# Dynamic Balance of the Head in a Flexible Legged Robot for Efficient Biped Locomotion
## 1 Introduction  
Robotics is a multidisciplinary research field, increasingly present in nowadays daily life. Robotics is modifying the classical design concepts, joining mechanics, electronics, and computer science knowledge, including life science considerations, resulting in the design of novel systems, which actively interact with the environment [1, 2, 3]. The design of such systems is innovative and includes many more aspects than classical systems design, e.g., variable stiffness actuators, soft robotics, dielectric elastomer actuators, artificial skins. The significant amount of possible interactions between different engineering fields increases the creativity and opens-up novel frontiers in design. Humanoid robotics is a novel research branch [4], oriented to conceive systems with human-like behaviour. However, the design of a machine, emulating the complexity of the human body and mind, is a big step to pursue; therefore, many systems are conceived to reduce the gaps between human and humanoid robots [5, 6, 7]. Among several human characteristics useful for humanoid robots, many research efforts were dedicated to replicate the locomotion mechanisms of humans and legged animals [8, 9], which are considered the reference systems of the bio-inspired robotic research [10]. The best compromise between locomotion performance and input power tothe actuators is desired to reproduce bio-inspired locomotion in robots. Energy harvesting approaches were proposed in [11; 12] to optimize the input power to actuators during the cyclic legged locomotion using passive elements. Calibrated actuations in biped and quadrupedal robots were adopted to improve the locomotion performance [2; 13]; algorithms for locomotion control were implemented on legged robots, such as the central pattern generator [14; 15] and the simplified Zero Moment Point (ZMP) approach [16] to perform biped locomotion in a planar surface [17]. In recent works [18; 19], the dynamic balance of biped robots is studied with the final aim to bypass the ZMP approach. However, the dynamic balance is an important problem in biped locomotion, and in order to solve the problem, some novel solutions have been conceived, e.g., wheels on feet [20; 21]. Recently, some of the present authors have proposed a novel biped wheeled robot with a flexible structure, named Rollo [22]. Rollo is a simple robot, merging the characteristics of biped wheeled robots with flexible legs to increase the performance and minimize the number of actuators. The robot flexibility helps the locomotion performance; however, an undesired oscillation of the robot head during the locomotion is generated, as shown in [23]. The head vibrations could reduce the dynamic balance of the robot, and a proper attenuation of the head oscillations is necessary to increase the safety of the human-robot interactions. Indeed, vibration damping [24] is a common problem for several engineering fields, e.g., civil [25; 26], automotive [27; 28], aerospace [29], and tooling machines [28]. Vibrations are usually reduced through active devices and control strategy [30; 31; 32] or passive damping [33; 34]. In robotics, the vibration damping is studied for increasing the bipedal or quadrupedal performance of robots, as shown in [2; 13; 35; 36]. This paper aims at reducing the head vibration level without modifying the properties of the robot. The robot head's oscillations are analyzed in detail, evaluating some causes of vibrations, e.g., the robot flexible structure and the friction with the ground. A solution to reduce vibrations is proposed and implemented in the robot model. In the proposed approach, the best compromise between performance and constraints is a passive solution, i.e., a tuned mass-spring-damper system (TMD). The tuned vibration absorber is probably the oldest and most commonly used passive device to attenuate vibrations. The first patent on this kind of device belongs to [PERSON] [37]; the device was later optimized for a single degree of freedom (sdof) system by [PERSON] [38].