# A Conditional Extension of the Park-Pham Theorem

Bryce Alan Christopherson  Department of Mathematics, University of North
Dakota, Grand Forks, ND 58202<EMAIL_ADDRESS>and  Darian
Colgrove  Department of Mathematics, University of North Dakota, Grand Forks,
ND 58202<EMAIL_ADDRESS>
###### Abstract.

Neglecting many motivating details for the Park-Pham theorem (previously known
as the Kahn-Kalai conjecture), the result starts with a finite set $X$, a non-
trivial upper set $\mathcal{F}\subseteq 2^{X}$, and a particular parameterized
family of random variables $X_{p}$, then proceeds to provide an upper bound on
the value $p_{c}(\mathcal{F})$ such that
$\mathbb{P}(X_{p_{c}(\mathcal{F})}\in\mathcal{F})=1/2$. A natural question to
ask is if there is an analog to the Park-Pham theorem for upper sets in finite
posets other than $2^{X}$ and other parameterized families of random variables
taking values in them. In this short note, we show that there is, with minor
adjustments, in at least one circumstance. This is done by producing a
conditional version of the $\epsilon$-dependent form of the Park-Pham theorem,
which may be interesting in its own right.

###### Key words and phrases:

Thresholds, Kahn-Kalai, Park-Pham

###### 2000 Mathematics Subject Classification:

Primary 06A07, 05C80; Secondary 60C05, 68R01

a
Dept. of Mathematics, University of North Dakota. Email:
<EMAIL_ADDRESS>
Dept. of Mathematics, University of North Dakota. Email:
<EMAIL_ADDRESS>
Conflict of Interest Statement: The authors declare none.
Funding Statement: This work received no specific grant from any funding
agency, commercial or not-for-profit sectors.
Data Availability Statement: Data sharing not applicable to this article as no
datasets were generated or analysed during the current study.

##  1\. Introduction

The Park-Pham theorem [7] (previously the Kahn-Kalai conjecture [4]) bounds
the probability that a given subset of a finite set, chosen at random from a
finite set of objects in a particular way, has a specified property under the
proviso that the property respects supersets. As it should be, this bound
involves (either directly or indirectly) how large the set is, how large the
smallest subset with the property is, and the way one randomly chooses
subsets. The precise statement of this requires some definitions (for a more
detailed introduction, see [6]), which we give in Section 2.

In this paper, we prove a conditional version of the $\epsilon$-dependent
Park-Pham theorem. Aside from the fact that it extends the Park-Pham theorem
in a limited way, the reason a conditional form of the $\epsilon$-dependent
version is interesting is as follows: Suppose one has a finite poset $P$ and
an upper set $U$ in it. Then, if one takes an injection $f:P\rightarrow 2^{X}$
for a finite set $X$, the conditional probability $\mathbb{P}\big{(}X_{p}\in
f(U)|X_{p}\in f(P)\big{)}$ is the same as $\mathbb{P}(Y_{p}\in U)$ for the
random variable $Y_{p}$ with distribution $\frac{\mu_{p}\circ
f}{\mu_{p}\big{(}f(P)\big{)}}$. That is, a conditional version will allow us
to show there is at least one case where the Park-Pham theorem can be extended
to a more general context of upper sets in finite posets and parameterized
families of random variables taking values in them. We discuss this point more
in Remark 3.1 following some background.

##  2\. Background

Given a finite set $X$, we say that a property of $X$ is a subset
$\mathcal{F}\subseteq 2^{X}$. A property of $X$ is said to be an upper set if
$B\in\mathcal{F}$ whenever $A\subseteq B$ for some $A\in\mathcal{F}$. For a
customary example, the subset of subgraphs $H$ of a given graph $G$ such that
$H$ contains a subgraph isomorphic to some target graph $K$ forms an upper
set, as the addition of vertices or edges to a graph preserves subgraph
inclusion [2]. A lower set is defined similarly, where $B\in\mathcal{F}$
whenever $B\subseteq A$ for some $A\in\mathcal{F}$. We say a property
$\mathcal{F}$ is nontrivial if $\mathcal{F}\neq 2^{X},\emptyset$. The notion
of an upper set and a lower set can be extended to posets $P$ other than
$2^{X}$ as well: $S\subseteq P$ is an upper set (resp. lower set) if $x\in S$
and $x\leq y$ implies $y\in S$ (resp. $x\in S$ and $y\leq x$ implies $y\in
S$).

We will regard thresholds for sequences of upper sets in $2^{X}$ particularly.
For $p\in[0,1]$ define the product measure $\mu_{p}$ on $X$ by
$\mu_{p}(S)=p^{|S|}(1-p)^{|X|-|S|}$ and extend this to a probability measure
on $2^{X}$ by $\mu_{p}(\mathcal{F}):=\sum_{S\in\mathcal{F}}\mu_{p}(S)$.

######  Definition 2.1 (Threshold).

Given a sequence of finite sets $X_{n}$ and a sequence of upper sets
$\mathcal{F}_{n}\subseteq 2^{X_{n}}$, we say that a function $p^{*}(n)$ is a
threshold for $\mathcal{F}_{n}$ if
$\mu_{p(n)}\big{(}\mathcal{F}_{n}\big{)}\rightarrow 0$ when
$\frac{p(n)}{p^{*}(n)}\rightarrow 0$ and
$\mu_{p(n)}\big{(}\mathcal{F}_{n}\big{)}\rightarrow 1$ when
$\frac{p(n)}{p^{*}(n)}\rightarrow\infty$.

Every sequence of nontrivial upper sets $\mathcal{F}_{n}$ has a threshold. One
choice in particular is the critical probability of $\mathcal{F}$, denoted
$p_{c}(\mathcal{F})$, which is the unique value $p\in[0,1]$ such that
$\mu_{p}(\mathcal{F})=\frac{1}{2}$. That is,
$p^{*}(n):=p_{c}\big{(}\mathcal{F}_{n}\big{)}$ is a threshold for the sequence
$\mathcal{F}_{n}$ [3].

The Kahn-Kalai conjecture [4] (now, the Park-Pham theorem [7]) concerns the
relationship between the threshold obtained from the critical probability
$p_{c}$ and the expectation threshold of an upper set, a quantity derived from
considering certain covers of upper sets. Given an upper set
$\mathcal{F}\subseteq 2^{X}$, we say that a subset $\mathcal{G}\subseteq
2^{X}$ is a cover of $\mathcal{F}$ if
$\mathcal{F}\subseteq\bigcup_{S\in\mathcal{G}}\langle S\rangle$, where
$\langle S\rangle$ denotes the upper set generated by $S$, i.e. $\langle
S\rangle=\left\\{T:S\subseteq T\right\\}$. For $p\in[0,1]$, we say that
$\mathcal{F}$ is $p$-small if there exists a cover $\mathcal{G}$ of
$\mathcal{F}$ such that $\sum_{S\in\mathcal{G}}p^{|S|}\leq\frac{1}{2}$. The
expectation threshold of an upper set $\mathcal{F}$, denoted $q(\mathcal{F})$,
is then defined to be the largest $p$ for which $\mathcal{F}$ is $p$-small.

Now, let $\mathcal{F}_{0}$ denote the set of minimal elements of
$\mathcal{F}$; i.e.,

$\mathcal{F}_{0}=\left\\{S\in\mathcal{F}:\enskip S^{\prime}\not\subseteq
S\textrm{ for all }S^{\prime}\in\mathcal{F}\setminus S\right\\}$

, let
$\ell_{0}(\mathcal{F})=\textrm{max}\left\\{|S|:S\in\mathcal{F}_{0}\right\\}$
and write
$\ell(\mathcal{F})=\textrm{max}\left\\{\ell_{0}(\mathcal{F}),2\right\\}$. The
Park-Pham theorem [7](previously the Kahn-Kalai conjecture [4]) bounds the
critical probability by a logarithmic factor of the expectation threshold:

######  Theorem 2.2 (Park-Pham Theorem).

There exists a universal constant $K$ such that for every finite set $X$ and
every nontrivial upper set $\mathcal{F}\subseteq 2^{X}$,

$q(\mathcal{F})\leq p_{c}(\mathcal{F})\leq
Kq(\mathcal{F})\log\ell(\mathcal{F}).$

A version of the above, given by Bell [1], finds an optimized value of $K$:

######  Theorem 2.3 (Bell’s Park-Pham theorem with optimal constant [1]).

For every finite set $X$ and every nontrivial upper set $\mathcal{F}\subseteq
2^{X}$,

$q(\mathcal{F})\leq p_{c}(\mathcal{F})\leq
8q(\mathcal{F})\log\left(2\ell_{0}(\mathcal{F})\right).$

It is worth noting that one can optimize the above further in certain cases.
For instance, it was shown in [8] that one can set $K\approx 3.998$ in the
case where $\ell(F_{n})\rightarrow\infty$.

######  Remark 2.4.

If $X_{p}$ is the random variable with distribution $\mu_{p}$, then Theorem
2.2 can be recast as $\mathbb{P}(X_{p}\in\mathcal{F})>1/2$ whenever
$p>Kq(\mathcal{F})\log\ell(\mathcal{F})$. This version makes it clear that
there should be a more general variant of the bound, which replaces
$\mathbb{P}(X_{p}\in\mathcal{F})>1/2$ with
$\mathbb{P}(X_{p}\in\mathcal{F})>1-\epsilon$ for any $\epsilon\in(0,1)$ and
modifies the lower bound for $p$ with some dependence on $\epsilon$. That this
is possible was also provided by Park and Pham alongside their original proof,
and was then later optimized by Bell:

######  Theorem 2.5 (Bell’s optimal $\epsilon$-dependent Park-Pham theorem
[1]).

For every finite set $X$, every nontrivial upper set $\mathcal{F}\subseteq
2^{X}$, and any $\epsilon\in(0,1)$,
$\mathbb{P}(X_{p}\in\mathcal{F})>1-\epsilon$ whenever
$p>48q(\mathcal{F})\log\left(\frac{\ell_{0}(\mathcal{F})}{\epsilon}\right)$.

##  3\. A Conditional Variant

In this paper, we will prove that a conditional version of Theorem 2.5 is
possible with some caveats. Throughout, $K$ will refer to the constant in
Theorem 2.2, $X$ will be a finite set, and $X_{p}$ will be the random variable
with distribution $\mu_{p}$ as in Remark 2.4.

######  Remark 3.1.

As noted in the introduction, a conditional form of the $\epsilon$-dependent
version of the Park-Pham theorem allows us to use the conditional probability
$\mathbb{P}\big{(}X_{p}\in f(U)|X_{p}\in f(P)\big{)}$ to model the
unconditional probability $\mathbb{P}(Y_{p}\in U)$ for the random variable
$Y_{p}$ with distribution $\frac{\mu_{p}\circ f}{\mu_{p}\big{(}f(P)\big{)}}$
(see Lemma 3.14 for a proof of this), where $P$ here is a finite poset, $U$ is
an upper set in it, and $f:P\rightarrow 2^{X}$ is an injection. That is, the
conditional version is a toy that lets us show there is at least one case
where the Park-Pham theorem can be extended to a more general context. The
main result is the following:

Theorem 3.15 Let $P$ be a finite poset, let $U\subseteq P$ be an upper set,
let $f:P\rightarrow 2^{X}$ be an injection, and suppose $Kq(\langle
f(U)\rangle)\log\ell_{0}(\langle f(U)\rangle)<1$. Writing $Y_{p}$ for the
random variable with distribution
$\frac{1}{\mu_{p}\big{(}f(P)\big{)}}(\mu_{p}\circ f)$, there exists a value
$0<a<1$ such that for any $\epsilon\in(a,1)$ a nonempty interval
$I\subseteq(0,1)$ can be found so that for all $p\in I$,
$\mathbb{P}\big{(}Y_{p}\in U\big{)}>1-\epsilon.$

Said another way, if the Park-Pham bounds for the upper set $\langle
f(U)\rangle$ tells us anything at all (i.e. $Kq(\langle
f(U)\rangle)\log\ell_{0}(\langle f(U)\rangle)<1$), then we get the
$\epsilon$-dependent result for the variables $Y_{p}$ relative to the upper
set $U$ in the poset $P$. The only difference is the slight change in the
acceptable values of $\epsilon$ (i.e. $(a,1)$ instead of $(0,1)$) and the
potential existence of both a lower and upper bound on the values $p$ such
that we can ensure $\mathbb{P}\big{(}Y_{p}\in U\big{)}>1-\epsilon$ (i.e. the
interval $I\subseteq(0,1)$ need not be of the form $(b,1)$ for some $0<b<1$).
This is sensible: If the poset $P$ does not have a greatest element, then two
nontrivial upper sets can have an empty intersection and $Y_{p}$ can become
‘stuck’ in the lower set generated by a maximal element of $P$ that does not
belong to $U$.

To produce our conditional variant of the $\epsilon$-dependent Park-Pham
theorem, we will first require a lemma so simple that it almost need not be
noted.

