## 4 Path Model Estimation
### IPSO-Based Path Estimation  
The detailed process of the IPSO-based path parameter estimation is shown in Figure 9, consisting of the following steps. Step 1: Acquiring the path pixel set  
Scan the binarized path image line by line, recording the pixel positions \(hl_{j}\) and \(lr_{j}\) of the left and right boundary points of the path. The blue center pixel set \(\mathrm{S_{C}}\) of the path is obtained. Step 2: Defining the fitness function  
Optimize the straight path parameters \(\beta_{0}\) and \(\beta_{1}\) based on the minimum mean squared error criterion. Equation (16) is defined as the fitness function of IPSO. Step 3: Generating the particle swarm  
In the \(h\)-dimensional parameter search space, the size of the particle swarm is \(r\), i.e., \(r\) particles constitute the population \(\mathbf{Q}=\left\{\mathbf{q}_{1},\mathbf{q}_{2},\ldots,\mathbf{q}_{r}\right\}\). The \(t^{\mathrm{th}}\) particle is an \(h\)-dimensional vector, representing the position vector of the particle in the search space, i.e., \(\mathbf{q}_{t}=\left[Q_{t1},Q_{t2},\ldots,Q_{th}\right]^{\mathrm{T}}\), which is a potential solution for the parameter estimation problem. The velocity vector of the \(t^{\mathrm{th}}\) particle is \(V_{t}=\left[V_{t1},V_{t2},V_{t3},\ldots,V_{th}\right]^{\mathrm{T}}\). The velocity range of the particle swarm is \(\left[V_{\min},\,V_{\max}\right]\), and its position range is \(\left[Q_{\min},\,Q_{\max}\right]\). Step 4: Initializing the particle swarm  
The parameters of the particle swarm are configured as follows: the population size \(r=20\), and the maximum number of evolution generation \(N_{\mathrm{IterMax}}=100\). The first and last points of the target path center point set \(\mathrm{S_{C}}\), i.e., \((x_{1},y_{1})\) and \((x_{M},y_{M})\), are chosen to estimate the initial parameters of the straight path, obtaining \(\beta_{0}=\beta_{0\mathrm{Int}}\) and \(\beta_{1}=\beta_{1\mathrm{Int}}\). Each individual of the particle swarm is initialized using the initial parameters, including the individual extremum \(\mathbf{P}_{t}=\left[P_{t1},\,P_{t2},P_{t3},\,\ldots,P_{th}\right]^{\mathrm{T}}\) and the global extremum \(\mathbf{P}_{g}=\left[P_{g1},\,P_{g2},P_{g3},\,\ldots,P_{gh}\right]^{\mathrm{T}}\) at the first each generation. The initial fitness value \(f_{\mathrm{int}}\) of each particle is calculated by means of Equation (16), used as the fitness value \(f_{t}^{s}\) of the individual extremum and \(f^{g}\) of the global extremum, i.e., \(f_{t}^{s}=f_{\mathrm{int}}\) and \(f^{g}=f_{\mathrm{int}}\). Step 5: Evolving the particle swarm  
Figure 9: IPSO-based path parameter estimation. For the \(r\) particles in the \(u^{\rm th}\) generation (\(u=1,\,2,\,\ldots\,,\,N_{\rm IterMax}\)), calculate the average value of all individual extrema by means of Equation(17):  
\[\overline{\mathbf{P}_{u}}=\frac{1}{r}\sum_{t=1}^{r}\mathbf{P}_{t} \tag{17}\]  
The inertial weight of the \(t^{\rm th}\) particle is obtained by means of Equation (18):  
\[\omega=1.1-\frac{\mathbf{P}_{g}}{\overline{\mathbf{P}_{u}}} \tag{18}\]  
Then, Equation (19) is used to compute the acceleration factor of the individual. \[c_{1}=c_{2}=1+\frac{\mathbf{P}_{g}}{\mathbf{P}_{t}} \tag{19}\]  
For the \(u^{\rm th}\) generation particle \(q_{t}(u)\), calculate the \(d^{\rm th}\) (\(d=1,\,2,\,\ldots\,,\,h\)) component of the velocity vector of the (\(u\)+1)\({}^{\rm th}\) generation according to Equation (20). \[V_{\rm td}(u+1)=\omega V_{\rm td}(u)+c_{1}r_{1}(P_{\rm td}(u)-Q_{\rm td}(u))+c _{2}r_{2}\left(P_{\rm gd}(u)-Q_{\rm id}(u)\right) \tag{20}\]  
where \(r_{1}\) and \(r_{2}\) are the random numbers uniformly distributed in the range of [0,1]. If \(V_{\rm td}(u+1)\) exceeds the speed range [\(V_{\rm min}\), \(V_{\rm max}\)], adjust it to the nearest speed boundary value. Moreover, Equation (21) is used to calculate the \(d^{th}\) component of its position vector at the (\(u\)+1)\({}^{\rm th}\) generation.