Continuous Wave Peristaltic Motion in a Robot Alexander S. Boxerbaum, Hillel J. Chiel, and Roger D. Quinn w

Abstract — We have developed several innovative designs for a new kind of robot that uses a continuous wave of peristalsis for locomotion, the same method that earthworms use, and report on the first completed prototypes. This form of locomotion is particularly effective in constrained spaces, and although the motion has been understood for some time, it has rarely been effectively or accurately implemented in a robotic platform. We present a technique using a braided mesh exterior to produce smooth waves of motion along the body of a worm-like robot. We also present a new analytical model of this motion and compare predicted robot velocity to a 2-D simulation and a working prototype. Because constant-velocity peristaltic waves form due to accelerating and decelerating segments, it has been often assumed that this motion requires strong anisotropic ground friction. However, our analysis shows that with uniform, constant velocity waves, the forces that cause accelerations within the body sum to zero. Instead, transition timing between aerial and ground phases plays a critical role in the amount of slippage, and the final robot speed. The concept is highly scalable, and we present methods of construction at two different scales.

This is the first draft of a paper that will be available in the International Journal of Robotics Research. Manuscript received January 2011. The final version of the paper includes a fourth author, Kendrick Shaw, who contributed a more rigorous analytical approach in Section II. This work was supported in part by the NSF grant, "Dynamical Coordination and Sequencing of Multifunctionality in Animals and Robots," (NSF IIS-1065489) and by Roger D. Quinn, Hillel J. Chiel, and Kenneth Loparo. Alexander S. Boxerbaum is with the Department of Mechanical and Aerospace Engineering, Case Western Reserve University, Cleveland, OH 44106-7222 USA (Phone: 216-952-2641, email: [email protected]). Hillel J. Chiel is with the Departments of Biology, Neurosciences, and Biomedical Engineering, Case Western Reserve University, Cleveland, OH 44106-7080 USA (email: [email protected]). Roger D. Quinn is with the Department of Mechanical and Aerospace Engineering, Case Western Reserve University, Cleveland, OH 441067222, USA (email: [email protected]).

Figure 1: A robot that creates peristaltic motion with a continuously deformable exterior surface. I. INTRODUCTION Soft-bodied invertebrates, such as leeches, worms, and slugs, have successfully colonized marine, terrestrial, and fossorial (underground) environments. Their robust and flexible behaviors are achieved by deploying muscle groups arranged in ordered configurations— longitudinally, circumferentially, or helically. These soft structures can be grouped into two categories: hydrostatic skeletons [Skierczynski, et al. 1996] and muscular hydrostats [Kier and Smith, 1985]. Hydrostatic skeletons have a central fluid-filled cavity. Contraction of a muscle component of the cavity induces an expansion of other parts of the cavity and of its surrounding muscle, allowing the fluid to act as a mechanical power transmission. Muscular hydrostats, such as tongues, trunks and tentacles have no central fluid filled cavity. These structures have higher power-to-mass ratios. Using dense sensor arrays, nervous systems coordinate the many degrees of mechanical freedom in the animal’s body to flexibly move in a variety of ways, including peristaltic crawling, anchor-and-extend, and swimming [Brusca and Brusca, 1990; Ekeberg and Griller,

1999]. The large and flexible behavioral repertoire that allows invertebrates to reshape their bodies could be extremely useful if transferred to a robotic platform with suitable control. A single robot could use these degrees of freedom to shape itself into an active sensing platform or a flexible manipulator, or to move through complex, tortuous regions (e.g., burrowing through rubble). Moreover, the flexibility of the body and its control could serve as a robust substrate for higher-order behaviors, such as the ability to learn better ways of achieving complex goals. Controlling such devices is highly challenging [Tesch, et al. 2009; Transeth, et al. 2009]. Whereas animals such as octopodes frequently use the full dexterity of their limbs in an extraordinary display of multifunctionality, robotic implementations typically simplify the problem by reducing or grouping the degrees of freedom [Menciassi, et al. 2004; Lee, et al. 2010, Trimmer, et al. 2006], and or by replacing continuously deformable soft bodies with rigid joints [Wang and Yan 2007; Omori, et al. 2009]. These simplifications for the sake of control come at a cost to multifunctional flexibility and performance. Vertebrates such as snakes and salamanders are also hyper-redundant, but not soft, and for this reason, robotic implementations inspired by them have had more success [Hirose 1993; Ijspeert, et al. 2007]. However, snakelike undulations do not work in the most confined spaces or where burrowing is required. We previously developed a worm robot using long artificial muscles in series (Fig. 2) [Mangan, et al. 2002]. The artificial muscle consisted of a braided mesh that was used to create a material with anisotropic strain properties. Compression along one axis caused expansion in another. In this case, the material was woven into a cylindrical shape and a bladder inflated the cylinder, pushing outward radially, which caused axial contraction. This robot had much in common with most robots attempting peristaltic motion: a small number of identical segments (often three) attached in series, each of which can alternately contract axially and expand radially [Dario, et al. 2004; Menciassi, et al. 2004; Wang and Yan 2007; Omori, et al. 2009; Seok, et al. 2010; Trivedi, et al. 2008 (review)]. In these robots, the area in between each segment is unactuated. This ubiquitous trend in robot design is consistent with the way peristaltic motion is explained in the literature, in which discrete segments are used to approximate continuous small muscle movements for clarity [Alexander 2003]. There also may be a tendency among engineers to reduce and simplify the design as much as possible. However, peristaltic motion is a wave of motion; a more accurate approximation leads to a more “fluid” and effective motion. The segmented approach would be more suited to modeling animals that do have large segments,

