# Local Region-to-Region Mapping-based Approach to
Classify Articulated Objects

Ayush Aggarwal∗, Rustam Stolkin, Naresh Marturi  Extreme Robotics Laboratory,
School of Metallurgy and Materials, University of Birmingham (UoB)
Birmingham, United Kingdom
Email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>This work was supported by the UK National Centre for Nuclear Robotics (NCNR).
Part funded by CHIST-ERA under Project EP/S032428/1 PeGRoGAM and in part
supported by Faraday Institution sponsored Recycling of Lithium Ion Batteries
(ReLiB) project (grant: FIRG005).

###### Abstract

Autonomous robots operating in real-world environments encounter a variety of
objects that can be both rigid and articulated in nature. Having knowledge of
these specific object properties not only helps in designing appropriate
manipulation strategies but also aids in developing reliable tracking and pose
estimation techniques for many robotic and vision applications. In this
context, this paper presents a registration-based local region-to-region
mapping approach to classify an object as either articulated or rigid. Using
the point clouds of the intended object, the proposed method performs
classification by estimating unique local transformations between point clouds
over the observed sequence of movements of the object. The significant
advantage of the proposed method is that it is a constraint-free approach that
can classify any articulated object and is not limited to a specific type of
articulation. Additionally, it is a model-free approach with no learning
components, which means it can classify whether an object is articulated
without requiring any object models or labelled data. We analyze the
performance of the proposed method on two publicly available benchmark
datasets with a combination of articulated and rigid objects. It is observed
that the proposed method can classify articulated and rigid objects with good
accuracy.

###### Index Terms:

Object Classification, Articulated Objects, Articulated Classification

##  I Introduction

Figure 1: The pipeline of the proposed registration-based object
classification method. For each set of frames $i,i+k\in N$, where $k$ is frame
skip, we first perform pre-processing on the depth map ${\bf D}$ and RGB image
${\bf C}$ for all the frames to obtain the point cloud ${\bf P}$ of the object
from the scene. These clouds are then downsampled to obtain ${\bf P}^{d}$.
Both the point clouds i.e. ${\bf P}_{i}^{d}$ and ${\bf P}_{i+k}^{d}$ are then
voxelised to obtain ${\bf V}$ voxels. For each voxel grid ${\bf V}_{m}$,
points in the voxel region are selected in both ${\bf P}_{i}^{d}$ and ${\bf
P}_{i+k}^{d}$, which is then used to perform registration to obtain a local
transformation ${\bf T}_{m}$. All the obtained local transformations are then
quantised and clustered to obtain the unique set of local transformations in
hash table $\mathcal{H}$. Based on this set, a decision $\rho_{i}$ on this set
is made to belong to one of the classes. Finally, we utilise a moving max-
count filter on the decisions vector $\bm{\rho}$ to classify the object.

Several regularly used household and industrial objects are uniquely
represented as articulated, i.e., objects composed of multiple rigid links
that are kinematically linked, e.g., doors, chains, clamps etc. Identifying
such types of objects in general scenes (also filled with other rigid objects)
is of prime interest for many robotic applications. Specifically, knowledge of
object properties like rigidity and articulation (in addition to geometric
visual features) helps in reliable tracking and pose estimation for robotic
and vision applications, e.g., robot tasked with clearing unknown objects in
case of hazardous decommissioning tasks [1]. It is highly challenging for a
robot to operate in the scenes with articulated objects without knowing these
specific object properties. Motivated from this, we propose a classification
technique to detect if an object being manipulated is either rigid or
articulated.

As mentioned, articulated objects are composed of multiple rigid structures.
When manipulated, each of these structures move differently relative to the
type of the joint they are linked with. These movements are classified into
three categories: revolute, prismatic and free-form, which also help in
kinematic modelling. In the literature, modelling, pose estimation, and
tracking of articulated objects has been studied extensively [2, 3, 4, 5, 6,
7, 8, 9, 10, 11]. In [2], authors presented a method to learn a kinematic
model of an articulated object from a given video sequence. In this method,
they first learn the possible object articulation model and then verify the
learned model on the testing sequence. The method is designed to handle free
form movements. However, it is constrained to objects with only two rigid
parts. Furthermore, the method relied on markers for accurate object position,
which may not be available in real scenarios. A marker-less object skeleton
estimation method from multi-view point cloud was proposed in [3], wherein the
authors utilise the generated skeleton curvatures to align and form the model
of the object. Although the method is marker-less, it requires a complete
point cloud of the object. In comparison, the proposed method can work on
single-view (marker-less) point clouds from a video sequence to classify the
articulated objects.

Several deep learning-based methods for estimating object articulation type
and articulation axis are proposed in the literature [4, 5, 6]. In [4], the
authors proposed an approach that predicts object and hand models to
understand object-hand interactions. Articulated object detection method using
RGB-D video sequence was proposed in [5], wherein the authors first detect the
object parts plane and axis over each image and then utilise the temporal
relation between the frames to predict the object bounding box, articulation
planes and axis. In [6], the authors used object RGB-D images and
corresponding part segmentations to predict the kinematic constraint between
the parts. These learning-based methods exhibit good performance and
robustness towards noise; however, they require accurately labelled training
data, which may not be always available in practical real-world scenarios.
Furthermore, with free-form articulated objects, there are numerous possible
object states, making labelling a time-consuming and extensive process. To
overcome this limitation, we propose an online classification method that can
directly infer the object’s type without the need for labelled data. In [7],
the authors proposed a part segmentation-based model creation and tracking
system on images, where they create shape-based models for each rigid
component of the object and track them to form a complete articulation model.
Finally, in [8], a three-step articulation state estimation method is proposed
by processing RGB-D images over an interaction sequence.

The aforementioned methods are designed to estimate the model and kinematics
of articulated objects. However, they all assume that the scene object is
articulated. We believe it is important to first analyse whether the object in
the scene is articulated before attempting to model or track it. To achieve
this, we perform a temporal local region-to-region registration on 3D point
clouds obtained from an observed sequence of object movements/interactions.
Using the video corresponding to the object’s movements, we first generate a
mask for the object of interest in each video frame using an off-the-shelf
object localization method. Then, using this generated mask and the
corresponding depth information, a point cloud of the object is created for
each frame. For each set of consecutive frame point clouds, we perform local
region registrations, which result in local transformations from one frame to
the other. These local transformations are then quantized (to reduce the
effect of cloud noise) and clustered to obtain the final set of unique
transformations between the local regions of the two frames. For a rigid
object, these transformations are the same for all local regions, while for
articulated objects, the local transformations may not be the same, and the
local regions of different articulated parts may move with different rotations
or translations.

Figure 2: The in-frame classification process. Given the pre-processed object
point cloud from two frames (e.g. Frame 1 and Frame 2), these clouds are at
first merged and then split into voxel grids (marked with coloured regions) to
object the local regions from both clouds. Each voxel grid is then selected
(marked with green colour) for local registration. The registrations are then
quantised and the regions with similar registration are clubbed together (as
shown in last image with arrows marking the magnitude and direction of
transformation). Finally, after counting the number of unique registration,
the motion of object in these two frames is classified as articulated or
rigid.

The key contributions of this paper are summarised as follows:

  * •

We propose a new model-free object classification method to identify if an
object is rigid or articulated by observing a sequence of object movements.

  * •

We design a constraint-free registration-based approach with local region-to-
region mapping to detect object with any type of articulation during
classification.

The main advantage of the proposed method is that it is a constraint-free
approach, which can classify any type of articulated object, without being
limited to specific types of articulations. The proposed method is a
parametric approach and does not require any training, making it independent
of any labelled data requirements. We analysed the performance of the proposed
method using two publicly available benchmark datasets containing a mixture of
rigid and articulated objects. The results obtained demonstrate the ability of
our approach to classify articulated objects.

