# Towards reliable and transparent vaccine phase III trials with smart
contracts

Ivan da Silva Sendin
School of Computer Science
Federal University of Uberlandia
<EMAIL_ADDRESS>
Rodrigo Sanches Miani
School of Computer Science
Federal University of Uberlandia
<EMAIL_ADDRESS>

###### Abstract

Transforming a vaccine concept into a real vaccine product is a complicated
process and includes finding suitable antigens and regulatory, technical, and
manufacturing obstacles. A relevant issue within this scope is the clinical
trial process. Monitoring and ensuring the integrity of trial data using the
traditional system is not always feasible. The search for a vaccine against
the coronavirus SARS-CoV-2 illustrates this situation. The scientific
credibility of findings from several vaccines’ clinical trials contributed to
distorted perceptions concerning the benefits and risks of the drug. This
scenario is ideal for applying technologies such as Blockchain and Smart
Contracts in healthcare issues. This paper proposes a protocol based on Smart
Contracts, named VaccSC, to enable transparency, accounting, and
confidentiality to Phase III of vaccine experiments. The protocol was
implemented in Solidity language, and results show that the VaccSC enables
double-blindness, randomization, and the auditability of clinical data, even
in the presence of dishonest participants.

##  1 Introduction

The COVID-19 pandemic that hit the world in 2019/2020 [6] is undoubtedly a
disruptive event. Basically, all aspects of our society have been affected -
and still will be - due to this pandemic. During the COVID-19 pandemic,
vaccine development underwent irreversible changes. Before 2019, it was
expected that a vaccine would take a few years or even a decade to develop and
be considered safe, efficient, and made available to the market [4]. Contrary
to initial expectations, several laboratories [7] released advanced studies of
COVID-19 vaccine candidates in less than a year of research. This speed may
raise doubts about the honesty and rigor of these scientific experiments.

Before regulatory approval, a vaccine candidate usually experiences three
phases of development in humans: Phase I, Phase II, and Phase III. During
Phase III large-scale trials, the human subjects are divided into two groups:
i) those who received the vaccine and ii) those who received a control, a
placebo, for example. The main outcome of this phase is the vaccine efficiency
with is calculated by comparing the incidence of disease in vaccinated
subjects and the incidence of disease in unvaccinated subjects (control
groups). Unfortunately, this process is not transparent to society as a whole,
which can compromise acceptance of the vaccine by some individuals [14, 11].

While transparency is important, the confidentiality of emerging data is also
relevant: vaccine developers should not have access to them with the risk of
changes in the trial design and compromising the final research results [12].
The recent technologies of Blockchain and Smart Contracts are primarily known
for adding a layer of ”trust” on the Internet services, as they offer
decentralization, immutability, and public access to stored data [25, 29].

The use of Blockchain and Smart Contracts in the health area is not new.
Several studies point out the possibility of using these technologies in
healthcare sustems[30, 22], diagnostics tracking[5], vaccine and medication
trial tracking[27, 2]. More recently, the so-called ”COVID-19 passports” were
addressed by these technologies [8], [10] [23]. The previously cited
approaches maintain trust ”out of the contract”. In other words, the data is
stored in the contract, and participants use it with a notary system.
Therefore, the contract cannot interfere with the integrity of the stored
information.

In this work we present a protocol based on smart contracts and Commitment
Schemas to enable transparency, accounting and confidentiality to Phase III of
vaccine experiments. Our work’s main contribution for the use of Smart
Contracts in vaccine trials is the use of Commitment Schemas to enable the
properties of double-blind, randomization and the auditability of clinical
data, even in the presence of dishonest participants. Also, the Smart
Contracts can determine whether the experiment has reached the desired
efficiency and decide on the vaccine approval.

The rest of paper is organized as follows. In Section 2 we outline the
theoretical background of our proposal which includes a description of
Commitment Schemes and Smart Contracts. We briefly describe the vaccine
development process in Section 3. Section 4 details the proposed protocol
called VaccSC. Section 5 summarizes related work and Section 9 concludes the
paper.

##  2 Background

###  2.1 Commitment Schemes and Coin Flipping

Commitment Schemas emerged in the early 1980s [3] and are used when a party
$\mathcal{A}$ needs to commit to a certain amount $v$ with another party
$\mathcal{B}$ without revealing the value of $v$ immediately.

A common construction of Commitment Schemes is done using cryptographic
hashing functions and has two phases:

Commit Phase

In this phase, $\mathcal{A}$ commits to the value $v$ generating a
nonce111Nonce is a pseudo-random number used only once in some cryptographics
protocol. $n$ and producing

$\mathrm{C}=\mathrm{H}(v|n),$

where $|$ indicates the concatenation operation and $\mathrm{H}$ is a
cryptographic hash function. The value $\mathrm{C}$ \- the commit - is then
made public;

Reveal Phase

To reveal the committed amount, $\mathcal{A}$ sends to $\mathcal{B}$ $v$ and
$n$, which calculates the hash value of the received data, and checks whether
the generated hash is equal to $\mathrm{C}$.

We say that $\mathrm{C}$ has the properties of bidding, since $\mathcal{A}$
cannot create $v^{\prime}$ and $n^{\prime}$ that produce the same
$\mathrm{C}$; and hidding because from $\mathrm{C}$ it is not possible to
determine $v$. The value $\mathrm{C}$ can be seen as a sealed version of the
value $n$ in the sense that once $\mathrm{C}$ has been published,
$\mathcal{B}$ does not knows $v$ and $\mathcal{A}$ can no longer change $v$
without being discovered. See [9] for a review and security recommendations.

Using the Commitment Schemes, one can build a protocol to establish a pseudo-
random number between parties that do not trust each other. This protocol is
known as ”coin flipping over telephone”. To generate a pseudo-random bit,
parts A and B follow the following steps:

  * •

$\mathcal{A}$ (and $\mathcal{B}$), privately, generate a pseudo-random bit
$R_{\mathcal{A}}$($R_{\mathcal{B}}$);

  * •

$\mathcal{A}$($\mathcal{B}$) commits and reveals
$R_{\mathcal{A}}$($R_{\mathcal{B}}$) using a Commitment Scheme;

  * •

Now, $\mathcal{A}$ and $\mathcal{B}$ knows both $R_{\mathcal{A}}$ and
$R_{\mathcal{B}}$, and both can calculate $R=R_{\mathcal{A}}\oplus
R_{\mathcal{B}}$;

  * •

The resulting bit $R$ is a pseudo-random bit shared by $\mathcal{A}$ and
$\mathcal{B}$.

One of the parties acting honestly is enough for the bit produced by the
protocol to be random[3].

###  2.2 Smart Contracts

Introduced by Nick Szabo in [26], Smart Contracts (SC) are programs whose
correct execution cannot be prevented. A SC can be compared to vending
machines: with a coin and the press of a button, the user gets the expected
product with a certain degree of confidence without dependence on other
entities.

Currently, the SC platforms are obtained through P2P networks that execute
contracts - also called dapps \- redundantly. The participating nodes in this
network receive financial incentives for the correct execution of these
contracts. The high degree of redundancy and the economic stimulus produces an
extremely reliable virtual computer.

The main SC platform currently available is Ethereum [28], which uses its own
cryptocurrency, called Ether, in its transactions and to reward participating
nodes. Ethereum is permissionless, meaning that anyone can access the network.
Users at the Ethereum platform are identified by an address that is
essentially a public key. Access to the system, such as transferring values or
invoking methods in contracts, is done through signed messages using the
private key corresponding to Ethereum addresses. The confidence in the correct
execution of programs and the extensive use of cryptographic protocols allow
SC to be secure, transparent, and auditable.

The applications of SC are countless: financial services, games, betting, and
network infrastructure are some examples222See https://etherscan.io/dapp for
an extensive list of dapps.. The capability to enforce rules, accurately
described by a programming language, allows the adoption of SC in other
application scenarios such as public administration [13] and regulated
activities [1].

##  3 Vaccine development proccess

The vaccine development goes through an elaborate development process after
the discovery stage [24]. It is common to divide this process into preclinical
and clinical stages. The clinical-stage is defined by several clinical trials
in human subjects organized by three phases. The objective of phase I is to
evaluate the vaccine’s safety in a small sample of human volunteers. During
phase II, the goal is to demonstrate immunogenicity vaccine by assessing the
impact of multiple variables on immune response, such as age, ethnicity,
gender, and presence of maternal or pre-existing antibodies. The sample of
human volunteers in phase II varies from hundreds to thousands of subjects.
Phase III trials are large-scale clinical trials and essential for
registration and approval to license the vaccine. If the phase III results
demonstrate efficacy and safety, the vaccine manufacturer can apply to the
national regulatory authority to approve and market the product. The primary
outcome is vaccine efficiency (VE), which can be defined as [16]:

$VE=(1-RR)*100\%$ (1)

where RR represents risk reduction (or relative risk). RR is the ratio between
the incidence of disease in vaccinated subjects and disease incidence in
unvaccinated subjects. Considering $AR_{1}$ the number of infected cases among
the subjects in the investigational vaccine group and $AR_{0}$ the number of
infected cases among the control subjects then

$RR=\frac{AR_{0}-AR_{1}}{AR_{0}}.$ (2)

This can be seen as a comparison of who received the vaccine and who received
a placebo. To illustrate how the RR is calculated we will use the [21]: a
threshold of 164 contamination confirmations was determined for this
experiment. If, for example, after 164 infections it is found that 120 were
from people who received the control and 44 from people who received the
vaccine, the RR of this vaccine would be approximately 0.63.

