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A review on locomotion robophysics: the study of movement at the intersection of robotics, soft matter and dynamical systems
Jeffrey Aguilar, Tingnan Zhang, Feifei Qian, Mark Kingsbury, Benjamin McInroe, Nicole Mazouchova, Chen Li, Ryan Maladen, Chaohui Gong, Matt Travers, Ross L. Hatton, Howie Choset, Paul B. Umbanhowar, Daniel I. Goldman
TL;DR
Robophysics addresses the need for principles of self-generated motion by studying simplified physical robots and models under controlled, systematic conditions. The review argues that this approach connects experiment, theory, and computation while informing locomotion physics, engineering, and biology.
Problem
The mapping from locomotor inputs to outputs is complicated and nonlinear, making intuitive principles difficult to develop even in systems with few degrees of freedom.
Method
The review examines simplified laboratory robots through systematic experimentation, parameter-space exploration, and dynamical-systems tools, alongside macroscopic modeling of complex substrates.
Results
Robophysics studies reveal conditions for optimal performance and failure and identify mechanical, geometric, and actuation features important for locomotion, including granular swimming mechanisms.
Takeaways & Limitations
Robophysics can aid biological-locomotion modeling and support movement through varied soft-matter environments by studying how robots control their own dynamics and those of substrates.
Takeaways & Limitations
Quasistatic continuum descriptions such as RFT are insufficient during fast locomotion when granular inertial effects significantly contribute to reaction forces.
Abstract
from arXiv · showhide
In this review we argue for the creation of a physics of moving systems -- a locomotion "robophysics" -- which we define as the pursuit of the discovery of principles of self generated motion. Robophysics can provide an important intellectual complement to the discipline of robotics, largely the domain of researchers from engineering and computer science. The essential idea is that we must complement study of complex robots in complex situations with systematic study of simplified robophysical devices in controlled laboratory settings and simplified theoretical models. We must thus use the methods of physics to examine successful and failed locomotion in simplified (abstracted) devices using parameter space exploration, systematic control, and techniques from dynamical systems. Using examples from our and other's research, we will discuss how such robophysical studies have begun to aid engineers in the creation of devices that begin to achieve life-like locomotor abilities on and within complex environments, have inspired interesting physics questions in low dimensional dynamical systems, geometric mechanics and soft matter physics, and have been useful to develop models for biological locomotion in complex terrain. The rapidly decreasing cost of constructing sophisticated robot models with easy access to significant computational power bodes well for scientists and engineers to engage in a discipline which can readily integrate experiment, theory and computation.
Introduction
Robots increasingly operate in human environments, but robust all-terrain locomotion remains far less capable than biological movement. The review proposes applying physics methods to simplified robotic models to identify fundamental principles of locomotion.
- Robotics has permeated society, yet graceful physical interaction with humans and complex environments remains an unresolved ambition.
- Biological systems outperform engineered devices across diverse terrain because morphology, control, and materials support robust navigation in varied substrates.
- Existing locomotion research has produced valuable models, but biologically detailed theories with tens to hundreds of parameters make fundamental principles difficult to isolate.
- The review argues that physics can identify necessary and sufficient conditions—a minimal feature set—for robust self-propelled locomotion in the real world.
- Physical robot models enable controlled, systematic studies of morphology and control that are often impossible with living systems.
1. Foundations of locomotion robophysics: key ingredients in a physics of moving systems
Locomotion robophysics studies self-deforming systems through simplified physical models, controlled experiments, parameter exploration, and dynamical-systems analysis. Its goal is to understand both successful and failed environment–locomotor interactions rather than optimize only successful performance.
- Locomotion robophysics pursues fundamental principles governing self-deforming entities interacting with complex environments.These systems are treated as nonequilibrium processes whose motion emerges from internally driven actuation.
- Simplified laboratory devices, theory, and systematically controlled experiments expose dynamics that are difficult to interpret in complex robots or natural systems.
- Robophysics examines failure alongside success, treating both as outcomes that reveal environment–locomotor interaction principles.
- Parameter-space exploration reveals transitions, bifurcations, and performance modes while enabling direct comparison between theory and experiment.
- Template-and-anchor modeling reduces complex locomotion to lower-order models while retaining connections to full robotic or biological systems.
- Sidewinding experiments showed that a relatively simple template can generate locomotion through superimposed vertical and horizontal curvature waves offset by π/2.
