The Effect of Tail Stiffness on a Sprawling Quadruped Locomotion

Josh Buckley 1, and Yasemin Ozkan-Aydin 2,∗

1Department of Biomedical Engineering, University of Galway, Ireland

2Department of Electrical Engineering, University of Notre Dame, IN, USA

###### Abstract

##

A distinctive feature of quadrupeds that is integral to their locomotion is
the tail. Tails serve many purposes in biological systems including
propulsion, counterbalance, and stabilization while walking, running,
climbing, or jumping. Similarly, tails in legged robots may augment the
stability and maneuverability of legged robots by providing an additional
point of contact with the ground. However, in the field of terrestrial bio-
inspired legged robotics, the tail is often ignored because of the
difficulties in design and control. This study will test the hypothesis that a
variable stiffness robotic tail can improve the performance of a sprawling
quadruped robot by enhancing its stability and maneuverability in various
environments. To test our hypothesis, we add a multi-segment, cable-driven,
flexible tail, whose stiffness is controlled by a single servo motor in
conjunction with a reel and cable system, to the underactuated sprawling
quadruped robot. By controlling the stiffness of the tail, we have shown that
the stability of locomotion on rough terrain and the climbing ability of the
robot are improved compared to the movement with a rigid tail and no tail. The
flexible tail design also provides passively controlled tail undulation
capabilities through the robot’s lateral movement, which contributes to
stability.

## Keywords:

quadruped locomotion, bio-inspired robotics, flexible tail, stability

##  1 Introduction

Figure 1: Examples of terrestrial-tailed animals and robots. The robotic
systems depicted in the figure were designed with inspiration drawn from the
anatomical and behavioral characteristics of the animals featured in the
central image. (A) Muddybot inspired by mudskipper fish to demonstrate the
function of a tail on the locomotion of granular surfaces [28], (B) RISE Ver.3
climbing robot [14], (C) Articulated-tailed quadruped robot [3], (D)
Prosthetic tail to extend human body functions [29], (E) Tailed hexapod robot
[19], (F) Tailbot, a lizard-inspired tailed quadruped robot [8].

Tails have been observed to serve a multitude of functions in various extinct
and extant animal species, ranging from mating and defense to aiding in
locomotion [15, 18, 36, 24, 28]. Some animals, such as lizards, are capable of
detaching their tails as a defensive mechanism to distract predators [1, 16,
30]. Tails also aid in maintaining balance [15, 41] and stability [43],
especially during climbing or navigating through dense vegetation, and provide
maneuvering [13] and grasping [12] abilities. For example, many species of
monkeys use their prehensile tail to grasp branches to support themselves
while feeding [37, 11].

In many cases, tails serve as a crucial propulsion system in both aquatic [23,
25, 9, 10] and terrestrial environments[15, 18, 36, 28]. Fish use their tails
to move through the water by undulating antagonistic muscles that control tail
oscillation [23]. Additionally, fish use their tails for steering and
maintaining balance, making quick movements and turning to avoid obstacles or
predators [22]. Similarly, the tail assists in locomotion activities of
terrestrial animals such as quadrupedal walking [21, 42, 16], climbing [8],
jumping [2, 17], aerial descent [44], and gliding[39]. Some legged animals,
such as kangaroos use their tails as a counterbalance during bipedal hopping,
but when moving slowly, during grazing and social interactions. Animals use
the tail as a “fifth leg”, providing an extra point of contact with the ground
and an additional propulsive force [32, 21].

These natural tail functionalities have served as a source of insight and
inspiration for the development of novel robotic systems that emulate and
replicate the natural movements and functionalities of the biological
organisms [3, 29]. The primary focus of the research conducted to date has
been with regard to tails that allow for inertial adjustments to provide
balance and stability [4, 31], or maneuverability when turning [35, 6, 19, 7,
35, 20], running [8], jumping [5] and climbing [14, 27]. There has been
limited research with respect to tails providing stability via continuous
ground support, and tails that directly aid in propulsion. In [40], a tail was
designed that periodically “taps” off the ground, providing a momentary ground
contact. This design provided some success in obstacle navigation and
releasing the robot when trapped.

Although there are several potential advantages of using tail-like appendages
on robots, there are challenges that need to be overcome to fully realize the
potential of tailed systems [26, 38]. For example, one of the challenges in
developing bio-inspired robotic tails is designing them to be flexible and
agile enough to provide the desired benefits without adding too much weight or
complexity to the overall system. In addition, we need to consider how to
integrate the tail into the overall control system of the robot and how to
power it. There is also a need for more sophisticated control algorithms and
sensors that can accurately sense and respond to the movements and forces
acting on the tail during locomotion.