The disease’s occurrence is the most common endpoint; however, the trial may
be based on other clinical endpoints, such as incidence of infection or
immunological correlations of protection. After completing Phase III trials
and following licensure of the product, Phase IV studies, also referred to as
postmarketing surveillance studies (PMS), are used to monitor the vaccine for
safety and population effectiveness.

The well-accepted standard for carrying out vaccination experiments is
conducted in the randomized controlled, double-blind trial system [15]. In
this system, the person responsible for applying the vaccine randomly chooses
what will be applied to each individual, not knowing its content (vaccine or
control). Also, the subject who receives the shot does not know what he/she is
receiving. This procedure aims to avoid bias in the vaccination process.
Eventually, a dishonest vaccine company would like to improve its product’s
efficiency by selecting a healthier subpopulation to receive the vaccination
and a less healthy population to receive control. Similarly, the person
receiving the shot should not know its content to avoid the experiment’s
placebo effect.

##  4 Proposed Protocol

In this session, we present VaccSC: a SC based protocol to track a Phase III
vaccine trial providing transparency and double-blind behavior.

For the sake of simplicity, the proposed protocol models the participation of
three types of entities:

Vaccine Developer

Responsible for distributing vaccines to clinics and protecting information
about their content. He is also responsible for deploying the contract, being
the only participant who can distinguish the real vaccine from the control,
creates the commits for the shots, which will be used as identifiers in the
contract;

Vaccine Clinic

Apply the shots and updates the VaccSC biding the shot with a specific
patient;

Patient

Receives the shot and notifies the contract when he/she becomes ill.

We assume that all participants have an Ethereum address and keep the
corresponding private key securely. This step can be achieved using a
smartphone, for example.

###  4.1 Random Patient/Shot Assignment

As stated earlier, the random association between the shot and the patient
prevents bias. The generation of random numbers on SC environment is
complicated and must be done carefully. In this protocol, we chose to use the
approach in which each participant generates a random number ($R_{1}$ and
$R_{2}$). The SC combines these numbers using bitwise XOR operation. With this
approach, one part being honest is enough for the number to be random. This
process is described in Figure 1.

Vaccine
---
Clinic
Patient
---
Vaccine Trial
---
Smart Contract
1234$R_{1}$Reveal $R_{1}$$R_{2}$Reveal $R_{2}$ Figure 1: Each participant
privately generates a random number and commits this number. After both
numbers are committed to the contract, each participant reveals the value
(steps 3 and 4). The SC combines these numbers - $R_{1}\otimes R_{2}$ \- to
associate one of the shots available to the patient.

###  4.2 Double Blindness

Here we describe how the SC guarantees Double Blindness property. The Vaccine
Clinic does not know what he is applying as much as the patient does not know
if he receives the vaccine or a control shot. Initially, Vaccine Developer
protects the information on the content of each shot that will be part of
Phase III as described in Section 2.1:

$C_{i}=\mathrm{H}(n_{i}|v_{i}),$

the value $C_{i}$ is used as a shot identifier by the contract and must be
physically associated with the shot. The values $n_{i}$ and $v_{i}$ are saved
and used later to reveal the shot’s content and calculate the vaccine
effectiveness.

Also, when deploying the contract, Vaccine Developer informs:

Number of participants:

corresponding to the number of available shots;

Infected Threshold:

number of patients who need to be infected before the committed values are
revealed and the vaccine’s efficiency is calculated;

Target efficiency:

minimum VE required in order to vaccine be approved;

Vaccination Clinics:

the Ethereum address of clinics that will apply the shots and update the
VaccSC.

The data structure that stores the necessary information to calculate vaccine
efficiency are shown in Table 1. The fields are presented in chronological
order in which they are changed during the execution of the contract.

Information | Type | Access and Control
---|---|---
Commit | hash | Immutable, crated by Vaccine Developer
Clinic | Address | Each clinic is added by Vaccine Developer
Patient | Address | Clinic and patient
gotSick | Boolean | Patient
Vaccine Type | Vaccine/Placebo |  VaccSC fills field with the reveal provided by Vacine Developer
Table 1: Data structure used by VaccSC to keep vaccine trial information.

###  4.3 Execution

Once the Vaccine Developer deployed the VaccSC, the protocol follows the
following steps:

  1. 1.

The Vaccine Developer delivers the shots to each Vaccination Clinic and makes
the association between the shots and the clinics in the contract;

  2. 2.

The clinic applies the shot and associates the shot identifier with a specific
patient. The patient confirms this to contract;

  3. 3.

If the patient becomes ill, he communicates the fact to the contract;

  4. 4.

When the infected threshold is reached, the contract creates an event
informing the Vaccine Developer;

  5. 5.

Using the Ethereum addresses of infected patients, Vaccine Developer selects
the corresponding commits. After that, the Vaccine Developer sends information
about patients who received the control to the contract. The contract
calculates the number of patients who received the vaccine;

  6. 6.

Vaccine efficiency can be determined (using Equation 2), and its approval
status can be obtained directly from the contract by anyone.

An overview of the protocol is shown in Figure 2.

Vaccine
---
Developer
Vaccine
---
Clinic
Patient
---
Vaccine Trial
---
Smart Contract
123456Vaccine ShotVaccinatesGot sickTrial FinishedRevealVaccine Status Figure
2: Protocol events associated with one shot. Step four occurs only once. Solid
lines indicates physical events whith contract updates, and dashed lines
indicates VaccSC interactions.

###  4.4 Security analysis

In the associating vaccines to patients phase, both participants are equally
responsible for generating the random number, and one of the participants
acting honestly is sufficient to produce a random number. If the two conspire
to choose a specific shot, they still cannot choose between vaccine or
control, as the Vaccine Developer protected this information.

The double blindness property has the following analysis:

Patient

The action expected by the patient is to communicate his own disease event to
the contract. Patients do not know about their vaccination status, so they
cannot effectively change the results of the contract;

Vaccine Clinic

As he applies the shot but does not know its content, he is also unable to
defraud the protocol;

Vacccine Developer

A dishonest developer can improve the efficiency of the vaccine by decreasing
the number of people who have been effectively vaccinated and have become ill.
Using the proposed protocol, he cannot do that because his action will
precisely reveal the patients who received a placebo, so the omission is
harmful to him. Therefore, the forging of ”control patients” does not occur
because the shot status is committed.

###  4.5 Availability

The VaccSC protocol was implemented in Solidity language, the client side
applications were developed using the Brownie suite333https://github.com/eth-
brownie and Python language. The contract is available for download at
https://github.com/ivansendin/Vaccine. Commit Scheme and Coin Tossing
libraries are available at https://github.com/ivansendin/CommitExamples.

##  5 Related Work

Early attempts to apply SC to conduct clinical experiments use SC to store
data securely, guaranteeing the data’s authenticity and integrity through the
access control and digital signatures provided by the platform. There are
several works available with this approach [17, 2, 18, 27, 19, 20].

[17] used SC as a solution to data manipulation issues common to clinical
trials. The authors propose a hierarchical arrangement of two core types of
smart contracts: i) a regulator contract, holding a data structure containing
clinical trial authorization (CTA) details, and ii) a trial contract deployed
by contract research organizations (CROs). The trial contract’s logic
effectively enforces aspects of the trial protocol, securing that neither
subjects nor measurements are added outside the trial timelines. At the same
time, the tamper-resistant characteristics of the blockchain prevent data
manipulation. Authors deployed contracts using a private Ethereum blockchain
with synthetic data representing the clinical trials of Tamiflu, an influenza
drug. They were able to query the state of data, such as the number of trials
underway, the number of subjects recruited, and the timestamp at which the
transaction was processed.

[2] attempted to clarify the three main principles of using blockchain in
clinical trials: timestamping, time-ordering and smart-contracting. Regarding
SC , authors focus on providing examples of how it can be applied to clinical
trials. According to them, SC can be applied in the entire process. The case
of randomization of a patient in a clinical trial before written consent is
briefly discussed as a potential application. The paper does not provide any
experiments or SC implementations.

Omar et al. [18] and [19] proposed a SC using Ethereum to tackle data
management challenges in clinical trials. The proposed SC initially captured
three stages of a clinical trial process: new drug application, clinical trial
initiation, and patient enrollment stages. [19] discussed patient monitoring
and severe adverse event occurrence. The smart contract was written in
Solidity and captures interactions among clinical trial stakeholders such as
the regulatory agency, drug sponsors, physicians, and patients. For instance,
for the new drug application scenario, the smart contract implements the
process in which the sponsor obtains permission from the FDA (Federal Drug
Agency) to test their drug in humans. The contract has a function that
requests the sponsor to upload the required documentation. When the
documentation status is set to true, the FDA announces the approval or
rejection of the request.

[27] developed a blockchain-based web portal to tackle challenges in the
current clinical trial process. The designed portal is accessible to all
interested parties, facilitating the interaction between patient and clinical
investigator. The portal also enables expediting adverse event reporting. The
authors modeled a phase II clinical trial during their experiments. They
simulated how a previously completed clinical trial (omalizumab drug) would
behave using blockchain software technologies. Despite arguing that forcing
all participating parties to use a service like that is still a challenge,
they showed that data entry, storage, and adverse event reporting could be
performed more robustly and securely.

