## 2 Rock-Drilling Robot Arm Modeling and Error Calculation
### Improvement of SBOA  
Population initialization is crucial for enhancing the search ability of stochastic algorithms. Due to its simplicity and inherent randomness, the SBOA also utilizes a random initialization method to determine the initial positions of the secretary birds within the search space:  
\[X_{i,j}=lb_{j}+r\times\left(ub_{j}-lb_{j}\right),i=1,2,\ldots,N,j=1,2,\ldots, Dim \tag{9}\]  
where \(X_{i}\) denotes the position of the i-th secretary bird, \(lb_{j}\) and \(ub_{j}\) represent the lower and upper bounds, respectively, and \(r\) denotes a random number between 0 and 1.  
However, the distribution of randomly generated populations in the solution space often tends to be uneven, leading to over-density in some regions and excessive sparsity in others. This uneven distribution reduces the algorithm's efficiency in utilizing the entire search space and affects population diversity. The good point set strategy [31], an optimization method aimed at improving search efficiency, has been successfully applied to enhance various optimization algorithms [32; 33; 34; 35]. To address these issues, this paper proposes an improved SBOA that initializes the secretary bird population using the good point set strategy. This approach results in a more uniform distribution of the initial population within the solution space and improves the algorithm's convergence speed. The good point set strategy enables more uniform coverage of the entire search space with fewer sample points.  
Suppose \(H_{s}\) is the unit cube in an s-dimensional Euclidean space. If \(r\in H_{s}\), it takes the following form:  
\[P_{n}(k)=\left\{\left(r_{1}^{(n)}\cdot k\right),\left(r_{2}^{(n)}\cdot k\right), \cdots,\left(r_{s}^{(n)}\cdot k\right),1\leq k\leq n\right\} \tag{10}\]  
Its deviation \(\varphi(n)\) satisfies \(\varphi(n)=c(r,\varepsilon)n^{-1+\varepsilon}\), where \(c(r,\varepsilon)n^{-1+\varepsilon}\) is a constant related only to \(r\) and \(\varepsilon\), with \(\varepsilon\) being an arbitrary positive number. Then, \(P_{n}(k)\) is referred to as the set of good points, and \(r\) is the good point, where \(\left(r_{s}^{(n)}\cdot k\right)\) represents the fractional part and \(n\) denotes the number of points, with \(\mathrm{r}=\{2\cos(2\pi k/p),1\leq k\leq s\}\), where \(p\) is the smallest prime number satisfying \((p-3)/2\geq s\).  
The mapping to the search space is as follows:  
\[x_{i}(j)=(ub_{j}-lb_{j})\cdot\{r_{j}^{(i)}\cdot k\}+lb_{j} \tag{11}\]  
where \(lb_{j}\) and \(ub_{j}\) represent the lower and upper bounds, respectively. To verify the uniformity of the point sets generated by the good point set strategy compared to randomly generated point sets, 100 initial points were generated using both the good point set strategy and the random strategy in 2D and 3D spaces, respectively. Their distributions are shown in Figure 4.  
It can be observed that the points generated by the good point set strategy, as illustrated in Figure 4a,c, are more uniformly distributed compared to those generated by the random strategy, as shown in Figure 4b,d. Consequently, the incorporation of the good point set  
Figure 4: Distribution of points generated by the two strategies. (**a**) Two-dimensional point distribution generated by the good point set strategy. (**b**) Two-dimensional point distribution generated by the randomized strategy. (**c**) Three-dimensional point distribution generated by the good point set strategy. (**d**) Three-dimensional point distribution generated by the randomized strategy.  
strategy effectively enhances population diversity, which facilitates global search in the early stages of the algorithm and accelerates convergence.