# Modeling, Characterization, and Control of Bacteria-inspired Bi-flagellated
Mechanism with Tumbling
††thanks:  We acknowledge financial support from the National Science
Foundation under Grant numbers CAREER-2047663 and CMMI-2101751.

Zhuonan Hao1, Sangmin Lim1, Mohammad Khalid Jawed1 1Department of Mechanical &
Aerospace Engineering, University of California, Los Angeles, 420 Westwood
Plaza, Los Angeles, CA 90095

###### Abstract

Multi-flagellated bacteria utilize the hydrodynamic interaction between their
filamentary tails, known as flagella, to swim and change their swimming
direction in low Reynolds number flow. This interaction, referred to as
bundling and tumbling, is often overlooked in simplified hydrodynamic models
such as Resistive Force Theories (RFT). However, for the development of
efficient and steerable robots inspired by bacteria, it becomes crucial to
exploit this interaction. In this paper, we present the construction of a
macroscopic bio-inspired robot featuring two rigid flagella arranged as right-
handed helices, along with a cylindrical head. By rotating the flagella in
opposite directions, the robot’s body can reorient itself through repeatable
and controllable tumbling. To accurately model this bi-flagellated mechanism
in low Reynolds flow, we employ a coupling of rigid body dynamics and the
method of Regularized Stokeslet Segments (RSS). Unlike RFT, RSS takes into
account the hydrodynamic interaction between distant filamentary structures.
Furthermore, we delve into the exploration of the parameter space to optimize
the propulsion and torque of the system. To achieve the desired reorientation
of the robot, we propose a tumble control scheme that involves modulating the
rotation direction and speed of the two flagella. By implementing this scheme,
the robot can effectively reorient itself to attain the desired attitude.
Notably, the overall scheme boasts a simplified design and control as it only
requires two control inputs. With our macroscopic framework serving as a
foundation, we envision the eventual miniaturization of this technology to
construct mobile and controllable micro-scale bacterial robots.

###### Index Terms:

bio-inspired robot, tumble control scheme

##  I Introduction

The study of flagellated bacteria and microorganisms has provided valuable
insights into the development of flagellated robots. [1, 2, 3]. These robots
mimic the locomotion of flagellated organisms, which rely on the intricate
interaction between their helical structures, known as flagella, and the
surrounding viscous fluid. By understanding and replicating these propulsion
mechanisms, flagellated robots can achieve functional movements such as
running, turning, and stopping. Additionally, natural observations have
highlighted that different types of bacteria, including uni-flagellated and
multi-flagellated species, rely on distinct propulsion mechanisms to achieve
specific forms of locomotion [4, 5].

Uni-flagellated bacteria possess a single flagellum filament protruding from
the side of the body, enabling them to swim through the rotary motion of the
flagellum relative to the cell body [6, 5]. Notably, previous investigations
have revealed that when the rotational frequency of the flagella exceeds a
threshold, buckling instability occurs, resulting in highly nonlinear swimming
trajectories [7, 8]. The body orientation of these robots can be controlled
simply by adjusting the spin speed of the flagellum, a mechanism that has been
widely employed in uni-flagellated robot designs to achieve motility [9, 10].
However, research on robotic propulsion inspired by multi-flagellated bacteria
is relatively limited.

Multi-flagellated bacteria exhibit locomotion through the interplay of their
flagella, involving phenomena such as bundle formation, tumbling, and
polymorphic transformations, all of which arise from different flagella
actuation [6, 11, 12]. Bundle formation occurs when two or more flagella spin
in the same direction, generating efficient longitudinal propulsion. The
propulsive force is approximately linearly related to the spin speed, and this
observation has been effectively utilized in bi-flagellated robots to enable
single-direction mobility [13, 14]. The presence of multiple flagella in these
robots offers benefits, suggesting alternative approaches for speed
enhancement beyond flagellum geometry optimization. However, a major
limitation arises in the area of turning or reorienting the body, preventing
these robots from swimming freely in space. Recent research has explored
changing the spin direction of one or more flagella, gradually reducing the
propulsion thrust and generating a turnover torque [4]. This results in rapid
tumble events and seemingly erratic body reorientation. Our study models the
tumbling event as a predictable phenomenon and aims to incorporate the
tumbling mechanism into a bi-flagellated robot to enhance steerability.

Figure 1: Snapshots from (a) experiments and (b) simulations. Side view of the
bi-flagellated rotating around an axis at $t\in\\{0,5,10\\}\text{s}$. Two
identical flagella rotate in opposite directions, i.e., clockwise (CW) and
counterclockwise (CCW), at an angular velocity of $\omega=280~{}\text{rpm}$.

To imitate the fluid-structure interplay between flagellum and low Reynolds
number flow, computational fluid dynamics model including Resistive Force
Theory (RFT), Slender Body Theory (SBT) [15], and Regularized Stokeslets
Segments (RSS) [16] are used to predict the motion of uni- and multi-
flagellated robots. RFT introduces the drag coefficient along the tangential
and perpendicular directions of the flagellum. The method is computationally
inexpensive but neglects the hydrodynamic interactions between flows induced
by different parts of the flagellum. An accurate quantitative analysis
requires a non-local hydrodynamics force model that accounts for the
interaction between the flow induced by distant parts of the filament. Both
SBT and RSS rely on the linearity of the Stokes equations for low Reynolds
number flow, which can accurately describe the evolution of flagellum dynamics
with long-ranged interactions in a viscous fluid.

To understand the physical phenomenon of flagellated locomotion, we couple the
rigid body dynamics with the hydrodynamics model to simulate the robot’s
trajectory. Position and orientation are observable states in the system,
which allow us to study periodical locomotion. As shown in Figure 1, the
experiment and simulation show good agreement. The proposed simulation
framework successfully reveals the interaction between two flagella from long-
ranged hydrodynamics.

Our contributions are as follows. We model and create a macroscopic bi-
flagellated robot to study how different actuation modes can switch robot
locomotion patterns. A simple bi-flagellated robot that exploits variation in
viscosity and structure in its tails can effectively reorient the body. A
framework comprising experiments and simulations is developed to study the
robot’s locomotion. The simulation tool can be used to generate data to
explore the parameter space of the tumbling phenomenon. Meanwhile, the robot’s
dynamics are fully described and can be used to formulate the control scheme.
The physics behind the tumbling locomotion is elaborated in detail. The
simplicity of the robot and the small number of moving parts can eventually
lead to the miniaturization of this robot.

