# End-to-end Multi-Instance Robotic Reaching from Monocular Vision

Zheyu Zhuang1, Xin Yu2, Robert Mahony1 This research was supported by the
Australian Research Council through the “Australian Centre of Excellence for
Robotic Vision” CE140100016.1 Zheyu Zhuang, Robert Mahony are with “Australian
Centre for Robotic Vision”, Research School of Engineering, The Australian
National University<EMAIL_ADDRESS>Xin Yu is with School of Computer
Science, The University of Technology Sydney<EMAIL_ADDRESS>
###### Abstract

Multi-instance scenes are especially challenging for end-to-end visuomotor
(image-to-control) learning algorithms. “Pipeline” visual servo control
algorithms use separate detection, selection and servo stages, allowing
algorithms to focus on a single object instance during servo control. End-to-
end systems do not have separate detection and selection stages and need to
address the visual ambiguities introduced by the presence of arbitrary number
of visually identical or similar objects during servo control. However, end-
to-end schemes avoid embedding errors from detection and selection stages in
the servo control behaviour, are more dynamically robust to changing scenes,
and are algorithmically simpler. In this paper, we present a real-time end-to-
end visuomotor learning algorithm for multi-instance reaching. The proposed
algorithm uses a monocular RGB image and the manipulator’s joint angles as the
input to a light-weight fully-convolutional network (FCN) to generate control
candidates. A key innovation of the proposed method is identifying the optimal
control candidate by regressing a control-Lyapunov function (cLf) value. The
multi-instance capability emerges naturally from the stability analysis
associated with the cLf formulation. We demonstrate the proposed algorithm
effectively reaching and grasping objects from different categories on a
table-top amid other instances and distractors from an over-the-shoulder
monocular RGB camera. The network is able to run up to $\sim$160 fps during
inference on one GTX 1080 Ti GPU.

##  I INTRODUCTION

Scenes that contain multi-instances are common in our daily life, for example;
cutlery sets on dining tables, books and stationary on desks, fruit hanging on
trees, etc. As robotic systems transition into more real-world and shared
autonomy environments, the ability to reach and grasp objects amid distracters
and in the presence of visually similar objects becomes critical. Pipeline
approaches follow the “detect, decide, then servo” paradigm. Grasping
algorithms such as [1, 2] first detect objects and estimate 6DoF poses for
each detected object. The selection module chooses which object to grasp and
the estimated grasp pose is provided to the servo module. The grasp action is
undertaken using the standard pose control for robotic manipulators. Such a
pipeline can be run in real-time if the computational requirements can be met
and a heuristic is used to ensure the algorithm does not jump between
different grasp targets unintentionally.

Convolutional neural networks have become the default algorithm for detection
and pose estimation from visual data. Algorithms that first regress an
intermediate representation, such as image keypoints, and then compute object
pose by solving a PnP problem [3, 4, 5], have achieved impressive performance
on popular monocular pose estimation datasets including LINEMOD [6] and
Occlusion LINEMOD [7]. However, existing algorithms are trained on single
instance datasets and it is unclear how the underlying architecture will adapt
to multi-instance. Tremblay et al. [8] propose a real-time pose estimation
network with multi-instance capability and showcase repeatable experiments of
robotic grasping. To authors understanding, this work [8] is the state-of-the-
art result in multi-instance visual reaching and grasping.

Figure 1: Architecture of the proposed closed-loop reaching algorithm. A
fully-convolutional network densely predicts a control Lyapunov function (cLf)
value $\widehat{\mathcal{V}}$ and control $\widehat{u}$ associated to each
foreground image grid cell. Non-optimal suppression is achieved by selecting
the control associated with the grid cell corresponding to the lowest cLf
value. The control is updated in real-time as the image and joint angles are
updated. The reaching trajectory terminates when the regressed Lyapunov value
is lower than a threshold.

While separating the computer vision pose estimation from servo control is
conceptually simple, this approach can lead to undesirable error propagation
between modules [8]. Furthermore, in dynamic scenes, the decision module needs
to associate and track the estimated poses between frames [9].

In contrast, end-to-end approaches are known to be robust to dynamic scene
changes and model errors [10]. Levine et al. [11] demonstrate a robot
accomplishing real-world tasks such as hanging clothes and screwing on a
bottle cap by executing control learnt from RGB inputs. James et al. [12]
transfer a recurrent network that learns a multi-stage task from simulation to
real-world with no real-samples. This work shows the robot sequentially reach,
grasp and place a red cube amid distracters. Zhang et al. [13] achieve closed-
loop reaching towards one unique target in a clutter from monocular RGB images
by learning a visuomotor policy from a pose-based controller. To increase the
network’s accuracy, [12] learns the target pose from the image feature vector
as an auxiliary task, and [13] pre-trains the CNN feature extractor by
regressing the target pose. These techniques apply only to the single instance
scenario. For multi-instance image-to-control learning algorithms, Zhuang et
al. [14] use similar fully-connected network architecture as in [13, 12].
However, a hot-swap scheme is proposed to attend network’s focus onto a single
instance during the final stages of servo control.

In this paper, we propose a fully-convolutional network that densely predicts
control action over a grid of image cells, associating the dominant visible
instance of the desired object category with the cell. This naturally gives
the network the ability to generalise over an arbitrary number of object
instances visible in different grid cells. A key innovation is that we
generate the desired control action from a control Lyapunov function
formulation, and separately regress the value of the Lyapunov function
alongside the corresponding control. The algorithm then uses the regressed
Lyapunov function value to select the control action associated with the
lowest Lyapunov value, ensuring that the closed-loop system is drawn to the
object that is easiest to grasp. This behaviour is encoded in the Lyapunov
control function design, and the selected control acts to continuously
decrease the Lyapunov function value, leading to successful grasp actions.

In summary, the key contributions of this paper are:

  * •

We demonstrate a real-time, closed-loop, image-to-control fully-convolutional
network for robotic reaching in cluttered and dynamic environments. The
proposed network achieves consistent high grasp success rate for different
object categories regardless of the presence of simultaneous instances and
visual clutter.

  * •

We showcase the simplicity and efficiency of utilising a control Lyapunov
function approach to deal with visual ambiguity associated with multi-instance
grasping.

  * •

We demonstrate that the proposed approach can be trained entirely on simulated
data and transfer effectively to real world scenarios.

##  II Formulation

This section presents the formulation of the control Lyapunov function and
corresponding velocity control for a reaching task.

The 6 DoF poses of target frame $\\{G\\}$ and end-effector frame $\\{H\\}$ are
represented by elements of the Special Euclidean Group $\mathrm{SE}(3)$.
Denote the pose of a frame $\\{B\\}$ with respect to a reference frame
$\\{A\\}$ as $\prescript{A}{}{\bm{X}}_{B}$, and its rotation matrix and
translation vector as $\prescript{A}{}{\bm{R}}_{B}\in\mathrm{SO}(3)$ and
$\prescript{A}{}{\bm{p}}_{B}\in\mathbb{R}^{3}$ respectively. The left
superscript is omitted if the pose is defined with respect to the world
reference frame. The absolute end-effector pose
$\prescript{}{}{\bm{X}}_{H}=\prescript{}{}{\bm{X}}_{H}(\bm{\theta})$ is a
function of joint angles $\bm{\theta}\in\mathbb{R}^{6\times 1}$, i.e., the
forward kinematics model of the manipulator.

###  II-A Symmetry-aware Control Lyapunov Function

A control Lyapunov function (cLf) for a reaching task is a continuously
differentiable scalar-valued positive-definite function
$\mathcal{V}(\bm{\theta})$ of the joint angles. $\mathcal{V}(\bm{\theta})$ is
zero only at the joint coordinates for the desired goal pose $\\{G_{i}\\}$ of
an object $i$. We formulate the cLf as:

$\displaystyle\mathcal{V}(\prescript{}{}{\bm{X}}_{H}):=\frac{1}{2}\|\prescript{}{}{\bm{X}}_{H}-\prescript{}{}{\bm{X}}_{G_{i}}\|^{2}_{kF},$
(1)

where $\|\cdot\|_{kF}$ denotes a preferentially weighted Frobenius norm. In
particular, we preferentially weight the translation component of the pose
matrix to balance the relative sensitivity of the homogeneous transform to
rotational and translation displacements. In this work, a preferential
weighting of 5-to-1, translation to rotation weighting, is determined
empirically.

