# One-shot Learning for Autonomous Aerial Manipulation

Claudio Zito1∗, and Eliseo Ferrante1
1 Technology Innovation Institute (TII), Abu Dhabi, United Arab Emirates
Claudio.Zitotii.ae
∗ Correspondent author

###### Abstract

This paper is concerned with learning transferable contact models for aerial
manipulation tasks. We investigate a contact-based approach for enabling
unmanned aerial vehicles with cable-suspended passive grippers to compute the
attach points on novel payloads for aerial transportation. This is the first
time that the problem of autonomously generating contact points for such tasks
has been investigated. Our approach builds on the underpinning idea that we
can learn a probability density of contacts over objects’ surfaces from a
single demonstration. We enhance this formulation for encoding aerial
transportation tasks while maintaining the one-shot learning paradigm without
handcrafting task-dependent features or employing ad-hoc heuristics; the only
prior is extrapolated directly from a single demonstration. Our models only
rely on the geometrical properties of the payloads computed from a point
cloud, and they are robust to partial views. The effectiveness of our approach
is evaluated in simulation, in which one or three quadropters are requested to
transport previously unseen payloads along a desired trajectory. The contact
points and the quadroptors configurations are computed on-the-fly for each
test by our apporach and compared with a baseline method, a modified grasp
learning algorithm from the literature. Empirical experiments show that the
contacts generated by our approach yield a better controllability of the
payload for a transportation task. We conclude this paper with a discussion on
the strengths and limitations of the presented idea, and our suggested future
research directions.

##  I Introduction

There is evidence that humans possess an internal model of physical
interactions that enables us to grasp, lift, pull, or push objects of diverse
nature in various contexts [13]. It is also evident that such internal models
are constructed over time as an accumulation of experience as opposed to an
inherent comprehension of physics. In this paper, we investigate an internal
model for enabling autonomous quadrotors equipped with cable-suspended passive
grippers to generate contacts with unknown payloads for aerial transportation.
By unknown payloads, we mean that our approach does not require a full CAD
model of the object or information relative to its physical properties, such
as its centre of mass (CoM) or friction coefficients. We only rely on
geometric features extrapolated from vision. As the human’s internal model,
our solution is not failure-free but provides a way to generate candidate
grasps even when no information is available.

The hype for aerial manipulation has reached a high-fever pitch. Unmanned
aerial vehicles (UAVs), such as quadrotors, have been recently at the centre
of attention of the scientific community as the next means of autonomy. Their
dynamic simplicity, manoeuvrability and high performance make them ideal for
many applications ranging from surveillance to emergency response. More
recently, such systems have been investigated for aerial transportation of
payloads by towed cables. Small-size single and multiple quadrotors have been
employed for load transportation and deployment by designing control laws for
minimum swing and oscillation of the payload [11, 12, 17]. Although in their
dawn, current approaches disregard the generation of the contact points.
Single drones assume point-mass loads to simplify the effects of the dynamics,
and multiple drones are manually attached nearby the vertices or edges of the
load, following the intuition of maximising the moments exercised on the
object. No aerial system is capable of generating on-the-fly contacts, and it
is not clear how the proposed controllers can cope with different payloads or
contact configurations.

On the other hand, we have witnessed a growing interest in robot grasping and
manipulation tasks in the last decade [2, 19]. Although we are still far from
robots freely manipulating arbitrary objects, several promising solutions have
been proposed over the years [20, 3]. Google employed a dozen robots
interacting in parallel with their own environment to learn how to predict
what happens when they move objects around [10]. However, collecting such a
large amount of data for any task is very hard, and many researchers have
focussed on more practical solutions. Additionally, the task of aerial
manipulation presents another significant challenge–failed attempts may
irreparably damage the payload and the AUVs. A more appealing approach is to
learn models in a one-shot or a few-shot fashion when possible. Such
approaches typically employ generative models which learn probability
densities from demonstrations. By providing one or a few examples of a good
solution, we substantially reduce the searching effort when facing novel
contexts. Furthermore, when the learning space is constructed over local
features, the models tend to have a good generalisation capability within and
across object categories [9].

In [9, 1], the authors formulate dexterous grasps for a humanoid robot as
contacts between the robot’s manipulative links and the object’s surface. Only
local surface properties of the object’s geometrics are required to learn the
model, such as curvatures on the (local) contact patches. The model learns
grasp types but does not encode task-dependent features or physical properties
of the objects, and it has only been demonstrated for pick-and-place tasks.
For different tasks beyond pick-and-place, the authors enable the possibility
of encoding handcrafted loss functions, which requires (a rarely available)
insight knowledge by the user on the specific task. In [18, 5], this approach
has been extended to enable a robot to make predictions over push operations
for novel objects. One-shot learning is utilised for identifying the contacts
between the pusher and the object, and between the object and the environment,
but an extra _motion model_ needs to be learned from real or simulated
experience in order to make the predictions. In practice, the motion models
encode the task. They are designed as conditional probabilities given the
initial contact models, following the general idea that making predictions on
familiar initial conditions will yield more robust solutions. A few dozens of
examples are sufficient to learn a motion model for planar push operations
since the environment constrains the motion. Such luxury is not available for
aerial manipulation, and it would require us to approximate the drone-payload
system dynamics to make any prediction.

Figure 1: Learned contact models. (left) shows a central contact with a flat
surface, i.e. top face of the Fedex box, (middle) shows a curved contact on a
teapot, and (right) a triangular configuration for three drones. The red
points represent the sub-sampled local area of the surface used to learn the
contact model. Black points represent the five highest of the local features
used to learn the task. The global reference frame of the object, e.g. at the
base of the teapot in (b), is not considered in the learning.

In contrast, this paper proposes one-shot learning for aerial manipulation in
which contacts are not solely learned from local features but also encode
intrinsic task-dependent knowledge without the need for handcrafted features,
as visible in Fig. 1. The underpinning idea is simple. Imagine having to
connect a load to a single drone. If the load is a box with uniform mass
distribution, probably the best thing we could do is attach the drone in the
middle of the top facing surface, just above the payload’s CoM. In order to
find the desired spot, we do not need to have a perfect model of the object or
see it in its completeness. A view on its entireness of the top facing surface
would be sufficient for estimating its centre. Nevertheless, counterexamples
come to mind where merely teaching the robot to attach the payload above its
CoM may not be a robust strategy; a package with overlapping ridges makes the
desired contact area irregular and unsuitable for reliable contact. If we have
taught the robot to connect over flat surfaces, we desire to maintain this
property on novel objects. Hence, our approach weights the contact’s local
shape versus its relative location over the visible payload’s surface. The
latter is implemented as a probability distribution over the distances between
the taught contact regions and sampled patches that describe the general shape
of the visible geometry of the payload, e.g., flat surfaces, edges and
corners. The main drawback of this approach is that we will need to learn a
new contact model for each desired task and contact type. However, the
framework is capable of generalising over different shaped payloads for the
same task and contact type as presented in Sec. III.

We evaluate our proposed approach in a set of empirical experiments and
compare the results against a baseline method. First, we have modified the
approach in [9] to cope with a single or multiple UAVs equipped with cable-
suspended passive grippers, in lieu of a dexterous manipulator. From the same
training data we learned contact models for the baseline and our approach.
Then, we use the learned models to generate candidate contact points for four
test payloads. The contacts have been evaluated in a simulated environment for
a transportation task, in which the UAVs need to lift the payload and
transport it along a desired trajectory.

The rest of the paper is structured as follows. In Sec. II, we introduce the
problem formulation in terms of object-centric representation of the contacts
and the robot mechanics. Section III describes how we learn the models needed
for aerial manipulation. Section IV presents how we infer contacts on novel
shapes. The experimental evaluation is presented in Sec. V and our results are
discussed in Sec. VI. We conclude with our final remarks about the strengths
and limitations of this work and future research directions.

##  II Problem formulation

Let us begin from defining our notation. Vectors will be consider column
vectors and written in bold letters. Matrix will be written as capital
letters. We also denote by $SE(3)=\mathbb{R}^{3}\times SO(3)$ the standard
Euclidian group in a three-dimensional space, and by
$SO(3)=\\{R\in\mathbb{R}^{3x3}|R^{\top}R=\mathbf{I}_{3x3},\det[R]=1\\}$ the
special orthogonal group representing rotations in the three-dimensional
space. To describe rigid body transformations or poses, we will use the
following format of denoting $v=(\bm{\mathbf{p}},\bm{\mathbf{q}})\in SE(3)$
where $\bm{\mathbf{p}}\in\mathbb{R}^{3}$ is the translational component and
$\bm{\mathbf{q}}\in SO(3)$ is the quaternion describing the rotational
component. Without losing generality, we abuse of the bold notation for the
quaternion since they are implemented as a column vector in $\mathbb{R}^{4}$.