##  II Proposed Method

In this section, we describe the proposed framework for articulated object
classification. For demonstration, we consider a $N$-frame video of an object
being manipulated where each frame consists of a depth map and an rgb image.
Then for each set of consecutive frames ${i,i+k}\in N$, we perform
registration between local regions of corresponding point clouds in order to
classify the motion between them as no motion, rigid motion, or articulated
motion. It is worth noting that the point cloud of a frame is generated using
the depth information of intended object regions. This process is presented in
detail in Sec. II-A. To perform this task, initially, each frame is passed
through a set of pre-processing steps for noise and background filtering.
Afterwards, the corresponding filtered point clouds of the frames are passed
through the classifier. The complete pipeline of our approach is shown in Fig.
1. Majorly, the pipeline is divided into two modules, i.e., the pre-processor
and the classifier. Each of these modules are described in the following
subsections.

###  II-A Pre-Processing

In general scenario, the depth map ${\bf D}_{i}$ and rgb image ${\bf C}_{i}$
of a given frame $i$ may consist of a variety of items other than the object
of interest, e.g. wall, table, etc. For the proof of concept, we assume that
in any sequence of frames only a single object is being manipulated and all
the remaining objects are stationary. Hence, all these other objects can be
considered as unwanted noise elements which impact the performance of the
classifier. Further, the presence of these objects creates a logical problem
in accurate classification, which is discussed in the next subsection.

Data: Depth map ${\bf D}_{i}$, rgb image ${{\bf C}_{i}}$ for $i$th frame,
object id $O$, Camera intrinsic ${\bf c}_{K}^{i}$, outlier noise standard
deviation $s$, voxel downsample size $v$, smoothening radius $r$

Result: Cropped Point Cloud ${\bf P}_{i}^{d}$

1 $p,bb=$ ObjectDetection(_${{\bf C}_{i}},O$_);

2 if  _$p\neq None$_ then

3 ${\bf D}_{i}^{c}=$ Crop(_${\bf D}_{i},{bb}$_);

4 ${\bf P}_{i}=$ ConvertToCloud(_${\bf D}_{i}^{c},{\bf c}_{K}^{i}$_);

5 ${\bf P}_{i}^{o}=$ OutlierNoiseRemoval(_${\bf P}_{i},s$_);

6 ${\bf P}_{i}^{v}=$ VoxelDownsample(_${\bf P}_{i}^{o},v$_);

7 ${\bf P}_{i}^{d}=$ MeanSmoothening(_${\bf P}_{i}^{v},r$_);

8

9else

10 Skip Frame

11 end if

Algorithm 1 Pre-Processing of input frame depth map and rgb image

In this pre-processing stage, we process each frame to remove the unwanted
elements and noise from the point cloud. At first, we perform object
localisation in the rgb image ${\bf C}_{i}$ of the frame using the well-known
Mask-RCNNv2 [12] mask generation model. We utilise the pre-trained weights
over Microsoft COCO dataset [13] and do not perform any fine-tuning on the
considered objects. From the Mask-RCNNv2, we obtain a list of detection score
and mask ($bb$) (no classifications utilised) over the objects in the frame
image. From these predictions, we create mask instances per object. To select
the intended object, we identify the mask instance, between two consecutive
frames, that is being manipulated. If an object is being manipulated, we
consider the respective mask. However, if no object is found in the frame, we
skip that frame and move to process the next set of frames. Considering the
case when the intended object is detected in ${\bf C}_{i}$, we crop out the
object and remove rest of the information from depth map to obtain ${\bf
D}_{i}^{c}$.

This process is performed for all the frames. Once the intended object region
is obtained, we convert the object depth map to 3D point cloud ${\bf P}_{i}$
using the depth camera intrinsic parameters ${\bf c}_{K}^{i}$. ${\bf P}_{i}$
is further processed to remove the outliers and noise using statistical noise
removal, to obtain ${\bf P}_{i}^{o}$. This processed point cloud is then
downsampled using a voxel size $v$. This downsampled point cloud ${\bf
P}_{i}^{v}$ is then passed through a mean smoothness filter with radius $r$ to
remove the sensor noise. The smoothened point cloud ${\bf P}_{i}^{d}$ is then
passed to the classifier module. Pre-processing steps are summarised in
Algorithm 1. Note that any other object localisation method [14, 15, 16, 17]
can also beused instead of Mask-RCNNv2 to obtain the object mask. Further, the
learning methods are only utilised for mask generation, and are not part of
the proposed algorithm.

###  II-B Classifier

The corresponding filtered point clouds of the pre-processed frames ${i}$ and
${i+k}$ are used by this module to classify if they belong to any of the
following three classes:

i) No Motion (NM)

– when there is no motion in object between the frames $i$ and $i+k$.

ii) Rigid Motion (RM)

– when there is a rigid motion in the object between the frames $i$ and $i+k$.

iii) Articulated Motion (AM)

– when there are multiple unique motions in the object between the frames $i$
and $i+k$.

This classification is performed for all the frame sets in the given video,
henceforth called as in-frame classification. The proposed in-frame
classification method is summarised in Algorithm 2. In this method, we first
split ${\bf P}_{i}^{d}$ and ${\bf P}_{i+k}^{d}$ into voxel grids ${\bf V}$
with the voxel size $x$. This gives us $M$ voxel grids with points in them.
Then, for each voxel grid ${\bf V}_{m},m\in M$, we consider the points from
${\bf P}_{i}^{d}$ and ${\bf P}_{i+k}^{d}$ in voxel grid ${\bf V}_{m}$ as ${\bf
P}_{i}^{d,m}$ and ${\bf P}_{i+k}^{d,m}$ respectively. With the assumption that
the object motion between two frames is not very large, we perform Iterative
Closest Point (ICP) [18] based registration between ${\bf P}_{i+k}^{d,m}$ and
${\bf P}_{i}^{d,m}$. As it is assumed that the points of both the voxels are
close, the initial transformation for ICP is provided as identity matrix ${\bf
I}$. This registration results in a homogeneous transformation matrix ${\bf
T}_{m}$ with rotation ${\bf R}_{m}$ and translation ${\bf t}_{m}$ components
for the $m$th voxel grid. The rotation matrix is converted to quaternion form
${\bf q}_{m}$ for further processing.

Data: source point cloud ${\bf P}_{i}^{d}$, target point cloud ${\bf
P}_{i+k}^{d}$, voxel size $x$, quaternion quantisation $\overline{q}$,
translation quantisation $\overline{t}$, moving window size $w$

Result: In-frame object class $\rho_{i}$

1 ${\bf V}=$ Voxelise(_${\bf P}_{i}^{d},{\bf P}_{i+k}^{d},x$_);

2 $M=|{\bf V}|$;

3 $m=0$;

4 $\mathcal{H}=\\{\\}$;

5 do

// Local voxel region selection

6 ${\bf P}_{i}^{d,m}=$ SelectVoxelPoints(_${\bf P}_{i}^{d}$_);

7 ${\bf P}_{i+k}^{d,m}=$ SelectVoxelPoints(_${\bf P}_{i+k}^{d}$_);

8 ${\bf T}_{m}=$ ICPRegistration(_${\bf P}_{i}^{d,m},{\bf P}_{i+k}^{d,m}$_);

9 ${\bf R}_{m}={\bf T}_{m}[:3,:3]$;

10 ${\bf t}_{m}={\bf T}_{m}[:3,3]$;

11 ${\bf q}_{m}=$ ToQuaternion(_${\bf R}_{m}$_);

// Quantisation operation

12 $\tilde{\bf q}_{m}=\lfloor\frac{{\bf
q}_{m}}{\overline{q}}\rfloor*\overline{q}$;

13 $\tilde{\bf t}_{m}=\lfloor\frac{{\bf
t}_{m}}{\overline{t}}\rfloor*\overline{t}$;

