# Energy-Efficient Wireless Hopping Sensor Relocation Based on Prediction of Terrain Conditions
## 2 Previous Work  
### Characteristics and Movement Models of Hopping Sensors  
Typical mobile sensors are based on wheels. This structure has the disadvantage that it is very difficult to move in rough terrain such as gravel, sand, and so on. In order to overcome the mobility problem, hopping sensors that mimic movements such as a grasshopper or a frog have been devised. In addition, the hopping sensor enables communication among sensor nodes in environments where it is difficult for a sensor to communicate each other because the distance between sensor nodes is farther than a normal communication radius of a sensor, or where there are various obstacles which block communication propagation. Up to now, various researchers have shown that the connectivity between hopping sensors could be improved by the jumping of each hopping sensor. The papers [21,22] have shown that the communication radius is increased about six times compared to the one on the ground when a hopping sensor jumps 1 m from the ground. The paper [23] actually implemented a hopping sensor using a separate launcher for jumping and measured the transmission radius varied according to the height of the jump. The changed transmission radii were compared with the results of [21], and it was confirmed that they had similar results.  
Compared to a wheel-based sensor node, a hopping sensor, which is moving on jump, might be inherently less accurate to migrate to the desired area. In fact, it would be more desirable to have a positive explanation for enabling movement in an environment that cannot be moved with wheels, rather than a negative idea that movement accuracy is poor. The paper [15] first attempted a performance analysis of how a hopping sensor node was affected by wind when jumping and moving. It is obvious that the movement of hopping model is more vulnerable to air resistance than that of wheel-based model. Mathematical modeling of this hopping movement can be explained as follows.  
As shown in Figure 1, the hopping sensor does not land exactly the targeted location (\(\mathbf{T}\)) after jumping, and it is more likely to land near it (i.e., actual landing location; \(\mathbf{L}\)). Let the variables \(\mathbf{T}\) and \(\mathbf{L}\) be vectors of the target and actual landing locations. The error vector \(\mathbf{D}\) with respect to the difference between the locations can be expressed as \(\mathbf{L}-\mathbf{T}\). Here, we assume that \(\mathbf{D}\) follows the two-dimensional standard normal distribution with means (0, 0), standard deviations (\(\sigma_{x}\), \(\sigma_{y}\)), and correlation \(\rho\). The probability density function can be defined as  
\[f_{XY}=\frac{1}{2\pi\sigma_{x}\sigma_{y}\sqrt{1-\rho^{2}}}e^{-\frac{1}{2\ (1-\rho^{2})}(\frac{\rho^{2}}{\sigma_{x}^{2}}+\frac{\rho^{2}}{\sigma_{y}^{2}} -\frac{2\pi xy}{\alpha xy})} \tag{1}\]  
Let us consider that the hopping range is 2, standard deviations are 0.3, and the correlation is zero. The movement of the hopping sensor is as shown in Figure 1. The movement of the red circle is an ideal movement, but the movement of the blue circle is a movement that reflects the mentioned constraints. It can be easily seen that the movement of the blue circle more appropriately reflects the movement in rough environment suitable for the hopping sensor than that of the red circle. Therefore, the above error vector of the movement would be preferably applied to the simulation in Section 4.