Single actuator wave-like robot (SAW): Design, modeling, and experiments

Abstract

In this paper, we present a single actuator wave-like robot, a novel bioinspired robot which can move forward or backward by producing a continuously advancing wave. The robot has a unique minimalistic mechanical design and produces an advancing sine wave, with a large amplitude, using only a single motor but with no internal straight spine. Over horizontal surfaces, the robot does not slide relative to the surface and its direction of locomotion is determined by the direction of rotation of the motor. We developed a kinematic model of the robot that accounts for the two-dimensional mechanics of motion and yields the speed of the links relative to the motor. Based on the optimization of the kinematic model, and accounting for the mechanical constraints, we have designed and built multiple versions of the robot with different sizes and experimentally tested them (see movie). The experimental results were within a few percentages of the expectations. The larger version attained a top speed of 57 cm s(-1) over a horizontal surface and is capable of climbing vertically when placed between two walls. By optimizing the parameters, we succeeded in making the robot travel by 13% faster than its own wave speed.

Full text


Abstract— In this paper, we present a single actuator
wave-like robot (SAW), a novel robot which can move
forward or backward by producing a continuously
advancing wave. The robot has a unique minimalistic
mechanical design and produces an advancing sine wave,
with a large amplitude, using only a single motor but with
no internal straight spine. The direction of locomotion is
determined by the direction of rotation of the motor. A
kinematic model of the robot is developed that accounts
for the two-dimensional mechanics of motion and yields
the speed of the links relative to the motor. Based on the
optimization of the kinematic model, and accounting for
the mechanical constraints, we have designed and built
multiple versions of the robot with different sizes and
experimentally tested them (see movie). The experimental
results were within a few percentages of the expectations.
The larger version attained a top speed of 23 cm/s over a
horizontal surface and is capable of climbing vertically
when placed between two walls. By optimizing the
parameters, we succeeded in making the robot travel by
13% faster than its own wave speed.
I. INTRODUCTION
In the last decades, multiple studies have analyzed the
locomotion of crawling robots inside tubes for maintenance
purposes and in biological vessels for medical applications.
In many of those applications, the robots must overcome
rough terrain characterized by anisotropic properties, high
flexibility, varying dimensions, and low friction coefficients
[1]-[4]. A key element in the design of small crawling robot
is a minimalist approach, i.e. small number of motors and
controllers, which allows for miniaturization. Two main
locomotion patterns have been investigated: worm-like
locomotion [5]-[26] and undulating locomotion which
resembles a continuously advancing wave [27]-[43]. Worm-
like robots advance by changing the distance between their
links [5]-[26]. There are two types of worm-like robots;
inchworm-like robots and earthworm-like robots. Inchworm-
like robots [5]-[16] are generally made of two cells
(sometimes three3 as in [15]) fitted with clamps to increase
or decrease the friction forces by changing the normal forces
or the coefficients of friction. Earthworm-like robots [17]-
[24] are made of a larger number of cells, often four or more.
Multiple mechanisms of locomotion were developed using
magnet coils [19], shape memory alloys [17], an external
electromagnetic field [20],[21] and inflatable cells [23].
The Authors are with the Mechanical Engineering Department of Ben
Gurion University P.O. Box 653 Be’er Sheva 8855630 Israel (e-mail:
zadavid@bgu.ac.il).
This research was partially supported by the Helmsley Charitable Trust
through the Agricultural, Biological and Cognitive Robotics Initiative of
Ben-Gurion University of the Negev.
Using the inflatable cells approach, Glozman et al. [24]
applied one actuator and a single air/water source to drive an
inflatable worm made of multiple elastic cells inside the
intestines of a swine. Novel designs of inchworm-like and
earthworm-like robots actuated by a single motor were
developed by Zarrouk et al. [25],[26]. This minimalist design
allowed us to reduce the size, weight, energy consumption,
and to increase the reliability of the robot.
Fig. . The novel single actuator wave like robot (SAW). The robots have a
spine that constrains the links to move around it, producing an advancing
wave like motion (see movie .(
Wave-like locomotion was successfully produced by hyper
redundant snake robots [27]-[34] only (even though,
kinematically speaking a single actuator is required). The
first documented attempt to produce wave like locomotion
dates back to the 1920s by artist Pyotr (Petr) Miturich [41]
who suggested a design comprising an assembly of gears.
But nearly 30 years later the problem remained unsolved.
Taylor et al. (1951-1952) [36],[37][36], who investigated
the locomotion of wave-like and spiral-like locomotion in
low Reynolds environment, expressed his inability to
develop a mechanism that will allow to produce those
motions in order to experimentally validate his analysis.
More recently, some progress was reported by producing
cyclic motion with a small number of actuators which to a
certain extent resembles a wave but is actually a rigid straight
spine Error: Reference source not found-[41]. Other attempts
included producing a wave by vibrating a rod [42],[43], but
this method results in relatively small amplitudes whose size
is a function of the damping.
In this paper, we present the first single actuator wave-like
robot (SAW) which can produce a nearly perfect sinusoidal
advancing wave-like motion (Fig. ). In Section II, we
describe the kinematics of the wave locomotion. In Section
III, we present our novel design for the wave-like robot and
model its kinematics in Section IV. The kinematical model
Single Actuator Wave-Like Robot (SAW): Design, Modeling, and
Experiments
David Zarrouk, Moshe Mann, Amotz Hess