// Hash table is generated with unique keys by clustering

14 if  _$[\tilde{\bf q}_{m}|\tilde{\bf t}_{m}]\in\mathcal{H}$_ then

15 $\mathcal{H}[\tilde{\bf q}_{m}|\tilde{\bf t}_{m}]=[\mathcal{H}[\tilde{\bf
q}_{m}|\tilde{\bf t}_{m}];{\bf V}_{m}]$;

16

17 else

18 $\mathcal{H}[\tilde{\bf q}_{m}|\tilde{\bf t}_{m}]=[{\bf V}_{m}]$;

19

20 end if

21 $m=m+1$;

22

23while  _$m\leq M$_ ;

24$K=|\mathcal{H}|$;

25 if  _$K\geq 2$_ then

26 $\rho_{i}=AM$;

27

28 else if  _$K=1$_ then

29 $\rho_{i}=RM$;

30

31 else

32 $\rho_{i}=NM$;

33

34 end if

Algorithm 2 Articulated and rigid object classification using point clouds

For each $m\in M$, ${\bf q}_{m}$ and ${\bf t}_{m}$ are quantised with
$\overline{q}$ and $\overline{t}$ to remove any noise errors in the
registration process.These quantised values are then joined and hashed into a
hash table $\mathcal{H}$, wherein for each key, we store the respective voxel
grids. Due to the possible articulated nature of the objects, it is observed
that some portion of an object may become occluded or go out of the frame
within consecutive frames (e.g. a box opening may occlude the top portion of
the box). To overcome this issue, in the proposed method, we skip the voxel
grids which do not have any correspondence match during ICP. Further, if a
match has low confidence score, we also skip those registrations. After these
skips, if more than $\alpha\%$ of the voxel grids are matched successfully, we
progress with the classification step, otherwise those frames are considered
to be non-reliant and the process moves to the next set of frames.

Utilising the prepared hash table $\mathcal{H}$, we perform in-frame
classification based on the following rules:

  * •

if number of keys in $\mathcal{H}>1$, then the motion is articulated and
classified as AM;

  * •

if the number of keys in $\mathcal{H}=1$, and the key corresponds to a
rotation / translation, the motion is RM;

  * •

if the number of keys in $\mathcal{H}=1$ and the key represents no motion,
i.e., $q\in 0^{1\times 4}$ and $t\in 0^{1\times 3}$, then it is NM.

The in-frame classification method is shown in Fig. 2 and is summarised in
Algorithm 2. All the in-frame classifications are stored in a list
$\bm{\rho}$, to obtain the overall classification of the object. We utilise a
moving average filter over $\bm{\rho}$ with a window of size $w$. After
filtering, if any of the elements in the list corresponds to AM, we classify
the overall object as articulated. Otherwise, if only RM motion is present in
the list, without any AM motion, we consider the object to be rigid. In the
end, if only NM motion is present, then the final classification is
nondeterministic due to no motion present.

From the algorithm 2, it can be observed that the proposed method is dependent
on the type of motion of local regions in ${\bf P}_{i}^{d}$. Hence, if any
large noise or background elements are present in ${\bf P}_{i}^{d}$, the
registration from these objects will result in no motion (as they are assumed
to be fixed). In case of rigid objects, presence of background elements will
result in two hash keys, hence classifying it as an articulated object. One
way to avoid this issue is to not consider no motion key in the hash map.
However, this approach conflicts with the presence of articulated objects in
the frame in a way that if some components of an articulated object are static
over the whole video sequence, this static component will also be avoided,
resulting in classifying the object as rigid. To rectify this issue, we masked
the intended object during pre-processing, as described in the previous
subsection, and only the point cloud information of the intended object is
considered in the proposed classification algorithm.

##  III Experimental Validations

Experiments are conducted over two benchmark datasets from the literature to
demonstrate the ability of the proposed method to classify if an object is
articulated or rigid. We evaluate the performance based on the mean accuracy
score of classification. Further, qualitative results are presented to
showcase the articulation regions detected by the proposed method from a given
observation.

###  III-A Dataset Description

We analyse the performance of the proposed method over RBO [19] and YCB video
dataset [20]. RBO dataset provide a set of interaction videos and point clouds
for multiple articulated objects. These interaction videos are captured in a
controlled environment with motion of objects. For rigid objects, to the best
of authors knowledge, no interaction-based datasets with point clouds are
available in the literature. However, to showcase the performance of the
proposed method for rigid objects, we consider the YCB video dataset which
provide multi-object video sequences with point clouds. From RBO dataset, we
consider the interaction sequences for $4$ articulated objects, which include,
globe, laptop, cabinet, and microwave. Whereas banana, gelatin box, mustard
bottle, and power drill are $4$ rigid objects considered from the YCB video
dataset for experiments. For each object, at least $10$ video sequences with
maximum $200$ frames is utilised for experimentation in this paper.

###  III-B Experimental Setup

As discussed in Sec. II, the proposed method rely on multiple parameters
during the pre-processing and classifier steps. The value of these parameters
are decided on multiple trials and statistical properties of the data. For
downsampling and smoothening, the value of parameters like $v$ and $r$ is
decided by utilising the point cloud properties. Downsample voxel size $v$ is
calculated as $\frac{1}{20}$ of the scene object diameter and $r$ is
calculated as $5\times v$. Statistical outlier removal parameter $s$ is set to
$0.5$ with nearby points threshold as $0.1$ times the number of points in the
object point cloud. Voxel size $x$ in the classifier is set as the
$\frac{1}{5}$th of the maximum distance between the points in the point cloud.
Other parameters like $\overline{q}$, $\overline{t}$, and $k$ are decided
after multiple trials over a sub-set of the considered data. In this work,
$\overline{q}=0.1$, $\overline{t}=0.1$, and $k=5$ are used for the
experiments.

TABLE I: Performance Analysis for RBO dataset. Object | Accuracy %
---|---
Laptop | $88.00$
Globe | $76.00$
Microwave | $100.00$
Cabinet | $100.00$
Average | $91.00$

###  III-C Performance Analysis

We analyse the performance of the proposed method both quantitatively and
qualitatively. For quantitative analysis, we measure the classification
accuracy and average classification probability, while for qualitative
analysis, we show the direction and magnitude of registrations between the
frames. The per object accuracy is measured as percentage of total number of
video sequences correctly classified to total number of video sequences
processed for each object. The average classification probability is
calculated as the mean probability of each frame set to be classified as AM or
RM. Registration direction and magnitude in qualitative results is represented
using the arrows with the angle of arrows representing the direction, and the
length of arrows representing the magnitude of transformation.

####  III-C1 Accuracy Analysis

The accuracy performance for both RBO and YCB datasets are present in Table I
and Table II. From Table I, it is observed that the proposed method is able to
classify Microwave and Cabinet object as articulated with $100\%$ accuracy.
For Laptop, $88\%$ accuracy is achieved. A decrease in accuracy is observed
due to the presence of mirror screen, which affect the depth sensor and make
data very noisy, resulting in registration failure. Further for Globe object,
$76\%$ accuracy is achieved, which, considering we are not utilising colour
information in registration and the symmetric nature of the object, is
significantly good. Overall, the proposed method show an accuracy of $91\%$ to
classify the articulated objects in the RBO dataset accurately.

TABLE II: Performance Analysis for YCB dataset. Object | Accuracy %
---|---
Power Drill | $89.47$
Gelatin Box | $86.36$
Banana | $95.45$
Mustard Bottle | $91.30$
Average | $90.65$

For rigid object from YCB dataset, the proposed method is able to classify
rigid objects with $90.65\%$ accuracy as shown in Table II. Power Drill,
Banana, and Mustard Bottle objects are classified with $89.47\%$, $95.45\%$,
and $91.30\%$ respectively. $86.36\%$ accuracy for Gelatin Box is achieved.
The lower accuracy is observed for Gelatin Box that is symmetric due to which
local registration is impacted. From this analysis, we demonstrate that the
proposed method is able to classify an object with high accuracy, without the
need of object model, labelled data, and articulation constraints.

