Issue 23

R. Vertechy et alii, Frattura ed Integrità Strutturale, 23 (2013) 47-56 ; DOI: 10.3221/IGF-ESIS.23.05 52 C ONTROL S YSTEM his section describes an interaction-force control system for the considered agonist-antagonist DE actuator. The controller is based on an optimum observer and on a suitable state-feedback law. In particular, for controller design purposes and owing to simulation results reported in the previous section, the actuator model has been linearized such that the considered prototype can be described via the lumped parameter system depicted in Fig. 7, where m ( m =105g) is the effective inertia of the actuator output (comprising the sensor mass too), K l ( K l = 66.5N/m) is a constant stiffness coefficient capturing the OFF-state quasi-static linear elastic response of the actuator, F int is the interaction force exchanged with the environment, F dist is a disturbance force accounting for the un- modelled non-linear and time-dependent mechanical response of the system (comprising DE film visco-elasticity), and F em,l is the “electric” force generated by the electrical activation of the agonist-antagonist DE films. Based on Eq. (2), for the considered DE actuator prototype     2 2 2 1 , em l l max max V V F K d d V V                           (13) where V max ( V max = 6.7kV) is the maximum voltage which can be placed between each pair of DE actuator electrodes. As a result, the DE actuator dynamics can be written as , int l dist em l F m K F F        (14) Figure 7 : Actuator lumped-parameter model. Optimal Actuator State Estimator Controller development requires the complete knowledge of the variables  ( along with its time derivative), F int and F dist . The considered actuator is equipped with a position and a force sensor that enable the straight measurement of  and F int ; however, no direct information is available for F dist . While this disturbance force could be determined via the visco-elastic model described before, for control purposes we have preferred to estimate it via a Kalman filter. Specifically, consider the augmented state-space system of the DE actuator dynamics  ele F      A B C  x x u y x w  (15.1) 1 2 0 1 1 1 0 0 0 0 1 0 0 0 0 0 0 1 0 , , , , 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 T K m m m m u u                                                   A= B= = C= u  (15.2) where T dist int F F         x (15.3)   T pos force w w  w (15.4) are respectively the state-space variable and the sensor noise vectors (with W force and W pos being the effective sensor variances), and u 1 and u 2 are white noise processes with variances U 1 and U 2 . In the system, F dist and F int have been considered as Wiener processes [9], that is as continuous functions which vary slowly with independent increments ( namely 1 dist F u   and 2 int F u   ). Then, the estimate T