[20] addressed the issue of centralized vaccine production supervision. The
authors proposed a decentralized blockchain-based method to enforce privacy
protection of vaccine production records. The blockchain structure is divided
into two levels. The first one is the private data of the pharmaceutical
company and consists of production records. The second one is public data. A
public blockchain is used to store this data. Every transaction in the block
represents a process for each batch of vaccine within the production process.
Each transaction also has its timestamp and signature of the company and the
hash value of the previous transaction to prevent production data fraud. The
authors were able to protect the privacy of the company’s production of
vaccines using this structure. The proposed framework was implemented using
the Hyperledger Fabric blockchain. The authors only evaluated the spatio-
temporal efficiency (impact of the transaction against latency, throughput,
and blockchain size) of the method.

##  6 Conclusion

We presented VaccSC: a protocol based on Smart Contracts that brings
transparency and reliability to Phase III of vaccine trials while maintaining
the double-blind property present in existing protocols. The protocol prevents
the Vaccine Developer from hiding relevant information to vaccine trials and
allows anyone to audit the trial data. The protocol achieves these conflicting
requirements using Commitment Schemas.

The presented contract models a simplified version of the real world, other
scenarios - such as the presence of observers to ratify the events - can be
easily added to the contract. Ultimately, the proposed protocol captures the
essence of Smart Contracts - ”Code is law” - and can be used as a deciding
element on the approval of a vaccine, whether by a country or by an
individual.

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# Observation of dipolar splittings in high-resolution atom-loss spectroscopy
of 6Li $p$-wave Feshbach resonances

Manuel Gerken  Binh Tran  Stephan Häfner  Physikalisches Institut, Universität
Heidelberg, Im Neuenheimer Feld 226, 69120 Heidelberg, Germany  Eberhard
Tiemann  Institut für Quantenoptik, Leibniz Universität Hannover, Welfengarten
1, 30167 Hannover, Germany  Bing Zhu<EMAIL_ADDRESS>Matthias
Weidemüller<EMAIL_ADDRESS>Physikalisches Institut,
Universität Heidelberg, Im Neuenheimer Feld 226, 69120 Heidelberg, Germany
Hefei National Laboratory for Physical Sciences at the Microscale and
Department of Modern Physics, and CAS Center for Excellence and Synergetic
Innovation Center in Quantum Information and Quantum Physics, University of
Science and Technology of China, Hefei, Anhui 230026, China

(August 27, 2024)

###### Abstract

We report on the observation of dipolar splitting in 6Li $p$-wave Feshbach
resonances by high-resolution atom-loss spectroscopy. The Feshbach resonances
at 159 G and 215 G exhibit a doublet structure of
$10\text{\,}\mathrm{m}\mathrm{G}$ and $13\text{\,}\mathrm{m}\mathrm{G}$,
respectively, associated with different projections of the orbital angular
momentum. The observed splittings agree very well with coupled-channel
calculations. We map out the temperature dependence of the atom-loss spectrum
allowing us to extrapolate resonance positions and the corresponding widths to
zero temperature. The observed dipolar splitting in fermionic lithium might be
useful for the realization of the quantum phase transition between the polar
and axial $p$-wave superfluid phases.

##  I Introduction

It is generally known that magnetic $p$-wave Feshbach resonances (FRs) in
ultracold atomic collisions exhibit a dipolar splitting, resulting in two
resonance features corresponding to $m_{l}=0$ and $m_{l}=\pm 1$ Ticknor  _et
al._ (2004); Strauss  _et al._ (2010). Here, $m_{l}$ is the projection of the
orbital angular momentum $l$ along the external magnetic field. This non-
degeneracy was predicted to give rise to a phase transition from a $p_{x}$ to
a $p_{x}+ip_{y}$ state in a $p$-wave superfluid (SF) Gurarie and Radzihovsky
(2007), which plays an essential role in SF liquid 3He Volovik (1992) and
neutron SFs inside neutron stars Page  _et al._ (2011); Shternin  _et al._
(2011). The dipolar splitting was observed in a spin-polarized gas of 40K
Ticknor  _et al._ (2004) and a large variety of quantum-gas mixtures Pilch
_et al._ (2009); Wang  _et al._ (2011); Repp  _et al._ (2013); Wang  _et al._
(2013); Dong  _et al._ (2016). It is attributed to effective spin-spin (_ss_)
interactions including the magnetic dipole-dipole interaction and the second-
order spin-orbit coupling Strauss  _et al._ (2010). Recently, the observation
of dipolar splitting has been extended to $d$-wave resonances Cui  _et al._
(2017); Yao  _et al._ (2019), and an additional splitting mechanism has been
discovered for $m_{l}=+1$ and $m_{l}=-1$ due to spin-rotation interaction in
$p$-wave FRs Zhu  _et al._ (2019); Häfner  _et al._ (2019).

Due to the favorable properties of its FRs, the fermionic isotope of lithium,
6Li, one of the only two stable fermionic isotopes among the alkali metals,
has been a major workhorse in experiments with quantum-degenerate Fermi gases.
Prime applications include the study of the BCS-BEC crossover Jochim  _et al._
(2003); Zwierlein  _et al._ (2003); Bartenstein  _et al._ (2004a); Chin  _et
al._ (2004); Bourdel  _et al._ (2004); Zwierlein  _et al._ (2004); Bartenstein
_et al._ (2004b), the Efimov effect Ottenstein  _et al._ (2008); Huckans  _et
al._ (2009); Wenz  _et al._ (2009); Williams  _et al._ (2009); Lompe  _et al._
(2010), the double SF Ferrier-Barbut  _et al._ (2014), and the Fermi-Hubbard
model in optical lattices Chin  _et al._ (2006); Parsons  _et al._ (2015);
Duarte  _et al._ (2015); Omran  _et al._ (2015); Greif  _et al._ (2016); Mitra
_et al._ (2018). Although dipolar splitting in 6Li $p$-wave FRs was predicted
to be $\sim$ $10\text{\,}\mathrm{m}\mathrm{G}$ Chevy  _et al._ (2005), it was
not resolved so far due to insufficient magnetic resolution in previous
experiments Zhang  _et al._ (2004); Schunck  _et al._ (2005); Nakasuji  _et
al._ (2013); Yoshida  _et al._ (2018); Waseem  _et al._ (2017, 2018). By using
an optical lattice $p$-wave Feshbach molecules in the $m_{l}=\pm 1$ component
were selectively formed without resolving the dipolar splitting Waseem  _et
al._ (2016), while the studies of the molecular binding energy and lifetime in
Refs. Fuchs  _et al._ (2008); Inada  _et al._ (2008); Maier  _et al._ (2010)
neglected this effect.

In this work we report high-resolution trap-loss spectroscopy on three 6Li
$p$-wave FRs in the two lowest-energy hyperfine states. Dipolar splittings of
about $10\text{\,}\mathrm{m}\mathrm{G}$ are resolved in two of the three
resonances and systematic effects of the sample temperature on the observed
splitting are investigated. We model the observations with a coupled-channels
(cc) calculation and find an excellent agreement between experiment and
modeling.

##  II Dipolar splitting

Figure 1: 6Li $p$-wave FRs in the $\ket{1}\oplus\ket{1}$,
$\ket{1}\oplus\ket{2}$, and $\ket{2}\oplus\ket{2}$ entrance channels observed
as atomic loss features. The figure shows the fractional remaining Li atoms in
spin-state $\ket{1}$ (solid squares) or $\ket{2}$ (open circles) after a
holding time of $500\text{\,}\mathrm{m}\mathrm{s}$,
$150\text{\,}\mathrm{m}\mathrm{s}$ and $100\text{\,}\mathrm{m}\mathrm{s}$ from
left to right, respectively. The resonances near $159\text{\,}\mathrm{G}$ in
the $\ket{1}\oplus\ket{1}$ channel and $214\text{\,}\mathrm{G}$ in the
$\ket{2}\oplus\ket{2}$ channel show doublet structures, where quantum numbers
$m_{l}$ are assigned according to theoretical modeling. The data of spin state
$\ket{2}$ for the resonance near $185\text{\,}\mathrm{G}$ is vertically
shifted to avoid overlapping with that of $\ket{1}$. The solid curves are fits
of multi-peak Gaussian functions from which the resonance positions $B^{e}$
and widths $w^{e}$ are extracted as listed in Table 1.

A detailed description of our experimental apparatus has already been given in
previous publications Repp  _et al._ (2013); Pires  _et al._ (2014). In brief,
we optically trap a laser-cooled sample of 6Li atoms in the two energetically
lowest hyperfine states $\ket{f=1/2,m_{f}=1/2}$ (labeled as $\ket{1}$) and
$\ket{f=1/2,m_{f}=-1/2}$ (labeled as $\ket{2}$) in a cigar-shaped optical
dipole trap. Here, $f$ and $m_{f}$ refer to the total angular momentum and its
projection along the external magnetic field. The sample is evaporatively
cooled within $\sim 5\text{\,}\mathrm{s}$ at a magnetic field of $\sim$
$890\text{\,}\mathrm{G}$. Both spin states can be selectively populated by
removing the other one with a short resonant light pulse. Finally, we end up
with $3\times 10^{4}$ atoms in each spin state at a temperature of $T\approx$
$0.14\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{K}$. The trapping frequencies
are determined to be $\omega=2\pi\times$
$(30,170,180)\text{\,}\mathrm{H}\mathrm{z}$, resulting in a peak atomic
density of $1.6(1)\times$$10^{11}\text{\,}\mathrm{c}\mathrm{m}^{-3}$ and
$T/T_{F}=0.53(4)$.

