## 1 Introduction  
Fast-steering mirror (FSM) systems, which enable the tip-tilt motion of a mirror, are widely utilized in high-precision opto-mechantronic devices for optical scanning and pointing applications, such as micro-scanning systems [1], free-space imaging systems [2], adaptive-optics laser systems [3], and beam stabilization systems [4]. Laser beam steering (LBS) systems based on FSMs are increasingly being integrated with moving platforms such as spacecraft, deep-space exploration satellites, airplanes, and other vehicles [5, 6, 7, 8]. FSM systems typically leverage either voice coil [9] or piezoelectric (PZT) actuation [10]. Voice coil actuators support a wide actuation range with moderate bandwidth and can accommodate larger mirror systems [11]. However, the wide actuation range results in a lower actuation resolution compared to piezo-actuated FSMs. Piezo-actuated FSM systems are often relatively compact, using small actuators to support higher system bandwidths and scan rates while operating over narrower scan ranges [12]. Thus, they are suitable for integration into high-speed scanning systems [13]. In addition, they have higher actuation resolutions than voice coil actuators. However, the design elements of piezo-actuated FSM systems include a complex flexible suspension structure that supports the mover and prepresses the actuators [14], mechanical amplification structures for a push-pull configuration [15, 16], and specific suspension designs for space-qualified FSMs used on target platforms [17]. Moreover, these systems typically require significant design effort, must be fabricated primarily via electrical discharge machining, and are generally rather challenging to optimize and manufacture [18]. FSM tip-tilt motion systems for LBS must be resilient to dynamic disturbances, such as atmospheric turbulence and mechanical vibrations [19; 20]. Atmospheric turbulence, which involves changes in air density between the target and the application platform, is the main cause of degradation in laser propagation and typically has a broadband frequency [21; 22; 23]. Mechanical vibration is the natural frequency of disturbances (typically narrowband) and can be generated by the laser emission source or by movement of the platform on which the source is mounted [24; 25]. Tip-tilt control is designed to accurately aim the laser beam at a target point while suppressing both broadband and narrowband jitter when using an FSM. Fast-convergent adaptive control algorithms for beam jitter suppression have been widely explored to achieve optimal control. Recent studies have introduced least-mean-square (LMS) and recursive least-square adaptive filtering algorithms [26; 27; 28]. Other jitter-control schemes proposed for PZT actuators include a vibration effect cancellation scheme [29], a notch filter to suppress the dominant resonant mode [30], and a feedforward controller [31]. An input-shaping controller and integral resonance controller have also been developed [32]. In this paper, we aim to design the mechanical structure of the PZT FSM to maximize performance metrics related to mechanical range and main resonant mode frequency (including the optical mirror). Due to the recent need to increase the reliability of piezo actuator FSM motion systems on target platforms under various environmental conditions, the proposed design simplifies the mechanical structure and is relatively easy to manufacture. Further, a high-precision tip-tilt controller was designed for the FSM system to enable the PZT actuators to maintain an extremely small rotational position under mechanical vibration disturbances [33; 34; 35]. To this end, it was necessary to design a high-precision FSM motion system that can suppress narrowband jitter using a tip-tilt controller while offering fast response and strong disturbance rejection [36; 37; 38]. Several methods have been formulated to enhance the dynamic disturbance-attenuation performance of control systems [39; 40].