Many objects of interest have geometric symmetries and there are a continuum
of equally valid grasp poses
$\prescript{}{}{\bm{X}}_{G_{i}}\in\mathcal{G}_{i}$ for an object $i$ described
by a set $\mathcal{G}_{i}\subset\mathrm{SE}(3)$. To address this we allow the
pose $\prescript{}{}{\bm{X}}_{G_{i}}$ in (1) to vary within the constraint set
$\mathcal{G}_{i}$ depending on the end-effector pose
$\prescript{}{}{\bm{X}}_{H}$. That is, for a given object $i$, then
$\prescript{}{}{\bm{X}}_{G_{i}}:=\prescript{}{}{\bm{X}}_{G_{i}}(\prescript{}{}{\bm{X}}_{H})$
is chosen as a function of the end-effector pose

$\prescript{}{}{\bm{X}}_{G_{i}}:=\operatorname*{arg\,min}_{\prescript{}{}{\bm{X}}_{G}\in\mathcal{G}_{i}}\left(\|\prescript{}{}{\bm{X}}_{H}-\prescript{}{}{\bm{X}}_{G}\|^{2}_{kF}\right).$

(a)

(b)

Figure 2:  (a) The proposed network architecture. (b) Visualisation of the
dynamic robustness of the reaching performance. The robot is undertaking a
real-time reaching trajectory, however, it is stopped every 3s and the scene
is rearranged. The Lyapunov value of the image grid cell is coded by colour as
shown by the colour-bar. Initially, target instance 1 has the lowest Lyapunov
value and the reaching trajectory is focusing on this instance. The control is
unchanged with the addition of another target instance 3. After target 1 is
removed, the reaching trajectory refocuses on target instance 2. The
introduction of any extra instances or distractors makes no impact on the
successful grasp achieved at time $t=9s$.

###  II-B Velocity Controller Design

The velocity controller is derived based on the differentiation of the cLf to
guarantee the decrease property of the cLf. This decrease property makes the
proposed non-optimal suppression possible.

To formulate the velocity control, we use the velocity Jacobian
$\bm{J}=\bm{J}(\bm{\theta})$ for the manipulator. Denoting the angular and
translational rigid body velocity of $\prescript{}{}{\bm{X}}_{H}$ expressed in
its body-fixed frame $\\{H\\}$ by $\bm{\omega}\in\mathbb{R}^{3\times 1}$ and
$\bm{v}\in\mathbb{R}^{3\times 1}$ respectively, one has
$(\bm{\omega},\bm{v})^{\top}=\bm{J}(\bm{\theta})\dot{\bm{\theta}}$. The
partial differential of cLf $\mathcal{V}$ with respect to the pose of the
world frame relative to end-effector $\prescript{H}{}{\bm{X}}$ frame is
derived from Eq. (1) as:

$\nabla{\mathcal{V}{(\prescript{}{}{\bm{X}}_{H})}}=\text{proj}^{k}_{\mathfrak{se}(3)}\left(\prescript{}{}{\bm{X}}_{H}^{\top}(\prescript{}{}{\bm{X}}_{H}-\prescript{}{}{\bm{X}}_{G})\right)\in\mathfrak{se}3,$

where $\nabla$ is differentiation with respect to $\mathrm{SE}(3)$,
$\text{proj}^{k}_{\mathfrak{se}(3)}$ is the matrix projection operator that
maps an arbitrary $4\times 4$ matrix to the nearest member of
$\mathfrak{se}(3)$ measured in Frobenius norm subject to the preferential
weighting $k$. In particular, for a general matrix

$D=\begin{pmatrix}D_{11}&D_{12}\\\
D_{21}&D_{22}\end{pmatrix}\in\mathbb{R}^{4\times 4},$

with $D_{11}\in\mathbb{R}^{3\times 3}$, $D_{12}\in\mathbb{R}^{3\times 1}$,
$D_{21}\in\mathbb{R}^{1\times 3}$ and $D_{22}\in\mathbb{R}$, the matrix
projection is defined as:

$\displaystyle\text{proj}^{k}_{\mathfrak{se}(3)}(D)=\begin{pmatrix}\frac{1}{2}(D_{11}-D_{11}^{\top})&kD_{12}\\\
0&0\end{pmatrix}\in\mathfrak{se}(3),$

where $k>0$ is the scaling factor that weights the translation control
sensitivity relative to the rotational sensitivity. Let $(\cdot)^{\vee}$
denote the linear readout mapping that takes a matrix in $\mathfrak{se}(3)$
and forms the associated $(\bm{\omega},\bm{v})\in\mathbb{R}^{6}$ vector of
angular and linear velocities. The proposed joint velocity control is

$\bm{u}:=-\bm{J}(\bm{\theta})^{-1}(\nabla{\mathcal{V}{}}(\prescript{}{}{\bm{X}}_{H}))^{\vee},$
(2)

where $\bm{\theta}$ are joint angles associated to
$\prescript{}{}{\bm{X}}_{H}$. This velocity controller design guarantees, in a
closed-loop system, executing the velocity control continuously decreases the
Lyapunov value, i.e.

$\dot{\mathcal{V}}=-\text{tr}(\nabla{\mathcal{V}{}}^{\top}\nabla{\mathcal{V}{}})=-\|\nabla{\mathcal{V}{}}\|^{2}_{kF}<0,$

where $\dot{\mathcal{V}}$ denotes the time derivative of $\mathcal{V}$.

##  III Learning the Control Lyapunov Function

The proposed network divides the input image into square grid cells. Each cell
predicts a binary foreground visibility score, a Lyapunov value and a control
vector. All grid cells corresponding to the same visible instance share
identical supervision labels. In this work, the resolution of the grid cell is
set to $1/8$ of the input image size.

###  III-A Network Architecture

The network architecture is illustrated in Fig. 2a. We adopt an image feature
extractor as the backbone of this network; the extracted image feature is
shared between the two output regressors. The architecture is built upon
ResNet18 [15] by an auto-encoder with skip connections. The Joint auto-encoder
comprises two fully-connected layers. It learns a higher dimensional latent
representation of the joint angles $\bm{\theta}\in\mathbb{R}^{6\times 1}$. The
latent joint representation is then tiled over the spatial dimension and
concatenated with image features and the coordinate tensor. The segmentation
regressor performs binary classification to separate the foreground from the
background, and the control regressor infers the cLf value
$\widehat{\mathcal{V}}$ and velocity control $\widehat{\bm{u}}$. Residual
block with $1\times 1$ convolution in [16] is used in two output regressors.

The spatial invariance property of convolutional kernels grants fully-
convolutional architectures the potential of generalising to an arbitrary
number of object instances. However, for an image-to-control task, the network
needs access to precise spatial information. Inspired by “CoordConv” in [17,
18], we introduce a coordinate tensor input. This is a two-channel tensor,
that carries the normalised UV coordinates of each grid cells. Coordinate
tensor is concatenated with the joint features and image features as the input
to the control regressor.

###  III-B Loss Functions

Binary cross entropy loss is used for the segmentation branch:

$\mathcal{L}_{\text{seg}}=\frac{1}{N}\sum_{n=1}^{N}\left(y_{n}\,\text{log}S(\widehat{\mathcal{C}}_{n})+(1-y_{n})\,\text{log}(1-S(\widehat{\mathcal{C}}_{n}))\right),\\\
$

where $n\in\\{1,\dots,N\\}$ denotes grid cell index, $S(\cdot)$,
$\widehat{\mathcal{C}}$ and $y$ represent the sigmoid function, segmentation
regressor output and binary visibility label respectively.