2. Robophysics as an enabler for progress in diverse areas
Robophysics connects robotic locomotion with dynamical systems, soft matter physics, and biological movement. Through simplified robots, systematic experimentation, and model comparison, it supports both new physical insight and improved understanding of locomotion.
- Dynamical and nonequilibrium systems: Robotic locomotion provides experimental systems for studying nonequilibrium dynamics, feedback control, stability, bifurcations, and related phenomena.
- Soft matter physics: Interactions with sand, snow, grass, and other soft substrates can reveal locomotor dynamics that motivate new soft matter physics.
- Biological movement and robustness: Robots can test biological hypotheses about morphology, neuromechanical control, and templates and anchors in physiologically relevant environments.
- The review uses simple robots to compare experiment, simulation, and analytical theory across challenging terrestrial environments.
- The review primarily focuses on steady rhythmic locomotion in dry homogeneous granular media, a narrow subset of possible behaviors and substrates.
- Geometric mechanics: Geometric mechanics offers a general language for relating cyclic self-deformations to body translation and rotation, including in dry granular media.
3. Locomotion robophysics in hard ground environments
Robophysics uses simplified robots, systematic parameter variation, and dynamical-systems analysis to reveal rich hard-ground locomotion dynamics and inform efficient walking and novel locomotion mechanisms.
- Hard-ground locomotion: SLIP-based robots demonstrate that even rigid, flat, frictional ground produces rich locomotion dynamics requiring systematic physical analysis.The review presents SLIP-inspired hopping and walking robots as simplified systems for studying biological gait dynamics and robot locomotion.
- Jumping: 20,160 automated jumps of a constrained 1-D SLIP hopper revealed single-jump and stutter-jump strategies shaped by resonance, transient dynamics, and hybrid phase transitions.The experiments varied sinusoidal forcing parameters, including phase and frequency, and used an analytical model to interpret the resulting strategies.
- Bipedal walking: Obstacle-laden hard-ground environments remain difficult for bipedal robots, despite improved performance in simpler scenarios.The review links this gap to the need to understand how grasping, manipulation, balance, and sensing interact during locomotion.
- Bipedal walking: Dynamic walking models and passive elements have enabled highly efficient robots, including a biped capable of traveling 65 km on one battery charge.ATRIAS uses the SLIP model to optimize walking gaits and reduce mechanical transport cost; passive dynamic walking is another studied mechanism.
- Parameter exploration: Systematic variation of foot, joint, gait, and undulation parameters has revealed contact physics, anisotropic friction, single-actuator steering, galloping, and climbing mechanisms.Centipede-like robots also showed speed-dependent undulation amplitude associated with an instability caused by a supercritical Hopf bifurcation.
4. Robophysics in air & water
Robophysics studies movement through air and water with physical robot models, computational fluid dynamics, and controlled measurements of fluid–locomotor interactions.
- Fluid locomotion: Experimental robots and Navier–Stokes simulations have both advanced understanding of locomotion in fluids.The review surveys robot models of fish-like propulsion and other fluid locomotion alongside computational approaches.
- Air: Robofly experiments, miniature flying robots, and wind-tunnel measurements have been used to study insect flight, wing morphology, and aerodynamic forces.Measurements included elevator deflection and force across wind speeds and angles of attack.
- Air and water: Figure 5 assembles robotic and simulated examples spanning robotic tuna, glass knifefish, flapping robots, insect flight, and giant danio fluid dynamics.The examples illustrate the range of physical and computational models used for locomotion in fluids.
5. Robophysics in granular media, movement on and within a substrate between solid and fluid
Robophysical studies of granular media use controlled robots, simulations, and biological comparisons to uncover how morphology, control, and substrate interactions govern locomotion. Across walking, sidewinding, flippering, and sand-swimming, simplified devices expose performance mechanisms and failure modes.
- Granular media as a robophysics system: Dry granular media provide controllable experimental states and relatively simple particle-based simulations for studying locomotion on flowable substrates.Researchers can vary compaction and model dissipative contacts between particles and body elements.
- Legged locomotion: SandBot speed depends sensitively on leg frequency, intra-cycle kinematics, ground compaction, and leg penetration during granular walking.Experiments and terradynamic simulations support systematic comparisons across gait frequencies, foot sizes, masses, and ground stiffnesses.
