Bio-inspiration and bio-mimetics are growing research fields. In the last years, the animal kingdom has served as an example in the development of several bio-inspired structures and mechanisms, especially in robotics. An example of this is legged locomotion, that allows agile movements, great stability on different terrains, high speed and energy efficient running .
Some of the first walking machines were ODEX one , PVII Quadruped Vehicle  and SILO4 , that achieved optimal gait patterns through a statically stable walking. These prototypes had important movement restrictions, since at least three of their legs had to be in stance with the ground.
In the field of running legged robots representative examples are KOLT , Scout II , BigDog , Star1ETH  and the MIT Cheetah 2 . However these prototypes do not take advantage of the flexibility of the trunk in order to achieve fast galloping speed and power efficiently as their natural counterparts do, like, e.g., cheetahs, horses and greyhounds .
In our current work, we focus on such key feature, and aim at designing legged robots with high energy efficiency and speed using a compliant backbone. In the literature, research devoted to understanding and developing the mechanics of bending of the trunk can be found. Three types of flexible trunks have been proposed: actuated, semi-actuated and passive.
In the class of robots with actuated trunk, the work presented in  showed how the performance of a quadrupedal can be improved actuated joints that allow flexion and extension of the trunk, compared to a rigid bodied robot. The quadruped robot presented in  was equipped with a tensegrity-based spine that helped maintaining movement and balance during gaits. Lastly, robot Stoch2  showed the advantages of actuated spine.
As far as legged robots with semi-actuated trunk is concerned, a comparison between three types of flexible spines in the Lynx-robot was performed in . The first of such spines was actuated for both flexion and extension movements, while the other two, made of a glass fiber rod with different stiffness, were actuated only for the flexion movement and the extension movement was passive.
Concerning legged robots with passive trunk, in  the effect of trunk flexibility on the dynamics of a quadruped robot running with a bounding gait was studied. This model was composed of two rigid bodies representing the hindquarters and forequarters, connected through a torsion spring. It was demonstrated that at a constant energy level, the trunk oscillation range and the average forward speed are inversely related. In  a robot with a flexible backbone whose stiffness could be changed varying the pressure of pneumatic actuators was used to study the stability of gait pattern for different the trunk stiffnesses. Finally, in  the difference between rigid and passive articulated trunk was investigated. The authors showed that the articulated trunk allowed longer strides and significantly affects the dynamics of the robot as well as its power efficiency.
In our current work we are investigating flexible backbones for legged robots, and how this can be exploited for fast and energy efficient running for quadruped robots. We have shown how a flexible spine can greatly help to achieve low power consumption storing and releasing energy during gait. Additionally, a dramatic energy saving can be obtained when the oscillations of the trunk reach a quasi-resonant regime.
It is important to highlight that in most rigid-bodied running quadrupedal robots the legs mass is considered negligible for the purpose of studying robots body dynamics. Such assumption allows a simpler dynamic modeling, and light legs allow faster movements, and therefore faster running. However, when it comes to flexible trunk, this plays a key role, since the motion of the masses of the legs (plus tail and head) and their contact with the ground generates the trunk bending [1, 20].
In a running gait, the center of mass of the leg reaches its lowest point at the middle of step. The kinetic energy and gravitational potential energy reaction force are stored as elastic energy during the stance phase, when the leg touches the ground, and recovered during the flight phase, when the leg leaves the ground (see Fig. 1, steps h, a, b).
The most studied model for robotic legs for running gaits is the SLIP (Spring Loaded Inverted Pendulum) model (see Figure 2, top). In the flight phase the spring has no effect and thus is not considered, so the dynamics of the leg is represented taking into account only the point mass.
One of the first works on the SLIP model is due to Marc Raibert , who showed that SLIP can describe the characteristics of running, trotting or hopping in one leg for bipeds and quadrupeds. Aspects such as stability, dynamics and energy efficiency can be taken into account in this model. Also, Fumiya Iida et al.  showed that walking can be described using the bipedal version of this model. Using the SLIP models, various types of quadruped robots have been development, such as KOLT , Scout II , BigDog , Start1ETH  and the MIT cheetah , which are capable of walking, trotting and galloping at high speeds in different terrains.
However, the SLIP model characterizes the dynamic formulation in a simple way, since it represents the robot’s leg as a point mass and a massless spring that extends towards to the ground. This neglects the inertia of the leg . Hence, this model falls short when it comes to the flight phase of the legs of a galloping robot, i.e. when the leg is not in stance with the ground, since it does not allow generating a force for bending the trunk. In most animals the mass of the leg is very important when performing the galloping movement, especially in quadrupeds with flexible trunks . As mentioned earlier, the mass of the legs helps bending the trunk, allowing it to store and release elastic energy, which allows smoother movements and a more energy efficient gallop [18–20].
Therefore, in order to study the effect of the legs’ masses in the dynamics, new models need to be developed. In this paper, we propose a Mass-Mass-Spring (MMS) leg model, as an alternative for quadruped robots with flexible trunk, and demonstrate that considering the mass of the leg in its dynamic modeling, it is possible to control the rotational force at the hip, and therefore induce a bending moment at the end of the trunk in the flight phase.
In the following we compare the proposed Mass-Mass-Spring (MMS) leg model with the Spring Loaded Inverted Pendulum (SLIP) model. As can be seen in Figure 2 (top), the SLIP model is composed of a point mass, M, which represents the hip, and a linear spring, k, that transmits the reaction forces between the ground and the hip, acting as energy storage during the stance phase. For more details about this model we refer the reader to [32–33]. The MMS model takes into account the mass of the leg, located at the knee joint, in addition to the mass of the hip, and a spring that establishes the contact between the mass of the leg and the ground, as shown in Fig. 2, (bottom). Thanks to the additional mass m, it is possible to model the forces that allow the trunk to bend in the flight phase.
 Such effect has been observed also in [21,22] and  for hopping robots with compliant legs, and on flexible backbones for fish-like robots . Also, a similar effect is also present in insects and birds, whose thoraxes contain compliant structures that accumulate and release energy during the flapping cycle at the benefit of consumption, and also flight stability, see, e.g. [25,26].