*4.2. Sensor Calibration*

Each slider is equipped with a XSens MTw inertial measurement unit (IMU). The sensor comprises a three-axis magneto-resistive magnetometer, an accelerometer, gyroscopes, and a barometer. For the following experiment, we only use the magnetometer. The sensor is attached to a wooden stick to reduce the influence of the metal wheels on the measurement. Although the sliders are equipped with sensors from the same product line of the same manufacturer, their absolute perception differs. Additionally, the sensors can still perceive the metal in the wheels of the robots. Therefore, we need to calibrate the sensors relatively to each other to perceive the environment equally using the approach proposed in [47].

The authors in [47] assume that the sensor readings of one sensor can be expressed as another sensor's reading through an affine transformation. To estimate the rotation and translation, multiple sensor readings of all sensors have to be acquired. These readings are then exploited to estimate the rotation and translation relative to one specific sensor by means of a least squares method. In this experiment, each magnetic field sensor reads at a position *x<sup>m</sup>* one measurement of the magnetic field per Euclidean axis. During the estimation, absolute values of these measurements are used. Figure 2a shows the absolute values of the sensor readings for multiple measurement locations of two sensors. The error of the sensor readings before and after calibration are presented in Figure 2b. The correction thus reduces the bias and the standard deviation of the error between both sensors.

**Figure 2.** (**a**) Absolute values of the magnetic field samples of two sensors. It is assumed that each sensor measured at the same locations. (**b**) Error of the absolute values of the magnetic field samples before and after the corrections. The calibrated sensor has now the same mean as the reference sensor, and the standard deviation of the error is reduced.

However, this calibration is only useful if the orientation of both sensors is constant during the experiment. As the sensors always measure in the same orientation, this assumption is fulfilled for our experiments. For further information on intrinsic calibration of inertial and magnetic sensors, the reader is referred to [48].

#### *4.3. Collecting Ground Truth Data*

In order to evaluate the performance of the distributed exploration, we also need to know the actual magnetic field in the laboratory—a ground truth data. For collecting the ground truth data, one slider measures the area of the Holodeck in a systematic fashion, where the distance between each measurement was set to be 5 cm such that in total, 8699 measurement points were collected. On each measurement position, multiple sensor readings are taken and averaged. The resulting ground truth is displayed in Figure 3.

**Figure 3.** Magnetic field intensity of the Holodeck collected for the experiment with real sensors. The measurements were made in 5 cm steps.

#### **5. Experimental System Design**

Our setup relies on ROS (https://www.ros.org/, accessed on 19 March 2022), which manages the communication between all software modules called *nodes*. On each slider, several ROS nodes are running such as the motor controller, which translates the measurement locations into velocity commands for each wheel, the path-planner, and the sensor.

As a path-planner, we use the popular A\* [49,50]. We implemented the A\* algorithm as a global and as a local planner, which is utilized for collision avoidance. Therefore, each slider does not only consider the global map but also a local map around its current position.

After receiving a new waypoint, the global path planner estimates a path in the global map from the current position to the goal avoiding the obstacles. If there is no other robotic system in its path, the goal is reached. However, if another slider enters the local frame while the robot is on its way toward the goal, the robot stops, and the path within the local frame is re-planned to avoid collisions. If the planner is not able to find a solution in the local frame within a given time, the global path planning is re-initiated, taking the current slider as an obstacle into account.

The whole system design for this experiment is shown in Figure 4. The distributed exploration criterion uses the computed map excluding the locations of the obstacles. In addition, the map information is used by the path-planner to find an obstacle-free path to the estimated measurement location *<sup>x</sup>*. Figure <sup>4</sup> also describes the process-flow of the whole system.

For comparison, we will use non-Bayesian SOF and SOE formulations as discussed in [14]. As in these formulations, the ADMM algorithm [33] was used for estimation, we will refer to them as ADMM for SOF and ADMM for SOE, respectively. For the Bayesian learning and algorithms discussed in this paper, we will refer to them as D-R-ARD for SOF and the D-R-ARD for SOE (see also Table 1).

**Table 1.** The algorithms that are used in this experiment and where they are introduced.


In the experiments, we will set the number of basis functions to *N* = 560, which also determines the size of the vector *w*. The basis functions are distributed in a regular grid. We consider Gaussian basis functions with a width set to *σ<sup>n</sup>* = 0.25 such that

$$\phi\_n(\mathbf{x}, \pi\_n) = \exp\left\{-\frac{||\mathbf{x} - \pi\_n||^2}{2\sigma\_n^2}\right\},\tag{57}$$

where *<sup>π</sup><sup>n</sup>* <sup>∈</sup> <sup>R</sup>*<sup>s</sup>* and *<sup>s</sup>* <sup>=</sup> *<sup>d</sup>*.

After initialization of the system, every agent takes a first measurement and incorporates it in its local measurement model to calculate the first estimate of the regression. Then, each algorithm requires that the intermediate estimated parameter weights are distributed to the neighbors (following Figure 4) to do an average consensus [40,41]. Consequently, each agent can proceed to estimate with the regression using the averaged intermediate parameter weights. When the distributed regression converged, the agents use the estimated covariance matrix in the distributed exploration step. In this step, the agents propose candidate positions to their neighbors and receive information to compute the D-optimality criterion locally. When the best next measurement locations are chosen, they are passed to the coordination part [51] to verify that all agents go to different positions. If the measurement location is considered as valid, an agent locally plans its path on the global frame to reach the goal. While approaching the goal, the agent checks if other agents entered into the local frame to avoid collisions. When all agents reached their goal, the agents take measurements and the process flow continues.

**Figure 4.** System design with additional path planner and map constraints. Each gray box represents interaction between other agents. In some boxes, the lower right indicates where this process belongs. This software setup is representative for the SOE distribution paradigm.

As evaluation metric, we chose the normalized mean square error (NMSE), which can be defined as

$$\varepsilon \stackrel{\triangle}{=} \frac{||y\_{\text{true}}(\mathbf{X}\_T) - \Phi(\mathbf{X}\_T, \Pi)\hat{w}||}{||y\_{\text{true}}(\mathbf{X}\_T)||},\tag{58}$$

where *<sup>y</sup>*true(*XT*) <sup>∈</sup> <sup>R</sup>*<sup>T</sup>* is the ground truth measured at *<sup>T</sup>* <sup>∈</sup> <sup>N</sup> positions *<sup>X</sup><sup>T</sup>* <sup>∈</sup> <sup>R</sup>*T*×*d*. Here, we set *T* = 560, and these locations are equal to the center positions of the Gaussian basis functions.