The paper is organized as follows. In section II, we demonstrate the
structural design and experimental setup of the bi-flagellated robot. Section
III presents a computational framework to describe the robot dynamics in a
viscous fluid. Section IV explores the optimal robot geometry for best
dynamics performance, and we validate our simulation results against the
experiments. Section V concludes our works and proposes the potential
directions for future study.

##  II Experimental design

###  II-A Robotic structure

The robot depicted in Figure 2 (a) consists of a cylindrical head and two
right-handed helical flagella with plates attached to the motor shaft. The
head has a radius of $r_{h}$ measuring 2.5 cm and a height of $h$ measuring
4.3 cm. Inside the head, two tiny brushed DC geared motors are located, each
rated at 6 V voltage and capable of a stall current of 1.5 A. The motors are
equipped with magnetic encoders and an IMU module. The motor shaft protrudes
from the head, and its rotation direction and speed are controlled via PWM
feed using a microcontroller. The flagella are manufactured using rapid
prototyping techniques with Polylactic acid (PLA), a type of 3D printing
material. The PLA flagella have a fixed cross-sectional radius of $r_{0}$
measuring 1.58 mm and a helix radius $R$ measuring 6.36 mm. To generate
sufficient experimental data for investigating the tumbling mechanism, the
helix length $l$ is varied between 63.6 and 127.2 mm, while the helix pitch
$\lambda$ is varied between 15.9 and 63.6 mm. The PLA material used for the
flagella is considered to be non-deformable, with a Young’s modulus $E$ of
4.107$\times 10^{9}$ Pa. These design and parameter variations allow for the
exploration of different configurations of the flagellated robot and provide a
range of experimental data to study the underlying mechanisms of tumbling.

Figure 2: Robot schematic and experimental setup. (a) The bio-inspired robot
is comprised of two components: (i) a cylindrical head with radius $r_{h}$ and
height $h$, and (ii) two helical flagella tails with radius $r$, length $l$,
pitch $\lambda$, and cross-section radius $r_{0}$. (b) The bi-flagellated
robot immerses into glycerin and rotates its body around a steering joint
(1-DOF), i.e., y-axis. The rotations around other axes and translations are
limited for.

###  II-B Experimental setup

Our experiments are designed to serve two main purposes: (i) to explore the
optimal structure of the flagellum and robot that generates maximum steering
efficiency, and (ii) to achieve controllable direction changes. The details of
our experimental setup are as follows.

Figure 2 (b) illustrates the experimental apparatus used to validate the
numerical simulations developed in the subsequent section. The platform
comprises four components: (i) a glycerin tank with dimensions of 122 cm
(length) $\times$ 45 cm (width) $\times$ 51.5 cm (height), (ii) the bi-
flagellated robot, (iii) a steering joint, and (iv) a positioning frame.
Glycerin, with a density $\rho$ of 1.26 g/ml and viscosity $\mu$ of 1
Pa$\cdot$s at 25∘ Celsius, is chosen as the surrounding viscous environment.
To facilitate quantitative comparison, we restrict the robot’s degree of
freedom (DOF) to a single-axis rotation using the steering joint. This setup
enables us to characterize the tumbling behaviors associated with different
flagellum designs. The DC geared motors located inside the robot’s head are
connected to an external microcontroller, allowing us to adjust the rotational
speed and direction. The test platform permits us to vary the rotation speed
$\omega$ within the range of 0 to 30 rad/s, ensuring that the low Reynolds
number condition is satisfied, i.e., $Re=\rho\omega Rr_{0}/\mu\leq 0.37$. This
condition guarantees that the fluid flow is predominately governed by
viscosity rather than inertia. By operating within the low Reynolds number
regime, we can accurately investigate the hydrodynamic interactions between
the flagella and the surrounding fluid.

##  III Numerical method of bi-flagellated locomotion

TABLE I: Parameters with symbolic representations Symbol | Value | Unit | Description
---|---|---|---
$E$ | 4.107$\times 10^{9}$ | Pa | Young’s modulus
$\rho$ | 1260 | kg/$\mathrm{m}^{3}$ | Density
$\mu$ | 1 | Pa$\cdot$s | Viscosity
$r_{0}$ | 1.58 | mm | Cross-sectional radius
$R$ | 6.36 | mm | Helix radius
$\lambda$ | 31.8 | mm | Helix pitch
$l$ | 95.4 | mm | Helix axial length
$d$ | 22 | mm | Flagellum spacing distance
$r_{h}$ | 25 | mm | Head radius
$h$ | 43 | mm | Head height
$r_{m}$ | 2 | mm | Mass center shift
$m_{h}$ | 0.1 | kg | Head mass
$g$ | 9.8 | m/$\mathrm{s}^{2}$ | Gravitational acceleration
$\omega$ | 0.1 | rad/s | Flagellum rotation speed
$\Delta l$ | 5 | mm | Discretization length
$\Delta t$ | 0.01 | s | Time step
$C_{t}$ | 4 | 1 | Translational drag coefficient
$C_{r}$ | 2.3 | 1 | Rotational drag coefficient

The bi-flagellated robot consists of two main components: the helical
flagellum and the cylindrical head. In order to simulate the locomotion of
this robot in a viscous fluid medium, we develop a numerical model that
combines three key components: (i) a kinematic representation for the bi-
flagellated mechanism, (ii) Regularized Stokeslet Segments (RSS) to model the
long-ranged hydrodynamics forces, and (iii) a forced-head hydrodynamics model
to capture the interaction between the flagellum and the head. This section is
structured as follows. In Section III-A, we provide a description of the
kinematic representation of the helical flagellum and cylindrical head. This
representation allows us to characterize the motion of the robot in terms of
its shape and orientation. Then, in Section III-B, we explain how we integrate
the kinematic representation with the RSS method to compute the hydrodynamic
forces exerted on the flagellum. The RSS method considers the interactions
between different segments of the flagellum and the surrounding fluid. Next,
in Section III-C, we detail the equations of motion (EOM) that govern the
dynamics of the bi-flagellated robot. These EOM are derived from the theory of
rigid body dynamics, taking into account the forces and torques acting on the
flagellum and the head. By solving these equations, we can simulate the motion
and behavior of the robot in response to the hydrodynamic forces. Finally, in
Section III-E, we discuss the geometric and physical conditions of the
problem, including the dimensions and material properties of the flagellum and
head. These conditions play a crucial role in determining the behavior and
performance of the bi-flagellated robot in the simulated fluid environment.

