Andre Seyfarth

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.

Stable Running with Segmented Legs

Spring-like leg behavior is found in both humans and animals when running. In a spring-mass model, running proves to be self-stable in terms of external perturbations or variations in leg properties (for example, landing angle). However, biological limbs are not made of springs, rather, they consist of segments where spring-like behavior can be localized at the joint level. Here, we use a two-segment leg model to investigate the effects of leg compliance originating from the joint level on running stability. Owing to leg geometry a non-linear relationship between leg force and leg compression is found. In contrast to the linear leg spring, the segmented leg is capable of reducing the minimum speed for self-stable running from 3.5 m s-1 in the spring-mass model to 1.5 m s-1 for almost straight joint configurations, which is below the preferred transition speed from human walking to running (≈2 m s-1). At moderate speeds the tolerated range of landing angle is largely increased (17° at 5 m s-1) compare with the linear leg spring model (2°). However, for fast running an increase in joint stiffness is required to compensate for the mechanical disadvantage of larger leg compression. This could be achieved through the use of non-linear springs to enhance joint stiffness in fast running.

Upright human gait did not provide a major mechanical challenge for our ancestors.

Habitual bipedalism is considered as a major breakthrough in human evolution and is the defining feature of hominins. Upright posture is presumably less stable than quadrupedal posture, but when using external support, for example, toddlers assisted by their parents, postural stability becomes less critical. In this study, we show that humans seem to mimic such external support by creating a virtual pivot point (VPP) above their centre of mass. A highly reduced conceptual walking model based on this assumption reveals that such virtual support is sufficient for achieving and maintaining postural stability. The VPP is experimentally observed in walking humans and dogs and in running chickens, suggesting that it might be a convenient emergent behaviour of gait mechanics and not an intentional locomotion behaviour. Hence, it is likely that even the first hominis may have already applied the VPP, a mechanism that would have facilitated the development of habitual bipedalism.

Dynamics of the long jump

A mechanical model is proposed which quantitatively describes the dynamics of the centre of gravity (c.g.) during the take-off phase of the long jump. The model entails a minimal but necessary number of components: a linear leg spring with the ability of lengthening to describe the active peak of the force time curve and a distal mass coupled with nonlinear visco-elastic elements to describe the passive peak. The influence of the positions and velocities of the supported body and the jumpers leg as well as of systemic parameters such as leg stiffness and mass distribution on the jumping distance were investigated. Techniques for optimum operation are identified: (1) There is a minimum stiffness for optimum performance. Further increase of the stiffness does not lead to longer jumps. (2) For any given stiffness there is always an optimum angle of attack. (3) The same distance can be achieved by different techniques. (4) The losses due to deceleration of the supporting leg do not result in reduced jumping distance as this deceleration results in a higher vertical momentum. (5) Thus, increasing the touch-down velocity of the jumpers supporting leg increases jumping distance.

Stable operation of an elastic three-segment leg.

Abstract. Quasi-elastic operation of joints in multi-segmented systems as they occur in the legs of humans, animals, and robots requires a careful tuning of leg properties and geometry if catastrophic counteracting operation of the joints is to be avoided. A simple three-segment model has been used to investigate the segmental organization of the leg during repulsive tasks like human running and jumping. The effective operation of the muscles crossing the knee and ankle joints is described in terms of rotational springs. The following issues were addressed in this study: (1) how can the joint torques be controlled to result in a spring-like leg operation? (2) how can rotational stiffnesses be adjusted to leg-segment geometry? and (3) to what extend can unequal segment lengths and orientations be advantageous? It was found that: (1) the three-segment leg tends to become unstable at a certain amount of bending expressed by a counter-rotation of the joints; (2) homogeneous bending requires adaptation of the rotational stiffnesses to the outer segment lengths; (3) nonlinear joint torque-displacement behaviour extends the range of stable leg bending and may result in an almost constant leg stiffness; (4) biarticular structures (like human gastrocnemius muscle) and geometrical constraints (like heel strike) support homogeneous bending in both joints; (5) unequal segment lengths enable homogeneous bending if asymmetric nominal angles meet the asymmetry in leg geometry; and (6) a short foot supports the elastic control of almost stretched knee positions. Furthermore, general leg design strategies for animals and robots are discussed with respect to the range of safe leg operation.

Swing leg control in human running

Humans can run within a wide range of speeds without thinking about stabilizing strategies. The leg properties seem to be adjusted automatically without need for sensory feedback. In this work, the dynamics of human running are represented by the planar spring mass model. Within this framework, for higher speeds, running patterns can be stable without control strategies. Here, potential strategies that provide stability over a broader range of running patterns are considered and these theoretical predictions are compared to human running data. Periodic running solutions are identified and analyzed with respect to their stability. The control strategies are assumed as linear adaptations of the leg parameters-leg angle, leg stiffness and leg length-during the swing phase. To evaluate the applied control strategies regarding their influence on landing behavior, two parameters are introduced: the velocity of the foot relative to the ground (ground speed matching) and the foots angle of approach. The results show that periodic running solutions can be stabilized and that control strategies, which guarantee running stability, are redundant. For any swing leg kinematics (adaptation of the leg angle and the leg length), running stability can be achieved by adapting the leg stiffness in anticipation of the ground contact.