- Legged locomotion: A universal scaling model related robotic and biological locomotor performance across 10 g to 2.5 kg to leg penetration ratio over varying stiffnesses and gait frequencies.The model was applied to lizards, geckos, and crabs, whose performance was likewise determined by leg penetration ratio.
- Sidewinding: Sidewinding robots revealed a trade-off on granular inclines: increasing contact length improves balance and reduces substrate stress, whereas excessive contact causes drag.Failure occurs through tipping from gravity or slipping when the substrate yields.
- Sidewinding: The sidewinding gait functions as a control template using posteriorly propagating horizontal and vertical waves to regulate contact with yielding granular media.Robophysical experiments helped anchor this template model of locomotion.
- Flipper-based terrestrial locomotion: FlipperBot experiments showed that wrist flexibility reduces interaction with disturbed ground, while extremely shallow insertion depths of 1.4 mm prevent forward movement.A fixed wrist induces a larger disturbed region and progressively reduces flipper-generated forces.
- Sand-swimming: Sand-swimming studies used a 7-link robot and simulations to examine lateral undulation inside 6 mm plastic particles, while control can create a localized frictional fluid.The frictional-fluid region extends roughly a body-width from the swimmer.
6. Heterogeneous terrestrial substrates
Robophysics extends locomotion studies to heterogeneous terrain by systematically varying obstacles, body morphology, and terrain properties. These studies connect local interactions and morphology to traversal outcomes and motivate automated terrain generation.
- Motivation: Heterogeneous environments challenge locomotion because natural terrain varies in obstacle geometry, orientation, spacing, and substrate properties.Understanding effective movement requires parameterizing and systematically creating these variations rather than probing them only by hand.
- Evolution of movement: Robophysics also links locomotion experiments to evolution by using robot models such as MuddyBot to investigate movement mechanisms in extinct terrestrial walkers.MuddyBot is morphologically inspired by skeletal reconstructions including Acanthostega.
- Body morphology and clutter: Reduced body roundness impaired traversal, while a rounded cockroach-inspired shell enabled a small legged robot to roll through densely cluttered beams.The cuboidal body failed under the same cluttered conditions.
- Automated terrain generation: SCATTER automates arbitrary terrain creation and testing using an air-fluidized bed, tilting actuators, and a jamming gripper that resets boulders and robots.The system is designed to vary sand compaction, substrate inclination, and obstacle arrangements reproducibly.
- Obstacle interactions: A robot’s scattering direction after contacting a boulder depended sensitively on the local surface inclination and initial contact position.For multiple boulders, trajectories can be statistically estimated by superposing scattering angles from individual interactions.
7. Wet terrestrial substrates
Wet terrestrial substrates introduce rheological changes and experimental-control challenges that require new robophysical preparation methods. Robotic digging and vibrational sieving demonstrate how controlled physical models can reveal mechanisms in saturated and partially wet media.
- Wet granular regimes: Increasing moisture from slightly wet to fully saturated causes dramatic changes in granular rheology and the structures that wet substrates can form.Dry grains form smooth piles, slightly wet sand can form sandcastles, and saturated media become slurry-like.
- Fully saturated substrates: RoboClam studies showed that strategically inducing local fluidization during burrowing decreases force requirements in fully saturated soil.The robot modeled the Atlantic razor clam’s two-anchor digging mechanics and used parameter variation and genetic algorithms to optimize digging.
- Experimental challenge: Partially wet granular mixtures are difficult to prepare with precise, uniform initial moisture and compaction conditions.This limits straightforward reproducible experimentation in many terrestrial soils.
- Experimental methods: Vibrational sieving creates wet granular media with controlled moisture contents and compaction for studies of fire-ant digging and lizard locomotion.The technique supports future questions about localized intrusion in wet substrates.
8. Computational tools in locomotion robophysics
Computational modeling in locomotion robophysics integrates simulations with experiments to predict forces and explain interactions across fluids, hard ground, and granular media. The section emphasizes that each substrate requires different computational approaches and has distinct limitations.
- Computational modeling complements experiments by enabling quantitative predictions and explanations of underlying locomotor mechanics.
- Modeling dry granular media, continuum descriptions: Granular locomotion can often use macroscopic empirical force relations such as resistive force theory because grain-level details are washed out in bulk responses.This simplification is challenged by fast locomotion or substrates with larger and more complex obstacles, where hard contacts and granular inertia matter.
