Abstract—While soft-bodied animals have an extraordinarily diverse set of robust behaviors, soft-bodied robots have not yet achieved this flexiblity. In this paper, we explore controlling a truly continuously deformable structure with a CPG-like network. Our recently completed soft wormlike robot with a continuously deformable outer mesh, along with a continuum analysis of peristalsis, has suggested the neural control investigated here. We use a Wilson-Cowan neuronal model in a continuum arrangement that mirrors the arrangement of muscles in an earthworm. We show that such a system is well suited to incorporate sensory input and can create both rhythmic and nonrhythmic activity. The system can be controlled using straightforward descending signals whose effects are largely decoupled and can modulate the properties from CPG-like behaviors to static waves. This approach will be useful for designing robotic systems that express multiple adaptive behavioral modes.

I. INTRODUCTION

he nervous systems of soft-bodied invertebrates, such as leeches, worms, and slugs allow these animals to exhibit

an extraordinarily diverse and robust repertoire of behaviors, including peristaltic crawling, anchor-and-extend, swimming, and dexterous manipulation. They achieve this by using distributed arrays of neurons to coordinate the many degrees of mechanical freedom of the animal’s body [1]–[3]. This large and flexible behavioral repertoire that allows invertebrates to reshape their bodies could be extremely useful if transferred to a robotic platform.

Controlling such so-called hyper-redundant devices [4], however, is challenging [5], [6]. Robotic implementations typically simplify the problem by reducing or grouping the degrees of freedom [7]–[9], or by replacing continuously deformable soft bodies with rigid joints [10], [11]. These simplifications for the sake of control come at a cost to multifunctional flexibility and performance. Vertebrates such as snakes, and salamanders while also hyper-redundant, are not soft-bodied, and for this reason, robotic implementations inspired by them have had more success [12]–[14].

Manuscript received March 28, 2011. This work was supported by the

NSF grant IIS-1065489 and by Roger D. Quinn and Hillel J. Chiel. 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: asb22@case.edu).

Andrew D. Horchler is with the Department of Mechanical and Aerospace Engineering, Case Western Reserve University, Cleveland, OH 44106-7222 USA (email: adh9@case.edu).

Kendrick M. Shaw is with the Department of Biology, Case Western Re-serve University, Cleveland, OH 44106-7080 USA (email: kms15@case.edu).

Hillel J. Chiel is with the Departments of Biology, Neurosciences, and Biomedical Engineering, Case Western Reserve University, Cleveland, OH 44106-7080 USA (email: hjc@case.edu).

Roger D. Quinn is with the Department of Mechanical and Aerospace Engineering, Case Western Reserve University, Cleveland, OH 44106-7222 USA (email: rdq@case.edu).

It is easier to control a series of linked rigid bodies than a soft body, not because of the mechanical difficulty of the task, but rather because of the classical control paradigms that are commonly used. These techniques typically involve knowing the exact kinematics of the body, deciding where it “should” be, and applying forces and moments to get it to the desired new position, all the while calculating the effects of those forces on the body and limbs [6]. Finding the equations of motion for such rigid bodies is much easier than for their soft-body counterparts [15], and this has placed a limitation on the state-of-the-art. However, animals, soft-bodied or not, do not rely on such techniques.

We are interested in peristaltic locomotion for a hyper-redundant wormlike robot. Most previous robots attempting peristaltic motion have much in common: a small number of identical segments (often as few as three) attached in series, each of which can alternately contract axially and expand radially [7], [10], [11], [16]–[18]. In these robots, the area between each segment is un-actuated. These robots also tend to slip, which has led investigators to conclude that friction was important for this mode of locomotion [7], [19]–[21]. This is also consistent with the fact that worms must generate strong ground friction forces in order to burrow. However, our recent analysis has shown that transition timing among segments plays a key role in slippage, and discrete segments exacerbate these effects [22], [23]. This led us to the development of a mechanical structure that is instead continuously deformable.

How can a continuous wave of peristaltic motion be created in a robot? Earthworms have continuous sheets of both longitudinal and circumferential muscle fibers that work together to create waves of peristaltic motion [24]. During forward locomotion, these two muscle groups are antagonistically coupled by segments of hydrostatic fluid, creating a hydrostatic skeleton [25], and are typically activated in alternation at a given location along the body. The circumferential muscles cause lengthwise elongation, and move the segments forward, whereas the longitudinal muscles shorten the segments. In our recent robot design, we use a braided mesh similar to that used in pneumatically-powered artificial muscles [26] to create this coupling between longitudinal and radial motion. The outer wall consists of a single continuous braided mesh (Fig. 1) with circumferential actuators spaced at intervals along the long axis. In lieu of longitudinal actuators, return springs are used along the length of the body to cause lengthwise contraction in the absence of activation. When the actuators are activated in series, a smooth continuous 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 [22], [23].