A weighted L1 loss $\mathcal{L}_{\text{ctrl}}$ is used for the control branch
and this should not be confused with the Lyapunov function $\mathcal{V}$. Only
the loss generated by foreground grid cells are penalised. The control
regressor’s loss function is formulated as:

$\mathcal{L}_{\text{ctrl}}=\frac{1}{N_{\text{pos}}}\sum_{n=1}^{N}y_{n}\left(\left|\mathcal{V}_{n}-\widehat{\mathcal{V}}_{n}\right|+\frac{1}{6}\left|\bm{u}_{n}-\widehat{\bm{u}}_{n}\right|\right),$
(3)

where ${N_{\text{pos}}}$ represents the total number of grid cells in the
foreground. The final loss function is:
$\mathcal{L}=\mathcal{L}_{\text{seg}}+\mathcal{L}_{\text{ctrl}}.$

###  III-C Non-optimal Suppression

Our network predicts a tuple of values
$(\widehat{\mathcal{V}},~{}\widehat{\bm{u}})$ at each grid cell associated
with targets present in that grid cell. An essential step in the algorithm is
to select the single input $\widehat{\bm{u}}$ that is the optimal control
action proposal for the robot to act on. In the pipeline algorithms, this
control choice is determined by the instance the algorithm is targeting to
grasp, a decision process that must be explicitly coded. For end-to-end
methods such as our system, this decision process emerges from the dynamic
behaviour of the system.

Since we have derived a control action from a global control Lyapunov function
formulation and regress the actual value of the Lyapunov function along with
the control proposal, these values provide an ideal metric of optimality for
choosing the control action. By executing the action associated with the
minimal Lyapunov value chosen across the segmentation masks, the control
proposal is adapted to maximally decrease that particular instance of the cLf.
It follows, that the closed-loop motion of the robot reinforces the initial
preference for a given instance. This motion decreases the cLf value
associated to this instance more quickly than the cLf value associated with
other instances. This increases the likelihood that the same instance will
generate the minimal Lyapunov regressor for the next image input. Indeed, in
the absence of errors and for static scenes, the choice of instance is locked
in by the initial minimisation.

For real-world dynamic scenarios, the closed-loop motion of the robot can be
seen as an analogue solver for a stochastic gradient descent algorithm that
both selects an instance to target and then computes the grasp pose. We refer
to this process as non-optimal suppression since the evolution of the closed-
loop acts to increase confidence in the chosen instance while the closed-loop
trajectory is tracked. The progressive nature of the decision process and its
integration into the servo task makes the formulation naturally robust to
dynamic variations in the scene. A reaching example with visualised non-
optimal suppression process is shown in Fig. 2b.

Image grid cells associated with the same object instance share identical
regression targets. In practice, it is nearly impossible for these cells to
produce identical values. Selecting the lowest Lyapunov value amongst those
cells in a given instance segmentation provides the proposed method to
suppressing similar control action proposals. Statical methods like clustering
and averaging may increase the accuracy of the inference. However, a frame-by-
frame clustering and averaging algorithm adds significant computation
overhead. Thus, we do use a cross-frame exponentially weighted moving
averaging term, known as momentum in machine learning literature, to filter
sudden changes in the closed-loop control action inference. The momentum term
is define as:

$\bar{\bm{u}}_{t}=\eta\bar{\bm{u}}_{t-1}+(1-\eta)\widehat{\bm{u}}_{t},$ (4)

where $\eta$ is a tuneable constant $\in[0,1]$ and $\widehat{\bm{u}}_{t}$ is a
current raw velocity control prediction from the network. This is equivalent
to applying a low pass filter to the raw control signal generated by the
network. The constant $\eta$ is set to 0.5 in our experiments. It leads to
smother reaching trajectories especially under high controller gain.

##  IV Implementation

We perform robotic grasping as the ultimate test for the proposed reaching
algorithm. A two-finger parallel gripper is attached to a UR5 manipulator. The
finger tips of the parallel gripper are padded with textured soft-silicone to
increase the contact area. A Realsense D435 is place over the shoulder of the
manipulator (see Fig.3). Only the RGB camera of Realsense is used for the
experiments. The camera’s frame rate is set to 60Hz and the resolution is
resized to $512\times 384$.

(a) Simulation

(b) Real-world

Figure 3: Lab and simulation environments: The first-person camera is
positioned as shown in Fig. 3b (marked with the white circle), pointing
towards the table workspace. The simulated environment is geometrically
identical to the physical layout in the lab. The simulated camera is
calibrated to simulate the real camera.

###  IV-A Data Collection

Generating large-scale, pre-labeled data with simulators is almost cost-free
compared to collecting the real-world data. Deploying a network trained purely
on synthetic data to the real-world has been proven feasible [8, 12]. In this
work, we replicate the real-world setting in Coppeliasim. The camera
extrinsics are calibrated to the manipulator base by observing a checkerboard
attached to the end-effector.

For each sample, the number of simultaneous instances is equally sampled
between one and three, and instances are simulated on the tabletop amongst a
collection of random distractors. The end-effector is positioned at a random 6
DoF pose within the workspace of the robot. The end-effector’s initial
simulated translation component is sampled on the surface of a quarter sphere,
whose radius is sampled from uniform distribution and constrained by the
manipulator’s usable workspace. The initial simulated rotation is sampled
based on axis-angle representation; the direction axis is sample inside a
downward-facing cone, and the angle is sample in $[-\pi,~{}\pi]$. Domain
randomisation [19] is only applied to all visible, non-object entities. In
order to increase data efficiency and improve the local convergence of the
closed-loop system, we increase the sampling density while the manipulator
approaches the neighbourhood of a target.

In this work, we collect three datasets for mugs, IKEA LACK Table Legs (table
leg for short), and potted meat (Spam) cans. Mug and Spam are from the YCB
dataset [20]. The “multi-table leg” dataset contains up to two simultaneous
instances; the “multi-spam” and “multi-mug” dataset contain up to three
instances. No spam was consumed in the development of this paper.

###  IV-B Network Training Details

Our training dataset contains approximately 55k simulated samples for each
object category. We train our networks with 90% of the training dataset while
using the remaining 10% for evaluation. The brightness, saturation, contrast
and hue of input images are randomly jittered at 10% of their maximum ranges
to alleviate the domain gap between simulated and real data. The weights of
ResNet18 backbone are randomly initialised. We use ADAM optimiser with the
batch size 64 for learning. The learning rate is initialised as $10^{-3}$ with
a decay rate 0.8 for every 5 epochs. The maximum training epoch is set to 80.

(a) With CoordConv

(b) Without CoordConv

Figure 4:  Visualisation of 32 reaching trajectories for two networks trained with and without CoordConv for the “multi-spam” dataset. For a fair comparison, experiments with each network share the same pre-sampled random initial end-effector poses. The test scene is static, and contains one Spam Can and distractors. The vertical axis indicates the control regression loss $\mathcal{L}_{\text{ctrl}}$ defined in Eq. (3) . |  Coord- Conv |  Grasp Success
---|---|---
|  | 1 | 2 | 3 | 4 | $\%$
Spam | with | 13 | 14 | 15 | 14 | 93.3
w/o | 12 | 11 | 12 | 11 | 76.7
LACK Leg | with | 14 | 14 | 14 | 12 | 90.0
w/o | 9 | 10 | 9 | 9 | 61.7
Mug | with | 14 | 14 | 15 | 14 | 95.0
w/o | 13 | 13 | 13 | 14 | 88.3
Avg. (%) | with | - | - | - | - | 92.8
w/o | - | - | - | - | 75.6
TABLE I: Statistical results of real-world grasping experiments using with and
without CoordConv. 60 reaching experiments (15 for one, two, three and four
simultaneous instances) are conducted for each object category.

##  V Grasping Experiments

In this section, we demonstrate the performance and robustness of the proposed
reaching method through multi-instance real-world grasping experiments. Mugs,
Table Legs, and Spam cans are used as testing objects.