Three 6Li $p$-wave FRs are identified by performing magnetic-field dependent
loss spectroscopy of Li samples either polarized at a single hyperfine state
($\ket{1}$ or $\ket{2}$) or in a mixture of both spin states, as shown in Fig.
1. After optimized holding times of $500\text{\,}\mathrm{m}\mathrm{s}$
($159\text{\,}\mathrm{G}$ FR), $150\text{\,}\mathrm{m}\mathrm{s}$
($185\text{\,}\mathrm{G}$ FR) and $100\text{\,}\mathrm{m}\mathrm{s}$
($214\text{\,}\mathrm{G}$ FR) the remaining number of Li atoms in each spin-
state is detected by absorption imaging. The points for each resonance are
taken in a random order of the magnetic field. Calibration of the magnetic
field is performed by rf-spectroscopy of the nuclear spin-flip transition
between the Li $\ket{1}$ and $\ket{2}$ state. The Breit-Rabi formula is then
used to infer the magnetic field strength. Its total uncertainty is derived
from calibration measurements to be $10\text{\,}\mathrm{m}\mathrm{G}$,
resulting from daily drifts, residual field curvature along the long axis of
the atom cloud, and calibration uncertainties.

In Fig. 1, doublet structures corresponding to $m_{l}=\pm 1$ and $m_{l}=0$
components in the 6Li $p$-wave FRs near $159\text{\,}\mathrm{G}$ and
$214\text{\,}\mathrm{G}$ are presented. Gaussian functions are fitted to the
data to extract the experimental loss peak positions $B^{e}_{m_{l}}$ and loss
widths $w^{e}_{m_{l}}$. Doublet splittings $\delta^{e}$ are defined as
distances between the two peaks. The results are listed together with the
theoretical resonance splitting $\delta^{t}$ obtained from a full cc
calculation in Table 1.

The observed doublet splittings are in excellent agreement with a cc
calculation including an effective spin-spin interaction, similar to that in
Ref. Knoop  _et al._ (2009). We measure a splitting of
$10(1)\text{\,}\mathrm{m}\mathrm{G}$ and $13(1)\text{\,}\mathrm{m}\mathrm{G}$
for the $159\text{\,}\mathrm{G}$ and $214\text{\,}\mathrm{G}$ $p$-wave FR,
respectively. In case of the $p$-wave FR in channel $\ket{1}\oplus\ket{2}$,
the small calculated splitting of $4\text{\,}\mathrm{m}\mathrm{G}$ is not
resolved in the experiment, since the splitting is considerably smaller than
the typical width of the loss peak. The assignments of quantum numbers $m_{l}$
in Fig. 1 are from the cc calculation. Our observations also agree very well
with previous predictions Chevy  _et al._ (2005); Fuchs  _et al._ (2008).

Table 1: 6Li $p$-wave FRs. The experimentally obtained resonance positions $B^{e}_{\pm 1}$, doublet splittings $\delta^{e}$ and widths $w^{e}_{m_{l}}$ are extracted by fitting multi-peak Gaussian functions to the loss spectra. The first number in brackets give their determination uncertainty and the second number the systematic uncertainty of $10\text{\,}\mathrm{m}\mathrm{G}$. The theoretical doublet splittings $\delta^{t}$ obtained from the cc scattering calculations for a relative kinetic energy of $k_{B}\times$ $140\text{\,}\mathrm{n}\mathrm{K}$, matching with the experimentally measured temperature, are given. |  $B^{e}_{\pm 1}$ (G) |  $w^{e}_{0}$ (mG) |  $w^{e}_{\pm 1}$ (mG) |  $\delta^{e}$ (mG) |  $\delta^{t}$ (mG)
---|---|---|---|---|---
$\ket{1}\oplus\ket{1}$ | 159.097(1)(10) | 6(2) | 17(2) | 10(1) | 10
$\ket{1}\oplus\ket{2}$ | 185.092(1)(10) | 10(1) | 10(1) | - | 4
$\ket{2}\oplus\ket{2}$ | 214.825(1)(10) | 7(2) | 10(1) | 13(1) | 13

##  III Effects of finite temperature

Figure 2: Temperature dependence of trap loss spectrum near the
$214\text{\,}\mathrm{G}$ 6Li $p$-wave FR. (a) Loss spectrum at different
sample temperatures. The base-lines of the profiles are shifted according to
$T$, as shown at the right axis. The data is fitted using a double-Gaussian
function to extract the loss peak positions $B_{m_{l}}^{e}$ and widths
$w^{e}_{m_{l}}$. The dotted vertical line shows the extrapolated peak position
of $B^{e}_{\pm 1}$ at zero temperature at $214.824\text{\,}\mathrm{G}$. (b)
$B^{e}_{m_{l}}$ (black symbols) and $w^{e}_{m_{l}}$ (red symbols) of the
observed loss features at different $T$ for the $m_{l}=\pm 1$ (solid circles)
and $m_{l}=0$ (open squares) components. $B^{e}_{m_{l}}$ are referenced to the
zero temperautre value of $B^{e}_{\pm 1}$ at $214.824\text{\,}\mathrm{G}$. The
lines are linear fits to the data.

The finite temperature of the gas causes systematic effects on Feshbach
resonances in atom-loss spectra, such as asymmetries, shifts, broadenings, and
saturation Ticknor  _et al._ (2004); Dong  _et al._ (2016); Zhang  _et al._
(2004); Waseem  _et al._ (2017, 2018). Averaging over the Maxwell-Boltzmann
distribution of the collision energy results in an asymmetric lineshape
Ticknor  _et al._ (2004); Chevy  _et al._ (2005) shifting the resonance center
and broadening the spectrum. Saturation of atom loss stems from the unitarity
limit at finite temperatures, rendering it impossible to resolve magnetic
resonance splittings below magnetic fields of order $3k_{B}T/2\delta\mu$
Waseem  _et al._ (2017, 2018), with $\delta\mu$ denoting the relative magnetic
moment between the molecular and atomic states. To ensure that our
measurements (magnetic field resolution of $\sim$
$1\text{\,}\mathrm{m}\mathrm{G}$) are not limited by such effects,
temperatures below $T\sim$ $0.7\text{\,}\mu\mathrm{K}$ near the
$214\text{\,}\mathrm{G}$ resonance ($\delta\mu=k_{B}\times$
$118(8)\text{\,}\mu\mathrm{K}\mathrm{/}\mathrm{G}$ Fuchs  _et al._ (2008)) are
required.

We investigate the temperature effects on the resonance lineshapes near the
$214\text{\,}\mathrm{G}$ $p$-wave FR. Evaporation is stopped at varying trap
depths resulting in $T\leq$ $0.3\text{\,}\mu\mathrm{K}$, and atomic loss
spectra are recorded, as shown in Fig. 2(a). Above $0.3\text{\,}\mu\mathrm{K}$
the line-shape becomes asymmetric and the doublet spitting is unresolvable due
to the increase of both the temperature-broadened width and the influence of
effects due to unitarity, as discussed above. The extracted loss peak
positions and widths are plotted in Fig. 2(b).

As expected Ticknor  _et al._ (2004), the resonance peak moves towards lower
magnetic field at a lower temperature. We notice that the observed loss width
$w^{e}$ is considerably larger than the expected temperature-broadened width
$k_{B}T/\delta\mu$, ranging between $1.1\text{\,}\mathrm{m}\mathrm{G}$ and
$2.5\text{\,}\mathrm{m}\mathrm{G}$ within the temperature range of the
experiments. To the lowest order, the observed temperature dependence of the
loss peak position and width can be well approximated by linear functions
(black and red lines in Fig. 2(b)). This yields for the resonance positions
slopes of $16(3)\text{\,}\mathrm{m}\mathrm{G}\mathrm{/}\mu\mathrm{K}$ and
$10(3)\text{\,}\mathrm{m}\mathrm{G}\mathrm{/}\mu\mathrm{K}$ for the $m_{l}=0$
and $m_{l}=\pm 1$ components, respectively. Although the slopes are slightly
different for the two $|m_{l}|$ values, the doublet splitting $\delta^{e}$
remains almost constant within the experimental uncertainty when $T$ is
varied. Extrapolation to zero temperature gives a splitting of
$13(1)\text{\,}\mathrm{m}\mathrm{G}$.

