B 105 12 Xin WangnlabelA Xiaolin HounlabelA Lan ChennlabelA Yoshihisa
KishiyamaelabelB Takahiro AsaielabelB 3 [labelA]The authors are with DOCOMO
Beijing Communications Laboratories, Co. Ltd., No. 2 Kexueyuan South Road,
Beijing 100191, China. [labelB]The authors are with NTT DOCOMO, INC., 3-6
Hikari-no-oka, Yokosuka-shi, Kanagawa-ken, 239-8536, Japan. 130 508

# Deep Learning-based Massive MIMO CSI Acquisition for 5G Evolution and 6G

(2022; 2022)

###### keywords:

Channel state information, deep learning, and downlink MIMO transmission

Channel state information (CSI) acquisition at the transmitter side is a major
challenge in massive MIMO systems for enabling high-efficiency transmissions.
To address this issue, various CSI feedback schemes have been proposed,
including limited feedback schemes with codebook-based vector quantization and
explicit channel matrix feedback. Owing to the limitations of feedback channel
capacity, a common issue in these schemes is the efficient representation of
the CSI with a limited number of bits at the receiver side, and its accurate
reconstruction based on the feedback bits from the receiver at the transmitter
side. Recently, inspired by successful applications in many fields, deep
learning (DL) technologies for CSI acquisition have received considerable
research interest from both academia and industry. Considering the practical
feedback mechanism of 5th generation (5G) New radio (NR) networks, we propose
two implementation schemes for artificial intelligence for CSI (AI4CSI), the
DL-based receiver and end-to-end design, respectively. The proposed AI4CSI
schemes were evaluated in 5G NR networks in terms of spectrum efficiency (SE),
feedback overhead, and computational complexity, and compared with legacy
schemes. To demonstrate whether these schemes can be used in real-life
scenarios, both the modeled-based channel data and practically measured
channels were used in our investigations. When DL-based CSI acquisition is
applied to the receiver only, which has little air interface impact, it
provides approximately 25% SE gain at a moderate feedback overhead level. It
is feasible to deploy it in current 5G networks during 5G evolutions. For the
end-to-end DL-based CSI enhancements, the evaluations also demonstrated their
additional performance gain on SE, which is 6% – 26% compared with DL-based
receivers and 33% – 58% compared with legacy CSI schemes. Considering its
large impact on air-interface design, it will be a candidate technology for
6th generation (6G) networks, in which an air interface designed by artificial
intelligence can be used.

##  1 Introduction

Massive multiple-input multiple-output (MIMO)[1] is a key technology for 5th
generation (5G) New Radio (NR). Channel state information (CSI) on the
transmitter side plays a key role in improving the spectrum efficiency of
downlink transmissions in massive MIMO systems. Such information is necessary
for downlink precoding schemes to improve the signal-to-noise ratio (SNR) at
dedicated receivers and eliminate multiuser interference (MUI), which enables
high-order multiuser MIMO (MU-MIMO) downlink transmissions.

Therefore, diverse technologies for CSI acquisition have been studied in both
academic research and the standardization of mobile networks. Limited feedback
schemes with codebook-based vector quantization or explicit channel
matrix/eigenvector feedback have been introduced based on legacy signal
processing technologies. A common challenge of these approaches is to find an
efficient way to represent CSI with a limited number of bits, which can be
accommodated by the feedback channel with an acceptable overhead.

Among them, limited feedback schemes[2] have been widely used, especially for
frequency division duplex (FDD) systems and time division duplex (TDD) systems
without radio frequency (RF) calibration. For limited feedback schemes, the
base station (BS) transmits reference signals to facilitate the user equipment
(UE) estimating the downlink channel, and the UE quantizes the CSI with a
specified codebook into a group of bits. These bits are fed back to the BS via
the uplink feedback channel and the BS reconstructs the CSI based on them. The
accuracy of the reconstructed CSI is constrained by the overhead introduced by
the downlink reference signal and uplink feedback. For massive MIMO systems
with a large number of antenna ports, more resources are necessary for
transmitting downlink reference signals, and it is also difficult to quantize
a large-scale CSI matrix into a limited number of bits without losing
accuracy.

For example, NR Release 15[3] specified two CSI limited feedback schemes,
namely, Type I and Type II CSI. Type I CSI serves as a low-overhead scheme
that quantizes the CSI into one of the codewords in the codebook, whereas Type
II CSI uses a linear combination codebook, where multiple codewords can be
selected from the codebook and the CSI is represented as a linear combination
of selected codewords. Both the index of the selected codeword and quantized
combination coefficients are sent back to the BS from UE. To control the
feedback overhead, NR Type II CSI uses coarse quantization of combination
coefficients and a large frequency domain granularity. Coarse quantization
introduces severe nonlinear noise into feedback CSI. In addition, the large
frequency domain granularity introduces information loss, which makes it
difficult to reconstruct an accurate CSI with smaller granularity.

As one of the key technologies in the new wave of artificial intelligence
(AI), deep learning (DL) has been successfully used in many applications, such
as image processing and natural language processing[4]. Benefiting from the
large-scale parallel computing power provided by modern processors, such as
graphics processing units (GPUs) or tensor processing units (TPUs), DL
networks trained with big data can usually find a better efficient solution to
intractable problems compared with legacy optimization methods, especially for
those in NP-hard families and those that are difficult to mathematically model
[5, 6]. Deep neural networks can learn hidden mathematical models behind big
data and fit nonlinear functions. One related example is image compression and
super resolution[7, 8, 9, 10], where a deep neural network can learn the
hidden structure inside images and then utilize this to compress and recover
images or reconstruct high-resolution images based on their low-resolution
versions.

Similar to image compression or super-resolution, a complex model also hides
behind the observed CSI, which is difficult to obtain practically and helpful
for CSI acquisition. Motivated by this observation, DL technology has recently
been introduced for CSI acquisition. In [11], CsiNet was introduced as an end-
to-end CSI compression and reconstruction method. Inspired by image super-
resolution, SRCNN[9] was used for channel estimation based on IEEE 802.11ad
reference signals as a receiver enhancement scheme[12]. Our previous work
adopted an advanced super-resolution network VDSR[10] to use cases of both
end-to-end CSI acquisition[13] and receiver enhancements based on NR Type II
feedback [14].

The progress of DL in CSI acquisition has attracted the attention of the
industrial community. The 3rd Generation Partnership Project (3GPP) will
initiate the study of this topic in the 2nd quarter of 2022[15]. The use cases
of AI for CSI and their potential performance gain in practical mobile
networks are the key objectives of this study. To address this issue, we
introduce our AI for CSI (AI4CSI) use cases and evaluate their performance
gains in NR networks. The contributions of this study are as follows.

  * •

Propose an enhanced end-to-end AI4CSI network based on the CsiNet in [11] with
better performance, and extend our AI4CSI receiver in [14] to explicit
feedback.

  * •

Investigate the spectrum efficiency (SE), feedback overhead, and computational
complexity of legacy and DL-based CSI feedback methods in a practical NR
network, including Type I and II limited feedback schemes specified in current
NR standards, explicit feedback schemes, and DL-based CSI for receiver
enhancements (AI4CSI Rx) and end-to-end enhancements (AI4CSI E2E).

  * •

Investigate the performance of AI4CSI schemes with measured channel data to
verify their feasibility for practical deployments.

Our study demonstrates the potential gain of AI4CSI schemes in a practical
mobile system with limited feedback channel capacity. When AI4CSI is applied
only to the receiver, which has little air-interface impact, it improves the
accuracy of CSI reconstruction and requires fewer downlink reference signals.
The performance gain in the SE of the downlink transmissions is approximately
25%. Therefore, it is feasible for deployment in current mobile networks
during network evolution. For AI4CSI E2E schemes, we show their significant
performance gain, especially for moderate feedback overhead, and that it is
feasible with practically measured channel data. This shows the prospects of
applying an AI/DL defined air-interface for CSI in future networks, such as
the 6th Generation (6G) mobile networks, where AI will be a native component
of the networks.

The remainder of this paper is organized as follows. Section 2 introduces the
proposed system. Section 3 describes the legacy and the proposed AI4CSI
schemes. Performance evaluations with model-based channel data and practical
channel measurements are presented in Sections 4 and 5. Finally, section 6
concludes the paper.

Notations: Vectors (lower case) and matrices (upper case) are presented in
boldface. $(\cdot)^{\mathrm{T}}$, $(\cdot)^{\mathrm{H}}$ and
$(\cdot)^{\mathrm{-1}}$ denote the transpose, conjugate transpose, and
inverse, respectively.

##  2 5G NR Systems

We studied the DL-based CSI feedback and reconstruction in a typical 5G NR
mobile network. Following LTE specifications, NR also uses OFDM-based waveform
and multiplexing schemes[16], in which the time-frequency domain resources are
divided into resource blocks (RBs), each of which comprises 14 OFDM symbols in
the time domain and 12 subcarriers in the frequency domain. These time-and
frequency-domain resources make up the resource grid of an OFDM system, where
all channels and reference signals (RSs) are mapped onto the resource grid and
then transmitted.

Figure 1: The diagram of CSI feedback and downlink transmissions. Figure 2: An
example mapping pattern of NR CSI-RS.

A typical closed-loop downlink transmission procedure with CSI feedback is
shown in Fig. 1. In an NR system, CSI-RSs are transmitted from the BS to the
UE to facilitate UE downlink channel estimation[17]. The current NR
specifications support up to 32 antenna ports for CSI-RS transmissions. The
signals of different ports are multiplexed using a combination of frequency-
division multiplexing (FDM), time-division multiplexing (TDM), and code-
division multiplexing (CDM). The frequency-domain density of NR CSI-RS is 1 or
0.5 resource elements per port per RB. An example of CSI-RS on an OFDM
resource grid is shown in Fig. 2.

UEs estimate the downlink channel based on the received CSI-RS and then
generate CSI feedback, which is transmitted to the BS via uplink feedback
channels. When CSI is available at the BS, it can use typical downlink MIMO
precoding, such as zero-forcing[18] to transmit data to multiple UEs with
spatial multiplexing.

Feeding downlink CSI back to BS efficiently has been discussed for decades ,
since MIMO technology was introduced into mobile systems. In the next section,
we first introduce the basic signal model of the mobile system and then
introduce several typical CSI acquisition schemes, legacy, and AI-based
schemes.

