# Optimization of Redundant Degrees of Freedom in Robotic Flat-End Milling Based on Dynamic Response
## 1 Introduction  
With the advantages of good flexibility, large working ranges, and low cost, industrial robots have a wide range of applications, such as welding, palletizing, grinding, drilling, and milling [1]. In contrast to traditional computerized numerical control (CNC) milling machines, industrial robots have a low static stiffness, which will result in lower positioning accuracy, while their lack of dynamic stiffness will result in vibrations, making them more susceptible to chatter during robotic machining [2]. Consequently, the insufficient stiffness of industrial robots is a major contributor to poor machining accuracy and negatively impacts surface quality. The limitations in stiffness and positioning accuracy restrict their application in milling operations [3]. Previous research has mainly focused on redundant angle optimization and positioning error prediction separately to resolve the low rigidity issue and improve machining performance, including the stiffness model and performance indexes [4, 5], dynamic analysis and posture planning [6, 7, 8], chatter identification and suppression [9, 10]. This article aims to address these issues by developing a dynamic response index model for optimizing redundant degrees of freedom in robotic milling. It combines the influence of both tool vibrations and robot postures on the quality of machined surfaces. Redundant degrees of freedom are considered crucial for improving precision and quality in robotic milling, accounting for the configuration-dependent dynamic behavior ofthe robot. Various optimization methods based on kinematic models have been proposed, including joint limits [11], singularity avoidances [12], collision avoidance [13], and trajectory smoothness [14]. [PERSON] [15] proposed an evaluation index based on recurrent neural networks to realize robot posture control, considering joint angle constraints, velocity constraints and periodic input interference constraints. [PERSON] [16] proposed an objective function to optimize the base coordinate position and end-effector posture of the robot. This function is defined as the sum of the squared distances between the preset and actual elements in the end-effector posture matrix. [PERSON] et al. [17] analyzed the error similarity in the absolute positioning error of the robot and quantified it through the semi-variance function. [PERSON] et al. [18] adopted a closed-loop error compensation method for robotic flank milling on complex-shaped surfaces. The process involves in situ measurement of the finished surface using a laser tracker, mapping the points to a reference coordinate system, and fitting a deterministic bicubic B-spline surface to extract systematic machining errors. To address the impact of irregular robot posture changes on machining efficiency, [PERSON] et al. [19] proposed a speed adjustment strategy to maximize the feed rate. This strategy operates under the constraints of robot joint velocity, effectively mitigating the influence of uneven robot posture changes and enhancing overall machining efficiency. To address the inherent weakness of robot stiffness, scholars have introduced a stiffness performance index to improve both the quality of machined surfaces and the stability in robotic machining. For instance, [PERSON] et al. [20] combined the stiffness ellipsoid and external forces to propose an end deformation index of the robot. This index was then utilized to generate a motion performance mapping graph that relates joint angles and external forces, enabling the optimization of the robot posture and workpiece placement through an analysis of six joint-mapping graphs. Addressing the static positioning and tracking errors of the Delta robot, [PERSON] [21] presented a quasi-static deformation compensation model. Considering stiffness as a constraint condition and the smoothness of the robot posture as the optimization objective function, [PERSON] et al.