###  II-A Mechanics

We consider a set of $N_{q}\geq 1$ quadrotors with cable-suspended magnetic
grippers. The inertia frame $\mathbb{I}$ is defined by the unit vectors
$\bm{\mathbf{e}}_{x}=[1,0,0]^{\top}$, $\bm{\mathbf{e}}_{y}=[0,1,0]^{\top}$,
and $\bm{\mathbf{e}}_{z}=[0,0,1]^{\top}\in\mathbb{R}^{3}$, and the third
vector is aligned opposite to the direction of gravity. For each quadrotor, we
define a body-fixed frame
$\mathcal{B}_{n}=[\bm{\mathbf{b}}_{x},\bm{\mathbf{b}}_{y},\bm{\mathbf{b}}_{z}]$
located at the centre of mass (CoM) of the quadrotor with its third axis
aligned upward.

The pose $b_{n}=(\bm{\mathbf{p}}_{n},\bm{\mathbf{q}}_{n})\in SE(3)$ describes
the location of the CoM of the $n$-th quadrotor,
$\bm{\mathbf{p}}_{n}\in\mathbb{R}^{3}$, and its rotation with respect to the
inertia frame, $\bm{\mathbf{q}}_{n}\in SO(3)$. The mass and the inertia matrix
for each quadrotor are denoted as $m_{n}\in\mathbb{R}$ and
$\mathbf{J}_{n}\in\mathbb{R}^{3x3}$, respectively. We also denote the control
input for the quadrotor as the pair $(f_{n},\bm{\mathbf{M}}_{n})$, where
$f_{n}\in\mathbb{R}$ is the total thrust and
$\bm{\mathbf{M}}_{n}\in\mathbb{R}^{3}$ the generated moment with respect to
its body frame. With respect to the inertia frame, the quadrotor can generate
a thrust $fR(\bm{\mathbf{q}})\bm{\mathbf{e}}_{z}\in\mathbb{R}^{3}$, where
$R(\bm{\mathbf{q}})\in SO(3)$ is the equivalent rotation matrix to the
quaternion $\bm{\mathbf{q}}$.

Each quadrotor is equipped with a cable of length $l_{n}$ and let
$\xi_{n}\in\mathcal{S}$ be the unit-vector representing the direction of the
$n$-th cable pointing outward from the quadrotor’s CoM to the gripper. Let
$L_{n}\in SE(3)$ be the pose for the end-effector’s link for the $n$-th
quadrotor with respect to the inertia frame. The contact points are defined
with respect to the payload’s reference frame, $z\in SE(3)$, where
$m_{z}\in\mathbb{R}$ and $\mathbf{J}_{z}\in\mathbb{R}^{3x3}$ represent the
mass and the inertia matrix, respectively.

###  II-B Surface features

Figure 2: Computation of a surface feature on a teapot. (left) the full point
cloud of the teapot with its reference frame. The normal of the sampled point
is show as a blue arrow. (middle) the principal curvature directions are drown
for the same point as red and green arrows. (right) the associated reference
frame for the surface feature with respect to the object’s reference frame.

Surface features encode the geometrical properties of an object and are
derived from a 3D point cloud, as shown in Fig. 2. We represent a surface
feature as a pair $s=(v,\bm{\mathbf{r}})\in SE(3)\times\mathbb{R}^{2}$, where
$v=(\bm{\mathbf{p}},\bm{\mathbf{q}})\in SE(3)$ represents the pose of the
surface feature $s$ and $\bm{\mathbf{r}}\in\mathbb{R}^{2}$ is a vector of the
surface descriptors. All poses denoted by $v$ are specified relative to the
inertia frame $\mathcal{I}$.

To compute the pose $v\in SE(3)$, we first estimate the surface normal at
point $\bm{\mathbf{p}}$ using a PCA-based method [6]. Surface descriptors
correspond to the local principal curvatures around point $p$ [16], which lie
on the tangential plane to the object’s surface and perpendicular to the
surface normal at $p$. Let $k_{1}\in\mathbb{R}^{3}$ be the direction of
highest curvature, and $k_{2}\in\mathbb{R}^{3}$ the direction of lowest
curvature perpendicular to $k_{1}$. Let us also define
$\bm{\mathbf{r}}=[r_{1},r_{2}]^{\top}\in\mathbb{R}^{2}$ as a 2D feature vector
representing the curvatures along directions $k_{1}$ and $k_{2}$,
respectively. The surface normal and principal directions form a body-fixed
frame for the surface point $\bm{\mathbf{p}}$ and enable us to compute the 3D
orientation $\bm{\mathbf{q}}$ that is associated to the point.

###  II-C Kernel density estimator

In this work, probability density functions are approximated via kernel
density estimation (KDE) [15], which are built around surface features (see
Sec. II-B). A kernel can be described by its mean point
$\mu^{s}=(\mu_{p},\mu_{q},\mu_{r})$ and bandwidth
$\sigma^{s}=(\sigma_{p},\sigma_{q},\sigma_{r})$:

$\displaystyle\mathcal{K}(s$ $\displaystyle|\,\mu^{s},\sigma^{s})\ =$ (1)
$\displaystyle\mathcal{N}_{3}(\bm{\mathbf{p}}\,|\,\mu_{p},\sigma_{p})\
\Theta(\bm{\mathbf{q}}\,|\,\mu_{q},\sigma_{q})\
\mathcal{N}_{2}(\bm{\mathbf{r}}\,|\,\mu_{r},\sigma_{r})$

where
$s=(v,\bm{\mathbf{r}})=(\bm{\mathbf{p}},\bm{\mathbf{q}},\bm{\mathbf{r}})$ is
the surface feature being compared against the kernel, $\mathcal{N}_{k}$ is an
$k$-variate Gaussian distribution, and $\Theta$ corresponds to a pair of
antipodal von Mises-Fisher distributions forming a distribution similar to
that of a Gaussian distribution for $SO(3)$ [4].

Given a set of $N_{s}$ surface features, the probability density in a region
of space is computed as the local density of features in that region, as

$P(\bm{\mathbf{p}},\bm{\mathbf{q}},\bm{\mathbf{r}})\equiv
P(s)\simeq\sum_{i=1}^{N_{s}}w_{i}\mathcal{K}(s|s_{i},\sigma^{s})$ (2)

where $s_{i}$ corresponds to the $i$-th surface feature acting as a kernel,
$w_{i}$ corresponds to its weighting with the constraint
$\sum_{i=1}^{N_{s}}w_{i}\\!=\\!1$, and $\sigma^{s}$ is a user-defined
bandwidth for the surface features.

##  III Learning contacts

We learn contacts from a single demonstration. We require a point cloud of the
payload and the configuration of the quadrotors and their passive grippers.
Partial views of the objects are sufficient for learning reliable models under
the assumption that the surface in contact is fully visible. Figure 1 (left)
shows the case in which only the top face of a box is visible at training
time. Although, learning contacts is computational efficient as demonstrated
in [9], the time computation grows linearly with the number of links in
contact, the number of triangles in the mesh representing the links, and the
surface points considered. Furthermore, a new model needs to be learned for
different drone configurations, contact types or tasks.

###  III-A Object and task model

The object model describes the composition of a point cloud in terms of its
surface features distribution. In the literature, this approach is used to
learn the features distribution only nearby the contact area, e.g., [9, 1,
18], while in [5] the CoM of the object to be pushed is also estimated as a
distribution from visible surface features. In contrast, we decouple the
object model in two densities. The first describes the surface features
distribution nearby the contacts, while the second encodes the location of
these near-the-contact features with respect to other features in the visible
point cloud, e.g., its corners and edges.

At training time, we observe a set of contacts between the quadrotors’
manipulative links and an object point cloud. Before learning a contact model
(see Sec. III-B), we collect a set of $N_{O}$ features,
$s_{i}=(v_{i},\bm{\mathbf{r}}_{i})\in SE(3)\times\mathbb{R}^{2}$, within the
surface in contact with the manipulative link. This enables us to compute a
joint probability distribution as

$O(v,\bm{\mathbf{r}})\equiv
P(v,\bm{\mathbf{r}})\simeq\sum_{i=1}^{N_{O}}w_{i}\mathcal{K}(v,\bm{\mathbf{r}}|s_{i},\sigma^{s}).$
(3)

further referred as the _object model_ as in [9], where the weight
$w_{i}=1/N_{O}$. Figure 1 shows in red the local area considered for
extrapolating the features for two different contact types: a) flat contact
and b) curved contact.

