## 4 Experiments and Results
### Results and Analysis  
Each experiment was started after placing the hTromo robot in a predefined position inside the area. We recorded the robot in action from the beginning to the end of the experiments. Once the tests were completed, the recorded videos were post-processed using the image processing algorithm detailed in Section 4 in order to generate the track map. Figure 16 presents the track map images of the hTromo robot that were generated following the first set of experiments that sought to validate the application of Theorem 1. The green colored shading represents the area covered by our hTromo robot. The percentage of area covered was computed using Equation (1) and is displayed on top of the tracked images. Figure 17 presents the tiled area during different stages of our first set of experiment. The figure indicates the actual position of the robot at a specific time point, and the associated track map at that instance overlaid with the completed tiling set. The robot path was executed according to the global tiling set specified in basic tromino theory. The results clearly show that our hTromo robot covered more than 95% of the test area in the first set of experiment validating the underline theory of tromino tiling. In the second set of an experiment involving Lemma 1 of Theorem 2, the area covered by our hTromo robot was found to be over 95%. Figure 16 presents the tiled area during different stages of our second set of experiment. We extended the testing area by adjusting the metal frames as mentioned in Section 4, in order to validate Lemma 2 of Theorem 2. The same process has been followed to generate the track map in order to compute total area covered. The results show that the thromo covered more than 91% of the defined testing area thereby validating the application of Lemma 2 shown in Figure 18.  
Figure 14: Defined test area with segmented square grids. Figure 15: Test area with parallelepiped structure with provision for camera mount. Figure 16: Experiments validating application of Theorem 1. Figure 17: Experiments validating application of Theorem 2, Lemma 1. Figure 18: _Cont._  
In the fourth and fifth set of experiments, obstacles were placed inside the testing area as a means of modifying the test space as required by the Lemma. The boundaries of the testing area were adjusted according to the arguments in Lemma 3 and 4 of Theorem 3. We computed the total area covered by excluding the pixels associated with the obstacles by utilizing Equation (2). Figure 19 depicts the area coverage process while validating the application of Lemma 3. Results clearly show that hTromo robot covered an area in excess of 97% of the total test area, thereby validating the application of the Lemma 3 of Theorem 3. Similarly, with experiments involving Lemma 4, the hTromo robot covered an area of over 95% of the total test area. Figure 20, presents the tiled area during the different stages of a fifth and final set of experiments. Figure 19: Tiling Process with Lemma 3 tile set. Figure 18: Experiments validating application of Theorem 2, Lemma 2. Furthermore, with the sixth set of experiments, we changed the test bed size to 168 \(\times\) 168 cm by adjusting the metal frames. We placed obstacles inside the test area in order to make the square space into an Aztec diamond. We again followed the same experimental procedure to generate the tack maps to compute the total area covered. Since we placed obstacles inside the test arena; we used Equation (2) to compute the total area covered. Figure 21 shows the area coverage process during the validation of Lemma 5. The results show that the hTromo robot covers more than 97% of the defined area, thereby validating the application of the Lemma 5 of Theorem 4. The experiments that involve Lemma 6 of Theorem 5 was tested in a 112 cm \(\times\) 112 cm area. The area coverage of hTromo robot under this experiment was computed using the Equation (2). Figure 22 shows the coverage process of the hTromo robot during the validation Lemma 6.