Figure 2: A sprawling quadruped robot combined with a multi-segment flexible
tail. A computer-aided design of the fully autonomous quadruped robot (about
20 cm long) that includes two body segments with a pair of directional
flexible legs connected via an XL-320 servo motor. A 7.4 V LiPo battery and
Robotis Open-CM 9.04 controller are placed into the box on the top of the body
servo (red box). Two XL-320 servos control the horizontal and vertical motions
of the coupled legs in a segment. A flexible, multi-segmented, cable-driven
tail is attached to the back of the robot.

In this paper, we study how the flexibility of the tail can enhance the
locomotion performance of a sprawled quadruped robot used in [34] in various
environments. We designed a 3D-printed, underactuated articulated tail whose
stiffness could be controlled through a cable-driven servo-actuated system.
Our tail design was inspired by the s-shaped tail of kangaroos, which is
composed of several caudal vertebrae and provides great control when hopping
and walking, as well as physical support when grazing [32]. During the
prototyping stages of the tail, it was found that an excess of vertebra-like
segments led to an overabundance of flexibility. Subsequently, the reduced
number of such segments is sufficient for emulating the characteristic
behavior of a biological vertebrate tail. This approach allowed us to strike a
balance between stability and flexibility, which was imperative for enhancing
the robot’s climbing capability. In evaluating the efficacy of our approach,
we subjected our quadruped robot with a flexible tail to various environments.
Specifically, we tested the robot’s performance on smooth surfaces, inclined
and obstacle environments, and natural terrains. To provide a baseline, we
compared the performance of the robot with that of the tailless and rigid-
tailed robots in each of these environments.

##  2 Materials and Methods

###  2.1 Quadruped Robot Design

In this study, we added a multisegmented tail to the quadruped robot that was
used in the previous studies [33, 34]. The robot has two body segments each
consisting of two directional flexible multi-jointed legs (total length = 12
cm) that are coupled via a rigid 1-DoF mechanism (Fig.2). Springs (spring
constant = 0.2 kg/cm, McMaster; product number, 9654K949) that are connected
between the upper and lower part of the leg joints allow the legs to bend when
encountering obstacles and then return to a neutral position passively once
the obstacle has been traversed [33].When the legs are at their neutral angle
(i.e. the leg coupling mechanism is parallel to the ground), the height of the
CoM of the robot is 5 cm. All the robot parts were 3D-printed with a Stratasys
F170 printer using ABS material.

The robot consists of a Robotis OpenCM 9.04 microcontroller, a lithium polymer
battery (11.1 V, 1000 mAh), and six Robotis Dynamixel XL-320 motors (stall
torque is 0.39 N.m): four for controlling the vertical and horizontal motion
of the coupled legs, one for controlling the rotation of the body, and one for
controlling tail stiffness. Because of the mechanical constraints, in the
horizontal plane, the legs can rotate from $-25^{o}$ to $30^{o}$, and the body
servo can rotate from $-30^{o}$ to $30^{o}$ from their neutral positions. The
servos that control the vertical motion of the legs can rotate $30^{o}$, which
raises the tip of the leg about 4 cm above the ground.

###  2.2 Tail Design

####  2.2.1 Rigid Tail

The rigid tail was designed to be of similar shape to the robot’s legs
(rectangular), but longer in size (90 mm in length, 3.5 mm thick, and 8 mm
wide, Fig. 3A). The top of the tail was connected to the rear servo motor. The
bottom of the tail was filleted, with the removal of sharp corners key to
ensuring the tail does not get stuck on obstacles as it moves. The primary
goal of this design was to provide an extra contact point at the back section
of the robot. Since the legs are coupled, only one of the legs of each segment
are on the ground at each time step [34], and therefore cannot create a
support triangle. Without a tail, the diagonal gait of the robot was
interrupted and led to instability during locomotion [34], with the body of
the tail striking the ground twice per gait cycle. This also led to foot
slippage and insufficient ground clearance, which leads to problems when the
robot attempts to climb and navigate obstacles.

This tail acted passively, but its lateral motion could also be controlled by
the rear servo motor which allows the tail to lift off the ground. This
mechanism could be useful in applications such as climbing or walking on rough
terrain. Were the robot to get stuck on an obstacle, repeated rotation of the
rigid tail from side to side could push the robot out of this position.

####  2.2.2 Flexible Tail - Mechanism and Actuation Design

The tail was comprised of five articulating vertebra-like segments of cuboidal
shape (Fig. 2). The fifth segment of the tail has a narrow, curved end, which
was in contact with the ground. This segment included a fillet on the interior
face, allowing for the tail to pass over obstacles and sharp ledges, without
causing the robot to get stuck. The tail was 3D printed using a Stratasys F170
printer with all parts intact. The links between each segment were designed in
order to eliminate the need for assembly. Dimensions of the linkage component
can be modified in order to alter the rigidity of the tail along with the
range of motion.

