Dynamics and stability of legged locomotion in the horizontal plane: a test case using insects

AquaClimber: a limbed swimming and climbing robot based on reduced order models

Many legged robots have taken insight from animals to run, jump, and climb. Very few, however, have extended the flexibility of limbs to the task of swimming. In this paper, we address the study of multi-modal limbed locomotion by extending our lateral plane reduced order dynamic model of climbing to swimming. Following this, we develop a robot, AquaClimber, which utilizes the model's locomotive style, similar to human freestyle swimming, to propel itself through fluid and to climb vertical walls, as well as transition between the two. A comparison of simulation and model results indicate that the simulation can predict how hand design, arm compliance, and driving frequency affect swimming speed and behavior. Using this reduced order model, we have successfully developed the first limbed aquatic-scansorial multi-modal robot.

Universal Features in Panarthropod Inter-Limb Coordination during Forward Walking

Terrestrial animals must often negotiate heterogeneous, varying environments. Accordingly, their locomotive strategies must adapt to a wide range of terrain, as well as to a range of speeds to accomplish different behavioral goals. Studies in Drosophila have found that inter-leg coordination patterns (ICPs) vary smoothly with walking speed, rather than switching between distinct gaits as in vertebrates (e.g., horses transitioning between trotting and galloping). Such a continuum of stepping patterns implies that separate neural controllers are not necessary for each observed ICP. Furthermore, the spectrum of Drosophila stepping patterns includes all canonical coordination patterns observed during forward walking in insects. This raises the exciting possibility that the controller in Drosophila is common to all insects, and perhaps more generally to panarthropod walkers. Here, we survey and collate data on leg kinematics and inter-leg coordination relationships during forward walking in a range of arthropod species, as well as include data from a recent behavioral investigation into the tardigrade Hypsibius exemplaris. Using this comparative dataset, we point to several functional and morphological features that are shared among panarthropods. The goal of the framework presented in this review is to emphasize the importance of comparative functional and morphological analyses in understanding the origins and diversification of walking in Panarthropoda. Introduction.

Impact of slope on dynamics of running and climbing

By combining biological studies and modeling work, the dynamics of running on horizontal terrain and climbing pure vertical surfaces have been distilled down to simple reduced order models. These models have inspired distinct control and design considerations for robots operating in each terrain. However, while the extremes are understood, the intermediate regions of moderate slopes have yet to be fully explored. In this paper, we examine how cockroaches vary their behavior as slope is changed from horizontal to vertical, with special care to examine individual leg forces when possible. The results are then compared with a lateral leg spring based (LLS, horizontal running) and Full-Goldman based (FG, vertical running) models. Overall, from the experimental data, there appears to be a continuous shift in the dynamics as slope varies, which is confirmed by similar behaviors exhibited by the LLS and FG models. Finally, by examining the stability and efficiency of the models, it is shown that there are stability limits related to the amount of energy added by the front versus rear legs. This corresponds to the shift in leg usage demonstrated by the biological experiments and may have significant implications for the design and control of multi-modal robotic systems.

Locomotion of Ants Walking up Slippery Slopes of Granular Materials

Many insects encounter locomotory difficulties in walking up sand inclines. This is masterfully exploited by some species for building traps from which prey are rarely able to escape, as the antlion and its deadly pit. The aim of this work is to tear apart the relative roles of granular material properties and slope steepness on the insect leg kinematics, gait patterns, and locomotory stability. For this, we used factorial manipulative experiments with different granular media inclines and the ant Aphaenogaster subterranea. Our results show that its locomotion is similar on granular and solid media, while for granular inclined slopes we observe a loss of stability followed by a gait pattern transition from tripod to metachronal. This implies that neither the discrete nature nor the roughness properties of sand alone are sufficient to explain the struggling of ants on sandy slopes: the interaction between sand properties and slope is key. We define an abnormality index that allows us to quantify the locomotory difficulties of insects walking up a granular incline. The probability of its occurrence reveals the local slipping of the granular media as a consequence of the pressure exerted by the ant's legs. Our findings can be extended to other models presenting locomotory difficulties for insects, such as slippery walls of urns of pitcher plants. How small arthropods walking on granular and brittle materials solve their unique stability trade-off will require a thorough understanding of the transfer of energy from leg to substrate at the particle level.

Effective locomotion at multiple stride frequencies using proprioceptive feedback on a legged microrobot

Limitations in actuation, sensing, and computation have forced small legged robots to rely on carefully tuned, mechanically mediated leg trajectories for effective locomotion. Recent advances in manufacturing, however, have enabled in such robots the ability for operation at multiple stride frequencies using multi-degree-of-freedom leg trajectories. Proprioceptive sensing and control is key to extending the capabilities of these robots to a broad range of operating conditions. In this work, we use concomitant sensing for piezoelectric actuation with a computationally efficient framework for estimation and control of leg trajectories on a quadrupedal microrobot. We demonstrate accurate position estimation (<16[Formula: see text] root-mean-square error) and control (<16[Formula: see text] root-mean-square tracking error) during locomotion across a wide range of stride frequencies (10 Hz-50 Hz). This capability enables the exploration of two bioinspired parametric leg trajectories designed to reduce leg slip and increase locomotion performance (e.g. speed, cost-of-transport (COT), etc). Using this approach, we demonstrate high performance locomotion at stride frequencies (10 Hz-30 Hz) where the robot's natural dynamics result in poor open-loop locomotion. Furthermore, we validate the biological hypotheses that inspired the trajectories and identify regions of highly dynamic locomotion, low COT (3.33), and minimal leg slippage (<10%).

