# Evaluating Robustness of Visual Representations for Object Assembly Task
Requiring Spatio-Geometrical Reasoning

Chahyon Ku1, Carl Winge1, Ryan Diaz1, Wentao Yuan2 and Karthik Desingh1
1Minnesota Robotics Institute (MnRI) and Department of Computer Science and
Engineering (CS&E), University of Minnesota, Minneapolis, MN 55414 US.
<EMAIL_ADDRESS>D Allen School of Computer
Science and Engineering, University of Washington, Seattle, WA 98195 US.
<EMAIL_ADDRESS>
###### Abstract

This paper primarily focuses on evaluating and benchmarking the robustness of
visual representations in the context of object assembly tasks. Specifically,
it investigates the alignment and insertion of objects with geometrical
extrusions and intrusions, commonly referred to as a peg-in-hole task. The
accuracy required to detect and orient the peg and the hole geometry in SE(3)
space for successful assembly poses significant challenges. Addressing this,
we employ a general framework in visuomotor policy learning that utilizes
visual pretraining models as vision encoders. Our study investigates the
robustness of this framework when applied to a dual-arm manipulation setup,
specifically to the grasp variations. Our quantitative analysis shows that
existing pretrained models fail to capture the essential visual features
necessary for this task. However, a visual encoder trained from scratch
consistently outperforms the frozen pretrained models. Moreover, we discuss
rotation representations and associated loss functions that substantially
improve policy learning. We present a novel task scenario designed to evaluate
the progress in visuomotor policy learning, with a specific focus on improving
the robustness of intricate assembly tasks that require both geometrical and
spatial reasoning. Videos, additional experiments, dataset, and code are
available at https://bit.ly/geometric-peg-in-hole.

##  I INTRODUCTION

Peg-in-hole assembly has been a longstanding and actively researched problem
within the field of robotics as an essential component of industrial robotics
that require reasoning over geometries, affordances, and contact models of
objects. Prior works have predominantly focused on single-arm top-down peg-in-
hole tasks, where the hole is attached to a fixture, while a robot arm picks
and inserts the peg in top-down action space [1, 2, 3, 4, 5, 6, 7, 8]. While
these approaches effectively reduce the complexity of the task to focus on
contact modelling and sub-millimeter-precision for industrial use cases, they
make assumptions unfit for robot manipulation in more unstructured
environments. For robots to adequately manipulate household items with parts
that fit together (e.g. containers and lids), we consider the following
requirements to be essential. To manipulate a diverse set of containers and
lids the robot will encounter, the robot should be able to look at and reason
about the intra-category variations of geometry (food containers vs. water
bottles) as well as the objects’ spatial (what is the relative transformation
between parts?) and geometric (what parts fit together in what way?)
relationship. To operate in unstructured indoor environments, the robot may
not assume objects are fixed to a flat surface but rather pick up both
relevant parts and assemble them using two arms in full SE(3) action space.
The robot also may not have multiple external cameras giving a near-complete
third-person-views of the task, the robot, and the objects.

Based on these requirements, we propose a novel geometric-peg-in-hole task,
inspired by traditional peg-in-hole but adapted for object variations and
unstructured environments, that evaluates the spatio-geometric reasoning
capabilities of visuo-motor policies. Our pegs and holes have geometric
variations in the extrusions such as plus-sign, minus-sign, or pentagon, which
require the robot to generalize over these intra-category variations without
prior knowledge, as well as the different amounts of rotational variation
required to put them together (see Fig. 2 with rotational symmetries). The
task starts out with the peg and hole in each gripper, but with randomized
grasp noise to test the policy’s ability to generalize over uncertainties
during grasp estimation and execution (see Fig. LABEL:fig:teaser center). As
both objects are held and controlled by the arms, the robot should not only
reason about the current and goal SE(3) transformations between the objects,
but also make use of all axes of SE(3) action space to insert the objects. On
top of the simulation environment created using PyBullet [9], we also design a
repeatable task setup in the real world, where the objects are always picked
up and put down in the same position and orientation on a fixture, such that
data collection and policy evaluation can be done without human intervention
(see supplementary video and Fig. LABEL:fig:teaser right).

Then, how can we train a robot to complete such a task? End-to-end learning
from demonstrations offers a flexible framework for training robust robots
without the need to explicitly model object and task representations. Taking
wisdom from the recent success of pre-trained representations in natural
language processing and computer vision, recent works such as R3M [10] and MVP
[11] propose robotics-specific visual representations trained from large-scale
datasets of ego-centric videos. To assess the geometric reasoning capability
of visual representations, we conduct a performance comparison. This
comparison involves evaluating the performance of from-scratch trained
baselines against various pre-trained representations in a image-to-action
imitation learning setup.

In summary, our main contributions are:

  * •

A novel dual-arm geometric-peg-in-hole task designed to evaluate spatio-
geometric reasoning capabilities of visual representations on 9 pairs of peg
and hole objects with randomized grasp in rotation and translation of all
axes.

  * •

A comprehensive analysis of eight visual representations, including two from-
scratch encoders and six pre-trained models, is conducted to train policies
within an imitation learning framework. This assessment quantitatively
measures their success in handling increasingly challenging task (grasp)
variations with two different imitation learning methods.

  * •

An evaluation of control representations (absolute vs. delta) and rotation
representations (quaternion vs. Gram-Schmidt) in the context of our task
learning.

  * •

A simulation environment, dataset, trained models, and evaluation code for
others to benchmark and extend upon this task.

##  II RELATED WORKS

###  II-A Imitation Learning for Assembly Tasks

End-to-end learning has demonstrated that robots can learn to control their
actuators directly from raw sensor observations [12, 13, 14, 15], avoiding the
need for a brittle modular pipeline with explicit state estimation.
Traditionally, imitation learning works such as Form2Fit [16] and Transporter
[17] formulated assembly tasks as a supervised classification task of
predicting SE(2) pick-and-place locations to train high-performing and data-
efficient imitation learning policies. These methods require a tabletop setup
with a clear camera-view of the objects, stable insertion targets, and 2
dimensional action spaces. More recently, Robomimic [18] proposed an imitation
learning benchmark with SE(3) action space, and trained imitation learning
policies for assembly-like tasks such as Tool Hang (insert hook onto base
frame, hang tool on hook) and Square (place square nut on a rod). While these
tasks featured SE(3) action space and rotation of grasped objects, they
focused on a single set of objects without geometric variation and on single-
arm top-down insertion. In this work, we propose a dual-arm SE(3) insertion
task that evaluates the robot’s ability to generalize over object variations
and reason about SE(3) transformations between the parts.

###  II-B Pretraining for Robotics

To avoid retraining the neural network for every task, more recent works have
leveraged state-of-the-art visual pretrained models such as ResNet [19]
pretrained on ImageNet [20], CLIP [21], MAE [22], and R3M [10] to obtain
visual features to train a policy network. CLIP [21] achieves state-of-the-art
results in tasks like ImageNet classification by performing contrastive
pretraining on a large-scale non-public image-caption dataset. R3M [10]
utilizes time contrastive loss and video-language alignment to pretrain a
ResNet [19] on egocentric videos sourced from the Ego4D dataset [23]. MAE [22]
achieves state-of-the-art transfer results in image classification by
pretraining visual transformers [24] on the ImageNet data. The pretraining
process involves using masked input reconstruction, also known as masked
autoencoding, as the supervision signal. MVP [11] utilizes masked autoencoding
from MAE to pretrain vision transformers [24] on egocentric videos sourced
from a combination of datasets, such as Epic Kitchens [23]. In this work, we
compare Non-pretrained ResNet-18 and ResNet-50 models with the above mentioned
pretrained models except for MVP (does not provide ViT/B or ResNet-50 variants
others commonly have).

