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.     

Empirical Study on Shapes of the Foot Pad and Walking Gaits for Water-Running Robots

Recently, various kinds ofbiomimetic robots have been studied. Among these biomimetic robots, water-running robots that mimic the characteristics of basilisk lizards have received much attention. However, studies on the performance with respect to different geometric parameters and gaits have been lacking. To run on the surface of water, a water-running robot needs suffi- cient force with high stability to stay above the water. We experimentally measured the performance of the foot pads with different geometric parameters and with various gaits. We measured and analyzed the forces in the vertical direction and rolling angles of five different foot pad shapes： a circular shape, square shape, half-spherical shape, open half-cylinder shape, and closed half-cylinder shape. Additionally, the rolling stabilities of three kinds of gaits： biped, trotting, and tripod, were also empirically analyzed. The results of this research can be used as a guideline to design a stable water-running robot.     

A study on continuous follow-the-leader (FTL) gaits: an effective walking algorithm over rough terrain

The follow-the-leader (FTL) gait is an effective walking algorithm for a legged system to traverse a rough terrain. In an FTL gait, all the legs simply place at the footprints made by the legs ahead of them. By this way the demand on foothold selection is significantly reduced. A special category of FTL gaits, called continuous FTL gaits, provide a smooth body motion during walking and enable the legged system to reach a higher speed. In this paper, a comprehensive study of continuous FTL gaits is presented. The equations for two types of continuous FTL gaits are formulated. The stability of these continuous FTL gaits is studied analytically and verified numerically. Strategies of forbidden area avoidance and special methods of large foot adjustment are introduced. The motion resulting from the use of these strategies and methods is simulated and checked using computer graphics.     

Regression techniques for the prediction of lower limb kinematics

This work presents a novel and extensive investigation of mathematical regression techniques, for the prediction of laboratory-type kinematic measurements during human gait, from wearable measurement devices, such as gyroscopes and accelerometers. Specifically, we examine the hypothesis of predicting the segmental angles of the legs (left and right foot, shank and thighs), from rotational foot velocities and translational foot accelerations. This first investigation is based on kinematic data emulated from motion-capture laboratory equipment. We employ eight established regression algorithms with different properties, ranging from linear methods and neural networks with polynomial support and expanded nonlinearities, to radial basis functions, nearest neighbors and kernel density methods. Data from five gait cycles of eight subjects are used to perform both inter-subject and intra-subject assessments of the prediction capabilities of each algorithm, using cross-validation resampling methods. Regarding the algorithmic suitability to gait prediction, results strongly indicate that nonparametric methods, such as nearest neighbors and kernel density based, are particularly advantageous. Numerical results show high average prediction accuracy (rho = 0.98/0.99, RMS = 5.63 degrees/2.30 degrees, MAD = 4.43 degrees/1.52 degrees for inter/intra-subject testing). The presented work provides a promising and motivating investigation on the feasibility of cost-effective wearable devices used to acquire large volumes of data that are currently collected only from complex laboratory environments.     

Gait selection and the ontogeny of quadrupedal walking in squirrel monkeys (Saimiri boliviensis)

Locomotor researchers have long known that adult primates employ a unique footfall sequence during walking. Most mammals use lateral sequence (LS) gaits, in which hind foot touchdowns are followed by ipsilateral forefoot touchdowns. In contrast, most quadrupedal primates use diagonal sequence (DS) gaits, in which hind foot touchdowns are followed by contralateral forefoot touchdowns. However, gait selection in immature primates is more variable, with infants and juveniles frequently using LS gaits either exclusively or in addition to DS gaits. I explored the developmental bases for this phenomenon by examining the ontogeny of gait selection in juvenile squirrel monkeys walking on flat and simulated arboreal substrates (i.e., a raised pole). Although DS gaits predominated throughout development, the juvenile squirrel monkeys nonetheless utilized LS gaits in one-third of the ground strides and in one-sixth of pole strides. Multiple logistic regression analyses showed that gait selection within the juvenile squirrel monkey sample was not significantly associated with either age or body mass per se, arguing against the oft-cited argument that general neuromuscular maturation is responsible for ontogenetic changes in preferred footfall sequence. Rather, lower level biomechanical variables, specifically the position of the whole-body center of mass and the potential for interference between ipsilateral fore and hindlimbs, best explained variation in footfall patterns. Overall, results demonstrate the promise of developmental studies of growth and locomotor development to serve as "natural laboratories" in which to explore how variability in morphology is, or is not, associated with variability in locomotor behavior.     

