Behaviour-based modelling of hexapod locomotion: linking biology and technical application

Walking in insects and most six-legged robots requires simultaneous control of up to 18 joints. Moreover, the number of joints that are mechanically coupled via body and ground varies from one moment to the next, and external conditions such as friction, compliance and slope of the substrate are often unpredictable. Thus, walking behaviour requires adaptive, context-dependent control of many degrees of freedom. As a consequence, modelling legged locomotion addresses many aspects of any motor behaviour in general. Based on results from behavioural experiments on arthropods, we describe a kinematic model of hexapod walking: the distributed artificial neural network controller walknet. Conceptually, the model addresses three basic problems in legged locomotion. (I) First, coordination of several legs requires coupling between the step cycles of adjacent legs, optimising synergistic propulsion, but ensuring stability through flexible adjustment to external disturbances. A set of behaviourally derived leg coordination rules can account for decentralised generation of different gaits, and allows stable walking of the insect model as well as of a number of legged robots. (II) Second, a wide range of different leg movements must be possible, e.g. to search for foothold, grasp for objects or groom the body surface. We present a simple neural network controller that can simulate targeted swing trajectories, obstacle avoidance reflexes and cyclic searching-movements. (III) Third, control of mechanically coupled joints of the legs in stance is achieved by exploiting the physical interactions between body, legs and substrate. A local positive displacement feedback, acting on individual leg joints, transforms passive displacement of a joint into active movement, generating synergistic assistance reflexes in all mechanically coupled joints.     

How locomotion sub-functions can control walking at different speeds?

Inspired from template models explaining biological locomotory systems and Raibert׳s pioneering legged robots, locomotion can be realized by basic sub-functions: elastic axial leg function, leg swinging and balancing. Combinations of these three can generate different gaits with diverse properties. In this paper we investigate how locomotion sub-functions contribute to stabilize walking at different speeds. Based on this trilogy, we introduce a conceptual model to quantify human locomotion sub-functions in walking. This model can produce stable walking and also predict human locomotion sub-function control during swing phase of walking. Analyzing experimental data based on this modeling shows different control strategies which are employed to increase speed from slow to moderate and moderate to fast gaits.     

Convergent evolution and locomotion through complex terrain by insects, vertebrates and robots

Arthropods are the most successful members of the animal kingdom largely because of their ability to move efficiently through a range of environments. Their agility has not been lost on engineers seeking to design agile legged robots. However, one cannot simply copy mechanical and neural control systems from insects into robotic designs. Rather one has to select the properties that are critical for specific behaviors that the engineer wants to capture in a particular robot. Convergent evolution provides an important clue to the properties of legged locomotion that are critical for success. Arthropods and vertebrates evolved legged locomotion independently. Nevertheless, many neural control properties and mechanical schemes are remarkably similar. Here we describe three aspects of legged locomotion that are found in both insects and vertebrates and that provide enhancements to legged robots. They are leg specialization, body flexion and the development of a complex head structure. Although these properties are commonly seen in legged animals, most robotic vehicles have similar legs throughout, rigid bodies and rudimentary sensors on what would be considered the head region. We describe these convergent properties in the context of robots that we developed to capture the agility of insects in moving through complex terrain.     

Insects Use Two Distinct Classes of Steps during Unrestrained Locomotion

Background

Adaptive, context-dependent control of locomotion requires modulation of centrally generated rhythmic motor patterns through peripheral control loops and postural reflexes. Thus assuming that the modulation of rhythmic motor patterns accounts for much of the behavioural variability observed in legged locomotion, investigating behavioural variability is a key to the understanding of context-dependent control mechanisms in locomotion. To date, the variability of unrestrained locomotion is poorly understood, and virtually nothing is known about the features that characterise the natural statistics of legged locomotion. In this study, we quantify the natural variability of hexapedal walking and climbing in insects, drawing from a database of several thousand steps recorded over two hours of walking time.

Results

We show that the range of step length used by unrestrained climbing stick insects is large, showing that step length can be changed substantially for adaptive locomotion. Step length distributions were always bimodal, irrespective of leg type and walking condition, suggesting the presence of two distinct classes of steps: short and long steps. Probability density of step length was well-described by a gamma distribution for short steps, and a logistic distribution for long steps. Major coefficients of these distributions remained largely unaffected by walking conditions. Short and long steps differed concerning their spatial occurrence on the walking substrate, their timing within the step sequence, and their prevalent swing direction. Finally, ablation of structures that serve to improve foothold increased the ratio of short to long steps, indicating a corrective function of short steps.