Suppose that there are $N_{t}$ antenna ports in the BS to serve $M$ UEs in a
cell, and there are $N_{r}$ antenna ports for each UE. For an OFDM waveform
with $K$ subcarriers, a general received CSI-RS at the UE can be written as

$\mathbf{r}_{m,k}=\mathbf{H}_{m,k}\mathbf{s}_{RS,k}+\mathbf{n}_{m,k},$ (1)

where $m=1,\cdots,M$ is the index of UE and $k$ is the index of the
subcarriers where CSI-RS is transmitted.
$\mathbf{H}_{m,k}\in\mathcal{C}^{N_{r}\times N_{t}}$ denotes the downlink
channel in the $k$-th subcarrier,
$\mathbf{s}_{RS,k}\in\mathcal{C}^{N_{t}\times N_{t}}$ is the transmitted CSI-
RS signal, and $\mathbf{n}_{m,k}\in\mathcal{C}^{N_{r}\times N_{t}}$ denotes
the thermal noise of the receiver. Without loss of generality, we assume that
the CSI-RSs for different antenna ports are orthogonal. Therefore,
$\mathbf{s}_{RS,k}$ is a diagonal matrix in which the transmitted signals of
CSI-RS appear as diagonal elements.

After receiving CSI-RS, UEs estimate the channel based on the received signal
and obtain the estimated channel $\hat{\mathbf{H}}_{m,k},\forall
k=1,\cdots,K$. Although CSI-RS has a low frequency-domain density, UEs can use
interpolation methods to obtain channels for all subcarriers, for example,
zero- or first-order interpolation as a simple implementation. or MMSE
interpolation using the known channel statistics.

NR specifications define the frequency-domain feedback granularity as a
subband comprising several RBs. For subband feedback, UEs can calculate the
subband equivalent channel as

$\hat{\mathbf{H}}_{m}^{(s)}=\mathbf{v}_{N_{r}}\left(\sum_{k=k_{s}}^{k_{e}}\hat{\mathbf{H}}_{m,k}^{\mathrm{H}}\hat{\mathbf{H}}_{m,k}\right),$
(2)

where $s$ is the index of the subband, $k_{s}$ and $k_{e}$ are the start and
end indices of subcarriers for a subband, and $\mathbf{v}_{N}(\cdot)$ denotes
$N$ eigenvectors corresponding to $N$ largest eigenvalues of a matrix.

The CSI is usually decompensated into several components for separate
feedback. Channel gain is usually reported as a factor of the channel quality
indicator (CQI), which is the signal-to-interference-and-noise ratio (SINR)
derived from the UE. The UE may also report a rank indicator (RI) that
indicates the expected MIMO transmission layer number according to channel
conditions.

Among these components, the one representing the channel matrix is the most
important. Because the channel gain is reported implicitly in CQI, the
normalized channel matrix, which can be called channel direction information
(CDI), should be reported to the BS from UE. There are many schemes that
report CDI to BS explicitly or implicitly, which is the focus of our study and
will be discussed in Section 3.

After receiving feedback from the UEs, the BS can reconstruct the channel
matrix and prepare the downlink transmissions. Define the channel matrix
recovered by the BS as $\tilde{\mathbf{H}}_{m,k}$. The downlink channel of all
serving UEs is then

$\tilde{\mathbf{H}}_{k}=\left[\tilde{\mathbf{H}}_{1,k}^{\mathrm{H}},\tilde{\mathbf{H}}_{2,k}^{\mathrm{H}},\cdots,\tilde{\mathbf{H}}_{M,k}^{\mathrm{H}}\right]^{\mathrm{H}},$
(3)

where the BS can compute the downlink precoder $\mathbf{P}_{k}$ by following
the steps in [18]. For a simplified example, when the UE has one mounted
antenna ($N_{r}=1$), the dimension of each $\tilde{\mathbf{H}}_{m,k}$ is
$1\times N_{t}$. The zero-forcing precoder can be obtained by

$\mathbf{P}_{k}=\tilde{\mathbf{H}}_{m,k}^{\mathrm{H}}\left(\tilde{\mathbf{H}}_{m,k}\tilde{\mathbf{H}}_{m,k}^{\mathrm{H}}\right)^{-1}.$
(4)

The received signal of downlink transmissions for all UE antennas on $k$-th
subcarriers is then

$\mathbf{r}_{k}=\mathbf{H}_{k}\mathbf{P}_{k}\mathbf{s}_{D,k}+\mathbf{n}_{k},$
(5)

where
$\mathbf{H}_{k}=[\mathbf{H}_{1,k}^{\mathrm{H}},\cdots,\mathbf{H}_{M,k}^{\mathrm{H}},]^{\mathrm{H}}$,
$\mathbf{s}_{D,k}\in\mathcal{C}^{M\times 1}$ represents the data transmitted
to the UE, and $\mathbf{n}_{k}$ denotes the thermal noise of the receiver.

##  3 CSI Acquisition Schemes

As discussed in Section 2, the most important part of the CSI feedback is CDI,
whose dimension increases with the number of antenna ports, and which becomes
a large-scale matrix in massive MIMO systems.

In legacy signal processing, the processing on CDI is categorized as a
quantization or compression scheme. The objective is to represent the CDI
through a limited number of bits without losing much information. Many schemes
have been discussed over the decades, since MIMO technologies have been used
in mobile systems. We introduce two types of legacy schemes in Section 3.1 and
several DL-based schemes in Section 3.2.

###  3.1 Legacy CSI Acquisition Schemes

####  3.1.1 Limited Feedback of Precoding Matrix

NR uses a limited feedback scheme that quantizes the precoding matrix expected
by the UE into an index called the precoding matrix index (PMI) and reports it
to the BS instead of reporting the CDI. The BS can then use the reported PMI
directly to downlink single-user MIMO (SU-MIMO) transmissions. However, for
MU-MIMO transmissions, the original CDI is required for the BS to calculate
the MIMO precoder. Fortunately, the precoder matrix selected by the UE is
usually highly correlated with the downlink channel, which means that we can
treat the PMI as a representation of the CDI and reconstruct the channel
matrix from it.

The detailed procedure for NR feedback and recovery is defined in [3]. The UE
first calculates the feedback content based on the subband equivalent channel
$\hat{\mathbf{H}}_{m}^{(s)}$ and sends the feedback content to the BS. Here,
we consider NR Type II feedback, which uses a two-stage linear-combination
codebook. The BS can reconstruct the subband level feedback channel based on
the UE CSI feedback, which can be represented as

$\tilde{\mathbf{H}}_{m}^{(s)}=\mathbf{W}_{m,1}\mathbf{W}_{m,2}^{(s)},$ (6)

where $\mathbf{W}_{m,1}$ consists of several DFT base vectors selected by the
UE to represent the CSI, which is wideband feedback according to NR
specifications, and $\mathbf{W}_{m,2}^{(s)}$ includes combination coefficients
for subband $s$. If all base vectors are included in $\mathbf{W}_{m,1}$ and
$\mathbf{W}_{m,2}^{(s)}$ is unquantized, there is no information loss in the
PMI. However, to control the feedback overhead, only a subset of the base
vectors is selected, and both the amplitudes and phases of the elements in
$\mathbf{W}_{m,2}^{(s)}$ are quantized into a few bits, which introduces
severe quantization noise.

The reconstructed channel is then used for downlink transmissions. Because the
feedback granularity is subband, the downlink precoder usually has to be
calculated with the same granularity. Both quantization noise and large
granularity degrade the SE of the downlink transmissions.

####  3.1.2 Explicit Feedback

Another legacy CSI approach is explicit CDI feedback since itself is more
expected for MU-MIMO precoding. CDI or its equivalent forms can be reported to
the BS directly in explicit feedback schemes.

As a typical explicit feedback scheme, we consider direct CDI feedback with
scalar quantization in this study. The real and imaginary parts of the nonzero
elements in $\hat{\mathbf{H}}_{m,k}$ are quantized into a specified number of
bits. Considering the feedback overhead constraint in practical systems, there
are typically insufficient resources to transmit all nonzero elements in
$\hat{\mathbf{H}}_{m,k}$. Therefore, subband level feedback is also used to
control the feedback overhead.

Note that the vector quantization schemes used for implicit feedback can also
be used for explicit feedback. In such a case, a similar performance is
expected because quantization noise is the dominant factor in the performance.

###  3.2 Deep Learning based CSI Schemes

####  3.2.1 DL-based CSI Reconstruction at Receiver (AI4CSI Rx)

With the constraint of feedback overhead, limited information can be
transmitted and received via feedback channels. To obtain a more accurate
reconstructed CSI from limited information, the reconstruction algorithm
should have detailed information on the inter-structure of the channel, which
depends on the detailed radio propagation procedure. Practical over-the-air
channels usually consist of many multi-path components, each of which is
composed of many rays. If radio propagation can be mathematically described
and accurately estimated, the CSI can be reconstructed accurately as well.
However, it is a challenging task to model and estimate radio propagation
using legacy signal processing technologies. The practical model is hidden
behind the observable channels. By training with big data, DL networks can
learn the hidden structure from a large number of channel samples and utilize
the information it learns to reconstruct the CSI. Therefore, we consider using
DL to reconstruct CSI from the inaccurate feedback of UEs.

DL networks have been used for image super-resolution, which reconstruct a
high-resolution image from its low-resolution copy[7]. During the training
stage of networks, the mapping between low-resolution and high-resolution
images or the inner structure information can be learned using a large set of
training data. A typical super-resolution network can process an image with
multiple channels, improve the resolution of the images, and filter noise. For
both implicit and explicit feedback, the feedback CSI is also two-dimensional
data with multiple channels, such as real and imaginary parts or amplitude and
phase parts. The CSI with large feedback granularity can be viewed as a low-
resolution copy, and quantization noise is introduced, which should be
filtered.

Figure 3: System structure for CSI reconstruction during training stage and
deployment stage. (a) AI4CSI Rx. (b) AI4CSI E2E.

Motivated by the similarity between CSI reconstruction and image super-
resolution, we adopted image super-resolution networks to reconstruct the CSI
with NR Type II feedback in [14]. General diagrams of the networks in the
training and deployment stages are presented in Fig. 3 (a), respectively.

The proposed AI4CSI Rx network is deployed on the BS side. Because both NR and
explicit feedback introduced in Section 3.1 can be viewed as low-resolution
versions of CSI, the proposed network can accept either of them as inputs.
After receiving UE feedback, the BS can reconstruct the channel matrix with
any legacy interpolation method and then convert the complex numbers into real
ones. The real and imaginary parts of the channel matrix are stored separately
in the tensors and treated as two channels of the input data for the DL
network.

Among the image super-resolution networks, we selected the VDSR network[10] to
reconstruct the CSI. It is a very deep convolutional neural network (CNN) with
16–20 convolution layers, using which inner structure information is extracted
from the training data. As the output of the network is an enhanced version of
the input, VDSR also applies a global residual network architecture that is
widely used for image processing.

