3.1.4. Distributed Computation of the D-Optimality Criterion for SOF

For SOF, (37) is unsuited for a distributed computation such that some changes have to be made. First, we define the following terms to facilitate the distributed formulation

$$H \triangleq \Phi \hat{\mathbf{I}} \Phi^T = K \times \frac{1}{K} \sum\_{k=1}^{K} \Phi\_k \hat{\mathbf{I}}\_k \Phi\_{k'}^T \tag{49}$$

$$d \triangleq \Phi \hat{\Gamma} \Phi(\tilde{\mathbf{x}}, \Pi) = \boldsymbol{K} \times \frac{1}{K} \sum\_{k=1}^{K} \Phi\_k \hat{\Gamma}\_k \Phi(\tilde{\mathbf{x}}, \Pi\_k) \,\tag{50}$$

$$v \triangleq \boldsymbol{\Phi}^T(\widetilde{\mathbf{x}}, \Pi) \widehat{\mathbf{f}} \boldsymbol{\Phi}(\widetilde{\mathbf{x}}, \Pi) = \boldsymbol{K} \times \frac{1}{\boldsymbol{K}} \sum\_{k=1}^{K} \boldsymbol{\Phi}\_k^T(\widetilde{\mathbf{x}}, \Pi\_k) \widehat{\mathbf{f}}\_k \boldsymbol{\Phi}(\widetilde{\mathbf{x}}, \Pi\_k), \tag{51}$$

where **Π***<sup>k</sup>* = [*π*1, ... , *πNk* ] *<sup>T</sup>* <sup>∈</sup> <sup>R</sup>*Nk*×*<sup>s</sup>* and **<sup>Γ</sup>***<sup>k</sup>* = [*γ*1, ... , *<sup>γ</sup>Nk* ] *<sup>T</sup>*. All terms in (49)–(51) can then be computed by means of an averaged consensus [40,41]. Next, we reformulate **Σ***<sup>w</sup>* with the help of the matrix-inversion-lemma as

$$
\Sigma\_{\overline{w}} = \widehat{\Gamma} - \widehat{\Gamma}\Phi^T(\Lambda^{-1} + \Phi\widehat{\Gamma}\Phi^T)^{-1}\Phi\widehat{\Gamma} = \widehat{\Gamma} - \widehat{\Gamma}\Phi^T(\Lambda^{-1} + H)^{-1}\Phi\widehat{\Gamma}.\tag{52}
$$

Now, (37) can be reformulated in a distributed setting for SOF as

$$\begin{split} f(\tilde{\mathbf{x}}, \tilde{\boldsymbol{\lambda}}) &= 1 + \tilde{\lambda}\boldsymbol{\Phi}^{T}(\tilde{\mathbf{x}}, \Pi) \Sigma\_{w} \boldsymbol{\Phi}(\tilde{\mathbf{x}}, \Pi) \\ &= 1 + \tilde{\lambda}\boldsymbol{\Phi}^{T}(\tilde{\mathbf{x}}, \Pi) \left( \hat{\mathbf{f}} - \hat{\mathbf{f}}\boldsymbol{\Phi}^{T}(\boldsymbol{\Lambda}^{-1} + \boldsymbol{H})^{-1}\boldsymbol{\Phi}\hat{\mathbf{f}} \right) \boldsymbol{\Phi}(\tilde{\mathbf{x}}, \Pi) \\ &= 1 + \tilde{\lambda}\boldsymbol{\Phi}^{T}(\tilde{\mathbf{x}}, \Pi) \hat{\mathbf{f}}\boldsymbol{\Phi}(\tilde{\mathbf{x}}, \Pi) - \tilde{\lambda}\boldsymbol{\Phi}^{T}(\tilde{\mathbf{x}}, \Pi) \hat{\mathbf{f}}\boldsymbol{\Phi}^{T}(\boldsymbol{\Lambda}^{-1} + \boldsymbol{H})^{-1}\boldsymbol{\Phi}\hat{\mathbf{f}}\boldsymbol{\Phi}(\tilde{\mathbf{x}}, \Pi) \\ &= 1 + \tilde{\lambda}\boldsymbol{\nu} - \tilde{\lambda}\boldsymbol{d}^{T}(\boldsymbol{\Lambda}^{-1} + \boldsymbol{H})^{-1}\boldsymbol{d}. \end{split} \tag{53}$$