such as caterpillars [Trimmer, et al. 2006]. A notable exception is an amoeba inspired robot that does not use peristaltic motion, but has a novel whole-body method of locomotion [Ingram and Hong 2005]. Our previous robot [Mangan, et al. 2002] moved much more slowly than expected (Fig. 2). It would often appear to slip backwards, or have difficulty progressing when an obstacle landed between actuators. The slipping, which may be common among all robots attempting peristaltic motion, has led investigators to conclude that friction was important for this mode of locomotion [Alexander 2003; Menciassi, et al. 2004; Zimmermann and Zeidis 2007; Zarrouk, et al. 2010]. Also, the robot’s power requirements were substantial: it required an off-board pressurized air supply. These issues were the impetus for re-evaluating our understanding of peristaltic motion and its implementation in a robotic platform.

Figure 2: (Left) A previous worm-like robot with discrete actuators surrounded by a braided mesh. (Right) The inner actuator core that inflates the mesh [Mangan, et al. 2002].

II. THEORY OF PERISTALTIC LOCOMOTION Peristaltic locomotion has several interesting, counter-intuitive properties. The waves of expansion and contraction flow in the opposite direction of the robot motion. This is a direct result of the anisotropic strain properties of the body. When a section leaves the ground, a new ground contact point forms directly behind it. The contracting section will accelerate outward axially, but that motion is constrained on the rear side by the new ground contact point, so the segment must move forward (see video, Supplementary Material, for examples of this motion). An analytical model of peristaltic motion would be useful in many ways. It would provide insights for increasing robot speed and efficiency, and might eventually play a key part in a high-

level control strategy. Both kinematic [Quillin 1999] and dynamic models [Quillin 2000; Alexander 2003] of earthworm locomotion have been previously developed. Quillin defines worm speed as:

Speed =

Stride Length Cycle Time

(1)

While this observational model accurately characterizes earthworm locomotion on flat ground, it does not capture or explain the causes of slippage, and therefore tends to overestimate the predicted speed of worm-like robots. For this reason, an observational measure of ‘efficiency’ is frequently used to describe the discrepancy, defined as the ratio of the effective stride length over the predicted stride length [Zarrouk, et al. 2010]. Recent work into the actual causes of slow robot speed and slippage have focused both on the kinematics and dynamics of segmented peristaltic motion [Boxerbaum, et al. 2009, 2010; Zarrouk, et al. 2010]. The analytical approach outlined here is aimed at finding general principles for improving robot locomotion by increasing the effective stride length. It is also possible that such principles may apply to the animal’s locomotion as well. A. Analytical approach to modeling position, velocity, and forces: Based on the observation that peristaltic robots that use long segments do not perform well, let us look at the other extreme, where a body is free to continuously deform. Consider a differential axial element on the front of the robot (Fig. 3). The element’s initial displacement from its original position is first just the axial strain of that element caused by the deformation wave (
(dl)). However, in the next moment, its displacement will increase both due to the new strain in the element and the axial strain of the next differential element entering the wave. Therefore, the total displacement (P) of the first element can be described as the integral of the strain as a function of length, l:

P(l) =


(l)dl

(2)

Figure 3: Illustration showing the new position of a point on the front of the robot (red dot) as the waveform travels through the body. The displacement as a function of time (P(t)) or position (P(l)) can be found by integrating the deformation wave function
. If the deformation wave as a whole has a constant velocity Vwave, the position of a point P in global coordinates can be found as a function of time (t) by using the following equalities: l = t Vwave ,

dl = dt Vwave

(3a, b)

