# Open X-Embodiment: Robotic Learning Datasets and RT-X Models

Open X-Embodiment Collaboration0
Abhishek Padalkar6, Acorn Pooley7, Ajinkya Jain10, Alex Bewley7, Alex Herzog7,
Alex Irpan7, Alexander Khazatsky18,
Anant Rai14, Anikait Singh7,22, Anthony Brohan7, Antonin Raffin6, Ayzaan
Wahid7, Ben Burgess-Limerick15, Beomjoon Kim12,
Bernhard Schölkopf13, Brian Ichter7, Cewu Lu17,5, Charles Xu22, Chelsea
Finn7,18, Chenfeng Xu22, Cheng Chi3,18,
Chenguang Huang24, Christine Chan7, Chuer Pan18, Chuyuan Fu7, Coline Devin7,
Danny Driess7, Deepak Pathak2,
Dhruv Shah22, Dieter Büchler13, Dmitry Kalashnikov7, Dorsa Sadigh7, Edward
Johns9, Federico Ceola11, Fei Xia7,
Freek Stulp6, Gaoyue Zhou14, Gaurav S. Sukhatme26, Gautam Salhotra26,10, Ge
Yan23, Giulio Schiavi4, Gregory Kahn22,
Hao Su23, Hao-Shu Fang17, Haochen Shi18, Heni Ben Amor1, Henrik I
Christensen23, Hiroki Furuta21, Homer Walke22,
Hongjie Fang17, Igor Mordatch7, Ilija Radosavovic22, Isabel Leal7, Jacky
Liang7, Jad Abou-Chakra15, Jaehyung Kim12,
Jan Peters19, Jan Schneider13, Jasmine Hsu7, Jeannette Bohg18, Jeffrey
Bingham7, Jiajun Wu18, Jialin Wu8,
Jianlan Luo22, Jiayuan Gu23, Jie Tan7, Jihoon Oh21, Jitendra Malik22, Jonathan
Tompson7, Jonathan Yang18,
Joseph J. Lim12, João Silvério6, Junhyek Han12, Kanishka Rao7, Karl
Pertsch22,18, Karol Hausman7, Keegan Go10,
Keerthana Gopalakrishnan7, Ken Goldberg22, Kendra Byrne7, Kenneth Oslund7,
Kento Kawaharazuka21, Kevin Zhang2,
Krishan Rana15, Krishnan Srinivasan18, Lawrence Yunliang Chen22, Lerrel
Pinto14, Liam Tan22, Lionel Ott4, Lisa Lee7,
Masayoshi Tomizuka22, Maximilian Du18, Michael Ahn7, Mingtong Zhang25, Mingyu
Ding22, Mohan Kumar Srirama2, Mohit Sharma2,
Moo Jin Kim18, Naoaki Kanazawa21, Nicklas Hansen23, Nicolas Heess7, Nikhil J
Joshi7, Niko Suenderhauf15, Norman Di Palo9,
Nur Muhammad Mahi Shafiullah14, Oier Mees24, Oliver Kroemer2, Pannag R
Sanketi7, Paul Wohlhart7, Peng Xu7, Pierre Sermanet7,
Priya Sundaresan18, Quan Vuong7, Rafael Rafailov7,18, Ran Tian22, Ria Doshi22,
Roberto Martín-Martín20,
Russell Mendonca2, Rutav Shah20, Ryan Hoque22, Ryan Julian7, Samuel
Bustamante6, Sean Kirmani7, Sergey Levine7,22,
Sherry Moore7, Shikhar Bahl2, Shivin Dass26,20, Shubham Sonawani1, Shuran
Song3, Sichun Xu7, Siddhant Haldar14,
Simeon Adebola22, Simon Guist13, Soroush Nasiriany20, Stefan Schaal10, Stefan
Welker7, Stephen Tian18, Sudeep Dasari2,
Suneel Belkhale18, Takayuki Osa21, Tatsuya Harada21,16, Tatsuya Matsushima21,
Ted Xiao7, Tianhe Yu7, Tianli Ding7,
Todor Davchev7, Tony Z. Zhao18, Travis Armstrong7, Trevor Darrell22, Vidhi
Jain7,2, Vincent Vanhoucke7, Wei Zhan22,
Wenxuan Zhou7,2, Wolfram Burgard27, Xi Chen7, Xiaolong Wang23, Xinghao Zhu22,
Xuanlin Li23, Yao Lu7, Yevgen Chebotar7,
Yifan Zhou1, Yifeng Zhu20, Ying Xu7, Yixuan Wang25, Yonatan Bisk2, Yoonyoung
Cho12, Youngwoon Lee22, Yuchen Cui18,
Yueh-Hua Wu23, Yujin Tang7,21, Yuke Zhu20, Yunzhu Li25, Yusuke Iwasawa21,
Yutaka Matsuo21, Zhuo Xu7, Zichen Jeff Cui14 0Corresponding email: open-x-
embodiment@googlegroups.com;1Arizona State University; 2Carnegie Mellon
University; 3Columbia University; 4ETH Zürich; 5Flexiv Robotics; 6German
Aerospace Center; 7Google DeepMind; 8Google Research; 9Imperial College
London; 10Intrinsic LLC; 11Istituto Italiano di Tecnologia; 12Korea Advanced
Institute of Science & Technology; 13Max Planck Institute; 14New York
University; 15Queensland University of Technology; 16RIKEN; 17Shanghai Jiao
Tong University; 18Stanford University; 19Technische Universität Darmstadt;
20The University of Texas at Austin; 21The University of Tokyo; 22University
of California, Berkeley; 23University of California, San Diego; 24University
of Freiburg; 25University of Illinois Urbana-Champaign; 26University of
Southern California; 27University of Technology, Nuremberg;

###### Abstract

Large, high-capacity models trained on diverse datasets have shown remarkable
successes on efficiently tackling downstream applications. In domains from NLP
to Computer Vision, this has led to a consolidation of pretrained models, with
general pretrained backbones serving as a starting point for many
applications. Can such a consolidation happen in robotics? Conventionally,
robotic learning methods train a separate model for every application, every
robot, and even every environment. Can we instead train “generalist” X-robot
policy that can be adapted efficiently to new robots, tasks, and environments?
In this paper, we provide datasets in standardized data formats and models to
make it possible to explore this possibility in the context of robotic
manipulation, alongside experimental results that provide an example of
effective X-robot policies. We assemble a dataset from $22$ different robots
collected through a collaboration between $21$ institutions, demonstrating
$527$ skills ($160266$ tasks). We show that a high-capacity model trained on
this data, which we call RT-X, exhibits positive transfer and improves the
capabilities of multiple robots by leveraging experience from other platforms.
More details can be found on the project website robotics-
transformer-x.github.io.