From the same point cloud, we collect a second set of $N_{T}$ features by
uniformly sampling the visible point cloud to compute a second joint
probability as

$T(v,\bm{\mathbf{r}})\equiv
P(v,\bm{\mathbf{r}})\simeq\sum_{j=1}^{N_{T}}w_{j}\mathcal{K}(v,\bm{\mathbf{r}}|s_{j},\sigma^{t}).$
(4)

further referred as the _task model_ , where
$s_{j}=(v_{j},\bm{\mathbf{r}}_{j})\in SE(3)\times\mathbb{R}^{2}$, $\sigma^{t}$
is the bandwidth and $w_{j}=r_{1j}/r$ is the weight associated with the
feature represented by the $j$-th kernel and it is proportional to its highest
curvature normalised with respect to the maximum curvature value available
between the $T_{s}$ sampled points, $r$. This enables us to give more
importance to salient features such as corners and edges.

Figure 3: Graphical representation of the task model (left) and the link pose
(right) used in the contact model. The task model is constructed as a rigid
body transformation $u_{ij}$ in $SE(3)$ from a task feature $s_{j}$ to a
surface feature $s_{i}$ in the contact region. The link pose is computed as a
rigid body transformation $u_{ni}$ in $SE(3)$ from a surface feature $s_{i}$
in the contact region to the reference frame of the robot’s link. All the
$SE(3)$ poses $s_{j}$, $s_{i}$, $L_{n}$ are computed with respect to the
inertia frame $\mathcal{I}$.

###  III-B Contact models

The contact model describes the relation between the robot’s end-effector and
the object surface. It is trained in a one-shot fashion from a demonstration.
A teacher presents to the robot the desired contact, either in simulation or
in reality, then the local surface in contact with the robot’s link is sampled
and a probability density function is learned.

We denote by $u_{ni}=(\bm{\mathbf{p}}_{ni},\bm{\mathbf{q}}_{ni})\in SE(3)$ the
pose $L_{n}$ of the $n$-th link relative to the pose $v_{i}$ of the $i$-th
surface feature from (3). We compute this as

$u_{ni}=v_{i}^{-1}\circ L_{n}$ (5)

where $\circ$ denotes the pose composition operator, and
$v_{i}^{-1}=(-q_{i}^{-1}p_{i},q_{i}^{-1})$ is the inverse of pose $v_{i}$.

The contact density $M_{n}(u,\bm{\mathbf{r}})$ closely resembles the surface
feature kernel function in (1), and is defined as follows:

$M_{n}(u,\bm{\mathbf{r}})\simeq\sum_{i=1}^{N_{c}}w_{ni}\mathcal{N}_{3}(\bm{\mathbf{p}}\,|\,\bm{\mathbf{p}}_{ni},\sigma_{p}^{c})\
\Theta(\bm{\mathbf{q}}\,|\,\bm{\mathbf{q}}_{ni},\sigma_{q}^{c})\
\mathcal{N}_{2}(\bm{\mathbf{r}}\,|\,r_{ni},\sigma_{r}^{c})$ (6)

where $N_{c}\leq N_{O}$ is a user defined parameter to allow downsampling to
save computational time and $w_{ni}$ corresponds to the kernel’s weighting
such that $\sum_{i=1}^{N_{c}}w_{ni}\\!=\\!1$.

###  III-C Quadrotor configuration model

The configuration model encodes the poses of the quadrotors and their grippers
as demonstrated during training. For the case of a single drone, the
configuration model merely describes the kinematic relation between the
quadrotor and its gripper. However, when $n>1$ this model enables us to reduce
the configuration space when transferring the contacts to another surface by
focusing only on those configurations that resemble the one in the training
example.

The poses of the $N_{b}\geq 1$ quadrotors are represented as
$h_{n}=(b_{n},L_{n})$ with $b_{n}=(p_{n},q_{n})\in SE(3)$ the pose of the
$n$-th drone and $L_{n}=(p_{L_{n}},q_{L_{n}})\in SE(3)$ the pose of its
manipulative link. We approximate the configuration density as

$H(h)\simeq\sum_{i=1}^{N_{b}}w_{i}\mathcal{N}_{3}(\bm{\mathbf{p}}_{L}\,|\,\bm{\mathbf{p}}_{L_{i}},\sigma_{p})\
\Theta(\bm{\mathbf{q}}_{L}\,|\,\bm{\mathbf{q}}_{L_{i}},\sigma_{q}^{c})\ $ (7)

where $w_{i}=e^{-\alpha||b-b_{i}||^{2}}$ depends on the similarity between the
drone’s pose, $b$, and the kernel’s one, $b_{i}$, while
$L=(\bm{\mathbf{p}}_{L},\bm{\mathbf{q}}_{L})$ is the link pose compared
against the $N_{b}$ link poses from the kernels,
$L_{i}=(\bm{\mathbf{p}}_{L_{i}},\bm{\mathbf{q}}_{L_{i}})$.

##  IV Transferring to novel surfaces

Once the models are learned we can transfer the contacts on novel payloads.
The aim is to find local features on the new object that are similar to the
ones presented in the training and to place the robot manipulative links
accordingly. For evaluating the approach, we learn several contact models for
different tasks and contact types–we call them _testing conditions_ as
introduced in Sec. V. However, when presented with a new point cloud, we
manually select the appropriate contact model for each testing conditions.
Automatic model selection can be formulated as an optimisation problem [14] or
a learning one [8], but we kept it as out of scope for this work.

Figure 4: Graphical representation of inferring a candidate link pose for a
novel surface. First, from the task model we sample a feature $s_{qj}$ on the
novel point cloud. We then apply the transformation $u_{ij}$ observed when
learning the model to identify a feasible contact region. From that region
(circled area in the middle image), we sample a surface feature from the
object model $O(v,\bm{\mathbf{r}})$. Finally we apply the transformation
$u_{ni}$ observed at training time to place the link $L_{n}$.

###  IV-A Query density

The query density represents the distribution of link poses over a novel point
cloud. By searching for local similarities in the query point cloud’s surface
features, we can estimate the relative link pose $u$ which transfers the
learned contact onto the new surface. This process is designed to be
transferable such that a model trained upon a single object can be applied to
a variety of previously unseen objects. Figure 4 shows a graphical
representation of the method: starting from a sampled task feature, we
identify a potential area on the visible surface where to seek for a good
contact.

We define the query density as $Q(L_{n},u,v,\bm{\mathbf{r}})$ where $v\in
SE(3)$ denotes a point on the objects surface expressed in the inertial frame,
$\bm{\mathbf{r}}\in\mathbb{R}^{2}$ is the surface curvature of such a point,
$u\in SE(3)$ denotes the pose of the link relative to a local frame on the
object, and $L_{n}$ is the pose of the $n$-th link with respect to the inertia
frame.

For each link, its pose distribution over a new point cloud is given by
marginalising $Q(L_{n},u,v,\bm{\mathbf{r}})$ with respect to $u$, $v$, and
$\bm{\mathbf{r}}$. Since $L_{n}=v\circ u$ is described in terms of $v$ and
$u$, and by assuming that the density in the inertia frame of the surface
point, $v$, and the distribution of link poses relative to a surface point,
$u$, are conditionally independent given the curvature on a point,
$\bm{\mathbf{r}}$, we factorise the query density as follows

$\displaystyle Q(L_{n})=$ $\displaystyle\int\int\int
P(L_{n},u,v,\bm{\mathbf{r}})dudvdr$ (8) $\displaystyle\int\int\int
P(L_{n}|u,v)P(u,\bm{\mathbf{r}},v)dudvdr$ $\displaystyle\int\int\int
P(L_{n}|u,v)P(u|\bm{\mathbf{r}})P(v|\bm{\mathbf{r}})P(\bm{\mathbf{r}})dudvdr$

where, by following [9], we implement $P(L_{n}|u,v)$ as a Dirac function, and
$P(u|\bm{\mathbf{r}})$ as $M_{n}(u|\bm{\mathbf{r}})$, the conditional
probability that the $n$-th link will be placed at pose $u$ with respect to
the surface feature, given that this surface feature has curvature
$\bm{\mathbf{r}}$. $P(v|\bm{\mathbf{r}})$ is implemented as
$O(v|\bm{\mathbf{r}})$, the probability of the observed curvature
$\bm{\mathbf{r}}$. Finally, $P(r)=M_{n}(\bm{\mathbf{r}})O(\bm{\mathbf{r}})$ is
chosen to reinforce that the contact model $M$ and the object model $O$ are
observing the same surface feature. $M_{n}(\bm{\mathbf{r}})$ is the
distribution of features in the contact model, and $O(\bm{\mathbf{r}})$ is the
distribution of feature $\bm{\mathbf{r}}$ in the new point cloud.

