Maneuverable and Efficient Locomotion of a Myriapod Robot with Variable Body-Axis Flexibility via Instability and Bifurcation.

Legged robots have remarkable terrestrial mobility, but are susceptible to falling and leg malfunction during locomotion. The use of a large number of legs, as in centipedes, can overcome these problems, but it makes the body long and leads to many legs being constrained to contact with the ground to support the long body, which impedes maneuverability. A mechanism for maneuverable locomotion using a large number legs is thus desirable. However, controlling a long body with a large number of legs requires huge computational and energy costs. Inspired by agile locomotion in biological systems, this study proposes a control strategy for maneuverable and efficient locomotion of a myriapod robot based on dynamic instability. Specifically, our previous study made the body axis of a 12-legged robot flexible and showed that changing the body-axis flexibility produces pitchfork bifurcation. The bifurcation not only induces the dynamic instability of a straight walk but also a transition to a curved walk, whose curvature is controllable by the body-axis flexibility. This study incorporated a variable stiffness mechanism into the body axis and developed a simple control strategy based on the bifurcation characteristics. With this strategy, maneuverable and autonomous locomotion was achieved, as demonstrated by multiple robot experiments. Our approach does not directly control the movement of the body axis; instead, it controls body-axis flexibility, which significantly reduces computational and energy costs. This study provides a new design principle for maneuverable and efficient locomotion of myriapod robots.

Contribution of Phase Resetting to Statistical Persistence in Stride Intervals: A Modeling Study.

Stride intervals in human walking fluctuate from one stride to the next, exhibiting statistical persistence. This statistical property is changed by aging, neural disorders, and experimental interventions. It has been hypothesized that the central nervous system is responsible for the statistical persistence. Human walking is a complex phenomenon generated through the dynamic interactions between the central nervous system and the biomechanical system. It has also been hypothesized that the statistical persistence emerges through the dynamic interactions during walking. In particular, a previous study integrated a biomechanical model composed of seven rigid links with a central pattern generator (CPG) model, which incorporated a phase resetting mechanism as sensory feedback as well as feedforward, trajectory tracking, and intermittent feedback controllers, and suggested that phase resetting contributes to the statistical persistence in stride intervals. However, the essential mechanisms remain largely unclear due to the complexity of the neuromechanical model. In this study, we reproduced the statistical persistence in stride intervals using a simplified neuromechanical model composed of a simple compass-type biomechanical model and a simple CPG model that incorporates only phase resetting and a feedforward controller. A lack of phase resetting induced a loss of statistical persistence, as observed for aging, neural disorders, and experimental interventions. These mechanisms were clarified based on the phase response characteristics of our model. These findings provide useful insight into the mechanisms responsible for the statistical persistence of stride intervals in human walking.

Contribution of Afferent Feedback to Adaptive Hindlimb Walking in Cats: A Neuromusculoskeletal Modeling Study.

Mammalian locomotion is generated by central pattern generators (CPGs) in the spinal cord, which produce alternating flexor and extensor activities controlling the locomotor movements of each limb. Afferent feedback signals from the limbs are integrated by the CPGs to provide adaptive control of locomotion. Responses of CPG-generated neural activity to afferent feedback stimulation have been previously studied during fictive locomotion in immobilized cats. Yet, locomotion in awake, behaving animals involves dynamic interactions between central neuronal circuits, afferent feedback, musculoskeletal system, and environment. To study these complex interactions, we developed a model simulating interactions between a half-center CPG and the musculoskeletal system of a cat hindlimb. Then, we analyzed the role of afferent feedback in the locomotor adaptation from a dynamic viewpoint using the methods of dynamical systems theory and nullcline analysis. Our model reproduced limb movements during regular cat walking as well as adaptive changes of these movements when the foot steps into a hole. The model generates important insights into the mechanism for adaptive locomotion resulting from dynamic interactions between the CPG-based neural circuits, the musculoskeletal system, and the environment.

Fast and Slow Adaptations of Interlimb Coordination via Reflex and Learning During Split-Belt Treadmill Walking of a Quadruped Robot.

