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.

A key problem in robotic locomotion is in finding optimal shape changes to effectively displace systems through the world. Variational techniques for gait optimization require estimates of body displacement per gait cycle; however, these estimates introduce error due to unincluded high order terms. In this paper, we formulate existing estimates for displacement, and describe the contribution of low order terms to these estimates. We additionally describe the magnitude of higher (third) order effects, and identify that choice of body coordinate, gait diameter, and starting phase influence these effects. We demonstrate that variation of such parameters on two example systems (the differential drive car and Purcell swimmer) effectively manages third order contributions.

The gaits of locomoting systems are typically designed to maximize some sort of efficiency, such as cost of transport or speed. Equally important is the ability to modulate such a gait to effect turning maneuvers. For drag-dominated systems, geometric mechanics provides an elegant and practical framework for both ends—gait design and gait modulation. Within this framework, “constraint curvature” maps can be used to approximate the net displacement of robotic systems over cyclic gaits. Gait optimization is made possible under a previously reported “soap-bubble” algorithm. In this work, we propose both local and global gait morphing algorithms to modify a nominal gait to provide single-parameter steering control. Using a simplified swimmer, we numerically compare the two approaches and show that for modest turns, the local approach, while suboptimal, nevertheless proves effective for steering control. A potential advantage of the local approach is that it can be readily applied to soft robots or other systems where local approximations to the constraint curvature can be garnered from data, but for which obtaining an exact global model is infeasible.

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.

The complex dynamics of agile robotic legged locomotion requires motion planning to intelligently adjust footstep locations. Often, bipedal footstep and motion planning use mathematically simple models such as the linear inverted pendulum, instead of dynamically-rich models that do not have closed-form solutions. We propose a real-time optimization method to plan for dynamical models that do not have closed form solutions and experience irrecoverable failure. Our method uses a data-driven approximation of the step-to-step dynamics and of a failure margin function. This failure margin function is an oriented distance function in state-action space where it describes the signed distance to success or failure. The motion planning problem is formed as a nonlinear program with constraints that enforce the approximated forward dynamics and the validity of state-action pairs. For illustration, this method is applied to create a planner for an actuated spring-loaded inverted pendulum model. In an ablation study, the failure margin constraints decreased the number of invalid solutions by between 24 and 47 percentage points across different objectives and horizon lengths. While we demonstrate the method on a canonical model of locomotion, we also discuss how this can be applied to data-driven models and full-order robot models.

In this work we present the design of a swimming robot that is inspired by the body shape modulation of small microorganisms. Amoebas are small single celled organisms that locomote through deformation and shape change of their body. To achieve similar shape modulation for swimming propulsion in a robot we developed a novel flexible appendage using tape springs. A tape spring is an elongated strip of metal with a curved cross-section that can act as a stiff structure when loaded against the curvature, while it can easily buckle when loaded with the curvature. We develop a tape spring appendage that is capable of freely deforming its perimeter through two actuation inputs. In the first portion of this paper we develop the kinematics of the appendage mechanisms and compare with experiment. Next we present the design of a surface locomoting robot that uses two appendages for propulsion. From the appendage kinematics we derive the local connection vector field for locomotion kinematics and study the optimal gait for forward swimming. Lastly, we demonstrate robot swimming performance in open water conditions. The novel appendage design in this robot is advantageous because it enables omnidirectional movement, the appendages will not tangle in debris, and they are robust to collisions and contact with structures.

Isolated mechanical systems—e.g., those floating in space, in free-fall, or on a frictionless surface—are able to achieve net rotation by cyclically changing their shape, even if they have no net angular momentum. Similarly, swimmers immersed in “perfect fluids” are able to use cyclic shape changes to both translate and rotate even if the swimmer-fluid system has no net linear or angular momentum. Finally, systems fully constrained by direct nonholonomic constraints (e.g., passive wheels) can push against these constraints to move through the world. Previous work has demonstrated that the displacement induced by these shape changes corresponds to the amount ofconstraint curvaturethat the gaits enclose. Properly assessing or optimizing the utility of a gait also requires considering the time or resources required to execute it: A gait that produces a small displacement per cycle, but that can be executed in a short time, may produce a faster average velocity than a gait that produces more displacement, but takes longer to complete a cycle at the same instantaneous effort. In this paper, we consider gaits under two instantaneous measures of effort. For each of these costs, we demonstrate that fixing the average instantaneous cost to a unit value allows us to transform the effort costs into time-to-execute costs for any given gait cycle. We then illustrate how the interaction between the constraint curvature and these costs leads to characteristic geometries for optimal cycles, in which the gait trajectories resemble elastic hoops distended from within by internal pressures.