Our large-scale prototype uses a cam mechanism at the end of the robot to power all twelve actuators with a single offset

A Controller for Continuous Wave Peristaltic Locomotion

Alexander S. Boxerbaum, Andrew D. Horchler, Kendrick M. Shaw, Hillel J. Chiel, and Roger D. Quinn

T

waveform. The resultant deformation wave has many of the properties that our analysis shows are desirable: it is periodic, it has a constant shape and speed, any given segment has a short ground phase, and the waveform moves fast. Our prototype achieved a speed of 6 body lengths per minute (calculated from length of braided mesh, excluding cam mechanism) (Fig. 2), a very fast speed for peristaltic locomotion. In comparison, earthworms travel at speeds of 1.2 to 3.6 body lengths per minute [24], and our own previous segmented robot traveled at 0.8 body lengths per minute [27].

However, with the twelve circumferential actuators effectively coupled into a single degree-of-freedom for initial testing, only wave speed can be modulated and the prototype is unable to adapt to the environmental changes. Our analysis highlights the importance of transition timing between aerial and ground phases (see Methods below for a summary), something that cannot be optimized without a more flexible control scheme. To attain this, it would be desirable to have independent control of each actuator and thus the ability to also modulate wave amplitude and wave frequency as functions of time and position along the body. This, of course, greatly complicates the control problem, and this paper presents the foundations of our approach, which may have more broad applications in soft robot control.

Many groups have used neuroethological studies and neurally-inspired control to tackle the problem of generating robust motor coordination for hyper-redundant systems. By coupling central pattern generator (CPG) controllers to different degrees of freedom, it is possible to create coordinated rhythmic behavior in hyper-redundant robots

(e.g., swimming in robotic lamprey [28] and knifefish [29], walking and swimming in a salamander robot [30], and undulatory locomotion in a snake robot [31]). The CPGs are based on limit cycle oscillators that drive each segment with a stable rhythmic pattern. Adjacent segments are then coupled via a phase offset to achieve waves along the length of the body.

The approach of Wadden, et al. [32] is similar to our own and forgoes discrete segments and coupled CPGs to develop a new model of undulatory swimming in the lamprey. Instead, a “continuous” pattern-generating network is created from a column of excitatory and inhibitory Hodgkin-Huxley neurons distributed along the length of the body. These spiking neurons interact more strongly with their local neighbors than with more distant neurons. The result is coordinated activity with a phase lag that is capable of producing a wide range of frequencies for forward or backward swimming.

While very biologically realistic, Hodgkin-Huxley model neurons are often more detailed than is needed for most engineering applications and require a great many parameters to be specified. Recently, we created a numerical method of modeling the primary visual cortex, based on the spatially-extended Wilson-Cowan model of neural population activity [33], [34], that incorporates stochastic phenomena and simulated sensory input from the retina. This allowed us to find non-symmetric steady-state solutions to the equations that also correspond to phosphenes [35]. Our numerical model also demonstrated the ability of the Wilson-Cowan equations to produce both static and moving waves with smooth transitions, with and without external input. These properties suggested that a similar dynamical system could be applied to generating continuous peristaltic locomotion.

II. METHODS

Designing neural control networks is very challenging, and there remains no formal approach to doing so [30]. An intuitive familiarity with both the desired behavior and the properties of neural control circuits is needed to find potential solutions. Our experience studying peristaltic motion has led to principles for effective motion. We will first derive equations quantifying those principles, and then develop a neural model for the adaptive control of peristaltic motion.

A. Theory of Peristaltic Motion

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, but that motion is constrained on the rear side by the new ground contact point, so the segment must move forward.

While both kinematic [24] and dynamic models [19], [36] of earthworm locomotion have been previously developed, these models do not capture or explain the causes of slippage, and therefore tend to overestimate the predicted speed of

Fig. 2. Stills from a video of the second prototype moving over 1.14 seconds. The blue lines indicate the smooth continuous rearward progression of waves. The robot is travelling at six body-lengths per minute (4 m/min) without any anisotropic friction devices.