Now,

P(t) =


( (t V

wave

) Vwave ) dt

(4)

Also, since the position of the point is the integral of the velocity with respect to time, the function inside the integrand of Equation 4 must describe the velocity of the point: (5)

VP (t) =
(t Vwave ) Vwave

These units are consistent because the output of the strain function is dimensionless. Equation 5 is somewhat surprising. It states that at any time, the speed of a point on the robot is directly proportional only to the strain at that specific location. More generally, all other considerations aside, increasing the local deformation (anisotropic strain) or increasing the wave speed will make the robot go faster. The analysis so far has assumed that the robot does not slip. However, it has been widely assumed that friction is required to maintain constant robot speed, i.e., an external force must be applied to keep the robot from moving backwards. We can look at forces within this analytical model by deriving the accelerations required to achieve the desired kinematics. First, let us convert Equation 5 back into positional form by applying (3b). VP (l) =
(l) Vwave

(6)

Taking the derivative with respect to time and applying (3b) again, we find the acceleration of any point on the robot:

AP (l) =

dV d(
(l) Vwave ) d(
(l)) = = (Vwave ) 2 dt dt dl

(7)

Equation 7 is consistent with Alexander’s prediction that since individual segments of the body are constantly being accelerated and decelerated, peristaltic motion requires energy on the order of the velocity squared [Alexander 2003]. However, this is only applies to the masses that are accelerating and decelerating. Thus, it would be possible for both earthworms and robots to maintain their momentum by controlling their central body cavity such that it has a near constant velocity. Such a technique would have clear energy savings. Now that we have an equation for the acceleration of any point on the robot, we can find the forces at the ground contact points. This will be the sum of all the forces acting on the differential segments that have left the ground. If we assume that the robot starts from rest, and the beginning of the deformation wave is a ground contact point, then we can integrate over the wave to find this force:

F=




A(l) dl

(8)

where
is the linear density, making (8) an extension of Newton’s Second Law. Applying (7), we get: 2 F = Vwave 




d ((l)) 2 dl = Vwave  (l) dl

(9)

Thus, the required ground reaction forces are proportional to the square of the velocity of the deformation wave, and the linear density. However, as long as the strain function returns to zero, that is, the differential segment returns to its initial length, there will be a time when the forces sum to zero. In other words, as long as ground contact occurs at the moment when the segment returns to its rest length, no ground contact forces will be required. If ground contact occurs early or late, that segment will still be moving (according to Equation 6) and the forces about the ground contact points will be non-zero (Equation 9). Therefore, ground contact timing is very important. This far, we have only assumed that (dl) returns to zero at some point. The deformation wave can be a-periodic and highly asymmetrical. However, according to the above analysis, there are still points in time when the forces about the ground contact point are not zero, i.e., when (dl) is not zero. Let us add the condition that (dl) is in fact a periodic function: (10)

 (l) =  (l +
)

where
is the period length in pre-deformed space. Up to this point, we have treated integrals as indefinite integrals, i.e., initial conditions were all zero. Here, let us evaluate the sum of the forces over any two points separated by one period. Because of Equation 10, we see that the forces must sum to zero: l +

2 F = Vwave
[ (l)] l = 0

(11)

Therefore, on flat ground, a continuously deformable robot with an even weight distribution, a whole number of waves, and traveling at a constant speed, will have no need for external forces, regardless of the shape of the waveform. In this way, the motion is analogous to a wheel rolling on flat ground: points along the circumference are accelerating, but the wheel rolls at a constant velocity and requires no external forces. An important condition for this is that the structure is continuously deformable. As others have shown, the forces of several discrete segments do not sum to zero [Alexander 2003, Zarrouk, et al. 2010]. This may provide insight into the ubiquitous problem of slip, and suggests a distinct advantage for robots with continuously deformable structures.

B. Adding slip to the analytical model: While the causes of slippage will be explored more fully later in this paper, here we introduce a modification to Equation (4) that can describe its effects. For a given robot design, the strain that is lost is typically the same over each step cycle, and can therefore be incorporated into the strain function by the subtraction of a constant value, Q:

P(t) = Vwave


((t V

wave

)  Q) dt

(12)

When comparing analytical results to data, Q can be chosen such that the velocity, which is proportional to the strain curve, dips below zero at the first ground contact, and comes back up at lift off (Fig. 16, bottom). This is consistent with the observation that the strain that occurs after ground contact contributes to moving the robot backwards rather than forwards (see results for examples). The area under the strain curve is also reduced substantially, thereby dramatically decreasing the displacement of a point on the robot. In terms of segmented robots, even after the displacement of one segment is diminished during liftoff, it will lose again and again as the next segments lift off. The constant Q may be used more broadly to assess the effectiveness of a given robot’s ability to use peristalsis, where lower values indicate faster motion relative to the theoretical top speed.

