pnasresearcharticle

Kamdar

A taller person usually swims faster thanks to the more pronounced increase of
hydrodynamic thrust with height relative to fluid drag in inertia-dominated
flows. However, this general positive size-speed correlation may not hold at
microscopic scales for bacteria, where viscous dissipation overwhelms inertia
in their swimming. By combining experiments, simulations and theory, we show
that the swimming speed of bacteria with multiple flagella remains constant
across the natural range of bacterial lengths, settling a long-lasting debate
over the size-speed relation of bacterial swimming. Our quantitative analyses
further elucidate the hydrodynamic origin of such an unexpected relation. Our
study sheds light on the functional benefit of multiflagellarity and provides
a useful guide for designing artificial microswimmers.

S.K and X.C conceived the project. S.K., D.G. and X.C. designed the research
and developed the models. S.K., D.G., S.G., Y.K., T.C. and X.C. performed
experiments. W.L., Y.K. and S.L. performed the IB simulations. M.T.-C. and
E.L. performed the SBT simulations. S.K., D.G. and X.C. co-wrote the
manuscript with inputs from all the authors. S.L. and X.C. supervised the
project. The authors declare no competing interests. 1These authors
contributed equally to this work. 2To whom correspondence should be addressed.
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
# Multiflagellarity leads to the size-independent swimming speed of
peritrichous bacteria

Shashank Kamdar  Department of Chemical Engineering and Materials Science,
University of Minnesota, Minneapolis, MN 55455, USA  Dipanjan Ghosh
Department of Chemical Engineering and Materials Science, University of
Minnesota, Minneapolis, MN 55455, USA  Wanho Lee  National Institute for
Mathematical Sciences, Daejeon 34047, Republic of Korea  Maria Tătulea-Codrean
Department of Applied Mathematics and Theoretical Physics, University of
Cambridge, Cambridge CB3 0WA, United Kingdom  Yongsam Kim  Department of
Mathematics, Chung-Ang University, Seoul 06974, Republic of Korea  Supriya
Ghosh  Department of Chemical Engineering and Materials Science, University of
Minnesota, Minneapolis, MN 55455, USA  Youngjun Kim  Department of Chemical
Engineering and Materials Science, University of Minnesota, Minneapolis, MN
55455, USA  Tejesh Cheepuru  Department of Chemical Engineering and Materials
Science, University of Minnesota, Minneapolis, MN 55455, USA  Eric Lauga
Department of Applied Mathematics and Theoretical Physics, University of
Cambridge, Cambridge CB3 0WA, United Kingdom  Sookkyung Lim  Department of
Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221, USA
Xiang Cheng  Department of Chemical Engineering and Materials Science,
University of Minnesota, Minneapolis, MN 55455, USA

###### Abstract

To swim through a viscous fluid, a flagellated bacterium must overcome the
fluid drag on its body by rotating a flagellum or a bundle of multiple
flagella. Because the drag increases with the size of bacteria, it is expected
theoretically that the swimming speed of a bacterium inversely correlates with
its body length. Nevertheless, despite extensive research, the fundamental
size-speed relation of flagellated bacteria remains unclear with different
experiments reporting conflicting results. Here, by critically reviewing the
existing evidence and synergizing our own experiments of large sample sizes,
hydrodynamic modeling and simulations, we demonstrate that the average
swimming speed of Escherichia coli, a premier model of peritrichous bacteria,
is independent of their body length. Our quantitative analysis shows that such
a counterintuitive relation is the consequence of the collective flagellar
dynamics dictated by the linear correlation between the body length and the
number of flagella of bacteria. Notably, our study reveals how bacteria
utilize the increasing number of flagella to regulate the flagellar motor
load. The collective load sharing among multiple flagella results in a lower
load on each flagellar motor and therefore faster flagellar rotation, which
compensates for the higher fluid drag on the longer bodies of bacteria.
Without this balancing mechanism, the swimming speed of monotrichous bacteria
generically decreases with increasing body length, a feature limiting the size
variation of the bacteria. Altogether, our study resolves a long-standing
controversy over the size-speed relation of flagellated bacteria and provides
new insights into the functional benefit of multiflagellarity in bacteria.

###### keywords:

Bacterial motility $|$ Size-speed relation $|$ Flagellar dynamics $|$
Multiflagellarity

This manuscript was compiled on August 28, 2024
www.pnas.org/cgi/doi/10.1073/pnas.XXXXXXXXXX

11]2

The motility of bacteria is an essential part of their strategy for survival
and competition in dynamic environments (1, 2, 3). This locomotion at
microscopic scales is directly relevant to macroscopic ecological and
biomedical processes such as bio-remediation, oceanic carbon cycle and spread
of infections, which are of great consequence to human well-being (4, 5, 6,
7). As a paradigmatic model for bacterial motility, flagellated bacteria swim
by rotating a helical rod-shaped propeller composed of either a single
flagellum (monotrichous bacteria) or a bundle of multiple flagella
(peritrichous bacteria) (8). Driven by individual motors, these flagella
collectively rotate to generate hydrodynamic thrust, which balances the fluid
drag experienced by bacteria and enables their swimming (9). At the same time,
the torque driving the flagellar rotation is offset by the torque driving the
rotation of the body to ensure zero net torque on a bacterium. The flagellar
motors carry these torques as a load following a characteristic torque-speed
relation (10). Although it is generally expected that the size of flagellated
bacteria affects their swimming speed due to the dependence of fluid drag on
body shape dictated by universal hydrodynamic principles (9), the relation
between the body size of a bacterium and its swimming speed is still under
debate. A consensus on even the qualitative trend of the relation has not yet
been reached, with studies reporting positive (11), negative (12) and no
correlation (13) between size and speed covering all the possible monotonic
trends.

It is surprising that a controversy over such a fundamental relation of the
kinematics of bacterial motility persists following the decades of research on
the hydrodynamics of swimming flagellated bacteria (see (9) and references
therein), especially considering the implication of the size-dependent
swimming speed on the evolution and ecology of bacteria (3). As one of the key
phenotypes of bacteria, bacterial body size shows natural variations even in a
genetically identical population growing in a constant homogeneous environment
(14). These variations arise from the stochastic nature of gene expression and
the fluctuation of generation times. Since the chemotactic ability of a
bacterium is positively correlated with its swimming speed (15, 16, 17), a
size-dependent swimming speed would indicate a size-dependent chemotactic
ability. Hence, the natural size variation of a bacterial population would
result in the diversity of chemotactic behaviors in the population. Such a
phenotypic diversity is known to strongly affect the overall fitness of the
population for selection and needs to be closely regulated in ecological
settings (18, 19).

Figure 1: Size-independent swimming speed. (A) The distributions of the width
$W$ and the length $L$ of the cell bodies of E. coli from a genetically
identical colony. Inset: A micrograph of a fluorescently labeled bacterium
with $W$ and $L$ indicated. The scale bar is 1 $\mu$m. (B) A violin plot
showing the distributions of the swimming speed $V$ of E. coli of different
lengths. The population average swimming speeds (the white dots) and the
standard deviations of the distributions (the black lines) are independent of
$L$.

Alongside body size, the number and the morphology of bacterial flagella also
profoundly influence the swimming speed of bacteria (20). Thus, potential
correlations between body size and flagellar architecture substantially
complicate the understanding of the size-speed relation and pose a major
challenge for resolving the controversy. Studies that independently
investigated the dependence of the swimming speed on the size of bacteria (11,
12, 13) or the architecture of flagella (21, 22, 23) failed to deliver a
coherent picture of the swimming behaviors of flagellated bacteria.
Recognizing the deficiency of the previous approach, here we combine existing
experimental evidence with our own high-throughput experiments, hydrodynamic
models and simulations and conduct a comprehensive study of the interrelation
between physical features (the body size and the architecture of flagella) and
swimming characteristics (the swimming speed and the angular speeds of body
and flagella) of Escherichia coli (E. coli), the most studied model of
peritrichous bacteria for bacterial motility. Our study reveals that the
linear correlation between the size of bacteria and the number of flagella
enforces load sharing among different flagellar motors, which regulates the
collective dynamics of multiple flagella and sets the size-independent
swimming speed of bacteria. As such, our work provides a unified resolution to
the long-standing controversy and sheds light on the hydrodynamic origin of
the size-speed relation of flagellated bacteria.

Our paper is organized as follows. First, we present our experimental findings
on how the swimming speed of bacteria depends on their body length and discuss
our results in the context of the existing literature. We then illustrate the
origin of the size-speed relation via two sets of low-Reynolds-number
hydrodynamic models and simulations. Finally, we consider the biological
implications of our results on the swimming efficiency of bacteria, the
selective advantage of multiflagellarity, and differences between peritrichous
and monotrichous bacteria in swimming and morphology.

## Results

### Size-speed relation of flagellated bacteria

To resolve the controversy over the relation between body size and swimming
speed of flagellated bacteria, we image the locomotion of a large number of E.
coli cells ($>25,000$) in bulk fluids away from system boundaries and measure
the swimming speed of bacteria, $V$, in the run phase of their classic run-
and-tumble motion (Materials and Methods) (8). The body length, $L$, of E.
coli displays a broad right-skewed distribution with a median of 2.87 $\mu$m
(Fig. 1A), which allows us to probe the correlation between $V$ and $L$ over
threefold bacterial size. In comparison, the distribution of body width is
much narrower, which we treat as a constant. At a given body length, the
swimming speed of individual bacteria varies strongly (Fig. 1B), owing to the
cell-to-cell variations in metabolism and the number and morphology of
flagella (24). Thanks to the large sample size of our experiments, we are able
to measure the population-average swimming speeds and the speed distributions
accurately despite this strong variation. Particularly, we find that both the
average swimming speeds and the speed distributions are independent of $L$,
exhibiting no correlation with the body length (Fig. 1B). This size-
independent swimming speed is quite counterintuitive: as the fluid drag
coefficient on a bacterial body increases by more than $60\%$ when the body
length increases from 2 to 6 $\mu$m (Eq. (9) in Supplementary Information
(SI)), theories and simulations of bacterial swimming would predict more than
$30\%$ decrease in $V$ when the architecture of the flagellar bundle is fixed
(Fig. S1).