##  I Introduction

A central lesson from recent advances in machine learning and artificial
intelligence is that large-scale learning from broad and diverse datasets can
enable capable AI systems by providing for general-purpose pretrained models.
In fact, large-scale general-purpose models typically trained on large and
diverse datasets can often outperform their _narrowly targeted_ counterparts
trained on smaller but more task-specific data. For instance, open-vocabulary
image classifiers (e.g., CLIP [1]) trained on large datasets scraped from the
web tend to outperform fixed-vocabulary models trained on more limited
datasets, and large language models [2, 3] trained on massive text corpora
tend to outperform systems that are only trained on narrow task-specific
datasets. Increasingly, the most effective way to tackle a given narrow task
(e.g., in vision or NLP) is to adapt a general-purpose model. However, these
lessons are difficult to apply in robotics: any single robotic domain might be
too narrow, and while computer vision and NLP can leverage large datasets
sourced from the web, comparably large and broad datasets for robotic
interaction are hard to come by. Even the largest data collection efforts
still end up with datasets that are a fraction of the size and diversity of
benchmark datasets in vision (5-18M) [4, 5] and NLP (1.5B-4.5B) [6, 7].
Perhaps more importantly, such datasets are often still narrow along some axes
of variation, either focusing on a single environment, a single set of
objects, or a narrow range of tasks. How can we overcome these challenges in
robotics and move the field of robotic learning toward the kind of large data
regime that has been so successful in other domains?

Inspired by the generalization made possible by pretraining large vision or
language models on diverse data, we take the perspective that the goal of
training generalizable robot policies requires X-embodiment training, i.e.,
with data from multiple robotic platforms. While each individual robotic
learning dataset might be too narrow, the union of all such datasets provides
a better coverage of variations in environments and robots. Learning
generalizable robot policies requires developing methods that can utilize
X-embodiment data, tapping into datasets from many labs, robots, and settings.
Even if such datasets in their current size and coverage are insufficient to
attain the impressive generalization results that have been demonstrated by
large language models, in the future, the union of such data can potentially
provide this kind of coverage. Because of this, we believe that enabling
research into X-embodiment robotic learning is critical at the present
juncture.

Following this rationale, our work has two primary goals: (1) Demonstrate that
policies trained on data from many different robots and environments enjoy the
benefits of positive transfer, attaining better performance than policies
trained only on data from each evaluation setup. (2) Provide datasets, data
formats and models for the robotics community to enable future research on
X-embodiment models.

We focus our work on robotic manipulation. Addressing goal (1), our empirical
contribution is to demonstrate that several recent robotic learning methods,
with minimal modification, can utilize X-embodiment data and enable positive
transfer. Specifically, we train the RT-1 [8] and RT-2 [9] models on $9$
different robotic manipulators. We show that the resulting models, which we
call RT-X, can improve over policies trained only on data from the evaluation
domain, exhibiting better generalization and new capabilities. Addressing (2),
we provide the Open X-Embodiment (OXE) Repository, which includes a dataset
with $22$ different robotic embodiments from $21$ different institutions that
can enable the robotics community to pursue further research on X-embodiment
models, along with open-source tools to facilitate such research. Our aim is
not to innovate in terms of the particular architectures and algorithms, but
rather to provide the model that we trained together with data and tools to
energize research around X-embodiment robotic learning.

Figure 0: The Open X-Embodiment Dataset. (a): the dataset consists of 60
individual datasets across $22$ embodiments. (b): the Franka robot has the
largest diversity in visually distinct scenes due to the large number of
Franka datasets, (c): xArm and Google Robot contribute the most number of
trajectories due to a few large datasets, (d, e): the dataset contains a great
diversity of skills and common objects.

##  II Related Work

Transfer across embodiments. A number of prior works have studied methods for
transfer across robot embodiments in simulation [10, 11, 12, 13, 14, 15, 16,
17, 18, 19, 20, 21, 22] and on real robots [23, 24, 25, 26, 27, 28, 29]. These
methods often introduce mechanisms specifically designed to address the
embodiment gap between different robots, such as shared action representations
[14, 30], incorporating representation learning objectives [17, 26], adapting
the learned policy on embodiment information [30, 31, 11, 18, 15], and
decoupling robot and environment representations [24]. Prior work has provided
initial demonstrations of X-embodiment training [27] and transfer [25, 32, 29]
with transformer models. We investigate complementary architectures and
provide complementary analyses, and, in particular, study the interaction
between X-embodiment transfer and web-scale pretraining. Similarly, methods
for transfer across human and robot embodiments also often employ techniques
for reducing the embodiment gap, i.e. by translating between domains or
learning transferable representations [33, 34, 35, 36, 37, 38, 39, 40, 41, 42,
43]. Alternatively, some works focus on sub-aspects of the problem such as
learning transferable reward functions [44, 17, 45, 46, 47, 48], goals [49,
50], dynamics models [51], or visual representations [52, 53, 54, 55, 56, 57,
58, 59] from human video data. Unlike most of these prior works, we directly
train a policy on X-embodiment data, without any mechanisms to reduce the
embodiment gap, and observe positive transfer by leveraging that data.

Large-scale robot learning datasets. The robot learning community has created
open-source robot learning datasets, spanning grasping [60, 61, 62, 63, 64,
65, 66, 67, 68, 69, 70, 71], pushing interactions [72, 73, 74, 23], sets of
objects and models [75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85], and
teleoperated demonstrations [86, 87, 88, 89, 90, 8, 91, 92, 93, 94, 95]. With
the exception of RoboNet [23], these datasets contain data of robots of the
same type, whereas we focus on data spanning multiple embodiments. The goal of
our data repository is complementary to these efforts: we process and
aggregate a large number of prior datasets into a single, standardized
repository, called Open X-Embodiment, which shows how robot learning datasets
can be shared in a meaningul and useful way.

Language-conditioned robot learning. Prior work has aimed to endow robots and
other agents with the ability to understand and follow language instructions
[96, 97, 98, 99, 100, 101], often by learning language-conditioned policies
[45, 102, 103, 104, 105, 40, 106, 8]. We train language-conditioned policies
via imitation learning like many of these prior works but do so using large-
scale multi-embodiment demonstration data. Following previous works that
leverage pre-trained language embeddings [107, 45, 108, 103, 40, 109, 110, 8,
111, 112] and pre-trained vision-language models [113, 114, 115, 9] in robotic
imitation learning, we study both forms of pre-training in our experiments,
specifically following the recipes of RT-1 [8] and RT-2 [9].