III. USING A BRAIDED MESH TO CREATE CONTINUOUS DEFORMATION How can a continuous wave of peristaltic motion be created? Earthworms have continuous sheets of both axial and circumferential muscle fibers that work together to create waves of peristaltic motion. During forward locomotion, these two muscle groups are coupled by segments of hydrostatic fluid, creating a hydrostatic skeleton, and are typically activated in alternation at a given location along the body. In the new robot design, we use a braided mesh similar to that used in pneumatically-powered artificial muscles [Quinn, et al. 2003] to create this coupling between axial and radial motion with a single hoop actuator. The robot is still cylindrical in shape, but the outer wall now consists of a single continuous braided mesh (Fig. 1). Any location along the braided mesh can be fully expanded or contracted. Hoop actuators are located at intervals along the long axis, close enough together that smooth, continuous waves can be formed. When these hoop actuators are activated in series, a waveform travels down the

length of the body. The result is a fluid motion more akin to peristaltic motion than that generated by previous robots [Boxerbaum, et al. 2009]. We have developed a new analytical model of peristaltic motion that can deal with continuously deformable structures. We now need to describe the deformation wave function,


(l), for the specific kinematics of our robot. Later in this paper, a specific waveform generator will be added to the model that approximates a cam mechanism that has been built to control the initial prototype robot.

Radially Expanded

Radially Contracted

Figure 4: A single element of the braided mesh is used to derive the anisotropic strain properties of the material. The dimension c is the input, the change in length due to the hoop actuator.

A. Basic Four-bar Mechanism Derivation of Strain: The mechanical strain that occurs with the simple braided mesh described above can be directly calculated from the geometry of four crossing strands (Fig. 4). We will assume the strands are rigid in order to treat them as a four-bar mechanism. However, there must be bending in these fibers in order for distinct waves to form. The scale of the weave is not important for this derivation, as it only describes the anisotropic properties of a continuous ideal material. The hoop actuator contracts along the vertical length d, changing its length to d'. The dimension

along e will expand by an amount that is a function of the initial shape of the diagonal element, defined here by the angle
. From the Pythagorean theorem and the law of sines, we have:

d'2 + e'2 = (2 f ) f sin(
/2)

=

2



e'=

d /2 sin(
/2  /2)

(2 f ) 

2


d'2

f =

(13)

d 2 cos( /2)

(14)

The change in length along d is due to the hoop actuator displacement, c: (15) The input c is often a periodic function that describes the contractions as a function of time or position. The two values d' and c must be scaled appropriately. For instance, if c is the total displacement of the hoop actuator over the circumference, then d' is the maximum circumference of the entire braided mesh. The above equations can be combined to find the new axial length e':

 2 d e'=  
(d
c) 2  cos( /2) 

(16)

Lastly, for the purposes of this analysis we will define the strain of the material as: (17) where

e = d tan(
/2)

(18)

Combining (16), (17), and (18) we now have an equation for the axial strain of the braided mesh as a function of the hoop actuator activation c and the geometry of the mesh defined by d and
:  2 d 
(d
c) 2
d tan( /2)  cos( /2)  = d tan( /2)

(19)

We will see that a strain function of this kind plays a critical role in determining the motion of the robot or animal. * * Author’s Note: The derivation of Equation 19 in Equations 13-18, could potentially be moved to an appendix or online companion, as the math is important to show, but rather straight-forward.

IV. 2-D DYNAMIC SIMULATION A simple 2-D dynamic simulation was created using Working Model 2D (Design Simulation Technologies, version 9.0) to evaluate this method of locomotion, and to capture the discrete nature of individual segments that are not represented in the analytical model. Each simulated body segment consists of a modified four-bar mechanism, where each bar is split into two pieces joined by a torsional spring (Fig. 5). This approximates the ability of the braided mesh to bend, an essential capability for wave formation. The number of segments tested ranged between six and twelve, each driven by a periodic function derived from a cam mechanism discussed later. One of the advantages of this simulation is easy access to a large amount of data, including the positions, velocities, and accelerations of points on the robot, including its center of mass.

Figure 5: A 2-D simulation of the robotic concept with two whole waves. Please see the attached video supplement for examples of the motion. Because this simulation does not have a continuous exterior wall, the ground contact transitions are not perfect. Here, the deformation of each segment is the result of interactions between many actuators, as is the case with true soft-body dynamics. By changing the number of waves over the length of the body, and therefore the number of actuators per wave, we were able to study the effects of segmentation and soft-body interactions on locomotion speed.