Figure 5: Inferring contacts on a new shape for aerial manipulation. The first
column shows the training examples use for learning the contact models. The
first, third, fifth and sixth rows show the transferring of the contacts for
our approach. The other rows shows the baseline method [9]. The first column
shows the UAVs configuration, the surface used for learning and the contact
features. Red points are the local features of the contacts, black points are
the task features. The other columns demonstrate the link pose computed on a
new surface. The associated drone’s pose is shown with a frame using the
conventional RGB colour map for x-, y-, and z- axes, respectively.

We extend this formulation with another random variable
$u_{ij}=(\bm{\mathbf{v}}_{ij},\bm{\mathbf{q}}_{ij})$ to encode the desired
pose of the selected $i$-th surface feature of the contact model relative to
the task model’s features defined in (4). Thus, $u_{ij}=v_{i}^{-1}\circ
v_{j}$, where $s_{j}=(v_{j},q_{j})$ belong to the task model. Then we
approximate the query density by sampling $N_{Q}$ kernels centred on weighted
link poses from (7), so that

$Q(L_{n})\simeq\sum_{k=1}^{N_{q}}w_{nk}\mathcal{N}_{3}(\bm{\mathbf{p}}\,|\,\bm{\mathbf{p}}_{k},\sigma_{p})\
\Theta(\bm{\mathbf{q}}\,|\,\bm{\mathbf{q}}_{k},\sigma_{q})$ (9)

where $(\bm{\mathbf{p}}_{k},\bm{\mathbf{q}}_{k})\in SE(3)$ describes the
$k$-th kernel. The variable

$w_{nk}=\frac{1}{Z}\sum_{i=1}^{N_{i}}P(L_{n}|u_{ni},v_{i})M(u_{ni}|\bm{\mathbf{r}}_{i})M(\bm{\mathbf{r}}_{i})O(v_{i}|\bm{\mathbf{r}}_{i})O(\bm{\mathbf{r}}_{i})(\sum_{j=1}^{N_{j}}P(u_{ij}|\bm{\mathbf{r}}_{i},\bm{\mathbf{r}}_{j})T(v_{j}|\bm{\mathbf{r}}_{j})T(\bm{\mathbf{r}}_{j}))$

weights the query density according to the contact model and the task model.
To compute the weighting, we randomly sample from the new point cloud $N_{i}$
surface features from the estimated contact surface and $N_{j}$ features
according to the task model so that $s_{j}\sim T(v_{j}|\bm{\mathbf{r}}_{j})$
is the observed feature $\bm{\mathbf{r}}_{j}$ on the new point cloud. Again,
we implement $P(u_{ij}|\bm{\mathbf{r}}_{i},\bm{\mathbf{r}}_{j})$ as a Dirac
function and $T(v_{j}|\bm{\mathbf{r}}_{j})T(\bm{\mathbf{r}}_{j})$ represents
the probability of the observed curvature on the new point cloud. The value
$Z$ is a normaliser, and the value $N_{i}$ and $N_{j}$ are maintained constant
across all the experiments.

###  IV-B Contact optimisation and selection

Let us denote by $\hat{h}=\\{(b_{i},L_{i})\\}_{i}^{n}$ the poses of the
quadrotors and their gripper in $SE(3)$. The objective of the contact
optimisation is to generate candidate contacts by maximising the product of
the query densities and the quadrotors configuration density, as follows:

$\operatorname*{arg\,max}_{\hat{b}}\mathcal{J}(\hat{h})=\operatorname*{arg\,max}_{(b_{i},L_{i})}\prod_{(b_{i},L_{i})\in\hat{b}}H(h_{i})Q(L_{i})$
(10)

where $\mathcal{J}$ is the likelihood of the candidate contact on the new
point cloud. We optimise the likelihood using simulated annealing (SA) [7].

##  V Results

Figure 6: The results summarise the transportation experiments. Each solution from Fig. 5 are used for 10 trials in a simulated environment in which the single or multiple UAVs lift the payload, transport it along a circular trajectory twice, and then hover the payload above ground for at least five seconds. The x-axis represent the time in second for all plots. The top row shows the averaged position and velocity error w.r.t. the desired payload’s trajectory (baseline and proposed method separately), and both averaged rotational errors together in the last plot. The bottom row show the UAV’s thrust (together) and moments (separately). For the three UAVs condition, the values in the bottom row have been averaged for each UAV. TABLE I: Experimental parameters. All of the parameters are kept constant over the experiments. We use diagonal inertia matrices for both the UAVs and payloads. The gains values are for the $x-$, $y-$ and $z-$ components respectively. UAV and gains parameters are taken from [12]. Contacts | UAV | Payload | Gains
---|---|---|---
Learning | $m_{n}$ | 250g | $m_{z}$ | 250g |  | Controller | Cable
$N_{O}$ | 500 | $\mathbf{J}_{n}^{xx}$ | 0.0006 | $\mathbf{J}_{z}^{xx}$ | 0.006 | pos | 6, 6, 10 | 1.45, 1.45, 3
$N_{T}$ | 50 | $\mathbf{J}_{n}^{yy}$ | 0.0005 | $\mathbf{J}_{z}^{yy}$ | 0.0059 | vel | 3, 3, 6 | 3.7, 3.2, 2.5
$N_{C}$ | 500 | $\mathbf{J}_{n}^{zz}$ | 0.001 | $\mathbf{J}_{z}^{zz}$ | 0.107 | rot | 0.25, 0.25, 0.08 | 0.5, 0.5, 0.02
Inferring | $l_{n}$ | 0.5m |  | ang | 0.025, 0.025, 0.005 | 0.16, 0.16, 0.04
$N_{i}$ | 500 | # prop | 4 |  | $\xi$ | 16, 16, 16 |
$N_{j}$ | 5 | max RPM | 16400 |  | $\omega$ | 6.7, 6.7, 3.1 |
$N_{Q}$ | 1000 | min RPM | 5500 |  |

The generalisation to novel objects of the contact models has been evaluated
in a set of simulated experiments on several conditions as follows. For a
single UAV, we learn two types of contacts: (i) a contact placed at the centre
of the visible upper surface of a box, and (ii) a contact on a curved surface.
For multiple drones, we learn a triangular formation for three UAVs for a
transportation task.

In all conditions, we learn each contact model in one single demonstration.
Each training object was loaded in the simulation as full or partial point
cloud, as presented in Fig. 1. We do not retrieve global information about the
object, such as its CoM. The UAVs were manually placed to generate the desired
contacts with their passive gripper over the visible point cloud. For all
conditions, we sampled $N_{O}=500$ local features to represent the contacts,
in both our approach and baseline method, and $N_{T}=50$ for the task model
(present only in our approach). Once a contact model was learned, it was
stored and labelled for future reference. Only kinematic and geometrical
information were used to learn the models; forces and dynamics are not needed
for learning.

Once the contact models are learned, a new full or partial point cloud is
presented to the system. In Fig. 5 we show the point clouds belonging to four
objects used for testing: a) 68cm x 61 cm x 28cm FedEx box (top two rows); b)
the Stanford bunny (third and fourth rows); the hollowed triangular shape used
to learn the three drones’ contact models (fifth and seventh rows); and a 68cm
x 68 cm x 28cm FedEx box (sixth and eighth rows). We then compare the best
five solutions generated for each condition with the ones computed by an
adapted version of the grasping algorithm presented in [9]. Since [9]
considers a robot manipulator equipped with a hand, we modify it to consider
the quadrotor configuration model. Therefore, the main difference between the
two approaches remains the task model.

Furthermore, each solution from Fig. 5 is also evaluated in a simulated
environment for a transportation task. The simulator is initialised with the
computed optimised contacts, and the UAVs will need to transport the payload
along a pre-planned circular trajectory with a 1m radius performed for twelve
seconds. The trajectory is computed such that the UAVs need to lift the
payload to the starting position of the trajectory, perform the circle twice
and then hover above the ground, maintaining the payload in the same position
as the last waypoint of the trajectory. The simulation uses the PCMPC
controller as in [12], also adapted by us to control a single UAV with a
payload that is not a point mass object. Additional, Gaussian noise with 1cm
standard deviation was applied to the perceived position of the payload, and
each transportation task for each contact was repeated ten times. The results
were averaged across each object and trial and presented in Fig. 6.

