Hartmut Geyer

Compliant leg behaviour explains basic dynamics of walking and running

The basic mechanics of human locomotion are associated with vaulting over stiff legs in walking and rebounding on compliant legs in running. However, while rebounding legs well explain the stance dynamics of running, stiff legs cannot reproduce that of walking. With a simple bipedal spring–mass model, we show that not stiff but compliant legs are essential to obtain the basic walking mechanics; incorporating the double support as an essential part of the walking motion, the model reproduces the characteristic stance dynamics that result in the observed small vertical oscillation of the body and the observed out-of-phase changes in forward kinetic and gravitational potential energies. Exploring the parameter space of this model, we further show that it not only combines the basic dynamics of walking and running in one mechanical system, but also reveals these gaits to be just two out of the many solutions to legged locomotion offered by compliant leg behaviour and accessed by energy or speed.

A movement criterion for running.

The adjustment of the leg during running was addressed using a spring-mass model with a fixed landing angle of attack. The objective was to obtain periodic movement patterns. Spring-like running was monitored by a one-dimensional stride-to-stride mapping of the apex height to identify mechanically stable fixed points. We found that for certain angles of attack, the system becomes self-stabilized if the leg stiffness was properly adjusted and a minimum running speed was exceeded. At a given speed, running techniques fulfilling a stable movement pattern are characterized by an almost constant maximum leg force. With increasing speed, the leg adjustment becomes less critical. The techniques predicted for stable running are in agreement with experimental studies. Mechanically self-stabilized running requires a spring-like leg operation, a minimum running speed and a proper adjustment of leg stiffness and angle of attack. These conditions can be considered as a movement criterion for running.

Swing-leg retraction: a simple control model for stable running

SUMMARY In running, the spring-like axial behavior of stance limbs is a well-known and remarkably general feature. Here we consider how the rotational behavior of limbs affects running stability. It is commonly observed that running animals retract their limbs just prior to ground contact, moving each foot rearward towards the ground. In this study, we employ a conservative spring-mass model to test the effects of swing-leg retraction on running stability. A feed-forward control scheme is applied where the swing-leg is retracted at constant angular velocity throughout the second half of the swing phase. The control scheme allows the spring-mass system to automatically adapt the angle of attack in response to disturbances in forward speed and stance-limb stiffness. Using a return map to investigate system stability, we propose an optimal swing-leg retraction model for the stabilization of flight phase apex height. The results of this study indicate that swing-leg retraction significantly improves the stability of spring-mass running, suggesting that swing-phase limb dynamics may play an important role in the stabilization of running animals.

Positive force feedback in bouncing gaits

During bouncing gaits (running, hopping, trotting), passive compliant structures (e.g. tendons, ligaments) store and release part of the stride energy. Here, active muscles must provide the required force to withstand the developing tendon strain and to compensate for the inevitable energy losses. This requires an appropriate control of muscle activation. In this study, for hopping, the potential involvement of afferent information from muscle receptors (muscle spindles, Golgi tendon organs) is investigated using a two–segment leg model with one extensor muscle. It is found that: (i) positive feedbacks of muscle–fibre length and muscle force can result in periodic bouncing; (ii) positive force feedback (F+) stabilizes bouncing patterns within a large range of stride energies (maximum hopping height of 16.3 cm, almost twofold higher than the length feedback); and (iii) when employing this reflex scheme, for moderate hopping heights (up to 8.8 cm), an overall elastic leg behaviour is predicted (hopping frequency of 1.4–3 Hz, leg stiffness of 9−27 kN m−1). Furthermore, F+ could stabilize running. It is suggested that, during the stance phase of bouncing tasks, the reflex–generated motor control based on feedbacks might be an efficient and reliable alternative to central motor commands.

Intelligence by mechanics.