##  III IMITATION LEARNING FRAMEWORK

Figure 1: Network architecture used in the imitation learning pipeline. The
network receives 3 images as input, produces image embeddings for each image
using a vision encoder of choice (pretrained or non-pretrained), concatenates
them all with robot proprioception, and outputs the next action through an MLP
head.

The goal of imitation learning is to train a policy
$\pi:\mathcal{O}\rightarrow\mathcal{A}$ that maps all observations to an
action that will progress the robot towards executing the task. Our dataset
$\mathcal{D}=\\{(O_{i},a_{i})\\}_{i=1,...N}$ consists of $N$ observation-
action pairs. Each observation $O_{i}=((I_{v})_{v\in V},s)$ is a tuple of RGB
images $I_{v}$ from view $v\in V$ and the current state of the robot
$s\in\mathcal{S}$. Each action $a_{i}\in\mathcal{A}$ is the action the expert
performs during demonstration.

Our architecture shown in Fig. 1 is inspired by imitation learning evaluation
frameworks from Robomimic [18], R3M [10], and MVP [11]. In our implementation,
the policy $\pi_{\theta}$ consists of three parts: the image preprocessor $f$,
the image encoder $g_{\theta}$, and the policy head $h_{\phi}$. The image
preprocessor $f:\mathcal{I}^{640\times 480}\rightarrow\mathcal{I}^{224\times
224}$ crops and resizes the original RGB image to a consistant size for fair
comparison of all vision encoder models (ViT-B/16 requires this size). The
image encoder $g_{\theta}:\mathcal{I}^{224\times
224}\rightarrow\mathbb{R}^{D}$ is a neural network that deterministically maps
an RGB image $I_{v}$ to a $D$-dimensional image embedding $e_{v}$. The policy
head
$h_{\phi}:\mathbb{R}^{D}\times...\times\mathbb{R}^{D}\times\mathcal{S}\rightarrow\mathcal{A}$
is a multi-layer perceptron on top of concatenated image embeddings
$(e_{v})_{v\in V}$ and robot state $s$ which produces the final output action
$a$. In summary,
$\pi_{\theta}(O)=h_{\phi}(g_{\theta}(f(I_{v_{1}})),...,g_{\phi}(f(I_{v_{|V|}})),s)=\hat{a}$.
We train either the parameters $\phi$ (frozen $g$) or both $\theta$ and $\phi$
(unfrozen $g$) by back-propagating the mean squared error loss
$\mathcal{L}=MSE(a,\hat{a})$.

We choose $\mathcal{S}\subseteq SE(3)\times SE(3)$ and $\mathcal{A}\subseteq
SE(3)\times SE(3)$ to be absolute gripper poses of both arms. For each arm,
there are 3 values representing xyz position and 6 values representing the
first two columns of the rotation matrix [25], so we represent both as a 18
dimensional vector. The reason for this choice is explained in Sec. V.

##  IV TASK SETUP

###  IV-A Objects

We generate 3D models of nine peg and hole object pairs using Blender [26]
(Fig. 2). Each object pair has one object (the “peg”) with an extrusion of a
specific geometrical shape, with the other object (the “hole”) having an
equivalently-shaped hole. The objects are categorized based on their
rotational symmetries (i.e. of the orders 1, 2, and 4). When rotating the
object by {$0\degree,90\degree,180\degree,270\degree$}, order 1 objects have 4
unique orientations (see Fig. 2, top row), order 2 objects have 2 unique
orientations, and order 4 objects have 1 unique orientation. The objects used
in the real-world environment are 3D printed directly from the generated 3D
models.

Figure 2: 9 peg and hole pairs with various shapes, such as arrow, key, U,
pentagon, minus-sign, hexagon, diamond, plus-sign, and circle, grouped into
categories with rotational symmetries of orders 1, 2, and 4.

###  IV-B Simulation Setup

The PyBullet [9] simulation environment (see Fig. 3) is used for data
collection and imitation learning evaluation. It consists of the calibrated
clone of our real world setup including two 6-DoF robot arms (Universal Robots
UR5e), with two parallel jaw grippers (2F-85 Robotiq grippers) and one top-
down view RGB camera (Intel RealSense LiDAR Camera L515 in the real-world).
Additionally, two wrist RGB cameras are added in the simulation setup to
experiment if multiple views help with the policy learning.

Figure 3: Task execution sequence in the simulation setup with top-view and
left and right wrist-views. Figure 4: Possible variations in grasping the
objects.

Task Initialization: In the beginning of the episode, peg and hole are
randomly held in the gripper based on the specific task variation. As shown in
Fig. 4, the in-hand pose of the cap and bottle are randomized from the base
pose in 4 ways: XT is the noise in the X-axis Translation of the gripper,
uniformly sampled from [-0.01m, 0.01m]. ZT is the noise in the Z-axis
Translation of the gripper, uniformly sampled from [-0.01m, 0.01m]. YR is the
noise in the Y-axis Rotation of the gripper, uniformly sampled from [-11.25∘,
11.25∘]. ZR is the random Z-axis Rotation of the gripper, uniformly sampled
from {0∘, 90∘, 180∘, 270∘}. Note that ZR is not noise but rather a discrete
set of rotations that requires the model to conduct geometric reasoning,
whereas XT, ZT, and YR are noisy perturbations.

Scripted Expert Demonstrations: The task starts in the “show” state (see Fig.
3): a fixed joint configuration that points both grippers towards the camera.
With the objects in hand, both arms simultaneously move towards the “align”
pose (see Fig. 3), where the objects are perfectly aligned with each other.
Afterwards, the arms make a simple linear cartesian move towards the “insert”
pose (see Fig. 3), where the objects are put together.

Demonstration Statistics: Roughly 44% of the demonstrations have order 1
objects, 33% have order 2 objects, and 22% have order 4 objects due to the
distribution of possible shapes in each category (see Fig. 2). Each
demonstration contains a minimum of 10 frames and a maximum of 40 frames. The
task is considered successful if the relative transform of the peg and the
hole are apart within a margin of 1cm in translation and $5^{\circ}$ in
orientation from the expected ground-truth relative transform of the peg and
the hole. While this is a relaxation of the precision required for industrial
peg-in-hole tasks, it still poses a significant challenges to vision-based
systems, as SOTA algorithms such as GDR-Net [27] are evaluated on 2cm accuracy
for tabletop object pose estimation.

###  IV-C Real World Setup

The real environment (as seen in Fig. LABEL:fig:teaser right) is identical to
the simulation environment with a few additions to allow for automatic data
collection and evaluation. Since objects cannot simply be spawned into the
real-life grippers like in simulation, we create a picking phase before the
episode and a placing phase after the episode. After the blocks are picked,
the robot moves to a predefined joint state to show the blocks to the camera.
From there, the robot runs either the scripted demonstration or rollout of the
trained policy as in the simulation. After the execution, the system always
places the peg and hole in a fixed pose such that the extrusion and intrusion
are face-down from the top-view inside 2 3D-printed fixtures to ensure precise
placement. Thus, the system can always assume that the peg and hole are picked
up from the same pose and can vary the gripper-object transformation as needed
for each task variation. For more details on the real world setup, refer to
the supplementary video.

##  V EXPERIMENTS

###  V-A Experimental Setup

Task Variations: We compare performance on the four grasp variations (XT, ZT,
YR, ZR) shown in Fig. 4 and their combinations XTZR, ZTZR, YRZR, and XZTYZR
(denoting all variations). The ZR variation requires the model to perform
geometric-spatial reasoning to succeed, whereas the other variations require
the model to be robust to perturbations in the grasping. Hence, complex
variation compositions involving ZR requires the model to do geometric-spatial
reasoning to perform the assembly task successfully.