V. ROBOTIC CONCEPTS AT A SMALL SCALE The kinematics of peristaltic motion are entirely scale invariant. At any given scale, a crosshatched mesh needs to be constructed with the correct stiffness, and a suitable actuation method found. Here, we briefly propose two methods of construction at a very small scale.

A. Shape Memory Alloys: A robot with a diameter on the order of one centimeter would have applications in medicine, including examination of the entire gastrointestinal tract, as well as applications in search and rescue environments and military reconnaissance. Shape Memory Alloys (SMAs) are a good candidate for actuation at this scale. Micro helix SMAs have strain ratios of up to 200% and can be actuated in 0.2 seconds [Menciassi, et al. 2004]. One challenge of working at such a small scale is how to get the SMA actuator to return to its initial position. It must cool down, and a force must be applied to restore it. Here, we propose a method of returning the SMA to its original state that relies on a hydrostatic fluid. This fluid could also serve to cool the SMAs, and speed up the cycle. We also propose that the wiring for the SMA actuators can also constitute the braided mesh (Fig. 6) [Boxerbaum, et al. 2009]. Seok, et al. recently address this problem by mechanically fixing the length of the robot, so one constricting SMA forces others open. This method succeeded in producing locomotion, with losses due to the uneven expansion cycle [Seok, et al. 2010].

Figure 6: SoftWorm robotic concept using shape memory alloys and a hydrostatic fluid as a return spring.

Figure 7: A cross sectional view of the SMA concept. The brown arrows indicate the flow of the exterior braided mesh. The blue arrows indicate the flow of the bolus of fluid that expands the contracted sections. The red arrows indicate expanding and contracting hoop actuators. In this implementation, shown in Fig. 7, a bolus of fluid (large blue arrows) moves between the outer skin and the inner payload of the robot by the sequential constriction of hoop SMA actuators (red inward-pointing arrows). As the fluid is squeezed at the trailing edge of the wave, it causes radial expansion at the leading edge of the wave (red outward-pointing arrows). The result is the generation of continuous peristaltic waves along the robot, causing it to move in the opposite direction of the wave (brown arrows). B. Hydrostatic Fluid Actuators: An alternative method of actuation at small scales is being explored as well. The braided mesh of the robot could be made of hollow tubing and serve as hydraulic lines for microhydraulic actuators at each hoop (Fig. 8). Hydraulic actuators are generally only effective as pushing actuators, requiring the natural state of the robot to be elongated and narrow. Expansion at one of the hoop actuators would be achieved by applying pressure at the end of the hydraulic line. This would also allow for mechanical coupling of the hoop actuators, and allow them to be driven by a single end-mounted motor. This setup could achieve faster waves, and therefore faster robot speeds than the SMA implementation, but it would require an effective microhydraulic piston to be developed.

Figure 8: Micro-hydraulic actuator concept. Here, the hoop actuators expand against a contractile force to create the wave motion.

VI. ROBOTIC CONCEPTS AT A LARGE SCALE Two large-scale prototypes of this new robotic concept have been completed and tested. With a maximum diameter of 25 cm, they are scaled to function in fresh water mains. Their hollow cores would allow servicing to be performed without shutting off water flow. Latex surgical tubing is used as a return spring along the opposing axis of the hoop actuators in these prototypes (Fig. 9).

Figure 9: The first prototype keeps its anisotropic strain properties by weaving bicycle brake cable sheathing and securing it with latex tubing that also acts as a return spring. This method of construction did not keep the mesh in good alignment after several trials. In this figure, a single hoop actuator has been installed and is in the constricted position. In this implementation, the braided mesh that provides the unique anisotropic strain properties has an elegant dual function. It is made of bicycle brake cable sheathing, which is hollow and rigid along its long axis. Steel cables run through the sheathing out to individual hoop actuators where there is a mechanism that interrupts the brake cable sheathing and routes