was used to optimize the design of the robot. Finally,
experiments performed with the robots which we built are
presented in Section V.
II. KINEMATICS OF A TRAVELLING WAVE AND COMPARISON
TO ROTATING HELIX.
In this section, we show that the projection of a rotating
helix forms an advancing sine wave.
A. Traveling wave
The simplest model of traveling wave is an advancing sine
wave, or harmonic wave. Its mathematical presentation is
( ) ( )
, siny x t A kx wt
= −
()
where x is the space coordinate, t is the time, y is the height
of the wave at point x and time t, and A is the amplitude. The
angular velocity w of the wave is related to the frequency by
2
w
f
π
=
()
and the wave length L of the traveling wave is related to the
wave number by
2
Lk
π
=
()
The travelling speed of the wave is thus
wave
w
V f L k
= × =
()
B. Mathematical model of helix and its projection.
A helical curve with its axis in the x direction is described
parametrically by
( )
( )
2
sin
cos
L
x a
y A a
z A a
π
= ×
=
=
()
where L is the length of the pitch and A is the radius of the
helix and a is the independent paramter. The two
dimensional projection of the helix on the X-Y plane ( z = 0)
yields the following sine function:
( )
2
2
sin sin
L
x a
x
y A a A L
ππ
= ×
 
= =  ÷
 
()
A 3D helix whose axis is parallel to the x direction and its 2D
projection on the X-Y plane are presented in Fig. . Its
projection is a sine wave, as seen from Eq. )(
Fig. . A helix and its projection on the X-Y plane. The projection of the helix
is a sine wave, where the amplitude is the radius of the pitch .
C. Rotating helix and comparison to traveling wave.
When the helix rotates around its axis (the x axis) at a
constant angular frequency w (counterclockwise) the
parametric equations of the helix (Eq. ()) are multiplied by
the rotation matrix around the x axis:
( )
( )
( ) ( ) ( )
( ) ( ) ( )
( )
( )
( )
, 1 0 0 2
, 0 cos sin sin
, 0 sin cos cos
2
sin
cos
La
x a t
y a t wt wt A a
z a t wt wt A a
La
A a wt
A a wt
π
π
 
 
    
   
= −  
    
   
    
 
 
 
 
= −
 
 
−
 
 
()
Inserting a = 2π/L*x into y demonstrates that the projection
of the rotating helix is an advancing sine wave given by:
2
siny A z wt
L
π
 
