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  1. Worm-like robots have demonstrated great potential in navigating through environments requiring body shape deformation. Some examples include navigating within a network of pipes, crawling through rubble for search and rescue operations, and medical applications such as endoscopy and colonoscopy. In this work, we developed path planning optimization techniques and obstacle avoidance algorithms for the peristaltic method of locomotion of worm-like robots. Based on our previous path generation study using a modified rapidly exploring random tree (RRT), we have further introduced the Bézier curve to allow more path optimization flexibility. Using Bézier curves, the path planner can explore more areas and gain more flexibility to make the path smoother. We have calculated the obstacle avoidance limitations during turning tests for a six-segment robot with the developed path planning algorithm. Based on the results of our robot simulation, we determined a safe turning clearance distance with a six-body diameter between the robot and the obstacles. When the clearance is less than this value, additional methods such as backward locomotion may need to be applied for paths with high obstacle offset. Furthermore, for a worm-like robot, the paths of subsequent segments will be slightly different than the path of the head segment. Here, we show that as the number of segments increases, the differences between the head path and tail path increase, necessitating greater lateral clearance margins. 
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    Earthworm-like peristaltic locomotion has been implemented in >50 robots, with many potential applications in otherwise inaccessible terrain. Design guidelines for peristaltic locomotion have come from observations of biology, but robots have empirically explored different structures, actuators, and control waveform shapes than those observed in biological organisms. In this study, we suggest a template analysis based on simplified segments undergoing beam deformations. This analysis enables calculation of the minimum power required by the structure for locomotion and maximum speed of locomotion. Thus, design relationships are shown that apply to peristaltic robots and potentially to earthworms. Specifically, although speed is maximized by moving as many segments as possible, cost of transport (COT) is optimized by moving fewer segments. Furthermore, either soft or relatively stiff segments are possible, but the anisotropy of the stiffnesses is important. Experimentally, we show on our earthworm robot that this method predicts which control waveforms (equivalent to different gaits) correspond to least input power or to maximum velocity. We extend our analysis to 150 segments (similar to that of earthworms) to show that reducing COT is an alternate explanation for why earthworms have so few moving segments. The mathematical relationships developed here between structural properties, actuation power, and waveform shape will enable the design of future robots with more segments and limited onboard power. 
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    Inspired by earthworms, worm-like robots use peristaltic waves to locomote. While there has been research on generating and optimizing the peristalsis wave, path planning for such worm-like robots has not been well explored. In this paper, we evaluate rapidly exploring random tree (RRT) algorithms for path planning in worm-like robots. The kinematics of peristaltic locomotion constrain the potential for turning in a non-holonomic way if slip is avoided. Here we show that adding an elliptical path generating algorithm, especially a two-step enhanced algorithm that searches path both forward and backward simultaneously, can make planning such waves feasible and efficient by reducing required iterations by up around 2 orders of magnitude. With this path planner, it is possible to calculate the number of waves to get to arbitrary combinations of position and orientation in a space. This reveals boundaries in configuration space that can be used to determine whether to continue forward or back-up before maneuvering, as in the worm-like equivalent of parallel parking. The high number of waves required to shift the body laterally by even a single body width suggests that strategies for lateral motion, planning around obstacles and responsive behaviors will be important for future worm-like robots. 
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