A mathematical model of adaptive behavior in quadruped locomotion

Gait Transition from Pacing by a Quadrupedal Simulated Model and Robot with Phase Modulation by Vestibular Feedback

We propose a method to achieve autonomous gait transition according to speed for a quadruped robot pacing at medium speeds. We verified its effectiveness through experiments with the simulation model and the robot we developed. In our proposed method, a central pattern generator (CPG) is applied to each leg. Each leg is controlled by a PD controller based on output from the CPG. The four CPGs are coupled, and a hard-wired CPG network generates a pace pattern by default. In addition, we feed the body tilt back to the CPGs in order to adapt to the body oscillation that changes according to the speed. As a result, our model and robot achieve stable changes in speed while autonomously generating a walk at low speeds and a rotary gallop at high speeds, despite the fact that the walk and rotary gallop are not preprogramed. The body tilt angle feedback is the only factor involved in the autonomous generation of gaits, so it can be easily used for various quadruped robots. Therefore, it is expected that the proposed method will be an effective control method for quadruped robots.

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.

Computational Modeling of Spinal Locomotor Circuitry in the Age of Molecular Genetics

Neuronal circuits in the spinal cord are essential for the control of locomotion. They integrate supraspinal commands and afferent feedback signals to produce coordinated rhythmic muscle activations necessary for stable locomotion. For several decades, computational modeling has complemented experimental studies by providing a mechanistic rationale for experimental observations and by deriving experimentally testable predictions. This symbiotic relationship between experimental and computational approaches has resulted in numerous fundamental insights. With recent advances in molecular and genetic methods, it has become possible to manipulate specific constituent elements of the spinal circuitry and relate them to locomotor behavior. This has led to computational modeling studies investigating mechanisms at the level of genetically defined neuronal populations and their interactions. We review literature on the spinal locomotor circuitry from a computational perspective. By reviewing examples leading up to and in the age of molecular genetics, we demonstrate the importance of computational modeling and its interactions with experiments. Moving forward, neuromechanical models with neuronal circuitry modeled at the level of genetically defined neuronal populations will be required to further unravel the mechanisms by which neuronal interactions lead to locomotor behavior.

Active Gaits Generation of Quadruped Robot Using Pulse-Type Hardware Neuron Models

In this chapter, the authors will propose the active gait generation of a quadruped robot. We developed the quadruped robot system using self-inhibited pulse-type hardware neuron models (P-HNMs) as a solution to elucidate the gait generation method. We feedbacked pressures at the robot system’s each foot to P-HNM and varied the joints’ angular velocity individually. We experimented with making the robot walk from an upright position on a flat floor. As a result of the experiment, we confirmed that the robot system spontaneously generates walk gait and trot gait according to the moving speed. Also, we clarified the process by which the robot actively generates gaits from the upright state. These results suggest that animals may generate gait using a similarly simple method because P-HNM mimics biological neurons’ function. Furthermore, it shows that our robot system can generate gaits adaptively and quite easily.

Three-dimensional walking of a simulated muscle-driven quadruped robot with neuromorphic two-level central pattern generators

We aim to design a neuromorphic controller for the locomotion of a quadruped robot with muscle-driven leg mechanisms. To this end, we use a simulated cat model; each leg of the model is equipped with three joints driven by six muscle models incorporating two-joint muscles. For each leg, we use a two-level central pattern generator consisting of a rhythm generation part to produce basic rhythms and a pattern formation part to synergistically activate a different set of muscles in each of the four sequential phases (swing, touchdown, stance, and liftoff). Conventionally, it was difficult for a quadruped model with such realistic neural systems and muscle-driven leg mechanisms to walk even on flat terrain, but because of our improved neural and mechanical components, our quadruped model succeeds in reproducing motoneuron activations and leg trajectories similar to those in cats and achieves stable three-dimensional locomotion at a variety of speeds. Moreover, the quadruped is capable of walking upslope and over irregular terrains and adapting to perturbations, even without adjusting the parameters.