TABLE III: Per scene sequence classification probabilities for RBO dataset. Object | Probability of rigid | Probability of articulated
---|---|---
Laptop | $0.1472$ | $0.8528$
Globe | $0.0041$ | $0.9959$
Microwave | $0.0251$ | $0.9749$
Cabinet | $0.0543$ | $0.9457$
Average | $0.0577$ | $0.9423$
TABLE IV: Per scene sequence classification probabilities for YCB dataset. Object | Probability of rigid | Probability of articulated
---|---|---
Power Drill | $0.8744$ | $0.1256$
Gelatin Box | $0.7380$ | $0.2620$
Banana | $0.9847$ | $0.0153$
Mustard Bottle | $0.8433$ | $0.1567$
Average | $0.8601$ | $0.1399$

####  III-C2 Probability Analysis

Table III and IV represents the probability of a set of frames to be
classified as articulated or rigid. As discussed in Sec. II, each frame set is
classified as AM, RM, or NM, while the final classification is based on the
max-count filtered output. In Table III, performance for objects from RBO
dataset is shown and it can be observed that probability for a set of frames
to be classified as AM is greater than for RM and NM. Similarly, for rigid
objects from YCB dataset, it can be observed from Table IV that the
probability for set of frames to be classified as RM is higher than for AM and
NM. This analysis shows that the proposed algorithm is identifying the type of
objects with higher probabilities even for a single set of frames, hence
demonstrating the effectiveness of the proposed method.

####  III-C3 Qualitative analysis

The local registration results for RBO and YCB datasets, along with the
respective rgb images for the two frames are shown in Fig. 3 and 4. In Fig. 3,
it can be observed that more than $1$ unique registration are obtained between
the frames and hence they are classified as articulated objects. Similarly, in
Fig. 4, it can be observed that a single unique transformation is obtained for
all the local regions; hence, classified as rigid objects.

Figure 3: Local region registration results for RBO dataset. The arrows
represent the direction and magnitude of local region registrations. It is
observed that there are multiple unique transformations between two frames for
all the objects and hence are classified as articulated. Figure 4: Local
region registration results for YCB dataset. The arrows represent the
direction and magnitude of local region registrations. It is observed that
there are multiple unique transformations between two frames for all the
objects and hence are classified as rigid.

##  IV Conclusion

A registration-based local region-to-region mapping approach for articulated
object classification has been proposed in this paper. We formulate the object
articulation classification as a temporal movement detection method, wherein,
we consider the object point cloud from two consecutive time frame observation
and identify unique local motions between the two frames. An object is then
classified to be articulated if the object is undergoing more than one unique
motion during the observations. Otherwise, the object is either in rigid
motion or no motion. Due to the model-free nature of the proposed method, it
is applicable to wide variety of articulated objects as compared to state-of-
the-art methods in the literature. Further, the proposed method has no
labelled data requirements. The experimental results with two benchmark
datasets demonstrated that the proposed method is able to classify the
articulated objects with high accuracy. In future, we plan to extend this work
for articulated object modelling and tracking.

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# Unequal torsional rotations driven by normal to modulated phase transition
in 4-biphenylcarboxy coupled L-phenylalaninate governed by non-covalent
constraints

Somnath Dey<EMAIL_ADDRESS>Department of Chemical Sciences, Indian
Institute of Science Education and Research (IISER) Kolkata, Mohanpur 741246,
India  Institute of Crystallography, RWTH Aachen University, Jägerstraße
17-19, 52066 Aachen, Germany  Supriya Sasmal  Department of Chemical Sciences,
Indian Institute of Science Education and Research (IISER) Kolkata, Mohanpur
741246, India  Saikat Mondal  Department of Chemical Sciences, Indian
Institute of Science Education and Research (IISER) Kolkata, Mohanpur 741246,
India  Santosh Kumar  Department of Chemical Sciences, Indian Institute of
Science Education and Research (IISER) Kolkata, Mohanpur 741246, India
Rituparno Chowdhury  Department of Chemical Sciences, Indian Institute of
Science Education and Research (IISER) Kolkata, Mohanpur 741246, India
Debashrita Sarkar  Department of Chemical Sciences, Indian Institute of
Science Education and Research (IISER) Kolkata, Mohanpur 741246, India  C.
Malla Reddy  Department of Chemical Sciences, Indian Institute of Science
Education and Research (IISER) Kolkata, Mohanpur 741246, India  Lars Peters
Institute of Crystallography, RWTH Aachen University, Jägerstraße 17-19, 52066
Aachen, Germany  Georg Roth  Institute of Crystallography, RWTH Aachen
University, Jägerstraße 17-19, 52066 Aachen, Germany  Debasish Haldar
Department of Chemical Sciences, Indian Institute of Science Education and
Research (IISER) Kolkata, Mohanpur 741246, India

###### Abstract

Amongst the derivatives of 4-biphenylcarboxylic acid and amino acid esters,
the crystal structure of 4-biphenylcarboxy-(L)-phenylalaninate is unusual
owing to its monoclinic symmetry within a pseudo-orthorhombic lattice. The
distortion is described by disparate rotational property around the chiral
centers ($\varphi_{\mathrm{chiral}}$ $\simeq$ -129 degrees and 58 degrees) of
the two molecules in the asymmetric unit. Each of these molecules comprise of
planar biphenyl moieties ($\varphi_{\mathrm{biphenyl}}$ = 0 degrees). Using
temperature dependent single crystal X-ray diffraction experiments we show
that the compound undergoes a phase transition below $T$ $\sim$ 124 K that is
characterized by a commensurate modulation wave vector, $\mathbf{q}$ =
$\delta(101)$, $\delta$ = $\frac{1}{2}$. The (3+1) dimensional modulated
structure at $T$ = 100 K suggests that the phase transition drives the
biphenyl moieties towards non coplanar conformations with significant
variation of internal torsion ($\varphi^{\mathrm{max}}_{\mathrm{biphenyl}}$
$\leq$ $20$ degrees). These intramolecular rotations lead to dimerization of
the molecular stacks that are described predominantly by intermolecular tilts
and small variations in intermolecular distances. Atypical of modulated
structures and superstructures of biphenyl and other polyphenyls,
$\varphi_{\mathrm{biphenyl}}$ of one independent molecule is two to four times
larger than the other. We suggest that while the rotations arise from
intramolecular steric factors, competing intermolecular
H–C${\cdot}{\cdot}{\cdot}$C–H contacts and weak C–H${\cdot}{\cdot}{\cdot}$O
hydrogen bonds govern the distinctively unequal torsional property of the
molecules.

##  I Introduction

Molecular biphenyl has been investigated extensively for its stability and
conformation in different thermodynamic states. At ambient conditions, the
differences in the conjugation states of the $\pi$-electrons are governed
primarily by the twist about the central C–C single bond in the order: 40 deg
- 45 deg in gas phase, 20 deg - 25 deg in solution and 0 deg (mutually
coplanar) in the solid state in centrosymmetric monoclinic space group
$P2_{1}/a$ [1, 2, 3, 4]. The planar conformation due to constraints from
intermolecular interactions is energetically unfavourable and steric hindrance
between the ortho hydrogen atoms is compensated for by out of plane dynamic
disorder and in plane displacements of those hydrogens away from each other
[4, 5, 6, 7, 8, 9] A recent study has also suggested the role of
intramolecular exchange energy between single bonded carbon atoms in
stabilizing the planar conformation [10]. Absorption and fluorescence studies
showed additional bands in their spectra at low temperatures ($T$) [11, 12].
$T$-dependent Raman spectroscopy and Brillouin scattering experiments have
suggested two phase transitions at $T_{c1}$ = 42 K and $T_{c2}$ = 17 K
respectively [13, 14, 15, 16]. The phase transition at $T_{c1}$ is continuous
and governed by a soft mode associated with the torsion about the central C–C
single bond followed by discontinuous changes at $T_{c2}$. Inelastic neutron
scattering experiments on its deuterated form confirmed the phase transitions
with appearance of additional satellite reflections [17]. The modulation wave
vector ($\mathbf{q}$) was determined to be $\mathbf{q}_{I}$ =
${\delta}_{a}\mathbf{a}^{*}\,+\,\frac{1}{2}(1-{\delta}_{b})\mathbf{b}^{*}$ and
$\mathbf{q}_{II}$ = $\frac{1}{2}(1-{\delta}_{b})\mathbf{b}^{*}$ at the
intermediate and low temperature phases respectively and were found to vary
with temperature, suggesting an incommensurate nature of the modulation [17].
The modulated structure of the low temperature phase II was described within
an acentric superspace group symmetry $Pa(0{\sigma}_{2}0)0$ [18, 19, 20] and
found to be essentially associated with small modulation of translation and
rotation ($\omega$) normal to the mean molecular plane, and a significant
torsion ($\varphi$) between the phenyl rings [21, 22, 23, 24]. Theoretical
studies have suggested that competition between intramolecular and
intermolecular forces drives the phase transition towards the incommensurately
modulated states [25, 26, 27, 28].