Model Variations: We compare the performance of 2 randomly initialized image
encoders and 6 pretrained frozen image encoders (3 ResNet [19] models and 3
vision transformer [24] models). Non-pretrained ResNet-18 and ResNet-50 are
randomly initialized and jointly trained with the policy head. ImageNet
ResNet-50, R3M ResNet-50, and CLIP ResNet-50 are frozen ResNet-50s initialized
with ImageNet [20], R3M [10], and CLIP [21] weights. ImageNet ViT-B/16, CLIP
ViT-B/16, and MAE ViT-B/16 are frozen ViT-B/16s initialized with ImageNet,
CLIP, and MAE [22] weights.

We use ResNet-50 and ViT-B/16 variants of the pretrained models, because they
are the only versions available for all above-mentioned pretraining schemes
and have been used for imitation learning in previous works [10, 11].

Object Set Variations: As shown in Fig. 2, we have objects with rotational
symmetries of Order-1, Order-2, and Order-4. Order-all denotes the union of
Order-1, Order-2, and Order-4, which is used for training all models.

###  V-B Simulation Results

In this section, we will compare the performance of models trained and tested
with simulation. Unless otherwise indicated, we use Non-pretrained ResNet-18.
Note: All models in the simulation experiments are trained with datasets
containing all 9 objects from Order-1, Order-2, and Order-4 to test the
models’ capability to generalize over geometric variations.

Comparison of Visual Representations: All visual representation, pretrained or
non-pretrained, are mostly successful in task variants with just grasp noise
perturbations, with success rates of 0.900 or above. However, we observe that
the non-pretrained models perform best on ZR that requires geometric-spatial
reasoning, with average 0.800 success rates, compared to the best performing
pretrained models (CLIP) with average 0.600 success rate (TABLE I).

| XT | ZT | YR | ZR
---|---|---|---|---
Non-pretrained ResNet-18 | 1.000 | 1.000 | 1.000 | 0.775
Non-pretrained ResNet-50 | 1.000 | 1.000 | 1.000 | 0.825
ImageNet ResNet-50 | 1.000 | 1.000 | 0.925 | 0.425
R3M ResNet-50 | 0.950 | 1.000 | 1.000 | 0.275
CLIP ResNet-50 | 1.000 | 1.000 | 0.975 | 0.625
ImageNet ViT-base | 0.950 | 1.000 | 0.975 | 0.450
CLIP ViT-base | 1.000 | 1.000 | 0.900 | 0.575
MAE ViT-base | 1.000 | 1.000 | 0.925 | 0.350
TABLE I: Success rates of all visual representations trained with 100
demonstrations of indicated task variation. Non-pretrained ResNets clearly
outperform pretrained models on ZR.

The performance on combinations of grasp variations (XTZR, ZTZR, and YRZR) is
much worse when trained using the same amount of data (100 episodes). Success
rates of the models range from 0.400 to 0.600 for Non-pretrained ResNet-18,
the best performing model. We hypothesize that the amount of data (100
episodes) is not sufficient enough to capture the features to successfully
complete the trajectory, and confirm that increasing the amount of
demonstrations drastically improves performance from 0.38 average success rate
to 0.65 and 0.70 average success rate (TABLE II).

| XTZR | ZTZR | YRZR | XZTYZR
---|---|---|---|---
Non-pretrained ResNet-18 | 0.825 | 0.825 | 0.675 | 0.275
Non-pretrained ResNet-50 | 0.425 | 0.775 | 0.300 | 0.075
ImageNet ResNet-50 | 0.225 | 0.225 | 0.175 | 0.050
R3M ResNet-50 | 0.150 | 0.275 | 0.05 | 0.050
CLIP ResNet-50 | 0.500 | 0.575 | 0.250 | 0.150
ImageNet ViT-base | 0.150 | 0.300 | 0.225 | 0.025
CLIP ViT-base | 0.300 | 0.250 | 0.200 | 0.050
MAE ViT-base | 0.375 | 0.25 | 0.175 | 0.050
TABLE II: Success rates of all visual representations trained with 1000
demonstrations of indicated task variation using all objects.

Comparison of Task and Object Variations: We provide a more fine-grained
evaluation of our best model (Non-pretrained ResNet-18) by running 3 different
evaluation runs of 40 randomized episodes and reporting the mean and standard
deviation over the 3 runs (TABLE III). “All” denotes the average of all
evaluation runs above (total 360 episodes). These models are trained on 1000
demonstrations, which includes all shapes, with proprioception and 3 views
(top + 2 wrist cameras).

Objects | XTZR | ZTZR | YZR | XZTYZR
---|---|---|---|---
circle | 0.85$\pm$0.07 | 1.00$\pm$0.00 | 0.83$\pm$0.05 | 0.43$\pm$0.07
plus | 0.93$\pm$0.04 | 1.00$\pm$0.00 | 0.77$\pm$0.01 | 0.38$\pm$0.04
minus | 0.80$\pm$0.03 | 0.98$\pm$0.00 | 0.44$\pm$0.10 | 0.33$\pm$0.10
diamond | 0.77$\pm$0.06 | 1.00$\pm$0.00 | 0.33$\pm$0.08 | 0.34$\pm$0.08
hexagon | 0.71$\pm$0.08 | 1.00$\pm$0.00 | 0.38$\pm$0.08 | 0.30$\pm$0.08
u | 0.37$\pm$0.08 | 0.54$\pm$0.06 | 0.12$\pm$0.06 | 0.17$\pm$0.03
pentagon | 0.34$\pm$0.10 | 0.56$\pm$0.04 | 0.10$\pm$0.07 | 0.18$\pm$0.07
arrow | 0.38$\pm$0.07 | 0.66$\pm$0.08 | 0.18$\pm$0.03 | 0.17$\pm$0.05
key | 0.38$\pm$0.07 | 0.66$\pm$0.08 | 0.17$\pm$0.04 | 0.19$\pm$0.05
all | 0.61$\pm$0.02 | 0.82$\pm$0.03 | 0.37$\pm$0.05 | 0.28$\pm$0.03
TABLE III: Success rates of Non-pretrained ResNet-18 trained on 1000
demonstrations including all objects. Mean and standard deviations over 3
different evaluations of 40 randomized rollouts.

We observe that the success rates are mostly consistent when evaluated on a
completely new set of 40 randomized episodes, with a standard deviation
falling around 0.05. We observe that while the success rate of rotation groups
(Order-4: circle and plus, Order-2: minus, diamond, and hexagon, Order-1: u,
pentagon, and arrow) decline consistently with increased degrees of symmetry.
However, there is no clear trend in the order of performance for objects
inside each group.

Comparison of Imitation Learning Methods: We compare our original setup, BC-
MLP with various image encoders, against BC-RNN [1], a popular SOTA BC
baseline that is shown to improve performance over BC-MLP. Instead of a MLP of
layer sizes [1024, 1024, 18] on top of the image encoder, BC-RNN has two LSTM
layers of hidden sizes [1024, 1024] and output size 18 that fuses image
embeddings and proprioception from 10 most recent time frames as implemented
in the original paper [1]. The tabulated results show success rates evaluated
over 40 rollouts. These models are all trained on 1000 demonstrations with
proprioception and 3 views (top + 2 wrist cameras).