Figure 1:  RT-1-X and RT-2-X both take images and a text instruction as input
and output discretized end-effector actions. RT-1-X is an architecture
designed for robotics, with a FiLM [116] conditioned EfficientNet [117] and a
Transformer [118]. RT-2-X builds on a VLM backbone by representing actions as
another language, and training action text tokens together with vision-
language data.

##  III The Open X-Embodiment Repository

We introduce the Open X-Embodiment Repository (robotics-
transformer-x.github.io) – an open-source repository which includes large-
scale data along with pre-trained model checkpoints for X-embodied robot
learning research. More specifically, we provide and maintain the following
open-source resources to the broader community:

  * •

Open X-Embodiment Dataset: robot learning dataset with _1M+ robot
trajectories_ from _$22$ robot embodiments_.

  * •

Pre-Trained Checkpoints: a selection of RT-X model checkpoints ready for
inference and finetuning.

We intend for these resources to form a foundation for X-embodiment research
in robot learning, but they are just the start. Open X-Embodiment is a
community-driven effort, currently involving $21$ institutions from around the
world, and we hope to further broaden participation and grow the initial Open
X-Embodiment Dataset over time. In this section, we summarize the dataset and
X-embodiment learning framework, before discussing the specific models we use
to evaluate our dataset and our experimental results.

###  III-A The Open X-Embodiment Dataset

The Open X-Embodiment Dataset contains 1M+ real robot trajectories spanning
$22$ robot embodiments, from single robot arms to bi-manual robots and
quadrupeds. The dataset was constructed by pooling 60  _existing_ robot
datasets from $34$ robotic research labs around the world and converting them
into a consistent data format for easy download and usage. We use the RLDS
data format [119], which saves data in serialized tfrecord files and
accommodates the various action spaces and input modalities of different robot
setups, such as differing numbers of RGB cameras, depth cameras and point
clouds. It also supports efficient, parallelized data loading in all major
deep learning frameworks. For more details about the data storage format and a
breakdown of all 60 datasets, see robotics-transformer-x.github.io.

###  III-B Dataset Analysis

Fig. analyzes the Open X-Embodiment Dataset. Fig. (a) shows the breakdown of
datasets by robot embodiments, with the Franka robot being the most common.
This is reflected in the number of distinct scenes (based on dataset metadata)
per embodiment (Fig. (b)), where Franka dominates. Fig. (c) shows the
breakdown of trajectories per embodiment. To further analyze the diversity, we
use the language annotations present in our data. We use the PaLM language
model [3] to extract objects and behaviors from the instructions. Fig. (d,e)
show the diversity of skills and objects. While most skills belong to the
pick-place family, the long tail of the dataset contains skills like “wiping”
or “assembling”. Additionally, the data covers a range of household objects,
from appliances to food items and utensils.

##  IV RT-X Design

To evaluate how much X-embodiment training can improve the performance of
learned policies on individual robots, we require models that have sufficient
capacity to productively make use of such large and heterogeneous datasets. To
that end, our experiments will build on two recently proposed Transformer-
based robotic policies: RT-1 [8] and RT-2 [9]. We briefly summarize the design
of these models in this section, and discuss how we adapted them to the
X-embodiment setting in our experiments.

###  IV-A Data format consolidation

One challenge of creating X-embodiment models is that observation and action
spaces vary significantly across robots. We use a coarsely aligned action and
observation space across datasets. The model receives a history of recent
images and language instructions as observations and predicts a 7-dimensional
action vector controlling the end-effector ($x$, $y$, $z$, roll, pitch, yaw,
and gripper opening or the rates of these quantities). We select one canonical
camera view from each dataset as the input image, resize it to a common
resolution and convert the original action set into a 7 DoF end-effector
action. We normalize each dataset’s actions prior to discretization. This way,
an output of the model can be interpreted (de-normalized) differently
depending on the embodiment used. It should be noted that despite this coarse
alignment, the camera observations still vary substantially across datasets,
e.g. due to differing camera poses relative to the robot or differing camera
properties, see Figure 1. Similarly, for the action space, we do not align the
coordinate frames across datasets in which the end-effector is controlled, and
allow action values to represent either absolute or relative positions or
velocities, as per the original control scheme chosen for each robot. Thus,
the same action vector may induce very different motions for different robots.

###  IV-B Policy architectures

We consider two model architectures in our experiments: (1) RT-1 [8], an
efficient Transformer-based architecture designed for robotic control, and (2)
RT-2 [9] a large vision-language model co-fine-tuned to output robot actions
as natural language tokens. Both models take in a visual input and natural
language instruction describing the task, and output a tokenized action. For
each model, the action is tokenized into 256 bins uniformly distributed along
each of eight dimensions; one dimension for terminating the episode and seven
dimensions for end-effector movement. Although both architectures are
described in detail in their original papers [8, 9], we provide a short
summary of each below:

RT-1 [8] is a 35M parameter network built on a Transformer architecture [118]
and designed for robotic control, as shown in Fig. 1. It takes in a history of
$15$ images along with the natural language. Each image is processed through
an ImageNet-pretrained EfficientNet [117] and the natural language instruction
is transformed into a USE [120] embedding. The visual and language
representations are then interwoven via FiLM [116] layers, producing 81
vision-language tokens. These tokens are fed into a decoder-only Transformer,
which outputs the tokenized actions.

Figure 2: RT-1-X mean success rate is $50\%$ higher than that of either the
Original Method or RT-1. RT-1 and RT-1-X have the same network architecture.
Therefore the performance increase can be attributed to co-training on the
robotics data mixture. The lab logos indicate the physical location of real
robot evaluation, and the robot pictures indicate the embodiment used for the
evaluation.

RT-2 [9] is a family of large vision-language-action models (VLAs) trained on
Internet-scale vision and language data along with robotic control data. RT-2
casts the tokenized actions to text tokens, e.g., a possible action may be “1
128 91 241 5 101 127”. As such, any pretrained vision-language model (VLM
[121, 122, 123]) can be finetuned for robotic control, thus leveraging the
backbone of VLMs and transferring some of their generalization properties. In
this work, we focus on the RT-2-PaLI-X variant [121] built on a backbone of a
visual model, ViT [124], and a language model, UL2 [125], and pretrained
primarily on the WebLI [121] dataset.

###  IV-C Training and inference details

Both models use a standard categorical cross-entropy objective over their
output space (discrete buckets for RT-1 and all possible language tokens for
RT-2).