Gaits of monkeys and horses: A methodological critique

In a previous paper it was shown that the traditional methodology used to analyze quadrupedal gaits is inadequate when applied to the gaits of monkeys. The traditional methods were modified for use with the monkey sample (Prost, '65). Was the modification a general improvement of the traditional approach or was it dictated by peculiarities of monkey locomotion? The traditional analysis receives its support primarily from its usefulness in handling horse gaits. If it can be shown that the modified methods improve the analysis for horses, then it can be claimed that the system is not a special case designed uniquely for use with monkeys. Motion pictures were taken of an American Saddle Bred, five-gaited horse performing several classic gaits. The films were analyzed using the modified methods developed to analyze monkey gaits. The analysis was superior to the traditional approach. All of the organizing principles which structure horse gaits were found to be identical to those which structure monkey gaits.     

Effects of a Foot Placement Constraint on Use of Motor Equivalence during Human Hopping

Humans can robustly locomote over complex terrains even while simultaneously attending to other tasks such as accurate foot placement on the ground. We investigated whether subjects would exploit motor redundancy across the joints of the leg to stabilize overall limb kinematics when presented with a hopping task that constrained foot placement position. Subjects hopped in place on one leg (2.2 Hz) while having to place their foot into one of three target sizes upon landing (0.250, 0.063, 0.010 m²). As takeoff and landing angles are critical to this task performance, we hypothesized smaller target sizes would increase the need to stabilize (i.e., make more consistent) the leg orientation through motor equivalent combinations of segment angles. As it was not critical to the targeting task, we hypothesized no changes for leg length stabilization across target size. With smaller target sizes, we saw total segment angle variance increase due to greater signal-dependent noise associated with an increased activation of leg extensor muscles (medial and lateral gastrocnemius, vastus medialis, vastus lateralis and rectus femoris). At smaller target sizes, more segment angle variance was aligned to kinematic deviations with the goal of maintaining leg orientation trajectory. We also observed a decrease in the variance structure for stabilizing leg length at the smallest target conditions. This trade-off effect is explained by the nearly orthogonal relationship between the two goal-equivalent manifolds for leg length vs. leg orientation stabilization. Our results suggest humans increasingly rely on kinematic redundancy in their legs to achieve robust, consistent locomotion when faced with novel conditions that constrain performance requirements. These principles may generalize to other human locomotor gaits and provide important insights into the control of the legs during human walking and running.     

Dynamic motion planning of 3D human locomotion using gradient-based optimization

Since humans can walk with an infinite variety of postures and limb movements, there is no unique solution to the modeling problem to predict human gait motions. Accordingly, we test herein the hypothesis that the redundancy of human walking mechanisms makes solving for human joint profiles and force time histories an indeterminate problem best solved by inverse dynamics and optimization methods. A new optimization-based human-modeling framework is thus described for predicting three-dimensional human gait motions on level and inclined planes. The basic unknowns in the framework are the joint motion time histories of a 25-degree-of-freedom human model and its six global degrees of freedom. The joint motion histories are calculated by minimizing an objective function such as deviation of the trunk from upright posture that relates to the human model's performance. A variety of important constraints are imposed on the optimization problem, including (1) satisfaction of dynamic equilibrium equations by requiring the model's zero moment point (ZMP) to lie within the instantaneous geometrical base of support, (2) foot collision avoidance, (3) limits on ground-foot friction, and (4) vanishing yawing moment. Analytical forms of objective and constraint functions are presented and discussed for the proposed human-modeling framework in which the resulting optimization problems are solved using gradient-based mathematical programming techniques. When the framework is applied to the modeling of bipedal locomotion on level and inclined planes, acyclic human walking motions that are smooth and realistic as opposed to less natural robotic motions are obtained. The aspects of the modeling framework requiring further investigation and refinement, as well as potential applications of the framework in biomechanics, are discussed.     

Single species models with logistic growth and dissymmetric impulse dispersal

In this paper, two classes of single-species models with logistic growth and impulse dispersal (or migration) are studied: one model class describes dissymmetric impulsive bi-directional dispersal between two heterogeneous patches; and the other presents a new way of characterizing the aggregate migration of a natural population between two heterogeneous habitat patches, which alternates in direction periodically. In this theoretical study, some very general, weak conditions for the permanence, extinction of these systems, existence, uniqueness and global stability of positive periodic solutions are established by using analysis based on the theory of discrete dynamical systems. From this study, we observe that the dynamical behavior of populations with impulsive dispersal differs greatly from the behavior of models with continuous dispersal. Unlike models where the dispersal is continuous in time, in which the travel losses associated with dispersal make it difficult for such dispersal to evolve e.g., [25], [26], [28], in the present study it was relatively easy for impulsive dispersal to positively affect populations when realistic parameter values were used, and a rich variety of behaviors were possible. From our results, we found impulsive dispersal seems to more nicely model natural dispersal behavior of populations and may be more relevant to the investigation of such behavior in real ecological systems.     