For the case when the criterion (46) is used for evaluaton of the D-optimality, the variable *<sup>q</sup>*(*<sup>π</sup>*) in (46) and the second additive term there have to be reformulated in a form suitable for SOF data splitting. For the former, we utilize the definitions in (49)–(51), together with (52) such that

$$\begin{split} q(\tilde{\pi}) &= \gamma^{-1} + \boldsymbol{\Phi}^{T}(\mathcal{X}, \tilde{\pi}) \Lambda \boldsymbol{\Phi}(\mathcal{X}, \tilde{\pi}) - \boldsymbol{\Phi}^{T}(\mathcal{X}, \tilde{\pi}) \Lambda \boldsymbol{\Phi} \Sigma\_{w} \boldsymbol{\Phi}^{T} \Lambda \boldsymbol{\Phi}(\mathcal{X}, \tilde{\pi}) \\ &= \gamma^{-1} + \boldsymbol{\Phi}^{T}(\mathcal{X}, \tilde{\pi}) \Lambda \boldsymbol{\Phi}(\mathcal{X}, \tilde{\pi}) - \boldsymbol{\Phi}^{T}(\mathcal{X}, \tilde{\pi}) \Lambda (H - H(\Lambda^{-1} + H)^{-1}H) \Lambda \boldsymbol{\Phi}(\mathcal{X}, \tilde{\pi}) \\ &= \gamma^{-1} + \boldsymbol{\Phi}^{T}(\mathcal{X}, \tilde{\pi}) \Lambda \boldsymbol{\Phi}(\mathcal{X}, \tilde{\pi}) - \boldsymbol{\Phi}^{T}(\mathcal{X}, \tilde{\pi}) \Lambda (\Lambda + H^{-1})^{-1} \Lambda \boldsymbol{\Phi}(\mathcal{X}, \tilde{\pi}) \\ &= \gamma^{-1} + \boldsymbol{\Phi}^{T}(\mathcal{X}, \tilde{\pi}) (\Lambda - \Lambda(\Lambda + H^{-1})^{-1} \Lambda) \boldsymbol{\Lambda} \boldsymbol{\Phi}(\mathcal{X}, \tilde{\pi}) \boldsymbol{\Phi}(\mathcal{X}, \tilde{\pi}) \\ &= \gamma^{-1} + \boldsymbol{\Phi}^{T}(\mathcal{X}, \tilde{\pi}) (\Lambda^{-1} + H)^{-1} \boldsymbol{\Phi}(\mathcal{X}, \tilde{\pi}). \end{split} \tag{54}$$

The other term in (46) is then reformulated similarly using the results (49)–(52) as

$$\begin{split} \mathbf{c}^{T}(\widetilde{\boldsymbol{\pi}})\boldsymbol{\Sigma}\_{w}\boldsymbol{\Phi}(\widetilde{\boldsymbol{\pi}},\boldsymbol{\Pi}) &= \boldsymbol{\Phi}^{T}(\boldsymbol{X},\widetilde{\boldsymbol{\pi}})\boldsymbol{\Lambda}\boldsymbol{\Phi}(\widetilde{\boldsymbol{\Gamma}}-\widetilde{\mathbf{I}}\boldsymbol{\Phi}^{T}(\boldsymbol{\Lambda}^{-1}+\boldsymbol{H})^{-1}\boldsymbol{\Phi}\widetilde{\boldsymbol{\Gamma}})\boldsymbol{\Phi}(\widetilde{\boldsymbol{\pi}},\boldsymbol{\Pi}) \\ &= \boldsymbol{\Phi}^{T}(\boldsymbol{X},\widetilde{\boldsymbol{\pi}})\boldsymbol{\Lambda}(\boldsymbol{\Phi}\widetilde{\boldsymbol{\Gamma}}\boldsymbol{\Phi}(\widetilde{\boldsymbol{\pi}},\boldsymbol{\Pi})-\boldsymbol{\Phi}\widetilde{\boldsymbol{\Gamma}}\boldsymbol{\Phi}^{T}(\boldsymbol{\Lambda}^{-1}+\boldsymbol{H})^{-1}\boldsymbol{\Phi}\widetilde{\boldsymbol{\Gamma}}\boldsymbol{\Phi}(\widetilde{\boldsymbol{\pi}},\boldsymbol{\Pi})) \\ &= \boldsymbol{\Phi}^{T}(\boldsymbol{X},\widetilde{\boldsymbol{\pi}})\boldsymbol{\Lambda}(\boldsymbol{d}-\boldsymbol{H}(\boldsymbol{\Lambda}^{-1}+\boldsymbol{H})^{-1})\boldsymbol{d}) \\ &= \boldsymbol{\Phi}^{T}(\boldsymbol{X},\widetilde{\boldsymbol{\pi}})\boldsymbol{\Lambda}(\boldsymbol{I}-\boldsymbol{H}(\boldsymbol{\Lambda}^{-1}+\boldsymbol{H})^{-1})\boldsymbol{d}. \end{split} \tag{55}$$