Models | XTZR | ZTZR | YRZR | XZTYZR
---|---|---|---|---
ResNet-18 MLP | 0.825 | 0.825 | 0.675 | 0.275
ResNet-18 RNN | 0.525 | 0.300 | 0.350 | 0.075
CLIP ResNet-50 MLP | 0.500 | 0.575 | 0.250 | 0.150
CLIP ResNet-50 RNN | 0.300 | 0.325 | 0.150 | 0.200
CLIP ViT-base MLP | 0.300 | 0.250 | 0.200 | 0.050
CLIP ViT-base RNN | 0.150 | 0.350 | 0.025 | 0.025
TABLE IV: Success rates of Non-pretrained ResNet-18, CLIP ResNet-50, and CLIP ViT-Base with MLP vs LSTM action decoders, trained on 1000 demonstrations including all objects. | Shape | XT | ZT | YR | ZR | XTZR | ZTZR | YRZR | XZTYZR
---|---|---|---|---|---|---|---|---|---
| plus | 0.9 | 1.0 | 0.7 | 1.0 | 0.6 | 0.7 | 0.2 | 0.3
One model per shape | minus | 1.0 | 1.0 | 1.0 | 0.7 | 0.4 | 0.5 | 0.3 | 0.0
| key | 1.0 | 1.0 | 0.5 | 0.0 | 0.0 | 0.3 | 0.0 | 0.1
| plus | 0.3 | 0.2 | 0.2 | 0.9 | 0.2 | 0.5 | 0.1 | 0.0
One model for all shapes | minus | 0.5 | 0.2 | 0.1 | 0.5 | 0.2 | 0.6 | 0.0 | 0.0
| key | 0.3 | 0.0 | 0.2 | 0.3 | 0.1 | 0.3 | 0.1 | 0.0
TABLE V: Real world results for one model per shape vs. one model for all shapes, trained from 10 demonstrations for each object Rotation-Loss | XTZR | ZTZR | YRZR | XZTYZR
---|---|---|---|---
Quaternion - MSE | 0.275 | 0.250 | 0.000 | 0.000
Quaternion - Frobenius | 0.225 | 0.375 | 0.100 | 0.000
6D - Frobenius | 0.575 | 0.625 | 0.200 | 0.125
6D - MSE | 0.825 | 0.825 | 0.675 | 0.275
TABLE VI: Success rates of the Non-pretrained ResNet-18 using a combination of
a certain rotation representation (either as a quaternion or as a 6d rotation
matrix completed by Gram-Schmidt orthonormalization).

Contrary to our expectations, we observe that the performance is worse for BC-
RNN on all tasks and models we ran these experiments on: Non-pretrained
ResNet-18, CLIP ResNet-50, and CLIP ViT-base. We hypothesize that BC-RNN
cannot use the history as well as it performed in Robomimic [18], because BC-
RNN performs better on longer tasks while our tasks are relatively short. To
be more specific, the 5 tasks in the RoboMimic have average episode lengths of
48 (Lift), 116 (Can), 151 (Square), 469 (Transport), and 480 (Tool Hang),
where the performance of BC-RNN improved for only the longer three tasks. In
comparison to their longer tasks, our tasks average around 10 to 40 frames.
While ALOHA [28], another recent work also proposes improvements in temporal
smoothing and consistency through use of transformers, we do not conduct
experiment with their setup assuming insignificant improvements from the same
reasons.

Comparison of Action Representation and Rotation Regression Targets: The
choice of action and rotation representations was crucial for learning the
policy as described in TABLE VI. First, we observed that position control of
the end-effector poses was the best action representation for the task, as
velocity control was too sensitive to the step size of the action and was
completely unable to correctly model trajectories that change sharply in the
velocity space. Second, we observed that using 6D representations for 3D
rotations proposed in [25], which regresses the first 2 columns of the
rotation matrix and performs Gram-Schmidt orthonormalization to produce the
full rotation matrix, greatly outperformed the standard quaternion
representation.

###  V-C Real World Results

We conduct real world experiments to verify if (1) our observations from the
simulation experiments are consistent with the real world (2) our setup can
generalize with fewer demonstrations on the real robot. We experiment with a
setup that is more similar to our simulation setup by training a single model
to generalize over all the shapes. We calculate the success rates by counting
“smooth insertions,” which does not include forcing the objects together such
that objects slip into each other even when they are not aligned. In other
words, we do not count unintended insertions that did not properly align the
objects towards success. Additionally, we also experiment with a setup that
has one model per shape.

We observe that one model for all shapes (TABLE V bottom) perform worse than
models trained and evaluated in simulation, because of fewer number of
demonstrations in the real world. In this setup, we were not able to draw
conclusive observations on the exact trend of performance, due to high
variance in success rates likely induced from limited number of train
demonstrations (30 vs. 100/1000/10000) and evaluation runs (10 vs. 40). It is
also worth noting that the real world setup does not have wrist cameras,
adding to the performance drop compared to simulated experiments. In our
additional experiments where we train one model per shape (TABLE V top), we
notice considerable improvements in success rates, suggesting that more
demonstrations are needed to generalize over multiple shapes. However, the
general trend of performance is consistent with our simulation setup: (a)
objects with more degrees of symmetry perform better (b) tasks with simpler
variations perform better than combinations (e.g. ZR performs better than
XTZR, ZTZR, YRZR, and XZTYZR for all shapes except the key).

###  V-D Additional Details & Analysis

Apart from the core experiments mentioned above, we also provide additional
details (Object Details, Training Hyperparameters) and experiments on the
website111Project webpage: https://bit.ly/geometric-peg-in-hole:

Frozen vs. Finetuning: We conduct additional experiments by fine-tuning the
pretrained image encoders with the action decoder to conclude that (a)
Pretrained ResNet-50 models perform better when trained unfrozen, but still
does not outperform from-scratch models and (b) Pretrained ViT-B/16 models are
more unstable when trained unfrozen and often performs worse than the frozen
counterpart.

Data Efficiency of Models: We compare the performane of all visual
representations trained with 100, 1000, and 10000 training episodes to observe
that the non-pretrained models consistently improves with more data while
frozen models saturate in performance at 1000 demos due to the limited
capacity of MLP action decoders.

With vs. Without Proprioception: We provide analysis of models trained without
proprioception to isolate the effect of visual representations and observe
that proprioception is crucial for task variations with Z rotation because the
policy network benefits from explicityly knowing the robot’s current state to
accurately step toward the task goal.

Colored Extrusions for Ease of Recognition: We compare the performance of
models trained on colored objects with more contrast between the base and the
extrusion.

##  VI CONCLUSION

In this paper, we propose a novel dual-arm geometric-peg-in-hole task and
evaluate the spatio-gemoetric reasoning capabilities of various visual
representations through an imitation learning framework. We observe that a
non-pretrained visual encoder with MLP policy decoder were most effective in
completing the task across all variations. For future work, we plan to explore
data-driven contact modelling by incorporating force feedback while also
extending the geometric variation of objects to more realistic objects.
Furthermore, we hypothesize that advanced imitation learning methods such as
DiffusionPolicy [29], which is shown to perform well in multimodal learning
tasks, may significantly improve performance on our task and leave it for
future work.

## ACKNOWLEDGMENT

We thank Bahaa Aldeeb and Alireza Rezazadeh for providing helpful feedback on
our initial draft and all other members of the Robotics: Perception and
Manipulation (RPM) Lab for their insightful discussions. This project is
partially funded by the URS and UROP Program at the University of Minnesota
and the MnRI Seed Grant from the Minnesota Robotics Institute.

