# A Stackelberg Trust-Based Human-Robot Collaboration Framework for Warehouse Picking
## 4 Stackelberg Trust-Based Human-Robot Collaboration Framework for Warehouse Picking
### Efficiency Reward Function Related to Fatigue  
When considering the reward function for human-robot collaborative efficiency, we incorporate human factors by treating human fatigue as a factor influencing the efficiency reward function. We model fatigue and recovery using the framework from [PERSON] et al. [37]. This model has been adapted for our operating scenario. The implementation incorporates their original data and parameter settings. For different task types, the robot does not show significant differences in execution time, while humans achieve higher execution accuracy on complex tasks but require more time. The regular time reward function for task execution is denoted as \(R_{t-0}\). Additionally, for humans, continuous operations lead to fatigue accumulation, which causes a decrease in execution efficiency. The fatigue accumulation reward function is denoted as \(R_{t-f}\)Considering the above factors, the efficiency reward function for human-robot picking is defined as follows:  
\[R_{time}=R_{t-0}+R_{t-f}, \tag{6}\]  
A reasonable explanation is provided here for the efficiency of human-robot operations. The regular time reward function \(R_{t-0}\) for different work states is shown in Table 1. Among them, the efficiency is highest when the robot executes tasks alone without human inspection. Efficiency decreases when the robot exits the task and the human executes it, as the subjectivity of the human results in slightly lower efficiency for complex tasks compared to normal tasks. The efficiency is lowest when the robot executes the task and the human inspects it. In this case, the value is set as the expected efficiency of correcting errors and performing inspection tasks when no errors are found. Therefore, when handling complex tasks with lower robot accuracy, the reward value in this state also decreases. In a long-term working environment, workers' accumulated fatigue leads to an increase in their working time. However, during non-working periods, their fatigue is alleviated, and they gradually recover. For the fatigue accumulation function of humans during continuous work  
\[F_{work}(t)=\Big{(}S\Big{(}k_{work}\Big{(}t-t_{0}-t_{half}\Big{)}\Big{)}-S \Big{(}k_{work}\cdot t_{half}\Big{)}\Big{)}\cdot(F_{max}-F_{rest}(t_{0}))+F_{ rest}(t_{0}), \tag{7}\]  
At the same time, when the human is not involved in tasks, their fatigue is alleviated. The fatigue recovery function at this point is  
\[F_{rest}(t)=F_{work}(t_{0})\cdot e^{-k_{rest}(t-t_{0})}, \tag{8}\]  
The fatigue function is triggered under different human states:  
\[F(t)=\left\{ \begin{array}{ll}F_{work}(t),&w_{t}^{h}=w_{Rexx}^{h},\\ F_{rest}(t),&w_{t}^{h}=w_{quil}^{h},\end{array} \right. \tag{9}\]  
where \(S(x)=\frac{1}{1+e^{-x}}\) is the sigmoid function, \(F(t)\) represents the fatigue level at time \(t\), and \(F_{work}(t)\) and \(F_{rest}(t)\) represent the fatigue levels under different continuous work states. The continuous fatigue function is discretized into different fatigue levels. \(k_{work}\) and \(k_{rest}\) are the coefficients of the fatigue growth rate and decay rate, respectively, representing the impact of work or rest states on fatigue levels; \(t_{0}\) indicates the most recent time point at which a switch between work and rest states occurred; \(t_{half}\) represents the time point when the fatigue levels reach half of the maximum; and \(F_{max}\) is the maximum fatigue level. In a healthy state, a worker's accumulated fatigue does not significantly affect their work time beyond a certain point. However, the higher the fatigue level is, the longer the time required to complete the same task. The fatigue function is shown in Figure 6. Therefore, the fatigue reward function is determined by the human's fatigue level and the basic task execution time:  
\[R_{t-f}=R_{t-0}\cdot\lambda F(t), \tag{10}\]  
where \(\lambda\) is the coefficient representing the impact of fatigue on execution efficiency.