During the training stage, we used a perfect channel matrix as the tag to
train the entire network. The mean square error (MSE) between the output of
the network and the perfect channel matrix was calculated and then
backpropagated to update the coefficients of the CSI reconstruction network.
During this stage, the CSI reconstruction network learns the structure between
the low-resolution CSI and the expected perfect CSI. After training, the
network was deployed to the BS.

Figure 4: Network architectures of AI4CSI networks (a) The balanced design.
(b) The Lean UE design.

We consider two alternatives to AI4CSI Rx. One is AI4CSI Rx (Type II), the CSI
reconstructed from NR Type II feedback, and the other is AI4CSI Rx (Explicit),
the CSI reconstructed from the explicit feedback. The performance of both
alternatives is investigated in Section 4.

####  3.2.2 DL-based End-to-end CSI Acquisition (AI4CSI E2E)

As introduced in Section 1, DL-based end-to-end enhancements have received
considerable research interest in recent years. The ultimate objective of
communication systems is to transmit information from one end (transmitter) to
another (receiver) with minimal errors. For CSI feedback and reconstruction,
the objective is to let the BS reconstruct a perfect CSI with feedback
information from the UE. Existing feedback schemes are usually designed
heuristically because it is difficult to jointly optimize all the factors in a
typical feedback scheme, such as codebook selection, quantization method,
quantization levels, and quantized bit allocation. DL provides an alternative
approach in which DL networks design UE feedback and BS reconstruction, which
is an end-to-end solution.

In AI/DL technologies, an auto-encoder based on self-supervised learning has
an end-to-end structure. The input of the network is also used as training
data to train the network output. Therefore, the network is trained to
generate outputs that are similar to the inputs. The autoencoder can be
divided into two parts: an encoder and decoder. The last layer of the encoder,
that is, the output layer of the encoder, can be viewed as a compression
layer. Its output is the feedback content generated by the auto-encoder, the
overhead of which can be controlled by restricting the number of nodes in this
layer. Under such overhead constraints, the task of the decoder network is to
recover the input with minimal error.

Based on this feature of the auto-encoder, existing works use it for CSI
feedback and reconstruction [11]. A diagram of the autoencoder-based CSI
schemes is shown in Fig. 3 (b). On the UE side, the encoder network can be
used to encode CDI into a small-scale vector, which is then quantized and
transmitted to the BS. Subsequently, the BS can apply the decoder network to
recover the CDI as shown in the deployment stage in Fig.3 (b). The output of
the encoder is a vector of real numbers that should be compressed and tends to
be uniformly distributed. We then use scalar quantization to quantize each
element of the vector into bits for feedback.

Following the principle of the auto-encoder and pioneering work on its
application to CSI feedback[11], we consider two types of end-to-end CSI
feedback and reconstruction networks. One is an enhanced network based on
CsiNet [11], the architecture of which is shown in Fig. 4 (a).

Our proposed enhanced network considers a balanced design between the encoder
and decoder, similar to CsiNet, in which the computational complexity of the
encoder and decoder is comparable. Inspired by the fact that most data
compression schemes transform the original data into a proper domain where the
transformed data are sparse, we add a full connection layer before the
original version of CsiNet to emulate the domain transformation operation.
Unlike the domain transformation in existing compression schemes, the
transformation in DL networks is obtained by training with big data rather
than a predefined one. It is expected that the DL network can find a better
domain to compress CSI than any domain used in legacy compression schemes. An
inverse domain transformation layer is also added at the decoder side.

Another AI4CSI E2E network to be evaluated has a lean-UE design. For AI4CSI
E2E, the encoder is deployed in the UE and the decoder is deployed in the BS.
Therefore, the complexity of the encoder is negligible to avoid increasing the
UE processing complexity. Motivated by this, we consider a network with a lean
UE design introduced in [13]. The architecture of the network is illustrated
in Fig. 4 (b). For the lean UE network, only one necessary compression layer
is maintained at the encoder side. On the decoder side, after the linear layer
to recover the dimension of the CSI, super-resolution networks are still used
to recover the full-resolution CSI and filter the noises.

The two implementations introduced in this subsection are named AI4CSI E2E
(balanced) and AI4CSI E2E (lean UE).

##  4 Performance Evaluations with Simulated Channels

###  4.1 Simulation Setup

We evaluate the performance of all the proposed algorithms introduced in
Section 3.1 and Section 3.2 in a typical NR network. The major simulation
assumptions are listed in Table 1, and follow those used in the study of NR
systems.

In our simulations, an array with 32 antenna elements was mounted on the BS
with a $4\times 4\times 2$ layout (horizontal $\times$ vertical $\times$
polarization), and one antenna was mounted on each UE. We consider four UEs
that periodically report their CSI every 5 ms. After collecting their CSI
reports, the BS transmits downlink data using ZF precoding[18] with the CSI it
receives, which is obtained in different ways, as introduced in Section 3.

In our evaluations, we considered different feedback configurations with
different frequency-domain granularity and quantization accuracy. For NR CSI
feedback schemes, the feedback overhead differs from these configurations, as
shown in Table 2. Note that we extended the range of the parameters specified
in the current NR specification[3] to generate feedback content with different
overheads.

Table 1: Simulation Assumptions Parameter | Values
---|---
Carrier Frequency | 4 GHz
Bandwidth | 100 MHz
Subcarrier Spacing | 30 kHz
Subcarrier Number | 3276 (273 RBs)
Channel Model | 3GPP NR CDL
BS Antenna Array | 32 $(4\times 4\times 2)$
UE Antenna | 1
UE Number | 4
UE velocity | 3 kmph
Maximum Doppler frequency | 11 Hz
Feedback Period | 5 ms
SNR | 25 dB
Feedback Overhead | 10 kbps — 8.8 Mbps (per UE)
CSI-RS Configuration | 1/2 — 1/64 RE per port per RB
Table 2: NR Type I/II Feedback Configurations Overhead | Subband | Codeword | Phase | Amplitude
---|---|---|---|---
(kbps) | Number | Number | Quantization | Quantization
10 | 18 | 1 | 3 bit | 3 bit
94 | 18 | 4 | 3 bit | 3 bit
330 | 36 | 6 | 3 bit | 3 bit
570 | 54 | 6 | 4 bit | 3 bit
1100 | 108 | 6 | 4 bit | 3 bit

For explicit channel matrix feedback, the feedback overhead is controlled by
feedback granularity, The density of the CSI-RS is also adjusted according to
the feedback granularity. In the current NR specification, the frequency
domain density of the CSI-RS can be as low as $1/2$ resource elements (RE) per
port per RB. In our evaluations, in addition to $1/2$ RE per port per RB, we
further reduced the density of the CSI-RS and lowered the density to $1/128$
RE per port per RB. The evaluation configurations for explicit channel
feedback are shown in Table 3. For all cases shown in Table 3, we use 5 bits
to quantize both the real and imaginary parts of the complex numbers. We also
evaluated the performance of implicit feedback methods with different CSI-RS
densities to test their super-resolution capability.

The feedback overhead of AI4CSI E2E scheme can be controlled by configuring
the encoder network. In this work, we consider that each output of the
compression layer is quantized into 5 bits, the same as the bit width used for
explicit feedback schemes. Therefore, we can flexibly adjust the overhead of
AI4CSI E2E schemes according to requirements.

We used the NR CDL channel models specified in [19] to generate the training,
verification, and test datasets for all schemes. There are multiple models
with different power-delay profiles defined in [19]. To avoid overfitting of
the DL network, our training dataset consisted of channel samples of different
models and root mean square (RMS) delay spreads (DS). More specifically, we
generated the datasets by mixing CDL-A, CDL-B, and CDL-C models, each of which
consists of 23–24 multipath components with different powers and delays. The
RMS DS of CDL models can be scaled to simulate different practical channel
conditions. In the simulations, the training data set has 100’000 channel
samples, each of which is generated by randomly selecting one of the three CDL
models, and then randomly selecting its RMS DS from short (30 ns), nominal
(100 ns), and long (300 ns). The DL network can learn more propagation
conditions from these training datasets. We also generate verification and
test data sets, each of which has 1’000 samples, following the same way.

Table 3: Explicit Channel Feedback Configurations Overhead | CSI-RS Density | Port Number
---|---|---
(kbps) | (RE/RB/Port) |
17 | 1/128 | 8
68 | 1/128 | 32
270 | 1/64 | 32
550 | 1/32 | 32
1100 | 1/16 | 32
Figure 5: SE of schemes with different feedback overhead. Figure 6: SE of
implicit feedback schemes with different CSI-RS density.

###  4.2 Spectrum Efficiency

Fig. 5 shows the performance of the baseline and AI4CSI schemes. First, we
discuss the tendency of legacy signal processing algorithms. As shown in the
figure, the achievable SE of the two legacy schemes tends to be linear with
the feedback overhead. For NR Type II feedback, the vector quantization used
in the schemes exhibits better performance at a low overhead level. However,
explicit feedback can provide a more accurate CSI with medium to high
overheads. There is a large area on the upper left of both curves, where a
better trade-off between the SE and feedback overhead can be achieved. This is
the objective of AI4CSI schemes.

It is illustrated in Fig. 5 that AI4CSI Rx schemes can recover the channel
matrix with higher accuracy compared with legacy schemes. Using the
reconstructed CSI, there is an SE gain of approximately 25% when AI4CSI Rx is
used with NR Type II feedback.

In Fig. 6, we plotted the SE of schemes with different CSI-RS densities. It is
shown that AI4CSI Rx provides $18\%$—$27\%$ SE gain. When the density of CSI-
RS is $1/64$ per port per RB, which is $1/32$ of the current specified
density, AI4CSI Rx provides similar spectrum efficiency as the legacy schemes
with current CSI-RS density. This demonstrates its capability of working with
very low frequency-domain granularity, which is benefitted from the capability
of super-resolution networks.

From the results shown in Fig. 5, AI4CSI Rx applied on explicit feedback can
also improve the SE compared with the baseline schemes. When explicit feedback
can provide a sufficiently accurate channel matrix, the super-resolution
network can recover CSI accurately enough for MU-MIMO transmissions, which
makes the SE approach the ideal CSI upper bound. Similar to the performance of
legacy implicit and explicit feedback schemes, the DL-based approach with
explicit channel matrix feedback is not superior to the same network with
implicit feedback. The two curves cross in the moderate feedback overhead
region. This hints at considering detailed feedback schemes and CSI-RS
configurations according to the acceptable feedback overhead that can be
accommodated in uplink channels.