Development of an 8DOF quadruped robot and implementation of Inverse Kinematics using Denavit-Hartenberg convention

Quadruped robots can mimic animal walking gait and they have certain advantages like walking on terrain and extremely rough surfaces. Obstacles can impede the movement of wheeled vehicles, where a quadruped can adapt to avoid obstacles by adjusting its height. A quadruped robot is designed and developed for in this paper, which could be controlled by the Android operating system. The Inverse Kinematics Solutions are derived for the developed structure using Denavit-Hartenberg convention and using those solutions the movements are simulated using a custom-made 3D software. An Android application is developed, which is able to control the robot using Bluetooth. The robot currently has following six different movements: front, back, left, right walking, clockwise and anti-clockwise rotation. The robot uses the ultrasound sensor to detect any obstacle closer than 300 cm (maximum) and if an impediment appears, the robot will automatically move parallel to the obstacle until it is avoided. Currently, it can move at a speed of 15.5 cm/s (approximately). To complete a full rotation of 360°, it takes 6 seconds. It can be used to develop and implement any autonomous path-planning algorithm.

Adaptive Control Strategies for Interlimb Coordination in Legged Robots: A Review

Walking animals produce adaptive interlimb coordination during locomotion in accordance with their situation. Interlimb coordination is generated through the dynamic interactions of the neural system, the musculoskeletal system, and the environment, although the underlying mechanisms remain unclear. Recently, investigations of the adaptation mechanisms of living beings have attracted attention, and bio-inspired control systems based on neurophysiological findings regarding sensorimotor interactions are being developed for legged robots. In this review, we introduce adaptive interlimb coordination for legged robots induced by various factors (locomotion speed, environmental situation, body properties, and task). In addition, we show characteristic properties of adaptive interlimb coordination, such as gait hysteresis and different time-scale adaptations. We also discuss the underlying mechanisms and control strategies to achieve adaptive interlimb coordination and the design principle for the control system of legged robots.

Mechanisms of left-right coordination in mammalian locomotor pattern generation circuits: a mathematical modeling view

The locomotor gait in limbed animals is defined by the left-right leg coordination and locomotor speed. Coordination between left and right neural activities in the spinal cord controlling left and right legs is provided by commissural interneurons (CINs). Several CIN types have been genetically identified, including the excitatory V3 and excitatory and inhibitory V0 types. Recent studies demonstrated that genetic elimination of all V0 CINs caused switching from a normal left-right alternating activity to a left-right synchronized “hopping” pattern. Furthermore, ablation of only the inhibitory V0 CINs (V0_D subtype) resulted in a lack of left-right alternation at low locomotor frequencies and retaining this alternation at high frequencies, whereas selective ablation of the excitatory V0 neurons (V0_V subtype) maintained the left–right alternation at low frequencies and switched to a hopping pattern at high frequencies. To analyze these findings, we developed a simplified mathematical model of neural circuits consisting of four pacemaker neurons representing left and right, flexor and extensor rhythm-generating centers interacting via commissural pathways representing V3, V0_D, and V0_V CINs. The locomotor frequency was controlled by a parameter defining the excitation of neurons and commissural pathways mimicking the effects of N-methyl-D-aspartate on locomotor frequency in isolated rodent spinal cord preparations. The model demonstrated a typical left-right alternating pattern under control conditions, switching to a hopping activity at any frequency after removing both V0 connections, a synchronized pattern at low frequencies with alternation at high frequencies after removing only V0_D connections, and an alternating pattern at low frequencies with hopping at high frequencies after removing only V0_V connections. We used bifurcation theory and fast-slow decomposition methods to analyze network behavior in the above regimes and transitions between them. The model reproduced, and suggested explanation for, a series of experimental phenomena and generated predictions available for experimental testing.