Using our data-intensive results as a benchmark, we revisit previous studies
that explored the size-speed relation of E. coli. Di Leonardo and co-workers
first measured the swimming speed of E. coli as a function of the body length
of bacteria near a fluid-air interface (11). They reported a weak increase of
$V$ with $L$. Nevertheless, the sample size of their measurements was small
($\sim 50$ bacteria in total), which explains the large uncertainties of their
measurements that may obscure the genuine trend of $V(L)$. Kaya and Koser
reported the swimming speed of E. coli near the solid boundaries of
microfluidic channels in external shear flow (13). They showed that $V$ is
independent of $L$, agreeing with our measurements. However, it is not clear
how solid boundaries and shear flow in their experiments affect the swimming
behaviors of bacteria (25, 26, 27). More recently, Guadayol et al. measured
the size-dependent swimming speed of both wild-type and filamentous E. coli
treated with antibiotics in bulk fluids (12). The authors reached a conclusion
that $V$ decreases with $L$ based on their results of filamentous bacteria
with $L$ up to 38 $\mu$m, although their measurements on wild-type bacteria do
show a constant swimming speed independent of $L$. For antibiotic-treated
filamentous bacteria, the body length of bacteria is larger than the length of
flagella ($\sim 6$ to 8 $\mu$m). Flagella distributed randomly along the
entire body surface cannot form a single bundle (28). Hence, the swimming
mechanism of filamentous bacteria is qualitatively different from that of
natural-sized wild-type bacteria (29). In this study, we focus on bacteria of
natural sizes, whose swimming is driven by the rotation of a single bundle.

To solve the puzzle of the size-independent swimming speed, we explore the key
role of multiflagellarity in bacterial swimming next.

### Correlation between flagellar number and bacterial length

Figure 2: Multiflagellarity. The number of flagella $N_{f}$ versus the body
length $L$ of E. coli. Gray disks are individual measurements, whereas black
disks and purple squares are the binned average values. The dashed lines are
the linear regressions of the binned data.

We measure the number of flagella, $N_{f}$, for bacteria of different $L$
using transmission electron microscopy (TEM) (Materials and Methods), which
shows a positive linear correlation (Fig. 2),

$N_{f}=1.34L+1.79.$ (1)

Here, $L$ is measured in units of microns. This positive correlation provides
the first clue towards the origin of size-independent swimming speed.

The linear relation of Eq. 1 is consistent with the previous finding that
flagella are randomly distributed along the length of the bacterial body (28).
A similar linear relation has also been observed by Mears (Fig. 2) (30).
However, a recent study found that bacteria under different growth conditions
show varying average body size but the same average number of flagella (31).
This discrepancy suggests that the finding of the study may not be applicable
to the cell-to-cell variation in a single population under the same growth
condition studied here in our experiments.

### Collective flagellar dynamics

Deciphering the origin of the size-independent swimming speed requires a
quantitative understanding of the collective dynamics of multiple flagella in
a bundle. Chattopadhyay and co-workers measured the angular speed of the
flagellar bundle $\omega_{f}$ as a function of the body length of E. coli
using an optical trap (32), where they found $\omega_{f}$ increases with $L$
(Fig. 3A). Such an increasing trend needs to be combined with the linear
correlation between $N_{f}$ and $L$ to form a coherent picture of the size-
speed relation of flagellated bacteria.

The motion of individual flagella in a bundle must be coordinated to ensure
the smooth rotation of the bundle as a whole. In addition to the geometric
constraints imposed by the helical shape (33), the inter-flagellar
interactions at the short distance of tens of nanometers in a bundle involve
complex hydrodynamic, elasto-hydrodynamic and steric interactions (34, 35, 21,
36, 37), which cannot be solved analytically. To overcome the difficulty, we
begin by examining two limits that allow for exact solutions: Model A
considers independently rotating non-interacting flagella, whereas Model B
deals with a single rotating bundle without relative inter-flagellar motions.
The two models reveal two opposing factors respectively, the competition of
which determines the collective dynamics of multiple flagella in a rotating
bundle. To further understand such a competition, we conduct two different
types of low-Reynolds-number hydrodynamic simulations, one following each of
the two analytical models. Starting from the opposite limits, the two
simulations converge and quantitatively match the experimental findings.

Figure 3: Collective dynamics of multiple flagella in a bundle. (A) Angular
speed of flagella $\omega_{f}$ as a function of body length $L$ of bacteria.
$\omega_{f}$ is normalized by $\omega_{f}$ at $L=2$ $\mu$m, $\omega_{f}^{0}$.
Empty symbols are for simulations, and solid disks for experiments extracted
from Ref. (32). Model A: an extended Purcell model with independently rotating
flagella. Model B: the classic Purcell model with a single rotating bundle.
(B) Swimming speed $V$ as a function of $L$. $V$ from the models is normalized
by $V$ at $L=2$ $\mu$m, $V_{0}$, whereas experimental $V$ extracted from Fig.
1B is normalized by the average $V$ over all $L$. The lines and symbols are
the same as those used in (A). The solid line is the prediction of Model B
using the experimental $\omega_{f}$ as input. The shaded region indicates the
uncertainty due to the experimental errors of $\omega_{f}$. (C) Arrangement of
flagella in the SBT simulations and mechanism for hydrodynamic enhancement of
rotation. Top: A schematic of the flagellar arrangement (not to scale). The
axes of the helical flagella are normal to the plane ($x$-$y$) and located
evenly around a circle of radius $R_{b}$ with an inter-flagellar distance
$d=2R_{b}\sin(\pi/N_{f})$. The number of flagella $N_{f}=4$ in the schematic.
The simulation results in (A) and (B) have $R_{b}=0.5$ $\mu$m and $N_{f}(L)$
from Eq. 1. Bottom: Two fictitious flagella (white dashed circles) are
hydrodynamically entrained into a higher rotation speed due to the flows
induced by two actual flagella (black dashed circles). The background color
and the streamlines indicate, respectively, the magnitude and direction of the
flow velocity $u$ induced by the actual flagella in the $x$-$y$ plane. Note
that the thick arrows, which represent the in-plane velocity of the flagella
along circular trajectories (dashed circles), $u_{f}=\omega_{f}R$, form acute
angles with the direction of local flow (blue streamlines). At fixed motor
torque, this results in the hydrodynamic enhancement of rotation for the
fictitious flagella. Here, $R$ is the helical amplitude of flagellar
filaments.

Model A: We first consider $N_{f}$ non-interacting flagella that rotate
independently (SI Sec. 1A). We extend the Purcell model of flagellated
bacteria and use resistive force theory (RFT) to calculate the drag
coefficients of flagellar filaments (38). In Model A, bacterial swimming speed
increases weakly with $L$, exhibiting an approximately size-independent
swimming speed (Fig. 3B). However, the angular speed of flagella $\omega_{f}$
is also independent of $L$, contrary to the increasing trend of
$\omega_{f}(L)$ observed in experiments (Fig. 3A) (32).

The increase of $\omega_{f}(L)$ is caused by the fluid-mediated inter-
flagellar interactions ignored in Model A, an effect we illustrate via
simulations of interacting flagella at different inter-flagellar distances (SI
Sec. 2A) (39). Specifically, we prescribe the arrangement of flagella by
placing $N_{f}$ rigid rotating helical filaments evenly around a circle of
control radius $R_{b}$ with the inter-flagellar distance
$d=2R_{b}\sin(\pi/N_{f})$ (Fig. 3C). The flagellar assembly and the body are
coupled through the force and torque balance, whereas the drag coefficients of
the flagellar assembly are calculated numerically using slender-body theory
(SBT). Figure 3C shows the flow field induced by two rotating flagella in the
plane ($x$-$y$) normal to the axes of the flagellar helices ($z$). The
trajectories of the rotating flagella within the plane are circular as
indicated by the black dashed circles. If two more flagella were added to the
assembly (the white dashed circles in Fig. 3C), they would experience less
viscous resistance from the already entrained fluid, because the projection of
the velocity of the original flow to the moving direction of the new flagella
is positive at the positions of the new flagella. Thus, coupled by
hydrodynamic interactions, the flagella rotate faster together than they would
on their own assuming a constant motor torque.

Quantitatively, we confirm that $\omega_{f}$ increases with $N_{f}$, which in
turn increases with $L$ due to the linear correlation between $N_{f}$ and $L$
(Fig. 3A). Such an increasing trend becomes more obvious as the hydrodynamic
coupling grows stronger with reducing $d$ (Fig. S3A). Our SBT simulations thus
demonstrate that the long-range hydrodynamic coupling between flagella
enhances flagellar rotation. As we ignore the torque density along the
filament centerline that would contribute a higher-order correction to the
flow, the simulations cannot capture the effect of viscous dissipation induced
by the relative rotation of flagella at short distances (38, 34). Hence, at
small $d$ comparable to the inter-flagellar distance of real bacterial
bundles, the SBT simulations overpredict the trend of $\omega_{f}(L)$ (Fig.
S3A).

Model B: To describe flagellar dynamics at small $d$, we apply the classic
Purcell model in the second limit, where $N_{f}$ flagella combine into a tight
bundle without relative motions (SI Sec. 1B). In Model B, $\omega_{f}$
increases substantially with $N_{f}$ (Fig. 3A), which sets the upper bound on
the flagellar angular speed of the SBT simulations as $d\to 0$ (Fig. S3A). The
strong increase of $\omega_{f}$ also leads to a strong increase of $V$ with
$L$ (Fig. 3B). Interestingly, if we insert the experimental $\omega_{f}(L)$ as
an input into the interrelations $V(\omega_{f})$ and $\omega_{b}(\omega_{f})$
predicted by Model B (Eqs. 14 and 15 in SI), we recover not only the size-
independent swimming speed (Fig. 3B) but also the dependence of the angular
speed of bacterial body on the body length, $\omega_{b}(L)$ (Fig. S2).