###  III-A Kinematic representation

####  III-A1 Helical flagellum

We model the flagella filament as a perfect helix with radius $R$, pitch
$\lambda$, and axial length $l$ (see Figure 2 (a)). A right-handed helix in
the Cartesian coordinate system is parameterized as a function of $s$, i.e.,

$\displaystyle\mathbf{r}(s)=\left[R\cos\frac{2\pi s}{L},R\sin\frac{2\pi
s}{L},\frac{\lambda s}{L}\right],0\leq s\leq l,$ (1)

where $L=\sqrt{(2\pi R)^{2}+\lambda^{2}}$ is the contour length of one helical
turn. In the case of a left-handed helix, the second term has a negative sign.
We employ the discretization method to model the kinematic of helical
filament. In the schematic of Figure 3 (a), each discrete helical curve
consists of $N+1$ nodes, i.e.,
$\mathbf{n}=[\mathbf{n}_{0},\mathbf{n}_{1},\cdots,\mathbf{n}_{N}]$. We take
the first two nodes $\mathbf{n}_{0}$ and $\mathbf{n}_{1}$ as the connection to
the head. Starting from $i=2$, the coordinate of the remaining nodes is
calculated by taking $s=(i-2)l/(N-2)$ in Equation 1.

The $N+1$ nodes correspond to $N$ edge vector $\mathbf{e}^{1}$,
$\cdots$,$\mathbf{e}^{N-1}$, such that
$\mathbf{e}^{i}=\mathbf{x}_{i+1}-\mathbf{x}_{i}$, where $i=1,\cdots,N-1$.
Hereby, we denote the node-associated quantities by superscripts and edge-
associated quantities by subscripts. Nodal positions constitute the $3N$ sized
DOF vector, i.e., $\mathbf{X}=[\mathbf{x}_{0},\cdots,\mathbf{x}_{N}]^{T}$,
where the superscripts $T$ denotes transposition.

Because the rigid flagellum can only rotate around a single fixed axis, i.e.,
z-axis, the angular velocity vector is specified as
$\boldsymbol{\omega}=[0,0,\omega_{z}]$. By defining the rotation axis
$\mathbf{x}_{\text{rotate}}=[0,0,1]$, we can obtain the linear velocity of
each node by
$\dot{\mathbf{x}}_{i}=\boldsymbol{\omega}\times(\mathbf{x}_{i}-\mathbf{x}_{\text{rotate}})$,
where $\times$ denotes the cross product of two vectors. With nodal
velocities, we can update the nodal positions at each time step. Rearranging
the derivative of DOF vector as
$\mathbf{U}=\dot{\mathbf{X}}=[\dot{\mathbf{x}}_{0},\cdots,\dot{\mathbf{x}}_{N}]^{T}$,
the variable is used to formulate the drag force in RSS (see details in
Section III-B).

####  III-A2 Cylindrical head

Without losing the generality, we use a single head node
$\mathbf{n}_{\text{head}}$ in Figure 3 (a.1) to represent the spatial
configuration of the bi-flagellated system. Concerning a prescribed fixed
coordinate system, denoted as inertial frame
$\mathbf{x}^{I}:x^{I}-y^{I}-z^{I}$, we can describe the translation and
rotation on the rotated coordinate system, designated as body frame
$\mathbf{x}^{B}:x^{B}-y^{B}-z^{B}$ attached to the head (see Figure 3 (b)). We
take the Euler angles to represent the orientation of the head, which are
typically denoted as yaw $\alpha$, pitch $\beta$, and roll $\gamma$ in $Z-Y-X$
convection. In the steering joint setup, we define pitch angle $\beta$ as the
angle between the axis $z_{I}$ and axis $z_{B}$ to describe the orientation of
the bi-flagellated system. Further, to model the free swimming motion after
removing the steering joint, we introduce quaternion for orientation
$\mathbf{q}=(q_{0},q_{1},q_{2},q_{3})$ (convertable to Euler angle) and axial
angular velocities along body frame
$\boldsymbol{\omega}^{B}=(\omega_{x}^{B},\omega_{y}^{B},\omega_{z}^{B})$, and
define DOF vector
$\mathbf{Q}=[\mathbf{x}^{I},\dot{\mathbf{x}}^{I},\mathbf{q},\boldsymbol{\omega}^{B}]$
to represent spatial information.

###  III-B Regularized Stokeslets Segments

Figure 3: Kinematic representation of bi-flagellated system. (a) Long-range
hydrodynamics. (a.1) Discrete schematic of the bi-flagellated robot. Each
helical flagellum is discretized into $N+1$ nodes. Superscript denotes the
helix index, and subscript denotes the node index. $\mathbf{n}_{0}^{1}$ and
$\mathbf{n}_{0}^{2}$ connect the flagellum with the head node, as the
application points of the forces generated by each helix, interacting with the
head node in rigid body dynamics. Inset: Notations associated with the flow
$\dot{\mathbf{x}}_{m}$ at point $\mathbf{x}_{m}$ generated by a line segment
$\mathbf{e}_{k}=\mathbf{x}_{k+1}-\mathbf{x}_{k}$. (a.2) Time series of
normalized hydrodynamics forces $\bar{F}$ along x, y, and z direction with
flagella spacing distance $d=0.022\text{ m}$ and rotation speed
$\omega_{1}=1\text{ rad/s}$ and $\omega_{2}=-1\text{ rad/s}$. The sinusoidal
wave pattern arises due to the long-ranged hydrodynamics interactions between
flows induced by different segments of the flagella. (b) The description of
the robot coordinate in the body frame and inertia frame with Euler’s angle
representation. The green arrow line $\mathbf{N}$ indicate the line of nodes.