Both AI4CSI E2E networks can improve the SE-feedback overhead tradeoff when
the feedback overhead is moderate, that is, approximately 100 kbps per UE. The
performance gain on the SE is 33% – 58% compared with the NR Type II feedback.
However, these two networks cannot outperform the NR feedback at very low
feedback overhead levels, that is, when the feedback overhead is approximately
10–30 kbps for each UE. Under such feedback overhead conditions, the number of
nodes in the compression layer of the network is constrained to 10–30. The
network must find a scheme to represent a large-scale channel matrix with only
10 real numbers. This is difficult and requires more channel data to train the
network for this objective.

Between the two different end-to-end networks, the network with the lean UE
design exhibits better performance in most cases. Although the complexity of
the UE network is significantly reduced, we used a large and very deep CNN on
the BS side. With the price of complexity, it achieves better performance than
a balanced design. This complexity issue is discussed in the following
subsections.

###  4.3 Computational Complexity

Following the simulation assumptions in Table 1, we analyzed the computational
complexity of each scheme. We used the number of float operations per second
(FLOPS) to denote the computational complexity of the algorithms. The
complexity is evaluated when the feedback overhead is 550 kbps (570 kbps for
NR Type II feedback because its feedback overhead cannot be adjusted
flexibly). The computational complexity of operations for pre-and post-
processing, such as basic matrix calculations and fast Fourier transforms, was
also evaluated.

Table 4: The computational complexity the schemes under investigation. Scheme | BS Complx. | UE Complx.
---|---|---
| (GFLOPS) | (GFLOPS)
NR Type II | $<0.1$ | 0.9
Explicit | $<0.1$ | $<0.1$
AI4CSI Rx (Type II) | 243 | 0.9
AI4CSI Rx (Explicit) | 243 | $<0.1$
AI4CSI E2E (Lean UE) | 243 | 0.5
AI4CSI E2E (Balanced) | 3 | 1

The computational complexities for each scheme are listed in Table 4. Because
the UE needs to calculate and select the codeword from the codebook with NR
CSI feedback schemes, where operations such as vector correlation or even
singular value decomposition may be involved, the complexity of NR Type II
feedback on the UE side cannot be ignored. However, the complexity on the BS
side is trivial because it only needs to combine the vectors reported by UE.
However, for practical systems, we usually expect that computational
complexity can be on the BS side with a lean UE design.

By enlarging the network scale as well as the complexity on the BS side,
AI4CSI E2E schemes can have a lean UE design and still have a performance gain
compared with legacy methods. The UE can easily compress the CSI with a
network that has only one layer. Even when considering a balanced complexity
design, the UE complexity is comparable with existing schemes, and the BS
complexity can be dramatically reduced.

##  5 Performance Evaluations with Real-life Channel Measurements

We evaluated the performance of AI4CSI schemes with channels generated by
models used in 3GPP, which is a kind of stochastic spatial correlation channel
model. However, such channels are more deterministic than those in real life
and can be easily managed by deep learning. The generalization performance of
deep learning for communications, that is, whether it is still superior to
legacy methods in practical deployments, is a major concern in using them in
practical systems. In this section, we provide initial trials of AI4CSI
schemes with real-life channel samples. Since AI4CSI E2E schemes have shown
their potential gain with moderate feedback overhead in the simulations, we
focus on their performance in the evaluations with real-life channels.

Table 5: MSE of end-to-end CSI feedback and recovery with realistic channel measurements. Scheme | BS Complx. | UE Complx. | SE
---|---|---|---
| (GFLOPS) | (GFLOPS) | (bps/Hz)
NR Type II | $<0.1$ | 0.9 | 5.7
AI4CSI E2E (Balanced) | 2 | 1 | 14.5
AI4CSI E2E (Lean UE) | 60 | 0.5 | 15.4
AI4CSI E2E (Lean UE) | 240 | 0.5 | 20.9
AI4CSI E2E (Lean UE) | 960 | 0.5 | 26.3

Although there is no common dataset for AI-based physical layer design in
mobile systems, we attempt to utilize some open datasets to test whether the
AI4CSI schemes can be deployed in real-life scenarios. The channel
measurements used for the evaluations shown in this section are introduced in
[20]. The channels are measured in an indoor office scenario on the 3.5 GHz
band with 100 MHz bandwidth. The MIMO configuration is $4\times 4$. The
measured channel dataset contains 320’000 samples that are measured at
different locations in the scenario. During the measurements, the receiver
moved at a speed of 1.5m/s.

Considering that the characteristics of the real-life channel are different
from the ones obtained with models, the AI4CSI networks were retrained with
real-life channels. In practice, it is usually possible to collect channel
samples from cells before the deployment of base stations and use them to
train the network. We randomly extracted 1’000 channel samples from all
320’000 samples to create a verification dataset, and another randomly
selected 1’000 samples to set up the test dataset. The remaining data were
used as the training dataset. Similar to the simulations, we evaluated the SE
of the AI4CSI E2E schemes to demonstrate their overall performance for
downlink transmissions.

The network architecture of AI4CSI networks is the same as the ones used in
the previous section. Because real-life channels are obtained with different
MIMO configurations, the hyperparameters of the networks were adjusted to
accommodate the dimensions of the data. Following the feedback configurations
in [20], we considered a 25.6 kbps feedback overhead (128 bits for each
feedback) for all cases. The configurations of NR Type II CSI were also
modified to generate similar feedback overheads. The subband number was
configured to 10, the number of combined codewords was three, and the
combination coefficients were quantized with 3 bits for amplitudes and 4 bits
for phases.

During the trials, we noticed that it requires larger network models to handle
the real-life channel measurements because the channel characteristics are
more diverse than that generated by models. The propagation conditions, that
is, line-of-sight or non-line-of-sight, and the distribution of multipath
components on different positions in the scenario, are variant. Therefore, in
addition to the networks used in Section 4, we also consider more complex BS
side networks with larger widths, that is, larger channel numbers of the
convolution layers, to check if more complex networks can handle real-life
data. Because we do not want to introduce additional complexity for the UE,
the networks on the UE sides are unchanged. The computation complexity of the
networks is shown in Table 5, which was evaluated with the assumption of a 5
ms feedback period (200 feedbacks per second). For AI4CSI E2E (balanced)
network, because it is not flexible enough to adjust the complexity owing to
its complex network architecture, we only tested one network size
configuration. For AI4CSI E2E (lean UE), the network on the BS side is a very
deep convolution neural network whose width can be adjusted flexibly. We
tested more cases of different complexities. The width of the convolution
layers in the network was 64 when the complexity was 60 GFLOPS and 128 and 256
when the complexity was 240 GFLOPS and 960 GFLOPS, respectively. Note that the
complexity of networks with the same width is smaller than that of the
networks used in the previous section owing to the different configurations of
MIMO and feedback overhead.

It is shown in Table 5 that AI4CSI E2E schemes are superior to NR Type II
feedback schemes in terms of the SE. During the evaluations, we found that if
more complex networks on the BS side are allowed,  that is, the computational
complexity is increased to 240 or 960 GFLOPS from 60 GFLOPS , the SE is
continuously and rapidly improved by the complexity of the networks. This
verifies the capability of AI4CSI schemes to handle practical data in real-
life deployments, although the cost of computational complexity should be
considered. For the balanced design of the network, although there is no easy
way to scale it, it still provides a better tradeoff between complexity and
performance compared to the lean UE design. Because the information is lost
after air interface transmission between the encoder and decoder, it is
natural that more processing at the UE can improve to the final performance.

In summary, from this trial with real-life channel data, we find that the
AI4CSI E2E schemes can still work as they do with channels generated by the
models. The feasibility of these schemes was verified.

##  6 Conclusion

In this study, we investigated DL-based CSI acquisition schemes for massive
MIMO with the practical settings of current 5G mobile networks. We
investigated the SE, feedback overhead, and computational complexity of our
proposed two types of DL-based schemes, that is, DL-based receiver
enhancements and end-to-end design, as well as NR CSI feedback and explicit
channel matrix feedback.

The evaluations demonstrated the promising gain of DL-based CSI acquisition
for 5G evolution and 6G. At a moderate feedback overhead level, that is,
approximately 100 kbps per UE, the DL-based receiver enhancement introduces
approximately 25% performance gain on the SE of downlink transmissions, and
the end-to-end design provides more performance gain on SE, up to 58%,
compared with legacy schemes. The performance gain of DL-based CSI acquisition
schemes was also tested with a dataset consisting of channel measurements in a
practical scenario, which verifies their feasibility for real-life
deployments. As DL-based receiver enhancement has little impact on the air
interface, it can be used in 5G networks with 5G evolution without much
standardization effort. DL-based end-to-end CSI acquisition redefines the
feedback air interface of the networks, where the feedback content is designed
by the DL network itself. To obtain an additional performance gain, a new CSI
feedback framework with native AI support, which enables CSI feedback and
reconstruction designed by end-to-end DL networks, can be considered in future
6G networks.