To account for the short-range viscous and steric interactions induced by the
relative motions of flagella in a bundle (33), we conduct a second set of
simulations in the framework of the immersed boundary (IB) method to examine
the formation of a flagellar bundle from $N_{f}$ initially-separated elastic
helical filaments (SI Sec. 2B, Supplementary Videos 1 and 2) (40). In our IB
simulations, each of the filaments is connected normal to the surface of a
spherocylindrical cell body via an elastic hook (Fig. S4). At each $L$, we
match the initial angular speed of separated filaments with $\omega_{f}(L)$
calculated from Model B by adjusting the flagellar motor torque (40). This
initial condition imitates the rotation of filaments in an ideal bundle
without short-range inter-flagellar interactions. We then obtain
$\omega_{f}(L)$ after the formation of the stable flagellar bundle when the
short-range interactions fully set in. Contrary to the rotation-enhancing
effect of the long-range hydrodynamic coupling, the short-range viscous and
steric interactions reduce $\omega_{f}(L)$ relative to the prediction of Model
B, leading to a balanced result that quantitatively agrees with experimental
measurements (Fig. 3A) (32).

Finally, both the SBT simulations at a chosen $R_{b}$ and the IB simulations
show an approximately size-independent swimming speed, agreeing with our
experiments (Fig. 3B).

## Discussion

### Motor performance

Figure 4: Flagellar motor performance. A schematic of the flagellar motor
performance curve connecting the motor torque $\tau_{m}$ and speed
$\omega_{m}$. $\tau_{m}$ reaches an approximate plateau $\tau_{m}^{0}$ at low
$\omega_{m}$. The knee speed $\omega_{knee}$ and the maximum speed
$\omega_{max}$ are indicated. Intersection of the performance curve with the
load line $\tau_{m}=S\omega_{m}$ determines the motor operating speed. A
reference load line at bacterial body length $L=2$ $\mu$m is indicated by the
brown solid line. In comparison, three load lines at $L=5$ $\mu$m are shown,
which represent the experiments and the two model calculations in Fig. 3
respectively following the same line styles. Inset: the slope of the load line
$S$ as a function of $L$ from the experiments and the two models. $S$ is
normalized by the slope at $L=2$ $\mu$m, $S_{0}$. The horizontal dotted line
indicates a size-independent load $S/S_{0}=1$.

The complex flagellar dynamics discussed above can be physically understood
from the motor performance curve characterizing the torque-speed relationship
of individual flagellar motors (Fig. 4) (8, 10). The torque output of a
flagellar motor $\tau_{m}$ is approximately constant with increasing motor
angular speed $\omega_{m}=|\omega_{f}–\omega_{b}|$ up to a knee speed
$\omega_{knee}\approx 1300$ rad/s at room temperature, beyond which $\tau_{m}$
decreases linearly and reaches zero at $\omega_{max}$. Flagellar motors
operate below $\omega_{knee}$ with a constant torque $\tau_{m}^{0}$ for
swimming bacteria in our study. The drag torque on the body, together with the
torque balance between the body and the bundle, specifies a linear load line
with its slope depending on $N_{f}$, the fluid viscosity and the geometry of
the body and the bundle (SI Sec. 1) (41). The intercept between the load line
and the motor performance curve determines the operating $\omega_{m}$ of
flagella. Note that, since the rotational drag coefficient of the body is
substantially larger than that of the flagella, $\omega_{b}\ll\omega_{f}$.
Thus, $\omega_{f}$ follows closely the trend of $\omega_{m}$.

The different scenarios of flagellar configurations in the Results section can
now be reconsidered in terms of motor performance. First, for fixed $N_{f}$,
the total load on the $N_{f}$ motors would increase with the body length due
to the higher drag on the body, which would raise the slope of the load line
on each motor and thus slow down $\omega_{m}$. However, with increasing
$N_{f}(L)$ (Eq. 1), the increased load is carried by more flagella. If all the
flagella rotated independently without forming a bundle (Model A), the linear
increase of $N_{f}(L)$ would approximately cancel the proportionally increased
load of the body, resulting in an incorrect prediction of nearly constant load
on each motor and therefore constant $\omega_{m}$ and $\omega_{f}$ (the red
dashed-dotted line in Fig. 4, see Eq. 8 in SI for the calculation). In
contrast, flagellar motors in a tight bundle (Model B) collectively share the
increased load of the body (Eq. 16 in SI), thereby reducing the load on each
motor. As a result, $\omega_{m}$ and $\omega_{f}$ increase with $L$. Since the
viscous dissipation between flagella at short distances further raises the
load carried by each flagellum, the load line of a tight bundle without short-
range inter-flagellar interactions (Model B, the blue dashed line in Fig. 4)
overtakes the experimental load line with such interactions (the black line in
Fig. 4). Nevertheless, the hallmark of the collective load sharing between
multiple flagella—the increasing trend of $\omega_{f}(L)$—is well preserved
experimentally (Fig. 3A). Thus, by bringing the flagella closely together in
an interacting bundle, bacteria effectively regulate the load shared by each
flagellum in the bundle, which leads to faster flagellar rotation
counterbalancing the higher rotational fluid drag on larger cell bodies and
sustaining a constant swimming speed.

### Other possible mechanisms

We rule out two alternative mechanisms as the possible explanation of the
size-independent swimming speed. First, the decrease of wobbling angle of
bacteria results in the increase of their swimming speed (42). Thus, a
negative correlation between the wobbling angle and the body length of
bacteria could give rise to the size-independent swimming speed of bacteria.
We measure the wobbling angle of bacteria of different lengths (Fig. S5) and
show that the decrease of wobbling angle is not sufficient to explain the
size-independent swimming speed (SI Sec. 3A).

Second, it has been suggested previously that the change of the morphology of
flagellar filaments with the length of bacteria may be the origin of the size-
independent swimming speed (13, 43). We examine the morphology of flagellar
filaments via experiments using fluorescently-labeled bacteria (Materials and
Methods) and simulations (Fig. S6). Both experimental and numerical results
show that the change of the morphology is too small to explain the size-
independent swimming speed (SI Sec. 3B).

Figure 5: Propulsion efficiency. Propulsion efficiency $\epsilon$ versus
bacterial body length $L$. The dashed line is the linear regression of the
data. The error bars represent standard errors. The solid line is the
prediction of the classic Purcell model (Model B) with a constant number of
flagella $N_{f}=4$. A maximum efficiency of $\epsilon=1.87\%$ is obtained when
$L=5.1$ $\mu$m (the star).

### Propulsion efficiency

The efficiency of propulsion of microorganisms has been studied extensively
because of its role in the optimization of the swimming kinematics and
morphology of microorganisms (44, 38, 32, 45, 46). It is defined as the ratio
of the rate of work required to drag a swimmer at given velocity $V$ to the
total mechanical power exerted by the swimmer on the fluid when swimming at
the same speed (44). Specifically, the propulsion efficiency of flagellated
bacteria such as E. coli can be calculated as (32)

$\epsilon=\frac{A_{b}V^{2}}{N_{f}\tau_{m}^{0}\omega_{m}},$ (2)

where $A_{b}$ is the translational drag coefficient of an ellipsoidal cell
body given by Eq. 9 in SI and $V$, $N_{f}$ and $\omega_{m}$ are all taken
directly from experiments. In addition, we adopt the constant motor torque
$\tau_{m}^{0}\approx 400$ pN$\cdot$nm based on our RFT calculation using
$\omega_{f}$ from experiments as input.

Although the swimming speed of bacteria is independent of their size, the
propulsion efficiency decreases linearly with $L$ (Fig. 5). As $A_{b}/N_{f}$,
$V$ and $\tau_{m}^{0}$ are approximately size-independent, $\epsilon(L)\sim
1/\omega_{m}(L)$. Driven by the increase of $\omega_{m}$ with $L$ (Fig. 3A),
$\epsilon$ decreases with increasing $L$. Our experimental finding contradicts
the prediction of previous studies where the correlation between $N_{f}$ and
$L$ was ignored. The original Purcell model with a constant $N_{f}$ shows a
non-monotonic trend of $\epsilon(L)$ (the solid line in Fig. 5). The existence
of a maximum $\epsilon$ has been used to rationalize the size ratio between
bacterial body and flagella (38, 32). The monotonic decrease of the propulsion
efficiency observed in our experiments challenges this explanation. We find
that bacteria of small sizes have the highest propulsion efficiency, which is
close to the maximum efficiency of the Purcell model and approaches the upper
bound for flagellated bacteria at $2-3\%$ (9).

Figure 6: Swimming speed and morphology of monotrichous bacteria. (A) The
swimming speeds $V$ of three species of monotrichous bacteria as a function of
their body length $L$. $V$ is normalized by the swimming speed at the smallest
body length of each species $V_{0}$. $L$ is normalized by the average body
width $W$ (SI Table 2). Inset: Comparison of experimental $V(L)$ with the
prediction of Model B with a single flagellum $N_{f}=1$ (the solid line) (SI
Sec. 1B) and simulations of monotrichous bacteria (empty symbols). $f(L)$ is a
scaling function, considering the specific geometries of the body and the
flagellum of each species (Eq. 17 in SI). The model prediction agrees well
with the SBT (red squares) and IB (blue triangles) simulations. (B) The
probability density functions of the length of bacterial body for monotrichous
bacteria V. natriegens (top) and peritrichous bacteria E. coli (bottom). The
dashed lines are single-peak fittings as a guide of the eye.

### Functional benefit of multiflagellarity

Over billions of years of evolution, peritrichous bacteria have developed a
set of extraordinary physical and biochemical mechanisms, regulating the
complex dynamics of multiple flagella to achieve swimming in bulk fluids (47).
Such a daunting task, however, is redundant for monotrichous bacteria with a
single rotating flagellum, which accomplish swimming while avoiding the high
growth cost for the synthesis of additional flagella. Under strict
evolutionary control, multiflagellarity must confer a selective advantage to
peritrichous bacteria. One intuitive answer would be that, with more flagella
and therefore stronger propulsion, bacteria swim faster. Thus, it came as a
surprise when the seminal experiments by Darnton and co-workers suggested that
additional flagella barely increase the swimming speed of bacteria (20). These
authors then pose an interesting question: “If they do not allow the cell to
swim faster, why does a cell have multiple flagella?”