Interlimb coordination plays an important role in adaptive locomotion of humans and animals. This has been investigated using a split-belt treadmill, which imposes different speeds on the two sides of the body. Two types of adaptation have been identified, namely fast and slow adaptations. Fast adaptation induces asymmetric interlimb coordination soon after a change of the treadmill speed condition from same speed for both belts to different speeds. In contrast, slow adaptation slowly reduces the asymmetry after fast adaptation. It has been suggested that these adaptations are primarily achieved by the spinal reflex and cerebellar learning. However, these adaptation mechanisms remain unclear due to the complicated dynamics of locomotion. In our previous work, we developed a locomotion control system for a biped robot based on the spinal reflex and cerebellar learning. We reproduced the fast and slow adaptations observed in humans during split-belt treadmill walking of the biped robot and clarified the adaptation mechanisms from a dynamic viewpoint by focusing on the changes in the relative positions between the center of mass and foot stance induced by reflex and learning. In this study, we modified the control system for application to a quadruped robot. We demonstrate that even though the basic gait pattern of our robot is different from that of general quadrupeds (due to limitations of the robot experiment), fast and slow adaptations that are similar to those of quadrupeds appear during split-belt treadmill walking of the quadruped robot. Furthermore, we clarify these adaptation mechanisms from a dynamic viewpoint, as done in our previous work. These results will increase the understanding of how fast and slow adaptations are generated in quadrupedal locomotion on a split-belt treadmill through body dynamics and sensorimotor integration via the spinal reflex and cerebellar learning and help the development of control strategies for adaptive locomotion of quadruped robots.

Fractal mechanism of basin of attraction in passive dynamic walking.

Passive dynamic walking is a model that walks down a shallow slope without any control or input. This model has been widely used to investigate how humans walk with low energy consumption and provides design principles for energy-efficient biped robots. However, the basin of attraction is very small and thin and has a fractal-like complicated shape, which makes producing stable walking difficult. In our previous study, we used the simplest walking model and investigated the fractal-like basin of attraction based on dynamical systems theory by focusing on the hybrid dynamics of the model composed of the continuous dynamics with saddle hyperbolicity and the discontinuous dynamics caused by the impact upon foot contact. We clarified that the fractal-like basin of attraction is generated through iterative stretching and bending deformations of the domain of the Poincaré map by sequential inverse images. However, whether the fractal-like basin of attraction is actually fractal, i.e., whether infinitely many self-similar patterns are embedded in the basin of attraction, is dependent on the slope angle, and the mechanism remains unclear. In the present study, we improved our previous analysis in order to clarify this mechanism. In particular, we newly focused on the range of the Poincaré map and specified the regions that are stretched and bent by the sequential inverse images of the Poincaré map. Through the analysis of the specified regions, we clarified the conditions and mechanism required for the basin of attraction to be fractal.

Contribution of Phase Resetting to Adaptive Rhythm Control in Human Walking Based on the Phase Response Curves of a Neuromusculoskeletal Model.

Humans walk adaptively in varying environments by manipulating their complicated and redundant musculoskeletal system. Although the central pattern generators in the spinal cord are largely responsible for adaptive walking through sensory-motor coordination, it remains unclear what neural mechanisms determine walking adaptability. It has been reported that locomotor rhythm and phase are regulated by the production of phase shift and rhythm resetting (phase resetting) for periodic motor commands in response to sensory feedback and perturbation. While the phase resetting has been suggested to make a large contribution to adaptive walking, it has only been investigated based on fictive locomotion in decerebrate cats, and thus it remains unclear if human motor control has such a rhythm regulation mechanism during walking. In our previous work, we incorporated a phase resetting mechanism into a motor control model and demonstrated that it improves the stability and robustness of walking through forward dynamic simulations of a human musculoskeletal model. However, this did not necessarily verify that phase resetting plays a role in human motor control. In our other previous work, we used kinematic measurements of human walking to identify the phase response curve (PRC), which explains phase-dependent responses of a limit cycle oscillator to a perturbation. This revealed how human walking rhythm is regulated by perturbations. In this study, we integrated these two approaches using a physical model and identification of the PRC to examine the hypothesis that phase resetting plays a role in the control of walking rhythm in humans. More specifically, we calculated the PRC using our neuromusculoskeletal model in the same way as our previous human experiment. In particular, we compared the PRCs calculated from two different models with and without phase resetting while referring to the PRC for humans. As a result, although the PRC for the model without phase resetting did not show any characteristic shape, the PRC for the model with phase resetting showed a characteristic phase-dependent shape with trends similar to those of the PRC for humans. These results support our hypothesis and will improve our understanding of adaptive rhythm control in human walking.

Active lift inversion process of heaving wing in uniform flow by temporal change of wing kinematics.

The transition of the vortex pattern and the lift generated by a heaving wing in a uniform flow was investigated numerically. As a fundamental problem constituting the insects' flight maneuverability, we studied the relationship between a temporal change in the heaving wing motion and the change in the global vortex pattern. At a Strouhal number that generates an asymmetric vortex pattern, we found that temporal angular frequency reduction causes inversion of both the global vortex pattern and the lift sign. The inversion is initiated by the transfer of the leading-edge vortex, which interferes with the vortex pattern generated at the trailing edge. Successful inversion is conditioned on the starting phase and the time interval of the frequency reduction. The details of the process during the transition are discussed.

Neuromusculoskeletal model that walks and runs across a speed range with a few motor control parameter changes based on the muscle synergy hypothesis.