Movements of left and right limbs in mammals during locomotion are controlled by distinct rhythm-generating neuronal circuits in the spinal cord. Complex interactions between these circuits provide flexible coordination of limb movements in different gaits. It was shown that interactions between left and right spinal circuits are mediated by commissural interneurons. Genetic ablation of a particular type of these interneurons, called V0, leads to switching from a regular, left-right alternating “walking” activity to a left-right synchronous “hopping” pattern. Moreover, the V0 commissural interneurons have excitatory and inhibitory subtypes that appear to play different roles in the left-right coordination depending on locomotor speed. In this theoretical study, we build a simplified mathematical model of spinal circuits that describes left and right rhythm generators interacting bilaterally via several types of commissural connections. Using this model, we simulate different experimental manipulations, analyze the resultant alternating and synchronous regimes of activity, and propose explanations for the results of experimental studies. We show that although both excitatory and inhibitory V0 commissural pathways support left-right alternation, the resultant locomotor pattern and gait depend on the balance between different commissural interactions, which in turn may depend on the level of neuronal excitation and locomotor speed.

A simple rule for quadrupedal gait generation determined by leg loading feedback: a modeling study

We discovered a specific rule for generating typical quadrupedal gaits (the order of the movement of four legs) through a simulated quadrupedal locomotion, in which unprogrammed gaits (diagonal/lateral sequence walks, left/right-lead canters, and left/right-lead transverse gallops) spontaneously emerged because of leg loading feedbacks to the CPGs hard-wired to produce a default trot. Additionally, all gaits transitioned according to speed, as seen in animals. We have therefore hypothesized that various gaits derive from a trot because of posture control through leg loading feedback. The body tilt on the two support legs of each diagonal pair during trotting was classified into three types (level, tilted up, or tilted down) according to speed. The load difference between the two legs led to the phase difference between their CPGs via the loading feedbacks, resulting in nine gaits (3²: three tilts to the power of two diagonal pairs) including the aforementioned.

Analysis of the gait generation principle by a simulated quadruped model with a CPG incorporating vestibular modulation

This study aims to understand the principles of gait generation in a quadrupedal model. It is difficult to determine the essence of gait generation simply by observation of the movement of complicated animals composed of brains, nerves, muscles, etc. Therefore, we build a planar quadruped model with simplified nervous system and mechanisms, in order to observe its gaits under simulation. The model is equipped with a mathematical central pattern generator (CPG), consisting of four coupled neural oscillators, basically producing a trot pattern. The model also contains sensory feedback to the CPG, measuring the body tilt (vestibular modulation). This spontaneously gives rise to an unprogrammed lateral walk at low speeds, a transverse gallop while running, in addition to trotting at a medium speed. This is because the body oscillation exhibits a double peak per leg frequency at low speeds, no peak (little oscillation) at medium speeds, and a single peak while running. The body oscillation autonomously adjusts the phase differences between the neural oscillators via the feedback. We assume that the oscillations of the four legs produced by the CPG and the body oscillation varying according to the current speed are synchronized along with the varied phase differences to keep balance during locomotion through postural adaptation via the vestibular modulation, resulting in each gait. We succeeded in determining a single simple principle that accounts for gait transition from walking to trotting to galloping, even without brain control, complicated leg mechanisms, or a flexible trunk.

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.

Simple robot suggests physical interlimb communication is essential for quadruped walking

Quadrupeds have versatile gait patterns, depending on the locomotion speed, environmental conditions and animal species. These locomotor patterns are generated via the coordination between limbs and are partly controlled by an intraspinal neural network called the central pattern generator (CPG). Although this forms the basis for current control paradigms of interlimb coordination, the mechanism responsible for interlimb coordination remains elusive. By using a minimalistic approach, we have developed a simple-structured quadruped robot, with the help of which we propose an unconventional CPG model that consists of four decoupled oscillators with only local force feedback in each leg. Our robot exhibits good adaptability to changes in weight distribution and walking speed simply by responding to local feedback, and it can mimic the walking patterns of actual quadrupeds. Our proposed CPG-based control method suggests that physical interaction between legs during movements is essential for interlimb coordination in quadruped walking.