Many answers to the question have been proposed over the last two decades
including the role of multiflagellarity in increasing tumbling efficiency
(20), enhancing swimming stability (23), improving search and exploring
ability (48), and promoting swarming (49, 50) and biofilm formation (51) on
solid substrates. We provide here an alternative answer based on the prominent
function of flagella in swimming: growing multiple flagella allow bacteria of
different sizes to swim at the same speed. As the size variation is
unavoidable, if bacteria of larger sizes had lower motility and therefore
weaker chemotactic ability (17), then a colony of bacteria migrating
collectively in chemotactic response to an attractant would disintegrate over
time and be sorted out spatially according to bacterial sizes. Such
“phenotypic sorting” decreases the phenotypic heterogeneity of the population,
which in turn reduces the overall fitness of the population (2, 19, 52). The
size-independent swimming speed enabled by multiflagellarity allows a
bacterial colony to migrate collectively while preserving its phenotypic
diversity in bacterial sizes.

### Monotrichous versus peritrichous bacteria

The functional benefit of multiflagellarity discussed above has important
implications on the swimming behavior and the morphology of monotrichous
bacteria. First, the essential role of multiflagellarity in maintaining a
constant swimming speed implies that the swimming speed of monotrichous
bacteria should decrease with body size. To verify the prediction, we measure
the size-speed relation for three species of monotrichous bacteria, Vibrio
alginolyticus, Vibrio natriegens (V. natriegens), and Rhodobacter sphaeroides
(Materials and Methods). In contrast to the constant swimming speed of E.
coli, all monotrichous species show negative correlations between $V$ and $L$
(Fig. 6A), qualitatively agreeing with our model prediction (Fig. 6A inset).

Second, as the decrease of the swimming speed limits the chemotactic ability
of monotrichous bacteria of large sizes (17), the size of monotrichous
bacteria should be more tightly regulated with a narrower distribution than
the size of peritrichous bacteria. Figure 6B compares the probability density
function of the size of V. natriegens and that of E. coli. Indeed, the
normalized size variation of V. natriegens, defined as the full width at half
maximum of the distribution divided by the most probable size of bacteria, is
$28\%$, about half of that of E. coli at $53\%$. Moreover, the probability to
find a V. natriegens bacterium larger than its most probable size is $55\%$.
This probability is significantly higher at $77\%$ for E. coli due to the
highly asymmetric shape of their size distribution. Thus, it is less likely to
find V. natriegens of large sizes compared with E. coli, consistent with our
prediction based on the trend of the swimming speed. Future studies on more
bacterial species in each category are certainly needed to fully verify our
hypothesis on the effect of multiflagellarity on the size variation of
bacteria.

### Bacterial culturing and optical microscopy

Wild-type E. coli strain BW25113 was fluorescently tagged by the insertion of
PKK PdnaA-GFP plasmids. A small amount of bacterial stock was inoculated in 2
ml Terrific Broth (TB) [tryptone $1.2\%$ (w/v), yeast extract $2.4\%$ (w/v)
and glycerol $0.4\%$ (w/v)] supplanted with $0.1\%$ (v/v) of 100 mg/l
ampicillin as an antibiotic. This bacterial solution was then incubated in an
orbital shaker at 37 ∘C for 12-16 hours (the late exponential phase). The
bacterial culture was further reinoculated and diluted 1:100 with fresh TB and
grown in the exponential phase for 6.5 hours in an orbital shaker at 30 ∘C. We
harvested motile cells using gentle centrifugation (800$g$, 5 minutes),
discarding the supernatant and resuspending the cells in motility buffer [0.01
M potassium phosphate, 0.067 M NaCl, $10^{-4}$ M EDTA, pH 7.0]. The suspension
was then washed twice and adjusted to the desired concentration by adding
motility buffer. The density of bacteria in motility buffer was measured using
a biophotometer via optical density at 600 nm (OD600) and controlled at
$0.6n_{0}$, where $n_{0}=8\times 10^{8}$ cells/ml is the cell concentration at
OD${}_{600}=1.0$. The low density ensured that bacteria do not show collective
swimming in our study (53, 54). The motile cells were transferred to closed
PDMS microchannels of height 150 $\mu$m, where they maintained robust motility
for 3 hours at room temperature of 22 ∘C. The measurements were performed at
the room temperature within 15 minutes of injecting the cells into the
microchannels to ensure constant motility over time. The imaging of bacterial
motions was conducted 25 to 30 $\mu$m above the coverslip to avoid potential
surface influence using a Nikon Plan APO $60\times$ oil objective (NA = 1.4)
on an inverted confocal microscope (Nikon Ti-Eclipse) with a 488 nm laser.
Images were acquired by an Andor zyla sCMOS camera at 30 frames per second.

To image the conformation of bacterial flagella, we cultured E. coli strains
AW HCB 1, AW HCB 1707 and AW HCB 1732 in Bacto-tryptone broth [10 g Bacto-
tryptone, 5 g NaCl, 1 L distilled H2O] that contains minimal free –NH2 groups
in medium (55). The culturing procedure was the same as that for the wild-type
strain. Harvested bacterial cells were washed free of the growth medium by
centrifugation (800$g$, 5 minutes) thrice using distilled water. The resulting
pellet was resuspended in 500 $\mu$L of distilled water and 25 $\mu$L of 1 M
sodium bicarbonate (NaHCO3) was added to raise its pH and allow the optimal
coupling of the dye to free –NH2 groups on the bacteria. 1 mg of Alexa Fluor
(488 or 532 nm) carboxylic acid succinimidyl esters dye was dissolved in 100
$\mu$L anhydrous DMSO. 25 $\mu$L of this dye was added per 500 $\mu$L of cell
suspension. The cell suspension was then stirred on a gentle gyro shaker for
60 minutes in the dark. After 60 minutes, the excess dye was washed out to
minimize background fluorescence by centrifugation thrice using distilled
water. The flagella were imaged using the same Nikon Plan APO $60\times$ oil
objective on a Nikon Ti-Eclipse inverted microscope equipped with a 100W super
high-pressure mercury lamp. Images obtained were further processed in ImageJ
to enhance contrast for analysis. From the images, we manually measured the
axial length $\Lambda$ and the pitch $p$ of the helical flagellar bundle of
bacteria (Figs. S6B and C).

### Cell tracking and analysis

Image stacks from confocal microscopy were processed using the Python library
skimage. A GFP-expressing bacterial cell body appears as a bright ellipsoid
against a dark background (Fig. 1A inset). The video frames were binarized
using local intensity thresholding following the Niblack algorithm (56), and
the position of the centroid, the orientation, major axis, and minor axis of
the cell bodies of all the bacteria were recorded. A standard particle
tracking algorithm was then used to link the positions of bacteria across
frames, which yielded the full swimming trajectory of individual bacteria.
Over the course of a trajectory, each individual bacterium showed variations
of their projected length appearing in the video frames. The maximum of the
cell length tracked over time was taken as the true length of the bacterium.
To reduce noise in tracking, trajectories less than 0.3 seconds were
discarded. Since the scanning time of confocal microscopy over the size of a
single bacterium was $\sim 0.1$ ms in our experiments, the errors on bacterial
length caused by the motion blur of bacterial swimming was neglectable. The
average swimming speed of a bacterium over its whole trajectory was obtained,
which was then correlated with the length of the bacterium (Fig. 1B). The
depth of field of the imaging plane of our confocal microscope was about 1
$\mu$m. Thus, as the tumbling of a bacterium typically resulted in the
bacterium moving out of the shallow imaging plane, our method dominantly
tracked bacterial motions in their run phase.

To obtain the swimming speed, $V$, we calculated the central finite difference
between the positions of bacteria. To measure bacterial wobbling angle
$\theta$, we first obtained the temporal evolution of the angle difference
between the orientation of bacterial body and the mean direction of swimming
given by the unit tangential vector along the bacterial trajectory,
$\delta(t)$. The amplitude of the oscillation of $\delta(t)$ was then averaged
over the trajectory of the bacterium under consideration, which gave the
wobbling angle of the bacterium $\theta$. The correlation between $\theta$ and
the length of bacteria $L$ was analyzed and shown in terms of the distribution
of wobbling angles at different $L$ in Fig. S5. The procedures for measuring
$V$ and $\theta$ are the same as those used in our previous study (42).

### Transmission electron microscopy (TEM)

Transmission electron microscopy (TEM) was used to image the number of
flagella of E. coli over a sample size of more than 60 bacteria. Samples for
TEM studies were prepared by drop casting bacterial suspensions of
concentration $0.1n_{0}$ onto a lacey carbon film coated copper TEM grid
(Pacific GridTech). Negative charge was added to the film surface in a Pelco
glow discharger prior to sample deposition to improve yield of bacteria on the
imaging area of the film. The grids were air dried for a few minutes prior to
the TEM. For visualization of flagella, high-angle annular dark-field
(HAADF)-scanning (S)TEM images were obtained on an aberration-corrected FEI
Titan 60-300 (S)TEM microscope that is equipped with a CEOS DCOR probe
corrector. The microscope was operated at 200 keV with a beam current of 80
pA. The STEM convergence angle used was 25.5 mrad with HAADF detector inner
and outer collection angles of 50 and 200 mrad, respectively. Flagellar number
was counted manually through the TEM images.

### Monotrichous bacteria

We cultured Vibrio natriegens, a monotrichous marine bacterium, in Lysogeny
Broth (LB) supplemented with v2 salts [204 mM sodium chloride, 4.2 mM
potassium chloride, and 23.14 mM magnesium chloride]. The suspension in the
growth medium was directly observed using the inverted Nikon microscope in the
bright-field mode discussed above. The cell size and swimming speed of the
bacteria were then tracked manually. The swimming speed of Rhodobacter
sphaeroides, a monotrichous fresh-water bacterium, was measured from videos
posted on the website of the Berg lab (57). Prof. Katja Taute measured and
kindly provided us the swimming speed of Vibrio alginolyticus—another
monotrichous marine bacterium—as a function of cell lengths (58).