The solutions for the hollowed triangular shape are from the complete point
cloud of the object, for which solution 3 for our approach and solution 4 for
the baseline suggested an upside-down contact. We discarded these solutions
from the transportation task since we could not achieve the required kinematic
of the contact. All the other solutions from the baseline methods that do not
provide contacts for each UAV, have been modified to connect the gripper to
the closest point on the payload. Considering the kinematically infeasible
solutions for the three drones condition, we had a total of 147 runs (3
contact models x 5 solutions x 10 trials $-$ 3 infeasible solutions) for the
proposed model and 147 for the baseline. In all conditions we use a set of
fixed parameters as shown in Tab. I. The next section will discuss our results
and conclude with our final remarks.

##  VI Conclusion

This paper proposed a framework for learning task-dependent contacts for
aerial manipulation. Our models are learned in a one-shot fashion and do not
require a complete CAD model of the payload or dynamic modelling. We build on
a contact-based formulation. Typically, such methods rely only on local
information, and task-dependent features must be handcrafted. In contrast,
thanks to the task model presented in Sec. III-A, we capture meta-information
regarding the task by merely looking at the geometrical features of the point
cloud, without the need for user-specific insight about the task.

In Fig. 5 the second row shows the solutions from [9], which tend to prefer
contacts in feature-rich areas of the new query (i.e. the edges), while our
approach forces the contacts to lay in the middle; ideal for transporting a
box with a uniform mass distribution by a single drone. More importantly, it
is consistent with the demonstrated example by the expert at training time; a
flat contact in the middle of the visible surface. The same principle is
visible in the third and fourth rows, where the contacts are transferred from
a teapot to a more complex shape: the Stanford bunny. Again, our approach
tends to lay the contacts as close as possible to the middle section of the
bunny, which minimises the payload’s oscillations during the experiments. The
CoM for the bunny is computed as the reference frame of the STL model, which
lays on the neck of the bunny, making contacts on its head the most effective
for transportation, and contacts along its middle section the second most
effective. This information is not available to the learned models. Therefore,
the first solution generated by the baseline on the bunny’s thigh is the worse
type of contact, and the UAV was incapable of controlling the payload
configuration accurately, which rotated on itself and swung along the entire
trajectory.

In the case of the three UAVs, the search space is more constrained by the
demonstrated configuration of the UAVs. The baseline, driven by the only aim
of maximising the similarity with respect to the observed contacts nearby the
gripper at training time, often gets stuck in local minima providing sub-
optimal solutions, in which one or two good contacts in feature-rich areas
lead to higher likelihoods than three reasonable ones. Our approach tends to
discard those local minima in the tested examples thanks to the task model by
constraining the choice of the selected contact points.

In Fig. 6 we observe the averaged difference in transporting the payloads
given the choice of the contacts. Our approach outperforms the baseline in
terms of accuracy in following the pre-planned payload trajectory and by
minimising the effort computed by the UAVs as thrust and moments. It is
interesting to observe that, in all conditions, our approach enables the UAV
to control the orientation of the payload in a more agile way. This is due to
the relative position of the contact points with respect to the payload’s
shape, which minimises oscillations during the transportation. This is clearly
visible by the lower standard deviation from the desired position and
orientation of the payload associated with our approach. Furthermore, the
velocity profile of the payload and the output to the controller (thrust and
moments) show that our contact points simplify the work of the controller in
following the desired path.

Although performed in simulation, the empirical evaluation shows the potential
of the proposed idea. Without physical and dynamic information about the
system, a full convergence to an optimal solution is impossible to achieve.
Nonetheless, such information is rarely available in reality, and this work is
the first step towards autonomous contact selection for aerial manipulation.

##  VII Strengths, limitations and future work

The presented work extends a (static) contact-based formulation of grasp
synthesis to a dynamic task. While previous efforts have focused on
integrating an approximation of such dynamics in the models, we investigate
how much knowledge about the task we can capture by disregarding the dynamics
altogether.

The probabilistic formulation could be considered as a soft simultaneous
optimisation of multiple experts. Each expert learns some of the
characteristics of the contacts within the feature space, e.g. the task model,
and weighs their opinion in the choice of the candidate contact at query time.
Any soft simultaneous optimisation of multiple criteria would work in
principle. However, the probabilistic representation allows us to draw contact
points from a continuous representation of the payload’s surface.
Nevertheless, the experts only weight geometrical properties, which do not
play a key role in defining the dynamic behaviour of the payload. We assume
that the demonstration encodes the contact type (e.g., flat contact), the task
(e.g., contact above the payload’s CoM for a transportation task) and the
visible surface of the object provides enough salient features to learn a good
task model. Although there is no need for handcrafting task features into the
model, the choice of the training data becomes, therefore, crucial.

Furthermore, our approach assumes that the contact types and tasks are
conditionally independent in our model. However, this is generally not true,
as seen in the transportation of a FedEx box and the teapot examples. Applying
the model learned on the FedEx over the Stanford bunny will lead to a poor
choice of contacts since flat surfaces are quite limited in the chosen
payload. Therefore, generalisation across contact types and tasks becomes
harder to achieve. This also limits the use of the same contact model for a
different task (e.g., flat contact next to the edge for dragging the payload)
which will require learning a new model.

With many models that need to be learned, we also need to solve the problem of
model selection when facing a novel payload. In our empirical evaluation, we
disregard this problem by providing the correct model manually at testing
time. Reinforcement learning techniques can be used for model selection, but
the task would need to be encoded in a more informative manner for this
technique to converge to a good reward function.

The choice of the principal curvatures as surface features is also limiting.
It is a baseline for determining the contact points and the general shape of
the payload, but other features may provide better information about the
intrinsic dynamic properties of the payload or the encoded task, such as
surface roughness or intensity features–or more likely a combination of
multiple features.

In order to achieve a fully autonomous aerial manipulation, all these problems
will need to be addressed. This work pioneers the problem of grasp synthesis
for aerial transportation, but with the increasing interest in aerial
manipulation and the technological maturity that we are witnessing, we would
suggest that it will become central to many applications. In future work, we
will focus on evaluating this framework on a real platform with payloads with
different dynamical properties, e.g., mass distribution. Since our approach is
a generative model, multiple candidate contacts will be provided for each
payload. Primitive dynamic motions could be used to evaluate the selected
contacts according to the observed behaviour of the payload.

## Author Contributions

CZ is the main author of this work. CZ has designed and implemented the idea,
evaluated the framework and written the manuscript. EF has supervised this
project and revised the manuscript and data.

## Acknowledgments

We thank Giuseppe Loianno and Marek Kopicki for the support in developing this
work.

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# Spin-orbit coupling and electron scattering in high-quality InSb1-xAsx
quantum wells

S. Metti  Elmore Family School of Electrical and Computer Engineering, Purdue
University, West Lafayette, Indiana 47907, USA  Birck Nanotechnology Center,
Purdue University, West Lafayette, Indiana 47907, USA  C. Thomas  Department
of Physics and Astronomy, Purdue University, West Lafayette, Indiana 47907,
USA  Birck Nanotechnology Center, Purdue University, West Lafayette, Indiana
47907, USA  D. Xiao  Department of Physics and Astronomy, Purdue University,
West Lafayette, Indiana 47907, USA  Birck Nanotechnology Center, Purdue
University, West Lafayette, Indiana 47907, USA  M. J. Manfra† Department of
Physics and Astronomy, Purdue University, West Lafayette, Indiana 47907, USA
Birck Nanotechnology Center, Purdue University, West Lafayette, Indiana 47907,
USA  Elmore Family School of Electrical and Computer Engineering, Purdue
University, West Lafayette, Indiana 47907, USA  School of Materials
Engineering, Purdue University, West Lafayette, Indiana 47907, USA  Microsoft
Quantum Lab West Lafayette, West Lafayette, Indiana 47907, USA

###### Abstract

InSb1-xAsx is a promising material system for exploration of topological
superconductivity in hybrid superconductor/semiconductor devices due to large
effective $g$-factor and enhanced spin-orbit coupling when compared to binary
InSb and InAs. Much remains to be understood concerning the fundamental
properties of the two-dimensional electron gas (2DEG) in InSbAs quantum wells.
We report on the electrical properties of a series of 30 nm InSb1-xAsx quantum
wells grown 40 nm below the surface with three different arsenic mole
fractions, $x=$ 0.05, 0.13, and 0.19. The dependencies of mobility on 2DEG
density and arsenic mole fraction are analyzed. For the $x=0.05$ sample, the
2DEG displays a peak mobility $\mu=2.4$ $\times$ $10^{5}$ cm2/Vs at a density
of $n=2.5$ $\times$ $10^{11}$ cm-2. High mobility, small effective mass, and
strong spin-orbit coupling result in beating in the Shubnikov de Haas
oscillations at low magnetic field. Fourier analysis of the Shubnikov de Haas
oscillations facilitates extraction of the Rashba spin-orbit parameter
$\alpha$ as a function of 2DEG density and quantum well mole fraction. For
$x=0.19$ at $n=3.1$ $\times$ $10^{11}$ cm -2, $\alpha\approx 300$ meV Å, among
the highest reported values in III-V materials.