Included in the design was a connection component, a box attached to the back
of the robot. This component allowed the tail to attach and detach from the
robot with ease. It also raised the attachment point of the tail to a point
much higher than the body. This was important as it allowed a longer tail to
be developed, leading to an increased range of motion, and thus more effective
robot locomotion. The connection component also included small loops at the
top and at the side, which helped guide the cabling when wrapping around the
reel.

Figure 3: Rigid and Flexible Tail. A. Rigid tail B. States of the flexible
tail. Tail stiffness is increasing from left to right, causing the height of
the rear of the body to increase (SI Movie).

To control the flexibility of the tail we designed a mechanism that consists
of a custom-made reel that is attached to the rear servo motor (Fig. 2 and
Fig. 3). Kevlar wire was then threaded through the front and rear channels of
the tail and wrapped around the reel. which was attached to the servo motor.
The function of the rear wire was to flex the tail inwards (towards the body),
while the front wire had the opposite function; it extends/raises the tail
upwards (away from the body). These two cables can be configured to act as an
antagonistic pair as follows. The two cables are wrapped around the reel in
opposite directions (rear cable in an anticlockwise direction, front cable in
a clockwise direction). When the servo motor is in its initial condition, the
tail is relaxed and very flexible. By rotating the servo motor anticlockwise,
the tension in the rear cable was increased and the front cable was relaxed.
This tension resulted in a force that drove the tail toward the body. As the
ground was obstructing the tail’s motion, the tail applied a force on the
ground which resulted in pushing the robot higher. It was noted that this
could be beneficial in climbing applications.

A different approach to this antagonistic pair configuration is to increase
the rotation range of the tail, allowing the tail to completely flex towards
the body. The front surface of the tail (surface furthest from the body) is
then in contact with the ground. The servo motor would then rotate in a
clockwise direction, essentially ’flicking’ the tail in the opposite direction
and extending the tail away from the body. This approach resulted in a greater
generated force and drove the body of the robot higher. This flicking
mechanism could be implemented periodically to help stop the robot from
getting stuck on ledges or objects.

Both of these approaches resulted in driving the robot upwards, increasing the
clearance height of the legs and improving climbing ability. However, these
designs had an adverse effect on the robot’s stability. To solve this, the
cabling was reconfigured so that both cables were wrapped around the reel in
the same direction (anticlockwise). The tension in the cables was carefully
tuned so that when the reel rotated anticlockwise, the tail would be pulled
into an approximate ’s’ shape (Fig.3-B), similar to the shape of vertebrate
tails observed in certain biological species such as the kangaroo [32]. With
an applied rotation of the tail servo motor of between 55 to 90 degrees
(anticlockwise), the distance between the segments was reduced, which
increases the rigidity (or decreases the flexibility) of the tail. To return
the tail to its fully relaxed, flexible state, the servo motor was rotated in
a clockwise direction. Tension in the cables would be released and would have
no effect on the motion of the tail, allowing it to act passively. Two
different implementations of this approach were tested, primarily the effect
of periodically stiffening and relaxing the tail during key phases of the gait
cycle, and the effect of using a touch sensor to sense when the tail should
stiffen and relax. More detail on the stiffening/relaxation timing can be
found in the Results Section 3.3.3.

##  3 Experimental Results

The effectiveness of the controlled flexible tail on the robot’s locomotion
was tested and compared to identical tests carried out on the robot with a
rigid tail and the robot with no tail. The experiments aimed to simulate
diverse and challenging terrains that the robot might encounter in real-world
scenarios. Four scenarios were chosen to test the robot’s locomotion
capabilities; (1) walking on a flat surface (indoor), (2) climbing a series of
steps (indoor), (3) walking on inclined surfaces (indoor), and (4) traversal
of rough, natural terrain (outdoor). For each indoor experiment, there were
five trials completed. The robot began at the same location for each trial,
and the body position was kept constant by returning the servo motors to their
starting position before each test was conducted. In all of the experiments,
the robot walked with a diagonal gait in which the diagonal legs are on the
ground at 50% of the stance phase while undulating its body, [34, 33]. The
experiments were captured using a side view and a vertical aerial view using
two Logitech C920 pro webcams.

In order to investigate the effectiveness of the flexible tail in facilitating
stable and adaptive locomotion in natural environments, a series of outdoor
experiments were conducted on diverse terrains including mulch, pebbles, and
sand. The experiments were designed to demonstrate the challenging and
unpredictable characteristics of natural environments and to evaluate the
robot’s ability to maintain balance and stability on such surfaces.

Figure 4: Locomotion on flat terrain with different tail configurations.
Experimental snapshots of the robot from side view with A no tail B. a rigid
tail C. a flexible tail with a stiff setting, D. the flexible tail with a
relaxed setting, E. the flexible tail with an alternating relax/stiff setting.
The yellow trajectories show the tip trajectory of the robots during walking
(SI Movie).  Figure 5: Locomotion on a smooth surface with no tail, rigid
tail, and flexible tail. The mean and standard deviation of body length (BL)
per cycle for locomotion on smooth flat (blue), $10^{o}$ (red), $15^{o}$
(yellow), and $20^{o}$ (purple) terrain with different tail configurations.
Each experiment was repeated at least five times with a minimum of four cycles
per run.