The fundamental property of flexibility in conformations have made biphenyl an
excellent candidate to tune in multifaceted properties in materials. Torsion
between the rings has been demonstrated to regulate conductivity of single
molecule biphenyl-dithiol junctions [29, 30, 31, 32], tune in thermopower as
function of twist angle [31], degeneracy of energy states on substrates [33]
and theoretically suggest wide band gap semiconducting properties of its
derivatives [34]. On the other hand, biphenyl derivatives have also been
reported to influence and increase the efficiency of photophysical properties
[35, 36]. Planar biphenyl molecule in solid state favours maximum
intramolecular conjugation of $\pi$ electrons as well as increases the
probability of interactions between delocalized electrons that could aid in
optimal stacking of molecules. Coupling reaction mechanism [37] was
successfully employed to synthesize 4-biphenylcarboxy protected amino acid
esters of L-serine, L-tyrosine, L-alanine, L-leucine and L-phenylalanine [38,
39]. In solid state, the compounds crystallize either in acentric orthorhombic
space group symmetry $P2_{1}2_{1}2_{1}$ or the monoclinic subgroup $P2_{1}$
[38, 39]. Crystal packing in these systems is determined by
${\pi}{\cdot}{\cdot}{\cdot}{\pi}$ stacking between the biphenyl fragments and
linear strong hydrogen bonds between the amino acid ester moieties.

We presumed that the biphenyl moieties in these chemically coupled systems
could influence the bioactive amino acid esters and vice-versa with respect to
evolution or suppression of translational and rotational degrees of freedom in
their crystal structures at some thermodynamic condition. Reanalyzing all
their crystal structures, the system of 4-biphenylcarboxy-(L)-phenylalaninate
attracted our attention because the structure appeared to be similar to the
L-tyrosine analogue albeit the monoclinic distortion (Table 1,[39]) and two
crystallographically independent formula units [$Z^{\prime}$ = 2 [40], Fig.
1(a)] in the crystal structure of the former. The torsion about the chiral
center is significantly different for the independent molecules while the rest
of the rotations are similar [Fig. 1(a), [39]]. Each of these molecules
consist of coplanar biphenyl moieties which are stacked along $\mathbf{a}$ and
$\mathbf{b}$ respectively while the amide groups are connected by
intermolecular N–H${\cdot}{\cdot}{\cdot}$O hydrogen bonds [Fig. 1(b),[39]].

In the present study, $T$-dependent phase transition of
4-biphenylcarboxy-(L)-phenylalaninate has been investigated using single
crystal X-ray diffraction experiments. The low temperature phase II is found
to be a $2a{\times}b{\times}2c$ superstructure of the high temperature (phase
I) structure. The superstructure is described within the $(3+1)D$ superspace
approach as a commensurately modulated structure [18, 41, 42, 43, 44].
Structural properties of the phase I and the modulated structure have been
tabulated and compiled within $t$-plots ($t$ = phase of the modulation). The
origin and stability of phase II is discussed in terms of intra- and
intermolecular HC${\cdot}{\cdot}{\cdot}$CH contacts and intermolecular
hydrogen bonds and plausible mechanism of the phase transition is suggested.

##  II Experimental

###  II.1 Temperature dependent single crystal X-ray diffraction

Single crystals of the compound used in this study were obtained from those
reported by Sasmal  _et al._ [39]. The crystals were protected in oil under
mild refrigeration. Single crystal X-ray diffraction (SCXRD) experiments were
performed on a Agilent SuperNova, Eos diffractometer employing CuK${\alpha}$
radiation. Temperature of the crystal was maintained by a open flow nitrogen
cryostat from Oxford Cryosystems. During cooling, visual inspection of
diffraction images revealed weaker reflections in addition to strong
reflections at low temperatures. Diffraction images collected at $T$ = 150 K,
140 K and 130 K to 114 K in steps of ${\Delta}T$ = 2 K showed that the weaker
diffuse features appear at $T$ = 124 K that condense into satellite
reflections at $T$ = 122 K (Table 1, Fig. S1 in [45]). The transition
temperature is significantly higher than that of molecular biphenyl
($T_{c,\mathrm{biphenyl}}$ = 42 K). On the other hand, related polyphenyls
p-terphenyl and p-quarterphenyl undergo phase transition towards
superstructure phases at much higher critical temperatures
($T_{c,\mathrm{terphenyl}}$ $\approx$ 190 K [46],
$T_{c,\mathrm{quarterphenyl}}$ $\approx$ 233 K [47]). Complete diffraction
data were collected at $T$ = 160 K and $T$ = 100 K respectively.

Table 1: Temperature dependence of lattice parameters and the components of the modulation wave vector, $\sigma_{1}$ and $\sigma_{3}$. See Table S1 in [45] for reflections used.  $T$ (K) |  $a$ (Å) |  $b$ (Å) |  $c$ (Å) |  $\beta$ (∘) | $\sigma_{1}$ | $\sigma_{3}$ |  $V$ (Å3)
---|---|---|---|---|---|---|---
200[39] | 5.0560(3) | 8.6622(4) | 42.242(3) | 90.349(4) |  |  | 1850.00(18)
160 | 5.0479(2) | 8.6330(4) | 42.1525(15) | 90.513(3) |  |  | 1836.87(13)
150 | 5.0498(7) | 8.6161(8) | 42.136(11) | 90.607(11) |  |  | 1833.2(5)
140 | 5.0484(6) | 8.6093(7) | 42.145(10) | 90.661(10) |  |  | 1831.6(5)
130 | 5.0451(7) | 8.6014(8) | 42.120(11) | 90.713(11) |  |  | 1827.6(6)
128 | 5.0446(7) | 8.6002(8) | 42.113(11) | 90.723(11) |  |  | 1826.9(6)
126 | 5.0440(6) | 8.5992(7) | 42.114(10) | 90.720(9) |  |  | 1826.5(5)
124 | 5.0443(6) | 8.5978(7) | 42.100(10) | 90.733(9) | 0.499(9) | 0.51(7) | 1825.7(5)
122 | 5.0440(7) | 8.5970(7) | 42.100(10) | 90.745(10) | 0.498(6) | 0.49(4) | 1825.5(5)
120 | 5.0433(7) | 8.5984(7) | 42.100(10) | 90.746(10) | 0.497(5) | 0.52(3) | 1825.5(5)
118 | 5.0441(7) | 8.5958(8) | 42.092(11) | 90.764(10) | 0.500(4) | 0.50(3) | 1824.8(5)
116 | 5.0432(6) | 8.5926(7) | 42.087(9) | 90.775(9) | 0.500(4) | 0.51(3) | 1823.6(5)
114 | 5.0422(6) | 8.5923(7) | 42.090(10) | 90.787(10) | 0.500(4) | 0.53(3) | 1823.3(5)
100 | 5.0377(2) | 8.5898(3) | 42.0432(14) | 90.884(3) | 0.5 | 0.5 | 1819.11(11)