††preprint: APS/123-QED

##  I Introduction

Small band gap zinc-blende III-V semiconductors (such as InAs and InSb) are
the subject of renewed interest as they may be utilized as a platform to
explore topological superconductivity. Strong spin-orbit coupling (SOC) and
proximity-induced superconductivity from a parent s-wave superconductor are
key ingredients needed to realize a topological phase [1, 2, 3, 4, 5, 6, 7, 8,
9, 10, 11]. Gate-controlled SOC is of fundamental and practical interest. The
ability to externally manipulate the strength of SOC in a 2DEG facilitates
realization of spin dependent devices, such as the spin field effect
transistor [12].

The ternary compound InSb1-xAsx is appealing compared to InSb and InAs because
it is predicted to have a higher Rashba SOC parameter $\alpha$ [13, 14, 15,
8]. The increase in SOC with arsenic mole fraction in InSb1-xAsx has been
measured in nanowires [16] and recently in a surface 2DEG [11]. Here, we focus
on 30 nm quantum wells buried under a 40 nm In0.88Al0.12Sb top barrier.
Arsenic mole fractions of $x$ = 0.05, $x$ = 0.13, $x$ = 0.19 are used in the
quantum well. Our buried quantum well design, compared to a surface layer,
allows for more straightforward comparisons of the Rashba parameter between
samples at nearly equal electric field and 2DEG density, as well as
identification of electron scattering mechanisms over a broader range of 2DEG
density. Spin-orbit coupling and the resultant zero-field spin-split bands
arises from broken inversion symmetry. Inversion asymmetry may be microscopic
bulk inversion asymmetry (BIA) or structural inversion asymmetry (SIA).
Dresselhaus SOC is associated with the lack of inversion symmetry in the
crystal lattice, e.g. zinc-blende semiconductors [17]. The Rashba mechanism is
due to asymmetry in the macroscopic potential induced by an external gate
and/or an asymmetric heterostructure design [18]. In heterostructures built
with small bandgap III-V zinc-blende semiconductors, the Rashba coupling is
usually the dominant k-linear SOC term. Values for the Rashba parameter
$\alpha$ for InSb and InAs quantum wells have been reported. Hatke et al. [19]
reported $\alpha\approx$ 120 meVÅ in deep InAs quantum wells at $n\approx$ 3
$\times$ 1011 cm-2; in modulation-doped InSb quantum wells Gilbertson et al.
[20] extracted $\alpha$ between 130-160 meVÅ for a density range of $n\approx$
3.15-3.30 $\times$ 1011 cm-2. We report Rashba coupling of $\alpha\approx$ 300
meVÅ in a InSb0.81As0.19 quantum well, among the highest measured in III-V
semiconductors [21, 22, 23, 20, 24, 25, 26, 27, 19, 28].

Table 1:  Arsenic mole fraction, 2DEG density at zero gate bias, mobility at $n=2.5$ $\times$ $10^{11}$ cm-2, and effective mass extracted by temperature dependent SdH oscillations for the samples used in this study. Mobility is measured at $T$ = 0.3 K width=textwidth | As mole fraction ($x$) |  $n$ at Vg = 0 V ($10^{11}$ cm-2) |  ${\mu}$ at $n=2.5\times 10^{11}$cm-2 (cm2/Vs) | m* (me)
---|---|---|---|---
Sample A | 0.05 | 0.6 | 2.4 $\times$ $10^{5}$ | 0.0178 $\pm$ 0.0004
Sample B | 0.13 | 1.4 | 2.0 $\times$ $10^{5}$ | 0.0142 $\pm$ 0.0002
Sample C | 0.19 | 2.3 | 1.4 $\times$ $10^{5}$ | 0.0139 $\pm$ 0.0002

##  II Experimental Details

We examine three heterostructures grown by molecular beam epitaxy (MBE) on
GaAs (001) substrates with 0.5∘ miscut towards (111)B. Three different arsenic
mole fractions are used in the InSb1-xAsx quantum well: $x$ = 0.05, 0.13, and
0.19, labeled Sample A, B, and C respectively (see Table 1). To mitigate
lattice-mismatch between the GaAs substrate and InSbAs quantum well and AlInSb
barriers, the heterostructure (Fig. 1a) is built upon a relaxed buffer layer
developed in previous work [6]. The InSb1-xAsx quantum well is grown by
keeping the In, Sb and As shutters open simultaneously; the arsenic mole
fraction is determined by the substrate temperature and the Sb/As flux ratio.

Figure 1: a) Schematic of the heterostructure used in this experiment. Arsenic
mole fractions are $x$ = 0.05, 0.13 and 0.19 b) Band edge profile for $x$ =
0.05, 0.13 and 0.19 at $V_{g}$ = 0 V. c) Magnetotransport measurement at $T$ =
0.3 K of Sample C at $V_{g}$ = 0.65 V with $n=4.6$ $\times$ 1011 cm-2 and
$\mu$ = 1.13 $\times$ 105 cm2/Vs.

Using a self-consistent Schrödinger-Poisson (SP) simulator (NEXTNANO3 [29]) we
calculate the band edge profile for the devices used in our experiments under
changing gate potentials. As highlighted in Fig. 1b, in comparison to undoped
InSb quantum wells [30, 28], these undoped InSbAs samples have a conductive
2DEG channel in the quantum well at zero gate bias; higher carrier density at
$V_{g}$ = 0 V is obtained with increasing arsenic mole fraction (see Table 1)
[31, 11].

Insulated-gate Hall bars were fabricated to study electrical transport
properties. Mesas were defined with a citric acid etch. The Hall bars
typically are 75 $\mu$m wide and 1 mm long. The Ohmic probes are separated by
150 $\mu$m along the mesa edge. Before depositing the Ohmic metal stack an in-
situ argon ion mill with a voltage of 250 V, a beam current of 8 mA, and an
accelerating voltage of 50 V was performed to etch the top barrier such that
the contact metal stack is deposited directly at the InSbAs quantum well.
Secondary ion mass spectroscopy was used to determine the position of the
quantum well. This milling process was followed by deposition of 20 nm Ti and
180 nm Au for the Ohmic contacts. Next 40 nm of Al2O3 was deposited as a
dielectric via atomic layer deposition at 100 ∘C. Finally, the gate electrodes
were fabricated with the deposition of 20 nm of Ti and 150 nm Au.

Magnetotransport measurements were performed at temperatures between $T$ = 0.3
K and $T$ = 10 mK, unless otherwise specified. Higher temperature measurements
were required for effective mass determination and investigation of magneto-
intersubband scattering. In Fig. 1c we show representative high magnetic field
data for Sample C. Well-defined integer quantum Hall states with vanishing
longitudinal resistance and quantized Hall resistance are evident. The density
calculated from the Hall slope matches the density extracted from analysis of
Shubniokov de Haas (SdH) oscillations, indicative of a single conducting
channel. We measured mobility as a function of 2DEG density for the three
wafers at $T$ = 0.3 K to identify the dominant scattering mechanisms and to
observe changes that may occur as the arsenic mole fraction is varied between
samples. For determination of SOC parameters, our measurements focused on high
resolution low magnetic field ($B\leq 1$ T) measurements of the longitudinal
resistance oscillations.

##  III Results and Discussion

###  III.1 Electron scattering in ternary InSbAs quantum wells

Scattering of 2D electrons has been thoroughly investigated in the binary
compounds InSb [32, 33, 34, 35, 28] and InAs [36, 19]. It was determined that
at cryogenic temperatures mobility is primarily limited by long-range Coulomb
scattering from remote and background impurities for 2DEG densities below 1
$\times$ 1012 cm-2. Here, we investigate mobility in ternary InSbAs quantum
wells. A ternary quantum well may have additional sources of scattering. Anion
atoms (Sb, As) will be randomly distributed based on the available sublattice
sites, generating short-range disorder [37]. Alloy disorder and interface
roughness scattering are sources of short-range scattering that may lower
mobility and modify the functional dependence of mobility on 2DEG density in
InSbAs channels [38, 39].