## References

  * [1] T.L. Marzetta, “Noncooperative cellular wireless with unlimited numbers of base station antennas,” IEEE Transactions on Wireless Communications, vol.9, no.11, pp.3590–3600, 2010.
  * [2] G. Dietl and G. Bauch, “Linear precoding in the downlink of limited feedback multiuser MIMO systems,” Global Telecommunications Conference, 2007. GLOBECOM ’07. IEEE, pp.4359–4364, 2007.
  * [3] “3GPP TS 38.214, 3rd Generation Partnership Project; Technical Specification Group Radio Access Network; NR; Physical layer procedures for data,” Dec 2018.
  * [4] O. Simeone, “A brief introduction to machine learning for engineers,” Foundations and Trends in Signal Processing, vol.12, no.3-4, pp.200–431, 2018\.
  * [5] T. O’shea and J. Hoydis, “An introduction to deep learning for the physical layer,” IEEE Transactions on Cognitive Communications and Networking, vol.3, no.4, pp.563–575, 2017.
  * [6] X. You, C. Zhang, X. Tan, S. Jin, and H. Wu, “AI for 5G: Research directions and paradigms,” Science China Information Sciences, vol.62, no.2, pp.1–12, 2019\.
  * [7] Z. Wang, J. Chen, and S.C. Hoi, “Deep learning for image super-resolution: A survey,” arXiv, 2019.
  * [8] N. Ahn, B. Kang, and K.A. Sohn, “Fast, accurate, and lightweight super-resolution with cascading residual network,” Lecture Notes in Computer Science, vol.11214, pp.256–272, 2018.
  * [9] J. Yang, J. Wright, T.S. Huang, and Y. Ma, “Image super-resolution via sparse representation,” IEEE Transactions on Image Processing, vol.19, no.11, pp.2861–2873, 2010.
  * [10] J. Kim, J.K. Lee, and K.M. Lee, “Accurate image super-resolution using very deep convolutional networks,” 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp.1646–1654, 2016.
  * [11] C.K. Wen, W.T. Shih, and S. Jin, “Deep learning for massive MIMO CSI feedback,” IEEE Wireless Communications Letters, vol.7, no.5, pp.748–751, 2018\.
  * [12] Q. Shi, Y. Liu, S. Zhang, S. Xu, S. Cao, and V. Lau, “Channel estimation for WiFi prototype systems with super-resolution image recovery,” ICC 2019 - 2019 IEEE International Conference on Communications (ICC), pp.1–6, 2019.
  * [13] T. Chen, J. Guo, C.K. Wen, S. Jin, G.Y. Li, X. Wang, and X. Hou, “Deep learning for joint channel estimation and feedback in massive MIMO systems,” arXiv preprint arXiv:2011.07242, 2020.
  * [14] X. Wang, X. Hou, L. Chen, Y. Kishiyama, and T. Asai, “Deep learning based CSI reconstruction with limited feedback for massive MIMO systems,” 2021 Thirteenth International Conference on Mobile Computing and Ubiquitous Network (ICMU), pp.1–5, 2021.
  * [15] Qualcomm, “RP-213599, Study on artifical intelligence (AI)/machine learning (ML) for NR air interface,” Dec 2021.
  * [16] “3GPP TS 38.200, 3rd Generation Partnership Project; Technical Specification Group Radio Access Network; NR; Physical layer; General description,” Dec 2017.
  * [17] “3GPP TS 38.211, 3rd Generation Partnership Project; Technical Specification Group Radio Access Network; NR; Physical channels and modulation,” Dec 2019.
  * [18] Q. Spencer, A. Swindlehurst, and M. Haardt, “Zero-forcing methods for downlink spatial multiplexing in multiuser MIMO channels,” IEEE Transactions on Signal Processing, vol.52, no.2, pp.461–471, 2004.
  * [19] “3GPP TS 38.901, 3rd Generation Partnership Project; Technical Specification Group Radio Access Network; Study on channel model for frequencies from 0.5 to 100 GHz,” Dec 2017.
  * [20] J. Guo, X. Li, M. Chen, P. Jiang, T. Yang, W. Duan, H. Wang, S. Jin, and Q. Yu, “AI enabled wireless communications with real channel measurements: Channel feedback,” Journal of Communications and Information Networks, vol.5, no.3, 2020.


# Optimization-based Trajectory Planning for Tethered Aerial Robots*

S. Martínez-Rozas1, D. Alejo1, F. Caballero1 and L. Merino1 *This work has
been supported by the Spanish Ministry of Science, Innovation and Universities
(COMCISE RTI2018-100847-B-C22, MCIU/AEI/FEDER, UE).1S. Martínez-Rozas, D.
Alejo, F. Caballero and L. Merino are with Service Robotics Laboratory,
Universidad Pablo de Olavide, Seville, Spain. Email:
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

This paper presents a non-linear optimization method for trajectory planning
of tethered aerial robots. Particularly, the paper addresses the planning
problem of an unmanned aerial vehicle (UAV) linked to an unmanned ground
vehicle (UGV) by means of a tether. The result is a collision-free trajectory
for UAV and tether, assuming the UGV position is static. The optimizer takes
into account constraints related to the UAV, UGV and tether positions,
obstacles and temporal aspects of the motion such as limited robot velocities
and accelerations, and finally the tether state, which is not required to be
tense. The problem is formulated in a weighted multi-objective optimization
framework. Results from simulated scenarios demonstrate that the approach is
able to generate obstacle-free and smooth trajectories for the UAV and tether.

##  I Introduction

During the last decade, UAVs and UGVs have been employed with success in a
great variety of tasks potentially dangerous for humans, such as disaster
management, abandoned mines mapping and inspection, or even sewer inspection
[1, 2]. The great manoeuvrability of UAVs enables accessing remote areas that
would be impossible to reach with UGVs. However, the limited flight time of
small UAVs (few tens of minutes) reduces the applicability of this technology.
To increase their endurance, tethered UAVs fixed to a base station have been
proposed, but the use of a tether usually limits the UAV range to a great
extent. On the other hand, UGVs have a much larger autonomy, but they may not
be capable of reaching certain areas of interest.

The idea to take the best of the ground and aerial robots was proposed in the
late 90s through the definition of marsupial configurations [3]. These
configurations consist of a team of robots, in which one acts as the provider
robot, i.e. giving energy, transportation, communication, etc, to another
robot. This paper follows this idea and considers the use of an UGV capable of
deploying a tethered UAV to provide the latter with energy and high bandwidth
communications through the cable, see Fig. 1. This configuration poses
significant challenges for trajectory planning, as the shape of the cable
should be taken into consideration to ensure collision-free navigation,
especially in confined or underground environments, such as tunnels or sewers.
We assume that there exists a device that allows us to control the length of
the tether within a given range. We use this degree of freedom to prevent the
tether from colliding with objects in the environment.

This paper proposes a method to generate optimal trajectories that allow to
safely deploy a tethered UAV in such environments, preventing it and the
tether from colliding against obstacles. The main contribution of the paper is
the development of a formulation that considers a realistic shape of the
tether that links the UAV and the UGV into the motion planning problem, in
order to ensure the safety of the system. We validate the approach with
simulation experiments in challenging realistic environments.

Figure 1: Example of the proposed robotic system. The UAV is linked by a power
tether to a UGV. The blue line denotes the computed trajectory considering the
tether constraint.

The rest of the paper is organized as follows. Section II analyzes the
existing works in the literature that address similar problems. Then, Sections
III and IV describe the proposed methods for trajectory planning and
trajectory initialization. The experimental results are discussed in Section
V. Finally, the paper is concluded in Section VI, where we summarize the
contributions of the paper and give insights of future research directions.

##  II Related Work

Global motion planning studies the problem of obtaining a feasible trajectory
that allows a system to evolve from an starting configuration to a goal
configuration [4]. Most methods assume a graph or grid that models the
connectivity of the environment and then use graph search algorithms like
Djikstra, A* and Theta* [5] to get an optimal path. To generate the graph,
randomized methods such as PRM, RRT and its optimal variant RRT* [6] have been
successfully used in high-dimensional spaces.

On the other hand, local planners aim to generate feasible trajectories in
real-time parting from a global plan generated by one of the previous methods.
They take into account the constraints in the motion of the particular
platform and can handle the presence of unexpected obstacles. For instance,
[7] uses sparse graph optimization for obtaining optimal trajectories in UGVs,
considering kinodynamic constraints and unmodeled obstacles. In this paper, we
use a similar approach for generating optimal trajectories from a given
initial path.

Regarding planning in marsupial systems, [8] proposes an interesting approach
that uses a topological map that distinguishes the areas that are reachable by
each robot in the system. The idea is to make profit of the marsupial
configuration for reaching areas that would not be reachable with a single
robot due to its size or autonomy. However, it does not consider tethers
linking the different robots.

The research on aerial robots linked with a tether usually focuses on control
aspects. [9] proposes the use of a tether to increase the safety in landing
maneuvers. Then, [10] considers it also to increase the stability when a UAV
flights in confined spaces and [11] analyzes the performance of several flight
primitives for a tethered UAV. On the other hand, research on the use of
tether is more mature in Unmanned Underwater Vehicles (UUVs) [12], as the
tether is used as a safety recovery device and as a communication link.
However, it is usually assumed that the tethers move in a free space and thus
collisions due to tethers are not considered.

Regarding motion planning in marsupial tethered systems, [13] presents a
hierarchical approach to solve the path planning problem in map generation
missions. It uses two independent RRT* for the UAV and UGV. In the first
place, UGV plans its route and then the UAV limits its range according to the
UGV plan and the tether length, but it does not consider the collisions due to
the tether. In [14] the authors propose a relative localization system for a
marsupial tethered UGV-UAV pair. They use the estimation of the angles, as
measured by the cable’s provider SDK and a catenary model to estimate the
relative position of the UAV w.r.t. the UGV. Then, the same authors in [15]
propose a PRM planner in which reachable space is constrained by the tether in
two ways: avoiding any contacts of the tether with obstacles, and allowing up
to two different contact points to extend the effective range. However,
experimental results showed that including contact points had an important
negative impact in the relative localization of the platform, as the angles
measured by the SDK are no longer a good estimation for the shape of the
tether. Furthermore, the planner considers a tensed tether at any time,
enabling collision checking based on ray-tracing. In contrast, we consider a
hanging tether whose length is an additional configuration variable. The
tether is thus not ensured to be tight and we have to model its shape as a
catenary curve [16].

##  III Approach

This section describes the optimization-based method. It is assumed that a map
is available in the form of 3D occupancy grid. The goal is to generate a
feasible trajectory for a UAV tethered to a UGV from the starting point
(located on the UGV) to a goal position in the free space, considering dynamic
constraints such as limited UAV velocities and accelerations, and including
the length of the tether as a control variable. The UGV is assumed to remain
static during the maneuver. The trajectory optimizer makes use of an existing
global path as initial solution. This path is computed using the Lazy Theta*
algorithm [5] (see Section IV for details).

###  III-A General Statement

Our state space is defined by the UGV position
$\mathbf{p}_{g}=(x_{g},y_{g},z_{g})$, the UAV position $\mathbf{p}=(x,y,z)$
and the tether length $l$. At any instant, the tether length
$l\geq\|\mathbf{p}-\mathbf{p}_{g}\|$ and the positions of the UAV and UGV
define the whole tether position through the catenary model [16], whose
parameters can be computed using the Bisection Numerical Method [17].

The objective is to determine the trajectory of the UAV $\mathbf{p}(t)$ and
the length of the tether $l(t)$ (the UGV position will be considered as fixed)
so that the UAV reaches a desired configuration while attaining a set of
constraints posed by the obstacles, the kinematic and dynamic capabilities of
the vehicles. In our approach, such constraints will be encoded as a set of
cost functions to minimize.

###  III-B Casting the problem as a sparse non-linear optimization

In this work, we consider that the UGV position is fixed, so that the
optimization problem reduces to estimating the UAV and tether length
trajectories on time. We furthermore discretize the trajectory, so that the
states of our problem become the set:

$O=\\{\mathbf{p}_{i},l_{i},\Delta t_{i}\\}_{i=1,...,n}$ (1)

where $\Delta t_{i}=t_{i}-t_{i-1}$ is the time increment between consecutive
states, allowing to consider the temporal aspects. Without loss of generality,
we consider the UGV to be located at the origin of coordinates. The UAV takes-
off from the UGV, so $\mathbf{p}_{0}=\mathbf{0}$ and $l_{0}=0$.