A musculoskeletal foot model for clinical gait analysis

Several full body musculoskeletal models have been developed for research applications and these models may potentially be developed into useful clinical tools to assess gait pathologies. Existing full-body musculoskeletal models treat the foot as a single segment and ignore the motions of the intrinsic joints of the foot. This assumption limits the use of such models in clinical cases with significant foot deformities. Therefore, a three-segment musculoskeletal model of the foot was developed to match the segmentation of a recently developed multi-segment kinematic foot model. All the muscles and ligaments of the foot spanning the modeled joints were included. Muscle pathways were adjusted with an optimization routine to minimize the difference between the muscle flexion–extension moment arms from the model and moment arms reported in literature. The model was driven by walking data from five normal pediatric subjects (aged 10.6±1.57 years) and muscle forces and activation levels required to produce joint motions were calculated using an inverse dynamic analysis approach. Due to the close proximity of markers on the foot, small marker placement error during motion data collection may lead to significant differences in musculoskeletal model outcomes. Therefore, an optimization routine was developed to enforce joint constraints, optimally scale each segment length and adjust marker positions. To evaluate the model outcomes, the muscle activation patterns during walking were compared with electromyography (EMG) activation patterns reported in the literature. Model-generated muscle activation patterns were observed to be similar to the EMG activation patterns.     

Simultaneous recording of longitudinal displacements of both feet during human walking

1. Recordings of longitudinal displacements of both feet have been performed by linking each foot to a length-voltage transducer by means of threads. The movement transmitted to the transducer was reduced by winding the thread around a sixteen strand pulley block. 2. The aspect of the displacement curves allows a direct analysis of the walk. Some typical curves of normal and pathological gaits are presented. Graphic measurements done on several right and left cycles of a 6 m walk episode in 50 adult subjects provided data to determine the mean and the standard deviation of spatial and temporal parameters of the walk. 3. An advantage of this method is to permit the measurement of the parameters of several successive cycles of both sides and so to detect changes in the length, the duration and the velocity of successive cycles of the same foot and of alternated cycles of both feet. This being important to characterize pathological gaits.     

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.     

Hexapodal gaits and coupled nonlinear oscillator models

The general, model-independent features of different networks of six symmetrically coupled nonlinear oscillators are investigated. These networks are considered as possible models for locomotor central pattern generators (CPGs) in insects. Numerical experiments with a specific oscillator network model are briefly described. It is shown that some generic phase-locked oscillation-patterns for various systems of six symmetrically coupled nonlinear oscillators correspond to the common forward-walking gaits adopted by insects. It is also demonstrated that transitions observed in insect gaits can be modelled as standard symmetry-breaking bifurcations occurring in such systems. The present analysis, which leads to a natural classification of hexapodal gaits by symmetry and to natural sequences of gait bifurcations, relates observed gaits to the overall organizational structure of the underlying CPG. The implications of the present results for the development of simplified control systems for hexapodal walking robots are discussed.     

Symmetrical gaits of primates

Walking and symmetrical running gaits of 26 genera of primates are analyzed using numerical and graphical methods described previously. The raw data are 1701 feet of 16 mm motion picture film mostly exposed at 64 frames per second. Adult monkeys and apes usually use the walking trot or diagonal-sequence walks. Individual monkeys occasionally use lateral-sequence walks resembling those that are usual for human infants. Human children moving on hands and feet use gaits ranging from the walking pace through the lateral-sequence walks to the walking trot. An infant macaque studied from age 17 hours to 96 days first walked with a lateral-sequence, diagonal-couplets gait and then gradually shifted to the diagonal-sequence, diagonal-couplets gait of the adult. Few non-primates use the diagonal-sequence walks which are typical of primates. Typical support sequences are figured. Relative placement of feet and consequent slight asymmetry are described.     

具周期脉冲的周期差分系统的全局性态

本文建立了具有常数脉冲和周期脉冲的周期差分系统,得到了常数脉冲系统全局稳定周期解存在的充分条件,并证明了周期脉冲的周期系统的周期解是全局吸引的。     

Gaits of Japanese macaques (Macaca fuscata) on a horizontal ladder and arboreal stability

