# Advanced turning maneuver of a many-legged robot using pitchfork bifurcation

Shinya Aoi1, Ryoe Tomatsu1, Yuki Yabuuchi1, Daiki Morozumi1, Kota Okamoto1,
Soichiro Fujiki2, Kei Senda1, and Kazuo Tsuchiya1
1 Dept. of Aeronautics and Astronautics, Graduate School of Engineering, Kyoto
University, Kyoto daigaku-Katsura, Nishikyo-ku, Kyoto 615-8540, Japan
2 Dept. of Physiology and Biological Information, School of Medicine, Dokkyo
Medical University, 880 Kita-Kobayashi, Mibu-machi, Shimotsuga-gun Tochigi
321-0293, Japan

###### Abstract

Legged robots have excellent terrestrial mobility for traversing diverse
environments and thus have the potential to be deployed in a wide variety of
scenarios. However, they are susceptible to falling and leg malfunction during
locomotion. Although the use of a large number of legs can overcome these
problems, it makes the body long and leads to many legs being constrained to
contact with the ground to support the long body, which impedes
maneuverability. To improve the locomotion maneuverability of such robots, the
present study focuses on dynamic instability, which induces rapid and large
movement changes, and uses a 12-legged robot with a flexible body axis. Our
previous work found that the straight walk of the robot becomes unstable
through Hopf bifurcation when the body axis flexibility is changed, which
induces body undulations. Furthermore, we developed a simple controller based
on the Hopf bifurcation and showed that the instability facilitates the
turning of the robot. In this study, we newly found that the straight walk
becomes unstable through pitchfork bifurcation when the body-axis flexibility
is changed in a way different from that in our previous work. In addition, the
pitchfork bifurcation induces a transition into a curved walk, whose curvature
can be controlled by the body-axis flexibility. We developed a simple
controller based on the pitchfork-bifurcation characteristics and demonstrated
that the robot can perform a turning maneuver superior to that with the
previous controller. This study provides a novel design principle for
maneuverable locomotion of many-legged robots using intrinsic dynamic
properties.

###### Index Terms:

Many-legged robot, Maneuverability, Instability, Pitchfork bifurcation, Curved
walk, Turning

##  I Introduction

Legged locomotion, such as that of animals, allows excellent terrestrial
mobility for traversing diverse environments. Legged robots thus have
potential to be deployed in a wide variety of scenarios, such as search and
rescue [21, 35], hazardous environment operation and exploration [9, 51], and
planetary exploration [6, 55]. Various legged robots with agile animal-like
locomotion have recently been developed [1, 4, 23, 25, 24, 30, 33, 36, 37, 40,
46]. However, most of these robots have four legs and falling, which may
result in the breakdown of mechanical and electrical components and from which
it is difficult to recover, is inevitable during locomotion. Furthermore,
damage to even one leg greatly degrades their locomotive performance [12]. The
use of a large number of legs prevents falling and allows a certain level of
leg malfunction to be tolerated [27, 34].

Although the use of a large number legs has advantages for legged robots, it
makes the body long and increases the difficulty of motion planning and
control due to the many intrinsic degrees of freedom and complex interaction
with the environment. In particular, many legs are physically constrained to
be in contact with the ground to support the long body, which can impede
maneuverability. While humans and quadrupeds lean their bodies to enhance
turning maneuvers, the underlying mechanism of agile locomotion using a large
number of legs remains unclear from biological and engineering viewpoints
[18]. Thus, maneuverable locomotion for robots with a large number of legs
remains challenging.

Conventional controllers precisely plan the motion of all degrees of freedom
of the robot (e.g., how the long body is bent, where each foot touches the
ground, and in what order the legs move) and control the robot to stabilize
the desired motion. However, this approach has huge computational and energy
costs, making it inefficient. To design a simple and efficient controller with
high locomotor performance, the fundamental dynamic principles embedded in the
robot dynamics, including the interaction with the environment, should be
fully utilized [1, 10, 29, 30].

For maneuverable locomotion of many-legged robots that overcomes the above
difficulties and the limitations of conventional approaches, the present study
focuses on dynamic instability, which induces rapid and large movement
changes, and uses a 12-legged robot whose body axis is flexible. Our previous
work [2] showed that although many ground contact legs can impede
maneuverability, they induce straight walk instability and body undulations
through Hopf bifurcation when the body-axis flexibility is changed;
bifurcation qualitatively changes a dynamical system by changing a parameter,
and Hopf bifurcation changes the stability of an equilibrium point and creates
a limit cycle [47]. Stability refers to the capability to resist and recover
from disturbances; straight walk instability is thus expected to allow the
robot to easily change walking direction. Therefore, we developed a simple
controller based on the straight walk instability induced by the Hopf
bifurcation to change walking direction without precise motion planning and
control, and demonstrated that the straight walk instability facilitates the
turning of the robot [3].

