Locomotion Stability

Low-friction surfaces challenge bipedal locomotion by limiting the contact forces available during stepping. Inspired by penguin waddling, we investigate how lateral torso motion and center of mass (COM) placement affect locomotion as surface friction changes. Using a five-actuator biped, we compare an upright-gait strategy with a penguin-inspired torso-over-stance-leg strategy across multiple COM placements in simulation and hardware. In the 3-D simulator MuJoCo, we sweep through sinusoidal leg and hip actuation parameters across four friction coefficients mu = 0.1, 0.3, 0.5, 0.7. In simulation, torso-over-stance-leg motion produces more successful controllers and higher forward speeds at low friction, with the highest speed occurring for the high-COM configuration. Hardware experiments show the same low-friction speed trend: at mu=0.12, torso-over-stance-leg motion increases forward speed and reduces cost of transport at both tested COM ratios, and the higher COM also improves both measures. The high-COM penguin configuration is the fastest and most energy efficient while maintaining low sideways foot motion. At mu=0.45, the COM trend reverses: the lower-COM configurations are faster and more energy efficient, while gait strategy has little effect on forward speed but still changes sideways foot motion. These results show that the effects of lateral torso motion and COM placement depend on the available friction, and that forward speed, energy use, and slip-related foot motion can be modulated with a penguin-inspired torso motion on hardware.

Learned quadruped locomotion policies commonly map observations directly to joint-level actions, leaving the periodic structure of locomotion implicit in the policy. We investigate an alternative representation in which each joint trajectory is expressed as a command-conditioned Fourier series and modified online using feedback from the robot state. A context network generates the Fourier coefficients and the weights of a per-step feedback network, whose outputs adjust joint offsets, harmonic gains, frequency, and phase during execution. In simulation, we examine this explicit frequency structure alongside the hidden activations of an MLP policy that directly outputs joint targets. The harmonic waveforms change frequency and shape with commanded speed. Dynamic mode decomposition of selected MLP rollouts reveals dominant activation modes near the foot-height oscillation frequency and its second harmonic, showing periodic structure without an explicit Fourier generator. On a Unitree Go2, the simulation-trained harmonic controller records a provisional onboard-estimated peak speed of 3.67 meter per second and carries added loads up to 5.883 kilogram in separate trials.

Passive mechanical interfaces offer a lightweight alternative to actuated manipulators for quadruped payload carrying, but their impedance directly couples the payload dynamics with the locomotion pattern. This paper analyzes how passive-arm stiffness-damping selection affects payload-carrying locomotion under different gait and payload conditions. We compare damped and underdamped passive-arm impedance configurations in simulation during flat-ground locomotion. For crawl gaits, where the support polygon remains well defined, the results show that underdamped impedance increases passive-joint oscillations and can reduce the ZMP margin with respect to the support polygon. Trot is retained as a dynamic excitation case for the passive arm, but it is not used for direct ZMP-margin stability comparison. The results are summarized in gait-payload-stiffness-damping maps, where ZMP-margin reduction is evaluated for crawl gaits and trot is retained only as a passive-arm excitation case.

Open-loop multilegged locomotion over rough terrain has been interpreted as matter transport over a noisy channel: leg-ground interactions are discrete basic active contacts, terrain deletes or perturbs those contacts, and spatial redundancy concentrates the resulting thrust and arrival time. That construction is repetition-like because every module carries the same scalar locomotion task. It consequently provides neither a positive task rate nor a decoder that changes with the surviving contact set. Here we formulate locomotion instead as a quantized finite-frame expansion with erasures. A d-dimensional body-level command is mapped into N>d heterogeneous local contact commands. Rough terrain erases or corrupts frame coefficients, while a contact-gated compliant morphology physically realizes the weighted active-subframe decoder. For a linear-Gaussian model, mechanical equilibrium is exactly the posterior mean, tangent stiffness is posterior precision, and mechanical compliance is posterior covariance. Equal-norm Parseval frames are shown to be minimax optimal against one missing contact, two-contact robustness is governed by frame coherence, and a harmonic frame gives a directly realizable gait family. For independently surviving contacts of probability q, random Gaussian gait frames admit exact reconstruction at every analog dimension rate R<q, with a binomial reliability exponent, whereas recovery of arbitrary commands is impossible for R>q. Residual contact noise yields an asymptotic per-mode amplification 1/(q-R) and a vanishing mechanical stiffness margin at the threshold. An information-locomotion inequality and an exact incremental-redundancy rule direct the next gait component toward the softest task-relevant unresolved mode. The resulting analog frame-coding theorem establishes a finite relative redundancy and converse as part of a fundamental limit theory of legged locomotion.