Using the Breit-Wigner theoretical approach described in Refs. Waseem  _et
al._ (2017, 2018) and assuming $k_{B}T$ as the smallest energy scale, we have
estimated the shift of the resonances for two- and three-body loss processes
to be $5k_{B}/2\delta\mu=$
$21.2\text{\,}\mathrm{m}\mathrm{G}\mathrm{/}\mu\mathrm{K}$ and
$k_{B}/\delta\mu=$ $8.5\text{\,}\mathrm{m}\mathrm{G}\mathrm{/}\mu\mathrm{K}$,
respectively. By comparing the experimental and theoretical slopes we infer
that both two- and three-body loss contributes to the observed temperature
dependence. Disentangling these loss channels by time-resolved measurements is
beyond the scope of this paper. In order to realize a $p$-wave SF, one would
have to mitigate loss by identifying appropriate temperature and density
regimes, or by using dimensional control Zhou and Cui (2017); Waseem  _et al._
(2017).

For the observed widths of the loss resonances, we obtain slopes of
$37(7)\text{\,}\mathrm{m}\mathrm{G}\mathrm{/}\mu\mathrm{K}$ and
$14(4)\text{\,}\mathrm{m}\mathrm{G}\mathrm{/}\mu\mathrm{K}$ for the $m_{l}=\pm
1$ and $m_{l}=0$ components, while the zero-temperature widths are
$8(1)\text{\,}\mathrm{m}\mathrm{G}$ and $6(1)\text{\,}\mathrm{m}\mathrm{G}$,
respectively. Contributions to the observed widths include the resonance
intrinsic width $\gamma$ Chevy  _et al._ (2005); Waseem  _et al._ (2017,
2018), the thermal broadening, and other experimental broadening effects like
the trap-induced density inhomogeneity and magnetic-field noise. However, from
the observed linearity and the fact that $w_{e}(T)>k_{B}T/\delta\mu$ in Fig.
2(b) one may infer that $\gamma\gg k_{B}T$ in our experiment. This conclusion,
however, is in disagreement with recent estimations of $\gamma$ extracted from
the temperature- and interaction-strength-dependent two- Waseem  _et al._
(2017) and three-body Waseem  _et al._ (2018) loss rate constants near the
$p$-wave resonances above the Fermi temperature.

##  IV Conclusion

In conclusion, we have performed high-resolution atom-loss spectroscopy of
$p$-wave Feshbach resonances in an optically trapped ultracold 6Li gas and
resolved the splittings of about $10\text{\,}\mathrm{m}\mathrm{G}$ between
$m_{l}=0$ and $m_{l}=\pm 1$ components. Our measurements agree excellently
with a full cc calculation including spin-spin interactions. In spin-polarized
Fermi gases near $p$-wave FRs, there have been predictions of the phase
transition from a polar $p_{x}$ state to an axial $p_{x}+ip_{y}$ state as well
as the topological transition from a gapless to a gapped $p_{x}+ip_{y}$ phase
Botelho and Melo (2005); Gurarie  _et al._ (2005); Cheng and Yip (2005). In
the past, stabilities of single-component Fermi gases and $p$-wave Feshbach
molecules near $p$-wave FRs were investigated, while the realization of
$p$-wave SF in quantum gases is still challenging due to severe inelastic
losses Regal  _et al._ (2003); Zhang  _et al._ (2004); Schunck  _et al._
(2005); Gaebler  _et al._ (2007); Fuchs  _et al._ (2008); Inada  _et al._
(2008); Maier  _et al._ (2010). Dimensional confining has been identified as a
promising route for stabilizing Fermi gases near $p$-wave FRs and realizing
$p$-wave SF pairing Zhou and Cui (2017); Waseem  _et al._ (2017). Resolving
the predicted dipolar splitting of FRs in 6Li provides a good starting point
for future investigations in this direction exploring the individual control
of $m_{l}=0,\pm 1$ scattering processes Günter  _et al._ (2005). Furthermore,
experimental studies of the $p$-wave contact can now be extended to 6Li, which
was, previously, only accessible in a 40K gas due to its larger splitting of
the $m_{l}$ components Luciuk  _et al._ (2016).

###### Acknowledgements.

We are grateful to J. Ulmanis, S. Jochim and S. Kokkelmans for inspiring
discussions. S.H. acknowledges support by the IMPRS-QD. This work is supported
in part by the Heidelberg Center for Quantum Dynamics, the DFG/FWF Research
Unit FOR2247 under Project No. WE2661/11-1, and the DFG Collaborative Research
Centre SFB1225 (ISOQUANT).

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# Distributed Optimal Control Framework for High-Speed Convoys: Theory and
Hardware Results

Namya Bagree  Charles Noren  Damanpreet Singh  Matthew Travers  Bhaskar
Vundurthy  Department of Mechanical Engineering, Carnegie Mellon University,
USA (e-mail: {nbagree, damanprs}@andrew.cmu.edu).  The Robotics Institute,
Carnegie Mellon University, USA (e-mail: {cnoren, mtravers,
pvundurt}@andrew.cmu.edu).

###### Abstract

Practical deployments of coordinated fleets of mobile robots in different
environments have revealed the benefits of maintaining small distances between
robots, especially as they move at higher speeds. However, this is counter-
intuitive in that as speed increases, reducing the amount of space between
robots also reduces the time available to the robots to respond to sudden
motion variations in surrounding robots. However, in certain examples, the
benefits in performance due to traveling at closer distances can outweigh the
potential instability issues, for instance, autonomous trucks on highways that
optimize energy by vehicle “drafting” or smaller robots in cluttered
environments that need to maintain close, line of sight communication, etc. To
achieve this kind of closely coordinated fleet behavior, this work introduces
a model predictive optimal control framework that directly takes non-linear
dynamics of the vehicles in the fleet into account while planning motions for
each robot. The robots are able to follow each other closely at high speeds by
proactively making predictions and reactively biasing their responses based on
state information from the adjacent robots. This control framework is
naturally decentralized and, as such, is able to apply to an arbitrary number
of robots without any additional computational burden. We show that our
approach is able to achieve lower inter-robot distances at higher speeds
compared to existing controllers. We demonstrate the success of our approach
through simulated and hardware results on mobile ground robots.

###### keywords:

Mobile robots, Autonomous robotic systems, Decentralized control, Nonlinear
predictive control, Field robotics, Convoy, Platooning, Multi-agent systems

##  1 Introduction

Fleets of mobile robots have shown practical benefits by moving closer to each
other at high speeds to optimize system resource utilization and improve
process efficiencies. An example of this is a group of trucks that reduce fuel
consumption through the reduction of aerodynamic drag by following at lower
inter-vehicle distances (Nahavandi et al., 2022; Turri et al., 2017). However,
reducing inter-vehicle distances reduces the amount of time available to
respond to sudden unexpected behaviors or motions of the other robots in the
fleet. This can lead to instabilities in the behavior of the overall system.
To address these problems, our work investigates non-linear control techniques
for autonomous robots that can maintain low inter-robot distance over various
environments at high speeds.

While the advantages of a fleet of robots exist in many domains (Nahavandi et
al., 2022), small mobile robots can also see performance improvements by
driving fast at close distances in unstructured regions. A practical challenge
in search and rescue missions is maintaining consistent communication between
the robots, especially in cluttered environments where the loss of direct line
of sight can hinder inter-robot communications (Scherer et al., 2022). This is
another example of a scenario where actively maintaining low inter-robot
separation distances is essential to the performance of the team. To this end,
in this work, we deploy such a small-scale system in both simulations as well
as hardware.

Figure 1: Robot convoy performing a search and rescue task

The primary focus of this work is to present a non-linear model based
controller that addresses many issues with the current state-of-the-art
methods for controlling fleets of mobile robots operating in cluttered
environments. Our controller takes information from all of the adjacent robots
and computes both the feedforward as well as the feedback components that help
it proactively plan as well as reactively adjust behavior to compensate for
any unexpected behaviors across the fleet.

In addition, we designed the control framework to operate in a decentralized
fashion. This allows us to extend our framework to an arbitrary number of
robots with a constant computational requirement for each robot. We
demonstrate the overall efficacy of our approach via experiments in simulation
as well as hardware, for a fleet of wheeled ground vehicles.

##  2 Problem Definition

Figure 2: Schematic for the convoy control problem. Lead and follow vehicles
are defined with respect to vehicle $i$. Error $e$ is the Euclidean distance
between a vehicle’s desired position ($d^{ref}$ behind the lead vehicle) and
its current location.

Consider the schematic illustrated in Fig. 2 for vehicle $i$. We assume an
increasing index along the trajectory and thus refer to the neighboring
vehicles as the lead vehicle and the follow vehicle denoted by $i-1$ and
$i+1$, respectively. Given $L$ agents in the fleet, the state and control
trajectories for the agent $i$ over a time horizon of length $N$ are given by
$X^{i}=[x_{0}^{i},x_{1}^{i},...,x_{N}^{i}]$ and
$U^{i}=[u_{0}^{i},u_{1}^{i},...,u_{N-1}^{i}]$, respectively, where
$i\in\mathcal{I}_{L}=\\{1,2,\cdots,L\\}$.

The objective of the convoy problem as we define it in this work is to design
a distributed convoy controller that generates a control output $u^{i}_{0}$ at
every time instant to get a fleet of robots to follow a predefined trajectory.
In particular, the desired objectives for each vehicle are to

  1. 1.

Maintain a fixed distance $d^{ref}$ along the desired path from its lead
vehicle;

  2. 2.

Simultaneously travel at desired speeds;

  3. 3.

React to disturbances without creating instabilities to the overall behavior
of the convoy.

