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

Bioinspir. Biomim. 11 (2016) 046004

doi:10.1088/1748-3190/11/4/046004

PAPER

RECEIVED

24 January 2016

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

REVISED

13 May 2016 ACCEPTED FOR PUBLICATION

David Zarrouk, Moshe Mann, Nir Degani, Tal Yehuda, Nissan Jarbi and Amotz Hess

7 June 2016

Mechanical Engineering Department of Ben Gurion University PO Box 653 Be’er Sheva 8855630, Israel

PUBLISHED

E-mail: zadavid@bgu.ac.il

1 July 2016

Keywords: crawling robot, wave like locomotion, minimally actuated, design Supplementary material for this article is available online

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.

1. 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 [5–25] and undulating locomotion which resembles a continuously advancing wave [26–42].

Worm-like robots advance by changing the distance between their links [5–25]. There are two types of worm-like robots; inchworm-like robots and earthworm-like robots. Inchworm-like robots [5–15] are

* 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.

© 2016 IOP Publishing Ltd

generally made of two cells (sometimes three 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 [16– 23] are made of a larger number of cells, often four or more. Multiple mechanisms of locomotion were developed using magnet coils [18], shape memory alloys [16], an external electromagnetic field [19, 20] and inflatable cells [22]. Using the inflatable cells approach, Glozman et al [23] 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 [24, 25]. This minimalist design allowed us to reduce the size, weight, energy consumption, and to increase the reliability of the robot.

Wave-like locomotion was successfully produced by hyper redundant snake robots [26–33] only (even though, kinematically speaking a single actuator is required). The first documented attempt to produce

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Figure 1. 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 dates back to the 1920s by artist Pyotr (Petr) Miturich [41] who suggested a design comprising of an assembly of gears. But nearly 30 years later the problem remained unsolved. Taylor et al [35, 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 a helix motion 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 with changing width [37–40]. Other attempts included producing a wave by vibrating a rod [41, 42], but this method results in relatively small amplitudes whose size is a function of the damping.

Inspired by wave-like locomotion of snakes and flagella swimming of miniature organisms, we present in this paper that produces a nearly perfect wave actuated by a single motor which we name SAW (single actuator wave-like robot) that is presented in figure 1.

SAW’s locomotion principle differs from snakes by such that it does not slide and that it does not have anisotropic coefficient of friction (COF) but can still over a variety of surfaces and climb over obstacles. In section 2, we describe the kinematics of the wave locomotion. In section 3, we present our novel design for the wave-like robot and model its kinematics in section 4. The kinematical model was used to optimize the design of the robot. Finally, experiments performed with the robots which we built are presented in section 5.

2. Kinematics of a traveling wave and comparison to rotating helix In this section, we show that the projection of a rotating helix forms an advancing sine wave.

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2.1. Traveling wave The simplest model of traveling wave is an advancing sine wave, or harmonic wave. Its mathematical presentation is y (x , t ) = A sin (kx - wt ) ,

(1)

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 w f= (2) 2p and the wave length L of the traveling wave is related to the wave number by 2p . k The traveling speed of the wave is thus w Vwave = f ⋅ L = . k L=

(3)

(4)

2.2. Mathematical model of helix and its projection A helical curve with its axis in the x direction is described parametrically by L ⋅ a, 2p y = A sin (a) , z = A cos (a) , x=

(5)

where L is the length of the pitch and A is the radius of the helix and a is the independent parameter. The twodimensional projection of the helix on the X–Y plane (z = 0) yields the following sine function: x=

L ⋅ a, 2p

y = A sin (a) = A sin

2px L

( )

(6)

.

A 3D helix whose axis is parallel to the x direction and its 2D projection on the X–Y plane are presented in

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Figure 2. 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.

Figure 3. 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.

figure 2. Its projection is a sine wave, as seen from equation (6).