Leg force interference in polypedal locomotion

The examination of gaits and gait changes has been the focus of movement physiology and legged robot engineering since the first emergence of the fields. While most examinations have focused on bipedal and quadrupedal designs, many robotic implementations rely on the higher static stability of three or more pairs of legs. Thus far, however, the effect of number of pairs of legs on locomotion dynamics has not been examined. Accordingly, the present approach aims to extend available theory to polypedal designs and examines how the number of active walking legs affects body dynamics when combined with changing duty factors and phase relations. The model shows that ground force interference of higher numbers of active pairs of walking legs can prevent effective use of bouncing gaits, such as trot, and their associated advantages, such as energy efficiency, because significantly higher degrees of leg synchronization are required. It also shows that small changes in the leg coordination pattern have a much higher impact on the center-of-mass dynamics in locomotor systems with many legs than in those with fewer legs. In this way, the model reveals coordinative constraints for specific gaits facilitating the assessment of animal locomotion and economization of robotic locomotion.

The effects of feedback on stability and maneuverability of a phase-reduced model for cockroach locomotion

In previous work, we built a neuromechanical model for insect locomotion in the horizontal plane, containing a central pattern generator, motoneurons, muscles actuating jointed legs, and rudimentary proprioceptive feedback. This was subsequently simplified to a set of 24 phase oscillators describing motoneuronal activation of agonist-antagonist muscle pairs, which facilitates analyses and enables simulations over multi-dimensional parameter spaces. Here we use the phase-reduced model to study dynamics and stability over the typical speed range of the cockroach Blaberus discoidalis, the effects of feedback on response to perturbations, strategies for turning, and a trade-off between stability and maneuverability. We also compare model behavior with experiments on lateral perturbations, changes in body mass and moment of inertia, and climbing dynamics, and we present a simple control strategy for steering using exteroceptive feedback.

Lateral undulation of the flexible spine of sprawling posture vertebrates

Sprawling posture vertebrates have a flexible spine that bends the trunk primarily in the horizontal plane during locomotion. By coordinating cyclical lateral trunk flexion and limb movements, these animals are very mobile and show extraordinary maneuverability. The dynamic and static stability displayed in complex and changing environments are highly correlated with such lateral bending patterns. The axial dynamics of their compliant body can also be critical for achieving energy-efficient locomotion at high velocities. In this paper, lateral undulation is used to characterize the bending pattern. The production of ground reaction forces (GRFs) and the related center of mass (COM) dynamics during locomotion are the fundamental mechanisms to be considered. Mainly based on research on geckos, which show unrestricted movement in three-dimensional space, we review current knowledge on the trunk flexibility and waveforms of lateral trunk movement. We investigate locomotion dynamics and mechanisms underlying the lateral undulation pattern. This paper also provides insights into the roles of this pattern in obtaining flexible and efficient walking, running, and climbing. Finally, we discuss the potential application of lateral undulation patterns to bio-inspired robotics.

The Gait Design and Trajectory Planning of a Gecko-Inspired Climbing Robot

Inspired by the dynamic gait adopted by gecko, we had put forward GPL (Gecko-inspired mechanism with a Pendular waist and Linear legs) model with one passive waist and four active linear legs. To further develop dynamic gait and reduce energy consumption of climbing robot based on the GPL model, the gait design and trajectory planning are addressed in this paper. According to kinematics and dynamics of GPL, the trot gait and continuity analysis are executed. The effects of structural parameters on the supporting forces are analyzed. Moreover, the trajectory of the waist is optimized based on system energy consumption. Finally, a bioinspired robot is developed and the prototype experiment results show that the larger body length ratio, a certain elasticity of the waist joint, and the optimized trajectory contribute to a decrease in the supporting forces and reduction in system energy consumption, especially negative forces on supporting feet. Further, the results in our experiments partly explain the reasonability of quadruped reptile's kinesiology during dynamic gait.

Speed dependent phase shifts and gait changes in cockroaches running on substrates of different slipperiness

Background

Many legged animals change gaits when increasing speed. In insects, only one gait change has been documented so far, from slow walking to fast running, which is characterised by an alternating tripod. Studies on some fast-running insects suggested a further gait change at higher running speeds. Apart from speed, insect gaits and leg co-ordination have been shown to be influenced by substrate properties, but the detailed effects of speed and substrate on gait changes are still unclear. Here we investigate high-speed locomotion and gait changes of the cockroach Nauphoeta cinerea, on two substrates of different slipperiness.

Results

Analyses of leg co-ordination and body oscillations for straight and steady escape runs revealed that at high speeds, blaberid cockroaches changed from an alternating tripod to a rather metachronal gait, which to our knowledge, has not been described before for terrestrial arthropods. Despite low duty factors, this new gait is characterised by low vertical amplitudes of the centre of mass (COM), low vertical accelerations and presumably reduced total vertical peak forces. However, lateral amplitudes and accelerations were higher in the faster gait with reduced leg synchronisation than in the tripod gait with distinct leg synchronisation.

Conclusions

Temporally distributed leg force application as resulting from metachronal leg coordination at high running speeds may be particularly useful in animals with limited capabilities for elastic energy storage within the legs, as energy efficiency can be increased without the need for elasticity in the legs. It may also facilitate locomotion on slippery surfaces, which usually reduce leg force transmission to the ground. Moreover, increased temporal overlap of the stance phases of the legs likely improves locomotion control, which might result in a higher dynamic stability.

Stability Analysis of Legged Locomotion Models by Symmetry-Factored Return Maps

We present a new stability analysis for hybrid legged locomotion systems based on the “symmetric” factorization of return maps. We apply this analysis to two-degrees-of-freedom (2DoF) and three-degrees-of-freedom (3DoF) models of the spring loaded inverted pendulum (SLIP) with different leg recirculation strategies. Despite the non-integrability of the SLIP dynamics, we obtain a necessary condition for asymptotic stability (and a sufficient condition for instability) at a fixed point, formulated as an exact algebraic expression in the physical parameters. We use this expression to characterize analytically the sensory cost and stabilizing benefit of various feedback schemes previously proposed for the 2DoF SLIP model, posited as a low-dimensional representation of running. We apply the result as well to a 3DoF SLIP model that will be treated at greater length in a companion paper as a descriptive model for the robot RHex.