Fig. 1. A robot that creates peristaltic motion with a continuously deformable exterior surface.

wormlike robots. Our recent analysis assumes a continuously deformable body, as opposed to large segments, to study the speed of peristaltic motion [23]. Here, we extend that analysis to forces.

Consider a differential axial element on the front of the robot (shown in Fig. 3 as Δl for clarity). Let us assume that the mechanical strain of an element can be defined by a deformation wave, ε(l tVwave), where l is the position of the element measured from the head of the worm, t is time, and Vwave is the speed of propagation of the deformation wave along the body, all in undeformed coordinates. If we assume that the strain is initially zero everywhere on the body, the head element’s initial displacement from its original position is just the axial strain of that element caused by the deformation wave, ε(l tVwave)Δl (Fig 3, l0). 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 (Fig 3, l2).

The total displacement of the front of the robot can be defined as

P t( ) = ε l − tVwave( ) dl0

tVwave∫ (1)

Equation (1) can be rewritten it in terms of s = l/Vwave, where s is the time the traveling wave reaches the point at l, and then substituting u = t s, yielding

P t( ) = ε (s− t)Vwave( )Vwave ds0

t

∫ = ε −uVwave( )Vwave du0

t

∫ (2)

The velocity of the front of the robot can be found by differentiating (2) with respect to time:

Vp (t) = ε −t Vwave( )Vwave (3)

By taking another derivative with respect to time, the acceleration between the front of the robot and the nearest ground contact point is

Ap (t) =dVpdt

=d

dtε −t Vwave( )Vwave = −Vwave

2 ′ ε −t Vwave( ) (4)

Note that posterior parts of the robot will experience these same velocities and accelerations after a delay of l/Vwave.

The sum of the forces that produce the accelerations given in (4) must be equal and opposite to the ground reaction forces at the ground contact points. Therefore the total ground reaction force, F, is

F = −Vwave2 ′ ε −(t − l /Vwave )Vwave( )( )ρ

0

tVwave∫ dl

= ρVwave2 ε(−tVwave )

(5)

where ρ is the linear density of the robot. Equations (3) and (5) have many implications. Developing speed in peristalsis has always been challenging. Equation (3) states that if slip is controlled, increasing the local deformation (anisotropic strain) or increasing the wave speed are the only two ways to make the robot go faster. Equation (5) states that the required forces are proportional to the square of the velocity of the deformation wave, and the linear density.

Note, however, that even if the deformation wave, ε, is not

periodic, it is likely to return to zero at some point, i.e, the differential segment returns to its initial length. When it does, the velocity of that segment will also return to zero according to (3). If this occurs at the moment of new ground contact, then the differential element will make contact at the instant when it has zero speed relative to the ground, and no slipping will occur. If, however, ground contact occurs at some other point in the cycle, the difference in velocities ensures an over-constrained system, and something must slip. Likewise with the forces, if ε returns to zero as new ground contact occurs, then, as (5) shows, the force at both ground contact points will also go to zero. In effect, the internal forces of the waveform must cancel. An important condition for this is that the structure is continuously deformable.

Looking at a single wave, one can see that (5) must hold in order to not have an over constrained system, but only for each wave independently. Thus, it is ideal that once a waveform has started, that it passes through the robot unchanged. The next waveform, however, can be different, but if its total displacement is different, then there will be forces acting on the body, and those forces will be proportional to acceleration of the body relative to the last wave’s resultant velocity.

These principles create certain tradeoffs and will directly inform the search for a suitable controller and the necessary sensory feedback. If one desires to go as fast as possible, then the analysis suggests that the speed of the wave and the strain rate must be maximized. The easiest way to maximize the strain rate in a soft body is to have longer waves, and less time in stance. However, this can frequently cause early ground

Fig. 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), can be found by integrating the deformation wave function, ε, which is defined in the undeformed coordinate space but shown in the deformed coordinate space.

contact, which has serious negative consequences to the motion. Thus, an ideal controller would sense that the body is sagging, or that the environment has caused early ground contact, and naturally react. If one is aware that a new ground contact point has formed early, then the waveform can be adjusted to prevent dramatic negative forces. These findings are consistent with 2-D simulations and data taken from the first prototype robot [23] and directly inform the search for a suitable controller and the necessary sensory feedback.

