--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].

In the fields of robotics and biomechanics, the integration of elastic elements such as springs and tendons in legged systems has long been recognized for enabling energy-efficient locomotion. Yet, a significant challenge persists: designing a robotic leg that perform consistently across diverse operating conditions, especially varying average forward speeds. It remains unclear whether, for such a range of operating conditions, the stiffness of the elastic elements needs to be varied or if a similar performance can be obtained by changing the motion and actuation while keeping the stiffness fixed. This work explores the influence of the leg stiffness on the energy efficiency of a monopedal robot through an extensive parametric study of its periodic hopping motion. To this end, we formulate an optimal control problem parameterized by average forward speed and leg stiffness, solving it numerically using direct collocation. Our findings indicate that, compared to the use of a fixed stiffness, employing variable stiffness in legged systems improves energy efficiency by 20 % maximally and by 6.8 % on average across a range of speeds.

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.

Many robotic systems locomote using gaits - periodic changes of internal shape, whose mechanical interaction with the robot's environment generate characteristic net displacements. Prominent examples with two shape variables are the low Reynolds number 3-link "Purcell swimmer" with inputs of 2 joint angles and the "ideal fluid" swimmer. Gait analysis of these systems allows for intelligent decisions to be made about the swimmer's locomotive properties, increasing the potential for robotic autonomy. In this work, we present comparative analysis of gait optimization using two different methods. The first method is variational approach of "Pontryagin's maximum principle" (PMP) from optimal control theory. We apply PMP for several variants of 3-link swimmers, with and without incorporation of bounds on joint angles. The second method is differential-geometric analysis of the gaits based on curvature (total Lie bracket) of the local connection for 3-link swimmers. Using optimized body-motion coordinates, contour plots of the curvature in shape space give visualization that enables identifying distance-optimal gaits as zero level sets. Combining and comparing results of the two methods enables better understanding of changes in existence, shape and topology of distance-optimal gait trajectories, depending on the swimmers' parameters.

For a class of biped robots with impulsive dynamics and a non-empty set of passive gaits (unactuated, periodic motions of the biped model), we present a method for computing continuous families of locally optimal gaits with respect to a class of commonly used energetic cost functions (e.g., the integral of torque-squared). We compute these families using only the passive gaits of the biped, which are globally optimal gaits with respect to these cost functions. Our approach fills in an important gap in the literature when computing a library of locally optimal gaits, which often do not make use of these globally optimal solutions as seed values. We demonstrate our approach on a well-studied two-link biped model.

The robotic locomotion community is interested in optimal gaits for control. Based on the optimization criterion, however, there could be a number of possible optimal gaits. For example, the optimal gait for maximizing displacement with respect to cost is quite different from the maximum displacement optimal gait. Beyond these two general optimal gaits, we believe that the optimal gait should deal with various situations for high-resolution of motion planning, e.g., steering the robot or moving in "baby steps." As the step size or steering ratio increases or decreases, the optimal gaits will slightly vary by the geometric relationship and they will form the families of gaits. In this paper, we explored the geometrical framework across these optimal gaits having different step sizes in the family via the Lagrange multiplier method. Based on the structure, we suggest an optimal locus generator that solves all related optimal gaits in the family instead of optimizing each gait respectively. By applying the optimal locus generator to two simplified swimmers in drag-dominated environments, we verify the behavior of the optimal locus generator.