## References

  * [1] H. Park, J. Park, D.-H. Lee, J.-H. Park, M.-H. Baeg, and J.-H. Bae, “Compliance-based robotic peg-in-hole assembly strategy without force feedback,” _IEEE Transactions on Industrial Electronics_ , vol. 64, no. 8, pp. 6299–6309, 2017.
  * [2] X. Li, R. Li, H. Qiao, C. Ma, and L. Li, “Human-inspired compliant strategy for peg-in-hole assembly using environmental constraint and coarse force information,” in _2017 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)_. IEEE, 2017, pp. 4743–4748.
  * [3] O. Azulay, M. Monastirsky, and A. Sintov, “Haptic-based and $se(3)$-aware object insertion using compliant hands,” _IEEE Robotics and Automation Letters_ , vol. 8, no. 1, pp. 208–215, 2022.
  * [4] A. S. Morgan, Q. Bateux, M. Hao, and A. M. Dollar, “Towards generalized robot assembly through compliance-enabled contact formations,” _arXiv preprint arXiv:2303.05565_ , 2023.
  * [5] H.-C. Song, Y.-L. Kim, and J.-B. Song, “Guidance algorithm for complex-shape peg-in-hole strategy based on geometrical information and force control,” _Advanced Robotics_ , vol. 30, no. 8, pp. 552–563, 2016.
  * [6] W. Gao and R. Tedrake, “kpam 2.0: Feedback control for category-level robotic manipulation,” _IEEE Robotics and Automation Letters_ , vol. 6, no. 2, pp. 2962–2969, 2021.
  * [7] B.-S. Lu, T.-I. Chen, H.-Y. Lee, and W. H. Hsu, “Cfvs: Coarse-to-fine visual servoing for 6-dof object-agnostic peg-in-hole assembly,” _arXiv preprint arXiv:2209.08864_ , 2022.
  * [8] K. Van Wyk, M. Culleton, J. Falco, and K. Kelly, “Comparative peg-in-hole testing of a force-based manipulation controlled robotic hand,” _IEEE Transactions on Robotics_ , vol. 34, no. 2, pp. 542–549, 2018.
  * [9] E. Coumans and Y. Bai, “Pybullet, a python module for physics simulation for games, robotics and machine learning,” http://pybullet.org, 2016–2019.
  * [10] S. Nair, A. Rajeswaran, V. Kumar, C. Finn, and A. Gupta, “R3m: A universal visual representation for robot manipulation,” _arXiv preprint arXiv:2203.12601_ , 2022.
  * [11] I. Radosavovic, T. Xiao, S. James, P. Abbeel, J. Malik, and T. Darrell, “Real-world robot learning with masked visual pre-training,” _CoRL_ , 2022.
  * [12] S. Levine, C. Finn, T. Darrell, and P. Abbeel, “End-to-end training of deep visuomotor policies,” _J. Mach. Learn. Res._ , vol. 17, no. 1, p. 1334–1373, jan 2016.
  * [13] A. Ghadirzadeh, A. Maki, D. Kragic, and M. Björkman, “Deep predictive policy training using reinforcement learning,” in _2017 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)_. IEEE, 2017, pp. 2351–2358.
  * [14] Y. Duan, M. Andrychowicz, B. C. Stadie, J. Ho, J. Schneider, I. Sutskever, P. Abbeel, and W. Zaremba, “One-shot imitation learning,” in _NIPS_ , 2017.
  * [15] D.-A. Huang, S. Nair, D. Xu, Y. Zhu, A. Garg, L. Fei-Fei, S. Savarese, and J. C. Niebles, “Neural task graphs: Generalizing to unseen tasks from a single video demonstration,” in _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 2019, pp. 8565–8574.
  * [16] K. Zakka, A. Zeng, J. Lee, and S. Song, “Form2fit: Learning shape priors for generalizable assembly from disassembly,” in _Proceedings of the IEEE International Conference on Robotics and Automation_ , 2020.
  * [17] A. Zeng, P. Florence, J. Tompson, S. Welker, J. Chien, M. Attarian, T. Armstrong, I. Krasin, D. Duong, V. Sindhwani, and J. Lee, “Transporter networks: Rearranging the visual world for robotic manipulation,” _Conference on Robot Learning (CoRL)_ , 2020.
  * [18] A. Mandlekar, D. Xu, J. Wong, S. Nasiriany, C. Wang, R. Kulkarni, L. Fei-Fei, S. Savarese, Y. Zhu, and R. Martín-Martín, “What matters in learning from offline human demonstrations for robot manipulation,” in _arXiv preprint arXiv:2108.03298_ , 2021.
  * [19] K. He, X. Zhang, S. Ren, and J. Sun, “Deep residual learning for image recognition,” in _Proceedings of the IEEE conference on computer vision and pattern recognition_ , 2016, pp. 770–778.
  * [20] J. Deng, W. Dong, R. Socher, L.-J. Li, K. Li, and L. Fei-Fei, “Imagenet: A large-scale hierarchical image database,” in _2009 IEEE Conference on Computer Vision and Pattern Recognition_ , 2009, pp. 248–255.
  * [21] A. Radford, J. W. Kim, C. Hallacy, A. Ramesh, G. Goh, S. Agarwal, G. Sastry, A. Askell, P. Mishkin, J. Clark, G. Krueger, and I. Sutskever, “Learning transferable visual models from natural language supervision,” _CoRR_ , vol. abs/2103.00020, 2021. [Online]. Available: https://arxiv.org/abs/2103.00020
  * [22] K. He, X. Chen, S. Xie, Y. Li, P. Dollár, and R. Girshick, “Masked autoencoders are scalable vision learners,” in _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 2022, pp. 16 000–16 009.
  * [23] K. Grauman, A. Westbury, E. Byrne, Z. Chavis, A. Furnari, R. Girdhar, J. Hamburger, H. Jiang, M. Liu, X. Liu, _et al._ , “Ego4d: Around the world in 3,000 hours of egocentric video,” in _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 2022, pp. 18 995–19 012.
  * [24] A. Dosovitskiy, L. Beyer, A. Kolesnikov, D. Weissenborn, X. Zhai, T. Unterthiner, M. Dehghani, M. Minderer, G. Heigold, S. Gelly, J. Uszkoreit, and N. Houlsby, “An image is worth 16x16 words: Transformers for image recognition at scale,” in _9th International Conference on Learning Representations, ICLR 2021, Virtual Event, Austria, May 3-7, 2021_. OpenReview.net, 2021. [Online]. Available: https://openreview.net/forum?id=YicbFdNTTy
  * [25] Y. Zhou, C. Barnes, J. Lu, J. Yang, and H. Li, “On the continuity of rotation representations in neural networks,” in _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 2019, pp. 5745–5753.
  * [26] Blender Foundation, “Blender, an open-source 3d computer graphics software.” [Online]. Available: www.blender.org
  * [27] G. Wang, F. Manhardt, F. Tombari, and X. Ji, “GDR-Net: Geometry-guided direct regression network for monocular 6d object pose estimation,” in _IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)_ , June 2021, pp. 16 611–16 621.
  * [28] T. Zhao, V. Kumar, S. Levine, and C. Finn, “Learning fine-grained bimanual manipulation with low-cost hardware,” _Robotics: Science and Systems (RSS)_ , 2023.
  * [29] C. Chi, S. Feng, Y. Du, Z. Xu, E. Cousineau, B. Burchfiel, and S. Song, “Diffusion policy: Visuomotor policy learning via action diffusion,” _Robotics: Science and Systems (RSS)_ , 2023.

