# Quadrupedal Robots' Gaits Identification via Contact Forces Optimization
## 1 Introduction  
Nature has always been a source of inspiration for engineers and scientists who tried to replicate or mimick natural bio-mechanisms. In fact, this can be a smart strategy, since any natural living being is a highly optimized system over millions of years of evolution [1, 2, 3]. Magnificent examples of engineering bio-inspired devices date from very ancient Magna Greece times, as the [PERSON]'s dove automaton, and a large number of automata within the [PERSON]'s tradition. The masterful [PERSON]'s machines, such as the lion or the knight automaton, or the flying machines, some inspired by the flapping wings of birds, are also astonishing examples. To date, the development of automatic systems, in particular quadrupedal robots, has found great motivation in the field of human assistance. To assist and cooperate with humans, robots must be able to move easily in workplaces originally designed according to human needs [4]. For example, climbing stairs, avoiding obstacles, or opening doors [5, 6]. Among the many technical difficulties to be faced and solved, one of the most important aspects is related to portability and its duration. This scenario justifies the need to optimize the power and energy of locomotion to make these systems more efficient, compact, characterized by limited energy consumption, and using small size and weight actuators. Interesting questions arise: is it possible to conceive more efficient quadrupeds than those proposed by nature? Is it possible to disclose a different kind of gaits for a quadrupedal mechanism as the product of a strict optimization process? For example, depending on speed, energy consumption, or jerk minimization. Many engineers and robot designers assume the gait of animals is optimal since they would have been able to survive the competition and natural selection [7, 8, 9, 10]. For this reason, the quadruped robot tends to be designed inspired by the examples of nature [11, 12]. However, biological kinematics of locomotion is not reproducible directly by a legged robot, since biological muscles are extraordinarily efficient components, and animals can produce energy for their actuation by grabbing nutritive elements from the environment, transformed into actuationenergy by a digestive process. This permits them to have a relatively light energy storage system. Moreover, artificial sensors are still inferior, in terms of number, miniaturization, and sensitivity against the ones owned by animals [13]. To date, designers tend to create robots that resemble nature as closely as possible, such as imposing specific gaits such as walking, trotting, or galloping and developing mathematical optimization processes that only concern sub-systems, to overcome the technological limitations [14; 15]. There are many different approaches for performing the optimization. In [16], for example, the authors use harmonic functions to describe foot trajectory and tune some parameters such as amplitude, phase, and frequency. In [17; 18] the authors generate leg trajectories for overcoming small terrain irregularities by optimizing consumed energy. In [19] the authors optimize contact forces to walk on a high-slope terrain, but the gait is imposed a priori. Evolutionary computation, such as genetic algorithms (GAs), is often a natural choice for the gait optimization of legged robots since it uses optimization methods based on evolutionary theory [20]. Also, the controls of the mentioned cases, tend to mimic or approach the natural movement of the quadrupeds found in nature. Typical controls are central pattern generators [21; 22], hybrid zero dynamics [23], or the usual PD [24]. The present work breaks from all the problems relating to technological limitations and lays the foundations of how it is possible to identify, a priori, optimal gaits through parametric optimization algorithms. The authors propose an algorithm that does not depend either on the kinematic chains of the legs or power actuators.