We use Regularized Stokeslets Segments (RSS) methods to model the viscous drag
force experienced by a helical flagellum in motion within a viscous fluid. The
relation between the velocity vector $\mathbf{U}$ of (size $3N$) at nodes set
$\mathbf{n}$ and the hydrodynamics force vector $\mathbf{F}$ of (size $3N$)
applied on them is linearly configured by a geometry-associated matrix
$\mathbf{A}$ (vector of size $3N\times 3N$), i.e.,

$\displaystyle\mathbf{U}=\mathbf{A}\mathbf{F},$ (2)

We describe the formulation of matrix $\mathbf{A}$ on the discretized helical
flagellum as follows.

The primary Green’s function of Stokes flow is the Stokeslets, which describes
the flow associated with a singular point force. Referring to Figure 3 (a),
RSS provides a relationship between the velocity $\dot{\mathbf{x}}_{m}$ at a
point $\dot{\mathbf{x}}_{m}$ and the forces applied by each node on the fluid
such that

$8\pi\mu\dot{\mathbf{x}}_{m}=\sum_{k=0}^{N-2}\left(\mathbf{A}_{1}^{k}\mathbf{f}_{\mathrm{k}}^{h}+\mathbf{A}_{2}^{k}\mathbf{f}_{\mathrm{k}+1}^{h}\right),$
(3)

where $\mathbf{f}_{\mathrm{k}}$ is the force vector of size 3 that represents
the force applied by the $k$-th node onto the fluid. This is equal and
opposite to the hydrodynamics force onto the $k$-th node. The matrix
$\mathbf{A_{1}^{k}}$ and $\mathbf{A_{2}^{k}}$ are

$\begin{array}[]{r}\mathbf{A}_{2}^{k}=\left|\mathbf{s}_{k}\right|\left(\left(T_{1,-1}^{k,k+1}+c^{2}T_{1,-3}^{k,k+1}\right)\mathbf{I}+T_{1,-3}^{k,k+1}\left(\mathbf{r}_{k}\mathbf{r}_{k}^{T}\right)+\right.\\\
\left.T_{2,-3}^{k,k+1}\left(\mathbf{r}_{k}\mathbf{s}_{k}^{T}+\mathbf{s}_{k}\mathbf{r}_{k}^{T}\right)+T_{3,-3}^{k,-3}\left(\mathbf{s}_{k}\mathbf{s}_{k}^{T}\right)\right),\\\
\mathbf{A}_{1}^{k}=\left|\mathbf{s}_{k}\right|\left(\left(T_{0,-1}^{\mathbf{k},\mathbf{k}+1}+c^{2}T_{0,-3}^{\mathbf{k},\mathbf{k}+1}\right)\mathbf{I}+T_{0,-3}^{k,k+1}\left(\mathbf{r}_{k}\mathbf{r}_{k}^{T}\right)+\right.\\\
\left.T_{1,-3}^{k,k+1}\left(\mathbf{r}_{k}\mathbf{s}_{k}^{T}+\mathbf{s}_{k}\mathbf{r}_{k}^{T}\right)+T_{2,-3}^{k,k+1}\left(\mathbf{s}_{k}\mathbf{s}_{k}^{T}\right)\right)-\mathbf{A}_{2}^{k},\end{array}$

where $\mathbf{x}_{m}$ is the point of measurement, $c$ is the regularization
parameter (from analysis in [16], $c=1.031\cdot r_{0}$), $\mathbf{I}$ is the
3-b-3 identity matrix, $\mathbf{r}_{k}=\mathbf{x}_{m}-\mathbf{x}_{k}$,
$\mathbf{r}_{k+1}=\mathbf{x}_{m}-\mathbf{x}_{k+1}$, and
$\mathbf{e}_{k}=\mathbf{x}_{k+1}-\mathbf{x}_{k}$ are the position vectors
between edge and point of measurement, and the scalar quantities denoted by
$T$, e.g., $T_{0,-1}^{k,k+1}$ are expressed as follow

$\displaystyle
T_{0,-1}^{k,k+1}=\left.\frac{1}{\left|\mathbf{s}_{k}\right|}\log\left[\left|\mathbf{s}_{k}\right|R+\left(\mathbf{x}_{\alpha}\cdot\mathbf{s}_{k}\right)\right]\right|_{0}^{1},$
$\displaystyle
T_{0,-3}^{k,k+1}=-\left.\frac{1}{R\left[\left|\mathbf{s}_{k}\right|R+\left(\mathbf{x}_{\alpha}\cdot\mathbf{s}_{k}\right)\right]}\right|_{0}^{1},$
$\displaystyle
T_{1,-1}^{k,k+1}=\left.\frac{R}{\left(\left|\mathbf{s}_{k}\right|\right)^{2}}\right|_{0}^{1}-\frac{\left(\mathbf{x}_{0}\cdot\mathbf{s}_{k}\right)}{\left(\left|\mathbf{s}_{k}\right|\right)^{2}}T_{0,-1}^{k,k+1}\text{,
}$ $\displaystyle
T_{1,-3}^{k,k+1}=-\left.\frac{1}{R\left(\left|\mathbf{s}_{k}\right|\right)^{2}}\right|_{0}^{1}-\frac{\left(\mathbf{x}_{0}\cdot\mathbf{s}_{k}\right)}{\left(\left|\mathbf{s}_{k}\right|\right)^{2}}T_{0,-3}^{k,k+1}\text{,
}$ $\displaystyle
T_{2,-3}^{k,k+1}=-\left.\frac{\alpha}{R\left(\left|\mathbf{s}_{k}\right|\right)^{2}}\right|_{0}^{1}+\frac{1}{\left(\left|\mathbf{s}_{k}\right|\right)^{2}}T_{0,-1}^{k,k+1}-\frac{\left(\mathbf{x}_{0}\cdot\mathbf{s}_{k}\right)}{\left(\left|\mathbf{s}_{k}\right|\right)^{2}}T_{1,-3}^{k,k+1},$
$\displaystyle
T_{3,-3}^{k,k+1}=-\left.\frac{\alpha^{2}}{R\left(\left|\mathbf{s}_{k}\right|\right)^{2}}\right|_{0}^{1}+\frac{2}{\left(\left|\mathbf{s}_{k}\right|\right)^{2}}T_{1,-1}^{k,k+1}-\frac{\left(\mathbf{x}_{0}\cdot\mathbf{s}_{k}\right)}{\left(\left|\mathbf{s}_{k}\right|\right)^{2}}T_{2,-3}^{k,k+1},$

where $\mathbf{x}_{\alpha}=\mathbf{x}_{k}-\alpha\mathbf{s}_{k}$, and
$R=\sqrt{|\mathbf{x}_{\alpha}|^{2}+c^{2}}$.