## References

  * [1] H. Park, J. Park, D.-H. Lee, J.-H. Park, M.-H. Baeg, and J.-H. Bae, “Compliance-based robotic peg-in-hole assembly strategy without force feedback,” _IEEE Transactions on Industrial Electronics_ , vol. 64, no. 8, pp. 6299–6309, 2017.
  * [2] X. Li, R. Li, H. Qiao, C. Ma, and L. Li, “Human-inspired compliant strategy for peg-in-hole assembly using environmental constraint and coarse force information,” in _2017 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)_. IEEE, 2017, pp. 4743–4748.
  * [3] O. Azulay, M. Monastirsky, and A. Sintov, “Haptic-based and $se(3)$-aware object insertion using compliant hands,” _IEEE Robotics and Automation Letters_ , vol. 8, no. 1, pp. 208–215, 2022.
  * [4] A. S. Morgan, Q. Bateux, M. Hao, and A. M. Dollar, “Towards generalized robot assembly through compliance-enabled contact formations,” _arXiv preprint arXiv:2303.05565_ , 2023.
  * [5] H.-C. Song, Y.-L. Kim, and J.-B. Song, “Guidance algorithm for complex-shape peg-in-hole strategy based on geometrical information and force control,” _Advanced Robotics_ , vol. 30, no. 8, pp. 552–563, 2016.
  * [6] W. Gao and R. Tedrake, “kpam 2.0: Feedback control for category-level robotic manipulation,” _IEEE Robotics and Automation Letters_ , vol. 6, no. 2, pp. 2962–2969, 2021.
  * [7] B.-S. Lu, T.-I. Chen, H.-Y. Lee, and W. H. Hsu, “Cfvs: Coarse-to-fine visual servoing for 6-dof object-agnostic peg-in-hole assembly,” _arXiv preprint arXiv:2209.08864_ , 2022.
  * [8] K. Van Wyk, M. Culleton, J. Falco, and K. Kelly, “Comparative peg-in-hole testing of a force-based manipulation controlled robotic hand,” _IEEE Transactions on Robotics_ , vol. 34, no. 2, pp. 542–549, 2018.
  * [9] E. Coumans and Y. Bai, “Pybullet, a python module for physics simulation for games, robotics and machine learning,” http://pybullet.org, 2016–2019.
  * [10] S. Nair, A. Rajeswaran, V. Kumar, C. Finn, and A. Gupta, “R3m: A universal visual representation for robot manipulation,” _arXiv preprint arXiv:2203.12601_ , 2022.
  * [11] I. Radosavovic, T. Xiao, S. James, P. Abbeel, J. Malik, and T. Darrell, “Real-world robot learning with masked visual pre-training,” _CoRL_ , 2022.
  * [12] S. Levine, C. Finn, T. Darrell, and P. Abbeel, “End-to-end training of deep visuomotor policies,” _J. Mach. Learn. Res._ , vol. 17, no. 1, p. 1334–1373, jan 2016.
  * [13] A. Ghadirzadeh, A. Maki, D. Kragic, and M. Björkman, “Deep predictive policy training using reinforcement learning,” in _2017 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)_. IEEE, 2017, pp. 2351–2358.
  * [14] Y. Duan, M. Andrychowicz, B. C. Stadie, J. Ho, J. Schneider, I. Sutskever, P. Abbeel, and W. Zaremba, “One-shot imitation learning,” in _NIPS_ , 2017.
  * [15] D.-A. Huang, S. Nair, D. Xu, Y. Zhu, A. Garg, L. Fei-Fei, S. Savarese, and J. C. Niebles, “Neural task graphs: Generalizing to unseen tasks from a single video demonstration,” in _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 2019, pp. 8565–8574.
  * [16] K. Zakka, A. Zeng, J. Lee, and S. Song, “Form2fit: Learning shape priors for generalizable assembly from disassembly,” in _Proceedings of the IEEE International Conference on Robotics and Automation_ , 2020.
  * [17] A. Zeng, P. Florence, J. Tompson, S. Welker, J. Chien, M. Attarian, T. Armstrong, I. Krasin, D. Duong, V. Sindhwani, and J. Lee, “Transporter networks: Rearranging the visual world for robotic manipulation,” _Conference on Robot Learning (CoRL)_ , 2020.
  * [18] A. Mandlekar, D. Xu, J. Wong, S. Nasiriany, C. Wang, R. Kulkarni, L. Fei-Fei, S. Savarese, Y. Zhu, and R. Martín-Martín, “What matters in learning from offline human demonstrations for robot manipulation,” in _arXiv preprint arXiv:2108.03298_ , 2021.
  * [19] K. He, X. Zhang, S. Ren, and J. Sun, “Deep residual learning for image recognition,” in _Proceedings of the IEEE conference on computer vision and pattern recognition_ , 2016, pp. 770–778.
  * [20] J. Deng, W. Dong, R. Socher, L.-J. Li, K. Li, and L. Fei-Fei, “Imagenet: A large-scale hierarchical image database,” in _2009 IEEE Conference on Computer Vision and Pattern Recognition_ , 2009, pp. 248–255.
  * [21] A. Radford, J. W. Kim, C. Hallacy, A. Ramesh, G. Goh, S. Agarwal, G. Sastry, A. Askell, P. Mishkin, J. Clark, G. Krueger, and I. Sutskever, “Learning transferable visual models from natural language supervision,” _CoRR_ , vol. abs/2103.00020, 2021. [Online]. Available: https://arxiv.org/abs/2103.00020
  * [22] K. He, X. Chen, S. Xie, Y. Li, P. Dollár, and R. Girshick, “Masked autoencoders are scalable vision learners,” in _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 2022, pp. 16 000–16 009.
  * [23] K. Grauman, A. Westbury, E. Byrne, Z. Chavis, A. Furnari, R. Girdhar, J. Hamburger, H. Jiang, M. Liu, X. Liu, _et al._ , “Ego4d: Around the world in 3,000 hours of egocentric video,” in _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 2022, pp. 18 995–19 012.
  * [24] A. Dosovitskiy, L. Beyer, A. Kolesnikov, D. Weissenborn, X. Zhai, T. Unterthiner, M. Dehghani, M. Minderer, G. Heigold, S. Gelly, J. Uszkoreit, and N. Houlsby, “An image is worth 16x16 words: Transformers for image recognition at scale,” in _9th International Conference on Learning Representations, ICLR 2021, Virtual Event, Austria, May 3-7, 2021_. OpenReview.net, 2021. [Online]. Available: https://openreview.net/forum?id=YicbFdNTTy
  * [25] Y. Zhou, C. Barnes, J. Lu, J. Yang, and H. Li, “On the continuity of rotation representations in neural networks,” in _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 2019, pp. 5745–5753.
  * [26] Blender Foundation, “Blender, an open-source 3d computer graphics software.” [Online]. Available: www.blender.org
  * [27] G. Wang, F. Manhardt, F. Tombari, and X. Ji, “GDR-Net: Geometry-guided direct regression network for monocular 6d object pose estimation,” in _IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)_ , June 2021, pp. 16 611–16 621.
  * [28] T. Zhao, V. Kumar, S. Levine, and C. Finn, “Learning fine-grained bimanual manipulation with low-cost hardware,” _Robotics: Science and Systems (RSS)_ , 2023.
  * [29] C. Chi, S. Feng, Y. Du, Z. Xu, E. Cousineau, B. Burchfiel, and S. Song, “Diffusion policy: Visuomotor policy learning via action diffusion,” _Robotics: Science and Systems (RSS)_ , 2023.


# Fractional Quantum Heat Engine

Ekrem Aydiner  Department of Physics, Faculty of Science, İstanbul University,
34134, İstanbul, Turkey<EMAIL_ADDRESS>
###### Abstract

In this work, we introduce the concept of the fractional quantum heat engine.
We examine the space-fractional quantum Szilard heat engine as an example to
show that the space-fractional quantum heat engines can produce higher
efficiency than the conventional quantum heat engines.