In the present study, we show that the pitchfork bifurcation of the straight
walk is caused by changes in the body-axis flexibility in a way different from
that in previous work [2, 3]; pitchfork bifurcation changes the stability of
an equilibrium point and creates two equilibrium points [47]. The pitchfork
bifurcation not only destabilizes straight walking, but also causes curved
walking, where the flexible body axis forms a curved shape. Furthermore, we
found that the curvature of curved walking can be controlled by the body-axis
flexibility. We developed a simple control strategy based on the pitchfork-
bifurcation characteristics, which improved the turning maneuver compared to
that achieved using Hopf bifurcation. This study also provides a design
principle for a simple and efficient control scheme to create maneuverable
locomotion for various robots using intrinsic dynamic properties.

##  II Robot

We used the many-legged robot developed in [2] and improved in [3]. The total
length and mass are 135 cm and 8.5 kg, respectively. The robot consists of six
body segment modules (modules 1–6), as shown in Fig. 1. Each module is
composed of a single body and one pair of legs and has the same length. The
body segments are passively connected by yaw joints (yaw joints 1–5) in which
torsional springs (the spring constant is $k_{i}$ ($i=1,\dots,5$)) and
potentiometers are installed at the axes. The yaw joint angles are zero when
the body segments are aligned. Each leg has two links connected by pitch
joints. The legs in the first module (module 1) have an additional link
connected by a yaw joint to supplement the control of the walking direction
during turning tasks. Each leg joint is manipulated by an encoder-equipped
motor. The first module has a laser range scanner with a viewing angle of 240
deg (Hokuyo, URG-04LX) to find the relative position of a target for turning.
The robot was controlled by an external host computer (Intel Pentium 4 2.8
GHz, RT-Linux) with 2-ms intervals, and walked on a flat wooden floor with a
vinyl floor mat to suppress slipping. The computer control signals and
electric power were provided via external cables, which were kept slack and
suspended to avoid influencing the locomotor behavior of the robot.

Figure 1: (A) Photograph and (B) schematic model of many-legged robot. The
robot consists of six modules, each of which has one body segment and one pair
of legs. The legs are controlled by two pitch joints so that the leg tips
follow a periodic trajectory, including the anterior extreme position (AEP)
and the posterior extreme position (PEP). Body segments are passively
connected by yaw joints with installed torsional springs. The legs in the
first module have additional yaw joints to change the walking direction. A
laser range scanner is installed on the first module to find a position
relative to a target.

To make the robot walk in a straight line, we controlled the legs using the
two pitch joints of each leg to follow the desired movement, which consists of
two parts, namely half of an elliptical curve that starts from the posterior
extreme position (PEP) and ends at the anterior extreme position (AEP), and a
straight line from the AEP to the PEP (Fig. 1B). In the straight line, the leg
tips moved from the AEP to the PEP at a constant speed parallel to the body.
We set the duration of the half elliptical curve to $0.29$ s, that of the
straight line to $0.31$ s, and the distance between the AEP and the PEP in
each leg to $3$ cm. The left and right legs in each module moved in antiphase,
and the relative phase between the ipsilateral legs on adjacent modules was
set to $2\pi/3$ rad. When the leg yaw joint angles of the first module were
fixed so that the leg tip trajectories were parallel to the body, the robot
was expected to walk in a straight line while keeping the body segments
parallel to each other because torsional springs were installed on the body-
segment yaw joints and all support-leg tips moved parallel to the body
segments at an identical speed.

##  III Pitchfork bifurcation of straight walk

###  III-A Experimental results

Our previous work [2, 3] revealed that when we used torsional springs with the
same spring constant for all body-segment yaw joints (yaw joints 1–5) and
changed the spring constant uniformly among the joints, the straight walk
became unstable through Hopf bifurcation, which induced body undulations. This
bifurcation was verified by a Floquet analysis, which investigates the
stability of solutions of linear differential equations with periodic
coefficients, with a simple physical model. In this study, we performed robot
experiments of walking in a straight line, where we changed the body-axis
flexibility in a way that was different from that in our previous work.
Specifically, we used the same torsional spring for yaw joints 2–5 with
$k_{i}=41$ Nmm/deg ($i=2,\dots,5$) and various torsional springs for yaw joint
1 with $k_{1}=15$, $17$, $21$, $28$, $41$, and $75$ Nmm/deg. We set all the
body segments parallel to each other as the initial conditions. The leg yaw
joints in the first module were fixed during the experiments and we did not
attempt to control the walking direction, that is, the walking direction was
an open loop.

Figure 2: Characteristics of curved walk for $k_{1}$ values below the
threshold value. Yaw joint angles for (A) straight walk with $k_{1}=41$
Nmm/deg and (B) curved walk with $k_{1}=15$ Nmm/deg (see Movie 1). (C) Average
angle of yaw joint 1 during curved walk for $1/k_{1}$ that indicates pitchfork
bifurcation. The data points and error bars correspond to the means and
standard errors, respectively, of the results of five experiments. (D) Radius
of curvature of body axis for $1/k_{1}$. The data points and error bars
correspond to the means and standard errors, respectively, of the results of
10 experiments. Photographs of (E) straight walk with $k_{1}>\hat{k}_{1}$, (F)
curved walk with small curvature with $k_{1}\sim\hat{k}_{1}$, and (G) curved
walk with large curvature with $k_{1}<\hat{k}_{1}$.