For each object category, we separately conduct 60 reaching experiments (15
for one, two, three and four simultaneous instances). The gripper closes when
the Lyapunov value is below a designated threshold. We set this threshold to
0.005 for all our experiments. In each scene, the end-effector pose is
randomly initialised in the workspace, and target instances and distractors
are shuffled. IKEA table legs have larger grasping error tolerance due to
their elongated geometric shape. Hence, for IKEA table legs, we only register
a successful grasp when the gripper closes within $\pm 3~{}cm$ from the
designated grasp point. The grasping strategy for mugs is designed as opening
the gripper when its tip is inside the cavity of mugs. That is we grasp from
the inside out for mugs.

As shown in Tab.I, the proposed method achieves on average $92.8\%$ grasp
success rate over 3 different object categories without leveraging additional
sim-to-real transfer techniques apart from domain randomisation. More
importantly, the proposed algorithm maintains its high accuracy when an
arbitrary number of simultaneous instances are present. The network also
generalises beyond the number of instances included in the training dataset
for three object categories. The proposed algorithm exhibits strong false
positive rejection capability. Most of false positives in the vision pipeline
correspond to Lyapunov values larger than the current minimum and are
automatically rejected by our non-optimal suppression algorithm. Occasional
non-persistent false positives with lower Lyapunov values do disrupt the
control, however, this effect is not significant and is strongly mitigated by
the momentum term added to the closed-loop controller design formulated in
Eq.(4).

The proposed method achieves comparable overall grasp success rate to the
state-of-the-art pose estimation based multi-instance grasping [8]. Note that,
in our experiments the degrees of freedom of the object poses are constrained
since they lie on the table. However, our system regresses a full 6 DoF
control and the relative pose of the initial end-effector with respect to the
targets are unconstrained. A further key point is that the proposed network
can run at up to 160 fps on a single GTX 1080 Ti GPU. The camera that we use
only runs at 60Hz reducing the control bandwidth used the experimental studies
reported, however, the computational complexity of the approach is not a
limiting factor for higher fps than 60Hz. The real-time computational
performance of the proposed algorithm is significantly ahead of state-of-the-
art methods that use pipeline architectures (approximately 10 fps in [8]).

##  VI Ablation Study

The CoordConv module that tiles the grid cell representation with coordinates
is a crucial component of the proposed network, enhancing stable and reliable
real-world reaching. Islam et al. [21] provide evidence to support that
spatial information can be implicitly learnt by applying zero-padding
operations. However, this implicitly learnt spatial information has proved to
be insufficient for accurate visuomotor reaching task. Including the
coordinate labels into the input to the following convolutional layers
provides a simple way to compute geometric information from the image.

We observe the two networks with and without CoordConv have nearly identical
training and evaluation convergence for the three synthetic datasets. However,
in the real-world, without the CoordConv module, grasping success rate drops
from 92.8% to 75.6% (Tab.I). The major cause of failures is the imprecise and
unstable final reaching trajectory convergence. That is, the network without
the CoordConv module is able to identify targets, roughly servo control
towards the goal, but lacks more accurate geometric information for reliable
grasping.

For further investigation, we additionally perform 32 sets of reaching
experiments on one static scene with a single instance. In each set of
experiments, the networks with and without CoordConv separately servo the
robot from identical initial end-effector pose to the target instance. The
end-effector pose is randomised among different sets. Trajectories of tool-
centre point are visualised in Fig.4a and Fig.4b. It is clear that the network
with CoordConv produces significantly smoother trajectories and more stable
and precise local reaching convergence.

The joint auto-encoder is another important source of information for the
network regression. This allows the network to learn a higher dimensional
latent representation of lower dimensional input (i.e. joint angles in
$\mathbb{R}^{6}$) prior to being concatenated with image feature and
coordinate tensor. Although two networks with and without the joint auto-
encoder exhibit similar training convergence on three synthetic datasets, the
inclusion of joint auto-encoder achieves better generalisation over the
evaluation datasets; approximately 47.9% 30.6% and 6.0% relative performance
improvements are observed on the “multi-spam”, “multi-table leg”, and“multi-
mug” datasets respectively.

##  VII Conclusion

We propose a fully-convolutional, image-to-control network for a multi-
instance robotic reaching task. In particular, we formulate control actions
based on a control Lyapunov function and regress both the proposed control and
the associated Lyapunov value output at a grid cell level in an end-to-end
manner. This provides us a natural structure to implement the proposed
effective and efficient non-optimal suppression strategy and reach and grasp
robustly in highly complex dynamic scenes. Our system is trained entirely on
synthetic data yet robust against the sim-to-real domain gap. Real-world
experiments on three different object categories demonstrate the system is
able to reach and with high accuracy of 92.8% amid distractors and an
arbitrary number of object instances.

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# New Uncertainty Principle for a particle on a Torus Knot

Madhushri Roy Chowdhury<EMAIL_ADDRESS>Physics and Applied Mathematics
Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata 700108, India
Subir Ghosh<EMAIL_ADDRESS>Physics and Applied Mathematics Unit,
Indian Statistical Institute, 203 B. T. Road, Kolkata 700108, India

###### Abstract

The present work deals with quantum Uncertainty Relations (UR) subjected to
the Standard Deviations (SD) of the relevant dynamical variables for a
particle constrained to move on a torus knot. It is important to note that
these variables have to obey the two distinct periodicities of the knotted
paths embedded on the torus. We compute generalized forms of the SDs and the
subsequent URs (following the Kennard-Robertson formalism). These quantities
explicitly involve the torus parameters and the knot parameters where
restrictions on the latter have to be taken into account. These induce
restrictions on the possible form of wave functions that are used to calculate
the SDs and URs and in our simple example, two distinct SDs and URs are
possible. In a certain limit (thin torus limit), our results will reduce to
the results for a particle moving in a circle.

An interesting fact emerges that in the case of the SDs and URs, the local
geometry of the knots plays the decisive role and not their topological
properties.

##  I Introduction

In recent times, the constrained motion of quantum particles on specified
space curves embedded on specific surfaces (see for example r1 ), in
particular, embedded on a torus knot r7 ; r8  \- a torus knot r2  -, has
become an important area of research. Motivations for considering torus knot
as the path are numerous: carbon nanorings made from carbon nanotubes that are
intimately connected to the edges of nanotubes, nanochains, graphene, and
nanocones r4  having a variety of notable features such as multi-resonant
properties, magneto-optical activity, paramagnetism, and ferromagnetism r5 .
Practical applications of these nanostructures are in optical communications,
isolators, traps for ions and atoms, and lubricants r6 .

The quantum mechanics of a particle on a torus knot was analyzed in r7 .
Further, topological aspects such as the Berry phase and Hannay angle feature
to this model were investigated in r8 ; in particular, curvature and torsion
effects of the knotted path were also taken into account.

Uncertainty Relations (UR) play a key role both in understanding and
applications of quantum mechanics. In the present work, we will shed some
light on the UR imposed on the quantum mechanics of particles moving on a
torus knot.

It has long been appreciated in quantum mechanics that sufficient care should
be taken in deriving Uncertainty Relations (UR) for periodic degrees of
freedom, the prime example being the UR for $\sigma_{\phi}\sigma_{L_{z}}$
where $\sigma_{\phi}$ and $\sigma_{L_{z}}$ stand for Standard Deviations (SD)
for angle $\phi$ and its conjugate momentum $L_{z}$ respectively. For
continuous variables ranging from $-\infty\rightarrow+\infty$, such as
position $x$ and momentum $p_{x}=-i\hbar\partial/\partial x$, satisfying
$[x,p_{x}]=i\hbar$, using conventional definitions for SD for a generic
variable $A,~{}\sigma_{A}=\sqrt{\langle(A-\langle
A\rangle)^{2}\rangle}=\sqrt{\langle A^{2}\rangle-\langle A\rangle^{2}}$, (with
$\langle A\rangle=\int_{-\infty}^{\infty}\psi(x)^{*}A\psi(x)dx$), one derives

$\sigma_{x}\sigma_{p_{x}}\geq\hbar/2.$ (1)

Clearly a naive replacement of $x,p_{x}$ by $\phi,L_{Z}$, obeying
$\left[\phi,\hat{L}_{z}\right]=i\hbar$ will lead to

$\sigma_{\phi}\sigma_{L_{z}}\geq\hbar/2$ (2)

which, however, is inconsistent since, for angular momentum eigen functions
$\sigma_{L_{z}}=0$ but the angle $\phi$, being periodic, is restricted to
$2\pi\geq\phi\geq 0$. The problem lies with the treatment of $\phi$ which is
continuous and in principle can run from $-\infty$ to $+\infty$ but at the
same time physics has to be the same for $\phi=\beta$ and $\phi=\beta+2n\pi,n$
an integer. Various inequivalent solutions have been proposed r9 . Recent
works in related areas are r10 .