Animals and robots encounter mud at the water-land interface. Like sand, mud can stay solid or flow like a fluid. Unlike sand, the yield strength of mud at which solid-fluid transitions occur depends on not only the amount of solid relative to fluid (water in mud, air in dry sand), but also how much coarse grains and fine clay are within the solid. Despite understanding of locomotion on/within dry sand dominated by coarse grains with repulsive normal forces and friction, little is known for mud dominated by fine clay with strong cohesion. Here, we developed methods to prepare uniform mud of controlled, variable yield strength and characterize and track its drift from water evaporation. Compared to other flowable substrates, mud strength measured by upward force during penetration is weaker and can vary more, and mud sticks more during extraction to pull downward, making it more challenging for locomotion.

Vision-Language Navigation (VLN) for humanoid robots poses challenges existing benchmarks fail to address: bipedal locomotion imposes physical constraints absent from wheeled agents, humanoid morphologies vary across platforms, and egocentric observations are distorted by locomotion-induced camera dynamics. We present HumanoidVLN, a physics-grounded simulator and benchmark for VLN across diverse humanoid embodiments. Built on NVIDIA Isaac Sim, our platform supports an extensible set of humanoid configurations, demonstrated on four robots (Unitree G1, Unitree H1, Internal-A, Internal-B) spanning 10-12 lower-body DoF and heights from 1.17m to 1.80m, via a hierarchical control stack combining a reinforcement learning locomotion policy with interchangeable PD or MPC path trackers. New robots and VLN models integrate with minimal effort; we demonstrate compatibility with NaVILA, DualVLN, StreamVLN, and JanusVLN. Environments are drawn from artist-designed scenes and 3D Gaussian Splatting reconstructions, filtered for navigable areas exceeding 100 square meters. Instructions are generated by a dual generator-reviewer plus paraphraser multi-agent pipeline with human-in-the-loop verification, yielding 933 collision-aware reference episodes, each paired with one fine-grained instruction and three coarse-grained stylistic variants (formal, natural, casual). Across four models and four embodiments, JanusVLN achieves the highest mean success rate of 43.55% and nDTW of 48.38. In a 20-episode sim-to-real pilot with DualVLN and the Unitree G1, navigation errors correlate strongly (r=0.935), with a mean absolute difference of 0.68m and mean trajectory similarity of 0.782 (+/-0.188) nDTW. These results highlight the interaction between VLN models, controllers, and humanoid embodiments under physical execution. Code, benchmark, and data will be released upon acceptance at https://humanoid-vln.github.io/.

Human movement is inherently variable, with variability structured according to task relevance: movements are typically more consistent at task-critical points and more flexible elsewhere. In human-robot interaction (HRI), however, model-based assistance strategies commonly assume deterministic human behavior and suppress such variability, potentially altering how interactions are experienced and lowering sense of agency. While movement variability is increasingly recognized as functionally meaningful, its deliberate preservation in assisted interaction, and its consequences for user experience, remain underexplored. In this paper, we empirically investigate how different assistance strategies shape human movement variability, task performance, and subjective interaction experience in a shared control setting. We introduce an autonomy-supportive shared control strategy that preserves users' natural movement structure. This approach is evaluated in a user study in which participants push an intelligent powered wheelchair under three conditions: no assistance, conventional variability-reducing assistance, and variability-preserving assistance. While task-relevant performance remained comparable across assisted modes, preserving natural movement variability led to more favorable interaction experiences. In particular, participants reported significantly higher perceived agency compared to conventional assistance and highest perceived usefulness. These findings suggest that variability-aware assistance can support both performance and user autonomy in physical human-robot collaboration. More broadly, the results highlight the importance of designing assistive robotic systems that respect the embodied structure of human movement rather than treating variability as noise to be neglected or eliminated.

