# A Stackelberg Trust-Based Human-Robot Collaboration Framework for Warehouse Picking
## 4 Stackelberg Trust-Based Human-Robot Collaboration Framework for Warehouse Picking
### Human-Robot Decision-Making Based on Multi-Round Trust  
In this section, we first provide a rationale for human-robot strategies. We first derive the human strategy based on human trust. Then, we generate the optimal robot strategy through the robot's multi-round trust evaluation of humans. The symbol interpretations in human-robot decision-making are given in Appendix D.  
In traditional decision-making models, decisions are often made based on precise calculations of the value function. However, in human-robot collaborative picking environments, humans typically struggle to accurately evaluate various benefits due to limitations in energy and computational capacity. Instead, human decisions are more influenced by their level of trust in robots. Trust serves as a cognitive simplification mechanism, which enables humans to make quick decisions in complex environments. While these decisions may not always be optimal, they still reflect a certain level of reliance on the robot's reliability. Human Strategy  
This study uses trust as a weighting factor in decision-making. Trust directly influences human confidence in the robot's actions, which is reflected in the decision-making process. For instance, when humans have high trust in robots, they are more likely to rely on the robot's actions and intervene in the picking tasks with a lower probability. In the decision-making process, a function \(f(\tau_{t},\mathbf{\hat{A}}_{t})\) is introduced to weigh the probability of human choices [20]:  
\[P(a_{t}^{h}|\tau_{t},w_{t}^{r},\mathbf{\hat{A}}_{t})=S(f(\tau_{t},\mathbf{\hat{A}}_{t })\cdot P_{0}(a_{t}^{h}|\tau_{t},w_{t}^{r})), \tag{11}\]  
where \(S(x)=\frac{1}{1+e^{-x}}\) is the sigmoid function, and \(P_{0}(a_{t}^{h}|\tau_{t},w_{t}^{r})\) represents the benchmark for execution probability under different task types and robot work states. The function \(f(\tau_{t},\mathbf{\hat{A}}_{t})\) is expressed as a step function, which acts as a trust-weighted function. This \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{\((w_{exec}^{r},w_{check}^{h})\)} & \multicolumn{2}{c}{\((w_{exec}^{r},w_{quil}^{h})\)} & \multirow{2}{*}{\((w_{quir}^{r},w_{Resxc}^{h})\)} \\ \cline{2-3}  & \(f_{t}\)=success & & \(f_{t}\)=fail \\ \hline
1 & 1 & - & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Reward function related to accuracy. Figure 6: Human fatigue function. function reflects how humans recognize different task types during the process and react accordingly. For instance, humans show a significantly strong willingness to intervene when robots handle more complex tasks compared to normal tasks. Of course, this correlates with their trust in the robot. Hence, task types are integrated into the trust-weighted function. The most crucial aspect of the trust-weighted function is the human's current trust state in the robot, which has a profound impact on the probability of intervention. From the human action probability matrix, human strategy \(\pi_{h}(P_{a_{t}^{h}})\) is derived. (2) Multi-Round Robot Strategy  
The collaborative picking process between humans and robots can be considered a sequential game, where participants take turns making decisions in a specific order. Each participant's decision can be influenced by prior events or the actions of other participants. A key feature of this type of game is that agents can observe previous actions, which influence their decisions and ultimately shape the outcome. The Stackelberg game is a typical sequential game that clearly defines a decision-making order, where the leader moves first and the follower moves subsequently. The leader makes decisions while fully anticipating the follower's response, which enables more rational decisions. After observing the leader's decisions, the follower adopts an optimal strategy based on their best interests. We aim to consider the impact of the robot's decision in the current round on subsequent rounds in a continuous working state; in other words, a multi-round chain reaction.