In the present paper, we will closely follow the framework proposed by Ogawa
and Nagasawa on  that suggests working in terms of variables that are
explicitly periodic functions of  $\phi$. In on  the problem of a quantum
particle on a circle and its corresponding URs are studied in detail. In the
present work, a non-trivial extension of the above, a quantum particle on a
Torus knot, will be considered. It is important to notice that although the
particle on a circle problem can be stated with one degree of freedom, it
requires the Cartesian $X-Y$-plane with a two-variable parameterisation to
correctly account for the restricted domain of $\phi$ on . In the same spirit,
in our model, although the particle on a torus knot path embedded on a torus
can be described by one variable $\phi$ (due to the torus knot constraint), it
requires the Cartesian $X-Y-Z$ space with a three-variable parameterisation to
correctly account for the restricted domain of $\phi$ r7 ; r8 . Furthermore,
the novelty also comes from the explicit presence of the torus knot parameters
in the URs that can influence the latter. These works pertain to a general
remark by Ogawa that ”The uncertainty broadens out the space!” oga  since due
to quantum correlations, the mean position of the particle is at a coordinate
that is not in the particle’s configuration space (similar to the centre of
mass of a ring being at the centre, outside the ring).

##  II Particle on $S_{1}$ (circle)

Let us make a quick digression to the problem of particle on $S_{1}$ on
before moving on to our more complex problem at hand, particle on a torus
knot; essentially same framework will be exploited in the latter problem.
Inherently periodic Cartesian coordinates $X=Acos\phi,Y=Asin\phi$ are used as
basic position variables instead of $\phi$. Since $X^{2}+Y^{2}=A^{2}$ and
$\langle X\rangle^{2}\leq\langle X^{2}\rangle,~{}\langle
Y\rangle^{2}\leq\langle Y^{2}\rangle$, the authors of on  define the ”Mean
Resultant Length” (MRL) $\mathcal{A}$

$\mathcal{A}=\sqrt{\langle X\rangle^{2}+\langle Y\rangle^{2}}$ (3)

agreeing to the inequality

$\mathcal{A}=\sqrt{\langle X\rangle^{2}+\langle Y\rangle^{2}}\leq\sqrt{\langle
X^{2}+Y^{2}\rangle}=A.$ (4)

Following the definition
$\langle\\{X,Y\\}\rangle=\int_{0}^{\infty}\\{X,Y\\}|\psi(\phi)|^{2}d\phi$ on ,
let us introduce $\langle\phi\rangle$ when $\mathcal{A}\neq 0$, as $\langle
X\rangle=\mathcal{A}~{}cos\langle\phi\rangle,~{}\langle
Y\rangle=\mathcal{A}~{}sin\langle\phi\rangle$. Hence for uniform amplitude
$\mid\psi\mid$ one has $\mathcal{A}=0$ indicating that the mean position of
the particle is at the origin of the circle.

To construct URs involving the SDs, using $\Delta X=X-\langle
X\rangle,~{}\Delta Y=Y-\langle
Y\rangle,~{}\Delta\hat{L}_{z}=\hat{L}_{z}-\langle\hat{L}_{z}\rangle$, one has

$\sigma^{2}_{X}=\langle(\Delta X)^{2}\rangle=\langle X^{2}\rangle-\langle
X\rangle^{2},~{}\sigma^{2}_{Y}=\langle(\Delta Y)^{2}\rangle=\langle
Y^{2}\rangle-\langle Y\rangle^{2}$ (5)

and the commutation relations

$\left[X,\hat{L}_{z}\right]=-i\hbar Y,~{}\left[Y,\hat{L}_{z}\right]=i\hbar X.$
(6)

To follow the Kennard-Robertson kr  method (see also sch ), a real parameter
$\lambda$ is introduced such that

$\mid\mid(\Delta
X-i\lambda\Delta\hat{L}_{z})\mid\psi\rangle\mid\mid^{2}=\int\left[\\{(\Delta
X-i\lambda\Delta\hat{L}_{z})\psi\\}^{\ast}\\{(\Delta
X-i\lambda\Delta\hat{L}_{z})\psi\\}\right]dx$ (7)

yielding

$\langle\psi\mid(\Delta X+i\lambda\Delta\hat{L}_{z})(\Delta
X-i\lambda\Delta\hat{L}_{z})\mid\psi\rangle\geq 0$ (8)

leading to

$\Rightarrow\langle\psi\mid(\Delta
X)^{2}\mid\psi\rangle+\lambda^{2}\langle\psi\mid(\Delta\hat{L}_{z})^{2}\mid\psi\rangle-i\lambda\langle\psi\mid[\Delta
X,\Delta\hat{L}_{z}]\mid\psi\rangle\geq 0$ (9)

Using $[\Delta X,\Delta\hat{L}_{z}]=[X,\hat{L}_{z}]=-i\hbar Y$, we get,

$\langle\psi\mid(\Delta
X)^{2}\mid\psi\rangle+\lambda^{2}\langle\psi\mid(\Delta\hat{L}_{z})^{2}\mid\psi\rangle-\hbar\lambda\langle\psi\mid
Y\mid\psi\rangle\geq 0.$ (10)

This inequality will hold for any real value of $\lambda$, provided the
discriminant is $\leq 0$ so that

$(n\hbar)^{2}\langle\psi\mid Y\mid\psi\rangle^{2}-4\langle\psi\mid(\Delta
X)^{2}\mid\psi\rangle\langle\psi\mid(\Delta\hat{L}_{z})^{2}\mid\psi\rangle\leq
0$ (11)

Finally, the URs are obtained as

$\sigma_{X}\sigma_{\hat{L}_{z}}\geq\frac{\hbar}{2}\mid\langle
Y\rangle\mid,~{}\sigma_{Y}\sigma_{\hat{L}_{z}}\geq\frac{\hbar}{2}\mid\langle
X\rangle\mid$ (12)

This approach will be followed by us in deriving the torus knot URs.

##  III Torus, Torus Knot and Uncertainty Relations

A torus is the surface of revolution generated by revolving a circle in 3D
space about an axis coplanar with the circle. If the axis of revolution does
not touch the circle, then the surface has a ring shape and it is called the
torus of revolution. Torus knots are a special class of closed curves wound on
the surface of a geometric torus in $\Re^{3}$. In a $(p,q)$ torus knot, $p$
denotes the number of windings in the toroidal direction(about the axis of
rotational symmetry of the torus) and $q$ denotes the number of windings in
the poloidal directions (about the cross-sectional axis of symmetry of the
torus). Both $p$ and $q$ are natural numbers and they are coprime meaning that
the only positive integer that is a divisor of both $p,q$ is $1$. A torus knot
is trivial iff either $p$ or $q$ is equal to $1$ or $-1$ r2 ; r7 ; r8 .

Let us define $\alpha=-\frac{q}{p}$ to be the winding number of (p,q) torus
knot, that connects the angles $\theta$ and $\phi$ through $p\theta+q\phi=0$,
where $\eta,\theta,\phi$ constitute the toroidal coordinates, as given below.
It is a measure of the complexity and unique identity of the (p,q) torus knot.
The trefoil knot or the (2,3) torus knot is the simplest non-trivial knot (see
Figure).

The toroidal coordinates are defined as,

$x=\frac{a~{}sinh\eta~{}cos\phi}{cosh\eta-
cos\theta},~{}y=\frac{a~{}sinh\eta~{}sin\phi}{cosh\eta-
cos\theta},~{}z=\frac{a~{}sin\theta}{cosh\eta-cos\theta}$ (13)

where $0<\eta<\infty$, $-\pi<\theta<\pi$ and $0<\phi<2\pi$.