Three-dimensional Translational Dynamics and Stability of Multi-legged Runners

The spring-loaded inverted pendulum (SLIP) is a simple, passively-elastic two-degree-of-freedom model for legged locomotion that describes the center-of-mass dynamics of many animal species and some legged robots. Conventionally, SLIP models employ a single support leg during stance and, while they can exhibit stable steady gaits when motions are confined to the sagittal plane, three-dimensional gaits are unstable to lateral toppling. In this paper it is shown that multiple stance legs can confer stability. Three SLIP-inspired models are studied: a passive bipedal kangaroo-hopper, an actuated insect model, and passive and actuated versions of a hexapedal robot model. The latter models both employ tripod stance phases. The sources of lateral stability are identified and, for the passive systems, analytical estimates of critical parameters are provided. Throughout, rotations are ignored and only center-of-mass translational dynamics are considered.

Running in Three Dimensions: Analysis of a Point-mass Sprung-leg Model

We analyze a simple model for running: a three-dimensional spring-loaded inverted pendulum carrying a point mass (3D-SLIP). Our formulation reduces to the sagittal plane SLIP and horizontal plane lateral leg spring (LLS) models in the appropriate limits. Using the intrinsic geometry and symmetries and appealing to the case of stiff springs, in which gravity may be neglected during stance, we derive an explicit approximate mapping describing stride-to-stride behavior. We thereby show that all left-right symmetric periodic gaits are unstable, deriving a particularly simple mapping for sagittal plane dynamics. Continuation to fixed points for the “exact” mapping confirms instability of these gaits, and we describe a simple feedback stabilization scheme for leg placement at touchdown.

The synergy between the insect-inspired claws and adhesive pads increases the attachment ability on various rough surfaces

To attach reliably on various inclined rough surfaces, many insects have evolved both claws and adhesive pads on their feet. However, the interaction between these organs still remains unclear. Here we designed an artificial attachment device, which mimics the structure and function of claws and adhesive pads, and tested it on stiff spheres of different dimensions. The results show that the attachment forces of claws decrease with an increase of the sphere radius. The forces may become very strong, when the sphere radius is smaller or comparable to the claw radius, because of the frictional self-lock. On the other hand, adhesive pads generate considerable adhesion on large sphere diameter due to large contact areas. The synergy effect between the claws and adhesive pads leads to much stronger attachment forces, if compared to the action of claw or adhesive pads independently (or even to the sum of both). The results carried out by our insect-inspired artificial attachment device clearly demonstrate why biological evolution employed two attachment organs working in concert. The results may greatly inspire the robot design, to obtain reliable attachment forces on various substrates.

Towards highly-tuned mobility in multiple domains with a dynamical legged platform

As mobile robots become more commonly utilized in everyday applications, the tasks they are given will often require them to quickly traverse unprepared and varied environments. While traditional mobile platforms may falter under such conditions, animals utilize distinct locomotion modalities such as running, jumping, or climbing, to adroitly negotiate a wide variety of challenging and changing terrains. Due to limitations including available on-board power, legged robots have struggled to match the speed of these animals, even in a single mode of transport. In this paper we experimentally investigate the synergies and trade-offs in developing a dynamical legged robot capable of both running and climbing. We utilize bio-inspired 'templates' or reduced-order models of motion to identify how the dynamics change from running to climbing and seek to identify a minimal set of robotic adjustments necessary to switch locomotion modalities. This template-based design methodology is explained and the resultant robot behavior in each domain is characterized. We show that using a trotting gait, the platform demonstrates running speeds of up to 0.67 ms(-1) on level ground and climbing speeds of up to 0.43 ms(-1)on near-vertical surfaces (and up to 0.16 ms(-1) on vertical surfaces) while exhibiting dynamical behaviors comparable to that of the inspirational models.

The leg stiffnesses animals use may improve the stability of locomotion

Despite a wide diversity of running animals, their leg stiffness normalized by animal size and weight (a relative leg stiffness) resides in a narrow range between 7 and 27. Here we determine if the stability of locomotion could be a driving factor for the tight distribution of animal leg stiffness. We simulated an established physics-based model (the actuated Spring-Loaded Inverted Pendulum model) of animal running and found that, with the same energetic cost, perturbations to locomotion are optimally corrected when relative leg stiffness is within the biologically observed range. Here we show that the stability of locomotion, in combination with energetic cost, could be a significant factor influencing the nearly universally observed animal relative leg stiffness range. The energetic cost of locomotion has been widely acknowledged as influencing the evolution of physiology and locomotion behaviors. Specifically, its potential importance for relative leg stiffness has been demonstrated. Here, we demonstrate that stability of locomotion may also be a significant factor influencing relative leg stiffness.

Designing responsive pattern generators: stable heteroclinic channel cycles for modeling and control

A striking feature of biological pattern generators is their ability to respond immediately to multisensory perturbations by modulating the dwell time at a particular phase of oscillation, which can vary force output, range of motion, or other characteristics of a physical system. Stable heteroclinic channels (SHCs) are a dynamical architecture that can provide such responsiveness to artificial devices such as robots. SHCs are composed of sequences of saddle equilibrium points, which yields exquisite sensitivity. The strength of the vector fields in the neighborhood of these equilibria determines the responsiveness to perturbations and how long trajectories dwell in the vicinity of a saddle. For SHC cycles, the addition of stochastic noise results in oscillation with a regular mean period. In this paper, we parameterize noise-driven Lotka-Volterra SHC cycles such that each saddle can be independently designed to have a desired mean sub-period. The first step in the design process is an analytic approximation, which results in mean sub-periods that are within 2% of the specified sub-period for a typical parameter set. Further, after measuring the resultant sub-periods over sufficient numbers of cycles, the magnitude of the noise can be adjusted to control the mean period with accuracy close to that of the integration step size. With these relationships, SHCs can be more easily employed in engineering and modeling applications. For applications that require smooth state transitions, this parameterization permits each state's distribution of periods to be independently specified. Moreover, for modeling context-dependent behaviors, continuously varying inputs in each state dimension can rapidly precipitate transitions to alter frequency and phase.