= −
 ÷
 
()
III. ROBOT DESIGN
In the previous section, we showed that the projection of a
rotating helix is an advancing sine wave. Our robot design,
which uses a single motor to produce an advancing wave,
follows the same concept. The robot is composed of four
main parts: the motor house, the motor, the helix, and the
series of links (Fig. ). The motor is attached to the motor
housing from one side and to the helix from the other side.
The links are attached to the motor house. As the motor
rotates the helix, the links cancel the rotation along the axis
of the helix and maintain the vertical motion. In this way, the
links act as a 2D projection of the helix of the robot.
The helix of the larger version is nearly 25 cm long and is
composed of two windings and a short extension to reduce its
diameter. Its external diameter is 5.2 cm and its radius is A =
2.1 cm. The links, presented in Fig. , are 7 cm wide 1.83 cm
high (r = 0.9 cm), and the distance between the joints of two
links is 1.2 cm (Llink = 1.2 cm). The smaller version is nearly
scaled down by a factor of nearly 2:1 and the smallest is
scaled down by a factor of 3:1. The helix and the links are

3D printed. The robot is fitted with a 6 Volt, 12 mm motor
with 300:1 gear ratio. Based on its catalog specifications, the
motors and gearbox produce a torque of 2.9 Kg-cm at 45
rpm. It is noted that in most of the experiments (except when
specified otherwise), we used a single ~4V lithium–ion
battery which is substantially lower than its nominal input (6
-9 Volts). The total weight of the larger robot including one
battery is 188 grams, whereas the smaller one weighed only
47 grams only.
Fig. . The different parts of the robot. The robot has a housing for the motor.
The helix is attached to the motor and rotates relative to the housing. The
links are attached to the housing and do not undergo roll rotation.
IV. KINEMATICS ANALYSIS
In this section, we model the kinematics of the links and
calculate their speed relative to the head of the robot (motor
housing) as a function of its frequency of locomotion f, the
wave length Lwave, the amplitude of the wave A, the length of
the link Llink, and its height r. If the links do not slide over the
surface, (as we experimentally found in section 5 - Fig. ), the
speed of the robot will be equal to the speed of the tips of the
links, along the axis direction. We define the advance ratio
(AR) as the speed of the robot Vrobot divided by the speed of
the travelling wave relative to the motor base Vwave:
cycle
robot
wave wave
L
V
AR V L
= =
()
where Vrobot is the speed of the robot, Lwave is the length of the
wave, and Lcycle is the net advance per cycle (one rotation of
the helix)
robot cycle
V f L
=
()
Fig. . The The geometry of the link. The two main parameters are the link
length Llink, the distance between two adjacent joint, r is the height of the link
and wtip is the width of the tip.
A. Kinematics of the links
As the wave advances, the links move both horizontally
and vertically. As the wave advances by Δx, the link will
rotate by Δα (see Fig. ).
( )
( )
( )
( )
( )
atan cos atan sin
x x
dA kx kA k x
dx
α
=∆
 
∆ = = − ∆
 ÷
 
()
Due to the rotation, the tip of the link will move by a distance
ΔX
( )
sinX r
α
∆ = ∆
()
If the speed of the wave is Vwave, the time required by the
wave to advance by a distance of Δx is
/
wave
t x V∆ = ∆
()
Therefore the expected speed of the link is
link
X
Vt
∆
=∆
()
Inserting Eqs. ()-() into Eq. () we obtain the speed of the tip
of the link
( )
( )
( )
sin tan sin
link wave
r a kA k x
V V
x
− ∆
=∆
()
If we assume small angles
1
α
∆=
then
( )
( )
2 2
sin
1tan
k x k x
a Ak x Ak x
α
∆ ≈ ∆


∆ ⇒ ∆ ≈ ∆


=
()
And finally, by inserting Eq. () into Eq. (), one obtains the
speed of the tips of the links as a function of the height r,
amplitude A, wave length Lwave, and wave speed Vwave.
2
2
2
link wave wave
wave
V rAk V rA V
L
π
 
≈ =  ÷
 
()
Alternatively, the speed of the wave can be calculated as a
function of the actuation frequency:

( )
2
2
2
link wave
wave
A
V rAk V r f
L
π
≈ =
()
Therefore the speed of the link is proportional to the ratio of
the amplitude divided by the wave length, to the height of the
links and to the actuation frequency. In theory, it would be
advantageous to increase A/Lwave and r to increase the speed.
However, increasing those values results in collision between
the tips of neighboring links. This collision is most likely to
occur when two links are symmetrically oriented towards
each-others such as links i-1 and i in Fig. case A. Assuming
zero width of the tips of the links, collision will occur when
/ 2
atan
link
L
r
α
  = ∆
 ÷
 