######  Lemma 3.2.

Let $(E,\mathcal{E})$ be a measurable space and let $X$ be a
$(E,\mathcal{E})$-valued random variable. Suppose $A,B\in\mathcal{E}$ where
$A\subseteq B$ and $\mathbb{P}(X\in A)\neq 0$. Then,

$\mathbb{P}(X\in\langle A\rangle)=\frac{\mathbb{P}\big{(}X\in A|X\in
B\big{)}\mathbb{P}\big{(}X\in B\big{)}}{\mathbb{P}\big{(}X\in A|X\in\langle
A\rangle\big{)}}.$

###### Proof.

If $\mathbb{P}(X\in A)\neq 0$, then $A\subseteq\langle A\rangle$ and
$A\subseteq B$ gives $\mathbb{P}(X\in\langle A\rangle)\neq 0$ and
$\mathbb{P}(X\in B)\neq 0$. This lets us write

$\mathbb{P}\big{(}X\in A|X\in\langle A\rangle\big{)}\mathbb{P}(X\in\langle
A\rangle)=\mathbb{P}\big{(}X\in A\big{)}=\mathbb{P}\big{(}X\in A|X\in
B\big{)}\mathbb{P}\big{(}X\in B\big{)}.$

∎

Alongside the previous basic result, we will temporarily need a small tool
(which is, to a large extent, just for notational convenience). Given nested
subsets $A\subseteq B\subseteq 2^{X}$, we can approximate how ’large’ $B$ is
relative to how ’far up’ $A$ is in $2^{X}$ by taking the ratio of the
following two measurements: (1) for each $p$, how likely one is to obtain an
element of $A$ relative to the upper set it generates (i.e. how many new
elements are gained when moving from $A$ to $\langle A\rangle$ tells us how
’far up’ $A$ is in $2^{X}$); and, (2), how ‘large’ $B$ is in $2^{X}$ by
measuring how likely one is to obtain an element of $B$ at the same value of
$p$. Here is the precise definition.

######  Definition 3.3.

Let $\emptyset\neq A\subseteq B\subseteq 2^{X}$ and write $X_{p}$ for the
random variable with distribution $\mu_{p}$. Let $r_{A,B}(p)$ denote the
quantity

$r_{A,B}(p)=\frac{\mathbb{P}\big{(}X_{p}\in A|X_{p}\in\langle
A\rangle\big{)}}{\mathbb{P}\big{(}X_{p}\in B\big{)}}.$

######  Example 3.4.

Let $B$ correspond to the shaded portion of the Haase diagram for
$2^{\left\\{a,b,c\right\\}}$ as depicted below

$\emptyset$$\left\\{a\right\\}$$\left\\{b\right\\}$$\left\\{c\right\\}$$\left\\{a,b\right\\}$$\left\\{a,c\right\\}$$\left\\{b,c\right\\}$$\left\\{a,b,c\right\\}$

Then, if $A=\big{\\{}\left\\{a\right\\},\left\\{c\right\\}\big{\\}}$, we have:
$r_{A,B}(p)=2-\frac{2}{2-p}$, which is less than $1$ for all $p\in(0,1)$.

Intuitively, if $r_{A,B}(p)$ is greater than $1$, then $A$ is much higher up
than $B$ is large, while the reverse says that $B$ makes up a large part of
$2^{X}$ with $A$ not too high up. This is the basic idea of how we will use
it: Since we have $A\subseteq B$ by assumption, Lemma 3.2 tells us that
$r_{A,B}(p)\mathbb{P}(X_{p}\in\langle A\rangle)=\mathbb{P}\big{(}X_{p}\in
A|X_{p}\in B\big{)}$. That is, if $r_{A,B}(p)>1$, then it is easier to
randomly ‘hit’ $A$ in $B$ than it is to ‘hit’ $\langle A\rangle$ in $2^{X}$,
while having $r_{A,B}(p)<1$ tells us the opposite. This gives us the following
bargain-bin inequality:

######  Lemma 3.5.

Let $X$ be a finite set, suppose $\emptyset\neq A\subseteq B\subseteq 2^{X}$,
and write $X_{p}$ for the random variable with distribution $\mu_{p}$.
Then,$\mathbb{P}\big{(}X_{p}\in A|X_{p}\in B\big{)}>\frac{1}{2}r_{A,B}(p)$
whenever $p>Kq(\langle A\rangle)\log\ell\left(\langle A\rangle\right)$.

###### Proof.

By the Park-Pham theorem [7] (or, for a specific $K$, by Bell’s version in
Theorem 2.3), we have $\mathbb{P}(X_{p}\in\langle A\rangle)>1/2$ whenever
$p>Kq(\langle A\rangle)\log\ell\big{(}\langle A\rangle\big{)}$. Since
$A\subseteq B$ and by the definition of $\mu_{p}$, it is clear that
$\mathbb{P}(X_{p}\in A)\neq 0$. By Lemma 3.2, we get

$\frac{1}{2}<\mathbb{P}(X_{p}\in\langle
A\rangle)=\frac{\mathbb{P}\big{(}X_{p}\in A|X_{p}\in
B\big{)}\mathbb{P}\big{(}X_{p}\in B\big{)}}{\mathbb{P}\big{(}X_{p}\in
A|X_{p}\in\langle A\rangle\big{)}}=\frac{\mathbb{P}\big{(}X_{p}\in A|X_{p}\in
B\big{)}}{r_{A,B}(p)}.$

∎

######  Remark 3.6.

It is straightforward to show that the above can be strengthened to the
$\epsilon$-dependent version if $r_{A,B}(p)\geq 1$ for all $p$. In this case,
from Definition 3.3, we have $\mathbb{P}\big{(}X_{p}\in A|X_{p}\in
B\big{)}\geq\mathbb{P}(X_{p}\in\langle A\rangle\big{)}$. So, if $p>48q(\langle
A\rangle)\log\left(\frac{\ell_{0}\big{(}\langle
A\rangle\big{)}}{\epsilon}\right)$, we immediately get
$\mathbb{P}\big{(}X_{p}\in A|X_{p}\in B\big{)}\geq\mathbb{P}(X_{p}\in\langle
A\rangle\big{)}>1-\epsilon$ by Theorem 2.5.

To contrast how the $\epsilon$-dependent form is obtained Remark 3.6, a less
trivial fact is that a partial result still holds when even when
$r_{A,B}(p)<1$.

######  Lemma 3.7.

Let $X$ be a finite set, suppose $\emptyset\neq A\subseteq B\subseteq 2^{X}$,
and write $X_{p}$ for the random variable with distribution $\mu_{p}$. For any
$\epsilon\in(0,1)$ satisfying

$\epsilon>1-r_{A,B}(p)\left(1-\ell_{0}(\langle
A\rangle)2^{-p/\big{(}Kq(\langle A\rangle)\big{)}}\right),$

we have

$\mathbb{P}\big{(}X_{p}\in A|X_{p}\in B\big{)}>1-\epsilon.$

###### Proof.

First, observe the following inequality holds:

$\displaystyle-\ell_{0}(\langle A\rangle)2^{-p/\big{(}Kq(\langle
A\rangle)\big{)}}$ $\displaystyle<0$ $\displaystyle 1-\ell_{0}(\langle
A\rangle)2^{-p/\big{(}Kq(\langle A\rangle)\big{)}}$ $\displaystyle<1$
$\displaystyle r_{A,B}(p)\left(1-\ell_{0}(\langle
A\rangle)2^{-p/\big{(}Kq(\langle A\rangle)\big{)}}\right)$
$\displaystyle<r_{A,B}(p)$ $\displaystyle-r_{A,B}(p)\left(1-\ell_{0}(\langle
A\rangle)2^{-p/\big{(}Kq(\langle A\rangle)\big{)}}\right)$
$\displaystyle>-r_{A,B}(p)$ $\displaystyle
1-r_{A,B}(p)\left(1-\ell_{0}(\langle A\rangle)2^{-p/\big{(}Kq(\langle
A\rangle)\big{)}}\right)$ $\displaystyle>1-r_{A,B}(p)$

Now, if $1>r_{A,B}(p)>0$, then $0<1-r_{A,B}(p)<1$, while if $1\leq
r_{A,B}(p)$, then $1-r_{A,B}(p)\leq 0$. So, for any choice of $\epsilon$
satisfying the conditions of the theorem, we have $1-r_{A,B}(p)<\epsilon<1$,
which gives that

$0<1-(1-\epsilon)\frac{1}{r_{A,B}(p)}<1.$

So, set
$\eta=1-(1-\epsilon)\frac{1}{r_{A,B}(p)}=\frac{1}{r_{A,B}(p)}\big{(}r_{A,B}(p)-(1-\epsilon)\big{)}$.
Taking $K$ to be $K=48$ as in Bell’s $\epsilon$-dependent version of the Park-
Pham theorem (i.e. Theorem 2.5), we get $\mathbb{P}(X_{p}\in\langle
A\rangle)>1-\eta$ whenever $p>Kq(\langle
A\rangle)\log\left(\frac{\ell_{0}(\langle A\rangle)}{\eta}\right)=Kq(\langle
A\rangle)\log\left(\frac{r_{A,B}(p)\ell_{0}(\langle
A\rangle)}{r_{A,B}(p)-(1-\epsilon)}\right)$. To see that this is satisfied,
observe the following:

$\displaystyle p$ $\displaystyle>Kq(\langle
A\rangle)\log\left(\frac{r_{A,B}(p)\ell_{0}(\langle
A\rangle)}{r_{A,B}(p)-(1-\epsilon)}\right)$ $\displaystyle
2^{p/\big{(}Kq(\langle A\rangle)\big{)}}$
$\displaystyle>\frac{r_{A,B}(p)\ell_{0}(\langle
A\rangle)}{r_{A,B}(p)-(1-\epsilon)}$ $\displaystyle\epsilon$
$\displaystyle>1-r_{A,B}(p)\left(1-\ell_{0}(\langle
A\rangle)2^{-p/\big{(}Kq(\langle A\rangle)\big{)}}\right).$

Now, from $S\subseteq P$, we have $A\subseteq B$, and by the definition of
$\mu_{p}$ it is clear that $\mathbb{P}(X_{p}\in A)\neq 0$. By Lemma 3.2, we
get

$\displaystyle(1-\eta)=(1-\epsilon)\frac{1}{r_{A,B}(p)}$
$\displaystyle<\frac{\mathbb{P}\big{(}X_{p}\in A|X_{p}\in
B\big{)}}{r_{A,B}(p)}=\mathbb{P}(X_{p}\in\langle A\rangle)$
$\displaystyle(1-\epsilon)$ $\displaystyle<\mathbb{P}\big{(}X_{p}\in
A|X_{p}\in B\big{)}.$

∎

######  Remark 3.8.

The values of $\epsilon$ satisfying the inequality in Lemma 3.7 are not very
optimal when $r_{A,B}(p)\geq 1$, since (as noted in Remark 3.6) Lemma 3.5 can
be strengthened to $\mathbb{P}(X_{p}\in A\mid X_{p}\in B)>1-\epsilon$ for any
$\epsilon\in(0,1)$ whenever $p$ exceeds the bound given in Theorem 2.5 for the
upper set $\langle A\rangle$.

######  Example 3.9.

Consider $A\subseteq B\subseteq 2^{\left\\{a,b,c\right\\}}$ as in Example 3.4.
Then, Lemma 3.7 tell us that $\mathbb{P}\big{(}X_{p}\in A|X_{p}\in
B\big{)}>1-\epsilon$ whenever

$\displaystyle\epsilon$ $\displaystyle>1-r_{A,B}(p)\left(1-\ell_{0}(\langle
A\rangle)2^{-p/\big{(}Kq(\langle A\rangle)\big{)}}\right)$
$\displaystyle\epsilon$
$\displaystyle>1-\left(2-\frac{2}{2-p}\right)\left(1-2^{-4p/K}\right)$

Taking $K=48$ as in Theorem 2.5, we get that the minimum of
$1-\left(2-\frac{2}{2-p}\right)\left(1-2^{-p/12}\right)$ on the interval
$[0,1]$ occurs when $p=0.582289$ and has a value of approximately $0.98051$.
Therefore, it is possible to select a value $p\in(0,1)$ such that
$\mathbb{P}\big{(}X_{p}\in A|X_{p}\in B\big{)}>1-\epsilon$ for any $\epsilon$
satisfying $0.98051<\epsilon<1$. Indeed, this value can be found by first
choosing such an epsilon and solving the inequality $p>48q(\langle
A\rangle)\log\left(\frac{r_{A,B}(p)\ell_{0}(\langle
A\rangle)}{r_{A,B}(p)-(1-\epsilon)}\right)$. For example, if $\epsilon=0.99$,
then taking $p$ to satisfy $0.195217<p<0.889027$ then ensures that
$\mathbb{P}\big{(}X_{p}\in A|X_{p}\in B\big{)}>0.01$.

In the previous example, since $A$ is not an upper set in $B$, it is sensible
that there should be an upper bound for $p$ as well as a lower bound. As the
next example shows, if we take an upper set in $B$, then this upper bound
should vanish.