When we used large spring constants for $k_{1}$, the robot kept walking in a
straight line as expected, and the body segments were aligned, with all body-
segment yaw joint angles being almost zero (Figs. 2A and E, see Movie 1).
However, when $k_{1}$ was set to below a threshold value, the body-segment yaw
joints showed non-zero angles with the same sign; that is, the body axis was
curved and the robot walked in a curved line (Figs. 2B, F, and G, see Movie
1). Specifically, the robot walked in a curved line for $k_{1}=15$, $17$,
$21$, and $28$ Nmm/deg, but not for $k_{1}=41$ and $75$ Nmm/deg. Furthermore,
both left- and right-curved walking appeared. Figure 2C shows the angles of
yaw joint 1 for $1/k_{1}$ averaged over 5 s during a curved walk (the angles
for the other body-segment yaw joints are shown in Fig. 3). Although the
angles slightly differ among yaw joints 1–5 partly due to feet slippage, these
data indicate that the body axis shows a curved shape. Furthermore, while the
fluctuation among the trials increases with $1/k_{1}$, the magnitude of these
angles increases with $1/k_{1}$. These results suggest that the presence of
pitchfork bifurcation depends on $k_{1}$. These angle data were fitted by the
square root of $1/k_{1}$ [47]. The bifurcation point was estimated to be
$k_{1}=34$ Nmm/deg ($1/k_{1}=0.030$ deg/Nmm).

Figure 3: Average absolute angles of body-segment yaw joints during curved
walk for $1/k_{1}$. The data points and error bars correspond to the means and
standard errors, respectively, of the results of 10 experiments.

The dependence of the body-segment yaw joint angles on $1/k_{1}$ (Fig. 2C and
Fig. 3) indicates the change of the curved shape of the body axis for the
curved walk. Figure 2D shows the radius of curvature $r$ of the body axis for
$1/k_{1}$ calculated as $r=5L/\sum_{i=1}^{5}|\theta_{i}|$, where $\theta_{i}$
is the angle of yaw joint $i$ ($i=1,\dots,5$) and $L$ is the length of the
body segments. This figure shows that we can control the curvature of the body
axis to perform a curved walk by adjusting $k_{1}$ through pitchfork
bifurcation.

The pitchfork bifurcation also depends on parameters other than $k_{1}$. To
investigate this, we used different values for the spring constant of yaw
joints 2–5, gait speed, and relative phase between the ipsilateral legs on
adjacent modules. Specifically, we changed $k_{2-5}$ from $41$ to $28$
Nmm/deg, the distance between the AEP and the PEP in each leg (Fig. 1B) from
$3$ to $1.8$ cm, or the relative phase from $2\pi/3$ to $\pi$, and performed
the same experiments shown in Fig. 2. As a result, these values also induced a
curved walk below a critical value of $k_{1}$. However, the estimated
bifurcation point and radius of curvature of the body axis changed as shown in
Fig. 4. Specifically, when the spring constant of yaw joints 2–5 and gait
speed decreased, the estimated bifurcation point decreased from $k_{1}=34$ to
23 and 25 Nmm/deg ($1/k_{1}=0.030$ to 0.043 and 0.040 deg/Nmm), respectively.
However, the radius of curvature remained almost unchanged. In contrast, while
the relative phase did not change the bifurcation point as much ($k_{1}=31$
Nmm/deg, $1/k_{1}=0.032$ deg/Nmm), it achieved a smaller radius of curvature.

Figure 4: Change in radius of curvature by different values for the spring
constant of yaw joints 2–5, gait speed, and relative phase between the
ipsilateral legs on adjacent modules. The data points and error bars
correspond to the means and standard errors, respectively, of the results of
five experiments.

###  III-B Verification by Floquet analysis with simple physical model

The robot experiments suggested that the presence of pitchfork bifurcation in
the straight walk depends on the spring constant $k_{1}$ (Figs. 2C, 3, and 4).
We verified this bifurcation from a theoretical viewpoint using a Floquet
analysis with a simple physical model, as done in our previous work [2]. The
model was simplified from the original high-dimensional mechanical model to
extract the fundamentals of locomotion dynamics (Fig. 5A). In particular, the
model was two-dimensional because the movements were designed to make the
robot walk without up-and-down, roll, or pitch motions of the body segments.
Furthermore, because an important role of legs in locomotion is to receive
reaction forces from the floor, we neglected the inertial force of the legs
and instead modeled the reaction forces at the leg tips based on the geometric
conditions. Specifically, we assumed that the leg tips move relative to the
body segments as designed and that the leg tips receive the friction forces
during the straight line from the AEP to the PEP (Fig. 1B), which are
proportional to the velocities relative to the floor.

