# Multi-Tag Fusion Localization Method Based on Geometric Constraints
## 4 Fusion of Multiple Observations  
In the context of multi-tag localization, the identification and removal of aberrant observations is a fundamental step. Yet, the integration of the subsequent reliable observations from multiple tags is an equally significant challenge. Traditional KF has long been a cornerstone for data fusion, operating under the assumption that the motion of the target conforms to specific patterns, such as uniform velocity or constant acceleration. This assumption facilitates the formulation of kinematic equations that predict system states. Within this established framework, observational data from detected tags are merged with these predicted states to enhance the accuracy of the localization process. However, the standard KF is limited in its capacity to integrate only two elements: the predicted motion and the actual observation. This limitation becomes particularly evident when attempting to incorporate the wealth of data from multiple reliable observations, a scenario that surpasses the filter's original design. To address this, a novel approach has been introduced. This approach diverges from the standard  
At the k-th fusion, where \(k\in[1,n-1]\), the fused value from the (k \(-\) 1)-th fusion, \(\mathbf{\hat{X}}_{k-1}\) is used as the current moment's predicted value, \(\overline{\mathbf{X}_{k}}\):  
\[\overline{\mathbf{X}_{k}}=\mathbf{\hat{X}}_{k-1} \tag{22}\]  
The initial value, \(\overline{\mathbf{X}_{1}}\) is given by:  
\[\overline{\mathbf{X}_{1}}=\mathbf{H}^{-}\mathbf{Z}_{1} \tag{23}\]  
The prediction error, denoted as \(\overline{e}_{k}\), is calculated as  
\[ \begin{array}{ll}\overline{e}_{k}&=X_{k}-\overline{\mathbf{X}_{k}}\\ &=X_{k}-\hat{\mathbf{X}}_{k-1}\\ &=X_{k-1}+w_{k-1}-\hat{\mathbf{X}}_{k-1}\\ &=e_{k-1}+w_{k-1}\end{array}  \tag{24}\]  
The prediction error covariance matrix \(\overline{\mathbf{P}}_{k}\) is computed as  
\[ \begin{array}{ll}\overline{\mathbf{P}}_{k}&=E[\overline{e}_{k}\overline{e}_{k}^ {\mathrm{T}}]\\ &=P_{k-1}+\mathbf{Q}_{k-1}\end{array}  \tag{25}\]  
where Q is the process noise covariance matrix. The initial value, \(\overline{\mathbf{P}}_{1}\), is given by  
\[\overline{\mathbf{P}}_{1}=\mathbf{P}_{t-1} \tag{26}\]  
Figure 5: Algorithm flowchart. the rear of the vehicle at a height of 1.15 m, the camera captured 6 to 8 tags per frame at a resolution of 1280 \(\times\) 720 pixels and a frame rate of 25 fps. This study designed multiple simulation scenarios, as depicted in Figure 7.  
Simulation Scenario 1: Visual tags are laid out on the ground. AprilTag's tag36 h 10 series is used for the visual tags, with each tag measuring 0.04 m \(\times\) 0.04 m and spaced 0.15 m apart. This initial setup establishes a baseline for evaluating the method's accuracy in an unobstructed environment. Simulation Scenario 2: Adds obstacles based on Scenario 1. The incorporation of obstacles is intended to simulate real-world conditions where line-of-sight to tags may be partially blocked, thereby testing the method's resilience to occlusions. Simulation Scenario 3: Similar to Scenario 1, but with tags measuring 0.03 m \(\times\) 0.03 m and spaced 0.1 m apart. Reducing the tag size and adjusting the spacing allows us to examine the method's sensitivity to changes in tag density and visibility range. Simulation Scenario 4: Adds obstacles based on Scenario 3. Further introducing obstacles with the smaller tags assesses the method's adaptability to challenging conditions with increased occlusion and reduced tag size. This study conducted a series of six experiments designed to evaluate the effectiveness of geometric constraints and to assess the positioning accuracy and stability of the proposed method, as detailed in Table 1. The simulation software provided the ground truth for the experiments, establishing a reliable benchmark for comparison. Due to the uniform scaling of the tag models, changes in tag size were accompanied by corresponding adjustments in the spacing between tags.