# Dynamic Balance of the Head in a Flexible Legged Robot for Efficient Biped Locomotion
## 4 Head Vibration Reduction of the Rollo Model: Results and Discussion
### Head Vibration Reduction  
The head vibration problem is related to the 3 rd mode shapes of the robot on the floor, which has a relatively small damping ratio \(\zeta=2.2\%\). If the damping ratio of this mode shape was increased, the vibration level of the Robot head would be substantially reduced. The structural modification must be focused on increasing the 3 rd mode shape's damping ratio only; a general increment of the system dissipations would downgrade the locomotion performance, dissipating more energy at each step and reducing the robot range. Head oscillation could be attenuated using an active control strategy; usually, a velocity feedback control achieves pretty good performances in attenuating selective harmonics [47, 48, 49]. An active control strategy requires a sensor and an actuator at the head location to measure the dynamics and provide the force to attenuate the head oscillation. The undesired vibration frequency is \(f_{fund}=4.1\) Hz; therefore, the active control logic sampling frequency could be around 200 Hz, i.e., not extraordinarily high and suitable for an iPad. Hence, the accelerometer sensor included in the iPad could be used as sensor because the iPad is located at the robot head, precisely where the vibrations should be measured. Therefore, the sensor is already included in the robot, without any structural modification. The control logic could also be implemented in the iPad. The force could be provided by an inertial actuator placed at the Rollo head [50]. The necessity of an actuator is the main disadvantage of the active control strategy; the main reasons are: (1) the size and weight of the inertial actuator able to provide the required force could exceed the mass limit constrain and (2) it should provide a continuous force during the robot locomotion, which means it requires continuous energy to be efficient which is even a more important problem. Optimisation strategies to minimize the actuator required power exist [51]; however, the energy used for the head vibration reduction substantially reduces the robot range. The energy required is an important drawback of active control in the specific application, which propounds for implementing a passive solution. Figure 8: Acceleration at the supporting head beam during locomotion. Figure 7: Rollo wheels displacement and applied forces during human-like locomotion. The requirements for the selective increment of damping ratio and passive damping design suggest the use of an optimized vibration absorber, placed on the supporting beam, to attenuate head vibrations. A dynamic vibration absorber consists of a mass \(m_{a}\) connected to the primary system through a spring, having stiffness \(k_{a}\), and a viscous damper, having damping coefficient \(c_{a}\). The dynamic vibration absorber adds a dof to the system, but if the dynamic vibration absorber is tuned on a system natural frequency, it reduces the original system oscillation amplitude. However, an additional natural frequency related to the added dof is generated. The optimal calibration of spring stiffness \(k_{a}\) and damper viscous damping coefficients \(c_{a}\) let to minimize the response amplitude of the system at an arbitrary undesired frequency \(f_{und}\). In the case of single degree of freedom system (sdorf), the optimal values of mass and stiffness are tuned on the undesired natural frequency \(f_{und}\) and optimized to minimize the amplitude of the two resonances in the inertance transfer function, as follow:  
\[f_{und}=\frac{1}{2\pi}\sqrt{\frac{k}{m}}=\frac{1}{2\pi}\sqrt{\frac{k_{a}}{m_{a }}},\quad c_{a}=\sqrt{km}\sqrt{\frac{3\mu^{3}}{2(1+\mu)^{3}}},\quad\mu=\frac{m _{a}}{m} \tag{5}\]  
where \(m\) and \(k\) are the mass and stiffness of the sdorf system. The extension of the tuned mass damper to multi degrees of freedom (mdof) systems makes the analytical computation of \(k_{a}\) and \(c_{a}\) dependent on the system properties and the number of dofs.