Turning in Worm-Like Robots: The Geometry of Slip Elimination Suggests Nonperiodic Waves

Inspired by earthworms, soft robots have demonstrated locomotion using segments with coupled length-wise elongation and radial contraction. However, peristaltic turning has primarily been studied empirically. Surface-dependent slip, which results in frictional forces that deform the body segments, makes accurate models challenging and limited to a specific robot and environment. Here, instead of modeling specific surfaces and segments, we take a geometric approach to analyzing the constraints that result from elimination of slip for the general case of peristaltic locomotion. Thus, our abstract two-dimensional model applies to many different mechanical designs (e.g., fluidic actuation, origami, woven mesh). Specifically, we show how turning is limited by segment range of motion, which means that more than one wave will be required to completely reorient the body in an environment where slip is not possible. As a result, to eliminate slip, segments must undergo nonperiodic shape changes. By representing segments as isosceles trapezoids with reasonable ranges of motion, we can determine control waves that in simulation do not require slip. These waves follow from an initial "reach" (i.e., kinematic movement range) of the second segment. A strategy for choosing the second segment reach is proposed based on evaluating long-term turn stability. To demonstrate the value of the approach, we applied the nonperiodic waveform (NPW) to our earthworm-inspired soft robot, Compliant Modular Mesh Worm with Steering (CMMWorm-S). With the NPW, the robot slips less when compared with a naive periodic waveform, where each segment of the robot has the same kinematic reach of each wave, as indicated by the difference between predicted and actual body position over multiple waves. Using an NPW for turning, we observe a decrease in prediction error compared with a naive periodic waveform by 66%. Thus, while our model ignores many factors (inertial dynamics, radial deformation, surface forces), the resulting turn strategies can improve kinematic motion prediction for planning. The theoretical constraints on NPWs that eliminate slip during turning will help robot designers make application-specific design choices about body stiffness, frictional properties, body length, and degrees of freedom.