Figure 5: Floquet analysis using simple two-dimensional model. (A) Simple
model. (B) Floquet exponents when $k_{1}$ was varied. Different colors
represent different exponents. The exponent in red crosses into the right-half
plane along the real axis, indicating pitchfork bifurcation. The change in
critical value of $k_{1}$ by (C) spring constant of yaw joints 2–5, gait
speed, and relative phase between the ipsilateral legs on adjacent modules,
and (D) number of modules.

The equations of motion of the simple model can be expressed as

$\displaystyle K(q)\ddot{q}+h(q,\dot{q})=u(q,\dot{q})+\lambda(q,\dot{q},t)$
(1)

where $q=[x\,y\,\theta_{0}\,\cdots\,\theta_{5}]^{\scriptsize{\mbox{T}}}$,
$[x\,y]$ and $\theta_{0}$ are the position and yaw angle of the first module,
respectively; $K(q)$ is the inertia matrix; $h(q,\dot{q})$ is the nonlinear
term; $u(q,\dot{q})$ is the torque term of the torsional springs; and
$\lambda(q,\dot{q},t)$ is the reaction force term. Because the leg tips move
periodically relative to the body segments, the reaction force $\lambda$
becomes a function of time $t$. The detailed description and derivation of the
equations of motion (1) are shown in our previous work [2]. During the
straight walk of the model, we can write
$\hat{q}=[vt+x_{0}\,y_{0}\,0\,\cdots\,0]^{\scriptsize{\mbox{T}}}$ and
$\dot{\hat{q}}=[v\,0\,0\,\cdots\,0]^{\scriptsize{\mbox{T}}}$, where $x_{0}$
and $y_{0}$ represent the initial position of the first module and $v$ is the
velocity of the leg tips relative to the body segments. For
$z^{\scriptsize{\mbox{T}}}=[q^{\scriptsize{\mbox{T}}}\,\dot{q}^{\scriptsize{\mbox{T}}}]$,
the linearization of the equations of motion (1) for a straight walk using
$z=\hat{z}+\delta z$ gives

$\displaystyle\delta\dot{z}=A(t)\delta z$ (2)

Because the movements of the leg tips are periodic with the gait cycle $\tau$,
$A(t+\tau)=A(t)$ is satisfied. The detailed description and derivation of the
linearized equation (2) are also shown in our previous work [2]. The
fundamental solution matrix $Z(t)$ of the linearized equation with periodic
coefficients (2) can be expressed as

$\displaystyle Z(t)=\Phi(t)e^{\Lambda t}$ (3)

where $\Phi(t+\tau)=\Phi(t)$ [19]. Because we can use an identity matrix for
$Z(0)$ and $\Phi(0)$, the integration of (2) from $t=0$ to $\tau$ yields

$\displaystyle
Z(\tau)=\Phi(\tau)e^{\Lambda\tau}=\Phi(0)e^{\Lambda\tau}=e^{\Lambda\tau}$ (4)

The Floquet exponents (eigenvalues of the constant matrix $\Lambda$) and
corresponding eigenvectors explain the behavior of the model. Specifically,
when all real parts of the exponents are negative, the straight walk of the
model is asymptotically stable. In contrast, if any real part of the exponents
is positive, the straight walk becomes unstable. In our previous work [2],
when all the spring constants of yaw joints 1–5 of the simple model were
decreased uniformly, one pair of the Floquet exponents crossed the imaginary
axis from the left-half plane and entered the right-half plane above a
critical value of the spring constant, which indicates an oscillatory
destabilization of the straight walk to produce body undulations and implied
Hopf bifurcation (strictly speaking, this corresponds to Neimark-Sacker
bifurcation when considering the periodicity of the gait cycle [26]).
Furthermore, the relative amplitude and phase between the components of the
destabilizing eigenvector were comparable to those of the yaw joint movements
during the robot experiments. The Floquet analysis using a simple model is
useful for verifying the bifurcation observed in the robot experiments.

Figure 5B shows the Floquet exponents of the simple model when $k_{1}$ was
varied while the other spring constants $k_{i}$ ($i=2,\dots,5$) remained
fixed, as done in the robot experiments. Except for the zero exponents, all
exponents lie in the left-half plane for large $k_{1}$. However, with
decreasing $k_{1}$, one exponent moves along the real axis and enters the
right-half plane above $k_{1}=12$ Nmm/deg. Although this critical value is
smaller than the bifurcation point estimated in the robot experiments in
Section III-A, the components of the destabilizing eigenvector for yaw joints
1–5 at the critical point were $0.64$, $0.36$, $0.48$, $0.46$, and $0.16$,
respectively, showing the same sign. This indicates that the straight walk is
destabilized to produce a curved walk (positive disturbance induces right-
curved walk and negative disturbance induces left-curved walk) and implies
pitchfork bifurcation. Furthermore, the ratio between the components of the
destabilizing eigenvector is consistent with the ratio between the yaw joint
angles during the curved walk of the robot experiments (Fig. 3). Although this
analysis does not estimate the radius of curvature after the bifurcation due
to the limitation of the linear analysis, these results verify the
destabilization of the straight walk and the emergence of a curved walk
through pitchfork bifurcation, as observed in the robot experiments.