Level locomotion in wood ants: evidence for grounded running

In order to better understand the strategies of locomotion in small insects, we have studied continuous level locomotion of the wood ant species Formica polyctena. We determined the three-dimensional centre of mass kinematics during the gait cycle and recorded the ground reaction forces of single legs utilising a self-developed test site. Our findings show that the animals used the same gait dynamics across a wide speed range without dissolving the tripodal stride pattern. To achieve higher velocities, the ants proportionally increased stride length and stepping frequency. The centre of mass energetics indicated a bouncing gait, in which horizontal kinetic and gravitational potential energy fluctuated in close phase. We determined a high degree of compliance especially in the front legs, as the effective leg length was nearly halved during the contact phase. This leads to only small vertical oscillations of the body, which are important in maintaining ground contact. Bouncing gaits without aerial phases seem to be a common strategy in small runners and can be sufficiently described by the bipedal spring-loaded inverted pendulum model. Thus, with our results, we provide evidence that wood ants perform 'grounded running'.

Environmental interaction influences muscle activation strategy during sand-swimming in the sandfish lizard Scincus scincus

Animals like the sandfish lizard (Scincus scincus) that live in desert sand locomote on and within a granular medium whose resistance to intrusion is dominated by frictional forces. Recent kinematic studies revealed that the sandfish utilizes a wave of body undulation during swimming. Models predict that a particular combination of wave amplitude and wavelength yields maximum speed for a given frequency, and experiments have suggested that the sandfish targets this kinematic waveform. To investigate the neuromechanical strategy of the sandfish during walking, burial and swimming, here we use high-speed X-ray and visible light imaging with synchronized electromyogram (EMG) recordings of epaxial muscle activity. While moving on the surface, body undulation was not observed and EMG showed no muscle activation. During subsurface sand-swimming, EMG revealed an anterior-to-posterior traveling wave of muscle activation which traveled faster than the kinematic wave. Muscle activation intensity increased as the animal swam deeper into the material but was insensitive to undulation frequency. These findings were in accord with empirical force measurements, which showed that resistance force increased with depth but was independent of speed. The change in EMG intensity with depth indicates that the sandfish targets a kinematic waveform (a template) that models predict maximizes swimming speed and minimizes the mechanical cost of transport as the animal descends into granular media. The differences in the EMG pattern compared with EMG of undulatory swimmers in fluids can be attributed to the friction-dominated intrusion forces of granular media.

Instantaneous kinematic phase reflects neuromechanical response to lateral perturbations of running cockroaches

Instantaneous kinematic phase calculation allows the development of reduced-order oscillator models useful in generating hypotheses of neuromechanical control. When perturbed, changes in instantaneous kinematic phase and frequency of rhythmic movements can provide details of movement and evidence for neural feedback to a system-level neural oscillator with a time resolution not possible with traditional approaches. We elicited an escape response in cockroaches (Blaberus discoidalis) that ran onto a movable cart accelerated laterally with respect to the animals' motion causing a perturbation. The specific impulse imposed on animals (0.50 [Formula: see text] 0.04 m s[Formula: see text]; mean, SD) was nearly twice their forward speed (0.25 [Formula: see text] 0.06 m s[Formula: see text]. Instantaneous residual phase computed from kinematic phase remained constant for 110 ms after the onset of perturbation, but then decreased representing a decrease in stride frequency. Results from direct muscle action potential recordings supported kinematic phase results in showing that recovery begins with self-stabilizing mechanical feedback followed by neural feedback to an abstracted neural oscillator or central pattern generator. Trials fell into two classes of forward velocity changes, while exhibiting statistically indistinguishable frequency changes. Animals pulled away from the side with front and hind legs of the tripod in stance recovered heading within 300 ms, whereas animals that only had a middle leg of the tripod resisting the pull did not recover within this period. Animals with eight or more legs might be more robust to lateral perturbations than hexapods.

Leg-adjustment strategies for stable running in three dimensions

The dynamics of the center of mass (CoM) in the sagittal plane in humans and animals during running is well described by the spring-loaded inverted pendulum (SLIP). With appropriate parameters, SLIP running patterns are stable, and these models can recover from perturbations without the need for corrective strategies, such as the application of additional forces. Rather, it is sufficient to adjust the leg to a fixed angle relative to the ground. In this work, we consider the extension of the SLIP to three dimensions (3D SLIP) and investigate feed-forward strategies for leg adjustment during the flight phase. As in the SLIP model, the leg is placed at a fixed angle. We extend the scope of possible reference axes from only fixed horizontal and vertical axes to include the CoM velocity vector as a movement-related reference, resulting in six leg-adjustment strategies. Only leg-adjustment strategies that include the CoM velocity vector produced stable running and large parameter domains of stability. The ability of the model to recover from perturbations along the direction of motion (directional stability) depended on the strategy for lateral leg adjustment. Specifically, asymptotic and neutral directional stability was observed for strategies based on the global reference axis and the velocity vector, respectively. Additional features of velocity-based leg adjustment are running at arbitrary low speed (kinetic energy) and the emergence of large domains of stable 3D running that are smoothly transferred to 2D SLIP stability and even to 1D SLIP hopping. One of the additional leg-adjustment strategies represented a large convex region of parameters where stable and robust hopping and running patterns exist. Therefore, this strategy is a promising candidate for implementation into engineering applications, such as robots, for instance. In a preliminary comparison, the model predictions were in good agreement with the experimental data, suggesting that the 3D SLIP is an appropriate model to describe human running in three dimensions. The prediction of stable running based on movement-related leg-adjustment strategies indicates that both humans and robots may not require external targets directing the movement to run in three dimensions based on compliant leg function. This new movement-based reference enables the control of 3D running because leg adjustment is less sensitive and gait stability is separated from directional stability.