B. Using a Wilson-Cowan Model to Create Adaptive Waves To control a soft wormlike robot, the circumferential and

longitudinal actuators must be coordinated to allow peristaltic locomotion, but also turning, burrowing, and adaptation to the surroundings. In our model, the longitudinal and circumferential actuators are controlled by separate arrays of simulated neuronal populations. These populations have a spatial configuration that mirrors the muscle architecture of a worm, and populations closer to each other have stronger connections. The simulation presented here consists of a column of 150 actuator pairs along the body length.

The dynamics of the system are based on the spatially-extended Wilson-Cowan model of the primary visual cortex [34]. In the classical model, excitatory and inhibitory populations are arranged in two 2-D arrays. Such a system has been shown to produce statically stable patterns, as opposed to oscillatory behaviors, and should not be confused with a Wilson-Cowan oscillator [37], which has two mutually inhibitory populations that are externally excited. Here, we

modify the classic cortex arrangement by replacing the single inhibitory population array with two separate arrays of inhibitory populations, Ia and Ip, whose connections are shifted anteriorly and posteriorly, respectively, to the excitatory populations, E. The state variables that drive one of the two sets of actuators (circumferential or longitudinal) can be described as:

τ eE = −E +σ wee E⊗ E( )−wiae Ia⊗ Ia( )−wipe I p⊗ I p( )−wiceC( )τ iIa = −Ia +σ wei E⊗ E( )−wii Ia⊗ Ic( )( )τ iI p = −I p +σ wei E⊗ E( )−wii I p⊗ Ic( )( )

(6)

where the activation levels of the vector E represent the local force of a given actuator. The vector, C = (0.5(Ia +Ip)) ⊗ Îc, is a coupling term that connects (6) with three additional mirrored equations that describe the state of the opposing set of actuators. With no other external input, an active region of E will decay at a rate of –E ⁄ τe, where τe is a time constant of the population. We assume that excitatory neurons have a quicker response time, so τe = 1 and τi = 1.5. The function, σ (x) = 1.05 ⁄ (1 + 20 exp(−7x)) − 0.05, is a generalized logistic, which has a y-intercept of σ(0) = 0; strong negative inputs yield weak bounded negative outputs, which contributes to the stability of the system. The ⊗ operator convolves a population vector with an associated kernel. Ê is a Gaussian kernel that represents the spatial connectivity between the excitatory populations, E, and the inhibitory ones. Îa and Îp are shifted Gaussian kernels (see below and Fig. 5) that represent the spatial connectivity between the anterior and posterior inhibitory populations, respectively, and the excitatory populations. Îc is a Gaussian kernel that represents the connectivity between the two inhibitory populations and the excitatory populations, but with zero offset. All kernels are normalized, and the total weights of the connections are determined by the scalar constants, w (wee = 3.2, wei = 2.4, wice = 1.6, wii = 0.6). These scalar constants provide the easiest way to tune the behavior of the system and are also shown in Fig. 4.

In the classical Wilson-Cowan model, a single inhibitory population has broader yet weaker connections than the excitatory population. In this case, the combined influence of the connections to the excitatory populations forms what has been called a “Mexican hat” distribution (Fig. 5a). In our variation, the two kernels, Îa and Îp, are respectively offset negatively and positively by equal and opposite amounts relative to zero mean and combine to form a “Mexican hat” as well (Fig. 5b). However, biasing the weights of the shifted inhibitory populations gives rise to a variety of additional spatial influences (Fig. 5c).

In the robot, the circumferential and axial actuator groups are antagonistically coupled through a braided mesh or hydrostatic fluid. Together, they create the desired local force difference to produce locomotion. Weak inhibition between the two actuator networks (Fig. 4, wice) is applied so that they remain phase-locked.

Fig. 4. The various connections and their weights are shown between populations, as described in (6). It is important to note that each synaptic connection here is applied to multiple populations through a Gaussian distribution to its neighbors. The excitatory populations (green) directly stimulate the actuators, while also locally stimulating two inhibitory networks (blue). These two networks in turn inhibit the excitatory network more broadly, but are shifted in opposite directions along the network.

Fig. 5. An excitatory (blue) and inhibitory (red) Gaussian kernel combine when excited to produce a “Mexican hat” influence (magenta) (a). Similar influences can be created with two offset inhibitory kernels (b). If the inhibitory populations have asymmetrical influence, a variety of patterns can emerge (c). In a and b, motion is not sustained without external influence. In c, waves of motion naturally occur.