Figure 2: a) $\mu$ vs. 2DEG density measured at $T$ = 0.3 K for Samples A, B
and C. Inset: $\mu$ vs. 2DEG density plotted in a log-log scale for Sample A
and linear fit (dashed line) for extraction of $\beta$. b) Total scattering
rate 1/$\tau$ vs $x(1-x)$ at $T$ = 0.3 K and $n$ = $2.5\times 10^{11}$cm-2,
where x is the arsenic mole fraction. The dashed line yields
1/$\tau_{\text{alloy}}$ = 45 ns-1 per $\%$ As.

Mobility ($\mu$) vs. 2DEG density measured at $T$=0.3 K for different arsenic
mole fractions is displayed in Fig. 2a. As the arsenic mole fraction increases
from Sample A to Sample C, a decrease in peak mobility is observed. In
addition, a change in the functional dependence of $\mu$ vs. 2DEG density is
evident. For Sample A at low density, mobility is a strongly increasing
function of density, typical of charged impurity scattering. The dependence
may be approximated by $\mu\propto n^{\beta}$ over the limited density range
of $n=1-2.2\times 10^{11}$cm-2, yielding an exponent of $\beta$ = 0.6, as seen
in the inset of Fig. 2a with data plotted on a log-log scale. $\beta$ = 0.6
indicates that long-range charge disorder dominates scattering in this density
regime, specifically Coulomb disorder near the quantum well [39]. In addition
to point defects randomly distributed in the vicinity of the quantum well, the
semiconductor/dielectric interface is a known source of fixed surface charge.
In our structures, this surface is 40 nm away from the quantum well. Below
$n=1\times 10^{11}$cm-2, the slope in the $\mu$ vs. $n$ data increases,
suggestive of a transition to a localized regime at lowest density [39].

We note that at a fixed density of 2.5 $\times 10^{11}$ cm-2, a decrease
(increase) in mobility (scattering rate) with increasing arsenic mole fraction
is observed, where $\mu=e\tau/m^{*}$. Mobility peaks at 2.4 $\times$ $10^{5}$
cm2/Vs for Sample A, followed by Sample B and Sample C with mobility of 2.0
$\times$ $10^{5}$ cm2/Vs and 1.4 $\times$ 105 cm2/Vs at the same 2DEG density
(Table 1). A recent work from Lei $et$ $al.$[28] reports mobility of 2.6
$\times$ 105 cm2/Vs at density 2.4 $\times$ $10^{11}$ cm-2 in an undoped InSb
quantum well, a value similar to that seen in Sample A, suggesting that alloy
disorder scattering is not yet dominant at $x=0.05$.

For Sample B and Sample C, the weaker dependence of mobility on 2DEG density
suggests that short-range scattering limits mobility at increased arsenic mole
fraction [39]. The progression of $\mu$ vs. $n$ from Sample A to B and C
indicates that alloy disorder significantly impacts mobility at mole fractions
above $x=0.05$.

According to Matthiessen’s rule, the total scattering rate ($1/\tau$) is
determined by summing the scattering rates due to all independent scattering
mechanisms. At fixed measurement temperature $T$ = 0.3 K, we assume that
1/$\tau=1/\tau_{\text{other}}+1/\tau_{\text{alloy}}$, where
$1/\tau_{\text{other}}$ includes contributions for all scattering mechanisms
other than alloy disorder scattering. The alloy scattering rate should be
linear with $m^{*}[\delta V]^{2}x(1-x)$, where $x$ is the As mole fraction,
$m^{*}$ is the effective mass and $[\delta V]^{2}$ is the alloy scattering
potential [40].

In order to assess the increase in scattering associated with the change in
arsenic mole fraction we plot the total scattering rate ($1/\tau$) vs $x(1-x)$
at $T$ = 0.3 K and $n=2.5\times 10^{11}$ cm-2 as seen in Fig. 2b. The linear
dependence of the total scattering rate with $x(1-x)$ yields a slope of
1/$\tau$ = 45 ns-1 per $\%$ As, which quantifies the impact of alloy
scattering as we increase arsenic mole fraction.

To the best of our knowledge, the alloy scattering rate in InSbAs quantum
wells has not been reported previously. Therefore, we compare the extracted
alloy scattering rate of 1/$\tau_{\text{alloy}}$ = 45 ns-1 per $\%$ As with
the alloy scattering rate of $1/\tau_{\text{alloy}}=24$ ns-1 per $\%$ As
extracted by Gardner et al. in the more widely studied AlxGa1-xAs ternary
[41]. The rate measured in InSb1-xAsx quantum wells is comparable to
AlxGa1-xAs. $1/\tau_{\text{alloy}}$ is expected to be proportional to $[\delta
V]^{2}$ and m∗. The alloy scattering potential for an InSb-InAs alloy is
theoretically predicted to be 0.82 eV compared to 0.12 eV for GaAs-AlAs alloy
[42] while $m^{*}$ is $\approx$ 0.010-0.019me for InSb1-xAsx ternary compared
to 0.067me for AlxGa1-xAs [41]. Therefore, the alloy scattering rate for
InSb1-xAsx is within a factor of 2 of AlxGa1-xAs, despite the significantly
higher alloy scattering potential, due to the compensatory impact of the lower
effective mass.

We performed temperature dependent magnetotransport measurements to extract
the values of the effective mass for all samples at a density of $n\approx$ 2
$\times$ 1011 cm-2 [43]. Theory and experiments have shown that the ternary
InSb1-xAsx has a lower effective mass with increasing arsenic mole fraction
[14, 13, 31, 11]. Our analysis yielded results consistent with this trend, as
displayed in Table 1.

Figure 3: a) Self-consistent calculation of energy eigenvalues and
wavefunctions of first and second subbands, when total density, $n=3.2$
$\times$ 1011 cm-2. b) Self-consistent calculation of band diagram and
wavefunction of the first subband at Vg = 0 V, with density $n=1.4$ $\times$
1011 cm-2 (Solid line) and at Vg = 0.5 V with density $n$ = 3.6 $\times$ 1011
cm-2 (Dashed line). As the gate voltage increases, we observe enhanced band
bending and asymmetry of the wavefunction within the quantum well.

Finally we note a decrease in mobility for all samples above $n\approx
2.5\times 10^{11}$ cm-2, as seen Fig. 2a. Simulations point to the occupation
of the second subband at a similar density for all samples, $n\approx$ 3
$\times$ 1011 cm-2, as seen in Fig. 3a. The drop in mobility in our data may
be attributed to the proximity of the Fermi level to the bottom of the second
subband, allowing a channel for intersubband scattering [44]. Another possible
contribution is interface roughness scattering, since the electron wavefuncion
is pulled closer to the top barrier with higher gate voltage, as seen in the
simulation shown in Fig. 3b.

###  III.2 Analysis of spin-orbit coupling in InSbAs quantum wells

Due to high mobility and small effective mass we are able to resolve beating
in the oscillatory longitudinal resistance at low magnetic field. Beating may
be driven by several mechanisms, including density inhomogeneity [45],
magneto-intersubband scattering [46, 47, 48, 49] or zero-field spin splitting
arising from strong spin-orbit coupling [50]. Magnetotransport measurements
taken with different contact configurations on our devices indicate that
beating does not result from density inhomogeneity. Intersubband scattering
may occur when the chemical potential is proximal to the second electric
subband. Landau level crossing enhances magneto-intersubband scattering at
magnetic field given by $B_{MIS}=({\Delta}E_{1,2}m^{*})/(e\hbar)$ [46, 47, 48,
49]. As noted previously, simulations indicate the occupation of the second
subband is possible at high 2DEG density, therefore we have measured
temperature-dependent magnetotransport to rule out magneto-intersubband
scattering.

We first note that the magnetotransport does not show signs of significant
parasitic conduction. As shown in Fig. 1b, clear quantum Hall plateaus are
resolved with the longitudinal resistance vanishing at 1.8 T. Furthermore, the
densities extracted from the Hall slope and from the SdH oscillations minima
are equal. This suggests minimal contribution from parallel conduction
channels in the investigated range of density, with only the lowest energy
subband contributing significantly to transport.