######  Example 3.10.

If, in Example 3.4 and Example 3.9, we had instead taken
$A^{\prime}=\left\\{\left\\{a\right\\},\left\\{c\right\\},\left\\{a,c\right\\}\right\\}$
so that $A^{\prime}$ becomes an upper set in $B$, then it is routine to show
that we get $r_{A^{\prime},B}(p)=1$. In this case, Lemma 3.7 tells us that
$\mathbb{P}\big{(}X_{p}\in A^{\prime}|X_{p}\in B\big{)}>1-\epsilon$ whenever

$\displaystyle\epsilon$
$\displaystyle>1-r_{A^{\prime},B}(p)\left(1-\ell_{0}(\langle
A^{\prime}\rangle)2^{-p/\big{(}Kq(\langle A^{\prime}\rangle)\big{)}}\right)$
$\displaystyle\epsilon$ $\displaystyle>1-\left(1-2^{-4p/K}\right)=2^{-4p/K}$

Taking $K=48$ again as in Theorem 2.5, this says $\epsilon>1-2^{-p/12}$. Since
$2^{-p/12}$ is minimized on $[0,1]$ at $p=1$ where it takes on the value
$2^{-1/12}\approx 0.943874$, it is possible to select a value $p\in(0,1)$ such
that $\mathbb{P}\big{(}X_{p}\in A^{\prime}|X_{p}\in B\big{)}>1-\epsilon$ for
any $\epsilon$ satisfying $0.943874<\epsilon<1$. As before, we find this value
by first choosing such an $\epsilon$ and solving the inequality $p>48q(\langle
A^{\prime}\rangle)\log\left(\frac{r_{A^{\prime},B}(p)\ell_{0}(\langle
A^{\prime}\rangle)}{r_{A^{\prime},B}(p)-(1-\epsilon)}\right)=12\log\left(\frac{1}{\epsilon}\right)$.
For example, if $\epsilon=0.99$ as before, then taking $p>0.173995$ ensures
that $\mathbb{P}\big{(}X_{p}\in A^{\prime}|X_{p}\in B\big{)}>0.01$.

######  Remark 3.11.

Example 3.10 shows that we do not always get an upper bound on $p$ like we did
in Example 3.9. This is as it should be. For any subset $A^{\prime}\subseteq
B\subseteq 2^{X}$ that is an upper set in $B$ and contains the maximal
elements of $B$, increasing $p$ can only make it more likely for $X_{p}$ to
land in $A^{\prime}$ when it is already in $B$. This also highlights what was
said in Remark 3.8: Indeed, the bounds we get from Theorem 3.7 are not very
good in cases like this where the $p$-ratio is never smaller than $1$. Here,
we can only ensure that $\mathbb{P}\big{(}X_{p}\in A^{\prime}|X_{p}\in
B\big{)}>1-\epsilon$ for, approximately, the range $0.943874<\epsilon<1$.
Using Lemma 3.5 instead, it is easy to see that we can actually do this for
any $\epsilon\in(0,1)$.

To finish, we note that our use of the quantity $r_{A,B}(p)$ has been a bit
silly so far: If we need to compute an expression involving $r_{A,B}(p)$ to
say something about $\mathbb{P}(X_{p}\in A|X_{p}\in B)$, we most likely have
not made our job any easier. The next result, however, shows that we do not
actually need to perform this computation. In fact, any time the Kahn-Kalai
bounds give us any information at all about $p_{c}(\langle A\rangle)$, it is
possible to bound the conditional probability below, up to a certain maximum
value, by choosing $p$ within some subinterval of $(0,1)$.

######  Lemma 3.12.

Let $X$ be a finite set and suppose $\emptyset\neq A\subseteq B\subseteq
2^{X}$. Whenever $Kq(\langle A\rangle)\log\ell_{0}(\langle A\rangle)<1$, there
is some $0<a<1$ such that for every $\epsilon\in(a,1)$ the inequality

$\epsilon>1-r_{A,B}(p)\left(1-\ell_{0}(\langle
A\rangle)2^{-p/\big{(}Kq(\langle A\rangle)\big{)}}\right)$

holds for all $p$ in some nonempty subinterval of $(0,1)$.

###### Proof.

Write $g(p):=1-r_{A,B}(p)\left(1-\ell_{0}(\langle
A\rangle)2^{-p/\big{(}Kq(\langle A\rangle)\big{)}}\right)$ and suppose there
is no pair of values $(p,\epsilon)\in(0,1)^{2}$ such that the inequality
$\epsilon>g(p)$ holds. Then, for all $p\in(0,1)$, it must be the case that
$g(p)\geq 1$. Suppose as well that $Kq(\langle A\rangle)\log\ell_{0}(\langle
A\rangle)<1$ and take $1>p>Kq(\langle A\rangle)\log\ell_{0}(\langle
A\rangle)$. Then, we have

$\displaystyle g(p)=1-r_{A,B}(p)\left(1-\ell_{0}(\langle
A\rangle)2^{-p/\big{(}Kq(\langle A\rangle)\big{)}}\right)$ $\displaystyle\geq
1$ $\displaystyle 1-\ell_{0}(\langle A\rangle)2^{-p/\big{(}Kq(\langle
A\rangle)\big{)}}$ $\displaystyle\leq 0$ $\displaystyle 2^{p/\big{(}Kq(\langle
A\rangle)\big{)}}$ $\displaystyle\leq\ell_{0}(\langle A\rangle)$
$\displaystyle p$ $\displaystyle\leq Kq(\langle A\rangle)\log\ell_{0}(\langle
A\rangle).$

But the last line is a contradiction, so $\textrm{min}_{p\in(0,1)}g(p)<1$.
Hence, for any $\epsilon\in(\textrm{min}_{p\in(0,1)}g(p),1)$, the continuity
of $g$ yields that $g^{-1}(-\infty,\epsilon)$ is open and has a nonempty
intersection with $(0,1)$. ∎

Dispensing now with the crutch of $r_{A,B}(p)$, we reach the following as a
corollary.

######  Corollary 3.13.

Suppose $\emptyset\neq A\subseteq B\subseteq 2^{X}$. If $Kq(\langle
A\rangle)\log\ell_{0}(\langle A\rangle)<1$, then there is some $0<a<1$ such
that for any $\epsilon\in(a,1)$, a nonempty subinterval interval
$I\subseteq(0,1)$ can be found such that for all $p\in I$,

$\mathbb{P}\big{(}X_{p}\in A|X_{p}\in B\big{)}>1-\epsilon.$

Using the following small lemma, the previous result gives an extension of the
$\epsilon$-dependent Park-Pham theorem:

######  Lemma 3.14.

Let $P$ be a poset and let $f:P\rightarrow 2^{X}$ be an injection, where $X$
is some sufficiently large finite set. Taking the sigma algebra generated by
$f$ and a $P$-valued random variable $Y_{p}$ with distribution
$\frac{1}{\mu_{p}\big{(}f(P)\big{)}}(\mu_{p}\circ f)$, then for
$S\in\sigma(f)$, we have

$\mathbb{P}(Y_{p}\in S)=\mathbb{P}\big{(}X_{p}\in f(S)|X_{p}\in f(P)\big{)}.$

###### Proof.

If $f:P\rightarrow 2^{X}$ is injective, then $f(A)\cap f(B)=f(A\cap B)$ for
all $A,B\subseteq P$. So,

$\displaystyle\mathbb{P}\big{(}X_{p}\in f(S)|X_{p}\in f(P)\big{)}$
$\displaystyle=\frac{\mathbb{P}\big{(}X_{p}^{-1}(f(S))\cap
X_{p}^{-1}(f(P))\big{)}}{\mathbb{P}\big{(}X_{p}^{-1}(f(P))\big{)}}$
$\displaystyle=\frac{\mathbb{P}\big{(}X_{p}^{-1}(f(S)\cap
f(P))\big{)}}{\mathbb{P}\big{(}X_{p}^{-1}(f(P))\big{)}}$
$\displaystyle=\frac{\mathbb{P}\big{(}X_{p}^{-1}(f(S\cap
P))\big{)}}{\mathbb{P}\big{(}X_{p}^{-1}(f(P))\big{)}}$
$\displaystyle=\mathbb{P}(Y_{p}\in S).$

∎

The immediate consequence of Lemma 3.14 and Corollary 3.13 is the following:

######  Theorem 3.15.

Let $P$ be a finite poset, let $U\subseteq P$ be an upper set, let
$f:P\rightarrow 2^{X}$ be an injection, and suppose $Kq(\langle
f(U)\rangle)\log\ell_{0}(\langle f(U)\rangle)<1$. Writing $Y_{p}$ for the
random variable with distribution
$\frac{1}{\mu_{p}\big{(}f(P)\big{)}}(\mu_{p}\circ f)$, there exists a value
$0<a<1$ such that for any $\epsilon\in(a,1)$ a nonempty interval
$I\subseteq(0,1)$ can be found so that for all $p\in I$,
$\mathbb{P}\big{(}Y_{p}\in U\big{)}>1-\epsilon.$

##  4\. Conclusion

In this paper, we have shown that the $\epsilon$-dependent Park-Pham theorem
can be generalized to a conditional form for subsets $\emptyset\neq A\subseteq
B\subseteq 2^{X}$, albeit with a smaller range of values $\epsilon$ and with a
potential upper and lower bound (instead of just a lower bound) on the values
$p$ for which we can ensure $\mathbb{P}(X_{p}\in A|X_{p}\in B)>1-\epsilon$.
Since one can always embed a finite poset in the power set of a finite set,
this shows that there is an analog to the Park-Pham theorem applicable to
random variables with distributions other than $\mu_{p}$ and upper sets $U$ in
posets other than $2^{X}$ (indeed, it demonstrates it for at least one case,
see Theorem 3.15 and the discussion in Remark 3.1).

A great deal of further work along this line is possible–for instance, using
the non-uniform version of the Park-Pham theorem recently proven in [5],
Theorem 3.15 could be further generalized to admit a wider range of random
variables. We end with a few questions: In what cases does one have the upper
bound on the interval $I$ in Theorem 3.15 disappear? Is there some easy-to-
check condition an upper set $\mathcal{F}\subseteq 2^{X}$ must satisfy to
ensure that $Kq(\mathcal{F})\log\ell_{0}(\mathcal{F})<1$? What are the most
minimally-restrictive requirements a parameterized family of random variables
$Y_{p}$ must satisfy to admit a Park-Pham-type theorem (i.e. a result
analogous to Theorem 3.15)? What is the appropriate generalization of
thresholds to upper sets in other posets and other parameterized families of
random variables?

## References

  * [1] Tolson Bell, _The park-pham theorem with optimal convergence rate_ , The Electronic Journal of Combinatorics (2023), P2–25.
  * [2] P Erdős and A Rényi, _On random graphs i_ , Publ. math. debrecen 6 (1959), no. 290-297, 18.
  * [3] Svante Janson, _łluczak t., rucinski a. random graphs_ , 2000.
  * [4] Jeff Kahn and Gil Kalai, _Thresholds and expectation thresholds_ , Combinatorics, Probability and Computing 16 (2007), no. 3, 495–502.
  * [5] Bryan Park and Jan Vondrák, _A simple proof of the non-uniform kahn-kalai conjecture_ , arXiv preprint arXiv:2306.12576 (2023).
  * [6] Jinyoung Park, _Threshold phenomena for random discrete structures_ , Notices of the American Mathematical Society 70 (2023), no. 10, 1615–1625.
  * [7] Jinyoung Park and Huy Pham, _A proof of the kahn–kalai conjecture_ , Journal of the American Mathematical Society 37 (2024), no. 1, 235–243.
  * [8] Van Vu and Phuc Tran, _A short proof of kahn-kalai conjecture_ , arXiv preprint arXiv:2303.02144 (2023).


# Nonlinear Cooperative Control of Double Drone-Bar Transportation System

Peng Zhang<EMAIL_ADDRESS>Yongchun Fang<EMAIL_ADDRESS>Xiao Liang<EMAIL_ADDRESS>He Lin<EMAIL_ADDRESS>Wei He
<EMAIL_ADDRESS>Institute of Robotics and Automatic Information
System,College of Artificial Intelligence
Tianjin Key Laboratory of Intelligent Robotics, Nankai University, Tianjin
300350, China

###### Abstract

Due to the limitation of the drone’s load capacity, various specific tasks
need to be accomplished by multiple drones in collaboration. In some
transportation tasks, two drones are required to lift the load together, which
brings even more significant challenges to the control problem because the
transportation system is underactuated and it contains very complex dynamic
coupling. When transporting bar-shaped objects, the load’s attitude, the
rope’s swing motion, as well as the distance between the drones, should be
carefully considered to ensure the security of the system. So far, few works
have been implemented for double drone transportation systems to guarantee
their transportation performance, especially in the aforementioned aspect. In
this paper, a nonlinear cooperative control method is proposed, with both
rigorous stability analysis and experimental results demonstrating its great
performance. Without the need to distinguish the identities between the leader
and the follower, the proposed method successfully realizes effective control
for the two drones separately, mainly owning to the deep analysis for the
system dynamics and the elaborate design for the control law. By utilizing
Lyapunov techniques, the proposed controller achieves simultaneous positioning
and mutual distance control of the drones, meanwhile, it efficiently
eliminates the swing of the load. Flight experiments are presented to
demonstrate the performance of the proposed nonlinear cooperative control
strategy.