###  3.1 Flat Terrain

The primary objective of the experiments was to evaluate the robot’s capacity
to maintain a straight trajectory while navigating flat, smooth terrain, which
is an essential requirement for various practical applications in the real
world. The velocity of the robot was also analyzed for different tail
configurations (no tail, rigid tail, and flexible tail), measured in terms of
the distance traveled per gait cycle, normalized to the robot’s body length
(BL/cycle). The experiments aimed to assess the impact of the tail design and
properties on the robot’s locomotion performance, with particular attention to
stability, speed, and accuracy.

In the experiments, black tape was placed on the surface perpendicular to the
“starting line” to qualitatively examine the stability of the robot by
observing the deviations from the path in each of the configurations. Aspects
such as foot clearance and rocking were examined to determine their effect on
the robot’s balance. Balance disturbances lead to a reduction in locomotion
speed, disruption to the gait cycle, and unexpected turning.

####  3.1.1 Locomotion with No Tail

This experiment was carried out as described in section 3.1. In each of the
five trials conducted, the robot was unable to meet the goal of traveling
along a straight line while on level ground. The robot deviated $2.4\pm
0.71^{o}$ per cycle from the straight path and traveled with a speed of
$0.45\pm 0.03$ BL/cycle (Fig. 5).

This test revealed many problems in the robot’s locomotion without a tail.
Firstly, the body of the tail struck the ground twice during each gait cycle.
The front of the robot pitched upwards during the swing phase causing the rear
of the robot to strike the ground before foot placement could occur (Fig. 4A).
Secondly, the rear legs of the robot did not have sufficient ground clearance
during the swing phase, resulting in the rear swinging leg dragging along the
ground. Finally, the front legs were found to have excessive ground clearance,
leading to increased motion perpendicular to the ground, rather than the
desired motion parallel to the ground. These three problems were inexplicably
linked to the upwards pitching of the robot, and caused the robot’s diagonal
gait to be asymmetrical and uneven. This impaired the robot’s balance, in turn
hindering the robot’s ability to travel along a straight line.

####  3.1.2 Locomotion with Rigid Tail

Five replications of the previous experiment were conducted with identical
protocols. Quantitative analysis showed a substantial enhancement in
locomotion speed ($0.61\pm 0.09$), a 34% increase compared to locomotion with
no tail (Fig.5). However, the robot still could not travel in a straight line
and deviated from the path $3.13\pm 1.72^{o}$ per cycle with a rigid tail.

The tail successfully raised the rear of the robot, eliminating any contact of
the body with the ground. Upon close examination of the experiments, it was
found that there was still an unpredictable and unwanted variance from the
gait cycle. This occurred during the swinging phase of the front right leg.
The front right leg made contact with the ground before the back left leg,
interrupting the diagonal gait of the robot. This caused the front of the
robot to pitch downwards, and the tail to lift off the ground. There was
further instability when the tail remade contact with the ground.

####  3.1.3 Locomotion with Flexible Tail

Prior to the experiments being run on the flexible tail, the different tail
states/configurations were evaluated. The tail could be configured in the
following three ways; (1) a relaxed flexible tail where there was no tension
being applied (Fig. 3B-left), (2) a more rigid tail where the servo motor was
applying constant tension (Fig. 3B-middle), or (3) an alternating flexible to
rigid tail, which would be controlled by the rotation of the servo motor.

It was found that the flexible tail with constant rigidity provided the best
stability (less vertical oscillation) to the robot while walking on flat,
smooth terrain (Fig.4C). A further improvement in the robot’s ability to
travel in a straight line was observed in this test. The robot was capable of
traveling along a straight path with $0.71\pm 0.98^{o}$ rotation/cycle and
$0.60\pm 0.04^{o}$ BL/cycle forward displacement whereas the other tension
settings; relaxed and alternating flexible/rigid tail, resulted in $1.48\pm
1.77^{o}$ and $2.5\pm 1.77^{o}$ rotations/cycle. The robot’s movements were
smoother and more consistent due to the lack of sudden lurches or pitching. It
was found that the side-to-side undulations of the tail, although passive and
slight, helped maintain ground contact throughout all stages of the gait
cycle. The front leg was still making contact with the ground just before the
opposite back leg, however, this no longer caused instability. This is because
the ground contact of the tail still provided sufficient support to stop rapid
pitching.