Tracked robots are widely used in unstructured environments; however, their obstacle traversal capability is fundamentally limited by a tradeoff between front-end reachability and locomotion stability. This study presents TRASER (Tracked Robot with Articulated Spine for Extended Reach), a reconfigurable tracked robot capable of relocating both its articulation point and internal mass. TRASER employs a tape-spring mechanism that localizes compliance to the bending region while maintaining high stiffness in the remaining body, thereby improving both front-end reachability and center-of-mass (CoM) shifting capability. Geometric and static models are developed to analyze the effects of articulation point and CoM position on step and ditch traversal performances. Experiments demonstrate step traversal, suspended-platform traversal, and ditch traversal of 74%, 66%, and 59% of the robot body length, respectively. To the best of our knowledge, these results represent the highest reported obstacle traversal capabilities among tracked mobile robots.

Transcutaneous spinal cord stimulation (tSCS) modulates spinal sensorimotor circuits primarily through activation of afferent networks. While prior work has emphasized locomotor performance and spinal excitability, how tSCS affects conscious proprioceptive perception and the extent to which such effects parallel changes in locomotor control remain unclear. We investigated the acute and training-related effects of tSCS on ankle proprioception and gait in unimpaired adults (n = 14), with an independent control group (n = 14) completing identical proprioceptive training without stimulation. Proprioception was quantified using a bilateral robotic assessment of dynamic ankle localization ability (Crisscross), gross motor output using maximum dorsiflexion strength, and gait during normal and tandem treadmill walking using spatiotemporal, trunk-sway, and mediolateral center-of-mass (CoM) excursion measures. Acute tSCS increased ankle proprioceptive error (p < 0.001) while dorsiflexion strength was unchanged (p = 0.30). Gait shifted toward a modestly more constrained locomotor pattern, characterized by reduced step width and ML CoM excursion (p < 0.05). With continued training under stimulation, proprioceptive error decreased and, unlike the control group, the tSCS group showed progressive improvement that persisted after stimulation ended. Sagittal-plane gait measures recovered toward or beyond baseline, whereas mediolateral measures remained constrained, revealing a direction-dependent reorganization of locomotor control. Together, these findings show that tSCS influences multiple aspects of the sensorimotor control loop, disrupting conscious proprioception while reshaping locomotor behavior, and that the nervous system can adapt to altered afferent input through training.

In this study, we propose a framework that incorporates subjective evaluations provided by a Vision-Language Model (VLM) into the fitness evaluation and selection processes of a genetic algorithm. As the target of evolution, we employ virtual soft robots with flexible morphologies and locomotion and present the VLM with sequence images representing the locomotion of two individuals. Selection is performed via pairwise comparisons based on subjective evaluation terms such as adorably and weirdly. The outcomes of these comparisons are used as selection pressure within the genetic algorithm, enabling the simultaneous evolution of morphology and locomotion. Experimental results demonstrate that subjective selection by the VLM accelerates population convergence compared to random selection, while also giving rise to distinctive morphologies and motions corresponding to each evaluation term. An auxiliary experiment with human participants further showed that, although individual pairwise choices only partly agreed with the VLM selections, the resulting morphological and locomotion tendencies were qualitatively similar and repeated human evaluations imposed noticeable fatigue. Moreover, the observation that similar evolutionary outcomes emerged across different evaluation terms suggests that the VLM does not apply these terms in a purely literal manner but instead decomposes them into multiple internal evaluation criteria when making judgments. This work visualizes the evolutionary process through which subjective linguistic expressions are mapped onto embodied phenotypes and provides a foundational framework for analyzing the structure of subjective judgment in VLMs. The proposed approach is expected to contribute to new developments in evolutionary computation and artificial life research based on subjective evaluation.

We present a novel and minimalistic design of an aerial-terrestrial robot DASH: Ducted Aerial Spring Hopper. The goal is to enable both aerial and ground locomotion capabilities on a unified mobile robot that is mechanically-minimalistic, locomotion-versatile, and energy-efficient. We propose an organic integration of ducted fan co-axial body with a springy leg at the bottom for realization. The ducted fan module provides thrust-vectoring as the main actuation for agile flying; when it is combined with the light-weight spring leg, the robot realizes highly efficient ground hopping with energy circulation. Moreover, to realize optimal locomotion with two modes, we employ a contact-implicit model predictive controller to automatically choose locomotion modes and actuation. We successfully validated the design and control of DASH through a range of tasks, including periodic hopping, aerial flight, and mode-free locomotion with autonomous mode transitions during obstacle traversal.