###### keywords:

Nonlinear control, Lyapunov techniques, Swing elimination, Cooperative
control.

††thanks: This work is supported in part by National Natural Science
Foundation of China under Grant 61873132 and Grant 61903200, in part by
Natural Science Foundation of Tianjin under Grant 16JCZDJC30300 and Grant
19JCQNJC03500, in part by the China Postdoctoral Science Foundation under
Grant 2020M670632, and in part by the Fundamental Research Funds for the
Central Universities Nankai University under Grant 63201194.

, , , ,

††thanks: Corresponding author at Institute of Robotics and Automatic
Information System, Nankai University, Tianjin 300353, China. Tel.: +86 22
23505706; fax: +86 22 23500172.

##  1 Introduction

Drones are now widely used for material transportation because of their
ability to take off and land vertically [1, 2, 3, 4, 5, 6, 7, 8, 9, 10].
Aerial transportation, which is not limited to terrain, plays an increasingly
important role in emergency cases [11, 12]. Therefore, relevant research have
always been concerned. When exploring drone delivery, researchers mainly adopt
two methods, one of which is to use clamping claws or mechanical arms for
transportation [13, 14, 15, 16]. In this mode, the attitude of the load is
highly controllable, which makes it convenient for obstacles avoidance.
Another way of transportation is to suspend the load by sling ropes [17, 18],
which ensures the flexibility of the drone itself and reduces additional cost
on claws and arms. However, the load suspended by the cable inevitably sways
under the drones, which may bring hidden danger to system security. For some
particular transportation tasks, such as medical supplies, the damage caused
by the sway of the load is unacceptable. Besides, a single drone usually does
not have the ability to lift a heavy or a large-sized load. To solve these
problems, it is a very natural idea to utilize two or even more drones to
complete the tasks by cooperative control. Noting that effective control of
the drone itself is already a very challenging work because the drone is
underactuated with fewer control inputs than its degrees of freedom (DOFs),
cooperative control for the double-drone transportation system is much more
difficult, and it is still a fairly open problem so far.

In the process of transportation, it is difficult to directly control the
movement of the lifted-objects, which reflects the underactuated
characteristics and brings many challenges. In recent years, the control for
various underactuated systems, such as cranes and underwater vehicles, has
received considerable interest and achieved satisfactory results by utilizing
nonlinear control methods [19, 20, 21, 22, 23, 24, 25, 26]. With the
application of advanced sensors, feedback control for the intractable drone
has recently attracted extensive attention[27, 28, 29, 30]. Some classic
control strategies are used to realize various tasks such as aggressive
maneuvers, dynamic trajectory generation, and so on. As for the drone
transportation system, the load cannot be directly controlled, and the cables
exacerbate the system nonlinearities and introduce complex dynamic couplings,
these factors make the control problem for the drone transportion system full
of all kinds of challenges. So far, researchers have done a lot of work to
suppress and eliminate the load sway efficiently, with effective methods
including geometric control[31], dynamic programming[32], differential
flatness[33], time-optimal motion planning[34], hierarchical control[35], and
so on. Regardless of the control strategy, the transportation capacity of a
single drone is always limited, it is thus urgently needed to implement
transportation tasks by multiple drones with high-performance cooperative
control strategy.

Compared with the control of a single drone with a suspended load, the study
of double drone-bar system presents many challenges in both modeling and
control, which are mainly summarized as follows: 1) The bar-load for the
double drone-bar transportation system usually has a large size or weight.
Thus, the load cannot be regarded as a point; instead, its size and attitude
should be considered, making it complicated to describe the swing dynamics
accurately. 2) High degree of freedom implies that more variables are needed
to describe the system, including some non-independent variables. Thus, how to
deal with the variables without approximation for the modeling and control
problem is a great challenge. 3) The proposed method must properly and
efficiently coordinate the two drones throughout the transportation process.
Swing elimination and cooperative control make the problem even more
complicated. Due to the aforementioned facts, fewer research results have been
reported for such systems.

Due to the aforementioned facts, few research results have been reported for
such systems. A vision-based collaborative control scheme is proposed in [36],
wherein the motion of the two drones can be well organized even without
communication. However, without careful consideration for swing suppression,
the speed of the drones has to be strictly limited within a certain range.
Some load attitude control method is proposed in [37], which drives the bar to
the desired pose and guarantees exponential stability. Nevertheless, [37]
ignores the information exchange and does not consider the collaboration
between the drones. Linearization is used to infer the equilibrium’s
exponential stability in [37], failing to establish a precise mathematical
description for the double drone-bar dynamics. Due to the two drones’
independent control, the advantages of information exchange are not fully
implemented, limiting the applications in the double drone transportation
system. Instead, G. Loianno _et al._ propose a coordinated approach that
enables each drone to construct its controller, utilizing not only its own
sensor, but also the measurements acquired by other drones [38], which
inspires this work. Unfortunately, the inherently rigid structure strictly
determines the relative position between the two drones, which badly reduces
the flexibility of the transportation system. The formation control law is
presented in[39] to endow the interconnected system with a continuum of
equilibria. [40] presents the control system for the transportation of a slung
load by one or several helicopters and successfully implements experimental
tests. H. Lee _et al._ employ the parameter-robust linear quadratic Gaussian
method to solve the problem of slung-load transportation, and prove the
stability of the system with Lyapunov analysis [41]. It is noted that, since
the load is regarded as point mass and there is no actual information exchange
between the drones, the methods proposed in [39, 40, 41] cannot guarantee the
smoothness and safety of the transportation process, and they cannot always
yield satisfactory experimental results.

In many cases, the load may be too large or too heavy to be transported by
only one drone. Two drones will be used to coordinate the delivery of these
goods. With the widespread application, this double drone-bar system’s
prospects are becoming broader, making the corresponding research very urgent.
To solve the above-mentioned practical problems, this paper uses the
Lagrangian modeling method to establish an accurate double drone-bar model.
Based on the obtained model, a nonlinear coordinated control strategy is
proposed to ensure the asymptotic stability of the required balance point.
Lyapunov technique and LaSalle’s invariance theorem strictly prove the
correctness of the problem without turning to any linearization or
approximation. Finally, extensive experiments are carried out to verify the
effectiveness and robustness of the proposed method.

The main contributions of this paper are summarized as follows:

  1. 1.

An accurate model is set up for the double drone-bar transportation system
without turning to any linearization or approximation, based on which, a
nonlinear cooperative controller is designed, which carefully considers the
complex nonlinearity and the strong coupling between the drones, and achieves
asymptotic stability as proven by Lyapunov techniques and LaSalle’s
invariance.

  2. 2.

The proposed controller works independently on the two drones, meanwhile, the
interconnection of the two drones by load is rigorously analyzed, which helps
to effectively suppress the swing and control the relative position of the
double drones.

  3. 3.

As supported by experimental results, even in the presence of various
disturbances, such as wind or manual disturbances, the proposed control method
presents strong anti-swing ability and good robustness for different
transportation scenarios, superior to existing control methods.

The remaining part of the paper is organized as follows: Section $2$ uses the
Lagrange modeling method to set up the model for the double drone-bar
transportation system. Section $3$ divides the transportation system into the
inner loop and the outer one, where Lyapunov method is employed to design an
advanced control for the outer loop, with LaSalle’s invariance principle
utilized to analyze its stability. In section $4$, experimental results are
included to verify the feasibility and superior performance of the proposed
method. Section $5$ provides summaries and conclusions of this paper.

Figure 1: Modeling of the drone-bar system

##  2 System Model Development

As illustrated in Fig. 1, there are two drones and a bar load in the
transportation system, which are connected by two cables of equal length. For
the following part of this paper, we refer to this system as drone-bar system.
The drones’ and the bar’s positions are denoted by
$\bm{\xi}_{1}(t)=[y_{1}(t),z_{1}(t)]^{T}=[y,z]^{T}$,
$\bm{\xi}_{2}(t)=[y_{2}(t),z_{2}(t)]^{T}=[y+l_{1}S_{1}+l_{2}S_{2}+aC_{3},z-l_{1}C_{1}+l_{2}C_{2}+aS_{3}]^{T}$
and
$\bm{\xi}_{3}(t)=[y_{3}(t),z_{3}(t)]^{T}=[y+l_{1}S_{1}+\frac{a}{2}C_{3},z-l_{1}C_{1}+\frac{a}{2}S_{3}]^{T}$,
respectively. $m_{1},m_{2},m_{3}$ stand for their masses.
$\bm{\Theta}(t)=[\theta_{1}(t),\theta_{2}(t),\theta_{3}(t)]^{T}$ denotes the
swing signals, and they are defined in the manner as shown in Fig. 1.
$f_{1}\in\mathbb{R}$ and $f_{2}\in\mathbb{R}$ are applied thrust force in the
inertial frame, and $J_{1}\in\mathbb{R}^{+},J_{2}\in\mathbb{R}^{+}$ represent
the moment of inertia of the two drones. In addition, we denote by
$l_{1}=l_{2}=l>0$ the cables’ length, by $a>0$ the bar’s length, by $g$ the
gravity acceleration, by $\phi_{1},\phi_{2}$ the rotating angle of the drones.
The model of this drone-bar system is built on a plane perpendicular to the
X-axis and constructed according to their geometric relationship. It is
essential in our implementation to decompose control design into two loops as
the inner loop and the outer one. The lifting system has seven degrees of
freedom, and their corresponding generalized forces in the outer loop
subsystem are denoted by $Q_{i}$: $i\in[1,5],i\in N^{+}$. Lagrange’s method is
utilized for system modeling, and after performing a certain amount of
mathematical operations, the outer loop dynamics of the double drone-bar
system can be derived as

$\displaystyle Q_{1}$
$\displaystyle={f_{1}}\sin{\phi_{1}}+{f_{2}}\sin{\phi_{2}},$ (1)
$\displaystyle Q_{2}$
$\displaystyle={f_{1}}\cos{\phi_{1}}+{f_{2}}\cos{\phi_{2}},$ (2)
$\displaystyle Q_{3}$
$\displaystyle={f_{2}}\sin{\phi_{2}}{l_{1}}{C_{1}}+{f_{2}}\cos{\phi_{2}}{l_{1}}{S_{1}},$
(3) $\displaystyle Q_{4}$
$\displaystyle={f_{2}}\sin{\phi_{2}}{l_{2}}{C_{2}}-{f_{2}}\cos{\phi_{2}}{l_{2}}{S_{2}},$
(4) $\displaystyle Q_{5}$
$\displaystyle=-f_{2}\sin\phi_{2}aS_{3}+f_{2}\cos\phi_{2}aC_{3}.$ (5)

In the transportation task, we need to drive the two drones to the desired
positions as $\bm{\xi}_{1d}=[y_{1d},z_{1d}]^{T}$,
$\bm{\xi}_{2d}=[y_{2d},z_{2d}]^{T}$, while suppressing the swing of the load.
Also, during the whole transportation process, the distance of the two drones
should be limited within a certain range, so that the drones will not collide
with each other when they are too close, or break the cables when trying to
fly away from each other. In this sense, the mathematical representation of
the control target is

$\displaystyle y_{1}\rightarrow y_{1d},y_{2}\rightarrow
y_{2d},z_{1}\rightarrow z_{1d},z_{2}\rightarrow z_{2d},$
$\displaystyle\theta_{1}\rightarrow\theta_{1d},\theta_{2}\rightarrow\theta_{2d},$
(6)

where the desired positions are constant. Generally speaking, the working
scenes of multiple drones are mostly used to transport heavier objects, and
most of the sling ropes used in the experiment are steel ropes that are not
easy to bend. Similar with the work presented in [34, 35, 42, 43, 44, 45], we
make the following reasonable assumption about the cables and the swing
angles:

######  Assumption 1.

Aggressive motion control is not taken into account. The two cables in the
transportation system are always in tension. Besides, the bar is always under
the two drones in the vertical direction, and the bar is not in a straight
line with either of the two cables, i.e.

$\displaystyle-\frac{\pi}{2}<\phi_{1},\phi_{2},\theta_{1}(t),\theta_{2}(t),\theta_{3}(t)<\frac{\pi}{2}\quad\quad\forall\,t\geq
0.$

##  3 Control Development

In this section, a nonlinear hierarchical control scheme is used to facilitate
the design procedure. As [17] shows, drone transportation systems have the
cascade property, thus controllers can be designed for the inner loop and the
outer one separately. The contribution of this paper mainly focuses on the
design of the outer loop controller, and the inner loop adopts the controller
presented in [17, 31, 33], which performs well in experimental tests.