The presence of residual flexibility in the tail, despite being in a rigid
state, emerged as a crucial element in augmenting the robot’s stability and
locomotion. This allowed the tail to automatically make slight adjustments
when traversing the terrain. When the foot strike occurred, the tail moved
towards the body momentarily. During the flight phase, the tail then quickly
moved in the opposite direction away from the body, until it was limited by
the tension of the reel. The subtle movements of the tail played a pivotal
role in ensuring that the tail remained in contact with the ground, preserving
the overall stability of the robot with seamless transitions between gait
phases. The robot’s stance position was now altered; the front of the robot
was rotated slightly downwards. During the swing phase, the robot pitched
marginally upwards until the body angle was approximately parallel to the
ground. Then, when the foot strike occurred, the front of the robot pitched
downwards to return to the stance position. The swing phase was then again
initiated, and the cycle repeated. This resulted in a smooth, consistent gait,
without any sudden and rapid pitching taking place, which would cause the
robot to change direction.

The constant stability experienced by the robot led to a further increase in
locomotion speed. It traveled at a rate of $0.60\pm 0.04$ BL/cycle similar to
the locomotion with the rigid tail.

###  3.2 Inclined Surface

In these experiments, we assess the robot’s ability to navigate on an inclined
surface with each tail configuration. The experiments involved placing a
cardboard runway (50 cm x 85 cm) at various inclinations, ranging from 10 to
25 degrees, and testing the robot’s ability to traverse the entire length of
the runway successfully. Five trials were performed for each incline, and the
robot was considered capable of locomotion on an incline if it could make it
to the end of the runway successfully (Fig.5).

####  3.2.1 Locomotion with No Tail

With no tail, the robot was unable to successfully travel along any incline.
It failed to make it to the end of the runway angled at 10 degrees to the
horizontal in five consecutive trials. During each of the trials, pitching
occurred, resulting in the body striking the ground as well as low rear leg
clearance. The cause of this rocking is identical to what is described in
section 3.1.1.

In the flat terrain tests, the pitching of the robot had only a minor effect
on its locomotion capabilities. However, when traveling on an inclined
surface, the instability caused by the pitching led to the constant slipping
of the robot’s rear feet. This made the robot’s travel speed extremely slow
($0.05\pm 0.02$ BL/cycle). It also resulted in the robot eventually rotating
90 degrees and walking off the runway, therefore failing to meet the success
criteria of the experiment.

####  3.2.2 Locomotion with Rigid Tail

The addition of a rigid tail allowed the robot to travel along a runway of
inclines 10 and 15 degrees with $0.45\pm 0.03$ and $0.37\pm 0.04$ BL/cycle,
respectively. It is noteworthy that, during locomotion along these inclines,
the robot experienced some degree of slippage. However, the hind legs of the
robot exhibited adequate ground clearance, thereby facilitating forward
movement. When the angle was increased to 20 degrees, the slippage of the feet
was greater than the distance stepped forward, therefore, the robot traveled
in the reverse direction.

Figure 6: Experimental snapshots of the robot climbing up a series of steps.
(A) Without a tail (the robot could not climb onto track), (B) With a rigid
tail (robot fell from track at the second step), and (C-D) With a flexible
tail that is periodically stiffened and relaxed. The stairs were made of 2.5
cm height foam blocks. The yellow trajectory shows the trajectory of the tip
of the robot during climbing (SI Movie).

####  3.2.3 Locomotion with Flexible Tail

The results obtained from our previous experiments have indicated that the
implementation of a flexible tail with constant tension settings yielded the
most optimal performance, we therefore conducted our inclined experiments
utilizing exclusively this tension setting. When tested with a flexible tail,
the robot successfully traveled along a runway of 10, 15, and 20 degrees with
$0.40\pm 0.03$, $0.37\pm 0.03$, and $0.27\pm 0.06$ BL/cycle, respectively.
When the incline of the runway was increased to 25 degrees, the robot could
not travel along it due to sliding. The flexibility of the tail was not
altered during this experiment and was maintained at high rigidity at all
times. This allowed the robot to have constant contact with the runway.

For inclines of 10 and 15 degrees, the robot’s motion was only slightly
inhibited by some minor slippage of the rear feet. It completed the tests
efficiently, while also traveling along the straight line indicated on the
runway. In contrast, the robot experienced more pronounced slippage of the
front and hind legs when traversing inclines of 20 degrees. Despite this, the
robot was able to complete the test, albeit at a slightly reduced speed
compared to the preceding trials. However, the robot’s performance was
severely impaired when the angle of the incline was further increased to 25
degrees. In this case, the robot was unable to make any net progress up the
incline, as it slid back to its previous point after each step. It remained
here indefinitely, not moving upwards along the track but also not moving
downwards and falling off. This suggests that if a change was made to increase
the grip of the feet, the robot may be able to successfully move along this
incline.

###  3.3 Stepped Terrain

The stepped terrain experiments were split into two sections: (1) upward and
(2) downward climbing. Both sections followed identical procedures and had the
same success criteria, as detailed below. The only difference between the two
sections was that in the upward climb test, the robot initiated from the
ground in front of the first step and ascended the steps, while in the
downward climb test, the robot commenced from the top step and descended the
’runway’ towards the ground, where the ground was considered as the sixth
’step’.

