve; in this
event, the gyroscopic properties of an AAA would similarly negligible drift,
similar to strictly vertical atomic accelerometers [26, 6, 7, 15, 27].

The size of the proposed AAA is suitable for navigation application of ships
and submarines, however it may become an obstacle for smaller vehicles. The
separation distances proposed in the long-term stability analysis, Figure 2,
of $l=1\;$m alludes to a sensor which requires precision and care in reducing
the positional uncertainty and misalignment to a necessary level (Figure 4),
otherwise the error will begin to diverge once a factor surpasses some
threshold value. Fortunately, these thresholds can be surpassed with modern
technology [5, 75, 76, 75]. It should be noted that the requisite separation
of $l=1$m is due to the choice of using $N=4$ three-axis sensors; as per Eq.
(5), one can reduce $l$ by increasing $N$. This is ultimately restricted by
the size of the individual sensors, and although atomic sensors are quite
large, significant progress has been made on devising compact hardware in the
past few decades [77, 78, 7] \- it is not far-fetched to assume that progress
will continue in size reduction.

Unlike most classical inertial sensors, atomic sensors operate within a very
narrow dynamic range. For example, atomic accelerometers typically operate
within a bandwidth of $10^{-4}-10^{-3}$ms-2 [19, 22]. This is a necessity so
that the atoms properly interact with the laser pulses [19, 25, 64];
consequently there is a trade-off between resolution and dynamic range [19].
The limited bandwidth of an AAA is not an obstacle for some potential
applications, such as stabilization [52, 53, 54], seismology [55, 56, 57, 58],
and perhaps even navigation of slower moving vehicles. However, for general
navigational purposes, the AAA would be expanded to a hybrid system containing
classical sensors. In fact, atomic inertial sensors are often hybridized with
classical sensors [15, 73, 42, 74, 22] to improve sensitivity and reduce the
amount of dead time between measurements. The bias drifts inherit to classical
sensors are then filtered out through the means of statistical methods, such
as a Kalman filter [15, 73, 79, 22]. By incorporating sufficiently accurate
classical sensors, one could devise an adaptive and predictive scheme to
ensure the AAA is kept within a functional dynamic range [21, 22].

In this manuscript, we have established the theoretical foundations of using
an AAA to measure rotational velocity. This has many interesting applications,
such as navigation [45, 46, 47, 48, 49], stabilization [52, 53, 54] and
seismology [55, 56, 57, 58]. Nonetheless, it is crucial to test the validity
with an experimental demonstration. The simplest method would consist of two
three-axis atomic accelerometers, which could measure a single vector
component of $\bm{\omega}$, which will suffice as a proof-of-principal
demonstration.

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## Acknowledgements

This research is supported by the National Research Foundation, Singapore and
A*STAR under its Quantum Engineering Programme (NRF2021-QEP2-03-P01 and
NRF2021-QEP2-03-P06) and the DSO National Laboratories. The authors would also
like to thank Fong En Oon, Kai Sheng Lee, Elizaveta Maksimova and Christoph
Hufnagel for fruitful discussions.

## Data Availability

The raw data used in Figures 2-4 can be downloaded from the following Github
repository: https://github.com/GyroEmulator/Raw-Data.

## Conflict of interest

The authors have no conflicts to disclose.