the cable around the circumference (Fig. 11). Two cables run through each sheath and split in opposite directions to meet on the far side, creating a hoop actuator that is controlled at one end of the robot. This doubles the stroke length of the actuator compared with a single cable wrapped around the whole circumference. In the first prototype, the mesh of brake cable sheathing keeps its shape because it is woven and secured with the latex tubing at most junctures (Fig. 9). This technique did not prove robust in repeated testing, and the uneven bending of the sheathing increased cable friction. The second prototype remedies this problem by using swivel joints at each crossing (Fig. 10). These joints are specialized to provide several functions. Some guide the hoop actuator cable around the circumference and transmit its forces to the mesh. Others secure the latex return spring along the long axis of the robot. In future versions, this latex return spring can be replaced with cable to add additional degrees of freedom and turning ability. Securing the mesh with swivel joints precludes the need to braid the mesh, and allows the mesh material to have a constant bending radius, which in turn reduces the friction of the brake cable running within the sheathing. Alternatively, encasing the mesh in a soft polymer skin would also preserve the alignment of the strands and act as a return spring. Each hoop actuator has a special swivel joint with a clamping mechanism (Fig. 10, A). This secures both strands of the hoop actuator, and allows the rest length of each actuator to be tuned. This addresses a mechanical limitation we discovered in the first prototype. The most distant actuators require more force to actuate because of their long runs through the brake cable sheathing. In the first prototype, the cable clamping mechanism was at one end of the robot, where it was pulled by a drive motor (Fig. 12), and the clamp was frequently not strong enough to resist these high forces. So, typically after a few waveforms, the cables slipped out, causing the most distant actuators to fail. Even with several failed actuators, the robot still moved forward at a slower speed.

A

B

C

Figure 10: The second prototype secures the braided mesh, the hoop actuators and the latex return spring with several hundred specialized swivel joints. They secure the end of the hoop actuator cables (A), guide and secure the latex return spring (B), and guide the hoop actuator cables (C).

Figure 11: A specialized swivel joint terminates the brake cable sheathing and routes the nested cable in two directions to form the hoop actuator around the circumference of the body. The rubber foot in the foreground provides traction with the ground.

Figure 12: The cam mechanism that drives all actuators and creates two traveling waves along the length of the robot. The location of the cable origin about the circumference indicates the phase shift relative to the other actuators.

Figure 13: Three waveforms created by the cam mechanism can be described using the law of cosines. The greater the distance between the cam head and the axis of rotation, the sharper the lower transition. In both prototypes, the steel actuator cables are pulled in sequence by a cam driven by a single drive motor at one end of the robot (Fig. 12). While future versions may have individually controlled actuators in order to study sensorimotor wave propagation and adaptive behavior, this

mechanism creates peristaltic motion with no computational overhead and with a waveform that provides good speed. In this way, forward and backward motion is controlled as a single degree of freedom using a single DC motor. The cam mechanism is designed to pull on the cables with a waveform that is roughly sinusoidal in both time and space. The exact waveform is a combination of both sine and cosine waves that has a near singularity due to the geometry (Fig. 13). The shape of the waveform can be adjusted easily by changing the length of the cam arm (Fig. 12, line b). In the current setup, two waves are present at all times. Closely paired cables visible in Figure 12 are routed to two hoop actuators spaced apart by half the length of the robot. Their proximity to each other on the perimeter of the cam indicates that these two actuators will have nearly identical states at any given time. With this style of cam mechanism, any whole number of waves along the body is possible. Both prototypes were designed such that two full waves propagated along the length of the body in order to prevent early ground contact, while still providing at least five hoop actuators per wave. In the first prototype, ten hoop actuators are distributed along the length of the robot, utilizing only half the available brake cable sheathings. The second prototype has twelve hoop actuators, but the mesh is constructed of only twelve strands in order to reduce weight and the number of swivel joints (Fig. 14).

Figure 14: Cable wiring diagram for the second prototype with twelve actuators and twelve strands. The green strands wrap clockwise relative to the cam mechanism, while the red strands wrap counterclockwise. Each actuator location is paired at the cam head with a location half the length of the robot away. We initially attempted to use polyester string as an actuator cable, because of its very small minimum bend radius. These strings repeatedly broke under loading. While Kevlar or Spectra string may still be good alternatives, we decided to use steel cable, specifically for its strength and its natural pairing with the brake cable sheathing. The larger minimum bend radius of steel cable meant that special care had to be taken in how the cables were routed. The final mechanism routes the cables such that the minimum bend radius is never less than 12 mm, sufficient to accommodate any steel cable small enough to fit in the brake cable sheathing. Video of all trials was recorded from an angle orthogonal to the direction of motion. The second prototype trials were processed using WINanalyse (Mikromak) software to extract position data.