2.3. 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 (equation (5)) are multiplied by the rotation matrix around the x axis: ⎡ x (a , t ) ⎤ ⎡1 ⎤ ⎡ L a 2p ⎤ 0 0 ⎢ ⎥ ⎥ ⎢ ⎥⎢ ⎢ y (a , t )⎥ = ⎢ 0 cos (wt ) - sin (wt )⎥ ⎢ A sin (a) ⎥ ⎣ 0 sin (wt ) cos (wt ) ⎦ ⎣ A cos (a)⎦ ⎣ z (a , t ) ⎦ ⎡ ⎤ L a 2p ⎢ ⎥ = ⎢ A sin (a - wt ) ⎥. ⎣ A cos (a - wt )⎦ (7)

Inserting a = 2π/L*x into y demonstrates that the projection of the rotating helix is an advancing sine wave given by: 3

⎛ 2p ⎞ y = A sin ⎜ x - wt ⎟ . ⎝L ⎠

(8)

3. 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 (figure 3). 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 (diameter of the ‘wire’ is 10 mm). The links, presented in figure 4, are

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Figure 4. 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.

7 cm wide 1.83 cm high (r = 0.915 cm), and the distance between the joints of two links is 1.2 cm (Llink = 1.2 cm). There is a 3 mm gap between the link and the helix. The smaller version is scaled down by a factor of nearly 1:2 and the smallest is scaled down by a factor of 1:3. The helix and the links are 3D printed.

The robot is fitted with a 6 V, 12 mm motor with 1:300 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 ∼4 V lithium-ion battery which is substantially lower than its nominal input (6–9 V). The total weight of the larger robot including one battery is 188 g, whereas the smaller one weighed only 47 g.

4. 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 the 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—figure 10), the speed of the robot will be equal to the horizontal speed of the tips. We define the advance ratio (AR) as the speed of the robot Vrobot divided by the speed of the traveling wave relative to the motor base Vwave AR =

L cycle Vrobot = , Vwave L wave

(9)

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) Vrobot = f L cycle .

(10)

4.1. Kinematics of the links During the motion, the links move both horizontally and vertically. As the wave advances by Δx, the link 4

will rotate by Δα (see figure 5).

Da

⎛d ⎞ a tan ⎜ (A cos (kx )) ⎟ ⎝ dx ⎠ = a tan ( - kA sin (kDx )). =

(x =Dx )

(11)

Due to this rotation, the tip of the link will move horizontally by a distance ΔX: DX = r sin (Da) .

(12)

If the speed of the wave is Vwave, the time required by the wave to advance by a distance of Δx is: Dt = Dx Vwave.

(13)

Therefore the expected speed of the link is: DX .

(14) Dt Inserting equations (11)–(13) into (14) we obtain the speed of the tip of the link: Vlink =

r sin (a tan ( - kA sin (kDx ))) Vwave.

Dx If we assume small angles Da  1 then: Vlink =

⎧ sin (kDx ) » kDx , Da  1  ⎨ ⎩ a tan (Ak 2Dx ) » Ak 2Dx .

(15)

(16)

And finally, by inserting equation (16) into (15), 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

⎛ 2p ⎞2 Vlink » rAk 2Vwave = rA ⎜ ⎟ Vwave. ⎝ L wave ⎠

(17)

Alternatively, the speed of the wave can be calculated as a function of the actuation frequency:

Vlink » rAk 2Vwave = (2p )2

A r f.

L wave

(18)

Therefore the speed of the link is proportional to the ratio of the amplitude divided by the wave length A/ Lwave, to the height of the links r and to the actuation frequency f. 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

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Figure 5. 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.

to occur when two links are symmetrically oriented towards each-other such as links i − 1 and i in figure 5 case A. Assuming zero width of the tips of the links, collision will occur when; ⎛L 2⎞ atan ⎜ link ⎟ = Da . ⎝ r ⎠

(19)

Inserting the value of Δα into from equation (11) into (19), it is possible to obtain the condition of collision as a function of the size of the links and the wave parameters

⎛ ⎞ ⎛ Llink ⎞2 ⎛ Llink ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ 2 ⎟ ⎝ 2 ⎠ ⎜ ⎟ . ⎜ ⎟ » - kA sin ⎜k 0.5 ⎟ 2 ⎛ ⎞ r ⎛ ⎞ L ⎜ ⎟ link 2 ⎜⎜ ⎜r + ⎜ ⎟ ⎟ ⎟ ⎝ ⎠ ⎝ 2 ⎠ ⎠ ⎟⎠ ⎝ ⎝

(20)

4.2. Simulating the 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

ò0

L tot = N

L

0.5 ⎛ ⎛d ⎞2 ⎞ ⎜ 1 + ⎜ y ( x ) ⎟ ⎟ dx , ⎝ dx ⎠⎠ ⎝

(21)

where N is the number of waves in the sine function (N = 2 in our robot). To calculate the positions of all 5