Humans walk and run, as well as change their gait speed, through the control of their complicated and redundant musculoskeletal system. These gaits exhibit different locomotor behaviors, such as a double-stance phase in walking and flight phase in running. The complex and redundant nature of the musculoskeletal system and the wide variation in locomotion characteristics lead us to imagine that the motor control strategies for these gaits, which remain unclear, are extremely complex and differ from one another. It has been previously proposed that muscle activations may be generated by linearly combining a small set of basic pulses produced by central pattern generators (muscle synergy hypothesis). This control scheme is simple and thought to be shared between walking and running at different speeds. Demonstrating that this control scheme can generate walking and running and change the speed is critical, as bipedal locomotion is dynamically challenging. Here, we provide such a demonstration by using a motor control model with 69 parameters developed based on the muscle synergy hypothesis. Specifically, we show that it produces both walking and running of a human musculoskeletal model by changing only seven key motor control parameters. Furthermore, we show that the model can walk and run at different speeds by changing only the same seven parameters based on the desired speed. These findings will improve our understanding of human motor control in locomotion and provide guiding principles for the control design of wearable exoskeletons and prostheses.

Advantage of straight walk instability in turning maneuver of multilegged locomotion: a robotics approach.

Multilegged locomotion improves the mobility of terrestrial animals and artifacts. Using many legs has advantages, such as the ability to avoid falling and to tolerate leg malfunction. However, many intrinsic degrees of freedom make the motion planning and control difficult, and many contact legs can impede the maneuverability during locomotion. The underlying mechanism for generating agile locomotion using many legs remains unclear from biological and engineering viewpoints. The present study used a centipede-like multilegged robot composed of six body segments and twelve legs. The body segments are passively connected through yaw joints with torsional springs. The dynamic stability of the robot walking in a straight line changes through a supercritical Hopf bifurcation due to the body axis flexibility. We focused on a quick turning task of the robot and quantitatively investigated the relationship between stability and maneuverability in multilegged locomotion by using a simple control strategy. Our experimental results show that the straight walk instability does help the turning maneuver. We discuss the importance and relevance of our findings for biological systems and propose a design principle for a simple control scheme to create maneuverable locomotion of multilegged robots.

Adaptation mechanism of interlimb coordination in human split-belt treadmill walking through learning of foot contact timing: a robotics study.

Human walking behaviour adaptation strategies have previously been examined using split-belt treadmills, which have two parallel independently controlled belts. In such human split-belt treadmill walking, two types of adaptations have been identified: early and late. Early-type adaptations appear as rapid changes in interlimb and intralimb coordination activities when the belt speeds of the treadmill change between tied (same speed for both belts) and split-belt (different speeds for each belt) configurations. By contrast, late-type adaptations occur after the early-type adaptations as a gradual change and only involve interlimb coordination. Furthermore, interlimb coordination shows after-effects that are related to these adaptations. It has been suggested that these adaptations are governed primarily by the spinal cord and cerebellum, but the underlying mechanism remains unclear. Because various physiological findings suggest that foot contact timing is crucial to adaptive locomotion, this paper reports on the development of a two-layered control model for walking composed of spinal and cerebellar models, and on its use as the focus of our control model. The spinal model generates rhythmic motor commands using an oscillator network based on a central pattern generator and modulates the commands formulated in immediate response to foot contact, while the cerebellar model modifies motor commands through learning based on error information related to differences between the predicted and actual foot contact timings of each leg. We investigated adaptive behaviour and its mechanism by split-belt treadmill walking experiments using both computer simulations and an experimental bipedal robot. Our results showed that the robot exhibited rapid changes in interlimb and intralimb coordination that were similar to the early-type adaptations observed in humans. In addition, despite the lack of direct interlimb coordination control, gradual changes and after-effects in the interlimb coordination appeared in a manner that was similar to the late-type adaptations and after-effects observed in humans. The adaptation results of the robot were then evaluated in comparison with human split-belt treadmill walking, and the adaptation mechanism was clarified from a dynamic viewpoint.

Investigation of adaptive split-belt treadmill walking by the hindlimbs of rats.

In this study, we investigated the adaptive behavior during hindlimb locomotion of rats on a split-belt treadmill. We measured and analyzed the movement of intact rats walking by the hindlimbs on the splitbelt treadmill with two conditions: symmetric and asymmetric belt speed. In addition, we conducted the dynamic simulation of a neuromusculoskeletal model of rat's hindlimb walking on a split-belt treadmill. We investigated the immediate modulations of the duty factors and relative phase between the right and left limbs depending on the conditions of the treadmill. The results of the simulation were qualitatively similar to those of the measurement experiment. Furthermore, these results were qualitatively similar to the measurement data of the humans and cats in the previous studies. This suggests that our model have the essential aspects to produce the adaptive split-belt treadmill walking in dynamics viewpoints.

