## 3 Proposed Work
### State Space Model
The size of the voxel representation is assumed to be a \(\times\) a \(\times\) a, the normal distribution points are defined as \(N_{ijk}\) in Equation (5), and \(x,y,z\) is calculated with respect to \(i,j,k\). \[N_{ijk}^{f}=N\bigg{(}\frac{a}{d}\bigg{(}\frac{1}{\theta_{v}}+\frac{1}{\theta_{ h}}\bigg{)},a^{2} \tag{5}\]

where

* \(d\) is Sensor distance from the grid;
* \(\theta_{v}\) is vertical resolution of 3D LiDAR;
* \(\theta_{h}\) is horizontal resolution of 3D LiDAR. Hence, the probability of occupancy for each grid is:

\[P(n)=\frac{N_{ijk}}{N_{ijk}^{f}} \tag{6}\]

Because each state in the grid space is either \"Free\", \"Occupied\", or \"Unknown\", the grid state is defined as \"F, O, U\". The minor object in the unknown state may not be detected by the sensor due to its small size. The \"Free\" or \"Occupied\" or \"Unknown\" or \"Conflict\" elements of sensor data in grid space are represented by the set F, O, U, C. Conflict refers to data that are redundant, such as an image or some noise. The following mass functions are shown for the grid space: \(m(F)\), \(m(O)\), \(m(U)\), \(m(C)\). All states have masses that satisfy \(m=1\). The occupancy probability which is in Equation 6 shows grid timing and the probability of occupancy for each grid, as well as the data fusion formula derived to satisfy the Dezert Smarandache Theory (DSmT). ### Learning Informative Path Planning

The researchers have introduced a new algorithm, named Learning Informative Path Planning (Algorithm 2), to optimize a path in a continuous space with certain constraints that pertain to path budgets and the motion capabilities of a robot designed for plant phenotyping. The utility of the location set A is assigned by the sensing quality function F(A). Given: finite set \(V\) of locationsWant: \(A^{*}\subseteq V\) such that

\[A^{*}=argmaxF(A)\]

\[|A|\leq k\]

We now describe LIPP (Learning Informative Path Planning algorithm) follows recursive Bayesian for updating and replanning the path periodically. An upper bound of B time steps on the path cost and specified starting and ending locations (\(s_{1}\)) are used to initialize the algorithm. The utility function which represents the \"conditional utility\" conditioned on observation \(Y_{s}\) applies a nonadaptive algorithm in the first time step. After that, the algorithm moves the robot to the next location on the chosen nonadaptive path and iteratively uses update in subsequent time steps with an updated starting location of \(s_{1}\) and a budget of \(B_{1}\), while maintaining the finishing location at sB. Selecting the sensing location will be computed by,

\[S^{*}:=argmaxF(A\cup\{S\})A:=A\cup\{S^{*}\}\]
```
Input:\(s_{L},s_{u},u,B\) Output: Sequence of selected locations \(\pi(xV)\)  Begin \(s\gets s_{1};\pi[1]=s;Bpp\gets B;obs\leftarrow\{\}\);  for \(1\leq t\leq B\)do \(y_{s}\leftarrow\mathit{observe}(s);obs\gets obs\cup\{y_{s}=y_{s}\}\); \(P_{t}\gets\mathit{update}(s,sB,Bpp,F(|obs))\); \(s\gets P_{t}[2];\pi[t+1]\gets s;Bpp\gets Bpp\) -1  return \(\pi\) end
```
**Algorithm 2**Learning Informative Path Planning algorithm

[PERSON]'s path planning techniques [30] and earlier work assumed that the world was fully known in advance. However, particularly in outdoor settings, planning is often initiated with information that is incorrect, incomplete, lacking sufficient resolution, or has changed since it was obtained. This type of planning is based on the \"free space assumption\", which states that until it is proven otherwise, the unknown world can be completely traversed. To allow replanning based on sensor data gathered during the mission, the algorithms must be modified. This refers not to replanning around small obstacles, which is the responsibility of the reactive local navigator, but to replanning to obtain information critical to the robot's ability to achieve its goal according to the previous plan. The common approach is as follows: the system creates a global path using the information available and uses obstacle avoidance to navigate around obstacles.