Now, $\eta=\eta_{0}$ fixes the toroidal surface on which the knot is wound and
by imposing the winding number condition $\theta=\alpha\phi$, we have only one
independent variable $\phi$. Thus a particular $\alpha$ torus knot is
characterized by

$x=\frac{a~{}sinh\eta_{0}~{}cos\phi}{cosh\eta_{0}-cos(\alpha\phi)},~{}y=\frac{a~{}sinh\eta_{0}~{}sin\phi}{cosh\eta_{0}-cos(\alpha\phi)}~{}z=\frac{a~{}sin(\alpha\phi)}{cosh\eta_{0}-cos(\alpha\phi)}$
(14)

where $0<\phi<2\pi p$. The major radius and the minor radius of a torus are
represented by R and d respectively. We define the parameter
$a^{2}=R^{2}-d^{2}$ and the aspect ratio to be $R/d=cosh(\eta_{0})$.

The required commutation relations will be

$[x,\hat{L}_{z}]=i\hbar\frac{\partial
x}{\partial\phi},~{}~{}[y,\hat{L}_{z}]=i\hbar\frac{\partial
y}{\partial\phi},~{}~{}[z,\hat{L}_{z}]=i\hbar\frac{\partial z}{\partial\phi}.$
(15)

It is easy to see that using the relations (14), the above commutation
relations will be very complicated. First, we use a compact notation

$x=\frac{a\beta cos\phi}{\gamma-cos(\alpha\phi)},~{}~{}y=\frac{a\beta
sin\phi}{\gamma-cos(\alpha\phi)},~{}~{}z=\frac{asin(\alpha\phi)}{\gamma-
cos(\alpha\phi)}$ (16)

where $\beta=sinh\eta_{0},\gamma=cosh\eta_{0}$ and

$x^{2}+y^{2}+z^{2}=a^{2}\frac{\beta^{2}+sin^{2}(\alpha\phi)}{\left(\gamma-
cos(\alpha\phi)\right)^{2}}.$ (17)

Since $q,p$ are integers but $\alpha=-q/p$ is not necessarily an integer,
$x,y,z$ are periodic under $\phi\rightarrow\phi+2p\pi$. In the present work,
we will restrict ourselves to positive $p,q$ values.

Figure: The $(2,3)$ Trefoil knot

To produce analytical results and for a comparison with the results obtained
for particle on circle on , we invoke the popular thin torus approximation
with $a>>0$ and a large aspect ratio, i.e., large $\eta_{0}$ so that to
$O(1/\eta_{0})$, $\beta={\sqrt{\gamma^{2}-1}}\approx\gamma-1/\gamma$. Thus we
now work with the parameterisations,

$x=a~{}cos\phi\left(1+\frac{cos(\alpha\phi)}{\gamma}\right),~{}~{}y=a~{}sin\phi\left(1+\frac{cos(\alpha\phi)}{\gamma}\right),~{}~{}z=a~{}\frac{sin(\alpha\phi)}{\gamma},$
(18)

with $L_{z}=-i\hbar\partial/\partial\phi$.By dropping the $1/\gamma$-terms, a
circle of radius $a$ is recovered. Note that the same periodicity
$\phi\rightarrow\phi+2p\pi$ is maintained. The relation (17) reduces to

$x^{2}+y^{2}+z^{2}=a^{2}(1+\frac{2}{\gamma}cos(\alpha\phi))$ (19)

(with $(a/\gamma)^{2}\approx 0$), which can further be rewritten as

$x^{2}+y^{2}+z^{2}=a^{2}\left(1+\frac{2}{\gamma}{\sqrt{1-\left(\frac{\gamma
z}{a}\right)^{2}}}~{}\right).$ (20)

Notice that for coordinates on the torus knot, in the thin torus
approximation, in (18), $z<<x,y$ since $z$ has a factor of $1/\gamma$. Finally
(20) simplifies to

$x^{2}+y^{2}+(1+\gamma)z^{2}=a^{2}\left(1+\frac{2}{\gamma}\right).$ (21)

Using (15) in the Kennard-Robertson approach, the commutators in this
approximation turn out to be

$[x,L_{z}]=-i\hbar\left(y+\frac{\alpha}{a}zx\right),~{}[y,L_{z}]=i\hbar\left(x-\frac{\alpha}{a}zy\right),~{}[z,L_{z}]=i\hbar\alpha\left(\frac{a}{\gamma}-\frac{\gamma}{2a}z^{2}\right).$
(22)

The cherished forms of URs for the particle on a $\alpha$-torus knot are given
by

$\sigma_{x}^{2}\sigma_{L_{z}}^{2}\geq\frac{\hbar^{2}}{4}\left<y+\frac{\alpha}{a}zx\right>^{2},~{}\sigma_{y}^{2}\sigma_{L_{z}}^{2}\geq\frac{\hbar^{2}}{4}\left<x-\frac{\alpha}{a}zy\right>^{2},~{}\sigma_{z}^{2}\sigma_{L_{z}}^{2}\geq\frac{\hbar^{2}\alpha^{2}}{4}\left<\frac{a}{\gamma}-\frac{\gamma}{2a}z^{2}\right>^{2}.$
(23)

It is worthwhile to note that for $\alpha=0$ the torus knot disappears and as
expected, our results reduce to the simpler results of a particle on a circle
(12) on . Another significant observation is that the URs depend on the local
geometric structure of the knots and not on their topological properties. It
is well known that the $(p,q)$ knot is topologically equivalent to the $(q,p)$
knot, which is not reflected in (23) since the results will change under
$\alpha\rightarrow 1/\alpha$.

These results are new and constitute an important part of our work.

###  III.1 Application: Standard Deviations

$L_{z}$-eigenfunctions $\psi_{n}$ will be of the form

$\psi_{n}=\frac{1}{{\sqrt{2p\pi}}}e^{i\frac{n}{p}\phi}$ (24)

for integer $n$. The $p$-dependence results from the specific periodicity
connected to the $p,q$ torus knot. Let us consider a superposition $\psi$ of
two such solutions

$\psi=\frac{1}{{2\sqrt{p\pi}}}(e^{i\frac{n}{p}\phi}+e^{i\frac{k}{p}\phi})$
(25)

and use it to compute the expectation values

$<x>=\int_{0}^{2p\pi}\psi^{*}~{}x~{}\psi~{}d\phi,~{}~{}<L_{z}>=\int_{0}^{2p\pi}\psi^{*}~{}L_{z}~{}\psi~{}d\phi$
(26)

We recall that $\alpha=-q/p$ and $n,k,q,p$ all are integers and with

$<x>=\int_{0}^{2p\pi}\left(\frac{1}{{2\sqrt{p\pi}}}\right)^{2}(e^{-i\frac{n}{p}\phi}+e^{-i\frac{k}{p}\phi})\left(a~{}cos\phi\left(1+\frac{cos(\alpha\phi)}{\gamma}\right)\right)(e^{i\frac{n}{p}\phi}+e^{i\frac{k}{p}\phi})d\phi$
(27)

and similarly for $<y>,<z>$, the results are

$<x>=\frac{a}{2\gamma}\delta^{1}_{\alpha}+\frac{a}{2}\delta^{1}_{\frac{|k-n|}{p}}+\frac{a}{4\gamma}\left(\delta^{1}_{\alpha+\frac{|k-n|}{p}}+\delta^{1}_{\alpha-\frac{|k-n|}{p}}\right),~{}~{}<y>=0,~{}~{}<z>=0,$
(28)

From the non-triviality condition of a torus knot, $p\neq 1,q\neq 1,p\neq
q,\alpha\neq 1$, the first term in RHS of $<x>$ vanishes thereby leading to

$<x>=\frac{a}{2}\delta^{1}_{\frac{|k-n|}{p}}+\frac{a}{4\gamma}\left(\delta^{1}_{\alpha+\frac{|k-n|}{p}}+\delta^{1}_{\alpha-\frac{|k-n|}{p}}\right),~{}~{}<y>=0,~{}~{}<z>=0.$
(29)