A humanoid that can walk should not relearn locomotion from scratch to jog or run. Yet current approaches often obtain gait diversity by prescribing gait schedules, imitating motion clips, training experts to switch between or distilling skills into one policy. These strategies can produce impressive behaviors, but offer limited flexibility across continuous speed commands, terrains, and morphologies. We study skill growth with GaitSpan, a framework that expands a pretrained, basic walking policy into faster locomotion. It treats walking as a seed skill: reusable motor structure for balance, support, body coordination, and contact transition that can be regenerated at new rhythms, extended into longer/higher strides, and corrected by residual adaptation. This expansion has three aspects: 1) rhythm generation, which modulates the frozen walking policy with multiple internal clocks and learns command-conditioned combinations of the resulting canonical actions; 2) stride shaping, which rewards dynamic locomotion patterns appropriate for higher commanded speeds using a physically grounded objective inspired by spring-loaded inverted pendulum dynamics; and 3) residual adaptation, which captures motion details not accounted for by rhythm generation or stride shaping. GaitSpan is the first to deliver a single command-conditioned humanoid policy that spans walking, jogging, and running-like regimes covering a continuous speed range, transfers across morphologies, and deploys zero-shot on unseen sim-to-sim, and real-world terrains. Compared with baselines either trained with multi-experts or imitation from humans, it learns faster and achieves stronger gait performance.

Vision-Language-Action (VLA) models have recently demonstrated strong generalization in robotic manipulation, yet their applicability to whole-body, contact-rich humanoid locomotion remains severely underexplored due to data scarcity, the absence of dynamically consistent demonstrations, and the difficulty of encoding optimality and safety in learning-based pipelines. This work introduces a unified framework WOLF-VLA that integrates whole-body optimal-control (OC) motion synthesis with large-scale multi-modal dataset to train VLAs capable of generating humanoid locomotion policies directly from natural-language instructions. We construct a comprehensive dataset of dynamically feasible humanoid trajectories across six locomotion-related task families, each parameterized by environmental variations, object colors, placements, and visual distractors. We train a VLA model using the collected joint trajectories, ego-centric visual observations and natural language instruction, yielding a policy that exhibits strong reasoning and robustness to initial-condition variability, and competitive performance across several tasks and environment settings. A systematic ablation study demonstrates the impact of each modality on the model performance. The full dataset, model checkpoints, and benchmarking simulation suite will be openly released, establishing a reproducible dynamically consistent benchmark for whole-body humanoid locomotion rich VLA control and enabling future research in scalable transfer of instruction-driven locomotion policies.

Lunar micro-rovers under 50 kg are a rapidly growing vehicle class, yet open, benchmarked tools for running optimization across design trades remain scarce. Rover sizing is a highly coupled problem with a feedback loop specific to surface vehicles: a heavier rover sinks deeper into loose regolith, deeper sinkage costs more power, more power requires a larger array, and the array adds mass. We present RoverDevKit, a tradespace exploration toolkit with an open evaluator that couples a closed-form wheel-soil model with bottom-up mass, solar-power, thermal-survival, and traverse models. It is fast enough to serve as the fitness function of a multi-objective genetic algorithm (NSGA-II): one search of 3000 mission evaluations takes under a minute on a single laptop core. We use it to map the trades among distance traveled, vehicle mass, and slope capability on mare, polar, highland, and crater-rim missions. Across those missions the limiting factor changes: energy storage at high latitude, slope traction on loose highland regolith, and traverse range on mare and crater-rim terrain. The wheel-soil model matches measured single-wheel drawbar pull on two independent datasets to within the 20-30% error typically reported for this class of model, and the mass model predicts published 5-50 kg rover masses to 10.5% median absolute error. A check against five published in-class designs places them near the designs the optimizer selects. Re-running the search after applying that wheel-soil comparison error leaves these conclusions unchanged.

Load transportation with quadruped robots is strongly affected by the dynamics of the physical interface between the robot and the load. Passive spring-based arms reduce weight and complexity compared to active manipulators, but their spring-damper dynamics can introduce oscillatory forces that degrade locomotion stability. This paper derives an extended Zero Moment Point (ZMP) formulation that includes passive payload-interface dynamics, relating stiffness, damping, and payload mass to the stability margin. The analysis shows that underdamped configurations can resonate with locomotion harmonics. Based on this insight, we augment a Single Rigid Body Dynamics model with passive subsystem dynamics and integrate it into a Model Predictive Control framework. In simulation, the proposed controller reduces stability violations by up to $10\times$, from $7.0\%$ to $0.7\%$, and increase locomotion efficiency by lowering horizontal ground reaction force effort by up to $15\%$ compared to a nominal baseline. Hardware experiments with a $2\,\mathrm{kg}$ payload show stable locomotion under pull-release disturbances where the nominal controller fails. The same model also enables end-effector tracking through passive arm dynamics without direct arm actuation.