Terrestrial locomotion in arachnids

In this review, we assess the current state of knowledge on terrestrial locomotion in Arachnida. Arachnids represent a single diverse (>100,000 species) clade containing well-defined subgroups (at both the order and subordinal levels) that vary morphologically around a basic body plan, yet exhibit highly disparate limb usage, running performance, and tarsal attachment mechanisms. Spiders (Araneae), scorpions (Scorpiones), and harvestmen (Opiliones) have received the most attention in the literature, while some orders have never been subject to rigorous mechanical characterization. Most well-characterized taxa move with gaits analogous to the alternating tripod gaits that characterize fast-moving Insecta - alternating tetrapods or alternating tripods (when one pair of legs is lifted from the ground for some other function). However, between taxa, there is considerable variation in the regularity of phasing between legs. Both large and small spiders appear to show a large amount of variation in the distribution of foot-ground contact, even between consecutive step-cycles of a single run. Mechanisms for attachment to vertical surfaces also vary, and may depend on tufts of adhesive hairs, fluid adhesives, silks, or a combination of these. We conclude that Arachnida, particularly with improvements in microelectronic force sensing technology, can serve as a powerful study system for understanding the kinematics, dynamics, and ecological correlates of sprawled-posture locomotion.

A bioinspired dynamical vertical climbing robot

This paper describes the inspiration for, design, analysis, and implementation of, and experimentation with the first dynamical vertical climbing robot. Biologists have proposed a pendulous climbing model that abstracts remarkable similarities in dynamical wall scaling behavior exhibited by radically different animal species. We numerically study a version of that pendulous climbing template dynamically scaled for applicability to utilitarian payloads with conventional electronics and actuation. This simulation study reveals that the incorporation of passive compliance can compensate for the scaled model’s poorer power density and scale disadvantages relative to biology. However, the introduction of additional dynamical elements raises new concerns about stability regarding both the power stroke and limb coordination schemes that we allay via mathematical analysis of further simplified models. Combining these numerical and analytical insights into a series of design prototypes, we document the correspondence of the various models to the scaled platforms and report that our final prototype climbs dynamically at vertical speeds up to 0.67 m/s (1.5 body-lengths per second, in rough agreement with our models’ predictions).

Experiments and theory of undulatory locomotion in a simple structured medium

Undulatory locomotion of micro-organisms through geometrically complex, fluidic environments is ubiquitous in nature and requires the organism to negotiate both hydrodynamic effects and geometrical constraints. To understand locomotion through such media, we experimentally investigate swimming of the nematode Caenorhabditis elegans through fluid-filled arrays of micro-pillars and conduct numerical simulations based on a mechanical model of the worm that incorporates hydrodynamic and contact interactions with the lattice. We show that the nematode's path, speed and gait are significantly altered by the presence of the obstacles and depend strongly on lattice spacing. These changes and their dependence on lattice spacing are captured, both qualitatively and quantitatively, by our purely mechanical model. Using the model, we demonstrate that purely mechanical interactions between the swimmer and obstacles can produce complex trajectories, gait changes and velocity fluctuations, yielding some of the life-like dynamics exhibited by the real nematode. Our results show that mechanics, rather than biological sensing and behaviour, can explain some of the observed changes in the worm's locomotory dynamics.

Multi-functional foot use during running in the zebra-tailed lizard (Callisaurus draconoides)

A diversity of animals that run on solid, level, flat, non-slip surfaces appear to bounce on their legs; elastic elements in the limbs can store and return energy during each step. The mechanics and energetics of running in natural terrain, particularly on surfaces that can yield and flow under stress, is less understood. The zebra-tailed lizard (Callisaurus draconoides), a small desert generalist with a large, elongate, tendinous hind foot, runs rapidly across a variety of natural substrates. We use high speed video to obtain detailed three-dimensional running kinematics on solid and granular surfaces to reveal how leg, foot, and substrate mechanics contribute to its high locomotor performance. Running at ~10 body length/s (~1 m/s), the center of mass oscillates like a spring-mass system on both substrates, with only 15% reduction in stride length on the granular surface. On the solid surface, a strut-spring model of the hind limb reveals that the hind foot saves about 40% of the mechanical work needed per step, significant for the lizard's small size. On the granular surface, a penetration force model and hypothesized subsurface foot rotation indicates that the hind foot paddles through fluidized granular medium, and that the energy lost during irreversible deformation of the substrate does not differ from the reduction in the mechanical energy of the center of mass. The upper hind leg muscles must perform three times as much mechanical work on the granular surface as on the solid surface to compensate for the greater energy lost within the foot and to the substrate.

Spikes alone do not behavior make: why neuroscience needs biomechanics

Neural circuits do not function in isolation; they interact with the physical world, accepting sensory inputs and producing outputs via muscles. Since both these pathways are constrained by physics, the activity of neural circuits can only be understood by considering biomechanics of muscles, bodies, and the exterior world. We discuss how animal bodies have natural stable motions that require relatively little activation or control from the nervous system. The nervous system can substantially alter these motions, by subtly changing mechanical properties such as body or leg stiffness. Mechanics can also provide robustness to perturbations without sensory reflexes. By considering a complete neuromechanical system, neuroscientists and biomechanicians together can provide a more integrated view of neural circuitry and behavior.