()
Inserting the value of Δα into from Eq. () into (), it is possible
to obtain the condition of collision as a function of the size of
the links and the wave parameters.
2
0.5
2
2
2
2sin
2
link
link
link
L
L
kA k
rL
r
 
 
 ÷
   ÷
 ÷
 ÷  
= −  ÷
 ÷  ÷
 
 ÷  
 ÷ +
 ÷ ÷
   ÷
 ÷
 ÷
 
 
 
()
Fig. . The rotation of the links during the adavnce of the wave. “A” marks
the beignning of the touchdown of link i and retraction of link i-1. In “B”,
the wave has advanced by Δx and link i is at the lowest point of the wave.
“C”, which occurs after the wave advances by a further Δx, marks the end of
the touching of link “i” and the beginning of the engagement of i+1.
A. Kinematics of the links
We assume that the links slide along the advancing wave
(rotating helix) while the first link is attached to the motor
housing. The number of links is determined by the length of
the wave Ltot divided by the length of the links.
( )
0.5
2
0
1
L
tot
d
L N y x dx
dx
 
 
= +
 ÷
 ÷
 ÷
 
 
∫
()
where N is the number of waves in the sine function (N = 2 in
our robot). Equation Chapter 1 Section 1To calculate the
positions of all the links of the robot, we sequentially solve
for the location of the endpoint of each link along the sine
wave. That is, we start with the location of the joint i of link
[xi yi] and solve for the x coordinate of the link’s endpoint
[xi+1 yi+1] by assuming that it is fastened to the sine wave
using the equation:
( ) ( )
2 2 2
1 1
sin( ) y
i i link
i i
x x A kx t L
ω
+ +
− + − − =
()
where Llink is the length of each link. Solving Eq. () returns
the position of the end point xi+1 of link i. The endpoint of
link “i” serves as the start point of link “i+1”, and so on until
the last link’s location is solved for. The location of each
link’s start point and end point provides complete
information of the link’s orientation, and is used to calculate
the location of the links’ tip [x_tipi y_tipi]:
1 1
1 1
_ 0
1
_ 0
2
0 0 0 0 1
i i i i i i
i i i i i i
x tip x x x x x
y tip y y y y y r
+ +
+ +
− −
         
         
= + − + − ×
         
         
         
()
The position of the links when the motor housing is fixed
was simulated using MATLAB™ (2013) program. Equations
() and () were solved at a rate of 500 times per cycle (results
and optimization are summarized in Fig. and in Ttable I).
The velocity is obtained by deriving the position as a
function of the time. In the simulation, we also accounted for
the width of the tip of the link, since in practice, the width
must be a few millimeters (in the simulation, we used wtip =
0.05 Lwave).
A two dimensional side projection of the simulated robot
is shown in Fig. . The robot consists of 25 rigid links
connected through revolute joints formed into a sine wave of
two spatial cycles. We focus here on the motion of link 5. As
link 5 approaches the lower bottom of the wave, it moves
slightly horizontally and rotates clockwise. Both of these
motions add up to move the bottom tip to the left, and
therefore the robot would move to the right.