Because the state equations use convolutions, boundary conditions need to be set at both the head and tail to define the values just outside the valid simulation space. Both zero-padded and reflective (mirror image) boundary conditions resulted in sustained excitatory activity where the waveform terminated. In some cases, this edge activity propagated through the simulation and caused a noticeable shift in the temporal frequency. A circular boundary condition, so that the head connects to the tail, eliminated these edge effects, but locked the simulation into patterns that have a whole number of waves over the length of the body. The boundary condition that qualitatively most reduced edge effects was a soft zero-padding that linearly forced only the excitatory populations to zero over the first and last 10% of the length of the simulation at the beginning of each time step. This is equivalent to saying that while the density of excitatory neurons remains constant over the length of the body, it decreases to zero at the head and tail.

The initial conditions of all populations were set to zero. Stable patterns would typically emerge from these initial conditions within 50 time-steps.

III. RESULTS

The architecture described in the Methods can produce a variety of waveforms (Fig. 6) by modulating the strength of the connections from the two sets of inhibitory populations to the excitatory populations. The waveform can travel both forward and backwards at a wide range of speed and even come to a complete stop (static wave), and can change its spatial frequency. In the top trace of Fig. 6, the vertical axis represents the position along the long axis of the body, and the color intensity denotes activity in either the circumferential or longitudinal actuator controllers. The horizontal axis is time for all traces. Thus, blue and red “lines” with a positive slope are moving towards the head over time (in fact, the slope is equal to the speed of the waveform itself). Both actuator

populations can produce waves on their own, but phase locking them with mild cross-inhibition enhances their regularity and also helps reduce boundary condition effects.

The third and fourth traces of Fig. 6 are of the difference and sum of the anterior and posterior inhibitory weights respectively. The local maxima and minima of both functions correlates with the properties of the waveform. When the difference is high, the temporal frequency is high, and the waves travel in the direction of least inhibition (Fig. 6A, B). When there is no difference, the waveform becomes static. The sum of the two inhibitory weights positively correlates to the spatial frequency (Fig. 6D, E). These correlations are largely decoupled with each other, allowing for a wide range of controllable waveforms.

IV. DISCUSSION

Based on our analysis of peristaltic motion, the shape of the waveform (spatial frequency) and the speed of the waveform (temporal frequency) are essential parameters for controlling behavior in a worm robot. Our controller can change these parameters in a robust and straightforward way. Of particular interest is the fact that any given waveform can be brought to a complete stop (zero temporal frequency) for arbitrary amounts of time. This would be useful in a robot when it needs to process sensory input before proceeding. It would also make it trivial to activate and de-activate sub-oscillations that could dramatically transform the resultant motion.

The system is robust to sudden changes in descending signals, regardless of the timing or magnitude of the change (Fig. 6A–C). While changes in the speed of the waveform can take place almost instantaneously, changes in the shape of the waveform itself are much slower. This could prove very beneficial in the robotic platform because it suggests the system will obey (5) from our analytical model. By preventing sudden changes to the shape of the waveform, the robot will not become over-constrained and slip.

Fig. 6. The first row shows the activation level along the length of the body of the two opposing actuator groups as a function of time. At t=0, there are 1.5 waveforms along the body (row two). The waveform initially travels towards the head, causing backwards motion, then switches between A and B, causing forward motion. The third row shows the normalized weights of the posterior and anterior inhibitions as they change with time as a result of descending signals. The fourth and fifth rows show the difference and sum respectively of the inhibitory weights. These quantities correlate well to properties of the resultant motion. A and B are the local maximum and minimum of the difference, and correspond to the greatest positive and negative temporal frequency. D and E are the local maximum and minimum of the sum, and correspond to the least and greatest spatial frequency. First 50 time-steps not shown.

Our analysis also shows that ground contact timing is critically important to peristaltic motion, and this system shows promise in being able to locally adapt to sudden changes in its state. Ground contact, force and strain sensors can be readily integrated into both the anterior and posterior inhibitory networks to modulate wave frequency, and create local adaptations to surroundings. 3-D turning can also be instantiated by treating the populations as matrices, rather than vectors. The system also appears to be robust to noisy sensory data.