We thank the late Professor Howard Berg for providing us with E. coli strains
and for fruitful discussion and suggestions. We also thank Katja Taute for
helping with the measurements of monotrichous bacteria, Linda Turner for
guiding us on fluorescent labelling, Xiao-lun Wu for insightful comments on
the trend of $\omega_{f}$ and Xinliang Xu for the suggestion on the drag
coefficients of bacterial body. This research was supported by the U.S.
National Science Foundation (NSF) (CBET-2028652) and by the IPRIME program,
University of Minnesota. S.K. acknowledges partial funding support from the
PPG foundation. M.T.-C. was supported by Clare College, Cambridge (Lynden-Bell
Research Fellowship). S.L. was supported by the U.S. NSF (DMS-1853591) and the
Charles Phelps Taft Research Center, University of Cincinnati. W.L. was
supported by the National Institute for Mathematical Sciences Grant of the
Korean government (B22920000). Y.K. was supported by National Research
Foundation of Korea Grant (2020R1F1A1A01074981). Part of this work was done at
the University Imaging Centers (UIC) and Characterization Facility of
University of Minnesota, which receives partial support from the U.S. NSF
through the MRSEC (DMR-2011401) and the NNCI (ECCS-2025124) programs.

[6]

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#  Consensus-Based Leader-Follower Formation Tracking for Control-Affine
Nonlinear Multiagent Systems*††thanks: *This work received support in part
from the Office of Naval Research (Grant No. N000141712622) and from a joint
award from Microsoft Corporation, 1 Microsoft Way, Redmond, WA 98052, USA and
the Maryland Robotics Center, 3156 Brendan Iribe Center, College Park, MD
20742, USA. ††thanks: 1C. Enwerem and J.S. Baras are with the Department of
Electrical & Computer Engineering and the Institute for Systems Research,
University of Maryland, College Park, MD 20742, USA. The work of C. Enwerem
was supported by the Microsoft Diversity in Robotics and Autonomy PhD
Fellowship. Emails: {enwerem<EMAIL_ADDRESS>
Clinton Enwerem1 John S. Baras1

###### Abstract

In the typical multiagent formation tracking problem centered on consensus,
the prevailing assumption in the literature is that the agents’ nonlinear
models can be approximated by integrator systems, by their feedback-linearized
equivalents, or by dynamics composed of deterministic linear and nonlinear
terms. The resulting approaches associated with such assumptions, however, are
hardly applicable to general nonlinear systems. To this end, we present
consensus-based control laws for multiagent formation tracking in finite-
dimensional state space, with the agents represented by a more general class
of dynamics: control-affine nonlinear systems. The agents also exchange
information via a leader-follower communication topology modeled as an
undirected and connected graph with a single leader node. By leveraging
standard tools from algebraic graph theory and Lyapunov analysis, we first
derive a locally asymptotically stabilizing formation tracking law. Next, to
demonstrate the effectiveness of our approach, we present results from
numerical simulations of an example in robotics. These results — together with
a comparison of the formation errors obtained with our approach and those
realized via an optimization-based method — further validate our theoretical
propositions.

###### Index Terms:

multiagent systems, consensus, formation control, control-affine nonlinear
systems

##  I Introduction

Multiagent formation tracking — the task of controlling a group of dynamic
units (or agents) to maintain a desired formation while following a reference
trajectory — has risen to become evidently one of the most popular topics in
cooperative control and related fields, owing to its numerous practical
applications. The formation tracking problem is distinct from the nominal
formation control problem (where the focus instead is only on the agents
converging to a desired formation) and has traditionally been tackled using
ideas from fields such as learning theory, graph theory, optimization, and
control theory, to name a handful. However, studies featuring methods from the
aforementioned areas typically consider the agents’ dynamic models as linear
systems — a common simplification from their general and more complex
nonlinear representations. Within the consensus control domain particularly,
several research articles [1, 2] have studied the formation tracking problem,
following the seminal work of [3]. These studies have been mostly adhoc,
however, extending the aforementioned ideas to only time-invariant linear
systems or systems with dynamics comprising deterministic linear and nonlinear
terms. Unfortunately, the control schemes designed under these approximations
are largely not generalizable and cannot be extended to the nonlinear case as
a result. This significant research gap thus motivates the need for results
applicable to a more general class of systems.

###  I-A Prior Work

Results demonstrating the application of consensus to formation control for
nonlinear systems began to appear fairly recently, starting in [4], which
featured a consensus-based event-triggered control scheme, with the agents
represented by nonlinear systems with a scalar control coefficient. Following
that, in a closely-related work [5], the authors laid down consensus-based
formation control laws for a class of nonlinear multiagent systems — with
models similar to those studied in [4]. More recently, consensus-based
formation control laws were given in [6], but for nonlinear systems with
linear-in-state drift terms. In our work, we derive original locally
asymptotically stable consensus-based formation tracking laws for a more
general class of nonlinear systems — affine-in-control systems with a state-
dependent drift term. Following [7, 8], we present general consensus control
rules that guarantee asymptotic decay of the formation error of the multiagent
system (MAS). In contrast to the aforementioned studies, we apply the
agreement protocol to the problem of formation tracking, where the agents are
modeled by general control-affine nonlinear systems.

###  I-B Main Contributions

We contribute the following to the state of the art in consensus-based
formation tracking: (i) an original consensus-based formation tracking control
scheme — for control-affine nonlinear systems — that guarantees asymptotic
convergence of each agent’s consensus error to the corresponding relative
state, thus preserving the formation (Section III), and (ii) numerical
formation tracking simulations that juxtapose the performance of our approach
with that obtained by an optimization-based method (Section V).

##  II Notation & Mathematical Preliminaries

Throughout, we denote vectors and matrices with boldface font (e.g., $\bm{x}$
and $\bm{A}$), with corresponding elements in regular italicized font, i.e.,
$x_{i}$ denotes the $i^{\text{th}}$ element of the vector $\bm{x}$, while
$A_{ij}$ is the element occupying the $i^{\text{th}}$ row and $j^{\text{th}}$
column of $\bm{A}$. $\bm{A}^{T}$ denotes the transpose of matrix $\bm{A}$,
$\times$ (in the context of three-dimensional vectors) represents the cross
product, and $\otimes$ is the standard Kronecker product operator. Unless
otherwise noted, $\bm{I}_{n}$ is the $n\times n$ identity matrix,
$\bm{\kappa}_{n}$ is the vector
$\begin{bmatrix}\kappa&\kappa&\dots&\kappa\end{bmatrix}^{T}\in\mathbb{R}^{n}$,
with $\kappa\in\mathbb{R}$, and
$\lvert\lvert\bm{x}\rvert\rvert=(\sum_{i=1}^{n}(x_{i})^{2})^{\frac{1}{2}}$ is
the vector Euclidean norm. $\text{\text{diag}}([\star_{i}]_{i\in{I}})$ is the
$|{I}|\times|{I}|$ diagonal matrix, with ${I}$ an index set. The
$i^{\text{th}}$ block, $\star_{i}$, can either be a scalar, vector, or matrix,
and will be clear from the context. $\mathbb{R}_{+}$ is the set
$\\{\alpha\in\mathbb{R}\ \lvert\ \alpha>0\\}$. To set the stage for the
theoretical framework and problem formulation to follow, we now introduce
several key lemmas, definitions, and theorems, to be invoked in the proofs to
come.

######  Definition 1 (Control-Affine Nonlinear System)

A control-affine nonlinear system is the system (here subscripted by $i$ to
represent the $i^{\text{th}}$ agent for notational cohesion):

$\displaystyle\dot{\bm{x}}_{i}$
$\displaystyle=f(\bm{x}_{i})+g(\bm{x}_{i})\bm{u}_{i}$ (1a)
$\displaystyle\bm{y}_{i}$ $\displaystyle=h(\bm{x}_{i}),$ (1b)

where $\bm{x}_{i}\in\mathcal{X}\subset\mathbb{R}^{n}$,
$\bm{u}_{i}\in\mathcal{U}\subseteq\mathbb{R}^{m}$, and
$\bm{y}_{i}\in\mathbb{R}^{q}$ are, respectively, the state, control, and
output vectors, with $m$ not necessarily equal to $q$. $f(\cdot)$, $g(\cdot)$,
and $h(\cdot)$ are, respectively, the drift and control vector fields, and the
output map — with appropriate dimensions — which are assumed to be
sufficiently-smooth functions ($\mathcal{C}^{n};n\geq 1$), with
$f(\bm{0}_{n})=\bm{0}_{n}$, so that $\bm{x}_{i}=\bm{0}_{n}$ is an open-loop
equilibrium point of (1), and $h(\bm{0}_{n})=\bm{0}_{q}$. $\mathcal{X}$ is a
compact set containing $\bm{0}_{n}$, while admissible control signals take
values in the set, $\mathcal{U}$, of piecewise continuous and absolutely-
integrable functions, i.e., the set $\\{\bm{u}_{i}(t)\ |\
\int_{0}^{t}{|\bm{u}_{i}(\tau)|d\tau}<\infty\\}$.

######  Definition 2 (Graph Theory)

An undirected, finite, graph (hereafter graph) $\mathcal{G}$ is the tuple
$(\mathcal{V},\mathcal{E})$, with $\mathcal{V}$ equal to the non-empty set
$\\{v_{1},v_{2},\dots,v_{k}\\}$ of $k$ distinct elements, called nodes or
vertices and $\mathcal{E}$ (the edge set) $\subseteq$
$\mathcal{V}\times\mathcal{V}=\\{\\{v_{i},v_{j}\\}\ \lvert\
v_{i},v_{j}\in\mathcal{V}\\}$. The graph is weighted if there exists a map
$w:\mathcal{E}\mapsto\mathbb{R}$ that associates to each edge, a real number,
denoted as $w_{ij}$. It is unweighted otherwise. A path is a sequence of
distinct vertices $v_{1},v_{2},\dots,v_{m}$, where every consecutive pair of
vertices (i.e., $v_{i}$ and $v_{i+1}$; $i=1,\dots,m$) is joined by an edge in
$\mathcal{E}$. A graph is connected if every node in the graph is connected to
every other node by a path; the graph is disconnected otherwise. The
neighborhood set, $N_{i}$, is the set $\\{v_{j}\in\mathcal{V}\ \lvert\
\\{v_{i},v_{j}\\}\in\mathcal{E},\ j\neq i\\}$. The graph adjacency and
Laplacian matrices (denoted, respectively, as $\bm{\mathcal{A}}(\mathcal{G})$
and $\bm{\mathcal{L}}(\mathcal{G})$) associated with $\mathcal{G}$, are the
matrices:

$\displaystyle\bm{\bm{\mathcal{A}}}(\mathcal{G})$
$\displaystyle=[\mathcal{A}_{ij}],\ \text{where}\
\mathcal{A}_{ij}=\begin{dcases}1,\ \text{if}\
\\{v_{i},v_{j}\\}\in\mathcal{E}\\\ 0,\ \text{otherwise}\end{dcases}$ (2a)
$\displaystyle\bm{\mathcal{L}}(\mathcal{G})$
$\displaystyle=[\mathcal{L}_{ij}],\ \text{where}\
\mathcal{L}_{ij}=\begin{dcases}\sum_{j\in N_{i}}\mathcal{A}_{ij},\ \text{if}\
i=j\\\ -\mathcal{A}_{ij},\ \text{otherwise}.\end{dcases}$ (2b)

Hereafter, for brevity, we shall drop the $\mathcal{G}$ argument in the
notations for the adjacency matrix and graph Laplacian.