The geometry matrix $\mathbf{A}$ is formulated by a rearrangement of block
matrices $\mathbf{A}_{1}^{k}$ and $\mathbf{A}_{2}^{k}$ on the corresponding
node index. At each time step in the simulation, knowing the position
$\mathbf{X}$ and velocity $\mathbf{U}$ of the node set $\mathbf{n}$, we can
construct the geometry matrix $\mathbf{A}$. Then we employ the force-velocity
relationship in Equation 2 to evaluate the hydrodynamics forces by
$\mathbf{\hat{F}}=\mathbf{A}\setminus\mathbf{U}$.

###  III-C Forced-head hydrodynamics model

To study the locomotion of a bi-flagellated system under external forces, we
build the forced-head hydrodynamics model by using the aforementioned
kinematic representation. A single head node $\mathbf{n}^{\text{head}}$ and
two interconnected nodes from helical flagellum $\mathbf{n}_{0}^{1}$ and
$\mathbf{n}_{0}^{2}$. in Figure 3 describe the configuration of the dynamical
system. Head node $\mathbf{n}_{\text{head}}$ accounts for the hydrodynamics
drag force and torque induced by the translation and rotation of the head. Two
connected nodes $\mathbf{n}_{0}^{1}$ and $\mathbf{n}_{0}^{2}$ with equal
distance $d/2$ to $\mathbf{n}^{\text{head}}$ account for the resultant of
hydrodynamics forces and torque generated by two helical flagella. In this
section, we analyze the forces and torques applied to the system and formulate
the equation of motion of bi-flagellated locomotion.

Hydrodynamics force on head. We use the Stokes law to compute the
hydrodynamics force on the robot head. As the head translates with velocity
$\dot{\mathbf{x}}^{\text{head}}$, the viscous fluid exerts a drag force to
resist the translation. Likewise, when the head rotates with angular velocity
$\boldsymbol{\omega}^{\text{head}}$, the viscous fluid applies torque to
resist that rotation. Stokes’ law model the hydrodynamics drag by considering
the object as a small sphere. For the cylindrical head, we can introduce two
prefactors $C_{t},C_{r}$ to account for the non-sphericity of the object.
Therefore, we model the translation drag force as

$\mathbf{f}_{t}^{\text{head }}=-C_{t}\cdot 6\pi\mu
r_{h}\dot{\mathbf{x}}^{\text{head}},$ (4)

where $r_{h}$ is the radius of the head, and the rotation drag torque as

$\mathbf{T}_{r}^{\text{head}}=-C_{r}\cdot 8\pi
d_{r}^{3}\boldsymbol{\omega}^{\text{head}},$ (5)

where $d_{r}$ is reference dimension to account for the non-spherical shape.
The value of $C_{t}$ and $C_{r}$ are determined by drop and rotation test. In
our model, $\mathbf{f}_{t}^{\text{head }}$ and $\mathbf{T}_{r}^{\text{head}}$
are applied on the head node $\mathbf{n}^{\text{head}}$.

Righting moment due to mass distribution. In our bi-flagellated system, the
head is conditioned to be neutral buoyancy. Therefore, the gravitational force
and buoyancy forces are balanced, i.e., $mg=\rho Vg$, where $m$ is the mass of
the head, $V$ is the volume of the head, $\rho$ is the density of fluid
medium, and $g$ is gravitational acceleration. However, the mass is not
uniformly distributed along the robot head. When the center of mass (COM) and
center of geometry (COG) is shifted by a distance $\textbf{r}_{m}$, a righting
moment tend to restore the robot to its previous attitude after any rotational
displacement. The moment can be modeled as

$\displaystyle\mathbf{T}_{m}^{\text{head}}=mg\textbf{r}_{m}\sin{\beta},$ (6)

where $\textbf{r}_{m}$ is displacement vector pointing from COM to COG, and
$\beta$ is the pitch angle.

Propulsive force from flagellum. In Section III-B, we evaluate the
hydrodynamics forces of each node along the discrete helical flagellum by the
method of RSS. In the bi-flagellated system, two flagella provide the
propulsion for the head. The propulsive force is equivalent to the resultant
forces of all nodes forces of each flagellum applied on the two connection
nodes $\mathbf{n}_{0}^{1}$ and $\mathbf{n}_{0}^{2}$, i.e.,
$\mathbf{f_{p}^{\text{tail}}}=\sum_{k=1}^{N}\mathbf{f_{k}}$.

For simplicity, we denote the resultant forces for two flagella as
$\mathbf{f_{p}^{1}}$ and $\mathbf{f_{p}^{2}}$. Figure 3(b) provides a time
evolution of the resultant forces when two flagella rotate in the opposite
direction. The force amplitude shows a sinusoidal pattern resulting from the
long-ranged coupling between two flagella. The opposite sign of forces along
the z-axis can cancel the propulsive effect but instead generate a turn-over
torque because of the spacing distance $d$, which is the fundamental mechanism
of the tumbling phenomenon. The torque applied equivalently on head node
$\mathbf{n}^{\text{head}}$ is given by

$\displaystyle\mathbf{T}^{\text{tail}}=(\mathbf{f_{p}^{1}}-\mathbf{f_{p}^{2}})\times\dfrac{d}{2},$
(7)

where $\mathbf{r_{1}}=\mathbf{x}_{0}^{1}-\mathbf{x}^{\text{head}}$ and
$\mathbf{r_{2}}=\mathbf{x}_{0}^{2}-\mathbf{x}^{\text{head}}$.

