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Locomotion is a behaviour resulting from the interaction of the nervous and musculoskeletal systems and the environment. However, the musculoskeletal systems of some terrestrial mammals present an intrinsic ability to realize a

By the transition from the sprawled reptilian leg configuration to the parasagittal mammalian limb, one additional segment was added in the fore and hindlimbs (scapula and elongation of the tarsus). Thus, scapula and femur, humerus and tibia, ulna and tarsus became functional correspondents in mammals

The locomotor apparatus evolved under the selective pressure of structural, (eco-) physiological and mechanical constraints. Among them, efficiency in energy consumption is certainly one of the most important constraints and is expressed in two dual ways to handle with energy during (loco-)motion.

First: reduction of muscular work as much as possible. Kustnezov

Second: storage or conversion of kinetic energy during one phase to recover it in another phase of the movement

The named strategies are a response to the physical constraints applied by interactions of the body with its environment. Mechanical interaction as one of the components of this interaction (beside physicochemical) is often explained as the necessity to support the body weight, i.e. to resist gravitational forces. But this is also true during locomotion. Inverse dynamic analysis demonstrates that the net torques created in the joints by muscles work mainly against gravity

The stability of locomotion particularly at key events – as escaping or hunting if stability and manoeuvrability are most important – is a second vector of this interaction. The study of the relationships between stability and morphology has to be embedded into one possible approach described by this contribution.

A standing quadrupedal animal is in static equilibrium. The feet, as points of ground contact, map a polygon called polygon of support. As long as the vertical projection of its centre of body mass remains inside the polygon, the animal is said to be statically stable. This is the case during slow walking, which is described as a succession of quasi-static equilibriums. Three legs are always in contact to the ground. The triangle of support changes periodically, and the vertical projection of the centre of mass then moves from one triangle to the next. At any time the animal is able to stop its forward motion if the speed is low and therefore the quantity of movement (mass time speed) is small. Cartmill et al.

Stability qualifies equilibrium. Equilibrium is said to be statically stable if a system returns to its initial equilibrium after a perturbation as for instance a pendulum after deflection from its vertical position. In case of a cyclic movement the motion of a system is said stable if it is able to go back to its nominal trajectory after perturbation. This restrictive definition of dynamic stability represents more than simply remaining on the feet. Since motion of a living organism underlies variability, the given definition is rarely realized in nature. In many studies on stability of motion systems, particularly in the robotics, a system is more pragmatically assumed stable if it reaches a neighbour stable state after a perturbation, even if it does not return exactly to its nominal state.

Locomotion is generally described as the behaviour that results from the interaction of the musculoskeletal system and the neural system and environment components. It is a complex task whose exhaustive description would require hundreds of variables. The aim of our modelling is to reduce the number of variables – i.e. the number of degrees of freedom (d.o.f.) – to a necessary minimum.

Due to the different physical mechanisms acting in slow and fast locomotion, the number of d.o.f. is reduced by addressing these two types of locomotion separately.

Applying Newton's first principle of dynamics, the forces exerted by the limbs on the ground (i.e. ground reaction forces) and measured by the use of ergo meters

During trotting and in-phase gaits, motions of the centre of mass of mammals occur mainly in the parasagittal plane. At trot, gait symmetry and resulting torsions of the spine around the longitudinal body axis lead this parasagittal motion of the CoM. At in-phase gaits, spine bending movements lead to extensive parasaggital movements of the pelvis, which contribute up to half of the stride length in small mammals

Most simulations studies of the last 15 years can be classified into three types, which differ in the description of the interaction between neuronal and mechanical systems and emerged after the publication of three basic articles: Taga

Tagas' purpose

Morphologists can learn from complex simulations if they address the right level of complexity. But the quality of the extrapolations made with very complex models (e.g.,

Thus, for the study of the evolution of musculoskeletal design, it seems to be more promising to comprehend a construction step by step and reconstruct it stepwise. In the field opened by Taga's work – i.e. coupling of one mechanical and one neuronal system – as a morphologist, the most interesting simulations are

McGeer

Blickhan

Farley et al. ^{–1}) – and thus the role of compliance – was already recognized as one determinant of locomotion

The stiffness of a musculoskeletal structure can be split into two major components: (

Dynamic stability of the spring-mass system – i.e. its ability to bounce stably – was studied first by Schwind and Koditchek

Approximations are numerous by spring-mass models. During forward movement, the spring-mass model describes only the stance phase of a spring leg, not the swing phase. The spring leg is assumed to be massless and from the point of view of the dynamics a massless object cannot realize a motion. Thus, in the simulation, the leg is artificially positioned with a given angle of attack during the flight phase of the spring-mass system before the next ground contact. This is an important restriction, but 85% of the mass is included in the head–trunk structure in small mammals. Limbs of small mammals are light indeed and therefore assuming massless legs is appropriate. Additionally, the model assumes that ground reaction forces keep the direction of the spring – towards the centre of mass – and this is true as well in humans as in quadrupeds at constant speed and during the middle stance phase

The comparison between the spring-mass system and galloping small mammals show additional descriptive limits. The simple spring-mass model trajectory undulates with one maximum and one minimum per cycle, whereas the centre of mass of galloping small mammals may have more than two extremums within each locomotor cycle. This gives rise to the development of extended spring-mass models in order to take the back dynamics into account (

The spring-mass model developed from the status of a simple mechanical model for human hopping to a neuromechanical template. Full and Koditschek

The introduction of the

The spring-mass model is a model at the level of organisms. Limb stiffness can be compared between species after normalization by mass and length. Blickhan and Full _{max}/

For comparative anatomists, global models as the spring-mass model are less informative since morphometric data normally are not involved in the models. It is possible to introduce them into a model, by distributing the elasticity (of the spring) onto the joints and thus by introducing a massless polysegmental kinematic connected with rotational springs (

Seyfarth et al. _{i}
_{i}
_{0})^{ν} (Eq. (1)). That is the torques–angle characteristics do not show hysteresis. Neglecting the torques at the foot pad, the torques equilibrium equation then simplifies into _{1} _{1} + _{2} _{2} = 0 (Eq. (2)) (

Assuming symmetrical loading the torques equilibrium equation (Eq. (2)) then simplifies into _{12}/_{23} = _{1}/_{3} (

Nowadays no quadruped model has been studied in the same detail. One prerequisite is a better experimental knowledge of the torque–aperture characteristics in quadruped mammals in order to limit the parameter space in the simulations. The experimental determination of the torque–aperture relationships (and following of the stiffness laws) at the level of each joint is accessible performing inverse dynamic calculations of the limbs. These experimental characteristics will also point out the level (distal versus proximal) at which energy dissipation occurs in the limbs during a cycle of gallop. Spring-mass models disregard up to now fully energy dissipation up to the theoretical work of Berkemeier

The influence of morphology onto stability cannot be understood intuitively and necessitates the use of numerical simulations. Stability

Actual small mammals, whose size is comparable to the oldest fossils of mammals, are a common point of interest for the paleontology and functional morphology. They are good candidates for the required experimental determination of the torque–aperture characteristics. Their body size enables for the most precise localization of all joint positions, including the most proximal using monoplanar videoradiography in a two dimensional frame. Moreover, their size enables for the measurement of the ground reaction forces over many locomotor cycles, a prerequisite for a valuable integration of the ground reaction forces and thus for the determination of the position of the centre of mass. Thus small mammals are appropriate to perform inverse dynamic calculations combining high-speed videoradiography (500 fps), force measurement and complementary electromyography. The synchronization of all these signals is – 100 years after Marey's effort to capture the instant in the movement – a realizable

(

Fig. 1. (