Kinematic and behavioral evidence for a distinction between trotting and ambling gaits in the cockroach Blaberus discoidalis

Earlier observations had suggested that cockroaches might show multiple patterns of leg coordination, or gaits, but these were not followed by detailed behavioral or kinematic measurements that would allow a definite conclusion. We measured the walking speeds of cockroaches exploring a large arena and found that the body movements tended to cluster at one of two preferred speeds, either very slow (<10 cm s(-1)) or fairly fast (∼30 cm s(-1)). To highlight the neural control of walking leg movements, we experimentally reduced the mechanical coupling among the various legs by tethering the animals and allowing them to walk in place on a lightly oiled glass plate. Under these conditions, the rate of stepping was bimodal, clustering at fast and slow speeds. We next used high-speed videos to extract three-dimensional limb and joint kinematics for each segment of all six legs. The angular excursions and three-dimensional motions of the leg joints over the course of a stride were variable, but had different distributions in each gait. The change in gait occurs at a Froude number of ∼0.4, a speed scale at which a wide variety of animals show a transition between walking and trotting. We conclude that cockroaches do have multiple gaits, with corresponding implications for the collection and interpretation of data on the neural control of locomotion.

Running over unknown rough terrain with a one-legged planar robot

The ability to traverse unknown, rough terrain is an advantage that legged locomoters have over their wheeled counterparts. However, due to the complexity of multi-legged systems, research in legged robotics has not yet been able to reproduce the agility found in the animal kingdom. In an effort to reduce the complexity of the problem, researchers have developed single-legged models to gain insight into the fundamental dynamics of legged running. Inspired by studies of animal locomotion, researchers have proposed numerous control strategies to achieve stable, one-legged running over unknown, rough terrain. One such control strategy incorporates energy variations into the system during the stance phase by changing the force-free leg length as a sinusoidal function of time. In this research, a one-legged planar robot capable of implementing this and other state-of-the-art control strategies was designed and built. Both simulated and experimental results were used to determine and compare the stability of the proposed controllers as the robot was subjected to unknown drop and raised step perturbations equal to 25% of the nominal leg length. This study illustrates the relative advantages of utilizing a minimal-sensing, active energy removal control scheme to stabilize running over rough terrain.

A single muscle's multifunctional control potential of body dynamics for postural control and running

A neuromechanical approach to control requires understanding how mechanics alters the potential of neural feedback to control body dynamics. Here, we rewrite activation of individual motor units of a behaving animal to mimic the effects of neural feedback without concomitant changes in other muscles. We target a putative control muscle in the cockroach, Blaberus discoidalis (L.), and simultaneously capture limb and body dynamics through high-speed videography and a micro-accelerometer backpack. We test four neuromechanical control hypotheses. We supported the hypothesis that mechanics linearly translates neural feedback into accelerations and rotations during static postural control. However, during running, the same neural feedback produced a nonlinear acceleration control potential restricted to the vertical plane. Using this, we reject the hypothesis from previous work that this muscle acts primarily to absorb energy from the body. The conversion of the control potential is paralleled by nonlinear changes in limb kinematics, supporting the hypothesis that significant mechanical feedback filters the graded neural feedback for running control. Finally, we insert the same neural feedback signal but at different phases in the dynamics. In this context, mechanical feedback enables turning by changing the timing and direction of the accelerations produced by the graded neural feedback.

A phase-reduced neuro-mechanical model for insect locomotion: feed-forward stability and proprioceptive feedback

In earlier work, we have developed an integrated model for insect locomotion that includes a central pattern generator (CPG), nonlinear muscles, hexapedal geometry and a representative proprioceptive sensory pathway. Here, we employ phase reduction and averaging theory to replace 264 ordinary differential equations (ODEs), describing bursting neurons in the CPG, their synaptic connections to motoneurons, muscle activation dynamics and sensory neurons, with 24 one-dimensional phase oscillators that describe motoneuronal activation of agonist-antagonist muscle pairs driving the jointed legs. Reflexive feedback is represented by stereotypical spike trains with rates proportional to joint torques, which change phase relationships among the motoneuronal oscillators. Restriction to the horizontal plane, neglect of leg mass and use of Hill-type muscle models yield a biomechanical body-limb system with only three degrees of freedom, and the resulting hybrid dynamical system involves 30 ODEs: reduction by an order of magnitude. We show that this reduced model captures the dynamics of unperturbed gaits and the effects of an impulsive perturbation as accurately as the original one. Moreover, the phase response and coupling functions provide an improved understanding of reflexive feedback mechanisms.

Insects running on elastic surfaces

In nature, cockroaches run rapidly over complex terrain such as leaf litter. These substrates are rarely rigid, and are frequently very compliant. Whether and how compliant surfaces change the dynamics of rapid insect locomotion has not been investigated to date largely due to experimental limitations. We tested the hypothesis that a running insect can maintain average forward speed over an extremely soft elastic surface (10 N m(-1)) equal to 2/3 of its virtual leg stiffness (15 N m(-1)). Cockroaches Blaberus discoidalis were able to maintain forward speed (mean +/- s.e.m., 37.2+/-0.6 cm s(-1) rigid surface versus 38.0+/-0.7 cm s(-1) elastic surface; repeated-measures ANOVA, P=0.45). Step frequency was unchanged (24.5+/-0.6 steps s(-1) rigid surface versus 24.7+/-0.4 steps s(-1) elastic surface; P=0.54). To uncover the mechanism, we measured the animal's centre of mass (COM) dynamics using a novel accelerometer backpack, attached very near the COM. Vertical acceleration of the COM on the elastic surface had a smaller peak-to-peak amplitude (11.50+/-0.33 m s(-2), rigid versus 7.7+/-0.14 m s(-2), elastic; P=0.04). The observed change in COM acceleration over an elastic surface required no change in effective stiffness when duty factor and ground stiffness were taken into account. Lowering of the COM towards the elastic surface caused the swing legs to land earlier, increasing the period of double support. A feedforward control model was consistent with the experimental results and provided one plausible, simple explanation of the mechanism.

Dynamics and kinematics of ant locomotion: do wood ants climb on level surfaces?