Limited transfer of newly acquired movement patterns across walking and running in humans

The two major modes of locomotion in humans, walking and running, may be regarded as a function of different speed (walking as slower and running as faster). Recent results using motor learning tasks in humans, as well as more direct evidence from animal models, advocate for independence in the neural control mechanisms underlying different locomotion tasks. In the current study, we investigated the possible independence of the neural mechanisms underlying human walking and running. Subjects were tested on a split-belt treadmill and adapted to walking or running on an asymmetrically driven treadmill surface. Despite the acquisition of asymmetrical movement patterns in the respective modes, the emergence of asymmetrical movement patterns in the subsequent trials was evident only within the same modes (walking after learning to walk and running after learning to run) and only partial in the opposite modes (walking after learning to run and running after learning to walk) (thus transferred only limitedly across the modes). Further, the storage of the acquired movement pattern in each mode was maintained independently of the opposite mode. Combined, these results provide indirect evidence for independence in the neural control mechanisms underlying the two locomotive modes.

Sensory feedback plays a significant role in generating walking gait and in gait transition in salamanders: a simulation study

Here, we investigate the role of sensory feedback in gait generation and transition by using a three-dimensional, neuro-musculo-mechanical model of a salamander with realistic physical parameters. Activation of limb and axial muscles were driven by neural output patterns obtained from a central pattern generator (CPG) which is composed of simulated spiking neurons with adaptation. The CPG consists of a body-CPG and four limb-CPGs that are interconnected via synapses both ipsilaterally and contralaterally. We use the model both with and without sensory modulation and four different combinations of ipsilateral and contralateral coupling between the limb-CPGs. We found that the proprioceptive sensory inputs are essential in obtaining a coordinated lateral sequence walking gait (walking). The sensory feedback includes the signals coming from the stretch receptor like intraspinal neurons located in the girdle regions and the limb stretch receptors residing in the hip and scapula regions of the salamander. On the other hand, walking trot gait (trotting) is more under central (CPG) influence compared to that of the peripheral or sensory feedback. We found that the gait transition from walking to trotting can be induced by increased activity of the descending drive coming from the mesencephalic locomotor region and is helped by the sensory inputs at the hip and scapula regions detecting the late stance phase. More neurophysiological experiments are required to identify the precise type of mechanoreceptors in the salamander and the neural mechanisms mediating the sensory modulation.

Hysteresis in the gait transition of a quadruped investigated using simple body mechanical and oscillator network models

We investigated the dynamics of quadrupedal locomotion by constructing a simple quadruped model that consists of a body mechanical model and an oscillator network model. The quadruped model has front and rear bodies connected by a waist joint with a torsional spring and damper system and four limbs controlled by command signals from the oscillator network model. The simulation results reveal that the quadruped model produces various gait patterns through dynamic interactions among the body mechanical system, the oscillator network system, and the environment. They also show that it undergoes a gait transition induced by changes in the waist joint stiffness and the walking speed. In addition, the gait pattern transition exhibits a hysteresis similar to that observed in human and animal locomotion. We examined the hysteresis mechanism from a dynamic viewpoint.