######  Definition 3 (Leader-Follower Formation Tracking)

Consider an interconnected system of $N+1$ identical agents — each with
dynamics described by (1a) — on the (connected) line graph depicted in Figure
1. For convenience, let the MAS comprise a single leader (with index $L$) and
$N$ followers. The formation tracking problem is to synthesize controls,
$\bm{u}_{i}$, under which the follower agents converge to states that respect
the inter-agent distance constraints imposed by a specified formation rule, as
the leader agent independently tracks a known reference trajectory,
$\bm{x}_{L}^{r}$. Formally, taking $\bm{\xi}_{i}\in\mathbb{R}^{n}$ to be the
desired (goal) state of the $i^{\text{th}}$ follower agent ($i=1,2,\dots,N$),
given by the formation specification, and $\bm{x}_{i}$ to be the
$i^{\text{th}}$ agent’s actual state, the formation tracking problem is to
find a $\bm{u}_{i}$ that drives the $i^{\text{th}}$ follower agent so that

$||\bm{x}_{j}(t)-\bm{x}_{i}(t)||=\lvert\lvert\bm{d}_{ij}\rvert\rvert=\delta_{ij},\
\forall\ j\ \in N_{i},\ \forall\ t\geq 0,$ (3)

and a $\bm{u}_{L}$ that drives the leader such that

$\lim_{t\rightarrow\infty}{\lvert\lvert\bm{x}_{L}(t)-\bm{x}_{L}^{r}(t)\rvert\rvert=0},$
(4)

where $\bm{d}_{ij}=(\bm{\xi}_{j}-\bm{\xi}_{i})\in\mathbb{R}^{n}$ is a constant
vector, with norm ${\delta}_{ij}\in\mathbb{R}_{+}$.

Recent studies [9] have argued for the representation of the interaction in
networked MAS using a three-layer multi-graph model as opposed to the
ubiquitous single-layer model that often portrays only communication or
information exchange. For completeness, therefore, we note here that Figure 1
depicts the connection of the agents on the communication layer; the
collaboration and information-sharing layers are taken to be subsumed in the
network.

$a_{L}$$a_{f_{1}}$$a_{f_{2}}$$a_{f_{N}}$

Figure 1: An unweighted line graph on $N+1$ vertices illustrating the leader-
follower network structure under consideration. For clarity, the nodes have
been renamed with the agent designations. Follower 1 ($a_{f_{1}}$), while
alike to the other followers, is the only agent with the leader in its
neighborhood set. Clearly, the network structure is a tree and not a complete
graph.

######  Theorem 1 (Mesbahi and Egerstedt [10])

For a connected goal formation, encoded by the graph,
$\mathcal{G}_{f}=(\mathcal{V},\mathcal{E}_{f})$, and a goal location set
$\mathcal{X}_{f}=\\{\bm{\xi}_{1},\bm{\xi}_{2},\dots,\bm{\xi}_{N}\\}$, where
$\bm{\xi}_{i}$ is the goal location of the $i^{\text{th}}$ agent, the
following formation control scheme

$\dot{\bm{x}}_{i}(t)=-\sum_{j\in
N_{i}}(\bm{x}_{i}(t)-\bm{x}_{j}(t))-(\bm{\xi}_{i}-\bm{\xi}_{j})$ (5)

will drive the MAS so that the agents converge to a constant displacement of
the target positions, $\bm{\tau}$, i.e., for all agents,
$\bm{x}_{i}(t)-\bm{\xi}_{i}\rightarrow\bm{\tau}\text{ as }t\rightarrow\infty$.

###### Assumption 1

The system in (1) is autonomous, hence associated notions of Lyapunov
stability apply.

###### Assumption 2

The network structure of the MAS is encoded by an unweighted, connected, and
static graph, i.e., neither $\mathcal{V}$ nor $\mathcal{E}$ are time-
dependent.

###### Assumption 3

The agents are taken to be represented by models with exact state information,
so we do not consider the influence of noise or exogenous disturbances.

###### Assumption 4

The leader is driven by an independent control ($\bm{u}_{\text{track}}$), so
that it asymptotically tracks its reference, $\bm{x}_{L}^{r}(t)\ \forall\
t\in[0,T]$, with $T$ finite.

###### Assumption 5

There exists a positive definite matrix $\bm{P}\in\mathbb{R}^{n\times n}$ such
that the following inequality holds:

$\displaystyle(\bm{x}_{i}-\bm{x}_{j})^{T}\bm{P}(f(\bm{x}_{i})-f(\bm{x}_{j}))\leq
0$ $\displaystyle\ \forall\ i,j=1,2,\dots,N\ \text{and}\ t\geq 0,$ (6)

which places bounds on the relative inter-agent states of the unforced
nonlinear dynamics for the multiagent system.

######  Assumption 6 ([7], Remark 1)

There exists a positive definite $N\times N$ matrix $\bm{M}$ such that:

$\bm{M}\bm{\mathcal{L}}=\bm{I}_{N},$ (7)

which translates to an existence of the graph Laplacian’s inverse and hence,
the connectivity of the associated graph (see Lemma 1).

##  III Main Result: Consensus-based Formation Maintenance Algorithm

Suppose Assumption 4 holds. We define the $i^{\text{th}}$ consensus error as:

$\bm{e}_{i}=\sum_{j\in N_{i}}{\mathcal{A}_{ij}(\bm{x}_{i}-\bm{x}_{j})}.$ (8)

Then, from Theorem 1, the inter-agent distance constraints (together with the
tracking requirement) can be achieved by respectively selecting the following
control law for the $i^{\text{th}}$ follower and leader:

$\displaystyle\bm{u}_{i}(t)=-g^{T}(\bm{x}_{i}(t))\bm{P}(\bm{e}_{i}(t)+\bm{d}_{i}),\
i=1,2,\dots,N,$ (9)
$\displaystyle\bm{u}_{L}(t)=-g^{T}(\bm{x}_{L}(t))\bm{P}(\bm{e}_{L}(t)+\bm{d}_{L})+\bm{u}_{\text{track}},$
(10)

where $\bm{d}_{i}=\sum_{j\in N_{i}}\bm{d}_{ij}$, and $\bm{P}$ satisfies (5).

###### Remark 1

We will show that, with (9), the follower agents reach consensus
(asymptotically) on the inter-agent distances specified by the formation rule.
That is, we can think of the consensus error as converging to the prescribed
agent-to-agent distances (as opposed to zero in the nominal consensus case),
which implies asymptotic decay of the formation error. Also notice that, since
$N_{L}=\\{a_{f_{1}}\\}$, the leader is driven by a control signal (10) that
balances the tracking and formation maintenance requirements.

To prove the validity of this result, we first define the following ensemble
notation (for the follower-only network):

$\bm{x}=\begin{bmatrix}\bm{x}_{1}^{T}&\bm{x}_{2}^{T}&\dots&\bm{x}_{N}^{T}\end{bmatrix}^{T}\in\mathbb{R}^{{N}n},$
(11)
$\bm{e}=\begin{bmatrix}\bm{e}_{1}^{T}&\bm{e}_{2}^{T}&\dots&\bm{e}_{N}^{T}\end{bmatrix}^{T}\in\mathbb{R}^{{N}n},$
(12)
$\bm{u}=\begin{bmatrix}\bm{u}_{1}^{T}&\bm{u}_{2}^{T}&\dots&\bm{u}_{N}^{T}\end{bmatrix}^{T}\in\mathbb{R}^{{N}m},$
(13)
$\bm{G}(\bm{x})=\text{\text{diag}}\Big{(}\begin{bmatrix}{g(\bm{x}_{1})}&{g(\bm{x}_{2})}&\dots&{g(\bm{x}_{N})}\end{bmatrix}\Big{)}\in\mathbb{R}^{{N}n\times{N}m},$
(14)
$\bm{F}(\bm{x})=\begin{bmatrix}f^{T}(\bm{x}_{1})&f^{T}(\bm{x}_{2})&\dots&f^{T}(\bm{x}_{N})\end{bmatrix}^{T}\in\mathbb{R}^{{N}n},$
(15)

and

$\bm{d}=\begin{bmatrix}\bm{d}_{1}^{T}&\bm{d}_{2}^{T}&\dots&\bm{d}_{{N}}^{T}\end{bmatrix}^{T}\in\mathbb{R}^{{N}n}.$
(16)

We can then write (9) as:

$\bm{u}(t)=-\bm{G}^{T}(\bm{x}(t))(\bm{I}_{N}\otimes\bm{P})(\bm{e}(t)+\bm{d}),$
(17)

with the equation

$\dot{\bm{x}}=\bm{F}(\bm{x})+\bm{G}(\bm{x})\bm{u}$ (18)

representing the ensemble dynamics of the MAS in block notation. With this
block formulation, it is easy to see that $\bm{e}$ in (12) satisfies the
following dynamical equation:

$\dot{\bm{e}}=(\bm{\mathcal{L}}\otimes\bm{I}_{n})\dot{\bm{x}}=(\bm{\mathcal{L}}\otimes\bm{I}_{n})(\bm{F}(\bm{x})+\bm{G}(\bm{x})\bm{u}).$
(19)

Next, we introduce key lemmas (mostly adapted from [7]) which will be useful
for proving the results to follow.

######  Lemma 1 ([10])

For a connected leader-follower graph on $N+1$ vertices with one leader and
$N$ followers, if $\bm{\mathcal{L}}\in\mathbb{R}^{N\times N}$ is the Laplacian
corresponding to the follower network, then $\bm{\mathcal{L}}$ is real,
symmetric, and positive definite, with its eigenvalues related by:
$0<\lambda_{1}\leq\lambda_{2}\leq\dots\leq\lambda_{N}$.