As a result, the exploration criterion can be re-formulated for SOF in the following form

$$\underset{\stackrel{\text{\tiny X}}{\rightleftharpoons}}{\arg\min} \log \left| \tilde{\boldsymbol{\Sigma}}\_{w} (\mathbf{X}, \Pi, \tilde{\mathbf{x}}, \tilde{\boldsymbol{\pi}}) \right| \equiv \tag{56}$$
 
$$\underset{\stackrel{\text{\tiny X}}{\rightleftharpoons}}{\arg\max} \log \left[ q(\tilde{\boldsymbol{\pi}}) f(\tilde{\mathbf{x}}, \tilde{\boldsymbol{\lambda}}) + \tilde{\boldsymbol{\lambda}} \Big( \boldsymbol{\phi}(\tilde{\mathbf{x}}, \tilde{\boldsymbol{\pi}}) - \boldsymbol{\Phi}^{T}(\mathbf{X}, \tilde{\boldsymbol{\pi}}) \boldsymbol{\Lambda} (\boldsymbol{I} - H(\boldsymbol{\Lambda}^{-1} + H)^{-1}) \boldsymbol{d} \right)^{2} \Big],$$

with *<sup>q</sup>*(*<sup>π</sup>*) defined in (54) and *<sup>f</sup>*(*<sup>x</sup>*, *λ*) given in (53).

## **4. Experimental Setup**

This section describes definition of the experimental setup, calibration of the sensors, and collection of ground-truth data for performance evaluation.

#### *4.1. Map Construction*

The following describes our experimental setup. We conducted the experiments indoor in our laboratory with two paper boxes as obstacles displayed in Figure 1a. Red lines in the figure represent the borders of the experimental area. We use two Commonplace Robotics (https://cpr-robots.com, accessed on 19 March 2022) ground-based robots with mecanum wheels; further in the text, we will refer to the robots as sliders due to their ability to move holonomically. To position the slider within the environment, the laboratory is equipped with 16 VICON (https://www.vicon.com/, accessed on 19 March 2022) Bonita cameras. For the experiment itself, we assume that the map is a priori known to the system. Thus, we need to record the map before the experiment. So, a single slider is equipped with a light detection and ranging (LIDAR) sensor. We use a Velodyne (https://velodynelidar.com/, accessed on 19 March 2022) *VLP-16* LIDAR and the corresponding robot operating system (ROS) package, which can be downloaded from the ROS repository. We construct the map while sending waypoints to the slider manually. The steering of the slider is done with the help of ROS' *navigation stack* [42] together with the *Teb Local Planner* [43]. The sensor output of the LIDAR and the slider position estimated by the VICON system are then used to generate a map with the *Octomap* [44] ROS package. Because we use the VICON position of the slider, which is accurate, this mapping procedure is simpler compared to simultaneous localization and mapping (SLAM) algorithms [45,46]. Figure 1b shows the constructed map, which is afterwards used in the experiment.

**Figure 1.** (**a**) The experimental setting with obstacles. The red line indicates the experimental area, where the slider can navigate. (**b**) The constructed map.
