# A Stackelberg Trust-Based Human-Robot Collaboration Framework for Warehouse Picking
## 4 Stackelberg Trust-Based Human-Robot Collaboration Framework for Warehouse Picking
### Trust Prediction Based on the POSG
#### 4.2.1 The POSG Model
\begin{split} T(\mathbf{\hat{A}}_{t+1}|\mathbf{\hat{A}}_{t},\tau_{t}, \upsilon_{t}^{r},w_{t}^{h})=\mathcal{N}(\mathbf{\hat{A}}_{s}\mathbf{\hat{A}}_{t}+\beta _{s},\sigma_{s}^{2})\\ \mathbf{\hat{A}}_{t}\,\sim\,\mathcal{N}(\mathbf{\hat{A}}_{t},\sigma_{s}^ {2}),\mathbf{\hat{A}}_{t+1}\,\sim\mathcal{N}(\mathbf{\hat{A}}_{t+1},\sigma_{s}^{2}) \end{split} , \tag{1}\]  
where \(\mathcal{N}(\mu,\sigma)\) represents a Gaussian distribution with mean \(\mu\) and standard deviation \(\sigma\). The linear parameters and standard deviation are state-dependent. This study selects trust-related parameters using heuristic methods. Additionally, it optimizes these parameters based on human behavioral science principles, which includes loss aversion and framing effects. The robot evaluates the human's trust state based on observed outcomes. **Definition 4**. : _Observation Path The robot's observation path is defined as follows:_  
\[\mathbf{\sigma}_{t}=b_{0}\tau_{0}a_{0}^{r}w_{0}^{a}a_{0}^{h}w_{0}^{h}\,\cdots a_{ t-1}^{r}w_{t-1}^{r}a_{t-1}^{h}w_{t-1}^{h}\,\upsilon_{t}^{h}, \tag{2}\]  
Figure 5: State transition diagram. _where \(o_{t-1}\to o_{t}\) represents the observation update at time \(t\). This process combines the prior observation \(o_{t-1}\) with the environmental state \(\tau_{t-1}\) at time \(t-1\). It incorporates both human and robot actions along with their respective state data. The complete sequence initiates from the initial belief \(b_{0}\)._  
Using this observable path, the robot infers the human's trust state through a belief state represented by a probability distribution. Definition 5: Belief The belief state in the POSG \(\mathcal{M}_{POSG}\) represents the robot's conditional probability distribution of the human's trust in each state:  
\[b_{t+1}(\mathbf{\hat{n}}_{t+1})=\frac{P(o_{t+1}|\mathbf{\hat{n}}_{t+1},\tau_{t},a_{t}^ {r},w_{t}^{r})\sum_{\mathbf{\hat{n}}_{t}}P\Big{(}\mathbf{\hat{n}}_{t+1}\Big{|}\mathbf{\hat{n }}_{t},\tau_{t},a_{t}^{r},w_{t}^{r},a_{t}^{h},w_{t}^{h}\Big{)}b_{t}(\mathbf{\hat{n }}_{t})}{P\big{(}o_{t+1}|\tau_{t},a_{t}^{r},w_{t}^{r},a_{t}^{h},w_{t}^{h},b_{t }\big{)}}, \tag{3}\]  
where \(P(o_{t+1}|\mathbf{\hat{n}}_{t+1},\tau_{t},a_{t}^{r},w_{t}^{r})\) represents the observation probability. It indicates the likelihood of observing a specific outcome \(o_{t+1}\) given the trust state \(\mathbf{\hat{n}}_{t+1}\) and the task \(\tau_{t}\). Additionally, it considers the robot's action \(a_{t}^{r}\) and the robot's work state \(w_{t}^{r}\). This explains the relationship between the new observation data and the trust state: \(\sum_{\mathbf{\hat{n}}_{t}}P(\mathbf{\hat{n}}_{t+1}|\mathbf{\hat{n}}_{t},\tau_{t},a_{t}^{r},w_{t}^{r},a_{t}^{h},w_{t}^{h})b_{t}(\mathbf{\hat{n}}_{t})\) is the total probability of transitioning from the previous trust state to the new trust state, which combines the transition function and prior belief. The normalization factor ensures that the updated belief is a valid probability distribution:  
\[ \begin{split} P(o_{t+1}|\tau_{t},a_{t}^{r},w_{t}^{r},a_{t}^{h},w_{ t}^{h},b_{t})=\sum_{\mathbf{\hat{n}}_{t+1}}P(o_{t+1}|\mathbf{\hat{n}}_{t+1},\tau_{t},a_{t}^{r},w_{t}^{r})\\ \sum_{\mathbf{\hat{n}}_{t}}P(\mathbf{\hat{n}}_{t+1}|\mathbf{\hat{n}}_{t},\tau_ {t},a_{t}^{r},w_{t}^{r},a_{t}^{h},w_{t}^{h})b_{t}(\mathbf{\hat{n}}_{t})\end{split} , \tag{4}\]  
The belief update step provides the robot with an updated belief \(b_{t+1}(h_{t+1})\) at each time step \(t\), which represents a new probability distribution over the human trust state \(h_{t+1}\). From this updated belief, the robot can further compute the predictive evaluation \(\mathbf{\hat{n}}_{t+1}^{pre}\), which is treated as a specific measure derived from the belief distribution to predict the human trust state:  
\[\mathbf{\hat{n}}_{t+1}^{pre}=\sum_{\mathbf{\hat{n}}}\mathbf{\hat{n}}\cdot b_{t+1}(\mathbf{ \hat{n}}), \tag{5}\]