######  Lemma 2 (Schur Complement Lemma [11])

For any real and symmetric matrix
$\bm{K}=\begin{bmatrix}\bm{K}_{11}&\bm{K}_{12}\\\
\bm{K}_{21}&\bm{K}_{22}\end{bmatrix}$, with $\bm{K}_{21}=\bm{K}^{T}_{12}$, the
following statements are equivalent: (i) $\bm{K}<0$; (ii) $\bm{K}_{11}<0$,
$\bm{K}^{T}_{22}-\bm{K}^{T}_{12}\bm{K}_{11}^{-1}\bm{K}_{12}<0$; (iii)
$\bm{K}_{22}<0$,
$\bm{K}^{T}_{11}-\bm{K}_{12}\bm{K}_{22}^{-1}\bm{K}^{T}_{12}<0$.

######  Lemma 3 ([7])

For a connected graph $\mathcal{G}$ with Laplacian $\bm{\mathcal{L}}$ and
adjacency matrix $\bm{\mathcal{A}}$ = [$\mathcal{A}_{ij}$], for any
$\bm{h}=[{\bm{h}}_{1}^{T}{\bm{h}}_{2}^{T}\dots\bm{h}_{N}^{T}]^{T}$ and
$\bm{k}=[{\bm{k}}_{1}^{T}{\bm{k}}_{2}^{T}\dots\bm{k}_{N}^{T}]^{T}$ in
$\mathbb{R}^{Nn}$,

$2\bm{h}^{T}(\bm{\mathcal{L}}\otimes\bm{I}_{n})\bm{k}=\sum_{i=1}^{N}\sum_{j=1}^{N}\mathcal{A}_{ij}(\bm{h}_{i}-\bm{h}_{j})^{T}(\bm{k}_{i}-\bm{k}_{j}).$
(20)

###### Lemma 4

If Assumptions 2 and 5 hold, then:

$(\bm{e}+\bm{d})^{T}(\bm{I}_{N}\otimes\bm{P})\bm{F(\bm{x})}\leq 0.$ (21)

###### Proof:

(Motivated by [7]) From (15), we can write

$\displaystyle(\bm{I}_{N}\otimes\bm{P})\bm{F(\bm{x})}$
$\displaystyle=\begin{bmatrix}f^{T}(\bm{x}_{1})\bm{P}&f^{T}(\bm{x}_{2})\bm{P}&\dots&f^{T}(\bm{x}_{N})\bm{P}\end{bmatrix}^{T}$
(22) $\displaystyle\triangleq\bm{F}^{\bm{P}}(\bm{x}).$ (23)

From the left-hand side of (21), it follows that:

$\displaystyle(\bm{e}+\bm{d})^{T}(\bm{I}_{N}\otimes\bm{P})\bm{F(\bm{x})}$
$\displaystyle=(\bm{e}+\bm{d})^{T}\bm{F}^{\bm{P}}(\bm{x})$
$\displaystyle=(\bm{x}^{T}(\bm{\mathcal{L}}\otimes\bm{I}_{n})+\bm{d}^{T})\bm{F}^{\bm{P}}(\bm{x})$
(24)

Invoking Lemma 3, we can write (III) as:

$\displaystyle(\bm{e}+\bm{d})^{T}(\bm{I}_{N}\otimes\bm{P})\bm{F(\bm{x})}$
$\displaystyle=\frac{1}{2}\sum_{i=1}^{N}\sum_{j=1}^{N}\mathcal{A}_{ij}(\bm{x}_{i}-\bm{x}_{j})^{T}\bm{P}(f(\bm{x}_{i})-f(\bm{x}_{j}))$
$\displaystyle\quad+\sum_{j\in
N_{i}}\bm{d}_{ij}^{T}\bm{P}(f(\bm{x}_{i})-f(\bm{x}_{j})),$ (25)

which is $\leq 0$ by Assumption 5 and since $\mathcal{A}_{ij}\geq 0\ \forall\
i,j={1,2,\dots,N}$ under Assumption 2. ∎

###### Theorem 2

Suppose Assumptions 2 and 5 hold. With the control defined in (9), the
followers in the MAS will eventually converge to states respecting (3).

###### Proof:

(Motivated by [7]) We begin the proof by selecting the Lyapunov function
candidate:

$V=\frac{1}{2}(\bm{e}(t)+\bm{d})^{T}(\bm{M}\otimes\bm{P})(\bm{e}(t)+\bm{d}),$
(26)

where $\bm{P}$ and $\bm{M}$ are as previously defined in Assumptions (5) and
(6), respectively. Then, omitting the time and $\bm{x}$ arguments for
conciseness, we can write the time derivative of $V$ along the trajectories of
the closed-loop system as:

$\displaystyle\dot{V}$
$\displaystyle=(\bm{e}+\bm{d})^{T}(\bm{M}\otimes\bm{P})\dot{\bm{e}}$ (27)
$\displaystyle=(\bm{e}+\bm{d})^{T}(\bm{M}\otimes\bm{P})(\bm{\mathcal{L}}\otimes\bm{I}_{n})(\bm{F}+\bm{G}\bm{u})$
(28)
$\displaystyle=(\bm{e}+\bm{d})^{T}(\bm{M}\bm{\mathcal{L}}\otimes\bm{I}_{n})(\bm{I}_{N}\otimes\bm{P})(\bm{F}+\bm{G}\bm{u}).$
(29)

By (17) and under Assumption 6, we can then write:

$\displaystyle\dot{V}$
$\displaystyle=(\bm{e}+\bm{d})^{T}(\bm{I}_{N}\otimes\bm{P})(\bm{F}+\bm{G}\bm{u})$
(30) $\displaystyle\dot{V}$
$\displaystyle=(\bm{e}+\bm{d})^{T}(\bm{I}_{N}\otimes\bm{P})\bm{F}$
$\displaystyle-(\bm{e}+\bm{d})^{T}(\bm{I}_{N}\otimes\bm{P})\bm{G}\bm{G}^{T}(\bm{I}_{N}\otimes\bm{P})(\bm{e}+\bm{d}).$
(31)

By Lemma 4 and since the second term of (31) is always negative, it follows
that $\dot{V}\leq 0$. Thus, the Lyapunov function (26) will decrease along the
trajectories of the closed-loop system, which implies (local) asymptotic
stability of the equilibrium point of (19). Since, with the change of
variables $\bm{z}(t)=\bm{e}(t)+\bm{d}$, it is easy to see that
$\dot{\bm{z}}\equiv\dot{\bm{e}}$. Thus, from (31) and (19), it follows that
$\bm{z}(t)\rightarrow\bm{0}_{n}$ as
$t\rightarrow\infty\implies||\bm{e}(t)||\rightarrow||\bm{d}||$ as
$t\rightarrow\infty$, thus completing the proof. ∎

###### Corollary 1

Define the formation error for (18) as $\bm{f}(t)$ = $\bm{d}-\bm{e}(t)$. Since
$||\bm{e}(t)||\rightarrow||\bm{d}||$ as $t\rightarrow\infty$, it immediately
follows that $||\bm{f}(t)||\rightarrow||\bm{d}-\bm{d}||=0$, as
$t\rightarrow\infty$, thus confirming asymptotic decay of the MAS’s formation
error.

###### Remark 2

For the popular case of the formation tracking problem, where the agents’
models are linear time-invariant systems — equivalent to setting
$f(\bm{x}_{i})=\bm{A}\bm{x}_{i}$ and $g(\bm{x}_{i})=\bm{B}$ in (1a), with
$\bm{A}$ and $\bm{B}$ constant matrices of appropriate dimensions (see [6] for
example) — the prevailing approach (assuming full controllability of the
linear system) is to define $\bm{u}_{i}$ in terms of some control gain matrix
$\bm{K}$, solve for a positive definite matrix $\bm{P}$ that satisfies the
linear system’s algebraic Riccati equation, and express $\bm{K}$ in terms of
$\bm{P}$, e.g., $\bm{K}=-\bm{B}^{T}\bm{P}^{-1}$. The interested reader can
consult [12] for a detailed treatment on the topic.

###### Theorem 3

Suppose a $\bm{P}$ satisfying (5) exists. Then, it must be the case that
$\bm{P}$ also satisfies the following linear matrix inequality (LMI)

$\begin{bmatrix}\bm{0}\in\mathbb{R}^{Nn\times
Nn}&\bar{\bm{P}}_{{Nn}}^{\frac{1}{2}}\\\
\bar{\bm{P}}_{{Nn}}^{\frac{1}{2}}&-\bm{\Delta}f\bm{D}^{T}\end{bmatrix}\leq 0,$
(32)

where $\bm{D}\in\mathbb{R}^{Nn\times N}$ is the matrix

$[d_{ij}],\ \text{with}\ d_{ij}=\begin{cases}\bm{0}_{n},\ \text{if}\ i=j\\\
\bm{d}_{ij},\ \text{otherwise,}\end{cases}$ (33)
$\bar{\bm{P}}_{{Nn}}=\bm{I}_{N}\otimes\bm{P}^{-1},\ \text{and}$ (34)

$\bm{\Delta}f\in\mathbb{R}^{Nn\times N}$ is the matrix

$[\delta f_{ij}],\ \text{with}\ \delta f_{ij}=\begin{cases}\bm{0}_{n},\
\text{if}\ i=j\\\ f(\bm{\xi}_{i})-f(\bm{\xi}_{j}),\
\text{otherwise.}\end{cases}$ (35)

###### Proof:

We begin the proof by noting (from (5)) that $\bm{P}$ must satisfy:

$\displaystyle(\bm{\xi}_{i}-\bm{\xi}_{j})^{T}\bm{P}(f(\bm{\xi}_{i})-f(\bm{\xi}_{j}))\leq
0$ (36) $\displaystyle\implies$
$\displaystyle\bm{d}_{ij}^{T}\bm{P}(f(\bm{\xi}_{i})-f(\bm{\xi}_{j}))\leq 0.$
(37)

By block diagonalization (as in (33) and (35)) and using the Kronecker
product, we can show that (37) is equivalent to

$\bm{D}^{T}(\bm{I}_{N}\otimes\bm{P})\bm{\Delta}f\leq 0,$ (38)

with $\bm{D}$ and $\bm{\Delta}f$ already defined. Invoking Lemma 2, it is
straightforward to show that (38) can be expressed as (32), which ends the
proof. ∎

###### Remark 3

In the preceding theorem, a change of variables (34) was necessary, to
simplify the notation and allow for an immediate invocation of the Schur
complement lemma. However, it is trivial to show that the original matrix of
interest ($\bm{P}$) can be readily obtained from the uppermost-left $n\times
n$ block of $\bar{\bm{P}}_{{Nn}}^{-1}$. We are also guaranteed a factorization
of the form
$\bar{\bm{P}}_{{Nn}}=\bar{\bm{P}}_{{Nn}}^{\frac{1}{2}}\bar{\bm{P}}_{{Nn}}^{\frac{1}{2}}$,
since $\bar{\bm{P}}_{{Nn}}\succ 0$ and $\otimes$ preserves positive
definiteness.

