## 2 Polyomino Tiling Theory  
Polyominoes are plain geometrical structures formed by endwise coupling of congruent squares [29]. Based on spatial orientation, geometrical transformation and chirality, each polyomino can be categorized into free polyominoes, one-sided Polyominoes, and fixed Polyominoes. For instance, the set domino, formed by the combination of two congruent square, can have single one-sided, single free and two fixed dominoes as its subsets. Correspondingly, triominoes (3-omino) can exist as two free, two one-sided, and six fixed triominoes [29]. In the case of tetromino that contains four constituent squares can form five free, seven one-sided, and nineteen fixed tetrominoes. In this paper, we used a Tromino inspired reconfigurable robot, hTromo capable of switching between one of the three forms. The Polyominoes tiling theory deals with the problem of partitioning or filling of a geometrical region using same or multiple sub-regions. Literature offers numerous work that discusses tiling theorems with proof for distinct polyomino set. With Tromino forming the inspiration for our hTromo robot, this paper presents our first attempt at applying Tromino tiling theory to coverage problem for a floor cleaning robot. Specifically, we apply five theorems proposed in [30; 31; 32]. An \(a\times b\) rectangle can be tiled with \(L\)- and \(I\)-trominoes if and only if the area of that rectangle is divisible by 3. Figure 1 shows tiles \(\tau 2\) and \(\tau 4\) a right-oriented L-tromino piece, and tiles \(\tau 1\) to \(\tau 3\), a left-oriented L-tromino piece. In the experiments performed, we either used pre-defined testbed space that was either a complete rectangle or modified rectangle based on the theorem considered where the concerned robot was able to achieve complete coverage. Figure 2 presents a sample case for Theorem 1 involving tiling using I- and L-tromino configurations. Let \(a\), \(b\) be the integers such that \(2\leq a\leq b\). An \(a\times b\) rectangle can be tiled with set '\(T\)' trominoes if and only if one of the following conditions holds:  
1. \(a\) = 3 and \(b\) is even;
2. \(a\
eq 3\) and \(ab\) is divisible by 3. T' is a set of four L-trominoes. Since the tromino has three constituent squares with unit area, any rectangle with an area that is a multiple of 3 can be covered using any of the trominoes pieces. This notion leads to the formulation of Condition 2 of Theorem 2. Let \(a\) and \(b\) be the dimensions of the rectangle to be tiled. Since the area of the rectangle has to be a multiple of 3, (\(a\times b\)) = 3\(n\), where \(n\in\{1,2,3\dots\dots\) }. This implies that either a or b must be divisible by 3. However according to condition 2, a cannot be 3. For example, the rectangles that have a dimension of 6 \(\times\) 5, 4 \(\times\) 9, 12 \(\times\) 6 can be tiled using the set of '\(T\)' trominoes. Condition 1 of Theorem 2 excludes certain rectangles that cannot be tiled using '\(T\)' trominoes. Specifically, according to this theorem, it is impossible to tile rectangles  
Figure 1: T’ set of Tromino pieces. Figure 2: Tiling a ’\(a\times b\)’ rectangle with I- & L-Trominoes. that have a dimension of 3 \(\times\) 5, 3 \(\times\) 7, 3 \(\times\) 9 using any arrangements of 'T' trominoes. The Lemmas 1 and 2 detailed below validates Theorem 2. **Lemma 1**. : _Let a = 3 and b \(\in\) {2, 4, 6} then a 3 \(\times\) b rectangle can be tiled using the arrangement of 'T' trominoes shown in Figure 2(a). Hence it is proved that the smallest rectangle that satisfies condition 1 Theorem 2 is (3 \(\times\) 2). Let b \(>\) 6, even numbers, and c \(\in\) {2, 4, 6}, then b = 3n + c, where n can have a positive even integer value. As such, it allows for splitting of a (3 \(\times\) b) rectangle into n (3 \(\times\) 2) rectangles and one (3 \(\times\) c) rectangle, as in Figure 2(b). This implies, if b \(\geq\) 2, even numbers, then a (4 \(\times\) b) rectangle can be tiled using a set of 'T' trominoes._  
**Lemma 2**.