###  3.1 Outer Loop Controller Design

First, define the following error signals:

$\displaystyle\bm{e}_{{\xi}_{1}}=[e_{y_{1}},e_{z_{1}}]^{T}=[y_{1}-y_{1d},z_{1}-z_{1d}]^{T},$
$\displaystyle\bm{e}_{{\xi}_{2}}=[e_{y_{2}},e_{z_{2}}]^{T}=[y_{2}-y_{2d},z_{2}-z_{2d}]^{T}.$
(7)

To facilitate subsequent controller development and analysis, define the
desired signals to satisfy:

$\displaystyle y_{2d}-y_{1d}=a,z_{1d}=z_{2d},\theta_{1d}=\theta_{2d}=0.$ (8)

Considering the underactuation property of the system, energy-based method is
adopted to design the cooperative control. To this end, calculate the storage
energy of the drone-bar system as:

$\displaystyle
E\\!=\\!\frac{1}{2}{\dot{\bm{q}}^{T}}\\!M\\!(\bm{q})\dot{\bm{q}}\\!+\\!m_{3}g\\!\left[\frac{1}{2}{l_{1}}(1\\!-\\!C_{1})\\!+\\!\frac{1}{2}{l_{2}}(1\\!-\\!C_{2})\right],$
(9)

where the state vector $\bm{q}\in\mathbb{R}^{5}$ and the inertia matrix
$M(\bm{q})\in\mathbb{R}^{5\times 5}$ in (9) are defined as

$\displaystyle\bm{q}(t)$
$\displaystyle=\left[y(t),z(t),\theta_{1}(t),\theta_{2}(t),\theta_{3}(t)\right]^{T},$
(10) $\displaystyle M(\bm{q})$
$\displaystyle=\left[{\begin{array}[]{*{20}{c}}{{M_{11}}}&0&{{M_{13}}}&{{M_{14}}}&{{M_{15}}}\\\
0&{{M_{22}}}&{{M_{23}}}&{{M_{24}}}&{{M_{25}}}\\\
{{M_{31}}}&{{M_{32}}}&{{M_{33}}}&{{M_{34}}}&{{M_{35}}}\\\
{{M_{41}}}&{{M_{42}}}&{{M_{43}}}&{{M_{44}}}&{{M_{45}}}\\\
{{M_{51}}}&{{M_{52}}}&{{M_{53}}}&{{M_{54}}}&{{M_{55}}}\end{array}}\right],$
(16)

with

$\displaystyle{M_{11}}\\!=\\!{M_{22}}\\!=\\!{m_{1}}\\!+\\!{m_{2}}\\!+\\!{m_{3}}\\!,\\!{M_{55}}\\!=\\!{m_{2}}{a^{2}}\\!+\\!{m_{3}}{\left({\frac{a}{2}}\right)^{2}},$
$\displaystyle{M_{13}}={M_{31}}={m_{2}}{l_{1}}{C_{1}}+{m_{3}}{l_{1}}{C_{1}},$
$\displaystyle{M_{14}}={M_{41}}={m_{2}}{l_{2}}{C_{2}},\,{M_{24}}={M_{42}}=-{m_{2}}{l_{2}}{S_{2}},$
$\displaystyle{M_{15}}={M_{51}}=-{m_{2}}a{S_{3}}-{m_{3}}\frac{a}{2}{S_{3}},$
$\displaystyle{M_{23}}\\!\\!=\\!\\!{M_{32}}\\!\\!=\\!\\!{m_{2}}{l_{1}}{S_{1}}\\!\\!+\\!\\!{m_{3}}{l_{1}}{S_{1}}\\!,\\!{M_{25}}\\!\\!=\\!\\!{M_{52}}\\!\\!=\\!\\!{m_{2}}a{C_{3}}\\!\\!+\\!\\!{m_{3}}\frac{a}{2}{C_{3}},$
$\displaystyle{M_{33}}={m_{2}}{l_{1}}^{2}+{m_{3}}{l_{1}}^{2},\,{M_{44}}={m_{2}}{l_{2}}^{2},$
$\displaystyle{M_{34}}\\!=\\!{M_{43}}\\!=\\!{m_{2}}{l_{1}}{l_{2}}{C_{1+2}}\\!,\\!{M_{45}}={M_{54}}\\!=\\!-{m_{2}}{l_{2}}a{S_{2\\!+\\!3}},$
$\displaystyle{M_{35}}={M_{53}}={m_{2}}{l_{1}}a{S_{1-3}}+{m_{3}}{l_{1}}\frac{a}{2}{S_{1-3}}.$

Taking the time derivative of $E$ results in

$\displaystyle\dot{E}\\!=$
$\displaystyle{\dot{y}_{1}}f_{1}\sin\phi_{1}\\!+\\!{\dot{y}_{2}}f_{2}\sin\phi_{2}\\!+\\!{\dot{z}_{1}}\\!\left[{{f_{1}}\cos{\phi_{1}}\\!-\\!\big{(}\\!{{m_{1}}\\!+\\!\frac{1}{2}{m_{3}}}\\!\big{)}g}\right]$
$\displaystyle+{\dot{z}_{2}}\left[{{f_{2}}\cos{\phi_{2}}-\big{(}{{m_{2}}+\frac{1}{2}{m_{3}}}\big{)}g}\right].$
(18)

Decomposing the applied thrust $f_{1}$, $f_{2}$ into $f_{1}\sin\phi_{1}$,
$f_{1}\cos\phi_{1}$, $f_{2}\sin\phi_{2}$, $f_{2}\cos\phi_{2}$ as shown in
(3.1), the control inputs are construsted as follows:

$\displaystyle{f_{1}}\sin{\phi_{1}}=$ $\displaystyle-
k_{p_{1}}{e_{{y_{1}}}}-{k_{d_{1}}}{\dot{y}_{1}}-k_{a_{1}}\left({{{\dot{\theta}}_{1}}^{2}+{{\dot{\theta}}_{2}}^{2}+{{\dot{\theta}}_{3}}^{2}}\right){\dot{y}_{1}}$
$\displaystyle-\frac{1}{2}{m_{3}}g\tan{\theta_{2d}}-\frac{\sigma\\!\rho
e_{y}}{\left(\rho\\!-\\!e_{y}^{2}\right)^{2}}$ (19)
$\displaystyle{f_{2}}\sin{\phi_{2}}=$ $\displaystyle-
k_{p_{2}}{e_{{y_{2}}}}-{k_{d_{2}}}{\dot{y}_{2}}-k_{a_{2}}\left({{{\dot{\theta}}_{1}}^{2}+{{\dot{\theta}}_{2}}^{2}+{{\dot{\theta}}_{3}}^{2}}\right){\dot{y}_{2}}$
$\displaystyle+\frac{1}{2}{m_{3}}g\tan{\theta_{2d}}+\frac{\sigma\\!\rho
e_{y}}{\left(\rho\\!-\\!e_{y}^{2}\right)^{2}}$ (20)
$\displaystyle{f_{1}}\cos{\phi_{1}}=$ $\displaystyle-
k_{p_{3}}{e_{{z_{1}}}}-{k_{d_{3}}}{\dot{z}_{1}}+\frac{1}{2}\left({2{m_{1}}+{m_{3}}}\right)g,$
(21) $\displaystyle{f_{2}}\cos{\phi_{2}}=$ $\displaystyle-
k_{p_{4}}{e_{{z_{2}}}}-{k_{d_{4}}}{\dot{z}_{2}}+\frac{1}{2}\left({2{m_{2}}+{m_{3}}}\right)g,$
(22)

where $k_{p_{1}}$, $k_{p_{2}}$, $k_{p_{3}}$, $k_{p_{4}}$, $k_{d_{1}}$,
$k_{d_{2}}$, $k_{d_{3}}$, $k_{d_{4}}$, $k_{a_{1}}$, $k_{a_{2}}$, $\sigma$ are
positive control gains. In addition, $e_{y}=e_{y_{1}}-e_{y_{2}}$ and $\rho$ is
a positive constant satisfying

$\displaystyle\rho>\left|e_{y_{1}}(0)-e_{y_{2}}(0)\right|^{2},$ (23)

with $e_{y_{1}}(0),e_{y_{2}}(0)$ being initial values. It is worth noting that
solving the problem of unknown drone and bar-load mass will overturn most of
the previous conclusions, making the difficult control problem more
complicated. Thus, we start with the _Exact Model Knowledge_ controller design
and theoretical analysis.

_Remark 1:_ The third term in the proposed control input (3.1)-(3.1) is to
enhance the coupling of the system. In general, the traditional model-free PD
controller cannot respond to load swing in a timely and efficient manner,
making it difficult to achieve satisfactory control effects. The designed term
will change accordingly to react efficiently to the swing when the bar is
oscillating, which enhances the coupling of the system and facilitates the
suppression of the swing.

_Remark 2:_ The last term in the proposed control inputs (3.1)-(3.1) is to
ensure the distance between the two drones within the desired range, and the
subsequent experiments can demonstrate such a capability. For instance, under
the influence of external disturbances, the positions of the two drones may be
too close, implying that the distance between the two drones in the horizontal
direction is about to reach the boundary of the desired range, thus
$e_{y}^{2}\rightarrow\rho^{-}$, directly making $\frac{\sigma\\!\rho
e_{y}}{\left(\rho\\!-\\!e_{y}^{2}\right)^{2}}\rightarrow\infty$. At this
point, control input (3.1)-(3.1) will drive the drones away from each other to
avoid collision. Similar conclusions can be drawn in the other cases. In this
way, owing to the introduction of the last term, the two drones will not
collide or go too far away from each other. In practical applications, we can
set a proper value for $\rho$ according to the demand of different
transportation process to achieve the desired control performance.

_Remark 3:_ We try to keep the desired position of the bar-load always within
the Y-Z plane during the flight. The error signal in the X-axis may come from
wind disturbances or other factors. Thus, we design a traditional PID
controller to realize the positioning of drones in the X-axis. As supported by
subsequent experimental results, the drones’ positioning error in the
X-direction is usually $\pm 1\,\mathrm{cm}$, and it is $\pm 3\,\mathrm{cm}$
even under wind disturbances.

_Remark 4:_ By invoking the theory on cascade systems, controllers can be
designed for the inner loop and the outer one separately. This manuscript is
concentrated on the outer loop subsystem, thus, the controller design
procedure and stability analysis are detailed. The inner loop dynamics on the
rotation motion of the quadrotor stay the same as those given in literatures
[46].

_Remark 5:_ By redefining the energy storage energy function given in (9) as
follows, the designed controller can be extended to a tracking controller:

$\displaystyle E\\!=$
$\displaystyle\frac{1}{2}{\dot{\bm{e}}^{T}}M(\bm{q})\dot{\bm{e}}\\!+\\!m_{3}g\left[\frac{1}{2}{l_{1}}(1-C_{1})\\!+\\!\frac{1}{2}{l_{2}}(1-C_{2})\right],$

where the tracking error vector is defined as
${\bm{e}}(t)=[e_{y1}(t),e_{z1}(t),e_{y2}(t),e_{z2}(t),\theta_{1}(t),\theta_{2}(t)]$,
where
$e_{y1}(t)=y_{1}(t)-y_{1d}(t),\,e_{z1}(t)=z_{1}(t)-z_{1d}(t),\,e_{y2}(t)=y_{2}(t)-y_{2d}(t),\,e_{z2}(t)=z_{2}(t)-z_{2d}(t)$.Then,
with the similar process in this manuscript, the tracking controller can be
designed. It can be proved that the system is asymptotically stable by using
Lyapunov technique and Barbalat’s lemma.

