# Analysis and Simulation of Polishing Robot Operation Trajectory Planning
## 2 Materials and Methods
### Spiral Arc Trajectory Planning of Quintic Polynomial Interpolation Algorithm
As we know, the quintic B-spline interpolation algorithm is a powerful tool that leverages the properties of a fifth-degree polynomial to construct smooth, continuous curves with controlled higher-order derivatives. Its local control feature further enhances its suitability for complex applications such as robot path planning, enabling precise and efficient trajectory planning in dynamic environments. In addition to its local control and continuity properties, the quintic B-spline also falls into a broader category of B-spline curves characterized by their order. The \(k\)-th B-splinehas the characteristics of order \(k-1\), and the quintic B-spline curve has the characteristics of fourth order. The \(k\)-th B-spline curve can be represented by segmented polynomials:  
\[P(t)=\sum_{i=0}^{n}P_{i}F_{i,k}(t)\hskip 14.226378 pti\in[t_{k-1},t_{n+1}] \tag{23}\]  
In the above equation,  
\[\left\{\begin{array}{l}F_{i,0}(t)=\left\{ \begin{array}{l}1\ t_{i}\leq t\leq t _{i+1}\\ 0\ \text{others}\\ F_{i,k}(t)=\frac{t-t_{i}}{t_{i+k-1}-t_{i}}F_{i,k-1}(t)+\frac{t_{i+k+1}-t}{t_{ i+k+1}-t_{i+1}}F_{i+1,k-1}(t)\\ \text{convention}\frac{0}{0}=0\end{array} \right.\]  
Let the robot joint position point sequence be represented by \((t_{i},P_{i})_{i}=0,1,\ldots,n\), and the total movement time is \(t=t_{f}-t_{0}\). To ensure the consistency between the starting end of the B spline and its actual data point, the repetition degree at both ends was taken as \(k+1=6\). Therefore, the corresponding knot value \(u_{k+t}=\text{u}_{5+1}\) (\(I=0,\,1,\,\ldots,\,n\)) requires \(n+5\) control vertices Q\(i\) (\(I=0,\,1,\,\ldots,\,n+4\)) and a corresponding junction point vector \(u=[u_{0},\,u_{1},\,\ldots,\,u_{n+10}]\). Follow the steps of the quintic B-spline interpolation for solving the trajectory of a spatial spiral circular arc. First, calculate the knot vector \(u=[u_{0},\,u_{1},\,\ldots,\,u_{n+10}]\); then, find \(n+5\) control points Q\(i\). The Q value of the control vertices can be obtained according to [PERSON] et al. [22]. Figure 7 shows the trajectory plots of each joint of the robot achieved by using the quintic B-spline interpolation algorithm. Figure 7a displays the displacement diagrams of the robot's six joints, which were obtained using the quintic B-spline interpolation algorithm. In Figure 7b, by zooming in on a specific segment of joint 4's displacement, we can observe the remarkable level of detail  
Figure 7: The trajectory diagrams of each joint obtained using a quintic B-spline interpolation algorithm. (**a**) Displacement Curve diagram for six join. (**b**) Enlarged view of displacement curve for Join 4. (**c**) Enlarged view of displacement curve for Join 6.  
in the joint trajectory planning. The curve exhibits an exceptional degree of smoothness, devoid of any jagged edges or abrupt changes that could potentially compromise the robot's performance. This high level of precision ensures that the joint movements are not only accurate but also fluid, contributing significantly to the robot's overall effectiveness and responsiveness. Similarly, Figure 7c, which focuses on a portion of joint 6's displacement, also demonstrates the smoothness of the curve, further reinforcing the advantages of the B-spline approach. The seamless transition between displacement points, without any abrupt changes or distortions, showcases the algorithm's ability to generate highly accurate and predictable motion trajectories. This is crucial for effective robot trajectory planning. Furthermore, the B-spline interpolation algorithm's adaptability and flexibility make it an ideal choice for handling complex robot motions. Its ability to approximate curves with high accuracy and minimal computational overhead allows for real-time implementation, enabling robots to quickly respond to changes in their environment or adjust their movements on the fly [23]. This capability is essential for robots operating in dynamic and unpredictable environments, where swift decision-making and precise execution are of utmost importance.