## 3 Proposed Work
### State Space Model  
The environmental condition cannot be directly measured. Through a set of observables, the state measures the effect on the outside world. The state space model, which consists of estimating and measuring the environment, describes the character of a dynamic system. Equation (1) depicts the estimation of the environment's state. \[X_{n+1}=A_{n}(x_{n+1},W_{n}) \tag{1}\]  
where  
* \(X_{n}\) denotes the current PC data from the environment;
* \(X_{n+1}\) denotes the subsequent PC data from the environment;
* \(W_{n}\) is the noise value in PC like NaN;
* \(A_{n}\) is the estimation function. Measurement of environment state is represented as Equation (2),  
\[Y_{n}=b_{n}(x_{n},V_{n}) \tag{2}\]  
where  
* \(Y_{n}\) the set of observable form LiDAR;
* \(b_{n}\) is measurement function between points;
* \(V_{n}\) is the measured noise. Let us assume the initial state \(X_{o}\) is uncorrelated with the dynamic noise \(W\), and the two sources of noise \((W_{n},V_{n})\) are statically independent. The Extended Kalman Filter (EKF) accommodate linear model. The main advantage of EKF is the small number of parameters it can object. In this work, the environment is captured using a LiDAR sensor, and the PC data are generated from it. To model the state of the LiDAR PC data, the EKF algorithm is employed. The EKF for LiDAR PC data is discussed in Algorithm 1. Three different states are identified: Free State, Occupied State, and Unknown State. An octree data design is utilized to store the sensor information and is effective for representing unknown features. The features are then extracted and stored in a voxel representation. The EKF algorithm is applied in both positioning systems and perception systems. ```
Input: observations \(\{y_{1},y_{2}\dots,y_{n}\}\) Output: State estimation and state prediction using PC data
1. The linearized matrix \(A_{n+1},n\) and \(B_{n}\) are computed from the nonlinear parameter \(a_{n}(x)\) and \(b_{n}(x)\). 2. The values \(a_{n}(x_{n/m})\) and \(b_{n}(x_{n/m})\) are obtained by substituting the filter state estimate and predict the statement estimate. 3. Examine the order of iteration. \begin{table}
\begin{tabular}{l l} \hline \hline
**Features** & **Values** \\ \hline Make & Velodyne LiDAR \\ Model & Puck LITE \\ Weight & 590 g \\ Output & Dual reoccurrence and 16 Channels \\ Range & 100 M \\ Capacity & Up to \(\sim\)600,000 points/s \\ FOV & 360\({}^{\circ}\) Horizontal, \(\pm\)15\({}^{\circ}\) Vertical \\ \hline \hline \end{tabular}
\end{table} 
Table 2: LiDAR Configuration. ### Grid Map Construction

An advanced motion planning system has been developed for autonomous vehicles, allowing them to navigate through dynamic and changing environments. The procedure of searching a secure and collision-free path from an initial location to a final destination is referred to as motion planning for autonomous vehicles. The path-planning problem is solved using a Non-Parametric (NP) method. The grid map of the sensed environment is generated using the Bayesian filtering method. Equation (3) represents the posterior function of the Bayesian filter, which can be calculated using Equations (1) and (2). \[\mathrm{P}(m/z_{1:t})=\frac{p(\mathrm{m}/z_{1:t-1})\mathrm{p}(Z_{t}/m)}{P(Z_{t} /Z_{1:t-1})} \tag{3}\]

where

* \(\mathrm{P}(m/z_{1:t})\) is the Posterior Points Occupy;
* \(p(\mathrm{m}/z_{1:t-1})\) is prior probability and \(\frac{p(Z_{t}/m)}{P(Z_{t}/Z_{1:t-1})}\) is likelihood. The log odds for each side of equation are found after simplifying Equation (3) by assuming a uniform prior distribution threshold of 0.5 for the unknown state. \[\mathrm{L}(m/z_{1:t})=L(\mathrm{m}/z_{1:t-1})+\mathrm{L}(Z_{t}/m) \tag{4}\]

Equation (4) is recursive, and \(\mathrm{L}(m/z_{1:t})\) is defined as the inverse of the sensor model. Thus, the grid map was incrementally updated while reading the new sensor data.