# Adaptive Deterministic Vibration Control of a Piezo-Actuated Active-Passive Isolation Structure
## 2 Analysis of a Piezo-Actuated Active-Passive Isolation Structure  
### Piezo-Actuated Active-Passive Isolation Structure Layout  
As shown in Figure 1, the piezo-actuated active-passive isolation structure is an improvement based on the stiff active damping structure [29], which can be applied as the support axes of the multi-DOF Stewart platform with a cubic hexapod configuration [30]. The payload plate and base plate of the Stewart platform are connected with the isolation structures via flexible joints which are used to avoid the influence of backlash and friction caused by the conventional joints. The piezoelectric actuator is used to execute the control output and the force sensor is adopted for the detection of residual vibration signal \(e\), which consists of a local force feedback system for active vibration control. The actuator and sensor are connected through a passive unit which is designed by a metal spring with a special structure to isolate the vibration source disturbance. The disturbance generated by the vibration source is transmitted to the force sensor through the passive unit, which leads to the residual vibration, and the primary path is formed from the vibration source to the force sensor. Then, the piezoelectric actuator actively compensates for the residual vibration feedback from the force sensor to form the secondary path.  
The piezo-actuated active-passive isolation structure can be equivalent to a single degree of freedom (DOF) dynamic model, then the dynamic model of the multi-DOF Stewart platform with the cubic hexapod configuration can be derived via the topological transformation of the Jacobi matrix. The cubic hexapod configu  
Figure 1: Structural diagram of a piezo-actuated active–passive isolation structure.  
supporting axes form a cube, and the two adjacent axes are orthogonal in space. Since the displacement of each active strut caused by microvibration is very small and the coupling between two adjacent axes is also small, the Jacobi matrix can be assumed to be invariant [10]. Meanwhile, the cubic hexapod configuration can realize the decoupling of geometric configuration and minimize the cross-coupling force effect of each axis. It can be seen that when the vibration isolation system is subjected to microvibration, the dynamics of the multi-DOF Stewart platform can be decoupled into six approximately identical independent single-DOF systems. Therefore, in order to control the vibration of the multi-DOF Stewart platform, in this paper, an adaptive active vibration controller is proposed for the single-DOF piezo-actuated active-passive isolation structure.  
The piezo-actuated active-passive isolation structure occupies little space, which is beneficial to the vibration isolation of the aerospace field. However, the structure consists of the flexible rod, the special-shaping spring, and the piezoelectric actuator in series; the dynamic behavior of each part is superimposed, which will make the isolation structure form a complex high-order dynamic system with multiple zeros and poles. Moreover, the deterministic vibration disturbance of on-orbit spacecraft has a time-varying characteristic. Therefore, an adaptive active vibration control system will be introduced based on the Youla parameterization approach.