We define the robotics data mixture used across all of the experiments as the
data from $9$ manipulators, and taken from RT-1 [8], QT-Opt [66], Bridge [95],
Task Agnostic Robot Play [126, 127], Jaco Play [128], Cable Routing [129],
RoboTurk [86], NYU VINN [130], Austin VIOLA [131], Berkeley Autolab UR5 [132],
TOTO [133] and Language Table [91] datasets. RT-1-X is trained on only
robotics mixture data defined above, whereas RT-2-X is trained via co-fine-
tuning (similarly to the original RT-2 [9]), with an approximately one to one
split of the original VLM data and the robotics data mixture. Note that the
robotics data mixture used in our experiments includes $9$ embodiments which
is fewer than the entire Open X-Embodiment dataset ($22$) – the practical
reason for this difference is that we have continued to extend the dataset
over time, and at the time of the experiments, the dataset above represented
all of the data. In the future, we plan to continue training policies on the
extended versions of the dataset as well as continue to grow the dataset
together with the robot learning community.

At inference time, each model is run at the rate required for the robot (3-10
Hz), with RT-1 run locally and RT-2 hosted on a cloud service and queried over
the network.

Evaluation Setting | Bridge | Bridge | RT-1 paper 6 skills
---|---|---|---
Evaluation Location | IRIS (Stanford) | RAIL Lab (UCB) | Google Robotic Lab
Robot Embodiment | WidowX | WidowX | Google Robot
Original Method |  LCBC [95] |  LCBC [95] | -
Original Method | $13\%$ | $13\%$ | -
RT-1 | $40\%$ | 30% | 92%
RT-1-X | $27\%$ | $27\%$ | $73\%$
RT-2-X (55B) | 50% | 30% | 91%
TABLE I:  Parameter count scaling experiment to assess the impact of capacity
on absorbing large-scale diverse embodiment data. For these large-scale
datasets (Bridge and RT-1 paper data), RT-1-X underfits and performs worse
than the Original Method and RT-1. RT-2-X model with significantly many more
parameters can obtain strong performance in these two evaluation scenarios.

##  V Experimental Results

Our experiments answer three questions about the effect of X-embodiment
training: (1) Can policies trained on our X-embodiment dataset effectively
enable positive transfer, such that co-training on data collected on multiple
robots improves performance on the training task? (2) Does co-training models
on data from multiple platforms and tasks improve generalization to new,
unseen tasks? (3) What is the influence of different design dimensions, such
as model size, model architecture or dataset composition, on performance and
generalization capabilities of the resulting policy? To answer these questions
we conduct the total number of 3600 evaluation trials across 6 different
robots.

###  V-A In-distribution performance across different embodiments

To assess the ability of our RT-X model variants to learn from X-embodiment
data, we evaluate their performance on in-distribution tasks. We split our
evaluation into two types of use cases: evaluation on domains that only have
small-scale datasets (Fig. 2), where we would expect transfer from larger
datasets to significantly improve performance, and evaluation on domains that
have large-scale datasets (Table I), where we expect further improvement to be
more challenging. Note that we use the same robotics data _training_ mixture
(defined in Sec. IV-C) for all the evaluations presented in this section. For
small-scale dataset experiments, we consider Kitchen Manipulation [128], Cable
Routing [129], NYU Door Opening [130], AUTOLab UR5 [132], and Robot Play
[134]. We use the same evaluation and robot embodiment as in the respective
publications. For large-scale dataset experiments, we consider Bridge [95] and
RT-1 [8] for in-distribution evaluation and use their respective robots:
WidowX and Google Robot.

For each small dataset domain, we compare the performance of the RT-1-X model,
and for each large dataset we consider both the RT-1-X and RT-2-X models. For
all experiments, the models are co-trained on the full X-embodiment dataset.
Throughout this evaluation we compare with two baseline models: (1) The model
developed by the creators of the dataset trained only on that respective
dataset. This constitutes a reasonable baseline insofar as it can be expected
that the model has been optimized to work well with the associated data; we
refer to this baseline model as the _Original Method_ model. (2) An RT-1 model
trained on the dataset in isolation; this baseline allows us to assess whether
the RT-X model architectures have enough capacity to represent policies for
multiple different robot platforms simultaneously, and whether co-training on
multi-embodiment data leads to higher performance.

Small-scale dataset domains (Fig. 2). RT-1-X outperforms Original Method
trained on each of the robot-specific datasets on 4 of the 5 datasets, with a
large average improvement, demonstrating domains with limited data benefit
substantially from co-training on X-embodiment data.

Row | Model | Size | History Length | Dataset | Co-Trained w/ Web | Initial Checkpoint | Emergent Skills Evaluation | RT-2 Generalization Evaluation
---|---|---|---|---|---|---|---|---
(1) | RT-2 | 55B | none | Google Robot action | Yes | Web-pretrained | $27.3\%$ | 62%
(2) | RT-2-X | 55B | none | Robotics data | Yes | Web-pretrained | 75.8% | 61%
(3) | RT-2-X | 55B | none | Robotics data except Bridge | Yes | Web-pretrained | $42.8\%$ | $54\%$
(4) | RT-2-X | 5B | 2 | Robotics data | Yes | Web-pretrained | $44.4\%$ | $52\%$
(5) | RT-2-X | 5B | none | Robotics data | Yes | Web-pretrained | $14.5\%$ | $30\%$
(6) | RT-2-X | 5B | 2 | Robotics data | No | From scratch | $0\%$ | $1\%$
(7) | RT-2-X | 5B | 2 | Robotics data | No | Web-pretrained | $48.7\%$ | $47\%$
TABLE II: Ablations to show the impact of design decisions on generalization
(to unseen objects, backgrounds, and environments) and emergent skills (skills
from other datasets on the Google Robot), showing the importance of Web-
pretraining, model size, and history.

Large-scale dataset domains (Table I). In the large-dataset setting, the
RT-1-X model does not outperform the RT-1 baseline trained on only the
embodiment-specific dataset, which indicates underfitting for that model
class. However, the larger RT-2-X model outperforms both the Original Method
and RT-1 suggesting that X-robot training can improve performance in the data-
rich domains, but only when utilizing a sufficiently high-capacity
architecture.

###  V-B Improved generalization to out-of-distribution settings

We now examine how X-embodiment training can enable better generalization to
out-of-distribution settings and more complex and novel instructions. These
experiments focus on the high-data domains, and use the RT-2-X model.

Unseen objects, backgrounds and environments. We first conduct the same
evaluation of generalization properties as proposed in [9], testing for the
ability to manipulate unseen objects in unseen environments and against unseen
backgrounds. We find that RT-2 and RT-2-X perform roughly on par (Table II,
rows (1) and (2), last column). This is not unexpected, since RT-2 already
generalizes well (see [9]) along these dimensions due to its VLM backbone.