The biomechanics of running in small animals have remained poorly characterized because of the difficulty of recording three-dimensional ground reaction forces. Available techniques limit investigations to animals with a body mass above 1 g. Here we present, for the first time, single-leg ground reaction forces of ants (body mass 10 mg), measured with a custom-built miniature force plate. We investigated forces and high-speed kinematics for straight level runs (average speed: 8.4 cm s(-1)) of Formica polyctena workers. The major finding was that the time course of ground reaction forces strongly differed from previous observations of larger insects. Maximum vertical force was reached during the first third of the tripod contact phase. During this period the body was decelerated predominantly by the front legs. Subsequently, the front legs pulled and accelerated the body. This 'climbing' type of stride may be useful on the bumpy and unstable substrates that the animals face in their natural habitats, and may therefore also occur on level ground. Propulsive forces were generated predominantly by the front and hind legs. Dragging of the gaster on the substrate resulted in a breaking momentum, which was compensated by the legs. Future investigations will reveal, whether the identified pattern is due to specialization.

Leg recirculation in horizontal plane locomotion

A protocol prescribing leg motion during the swing phase is developed for the planar lateral leg spring model of locomotion. Inspired by experimental observations regarding insect leg function when running over rough terrain, the protocol prescribes the angular velocity of the swing-leg relative to the body in a feedforward manner, yielding natural variations in the leg touch-down angle in response to perturbations away from a periodic orbit. Analysis of the reduced order model reveals that periodic gait stability and robustness to external perturbations depends strongly upon the angular velocity of the leg at touch-down. While the leg angular velocity at touch-down provides control over gait stability and can be chosen to stabilize unstable gaits, the resulting basin of stability is much smaller than that observed for the original lateral leg spring model with a fixed leg touch-down angle. Comparisons to experimental leg angular velocity data for running cockroaches reveal that while the proposed protocol is qualitatively correct, smaller leg angular accelerations occur during the second half of the swing phase. Modifications made to the recirculation protocol to better match experimental observations yield large improvements in the basin of stability.

Dynamics and stability of lateral plane locomotion on inclines

An actuated, lateral leg spring model is developed to investigate lateral plane locomotion dynamics and stability on inclines. A single actuation input, the force-free leg length, is varied in a feedforward fashion to explicitly and implicitly match prescribed lateral and fore-aft force profiles, respectively. Forward dynamic simulations incorporating the prescribed leg actuation are employed to identify periodic orbits for gaits in which the leg acts to either push the body away from or pull the body towards the foot placement point. Gait stability and robustness to external perturbation are found to vary significantly as a function of slope and velocity for each type of leg function. Results of these analyses suggest that the switch in leg function from pushing to pulling is governed by gait robustness, and occurs at increasing inclines for increasing velocities.

A model for insect locomotion in the horizontal plane: feedforward activation of fast muscles, stability, and robustness

We develop a neuromechanical model for running insects that includes a simplified hexapedal leg geometry with agonist-antagonist muscle pairs actuating each leg joint. Restricting to dynamics in the horizontal plane and neglecting leg masses, we reduce the model to three degrees of freedom describing translational and yawing motions of the body. Muscles are driven by stylized action potentials characteristic of fast motoneurons, and modeled using an activation function and nonlinear length and shortening velocity dependence. Parameter values are based on measurements from depressor muscles and observations of kinematics and dynamics of the cockroach Blaberus discoidalis; in particular, motoneuronal inputs and muscle force levels are chosen to approximately achieve joint torques that are consistent with measured ground reaction forces. We show that the model has stable double-tripod gaits over the animal's speed range, that its dynamics at preferred speeds matches those observed, and that it maintains stable gaits, with low frequency yaw deviations, when subject to random perturbations in foot touchdown and lift-off timing and action potential input timing. We explain this in terms of the low-dimensional dynamics.

Maneuvers during legged locomotion

Maneuverability is essential for locomotion. For animals in the environment, maneuverability is directly related to survival. For humans, maneuvers such as turning are associated with increased risk for injury, either directly through tissue loading or indirectly through destabilization. Consequently, understanding the mechanics and motor control of maneuverability is a critical part of locomotion research. We briefly review the literature on maneuvering during locomotion with a focus on turning in bipeds. Walking turns can use one of several different strategies. Anticipation can be important to adjust kinematics and dynamics for smooth and stable maneuvers. During running, turns may be substantially constrained by the requirement for body orientation to match movement direction at the end of a turn. A simple mathematical model based on the requirement for rotation to match direction can describe leg forces used by bipeds (humans and ostriches). During running turns, both humans and ostriches control body rotation by generating fore-aft forces. However, whereas humans must generate large braking forces to prevent body over-rotation, ostriches do not. For ostriches, generating the lateral forces necessary to change movement direction results in appropriate body rotation. Although ostriches required smaller braking forces due in part to increased rotational inertia relative to body mass, other movement parameters also played a role. Turning performance resulted from the coordinated behavior of an integrated biomechanical system. Results from preliminary experiments on horizontal-plane stabilization support the hypothesis that controlling body rotation is an important aspect of stable maneuvers. In humans, body orientation relative to movement direction is rapidly stabilized during running turns within the minimum of two steps theoretically required to complete analogous maneuvers. During straight running and cutting turns, humans exhibit spring-mass behavior in the horizontal plane. Changes in the horizontal projection of leg length were linearly related to changes in horizontal-plane leg forces. Consequently, the passive dynamic stabilization associated with spring-mass behavior may contribute to stability during maneuvers in bipeds. Understanding the mechanics of maneuverability will be important for understanding the motor control of maneuvers and also potentially be useful for understanding stability.