Most primates use diagonal sequence (DS), diagonal couplets (DC) gaits when they walk or run quadrupedally, and it has been suggested that DSDC gaits contribute to stability in their natural arboreal habitats compared to other symmetrical gaits. However, this postulate is based solely on studies of primate gaits using continuous terrestrial and arboreal substrates. A particular species may select suitable gaits according to the substrate properties. Here, we analyzed the gaits of Japanese macaques moving on a horizontal ladder with rung intervals ranging from 0.40 to 0.80 m to elucidate the relative advantages of each observed form of gait. The rung arrangement forced our macaques to choose either diagonal coupling or DS gaits. One macaque consistently used diagonal coupling (i.e., DSDC and LSDC gaits) across narrow and intermediate rung intervals, whereas the other macaque used DS gaits (i.e., DSDC and DSLC gaits). At wider rung intervals, both macaques shifted to a two‐one sequence (TOS), which is characterized by two nearly simultaneous touchdowns of both forelimbs and one touchdown of each hind limb in a stride. The transition to the TOS sequence increased the duration of support on multiple limbs, but always included periods of a whole‐body aerial phase. These results suggest that Japanese macaques prefer DSDC gaits, because the diagonal coupling and DS contribute separately to stability on complex supports compared to the lateral coupling and lateral sequence. We also postulate that stability triggers the transition from symmetrical gaits to the TOS sequence. Am J Phys Anthropol, 2009. © 2008 Wiley‐Liss, Inc.     

Hydrodynamic stability in posthatchling loggerhead (Caretta caretta) and green (Chelonia mydas) sea turtles

Swimming animals may experience a wide range of destabilizing forces resulting from the movements of their propulsors. These forces often cause movements in directions other than the intended trajectory (i.e., recoil motions), potentially increasing locomotor costs. We quantified rectilinear swimming stability for posthatchling loggerhead (Caretta caretta) and green turtles (Chelonia mydas). Sea turtles predominantly swim via “aquatic flight”, which is characterized by synchronous dorsoventral flapping of their forelimbs. We tested four predictions about the effects of “aquatic flight” on stability: (1) it would produce little lateral recoil; (2) lateral recoil motions would be non-cyclic; (3) vertical recoil motions would be larger than lateral recoil motions; and (4) vertical recoil motions would be cyclic. Additionally, because posthatchling loggerheads possess dorsal keels on the shell that are absent in green turtles, we evaluated whether such keels might improve stability in swimming turtles. While our expectations for patterns of cyclicity in recoil motions (predictions 2 and 4) were met, our expectations for differences in their absolute and relative magnitudes (predictions 1 and 3) were not. We suggest that lateral recoil motions were greater than predicted due to slight asynchronies between the motions of the left and right foreflippers. Additionally, although minimum lateral recoil motions were smaller than minimum vertical recoil motions, maximum recoil motions were greater in the lateral direction, so that average recoil did not differ significantly between these directions. Finally, because loggerheads did not display higher levels of stability compared to green turtles, there is little evidence to support a stabilizing role for dorsal keels in loggerhead turtles.     

A generic analytical foot rollover model for predicting translational ankle kinematics in gait simulation studies

The objective of this paper is to develop an analytical framework to representing the ankle–foot kinematics by modelling the foot as a rollover rocker, which cannot only be used as a generic tool for general gait simulation but also allows for case-specific modelling if required. Previously, the rollover models used in gait simulation have often been based on specific functions that have usually been of a simple form. In contrast, the analytical model described here is in a general form that the effective foot rollover shape can be represented by any polar function ρ=ρ(φ). Furthermore, a normalized generic foot rollover model has been established based on a normative foot rollover shape dataset of 12 normal healthy subjects. To evaluate model accuracy, the predicted ankle motions and the centre of pressure (CoP) were compared with measurement data for both subject-specific and general cases. The results demonstrated that the ankle joint motions in both vertical and horizontal directions (relative RMSE ～10%) and CoP (relative RMSE ～15% for most of the subjects) are accurately predicted over most of the stance phase (from 10% to 90% of stance). However, we found that the foot cannot be very accurately represented by a rollover model just after heel strike (HS) and just before toe off (TO), probably due to shear deformation of foot plantar tissues (ankle motion can occur without any foot rotation). The proposed foot rollover model can be used in both inverse and forward dynamics gait simulation studies and may also find applications in rehabilitation engineering.     

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.     

Rectifying postures reconstructed from joint angles to meet constraints.

Postures are often described and modeled using angles between body segments rather than joint coordinates. Models can be used to predict these angles as a function of anthropometry and postural requirements. Postural representation, however, requires the joint coordinates. The use of conventional forward kinematics to derive joint coordinates from predicted angles may violate task constraints, such as the placement of a hand on a target or a foot on a pedal. Errors arise because the anthropometry or other motion characteristics of a subject, for which the prediction is to be made, may differ from the data from which the prediction model was derived. We describe how to rectify model-predicted postures to exactly satisfy such task constraints. We require that the model used for predicting the angles also produce estimates of the variation in these predictions. We show how to alter the initial angle predictions, with the amount of perturbation at each angle dependent on the accuracy of its estimation, so as to exactly satisfy the joint coordinate constraints. Finally, we show in an empirical example that this correction usually produces better overall predictions of posture than those obtained initially.