# 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
Therefore, in the case of mdof systems, the spring stiffness \(k_{a}\) and damper viscous coefficient \(c_{a}\) are minimizing the amplitude of the resonances of the transfer function between the input force to the system in operative condition and the point of the system which has undesired oscillations [52]. During the robot motion, the input forces are at the wheel-floor contacts, and the reference output is the center of the head supporting beam, where the robot head is located. The system input forces are the traction forces produced by the wheel-floor contacts; Rollo has four driving wheels, two wheels for each foot. The total vibrations at the robot head can be obtained using the superposition principle of the vibration produced by each wheel, and since the traction forces are all the same during straight motion, the sum of the transfer functions between each wheel and the head location had to be optimized. The value of the vibration absorber mass \(m_{a}\) is chosen considering the mass constraint given in Section 2, i.e., \(m_{a}=0.01m_{tot}=0.3\) kg, where \(m_{tot}=30\) kg is the total weight of the robot considering structure and accessories. The spring stiffness \(k_{a}\) and viscous damping coefficient \(c_{a}\) are obtained from the optimisation procedure. The objective function to minimize is the sum of the resonances amplitude in the transfer function between the driving wheels and the head location [43], in the frequency range \(2\div 6\) Hz, which includes the undesirable resonance frequency \(f_{und}=4.1\) Hz. The objective function trend changing \(k_{a}\) and \(c_{a}\) is shown in Figure 9a. The optimal stiffness \(k_{a}\) and damping \(c_{a}\), which contemporary minimize the response amplitude of both the Rollo head and the dynamics of the vibration absorber added mass \(m_{a}\) are \(k_{a}=158.2\) N/m and \(c_{a}=3.2\) N/(m/s), i.e., at the green point in Figure 9a. The natural frequencies and damping ration of the two resonance are \(f_{3}=3.3\) Hz, \(\xi_{3}=12.7\%\) and \(f_{4}=4.6\) Hz, \(\xi_{4}=14.6\%\). The tuned mass damper increases the damping ratio of the undesired dynamics more than one order of magnitude with respect to the original design. The driving wheel-head transfer function obtained with the proposed solution is compared to the original behaviour of [PERSON]'s head, in Figure 9b: the tuned mass damper splits the original resonances into two peaks, as expected, the optimal value obtained for \(k_{a}\) and \(c_{a}\) let to minimize the amplitude of the two peaks. The acceleration amplitude of the head due to excitation at wheel location is reduced by one order of magnitude, which is an excellent improvement. The forward motion simulation performed in Section 4 is performed again with the implementation of the proposed tuned mass damper. The robot head vibrations are compared to the original design in Figure 10. The amplitude of vibrations of robot head generated at each step are successfully reduced: the acceleration peaks decrease, and the vibrations are attenuated quickly. An objective evaluation of the TMD performance is given by the RMS of the head acceleration: in the robot original design the RMS was 0.55 m/s2, while in the improved version it is 0.34 m/s2. The human-like locomotion is not affected by the proposed modification, and this is visible looking at the robot foot displacements. In Figure 10(a), the foot displacements obtained with the improved robot model are compared to the foot displacements of the original design. The two simulations are performed with the same input forces, i.e., the foot engine generate the same torques, and the foot displacements are almost the same in the two simulations. Hence, the dynamic absorber does not affect the locomotion, as it was required by the admissible modification constraints. The elongation of the dynamic  
Figure 10: Acceleration of the supporting head beam during human-like locomotion using the dynamic adsorber. Figure 9: Optimisation of the acceleration at the supporting head beam during locomotion.