###  3.2 Stability Analysis

######  Theorem 1.

For the nonlinear underactuated double drone-bar system, the designed
controller given by (3.1)-(22) guarantees that the desired equilibrium point
depicted in (2), is asymptotically stable, in the sense of

$\displaystyle\lim_{t\to\infty}\bm{e}_{\xi_{1}}=\bm{0},\lim_{t\to\infty}\bm{e}_{\xi_{2}}=\bm{0},\lim_{t\to\infty}{\bm{\Theta}}=\bm{0}$
(24)

###### Proof.

First of all, based on the analysis for the storage energy of the system, the
following Lyapunov candidate function is chosen:

$\displaystyle V(t)\\!=$
$\displaystyle\frac{1}{2}{\dot{\bm{q}}^{T}}\\!M\\!(\bm{q})\dot{\bm{q}}\\!\\!+\\!\frac{1}{2}k_{p_{1}}e_{{y_{1}}}^{2}\\!\\!+\\!\frac{1}{2}k_{p_{2}}e_{{y_{2}}}^{2}\\!\\!+\\!\frac{1}{2}k_{p_{3}}e_{{z_{1}}}^{2}\\!\\!+\\!\frac{1}{2}k_{p_{4}}e_{{z_{2}}}^{2}$
$\displaystyle\\!+\\!m_{3}g\left[\frac{1}{2}{l_{1}}(1-C_{1})\\!+\\!\frac{1}{2}{l_{2}}(1-C_{2})\right]\\!+\\!\frac{\sigma
e_{y}^{2}}{2\left(\rho\\!-\\!e_{y}^{2}\right)}$
$\displaystyle\\!+\\!\frac{1}{2}{m_{3}}ge_{y}\tan{\theta_{2d}},$ (25)

Taking the time derivative of $V(t)$, and substituting (3.1)-(22) into
$\dot{V}(t)$ yields

$\displaystyle\dot{V}(t)=$ $\displaystyle-
k_{a_{1}}\left({{{\dot{\theta}}_{1}}^{2}+{{\dot{\theta}}_{2}}^{2}+{{\dot{\theta}}_{3}}^{2}}\right){{\dot{y}}_{1}}^{2}-k_{a_{2}}\left({{{\dot{\theta}}_{1}}^{2}+{{\dot{\theta}}_{2}}^{2}+{{\dot{\theta}}_{3}}^{2}}\right){{\dot{y}_{2}}^{2}}$
$\displaystyle-{k_{d_{1}}}{\dot{y}_{1}}^{2}-{k_{d_{2}}}{\dot{y}_{2}}^{2}-{k_{d_{3}}}{\dot{z}_{1}}^{2}-{k_{d_{4}}}{\dot{z}_{2}}^{2}.$
(26)

After some mathematical calculation, the following result can be concluded
from (3.2):

$\displaystyle\dot{V}(t)\leq
0\Rightarrow{V\left({{t}}\right)}\leq{V\left(0\right)}<+\infty,$ (27)

where the fact of $\rho>\left|e_{y_{1}}(0)-e_{y_{2}}(0)\right|^{2}$ shown in
(23) is utilized. More specifically, suppose
$\left|e_{y_{1}}(t)-e_{y_{2}}(t)\right|^{2}$ tends to exceed the boundary of
$\rho$ from interior, then there exists some time making
$\left|e_{y_{1}}(t)-e_{y_{2}}(t)\right|^{2}\\!\rightarrow\\!\rho^{-}\Rightarrow\frac{\sigma
e_{y}^{2}}{2\left(\rho\\!-\\!e_{y}^{2}\right)}\\!\rightarrow\\!+\infty\Rightarrow
V(t)\\!\rightarrow\\!+\infty$, which conflicts with the conclusion in (27).
Therefore, it is concluded that $\frac{\sigma
e_{y}^{2}}{2\left(\rho\\!-\\!e_{y}^{2}\right)}\geq 0$, which further indicates
$\frac{1}{2}{m_{3}}ge_{y}\tan{\theta_{2d}}\geq-\frac{1}{2}{m_{3}}g\sqrt{\rho}|\tan{\theta_{2d}}|$.
$M(\bm{q})$ in (3.2) is a positive definite matrix, thus
$(\frac{1}{2}{\dot{q}^{{T}}}M\dot{q})\geq 0$.
$k_{p_{1}},k_{p_{2}},k_{p_{3}},k_{p_{4}}$ are positive gains, hence, the term
$(\frac{1}{2}k_{p_{1}}e_{y_{1}}^{2}+\frac{1}{2}k_{p_{2}}e_{y_{2}}^{2}+\frac{1}{2}k_{p_{3}}e_{z_{1}}^{2}+\frac{1}{2}k_{p_{4}}e_{z_{2}}^{2})\geq
0$. Besides, it is obvious that
$m_{3}{g}[\frac{1}{2}{l_{1}}({1-C_{1}})+\frac{1}{2}{l_{2}}(1-{C_{2}})]\geq
m_{3}g[{\frac{1}{2}{l_{1}}\left({1-1}\right)+\frac{1}{2}{l_{2}}\left({1-1}\right)}]$.
Consequently, because the terms contained in $V(t)$ are all lower bounded, the
function $V(t)$ itself is also lower bounded. Utilizing this fact, together
with the conclusion in (27), one can obtain that

$\displaystyle V\in{\mathcal{L}_{\infty}}\Rightarrow$
$\displaystyle{{{\dot{y}}}_{{1}}},{\dot{z}_{{1}}},{\dot{\theta}_{{1}}},{\dot{\theta}_{2}},{\dot{\theta}_{3}},{e_{{y_{{1}}}}},{e_{{y_{2}}}},{y_{{1}}},{z_{1}}\in{\mathcal{L}_{\infty}}$
$\displaystyle\frac{\sigma
e_{y}^{2}}{2\left(\rho\\!-\\!e_{y}^{2}\right)}\in{\mathcal{L}_{\infty}}.$ (28)

To prove the asymptotic stability of the designed outer loop controller, the
set
$\bm{\Omega}=\left\\{{\left({\bm{q},\dot{\bm{q}}}\right)\left|{\dot{V}=0}\right.}\right\\}$
is defined, and $\bm{\Psi}$ is the largest invariant set in $\bm{\Omega}$. For
clarity, the proof process is divided into the following two steps.

Step 1: In this part, it will be shown that $e_{z_{1}}=e_{z_{2}}=0$. According
to equation (3.2), one can obtain that

$\displaystyle\dot{V}\\!=\\!0\Rightarrow{\dot{y}_{1}}\\!=\\!{\dot{y}_{2}}\\!=\\!{\dot{z}_{1}}\\!=\\!{\dot{z}_{2}}\\!=\\!0,{\ddot{y}_{1}}\\!=\\!{\ddot{y}_{2}}\\!=\\!{\ddot{z}_{1}}\\!=\\!{\ddot{z}_{2}}\\!=\\!0.$
(29)

According to the expressions in (1), adding (3.1) and (3.1) yields

$\displaystyle-k_{p_{1}}{e_{{y_{1}}}}$ $\displaystyle-
k_{p_{2}}{e_{{y_{2}}}}={f_{1}}\sin{\phi_{1}}+{f_{2}}\sin{\phi_{2}}.$ (30)

Substituting $\ddot{y}(t)=0$ into (1) and integrating both sides of (30) with
respect to time, one can obtain that

$\displaystyle\left({{m_{2}}\\!+\\!{m_{3}}}\right){l_{1}}{C_{1}}{\dot{\theta}_{1}}\\!+\\!{m_{2}}{l_{2}}{C_{2}}{\dot{\theta}_{2}}\\!-\\!\left({{m_{2}}\\!+\\!\frac{1}{2}{m_{3}}}\right)a{S_{3}}{\dot{\theta}_{3}}$
$\displaystyle=$
$\displaystyle\left({-k_{p_{1}}{e_{{y_{1}}}}\\!-\\!k_{p_{2}}{e_{{y_{2}}}}}\right)t\\!+\\!{\beta_{1}},$
(31)

wherein $\beta_{1}$ denotes a constant to be determined. If
$(-k_{p_{1}}{e_{{y_{1}}}}-k_{p_{2}}{e_{{y_{2}}}})\neq 0$, when
$t\rightarrow\infty$, both sides of (3.2) tend to infinity, which has obvious
contradiction with the inference
${\dot{\theta}_{1}},{\dot{\theta}_{2}},{\dot{\theta}_{3}}\in{\mathcal{L}_{\infty}}$
in (29). Then the following conclusions are drawn:

$\displaystyle-
k_{p_{1}}{e_{{y_{1}}}}-k_{p_{2}}{e_{{y_{2}}}}=0\Rightarrow{e_{{y_{1}}}}=-\frac{{k_{p_{2}}}}{{k_{p_{1}}}}{e_{{y_{2}}}}.$
(32)

Substituting (32) into (3.2) and integrating both sides of equation (32) with
time yields

$\displaystyle\left(\\!{{m_{2}}\\!\\!+\\!\\!{m_{3}}}\\!\right){l_{1}}{S_{1}}\\!\\!+\\!\\!{m_{2}}{l_{2}}{S_{2}}\\!\\!+\\!\\!a{C_{3}}\\!\left(\\!{{m_{2}}\\!+\\!\frac{m_{3}}{2}}\\!\right)\\!=\\!{\beta_{1}}t\\!\\!+\\!\\!{\beta_{2}},$
(33)

wherein $\beta_{2}$ denotes a constant to be determined. If $\beta_{1}\neq 0$,
with similar analysis for (3.2)-(32), one can draw the following conclusion:

$\displaystyle\left(\\!{{m_{2}}\\!\\!+\\!\\!{m_{3}}}\\!\right){l_{1}}{C_{1}}{\dot{\theta}_{1}}\\!\\!+\\!\\!{m_{2}}{l_{2}}{C_{2}}{\dot{\theta}_{2}}\\!\\!-\\!\\!a{S_{3}}{\dot{\theta}_{3}}\left(\\!m_{2}\\!+\\!\frac{m_{3}}{2}\\!\right)\\!=\\!0.$
(34)

Taking the time derivative of $\bm{\xi}_{2}(t)$, we obtain the velocity
$\dot{\bm{\xi}}_{2}(t)$ of the drone on the right. Combining with the equation
(34), the following equations can be obtained:

$\displaystyle\begin{cases}\\!\left(\\!{{m_{2}}\\!\\!+\\!\\!{m_{3}}}\\!\right)\\!{l_{1}}\\!{C_{1}}{\dot{\theta}_{1}}\\!\\!+\\!\\!{m_{2}}\\!{l_{2}}\\!{C_{2}}{\dot{\theta}_{2}}\\!\\!-\\!\\!a{S_{3}}{\dot{\theta}_{3}}\\!\\!\left(\\!m_{2}\\!\\!+\\!\\!\frac{m_{3}}{2}\\!\right)\\!\\!=\\!0,\\\
{l_{1}}{C_{1}}{{\dot{\theta}}_{1}}+{l_{2}}{C_{2}}{{\dot{\theta}}_{2}}-a{S_{3}}{{\dot{\theta}}_{3}}=0,\\\
{l_{1}}{S_{1}}{{\dot{\theta}}_{1}}-{l_{2}}{S_{2}}{{\dot{\theta}}_{2}}+a{C_{3}}{{\dot{\theta}}_{3}}=0.\end{cases}$
(35)

Lemma 1:
$\dot{\theta}_{1}\\!=\\!\dot{\theta}_{2}\\!=\\!\dot{\theta}_{3}\\!=\\!0$ is
the only solution for (35).
Proof: please see the appendix for details.

According to Cramer’s Rule, one can obtain that

$\displaystyle{\dot{\theta}_{1}},{\dot{\theta}_{2}},{\dot{\theta}_{3}}=0\Rightarrow{\ddot{\theta}_{1}},{\ddot{\theta}_{2}},{\ddot{\theta}_{3}}=0.$
(36)

According to (36) and the expressions in (2), adding (21) and (22) yields

$\displaystyle{f_{1}}\cos{\phi_{1}}+{f_{2}}\cos{\phi_{2}}=\left({{m_{{1}}}{{+}}{m_{{2}}}{{+}}{m_{{3}}}}\right){{g}}$
$\displaystyle\Rightarrow$ $\displaystyle-
k_{p_{3}}{e_{{z_{1}}}}-k_{p_{4}}{e_{{z_{2}}}}=0.$ (37)

It is worth noting from (29) that when
$\dot{V}=0,\ddot{z}_{1}=\ddot{z}_{2}=0$, both of the drones do not provide any
acceleration in the vertical direction, thus, they only bear the load and
their own gravity. According to Assumption1, we can set the following
restrictions on $f_{1}\cos\phi_{1},f_{2}\cos\phi_{2}$ :

$\displaystyle{m_{1}}g<{f_{1}}\cos{\phi_{1}}<\left({{m_{1}}{+}{m_{3}}}\right)g,$
$\displaystyle{m_{2}}g<{f_{2}}\cos{\phi_{2}}<\left({{m_{2}}{+}{m_{3}}}\right)g.$
(38)

Substituting the results of (36) into (3)-(5), the following result can be
obtained :

$\displaystyle\begin{cases}\left({{m_{2}}+{m_{3}}}\right)g{S_{1}}={f_{2}}\sin{\phi_{2}}{C_{1}}+{f_{2}}\cos{\phi_{2}}{S_{1}},\\\
-{m_{2}}g{S_{2}}={f_{2}}\sin{\phi_{2}}{C_{2}}-{f_{2}}\cos{\phi_{2}}{S_{2}},\\\
{m_{2}}g{C_{3}}\\!\\!+\\!\\!\frac{1}{2}\\!{m_{3}}g{C_{3}}\\!=\\!-\\!{f_{2}}\\!\sin{\phi_{2}}\\!{S_{3}}\\!\\!+\\!\\!{f_{2}}\\!\cos{\phi_{2}}{C_{3}}.\end{cases}$
(39)

Substituting (22) into (39), one can obtain that

$\displaystyle\begin{cases}\tan{\theta_{1}}\left({\frac{1}{2}{m_{3}}g+k_{p_{4}}{e_{{z_{2}}}}}\right)={f_{2}}\sin{\phi_{2}},\\\
\tan{\theta_{2}}\left({\frac{1}{2}{m_{3}}g-k_{p_{4}}{e_{{z_{2}}}}}\right)={f_{2}}\sin{\phi_{2}},\\\
-k_{p_{4}}{e_{{z_{2}}}}={f_{2}}\sin{\phi_{2}}\tan{\theta_{3}}.\end{cases}$
(40)

Lemma 2: The only solution for (40) is ${e_{{z_{2}}}}=0$.
Proof: Please see appendix for details.

