A review on locomotion robophysics: the study of movement at the intersection of robotics, soft matter and dynamical systems

• Robust all-terrain mobility is essential for robots operating with humans, yet engineered systems still perform poorly in natural environments compared with biological organisms.Low-cost actuators, sensors, computing, and storage are enabling increasingly physical robots, but current mobility is strongest in relatively simple settings such as paved roads.Source: Introduction · pp. 3–4
• Animals navigate diverse media through evolved morphology, control, and materials, whereas biological and robotics research has not yet isolated the general principles that make locomotion robust.Complex biological models can provide insight, but their tens to hundreds of parameters make fundamental principles difficult to identify; fluid-locomotion studies have emphasized fluid physics more than general movement control.Source: Introduction · p. 4
• The review proposes applying physics methods to identify the necessary and sufficient conditions—a “minimal feature set”—for robust self-propelled locomotion in the real world.This program is presented as a complement to robotics engineering and computer science, analogous to how studying technological devices helped establish general sciences such as thermodynamics.Source: Introduction · pp. 3–4
• Robotic physical models enable controlled, systematic studies of organismal morphology and control, while also supporting more complex robots and revealing emergent phenomena in classical dynamical systems.Such models have been used to study movement through air, water, land, and granular media over the past 25 years.Source: Introduction · p. 5
• A complementary dynamical-systems language emerged from Shapere and Wilczek’s gauge-potential framework for shape-changing locomotion in highly dissipative media.Engineers developed the framework into predictive tools for relatively simple environments and, more recently, a complex medium.Source: Introduction · p. 5
• Recent engineering advances have expanded robot mobility from relatively simple environments toward increasingly complex terrains, including through RHex and BigDog research.Figure 1 also situates these developments among humanoid robots, delivery and video-capture drones, and other mobile systems.Source: Figure 1 · p. 5

1. Foundations of locomotion robophysics: key ingredients in a physics of moving systems

• Locomotion robophysics studies fundamental principles governing movement and control of self-deforming entities interacting with complex environments as non-equilibrium systems.Motion emerges through internally driven actuation and effective interaction, extending experimental robotics toward novel locomotor principles.Source: §1 · p. 6

1.1. Integration of science and engineering: creation leads to understanding and vice versa

• Robophysics links simplified laboratory models with engineered devices and natural environments, so each reveals phenomena and principles that inform the other.Real-world testing can expose phenomena absent from controlled settings, while simple models clarify dynamics that complex robots or animals obscure.Source: §1.1 · p. 6

1.2. Simplification aids understanding

• Simplified robophysical devices isolate minimal geometry and control needed to reproduce complex locomotor dynamics, enabling systematic tests of hypotheses and transferable models.Examples include SLIP-based hopping studies, a reduced RHex model yielding resistive force theory for a larger robot, and single-wheel terramechanics.Source: §1.2 · p. 7

1.3. Embracing poor performance and systematically surveying parameter space

• Robophysics treats failures and successes alike, using broad parameter surveys to reveal performance mechanisms, transitions, bifurcations, and robustness across environments.Systematic studies explained RHex sensitivity on granular media, enabled snake climbing on sandy slopes through contact-length modulation, and identified added mass as a sand-jumping failure mechanism; automated robots now support high-throughput exploration.Source: §1.3 · pp. 8–9

1.4. The importance of feedback and control templates

• Template-and-anchor control combines reduced-order behavioral models with detailed mechanisms, while systematic experiments identify templates and the sensors and actuators needed for autonomous behavior.Sidewinding is captured by vertical and horizontal curvature waves offset by π/2; modulating the template enables climbing and turning, while slip and inclination sensing can augment actuator encoders on slopes.Source: §1.4 · pp. 9, 11, Figure 2 · p. 10

2. Robophysics as an enabler for progress in diverse areas

• Robophysics links controlled robotic experiments with physics methods to discover general principles of locomotion, then test them in robust devices or organisms.Robotics can also accelerate progress in physics through automation.Source: §2 · p. 11
• Robotic locomotion provides experimental systems for studying nonequilibrium and dynamical systems, including feedback control, stability, bifurcations, limit cycles, chaos, and geometric mechanics.Robophysical systems can test theoretical predictions, challenge concepts, and probe how gross behaviors depend on lower-level parameters in hierarchically complex systems.Source: §2 · pp. 11–12
• Robophysics advances soft-matter physics by using locomotion to reveal robot–substrate interactions in sand, snow, grass, and other environments whose terrestrial equations of motion remain unknown.Experiments can expose novel interactions and motivate systematic empirical and theoretical studies.Source: §2 · p. 12
• Sand-jumping experiments with a 1-D legged robot revealed transient granular dynamics associated with jamming and added mass that significantly influence locomotor performance.These findings motivate a “terradynamics” of land locomotion and could support robust exploration of disaster sites, comets, and other celestial objects.Source: §2 · p. 13
• Robots serve as simplified physical models for testing biological hypotheses about morphology, neuromechanical control, templates, anchors, and robustness in complex environments.They enable systematic increases in locomotor and environmental complexity while complementing mathematical models.Source: §2 · pp. 13–14
• Robophysics aims to improve real-world robots by deriving design principles in simplified laboratory systems before applying lessons from dynamics, soft matter, and controls to hardened devices.This approach is presented as a potentially more practical and economical alternative to repeatedly building field-ready iterations that barely improve.Source: §2 · p. 14