##  3 Literature Review

To provide context for our control approach, we review relevant prior works
that address the convoy control problem defined in Section 2. Despite several
use cases for convoys (Nahavandi et al., 2022), we focus our discussion on
works for unstructured and cluttered environments.

One of the earliest works (Yazbeck et al., 2014) in convoy control mimics a
leader-follower behavior where each vehicle estimates and stores the path of
its predecessor as a set of points. The follower then estimates the
predecessor’s path curvature around a selected target and follows the
trajectory. (Nestlinger et al., 2022) extend this work to store position
measurements over time and apply a spline-approximation technique to obtain a
smooth reference path for the underlying motion controllers. However, these
methods force the robot to track the exact positions of its predecessors,
restricting system flexibility around obstacles.

Albrecht et al., 2020 provides a framework to switch between exact pose
tracking and flexible path search and tracking based on the environment. This
allows the trajectory following convoy algorithms to operate in real-world
conditions. However, there is no interaction between the outputs of the two
tracking methods, reducing the controller performance due to conflicting
switch decisions when operating in a cluttered environment.

In contrast to (Yazbeck et al., 2014; Nestlinger et al., 2022; Albrecht et
al., 2020), there exist prior works of convoy control wherein they
specifically provide an integrated obstacle avoidance module to operate a
fleet of robots in unstructured environments. For instance, in (Zhao et al.,
2017), the authors define a robust solution using an adaptive inter-robot
distance control and leader pose ($x_{0}^{i-1}$) estimation. However, there
are no common velocity-based control parameters and no prediction on future
paths for adjacent agents ($X^{i-1},X^{i+1}$). This leads to higher angular
variation as inter-robot distances reduce, resulting in wavy motions and
higher error in tracking at high speeds.

An additional prior work that incorporates specific obstacle avoidance in the
control framework is presented by (Shin et al., 2020). Here, the authors
approach the convoy problem with a controller that incorporates a passivity-
based MPC method that explicitly integrates a traversability map in the
planner. However, this framework doesn’t take feedback from the following
vehicles ($i+1$), which can lead to high inter-robot distances on cluttered
terrains. They also rely on continuous communication between the vehicles and
a base node to operate.

One other work that doesn’t apply the “follow-the-leader” framework as
discussed earlier, is presented by (Turri et al., 2017). The authors in this
work take a centralized approach to designing their controller, focusing
primarily on straight-line velocity profiles. The objective of the
optimization in their work is to improve fuel efficiency while ensuring
desired safe following distances between vehicles. The computational cost
increases quadratically as you add new agents and the overall approach needs
to be solved on a single computer, creating a single point of failure. This
also requires high-rate continuous communications between the robots and the
base node for safe convoying.

##  4 Convoy Behavior

(a) Environmental situation where robot $i+1$ is stuck

(b) Fleet response demonstrating the accordion problem

Figure 3: Comparison of convoys when following robots are neglected in the
framework (Case A) vs when they are not (Case B).

Our convoy system is designed to include the best aspects of an explicit
“follow-the-leader” behavior and a completely centralized controller. We
display a scenario in Fig. 3(a) to better understand the comparison of a
“follow-the-leader” (Case A) behavior against ours (Case B). Agent $i+1$ has
been forced to slow down while navigating a cluttered environment. A spring-
damper-mass system was defined for three agents to simulate both behaviors.
Each agent is only affected by its lead agent in Case A and both its adjacent
agents in Case B. Fig. 3(b) shows that Case B stabilizes faster and reduces
accordion-like effects between the agents. This also ensures lower inter-robot
distances through the environment, assisting real-time communication between
the agents.

The high-level controller behaviors listed in Section 2 can be expressed in an
optimal control framework. The optimal control framework presented in this
work expresses the behaviors described in (1) and (2) as terms in the
controller objective. The proposed controller’s cost structure encodes agent
reactivity to disturbances described in behavior (3). We now describe the
proposed controller in detail.

###  4.1 Convoy Controller

For the $i$th robot in the convoy, the discrete-time optimal control problem
framework is posed as: ∀i ∈I_L C_traj(X^i, U^i) + C_convoy(X^i, X^i-1, X^i+1)
x^i_k+1 = f(x^i_k,u^i_k, Δt), ∀k = 0,…,N-1 x^i_0=x^i(0), u^i_0=u^i(0), ∀i
∈I_L, where:

$\begin{split}&C_{traj}(X^{i},U^{i})=\sum_{k=0}^{N-1}(x_{k}^{i}-x^{i}_{traj,k})^{T}Q(x_{k}^{i}-x^{i}_{traj,k})\\\
&+u_{k}^{(i)T}Ru_{k}^{i}+(x_{N}^{i}-x^{i}_{traj,N})^{T}Q_{f}(x_{N}^{i}-x^{i}_{traj,N}),\end{split}$
(1) $\begin{split}C_{convoy}&(X^{i-1},X^{i},X^{i+1})=\\\
&\sum_{k=0}^{N-1}(x^{i}_{k}-x^{ref,i-1,i}_{k})^{T}Q_{lead}(x^{i}_{k}-x^{ref,i-1,i}_{k})\\\
&+(x^{i}_{k}-x^{ref,i,i+1}_{k})^{T}Q_{follow}(x^{i}_{k}-x^{ref,i,i+1}_{k}),\end{split}$
(2)

and where: $x_{k}\in\mathbb{R}^{m}$, $u_{k}\in\mathbb{R}^{n}$. Additionally,
$Q,Q_{f},Q_{lead},$ $Q_{follow}\in\mathbb{R}^{m\times m}$ are symmetric
positive definite matrices, and $R\in\mathbb{R}^{n\times n}$ is a positive
definite matrix. The state-transition model for the $i$th robot is defined as
$f(x_{k}^{i},u_{k}^{i},\Delta t)$. Superscript $ref$ refers to the reference
point defined between the agents with the indices following $ref$ (i.e. either
$i-1,i$ or $i,i+1$). The computation of this point is described in the
following section. As the run-time cost is evaluated over the same time
interval, the run-time costs in (1) and (2) may be collapsed into a single
quadratic cost expression. This new expression is defined as

$\displaystyle g(x_{k},u_{k})=\text{ }$
$\displaystyle(x_{k}^{i}-Q_{T}^{-1}y_{T})Q_{T}(x_{k}^{i}-Q_{T}^{-1}y_{T})$ (3)
$\displaystyle\text{ }-y_{T}^{T}Q_{T}y_{T}+Z_{T}+u_{k}^{(i)T}Ru_{k}^{i},$

where:

$\displaystyle Q_{T}$ $\displaystyle=Q+Q_{lead}+Q_{follow}$ $\displaystyle
y_{T}$
$\displaystyle=Qx_{traj,k}^{i}+Q_{lead}x_{k}^{ref,i-1,i}+Q_{follow}x_{k}^{ref,i,i+1}$
$\displaystyle Z_{T}$
$\displaystyle=(x_{traj,k}^{i})^{T}Qx_{traj,k}^{i}+(x_{k}^{ref,i-1,i})^{T}Q_{lead}(x_{k}^{ref,i-1,i})$
$\displaystyle+(x_{k}^{ref,i,i+1})^{T}Q_{follow}(x_{k}^{ref,i,i+1}).$

This operation is detailed further in Appendix A. A similar combination of
quadratic expressions can be performed on the terminal cost, yielding
$(Q_{F},y_{F},Z_{F})$, respectively. These parameters may be used to rephrase
the terminal cost:

$\phi(X_{N})=(x_{N}^{i}-Q_{F}^{-1}y_{F})Q_{F}(x_{N}^{i}-Q_{F}^{-1}y_{F})-y_{F}^{T}Q_{F}y_{F}+Z_{F}.$

Thus the cost function may be rephrased as:

$J=C_{traj}+C_{convoy}=\sum_{k=0}^{N-1}\\{g(x_{k},u_{k})\\}+\phi(X_{N}).$

Furthermore, by linearizing the system around $x_{k}$,  $u_{k}$ and defining
$A_{k}=\frac{\partial}{\partial x_{k}}f(x_{k},u_{k})$ and
$B_{k}=\frac{\partial}{\partial u_{k}}$, the optimization problem can be
interpreted and solved online as an iterative Linear Quadratic Regulator Li
and Todorov, 2004. This yields a control law:

$\displaystyle u_{k}=$
$\displaystyle-[R_{k}+B_{k}^{T}P_{k+1}B_{k}]^{-1}B_{k}^{T}P_{k+1}A_{k}(x_{k}^{i}-Q_{T}^{-1}y_{T})$
$\displaystyle=$ $\displaystyle-K_{k}(x_{k}^{i}-Q_{T}^{-1}y_{T}),$

with $P_{k}$ representing the solution to the Riccati Equation and $K_{k}$
being the optimal control gain matrix.

###  4.2 Controller Implementation and Design Discussion

The first component of the cost function is a quadratic trajectory tracking
cost penalizing deviations from a given convoy trajectory: $x_{traj}$. This
cost is defined in (1). In this work, the reference path $x_{traj}$ is
provided to the controller as either a pre-defined path or created for each
individual robot from observing the motion of other agents in the convoy e.g.,
“follow-the-leader” style approaches (Nestlinger et al., 2022; Yazbeck et al.,
2014).