V. CONCLUSIONS

This paper reports on a new controller for peristaltic motion in a robot that uses the Wilson-Cowan model to simulate large excitatory and inhibitory neuronal populations. Our approach to designing the initial architecture is based on careful observations of peristaltic motion in worms, our analytical model, and our first robot prototype. The simulated neural architecture shows many desirable characteristics. It is robust to sudden changes in descending signals. It can quickly and accurately adjust both the shape of the deformation wave, as well as the speed of the wave, two key parameters to controlling peristaltic locomotion. This arrangement can easily tune its period and even come to a complete stop for indefinite amounts of time. The controller is also amenable to large amounts of sensory input. We believe that this kind of system will be well suited to controlling the continuously deforming outer body of our novel robot.

ACKNOWLEDGMENT

This work would not have been possible without the support and encouragement of Nicole Kern and others.

REFERENCES [1] R. C. Brusca and G. J. Brusca, Invertebrates. Sinauer Associates,

Sunderland, MA. [2] Ö Ekeberg and S. Griller, “Simulations of neuromuscular control in

lamprey swimming,” Philos. Trans. R. Soc. Lond. B. Biol. Sci. 354:895–902, 1999.

[3] C. L. Huffard, “Locomotion by Abdopus aculeatus (Cephalopda: Octopodidae): walking the line between primary and secondary defenses,” J. Exp. Biol., 209:3697–3707, 2006.

[4] J. Ostrowski and J. Burdick, “Gait Kinematics for a Serpentine Robot,” in Proc. IEEE Int. Conf. Robotics and Automation (ICRA), Minneapolis, MN, 2:1294–1299, 1996.

[5] M. Tesch, K. Lipkin, I. Brown, R. Hatton, A. Peck, J. Rembisz, and H. Choset, “Parameterized and Scripted Gaits for Modular Snake Robots,” Advanced Robotics, 23(9):1131–1158, 2009.

[6] A. A. Transeth, K. Y. Pettersenand, and P. Liljeback, “A survey on snake robot modeling and locomotion,” Robotica, 27:999–1015, 2009.

[7] A. Menciassi, A. Gorini, G. Pernorio, and P. Dario, “A SMA Actuated Artificial Earthworm,” in Proc. Int. Conf. Robotics and Automation (ICRA), 2004.

[8] D. P. Tsakiris, M. Sfakiotakis, A. Menciassi, G. La Spina, and P. Dario, “Polchaete-like Undulatory Robotic Locomotion,” in Proc. IEEE Int. Conf. Robotics and Automation (ICRA), Barcelona, Spain, 2005, 3018–3023.

[9] B. Trimmer, A. Takesian, and B. Sweet, “Caterpillar locomotion: a new model for soft-bodied climbing and burrowing robots,” in Proc. 7th Int. Symp. Technology and the Mine Problem, Monterey, CA, 2006.

[10] K. Wang and G. Yan, “Micro robot prototype for colonoscopy and in vitro experiments,” J. Med. Eng. & Tech., 31(1):24–28, 2007.

[11] H. Omori, T. Nakamura, and T. Yada, “An underground explorer robot based on peristaltic crawling of earthworm,” Industrial Robot,

36(4):358–364, 2009. [12] S. Hirose, Biologically Inspired Robots: Snake-Like Locomotors and

Manipulators. Oxford University Press, Oxford, 1993. [13] A. J. Ijspeert, A. Crespi, D. Ryczko, and J. M. Cabelguen, “From

swimming to walking with a salamander robot driven by a spinal cord model,” Science, 315(5817):1416–1420, 2007.

[14] R. L. Hatton and H. Choset, “Generating gaits for snake robots: annealed chain fitting and keyframe wave extraction,” Auton. Robot, 28:271–281, 2010.

[15] M. W. Hannan and I. D. Walker, “Analysis and Initial Experiments for a Novel Elephant’s Trunk Robot,” in Proc. IEEE Int. Conf. Intelligent Robots and Systems (IROS), 2000, 330–337.

[16] P. Dario, P. Ciarletta, A. Menciassi, and B. Kim, “Modeling and experimental validation of the locomotion of endoscopic robots in the colon,” Int. J. Robotics Research, 23(4–5):549–556, 2004.

[17] S. Seok, C. D. Onal, R. Wood, D. Rus, and S. Kim, (2010). “Peristaltic Locomotion With Antagonistic Actuators in Soft Robotics,” in Proc. IEEE Int. Conf. Robotics and Automation (ICRA), 2010.