Specific examples to be addressed in this review

• The review uses simple robot models to compare experiments, simulations, and analytical theory, focusing mainly on terrestrial locomotion and steady rhythmic interactions while also discussing paleorobotics, biomechanics, and validated computation.Its examples are illustrative rather than exhaustive.Source: Specific examples to be addressed in… · pp. 14–15
• The review covers hard ground, air and water, granular and heterogeneous or wet substrates, computational tools, geometric mechanics, and future feedback-controlled locomotion in changing environments.Geometric mechanics is proposed as a general language for relating cyclic self-deformations to body translations and rotations, including in dry granular media.Source: Specific examples to be addressed in… · pp. 15–16

Hard-ground jumping

• SLIP-based robots established a lineage from Raibert’s first hopper to robust 3-D balancing robots, while later studies systematically compared jumping strategies and periodic-orbit stability.These studies varied physical and actuation parameters and used Poincare maps, Jacobians, and experiments to analyze stable hopping.Source: §3 · pp. 16–17
• 20,160 automated jumps revealed single- and stutter-jump strategies, whose optimal timing depended on resonance, transient dynamics, and hybrid aerial–ground transitions.The simplified 1-D SLIP hopper enabled systematic variation of forcing phase and frequency and exposed dynamics difficult to learn with more complex robots.Source: §3 · pp. 16–17, Figure 3 · p. 17

Bipedal walking

• Obstacle-rich hard-ground bipedal tasks remain failure-prone because motion planning is sensitive to obstacle type, sensing uncertainty, and deviations from assumed models.Simpler constrained settings support systematic analysis of dynamically stable and passive walking using pendulum-based models.Source: §3 · p. 18
• Passive-dynamics principles enabled highly efficient walkers, including a bipedal robot traveling 65 km on one battery charge, but complex obstacles require integrating grasping, manipulation, balance, and sensing.ATRIAS 2.1 used the SLIP model to optimize walking gaits and minimize mechanical transport cost.Source: §3 · p. 18

Multilegged locomotion

• Systematic parameter variation in multilegged robots uncovered contact and locomotion mechanisms, including anisotropic friction, single-actuator steering, scaled-impulse galloping, climbing, and speed-dependent undulation transitions.A centipede-like robot’s undulation amplitude increased with speed through a supercritical Hopf bifurcation, while a salamander robot modeled aquatic-to-terrestrial gait transitions.Source: §3 · pp. 18–19

Water locomotion

• Robotic fish studies systematically examined propulsion, maneuverability, stability, fin compliance, motion frequency, swimming speed, and turning ratio in water.These experiments built on extensive fluid–structure interaction research.Source: §4 · p. 19

Air locomotion

• Robofly and other flapping-wing robots revealed principles of insect flight and hovering, including how wing morphology, wind speed, and angle of attack affect forces and performance.Systematic wind-tunnel measurements produced empirical force models for flapping-winged fliers.Source: §4 · p. 20

Fluid–structure interactions

• Experiments with robotic fluid locomotors show that solving Navier–Stokes equations alone is insufficient to explain emergent interactions such as leading-edge vortices during insect flight.Robotic systems enable computational fluid-dynamics modeling while exposing interaction effects that arise during specific locomotion modes.Source: §4 · pp. 19–20

Aerial robotics

• Lower sensing and actuation costs have enabled affordable consumer-scale aerial robots, while improved aerodynamic understanding, control schemes, and designs are expected to extend these advances.Applications include reconnaissance, combat, crop monitoring, package delivery, and aerial footage.Source: §4 · pp. 19–20

5.1. Legged locomotion

• Legged robots on granular media are governed by leg actuation and penetration: a rotary-walking model and universal scaling relate speed across robots and animals to leg penetration ratio, while minimizing substrate yielding preserves performance.The model applies across 10 g–2.5 kg robotic and biological locomotors, and RFT simulations agree with experiments across robot sizes and substrate stiffnesses.Source: §5.1 · pp. 21–22, Figure 6 · p. 22