The convoy cost $C_{convoy}$ penalizes deviance from the convoy structure over
the future time horizon. This cost is defined in (2) where
$x^{ref,i-1,i}_{k},x^{ref,i,i+1}_{k}$ are the reference positions of the
$i^{th}$ given the positions of the $i-1$ and $i+1$ cars, respectively, and
$Q_{lead},\text{ }Q_{follow}$ are tunable positive semi-definite constant
matrices. The state vector includes the $x$ and $y$ coordinates, vehicle
orientation $\psi$ and velocity $v$. The control vector includes acceleration
$a$ and steering angle $\delta$.

The computation of $x^{ref}$ is based on the desired inter-robot distance,
$d^{ref}$. This desired inter-robot distance is defined as

$d^{ref}=\lambda_{1}v_{t}+\lambda_{2}(v_{i}-v_{i-1})+K,$

where $v_{t}$ is the desired target velocity and $v_{i}$ corresponds to
current velocity for agent $i$. $\lambda_{1}$ and $\lambda_{2}$ are tunable
parameters where $\lambda_{1}$ and $\lambda_{2}$ are non-negative values. $K$
is a constant minimum inter-robot distance for safe operation. Only the
current velocities of vehicle $i$ and the prior robot in the convoy structure
are considered for this work.

As shown in Fig. 2, the reference positions for agents $i-1$ and $i+1$ over
horizon $1:N$ are recovered by performing an open-loop forward rollout using
the linearized dynamics at the $i-1$ or $i+1$ agent’s state. The $i-1$ and
$i+1$ agents’ current velocity and steering are assumed to be constant over
the rollout. The reference positions for (2) are set by moving backward
$d^{ref}$ along the convoy trajectory from the predicted $i-1$ agent positions
$X^{i-1}$ and moving the same distance ahead of the $i+1$ agent positions
$X^{i+1}$.

The reactivity described in (3) of 2 aims to prevent collisions between agents
due to sudden variations in speed. To enable this behavior, the controller
computes a weighting factor, $w_{convoy}$, between the costs (1) and (2)
during run-time. This factor is based on the desired convoy spacing and the
current Euclidean distance $dist_{i,j}$ between agents $i$ and $j$. These
weights are multiplied to the $Q$ matrices in (2) and the weighting factor is
computed as:

$Q_{lead}=w_{i,i-1}*Q_{lead}$ $Q_{follow}=w_{i,i+1}*Q_{follow}$
$w_{i,j}=\begin{cases}1+\frac{w_{far}\times(dist_{i,j}-d^{ref})}{dist_{i,j}}&\text{if
}dist_{i,j}\geq d^{ref}\\\ \
1+\frac{w_{near}\times(d^{ref}-dist_{i,j})}{dist_{i,j}}&\text{otherwise.}\end{cases}$

The proposed formulation allows the robot to track its predecessor and
follower through the predicted $x^{ref}$ terms while also tracking its desired
planned path. The proposed additional convoy cost provided in the optimal
control formulation may be interpreted as a modification of the local
linearization point used in the LQR. This modification of the coordinate
transfer from the reference path ($x_{traj}$) through both the user-defined
weightings $(Q,Q_{lead},Q_{follow})$ and reference trajectories of the leading
($X^{lead}$) and following ($X^{follow}$) robots in the fleet.

Algorithm 1 Convoy controller with obstacle avoidance for robot $i$

Input: Robot states $x_{0}^{i-1}$, $x_{0}^{i}$, $x_{0}^{i+1}$
Output: Control sequence $U^{i}$

while Robots are in convoy, $i\in\mathcal{I}_{L}$ do

Run convoy controller, Section 4.1

if Output path $X^{i}$ is obstacle-free then

return Control sequence $U^{i}$

else

Define $D_{LookAhead}$ based on velocity $v_{t}$

Calculate desired velocity $v_{T}$ and direction $\theta_{T}$

Run the local planner Zhang et al., 2020

Get the modified obstacle-free path $X^{i}$

Send $X^{i}$ into a path following iLQR controller

return Control sequence $U^{i}$

end if

end while

###  4.3 Local Planner and Trajectory Controller

In the absence of an obstacle free path, Algorithm 1 ensures high-speed
following and maintains the convoy formation with the help of an additional
velocity scaling term:

$v_{T}=(1+\alpha)v_{tc}$ $\alpha=\lambda_{3}(d_{1}-d^{ref})-\lambda_{4}d_{2}$

Where $\lambda$’s are tunable parameters, $v_{tc}$ is the desired target
velocity from the convoy controller, $v_{T}$ is the modified desired target
velocity for the planner, $v_{r}$ is the robot’s velocity, $v_{l}$ is the
leader’s velocity, $d_{1}$ is the distance between the robot and the leader
along the trajectory and $d_{2}$ is the distance between the robot and the
follower along the trajectory.

The desired direction is selected based on the output of the convoy
controller. This is done to track the desired controller path to the greatest
extent before the robot switches back into the convoy controller mode. The
direction $\theta_{T}$ is defined via the look ahead distance $D_{LookAhead}$
along the optimal trajectory generated by the convoy controller.

The local planner takes these inputs and generates a feasible path that avoids
obstacles while tracking outputs from the convoy controller. This work is
based on the planner defined in (Zhang et al., 2020). The selected path is
then sent to an iLQR trajectory following controller to generate the control
sequence.

(a) Straight Line

(b) Low Curvature

(c) High speed $\infty$ loop

(d) Tight turns

(e) Tunnel environment

(f) Race track

Figure 4: Simulation Environments

(a) Straight Line

(b) Low Curvature

Figure 5: Simulation Results

(a) 4 m/s

(b) 5 m/s

(c) 6 m/s

(d) 7 m/s

(e) 8 m/s

Figure 6: Speed Variations in an $\infty$ loop

##  5 Simulation Results

The performance of the presented controller (“Convoy Controller”) is
characterized in a variety of simulated environments shown in Fig. 7. The
underlying simulator is Gazebo (Koenig and Howard, 2004). As discussed in
Section 3, there are many different approaches to enforcing convoy structure
at the control and planning level. To develop a comparison between the
presented methodology and existing literature, a baseline “Base Controller” is
developed by combining the local planning and distance variation behaviors
from (Zhao et al., 2017) with additional modifications from (Bayuwindra et
al., 2020; Albrecht et al., 2020; Mohamed-Ahmed et al., 2019). This
combination creates a decentralized controller, similar to our proposed
method, and outperforms the individual performance of each work separately
with respect to minimizing inter-agent distances without breaking convoy
formation.

Two error metrics are used to compare the results between both controller
performances. The first metric is as follows:

$e_{m_{1}}=\begin{cases}abs((d_{i-1,i+1}/2)-d_{i-1,i})&i\in(1,L)\\\ \
abs((d_{L-2,L}/2)-d_{L-1,L})&i=L\end{cases}$

This metric aims to understand how well a robot is able to maintain a position
midway between its adjacent robots. Variation in this metric can help
understand accordion-like behaviors that negatively affect fleet performance
with continuous acceleration and braking requirements. However, this error
metric can maintain a low value with all robots maintaining an equally large
following distance, which is not desirable. To that end, we define a second
error metric as follows:

$e_{m_{2}}=\text{abs}(d_{i-1,i}-d_{desired})$

This metric measures distance from the desired gap between robots and, along
with the first metric, can provide a thorough understanding of the convoy
performance.

(a) Tight Turns

(b) Tunnel

(c) Race Track

Figure 7: Simulation Results - Edge Cases

An initial comparison is done on a straight line and low curvature turn path.
Next, an $\infty$ loop with a 20m radius is simulated at speeds varying
between 4 and 8 m/s in order to test controller performance in continuous
turns. Finally, runs are performed on tight turns, a tunnel environment and a
racetrack. Additional constraints on agent operability and speed are added on
these runs to understand fleet performance in edge scenarios.

###  5.1 Straight line and low curvature runs

The first set of simulations was performed on a group of robots operating on
simple paths while starting from scattered initial locations. The aim of this
experiment is to understand how quickly various controllers settle and how the
error values vary during that period. As can be seen in Fig. 5, our convoy
controller is able to settle faster while maintaining lower values on both
error metrics. This displays faster convoy structure formation through our
framework, as each robot has a cost associated with all of its adjacent
agents.

###  5.2 High speed $\infty$ loop

We run robots in an $\infty$ loop at various speeds to understand system
capability limits. Fig. 6 provides error metric graphs run at various target
speeds between 4 - 8 m/s. It can be seen through the figures that at initial
speeds, error metrics for both controllers settle, while our control achieves
this faster. However, as speeds increase, the difference in performance
between our convoy controller and the base controller increases until a point
when the base controller breaks and isn’t able to maintain a convoy structure.

###  5.3 Tight turns

A series of waypoints are sent to the lead robot, the other robots have no
information about these points. On running the base controller, the following
agents tend to overshoot due to the presence of sharp turns. This behavior is
seen even if we tune down the look-ahead value to zero (Bayuwindra et al.,
2020), depicting an inherent unwillingness to follow through with such sharp
turns at high speeds. The error metric comparison between controllers can be
seen in Fig. 7(a). Our controller performs better where usual convoying
techniques overshoot and has difficulty tracking such tight turns.