In summary, the external forces and torques applied on the head includes
$\mathbf{f}_{t}^{\text{head}},\mathbf{T}_{r}^{\text{head}},\mathbf{T}_{m}^{\text{head}},\mathbf{f_{p}^{\text{tail}}},\mathbf{T}^{\text{tail}}$.
The governing equations of pivot steering in terms of pitch angle $\beta$ is:

$I_{y}\mathbf{\ddot{\beta}}=T_{rz}^{\text{head}}+T_{mz}^{\text{head}}+T^{\text{tail}}_{z},$
(8)

where the subscript $z$ represents the torch component along z direction, and
the governing equation of free swimming in terms of the DOF vector
$\mathbf{Q}$ is

$\displaystyle m\begin{bmatrix}0\\\ \ddot{\mathbf{x}}^{I}\end{bmatrix}$
$\displaystyle=\mathbf{q}\otimes\begin{bmatrix}0\\\
\mathbf{f}_{t}^{\text{head}}+\mathbf{f_{p}^{1}}+\mathbf{f_{p}^{2}}\end{bmatrix}\otimes\mathbf{q}^{*},$
(9) $\displaystyle\dot{\mathbf{q}}$
$\displaystyle=\frac{1}{2}\mathbf{q}\otimes\begin{bmatrix}0\\\
\mathbf{\omega}^{B}\end{bmatrix},$
$\displaystyle\mathbf{J}\dot{\mathbf{\omega}}^{B}$
$\displaystyle=-\mathbf{\omega}^{B}\times\mathbf{J}\mathbf{\omega}^{B}+\mathbf{T}_{r}^{\text{head}}+\mathbf{T}_{m}^{\text{head}}+\mathbf{T}^{\text{tail}},$

where $\mathbf{q}^{*}$ is the conjugate of $\mathbf{q}$, and
$\mathbf{J}=\text{diag}(I_{x},I_{y},I_{z})$ is matrix of moment of inertia.

###  III-D Control scheme of pivot tumbling

To study the controllability of tumbling, we rewrite Equation 8 as the state
space model by defining the state vector
$\mathbf{x}\triangleq[\beta,\dot{\beta}]$ and control input vector
$u\triangleq[\mathbf{f^{1}_{p}},\mathbf{f^{2}_{p}}]$ (assuming small $\beta$,
such that $\sin(\beta)\approx\beta$ holds):

$\dot{\mathbf{x}}(t)=\mathbf{A}\mathbf{x}(t)+\mathbf{B}\mathbf{u}(t),\quad\mathbf{x(0)}=\mathbf{x}_{0},$

where

$\mathbf{A}=\begin{bmatrix}0&1\\\ -\dfrac{mgr_{m}}{I_{y}}&-\dfrac{8\pi
C_{r}\mu
h^{3}}{I_{y}}\end{bmatrix},\mathbf{B}=\frac{d}{2I_{y}}\begin{bmatrix}0&0\\\
1&-1\end{bmatrix}$

The system is structurally stable because the eigenvalues of matrix
$\mathbf{A}$ have negative real part. The states of the system asymptotically
converges to the steady condition

$\beta_{\text{ss}}=\dfrac{(\mathbf{f^{1}_{p}}-\mathbf{f^{2}_{p}})d}{2mg\textbf{r}_{m}},\quad\dot{\beta}_{\text{ss}}=0$
(10)

Therefore, to realize the desired pitch angle $\beta_{\text{ref}}$, we require
$\mathbf{f^{1}_{p}}$ and $\mathbf{f^{2}_{p}}$ to satisfy below conditions

$\displaystyle\mathbf{f^{1}_{p}}+\mathbf{f^{2}_{p}}$
$\displaystyle=0,\text{\quad(max torque)}$ (11)
$\displaystyle\dfrac{(\mathbf{f^{1}_{p}}-\mathbf{f^{2}_{p}})d}{2mg\textbf{r}_{m}}$
$\displaystyle=\beta_{\text{ref}},\text{\quad(steady state value)}$
$\displaystyle\|\mathbf{f^{1}_{p}}\|,\|\mathbf{f^{2}_{p}}\|$
$\displaystyle\leq\mathbf{f_{\max}}.\text{\quad(effective propulsion)}$

However, to implement a actual control for the propulsion $\mathbf{f^{1}_{p}}$
and $\mathbf{f^{2}_{p}}$, we need more knowledge on the mechanism of
flagellated propulsion. In Section IV, we characterize the propulsion with
flagellum geometry and rotation speed.

###  III-E Definition of the problem

The general framework introduced above for the forced-head dynamics is now
applied to generating bi-flagellated locomotion. We provide specifics on the
geometry and physical parameters of this problem. The flagellum is chosen to
be a rigid right-handed helical filament with Young’s modulus $E$. The
geometrical parameters describing the helix structure include helix pitch
$\lambda$, helix radius $R$, axial length $l$, and cross-sectional radius
$r_{0}$. The values of the parameters in Table I are chosen to match the
laboratory experiments described in Section II. We use a dimensionless scheme
to generalize the results, except for several fundamental variables, to make a
valid comparison between macroscopic and microscopic mechanisms [17]. The
procedures are introduced as follows.

The helical flagellum is connected to the head at one extremity, where it is
rotated counterclockwise with a prescribed angular velocity $\omega$. Two
flagella are spaced with a specific distance $d$. Hereafter, we normalize the
spacing distance by the helix radius, $R$, such that the normalized distance
is $\bar{d}=d/R$, where the overbar symbol $\bar{\cdot}$ denotes the
normalized variables. Likewise, the geometrical parameters that describes the
flagellum structure include $\bar{\lambda}=\lambda/l$ and $\bar{R}=R/\lambda$.
The input for the bi-flagellated system is angular velocity $\omega$ as a
function of time. The propulsive thrust $F$ is the z component of
$\mathbf{f_{p}^{\text{tail}}}$ and turnover torque $T$ is the y component of
$\mathbf{T}^{\text{tail}}$, with the normalization as $\bar{F}=F/(\mu\omega
RL)$ and $\bar{T}=T/(\mu\omega R^{2}L)$, where $\mu$ denote the viscosity of
fluid. This dimensionless representation allows for generality across length
scales in interpreting our findings.