We also investigated the parameter dependence of the pitchfork bifurcation.
Figure 5C shows how the critical value of $k_{1}$ changes as the spring
constant of yaw joints 2–5, gait speed, and relative phase between the
ipsilateral legs on adjacent modules change in the same way as in the robot
experiments in Fig. 4. In addition, Fig. 5D shows the change in the critical
value when the number of the body segments is increased. When the spring
constant of yaw joints 2–5 and gait speed decrease, the critical value
decreases. However, the relative phase does not change the critical value as
much. These results are consistent with those in the robot experiments (Fig.
4). The number of body segments also does not change the critical value very
much. Although we investigated the effects of only changes in the spring
constant of yaw joints 2–5, gait speed, relative phase, and number of body
segments, this Floquet analysis can investigate other parameters, such as the
length and mass, as performed in our previous work [2] for Hopf bifurcation.

##  IV Turning maneuverability

###  IV-A Turning strategy based on pitchfork bifurcation

To investigate the maneuverability of the robot achieved with the aid of
pitchfork bifurcation, we focused on a turning task in which the robot
approached a target located on the floor in a direction different from that to
which the robot was oriented, as performed in our previous work [3], which
investigated the maneuverability with the aid of Hopf bifurcation. For a
target at any location (relative angle $\psi$ and distance $R$), there exists
a unique radius of curvature $\hat{r}$ of the curved walk with which the robot
will approach the target (Fig. 6A). Because the radius of curvature $r$ of the
body axis induced by the pitchfork bifurcation monotonically decreases with
$1/k_{1}$ (Fig. 2D), $k_{1}=\hat{k}_{1}$ is uniquely determined so that
$r=\hat{r}$. This means that when we use $k_{1}=\hat{k}_{1}$, the robot
spontaneously approaches the target due to the pitchfork bifurcation
characteristics, which is an optimal strategy for turning. However, this
strategy is feedforward, depending on the initial conditions of the robot and
target, that is, an open loop for the walking direction. In particular, the
direction in which the robot turns (left or right) depends on the initial
robot conditions because of the pitchfork bifurcation characteristics, and
thus this strategy does not guarantee the success of the turning tasks.
Therefore, we also used a supplementary turning controller developed in our
previous work [3], which uses the laser range scanner of the first module to
measure the relative target angle and manipulates the leg yaw joints of the
first module to approach the target based on the measured target angle (see
Appendix B). This supplementary controller is a closed loop for the walking
direction and allows the robot to approach the target even when
$k_{1}\neq\hat{k}_{1}$.

Figure 6: Turning task. (A) Radius of curvature $\hat{r}$ of curved walk with
which the robot approaches a target (relative angle $\psi$ and distance $R$).
(B) Trajectory of the first module on the floor, (C) target distance, and (D)
relative target angle for five experiments for three spring constants with
$\psi=45^{\circ}$, $R=1.3$ m, $\hat{r}=0.88$ m, and $1/\hat{k}_{1}=0.048$
deg/Nmm (see Movie 2). (E) Evaluation criteria $\varepsilon_{1}$,
$\varepsilon_{2}$, and $\varepsilon_{3}$ for $1/k_{1}$. The data points and
error bars correspond to the means and standard errors, respectively, of the
results of five experiments. Photographs of (F) initial conditions, (G)
unsuccessful approach with $k_{1}>\hat{k}_{1}$, (H) successful approach with
$k_{1}\sim\hat{k}_{1}$, and (I) unsuccessful approach with
$k_{1}<\hat{k}_{1}$.

###  IV-B Experimental results

For the initial conditions, we used $\psi=45^{\circ}$ and $R=1.3$ m for the
relative angle and distance between the first module and the target,
respectively, which yielded $\hat{r}=0.88$ m and $\hat{k}_{1}=21$ Nmm/deg
($1/\hat{k}_{1}=0.048$ deg/Nmm), and set all body-segment yaw joint angles to
zero (Fig. 6F). Figure 6B shows the trajectory of the first module on the
floor during the turning task for three torsional spring constants, namely
$k_{1}=15$ ($<\hat{k}_{1}$), $21$ ($\sim\hat{k}_{1}$), and $41$ Nmm/deg
($>\hat{k}_{1}$). Figures 6C and D show the time profiles of the target
distance and relative target angle with respect to the walking direction,
respectively, for these three spring constants. When the distance was less
than 0.15 m, we assumed that the robot reached the target and this task was
successfully completed. For $k_{1}=41$ Nmm/deg ($>\hat{k}_{1}$), the robot
hardly changed walking direction and the first module trajectory bulged
outward. As a result, the robot could not reach the target (Fig. 6G, see Movie
2). For $k_{1}=15$ Nmm/deg ($<\hat{k}_{1}$), although the robot could quickly
change walking direction, it moved in directions away from the target due to
the small radius of curvature created by pitchfork bifurcation and could not
reach the target (Fig. 6I, see Movie 2). In contrast, for $k_{1}=21$ Nmm/deg
($\sim\hat{k}_{1}$), the robot reached the target through the optimal curved
walk generated by pitchfork bifurcation (Fig. 6H, see Movie 2).