Emergent skills evaluation. To investigate the transfer of knowledge across
robots, we conduct experiments with the Google Robot, assessing the
performance on tasks like the ones shown in Fig. 3. These tasks involve
objects and skills that are not present in the RT-2 dataset but occur in the
Bridge dataset [95] for a different robot (the WidowX robot). Results are
shown in Table II, Emergent Skills Evaluation column. Comparing rows (1) and
(2), we find that RT-2-X outperforms RT-2 by $\sim 3\times$, suggesting that
incorporating data from other robots into the training improves the range of
tasks that can be performed even by a robot that already has large amounts of
data available. Our results suggest that co-training with data from other
platforms imbues the RT-2-X controller with additional skills for the platform
that are not present in that platform’s original dataset.

Our next ablation involves removing the Bridge dataset from RT-2-X training:
Row (3) shows the results for RT-2-X that includes all data used for RT-2-X
except the Bridge dataset. This variation significantly reduces performance on
the hold-out tasks, suggesting that transfer from the WidowX data may indeed
be responsible for the additional skills that can be performed by RT-2-X with
the Google Robot.

###  V-C Design decisions

Lastly, we perform ablations to measure the influence of different design
decisions on the generalization capabilities of our most performant RT-2-X
model, which are presented in Table II. We note that including a short history
of images significantly improves generalization performance (row (4) vs row
(5)). Similarly to the conclusions in the RT-2 paper [9], Web-based pre-
training of the model is critical to achieving a high performance for the
large models (row (4) vs row (6)). We also note that the $55B$ model has
significantly higher success rate in the Emergent Skills compared to the $5B$
model (row (2) vs row (4)), demonstrating that higher model capacity enables
higher degree of transfer across robotic datasets. Contrary to previous RT-2
findings, co-fine-tuning and fine-tuning have similar performance in both the
Emergent Skills and Generalization Evaluation (row (4) vs row (7)), which we
attribute to the fact that the robotics data used in RT-2-X is much more
diverse than the previously used robotics datasets.

##  VI Discussion, Future Work, and Open Problems

We presented a consolidated dataset that combines data from $22$ robotic
embodiments collected through a collaboration between $21$ institutions,
demonstrating $527$ skills ($160266$ tasks). We also presented an experimental
demonstration that Transformer-based policies trained on this data can exhibit
significant positive transfer between the different robots in the dataset. Our
results showed that the RT-1-X policy has a $50\%$ higher success rate than
the original, state-of-the-art methods contributed by different collaborating
institutions, while the bigger vision-language-model-based version (RT-2-X)
demonstrated $\sim 3\times$ generalization improvements over a model trained
only on data from the evaluation embodiment. In addition, we provided multiple
resources for the robotics community to explore the X-embodiment robot
learning research, including: the unified X-robot and X-institution dataset,
sample code showing how to use the data, and the RT-1-X model to serve as a
foundation for future exploration.

Figure 3: To assess transfer _between_ embodiments, we evaluate the RT-2-X
model on out-of-distribution skills. These skills are in the Bridge dataset,
but not in the Google Robot dataset (the embodiment they are evaluated on).

While RT-X demonstrates a step towards a X-embodied robot generalist, there
are many more steps needed to make this future a reality. Our experiments have
a number of limitations: it does not consider robots with very different
sensing and actuation modalities, it does not study generalization to new
robots, and it does not provide a decision criterion for when positive
transfer does or does not happen. Studying these questions is an important
direction for future work. We hope that this work will serve not only as an
example that X-robot learning is feasible and practical, but also provide the
tools to advance research in this direction in the future.

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# Modeling of 2D self-drifting flame-balls in Hele-Shaw cells

Jorge Yanez
Institute for Thermal Energy Technology and Safety
Karlsruhe Institute of Technology
Hermann-von-Helmholtz-Platz, 1
Karlsruhe, Germany.
<EMAIL_ADDRESS>
&Leonid Kagan
School of Mathematical Sciences,
Tel Aviv University
Tel Aviv, Israel.
<EMAIL_ADDRESS>
Mike Kuznetsov
Institute for Thermal Energy Technology and Safety
Karlsruhe Institute of Technology
Hermann-von-Helmholtz-Platz, 1
Karlsruhe, Germany.
<EMAIL_ADDRESS>&Gregory Sivashinsky
School of Mathematical Sciences,
Tel Aviv University
Tel Aviv, Israel.
<EMAIL_ADDRESS>
Corresponding author

# On disintegration of lean hydrogen flames in narrow gaps

Jorge Yanez
Institute for Thermal Energy Technology and Safety
Karlsruhe Institute of Technology
Hermann-von-Helmholtz-Platz, 1
Karlsruhe, Germany.
<EMAIL_ADDRESS>
&Leonid Kagan
School of Mathematical Sciences,
Tel Aviv University
Tel Aviv, Israel.
<EMAIL_ADDRESS>
Mike Kuznetsov
Institute for Thermal Energy Technology and Safety
Karlsruhe Institute of Technology
Hermann-von-Helmholtz-Platz, 1
Karlsruhe, Germany.
<EMAIL_ADDRESS>&Gregory Sivashinsky
School of Mathematical Sciences,
Tel Aviv University
Tel Aviv, Israel.
<EMAIL_ADDRESS>
Corresponding author

###### Abstract

The disintegration of near limit flames propagating through the gap of Hele-
Shaw cells has recently become a subject of active research. In this paper,
the flamelets resulting from the disintegration of the continuous front are
interpreted in terms of the Zeldovich flame-balls stabilized by volumetric
heat losses. A complicated free-boundary problem for 2D self-drifting near
circular flamelets is reduced to a 1D model. The 1D formulation is then
utilized to obtain the locus of the flamelet velocity, size, heat losses and
Lewis numbers at which the self-drifting flamelets may exist.

_Keywords_ Hydrogen flames $\cdot$ Diffusive-thermal instability $\cdot$
Flame-balls

##  1 Introduction

Hydrogen flames have gained increasing attention over the last years as a
clean and efficient energy solution [1]. Apart from their technological
relevance, hydrogen flames are remarkably rich dynamically. Because of the
high diffusion coefficient of molecular hydrogen, lean hydrogen-air flames are
known to experience diffusive-thermal instability [2,3] manifesting themselves
in the formation of a cellular structure in a state of chaotic self-motion.
Moreover, as has recently been discovered by Veiga-López et al. [4], lean
hydrogen-air flames evolving in narrow gaps of Hele-Shaw chambers, and where
heat-loss effects become important, continuous cellular flames may
disintegrate forming self-drifting cup-like flamelets leaving tree-like or
sprout-like traces (Fig. 1).