For each $\mathbf{p}_{i}$ and $l_{i}$, and considering that the UGV position
is fixed and known, there is a tether configuration $TE_{i}$, given by the
catenary model mentioned above. We also discretize the resultant tether model
into a set of $m$ positions $\mathbf{p}_{te}=(x_{te},y_{te},z_{te})$:

$TE_{i}=\\{\mathbf{p}_{te,j}\\}_{j=0,...,m-1}$ (2)

Based on the discretization of the states, our problem consists of determining
the values of the variables in $O$ that optimize a weighted multi-objective
function $f(O)$:

$O^{*}=\arg\min_{O}f(O)=\arg\min_{O}\sum_{k}\gamma_{k}*f_{k}(O)$ (3)

$O^{*}$ denotes the optimized collision-free path for UAV and tether from
start to goal point. The different cost terms are of the form
$f_{k}(O)=\delta_{k}(O)^{T}\Omega_{k}\delta_{k}(O)$, that is, they are
quadratic functions of non-linear _factors_ $\delta_{k}(O)$. $\gamma_{k}$ is
the weight for each component $f_{k}(O)$ of the objective. Each component
encodes a different constraint or optimization objective of our problem, and
will be presented next. All factors of the cost function are local with
respect to $O$, as they only depend on a few number of consecutive
configurations. Consequently, our optimization problem can be represented by a
sparse factor graph (see Fig. 2) and solved with non-linear sparse
optimization algorithms. In particular, we use _g 2o_ [18] as our optimization
back-end.

###  III-C Constraints and Objective Function

We introduce constraints in the graph generation such as tether length,
velocity and acceleration limits and equidistance among consecutive robot
poses. These constraints are introduced as penalty costs in the objective
function. In addition, we aim to maximize the distance to obstacles and to
minimize the length and execution time of the trajectory.

Figure 2: The optimization problem can be represented as a factor graph.
Circles represent optimization variables $\mathbf{p}_{i},l_{i},\Delta t_{i}$,
and squares, factors (the different components of the cost function). The
figure represent the factors for 4 consecutive positions (the links to factors
and variables out from this time slice are not shown).

####  III-C1 Equidistance among vertices trajectory

The optimization process might move vertices within the planning space in
order to minimize the costs of the different constraints. To prevent grouping
vertices in low-cost areas, we force them to keep a given distance with
respect to previous and subsequent vertices.

This is a multi-edge constraint that relates four consecutive UAV position
vertices. It calculates the distance between each of the pairs of consecutive
vertices and then computes the arithmetic average distance $\alpha$ among
them. The error $\epsilon$ is estimated between $\alpha$ and each distance
between vertices.

$\delta_{eq,i}=\begin{bmatrix}\alpha-\|p_{i-2}-p_{i-1}\|\\\
\alpha-\|p_{i-1}-p_{i}\|\\\ \alpha-\|p_{i}-p_{i+1}\|\end{bmatrix}$ (4)

####  III-C2 UAV obstacle avoidance

This factor penalizes that the trajectory gets closer than a safety distance
$\rho_{a}$ to the obstacles. This is an unary edge related with every position
vertex. The distance $d_{i,o}$ from a vertex $\mathbf{p}_{i}$ to the closest
obstacle is computed by a KD-Tree search over the 3D obstacle point-cloud. If
the distance $d_{i,o}$ is inferior than the safety distance $\rho_{a}$ a high
error is computed, otherwise the error is zero.

$\delta_{o,i}=\left\\{\begin{array}[]{cc}e^{\rho_{a}-\beta*d_{i,o}}&,if\ \
d_{i,o}\ <\ \rho_{o}\\\ 0&,otherwise\end{array}\right.$ (5)

where $\beta$ is a constant parameter. The error increases exponentially as
the distance to the obstacle gets closer to $0$.

####  III-C3 UAV kinematics

We also introduce a multi-edge that relates three consecutive UAV position
vertices. The aim of this edge is to smooth the optimized trajectory, avoiding
abrupt direction changes. Two vectors are calculated, the first between the
first and second consecutive vertices, $i-1$ and $i$, and the second between
the second and third consecutive vertices, $i$ and $i+1$. Then, if the angle
between the two vectors
$\arccos(\frac{(\mathbf{p}_{i}-\mathbf{p}_{i-1})^{T}(\mathbf{p}_{i+1}-\mathbf{p}_{i})}{\|\mathbf{p}_{i}\|\|\mathbf{p}_{+1}\|})$
is larger than a bound $\rho_{\theta}$, an error is calculated:

$\delta_{\theta,i}=\frac{1}{(\mathbf{p}_{i}-\mathbf{p}_{i-1})^{T}(\mathbf{p}_{i+1}-\mathbf{p}_{i})}$
(6)

####  III-C4 UAV trajectory duration time

We include an unary-edge related to time vertex. The edge is in charge of
minimizing the trajectory time. The error is computed taking into account the
time between two consecutive positions of the initial path and the current
time trajectory.

$\delta_{t,i}=\Delta t^{[0]}_{i}-\Delta t_{i}$ (7)

where $\Delta t^{[0]}_{i-1,i}$ corresponds to the initial time difference
reference for vertices $i-1$ and $i$ and $\Delta t_{i}$ is the current value
of the time difference vertex.

####  III-C5 UAV velocity

It corresponds to a multi-edge that relates two consecutive position vertices
and one time vertex. It is in charge to keep constant the speed during the
optimized trajectory. Then, the error corresponds to the subtraction of the
current and the desired velocity $\rho_{v}$ (a parameter of the algorithm).

$\delta_{v,i}=\frac{\|\mathbf{p}_{i+1}-\mathbf{p}_{i}\|}{\Delta{t}_{i+1}}-\rho_{v}$
(8)

####  III-C6 UAV acceleration

It corresponds to a multi-edge that relates three consecutive UAV position
vertices and two consecutive time vertices. It is in charge of keeping the
module of linear acceleration close to zero (to minimize control efforts) for
the optimized trajectory. To calculate the error two values of velocity and
time are necessary.

$\delta_{a,i}=\frac{\|\mathbf{v}_{i}\|-\|\mathbf{v}_{i-1}\|}{\Delta{t}_{i}+\Delta{t}_{i+1}}$
(9)

where $\mathbf{v}_{i}=\frac{\mathbf{p}_{i+1}-\mathbf{p}_{i}}{\Delta{t}_{i+1}}$
is the velocity between consecutive poses.

####  III-C7 Tether obstacle avoidance

This is a binary-edge that relates the UAV position vertex and the tether
length vertex. Its main objective is to avoid the tether collision. Two error
components are calculated. The first avoids positioning the tether in places
where it collides. If the distance from tether points to obstacles $d_{i,to}$
is lower that a safety distance $\rho_{t}$ then the error is computed,
otherwise, this component is zero. The error favors longer lengths of the
tether. The distance to obstacles is computed by using kd-tree search for the
points of the tether discretized model $TE_{i}$.

The second component forces the tether to stretch if it is free from
collision, that is, when $d_{i,to}$ is greater than the security distance
$\rho_{l}$.

$\displaystyle\delta_{l,i}[0]=$
$\displaystyle\left\\{\begin{array}[]{cc}10^{4}\cdot
e^{10(\|\mathbf{p}_{i}\|-l_{i})}&,if\ \ d_{i,to}\ <\ \rho_{l}\\\ \
0&,otherwise\end{array}\right.$ (12) $\displaystyle\delta_{l,i}[1]=$
$\displaystyle\left\\{\begin{array}[]{cc}0&,if\ \ d_{i,to}\ <\ \rho_{l}\\\
(l_{i}-\|\mathbf{p}_{i}\|)&,otherwise\end{array}\right.$ (15)

where $\|\mathbf{p}_{i}\|$ is the distance between UAV and UGV and $l_{i}$ the
tether length. The relative weight is much larger for the collision case as
this is a hard constraint to prioritize tether safety.

##  IV Problem initialization

Non-linear optimization problems need of a proper initialization in order to
reach the optimal solution. Using naive solutions as, for instance, a straight
line between starting and goal positions, could make the optimizer converge
into a local minimum.

We need to initialize the trajectory parameters defined in (1) and also the
number of states $n$. Ideally, the initial solution must be feasible, that is,
the different states must be reachable by the UAV without colliding with
obstacles (both the UAV and the tether). Unfortunately, there are no
approaches in the state of the art dealing with this problem. Here, we have
adapted the Lazy Theta* [5] algorithm in order to properly consider the
constraints derived from the tether connecting UAV and UGV. Once a feasible
sequence of states is computed by the planning algorithm, time information is
added to the computed path as initial solution for the trajectory. Next
paragraphs describe these steps with detail.

###  IV-A Path estimation

As previously mentioned, a Lazy Theta* [5] path planner has been adapted in
order to include the constraints of the tether into the plan computation
without expanding the dimension of the planning state in order to keep
computation as low as possible.

Thus, the function that checks the feasibility of a given state has been
updated to also consider the tether. Once a state is considered as feasible, a
second check is performed to evaluate the tether at that state using the
function described in Algorithm 1. It starts with the minimal possible tether
length and increases the length while the tether is in collision with an
obstacle at any point. When no collision is detected and the tether does not
reach the maximum length $l_{max}$, the state $\mathbf{p}$ is considered as
feasible for the catenary.

The function $isCatenaryInCollision(\mathbf{p},\mathbf{p}_{g},l)$ in Algorithm
1 computes the catenary curve between $\mathbf{p}$ and the UGV configuration
$\mathbf{p}_{g}$ subject to the tether length $l$, and discretizes it. For
each point of the catenary, the function computes the distance to the closest
obstacle in the planning space. If this distance if over a given threshold for
all the points of the catenary, the configuraton
$\\{\mathbf{p},\mathbf{p}_{g},l\\}$ is considered as free of collision.

checkCatenaryFeasibility(p,pg)  __

$l=||\mathbf{p}-\mathbf{p}_{g}||$;

while  _isCatenaryInCollision( $\mathbf{p}$,$\mathbf{p}_{g},l$) &&
$l<l_{max}$_ do

$l=l+\epsilon$;

end while

if  _$l <l_{max}$_ then

return TRUE;

else

return FALSE;

end if

end

Algorithm 1 Catenary feasibilty check between state $\mathbf{p}$ and UGV
configuration $\mathbf{p}_{g}$. Return TRUE if $\mathbf{p}$ is feasible.
Figure 3: Scenarios considered for validation.(a) S1: Open space with arc as
obstacle. (b) S2: Narrow/constrained space with denied access to UGV. (c) S3:
Confined space with outlet duct for UAV. (d) S4: Collapsed Fire Station. (e)
S5: Open space gas station.