Figure 7: Up/down stair climbing with different tail configurations. (A) Mean
and standard deviation of stairs climbed up (blue) and down (red) per cycle
without a tail, with a rigid tail, and with a flexible tail over five trials.
Inset shows the dimensions of the stairs. (B) Mean and standard deviation of
the rotation of the body per cycle for the same experiments.

The stair setup comprised six steps of uniform height (2.5 cm) and width (25
cm) made of foam boards (Foamular, HomeDepot). The primary objective was to
assess the robot’s climbing ability with different tail configurations, with
the ultimate goal of climbing as many steps as possible without falling off
the runway or failing to make it to the end. The criterion for a successful
climb was that all four feet of the robot should make contact with the steps
of the stairs. The total number of steps climbed was then counted over five
trials for each tail configuration, with a maximum score of 30 possible, and
then normalized by the total number of step cycles.

####  3.3.1 Upward Climb with No Tail

Without a tail, the robot failed to climb any steps, meaning a result of 0/30,
and rotated $30\pm 5^{o}$ per cycle. It was observed during the previous
experiment that on flat terrain the front of the robot pitched up during
locomotion, as well as the rear legs’ ground clearance was extremely small.
These two factors impaired the robot while climbing. During each trial, the
robot placed its front legs on the first step. It moved forward on that step
until the rear legs reached the edge of the step. When this happened, the rear
legs could not lift high enough to clear the ledge. The body was also making
contact with the step. As a result, the robot was stuck indefinitely. The
result of this experiment was significant, as it showcased the robot’s
inability to climb or navigate over obstacles without getting stuck.

####  3.3.2 Upward Climb with Rigid Tail

There was an improvement in the robot’s climbing ability when a rigid tail was
added. The robot successfully climbed 8/30 steps with $0.2\pm 0.08$
stairs/cycle over five trials (Fig.7A) with a $5.34\pm 1.75^{o}$
rotation/cycle. With the addition of the rigid tail, the robot could climb
onto the first step in every trial. The robot no longer pitched upwards during
locomotion, meaning the hind legs had sufficient ground clearance to make it
onto the first step. The robot encountered difficulties when moving forward on
the second step and when attempting to climb onto the next step (Fig.6B). This
was due to the lack of contact between the tail and ground, as the robot’s
legs were on a surface with higher elevation compared to the tail, which led
to the loss of support and stabilization. Consequently, the robot’s gait
reverted to its previous state without a tail, where the front end would
incline upwards, and the rear legs would have limited, to no ground clearance.
When the tail approached the step’s edge, it became stuck, obstructing the
robot’s progress. Despite being immobilized, the robot continued moving, but
without any forward motion along the track. The front legs continued to extend
forward, but the tail constrained the robot in place.

In most trials, the robot continued to move forward until, due to slipping of
the feet, it began to rotate to one side and ultimately move off the track. In
two instances, sudden jerking forward caused by the robot’s instability
allowed the robot’s tail to become free and climb to the next step and
continue climbing, although this required numerous gait cycles and a
considerable amount of time. It was observed that the robot had already
started rotating before being freed, leading to the movement toward the
track’s edge. Also, when the robot was attempting to climb while on the track,
one of the hind legs made insufficient clearance to climb over the step,
resulting in additional rotation and ultimately causing the robot to fall off
the track.

It was evident in this test that the length of the robot’s tail is a critical
factor in determining its climbing ability. It must be noted that although the
robot can climb better with a rigid stick-type leg, this rigid leg can easily
get caught on obstacles, not just when climbing. It also does not offer
sufficient ground support because the contact area of the leg is small.
Increasing the tail’s length increases the clearance height of the rear of the
robot and provides more support during climbing, which can enhance the robot’s
stability and balance. However, longer tails are more likely to get caught on
objects or ledges, leading to potential failures during locomotion. Meanwhile,
shorter tails decrease the probability of getting stuck but also decrease the
support provided during climbing and flat ground traversal, leading to reduced
stability and insufficient ground clearance.

####  3.3.3 Upward Climb with Flexible Tail

Making the tail longer and more flexible allows the robot to adapt the height
of its backside to the environment and provide more support during climbing.
Since the robot with the rigid tail could not climb the stairs, here we
programmed the flexible tail to periodically stiffen and relax during key
phases of the gait cycle. Tail stiffening occurred during the middle of the
flight phase of the front left leg i.e. when it was reaching forward. Tail
relaxation occurred at the same instant during the flight phase of the front
right leg. The robot successfully climbed 30 steps out of a possible 30, with
$0.45\pm 0.04$ stairs/cycle and $0.66\pm 0.41^{o}$ rotation/cycle, over 5
trials (Fig.7).