Traditional geometric mechanics models used in locomotion analysis rely heavily on systems having symmetry in SE(2) (i.e., the dynamics and constraints are invariant with respect to a system’s position and orientation) to simplify motion planning. As a result, the symmetry assumption prevents locomotion analysis on non-flat surfaces because the system dynamics may vary as a function of position and orientation. In this paper, we develop geometric motion planning strategies for a mobile system moving on a position space whose manifold structure is a cylinder: constant non-zero curvature in one dimension and zero curvature in another. To handle this non-flat position space, we adapt conventional geometric mechanics tools - in particular the system connection and the constraint curvature function - to depend on the system orientation. In addition, we introduce a novel constraint projection method to a variational gait optimizer and demonstrate how to design gaits that allow the example system to move on the cylinder with optimal efficiency.

We propose a method to generate actuation plans for a reduced order, dynamic model of bipedal running. This method explicitly enforces robustness to ground uncertainty. The plan generated is not a fixed body trajectory that is aggressively stabilized: instead, the plan interacts with the passive dynamics of the reduced order model to create emergent robustness. The goal is to create plans for legged robots that will be robust to imperfect perception of the environment, and to work with dynamics that are too complex to optimize in real-time. Working within this dynamic model of legged locomotion, we optimize a set of disturbance cases together with the nominal case, all with linked inputs. The input linking is nontrivial due to the hybrid dynamics of the running model but our solution is effective and has analytical gradients. The optimization procedure proposed is significantly slower than a standard trajectory optimization, but results in robust gaits that reject disturbances extremely effectively without any replanning required.

In the literature, researchers studying the lateral dynamics of tractor-trailer systems have each developed their own reduced-order lateral dynamic model, each with their own assumptions, state representations, and derivation methods. Little to no work has been performed to compare the accuracy of these models to each other, nor to validate their results against high-fidelity multi-body simulations or real truck data. The purpose of this paper is to identify several reduced-order lateral dynamic models for tractor-trailer systems present in the current literature, simulate their estimated states through common driving maneuvers, and then compare their estimates against reference data from a high-fidelity multi-body dynamic model. From this comparison, we identify the reduced-order model that maintains the least error from the reference data as the best representation of a real tractor-trailer system. The results of our comparison will be useful to researchers interested in using a reduced-order dynamic model from the literature as the basis for articulation angle estimation, model-predictive control, or adaptive control algorithms for autonomous tractor-semitrailer systems.

In this paper, we present a set of geometric principles for understanding and optimizing the gaits of drag-dominated kinematic locomoting systems. For systems with two shape variables, the dynamics of gait optimization are analogous to the process by which internal pressure and surface tension combine to produce the shape and size of a soap bubble. The internal pressure on the gait curve is provided by the flux of the curvature of the system constraints passing through the surface bounded by the gait, and surface tension is provided by the cost associated with executing the gait, which when executed at optimal (constant-power) pacing is proportional to its pathlength measured under a Riemannian metric. We extend these principles to work on systems with three and then more than three shape variables. We demonstrate these principles on a variety of system geometries (including Purcell's swimmer) and for optimization criteria that include maximizing displacement and efficiency of motion for both translation and turning motions. We also demonstrate how these principles can be used to simultaneously optimize a system's gait kinematics and physical design.