###  IV-B Trajectory estimation

The optimal nature of the Lazy Theta* algorithms prevents the creation of way-
points when there exists direct line of sight with respect to the previous
way-point. As a result, the computed path tends to be sampled frequently in
complex areas, while open areas are traversed with a straight line.

However, we are interested in sampling the trajectory at regular intervals, so
that kinematic constraints and obstacle costs can be better evaluated.
Actually, one of the constraints imposed by the optimizer is the equidistance
among trajectory way-points (Section III-C1).

Thus, the computed Lazy Theta* path is interpolated at fixed intervals between
way-points, adding points in the middle to fulfil the equidistance constraint.
Fortunately, the path planner guarantees that all points in the line that
connects two way-points are feasible because they comply with the line-of-
sight constraint of the Lazy Theta*. The length of the tether in the
interpolated points are computed using Algorithm 1.

Finally, this solution does not consider any kinematic constraints, it is just
a collision-free path from the initial UAV position to the goal. Therefore,
the values of $\Delta t_{i}$ in (1) are not yet initialized. To do so, a
constant velocity model is imposed over the path, so that $\Delta t_{i}$ can
be computed from sequenced way-points considering the Euclidean distance
between them.

The solution provided by this method is optimal in terms of the traversed
distance by the UAV considering catenary constraints, but the catenary length
is not optimal at each state, just feasible. This solution neither consider
kinematic constraints on the trajectory or distance to obstacle. All these
parameters will be computed/refined in the optimization process.

##  V Experimental Results

A set of experiments in simulated environments has been conceived to validate
the feasibility of the proposed approach. While the algorithm has been tested
on simulated environments, the full approach has been implemented as a planner
in Robot Operating System (ROS).

The trajectory planner has been implemented in C++, and it is publicly
available111https://github.com/robotics-upo/marsupial_optimizer, branch: g2o.
_g 2o_ [18] has been selected as the engine for graph optimization. The
solution makes use of classical approaches to solve computationally demanding
tasks, such as kd-trees for distance to closest point computation.

The approach has been validated in five different scenarios (S) with different
complexity, i.e. confined spaces (S3 and S4), open spaces (S1 and S5) and
narrow/constrained spaces (scenario S2), shown in the snapshot from Figure 3.
The accompanying video summarizes each of five experiments in the different
scenarios for better understanding. For each scenario, the robot has been
placed in different initial positions and several goals have been sent to the
planner. All the experiments were executed with the same set of parameters,
setting the maximum number of iterations of the optimizer to 100 and the rest
parameters to:

  * •

Weighting factors (in brackets the associated constraint): $\gamma_{eq}$ = 0.6
(1), $\gamma_{o}$ = 0.8 (2), $\gamma_{\theta}$ = 0.4 (3), $\gamma_{t}$ = 0.005
(4), $\gamma_{v}$ = 0.06 (5), $\gamma_{a}$ = 1.0 (6), $\gamma_{l}$ = 0.9 (7).

  * •

Avoiding collision constr.: $\beta$ = 4.0, $\rho_{o}$ = 1.0, $\rho_{l}$ = 0.2.

  * •

Kinematic constraints: $\rho_{\theta}$ = $\frac{\pi}{6}$, $\rho_{v}$ = 2.0.

The results of the experiments are detailed in Tables I and II. The metrics
presented are the following:

  * •

SIG: Scenario - Initial Position - Goal Position

  * •

LIP: Length of initial path [m]

  * •

LTO: Length of the optimized trajectory [m]

  * •

TIP: Time of the initial path [s]

  * •

TOT: Time of the optimized trajectory [s]

  * •

DOI: Distance obstacles-initial path [m]

  * •

DOO: Distance obstacles-optimized trajectory [m]

  * •

DCOI: Dist. catenary-obstacle solution initial [m]

  * •

DCOO: Dist. catenary-obstacles solution optimized[m]

  * •

VTO: Velocity of the optimized trajectory [m/s]

  * •

ATO: Acceleration of the optimized trajectory [m/s2]

  * •

TCI: Time compute initial solution

  * •

TCO: Time compute optimized solution

In order to benchmark the solution, we use as baseline the value of the
different metrics prior to optimization, that is, the solution of the
interpolated Lazy Theta* detailed in Section IV. Next paragraphs evaluate the
different metrics.

TABLE I: Results for experiments in simulated environments.

SIG LIP LTO TIP TOT Mean DOI Min DOI Mean DOO Min DOO Mean DCOI Min DCOl Mean
DCOO Min DCOO Mean VTO Max VTO Mean ATO Max ATO S1.1.1 3,97 3,98 1,99 1,99
0,84 0,51 0,84 0,54 0,70 0,28 0,69 0,28 2,0030 2,0120 0,0010 0,0140 S1.1.2
4,95 5,22 2,48 2,54 0,88 0,51 0,97 0,83 0,65 0,21 0,35 0,17 2,0550 2,0630
0,0010 0,0070 S1.1.3 5,34 5,89 2,67 2,81 1,01 0,51 1,18 0,90 0,72 0,23 0,69
0,23 2,0930 2,0960 -0,0010 0,0010 S1.1.4 6,27 6,77 3,13 3,30 0,91 0,49 1,13
0,78 0,81 0,44 0,80 0,33 2,0500 2,0510 0,0000 0,0000 S1.1.5 10,00 10,13 5,00
5,03 1,55 0,48 1,59 0,73 1,00 0,24 0,45 0,22 2,0160 2,0280 -0,0030 0,0100
S1.1.6 11,59 11,73 5,80 5,83 1,35 0,49 1,39 0,73 1,04 0,21 0,36 0,22 2,0140
2,0290 -0,0030 0,0040 S1.2.1 7,03 7,67 3,51 3,68 0,96 0,43 1,12 0,88 0,71 0,21
0,58 0,08 2,0830 2,0870 0,0010 0,0020 S1.2.3 8,38 9,03 4,19 4,33 1,04 0,43
1,21 0,93 0,69 0,18 0,60 0,07 2,0860 2,1270 0,0080 0,0240 S1.2.5 9,75 9,85
4,88 5,01 2,40 1,37 2,44 1,71 1,47 0,56 1,51 0,56 1,9650 1,9650 0,0000 0,0000
S1.2.6 11,69 11,82 5,84 6,00 2,01 1,54 2,10 1,65 1,58 0,58 1,58 0,58 1,9700
1,9710 0,0000 0,0000 S2.1.1 8,93 9,64 4,47 4,67 0,80 0,38 1,15 0,88 0,68 0,23
0,79 0,23 2,0630 2,0670 -0,0010 0,0000 S2.1.2 9,00 9,58 4,50 4,76 1,10 0,38
1,31 0,88 0,83 0,21 0,89 0,21 2,0130 2,0170 -0,0010 0,0010 S2.1.3 9,25 9,72
4,63 4,72 1,26 0,38 1,41 0,88 0,93 0,22 0,99 0,22 2,0580 2,0630 -0,0010 0,0000
S2.1.4 5,91 6,76 2,96 3,24 0,72 0,39 1,02 0,73 0,70 0,20 0,79 0,21 2,0830
2,0840 0,0000 0,0010 S2.1.5 6,00 7,22 3,00 3,45 0,79 0,38 1,16 0,88 0,81 0,21
0,92 0,21 2,0960 2,0980 -0,0010 0,0010 S2.1.6 6,47 7,39 3,24 3,54 0,79 0,38
1,09 0,88 0,89 0,21 1,92 0,20 2,0850 2,1490 0,0190 0,0440 S2.2.1 9,00 9,62
4,50 4,77 0,92 0,38 1,14 0,90 0,86 0,21 0,95 0,21 2,0180 2,0290 -0,0030 0,0000
S2.2.2 8,93 9,35 4,47 4,63 1,18 0,38 1,34 0,95 0,95 0,23 1,01 0,23 2,0210
2,0260 -0,0020 0,0000 S2.2.3 9,06 9,65 4,53 4,72 1,25 0,38 1,43 0,97 0,93 0,20
1,01 0,20 2,0420 2,0480 -0,0010 0,0000 S2.2.4 6,00 7,09 3,00 3,39 0,82 0,38
1,14 0,88 0,89 0,20 1,01 0,21 2,0900 2,0960 0,0010 0,0030 S2.2.5 5,97 6,97
2,99 3,30 0,83 0,38 1,13 0,95 0,90 0,21 1,01 0,20 2,1100 2,1260 0,0040 0,0130
S2.2.6 6,13 6,58 3,06 3,17 0,83 0,38 0,98 0,84 0,90 0,21 1,04 0,20 2,0790
2,0910 0,0030 0,0140 S2.3.1 9,25 9,73 4,63 4,72 1,04 0,38 1,19 0,90 0,93 0,22
0,99 0,22 2,0600 2,0740 -0,0040 0,0000 S2.3.2 9,06 9,65 4,53 4,73 1,21 0,38
1,39 0,97 0,93 0,20 1,01 0,20 2,0420 2,0480 -0,0010 0,0030 S2.3.3 8,93 9,34
4,47 4,63 1,22 0,38 1,39 0,94 0,95 0,23 1,01 0,23 2,0190 2,0240 -0,0020 0,0000
S2.3.4 6,40 7,34 3,20 3,49 0,82 0,38 1,11 0,88 0,90 0,21 1,01 0,21 2,1010
2,1040 0,0000 0,0020 S3.1.1 8,00 8,00 4,00 4,00 0,91 0,86 0,91 0,86 0,50 0,21
0,49 0,21 2,0000 2,0050 0,0000 0,0170 S3.1.2 12,60 12,60 6,30 6,30 0,91 0,86
0,92 0,86 0,44 0,14 0,43 0,12 2,0000 2,0050 0,0010 0,0110 S3.2.1 5,40 5,40
2,70 2,70 0,90 0,84 0,91 0,86 0,54 0,25 0,52 0,21 1,9990 2,0030 0,0010 0,0150
S3.2.2 10,00 10,00 5,00 5,00 0,91 0,84 0,92 0,85 0,47 0,18 0,47 0,18 2,0000
2,0030 0,0000 0,0140 S3.2.3 5,63 5,64 2,82 2,82 0,89 0,61 0,89 0,62 0,70 0,26
0,69 0,25 2,0000 2,0030 0,0000 0,0080 S3.2.4 7,81 7,82 3,91 3,91 0,97 0,61
0,98 0,63 0,67 0,20 0,66 0,20 2,0000 2,0030 0,0000 0,0080 S4.1.1 7,83 7,83
3,91 3,91 1,57 0,89 1,57 0,89 1,19 0,55 1,20 0,55 2,0000 2,0010 0,0000 0,0060
S4.1.2 11,42 11,57 5,71 5,73 2,23 0,89 2,29 0,89 1,64 0,55 1,64 0,56 2,0200
2,0200 0,0000 0,0000 S5.1.1 10,32 10,33 5,16 5,16 1,31 0,85 1,31 0,85 1,02
0,38 1,02 0,35 2,0010 2,0030 0,0000 0,0010 S5.1.2 10,41 10,63 5,21 5,34 1,52
1,02 1,70 1,28 1,37 0,49 1,44 0,48 1,9910 1,9910 0,0000 0,0000 S5.1.3 14,20
14,51 7,10 7,22 1,40 0,44 1,51 0,98 1,09 0,32 1,06 0,32 2,0110 2,0130 0,0000
0,0010 S5.1.4 13,94 14,51 6,97 7,19 1,36 0,45 1,59 0,98 1,00 0,23 0,99 0,17
2,0160 2,0210 0,0000 0,0020