###### keywords:

Fractional calculus , Quantum heat engine , Szilard engine

††journal: Journal of LaTeX Templates

##  1 Introduction

The quantum version of the Szilárd engine was proposed by Kim et al. [1] as
the counterpart of the classical one [2]. In many theoretical studies, it has
been shown that the positive work and efficiency can be obtained by using
quantum Sizilard engine [3, 4, 5, 6, 7, 8]. On the other hand, many
experimental realization have been implemented both of quantum engine and
quantum Szilard engine [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21,
22]. Recently, a similar version of the quantum engine has been studied by
Thomas et al. [23] for the infinite well potential. In this study, authors
show that the work and efficiency of this quantum Szilárd engine are positive
for single and many particles.

However, we know quantum heat engines and information theory have not been
comprehensively studied by using fractional methods in the literature so far.
The first study related to the fractional heat engine was suggested in Ref.
[24]. In this study, the work extraction and efficiency of the quantum Szilard
engine has been discussed for fractional power-law potential. The fractional
dynamics play an important role in physics and other science areas [25, 26,
27, 28, 29, 30, 31, 32, 33, 34]. Therefore, it is important to discuss and
understand the role of the fractional dynamics in the quantum heat engines. In
the previous study [24], we obtained the work and efficiency of the model
without using fractional calculus tools. However, in the present study, we
discuss the fractional quantum Szilard engine as an example and we show that
fractional exponent plays an important role on the work and efficiency.
Therefore, in this study we consider quantum Szilard engine and we discuss the
work and efficiency base on the space-fractional Schrödinger equation for
infinite well potential.

This paper is organized as follows: In Section 2, we summarize the solution of
the space-fractional Schrödinger equation for the infinite potential well. In
Section 3, we discussed the theoretical framework of the thermodynamics cycles
for the fractional quantum Szilard engine. In Section 4, we obtain work and
efficiency of the fractional quantum Szilard engine. We also discussed the
effects of the space-fractional exponent on the work and the efficiency.
Finally, in the last Section, we summarize the obtained results and give a
brief discussion.

##  2 Fractional Theory for Infinite Well

We consider here one-dimensional fractional Schrödinger equation for the
infinite well potential. This problem was discussed by Laskin in a several
papers [32, 33, 34]. Here we briefly review the fractional Schrödinger
equation and its results for infinite well potential. In the Refs. [32, 33,
34, 35], the Hamiltonian of the system is given by

$\displaystyle H_{\alpha}\psi_{n}(x)=E_{\alpha n}\psi_{n},(x)\quad
n=0,1,2,...$ (1)

where $\alpha$ denotes the fractional index $0<\alpha\leq 2$. For the infinite
square well with length of $2a$, the space-fractional Schrödinger equation is
given by

$\displaystyle
D_{\alpha}\left(-\hbar^{2}\frac{d^{2}}{dx^{2}}\right)^{\alpha/2}\psi_{n}(x)+V_{i}\psi_{n}(x)=E_{\alpha}\psi_{n}(x)$
(2)

where the coefficient $D_{\alpha}=\chi mc^{2}/(mc)^{\alpha}$ with $\chi$ a
positive real number, and $c$ is the speed of the light. When $\alpha=2$,
taking $\chi=1/2$ and hence $D=1/(2m)$ [35]. The first term in Eq. (2)
corresponds to the fractional kinetic energy

$\displaystyle
T_{\alpha}=D_{\alpha}|p|^{\alpha}=\frac{1}{2}mc^{2}\left(\frac{|p|}{mc}\right)^{\alpha}=D_{\alpha}\left(-\hbar^{2}\frac{d^{2}}{dx^{2}}\right)^{\alpha/2}.$
(3)

On the other hand, the potential for infinite well is defined as

$\displaystyle V_{i}=\left\\{\begin{array}[]{ll}0&\mbox{if $|x|<a$}\\\
+\infty&\mbox{if $|x|>a$}\end{array}\right.$ (6)

the potential takes infinite value at the $x=\pm a$. Finally, the solutions of
the fractional Schrödinger equation are given by

$\displaystyle\psi_{n}(x)=\left\\{\begin{array}[]{ll}\frac{1}{\sqrt{2}}\sin\frac{n\pi}{2a}x+a&\mbox{if
$|x|<a$}\\\ 0&\mbox{if $|x|>a$}\end{array}\right.$ (9)

and

$\displaystyle E_{\alpha
n}=D_{\alpha}\left(\frac{n\pi\hbar}{2a}\right)^{\alpha}$ (10)

where $\psi_{n}(x)$ denotes the wave function solution and $E_{\alpha n}$ is
the energy eigenvalues of Eq. (2). Here we note that some authors claim that
the solution does not satisfies ground state solution of the space-fractional
Schrödinger equation (2). However, it is shown that this problem can be solved
by assuming that the solution is the limit of the finite square well problem
[35]. Thus, the solution

$\displaystyle\lim_{V_{0}\rightarrow\infty}\psi_{n}^{finite}(x)=\psi_{n}(x)\qquad\lim_{V_{0}\rightarrow\infty}E_{\alpha
n}^{finite}=E_{\alpha n}$ (11)

satisfies the ground state requirement of the infinite square well [35].

To carry out the work and efficiency of the fractional quantum Szilard engine
we need the canonical partition function and heat exchanges for all stages.
Therefore, in the next, we discuss these quantities for the infinite well
potential given in Eq.(6) following the method given in Ref. [23].

##  3 Thermodynamics Cycles of the Fractional Engine

In the Fig. 1, box A with a single particle in the infinite potential
interacts with a heat bath with a higher temperature of $T_{h}$. However, in
AB isothermal stage, a thin potential barrier is completely inserted in the
box at the same temperature and the box is divided by the quasi-static
insertion process into two half side as in box B which behaves like the two
double well potentials.

Figure 1: Work and efficiency of the Szilard quantum heat engine for the
infinite well in a box of length $2a$. In this schema boxes A and B interact a
heat bath reservoir, boxes C and D interact a cold bath reservoir. The stages
AB and CD represent the isothermal insertion and removal processes. The stages
BC and DA denote the isochoric processes.

In BC isochoric stage, the system is contacted to a heat bath with a lower
temperature of $T_{c}$. However, in CD isothermal stage, the inserted thin
barrier is again removed. After CD process, the particle is again under a
single infinite well potential in box D with a cold bath temperature of
$T_{c}$. In the final DA isochoric stage, the system is contacted to a heat
bath with a higher temperature of $T_{h}$ and cycle is completed.

The energy eigenvalue for the particle in box A and D is given by

$\displaystyle E_{\alpha
n}^{A,D}=D_{\alpha}\left(\frac{n\pi\hbar}{2a}\right)^{\alpha}.$ (12)

The canonical partition functions $Z_{A}(\alpha)$ and $Z_{D}(\alpha)$ for the
particle in boxes A and D

$\displaystyle
Z_{A}(\alpha)=\sum_{n=1}^{\infty}\exp\left[-\frac{D_{\alpha}}{k_{B}T_{h}}\left(\frac{n\pi\hbar}{2a}\right)^{\alpha}\right].$
(13)

and

$\displaystyle
Z_{D}(\alpha)=\sum_{n=1}^{\infty}\exp\left[-\frac{D_{\alpha}}{k_{B}T_{c}}\left(\frac{n\pi\hbar}{2a}\right)^{\alpha}\right]\
.$ (14)

On the other hand, the energy eigenvalue for boxes B and C is given as

$\displaystyle E^{B}_{\alpha
n}=D_{\alpha}\left(\frac{2n\pi\hbar}{2a}\right)^{\alpha}.$ (15)

The correspondence partition functions $Z_{B}(\alpha)$ and $Z_{C}(\alpha)$ are
given as

$\displaystyle
Z_{B}(\alpha)=\sum_{n=1}^{\infty}2\exp\left[-\frac{D_{\alpha}}{k_{B}T_{h}}\left(\frac{2n\pi\hbar}{2a}\right)^{\alpha}\right].$
(16)

and

$\displaystyle
Z_{C}(\alpha)=\sum_{n=1}^{\infty}2\exp\left[-\frac{D_{\alpha}}{k_{B}T_{c}}\left(\frac{2n\pi\hbar}{2a}\right)^{\alpha}\right]$
(17)

where the pre-factor 2 in Eqs. (16) and (17) is written since the boxes are
divided into two by the barrier which leads to energy levels two-fold
degenerate.