Conclusions

Statistical and functional differences suggest that short and long steps are physiologically distinct classes of leg movements that likely reflect distinct control mechanisms at work.     

A Neuro-Mechanical Model of a Single Leg Joint Highlighting the Basic Physiological Role of Fast and Slow Muscle Fibres of an Insect Muscle System

In legged animals, the muscle system has a dual function: to produce forces and torques necessary to move the limbs in a systematic way, and to maintain the body in a static position. These two functions are performed by the contribution of specialized motor units, i.e. motoneurons driving sets of specialized muscle fibres. With reference to their overall contraction and metabolic properties they are called fast and slow muscle fibres and can be found ubiquitously in skeletal muscles. Both fibre types are active during stepping, but only the slow ones maintain the posture of the body. From these findings, the general hypothesis on a functional segregation between both fibre types and their neuronal control has arisen. Earlier muscle models did not fully take this aspect into account. They either focused on certain aspects of muscular function or were developed to describe specific behaviours only. By contrast, our neuro-mechanical model is more general as it allows functionally to differentiate between static and dynamic aspects of movement control. It does so by including both muscle fibre types and separate motoneuron drives. Our model helps to gain a deeper insight into how the nervous system might combine neuronal control of locomotion and posture. It predicts that (1) positioning the leg at a specific retraction angle in steady state is most likely due to the extent of recruitment of slow muscle fibres and not to the force developed in the individual fibres of the antagonistic muscles; (2) the fast muscle fibres of antagonistic muscles contract alternately during stepping, while co-contraction of the slow muscle fibres takes place during steady state; (3) there are several possible ways of transition between movement and steady state of the leg achieved by varying the time course of recruitment of the fibres in the participating muscles.     

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.     

Rules to limp by: joint compensation conserves limb function after peripheral nerve injury

Locomotion persists across all manner of internal and external perturbations. The objective of this study was to identify locomotor compensation strategies in rodent models of peripheral nerve injury. We found that hip-to-toe limb length and limb angle was preferentially preserved over individual joint angles after permanent denervation of rat ankle extensor muscles. These findings promote further enquiry into the significance of limb-level function for neuromechanical control of legged locomotion.     

Adapting stiffness and attack angle through trial and error to increase self-stability in locomotion

Biological systems are outperforming machines in legged locomoting under almost any conditions. This is partly due to their capability of learning from failure and adapting their control approach and morphological features. This paper proposes an approach that extends the spring-loaded inverted pendulum (SLIP) model with the capability to adapt its attack angle (control) and stiffness (morphology) based on previous locomotion attempts. A set of different update rules, i.e., how this experience is used to adapt, are systematically investigated. The results suggest that modifying either attack angle, or stiffness, or both is beneficial with respect to achieve stable locomotion. Particularly, if the current system configuration (control and morphology) outperforms the previous one, the results suggest that increasing the angle and decreasing the stiffness of the system leads to more stable solutions. Consequently, the basic SLIP model extended by the proposed learning capabilities is able to reach stable locomotion over a much wider range of parameter combinations simply through trial and error.     

An analytical estimation of the energy cost for legged locomotion

Legged locomotion requires the determination of a number of parameters such as stride period, stride length, order of leg movements, leg trajectory, etc. How are these parameters determined? It has been reported that the locomotor patterns of many legged animals exhibit common characteristics, which suggests that there exists a basic strategy for legged locomotion. In this study we derive an equation to estimate the cost of transport for legged locomotion and examine a criterion of the minimization of the transport cost as a candidate of the strategy. The obtained optimal locomotor pattern that minimizes the cost suitably represents many characteristics of the pattern observed in legged animals. This suggests that the locomotor pattern of legged animals is well optimized with regard to the energetic cost. The result also suggests that the existence of specific gait patterns and the phase transition between them could be the result due to optimization; they are induced by the change in the distribution of ground reaction forces for each leg during locomotion.     