To quantitatively clarify the turning performance dependence on $k_{1}$, we
employed three evaluation criteria, namely $\varepsilon_{1}$,
$\varepsilon_{2}$, and $\varepsilon_{3}$. For the criterion $\varepsilon_{1}$,
we used the distance of the target at 23 s (the earliest time at which the
task is successfully completed) to evaluate how quickly and successfully the
robot approached the target. For the criterion $\varepsilon_{2}$, we used the
absolute value of the relative target angle with respect to the walking
direction at 23 s to evaluate how quickly and successfully the robot was
oriented to the target. For the criterion $\varepsilon_{3}$, we used the
amount of the control input during the task from the supplementary turning
control in the leg yaw joints of the first module (Appendix B) to evaluate how
efficiently the robot performs turning. Specifically,
$\varepsilon_{3}=\int(\hat{\psi}_{1}^{2}+\hat{\psi}_{2}^{2})dt$. Figure 6E
shows the results for $1/k_{1}$. All criteria showed minimum values around
$k_{1}=\hat{k}_{1}$, which means that the turning strategy using pitchfork
bifurcation achieved the best performance and that the robot made the best use
of the curved walk induced by pitchfork bifurcation to complete the turning
task.

To verify the performance of the proposed controller using pitchfork
bifurcation, we additionally performed the same experiment as that in Fig. 6B
but using different initial conditions of the target, namely $\psi=40^{\circ}$
and $R=1.5$ m, which yielded $\hat{r}=1.2$ m and $\hat{k}_{1}=26$ Nmm/deg
($1/\hat{k}_{1}=0.039$ deg/Nmm). Figures 7A, B, and C show the evaluation
criteria $\varepsilon_{1}$, $\varepsilon_{2}$, and $\varepsilon_{3}$,
respectively, for $1/k_{1}$. All criteria show minimum values around
$k_{1}=\hat{k}_{1}$, which means that the turning strategy using pitchfork
bifurcation achieved the best performance, in the same way as shown in Fig.
6E. The results show similar trends, which verifies the performance of the
proposed controller.

Figure 7: Evaluation criteria (A) $\varepsilon_{1}$, (B) $\varepsilon_{2}$,
and (C) $\varepsilon_{3}$ for $1/k_{1}$ for different conditions
($1/\hat{k}_{1}=0.039$ deg/Nmm). The data points and error bars correspond to
the means and standard errors, respectively, of the results of five
experiments.

###  IV-C Contribution of supplementary turning control

As demonstrated above, the robot achieved maneuverable and efficient turning
to approach a target using the curved walk induced by the pitchfork
bifurcation. However, when the robot starts the approach with the body axis
straight, it takes some time for the convergence to a curved walk, as shown in
Fig. 2B. In addition, because the initial straight walk corresponds to the
unstable solution of the pitchfork bifurcation, it is unclear whether the
robot will turn to the left or right, as shown in Fig. 2C (the disturbance
determines the turning direction and the robot continues walking straight
unless disturbed). Because the use of the optimal spring constant
$\hat{k}_{1}$ does not necessarily guarantee the success of the approach to
the target, we also used the supplementary turning controller in the leg yaw
joints in the first module.

To clarify the contribution of this supplementary controller, we also
performed experiments without using the supplementary controller for
$k_{1}=21$ Nmm/deg ($\sim\hat{k}_{1}$). The experimental conditions were
identical to those in Fig. 6B. Figure 8 compares the trajectory of the first
module on the floor during the turning task with and without the supplementary
controller. In all the trials with the supplementary controller, the robot
successfully approached the target. In contrast, the robot without the
supplementary controller failed in many trials, because the robot started
walking in a straight line and took some time to converge to walking in a
curved line, and it sometimes curved to the different direction from the
target, as expected.

Figure 8: Comparison of the trajectory of the first module on the floor during
the turning task with and without the supplementary turning controller. Nine
trials without the supplementary controller and five trials with the
controller (Fig. 6B) are shown.

###  IV-D Comparison with previous strategy

To examine how the turning performance was improved by pitchfork bifurcation,
we also performed experiments using the turning strategy based on Hopf
bifurcation used in our previous work [3] and compared the performance. For
Hopf bifurcation, we used the same spring constant among the body-segment yaw
joints and employed five spring constants ($k_{i}=8.7$, $11$, $15$, $21$, and
$41$ Nmm/deg, $i=1,\dots,5$) to evaluate the turning performance for $k_{i}$,
where the Hopf bifurcation point is about $k_{i}=18$ Nmm/deg
($1/\hat{k}_{i}=0.057$ deg/Nmm), as obtained in our previous work [3]. The
experimental conditions were identical to those in Fig. 6E except for the
spring constants of the body-segment yaw joints. Figures 9A, B, and C compare
the turning performance in terms of the criteria $\varepsilon_{1}$,
$\varepsilon_{2}$, and $\varepsilon_{3}$, respectively, between the strategies
based on pitchfork and Hopf bifurcations. All criteria for Hopf bifurcation
showed minimum values in the unstable region, as observed in our previous work
[3]. However, the minimum values of pitchfork bifurcation are lower than those
of Hopf bifurcation for all criteria. This means that the turning strategy
based on pitchfork bifurcation created by tuning the body-axis flexibility is
superior to that based on Hopf bifurcation, which was developed in our
previous work [3].