Cerebellar circuitry as a neuronal machine

Shortly after John Eccles completed his studies of synaptic inhibition in the spinal cord, for which he was awarded the 1963 Nobel Prize in physiology/medicine, he opened another chapter of neuroscience with his work on the cerebellum. From 1963 to 1967, Eccles and his colleagues in Canberra successfully dissected the complex neuronal circuitry in the cerebellar cortex. In the 1967 monograph, "The Cerebellum as a Neuronal Machine", he, in collaboration with Masao Ito and Janos Szentágothai, presented blue-print-like wiring diagrams of the cerebellar neuronal circuitry. These stimulated worldwide discussions and experimentation on the potential operational mechanisms of the circuitry and spurred theoreticians to develop relevant network models of the machinelike function of the cerebellum. In following decades, the neuronal machine concept of the cerebellum was strengthened by additional knowledge of the modular organization of its structure and memory mechanism, the latter in the form of synaptic plasticity, in particular, long-term depression. Moreover, several types of motor control were established as model systems representing learning mechanisms of the cerebellum. More recently, both the quantitative preciseness of cerebellar analyses and overall knowledge about the cerebellum have advanced considerably at the cellular and molecular levels of analysis. Cerebellar circuitry now includes Lugaro cells and unipolar brush cells as additional unique elements. Other new revelations include the operation of the complex glomerulus structure, intricate signal transduction for synaptic plasticity, silent synapses, irregularity of spike discharges, temporal fidelity of synaptic activation, rhythm generators, a Golgi cell clock circuit, and sensory or motor representation by mossy fibers and climbing fibers. Furthermore, it has become evident that the cerebellum has cognitive functions, and probably also emotion, as well as better-known motor and autonomic functions. Further cerebellar research is required for full understanding of the cerebellum as a broad learning machine for neural control of these functions.

'Initial state' coordinations reproduce the instant flexibility for human walking

An important feature of human locomotor control is the instant adaptability to unpredictable changes of conditions surrounding the locomotion. Humans, for example, can seamlessly adapt their walking gait following a sudden ankle impairment (e.g., as a result of an injury). In this paper, we propose a theoretical study of the mechanisms underlying flexible locomotor control. We hypothesize that flexibility is achieved by modulating the posture at the beginning of the stance phase-the initial state. Using a walking model, we validate our hypothesis through computer simulations.

Interlimb coordination during locomotion: what can be adapted and stored?

Interlimb coordination is critically important during bipedal locomotion and often must be adapted to account for varying environmental circumstances. Here we studied adaptation of human interlimb coordination using a split-belt treadmill, where the legs can be made to move at different speeds. Human adults, infants, and spinal cats can alter walking patterns on a split-belt treadmill by prolonging stance and shortening swing on the slower limb and vice versa on the faster limb. It is not known whether other locomotor parameters change or if there is a capacity for storage of a new motor pattern after training. We asked whether adults adapt both intra- and interlimb gait parameters during split-belt walking and show aftereffects from training. Healthy subjects were tested walking with belts tied (baseline), then belts split (adaptation), and again tied (postadaptation). Walking parameters that directly relate to the interlimb relationship changed slowly during adaptation and showed robust aftereffects during postadaptation. These changes paralleled subjective impressions of limping versus no limping. In contrast, parameters calculated from an individual leg changed rapidly to accommodate split-belts and showed no aftereffects. These results suggest some independence of neural control of intra- versus interlimb parameters during walking. They also show that the adult nervous system can adapt and store new interlimb patterns after short bouts of training. The differences in intra- versus interlimb control may be related to the varying complexity of the parameters, task demands, and/or the level of neural control necessary for their adaptation.

A Nonlinear Dynamics Approach to Human Movement

Nonlinear dynamics and dynamical systems approaches and methodologies are increasingly being implemented in biomechanics and human movement research. Based on the early insights of Nicolai Bernstein (1967), a significantly different outlook on the movement control “problem” over the last few decades has emerged. From a focus on relatively simple movements has arisen a research focus with the primary goal to study movement in context, allowing the complexity of patterns to emerge. The approach taken is that the control of multiple degrees-of-freedom systems is not necessarily more difficult or complex than that of systems only comprising a few degrees of freedom. Complex patterns and dynamics might not require complex control structures. In this paper we present a tutorial overview of the mathematical underpinnings of nonlinear dynamics and some of its basic analysis tools. This should provide the reader with a basic level of understanding about the mathematical principles and concepts underlying pattern stability and change. This will be followed by an overview of dynamical systems approaches in the study of human movement. Finally, we discuss recent progress in the application of nonlinear dynamical techniques to the study of human locomotion, with particular focus on relative phase techniques for the assessment of coordination.