##  IV Results and discussion

The bi-flagellated system can undergo different motions by varying the
actuation modes of the two motors. In this section, we study the mechanism of
tumbling behavior. We first explore the parameter space that enhances the
direction change from flagellum geometry and robot structure. Then we show a
bi-flagellated robot that can efficiently reorient the body by tumbling.

###  IV-A Four distinct locomotion patterns

Figure 4: Locomotion patterns of the bi-flagellated robot. The robot can turn
and translate when rotating two flagella with different modes. Denote
$\omega>0$ when flagellum rotating counterclockwise. (a.1) Left turn when
$\omega_{1}>0,\omega_{2}<0$, (a.2) Right turn when
$\omega_{1}<0,\omega_{2}>0$, (a.3) upward translation when
$\omega_{1}<0,\omega_{2}<0$, (a.4) downward translation when
$\omega_{1}>0,\omega_{2}>0$. (b.1)-(b.4) The hydrodynamics force applied at
two flagella with corresponding flagella rotation modes (a.1)-(a.4). The light
red arrows represent the force direction applied on the node, and the dark red
arrows represent the resultant force direction applied on the extremity of the
flagellum.

Since two flagella are rotated by two motors separately, we can propose four
different actuation combinations according to the rotation direction of the
two motors, i.e.,

  1. 1.

$\omega_{1}>0$, $\omega_{2}<0$

  2. 2.

$\omega_{1}<0$, $\omega_{2}>0$

  3. 3.

$\omega_{1}<0$, $\omega_{2}<0$

  4. 4.

$\omega_{1}>0$, $\omega_{2}>0$

where we denote $\omega_{i}>0$ as the counterclockwise rotation from the
overlook view. The magnitude of angular velocities is equal regardless of the
rotation direction, i.e., $|\omega_{1}|=|\omega_{2}|$. By the numerical
simulation tool in Section III, we explore all the possible locomotion
patterns from the above combinations.

Figure 4 (a) demonstrates four different locomotion induced by two flagella,
including left turn, right turn, upward translation, and downward translation.
The corresponding plots of the force field can interpret the dynamics
mechanism behind each motion in Figure 4 (b). For the right-handed helix, a
counterclockwise rotation yield a resultant force pointing obliquely downward,
while the clockwise rotation yield an upward force, as the dark red arrows
show. Rotating two flagella with the same direction ensures the two resultant
forces point up or down simultaneously but not precisely in the same direction
due to the hydrodynamics coupling. On the contrary, rotating two flagella in
the opposite direction make the resultant forces work as a force couple that
is subject to generating a turnover torque. If the torque is large enough to
overcome the intrinsic inertial of the head, directional change takes place
with time evolution.

Figure 5: Geometrical parameters determine the magnitude of propulsive thrust and turnover torque. (a) Dependence of normalized propulsive force $\bar{F}$ (log-scaled) on both normalized pitch $\lambda/l$ and normalized radius $R/\lambda$. (b) Dependence of normalized turnover torque $\bar{T}$ (log-scaled) on both normalized pitch $\lambda/l$ and normalized radius $R/\lambda$. The upward triangle symbols in (a) and (b) correspond to Table II. The cycled red marker corresponds to the bi-flagellated robot. The red and green pentagram markers corresponds to $l=1600R$ and $l=6.25R$, respectively. (c) Normalized turnover torque $\bar{T}$ as a function of normalized flagella spacing distance $d/R$. (d) Normalized propulsive thrust $\bar{F}$ as a function of normalized flagella spacing distance $d/R$. Inset in (c) and (d): the normalized resultant force of two flagella as a function of normalized flagella spacing distance $d/R$. The non-linear relationship emerges when two flagella are in proximity, i.e., $d/R<5$, due to hydrodynamics coupling between two flagella.  TABLE II: Parameters of flagellum for five species of bacteria Label | Microorganism | $\lambda/l$ | $R/\lambda$
---|---|---|---
1 | Caulobacter crescentus [18] | 0.1667 | 0.1205
2 | Escherichia coli [19] | 0.3571 | 0.0909
3 | Rhizobium lupini [20] | 0.2500 | 0.1852
4 | Salmonella [21] | 0.2500 | 0.0909
5 | Vibrio alginolyticus [22] | 0.3243 | 0.1167
$\color[rgb]{1,0,0}\circ$ | Bi-flagellated robot | 0.33 | 0.2

###  IV-B Design space for optimal flagellum geometry

Figure 6: Comparison of simulation and experiment for fixed-body rotation. (a)
Time evolution of pitch angle $\beta$ when angular speed of flagellum varying
from 6.28 rad/s to 20.94 rad/s with $\lambda/l=3,R/\lambda=5,d/R=3.5$. (b) The
steady state values of pitch angle $\beta_{\text{max}}$, as a function of
angular velocity $\omega$, with a parameter set:
$\lambda/l=3,R/\lambda=5,d/R=3.5$. (c) The steady state values of pitch angle
$\beta_{\text{max}}$, as a function of normalized radius $R/\lambda$, with a
parameter set: $l/R=15,d/R=3.5$, and $\omega$ = 20.9 rad/s. (d) The steady
state values of pitch angle $\beta_{\text{max}}$, as a function of normalized
pitch $R/\lambda$, with a parameter set: $R/\lambda=5,d/R=3.5$, and $\omega$ =
20.9 rad/s.

Thus far, our findings on the bi-flagellated actuation and corresponding
locomotion patterns have brought insight into the run-and-turn behaviors of
bacteria. Next, we perform a broader exploration of the parameter space for
the structure of the helical flagellum and robot, emphasizing the ranges
relevant to natural multi-flagellated cells. We use normalized propulsive
thrust $\bar{F}$ and turnover torque $\bar{T}$ to represent propulsion and
steering efficiency. $\bar{F}$ characterize propulsion ability by a single
helical flagellum. In contrast, $\bar{T}$ represents the ability of direction
change formed by the force couple accounting for the long-ranged hydrodynamics
effect. A systematic parametric study is performed to quantify the dependence
of these variables on the geometry and structure parameters.