On the other hand, the heat exchanges $Q_{AB}$, $Q_{BC}$, $Q_{CD}$ and
$Q_{DA}$ for all stages of the thermodynamical cycle can be obtained by using
the partition functions. For this isothermal process AB, the heat exchanged
$Q_{AB}$ can be obtained as

$\displaystyle
Q_{AB}=U_{B}-U_{A}+k_{B}T_{h}\ln\frac{Z_{B}(\alpha)}{Z_{A}(\alpha)}$ (18)

where $U_{A}$ and $U_{B}$ are the internal energies of the box A and B,
respectively, which can be obtained from

$\displaystyle U_{A,B}=-\frac{\partial Z_{A,B}(\alpha)}{\partial\beta_{h}}$
(19)

where $\beta_{h}=1/k_{B}T_{h}$. Another isothermally stage CD leads to heat
exchange $Q_{CD}$ which is written as

$\displaystyle
Q_{CD}=U_{D}-U_{C}+k_{B}T_{c}\ln\frac{Z_{D}(\alpha)}{Z_{C}(\alpha)}$ (20)

where $U_{C}$ and $U_{D}$ are internal energies for box C and D, respectively
which can be found from

$\displaystyle U_{C,D}=-\frac{\partial Z_{C,D}(\alpha)}{\partial\beta_{c}}$
(21)

where $\beta_{c}=1/k_{B}T_{c}$ is the inverse temperature. On the other hand,
the heat exchanges for two isochoric processes BC and DA are given below. The
amount of exchanged heats $Q_{BC}$ and $Q_{DA}$ during isochoric process BC
and DA are given by

$\displaystyle Q_{BC}=U_{C}-U_{B}$ (22)

and

$\displaystyle Q_{DA}=U_{A}-U_{D}\ .$ (23)

By using the partition functions and heat exchanges we can compute work $W$
and efficiency $\eta$.

##  4 Work and efficiency of the fractional heat engine

The work for the Striling like cycle can be obtained from

$\displaystyle
W(\alpha)=k_{B}T_{h}\ln\frac{Z_{B}(\alpha)}{Z_{A}(\alpha)}-k_{B}T_{c}\ln\frac{Z_{C}(\alpha)}{Z_{D}(\alpha)}$
(24)

and the efficiency is given by

$\displaystyle\eta(\alpha)=1+\frac{Q_{BC}+Q_{CD}}{Q_{DA}+Q_{AB}}$ (25)

which is defined in terms of the heat exchanges. In the calculation, for
simplicity we neglect some extra energy cost for instance needed energy for
insertion and removal of the barrier and needed the energy for the coupling or
decoupling of the system to the heat baths.

Figure 2: Work and efficiency of the Szilard quantum heat engine for the non-
fractional infinite well. Here, we have set the constants as $m=9.11\times
10^{-31}$ kg, $k_{B}=1.380649\times 10^{-23}$ J/K, $h=6.62607015\times
10^{-34}$ Js $T_{h}=2$ K and $T_{c}=1$ K.

As a result, by using Eq. (24) and Eq. (25) we numerically obtained the work
$W$ and the efficiency $\eta$ of a Stirling-like cycle presented in Fig. 1 for
the $\alpha=2.0$. We note that for the $\alpha=2$, Eq. (10) reduce to

$\displaystyle E_{n}=\frac{n^{2}\pi^{2}\hbar^{2}}{8ma^{2}}$ (26)

which is the energy eigenvalue of a particle in the infinite quantum well. The
obtained numerical results for the $\alpha=2.0$ are given in Fig.2. Both
thermodynamic quantities in Fig.2 are plotted as a function of the parameter
$a$ which denotes the width of the the potential well. Here, we have set the
constants as $m=9.11\times 10^{-31}$ kg, $k_{B}=1.380649\times 10^{-23}$ J/K,
$h=6.62607015\times 10^{-34}$ Js, $T_{h}=2$ K and $T_{c}=1$ K.

Figure 3: Plot of the work $W$ of the Szilard quantum heat engine versus
length $a$ (in nano-meters) for the various fractional exponents. Here, we
have set the constants as $m=9.11\times 10^{-31}$ kg, $k_{B}=1.380649\times
10^{-23}$ J/K, $h=6.62607015\times 10^{-34}$ Js, $T_{h}=2$ K and $T_{c}=1$ K.
Figure 4: Plot of the efficiency $\eta$ of the Szilard quantum heat engine
versus length $a$ (in nano-meters) for the various fractional exponents. Here,
we have set the constants as $m=9.11\times 10^{-31}$ kg, $k_{B}=1.380649\times
10^{-23}$ J/K, $h=6.62607015\times 10^{-34}$ Js $T_{h}=2$ K and $T_{c}=1$ K.

For $\alpha=2.0$ which denotes the non-fractional infinite well, the obtained
results in Fig. 2 is compatible with the results in Ref. [23]. However, one
can see that these solutions clearly deviates in the case of the fractional
well as seen from Figs. 3 and 4.

To discuss the effect of the fractional parameter on the work and efficiency
we examine the solutions for the $\alpha<2.0$.

In Fig. 3 we presented the work curves versus the well width parameter $a$ for
various fractional exponent $\alpha$. One can see from the figure that work
$W$ is positive and maximum value of work is compatible te result of Fig. 2
and the result in Ref. [23]. When the $a$ increases the extracted work
smoothly decreases. However, interestingly, we see that when the space-
fractional exponent $\alpha$ decreases, the work persistently lives over
longer distances $a$.

Similarly, we presented the efficiency curves versus the well length parameter
$a$ for various fractional exponent $\alpha$ in In Fig. 4. One can see a
similar behaviour in the efficiency curves. In fact, the efficiency is
positive for large distance values for lower values of the fractional exponent
$\alpha$. Both results in Figs. 3 and 4 indicate that space-fractional quantum
engine behaves ve different from conventional quantum Szilard engine.

As a result, in this study, we obtain work and efficiency depending on the
potential width of the space-fractional quantum Szilard engine with single-
particle for infinite well. We show that the fractional exponent plays
important role on the work and efficiency. Additionally, we see that work and
efficiency live for large value $a$ when fractional exponent $\alpha$ values
become smaller which indicates an increase disorder in the system in the
fractional quantum Szilard engine

##  5 Conclusion

The classical and quantum version of the Szilard engine has been studied in
detail theoretically and experimentally [1, 2, 3, 4, 5, 6, 7, 8, 23, 9, 10,
11, 12, 13, 14, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24]. These studies show
that it is possible to set quantum Szilard heat engines which can extract
positive work in nano-size without violate the second law of thermodynamics.
On the other hand, these studies prove the connection between thermodynamic
entropy and information entropy.

Nowadays, the interest in these engines is increasing due to their potential
technological applications. Obtaining higher work and efficiency in such
engines are an important research topic. Therefore, alternative searches have
gained importance. In this work, we show that space-fractional quantum heat
engines can produce higher efficiency and more robust than conventional
quantum heat engines. To prove this, we examined the quantum Szilard heat
engine as an example and we presented here the obtained interesting results.
In this study, we have not only shown that the high efficiency can be
achieved, but also introduced the concept of the fractional quantum heat
engine to the literature.

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