--Salps are marine animals consisting of chains of jellyfish-like units. Their capacity for effective underwater undulatory locomotion through coordinating multi-jet propulsion has aroused significant interest in the field of robotics and inspired extensive research including design, modeling, and control. In this paper, we conduct a comprehensive analysis of the locomotion of salp-like systems using the robotic platform "LandSalp" based on geometric mechanics, including mechanism design, dynamic modeling, system identification, and motion planning and control. Our work takes a step toward a better understanding of salps' underwater locomotion and provides a clear path for extending these insights to more complex and capable underwater robotic systems. Furthermore, this study illustrates the effectiveness of geometric mechanics in bio-inspired robots for efficient data-driven locomotion modeling, demonstrated by learning the dynamics of LandSalp from only 3 minutes of experimental data. Lastly, we extend the geometric mechanics principles to multi-jet propulsion systems with stability considerations and validate the theory through experiments on the LandSalp hardware. These creatures are capable of efficient underwater undulatory locomotion by coordinating multi-jet propulsion. The structure and locomotion patterns of salps are closely related, which has attracted widespread interest in both biological and ecological research [1-5]. In the field of robotics, salps have attracted increasing attention due to their jet propulsion by expelling water through contraction, efficient underwater locomotion, and multi-unit coordination. Salps and jellyfish have inspired numerous robotic studies on the design of jet propulsion soft robots [6-12] and multi-robot coordination [13-17]. However, in the field of motion planning and control, most studies primarily consider undulatory locomotion by self-propulsion via body deformation [18-23], with only a few works involving underwater locomotion using jet propulsion [24-26]. This work was supported in part by ONR A ward N00014-23-1-2171. All the authors are with the Collaborative Robotics and Intelligent Systems (CoRIS) Institute at Oregon State University, Corvallis, OR USA. The units composing biological salps are called "zooids" (i.e., pseudoan-imals or not-quite-animals) because they exhibit many properties of animals but are not independent organisms from the colony. To discuss the general properties of multi-jet locomotion without making claims about the biological systems that inspire them, we use the terminology "chains" and "units" throughout this paper. The salp picture is reproduced from [27].

Recent advancements in soft actuators have enabled soft continuum swimming robots to achieve higher efficiency and more closely mimic the behaviors of real marine animals. However, optimizing the design and control of these soft continuum robots remains a significant challenge. In this paper, we present a practical framework for the co-optimization of the design and control of soft continuum robots, approached from a geometric locomotion analysis perspective. This framework is based on the principles of geometric mechanics, accounting for swimming at both low and high Reynolds numbers. By generalizing geometric principles to continuum bodies, we achieve efficient geometric variational co-optimization of designs and gaits across different power consumption metrics and swimming environments. The resulting optimal designs and gaits exhibit greater efficiencies at both low and high Reynolds numbers compared to three-link or serpenoid swimmers with the same degrees of freedom, approaching or even surpassing the efficiencies of infinitely flexible swimmers and those with higher degrees of freedom.

Geometric motion planning offers effective and interpretable gait analysis and optimization tools for locomoting systems. However, due to the curse of dimensionality in coordinate optimization, a key component of geometric motion planning, it is almost infeasible to apply current geometric motion planning to high-dimensional systems. In this paper, we propose a gait-based coordinate optimization method that overcomes the curse of dimensionality. We also identify a unified geometric representation of locomotion by generalizing various nonholonomic constraints into local metrics. By combining these two approaches, we take a step towards geometric motion planning for high-dimensional systems. We test our method in two classes of high-dimensional systems - low Reynolds number swimmers and free-falling Cassie - with up to 11-dimensional shape variables. The resulting optimal gait in the high-dimensional system shows better efficiency compared to that of the reduced-order model. Furthermore, we provide a geometric optimality interpretation of the optimal gait.

Inertia-dominated mechanical systems can achieve net displacement by 1) periodically changing their shape (known as kinematic gait) and 2) adjusting their inertia distribution to utilize the existing nonzero net momentum (known as momentum gait). Therefore, finding the gait that most effectively utilizes the two types of locomotion in terms of the magnitude of the net momentum is a significant topic in the study of locomotion. For kinematic locomotion with zero net momentum, the geometry of optimal gaits is expressed as the equilibria of system constraint curvature flux through the surface bounded by the gait, and the cost associated with executing the gait in the metric space. In this paper, we identify the geometry of optimal gaits with nonzero net momentum effects by lifting the gait description to a time-parameterized curve in shape-time space. We also propose the variational gait optimization algorithm corresponding to the lifted geometric structure, and identify two distinct patterns in the optimal motion, determined by whether or not the kinematic and momentum gaits are concentric. The examples of systems with and without fluid-added mass demonstrate that the proposed algorithm can efficiently solve forward and turning locomotion gaits in the presence of nonzero net momentum. At any given momentum and effort limit, the proposed optimal gait that takes into account both momentum and kinematic effects outperforms the reference gaits that each only considers one of these effects.

Legged robots leverage ground contacts and the reaction forces they provide to achieve agile locomotion. However, uncertainty coupled with contact discontinuities can lead to failure, especially in real-world environments with unexpected height variations such as rocky hills or curbs. To enable dynamic traversal of extreme terrain, this work introduces 1) a proprioception-based gait planner for estimating unknown hybrid events due to elevation changes and responding by modifying contact schedules and planned footholds online, and 2) a two-degree-of-freedom tail for improving contact-independent control and a corresponding decoupled control scheme for better versatility and efficiency. Simulation results show that the gait planner significantly improves stability under unforeseen terrain height changes compared to methods that assume fixed contact schedules and footholds. Further, tests have shown that the tail is particularly effective at maintaining stability when encountering a terrain change with an initial angular disturbance. The results show that these approaches work synergistically to stabilize locomotion with elevation changes up to 1.5 times the leg length and tilted initial states.