Let us rewrite $<x>$ as

$<x>=\frac{a}{2}\delta^{p}_{|k-n|}+\frac{a}{4\gamma}\left(\delta^{(p+q)}_{|k-n|}+\delta^{(p+q)}_{-|k-n|}\right),~{}~{}<y>=0,~{}~{}<z>=0.$
(30)

Furthermore, since $p,q$ are positive integers, we finally obtain

$<x>=\frac{a}{2}\delta^{p}_{|k-n|}+\frac{a}{4\gamma}\delta^{(p+q)}_{|k-n|},~{}~{}<y>=0,~{}~{}<z>=0.$
(31)

Expectation value of $<L_{z}>$ turns out to be

$<L_{z}>=\int_{0}^{2p\pi}\frac{1}{{2\sqrt{p\pi}}}(e^{-i\frac{n}{p}\phi}+e^{-i\frac{k}{p}\phi})\left(-i\hbar\frac{\partial}{\partial\phi}\right)\frac{1}{{2\sqrt{p\pi}}}(e^{i\frac{n}{p}\phi}+e^{i\frac{k}{p}\phi})d\phi$
$=\frac{(n+k)}{2p}\hbar,~{}~{}k\neq n.$ (32)

It will be interesting to compare the above results with those obtained in on
for particle on a circle,

$<X>=\frac{1}{2}\delta^{1}_{|n-k|},~{}~{}<Y>=0,~{}~{}<L_{z}>=\frac{(n+k)}{2}\hbar$
(33)

(i) For the torus nearly reducing to a circle (extremely thin torus) $a\approx
R,1/\gamma\approx 0$ and $p=1$ (no non-trivial knot) our results (29) reduces
to the circular case (33), as discussed in on .
(ii) The two expressions for $L_{z}$ in (32) and (33) become identical.
(iii) For a knot, $p,q$ are integers ($\neq 1$), $<L_{z}>$ for particle on a
knot (32) is smaller than $<L_{z}>$ for particle on a circle (33). For all
$<X>=<Y>=0$ or $<x>=<y>=<z>=0$ the particle is effectively at the centre of
the circle or torus respectively. To obtain a non-zero mean position, the
superposition in $\psi$ has to follow restrictions: on the circle $|n-k|=1$
and two possibilities on the torus $|n-k|=p$ or $|n-k|=p\pm q$. Notice the
knot parameters are explicitly entering the outcomes in a non-trivial way.
Obviously numerical values will also differ due to the torus and circle
parameters in the expressions.

In order to compute Standard Deviations (SD)

$\sigma_{x}^{2}=<x^{2}>-<x>^{2},~{}~{}\sigma_{y}^{2}=<y^{2}>-<y>^{2},~{}~{}\sigma_{z}^{2}=<z^{2}>-<z>^{2}$
(34)

we first calculate

$<x^{2}>=\frac{a^{2}}{2},~{}<y^{2}>=\frac{a^{2}}{2}+\frac{a^{2}}{4\gamma^{2}},~{}<z^{2}>=\frac{a^{2}}{2\gamma^{2}}.$
(35)

It is somewhat surprising to find that the above results do not depend on the
knot parameters (as these conspire to get cancelled), at least in the present
level of approximation and the particular form of $\psi$ considered.

From the result

$<L_{z}^{2}>=\frac{(n^{2}+k^{2})\hbar^{2}}{2p^{2}}$ (36)

we get

$\sigma_{L_{z}}^{2}=\frac{(n^{2}+k^{2})\hbar^{2}}{2p^{2}}-\left(\frac{(n+k)}{2p}\hbar\right)^{2}=\frac{\hbar^{2}}{4p^{2}}(n-k)^{2}.$
(37)

Considering the two possible restrictions

$(I)~{}~{}|n-k|=p:~{}\sigma_{x}\approx\frac{a}{2},~{}\sigma_{y}\approx\frac{a}{{\sqrt{2}}},~{}\sigma_{z}=\frac{a}{\sqrt{{2}}\gamma},~{}\sigma_{L_{z}}=\frac{\hbar}{2}$
(38)
$(II)~{}~{}|n-k|=p+q:~{}\sigma_{x}\approx\frac{a}{{\sqrt{2}}},~{}\sigma_{y}\approx\frac{a}{{\sqrt{2}}},~{}\sigma_{z}=\frac{a}{\sqrt{{2}}\gamma},~{}\sigma_{L_{z}}=\frac{\hbar(p+q)}{2p}.$
(39)

Interestingly, $\sigma_{L_{z}}$ for the choice (I) is the same as the circular
case for $|n-k|=1$ on .

We exhibit the results obtained so far in a table,

expec. values | $(I):p=|n-k|$ | $(II):p+q=|n-k|$
---|---|---
$<x>$ | $\frac{a}{2}$ | $\frac{a}{4\gamma}$
$<y>$ | 0 | 0
$<z>$ | 0 | 0
$<x^{2}>$ | $\frac{a^{2}}{2}$ | $\frac{a^{2}}{2}$
$<y^{2}>$ | $\frac{a^{2}}{2}+\frac{a^{2}}{4\gamma^{2}}$ | $\frac{a^{2}}{2}+\frac{a^{2}}{4\gamma^{2}}$
$<z^{2}>$ | $\frac{a^{2}}{2\gamma^{2}}$ | $\frac{a^{2}}{2\gamma^{2}}$
$<\sigma_{x}>$ | $\frac{a}{2}$ | $\frac{a}{{\sqrt{2}}}$
$<\sigma_{y}>$ | $\frac{a}{{\sqrt{2}}}$ | $\frac{a}{{\sqrt{2}}}$
$<\sigma_{z}>$ | $\frac{a}{{\sqrt{2}\gamma}}$ | $\frac{a}{{\sqrt{2}\gamma}}$
$<L_{z}>$ | $\frac{\hbar(n+k)}{2p}$ | $\frac{\hbar(n+k)}{2p}$
$<L^{2}_{z}>$ | $\frac{(n^{2}+k^{2})\hbar^{2}}{2p^{2}}$ | $\frac{(n^{2}+k^{2})\hbar^{2}}{2p^{2}}$
$<\sigma_{L_{z}}>$ | $\frac{|n-k|\hbar}{2p}=\frac{\hbar}{2}$ | $\frac{(p+q)\hbar}{2p}$

Table: Expectation values and Standard Deviations

###  III.2 Application: Uncertainty Relations

Let us return to the URs derived earlier in (23)

$\sigma_{x}\sigma_{L_{z}}\geq\frac{\hbar}{2}\left<y+\frac{\alpha}{a}zx\right>=\frac{\hbar}{2}\left<y-\frac{q}{pa}zx\right>,$
(40)
$\sigma_{y}\sigma_{L_{z}}\geq\frac{\hbar}{2}\left<x-\frac{\alpha}{a}zy\right>=\frac{\hbar}{2}\left<x+\frac{q}{pa}zy\right>,$
(41)
$\sigma_{z}\sigma_{L_{z}}\geq\frac{\hbar\alpha}{2}\left<\frac{a}{\gamma}-\frac{\gamma}{2a}z^{2}\right>=-\frac{q}{p}\frac{\hbar}{2}\left<\frac{a}{\gamma}-\frac{\gamma}{2a}z^{2}\right>.$
(42)

To explicitly calculate the above, in the LHS we need the following:

$<zx>=0,$ (43)
$<zy>=\frac{a^{2}}{2\gamma}\delta^{1}_{\alpha}+\frac{a^{2}}{4\gamma^{2}}\delta^{1-\alpha}_{\alpha}+\frac{a^{2}}{4\gamma}\left(\delta^{\alpha}_{1+\frac{k-n}{p}}+\delta^{\alpha}_{1-\frac{k-n}{p}}\right)$
$+\frac{a^{2}}{8\gamma^{2}}\left(\delta^{\alpha}_{1+\alpha+\frac{k-n}{p}}+\delta^{\alpha}_{1+\alpha-\frac{k-n}{p}}+\delta^{\alpha}_{1-\alpha+\frac{k-n}{p}}+\delta^{\alpha}_{1-\alpha-\frac{k-n}{p}}\right).$
(44)