Legged insects select the optimal locomotor pattern based on the energetic cost

 The gait transition in legged animals has attracted many researchers, and its relation to metabolic cost and mechanical work has been discussed in recent decades. We assumed that the energetic cost during locomotion is given by the sum of positive mechanical work and the heat energy loss that is proportional to the square of joint torque and examined the optimal locomotor pattern based on the energetic cost in a simple dynamical model of a hexapod by computer simulations. The obtained results well agree with characteristics in the locomotor patterns in legged animals; for example, the leg protraction time, step length, and the metabolic cost of transport are almost constant for many velocities, the leg cycling period decreases with velocity, and the energetic cost of locomotion induced by carrying loads linearly increases with mass loaded. This correspondence of the results of calculation to experimental results suggest that the heat energy loss for torque generation is proportional to the square of the torque during locomotion, and that the locomotor pattern in legged animals is highly optimized based on the energetic cost. Received: 22 December 1998 / Accepted: 14 April 2000     

Experiments and models of sensorimotor interactions during locomotion

During locomotion sensory information from cutaneous and muscle receptors is continuously integrated with the locomotor central pattern generator (CPG) to generate an appropriate motor output to meet the demands of the environment. Sensory signals from peripheral receptors can strongly impact the timing and amplitude of locomotor activity. This sensory information is gated centrally depending on the state of the system (i.e., rest vs. locomotion) but is also modulated according to the phase of a given task. Consequently, if one is to devise biologically relevant walking models it is imperative that these sensorimotor interactions at the spinal level be incorporated into the control system.     

Spinal motor control system incorporates an internal model of limb dynamics

The existence and utilization of an internal representation of the controlled object is one of the most important features of the functioning of neural motor control systems. This study demonstrates that this property already exists at the level of the spinal motor control system (SMCS), which is capable of generating motor patterns for reflex rhythmic movements, such as locomotion and scratching, without the aid of the peripheral afferent feedback, but substantially modifies the generated activity in response to peripheral afferent stimuli. The SMCS is presented as an optimal control system whose optimality requires that it incorporate an internal model (IM) of the controlled object's dynamics. A novel functional mechanism for the integration of peripheral sensory signals with the corresponding predictive output from the IM, the summation of information precision (SIP) is proposed. In contrast to other models in which the correction of the internal representation of the controlled object's state is based on the calculation of a mismatch between the internal and external information sources, the SIP mechanism merges the information from these sources in order to optimize the precision of the controlled object's state estimate. It is demonstrated, based on scratching in decerebrate cats as an example of the spinal control of goal-directed movements, that the results of computer modeling agree with the experimental observations related to the SMCS's reactions to phasic and tonic peripheral afferent stimuli. It is also shown that the functional requirements imposed by the mathematical model of the SMCS comply with the current knowledge about the related properties of spinal neuronal circuitry. The crucial role of the spinal presynaptic inhibition mechanism in the neuronal implementation of SIP is elucidated. Important differences between the IM and a state predictor employed for compensating for a neural reflex time delay are discussed. Received: 8 February 2000 / Accepted: 24 March 2000     

Suspending loads decreases load stability but may slightly improve body stability

Here, we seek to determine how compliantly suspended loads could affect the dynamic stability of legged locomotion. We theoretically model the dynamic stability of a human carrying a load using a coupled spring-mass-damper model and an actuated spring-loaded inverted pendulum model, as these models have demonstrated the ability to correctly predict other aspects of locomotion with a load in prior work, such as body forces and energetic cost. We report that minimizing the load suspension natural frequency and damping ratio significantly reduces the stability of the load mass but may slightly improve the body stability of locomotion when compared to a rigidly attached load. These results imply that a highly-compliant load suspension could help stabilize body motion during human, animal, or robot load carriage, but at the cost of a more awkward (less stable) load.     

How well can spring-mass-like telescoping leg models fit multi-pedal sagittal-plane locomotion data?

Idealized mathematical models of animals, with point-mass bodies and spring-like legs, have been used by researchers to study various aspects of terrestrial legged locomotion. Here, we fit a bipedal spring-mass model to the ground reaction forces of human running, a horse trotting, and a cockroach running. We find that, in all three cases, while the model captures center-of-mass motions and vertical force variations well, horizontal forces are less well reproduced, primarily due to variations in net force vector directions that the model cannot accommodate. The fits result in different apparent leg stiffnesses in the three animals. Assuming a simple fixed leg-angle touch-down strategy, we find that the gaits of these models are stable in different speed-step length regimes that overlap with those used by humans and horses, but not with that used by cockroaches.     

