# Path Planning of Intelligent Mobile Robots with an Improved RRT Algorithm
## 2 Improved RRT Path Planning Algorithm
### Collision Detection and Kalman Filter Redundancy Point Elimination  
Although the algorithm has been improved by the variable steering angle strategy proposed in this paper, the path searched for the first time may still contain more redundant points, which produces oscillations and unnecessary bit position adjustments during path planning. In this paper, based on the Kalman filtering algorithm, the current node passes through the collision detection, and then the step lengths between the node \(p_{rand}\) and the node \(p_{new}\) are compared, in which nodes with larger step lengths are filtered out through the Kalman gain until nodes with larger step lengths are no longer passed through this point for searching. There is no more searching through this point at this stage, and instead the prediction state of the next steering is adjusted. Figure 2: Schematic diagram of variable steering angle. Kalman Filtering is an autoregressive filtering method based on the minimum variance criterion, and its core principle originates from Bayesian filtering theory. The mathematical expressions are shown in Equations (5) and (6). \[x_{t}=Ax_{t-1}+Bi_{t} \tag{5}\]  
\[y_{t}=Hx_{t}+o_{t} \tag{6}\]  
In the above equation, \(x_{t}\) is the state of the environment in which the robot is located at time t, A is the state transition matrix, B is the input control matrix, y_t is the observation value of the environment by the robot at time t, \(i_{t}\) is the input noise, and \(o_{t}\) is the observation noise. H is the observation matrix, as shown in Equation (7). \[H= \begin{bmatrix}h_{11}&h_{12}&\dots&h_{1n}\\ h_{21}&h_{22}&\dots&h_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ h_{p1}&h_{p2}&\dots&h_{pn}\end{bmatrix}  \tag{7}\]  
The observation matrix maps the system state \(x_{t}\) to the observation space and describes how the system state affects the measurements. The dimension of H is \(p\times n\), where p is the number of observation variables and n is the number of state variables. The observation matrix H describes which state variables in the state vector \(x_{t}\) contribute to the measurements. The robot predicts the next state and error covariance based on a known dynamic model and then calculates the Kalman gain based on the predicted error covariance and measurement noise. The Kalman gain formula is shown in Equation (8). \[K_{n}=\frac{P_{n}H^{\mathrm{T}}}{HP_{n}H^{\mathrm{T}}+R} \tag{8}\]  
In the above equation, R represents the covariance matrix of the node measurement noise, P\({}_{n}\) is the covariance matrix of the nth node a priori estimation, H\({}^{\mathrm{T}}\) is the transpose of the measurement matrix H, and K\({}_{n}\) is the Kalman gain of the nth time. Under the assumption of constant time, the robot's state is updated based on the measurements, and the target movement point is determined, significantly reducing the computational time complexity of the random tree during the search. The flowchart of the Kalman filter algorithm is shown in Figure 3.  
Figure 3: Flow chart of Kalman filtering. Redundant points in path planning refer to unnecessary nodes between the starting point and the target point. Removing these points can make the path simpler while reducing the number of node observation pose, thereby improving search efficiency. Based on the fundamental principles of the Kalman filter algorithm, this paper estimates the node states through the mapping of the observation matrix H, predicting the future position of the nodes and evaluating the necessity of the nodes. The principle of redundant node elimination based on the Kalman filter is shown in Figure 4.  
As shown below, the black node Start is the starting point, Final is the target point, and points \(p_{1}\sim p_{7}\) are the path planning points, with a total of seven points.