## 5 Proposals
### Model Environment Uncertainties  
The uncertainties are primarily caused by human-driven vehicles such as forklifts. Let \(Z_{t}=\{z_{1}^{1},\ldots,z_{t}^{K}\}\) be the observations of all forklifts at time step \(t\) by the door control system and let \(z_{t}^{k}=<z_{t}^{k},dir^{k}>\) be the state of forklift \(k\). Here, \(z_{t}^{k}\) is the observed position and \(dir^{k}\) is the observed direction, where \(dir^{k}\in\{\leftarrow,\rightarrow\}\). Each forklift can take two actions, \(forward\) and \(wait\). The direction of a forklift remains fixed throughout its operation. The \(wait\) action is reserved for emergency situations only. The probabilistic model is illustrated in Figure 2. Each forklift is human-operated, and the driver controls its speed within a specified range. Suppose that a forklift is observed at position \(v\) at time step \(t\), as shown in Figure 2, where \(z_{t}^{k}=v\) and \(dir^{k}\) is \(\leftarrow\). It moves to \(z_{t+1}^{k}\) with \(forward\) action under \(p_{f}\). The state transition is written as \((z_{t}^{k},forward,z_{t+1}^{k})\). Note that the transition is not deterministic. Due to speed variations, the forklift may move forward by one or two steps, i.e., \(z_{t+1}^{k}=v_{1}\) or \(v_{2}\).  
Figure 1: Architecture of the proposed planner. Red circles are the workstations where the agent has to pick up workpieces. The purple triangles are the destinations of the workpieces. \(\#1,\#2,\ldots\) are the doors through which the agent should pass to deliver the collected workpieces.  
Figure 2: Motion model and state transition of a forklift.  
Each door \(n\) is associated with a designated barrier area \(B_{n}\), and the door control system evaluates the risk level of \(B_{n}\) to determine whether to grant or deny an open request. As illustrated in Figure 3, when the agent attempts to pass through a corridor via door \(n_{1}\) to reach another workshop, it sends an open request to the door controller upon approaching the door. The door controller evaluates the risk level of \(B_{n_{1}}\); if it confirms safety, it responds with a confirmation message \"Y\" and opens door \(n_{1}\). In contrast, if a forklift is detected approaching the corridor near door \(n_{2}\), then the area is considered unsafe. As a result, the controller denies the request and responds with a refusal message \"N\", keeping the door closed.  
In the following, we show the details of the risk evaluation for barrier area \(B_{n}\) of each door \(n\). Here, \(B_{n}=\{v_{1}^{n},v_{2}^{n},\ldots\}\) is the set of vertices around the door \(n\). Suppose that a forklift \(k\) enters the corridor at \(t_{0}\) and that the observation is \(z_{t_{0}}^{k}=<s_{t_{0}}^{k},dire^{k}>\). We simulate its path until it leaves the corridor and calculate the conditional probability of each vertex \(v\in B_{n}\) occupied by forklifts:  
\[p_{k,t}(v|Z_{t_{0}})=\frac{\text{visited times of }v}{\text{simulation times}} \tag{5}\]  
where \(t_{0}<t<T_{k}\), \(v\in B_{n}\), and \(T_{k}\) is the time step at which forklift \(k\) leaves the corridor. The risk level of \(B_{n}\) at time step \(t\) is evaluated by  
\[r(B_{n},t)=\sum_{v\in B_{n}}\sum_{k=1}^{K}p_{k,t}(v|Z_{t_{0}}). \tag{6}\]  
Thus, we obtain the risk distribution \(R_{isk}=\{\{r(B_{1},t_{0}+1),\ldots,r(B_{1},T^{*})\},\ldots,\{r(B_{N},T^{*}), \ldots,r(B_{N},T^{*})\}\}\), where \(T^{*}\) is the time step at which the last forklift leaves the corridor.