(a)

(b)

(c)

(d)

(e)

(f)

Figure 1: Hydrogen-air flamelets propagating through narrow gaps: (a) $d=5mm$,
$\%H_{2}=5.0$; (b) $d=3mm$, $\%H_{2}=7.5$; (c) $d=4mm$, $\%H_{2}=7.0$; (d)
$d=5mm$, $\%H_{2}=6.0$; (e) $d=2mm$, $\%H_{2}=9.75$; (f) $d=2mm$,
$\%H_{2}=9.75$. Regimes: (a)(b) one-headed finger, (c)(d)(e)(f) one-headed
branching.

These unexpected propagation modes extend flammability limits beyond those of
the planar flames. This is particularly important for the safety studies of
hydrogen-powered devices [5] as it implies that hydrogen flames may propagate
in gaps much narrower than initially anticipated.

##  2 Reaction-diffusion model

Some of the above features of the Hele-Shaw flames may be successfully
captured by an ultra-simple constant-density, buoyancy-free, reaction-
diffusion model [6]. In suitably scaled variables and parameters it reads,

$\displaystyle\dfrac{\partial T}{\partial t}=\dfrac{\partial^{2}T}{\partial
x^{2}}+\dfrac{\partial^{2}T}{\partial y^{2}}+(1-\sigma)\Omega-q(T-\sigma),$
(1) $\displaystyle\dfrac{\partial C}{\partial
t}=\dfrac{1}{Le}\left(\dfrac{\partial^{2}C}{\partial
x^{2}}+\dfrac{\partial^{2}C}{\partial y^{2}}\right)-\Omega,$ (2)
$\displaystyle\Omega=\dfrac{1}{2Le}(1-\sigma)^{2}N^{2}C\exp{\left[N\left(1-\dfrac{1}{T}\right)\right]},$
(3)

where $T=$ temperature in units of $T_{b}$, the adiabatic temperature of
combustion products; $C=$ concentration of the deficient reactant in units of
$C_{0}$, its value in the fresh mixture; $x,y,t=$ spatio-temporal coordinates
in units of $\ell_{th}=D_{th}/U_{b}$ and $\ell_{th}/U_{b}$, respectively;
$\ell_{th}=$ thermal width of the flame; $U_{b}=$ speed of a planar adiabatic
flame; $D_{th}=$ thermal diffusivity of the mixture, $\sigma=T_{0}/T_{b}$,
where $T_{0}=$ temperature sustained at the walls of the Hele-Shaw cell;
$Le=D_{th}/D_{mol}$ is the Lewis number; $D_{mol}=$ molecular diffusivity of
the deficient reactant; $N=T_{a}/T_{b}=$ scaled activation energy; $T_{a}=$
activation temperature; $\Omega=$ appropriately normalized reaction rate to
ensure that at large $N$ the scaled speed of the planar adiabatic flame is
close to unity; $q=$ scaled heat loss intensity specified as
$(\pi\ell_{th}/d)^{2}$; $d=$ width of the Hele-shaw cell. The adopted
expression for $q$ stems from the 1D heat equation, $T_{t}=T_{zz}$, considered
over the Hele-Shaw gap and subjected to isothermal boundary conditions,
$T(z=\pm d/2\ell_{th},t)=\sigma$. Hence,
$T-\sigma\sim\exp{\left[-\left(\pi\ell_{th}/d\right)^{2}t\right]}\cos{\left[\left(\pi\ell_{th}/d\right)z\right]}$.
The exponential rate of the temperature decay is then extrapolated over the
quasi-2D formulation of (1)-(2) in the $(x,y)$ plane.

Figures 2 and 3 display the results of numerical simulations of the effect of
heat losses on the breaking up of the flame in 2D. The problem has also been
explored by Martínez-Ruiz et al. [7] and Dejoan [8] through large-scale
numerical simulations of a model accounting both for the gas thermal
expansion, buoyancy and momentum loss.

(a)

(b)

Figure 2: Reaction rate $\Omega=0.4$ at consecutive equispaced instants for
one-headed flamelets under heat losses. Simulation of the reactive-diffusive
system (1)-(3) for $N=10$; $Le=0.25$; $\sigma=0.2$; $q=0.14$, $0<t<740$ (a);
$q=0.20$, $0<t<1000$ (b); $0<x,y<480$. Initial conditions,
$T(x,y,0)=\sigma+(1-\sigma)\exp{[-(x^{2}+y^{2})/\ell^{2}]}$, $\ell=0.25$,
$C(x,y,0)=1$, and adiabatic boundary conditions.

(a)

(b)

Figure 3: Distribution of the deficient reactant concentration, $C$, at
$t=740$ (a) and $t=1000$ (b). For conditions see the legend for Fig.2. Lighter
shading corresponds to lower concentration.

Near the quenching point, $q=0.22$, higher heat losses lead to a smaller drift
velocity and flamelet size (Fig. 2), which is in line with experimental
findings (Figs. 4, 5). Note that $q=0.22$ significantly exceeds the quenching
point $q=0.018$ for the planar flame, other conditions being identical
(Fig.6).

(a)

(b)

Figure 4: Drift velocity for downward (DW) and upward (UW) propagation. See
also Kuznetsov et al. [9 Fig. 7]. The flame-ball regime corresponds to
ultralean flames ($\%{H_{2}}<20$).  Figure 5: Flamelet diameter vs. gap size
for one-headed flamelets. Figure 6: Drift velocity $V$ vs. heat loss intensity
$q$ for a planar flame.

##  3 Reduction to a free-boundary problem

The fingering patterns of Figs.1-3 are traces of self-drifting near circular
reactive spots. They may be regarded as a 2D version of familiar self-drifting
3D flame-balls [10, 11] (see also comment ii) of Sec. 5). This observation in
turn suggests the possible existence of individual self-drifting 2D flame-
balls described by time-independent reaction-diffusion equations,

$\displaystyle V\dfrac{\partial
T}{\partial\eta}=\dfrac{\partial^{2}T}{\partial\xi^{2}}+\dfrac{\partial^{2}T}{\partial\eta^{2}}+(1-\sigma)\Omega-q(T-\sigma),$
(4) $\displaystyle V\dfrac{\partial
C}{\partial\eta}=\frac{1}{Le}\left(\dfrac{\partial^{2}C}{\partial\xi^{2}}+\dfrac{\partial^{2}C}{\partial\eta^{2}}\right)+(1-\sigma)\Omega,$
(5)

subjected to boundary conditions,

$T(\xi^{2}+\eta^{2}\to\infty)=\sigma,\;C(\xi^{2}+\eta^{2}\to\infty)=1.$ (6)