We now present the following algorithm for formation maintenance. $T_{\Sigma}$
and $\bm{x}_{i}(0)$ are the total number of unity-spaced time steps (in
[$0,T$], for a given interval) and the initial state of the $i^{\text{th}}$
follower agent, respectively.

Algorithm 1 Formation Maintenance Algorithm

0Inputs:  $\bm{D},\bm{\Delta}f,N,N_{i},\bm{\mathcal{A}},\delta
t,g(\cdot),f(\cdot)$, $\bm{x}_{i}(0)$, $\bm{x}_{L}(k)$

-1Outputs : $\bm{P},\bm{e}_{i}(k),\bm{x}_{i}(k);\ i={1,2,\dots,N}$

-2Solve (32) for $\bar{\bm{P}}_{{Nn}}$ and obtain $\bm{P}$ (see Remark 3)

-3$k\leftarrow 0$ to  $T_{\Sigma}$

-4$i\leftarrow 1$ to  $N$

-5Calculate $\bm{e}_{i}$ from (8)

-6Substitute $\bm{P}$ and $\bm{e}_{i}$ from above steps in (9)

-7$\bm{x}_{i}(k+1)\leftarrow f(\bm{x}_{i}(k))+g(\bm{x}_{i}(k))\bm{u}_{i}(k)$

-8

-9

##  IV Formation Specification

This section follows our previous work [13]; however, here, the position of
$i^{\text{th}}$ agent is in $\mathbb{R}^{3}$ and not the plane. As in [13], we
consider a triangular formation (with one leader and three followers) and also
assume this formation to be feasible and rigid (in the sense of [10], §6), and
invariant to homogeneous transformations, with appropriate dimensions.

##  V Simulation Example

As an example, consider the following dynamics for a six degrees-of-freedom
quadrotor, adapted from [14] and described here (for brevity) using the
Newton-Euler equations:

$\begin{drcases}\dot{\mathbf{p}}_{\mathcal{I}}&=\bm{v}_{\mathcal{I}}\\\
\dot{\bm{v}}_{\mathcal{I}}&=\frac{1}{m}\bm{f}_{\mathcal{I}}\\\
\dot{\bm{\zeta}}_{\mathcal{I}}&=\bm{T}_{{}_{\mathcal{B}}}^{{}_{\mathcal{I}}}\bm{\omega}_{\mathcal{B}}\\\
\dot{\bm{\omega}}_{\mathcal{B}}&={\bm{J}}^{-1}(\bm{\tau}_{\mathcal{B}}-\bm{\omega_{\mathcal{B}}}\times\bm{J\omega_{\mathcal{B}}}),\end{drcases}$

(39)

where the variables subscripted by ${\mathcal{I}}$ represent quantities in the
inertial frame, while those with a ${\mathcal{B}}$ subscript pertain to the
quadrotor’s body frame.
$\bm{p}_{\mathcal{I}}=\begin{bmatrix}x&y&z\end{bmatrix}^{T}$ and
$\bm{v}_{\mathcal{I}}=\begin{bmatrix}{v_{x}}&{v_{y}}&{v_{z}}\end{bmatrix}^{T}$
are, respectively, vectors of the $X$, $Y$, and $Z$ positions and linear
velocities of the quadrotor,
$\bm{\zeta}_{\mathcal{I}}=\begin{bmatrix}\phi&\theta&\psi\end{bmatrix}^{T}$ is
the quadrotor’s attitude vector (of roll, pitch, and yaw angles), while
$\bm{\omega}_{\mathcal{B}}=\begin{bmatrix}p&q&r\end{bmatrix}^{T}$ is the
vector of corresponding body-frame attitude rates. The parameters $m$,
$\bm{J}=\text{\text{diag}}(\begin{bmatrix}{I_{xx}}&{I_{yy}}&{I_{zz}}\end{bmatrix})$,
and $g$, represent the mass, inertia matrix, and acceleration due to gravity,
respectively. Finally,
$\bm{f}_{\mathcal{I}}=\begin{bmatrix}{f_{\mathcal{I}}^{t}}&{f_{\mathcal{I}}^{t}}&{f_{\mathcal{I}}^{t}}\end{bmatrix}^{T}$
and
$\bm{\tau}_{\mathcal{B}}=\begin{bmatrix}{\tau_{x}}&{\tau_{y}}&{\tau_{z}}\end{bmatrix}^{T}$
are, respectively, vectors of the total thrust force ($f^{t}_{\mathcal{I}}$)
and torques (about the roll, pitch, and yaw axes) applied to the quadrotor,
while $\bm{T}_{{}_{\mathcal{B}}}^{{}_{\mathcal{I}}}\in\mathbb{R}^{3\times 3}$
is a matrix that transforms the body-frame angular velocities to the inertial
frame.

(a)

(b)

Figure 2: (2(a)) Leader loop trajectory vs. reference. (2(b)) Leader
trajectory tracking with triangular formation maintenance.

It has been shown in [15] that the model in (39) can be expressed as a
control-affine nonlinear system (1a), but we will skip the details to adhere
to page limits. To simulate the quadrotor dynamics, we select the parameters
of the Crazyflie 2.1 quadcopter (Table I), adapted from [16].

TABLE I: Quadrotor Parameters

Parameter | Mass | Arm Length | $I_{xx}$ |  $I_{yy}$ | $I_{zz}$
---|---|---|---|---|---
kg | m |  $10^{-6}$ kg$\cdot$ m2
Value | 0.028 | 0.046 | 6.4893 | 16.4562 | 29.5435
|  |  |  |  |

With some calculations, it can be verified that $\bm{P}=\bm{I}_{12}$ satisfies
(38). For numerical simulation, we take the leader’s trajectory tracking law,
$\bm{u}_{\text{track}}$ (see Assumption 4), to be a trajectory-error
minimizing control law, after the manner of [13]. We also select a loop
(figure eight) trajectory and set $\bm{x}_{1}(0)=\bm{x}_{1\text{rnd}}$,
$\bm{x}_{2}(0)=\bm{x}_{2\text{rnd}}$, $\bm{x}_{3}(0)=\bm{x}_{3\text{rnd}}$,
$\delta_{L1}=1$ m, and $\delta_{12}=\delta_{13}=0.8$ m.
$\bm{x}_{1\text{rnd}}$, $\bm{x}_{2\text{rnd}}$, and $\bm{x}_{3\text{rnd}}$ are
random initial state vectors for the first, second, and third followers,
respectively.

##  VI Numerical Results & Discussion

As a prelude to discussing the consensus-based formation tracking results,
Fig. 2(a) shows the trajectory tracking performance of the leader. Here, we
see that the independent optimal control algorithm drives the leader (from an
arbitrary initial state) so that it accurately tracks the desired trajectory,
thus satisfying Assumption 4. In Fig. 2(b), the trajectory tracking and
formation maintenance results are presented. Here, it is clear that under the
consensus-derived control law, the trajectories of the followers closely track
that of the leader resulting in tight trajectory tracking with formation
persistence. Finally, to give a numerical sense of the formation tracking
performance of our proposed control scheme, we present the tracking and
formation errors of the MAS on Table II, with corresponding plots depicted on
Figs. 3(a) and 3(b). These errors are given in terms of the root-mean-square
error (RMSE) between the leader and reference trajectories and the RMSE
between the desired and actual inter-agent distances for each follower agent,
respectively. As expected, the consensus protocol yields almost negligible
formation errors, when viewed alone and also in comparison with errors
obtained via a formation tracking approach based solely on optimization.

TABLE II: Tracking and Formation Errors (RMSE)

Agent | Leader | Follower 1 | Follower 2 | Follower 3
---|---|---|---|---
RMSE (consensus) | 0.186 | 9.289$\times 10^{-17}$ | 7.692$\times 10^{-17}$ | 5.207$\times 10^{-17}$
RMSE (optimization) | 0.186 | 0.0302 | 7.692$\times 10^{-17}$ | 0.018
|  |  |  |

(a)

(b)

Figure 3: (3(a)) Time-evolution of the trajectory tracking error, given here
by the errors (RMSE) in each state variable of the trajectory vector:
$e_{x},e_{y},e_{z},$ and $e_{\psi}$. (3(b)) Time-evolution of the formation
error given by the RMSE between the desired and actual inter-agent distances,
for the first ($e_{F1}$), second ($e_{F2}$), and third ($e_{F3}$) follower
agents. Results obtained with consensus and those obtained via an exclusively
optimization-based method [13] are also pictured.

##  VII Conclusions

In sum, we presented results on consensus-based formation tracking for a
general class of nonlinear systems — control-affine systems with a state-
dependent drift term — and showed that, even with the MAS sharing information
via a network topology encoded by a tree, precise formation tracking was still
achieved. While our method delivers excellent formation tracking for a
nonlinear system with a high-dimensional state space, such near-perfect
performance is expected since we have assumed perfect knowledge of the agents’
states. Thus, for the more interesting case where agents have uncertain
dynamics, the development of similar control laws remains an open challenge.
We look forward to exploring this direction in future research.

## Acknowledgement

C. Enwerem thanks Erfaun Noorani of the Electrical and Computer Engineering
Department at the University of Maryland, College Park, for helpful technical
discussions and comments on the content and presentation of this work.

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