Figure 9: Comparison of evaluation criteria (A) $\varepsilon_{1}$, (B)
$\varepsilon_{2}$, and (C) $\varepsilon_{3}$ between the turning strategies
based on pitchfork bifurcation and Hopf bifurcation. The data points and error
bars correspond to the means and standard errors, respectively, of the results
of five experiments.

##  V Conclusion and discussion

In this study, we found that the straight walk of a many-legged robot with
flexible body axis becomes unstable through pitchfork bifurcation when the
body-axis flexibility is changed. The straight walk transitioned into curved
walk, whose curvature depended on the body-axis flexibility. We developed a
simple controller based on the pitchfork-bifurcation characteristics and
demonstrated that the robot achieves high turning maneuver superior to the
previous controller based on the Hopf bifurcation.

Maneuverability is related to the ability to change movement direction. When
the movement direction is destabilized during locomotion, the instability
provides driving forces to rapidly change the movement direction and thus
enhances maneuverability. Some military aircraft, such as the F-16, are
designed to be aerodynamically unstable to enhance maneuverability [8, 28].
The use of dynamic instability is thus useful from an engineering viewpoint.

The strategy of using movement direction instability to enhance
maneuverability is also used by animals. Because the instability is determined
by the body dynamics during interaction with the environment, it is prominent
in locomotion generated through aerodynamics and hydrodynamics, such as the
locomotion of flying insects [15, 38, 49] and sea animals [16, 17, 53]. It
also appears in legged locomotion. When the center of mass is high, as in
mammals, whose legs are under the body, leaning the body to the left or right
induces instability and helps turning [11, 42]. However, when the center of
mass is low, as in reptiles and arthropods, whose legs are away from the side
of the body, locomotor behavior is almost two-dimensional because the center
of mass moves in a horizontal plane. Therefore, the effect of body leaning is
small and thus such a turning strategy cannot be used, which implies that the
stability of the walking direction in the horizontal plane becomes more
crucial. It has been suggested that cockroaches manipulate the position of the
reaction forces from the floor entering the body to control the stability of a
straight walk in a horizontal plane and that the straight walk instability
helps their turning [39, 42].

Various bioinspired robots that use their body axis for propulsion, such as
snake robots [7, 45] and fish robots [13, 31, 43], have high maneuverability.
However, legged robots still have difficulty in achieving highly maneuverable
locomotion. This is partly because their interaction with the environment
(i.e., foot contact with the ground) is intermittent due to the repetition of
foot-contact and foot-off phases in leg movement. Although this intermittency
allows the traversal of diverse environments, it can make the robot lose
balance. Therefore, the control design of legged robots has focused on the
avoidance of balance loss using dynamic criteria, such as a supporting polygon
[20] and a zero-moment point [52], and maneuverability has not been well
investigated. Although increasing the number of legs prevents balance loss, it
also increases the number of contact legs, which impedes maneuverability.
Moreover, the number of degrees of freedom to be controlled increases, making
both motion planning and control difficult. In addition, many-legged robots
generally use actuators for controlling not only the leg joints but also body-
segment joints [48, 54], which requires huge computational and energy costs.
In contrast, our robot has passive body-segment joints, which do not directly
control the movement of the body axis, and instead determines the body-axis
flexibility to induce a curved walk by the pitchfork bifurcation in the robot
dynamics. The generation of robot movements with dynamics rather than
actuators is crucial for efficient locomotion [10], and our strategy greatly
reduces the computational and energy costs. Both the pitchfork and Hopf
bifurcations induce the straight walk instability and thus contribute to the
maneuverability. However, because the pitchfork bifurcation causes a curved
walk used for turning, unlike the Hopf bifurcation that causes body
undulations, the pitchfork bifurcation makes a greater contribution.
Furthermore, when such bifurcation is introduced into the locomotion dynamics
of other robots, such as snake and fish robots, through the mechanical and
control design, it would contribute to improving their maneuverability.