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: (x i + 1 - x i )2 + (A sin (kx i + 1 - wt ) - yi )2 = Llink 2, (22)

where Llink is the length of each link. Solving equation (22) 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]:

⎡ x _tipi ⎤ ⎡ x i+1 - x i ⎤ ⎡xi⎤ ⎢ ⎥ ⎢ yi ⎥ + 1 ⎢ yi + 1 - yi ⎥ = y _tip i⎥ ⎢ ⎥⎦ ⎢⎣ 0 ⎥⎦ 2 ⎢⎣ 0 ⎣ 0 ⎦ ⎡ x i+ 1 - x i ⎤ ⎡ 0⎤ + ⎢ yi + 1 - yi ⎥ ´ ⎢ 0⎥ r . ⎢⎣ ⎥⎦ ⎢⎣ ⎥⎦ 1 0

(23)

The position of the links when the motor housing is fixed was simulated using MATLAB™ (2013).

Equations (22) and (23) were solved at a rate of 500 times per cycle (results and optimization are summarized in figure 7 and in table 1). The velocity is obtained

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Table 1. The advance ratio obtained from the Optitrack.

Lwave (cm)

A/ Lwave

r/Lwave

Aver. AR Lcycle/ Lwave

STD AR

ΔAR/ Lcycle

1.7% 1.7% 4.3%

2% 5% 2%

7.7% 4.4%

8% 8%

Large SAW 10.4 10 10.4

0.2 0.1 0.2

0.088 0.092 0.168

0.71 0.33 1.13

Small SAW 5 5

0.2 0.1

0.092 0.092

0.76 0.38

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 figure 6. 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.

4.3. 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 figure 7, we present the AR as a function of the amplitude for three different values of links heights (r/ Lwave) 0.1, 0.15 and 0.2. For larger values, the links collide with each-others and a much smaller r is mechanically impractical (minimum diameter of the helix, thickness of the links and the gaps between the links and the helix). The collision is 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 0.77 in all three cases because of the collisions between the tips of neighboring links. The values obtained by the simulation were a few percent different than the experimental results but were about 10%– 15% lower than the approximated values obtained from equation (18).

We note that increasing the value of the L_link may allow to increase the value of r but the links become loose and the helix will contact the surface.

5. Experiments In section 4, 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

Figure 6. The simulation of the robot. The ‘motor housing’ is rigidly fixed. As the wave adavances from right to left, the lower tips of the links which will be in contact with the surface move slightly towards the left and rotate clockwise.

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Figure 7. The advance ratio (AR) 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.

Figure 8. 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.

their width. In this section, we 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 0.1 mm. We designed a special link for holding the reflective marker at the lower tip of the link (figure 8). 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.

5.1. 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

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Figure 9. 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.

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Figure 10. (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 sliding occured between the links and the surface. (Bottom) The horizontal speed of the links.

are presented in figure 9. The motion is very cyclic with very little difference between one cycle 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 (about two millimeters from each side).

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.

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 figure 10 (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 did not occur also when the robot was run at higher speeds (to slide, the acceleration must be more than 3 m s−2 which is gravity*COF). Furthermore, the links do not appear to losing contact with the surface due to inertial effects.

Table 1, summarizes the results of multiple experiments that we performed with the bigger version and the smaller version (1:2) of the robot. 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 (A/Lwave). Following the predictions of the simulation, we designed special links with larger r that allowed the robot to advance by 13% faster than the speed of the wave (AR = 1.13).

(See section 5.4 on how we managed to overcome this limitation.) We note here that to the best of our knowledge, traveling faster than the speed of the wave

8

(which at first glance may appear impossible), has never been previously reported in literature.

The energy requirement for crawling was measured.

Using 8 V input, the robot consumed 0.18 A current and crawled at about 15 cm s−1. Based on those results, the cost of transport of SAW (defined as the input power divided by the weight times the speed) is 3.8.

