## 4 Proposed Geometric Similarity Evaluation Methodology
### Geometric Similarity Evaluation  
To ensure the credibility of XILS platforms, previous studies have introduced parameter-based and scenario-based evaluation metrics. Herein, we present a new dynamics-based metric. A simulation is deemed credible only when each of these metrics satisfies its predefined evaluation criterion. However, evaluating a single scenario separately under multiple speed conditions increases the number of required tests exponentially, thereby extending both the duration of the process and raising the resource demand. To overcome this challenge, we propose a methodology that quantitatively analyzes the geometric similarity between different speed conditions within the same scenario, which enables the credibility of the platform under untested conditions to be predicted from the results of a single-speed trial. By allowing the overall reliability to be inferred from one experiment, this approach significantly reduces the number of tests needed -- and thus the associated time and cost--while also revealing potential compromising factors during the inference process. Hence, this integrated evaluation framework can enhance both the credibility and efficiency of XILS assessments.  
Geometric similarity evaluation leverages two scenario-based reliability metrics, namely applicability (\(A_{k}\)) and correlation (\(C_{k}\)), together with a dynamics-based fidelity metric, namely dynamic correlation (\(D_{k}\)). Parameter-based metrics are deliberately excluded from this analysis as they are unsuitable for inter-scenario comparisons and cannot be used to compare different speed conditions within the same scenario. Each metric is rendered as a triangular polygon on a spider chart for each speed condition, which enables a quantitative assessment of geometric similarity across low, medium, and high speeds within a single scenario. By jointly evaluating the structural characteristics of these spider-chart polygons and the interactions among the three metrics, this geometric analysis facilitates a more precise and efficient credibility assessment of XILS platforms. The proposed geometric similarity indices are formulated in Equations (41)-(44):  
\[Y_{k}^{L}=\Big{[}D_{k}^{L},\,A_{k}^{L},\,C_{k}^{L}\Big{]}, \tag{41}\]  
\[Y_{k}^{M}=\Big{[}D_{k}^{M},\,A_{k}^{M},\,C_{k}^{M}\Big{]}, \tag{42}\]  
\[Y_{k}^{H}=\Big{[}D_{k}^{H},\,A_{k}^{H},\,C_{k}^{H}\Big{]}, \tag{43}\]  
\[G_{k}=\frac{1}{3}(y_{S}(Y_{k}^{L},Y_{k}^{M})+y_{S}(Y_{k}^{M},Y_{k}^{H})+y_{S}( Y_{k}^{L},Y_{k}^{H})), \tag{44}\]  
where \(Y_{k}^{L}\), \(Y_{k}^{M}\), and \(Y_{k}^{H}\) denote the evaluation-metric vectors for the \(k\)-th scenario under low-speed (I), medium-speed (M), and high-speed (H) conditions, respectively. \(G_{k}\) is the geometry-based similarity score for the \(k\)-th scenario across its speed conditions, representing the average geometric similarity among the three speed levels; a higher \(G_{k}\) indicates greater structural likeness across speed variations. Finally, as shown in Equation (45), the effectiveness of XILS for the same scenario \(E_{k}\) is evaluated based on whether these similarity scores satisfy a predefined evaluation criterion:  
\[E_{k}=G_{k}\geq\alpha_{G}\,k\in\{1,2,\ldots,K\}, \tag{45}\]  
where \(\alpha_{G}\) denotes the geometric similarity evaluation criterion for \(G_{k}\). As expressed in Equations (46)-(50), \(\alpha_{G}\) represents the minimum guaranteed level of geometric similarity among the three speed conditions within a given scenario. If \(G_{k}\) equals or exceeds \(\alpha_{G}\), the XILS platform's credibility  
scenario, \(\sigma_{G}\) denotes the average of the maximum deviations in the geometric similarity indices across all scenarios. Thus, we can conclude that for all scenarios where \(G_{k}\) exceeds \(\alpha_{G}\), the geometric similarity between different speed conditions is sufficient, which allows us to assess the credibility of a given scenario without performing individual experiments for different speed levels.