## 4 FastSLAM-MO-PSO  
The FastSLAM-MO-PSO algorithm is designed to address the intricate interplay between direct measurement accuracy and environmental constraints in the domain of SLAM. Recognizing that traditional SLAM methods often struggle to balance the precision of direct measurements with the broader implications of environmental factors, this algorithm introduces a multi-objective optimization framework to concurrently optimize these aspects. The core motivation behind this design is the acknowledgment that in SLAM, the accuracy of a robot's pose estimation is not solely dependent on the measurement data, but is also significantly influenced by the environmental context. For instance, the presence of unexpected obstacles or the variability in sensor range can drastically affect the mapping outcome. By integrating MO-PSO with FastSLAM, the algorithm aims to achieve a more holistic optimization that considers both the fidelity of direct measurements and the adaptability to environmental constraints. The FastSLAM-MO-PSO algorithm implements this dual focus through a sophisticated measurement probability model, where two distinct objectives are defined:  
Direct Measurement Objective: This objective encapsulates the probability of accurate range measurements considering local measurement noise, the likelihood of detecting unexpected objects, and the scenario of not hitting any object or measurement failures. It is designed to maximize the probability of obtaining accurate and reliable direct measurements. The Direct Measurement Objective focuses on the accuracy and reliability of measurements obtained from the sensor. In our implementation, this objective is mathematically formalized by incorporating the measured values \(z\) and the true values \(z_{\text{star}}\). The calculation begins with the probability of measuring the correct range, denoted as _prob_hit_, which is computed using a normal distribution to account for local measurement noise:  
\[prob\_hit=\frac{1}{\sqrt{2\pi\sigma_{\text{hit}}^{2}}}\exp\left(-\frac{\left( \text{z}-z_{\text{star}}\right)^{2}}{2\sigma_{\text{hit}}^{2}}\right) \tag{5}\]  
where \(\sigma_{\text{hit}}\) = 20 represents the standard deviation of the measurement noise, indicating how much the measured values can deviate from the true value due to inaccuracies inherent in the sensor. Next, we calculate the probability of hitting unexpected objects, represented by _prob_short_. This probability is modeled using an exponential decay function:  
\[prob\_short=\lambda_{\text{short}}\times\exp(-\lambda_{\text{short}}\times \text{z}) \tag{6}\]  
In the above context, \(\lambda_{\text{short}}\) = 0.15 is the rate parameter of the exponential distribution, which determines how quickly the probability of hitting unexpected objects decreases with increasing distance. The overall Direct Measurement Objective is expressed as:  
\[obj1=p_{hit}\times prob\_hit+p_{short}\times prob\_short \tag{7}\]  
where \(p_{hit}\) = 0.8 reflects the prior probability of obtaining accurate measurements, suggesting a strong reliability of the sensor. The value \(p\_short\) = 0.05 indicates the prior probability of this event. Environmental Constraint Objective: This objective accounts for the probability of random measurements within the sensor's range and the constraints imposed by the maximum sensor range. It ensures that the mapping process is aware of and adapts to the environmental boundaries. We consider the probability of not hitting anything or experiencing measurement failures, denoted as _prob_max_:  
\[prob\_max= \begin{cases}1&if\ z=radar\_range\\ 0&otherwise\end{cases}  \tag{8}\]  
Finally, we incorporate the probability of random measurements, which we assume to be uniformly distributed within the radar's range:  
\[prob\_rand=\frac{1}{radar\_range} \tag{9}\]  
For the Environmental Constraint Objective, we aim to ensure that the SLAM algorithm is aware of its operational boundaries.