# On Sharing Spatial Data with Uncertainty Integration Amongst Multiple Robots Having Different Maps
## 6 Experimental Results  
This section presents the results of the experiments. The extended work discuss information sharing in heterogeneous maps made with different sensors and tests the proposed method under pressure with dynamic obstacles in the vicinity of robots. We used Pioneer-P3 DX [44] and Kobuki Turtlebot [45] robot shown in Figure 9a. Both the robots are wheeled differential drive robots and the motion model is explained in our previous work [12]. Both the robots used ROS [46] on Ubuntu computer and were on the same network to communicate with each other. Figure 8: Obstacle confidence decay function with uncertainty integration. With more positional uncertainty, the confidence falls below the threshold confidence fast. In all the cases, \(c_{th}=0.55\) and \(t_{2}=600\) s.  
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1\(\#\)\(x_{t}\): robot state, \(v_{t},\omega_{t}\): translation and rotational velocity. \(x_{t}=[x\;\;y\;\;\theta]^{T}\)
2\(\#\) EKF uses Jacobian to handle non-linearity. \(G_{t}\): Jacobian of motion function w.r.t state \(G_{t}\leftarrow \begin{bmatrix}1&0&-\frac{\Sigma_{t}}{\omega_{t}}cos\theta+ \frac{\Sigma_{t}}{\omega_{t}}cos(\theta+\omega_{t}\Delta t)\\ 0&1&-\frac{\Sigma_{t}}{\omega_{t}}sin\theta+\frac{\Sigma_{t}}{\omega_{t}}sin( \theta+\omega_{t}\Delta t)\\ 0&0&1\end{bmatrix} \)
3\(\#\)\(V_{t}\): Jacobian of motion w.r.t control \(V_{t}= \begin{bmatrix}-sin\theta+sin(\theta+\omega_{t}\Delta t)&\frac{\Sigma_{t} (sin\theta-sin(\theta+\omega_{t}\Delta t))}{\omega_{t}^{2}}+\frac{\Sigma_{t}( cos(\theta+\omega_{t}\Delta t)\Delta t)}{\omega_{t}}\\ \frac{cos\theta-cos(\theta+\omega_{t}\Delta t)}{\omega_{t}^{2}}&-\frac{\Sigma_ {t}(cos\theta-cos(\theta+\omega_{t}\Delta t))}{\omega_{t}^{2}}+\frac{\Sigma_{t} (sin(\theta+\omega_{t}\Delta t)\Delta t)}{\omega_{t}}\\ 0&\Delta t\end{bmatrix} \)
4\(\#\)\(M_{t}\): Covariance of noise in control space. \(\alpha_{1},\cdots,\alpha_{4}\): Error-specific parameters. \(M_{t}= \begin{bmatrix}\alpha_{1}v_{t}^{2}+\alpha_{2}\omega_{t}^{2}&0\\ 0&\alpha_{3}v_{t}^{2}+\alpha_{4}\omega_{t}^{2}\end{bmatrix} \)
5\(\#\)\(\mu_{t}\): Prediction updates in state. \(\mu_{t}=\mu_{t-1}+ \begin{bmatrix}\frac{-\Sigma_{t}}{\omega_{t}}sin\theta+ \frac{\Sigma_{t}}{\omega_{t}}sin(\theta+\omega_{t}\Delta t)\\ \frac{\Sigma_{t}}{\omega_{t}}cos\theta-\frac{\Sigma_{t}}{\omega_{t}}cos( \theta+\omega_{t}\Delta t)\\ \omega_{t}\Delta t\end{bmatrix} \)
6\(\#\)\(\Sigma_{t}\): Prediction updates in covariance. \(\Sigma_{t}=G_{t}\Sigma_{t-1}G_{t}+V_{t}M_{t}V_{t}^{T}\)
7\(\#\)\(Q_{t}\): Covariance of the sensor noise. \(Q_{t}= \begin{bmatrix}\sigma_{t}^{2}&0\\ 0&\sigma_{\phi}^{2}\\ 0&0&\sigma_{\phi}^{2}\end{bmatrix} \)
8\(\#\)\([m_{ix}\;m_{iy}]^{T}\): coordinates of the \(i\)th landmark. \(z_{t}^{i}\): measurement. \(q\): squared distance. \(q=(m_{k,x}-\mu_{t,x})^{2}+(m_{k,y}-\mu_{t,y})^{2}\) \(z_{t}^{k}= \begin{bmatrix}atan2(m_{k,y}-\mu_{t,y}m_{k,x}-\mu_{t,x})-\mu_{t,y}\\ m_{k,5}\end{bmatrix} \)
9\(\#\)\(H_{t}\): Jacobian of measurement with respect to state. \(H_{t}^{k}= \begin{bmatrix}-\frac{m_{ix}-\mu_{t,x}}{\sqrt[4]{\theta}}&-\frac{m_{ix }-\mu_{t,y}}{\sqrt[4]{\theta}}&0\\ \frac{m_{ix}-\mu_{t,y}}{\theta}&-\frac{m_{ix}-\mu_{t,y}}{\theta}&-1\\ 0&0&0\end{bmatrix} \)
10\(\#\)\(S_{t}\): Measurement covariance matrix. \(\mathbb{S}_{t}^{k}=H_{t}^{k}\Sigma_{t}[H_{t}^{k}]^{T}+Q_{t}\). 11\(\#\)\(j(i)\): likely correspondence after applying maximum likelihood estimate. \(j(i)=\text{argmax}\;\;\frac{1}{\sqrt{\text{det}(2\pi\mathbb{S}_{t}^{k})}}\)\(e^{-\frac{1}{2}\left(z_{t}^{i}-\frac{\beta_{t}^{k}}{2}\right)^{T}[\mathbb{S}_{t} ^{k}]^{-1}\left(z_{t}^{i}-\frac{\beta_{t}^{k}}{2}\right)}\)
12\(\#\)\(K_{t}\): Kalman gain, \(\mu_{t}\): state, \(\Sigma_{t}\): covariance. \(K_{t}^{i}=\Sigma_{t}[H_{t}^{(i)}]^{T}[S_{t}^{(i)}]^{-1}\) \(\mu_{t}=\mu_{t}+K_{t}^{i}(z_{t}^{i}-z_{t}^{i(t)})\) \(\Sigma_{t}=(I-K_{t}^{i}H_{t}^{(i)})\Sigma_{t}\)
13\(\#\) Apply Singular Value Decomposition and get Eigen-values \(\lambda_{i}\): \(\lambda_{1},\cdots\)\(\lambda_{n}=\text{svd}\left(\Sigma_{t}\right)=\text{svd}\left(