Research on the biomechanics of animal and human locomotion provides insight into basic principles of locomotion and respective implications for construction and control. Nearly elastic operation of the leg is necessary to reproduce the basic dynamics in walking and running. Elastic leg operation can be modelled with a spring-mass model. This model can be used as a template with respect to both gaits in the construction and control of legged machines. With respect to the segmented leg, the humanoid arrangement saves energy and ensures structural stability. With the quasi-elastic operation the leg inherits the property of self-stability, i.e. the ability to stabilize a system in the presence of disturbances without sensing the disturbance or its direct effects. Self-stability can be conserved in the presence of musculature with its crucial damping property. To ensure secure foothold visco-elastic suspended muscles serve as shock absorbers. Experiments with technically implemented leg models, which explore some of these principles, are promising.

A neural circuitry that emphasizes spinal feedback generates diverse behaviours of human locomotion

It is often assumed that central pattern generators, which generate rhythmic patterns without rhythmic inputs, play a key role in the spinal control of human locomotion. We propose a neural control model in which the spinal control generates muscle stimulations mainly through integrated reflex pathways with no central pattern generator. Using a physics‐based neuromuscular human model, we show that this control network is sufficient to compose steady and transitional 3‐D locomotion behaviours, including walking and running, acceleration and deceleration, slope and stair negotiation, turning, and deliberate obstacle avoidance. The results suggest feedback integration to be functionally more important than central pattern generation in human locomotion across behaviours. In addition, the proposed control architecture may serve as a guide in the search for the neurophysiological origin and circuitry of spinal control in humans.

The 3-D Spring–Mass Model Reveals a Time-Based Deadbeat Control for Highly Robust Running and Steering in Uncertain Environments

Over the past three decades, the spring-mass model has developed into the basic behavior model to study running in animals and robots. In the planar version, this model has helped to reveal and understand the passive stabilization of running in the horizontal and sagittal planes, and to derive from this knowledge control strategies for running robots. However, only few attempts have been made to transfer the knowledge to 3-D locomotion. Here, we show that the 3-D spring-mass model reveals a deadbeat control that does not require feedback about the actual ground level to produce highly robust running and steering in uncertain environments. The control naturally extends the time-based control derived for the planar version of this model and allows it to navigate rough terrain, while stabilizing running and steering. Using this control strategy, we demonstrate in simulation that a human-like system running at 5 ms-1 tolerates frequent ground disturbances up to 30% of the leg length. Moreover, we find that the control outperforms a classical leg-placement strategy in terms of turning rate and disturbance rejection if the relative errors in system energy and the other model parameters stay small ( 10%). Our results suggest that the time-based control can be a powerful alternative for leg-placement strategies in highly maneuverable running robots.

SPRING-LEGGED LOCOMOTION ON UNEVEN GROUND: A CONTROL APPROACH TO KEEP THE RUNNING SPEED CONSTANT

We present a feedforward control for spring-legged systems in uneven terrain which keeps the running speed constant. The control uses the unique transformation between ground level and flight time to automatically adapt the system parameters during flight to the actual ground level without having to detect it. We further demonstrate how this control for constant speed running can simultaneously be combined with other control strategies in spring-legged systems to adapt their behavior to a desired locomotion task.

Running and Walking with Compliant Legs

It has long been the dream to build robots which could walk and run with ease. To date, the stance phase of walking robots has been characterized by the use of either straight, rigid legs, as is the case of passive walkers, or by the use of articulated, kinematically-driven legs. In contrast, the design of most hopping or running robots is based on compliant legs which exhibit quite natural behavior during locomotion.

Extension and customization of self-stability control in compliant legged systems.

Several recent studies on the control of legged locomotion in animal and robot running focus on the influence of different leg parameters on gait stability. In a preceding investigation self-stability controls showing deadbeat behavior could be obtained by studying the dynamics of the system in dependence of the leg orientation carefully adjusted during the flight phase. Such controls allow to accommodate disturbances of the ground level without having to detect them. Here we further this method in two ways. Besides the leg orientation, we allow changes in leg stiffness during flight and show that this extension substantially improves the rejection of ground disturbances. In a human like example the tolerance of random variation in ground level over many steps increased from 3.5% to 35% of leg length. In single steps changes of about 70% leg length (either up or down) could be negotiated. The variable leg stiffness not only allows to start with flat leg orientations maximizing step tolerances but also increase the control subspace. This allows to customize self-stability controls and to consider physical and technical limitations found in animals and robots.