Magneto-intersubband scattering occurs when the Landau levels of two bands
cross, causing oscillations in the magnetotransport at frequency
$B_{MIS}=({\Delta}E_{1,2}m^{*})/(e\hbar)$. If this value is close to the
frequency of the underlying SdH oscillations, beating may result. To
understand if this mechanism was active in our devices, measurements of the
longitudinal resistance were performed over a broad range of temperature.
Magneto-intersubband scattering is relatively insensitive to changes in
temperature, whereas damping of the SdH oscillations with temperature occurs
rapidly as $A(T)\propto x$/sinh$(x)$ with
$x=(2{\pi}^{2}k_{b}T/{\hbar}{\omega}_{c})$ where ${\omega}_{c}=eB/{m}^{*}$
[49, 48, 27]. We performed temperature dependent measurements for Sample B and
Sample C at the highest achievable density. We observe rapid damping of the
oscillations (Fig. 4a) with increased temperature, also reflected in the fast
Fourier transforms (FFT). The amplitudes of the two frequency peaks clearly
decrease with increasing temperature (Fig. 4b), indicating that their origin
is SdH oscillations rather than magneto-intersubband scattering.

Figure 4:  a) Magnetotransport as a function of temperature for $x$ = 0.13 at
density of $n=3.6\times$ 1011 cm-2. b) Fast Fourier transforms for the data
shown in a). The damping of the FFT peaks as a function of temperature
indicates that the peaks are generated by SdH oscillations and not by magneto-
intersubband scattering.

###  III.3  Rashba spin-orbit coupling

The beating observed in low-field transport in our samples is attributed to
zero field spin-splitting due to strong Rashba spin-orbit coupling. Zero-field
spin-splitting ($\Delta_{so}$) is given by $2\alpha k_{F}$, where $k_{F}$ is
the Fermi wave vector defined as $k_{F}=\sqrt{2\pi n}$ and $\alpha$ is the
Rashba coupling parameter. The presence of SOC results in beating in the SdH
oscillations as seen in Fig. 5. Beating occurs because two occupied spin bands
have two slightly different densities of electrons, giving rise to two closely
spaced frequencies of SdH oscillations. This allows extraction of $\alpha$
from oscillatory low-field transport [51, 52, 21, 23, 53, 19]. Fourier
analysis of longitudinal magnetoresistance facilitates extraction of the
Rashba parameter $\alpha$ as a function of 2DEG density and arsenic mole
fraction in our devices.

We assume the Rashba coupling is the major contribution to SOC in our devices.
Due to the lack of microscopic inversion symmetry in zinc-blende crystals, the
$k^{3}$ Dresselhaus coupling also contributes to SOC in zinc-blende InSbAs
quantum wells [54, 25]. However, Rashba coupling is known to dominate in
highly asymmetric quantum wells such as those studied here [55, 20]. The
increased electrostatic asymmetry with increased gate voltage is confirmed by
self-consistent calculation of the band diagram as shown in Fig. 3b.

The Rashba parameter depends on the electric field in the quantum well and in
the top and bottom barriers when the wavefunction has finite value in the
barriers. The conduction band discontinuities at the top and bottom interfaces
also contribute when the wavefunction has differing amplitude at the top and
bottom interfaces [52, 53].

$\alpha\propto
a_{0}<E_{QW}>+b^{\prime}(<E_{u}>+<E_{l}>)-b(|\Psi_{u}|^{2}-|\Psi_{l}|^{2})\\\
$ (1)

The parametric dependence of $\alpha$ is shown in Eq. 1, where the first term
is the expectation value of the electric field in the quantum well weighted by
$a_{0}$, which is a material specific parameter determined by interband matrix
elements and energy differences [56]. $a_{0}$ is expected to increase with
increasing arsenic mole fraction in the InSbAs ternaries we study. The second
term contains the expectation value of the electric field in the upper and
lower barriers weighted by $b^{\prime}$ which characterizes Rashba coupling in
the barriers. The third term accounts for the contributions from the
wavefunction at the upper ($|\Psi_{u}|^{2}$) and lower interface
($|\Psi_{l}|^{2}$), where $b$ parameterizes band discontinuity at the
interfaces.

The Rashba parameter is determined once $\Delta n$, the density difference
between the two electron spin bands, is known [52]. This is accomplished by
taking a fast Fourier transform of the low-field longitudinal resistance,
which yields the frequencies of the two oscillations, $f_{1}$ and $f_{2}$. The
Rashba parameter is given by Eq. 2 [52].

$\alpha=\frac{\Delta n\hbar^{2}}{m^{*}}\sqrt{\frac{\pi}{2(n_{tot}-\Delta
n)}}\\\ $ (2)

In Eq. 2, $n_{tot}=n_{1}+n_{2}$, $\Delta n=n_{1}-n_{2}$ and $m^{*}$ is the
effective mass. We use the effective mass values measured for our samples as
shown in Table 1.

Figure 5:  SdH oscillations as a function of density for $x$ = 0.19. A shift
of the 2nd node to higher magnetic field is observed as the density increases.

The increase of the Rashba coupling with larger arsenic mole fraction is
evident in the raw magneto-resistance data plotted in Fig. 6a. The number of
oscillations between two nodes changes from approximately 25 to 13 from $x$ =
0.13 to $x$ = 0.19. Since the number of oscillations is inversely proportional
to $\Delta_{so}$, this indicates a higher $\alpha$ for the higher arsenic mole
fraction sample [21, 50]. The evolution of the Rashba parameter with density
for all arsenic mole fractions is depicted in Fig. 6b. A noticeable increase
of the Rashba coupling with increased arsenic mole fraction is evident. If we
compare the Rashba coupling extracted at the same density, $n\approx$ 2.9
$\times$ 1011 cm-2, the values for Sample A to C are respectively
$\alpha\approx$ 100, 150, 280 meV Å. Because the samples have the same top and
bottom barriers, nearly equal electric field in the quantum well and matching
electron distributions (as inferred from simulations), the increase in SOC may
be attributed to an increase in the material specific Rashba coupling
parameter $a_{0}$ with higher arsenic concentration. This is consistent with
predictions [13, 14, 15, 8].

For Sample B and Sample C, the Rashba coupling increases from $n\approx$ 2.5
and 3.5 $\times$ $10^{11}$ cm-2. This enhancement is consistent with
simulations, which indicate an increase in electric field and wavefunction
asymmetry towards the top barrier, as also seen in Fig. 2b. However, for both
Samples B and C, the Rashba parameter decreases after $n$ $\approx$ 3.5
$\times$ $10^{11}$ cm-2. It is worth noting that the decrease in $\alpha$
occurs at approximately the same density at which the second electric subband
begins to be occupied in the numerical simulations, although contribution from
a second subband to the magnetotransport is not apparent in the data. From the
simulations, the second subband wavefunction penetrates significantly more
into the top barrier than the first subband. A possible explanation for the
decrease in $\alpha$ with higher density is the increased propensity of
electrons to reside in the top barrier, which has lower intrinsic SOC than the
quantum well. However, more investigation is needed to quantitatively
understand this trend.

Figure 6:  a) Direct comparison of beating in SdH oscillations with $x$ = 0.13
and $x$ = 0.19 at $n=3.7$ $\times$ $10^{11}$ cm-2. Comparison at the same
density allows for comparison at similar electric field. The number of
oscillations between two nodes is inversely proportional to the value of
$\Delta_{so}$. For $x$ = 0.13 we observe a larger number of oscillations
compared to $x$ = 0.19, confirming a lower value of zero field spin splitting
energy. b) Extracted Rashba parameter as a function of 2DEG density for all
samples.

##  IV Conclusions

In this work we have detailed the low temperature transport properties of a
series of 2DEGs confined in InSbAs quantum wells with different arsenic mole
fractions. We extracted an alloy scattering rate of 1/$\tau$ = 45 ns-1 per
$\%$ As, determined the dependence of mobility on 2DEG density and arsenic
mole fraction, and detailed the evolution of dominant scattering mechanisms.
Analysis of SdH oscillations enabled extraction of the Rashba coupling
strength over a wide range of 2DEG density for all mole fractions. At
comparable density and nearly equal electric field, we observed an increase of
$\alpha$ with arsenic mole fraction. For $x$ = 0.19 $\alpha$ is $\approx$ 300
meVÅ, one of the highest values reported for III-V heterostructures. These
results agree with theoretical predictions of enhancement of SOC with arsenic
mole fraction [15, 13, 14] and suggest that InSb1-xAsx 2DEGs are a promising
platform for exploration of topological superconductivity by incorporating
these ternary quantum wells in a shallow structure coupled to an s-wave
superconductor.

$\dagger$ Author to whom correspondence should be addressed:
<EMAIL_ADDRESS>

We thank Dr. James Nakamura for suggestions for manuscript improvement and
Georg Winkler, Sam Teicher and Fabiano Corsetti for fruitful discussion. This
work was supported by Microsoft Quantum.

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

The authors have no conflicts to disclose.

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