5.2. Sidewinding

• Sidewinding robots test a biological two-wave template showing that slope-dependent contact length balances stability against substrate yielding and drag, while wave modulation enables animal-like turning without significant slip.Increasing contact length prevents tipping and reduces substrate stress, but excessive contact increases drag; differential and reversal turns arise from distinct wave modulations.Source: Figure 7 · p. 23, §5.2 · p. 24

5.3. Flipper-based terrestrial locomotion

• FlipperBot demonstrates that flexible-wrist kinematics can reduce failure on disturbed sand by controlling granular-state changes, whereas fixed wrists weaken previously disturbed substrate and shallow 1.4 mm insertion prevents forward motion.The robot models hatchling behavior, which uses a bent wrist on sand and a rigid wrist on hard ground to achieve comparable forward speeds.Source: §5.3 · p. 25

5.4. Sand-Swimming

• A seven-link sand-swimming robot and validated granular simulation show that lateral undulation is fastest with a single-period body wave and amplitude ratio A/λ ≈0.2, using a localized frictional fluid rather than solidified ground.The frictional fluid extends roughly one body-width from the swimmer, contrasting with surface locomotion that solidifies substrate for push-off.Source: §5.4 · pp. 25–26

Robophysics applied to evolution of movement

• Robotic models can extend locomotion physics to extinct terrestrial walkers because morphology alone is insufficient, requiring systematic variation of kinematics in anatomically inspired bodies.This approach combines skeletal morphology with locomotor strategies observed in modern analogue organisms to investigate early walker mechanisms.Source: §6 · p. 27, Figure 9 · p. 27

Cluttered beam traversal

• A rounded cockroach-inspired shell enabled a small legged robot to traverse densely cluttered beams via body rolling, whereas its cuboidal body became stuck (Fig. 11b).The robot reproduced the insects’ reduced traversal performance with less rounded body shells without adding sensors or changing control.Source: §6 · p. 27, Figure 11 · p. 29

Automated heterogeneous terrain

• SCATTER automated precise variation of substrate compaction, inclination, obstacle geometry, distribution, and mobility while conducting approximately 200 tests per day without human intervention.The system combined an air-fluidized bed, tilting actuators, a universal jamming gripper, and high-speed 3D tracking.Source: §6 · p. 28
• Obstacle interactions acted as attractive or repulsive scatterers whose strength depended sensitively on local boulder-surface inclination and contact position, enabling long-term trajectory estimates by superposition.The scattering framework generalized across obstacle geometry, orientation, and roughness and informed future arbitrary-terrain systems.Source: §6 · pp. 28–29

Snake locomotion on irregular substrates

• Force-based control produced more robust snake-robot motion than position-based control on a reconfigurable peg board, whose performance degraded under small parameter perturbations.Force control adapted body wavelength by stretching or shrinking, reducing backward slip and jamming across varying peg spacings.Source: Figure 11 · p. 29, §6 · p. 30

• Moisture content causes dramatic rheological changes in wet granular substrates, producing structures ranging from smooth dry-grain piles to sandcastles and saturated slurries.Source: §7 · p. 30
• Vibrational sieving creates wet granular media with controlled moisture content and compaction, enabling studies of fire-ant digging and lizard locomotion.The technique addresses the difficulty of preparing partially saturated granular mixtures with precise, homogeneous initial conditions.Source: §7 · pp. 30–31
• Drag measurements reveal complex wet-substrate dynamics, motivating soft-matter questions and robophysical principles for robots traversing arbitrary substrates.Source: §7 · p. 31
• Local fluidization during RoboClam’s two-anchor burrowing cycle reduces resistive forces, enabling deep digging in fully saturated mud.Systematic parameter variation and genetic-algorithm optimization identified the fluidization strategy; the mechanism and sequence are illustrated in Fig. 12.Source: §7 · p. 30, Figure 12 · p. 31

8. Computational tools in locomotion robophysics

• Computational tools enable quantitative predictions and mechanical explanations, but hard-ground contact and collision remain difficult to model accurately.Numerical methods support aerial, aquatic, and terrestrial locomotion studies, while discrete contact events create substantial modeling challenges.Source: §8 · pp. 31–32

8.1. Simulating hard ground

• Hard-ground simulation uses compliant regularization or complementarity formulations, but finite time steps, oscillations, frictional nonlinearities, and numerical instabilities remain challenges.LCP/NCP methods enforce nonpenetration constraints and can use optimization techniques, while cone-constraint formulations address friction in engines such as Chrono::Engine.Source: §8.1 · p. 32