In Figure 5 (a) and (b), we plot the magnitude of $\bar{T}$ and $\bar{F}$
(colorbar) in log-scale on the normalized pitch and normalized radius at
$\omega=20.94$ rad/s. We set helix radius $R$ as a constant value of 0.0064 m
and compute the helix pitch $\lambda$ and length $l$ per the corresponding
value of normalized values. In the plot of torque, we make the distance
between two flagella as constant $d=3R$ to eliminate the effect of spacing
distance $d$. Two phase diagrams have commonalities in the geometrical
parameters. We find that the two quantities increases as the normalized pitch
and radius increase. The highest value locate at the left-bottom region,
indicating the maximum torque and thrust. However, the region represents an
elongated flagellum concerning the radius, i.e., $L=1600R$, which is less
common in bacteria due to its poor maneuverability and high energy cost. From
the parameter distribution of several species of bacteria in Table II, as the
upward triangles, we learn that the optimal geometry locates on the right-
bottom region is a trade-off between dynamic performance and dimensionality.
Therefore, as the red marker shows, we take $\lambda/l=0.33$ and
$R/\lambda=0.2$ as representative values.

In Figure 5 (c) and (d), we plot the magnitude of $\bar{T}$ and $\bar{F}$ as a
function of normalized space distance $d/R$, at the representative geometrical
values. As $d/R$ increases, $\bar{T}$ and $\bar{F}$ monotonically increase,
but non-linearity of the curve due to hydrodynamics occurs when two flagellum
is in proximity. Intriguingly, from the insets of two plots, we see that the
long-ranged hydrodynamics exert different effects when two flagella rotate
differently. Hydrodynamics escalates the force magnitude when they rotate in
the opposite direction but decreases the magnitude when they rotate in the
same direction. Given torque is a product of force and arm, the magnitude
almost remains the same when the normalized spacing distance $d/R$ increases
from 2 to 3.5. The result implies the maximum propulsive force and turnover
torque occur when two flagella are spaced an infinite distance. However, the
long spacing distance is not preferable for either the bacteria or bio-
inspired robot. The multi-flagellated system must compromise with dimension
and propulsion efficiency. Here, we take $d/R=3.5$ in our bi-flagellated
system to ensure proper size of the head.

###  IV-C Analysis on tumbling process

Toward validating the numerical simulations presented in Section III, we now
perform a direct quantitative comparison with experimental results using the
apparatus described in Section II. Emphases are given to the evolution of
pitch angle $\beta$, as an accumulative result of applied forces and torques.

In this section, we investigate how pitch angle $\beta$ evolved with different
flagellum parameters, including normalized pitch $\lambda/l$ and rotation
speed $\omega$. In Figure 6(a), we plot $\beta$ as a function of time $t$,
with the reference assumed $\lambda/l=3,R/\lambda=5,d/R=3.5$. The pitch angle
keeps increasing with time and reaches a maximum value, denoted as
$\beta_{\text{max}}$. The magnitude of $\beta_{max}$ is proportional to the
rotation speed of the flagella, which is ensured by the steady condition in
Equation 10. Therefore, though without measurement the value of turnover
torque in experiment, we can use the $\beta_{\text{max}}$, a.k.a.,
$\beta_{\text{ss}}$ as an indicator of torque magnitude. Figure 6 (b) plot the
relationship between $\beta_{\text{ss}}$ and rotation speed $\omega$, and show
an excellent agreement between experiments and simulations. We learn that the
magnitude of torque generated by forces $\mathbf{f^{1}_{p}}$ and
$\mathbf{f^{2}_{p}}$ is linear with the rotation speed of the flagellum.

We employ our numerical simulations to explore the effect of $\lambda/l$ with
a comparison with experiments when keeping the angular velocity of flagellum
$\omega=20.94$ rad/s, for which we showed in the previous results that there
is a significant direction change effect. Figure 6(c) show a good match
between the experiment and simulation of the tendency of $\beta_{\text{max}}$
on $\lambda/l$. The agreement validate the result in Figure 5(a) about the
relationship between turnover torque and flagellum geometry.

###  IV-D Attitude control of bi-flagellated robot

Figure 7: Attitude control of bi-flagellated robot. (a) Relationship between
propulsion force and flagellum rotation speed $\omega$ with
$d/R=0.35,R/\lambda=0.2$, and $\lambda/l=0.33$. The slopes $K_{1}=0.0006$
Ns/rad and $K_{2}=-0.0006$ Ns/rad. (b) The evolution of pitch angle $\beta$
when apply the attitude control scheme with a reference
$\beta_{\text{ref}}=35^{\circ}$.

The previous sections shows that the propulsion force and turnover torque are
associated with flagellum structure $d/R,R/\lambda,\lambda/l$ and rotation
speed $\omega$. As for the control problem, it is not feasible to change the
structure related parameters to vary the propulsion force and torque in
process. Therefore, we take the rotation speed of two flagella
$\omega_{1},\omega_{2}$ as the actual control variable. Figure 6(b) illustrate
the torque is linear with the rotation speed for given flagellum structure.
Through our computation framework, we evaluate the propulsion force by
$\mathbf{f^{1}_{p}}=K_{1}\omega_{1},\mathbf{f^{2}_{p}}=K_{2}\omega_{2}$ as
Figure 7(a). This allows us to realize the control scheme mentioned in Section
III-D. We showcase a tracking for a constant attitude angle
$\beta_{\text{ref}}=35^{\circ}$ in both simulation and experiment. By solving
Equation 11, we obtain
$\omega_{1}=201.6\text{rpm},\omega_{2}=-201.6\text{rpm}$. Then we set the
rotation speed of two flagellum with the values and the pitch angle $\beta$
evolved as Figure 7(b).

##  V Conclusions and future work

In conclusion, we present a bi-flagellated mechanism and a numerical
simulation framework for studying bacteria tumbling behavior. A dimensionless
scheme generalize our results to the bacteria level. The framework are used to
explore the relationship between the steering ability and the structural
parameters of the bi-flagellated system. The attitude control scheme ensure us
to control the orientation of the robot.

Directions for future work include: (i) formulation of optimal control policy
for free swimming robot, and (ii) develop the simulation tools for soft and
elastic flagellum, accounting for the contact effect between flagella.

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