In the present study, we investigated the contribution of the pitchfork
bifurcation to maneuverability in robot experiments where the robot performs
turning only once on a flat floor. It is important to clarify the contribution
of the pitchfork bifurcation in more complex scenarios and environments in the
future. In particular, consecutive turnings are crucial for complex scenarios.
However, when we consider the experiments where the robot sequentially
approaches multiple targets placed at different locations on the floor, that
is, the robot performs multiple consecutive turnings, the initial conditions
of the targets, such as the relative angle and distance, will differ in each
turning. Different initial conditions of the targets induce different radiuses
of curvature for optimal turning, which require different body-axis
flexibilities. Because our robot uses torsional springs to determine the body-
axis flexibility, it is impossible to change the flexibility during the
experiment. In future studies, we would like to improve our robot and
controller, for example, by incorporating a variable flexibility mechanism in
the body axis to conduct more complex tasks. In addition, various turning
strategies have been developed in quadruped and hexapod robots to modulate the
leg movements for turning using bioinspired approaches based on central
pattern generators and sensory systems [5, 22, 32, 44, 50, 57] and
optimization techniques [14, 41, 56]. We would like to improve our
instability-based strategy in the body axis movement based on these strategies
to enhance turning maneuverability of multi-legged robots in the future.

##  Appendix A Supplementary movies

We recorded two supplementary movies to show the pitchfork bifurcation of a
straight walk and turning performance in the robot experiments:

  1. Movie 1:

Straight walk, curved walk with a small curvature, and curved walk with a
large curvature using a large spring constant, small spring constant, and very
small spring constant, respectively, for the torsional spring in yaw joint 1.

  2. Movie 2:

Unsuccessful approaches using a spring constant larger and smaller than the
optimal value for yaw joint 1, and successful approach using a spring constant
close to the optimal value.

##  Appendix B Supplementary turning control by leg joints

The optimal turning strategy is feedforward, depending only on the initial
relative position between the robot and target. In addition, the direction in
which the robot turns (left or right) depends on the initial robot conditions
because of the property of pitchfork bifurcation. To guarantee a successful
approach to the target, we used a supplementary feedback-based turning
controller, which was developed in our previous work [3]. Specifically, we
used the relative target angle $\psi$ of the first module measured by the
laser range scanner and the leg yaw joints $\psi_{1}$ and $\psi_{2}$ of the
first module. We determined the desired angles $\hat{\psi}_{1}$ and
$\hat{\psi}_{2}$ of $\psi_{1}$ and $\psi_{2}$ for each gait cycle
[$t_{i}^{n}\leq t<t_{i}^{n}+T$, where $t=t_{i}^{n}$ is the time when the
desired leg tip is at the PEP for the $n$th gait cycle, and $T$ is the gait
cycle duration ($=0.6$ s)] using

$\displaystyle\hat{\psi}_{i}(t)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}\hat{\psi}_{i}(t_{i}^{n})&t_{i}^{n}\leq
t<t_{i}^{n}+t_{\scriptsize{\mbox{start}}}\\\
\hat{\psi}_{i}(t_{i}^{n})+\Delta_{i}\displaystyle\frac{t-t_{i}^{n}-t_{\scriptsize{\mbox{start}}}}{t_{\scriptsize{\mbox{end}}}-t_{\scriptsize{\mbox{start}}}}&t_{i}^{n}+t_{\scriptsize{\mbox{start}}}\leq
t\leq t_{i}^{n}+t_{\scriptsize{\mbox{end}}}\\\
\hat{\psi}_{i}(t_{i}^{n}+t_{\scriptsize{\mbox{end}}})&t_{i}^{n}+t_{\scriptsize{\mbox{end}}}<t<t_{i}^{n}+T\end{array}\right.$
(8) $\displaystyle\Delta_{i}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{l}\psi(t_{i}^{n}+t_{\scriptsize{\mbox{start}}})-\hat{\psi}_{i}(t_{i}^{n}+t_{\scriptsize{\mbox{start}}})\\\
\hskip
56.9055pt|\psi(t_{i}^{n}+t_{\scriptsize{\mbox{start}}})-\hat{\psi}_{i}(t_{i}^{n}+t_{\scriptsize{\mbox{start}}})|<5^{\circ}\\\
5^{\circ}\hskip 48.36967pt\mbox{otherwise}\end{array}\right.$ (12)

where $t_{\scriptsize{\mbox{start}}}$ and $t_{\scriptsize{\mbox{end}}}$ were
set to $40$% and $80$%, respectively, of the duration of the half elliptical
curve of the leg tip trajectory ($=0.12$ and $0.23$ s) determined
experimentally. This controller means that each leg changed its yaw direction
toward the target only during the swing phase with $5^{\circ}$ of the maximum
turning angle for one gait cycle. We also limited the maximum angle of the leg
yaw joint to $5^{\circ}$ during the turning task. This supplementary control
did not aim to make the robot follow the optimal curved path generated by the
turning strategy using pitchfork bifurcation. Instead, it was designed so that
the first module modulated the walking direction based on the target
direction, which solves the problems related to the feedforward property of
the optimal turning strategies and the turning direction due to initial robot
conditions, and furthermore allows the robot to approach the target even when
$k_{1}\neq\hat{k}_{1}$.

## Acknowledgment

This study was supported in part by JSPS KAKENHI Grant Numbers JP17H04914,
JP19KK0377, and JP20H00229; JST FOREST Program Grant Number JPMJFR2021; and
the Inamori Foundation.

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