When the tail is stiffened, it allows the robot to climb to a higher step by
providing support in the form of constant ground contact, resulting in a large
clearance distance between the legs and the ground. While the robot is on the
higher step, the body remains parallel, resulting in smoother locomotion.
However, due to the length of the tail, it can get stuck when it reaches the
edge of a step, and the robot ceases to move forward.

To prevent this, the tail relaxes by releasing the tension on the cables. This
allows the tail to move over the step and for the robot to continue climbing.
Due to the periodic nature of the stiffening and relaxation of the tail, in
some cases, the tail may inhibit movement and the robot may appear stuck
momentarily. This occurs when the tail, in its stiffened state, meets a step.
The robot then attempts to move forward for a brief period. During the forward
swing, the tail is relaxed, and the robot accelerates rapidly forward. The
feet of the robot slides along the surface and as a result, the robot moves a
further distance forward than a typical cycle. The stiffening of the tail also
provides thrust to the robot, pushing it upward and forward at a specific
instant during the gait cycle, to increase locomotion effectiveness.

####  3.3.4 Downward Climb with No Tail

When the robot has no tail attached, it performs better at climbing downwards
compared to climbing upwards. The robot successfully traveled down 15 steps
out of a possible 30 with $0.53\pm 0.06$ stairs/cycle and $2.12\pm 0.47^{o}$
rotation/cycle, over six trials. From previous experiments it was observed,
that when traveling along flat terrain the robot pitches upwards, causing the
rear to make contact with the ground. This still occurred when climbing
downwards, but when the robot moved to a lower step, it violently pitched
downwards, causing the front to hit the ground or runway. This caused further
disturbance to the robot’s gait, leading to instability and inconsistencies in
the robot’s direction of travel.

####  3.3.5 Downward Climb with Rigid Tail

The robot successfully traveled down 21 steps with $0.62\pm 0.27$ stairs/cycle
and $0.92\pm 0.71^{o}$ rotation/cycle when a rigid tail was attached. The
robot’s locomotion patterns were made more consistent by the addition of a
tail. However, upon each foot placement, the robot pitched downwards and the
tail briefly lost contact with the supporting surface. This caused problems
when moving onto a lower step. In those cases, the robot pitched to an even
more extreme angle, resulting in the front of the robot making contact with
the surface. Similar to the previous experiment, these collisions with the
runway resulted in negative consequences such as misdirection and less
efficient locomotion, albeit to a lesser degree.

####  3.3.6 Downward Climb with Flexible Tail

With the addition of the flexible tail, the robot climbed down 30 out of a
possible 30 steps with $0.71\pm 0.04$ stairs/cycle and $1.3\pm 0.7^{o}$
rotation/cycle. Different to the upward climbing where the tail was
periodically stiffened and relaxed, in these experiments the tail was
configured so that it was fully flexible i.e., there was no tension in the
cables and there was no periodic motion of the reel. This configuration of the
tail made the robot more adaptable to its terrain. When stepping off each
step, the tail extends to the point where it is almost entirely vertical. At
this instant, the bottom of the tail is in direct contact with the ground (the
narrow end piece of the tail). Simultaneously, the front leg springs bend, and
the front of the robot makes contact with the ground. However, as the tail is
still supporting the robot, there is the little impact felt by it and there
are no deviations from the path. At all other stages, the tail is flexed away
from the body of the robot. This allows the tail to continue to offer support
when the legs of the robot have moved to a lower step.

###  3.4 OUTSIDE EXPERIMENTS

To demonstrate the tail function on locomotion, outside experiments were
carried out on the following terrains: a pebbled surface, mulch ground with
plant litter (twigs, leaves, etc.), and sand. The key parameter to be analyzed
in this experiment was how the robot reacted to uneven terrains, and if the
addition of a flexible tail improved these interactions This was done
qualitatively. In this section, tests were only performed on the robot with no
tail and with a flexible tail.

Figure 8: Outside demonstrations. (A) Side view of the robot while walking on
mulch surface. Yellow trajectory shows the trajectory of the tip of the tail
during walking (t=0-30 s), (B) Locomotion on a pebbled surface. (C) Side view
of the robot while walking on sand. At t=0 and 20s the tail of the robot is in
its relaxed state, and at t=10s it is in its rigid state. The red trajectory
shows the tail trajectory. Also, the footfalls of the robot are highlighted
with red arrows, (D) Back view of the robot and its tail trajectory while
walking on the sand (SI Movie).

####  3.4.1 Mulch Terrain

Without a tail, the robot was able to traverse on mulch without being
significantly affected by the softness of the surface (Fig.8A). However, when
the robot encountered organic debris like leaves and fallen twigs, the legs of
the robot would drag the debris along, resulting in a reduction to the robot’s
velocity. When a flexible tail was added, which exhibited periodic stiffening
and relaxation, the robot was more resilient to the aforementioned obstacles
(SI movie). The tail’s flexibility allowed it to absorb the impacts of the
obstacles and facilitate the robot’s smooth movement.