Deploying legged robots in extra-terrestrial environments includes many challenges due to complex terrain interactions, energy, and thermal constraints. For effective mechanical design of a lunar exploration quadrupedal robot, careful consideration of motor torques, energy expenditure, and cost of transport is required. The lunar surface is composed of granular regolith, which impacts the locomotion of legged robots and their performance. Locomotion algorithms trained with rigid contact assumptions are also ineffective when applied to environments with soft contacts, such as granular surfaces, which can result in instability and poor tracking. In this report, the physical modelling of the granular lunar surface-robot foot contacts is applied to a simulation environment with locomotion trained using Reinforcement Learning. A comparison is conducted between the policy trained on rigid contact and soft contact environments, analysing the gait and locomotion performance metrics. The analysis demonstrates that soft contacts simulating regolith surfaces pose additional challenges for Reinforcement Learning based training, result in a qualitatively different gait, and increase the overall energy expenditure.

Motor thermal management is often overlooked in the context of electrically-actuated robots, particularly legged robots, but motor overheating is a key factor that limits long-duration locomotion especially under payload conditions. This paper integrates a whole-body thermal model of a quadruped robot into the reinforcement learning pipeline to update motor temperatures, and proposes a two-stage training framework for motor thermal management. In this framework, a nominal policy is first pre-trained as a locomotion baseline capable of traversing diverse terrains. A residual policy is then trained on top of the nominal policy to provide corrective actions based on the robot's thermal state, ensuring high performance under low-temperature conditions and preventing motor overheating under high-temperature conditions. Simulation results demonstrate that the proposed policy achieves an effective balance between motor thermal safety and locomotion performance. Real-world experiments on a Unitree A1 quadruped robot further validate the approach: under a 3 kg payload, the robot achieves stable locomotion across multiple terrains for over 13 minutes, while the nominal policy alone leads to motor overheating in about 5 minutes.

Safe manipulation-oriented navigation for humanoid robots requires scene memory that remains reliable under locomotion-induced perceptual distortion, environmental changes, and interaction-level geometric safety constraints. Existing semantic mapping and scene-graph systems are difficult to deploy directly in this setting because they often assume stable camera trajectories, static environments, or coarse object geometry. We introduce the Multi-modal Interactive Field (MIF), a humanoid-oriented system that integrates confidence-aware semantic 3D Gaussian Splatting, discrepancy-triggered spatial memory updates, and task-driven geometric reconstruction within a closed-loop perception-adaptation pipeline. MIF couples three fields: an uncertainty-aware 3DGS Appearance Field that suppresses gait-induced blur, a Spatial Field that maintains topological memory, and a Geometry Field that supports Interaction Pose Safety (IPS) before manipulation. A discrepancy detection score is introduced to separate locomotion-induced false-positive changes from persistent changes and updates only locally inconsistent regions. On a Unitree-G1 humanoid in a real dynamic office, MIF improves relocation success in non-static environments from 12% to 94% compared with static scene-graph memory, while reducing semantic memory footprint by 91.4% through feature distillation for practical online operation. Project page and code: https://ziya-jiang.github.io/MIF-homepage/

Systems moving in low Reynolds number fluid regimes are known to be governed by a motility map'' which linearly relates their shape change rates to they body frame velocity moving through the fluid. A consequence of this is Purcell's Scallop Theorem'' -- a locomotion system that undergoes shape changes that follow the same path forward and backward in time (reciprocal body deformations) cannot achieve net displacement, regardless of pacing of those changes.We show that linear-in-velocity motility maps extend to any power law viscosity (a.k.a. Ostwald--de Waele fluid), and therefore to many biological fluids in intermediate shear ranges. We also show that the linear-in-velocity property can be violated in Carreau-Yasuda fluids to produce net motion using an inchworm'' model consisting of two unequal masses with unequal drag coefficients performing reciprocal motions. Interestingly, the direction of motion can be switched by changing speeds. Our results show that the linear motility map of geometric mechaincs can be used to analyze and design locomotion in power-law fluids, and that some nonlinear drag relationships such as Carreau-Yasuda can be exploited to generate net locomotion in seeming violation of the scallop theorem''.

