The Omega Turn: A General Turning Template for Elongate Robots

TL;DR

The Omega Turn paper addresses turning robustly for elongate limbless robots operating in cluttered environments by introducing a turning template that is a superposition of two traveling waves with a fixed forward-wave frequency. Through a hierarchical geometric framework and height-function optimization, the authors design the omega turn gait and validate it with numerical simulations and robophysical experiments, demonstrating superior angular displacement and obstacle clearance. Key contributions include linking omega-turn kinematics to nematode behavior, generalizing the approach to elongate multi-legged systems, and implementing a compliant control scheme that improves performance in complex environments, including granular media and outdoor terrains. The work offers a practical, biology-informed turning strategy with broad applicability to search-and-rescue and industrial inspection tasks, and it highlights future directions for integrating smooth transitions between omega turns and forward gaits for real-time motion planning.

Abstract

Elongate limbless robots have the potential to locomote through tightly packed spaces for applications such as search-and-rescue and industrial inspections. The capability to effectively and robustly maneuver elongate limbless robots is crucial to realize such potential. However, there has been limited research on turning strategies for such systems. To achieve effective and robust turning performance in cluttered spaces, we take inspiration from a microscopic nematode, C. elegans, which exhibits remarkable maneuverability in rheologically complex environments partially because of its ability to perform omega turns. Despite recent efforts to analyze omega turn kinematics, it remains unknown if there exists a wave equation sufficient to prescribe an omega turn, let alone its reconstruction on robot platforms. Here, using a comparative theory-biology approach, we prescribe the omega turn as a superposition of two traveling waves. With wave equations as a guideline, we design a controller for limbless robots enabling robust and effective turning behaviors in lab and cluttered field environments. Finally, we show that such omega turn controllers can also generalize to elongate multi-legged robots, demonstrating an alternative effective body-driven turning strategy for elongate robots, with and without limbs.

Figures (11)

• Figure 1: The bio-inspired omega turn allows for agile limbless robot turning. (a) The omega ($\Omega$) shaped turning behavior of the nematode worm C. elegans in a gait cycle. Limbless robot reorientation on various types of terrain: (b) flat hard ground, (c) rough grassland, and (d) a pile of rocks. (e) Generalization of the omega turn to elongate multi-legged robots.
• Figure 2: Turning behaviors in C. elegans. The illustration of (a) omega ($\Omega$) turn and (b) small angle turn. (top) Snapshots of C. elegans during turning behaviors. (ii) The body curvature profile with respect to time (x-axis) and spatial position (y-axis). The unit of body curvature is $BL^{-1}$. (c) Principle component analysis. (c.1) The dominant principle components during C. elegans turning behaviors. (c.2) The variance explained by principle components. 4 principle components are sufficient to explain over 80% of the variance.
• Figure 3: The rotational height functions on three 2-dimensional sub-shape spaces. (top) Shape space, (mid) feasibility map over shape space, and (bottom) height function over shape space (unit of height function: rad$^{-1}$). We illustrate two example of infeasible shape: exceed the joint angle limit and self-collision. (a) $\{[\tau_o \ A_o], \tau_o\in S^1, A_o\in \mathbb{R}^1\}$ (b)$\{[\tau_f \ A_f], \tau_f\in S^1, A_f\in \mathbb{R}^1\}$(c) $\{[\tau_f \ \tau_o], \tau_f\in S^1, \tau_o\in S^1\}$. The red and black colors represent the positive and negative values of the height function on the top figures. The black regions in the bottom figures represents the shapes that lead to self-collision. The blue curve shows the gait paths $f_1$, $f_2$ and $f_3$, designed to maximize the surface integral while not passing through the collision regions The surface integrals in (a) and (b) is the integral of surface enclosed by the gait path and the dashed line; in (c) is the integral of surface enclosed in the lower right corner (shadow by solid line) minus the surface enclosed in the upper left corner (shadow by dashed line)
• Figure 4: Effectiveness of omega turn. (a) Time evolution of the angular displacement in the simulation and the robot experiments during an omega turn. Each point represents the average over three trials. Error bars correspond to standard deviation in all plots/graphs. A sequence of video frames of the robot depicts the time evolution of the robot's body shape in 10 seconds. (b) The angular displacement for the turning gaits over a range of turning wave spatial frequencies ($k_t$) on flat ground. Error bars indicate the standard deviation. Omega turns have the largest angular displacement both in simulation and reality. (c) The area swept by the body for the turning gaits with varied turning wave spatial frequency $k_t$. The results are normalized by the robot body length squared (BL$^2$). The time evolution of robot's configurations executing the designed gaits over a period are shown in the red dashed boxes, where the gait fraction is indicated by colors from the beginning (blue) to the end (red).
• Figure 5: Amplitude modulation of turning gaits. The omega turn ($k_o = 1$, highlighted) displays the largest tunable range of angular displacement. Three time-lapse frames of robophysical experiments depicts the courses of turning with joint amplitude $60^\circ, 75^\circ$ and $90^\circ$ in one gait cycle.