[18] D. Trivedi, C. D. Rahn, W. M. Kier, and I. D. Walker, “Soft robotics: Biological inspiration, state of the art, and future research,” Applied Bionics and Biomechanics, 5(3):99–117, 2008.

[19] R. M. Alexander, Principles of Animal Locomotion. Princeton University Press, Princeton, NJ, 2003, p. 88.

[20] K. Zimmermann and I. Zeidis, “Worm-Like Motion as a Problem of Nonlinear Dynamics,” J. Theoret. Appl. Mech., 45(1):179–187, 2007.

[21] D. Zarrouk, I. Sharf, and M. Shoham, “Analysis of Earthworm-like Robotic Locomotion on Compliant Surfaces,” in Proc. IEEE Int. Conf. Robotics and Automation (ICRA), 2010.

[22] A. S. Boxerbaum, H. J. Chiel, and R. D. Quinn, “Softworm: A Soft, Biologically Inspired Worm-Like Robot,” in Neuroscience Abstracts, Society for Neuroscience Annual Meeting, Chicago, IL, 2009.

[23] A. S. Boxerbaum, H. J. Chiel, H. J., and R. D. Quinn, “A New Theory and Methods for Creating Peristaltic Motion in a Robotic Platform,” in Proc. Int. Conf. Robotics and Automation (ICRA), 2010, 1221–1227.

[24] K. J. Quillin, “Kinematic scaling of locomotion by hydrostatic animals: ontogeny of peristaltic crawling by the earthworm lumbricus terrestris,” J. Exp. Biol., 202:661–674, 1999.

[25] W. M. Kier and K. K. Smith, “Tongues, tentacles and trunks: the biomechanics of movement in muscular-hydrostats,” Zool. J. Linn. Soc., 83:307–324, 1985.

[26] R. D. Quinn, G. M. Nelson, R. E. Ritzmann, R. J. Bachmann, D. A. Kingsley, J. T. Offi, and T. J. Allen, “Parallel Strategies for Implementing Biological Principles Into Mobile Robots,” Int. J. Robotics Research, 22(3):169–186, 2003.

[27] E. V. Mangan, D. A. Kingsley, R. D. Quinn, and H. J. Chiel, “Development of a peristaltic endoscope,” in Proc. IEEE Int. Conf. Robotics and Automation (ICRA), 2002, 347–352.

[28] J. Ayers, W. Cricket, and O. Chris, “Lamprey Robots,” in Proc. Int. Symp. Aqua Biomechanisms, 2000.

[29] D. Zhang, D. Hu, L. Shen, and H. Xie, “Design of an artificial bionic neural network to control fish-robot’s locomotion,” Neurocomputing, 71:648–654, 2008.

[30] A. J. Ijspeert, “Central pattern generators for locomotion control in animals and robots: A review,” Neural Networks, 21:642–653, 2008.

[31] T. Matsuo, T. Yokoyama, D. Ueno, and K. Ishii, “Biomimetic Motion Control System Based on a CPG for an Amphibious Multi-Link Mobile Robot,” J. Bionic Eng. Suppl. (2008) 91–97.

[32] T. Wadden, J. Hellgren, A. Lansner, and S. Grillner, “Intersegmental coordination in the lamprey: simulations using a network model without segmental boundaries,” Biol. Cybernetics, 76:1–9, 1997.

[33] H. R. Wilson and J. D. Cowan, “Excitatory and Inhibitory Interactions in Localized Populations of Model Neurons,” Biophysical Journal, 12(1):1–24, 1972.

[34] G. B. Ermentrout and J. D. Cowan, “A Mathematical Theory of Visual Hallucination Patterns,” Biol. Cybernetics, 34:137–150, 1979.

[35] A. S. Boxerbaum, S. Skentzos, P. Thomas, “Simulation of geometric hallucinations on the visual cortex,” in Neuroscience Abstracts, Society for Neuroscience Annual Meeting, Washington, DC, 2008.

[36] K. J. Quillin, “Ontogenetic Scaling of Burrowing Forces in the Earthworm Lumbricus terrestris,” J. Exp. Biol., 203:2757–2770, 2000.

[37] T. Ueta and G. Chen, “On synchronization and control of coupled Wilson-Cowan neural oscillators,” Int. J. Bifurcation Chaos, 13(1):163–175, 2003.

[38] S. Grillner, “The Motor Infrastructure: From Ion Channels to Neronal Networks,” Nature Reviews Neuroscience, 4:573–586, 2003.