# On Sharing Spatial Data with Uncertainty Integration Amongst Multiple Robots Having Different Maps
## 4 Obstacle Removal and Update in T-Node Map  
This section briefly discusses about modeling the transience of obstacles first proposed in our previous work [12]. The new contribution of this extended work lies in integrating the positional uncertainty in the decay mechanism which is discussed in the next Section 4.1. The obstacles in the passages may be permanent or temporary and removed after some time. Regardless of the transience of the obstacles, all the robots update their respective T-node map. Newly added obstacles can only be re-confirmed if a robot actually visits the area near the obstacle. Hence, once an obstacle information on a particular node has been updated and communicated to other robots, robots update their map and if that obstacle is found again, the timestamp is updated. However, there is no upper time bound of when a robot would actually visit the particular location and update its map. The obstacle might already have been removed by that time. This problem needs to be modeled mathematically.  
A timestamp (T\({}_{s}\)) is maintained for each obstacle representing the time at which the obstacle was last seen. If an obstacle has recently been added to the map and a short time has elapsed since its addition, then the probability that it has not been removed is high. On the contrary, if a lot of time has elapsed since the addition of the obstacle, the probability that it still exists in the map is less. We model a confidence (\(c\)) measure which represents this probability \(0\leq c_{th}\leq 1\). The maximum value of confidence is 1 and its value decreases with time. The robots assumes that the obstacle still exists in the map until the corresponding confidence has not dropped to below a threshold confidence (C\({}_{th}\)). Depending on the nature of the environment, a threshold time (\(t_{th}\)) is chosen in which the confidence decays to \(c_{th}\) value and the time in which the confidence decays to zero is (\(t_{z}\)). To model the confidence decay, the following family of curves are chosen.  
\[c=1-\left(\frac{t_{th}}{t_{z}}\right)^{n} \tag{1}\]  
The curves given by Equation (1) have the desired characteristic that for higher values of \(n\), the curve flattens out more and delays confidence decay until the threshold time (\(t_{th}\)) and after that it decays quickly to zero in \(t_{z}\) time. For a given \(c_{th}\), \(t_{th}\) and \(t_{z}\), the value of the degree of the curve (\(n\)) can be found by solving Equation (1) as,  
\[\begin{split}\left(\frac{t_{th}}{t_{z}}\right)^{n}=1-c_{th},& \quad(0\leq c_{th}\leq 1),\\ n\ \log\left(\frac{t_{th}}{t_{z}}\right)=\log(1-c_{th}),\\ \implies n=\frac{\log(1-c_{th})}{\log\left(\frac{t_{th}}{t_{z}} \right)}.\end{split} \tag{2}\]  
Figure 7 shows the curves for the decay function given by Equation (1) for various values of \(n\). The Ufactor in Figure 7 shows the uncertainty factor which is discussed in Section 4.1. The various curves have been generated for \(c_{th}=0.55\) and \(t_{z}=600\) s, for varying values of \(t_{th}\) between \(300\) s to  
Figure 6: T-node map generation. (**a**) Binary grid-map. (**b**) Skeleton map. (**c**) Clustering within \(\delta\) distance. (**d**) Clustered node \(n\).  
600 s. It can be seen that the corresponding values of \(n\) can be found for different \(t_{th}\) according to Equation (2). Moreover, as the value of \(n\) increases, the decay curves flatten out more taking more time to reach the threshold time and then quickly decrease to zero.  
For a given instantaneous value of confidence \(c\), the elapsed time \(t\) is calculated from Equation (1) as,  
\[t=e^{\frac{1}{\hbar}\log(1-c)+\log(t_{z})}. \tag{3}\]  
The time remaining (\(t_{rem}\)) to reach the threshold time (\(t_{th}\)) is,  
\[t_{rem}=t_{th}-e^{\frac{1}{\hbar}\log(1-c)+\log(t_{z})}. \tag{4}\]