Fig. . The simulation of the robot. The “motor housing” is rigidly fixed. As
the wave adavances from left to rightright to left, the lower tips of the links
which will be in contact with the surface move slightly towards the left and
rotate clockwise.
B. Expected robot advancement speed
The simulation allowed us to visually gain insights into
the motion of the links and optimize the design of the robot.
If no sliding occurs, the speed of the robot will be equal (but
to the opposite direction) to the horizontal speed of the links
contacting the surface. Therefore, the simulation calculates
the position of the different links at all times and detects
which of the links is the lowest, i.e. expected to be in contact
with the ground. Averaging the speed of the lowest tips yields
the expected speed of the robot.
In Fig. , we present the advance ratio as a function of the
amplitude for 3 different values of links heights (r/Lwave).
Since practically speaking, the tips of the links have a width
wtip, we determine the maximum value of the amplitude
which results in the collision marked with *. By assuming
that the width of the tips is 5% of the wave length, we found
that the maximum AR is limited to nearly 0.7 in all three
cases because of the collisions between the tips of
neighboring links. See our the Aappendix on how we
managed to overcome this limitation.
Fig. . The advance ratio as a function of the amplitude for three different
heights. The asterix (*) marks the limit for which two neighboring links will
collide with each others.
V. EXPERIMENTS
In Section IV, we calculated the speed of the links as a
function of the different robot parameters such as the length
of the wave, the distance between the links, and their width.
In this section, we will experimentally measure the speed of
our 3D printed robot and compare it to the results of the
simulation. The position of the robot is measured using a 12
cameras Optitrack setup with a frequency of 120 Hz. The
accuracy of the system is nearly 0.1mm. We designed a
special link for holding the reflective marker at the lower tip
of the link (Fig. ). The special link has a side attachment for
the marker in which the center of the marker is on the axis of
the contact line with the surface. Using this link, the marker
remained on the side of the surface and would not interfere
with the experiment. The speed is determined by deriving the
position as a function of the time.
Fig. . A special link was manufactured to hold the reflective marker. The
center of the reflective is along the axis of the tip of the link.
A. Speed of the links
In our first experiment, we determined the trajectory of the
lower tip of one of the links (using the special link) when the
robot motor house was rigidly fixed. The trajectory and the
orientation of the tip of 8 cycles are presented in Fig. . The
motion is very cyclic with very little difference between one
cycle to and another. During a cycle, the tip moves vertically
by nearly 4 cm. This result is slightly less than expected (2*A
= 4.2 cm) and probably due to slight spacing between the
links and the helix. The horizontal motion is nearly 2 cm.
However, the motion of the link from the onset of contact
until disconnecting from the surface is nearly 1 cm.

Fig. . The motion of the bottom tip of the link during 8 cycles when the
robot is not moving. The arrow show the direction of motion.
In the second experiment, the robot was free to advance
and the position of the links was measured using the
Optitrack setup. We performed the experiments over
plywood and over aluminum which has a lower COF with
the links (nearly 0.3 whereas the COF over plywood is nearly
0.4).The results of the trajectory and the speed along the x
axis are presented in Fig. (over aluminum surface). The
trajectory shows that the link touches the surfaces at a single
contact point. Therefore the link is not sliding over the
surface and its relative speed to the surface is zero. Sliding
didn’t occur also when the robot was run at higher speeds.
Fig. . (Top) The position of the lowest tip of a link during horizontal
locomotion over alumium surface. The single point contact at each cycle
proves that no slide between the links and the surface. (Bottom) The
horizontal speed of the links.
Table 1, summarizes the results of multiple experiments that
we performed by the smaller and bigger versions of our
robots. The data is the average of at least 12 cycles. The
results were compared to the simulations and found to be
within a few percentage of each-others. The larger and
smaller versions performed nearly similarly for two different
amplitude to wave length ratios. Following the predictions of
the simulation, we designed special links with large height
that allowed the robot to advance by 13% faster than the
speed of the wave (AR = 1.13). We note here that a short
survey that we made with multiple roboticists found that they
all believed it is impossible to advance faster than the speed
of the wave!
Table I. The advance ratio as a function of the slope.
Lwave A/Lwave r/Lwave Aver. AR STD ΔAR/Lcycle
[cm] [cm] [cm] Lcycle/Lwave AR
Large SAW
10.4 0.2 0.088 0.71 1.7% 2%
10 0.1 0.092 0.33 1.7% 5%
10.4 0.2 0.168 1.13 4.3% 2%
Small SAW
5 0.2 0.092 0.76 7.7% 8%
5 0.1 0.092 0.38 4.4% 8%
B. High speeds
We performed two more experiments to reach higher
speeds with the robot. In the first experiment, we used the
same set up, but we powered the robot with two Lithium-ion
batteries in series producing 8 Volts instead of a single
battery. The robot reached a speed of 10.4 cm/s. In the
second experiment, we attached three batteries in series (12
Volts) and used a 1:100 gear ratio (instead of 1:300). In this
configuration, the robot crawled at 23 cm/s. It is noted that
even at high speed no sliding was observed between the
robot and the surface.
C. Crawling over slopes and vertically between walls
We also the tested ability of the robot to climb by placing
the robot between two layers of polyurethane foam whose
COF with the links of the robot is nearly 0.4. The robot was
powered by two Litium ion battery (as it was not able to
move using a single one) and climbed at a speed of 8.2 cm/s.
The experiment is presented in Fig. . Note that in this
experiment, the two walls must be precisely distanced from
each other (up to a few millimeters of accuracy) in order to
achieve enough normal force for climbing, but without
overly pressing on the robot as it will stall.
Fig. . The robot climbing vertically between two walls. Using 8V input,
the robot reached a speed of 8.2 cm/s.
VI. CONCLUSION
In this article, we developed a novel robot which generates
an advancing wave, that is nearly identical to a sine wave, by
rotating a helix that moves the links. The robot design is