It then follows from Lemma 1 that

$\displaystyle{e_{{z_{2}}}}=0,{\theta_{1}}={\theta_{2}}{{,}}{\theta_{3}}={{0}}\Rightarrow{e_{{z_{1}}}}=0.$
(41)

Step 2: In this part, it will be further proved that
$\theta_{1}=\theta_{1d},\theta_{2}=\theta_{2d},e_{y_{1}}=e_{y_{2}}=0$.
Firstly, substituting the results of (3.1) into the second equation of (40),
and combining the conclusions in (29) and (41), the following result can be
obtained after simplification:

$\displaystyle-k_{p_{2}}{e_{{y_{2}}}}+\frac{\sigma\\!\rho
e_{y}}{\left(\rho\\!-\\!e_{y}^{2}\right)^{2}}=\frac{1}{2}{m_{3}}g(\tan{\theta_{2}}-{\tan}\theta_{2d}).$
(42)

Regarding the right half part of (42), the following inference can be made
based on (8) and (41):

$\displaystyle{\mathrm{sgn}}\\!\left(\\!{\tan{\theta_{2}\\!\\!-\\!\\!\tan\theta_{2d}}}\\!\right)\\!=\\!{\mathrm{sgn}}\Big{(}\\!{\big{(}\\!{1\\!\\!+\\!\\!\frac{{k{p_{2}}}}{{k{p_{1}}}}}\big{)}{e_{{y_{2}}}}}\\!\Big{)}\\!=\\!{\mathrm{sgn}}\\!\left(\\!{{e_{{y_{2}}}}}\\!\right).$
(43)

For the left side of (42), based on (32), the following inference can be made:

$\displaystyle-k_{p_{2}}{e_{{y_{2}}}}\\!+\\!\frac{\sigma\\!\rho
e_{y}}{\left(\rho\\!-\\!e_{y}^{2}\right)^{2}}\\!=\\!-{e_{{y_{2}}}}\bigg{(}k_{p_{2}}\\!+\\!\frac{\sigma\\!\rho(1+\frac{k_{p_{2}}}{k_{p_{1}}})}{\left[\rho\\!-\\!(e_{y_{1}}-e_{y_{2}})^{2}\right]^{2}}\bigg{)}$
$\displaystyle\Rightarrow$
$\displaystyle\,\,\,\,{\mathrm{sgn}}\bigg{(}-k_{p_{2}}{e_{{y_{2}}}}+\frac{\sigma\\!\rho
e_{y}}{\left(\rho\\!-\\!e_{y}^{2}\right)^{2}}\bigg{)}-{\mathrm{sgn}}\left({{e_{{y_{2}}}}}\right).$
(44)

Therefore, we can conclude that the two sides of the equation (42) are
opposite, making ${e_{{y_{1}}}}=0$ as its only solution. Some similar analysis
can be implemented for the signal ${e_{{y_{2}}}}$, and the following
conclusions can be then drawn:

$\displaystyle{e_{{y_{1}}}}={e_{{y_{2}}}}=0.$ (45)

Combining the results presented in (29), (3.2), (41), (45), there is a unique
solution of $\dot{V}=0$. Therefore, we can obtain that

$\displaystyle\begin{cases}{y_{1}}={y_{1d}},{y_{2}}={y_{2d}},{z_{1}}={z_{2}}={z_{1d}}={z_{2d}},\\\
{\theta_{1}}={\theta_{2}}={\theta_{1d}}={\theta_{2d}},{\theta_{3}}=0.\end{cases}$
(46)

This calculation implies that $V=0$ is the unique minimum value at the desired
equilibrium point. Based on this conclusion, the results of (3.2)-(45) are
utilized again to conclude that the closed-loop system is stable, and the
maximum invariant set $\bm{\Psi}$ only contains the desired equilibrium point.
According to LaSalle’s invariance theorem, it is concluded that the
equilibrium point is asymptotically stable. ∎

##  4 Hardware Experiments

In this section, two groups hardware experimental results are provided to
verify the efficiency of the proposed method, especially its robustness
against uncertain parameters and external disturbances. The relevant
experimental video has been taken and uploaded to YouTube, specifically posted
at (https://youtu.be/hHg66bE2PRA).

Figure 2: Experiment testbed.

The self-built hardware experimental testbed for the drone-bar system is shown
in Fig. 2. The poses of the two drones and the bar load are obtained through
the Qualisys Motion Capture System. Data of 8 cameras are sent to the ground
station via the Robot Operating System (ROS) based transmission protocol under
the LAN. The corresponding control input and torque are calculated on the
ground station and then sent to the onboard computer via the WIFI with the 5G
band. The flight controller is PIXHAWK, which is connected between the onboard
computer and PIXHAWK by mavros based communication protocol. The Raspberry Pi
runs the 64-bit Ubuntu-mate 16.04 operating system. The utilization of 5G band
wireless network makes its network latency approximately 1-3 milliseconds,
which is the best data transmission scheme after a lot of debugging tests. The
physical parameters of the experimental testbed are given as follows:

$\displaystyle m_{1}=m_{2}=1.5\,{\rm kg},\,m_{3}=0.3\,{\rm kg},\,g=9.8\,{\rm
m/s^{2}}.$ $\displaystyle a=1.2\,{\rm m},l_{1}=l_{2}=0.9\,{\rm m}.$ (47)

Then, based on the experimental platform, two groups of experiments is carried
out to verify the effectiveness of the proposed method.

###  4.1 Experiment Group 1

Table 1: Parameters for Experiment 1 Exp 1 | Initial positions${\rm(m)}$ | Desired positions${\rm(m)}$
---|---|---
Test 1 | $\quad\quad\quad[y_{1},z_{1}]=[-0.1,1.3],\,[y_{2},z_{2}]=[\,\,\,\,1.5,1.3]\quad\quad\quad$ | $[y_{1d},z_{1d}]=[-1.3,1.8],\,[y_{2d},z_{2d}]=[-0.1,1.8]$
Test 2 | $[y_{1},z_{1}]=[-1.3,1.5],\,[y_{2},z_{2}]=[-0.2,1.5]$ | $[y_{1d},z_{1d}]=[\,\,\,\,0.0,1.9],\,[y_{2d},z_{2d}]=[\,\,\,\,1.5,1.9]$

In this group of experiments, the proposed controller (3.1)-(22) allows the
two drones to reach the desired positions and then hover at that location. To
achieve satisfactory convergence speed and anti-swing objectives, we repeat
several times for each experiment to adjust the parameters based on
experience. The control gains are listed as:

$\displaystyle k_{p_{1}}=k_{p_{2}}=5.2,k_{d_{1}}=k_{d_{2}}=6.0,$
$\displaystyle k_{p_{3}}=k_{p_{4}}=6.0,k_{d_{3}}=k_{d_{4}}=8.0,$
$\displaystyle\delta=4.0,\rho=2.0,k_{a_{1}}=k_{a_{2}}=0.75.$ (48)

Also, we implement some comparative experiments with the classic PD control as
the comparative controller, whose control gains are set as
$k_{p_{1}}=k_{p_{2}}=5.2,k_{d_{1}}=k_{d_{2}}=6.0,k_{p_{3}}=k_{p_{4}}=6.0,k_{d_{3}}=k_{d_{4}}=8.0$
after sufficient tuning. To further verify the robustness of the proposed
control method against system uncertainties and various disturbances, the
following three tests are carried out.

figure subfloat

(a)  Positions of the two drones for Exp 1-Test 1. (b)  Control input & Swing
angles.

Figure 3: Results for Experiment 1-Test 1: Swing Elimination in transportation
process. (black dashed lines: desired positions. red dotted-dashed lines: PD
controller. blue solid lines: proposed controller.).

figure subfloat

(a)  Positions of the two drones for Exp 1-Test 2. (b)  Control input & Swing
angles.

Figure 4: Results for Experiment 1-Test 2: Robustness against system
uncertainties. (black dashed lines: desired positions. red dotted-dashed
lines: PD controller. blue solid lines: proposed controller.).

Test 1: (Swing Elimination in transportation process). It is seen from Table 1
that the initial and desired positions are set to verify whether the proposed
method can meet the basic point-to-point transportation requirements. The
maximum wind speed is up to $3\,\mathrm{m/s}$ measured by a digital anemometer
AS8336 in the center of the experiment site, and the wind is in the negative
direction of the Y-axis. In the actual experiment, the bar will continuously
swing under the action of wind, which is particularly apparent in the PD
controller, making the load’s swing away from zero naturally.

Test 2: (Robustness against system uncertainties). To further verify the
robustness of the proposed control schemes against parameter uncertainties,
the bar’s physical parameters and ropes’ length are changed to
$m_{3}=0.5\,{\rm kg},a=1.5\,{\rm m},l_{1}=l_{2}=1.2\,{\rm m}$, yet the other
system parameters are still the same with Test 1. The initial and desired
positions are shown in Table 1. Like Test 1, this test is also conducted under
wind disturbances.

Test 3: (Robustness against external disturbances). In the state of drones
hovering, we test the anti-swing performance of the proposed method by
manually adding the disturbance. The bar is disturbed purposely every $20$
seconds from different directions.

figure subfloat

(a)  Positions of the two drones for Exp 1-Test 3. (b) Control input & Swing
angles.

Figure 5: Results for Experiment 1-Test 3: Robustness against external
disturbances. (red dotted-dashed lines: PD controller. blue solid lines:
proposed controller.).

The experimental results of the three tests are provided in Fig. 3-5,
specifically. It is worth noting that the forces of the two control methods in
the vertical direction are almost the same. Thus no comparison is shown here
to save space. For Test 1, it is seen from Fig. 3 that the settling time of
the proposed method is about only $20\%$ of that of the PD controller, which
indicates the tremendous advantages of the proposed method in transient
performance. For Test 2, one can find from Fig. 4 that the curves show a
better anti-swing performance of the proposed control by comparing it with the
classic PD controller, which is similar to the conclusion obtained in Test 1.
For Test 3, we can observe from Fig. 5 that disturbances manually added from
different directions are effectively rejected by the proposed control scheme.
Further, when the disturbance is added, the control inputs $f_{1}{\rm
sin}\phi_{1},f_{2}{\rm sin}\phi_{2}$ produce significant change to suppress
the bar swing, which makes the convergence speed of $\theta_{1},\theta_{2}$
faster than that of PD controller.

In summary, both controllers have achieved satisfactory drones positioning
results, yet, the proposed control scheme (3.1)-(22) presents better swing
suppression effect and good robustness concerning system uncertainties and
external disturbances such as wind disturbances and manually interference.

###  4.2 Experiment Group 2

Table 2: Parameters of Test 1 in Experiment 2 Experiment 2-Test 1 | Initial positions${\rm(m)}$ | 1st change${\rm(m)}$ | 2nd change${\rm(m)}$ | 3rd change${\rm(m)}$
---|---|---|---|---
$[y_{1},z_{1}]$ | $[-0.6,1.7]$ | $[-0.4,1.7]$ | $[-0.8,1.7]$ | $[-0.6,1.7]$
$[y_{2},z_{2}]$ | $[\,\,\,\,0.6,1.7]$ | $[\,\,\,\,0.4,1.7]$ | $[\,\,\,\,0.8,1.7]$ | $[\,\,\,\,0.6,1.7]$

This group of experiments are performed to verify the control effect of the
last term $\frac{\sigma\\!\rho e_{y}}{\left(\rho\\!-\\!e_{y}^{2}\right)^{2}}$
in the proposed control inputs (3.1)-(3.1). In addition, we implement some
comparative experiments with the classic PD control as the comparative
controller, whose control gains are set as
$k_{p_{1}}=k_{p_{2}}=5.2,k_{d_{1}}=k_{d_{2}}=6.0,k_{p_{3}}=k_{p_{4}}=6.0,k_{d_{3}}=k_{d_{4}}=8.0$
after sufficient tuning. Here, two comparative tests are carried out:

Test 1: (Relative distance test). In the horizontal direction, the desired
positions of the two drones are changed every $10$ seconds, which is recorded
in Table $2$.

Figure 6: Results for Experiment 2-Test 1: Relative distance test. [(black
dashed lines: desired positions. red dotted-dashed lines: PD controller. blue
solid lines: proposed controller.)] Figure 7: Results for Experiment 2-Test 2:
Robustness against external disturbances. [(black dashed lines: desired
positions. blue solid lines: proposed controller.)]

Test 2: (Response to the external disturbances brought by the last term). In
this test, external disturbance is added to make the two drones closer to or
away from each other. The change of $f_{1}{\rm sin}\phi_{1},f_{2}{\rm
sin}\phi_{2}$ is recorded to analyze the effectiveness of the proposed control
strategy (3.1)-(3.1).