- Simulating hard ground: Hard-ground simulation is difficult because nearly instantaneous collisions require small time steps, while compliance models can produce numerical oscillations and instabilities.Complementarity formulations impose nonpenetration constraints within the equations of motion, offering an alternative contact-modeling strategy.
- Simulating fluids: Fluid locomotion commonly uses computational fluid dynamics because Navier–Stokes equations are analytically tractable only for the simplest flows.CFD can model thrust and drag with appropriate boundary conditions, but turbulent-flow accuracy and complex geometries remain computationally costly.
- Simulating granular media, DEM: Discrete element method simulations model granular media as interacting particles, but large systems make contact-solving costs substantial.LCP solving scales as order N^2 per iteration, whereas DEM with grid-based collision detection can scale as order N in many practical problems.
- Modeling dry granular media, continuum descriptions: RFT predicts quasistatic granular locomotion across robots and animals, but it becomes insufficient for fast gaits because granular inertial effects contribute significantly to reaction forces.Extending RFT to dynamic three-dimensional motion requires measurements of granular kinematics and dynamics across high-speed interactions.
9. Geometric mechanics: a language describing the building blocks for locomotion robophysics
Geometric mechanics provides a quantitative language linking internal shape changes to body motion, and recent advances extend it from idealized models to locomotion in granular media. Empirical connections and curvature fields enable prediction and optimization of complex-medium gaits.
- Foundations of geometric mechanics: Geometric mechanics relates shape changes to body translations and rotations through a reconstruction equation and geometric phase.The framework originated in applications of classical mechanics, field theory, and quantum mechanics to microorganism locomotion.
- Open experimental need: Experimental tests of geometric locomotion theories remain few because early work focused mainly on mathematical analyses and idealized simple models.Later physical robotic implementations began testing dynamics in more realistic settings.
- Foundations of geometric mechanics: The kinematic reconstruction equation ξ = A(α) · ˙α maps shape velocity to body velocity, with the local connection encoding environmental constraints.Here ξ is body velocity, ˙α is shape velocity, and A is the local connection.
- Geometric mechanics in granular media: Empirical local connections derived by perturbing joint angles in shape space made geometric mechanics applicable to three-link swimmers in granular media.The approach uses resistive force theory simulations when complicated environmental models prevent straightforward analytical derivation.
- Geometric mechanics in granular media: Constraint Curvature Functions extend geometric methods to large-amplitude gaits by converting net displacement into area integrals that can be computed and visualized.These functions are constructed from local connection vector fields and are closely related to their curls.
- Geometric mechanics in granular media: Geometric methods predicted optimal gaits and enabled novel behaviors, including turning-in-place, for a three-link swimmer in a complex medium.A butterfly gait following CCF zerosets outperformed circular gaits, whose displacement initially increased with enclosed area before declining.
10. Conclusions and Outlook
The review presents locomotion robophysics as a physics-based approach using simplified robots, systematic experiments, and modeling to uncover principles of movement. It highlights applications to biology, soft matter, collective locomotion, and future robot design while noting the review’s focus on locomotion.
- Robophysics approach: Locomotion robophysics studies simplified laboratory robots through systematic experimentation and parameter-space exploration to discover principles of self-generated motion.The approach emphasizes understanding both successful and failed locomotion rather than focusing only on hardened field-ready devices.
- Applications to soft matter: Robophysics can advance soft matter physics by using robot–substrate interactions to reveal locomotor dynamics in sand, snow, grass, and other non-Newtonian or deformable terrains.Automated trackways such as SCATTER can measure substrate properties during robotic interaction, complementing rheometers.
- Applications to biology: Robophysical models help biology by systematically varying movements and revealing mechanical, geometric, and actuation features associated with animal locomotion.Examples include granular fluidization in sandfish swimming, flipper rigidity in sea turtle locomotion, and resonance in hopping.
- Collective locomotion: A robophysics of collective locomotion is needed because future robot teams will operate in environments more complex than the sterile settings typical of current swarming studies.Examples of intended tasks include building houses, digging tunnels, and cleaning drainpipes.
- Scope boundary: The review primarily focuses on locomotion robophysics because that is the authors’ area of expertise.Its scope is therefore narrower than the broader robophysics discipline the authors envision.
- Outlook: Robophysics links high-quality systematic experiments with theory and modeling, allowing principles developed in simplified systems to inform more robust robots and biological systems.The review frames this as a mutually beneficial intersection of engineering, physics, and biology.