The apparent “distance” between two configurations of a system and the “length” of trajectories through its configuration space can be significantly distorted by plots that use “natural” or intuitively selected coordinates. This effect is similar to the way that a latitude-longitude plot of the Earth distorts the size and shape of the continents. In this paper, we explore how ideas from cartography can be used to identify system parameterizations that better reflect the effort costs of changing configuration. We then apply these new parameters to provide geometric insight about two aspects of moving in dissipative environments such as low Reynolds number fluids: The shape of the optimal gait cycle for a three-link swimmer and the fundamentally superior efficiency of a serpenoid swimmer as compared to the classic three-link system.

Jumping spiders are capable of targeted jumps by using their front legs to guide the release of energy from their rear legs. In this paper, we present a simplified model of the jumping spider based on the anatomy of the real spider. The immediate goal of this model is to understand how the geometry of the legs affects the jumping motion, with the further goal of using this geometry in the future development of jumping robots. Through a set of simulations, dynamic analysis, and experiments with a physical realization of our model, we identify several features of the spiders' jumping mechanism, most notably that “vaulting” with the front legs allows the system to generate flatter take-off trajectories than could be achieved by simple aiming of a spring-release mechanism.

The spring-mass locomotion paradigm is showing great promise as a template for agile and efficient robots. Efficiency and stability are enabled by the passive generation of locomotion patterns, rather than enforced by the control system. However, as leg designs develop more articulation and complexity, a problem arises: how do we implement a chosen set of passive dynamics in complex hardware? We present compliance and impact inertia analyses in a “visually tactile” way, allowing complex and redundant mechanism designs to be easily evaluated. The patterns shown here begin a framework for the comprehensive design of agile, highly dynamic robots.

A casting manipulator is a robot that launches a payload as a projectile then alters the ballistic trajectory by pulling on an attached tether. Prior work in casting manipulation has focused on applying impulses through the tether to place the payload at a specific location and then retract it. In the spirit of whole-body and whole-arm manipulation, we propose that the tether itself can be used to act on the environment, rather than serving only as a means of directing the projectile. As a first step in this direction, we consider the case of throwing an end-weighted line to wrap a target above and away from the launch point. We explore the necessary conditions for successfully wrapping a target, including the various types of wrapping and the conditions required to make these wraps.

Planning robot motions often requires a notion of the “distance” between configurations or the “length” of a trajectory connecting them in the configuration space. If these quantities are defined so as to correspond to the effort required to change configurations, then they would likely differ from the Euclidean distance or arclength in the system's configuration parameters, distorting the visual representation of the relative costs of executing the motions. This problem is fundamentally similar to that of producing map projections with minimal distortion in cartography. A separate problem is that of nonlinear dimensionality reduction (NLDR), which, given a set of data, projects it into a lower-dimensional space while seeking to retain the geometric relationship between data points. In this paper, we show that NLDR can be applied to the kinematic cartography problem, allowing us to generate system parameterizations in which distance and arclength correspond to the effort of motion.

Highly articulated robot locomotion systems, such as snake robots, present special motion planning challenges. They possess many degrees of freedom, and therefore are modeled by a high dimensional configuration space which must be searched to plan a path. Kinematic and dynamic constraints further complicate the selection of effective controls. Finally, snake robots often have multiple modes of interaction with the terrain as contacts are made and broken, leading to complex and imperfect motion models. We believe that the space of useful controls that provides desirable motions, however, is much smaller. Useful net motions for such systems are often generated via gaits, or cyclic motions in the shape space. Gaits transform a high-dimensional continuum search into a relatively tractable discrete search. In this paper, we put forward a framework which allows a planner to generate paths in a low dimensional work space and select among gaits, pre-planned motions in the robot's shape space. The contribution of this paper rests on the “virtual chassis” which is a choice of body frame for the snake robot that allows the planner to efficiently select among and plan with gaits to direct the robot along the work space path. We demonstrate this planner running on a simulated snake robot navigating through a variety of clutter scenarios. The virtual chassis also has the benefit of allowing us to generalize notions of controllability to gait motions.