For the two tests in Experiment 2, system parameters and control gains of the
proposed methods are set the same as those in Experiment 1-Test 1. The forces
and drones’ positions of the two methods in the vertical direction are almost
the same. Thus no comparison is shown here to save space. For Test 1, it is
seen from Fig. 6 that both the proposed control scheme and classic PD method
achieve satisfactory positioning control, which is the same as the conclusion
of Experiment 1. However, the proposed method is more sensitive to the
distance between the drones, which enables it to implement more efficient
control for the distance. When the desired positions are changed, the error
$e_{y}=e_{y_{1}}-e_{y_{2}}$ will cause the term $\frac{\sigma\\!\rho
e_{y}^{2}}{\left(\rho\\!-\\!e_{y}^{2}\right)^{2}}$ to generate a large
momentary thrust, which pushes the drones back to the desired positions more
quickly than PD method. During the return process, the term
$\frac{\sigma\\!\rho e_{y}^{2}}{\left(\rho\\!-\\!e_{y}^{2}\right)^{2}}$
continues to decay until it reaches zero. For Test 2, we constantly decrease
and then increase the distance between the two drones in the horizontal
direction and record the curves of $f_{1}{\rm sin}\phi_{1},f_{2}{\rm
sin}\phi_{2}$. As Fig. 7 shows, in the process of increasing the distance,
there will be a moment when the force is large enough to pull the two drones
back to the desired positions. Considering that the rotating speed of
propellers has a physical upper limit, we thus choose larger $\rho$ to avoid
saturation.

##  5 Conclusion

For the double drone-bar system, this paper focuses on the elimination of the
bar-load swing angles and the coordination transportation. Specifically, a
precise model set up by the Lagrangian modeling method facilitates the
subsequent research. Besides, the proposed coordination controller
successfully stabilizes the whole system and guarantees the asymptotic
stability of the desired equilibrium point without any linearization or
approximations. Many experiments are carried out indoors to verify the
effectiveness and robustness of the controller. As the results show, the
proposed method has better performance in swing suppression and coordination
transportation. In future work, the experimental scene will be extended to the
outdoor environment to solve more realistic problems encountered in
transportation.

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##  Appendix A Proof of Lemma 1

It is learned that $\dot{\theta}_{1}=\dot{\theta}_{2}=\dot{\theta}_{3}=0$ is
one of the solutions for (35). To show that it is the unique solution, we
rewrite (35) into the following matrix form:

$\displaystyle
A\cdot\left[\dot{\theta}_{1},\,\dot{\theta}_{2},\,\dot{\theta}_{3}\right]^{T}=0,$
(49)

where the coefficient matrix $A\in\mathbb{R}^{3\times 3}$ is defined as:

$\displaystyle
A=\left[\begin{array}[]{ccc}({{m_{2}}+{m_{3}}}){l_{1}}{C_{1}}&{m_{2}}{l_{2}}{C_{2}}&-(m_{2}+\frac{m_{3}}{2})a{S_{3}}\\\
{l_{1}}{C_{1}}&{l_{2}}{C_{2}}&-a{S_{3}}\\\
{l_{1}}{S_{1}}&-{l_{2}}{S_{2}}&a{C_{3}}\end{array}\right].$ (53)

After some calculation, the determinant of $A$ is obtained as:

$\displaystyle|A|=\frac{m_{3}}{2}l_{1}l_{2}a\cdot(C_{1}C_{2+3}+C_{2}C_{1-3}).$
(54)

According to Assumption 1, the range of the signals
$\theta_{1},\theta_{2},\theta_{3}$ is $(-\frac{\pi}{2},\frac{\pi}{2})$. When
$|\theta_{2}+\theta_{3}|=\frac{\pi}{2}$, the bar and the right rope will be in
a straight line(see Fig.1), which leads to an apparent contradiction. In
addition, due to the effect of gravity, it is impossible that
$|\theta_{2}+\theta_{3}|>\frac{\pi}{2}$, thus $C_{1}C_{2+3}>0$. With some
similar analysis, one further derives that $C_{1}C_{1-3}>0$. Therefore, one
can conclude that under the circumstance of Assumption 1, $|A|>0$, which
implies that $\dot{\theta}_{1}=\dot{\theta}_{2}=\dot{\theta}_{3}=0$ is the
only solution for (35).

##  Appendix B Proof of Lemma 2

It is learned from (3.1) that $e_{z_{2}}=z_{2}-z_{2d}$. Subsequently, it will
be demonstrated with reduction to absurdity that $e_{z_{2}}=0$ always holds
for (40). This part of ananlysis is split into the following three cases.

###  B.1 Suppose $e_{z_{2}}=0$

If $e_{z_{2}}=0$, it can be concluded from (40) that

$\tan\theta_{1}=\tan\theta_{2},\,\,\tan\theta_{3}=0.$ $None$

According to Assumption 1, the range of the signals
$\theta_{1},\theta_{2},\theta_{3}$ is $(-\frac{\pi}{2},\frac{\pi}{2})$.
Therefore, one always has $\theta_{1}=\theta_{2},\theta_{3}=0$ for
$e_{z_{2}}=0$.

###  B.2 Suppose $e_{z_{2}}>0$

In section $2$, the drones’ positions are denoted by
$\bm{\xi}_{1}(t)=[y_{1}(t),z_{1}(t)]^{T}=[y,z]^{T}$ and
$\bm{\xi}_{2}(t)=[y_{2}(t),z_{2}(t)]^{T}=[y+l_{1}S_{1}+l_{2}S_{2}+aC_{3},z-l_{1}C_{1}+l_{2}C_{2}+aS_{3}]^{T}$.
If $e_{z_{2}}>0$, it can be concluded from (8) and (3.2) that

$\displaystyle{e_{{z_{1}}}}=-\frac{k_{p_{4}}}{k_{p_{3}}}{e_{{z_{2}}}},\,z_{1d}=z_{2d},\,l_{1}=l_{2}=l$
$None$ $\displaystyle\Rightarrow$
$\displaystyle\,\,e_{z_{1}}<0,\,z_{2}-z_{1}>0$ $\displaystyle\Rightarrow$
$\displaystyle\,\,l(C_{2}-C_{1})+aS_{3}>0.$

Under the condition of $e_{z_{2}}>0$ and
$-\frac{\pi}{2}<\theta_{1},\theta_{2},\theta_{3}<\frac{\pi}{2}$, one may
deduce from (40) that

$\displaystyle|\tan{\theta_{1}}|\cdot|{\frac{1}{2}{m_{3}}g+k_{p_{4}}{e_{{z_{2}}}}}|=|\tan{\theta_{2}}|\cdot|{\frac{1}{2}{m_{3}}g-k_{p_{4}}{e_{{z_{2}}}}}|$
$None$ $\displaystyle\Rightarrow$
$\displaystyle\,\,|\tan{\theta_{1}}|<|\tan{\theta_{2}}|\,\,\Rightarrow\,\,|\theta_{1}|<|\theta_{2}|$
$\displaystyle\Rightarrow$ $\displaystyle\,\,C_{1}>C_{2},$

which further leads to

$\displaystyle l(C_{2}-C_{1})+aS_{3}>0,\,C_{1}>C_{2}$ $None$
$\displaystyle\Rightarrow$ $\displaystyle\,\,aS_{3}>l(C_{1}-C_{2})>0$
$\displaystyle\Rightarrow$ $\displaystyle\,\,\theta_{3}>0.$

It is concluded from (40) that

$\displaystyle-k_{p_{4}}{e_{{z_{2}}}}={f_{2}}\sin{\phi_{2}}\tan{\theta_{3}}$
$None$ $\displaystyle\Rightarrow$
$\displaystyle\,\,{f_{2}}\sin{\phi_{2}}\tan{\theta_{3}}<0,\,\Rightarrow\,{f_{2}}\sin{\phi_{2}}<0$
$\displaystyle\Rightarrow$
$\displaystyle\,\,\tan{\theta_{1}}({\frac{1}{2}{m_{3}}g+k_{p_{4}}{e_{{z_{2}}}}})<0$
$\displaystyle\Rightarrow$
$\displaystyle\,\,\tan{\theta_{1}}<0,\,\Rightarrow\,\theta_{1}<0.$

where (A.4) is utilized. Regarding (A.5), one further makes the following
deductions according to (8), (3.1), (29), (30) and (32) that

$\displaystyle-{f_{2}}\sin{\phi_{2}}={f_{1}}\sin{\phi_{1}}=-k_{p_{1}}{e_{{y_{1}}}}-\frac{\sigma\\!\rho
e_{y}}{\left(\rho\\!-\\!e_{y}^{2}\right)^{2}}\,>\,0$ $None$
$\displaystyle\Rightarrow$
$\displaystyle\,\,{f_{1}}\sin{\phi_{1}}=-{e_{{y_{1}}}}\bigg{(}k_{p_{1}}+\frac{\sigma\\!\rho(1+\frac{k_{p_{1}}}{k_{p_{2}}})}{\left[\rho\\!-\\!(e_{y_{1}}-e_{y_{2}})^{2}\right]^{2}}\bigg{)}\,>\,0$
$\displaystyle\Rightarrow$
$\displaystyle\,\,e_{y_{1}}\,<\,0\,\Rightarrow\,e_{y_{2}}\,>\,0.$

Furthermore, because of the restrictions on $f_{1}\cos\phi_{1}$,
$f_{2}\cos\phi_{2}$ in (3.2), it is inferred from (21), (29), (3.2) and (A.6)
that

$\displaystyle{m_{1}}g<{f_{1}}\cos{\phi_{1}}<\left({{m_{1}}{+}{m_{3}}}\right)g$
$None$ $\displaystyle\Rightarrow$
$\displaystyle\,\,-k_{p_{3}}{e_{{z_{1}}}}+({{m_{2}}+\frac{1}{2}{m_{3}}})g\,<\,\left({{m_{1}}{+}{m_{3}}}\right)g$
$\displaystyle\Rightarrow$
$\displaystyle\,\,-k_{p_{3}}{e_{{z_{1}}}}<\frac{1}{2}m_{3}g\Rightarrow\,k_{p_{4}}{e_{{z_{2}}}}<\frac{1}{2}m_{3}g$
$\displaystyle\Rightarrow$
$\displaystyle\,\,\frac{1}{2}m_{3}g-k_{p_{4}}{e_{{z_{2}}}}>0.$

Based on the results in (A.5) and (A.7), the following conclusion is derived
from (40) that

$\displaystyle{f_{2}}\sin{\phi_{2}}<0,\,\frac{1}{2}m_{3}g-k_{p_{4}}{e_{{z_{2}}}}>0,$
$None$
$\displaystyle\tan{\theta_{2}}({\frac{1}{2}{m_{3}}g-k_{p_{4}}{e_{{z_{2}}}}}t)={f_{2}}\sin{\phi_{2}}$
$\displaystyle\Rightarrow$
$\displaystyle\,\,\tan{\theta_{2}}<0\,\Rightarrow\,{\theta_{2}}<0.$

According to the analysis from (A.2) to (A.8), suppose $e_{z_{2}}>0$, one can
obtain that $\theta_{1}<0,\theta_{2}<0,e_{y_{1}}<0,e_{y_{2}}>0$. When
$e_{y_{1}}<0,e_{y_{2}}>0$, the distance between the two drones in horizontal
direction can be derived as:

$\displaystyle e_{y_{2}}-e_{y_{1}}=(y_{2}-y_{1})-(y_{2d}-y_{1d})>0$ $None$
$\displaystyle\Rightarrow$
$\displaystyle\,\,y_{2}-y_{1}>y_{2d}-y_{1d}\,\Rightarrow\,y_{2}-y_{1}>a.$

When $\theta_{1}<0,\theta_{2}<0$, the distance between the two drones in the
horizontal direction can be derived as:

$\displaystyle
y_{2}-y_{1}=l_{1}S_{1}+l_{2}S_{2}+aC_{3}<aC_{3}-|l_{1}S_{1}|-|l_{2}S_{2}|$
$None$ $\displaystyle\Rightarrow$ $\displaystyle\,\,y_{2}-y_{1}<a,$

where $\bm{\xi}_{1}(t)=[y_{1}(t),z_{1}(t)]^{T}=[y,z]^{T}$ and
$\bm{\xi}_{2}(t)=[y_{2}(t),z_{2}(t)]^{T}=[y+l_{1}S_{1}+l_{2}S_{2}+aC_{3},z-l_{1}C_{1}+l_{2}C_{2}+aS_{3}]^{T}$
are utilized. It is obvious that (A.9) and (A.10) are contradictory, thus
$e_{z_{2}}>0$ does not exist.

###  B.3 Suppose $e_{z_{2}}<0$

The proof steps for $e_{z_{2}}<0$ and $e_{z_{2}}>0$ are roughly the same, thus
similar deductions can be performed when $e_{z_{2}}<0$. It is impossible that
$e_{z_{2}}<0$ as well.

According to the above analysis, the only solution for (40) is $e_{z_{2}}=0$.