## 2 Rock-Drilling Robot Arm Modeling and Error Calculation
### Sboa  
In this paper, the newly proposed optimization algorithm SBOA (introduced in 2024) is analyzed and studied. The search accuracy, convergence speed, and stability of SBOA are evaluated using the CEC2017 and CEC2022 benchmark test suites [30]. Inspired by the predation strategy of secretary birds searching for prey in harsh environments and their escape tactics to avoid predators, SBOA was designed to address complex optimization problems. In SBOA, each secretary bird's position represents a candidate solution to the optimization problem, and the algorithm not only demonstrates a high search efficiency but also identifies high-quality solutions. In the secretary bird's predation strategy, the bird searches across a wide area; a behavior abstracted to simulate the global optimization process, where the algorithm performs an extensive search within the solution space. This global search strategy allows the algorithm to explore potential high-quality solutions throughout a broad region of the solution space. The predation strategy of the secretary bird is divided into three phases: finding prey, attacking prey, and consuming prey. These correspond to the phases \(t<\frac{1}{3}T\), \(\frac{1}{3}T<t<\frac{2}{3}T\), and \(\frac{2}{3}T<t<T\) within the algorithm, where t represents the current iteration number and T denotes the maximum number of iterations, as shown in Figure 3.  
During the prey-finding phase \(t<\frac{1}{3}T\), for optimization algorithms, the initial phase requires stronger exploration capabilities to collect adequate information about the entire search space. Therefore, a differential evolution strategy is applied to strengthen exploration during this stage. Differential evolution leverages differences between individuals to generate new solutions, enhancing both the diversity of the algorithm and its global search capability. The position of the secretary bird in this phase is updated using Equation (4):  
\[x_{i,j}^{newP1}=x_{i,j}+(x_{random\_1}-x_{random\_2})\times R_{1} \tag{4}\]  
where \(x_{i,j}^{newP1}\) denotes the value of its j-th dimension, \(x_{random\_1}\) and \(x_{random\_2}\) are random candidate solutions for the first-stage iteration, and \(R_{1}\) represents a randomly generated array of dimension \(1\times Dim\) in the interval [0, 1], where \(Dim\) is the dimension of the solution space. Once the secretary birds spot potential prey, they initially hover around it, exhausting its patience and causing it to lower its guard. Similarly, by treating the best current individual as the prey, the secretary birds update their positions to approach it. Brownian motion \(RB\) is introduced to model the stochastic motion of the secretary bird during the prey consumption phase \(\frac{1}{3}T<t<\frac{2}{3}T\). Equation (5) is used to update the position of the secretary bird during this phase:  
\[x_{i,j}^{newP1}=x_{best}+\exp((\frac{t}{T})^{\wedge 4})\times(RB-0.5)\times(x_{ best}-x_{i,j}) \tag{5}\]  
where \(x_{best}\) denotes the current best value, \(RB=randn(1,Dim)\) represents the Brownian motion formula, and \(randn(1,Dim)\) is a randomly generated array of dimension \(1\times Dim\) from a standard normal distribution: mean = 0, standard deviation = 1. After continuous exertion, the secretary bird initiates its attack once the prey is exhausted. During the prey attack phase \(\frac{2}{3}T<t<T\), the secretary bird uses the Levy flight strategy to simulate various attack methods, thereby enhancing the algorithm's global search capability and improving its convergence accuracy. The rapid approach of the bird  
Figure 3: Secretary bird predation strategy. to its prey signifies that the candidate solution is nearing the current optimal solution. The position of the secretary bird during this phase is updated using Equation (6):  
\[x_{i,j}^{newP1}=x_{best}+(1-\frac{t}{T})^{\wedge(2\times\frac{t}{T})}\times x_{i, j}\times RL \tag{6}\]  
where \(RL=0.5\times Levy(Dim)\), and \(Levy(Dim)\) denotes the Levy flight distribution function.