simple, lightweight, cheap, and requires only a single motor
only to produce the wave. The direction of wave propagation
is determined by the sign of the voltage being applied to the
motor. We developed two prototypes:. tThe larger one with a
wave length of 10 cm thTand weighs only 188 gramsand a
1:2 smaller version weighing 47 grams. Both prototypes
proved to be highly reliable (considering that they are 3D
printed prototypes). During all of our experiments,
practically almost no maintenance was required.
We studied the kinematics of the links and developed a
simple model that explains how the motion is produced. The
model also predicts the approximated speed of the lower tips
of the links as a function of the wave length and amplitude
and size of the links. We also developed a simulation which
calculates the speed and visually presents the locomotion and
detects where collisions between the links will occur. The
simulation allowed us to visually comprehend the locomotion
mechanics and optimize the robot. We introduced the
advance ratio (AR) as the speed of the robot divided by the
speed of the wave. We found that in general, the AR is
smaller than 1, but by increasing the height of the links, the
advance ratio can be larger than 1 (A short survey between
roboticist showed us that they all believed that AR = 1 is the
maximum possible speed).
We measured the speed of the robot and the speed of the
lower tip of a link using an Optitrack system. By measuring
the speed of the lower tip of the link, we found that it
contacts the surface in at a single point, implying that not
sliding does not occurs. We performed multiple experiments
and found that they are all within a few percentages from the
expected speed by the simulation. By applying 12 Volts to
the motors, the robot moved at up to 23 cm/s and no sliding
was detected even in this case. The robot was also capable of
climbing vertically when finely placed between two surfaces
polyurethane foam at a speed of 8.2 cm/s.
Our future work will focus on analyzing the locomotion
of this type of robots over compliant and slippery surfaces.
VII. APPENDIX
A. Turning using steering wheels
We added steering wheels to the front of the robot as seen
in Fig. . The robot is now controlled using a two channel
joystick (extracted from an RC toy car - see movie). We
performed multiple experiment in crawling straight and
turning and captured the position using our Optitrack set up.
The results show that the robot can turn to either direction and
that the radius of turning was nearly 0.3 m.
Fig. . The robot with steering wheels. The direction of turning is
controlled by a second motor.
The results are presented in Fig. in which the name of the
robot is written (SAW). All the letters were completed in a
single run with no external intervention.
Fig. . The position of the robot with steering wheels. The robot wrote his
name.
B. Increasing the height to travel faster than the speed of
the wave
To increase the speed of the robot beyond the speed of the
wave, we developed three sets of links with different tips
which do not collide with each-others (see Fig. ). The height
r was nearly 1.75 cm. Using those links, the robot achieved a
speed which is 13% larger than the wave speed.
Fig. . The specially designed links that do not collide with each others.
C. Miniaturization
To Tthe single motor design allows for further
miniaturization of the robot. Our smallest version (Fig. 15) is
12 cm long and 3 cm wide and weighs 30 grams including the
motor and battery. It was tested and shown to be crawling at
nearly 3cm/s (see movie). Further miniaturization of the robot
is possible and depends on more precise manufacturing.

Fig. . The smallest version of the robot. The length is nearly 12 cm and
the width is about 3cm.
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