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. [10] show on the basis of a large set of footfall patterns that quadruped mammals prefer limb coordinations that maximize the time of existence of such a polygon of support during walking. If speed is increased, a preference for limb coordinations that involve diagonal pairs of limbs is given. During trotting, the polygon is reduced to a diagonal line connecting one hand and the ipsilateral foot. Thus, trotting animals have already leaved the frame of a succession of quasi-static equilibrium state to meet dynamic equilibrium. Idem gallop, bound, half-bound.

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 [11] can be doubly integrated numerically. The integration leads to the motion of the centre of mass (c.o.m.) of the body.

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 [21], [23], [24] and [51]. Therefore, the lateral excursion of the centre of mass is smaller than its vertical ones. Thus, modelling a quadruped mammal during fast locomotion is reasonable in a two-dimensional plane, whereas a two-dimensional model of slow locomotion will be a partial description of the reality. It is assumed that models deal with straightforward locomotion.

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 [57] modelled a rough musculoskeletal and a rough nervous system separately, whose interaction produces a stable cyclic motion. McGeer [31] studies the passive walking of a structure along an inclined plane, and investigates the passive properties of the mechanics disregarding voluntary the role of the nervous system. In direct line to the works of Cavagna and Taylor, Blickhan [7] proposed a synthetic model taking into account elastic energy storage during running: the spring-mass model.

The stiffness of a musculoskeletal structure can be split into two major components: (i) a passive component linked to the intrinsic properties of the materials and (ii) an active component (depending on muscle activation and geometry). Muscles force does not necessary result in movement. If agonist and antagonist muscles are activated synchronously so that torques of agonist and antagonist muscles remain in equilibrium, no movement will occur in the articulation, but its stiffness will be increased. Feldman [19] demonstrates that the control of only one parameter is sufficient to control joint stiffness and aperture.

Dynamic stability of the spring-mass system – i.e. its ability to bounce stably – was studied first by Schwind and Koditchek [53]. They derived an analytical criterion from the linearised (i.e. approximated) equation of motion. Seyfarth et al. [54] performed dynamic simulation of a non-linearised model and demonstrate its ability to bounce stably for some combinations of the forward speed, the spring-leg stiffness and the angle of attack, which is the angle spanned by spring leg with the ground line at the instant of touch down. That is within a range of values of stiffness and angle of attack and mass position a stable behaviour can be generated (Fig. 1D). Thus small irregularities of the ground that would result in small changes of the mass position relative to the ground can be overcome without resetting the values of the system, i.e. without control. This concept of a mechanical stabilisation without external control is known in physics as self-stabilization.

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 [6].

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 (Fig. 1C). Hackert and Fischer [35] propose to see the flexion of the spine in small mammals and the associated forward displacement of the centre of mass as a way to adjust the angle of attack of forelimbs and thus to use mechanical self-stabilization mechanism. Back flexion co-determines the position of the centre of mass and back stiffness sets the rough instant of touch down of the forelimb by controlling the downwards motion of the centre of mass during hindlimbs' stance phase. In respect to the simple spring-mass system, Seyfarth et al. [56] implement a rotation of the spring leg during its late swing phase when the mass has reached the apex of its trajectory. With this, the authors show that a tuning of the instant and verticality of the leg at touch down may influence positively the dynamic stability of the system. This may explain the retraction of the forelimb observed in galloping mammals before touch down [24].

The spring-mass model developed from the status of a simple mechanical model for human hopping to a neuromechanical template. Full and Koditschek [25] proposed to face the spring mass template like an asymptotical limit to which running systems tend to. Moreover, the authors conceptualized a method (rather new in the field of motion studies) to handle with complex system involving a lot of degrees of freedom. First: reduce the number of degrees of freedom by projecting the problem onto one plane, second then model it in this plane, then progressively include new additional degrees of freedom. This method is comparable to the early stage of movement learning in child [5], when degrees of freedom are introduced and the stiffness of the limbs reduced stepwise during growth [49].