Determination of lattice parameters and data reductions were performed using
the software suite CrysAlisPro [48] (Table 1, Table S1 in [45]). Satellite
reflections of first order ($m$ = 1) observed below $T_{c}$ could be indexed
with modulation wave-vector $\mathbf{q}$ = (${\sigma}_{1}$,0,${\sigma}_{3}$),
${\sigma}_{1}$ = ${\sigma}_{3}$ $\simeq$ $\frac{1}{2}$ with respect to the
basic monoclinic lattice. Here, the $\mathbf{q}$ = $\frac{1}{2}$(101) is
perpendicular to the $\mathbf{b}$ axis consistent with monoclinic symmetry
while in molecular biphenyl $\mathbf{q}_{I}$ violates monoclinic symmetry and
$\mathbf{q}_{II}$ is parallel to $\mathbf{b}$ [17]. Using the plugin program
NADA [49] in CrysAlisPro, deviation of the $\sigma$’s as function of $T$ from
a rational value of 0.5 were found to be within their e.s.ds (Table 1),
indicating a commensurate nature of the modulation. Reflections at $T$ = 100 K
were indexed by four integers ($hklm$) using a basic monoclinic $b$-unique
lattice (Table 1, Table S1 in [45]) and modulation wave vector, $\mathbf{q}$ =
($\frac{1}{2}$,0,$\frac{1}{2}$) and data integration was performed. Empirical
absorption correction was performed using AbsPack program embedded in
CrysAlisPro. The monoclinic lattice as well as the reflection conditions
suggest the superspace group $P2_{1}(\sigma_{1}0\sigma_{3})0$ with
$\sigma_{1}$ = $\sigma_{3}$ = $\frac{1}{2}$ [19, 20].

###  II.2 Structure refinement of the modulated structure

The crystal structure of the room temperature phase (phase I hereon) was
redetermined at $T$ = 160 K using Superflip [50] and refined using Jana2006
[51] and Jana2020 [52]. Atoms were renamed with suffixes ‘a’ and ‘b’ for the
two independent molecules ‘A’ and ‘B’ [Fig. 1(a)]. Anisotropic atomic
displacement parameters (ADPs) of all non-hydrogens atoms were refined.
Hydrogen atoms (H) were added to carbon and nitrogen atoms using riding model
in ideal chemical geometry with constraints for isotropic ADPs [$U_{iso}$(H) =
1.2$U_{eq}$(N), $U_{iso}$(H) = 1.2$U_{eq}$(Caromatic) and $U_{iso}$(H) =
1.5$U_{eq}$(Csp3)]. Since ${\beta}$ is close to 90∘, the integrated data was
tested for twinning employing twofold rotation parallel $\mathbf{a}$ as twin
law. This twin law is a true symmetry element in case of a hypothetical
orthorhombic lattice with point group symmetry $222$ [53, 54]. The fit of the
structure model improved (compare $R_{F}^{obs}$ = 0.0463 to 0.0408) and volume
of the second component refined to $\sim$ 2% (Table S2 in [45]). Finally,
positions of the H atoms of NH groups and parameter corresponding to isotropic
extinction correction was refined that further improved $R_{F}^{obs}$ values
(= 0.0393, Table S2 in [45]). The crystal structure reproduced the values for
intramolecular rotations reported those for the structure at $T$ = 200 K
[$\varphi_{\mathrm{chiral}}$ = $\varphi_{1}$ (hereon) and $\psi$ in Fig.
1(a)]. In addition, we also observe that the coplanar biphenyl rings are
significantly rotated with respect to the amide groups [at $T$ = 200 K:
$\varphi_{2}$ = 32.8 deg and 31.2 deg [39], at $T$ = 160 K in Fig. 1(a)] which
also remains invariant as function of temperature.

Figure 1: (a) Two independent formula units of
4-biphenylcarboxy-(L)-phenylalaninate (C23H21NO3) with atomic labels of non-
hydrogen atoms (‘a’ and ‘b’ for molecule ‘A’ and ‘B’ respectively) in phase I
at $T$ = 160 K. ${\varphi}_{\mathrm{A}}^{1}$ = -130.1 deg,
${\varphi}_{\mathrm{B}}^{1}$ = 56.1 deg;
${\mid}{\varphi}_{\mathrm{A}}^{2}{\mid}$ = 32.8 deg,
${\mid}{\varphi}_{\mathrm{B}}^{2}{\mid}$ = 30.8 deg;
${\varphi}_{\mathrm{A}}^{3}$ = ${\varphi}_{\mathrm{B}}^{3}$ = 0 deg;
${\psi}_{\mathrm{A}}$ = 36.4 deg, ${\psi}_{\mathrm{B}}$ = 36.2 deg. Viewing
direction along $[\overline{1}00]$. (b) View of corresponding layered
structure along $[111]$ emphasizing biphenyl stacks (AA)n and (BB)n along
$\mathbf{a}$, (ABAB)n along $\mathbf{b}$ and N–H${\cdot}{\cdot}{\cdot}$O bonds
(dashed orange) along $[\mp 100]$ directions. Phenyl rings of the ester groups
(transparent) stack only along $\mathbf{a}$.

The modulated structure of phase II at $T$ = 100 K was refined using Jana2006
and Jana2020. Fractional co-ordinates of all non-hydrogen atoms from the
crystal structure at $T$ = 160 K were used as starting model and the average
structure was refined against main reflections. In successive steps, an
incommensurate (IC) model described by one harmonic wave for displacive
modulation describing the atomic modulation functions (AMFs) and basic
parameters for anisotropic ADPs for all atoms was refined against main and
satellite reflections that resulted in good fit to the diffraction pattern
($R_{F}^{obs}$ = 0.0425). However, ADPs of four non-hydrogen atoms were found
to be non-positive definite. Since the components of $\mathbf{q}$,
${\sigma}_{1}$ and ${\sigma}_{3}$ are rational, three commensurately modulated
structures were pursued by fixing the initial phase of the modulation to
values, $t_{0}$ = $0$, $\frac{1}{4}$ and $\frac{1}{8}$ respectively. While the
former two $t_{0}$ values describe monoclinic $B2_{1}$ space group symmetry
for the equivalent 3D $2a\times{b}\times 2c$ superstructure, the third
corresponds to triclinic $B1$ symmetry. The commensurately modulated structure
(C) model corresponding to $t_{0}$ = $\frac{1}{4}$ resulted in the best fit to
the diffraction data ($R_{F}^{obs}$ = 0.0426) including ADPs of all atoms
positive definite. As the atomic modulation functions (AMFs) have sinusoidal
character, the residual values are similar to the IC model (Fig. 2, Fig. S2-S4
and Table S2 in [45]) However, the C model at $t_{0}$ = $\frac{1}{4}$ is
described with either cosine or sine waves for the AMFs (equal to number of
refinable fractional coordinates in the equivalent superstructure) reducing
significantly the number of refinable parameters as compared to the IC model
(compare $N_{\mathrm{C}}$ = 649 to $N_{\mathrm{IC}}$ = 811, further tests in
[45]). The final C model was further improved by refining the parameter
corresponding to isotropic extinction correction and AMFs and positions of
hydrogen atoms of NH groups ($R_{F}^{obs}$ = 0.0419, Table S2 in [45]).

Figure 2: ($x_{si}$,$x_{s4}$)-sections of Fourier map centered on carbon atoms
(black) (a) C23a of molecule ‘A’ and (b) C23b of molecule ‘B’ respectively.
The contour line and the width of the maps are 0.5 eÅ-3 and 2.5 Å
respectively.