Neuromechanical models for insect locomotion: Stability, maneuverability, and proprioceptive feedback

We describe a hierarchy of models for legged locomotion, emphasizing relationships among feedforward (preflexive) stability, maneuverability, and reflexive feedback. We focus on a hexapedal geometry representative of insect locomotion in the ground plane that includes a neural central pattern generator circuit, nonlinear muscles, and a representative proprioceptive sensory pathway. Although these components of the model are rather complex, neglect of leg mass yields a neuromechanical system with only three degrees of freedom, and numerical simulations coupled with a Poincaré map analysis shows that the feedforward dynamics is strongly stable, apart from one relatively slow mode and a neutral mode in body yaw angle. These modes moderate high frequency perturbations, producing slow heading changes that can be corrected by a stride-to-stride steering strategy. We show that the model's response to a lateral impulsive perturbation closely matches that of a cockroach subject to a similar impulse. We also describe preliminary studies of proprioceptive leg force feedback, showing how a reflexive pathway can reinforce the preflexive stability inherent in the system.

Improving horizontal plane locomotion via leg angle control

The lateral leg spring model has been shown to accurately represent horizontal plane locomotion characteristics of sprawled posture insects such as the cockroach Blaberus discoidalis. While passively stable periodic gaits result from employing a constant leg touch-down angle for this model, utilizing a similar protocol for a point mass model of locomotion in three dimensions produces only unstable periodic gaits. In this work, we return to the horizontal plane model and develop a simple control law that prescribes variations in the leg touch-down angle in response to external perturbations. The resulting control law applies control once per stance phase, at the instant of leg touch-down, and depends upon previous leg angles defined in the body reference frame. As a result, our control action is consistent with the neural activity evidenced by B. discoidalis during locomotion over flat and rough terrain, and utilizes variables easily sensed by insect mechanoreceptors. Application of control in the lateral leg spring model is shown to improve stability of periodic gaits, enable stabilization of previously unstable periodic gaits, and maintain or improve the basin of stability of periodic gaits. The magnitude of leg touch-down angle variations utilized during stabilization appear consistent with the natural variations evidenced by single legs during locomotion over flat terrain.

Running on uneven ground: leg adjustment to vertical steps and self-stability

Human running is characterized by comparably simple whole-body dynamics. These dynamics can be modelled with a point mass bouncing on a spring leg. Theoretical studies using such spring–mass models predict that running can be self-stable. In simulations, this self-stability allows for running on uneven ground without paying attention to the ground irregularities. Whether humans actually use this property of the mechanical system in such an irregular environment is, however, unclear. One way to approach this question is to study how the leg stiffness in stance and the leg orientation in flight are changed in response to ground perturbations. Here, for 11 human subjects we studied two consecutive contacts during running on uneven ground with a force plate of adjustable height (step of +5, +10 and +15 cm). We found that runners adjust their leg stiffness to the height of a vertical step. The adjustment is characterized by a 9% increase in leg stiffness in preparation for the perturbation and by a systematic decrease in proportion to the step height. At the highest vertical step (+15 cm), leg stiffness was reduced by about 26%. We also observed that the angle of attack decreased from 68 deg. to 62 deg. with increasing ground height. These leg adjustments are in accordance with the predictions of a stable spring–mass system. Furthermore, we could describe the identified leg forces and leg compressions with a simple spring–mass simulation for a given body mass, leg stiffness, angle of attack and initial conditions. We compared the experimental findings with the self-stabilizing properties of the spring–mass model, and discuss how humans use a combination of strategies that include purely mechanical self-stabilization and active neuromuscular control. Finally, beyond self-stability, we suggest that control may apply to smooth centre of mass kinematics.

Neuromechanical response of musculo-skeletal structures in cockroaches during rapid running on rough terrain

A musculo-skeletal structure can stabilize rapid locomotion using neural and/or mechanical feedback. Neural feedback results in an altered feedforward activation pattern, whereas mechanical feedback using visco-elastic structures does not require a change in the neural motor code. We selected musculo-skeletal structures in the cockroach (Blaberus discoidalis)because their single motor neuron innervation allows the simplest possible characterization of activation. We ran cockroaches over a track with randomized blocks of heights up to three times the animal's `hip' (1.5 cm),while recording muscle action potentials (MAPs) from a set of putative control musculo-skeletal structures (femoral extensors 178 and 179). Animals experienced significant perturbations in body pitch, roll and yaw, but reduced speed by less than 20%. Surprisingly, we discovered no significant difference in the distribution of the number of MAPs, the interspike interval, burst phase or interburst period between flat and rough terrain trials. During a few very large perturbations or when a single leg failed to make contact throughout stance, neural feedback was detectable as a phase shift of the central rhythm and alteration of MAP number. System level responses of appendages were consistent with a dominant role of mechanical feedback. Duty factors and gait phases did not change for cockroaches running on flat versus rough terrain. Cockroaches did not use a follow-the-leader gait requiring compensatory corrections on a step-by-step basis. Arthropods appear to simplify control on rough terrain by rapid running that uses kinetic energy to bridge gaps between footholds and distributed mechanical feedback to stabilize the body.

A hexapedal jointed-leg model for insect locomotion in the horizontal plane

We develop a simple model for insect locomotion in the horizontal (ground) plane. As in earlier work by Seipel et al. (Biol Cybern 91(0):76-90, 2004) we employ six actuated legs that also contain passive springs, but the legs, with "hip" and 'knee' joints, better represent insect morphology. Actuation is provided via preferred angle inputs at each joint, corresponding to zero torques in the hip and knee springs. The inputs are determined from estimates of foot forces in the cockroach Blaberus discoidalis via an inverse problem. The head-thorax-body is modeled as a single rigid body, and leg masses, inertia and joint dissipation are ignored. The resulting three degree-of-freedom dynamical system, subject to feedforward joint inputs, exhibits stable periodic gaits that compare well with observations over the insect's typical speed range. The model's response to impulsive perturbations also matches that of freely-running cockroaches (Jindrich and Full, J Exp Biol 205:2803-2823, 2002), and stability is maintained in the face of random foot touchdowns representative of real insects. We believe that this model will allow incorporation of realistic muscle models driven by a central pattern generator in place of the joint actuators, and that it will ultimately permit the study of proprioceptive feedback pathways involving leg force and joint angle sensing.