Acceleration of reinforcement learning by policy evaluation using nonstationary iterative method.

Typical methods for solving reinforcement learning problems iterate two steps, policy evaluation and policy improvement. This paper proposes algorithms for the policy evaluation to improve learning efficiency. The proposed algorithms are based on the Krylov Subspace Method (KSM), which is a nonstationary iterative method. The algorithms based on KSM are tens to hundreds times more efficient than existing algorithms based on the stationary iterative methods. Algorithms based on KSM are far more efficient than they have been generally expected. This paper clarifies what makes algorithms based on KSM makes more efficient with numerical examples and theoretical discussions.

Hysteresis in the metachronal-tripod gait transition of insects: a modeling study.

Locomotion in biological systems involves various gaits, and hysteresis appears when the gaits change in accordance with the locomotion speed. That is, the gaits vary at different locomotion speeds depending on the direction of speed change. Although hysteresis is a typical characteristic of nonlinear dynamic systems, the underlying mechanism for the hysteresis in gait transitions remains largely unclear. In this study, we construct a neuromechanical model of an insect and investigate the dynamic characteristics of its gait and gait transition. The simulation results show that our insect model produces metachronal and tripod gaits depending on the locomotion speed through dynamic interactions among the body mechanical system, the nervous system, and the environment in a self-organized manner. They also show that it undergoes the metachronal-tripod gait transition with hysteresis by changing the locomotion speed. We examined the hysteresis mechanism in the metachronal-tripod gait transition of insects from a dynamic viewpoint.

A stability-based mechanism for hysteresis in the walk-trot transition in quadruped locomotion.

Quadrupeds vary their gaits in accordance with their locomotion speed. Such gait transitions exhibit hysteresis. However, the underlying mechanism for this hysteresis remains largely unclear. It has been suggested that gaits correspond to attractors in their dynamics and that gait transitions are non-equilibrium phase transitions that are accompanied by a loss in stability. In the present study, we used a robotic platform to investigate the dynamic stability of gaits and to clarify the hysteresis mechanism in the walk-trot transition of quadrupeds. Specifically, we used a quadruped robot as the body mechanical model and an oscillator network for the nervous system model to emulate dynamic locomotion of a quadruped. Experiments using this robot revealed that dynamic interactions among the robot mechanical system, the oscillator network, and the environment generate walk and trot gaits depending on the locomotion speed. In addition, a walk-trot transition that exhibited hysteresis was observed when the locomotion speed was changed. We evaluated the gait changes of the robot by measuring the locomotion of dogs. Furthermore, we investigated the stability structure during the gait transition of the robot by constructing a potential function from the return map of the relative phase of the legs and clarified the physical characteristics inherent to the gait transition in terms of the dynamics.

Contributions of phase resetting and interlimb coordination to the adaptive control of hindlimb obstacle avoidance during locomotion in rats: a simulation study.

Obstacle avoidance during locomotion is essential for safe, smooth locomotion. Physiological studies regarding muscle synergy have shown that the combination of a small number of basic patterns produces the large part of muscle activities during locomotion and the addition of another pattern explains muscle activities for obstacle avoidance. Furthermore, central pattern generators in the spinal cord are thought to manage the timing to produce such basic patterns. In the present study, we investigated sensory-motor coordination for obstacle avoidance by the hindlimbs of the rat using a neuromusculoskeletal model. We constructed the musculoskeletal part of the model based on empirical anatomical data of the rat and the nervous system model based on the aforementioned physiological findings of central pattern generators and muscle synergy. To verify the dynamic simulation by the constructed model, we compared the simulation results with kinematic and electromyographic data measured during actual locomotion in rats. In addition, we incorporated sensory regulation models based on physiological evidence of phase resetting and interlimb coordination and examined their functional roles in stepping over an obstacle during locomotion. Our results show that the phase regulation based on interlimb coordination contributes to stepping over a higher obstacle and that based on phase resetting contributes to quick recovery after stepping over the obstacle. These results suggest the importance of sensory regulation in generating successful obstacle avoidance during locomotion.

Effects of structural flexibility of wings in flapping flight of butterfly.

The objective of this paper is to clarify the effects of structural flexibility of wings of a butterfly in flapping flight. For this purpose, a dynamics model of a butterfly is derived by Lagrange's method, where the butterfly is considered as a rigid multi-body system. The panel method is employed to simulate the flow field and the aerodynamic forces acting on the wings. The mathematical model is validated by the agreement of the numerical result with the experimentally measured data. Then, periodic orbits of flapping-of-wings flights are parametrically searched in order to fly the butterfly models. Almost periodic orbits are found, but they are unstable. Deformation of the wings is modeled in two ways. One is bending and its effect on the aerodynamic forces is discussed. The other is passive wing torsion caused by structural flexibility. Numerical simulations demonstrate that flexible torsion reduces the flight instability.