##  III Results and discussion

###  III.1 Structural phase transition and unequal distortion of molecules

In the present case, the monoclinic symmetry is retained below $T_{c}$ unlike
monoclinic to triclinic distortion at disorder–order phase transition of
$p$-terphenyl [55] and $p$-quarterphenyl [56]. In the final commensurately
modulated structure model with $t_{0}$ = $\frac{1}{4}$, sections corresponding
to $t$ = $\frac{1}{4}$ and $\frac{3}{4}$ (Fig. 3, Fig. S5 in [45]) are
physically relevant that corresponds to atomic positions in the equivalent
twofold superstructure in 3D (Fig. 4, Fig. S6 in [45]). Crystal structures of
phase I and phase II have group-subgroup relations and the doubling of the $a$
and $c$ axes describes the additional $B$-centering of the superstructure in
II. The superstructure derived using Jana2006 comprises of four molecules in
the asymmetric unit ($Z^{\prime}$ = 4); two each corresponding to molecule ‘A’
and ‘B’ of phase I (Fig. 4). The covalent bond distances are similar for the
independent set of molecules and are practically unaffected by modulation
(Table S6 in [45]). In the present study, discussion is based on the modulated
structure in order to establish unique relations between phase I and II
respectively [57, 58, 59, 60, 24, 61].

The modulated structure suggests that the phase transition is dominated by
evolution of internal torsional degrees of freedom (${\varphi}^{3}$ $>$
$0^{\circ}$) within the biphenyl moieties [Fig. 3(a)]. The twists about the
central C–C bond are significantly different for the two molecules where the
torsional modulation of ‘A’ are 2-4 times larger than those of ‘B’ (dihedral
angle: $\mid{\varphi}_{A}^{3}\mid$ = 15.6 deg, 20.5 deg;
$\mid{\varphi}_{B}^{3}\mid$ = 4.1 deg, 9.3 deg). These torsions are described
by highly anisotropic AMFs ($u$) along the three basis vectors where the
maximum amplitude are along $\mathbf{b}$ for the carbon atoms of biphenyl
(Fig. 2 and Table S4 in [45]). Notably, the rotations in the present structure
are significantly larger than those reported for molecular biphenyl ($\varphi$
$\simeq$ $\pm$ 5.5 deg in ref.[22, 21, 21]) but smaller than those in the low
temperature superstructure of $p$-terphenyl and $p$-quarterphenyl (Maximum
$\varphi_{\mathrm{terphenyl,quarterphenyl}}$ $\simeq$ 23 deg in ref.[55, 62,
56]). Another distinctive property of the modulated structure is the unequal
modulation for the two different moieties where $u_{\mathrm{biphenyl}}$ $>$
$u_{\mathrm{phenylalaninate}}$ (Table S4 in [45]). Therefore, the variation in
torsions $\varphi^{1}$ and $\psi$ are smaller [Fig. 3(c),(d)]. A possible
reason for the weaker modulations of the atoms around the chiral centers is
the directional strong intermolecular N–H${\cdot}{\cdot}{\cdot}$O bonds makes
large intramolecular rotations unfavorable. Note that the observed changes in
the rotations $\varphi^{2}$ [Fig. 3(b)] of molecule ‘A’ are predominantly
described by strong modulations of the molecule’s biphenyl moiety.
Nevertheless, the unequal values at the relevant $t$-sections of $\varphi^{3}$
are correlated to those of $\varphi^{2}$ [compare Fig. 3(a) and Fig. 3(b)]
suggesting the chemical influence of the amide groups on the phenyl rings and
vice-versa.

Figure 3: $t$-plots of intramolecular rotations of molecule ‘A’ (blue) and ‘B’
(red) as well as intermolecular tilts and distances between stacks of
4-biphenylcarxy-(L)-phenylalaninate. (a) Dihedral angle,
${\mid}{\varphi}^{3}{\mid}$ represents internal torsion within biphenyl; (b)
Dihedral angle, ${\mid}{\varphi}^{2}{\mid}$ represents torsion between the
inner ring of biphenyl and the amide groups; (c) ${\varphi}^{1}$ represents
torsion of the amide groups with respect to the -COOCH3 groups; (d) ${\psi}$
represent torsion of -COOCH3 groups with respect to amide groups. (e)
${\mid}\theta_{\mathrm{AA/BB}}{\mid}$ represent tilt between biphenyl rings of
‘A’ and ‘A’ii (blue); and of ‘B’ and ‘B’ii (red) and
${\mid}\theta_{\mathrm{AB/BA}}{\mid}$ (dashed-dotted black curve) represent
tilt between inner aromatic rings of biphenyl (bonded to amide groups) of ‘A’
and outer ring of ‘B’ and vice-versa. (f) Intermolecular distances ($d$)
between biphenyl rings of ‘A’ and ‘A’ii (blue), between those of ‘B’ and ‘B’ii
(red) and between those of ‘A’ and ‘B’ (black). Horizontal dashed lines
represent those angles and distances in phase I (${\mid}{\varphi}^{3}{\mid}$ =
${\mid}{\theta}_{AA/BB}{\mid}$ = 0 deg). Vertical dashed lines indicate $t$
values corresponding to angles and distances in the 3D superstructure.  Figure
4: Comparison of structures in phase I and phase II across the phase
transition highlighting the effect of internal torsion ($\varphi^{3}$) within
biphenyl on the stacking arrangements along $\mathbf{a}$. The tilt between the
biphenyl stacks, $\theta_{\mathrm{AA/BB}}$ are different for the inner rings
(bonded to amide rings) and the outer rings [corresponding values in $t$-plot
in Fig. 3(e)]. See full unit cells in Fig. S6 in [45]. View along
[$\overline{1}$00].

In the modulated structure, the biphenyl moieties in (AA)n and (BB)n stacks
are tilted with respect to each other [Fig. 3(e)] which are parallel in phase
I. These tilts ($\theta_{\mathrm{AA/BB}}$) are of the order of the internal
twists, $\varphi^{3}$ of the independent biphenyl moieties [compare Fig. 3(e)
to Fig. 3(a)] The orientation between the biphenyl moieties within the (ABAB)n
stacks also vary with $\Delta\theta_{\mathrm{AA/BB}}$ $\simeq$ 12 deg where
the value is intermediate to ${\varphi}_{A}^{3}$ and ${\varphi}_{B}^{3}$
[compare Fig. 3(e) to Fig. 3(a)]. In addition, intermolecular distances
between the biphenyl moieties within the stacks at the two $t$-sections are
different and vary up to $\Delta{d}_{\mathrm{AA/BB}}$ $\simeq$ 0.05 Åand
$\Delta{d}_{\mathrm{ABAB}}$ $\simeq$ 0.02 Å[Fig. 3(f)]. However, these
variations in $d$ are small compared to the molecular tilts, $\theta$. It
could therefore be argued that the dimerization of biphenyl molecular stacks
below $T_{c}$ are predominantly governed by distortion described by molecular
rotations rather than intermolecular distances. On the other hand,
intermolecular distances between aromatic rings of L-phenylalaninate vary
similarly to the biphenyls albeit significantly smaller interstack rotations,
$\theta$ $<$ 3 deg (Fig. S5 in [63]).

The increased distortions are also accompanied by suppression of dynamic
disorder below $T_{c}$. For example, the carbon atoms at ortho (C14, C16, C19,
C23) and meta (C13, C17, C20, C22) positions are strongly displaced (Fig. 5,
Table S4 and S5 in [45]). Subsequently, the ADPs are significantly reduced as
compared to phase I (Fig. 5, Table S5 in [45]). Notably, decrease of the ADPs
($U_{eq}$) from $T$ = 160 K to $T$ = 100 K is larger for those of molecule ‘A’
than those for ‘B’, while the square of the amplitude of modulations ($u^{2}$)
are greater for ‘A’ than those for ‘B’ [compare Fig. 5(b) to Fig. 5(a)].