8.2. Simulating fluids

• CFD numerically resolves thrust and drag in aerial and aquatic locomotion, with parallel computing enabling realistic large-scale simulations despite turbulence, cost, and geometry challenges.Examples include optimized anguilliform swimming, experimentally matching sunfish flow fields, fruit-fly flight control, and jellyfish propulsion agreeing with experimental data.Source: §8.2 · p. 33

8.3. Simulating granular media, DEM

• DEM accurately models dense granular locomotion when collision parameters are tuned and time steps are small, while coupling with multibody dynamics predicts robot and sandfish behavior.The coupled approach predicted robot undulation efficiency across 1–4 Hz and tested sandfish strategies, while simulations also exposed forces and locomotor transitions unavailable from experiments alone.Source: §8.3 · pp. 33–34

8.4. Modeling dry granular media, continuum descriptions

• Granular RFT predicts slow locomotion and neuromechanical phase lag, but fast locomotion involves significant granular inertial effects that make quasistatic RFT ineffective, while added-mass reactive theory reproduced jump heights.RFT torque-onset timing agreed well with sandfish electromyography. A 1-D reactive force theory incorporating added mass reproduced experimental jump heights in simulation.Source: §8.4 · pp. 36–38

Historical foundations

• Geometric mechanics originated with Shapere and Wilczek’s application of gauge-theoretic ideas and geometric phase to microorganism locomotion.Later work generalized the language to systems with nontrivial momentum evolution,,.Source: §9 · pp. 39–40

Experimental integration and outlook

• Because few experimental tests existed, separating gait analysis from model construction through empirical or simulated differential equations made physical experiments more feasible.The framework is proposed as a general robophysics language, with future applications to hybrid aerial–substrate locomotion and control-template search.Source: §9 · pp. 39, 40, 43

Core framework

• The reconstruction equation relates internal shape changes to body velocity, while Lie bracket averaging estimates geometric phase from cyclic gaits across diverse locomoting systems.Its local connection A maps shape velocity to body velocity as ξ = A(α) · ˙α.Source: §9 · pp. 40–41

Methodological advances

• Recent advances make geometric analysis viable in non-ideal environments by deriving local connections empirically and improving large-gait accuracy with Constraint Curvature Functions.Local connections can be obtained by perturbing joint angles across shape space using RFT simulation, while CCF area integrals compute displacement for large-amplitude deformations.Source: §9 · p. 41

Applications to granular swimming

• Empirical geometric methods enabled prediction of optimal gaits and creation of turning-in-place behaviors for locomotion in complex media.For a three-link swimmer, integrating curvature over a gait region approximates net forward displacement and reveals performance changes with gait size.Source: §9 · p. 41, Figure 16 · p. 42
• Constraint Curvature Functions predict displacement from gait area and identify an optimal “butterfly” gait that outperforms circular gaits.Displacement initially scales with the enclosed area, then declines when the gait spans opposite-sign curvature regions.Source: §9 · pp. 41–42, Figure 16 · p. 42

10. Conclusions and Outlook

• Robophysics prioritizes simplified physical robots, systematic experimentation, and parameter-space exploration to discover principles of self-generated motion.Physical devices complement theory and simulation by exposing insights across dynamical systems, soft matter, biology, and engineering robotics.Source: §10 · p. 43
• Robophysics should address diverse substrates, including cohesive soils, mud, snow, leaf litter, heterogeneous obstacles, slopes, and transitions between environments.Robots must control both their own dynamics and the environments they traverse; systems such as SCATTER can measure soft-matter properties during realistic robot–media interactions.Source: §10 · pp. 43–45
• Robophysics aids biology by systematically varying robotic movements to identify optimal performance, failure conditions, and the mechanical, geometric, and actuation features underlying animal locomotion.Examples include granular-fluidization mechanisms in sandfish swimming, flipper rigidity in hatchling sea turtles, resonance and stutter jumps, and localized fluidization in razor-clam burrowing.Source: §10 · p. 44
• Robophysical principles could enable top-down engineering of non-organic life-like movement and provide a laboratory-based way to investigate what it means for life to exist.The proposal complements bottom-up synthetic biology by constraining engineered systems through discovered principles of locomotion.Source: §10 · p. 45
• Collective robophysics is needed for robots operating in crowded conditions and complex substrates, where new principles must explain collective flow, clogging, and emergent task performance.Hydrodynamic theories of active matter, self-propelled-particle analogies, and the physics of glasses, supercooled fluids, and jammed solids may help explain such systems.Source: §10 · p. 45
• The review’s locomotion focus reflects the authors’ expertise, while the broader vision is a robophysics related to cybernetics and applicable wherever robots require effective action in complex environments.The discipline emphasizes high-quality systematic experiments together with theory and modeling, extending cybernetics’ concern with control and communication in organisms and machines.Source: §10 · pp. 45–46