VII. RESULTS A. Dynamic Simulation and Analytical Model: The analytical model assumes that the soft continuously deformable structure behaves exactly as it should, regardless of the number of segments. However, in soft structures, the actuation of one segment can directly affect another. To study this, we compared three simulations of a twelve-segmented worm robot that had one, two, or three waves over the length of the body at any time (Fig. 15). Whereas the kinematics of a rigid mesh predict that all three simulations should have the same speed, the simulated speed decreases by nearly a factor of two as the number of waves along the body increases from one to three. This can be attributed to the way in which the soft structure causes an interaction between segments. As number of segments per wave goes from 12 to 4, the ratio of actual maximum to minimum length goes from 3.2 to 1.9, corresponding to a decrease in speed of about half as that predicted by the analytical model. More waves also come at a higher energetic cost, as discussed in [Alexander 2003], if the robot cannot capture the energy of the decelerating segment.

Figure 15: The simulated horizontal displacement of a single segment as a function of time for three different waveforms. The body length is an average of 1.5 meters, giving average speeds of 0.17, 0.12, and 0.078 body-lengths/s. Whereas the one wave robot has a maximum to minimum horizontal strain ratio of 3.2, the three wave robot has a ratio of a little more than half 1.9, corresponding to a decrease in speed of about half as predicted by the analytical model. Note that the one wave per body-length is the fastest even though it has the most retrograde motion. This increase in retrograde motion is due to the fact that a single waveform cannot prevent ground contact during the flight phase.

When we adjust for the benefits of a single wave in terms of its ability to better approximate a high-strain waveform, there is another inefficiency that becomes apparent. The single waveform has significant negative horizontal motion during stance phase, which can be modeled with a higher Q value in the analytical model. Careful examination of the video suggests this is due to the fact that a single wave cannot effectively keep the swing phase segments off the ground. As segments contact the ground at inappropriate times, they apply forces on the stance phase segments that counteract forward motion, as our analytical model predicted. This negative motion is reduced when there are two proper ground contact points. Figure 16 shows the position and velocity of a point on the simulated 2-D robot compared to the analytical model. The Q factor can accurately capture the effects of early ground contact. While the velocity of the simulation changes with each new ground contact, the running average is well-described by the analytical model. The velocity curve in Fig. 16 also illustrates the importance of the swing-stance transition. The position is the integration of velocity over time, so the area shaded in orange (Fig. 16, bottom) is equal to the distance lost due to poor transitions between swing and stance. A 16% error in the swing-stance transition results in a reduction of speed by a factor of two.

Figure 16: Comparison of the analytical model (dashed lines) to the dynamic simulation (solid lines). The position is the integration of velocity over time, so the area shaded in orange is equal to the distance lost due to poor transitions between swing and stance. The red dashed line is the position predicted by the analytical model if the robot has ideal swing-stance transitions and Q = 0.

B. Current Prototypes: The first prototype generated the desired waveforms successfully for short periods of time. A speed of 0.97 m/min was achieved over a distance of 0.9 meters. The speed was intentionally slow in order to help diagnose problems. After that distance, individual actuator cables began to slip, causing non-uniform wave propagation. The robot still moved forward until four of the ten actuators had become non-functional. This suggests that in more rugged environments, the device will be robust to individual actuator failures. The second prototype addressed a number of issues that arose during testing of the first prototype, most of which involved refining the method of guiding the actuator cables. The swivel joints at each mesh juncture greatly improved the reliability of the device and increased the speed of the robot by significantly reducing the friction along the actuator cables. By moving the

cable clamping location from the cam mechanism to the far side of the hoop actuator, cable slippage was completely eliminated. This allowed the drive motor to operate at close to its maximum speed, allowing the robot to travel at 6 body-lengths per minute (4 m/min) (Fig.17). Fluid waves of motion were observed with little retrograde motion. 91% of the total step length was preserved. Because the body was rigid enough to prevent early ground contact and the task did not involve burrowing, anisotropic friction forces were not necessary to achieve good speeds.

Figure 17: Stills from a video of the second prototype moving over 1 second. The black lines indicate the smooth continuous rearward progression of waves. The analytical model was also compared to motion capture data taken from the second prototype trials. Using dimensions taken directly from the robot, the model describes the kinematics of motion very accurately (Fig. 18). A Q value of 0.5 was chosen to adjust the average slope in the position data, slightly higher than the value of 0.35 used when comparing the analytical model to the simulation. The velocity curve fit is not as accurate during ground transitions. While the kinematics of the cam mechanism suggest a short period of large negative velocity, the robot experiences a longer period of small negative velocity. The negative velocities still occur at the critical time when the kinematic strain rate is the highest. A robot that could sense early ground contact and respond to reduce it would likely see significant increases in speed, shown in Figure 18 as a Q value of 0.2.