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

 Motivated by experimental studies of insects, we propose a model for legged locomotion in the horizontal plane. A three-degree-of freedom, energetically conservative, rigid-body model with a pair of compliant virtual legs in intermittent contact with the ground allows us to study how dynamics depends on parameters such as mass, moment of inertia, leg stiffness, and length. We find periodic gaits, and show that mechanics alone can confer asymptotic stability of relative heading and body angular velocity. We discuss the relevance of our idealized models to experiments and simulations on insect running, showing that their gait and force characteristics match observations reasonably well. We perform parameter studies and suggest that our model is relevant to the understanding of locomotion dynamics across species. Received: 17 April 2001 / Accepted in revised form: 20 November 2001     

Joint stiffness of the ankle and the knee in running

The spring-mass model is a valid fundament to understand global dynamics of fast legged locomotion under gravity. The underlying concept of elasticity, implying leg stiffness as a crucial parameter, is also found on lower motor control levels, i.e. in muscle-reflex and muscle-tendon systems. Therefore, it seems reasonable that global leg stiffness emerges from local elasticity established by appropriate joint torques. A recently published model of an elastically operating, segmented leg predicts that proper adjustment of joint elasticities to the leg geometry and initial conditions of ground contact provides internal leg stability. Another recent study suggests that in turn the leg segmentation and the initial conditions may be a consequence of metabolic and bone stress constraints. In this study, the theoretical predictions were verified experimentally with respect to initial conditions and elastic joint characteristics in human running. Kinematics and kinetics were measured and the joint torques were estimated by inverse dynamics. Stiffnesses and elastic nonlinearities describing the resulting joint characteristics were extracted from parameter fits. Our results clearly support the theoretical predictions: the knee joint is always stiffer and more extended than the ankle joint. Moreover, the knee torque characteristic on the average shows the higher nonlinearity. According to literature, the leg geometry is a consequence of metabolic and material stress limitations. Adapted to this given geometry, the initial joint angle conditions in fast locomotion are a compromise between metabolic and control effort minimisation. Based on this adaptation, an appropriate joint stiffness ratio between ankle and knee passively safeguards the internal leg stability. The identified joint nonlinearities contribute to the linearisation of the leg spring.     

Self-organized control of bipedal locomotion by neural oscillators in unpredictable environment

A new principle of sensorimotor control of legged locomotion in an unpredictable environment is proposed on the basis of neurophysiological knowledge and a theory of nonlinear dynamics. Stable and flexible locomotion is realized as a global limit cycle generated by a global entrainment between the rhythmic activities of a nervous system composed of coupled neural oscillators and the rhythmic movements of a musculo-skeletal system including interaction with its environment. Coordinated movements are generated not by slaving to an explicit representation of the precise trajectories of the movement of each part but by dynamic interactions among the nervous system, the musculo-skeletal system and the environment. The performance of a bipedal model based on the above principle was investigated by computer simulation. Walking movements stable to mechanical perturbations and to environmental changes were obtained. Moreover, the model generated not only the walking movement but also the running movement by changing a single parameter nonspecific to the movement. The transitions between the gait patterns occurred with hysteresis.     

Bio-Inspired Controller for a Robot Cheetah with a Neural Mechanism Controlling Leg Muscles

The realization of a high-speed running robot is one of the most challenging problems in developing legged robots.The excellent performance of cheetahs provides inspiration for the control and mechanical design of such robots.This paper presents a three-dimensional model of a cheetah that predicts the locomotory behaviors of a running cheetah.Applying biological knowledge of the neural mechanism,we control the muscle flexion and extension during the stance phase,and control the positions of the joints in the flight phase via a PD controller to minimize complexity.The proposed control strategy is shown to achieve similar locomotion of a real cheetah.The simulation realizes good biological properties,such as the leg retraction,ground reaction force,and spring-like leg behavior.The stable bounding results show the promise of the controller in high-speed locomotion.The model can reach 2.7 m·s- 1 as the highest speed,and can accelerate from 0 to 1.5 m·s -1 in one stride cycle.A mechanical structure based on this simulation is designed to demonstrate the control approach,and the most recently developed hindlimb controlled by the proposed controller is presented in swinging-leg experiments and jump-force experiments.