###  5.4 Tunnel environments

An agent might get stuck during operation, especially in indoor environments
at high speeds. To simulate this, we initially stop the first robot in place
for 8 seconds and then revert to normal conditions. We simulate this through
narrow tunnels. In Fig. 7(b), the error metrics are capped in our approach,
whereas they continuously rise with the base controller as the last robot
isn’t able to get back into the convoy. Due to the cost term corresponding to
the immediate lead and follower in the convoy controller cost definition, as
soon as one of the agents isn’t maintaining desired motion, the inter-robot
gaps increase and the corresponding cost term shoots up. This causes the
remaining agents to slow down and come to a stop until the slowed-down
follower agent rejoins, capping the error metric.

###  5.5 Race track

We simulate a constrained outdoor environment at 4 m/s. We also add an
additional constraint that one of the robots isn’t able to achieve speeds
higher than 3 m/s, which would mimic a robot with operational issues or a
heterogeneous set of robots. We run this setup on a race track, shown in Fig.
4(f). The error metrics are seen in Fig. 7(c). The initial performance between
the controllers is similar until the point where the leader makes a turn after
the straight section on the race track. On the straight section, the base
controller notices a continuous drop in performance that it is unable to
recover past a turn, and the slow robot can no longer track the lead along the
race track. Our system, on the other hand, adapts to this robot and, despite
losing a little performance, is able to maintain an upper limit on the error
metric and continue on the desired path.

In summary, addressing all requirements from the problem definition in Section
2, our convoy controller achieves lower inter-robot distances, quickly adapts
to variations in environments, and achieves faster convergence for error
metrics when compared with the base controller.

(a) Straight Line

(b) Column Room

(c) Long Corridor

Figure 8: Test Environments

(a) Puddles

(b) Carpets

(c) Stones

(d) Broken Floor

Figure 9: Ground Conditions | Average Error Metric 1 (m) | Average Error Metric 2 (m)
---|---|---
Environment | Base Controller | Our Controller | Base Controller | Our Controller
Straight Line | 1.35 | 0.31 | 1.08 | 0.44
Sine Curve | 1.12 | 0.30 | 0.80 | 0.47
$\infty$ loop | 1.69 | 0.32 | 2.82 | 0.73
Tight Turn | 0.72 | 0.18 | 0.65 | 0.37
Tunnel | 10.20 | 1.08 | 10.94 | 2.24
Race Track | 3.77 | 1.45 | 2.29 | 1.02
Column Room | 1.91 | 0.97 | 5.61 | 0.95
Long Corridor | 3.78 | 0.89 | 5.14 | 0.68
Table 1: Overall results comparison

##  6 Hardware Implementation

###  6.1 System Overview

Each robot in our setup (Fig. 1) consists of Traxxas remote-controlled trucks
fitted with a communication node, LiDAR and IMU sensors, a Jetson AGX Xavier
and a motor controller.

For our perception framework, we make use of Super Odometry Zhao et al., 2021,
which is an IMU-centric pipeline that provides estimates of each agent’s
odometry. To ensure neighboring robots don’t create drifts in odometry in low
LiDAR feature regions such as narrow corridors, the LiDAR data was filtered
using LiDAR masks in the directions of the forward and rear robots. This
ensures robust perception performance in all environments.

A base station is used to provide input commands to the convoy leader, either
via a joystick or through a waypoint-sharing interface. All systems are run
over ROS and use the DDS protocol for real-time inter-robot communication.

(a) Metric 1 - Straight Line

(b) Metric 2 - Straight Line

(c) Metric 1 - Column Room

(d) Metric 2 - Column Room

(e) Metric 1 - Long Corridor

(f) Metric 2 - Long Corridor

Figure 10: Test Results

###  6.2 Results

####  6.2.1 Straight Line:

As discussed in Section 5.1, we test similar conditions on the three ground
robots. It can be seen that the error metrics in Fig. 10(a) and 10(b) that the
metrics vary similarly to those in Fig. 5(a) and 5(b) where our convoy
controller is able to achieve steady state values faster than the existing
base controller. This also re-validates simulation results on a real system.

####  6.2.2 Column room:

Such an environment (Fig. 8(b)) is able to test convoy agility due to the
continuous obstacle avoidance requirements via low-curvature maneuvers while
maintaining a stable convoy structure during high-speed maneuvers. The room
has dimensions of 35 m x 20 m. The operator commands a general direction of
motion to the lead vehicle and all three robots make use of the convoy
controller to follow closely while achieving speeds of 4 m/s. The robots make
continuous turns to not hit walls, columns and other obstacles. The robots run
in various directions around this room to cover a total distance of over 600
m. The error metrics between our and the base controller are shown in Fig.
10(c) and 10(d). Despite having to continuously accelerate, decelerate and
turn in such an environment, the convoy controller is able to maintain average
gaps of just over 5 m while operating at 4 m/s compared to 9 m following
distances through the base controller. We also run the same path with a single
robot and notice an increase in average speed by under 5%, displaying the fact
that our framework doesn’t reduce overall robot capabilities while increasing
agents.

####  6.2.3 Long corridors:

In narrow long corridors with doorways (Fig. 8(c)), we illustrate high-speed
convoy performance while adapting to turns and doorways. These environments
contain debris as shown in Fig. 9, testing robustness in our control system
and mimicking search and rescue operations. The performance difference can be
seen in Fig. 10(e) and 10(f). The average inter-robot distances are just under
4 m at high speeds, which is significantly lower than the lowest gaps of 9-15
m seen at comparable speeds in other research papers (Zhao et al., 2017; Shin
et al., 2020) and the base controller. The deviations in the metrics only take
place when the robots make a turn into another corridor, move over rough
grounds, or navigate through doorways. The performance drop of a convoy
against a single robot is less than 2%, indicating almost similar time to
achieve the goal. The robots were able to navigate over 3 km in such
environments without running into each other or leaving a robot behind,
displaying a robust and safe system.

##  7 Conclusions

We have designed an optimal decentralized control system to run on a multi-
robot convoy. Our design incorporates future predictions on the adjacent
robots and solves a cost minimization incorporating robot states and controls,
allowing the agents to operate closer to each other at high speeds. This, in
turn, enables robots to operate well in environments that require continuous
sharp turns and variations in speeds. The framework operates in a
decentralized manner, resulting in no additional computational requirements
while increasing the number of robots being deployed. We have been able to
show, through simulation and hardware experimentation, the improvement in
performance against the current state-of-the-art methods and have been able to
cut down following distances against state-of-the-art by half. The system can
potentially be improved in the future by incorporating environment-dependent
control optimizations and integrating hybrid communication control methods
with increased data transfer to reduce state prediction computational loads.

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##  Appendix A Quadratic Form Expansion

Given vectors $x\in\mathbb{R}^{n}$, $x_{a}\in\mathbb{R}^{n}$,
$x_{b}\in\mathbb{R}^{n}$, $x_{c}\in\mathbb{R}^{n}$ and symmetric invertible
matrices: $Q_{a}\in\mathbb{R}^{n\times n}$, $Q_{b}\in\mathbb{R}^{n\times n}$
and $Q_{c}\in\mathbb{R}^{n\times n}$, in the expression

$\displaystyle(x-x_{a})^{T}Q_{a}(x-x_{a})$
$\displaystyle+(x-x_{b})^{T}Q_{b}(x-x_{b})$ (4)
$\displaystyle+(x-x_{c})^{T}Q_{c}(x-x_{c}),$

the terms can be combined into a single quadratic form via expansion and the
use of multivariate completing the square. Consider the expansion of a general
single-term quadratic form such as those above. WLOG, this is performed for
form $a$ and is given as

$(x-x_{a})^{T}Q_{a}(x-x_{a})=x^{T}Q_{a}x-2x_{a}^{T}Q_{a}x+x_{a}^{T}Q_{a}x_{a}.$
(5)

By performing the expansion expressed in Equation 5 on each single-term
quadratic form in 4 and combining like terms, a single general quadratic form
in $x$ can be generated as

$\displaystyle x^{T}(Q_{a}+Q_{b}$
$\displaystyle+Q_{c})x-2(x_{a}^{T}Q_{a}+x_{b}^{T}Q_{b}+x_{c}^{T}Q_{c})x$ (6)
$\displaystyle+(x_{a}^{T}Q_{a}x_{a}+x_{b}^{T}Q_{b}x_{b}+x_{c}^{T}Q_{c}x_{c}).$

Making the following simplifications:

$\displaystyle Q_{T}$ $\displaystyle=Q_{a}+Q_{b}+Q_{c}$ $\displaystyle y_{T}$
$\displaystyle=Q_{a}x_{a}+Q_{b}x_{b}+Q_{c}x_{c}$ $\displaystyle Z_{T}$
$\displaystyle=x_{a}^{T}Q_{a}x_{a}+x_{b}^{T}Q_{b}x_{b}+x_{c}^{T}Q_{c}x_{c},$

(6) may be equivalently expressed as

$x^{T}Q_{T}x-2y_{T}^{T}x+Z_{T}.$

By performing a multivariate complete the square, this expression may be
rearranged into a quadratic form with the term $x$ and a remainder term:

$\displaystyle(x-Q_{T}^{-1}y_{T})Q_{T}(x-Q_{T}^{-1}y_{T})-y_{T}^{T}Q_{T}y_{T}+Z_{T}.$