Figure 5: Scatter plots of equivalent value of anisotropic ADPs ($U_{eq}$) and
square of the amplitude of modulations ($u^{2}$) of the carbon atoms of
biphenyl moieties (C12 through to C23, see Fig. 1.) of molecule ‘A’ (blue) and
‘B’ (red). (a) $U_{eq}$ of the carbon atoms at $T$ = 160 K (open circles) and
at $T$ = 100 K (full circles). (b) $u^{2}$ (diamonds) of the corresponding
carbon atoms of molecule ‘A’ and ‘B’. See Table S5 in [45].

###  III.2 Competitive forces governing modulations

Structural studies in the 3D phase of molecular biphenyl have suggested that
the ortho-hydrogen atoms are displaced away in the plane of the rings to
minimize steric hindrance [3, 4, 6]. On the other hand, dynamic disorder
predominantly governed by torsional vibrations around the long molecular axis
[22] is predicted to balance the planar conformation of biphenyl favourable
for crystal packing [9]. As short as 1.98 Å in phase I (Table 2), these
contacts are shorter than the predicted values for twice van der Waals radius
for hydrogen ($r$ = 1.1–1.2 Å [64, 65]). In the modulated structure, we
observe that the distances between the ortho-hydrogen atoms are marginally but
consistently larger than those in phase I (Table 2) that could suggest that
the torsional modulations aid in minimization of the presumed steric hindrance
below $T_{c}$ [63, 66].

Table 2: Comparison of non-bonded hydrogen${\cdot}{\cdot}{\cdot}$acceptor and hydrogen${\cdot}{\cdot}{\cdot}$hydrogen distances (Å) involved in hydrogen bonds and steric factors in phase I ($T_{1}$) and phase II ($T_{2}$; $t$ = $\frac{1}{4}$; $\frac{3}{4}$) respectively. Symmetry codes: Phase I– (i) $x-1,y,z$; (ii) $x+1,y,z$; (iii) $-x+1,y+\frac{1}{2},-z+1$; (iv) $-x+2,y-\frac{1}{2},-z$. Phase II– (i) $x-1,y,z,t$; (ii) $x+1,y,z,t$; (iii) $-x+1,y+\frac{1}{2},-z+1,-t$; (iv) $-x+2,y-\frac{1}{2},-z,-t$.  Interaction class | Atom group labels | Phase | distances (Å)
---|---|---|---
N–H${\cdot}{\cdot}{\cdot}$O | H1n1a${\cdot}{\cdot}{\cdot}$O3ai | I | 2.00
|  | II | 1.97; 1.95
| H1n1b${\cdot}{\cdot}{\cdot}$O3bii | I | 2.00
|  | II | 1.98, 2.00
C–H${\cdot}{\cdot}{\cdot}$O | H1c10a${\cdot}{\cdot}{\cdot}$O2aiii | I | 2.72
|  | II | 2.72; 2.68
| H1c10b${\cdot}{\cdot}{\cdot}$O2biv | I | 2.78
|  | II | 2.74; 2.73
Intra H${\cdot}{\cdot}{\cdot}$H | H1c14a${\cdot}{\cdot}{\cdot}$H1c19a | I | 1.98
|  | II | 2; 2
| H1c14b${\cdot}{\cdot}{\cdot}$H1c19b | I | 1.98
|  | II | 2; 2
| H1c16a${\cdot}{\cdot}{\cdot}$H1c23a | I | 2.03
|  | II | 2.1; 2.2
| H1c16b${\cdot}{\cdot}{\cdot}$H1c23b | I | 2.02
|  | II | 2.03; 2.06
Inter H${\cdot}{\cdot}{\cdot}$H | H1c21a${\cdot}{\cdot}{\cdot}$H1c7b | I | 2.59
|  | II | 2.6; 2.6
| H1c21b${\cdot}{\cdot}{\cdot}$H1c7aii | I | 2.45
|  | II | 2.4; 2.5

A peculiar property of the modulated structure under discussion is the
significant difference in the torsional amplitude ${\varphi}^{3}$ of the
independent molecules. This aspect cannot be explained solely based on the
intramolecular steric factors. Analysis of the crystal packing shows that each
of these independent biphenyl moieties maintains close intermolecular
CH${\cdot}{\cdot}{\cdot}$HC contacts with the phenyl rings of
L-phenylalaninate in AB and BA fashion (Fig. 4). These distances are
significantly longer (intermolecular $d_{\mathrm{H{\cdot}{\cdot}{\cdot}H}}$
$\geq$ 2.4 Å, Table 2) compared to the intramolecular
H${\cdot}{\cdot}{\cdot}$H distances. On the other hand, the aromatic rings of
L-phenylalaninate interact with adjacent oxygen atoms of –COOCH3 via
C–H${\cdot}{\cdot}{\cdot}$O hydrogen bonds (Fig. 4 and Table 2). These
hydrogen bonds are weaker [67] but highly directional
($\angle$(C–H${\cdot}{\cdot}{\cdot}$O) = 159–164 deg) with very little
variation in the distances. Interestingly, those H${\cdot}{\cdot}{\cdot}$H
distances involving biphenyl moieties of ‘B’ are consistently smaller than
those of ‘A’ in both phases (Table 2). We argue that in the presence of both
the van der Waals interactions and weak C–H${\cdot}{\cdot}{\cdot}$O bonds, the
larger distortions of ‘A’ is favored by weaker CH${\cdot}{\cdot}{\cdot}$HC
interactions while that is suppressed in ‘B’. The four different values of
intramolecular torsion $\varphi^{3}$ within the biphenyl moieties is
distinctively governed by intra- and intermolecular non-bonded
H${\cdot}{\cdot}{\cdot}$H contacts as well as weak hydrogen bonds.

##  IV Conclusions

The single crystal to single crystal phase transition of
4-biphenylcarboxy-(L)-phenylalaninate below $T$ = 124 K drives the 3D
structure directly to a locked-in twofold superstructure. The commensurately
modulated structure at $T$ = 100 K is accompanied by pronounced amplitudes of
torsion within biphenyl that are characteristic of modulated and
superstructures of other polyphenyls. The phase transition temperature is
significantly higher than that in biphenyl yet significantly smaller than for
p-terphenyl and p-quarterphenyl. Consistent with the $T_{c}$, the maximum
amplitude of torsion is also intermediate and in the order
${\varphi}_{\mathrm{quarterphenyl}}$ $\geq$ ${\varphi}_{\mathrm{terphenyl}}$
$>$ ${\varphi}_{\mathrm{4-biphenylcarboxy-L-phenylalaninate}}$ $>$
${\varphi}_{\mathrm{biphenyl}}$.

Topologically separated, conformations of both the weaker
C–H${\cdot}{\cdot}{\cdot}$O bonds and stronger N–H${\cdot}{\cdot}{\cdot}$O
bonds are rigid and that underlines their role in stabilizing the crystal
packing in both phases. A unique property of the present polyphenyl coupled
amino acid ester is the distinctively unequal torsional amplitude
(${\varphi}_{A}$ $>$ ${\varphi}_{B}$) within the independent molecules which
is governed by multiple level of competitions involving unequal van der Waals
constraints in presence of weak hydrogen bonds between the biphenyl and
L-phenylalaninate moieties. The unusual nature of the phase transition is
described by the fact that unequal displacive modulations of the two molecules
are complimented by unequal suppression of the dynamic disorder of their atoms
below $T_{c}$. This difference in the torsion as well as the phase transition
conditions highlights how conjugation of polyphenyls can possibly be
influenced by amino acid esters which in turn influences supramolecular
assemblies.

###### Acknowledgements.

We thank Prof. Sreenivasan Ramakrishnan, Dr. Vaclav Petricek, Dr. Sitaram
Ramakrishnan, Prof. Venkataramanan Mahalingam and Dr. Saumya Mukherjee for
helpful comments and fruitful discussions. Financial support from SERB-DST
(DST-SERB:PDF/2018/002502) and Alexander von Humboldt foundation is gratefully
acknowledged.

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