####  3.4.2 Pebbled Terrain

With no tail, the robot traveled extremely slowly ($\sim$ zero speed) over
pebbled surfaces. The robot’s legs repeatedly got stuck in gaps between
stones, due to the low ground clearance of the legs and the pitching of the
body. This was compounded by the robot’s eventual entrapment, which resulted
from one of its legs becoming firmly lodged between stones, rendering it
immobile.

When the robot was tested with a flexible tail that exhibited periodic
stiffening and relaxation, the locomotion velocity was increased to $0.26\pm
0.04$ BL/cycle (SI movie). The legs no longer got stuck due to the movement of
the tail and the support it provided. Although the tail occasionally became
stuck, the periodic relaxation of the tail allowed it to become dislodged
quickly without impeding the robot’s overall performance (Fig.8B).

####  3.4.3 Sand

When traveling on sand, the robot’s rear legs did not lift off the surface as
expected, but instead were dragged along through the sand. This unexpected
movement of the robot’s legs resulted in disruptions to the gait cycle and
subjected the rear legs to increased force and twisting, which the robot was
not designed to handle (SI movie). Continued use of the robot on sand would
likely lead to fracturing of the legs or body, as well as damage to the rear
motor due to the excessive forces being exerted.

In contrast to the issues encountered when the robot traversed a sandy terrain
without a tail, the addition of a flexible tail configured to periodically
stiffen and relax proved to be an effective solution (Fig.8C). The tail’s
ability to provide constant support and its side-to-side passive undulation
was clearly visible in the tracks left by the robot(Fig.8C-D). Specifically,
the tail enabled the robot to traverse small divots and uneven surfaces, which
it was previously unable to navigate effectively (SI movie).

##  4 Conclusions and Future Works

This paper describes the development and implementation of a multi-segment,
flexible tail with variable rigidity controlled by a cable-driven mechanism.
The performance of a sprawling quadruped robot was shown to be significantly
improved by conducting experiments that compared the locomotion of the robot
with different tail configurations, over several indoor and outdoor
environments.

The constant ground support provided by the flexible tail is key in
maintaining stable locomotion. This ensures a predictable gait cycle that
stops unexpected turning and slipping as experienced by the robot in the other
two tail configurations. This leads to an increase in the locomotion speed and
efficiency of the robot. Moreover, the constant ground support also enables
the robot to travel along an inclined surface with an angle of up to 20
degrees to the horizontal. The variable stiffness of the tail plays a
significant role in enhancing the robot’s climbing ability. The versatility
and adaptability of the tail allows the robot to successfully overcome
obstacles in its path when climbing, without the need for external sensing.

All of the aforementioned improvements to the robot’s locomotion were observed
when testing on natural terrain occurred. The adaptability of the flexible
tail greatly improved the robot’s locomotion capabilities when traveling on
pebbled, mulch, and sandy terrain.

A limitation of the open-loop system employed in conjunction with the flexible
tail is its inability to change the state of the flexible tail automatically.
The configuring of the tail i.e. full flexibility, constant rigidity, or
periodic rigidity, must take place before the beginning of the experiments,
based on the terrain. However, future work will focus on the integration of
various sensors to automatically change the state of the tail according to the
terrain properties. This will allow the robot to adapt to its terrain and
change its gait to effectively navigate obstacles, with the goal of applying
the robot autonomously in outdoor, real-world environments. Our preliminary
testing of the incorporation of touch sensors at the end of the tail has shown
promising results in aiding the robot during stair climbing, with the relaxing
and stiffening of the tail determined by the sensor (SI Movie).

With the implementation of a closed-loop system, the robot would travel with
the tail in its rigid state, when the surface is smooth or flat. If an
obstacle is sensed in front of the robot, the tail would periodically relax
and stiffen to prevent entrapment. Similarly, if the surface underneath the
robot were to fall off, the tail would fully relax, providing support while
the robot pitched downwards. These advances may lead to the autonomous
application of the legged robot in outdoor, real-world environments, where it
can be used for various tasks, such as search and rescue, environmental
monitoring, and inspection.

## Author Contributions

JB designed the different tail configurations, built the robot, and performed
the experiments, YOA designed and supervised the project, and both authors
analyzed the data and wrote the paper.

## Funding

Josh Buckley is supported by The Naughton Undergraduate Fellowship program.

## Acknowledgments

We would like to thank Callie King for her assistance in the assembly of the
robot body and stair setup. We would also like to express our appreciation to
the members of Notre Dame MiNiRo-Lab for their insightful discussions.

## Supplemental Data

The supplementary movie can be found in the submission files.

## Conflict of interest

The authors declare that the research was conducted in the absence of any
commercial or financial relationships that could be construed as a potential
conflict of interest.

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