5.2. High speeds We performed multiple experiments to increase the speed of the robot by increasing the power, lowering the gear ratio (1:100), and increasing the voltage. We measured the average speed of the robot from a two second run using the Optitrack. When powered with a 12 V inputs, the robot traveled at 57 cm s−1 or 5.6 wave lengths per second. It is noted that even at high speed no sliding is believed to have occurred during the steady state as the AR was 0.75 which is similar to the AR at the lower speed experiments. Furthermore, the robot did not lose contact with the surface due to centrifugal accelerations since the center of mass of the wave is (theoretically) on the axis of the helix. We did find some sliding during the acceleration stage which occurred during the first 3 cycles. The efficiency of locomotion which we define as the actual advance in a cycle divided by the advance in the steady state was 0.23, 0.65 and 0.8 respectively for the first, second and third cycles .

5.3. 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−1. The experiment is presented in figure 11. 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

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Figure 11. The robot climbing vertically between two walls. Using 8 V input, the robot reached a speed of 8.2 cm s−1.

Figure 12. The specially designed links that do not collide with each others.

Figure 13. Experimental results with the larger tips. The distances are normalized by the wave length.

achieve enough normal force for climbing, but without overly pressing on the robot as it will stall.

5.4. 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 9

different tips which do not collide with each-others (see figure 12). The length of the link Llink is 1.4 cm and the height r is 1.75 cm.

Using those links, the robot achieved a speed which is 13% larger than the wave speed. The results of one of the experiments are presented in figure 13. In nearly 6 cycles, the robot advances by 7 wave lengths.

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Figure 14. The smallest version of the robot. The length is nearly 12 cm and the width is about 3 cm.

Figure 15. The robot with steering wheels. The direction of turning is controlled by a second motor.

5.5. Miniaturization The single motor design allows for further miniaturization of the robot. Our smallest version (figure 14) is 12 cm long and 3 cm wide and weighs 30 g including the motor and battery. It crawled at nearly 8 cm s−1 (see movie). Further miniaturization of the robot is possible and depends on more precise manufacturing.

5.6. Turning using steering wheels We added steering wheels to the front of the robot as seen in figure 15. 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 of the motor house 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.

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Figure 16. The position of the motor house of the robot with steering wheels. The robot wrote his name.

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Figure 17. The robot climbing over obstacles.

The results are presented in figure 16 in which the name of the robot is written (SAW). All the letters were completed in a single run with no external intervention.

5.7. Crawling over obstacles In order to prove the robot’s ability to crawl over challenging terrain, we tested in climbing over multiple obstacles. One of those obstacles, composed of two 8 cm high rises is presented in figure 17. The surface is made of carpet to ensure high COF and the robot was capable of repeatedly overcoming this double obstacle.

A second experiment in climbing over a larger obstacle is presented in the attached movie.

6. 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.

This type of locomotion is inspired by the wave-like locomotion of snake and flagellar swimming but, mechanically speaking, is different than snakes since it does not slide relative to the ground and that it does not have an anisotropic COF (the robot can change its direction of motion by simply changing the wave motion). The robot design is simple, lightweight, 11

cheap, and requires only a single motor to produce the wave. The direction of wave propagation is determined by the sign of the voltage being applied to the motor. We developed three prototypes: the larger one with a wave length of 10 cm weighs only 188 g, a 1:2 smaller version weighing 47 g and a 1:3 version weighing 30 g. All prototypes proved to be relatively reliable (considering that they are 3D printed prototypes). During all of our experiments, 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 of the links relative to the motor 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 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 AR can be larger than 1 (this result may seem physically impossible at first glance). By adding wheels, the robot can be

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steered with a turning radius of 0.3 m and it can crawl over challenging terrain and climb over obstacles.

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 at a single point, implying that sliding does not occur and that the links do not disconnect from the surface. We performed multiple experiments and found that they are all within a few percentages from the expected speed by the simulation and the cost of transport of the robot is 3.8. The experiments also proved that the robot can advance faster than its own wave (13% in our experiments). The robot reached a top speed of 57 cm s−1 and no sliding was detected even at this speed during steady state. However, some sliding was observed during acceleration. The robot was also capable of climbing vertically when finely placed between two surfaces polyurethane foam at a speed of 8.2 cm s−1. Our future work will focus on analyzing the locomotion of this type of robots over compliant and slippery surfaces and designing swimming robots. For that purpose, we are considering the possibility of designing a variable gait system which allows to change the amplitude and wave length, therefore allowing the robot to optimize its energy efficiency and crawling performance over different surfaces.

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