Soft mobile robots have emerged as a promising area of research with potential applications in various disciplines including but not limited to search-and-rescue, service, surveillance, explorations, and manufacturing. In this article, we provide a comprehensive review of the current state of soft mobile robot research, focusing on wheelless terrestrial locomotive systems. We include past and present developments in locomotion strategies, actuation methods, modeling approaches, and control systems. Further, we identify key research challenges that must be overcome to enable the widespread adoption of soft mobile robots in various applications. Overall, this article provides a valuable resource for researchers and practitioners interested in the field of soft mobile robots and soft robotics.

Quadruped robots are often designed with rigid feet to simplify control and maintain stable contact during locomotion. While this approach is straightforward, it limits the ability of the legs to absorb impact forces and reuse stored elastic energy, leading to higher energy expenditure during locomotion. To explore whether compliant feet can provide an advantage, we integrate foot compliance into a reinforcement learning (RL) locomotion controller and study its effect on walking efficiency. In simulation, we train eight policies corresponding to eight different spring stiffness values and then cross-evaluate their performance by measuring mechanical energy consumed per meter traveled. In experiments done on a developed quadruped, the energy consumption for the intermediate stiffness spring is lower by ~ 17% when compared to a very stiff or a very flexible spring incorporated in the feet, with similar trends appearing in the simulation results. These results indicate that selecting an appropriate foot compliance can improve locomotion efficiency without destabilizing the robot during motion.

LIPM is everywhere in legged-locomotion control, but almost always as a modeling choice rather than as something the controller's cost actually prefers. This note tries to make that link more explicit. Working from a small centroidal OCP that penalizes the rate of angular momentum, we look at what its optimum tends to look like. Three things come out. With full-rank stance, the optimum drifts toward a pendular force pattern at a rate determined by the SVD of the moment Jacobian; the constant is set by foot-span geometry and matches the experiments to within 16%. With N=2 stance, as in trot, the friction cone introduces a lower bound on $\|\dot{H}_G\|$ that no amount of weight tuning fixes; we also see a non-smooth feasibility kink at a critical horizontal acceleration that we can write in closed form. Adding a task term that asks for a nonzero $\dot{H}_G$ moves the optimum off the pendular set in a predictable way. None of this is far from the classical ZMP/DCM picture. We test these claims on a point-mass quadruped and on the Unitree Go1 in MuJoCo (open-loop QP and a torque-level closed-loop controller), and we note where the asymptotic story stops being a good description of what the closed loop actually does.

Legged robots have demonstrated remarkable agility on rigid, stationary ground, but their locomotion reliability remains limited in non-inertial environments, where the supporting ground moves, tilts, or accelerates. Such conditions arise in ground transportation, maritime platforms, and aerospace settings, and they introduce persistent time-varying disturbances that break the stationary-ground assumptions underlying conventional legged locomotion. This survey reviews the state of the art in modeling, state estimation, and control for legged robots in non-inertial environments. We summarize representative application domains and motion characteristics, analyze the root causes of locomotion performance degradation, and review existing methods together with their key assumptions and limitations. We further identify open problems in robot-environment coupling, observability, robustness, and experimental validation, and discuss future directions in autonomy, system-level design, bio-inspired strategies, safety, and testing. The survey aims to clarify the technical foundations of this emerging area and support the development of reliable legged robots for real-world dynamic environments.

Nature suggests that exploiting the elasticities and natural dynamics of robotic systems could increase their locomotion efficiency. Prior work on elastic snake robots supports this hypothesis, but has not fully exploited the nonlinear dynamic behavior of the systems. Recent advances in eigenmanifold theory enable a better characterization of the natural dynamics in complex nonlinear systems. This letter investigates if and how the nonlinear natural dynamics of a kinematic elastic snake robot can be used to design efficient gaits. Two types of gaits based on natural dynamics are presented and compared to a state-of-the-art approach using dynamics simulations. The results reveal that a gait generated by switching between two nonlinear normal modes does not improve the locomotion efficiency of the robot. In contrast, gaits based on non-brake periodic trajectories (non-brake orbits) are perfectly efficient in the energy-conservative case. Further simulations with friction reveal that, in a more realistic scenario, non-brake orbit gaits achieve higher efficiency compared to the baseline gait on the rigid system. Overall, the investigation offers promising insights into the design of gaits based on natural dynamics, fostering further research.