## 1 Introduction  
In modern manufacturing, robotic arms play a crucial role in adaptive production systems. However, due to their structural compliance, these robotic arms often experience significant chatter vibrations, which can adversely affect machining stability and precision. Addressing this issue is essential to ensure high productivity and accuracy in automated machining environments. Force-controlled end-effectors have been shown to mitigate these vibrations and improve machining performance [1]. Additionally, recent studies have proposed novel disturbance compensation methods, integrating reaction force observers and extremum-seeking algorithms to enhance machining robustness and precision [2]. Machining efficiency and stability are critical for achieving high productivity and precision. One of the key challenges in machining processes is the mitigation of chatter, an unstable vibration phenomenon that adversely affects surface finish, tool life, and dimensional accuracy. To address this issue, researchers have extensively investigated chatter suppression methods, including process damping, spindle speed optimization, and adaptive control techniques [3]. Among these, optimizing the feed rate has gained significant attention due to its direct impact on cutting forces, stability, and machining efficiency. Cutting force modeling is essential for understanding the dynamic behavior of machining systems and ensuring process stability. Analytical approaches such as the mechanistic model proposed by [PERSON] [4] have effectively estimated cutting forces by accounting for tool geometry and material interaction. [PERSON] and [PERSON] [5] further extended this by introducing a time-domain simulation method that enhances force prediction accuracy under low radial immersion conditions, which are common in dynamic milling. Although these models are physically interpretable, they may still face limitations when applied to complex robotic environments. To address this, [PERSON] et al. [6] proposed a machine learning-based instantaneous force model using a neural network trained on experimental data, which demonstrated high accuracy in end milling tasks. This trend highlights the complementary roles of physics-based and data-driven methods in developing robust force models for chatter prediction and machining optimization. Once a validated cutting force model is established, it becomes possible to analyze the influence of feed rate variations on chatter intensity under different axial and radial depths of the cut. Previous studies have demonstrated that optimizing feed rate can significantly improve machining stability by shifting the system away from unstable operating regions [7]. [PERSON] et al. further emphasized the importance of spindle speed, feed rate, and machining time on surface roughness and burr formation in micro-milling operations [8]. In addition, the work by [PERSON] et al. explored the application of mechanistic cutting force models specifically for robotic deburring, highlighting the interplay between robot compliance and process dynamics [9]. Nevertheless, determining the optimal feed rate for a given machining condition remains a complex task due to the nonlinear interactions between cutting parameters and process stability. Recent research has emphasized the need for integrated data-driven and physics-informed approaches. For instance, [PERSON] et al. proposed an active contact force actuator design for chatter suppression in robotic milling, demonstrating improved machining stability through active force control [10]. Similarly, [PERSON] et al. introduced a continuous feed rate scheduling strategy synchronized with spindle speed variations, effectively reducing chatter while enhancing productivity [11]. These advancements underscore the growing role of intelligent control and adaptive scheduling in machining optimization. Moreover, the study by [PERSON] et al.