Several efforts have recently been made to relate the displacement of swimming three-link systems over strokes to geometric quantities of the strokes. In doing so, they provide powerful, intuitive representations of the bounds on a system's locomotion capabilities and the forms of its optimal strokes or gaits. While this approach has been successful for finding net rotations, noncommutativity concerns have prevented it from working for net translations. Our recent results on other locomoting systems have shown that the degree of this noncommutativity is dependent on the coordinates used to describe the problem and that it can be greatly mitigated by an optimal choice of coordinates. Here, we extend the benefits of this optimal-coordinate approach to the analysis of swimming at the extremes of low and high Reynolds numbers.

Sidewinding is an efficient translation gait used by snakes and snake robots over flat ground, and resembles a helical tread moving over a core cylindrical geometry. Most sidewinding research has focused on straight-line translation of the snake, and less on steering capabilities. Here, we offer a new, geometrically intuitive method for steering this gait: Tapering the core cylinder into a cone, such that one end moves faster than the other, changing the heading of the robot as it drives forward. We present several design tools for working with this cone, along with experimental results on a physical robot turning at different rates.

A mobile system's maneuverability describes the scale and span of the velocities with which it can move. In this paper, we present a new geometric framework for describing the maneuverability of kinematic locomoting systems, inspired by the manipulability analysis of robotic arms. This framework describes both the local maneuverability in the neighborhood of each shape the system can assume and the cyclic maneuverability achieved by executing gaits from a library. Additionally, the gait-level analysis includes tools that direct the search for gaits whose inclusion into the library will usefully improve the maneuverability. Throughout, we provide examples based on a swimming system operating at low Reynolds number.

Sidewinding is an efficient translation gait used by snakes over flat ground. When implemented on snake robots, it retains its general effectiveness, but becomes unstable on sloped surfaces. Flattening the sidewinding motion along the surface to provide a more stable base corrects for this instability, but degrades other performance characteristics, such as efficiency and handling of rough terrain. In this paper, we identify stability conditions for a sidewinder on a slope and find a solution for the minimum aspect ratio of the sidewinding pattern needed to maintain stability. Our theoretical results are supported by experiments on snake robots. In constructing our stability analysis, we present a new, tread-based model for sidewinding that is both consistent with previous models and provides new intuition regarding the kinematics of the gait. This new interpretation of sidewinding further admits a symmetry-based model reduction that simplifies its analysis. Additionally, an intermediate stage of the theoretical work contains a comprehensive analysis of the behavior of an ellipse in rolling contact with a sloped surface.

Gait evaluation techniques that use Stokes's theorem to integrate a system's equations of motion have traditionally been limited to finding only the net rotations or small translations produced by gaits. Recently, we have observed that certain choices of generalized coordinates allow these techniques to be extended to gaits that produce large translations. In this paper, we present a method for finding the optimal coordinate choice for this purpose, based on a Hodge-Helmholtz decomposition of the system constraints, and demonstrate the efficacy of the Stokes's theorem approach over a wide variety of gaits when using the optimized coordinate choice.

Snake robots have many degrees of freedom, which makes them both extremely versatile and complex to control. In this paper, we address this complexity by introducing two algorithms. Annealed chain fitting efficiently maps a continuous backbone curve to a set of joint angles for a snake robot. Keyframe wave extraction takes joint angles fit to a sequence of backbone curves, and identifies parameterized periodic functions which produce those sequences. Together, they allow a designer to conceive a gait in terms three-dimensional shapes and translate them into easily manipulated wave functions. We validate the algorithms by using them to produce rolling gaits for crawling and climbing.

Many factors such as size, power, and weight constrain the design of modular snake robots. Meeting these constraints requires implementing a complex mechanical and electrical architecture. Here we present our solution, which involves the construction of sixteen aluminum modules and creation of the Super Servo, a modified hobby servo. To create the Super Servo, we have replaced the electronics in a hobby servo, adding such components as sensors to monitor current and temperature, a communications bus, and a programmable microcontroller. Any robust solution must also protect components from hazardous environments such as sand and brush. To resolve this problem we insert the robots into skins that cover their surface. Functions such as climbing the inside and outside of a pipe add a new dimension of interaction. Thus we attach a compliant, high-friction material to every module, which assists in tasks that require gripping. This combination of the mechanical and electrical architectures results in a robust and versatile robot.