Pertaining to the two choices, $<zy>$ in (44) reduces to

$(I)~{}\rightarrow~{}|n-k|=p~{},<zx>=0,<zy>=\frac{a^{2}}{8\gamma^{2}};~{}~{}(II)~{}\rightarrow~{}|n-k|=p+q~{},<zx>=0,<zy>=\frac{a^{2}}{4\gamma}.$
(45)

Our aim is to compute the LHS and RHS independently for each of the URs
(40,41, 42 ) and check their validity.
(40): Our first result is that the UR involving $\sigma_{x}\sigma_{L_{z}}$ is
trivially satisfied since the RHS is zero (to the approximation we are
working).
(41): Analysis for the UR involving $\sigma_{y}\sigma_{L_{z}}$ for the two
possible options:
Choice (I): LHS
=$\frac{a\hbar}{2{\sqrt{2}}};~{}~{}RHS=\frac{a\hbar}{4}(1+\frac{q}{4p\gamma^{2}})\approx\frac{a\hbar}{4}$.
Clearly, the inequality is satisfied for choice I.
Choice (II): LHS
=$\frac{a\hbar}{2{\sqrt{2}}}(1+\frac{q}{p});~{}~{}RHS=\frac{a\hbar}{8\gamma}(1+\frac{q}{p})$
A simple algebra reveals that $LHS-
RHS\approx\frac{a\hbar}{8}(1+\frac{q}{p})(2{\sqrt{2}}-\frac{1}{\gamma})$.
Clearly the inequality is satisfied for choice II as well (since $1/\gamma$ is
small).

(42): Analysis for the UR involving $\sigma_{z}\sigma_{L_{z}}$ for the two
possible options:
Choice (I): LHS =$\frac{a\hbar}{2{\sqrt{2}}\gamma};~{}~{}RHS=-\frac{3a\hbar
q}{s\gamma p}~{}\rightarrow LHS-
RHS=\frac{a\hbar}{8\gamma}(2{\sqrt{2}}+\frac{3q}{p})$. Clearly, the inequality
is satisfied for choice I.
Choice (II): LHS
=$\frac{a\hbar}{2{\sqrt{2}}\gamma}(1+\frac{q}{p});~{}~{}RHS=-\frac{3a\hbar
q}{8\gamma p}~{}\rightarrow LHS-RHS=\frac{a\hbar
q}{8\gamma}\left(2{\sqrt{2}}(1+\frac{q}{p})+\frac{3q}{p}\right)$
Clearly, the inequality is satisfied for choice II as well.

To conclude this section, we note that we have derived new structures of
Uncertainty Relations applicable to a quantum particle on a torus knot. These
are non-trivial extensions of similar results for a particle on a circle,
discussed in on . We have considered a specific form of $\psi$ to explicitly
show that all the URs are satisfied. These constitute part of our main
results.

###  III.3 Application: Mean Resultant Length

Following on  let us define a Mean Resultant Length (MRL)

$R\equiv{\sqrt{<x>^{2}+<y>^{2}+(1+\gamma)<z>^{2}}}\leq{\sqrt{1+\frac{2}{\gamma}}}~{}a\approx\left(1+\frac{1}{\gamma}\right)a$
(46)

which holds because for each $j,~{}<x_{j}>^{2}<<x_{j}^{2}>$ and hence

$R={\sqrt{<x>^{2}+<y>^{2}+(1+\gamma)<z>^{2}}}\leq{\sqrt{<x^{2}>+<y^{2}>+(1+\gamma)<z^{2}>}}$
$={\sqrt{<x^{2}+y^{2}+(1+\gamma)z^{2}>}}={\sqrt{1+\frac{2}{\gamma}}}~{}a\approx\left(1+\frac{1}{\gamma}\right)a.$
(47)

A direct calculation of $R$ in (46) using $<x>,<y>,<z>$ derived above, we
find, for choice (I)

$R_{I}=<x>_{I}=\frac{a}{2},$ (48)

and for choice (II)

$R_{II}=<x>_{II}=\frac{a}{4\gamma}.$ (49)

Clearly, both choices satisfy the inequality (46).

Finally we combine the three URs in (23) to obtain

$\sigma^{2}_{R}\equiv\left(\sigma_{x}^{2}+\sigma_{y}^{2}+\left(1+\frac{1}{\gamma}\right)\sigma_{z}^{2}\right)\sigma_{L_{z}}^{2}\geq\frac{\hbar^{2}}{4}\left<y+\frac{\alpha}{a}zx\right>^{2}+\frac{\hbar^{2}}{4}\left<x-\frac{\alpha}{a}zy\right>^{2}+\left(1+\frac{1}{\gamma}\right)\frac{\hbar^{2}\alpha^{2}}{4}\left<\frac{a}{\gamma}-\frac{\gamma}{2a}z^{2}\right>^{2}.$
(50)

However, it is more natural to define a quantity $\tilde{\sigma}_{R}$, given
by,

$\tilde{\sigma}_{R}=\frac{{\sigma}_{R}}{R}$ (51)

that turn out to be (to our $O(1/\gamma)$ approximation), for the two options,

$\tilde{\sigma}_{R(I)}\sigma_{L_{z}}\geq\frac{\hbar}{2},$ (52)
$\tilde{\sigma}_{R(II)}\sigma_{L_{z}}\geq\frac{\hbar}{2}\left(1+\frac{q}{p}+\left(\frac{3q}{p}\right)^{2}\right).$
(53)

Comparison between the above two URs suggests that, whereas for option (I),
the normal UR is maintained for torus knots (in our approximate scheme), in
option (II) the lower bound of the UR increases, suggesting that the motion of
the quantum particle in a (torus) knot tend to increase the overall SDs and
hence uncertainties.

##  IV Conclusion and future outlook

Let us summarize our framework and findings. We have studied Uncertainty
Relations associated with a quantum particle on a torus knot. The motivation
is two-fold: on one hand, the treatment of quantum mechanics of particles on a
periodic path needs to be addressed very carefully. There are very few works
in literature that study periodic motion in more complicated paths than a
circular path. On the other hand, choosing a path embedded on a torus surface
has many practical applications, apart from its theoretical interest; it is a
nontrivial extension of the path on a circle, (we have followed the framework
of the recent work on ), being in a higher dimensional configuration space and
topological intricacies due to the presence of paths with a knot.

Commutation relations between the operators in question generate the URs. For
a periodic system, only periodic degrees of freedom can be used. For circular
motion, a Cartesian parameterisation in $X-Y$ is sufficient but for the torus
knot motion, a Cartesian parameterisation in three-dimensional $X-Y-Z$ space
is necessary. As a result more involved URs are generated, which, however, can
be reduced to the circular case under suitable (thin torus) approximation.

The periodic paths on a torus have two periodicities (characterized by the
integers $p,q$) compared to paths on a circle with a single periodicity, with
$p,q$ obeying certain constraints for a knotted path. For explicit
computation, we consider as an example a superposition of two eigenfunctions
as a generic wave function and verify the validity of the URs. Interestingly,
we show that, even in this simple example, restrictions on $p,q$ yield two
distinct superpositions are possible, giving rise to inequivalent URs. In all
cases, the new Standard Deviations and URs involve (torus and knot parameter)
corrections over the conventional values.

An interesting fact emerges that in the case of the SDs and URs, the local
geometry of the knots plays the decisive role and not their topological
properties. As an example note that the $(p,q)$ and $(q,p)$ torus knots are
topologically equivalent but with different local geometries and our results
are not invariant under the interchange of $p,q$.

The generalized results revealed here can be tested in experimental setups
involving particle motion in fibres along knotted paths. In a theoretical
context, it will be interesting to extend the present work in the case of
fuzzy or non-commutative torus knots.

##  V Acknowledgement

It is indeed a pleasure to thank Professor Naohisa Ogawa for numerous helpful
discussions.

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