# On Sharing Spatial Data with Uncertainty Integration Amongst Multiple Robots Having Different Maps
## 3 'T-Node' Map Representation  
A T-node representation of the map has been proposed in our previous work [12]. We briefly explain the T-node map and how obstacles are represented in it. Later, we describe how path planning is done on the node map. It is assumed that each robot is also assigned a unique robot-id (R\({}_{\text{id}}\)) and the robots are on the same network to exchange messages with each other. A node is defined as a point of turn in a path of the map. The paths are represented as a network of these nodes in the map. Figure 4a shows the node representation of the environment shown in Figure 1. Notice that, the nodes \(n_{1},n_{2},\cdots,n_{12}\) are the points of turns in the map. The terminal nodes are shown in red color in Figure 4a. Nodes are connected to each other through edges. Figure 4b shows the T-node map with an obstacle placed between the nodes \(n_{9}\) and \(n_{3}\). The distance between  
Figure 3: Correspondence problem due to the different types of maps [31] of the same environment. (a) Graph map. (b) Dense 3D map. (c) Grid map. Figure 2: Correspondence problem due to the scale and rotational differences between maps. the nodes \(n_{9}n_{3}\) is \(L\) and the distance of the obstacle from node \(n_{3}\) is \(x\), which can easily be estimated using an on-board distance sensor. A table stores T-node map's information viz. traversable/blocked edges (paths) and any changes at the edges. All the robots have access to this table. In the context of Figure 4b, the corresponding information is shown in Table 2. The table contains a set of four information about each path: (1) binary information of whether a new obstacle is found on an edge, (2) a binary information if the path is blocked and cannot be traversed, (3) details of the obstacle if the path is changed and (4) the timestamp (T\({}_{\text{s}}\)) when the information was updated. The details of the information will vary according to the type of the sensor used. For example, in case of Lidar, the obstacle information will contain: the obstacle coordinates from the node (d\({}_{\text{v}}\), d\({}_{\text{y}}\)), dimensions of the obstacle like width (\(w_{\text{obs}}\)) and height (h\({}_{\text{obs}}\)) and the positional uncertainty associated in estimating the obstacle (\(\sigma_{x},\sigma_{y}\)). The uncertainty information comes from the SLAM module used in the robot. As shown in T-node map of Figure 4b, only one of the edges \(n_{3}n_{9}\) is obstructed. This information is reflected in Table 2. It is possible that a new obstacle is found on a path, however the path could be still be traversed. A blocked path cannot be traversed by the robot. Each robot has a copy of this table which has small memory requirement as the meta-data for only the changed paths are required. Moreover, information is communicated to other robots only when some path information is changed. A T-node representation makes it easier for a robot to share information with other robots. The local maps maintained by the two robots might differ by some rotation, translation or scale. As an example, Figure 5a shows the section of the map of Figure 4b with obstacles. Figure 5b shows a scaled version, Figure 5c a rotated version and Figure 5d a scaled and rotated version of Figure 5a. However, the nodes on the paths remains the same and information that there is an obstacle on one of the edges is still conveyed clearly from Table 2 which maintains the details of the obstacles. In addition, with a T-node representation a global map is not required. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Node Path** & **New Obstacle** & **Path Blocked** & **Meta-Data** \\ \hline \(n_{1}n_{2}\) & 0 & 0 & - \\ \hline \(\cdots\) & \(\cdots\) & \(\cdots\) & \(\cdots\) \\ \hline \(n_{3}n_{9}\) & 1 & 1 & \(\{\text{ d:(d${}_{\text{v}}$,d${}_{\text{y}}$). }w:w_{obs}\text{h:}h_{obs}  
Even for a large number of nodes, only those edges which are changed is communicated to the robots. The small data size ensures fast and reliable communication with small communication bandwidth.