# A Stackelberg Trust-Based Human-Robot Collaboration Framework for Warehouse Picking
## 5 Experiment
### Formal Modeling and Verification
#### 5.3.2 Model Verification  
Our goal is to optimize the efficiency of human-robot collaborative picking while considering human factors to reduce human fatigue during work. We formalized the properties using the extended reward PCTL formula described in Equation (15) and validated our framework by changing the probability distribution of task types in the environment. To match the characteristics of continuous human-robot collaboration in industrial environments, the experiment simulates 100 consecutive rounds of collaboration. The PRISM model-checking tool is used for simulation, with a sample size of 10,000 to meet the verification requirements of the system. Trust  
First, we used the probabilistic formula \(P_{max=7}[F(\text{end})\&(trust=trust_{n})]\) to verify the reasonableness of human trust dynamics and the effectiveness of the trust evaluation in the proposed model. The results are shown in Figure 11, where \(trust_{n}\) represents different trust states, and the corresponding reachable probability value \(P_{trust_{n}}\) is calculated based on the probabilistic formula. Finally, the expected trust value is computed as \(trust=\sum P_{trust_{n}}\cdot trust_{n}\), yielding the resulting data. Two groups of robots with different performance levels were set up as a comparative experiment, and the task type distribution probability in the environment was varied. For example, a horizontal axis label of [0.7, 0.3] indicates that regular tasks occur with a probability of 0.7 and complex tasks with a probability of 0.3. In the figure, the solid line represents the average trust probability when humans collaborate with high-performance robots, while the dashed line represents low-performance robots. It is evident that in the trust-independent model, human trust varies with changes in the task-type distribution in the environment. In contrast, in the trust-based model, trust is proportional to the robot's performance. Efficiency  
In subsequent experiments, we based our analysis on the high-performance robot. Using the reward formula \(R\{\text{``time''}\}_{max=7}[F^{\text{``}}\text{end}^{\text{``}}]\), the system's picking efficiency was verified and compared. The results are shown in Figure 12, which represents the reward function related to human-robot collaborative efficiency. Figure 11: Trust dynamics. Similarly, to make the data more intuitive, a comparison of fatigue levels is presented in Table 4. Our framework significantly alleviates human fatigue during work compared to the other two models. From the average values shown in the table, the multi-round trust strategy model reduces worker fatigue by approximately 59.40% compared to the no-trust model. Additionally, it achieves a substantial reduction of 29.10% in worker fatigue compared to the single-round trust strategy model. (4) Accuracy  
Next, we applied the reward formula \(R\{^{\text{*}}\text{c\_rate}^{\text{*}}\}_{max=7}[F^{\text{*}}\text{end}^{\text {*}}]\) to validate and compare the picking accuracy of the system. The results are shown in Figure 14, which represents the reward function related to the accuracy of the human-robot collaborative work. The environmental settings follow those established in the efficiency experiments. In the figure, the black dashed line represents the system accuracy requirement set in the experiment, \(E_{p}=90\%\). As shown, when the environment contains a higher proportion of complex tasks, the no-trust model achieves relatively higher accuracy. This is because the no-trust model is configured to avoid robot execution of complex tasks, leaving these tasks primarily to human workers. While this setup improves accuracy in such cases, it negatively impacts work efficiency and poses potential health risks to workers due to increased workload and fatigue. Similarly, to make the data more intuitive, a comparison of accuracy data is presented in Table 5, where the data is presented using percentage differences.