Here $\xi=x$, $\eta=y+Vt$, and $V$ is an eigen-value of the problem. For
theoretical analysis it is technically advantageous to employ the familiar
large-activation-energy $(N\gg 1)$ near-equidiffusive $(Le^{-1}-1\ll 1)$
approach where the reaction-diffusion system (4) and (5) is reduced to a free-
boundary problem [12]. There, the reaction rate $\Omega$ transforms into a
localized source distributed along the interface $\eta=R(\theta)\cos{\theta}$,
$\xi=R(\theta)\sin{\theta}$. Eqs. (4) (5) translate into the familiar set of
equations for the reduced temperature $\Theta$ and excess-enthalpy $S$,

$\displaystyle
V\dfrac{\partial\Theta^{(0)}}{\partial\eta}=\dfrac{\partial^{2}\Theta^{(0)}}{\partial\xi^{2}}+\dfrac{\partial^{2}\Theta^{(0)}}{\partial\eta^{2}},$
(7) $\displaystyle V\dfrac{\partial
S}{\partial\eta}=\dfrac{\partial^{2}\left(S-\alpha\Theta^{(0)}\right)}{\partial\xi^{2}}+\dfrac{\partial^{2}\left(S-\alpha\Theta^{(0)}\right)}{\partial\eta^{2}}-\nu\Theta^{(0)}$
(8)

Here $\Theta=(T-\sigma)/(1-\sigma)$; $S=(\Theta+C-1)\beta$; the Zeldovich
number $\beta=(1-\sigma)N$, is assumed to be large;
$\alpha=\left(Le^{-1}-1\right)\beta$, $\nu=q\beta$ are assumed to be finite;
$\Theta^{(0)}=\Theta(\beta\to\infty)$.

At the reactive interface, $\xi^{2}+\eta^{2}=R^{2}(\theta)$, the following
conditions are held,

$\displaystyle\left[S\right]^{+}_{-}=0,\;\Theta^{(0)}=1,$ (9)
$\displaystyle\left[\dfrac{\partial S}{\partial
n}\right]^{+}_{-}=\alpha\left[\dfrac{\partial\Theta^{(0)}}{\partial
n}\right]^{+}_{-},$ (10)
$\displaystyle\left[\dfrac{\partial\Theta^{(0)}}{\partial
n}\right]^{+}_{-}=-\left(1+\dfrac{1}{R^{2}}\left(\dfrac{\partial
R}{\partial\theta}\right)^{2}\right)^{1/2}\exp{\left(\dfrac{S}{2}\right)}.$
(11)

Here $\partial/\partial n$ is the normal derivative. And,

$\displaystyle\Theta^{(0)}\to 0,\;S\to 0,$ (12)

at $\xi^{2}+\eta^{2}\to\infty$. Within the flame-ball interface the deficient
reactant is assumed to be fully consumed,

$C\left(\xi^{2}+\eta^{2}\leq R^{2}(\theta)\right)=0.$ (13)

Finally we observe that Eq. (7) allows for the exact solution which in polar
coordinates ($\eta=r\cos{\theta},\xi=r\sin{\theta}$) reads, [13, 14],

$\displaystyle\Theta^{(0)}(r,\theta)=\sum_{m}\exp{\left(kr\cos{\theta}\right)}\cos{\left(m\theta\right)}\cdot$
$\displaystyle\cdot\left[A_{m}K_{m}(kr)+B_{m}I_{m}(kr)\right]+H.$ (14)

Here $K_{m}$, $I_{m}$ are modified Bessel functions of first and second kind,
$k=V/2$, and $A_{m}$, $B_{m}$, $H$ are parameters to be determined by
conditions at $r=0$, $r=\infty$ and $r=R(\theta)$.

##  4 Reduction to a 1D model

###  4.1 General

The 2D free-boundary problem (7) - (13) is still too difficult for a
straightforward analytical treatment. We therefore propose to consider its
low-mode allocation-like reduction to a 1D model that we believe will keep
enough contact with the original 2D formulation. More specifically, in the 1D
approach $\Theta,C,S$ are projected on the $\eta$ \- axis running through the
center of the flame ball (Fig. 7).

Figure 7: Up: Top sketch view of the flamelet in a plane parallel both to the
paper and to the plates of the Hele-Shaw cell. Down: Profiles of temperature,
$T$, and concentration, $C$, through the center of the flame-ball.

There the interior of the flame-ball covers the interval $-R<\eta<R$. In the
expansion (14) we will keep only the leading order terms corresponding to
$m=0$. Moreover, we will set $\theta=0$ for $\eta>0$ and $\theta=\pi$ for
$\eta<0$. Conditions (9) - (13) on the flame-ball interface $\eta=\pm R$ and
$\eta=\pm\infty$ thus become,

$\displaystyle\left[S\right]^{+}_{-}=0,\;\Theta^{(0)}=1,$ (15)
$\displaystyle\left[\dfrac{\partial\Theta^{(0)}}{\partial\eta}\right]^{+}_{-}+\exp{\left(\dfrac{S}{2}\right)}=0,$
(16) $\displaystyle\left[\dfrac{\partial
S}{\partial\eta}\right]^{+}_{-}=\alpha\left[\dfrac{\partial\Theta^{(0)}}{\partial\eta}\right]^{+}_{-},$
(17)
$\displaystyle\Theta^{(0)}\left(\eta=\pm\infty\right)=0,\;S\left(\eta=\pm\infty\right)=0.$
(18)

And

$C\left(-R<\eta<R\right)=0.$ (19)

For the leading order approximation, $\beta\gg 1$,

$\displaystyle\Theta^{(0)}\left(\eta>R\right)=\exp{\left[k(\eta-R)\right]}\dfrac{K_{0}(k\eta)}{K_{0}(kR)},$
(20)
$\displaystyle\Theta^{(0)}\left(\eta<-R\right)=\exp{\left[k(\eta+R)\right]}\dfrac{K_{0}(-k\eta)}{K_{0}(kR)},$
(21) $\displaystyle\Theta^{(0)}\left(-R<\eta<R\right)=1,$ (22) $\displaystyle
C^{(0)}\left(\eta\right)=1-\Theta^{(0)}\left(\eta\right).$ (23)

For the higher order uniform approximation needed for evaluation of
$S=(\Theta+C-1)\beta$,

