# Analysis and Simulation of Polishing Robot Operation Trajectory Planning
## 2 Materials and Methods
### Spiral Arc Trajectory Planning of Quintic Polynomial Interpolation Algorithm
In addition, we introduce a concise quantitative comparison for Figures 10 and 11 for clear evaluation: (1) No Interpolation: The total energy consumption for the no-interpolation method is 646.25275 watt-hours, which reflects the most energy-efficient approach among the three methods evaluated; (2) Quintic polynomial interpolation: This method shows the highest energy consumption at 805.16337 watt-hours, indicating that while it may offer smoother trajectories, it does so at the cost of higher energy use. (3) Quintic B-spline interpolation: The energy consumption for the quintic B-spline method is 630.00482 watt-hours, positioning it as slightly more energy-efficientthan the no-interpolation method, while potentially offering better control and smoothness in trajectory planning. From the comprehensive analysis of the graphical results, it becomes evident that when tasked with grinding a consistent spatial spiral curve, the quintic polynomial interpolation algorithm, while effective, exhibits certain limitations. Specifically, it necessitates the longest execution time, indicating a less efficient trajectory planning process. Furthermore, the total motor power curve exhibits the most pronounced fluctuations, suggesting that the robot's motor system is subjected to increased stress and potentially higher wear rates. Consequently, this algorithm consumes the greatest amount of total motor energy, impacting operational costs and potentially reducing the robot's overall energy efficiency. In contrast, the quintic B-spline interpolation algorithm emerges as the optimal choice for this specific application. By consuming the least amount of time for trajectory planning, it significantly enhances the robot's productivity and response speed. Moreover, the smooth variation in the total motor power curve demonstrates the algorithm's capability to minimize stress on the motor system, promoting longer component lifespans and reduced maintenance requirements. This smoothness also translates into reduced energy consumption, making the robot's operation more environmentally sustainable and cost-effective. The planning effect of not using interpolation falls between the two algorithms. While this approach may suffice for certain applications with less stringent requirements, it fails to match the efficiency and precision offered by the quintic B-spline interpolation algorithm. When the trajectory planning program derived from the B-spline interpolation algorithm is integrated into the robot experimental platform, the results are truly remarkable. The robot seamlessly follows the planned trajectory, executing the grinding task with stability and precision. Figure 12 is a diagram demonstrating the robot's operation, which vividly illustrates the robot's smooth and vibration-free operation and serves as a testament to the algorithm's effectiveness. Figure 11: Total energy output of the motor for three different results. Figure 10: Total motor power for three results. In conclusion, the adoption of the quintic B-spline interpolation algorithm for trajectory planning of robots engaged in spatial spiral curve grinding offers a trifecta of benefits: a shortened running time, reduced energy consumption, and the elimination of vibrations. These advantages not only elevate the robot's performance but also contribute to a more sustainable and cost-effective manufacturing process. While our study focused exclusively on a spiral trajectory, we chose this due to its complexity and relevance in precision-demanding applications such as 3D printing and robotic assembly. To substantiate the choice and effectiveness of the interpolation methods used--quintic polynomial and B-spline--across various trajectory types, we refer to several studies that have successfully applied these techniques in broader contexts. For instance, Ref. [24] demonstrates the efficacy of B-spline interpolation in creating smooth and precise trajectories for industrial robots.