Q = 0.2 Q = 0.5

Q = 0.8

Q = 0.2 Q = 0.5 Q = 0.8

Figure 18: Comparison of the analytical model (Dark Blue) to trials of the second prototype (Connected Blue Dots). The analytical model is based on dimensions taken from the robot and the kinematics described earlier. A Q factor of 0.5 was chosen to align the position function, which also gave the best correlation in the velocity function. Q factors of 0.2 (Red) and 0.8 (Green) are also shown for comparison. VIII. DISCUSSION The analytical model provides many insights into peristaltic locomotion. If certain conditions are met, the dynamics of peristaltic locomotion do not require external forces (Eqs. 9 and 11). We are forced to look for a better understanding of why robots slip. The model also suggests that ground contact timing plays a critical role. Only if a segment touches the ground in its fully expanded state can there be no ground contact forces (Eq. 9) or velocity discrepancies (Eq. 6). Chief among our assumptions of this analysis is the continuity condition. The internal forces only sum to zero if the deformation is continuous. Otherwise, a host of additional conditions

need to be met, which are generally impractical to implement or control. Thus, robots with a small number of segments require substantial ground reaction forces (friction) or they will slip. Given a continuously deformable robot, the analytical model defines only two ways to move faster: by building waveforms with higher strain rates or by generating a faster wave (Eq. 5). The shape of the waveform deformation is limited by the need to have ground contact, and to prevent premature ground contact in the segments about to touch down. Premature ground contact was frequently observed in our 2-D simulation, even in the rearmost segments. When the ground contact point switches from one segment to the next, the second segment often contacts the ground before it has fully expanded radially. Therefore, after ground contact, it will continue to expand radially, instead of contracting as part of the next cycle. This means that the wave gets unnaturally stretched due to too many kinematic constraints, and at least one of the ground contact points must slip. In this situation, anisotropic frictional properties are beneficial by forcing the robot to slip forwards, rather than backwards. Problems can also occur during the ground-to-aerial transition. The analytical model shows that the acceleration of the segment would be greatest during the very beginning of the aerial phase. Fig. 19 is derived from (19) and shows that given a set displacement, c, the initial angle,


, is a critical factor in the amount of axial strain that is achieved. However, in robots with distinct segments that are controlled to have only one segment on the ground at a time, the transition between segments causes the ground-to-aerial segment to stay on the ground too long, during the time that it would be accelerating the most. This problem can be mitigated by having a continuously deforming outer mesh, or by keeping at least two segments on the ground at each ground contact location. While the most strain is achieved with small start angles, the forces required to move are high, due to the low mechanical advantage. Because the mesh is soft and flexible, this can be impractical. The braiding along the hoop actuators will not transfer the forces to the immediately adjacent mesh before buckling. It would be advantageous to have the smallest initial angle possible that does not induce buckling.

Figure 19: Strain as a function of the initial angle
, with a fixed displacement. Our second prototype achieved a speed of six body lengths per minute, a very fast speed for peristaltic locomotion. Comparatively, earthworms travel at speeds of 1.2 to 3.6 body lengths per minute (Quillin 1999), and our previous robot traveled at 0.8 body lengths per minute (Mangan, et al. 2002). While the Q value of 0.5 indicates faster speeds are possible, very little slippage occurred. Instead, the losses in speed are more likely due to the application of predictions from rigid kinematics to soft body dynamics. As the flexible 2-D simulation showed, bending of the structure reduces the effective strain rates. Lastly, early ground contact may have played a role in reducing speed, as gravity tends to pull the robot towards the ground prematurely. Nonetheless, these obstacles were substantially overcome by a method of actuation that can produce very high frequency waveforms with ease. IX. CONCLUSIONS We have developed a novel analysis of peristaltic motion, have captured its essential predictions in an analytical model, and have tested the predictions of the model by building prototypes and comparing the model predictions to the empirical results. We found that our previous robot, and nearly all other robots that claim to use peristaltic motion, move much more slowly than predicted because of the kinematics and dynamics caused by very long actuators that greatly exaggerate the segmentation of the robot. Our study of the kinematics of peristaltic motion, both in simulation and using analytical tools, suggests a new design of a worm-like robot with a continuously deforming outer mesh. We presented several methods of constructing such a robot with a continuously deforming exterior at different scales, and reported on the completion of two large-scale prototypes and their locomotion capabilities. Several novel mechanisms allow for simplification of the control problem by coupling the degrees of freedom. Both an analytical model and simulation effectively describe the motion of the prototypes and suggest several principles that ensure effective locomotion. While much attention has focused on the need for friction in peristalsis, we show

that ground contact timing is instead a critical factor in unconstrained environments. This approach shows promise of great improvements of speed and performance over previous wormlike robotic platforms.

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