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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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