The introduction of the neuromechanical parameter ‘stiffness’ leads to a level of description that enables provisory to ‘disregard’ how the stiffness is generated, i.e. to disregard the muscles, their insertion and activation. The number of degree of freedom of the biomechanical model becomes then considerably lower.

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 [8] introduce the relative stiffness k = (F _max/m g)/(Δl/l) and found a value around 8 for animals as different as insects and horses. This conveys that animals segmented limbs have comparable material characteristics.

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 (Fig. 1C). The stiffness is then the ratio variation of the joint torque over variation of the joint aperture, i.e. the slope of the tangent at the torque-aperture characteristics.

Seyfarth et al. [55] studied the stability of a massless three segmented leg very much in detail (for equations, see [55]) in order to describe its stabilizing behaviour in dependence to its initial configuration (zigzag or arc-like) and stiffness laws (constant or variable). The model assumes that the rotational springs act at the joints in a quasi elastic mode without dissipation, i.e. joint torques have the form T_i  = k_i  (ϕ – ϕ 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 T ₁ d ₁ + T ₂ d ₂ = 0 (Eq. (2)) (Fig. 1B). Seyfarth et al. assumed a symmetric loading of the limb during stance phase: the compression of a zigzag-like kinematical chain depends on the stiffness of its joints. With similar values of joints stiffness, a load results in a flexion of all joint angles at a time to the same extent, else the segmental chain would collapse first at the joint of less stiffness than others. But studies on the intralimb kinematics of small mammals [24] and [27] and birds [1], [58] and [59] during running show that changes occur in all joints at the same time and thus point to symmetrical loading and stiffness equilibrium.

Assuming symmetrical loading the torques equilibrium equation (Eq. (2)) then simplifies into k ₁₂/k ₂₃ = l ₁/l ₃ (Fig. 1B), whereas the length of the second segment vanish from the equations [49]. Numerical simulations of in place hopping of this elastic limb model lead to the following result: stability is enhanced if a short first segment corresponds to a long third segment and reciprocally [49]. Fig. 1E presents limb geometry of some terrestrial birds. Birds seem to follow this simple ‘rule’.

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 [4]. Inverse dynamic analysis is in small quadruped mammals a rather difficult but realizable task [57] since it is necessary to record separately the force exerted by each limbs. Outgoing from the torque–aperture relationships numerical simulations of a segmented biomechanical model of a quadruped with few degrees of freedom could be calculated looking for ‘optimal’ limb proportion with regard to the robustness of stability, i.e. the degree of disturbance the structure will accept to overcome without falling. The robustness of stability can be considered as an optimization criterion in the frame of the mechanical self-stabilization hypothesis.

The influence of morphology onto stability cannot be understood intuitively and necessitates the use of numerical simulations. Stability emerges for some value of the parameters – segment's length, articular stiffness…– and is fundamentally unpredictable. The hypothesis that intrinsic stability of the musculoskeletal mechanics exerted an evolutionary pressure that lead to the real body proportions was recently sustained by some simulation studies [33] and [54] that found a good correspondence between the values of the parameters that stabilize spring-mass models of humans and small mammals running, on the one hand, and the experimental values of stiffness and angle of attack, on the other hand. If stability generally results from the common action of the nervous and mechanical system, the nervous system can find in a self-stabilized mechanics a way to delegate and thus to improve one part of its work, particularly at high speed, where the reaction times have to be kept short. This would accord well with the decentralized structure of neural control (see Viala [60] and Jamont [40] in this issue).

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 [22] and [29] but exceptional task in integrative animal motion studies. Dynamic simulations on ‘hybrid’ models of model B and C (Fig. 1) should reveal whether the enhancement of the robustness of the mechanical self-stabilisation during locomotion acted as one determinant factor during evolution leading to actual body proportions.