TABLE II: Computational time in each of the simulated scenarios SIG | 1.1.1 | 1.1.2 | 1.1.3 | 1.1.4 | 1.1.5 | 1.1.6 | 1.2.1 | 1.2.3 | 1.2.5 | 1.2.6 | 2.1.1 | 2.1.2 | 2.1.3
---|---|---|---|---|---|---|---|---|---|---|---|---|---
| Mean TCI
---
0,84 | 24,38 | 2,64 | 1,02 | 32,83 | 40,64 | 7,66 | 21,80 | 0,48 | 0,64 | 0,51 | 0,62 | 0,70
| Mean TCO
---
1,35 | 0,99 | 2,38 | 3,91 | 4,49 | 10,10 | 1,47 | 4,98 | 9,71 | 14,97 | 9,21 | 12,16 | 10,14
SIG | 2.1.4 | 2.1.5 | 2.1.6 | 2.2.1 | 2.2.2 | 2.2.3 | 2.2.4 | 2.2.5 | 2.2.6 | 2.3.1 | 2.3.2 | 2.3.3 | 2.3.4
| Mean TCI
---
41,10 | 24,48 | 30,18 | 0,69 | 0,67 | 0,77 | 21,26 | 36,72 | 36,75 | 0,74 | 0,76 | 0,69 | 49,48
| Mean TCO
---
4,83 | 5,21 | 7,70 | 12,12 | 12,55 | 14,47 | 5,06 | 7,32 | 5,18 | 11,50 | 12,99 | 10,14 | 7,17
SIG | 3.1.1 | 3.1.2 | 3.2.1 | 3.2.2 | 3.2.3 | 3.2.4 | 4.1.1 | 4.1.2 | 5.1.1 | 5.1.2 | 5.1.3 | 5.1.4 |
| Mean TCI
---
0,37 | 0,38 | 0,38 | 0,38 | 0,58 | 1,21 | 8,15 | 1,16 | 0,46 | 0,84 | 1,42 | 49,07 |
| Mean TCO
---
4,36 | 7,85 | 2,41 | 6,15 | 4,56 | 7,71 | 10,75 | 21,30 | 12,61 | 18,95 | 25,34 | 25,12 |

###  V-A Trajectory length and elapsed time

The initial trajectories are generated by our Lazy Theta* algorithm, which is
focused on minimizing the length of the path and, assuming constant speed, the
time spent. Thus, the lengths of the optimized trajectories cannot outperform
the initial ones in these terms, and sometimes there are noticeable
differences as it considers another objectives such as obstacle clearance and
kinodynamic constraints.

The good news are that the differences rarely exceed 10% of the optimal length
and that the generated trajectories are smoother and safer than initial path.

###  V-B Distance to obstacles

The simulation results (DOI, DOO, DCOI, DCOO) confirm how the optimized
trajectory tends to move away from obstacles when the trajectory itself or the
tether is closer than the safety distance. As seen in the previous section,
this comes at a cost. To increase the clearance to the obstacles, the length
of the path should also increase with respect to the initial path.

###  V-C Trajectory kinematics

The kinematic constraints allows us to obtain a smoother trajectory with
respect to the initial path. Figure 4 shows an example in which the benefits
of the trajectory smoothing (blue line) can be clearly seen in comparison to
the initial path (green line). The results in Table I also show how the
velocity and acceleration (VTO, ATO) are close to the desired values (2.0
$m/s$ and 0 $m/s^{2}$ respectively).

Figure 4: Show how the optimized trajectory (blue line) is smoother than
initial path (green line). Tether configuration is shown as dotted curved
between planned way-point and UGV.

###  V-D Computation time

Table II presents a summary of the computational time required to estimate
both the initial and optimized trajectories. We used a Intel Core i7 2.20GHz
with 16 GB of RAM. The results indicate that computation times of the Lazy
Theta* algorithm presents a great variability, from fractions of second to 40
seconds, even when dealing with problems in the same scenario. On the other
hand, the optimization algorithm gives results ranging from one to twenty-five
seconds. As a general tendency, we have verified that the computation time
growths with larger trajectories and with increasing clutter. Nevertheless, we
consider this timing is acceptable for this type of complex planning problems.

##  VI Conclusions and Future Work

This paper presented a method for trajectory planning of a tethered UAV based
on optimization. To the best of our knowledge, this is the first planning
approach of this kind for this type of problem that considers the curvature of
a non-tight tether. The method has been intensively tested on simulated
scenarios and the results are promising as the proposed approach is able to
optimize the UAV trajectory together with its kinematics constraints taking
into account the limitations of the tether. Solutions are obtained quickly, in
a matter of seconds in most cases, in a regular computer.

Unfortunately, this solution does not consider the movement of the UGV to
compute trajectories of the whole system. Thus, future works will consider
extending this work to include the UGV as a moving system (marsupial system)
which trajectory can be also optimized together with the UAV and the tether,
as well as the tether deployment dynamics.

## References

  * [1] D. Alejo, F. Chataigner, D. Serrano, L. Merino, and F. Caballero, “Into the dirt: Datasets of sewer networks with aerial and ground platforms,” _Journal of Field Robotics_ , vol. n/a, no. n/a, 2020. [Online]. Available: https://onlinelibrary.wiley.com/doi/abs/10.1002/rob.21976
  * [2] R. Murphy, J. Dufek, T. Sarmiento, G. Wilde, X. Xiao, J. Braun, L. Mullen, R. Smith, S. Allred, J. Adams, A. Wright, and J. Gingrich, “Two case studies and gaps analysis of flood assessment for emergency management with small unmanned aerial systems,” in _2016 IEEE International Symposium on Safety, Security, and Rescue Robotics (SSRR)_ , 2016, pp. 54–61.
  * [3] R. Murphy, M. Ausmus, M. Bugajska, T. Ellis, T. Johnson, N. Kelley, J. Kiefer, and L. Pollock, “Marsupial-like mobile robot societies,” in _AGENTS ’99_ , 1999.
  * [4] S. M. LaValle, _Planning Algorithms_. USA: Cambridge University Press, 2006.
  * [5] A. Nash, S. Koenig, and C. A. Tovey, “Lazy theta*: Any-angle path planning and path length analysis in 3d.” in _Twenty-Fourth AAAI Conference on Artificial Intelligence, Atlanta, GA, USA, 11–15 July_. AAAI Press, 2010.
  * [6] S. Karaman and E. Frazzoli, “Sampling-based algorithms for optimal motion planning,” _The International Journal of Robotics Research_ , vol. 30, no. 7, pp. 846–894, 2011. [Online]. Available: https://doi.org/10.1177/0278364911406761
  * [7] C. Rösmann, W. Feiten, T. Wösch, F. Hoffmann, and T. Bertram, “Efficient trajectory optimization using a sparse model,” in _2013 European Conference on Mobile Robots_ , 2013, pp. 138–143.
  * [8] P. G. Stankiewicz, S. Jenkins, G. E. Mullins, K. C. Wolfe, M. S. Johannes, and J. L. Moore, “A motion planning approach for marsupial robotic systems,” in _2018 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), Madrid, 2018_. IEEE, 2018, pp. 1–9.
  * [9] L. A. Sandino, D. Santamaria, M. Bejar, A. Viguria, K. Kondak, and A. Ollero, “Tether-guided landing of unmanned helicopters without gps sensors,” in _2014 IEEE International Conference on Robotics and Automation (ICRA)_ , 2014, pp. 3096–3101.
  * [10] M. Schulz, F. Augugliaro, R. Ritz, and R. D’Andrea, “High-speed, steady flight with a quadrocopter in a confined environment using a tether,” in _2015 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)_ , 2015, pp. 1279–1284.
  * [11] X. Xiao, J. Dufek, and R. Murphy, “Benchmarking tether-based uav motion primitives,” in _2019 IEEE International Symposium on Safety, Security, and Rescue Robotics (SSRR)_ , 2019, pp. 51–55.
  * [12] M. Laranjeira, C. Dune, and V. Hugel, “Catenary-based visual servoing for tether shape control between underwater vehicles,” _Ocean Engineering_ , vol. 200, p. 107018, 2020. [Online]. Available: http://www.sciencedirect.com/science/article/pii/S0029801820300949
  * [13] C. Papachristos and A. Tzes, “The power-tethered uav-ugv team: A collaborative strategy for navigation in partially-mapped environments,” in _22nd Mediterranean Conference on Control and Automation_ , 2014, pp. 1153–1158.
  * [14] J. D. X. Xiao, Y. Fan and R. Murphy, “Indoor uav localization using a tether,” in _2018 IEEE International Symposium on Safety, Security, and Rescue Robotics (SSRR), Philadelphia, PA, 2018_. IEEE, 2018, pp. 1–6.
  * [15] X. Xiao, J. Dufek, M. Suhail, and R. Murphy, “Motion planning for a uav with a straight or kinked tether,” in _2018 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), Madrid, 2018)_. IEEE, 2018, pp. 8486–8492.
  * [16] E. Lockwood, _A Book of Curves_ , 1st ed. Cambridge University Press, ISBN: 103156, 1961.
  * [17] J. D. Burden, Richard L; Faires, _Numerical analysis_ , 3rd ed. Cambridge University Press, ISBN: 1036756653, 1985\.
  * [18] R. Kümmerle, G. Grisetti, H. Strasdat, K. Konolige, and W. Burgard, “G2o: A general framework for graph optimization,” in _2011 IEEE International Conference on Robotics and Automation (ICRA), Shanghai, 2011_. IEEE, 2011, pp. 3607–3613.