$\displaystyle\Theta\left(\eta>R\right)=\left[1+\beta^{-1}S(R)\right]\cdot$
$\displaystyle\cdot\exp{\left[k(\eta-R)\right]}\dfrac{K_{0}\left((k^{2}+q)^{\frac{1}{2}}\eta\right)}{K_{0}\left((k^{2}+q)^{\frac{1}{2}}R\right)},$
(24)
$\displaystyle\Theta\left(\eta<-R\right)=\left[1+\beta^{-1}S(-R)\right]\cdot$
$\displaystyle\cdot\exp{\left[k(\eta+R)\right]}\dfrac{K_{0}\left(-(k^{2}+q)^{\frac{1}{2}}\eta\right)}{K_{0}\left((k^{2}+q)^{\frac{1}{2}}R\right)},$
(25)
$\displaystyle\Theta\left(-R<\eta<R\right)=1+\beta^{-1}S\left(-R<\eta<R\right),$
(26) $\displaystyle S\left(-R<\eta<R\right)=A+B\exp{(k\eta)}I_{0}(k|\eta|)$
$\displaystyle\mspace{145.0mu}-\nu\eta/2k,$ (27)

where,

$\displaystyle A=\left[S(R)+S(-R)\right]$
$\displaystyle-\dfrac{1}{2}BI_{0}(kR)\left[\exp{(kR)}+\exp{(-kR)}\right]$ (28)
$\displaystyle B=\dfrac{\nu
R+S(R)-S(-R)}{kI_{0}(kR)\left[\exp{(kR)}-\exp{(-kR)}\right]},$ (29)
$\displaystyle
C(\eta>R)=1-\exp{\left[kLe(\eta-R)\right]}\dfrac{K_{0}(kLe\eta)}{K_{0}(kLeR)},$
(30) $\displaystyle
C(\eta<\\!-R)=1-\exp{\left[kLe(\eta+R)\right]}\dfrac{K_{0}(-kLe\eta)}{K_{0}(kLeR)},$
(31) $\displaystyle C(-R<\eta<R)=0.$ (32)

The uniform approximation employed in the above equations allows meeting the
boundary conditions $\Theta(\eta\to\infty)=0$, $S(\eta\to\infty)=0$, behind
the self-drifting flamelet, where $\eta\sim\beta$.

###  4.2 Results

Eqs. (20) - (32) meet the continuity conditions (15) at $\eta=\pm R$.
Substituting Eqs. (20) - (32) into jump conditions (16), (17) one ends up with
four algebraic relations for four unknown parameters $V=2k$, $R$, $S(R)$,
$S(-R)$ as functions of $\alpha$, $\nu$, $\beta$ (Figs. 9, 9, 10). More
technical details and the numerical procedure employed for the emerging
algebraic problem are presented in the earlier version of the paper [15].

Figure 8: Locus of the flamelet radii, $R$, as a function of the heat losses,
$\nu$, for different Lewis number parameters, $\alpha$.

Figure 9: Locus of the velocities, $V$, as a function of the heat losses,
$\nu$, for different Lewis number parameters, $\alpha$.

Figure 10: Locus of the flamelet radii, $R$, as a function of velocity, $V$,
for different Lewis number parameters, $\alpha$.

One can draw several conclusions from the results obtained.

  1. (a)

Parameter $\alpha$ strongly affects $V(\nu)$, $R(\nu)$ \- dependencies. The
heat loss intensity at the quenching (turning) point, $\nu_{q}$, increases
with the decrease of the Lewis number, $Le\simeq 1-\alpha/\beta$.

  2. (b)

At $\nu<\nu_{q}$ the multiplicity of $V(\nu)$, $R(\nu)$ is observed. A similar
non-uniqueness is known to occur in planar ($R=\infty$) non-adiabatic flames
which leads to the $\alpha$-independent relation between $V$ and $\nu$,
$V^{2}\ln{\left(1/V\right)}=\nu$, [16].

  3. (c)

Once the solutions for $V(\nu)$, $R(\nu)$ for a certain $\alpha$ are
available, one can plot spatial profiles of all state variables involved. For
brevity we depict here only $\Theta(\eta)$ for the upper and lower branches of
$V(\eta)$ (Figs. 12, 12). A significant drop of temperature $\Theta$ at the
rear side ($\eta=R$) of the self-drifting flame-ball might explain the
horseshoe shape of the advancing flamelet. The latter is often perceived as a
local extinction (opening ) of the front.

Figure 11: Profiles of $\Theta$ for $\alpha=1$, $\beta=10$ and different
values of $\nu$ for the upper branch of $V(\nu)$.

Figure 12: Profiles of $\Theta$ for $\alpha=1$, $\beta=10$ and different
values of $\nu$ for the lower branch of $V(\nu)$.

##  5 Concluding remarks

  1. (i)

In constructing a workable approximate solution we truncated the series
involving an infinite number of terms, as in Eq. (14), setting $m$ at zero, to
deal with sums involving only a finite number of terms.

  2. (ii)

There is an important distinction between the 3D and effectively 2D situation
typical of large aspect ratio Hele-Shaw cells. The 3D case allows for
stationary spherical flame-balls which may bifurcate into a self-drifting mode
[10, 11]. In the 2D case, the non-drifting circular flame-balls are ruled out.
They cannot meet boundary conditions at infinity (due to the logarithmic tail
of the associated concentration profiles [7]). So, in narrow gaps 2D self-
drifting flame-balls should emerge not as a bifurcation but rather as the only
way for the 2D flame-balls to exist.

  3. (iii)

In the present study the pattern of the flame disintegration is controlled by
the heat loss intensity $q=(\pi\ell_{th}/d)^{2}$, which in turn may be
controlled by the width of the Hele-Shaw gap ($d$) or the mixture composition
(% $H_{2}$ or $\ell_{th}$) [4, 9, 17].

  4. (iv)

The reaction-diffusion model of Sec. 2 does not seem capable of reproducing
two-headed flamelets. The latter is often observed experimentally [4, 18] as
well as in simulations of more sophisticated models that consider the burned
gas thermal expansion [7, 19] and are therefore susceptible to Darrieus-Landau
instability. At the moment the issue of two-headed flamelets remains
unexplained.

  5. (v)

The present work is devoted to estimates of the propagation velocity of an
individual flame-ball in a flat channel. It would be interesting to extend the
analysis over the collective propagation of a group of flame-balls appearing
in Figs. 1-3, and where the flamelets are competing for common fuel and mutual
heating of one another. A mean-field type of approach as the developed by
D’Angelo and Joulin [20] or Williams and Grcar [21] seems particularly
promising.

  6. (vi)

In the present study flame-balls emerge in gaseous premixtures as an extreme
case of diffusive-thermal instability where invariably $Le<1$. A similar
pattern is observed in smoldering burning of thin solid sheets with and
without imposed air flows [22-28]. There, similar to the gaseous systems, the
effective Lewis number is considerably below unity [28]. It would be
interesting to extend the 1D approach of this paper also to the smoldering
problem.

Declaration of competing interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

Acknowledgements

This research was supported in part by the US-Israel Binational Science
Foundation (Grant 2020-005).

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