# Research on Decision-Making Methods for Autonomous Navigation in Inland Tribitary Waterways
## 2 Analysis of the Autonomous Navigation Process
### Autonomous Route Tracking Without the Risk of Collision  
#### 2.1.1 Ship Motion Model  
During ship navigation and collision avoidance, it is generally unnecessary to consider the ship's roll, pitch, and heave motions. Instead, the focus lies on the ship's sway, surge, and yaw motions. To accurately simulate the ship's motion, the three-degree-of-freedom (3-DOF) MMG (Maneuvering Modeling Group), which takes into account the effects of wind, waves, and currents, is employed [44], which can be mathematically expressed as Equation (1).  
\[\left\{\begin{array}{l}(m+m_{x})\cdot\dot{u}+(m+m_{x})\cdot\!r\cdot\!v=X_{H}+X_ {P}+X_{R}+X_{E}\\ (m+m_{y})\cdot\!\dot{v}-\left(m+m_{y}\right)\cdot\!u\cdot\!v=Y_{H}+Y_{P}+Y_{R}+ X_{E}\\ (l_{zz}+l_{zz})\cdot\!\dot{r}=N_{H}+N_{P}+N_{R}+N_{E}\end{array}\right. \tag{1}\]  
The vessel dynamics are characterized by mass parameters where \(m\) denotes total mass, with \(m_{x}\) and \(m_{\gamma}\) representing longitudinal and transverse added mass, respectively. Accelerations are defined by \(\dot{u}\) (longitudinal), \(\dot{v}\) (transverse), and \(\dot{r}\) (angular). Inertial properties are quantified through \(Izz\) (moment of inertia) and \(Jzz\) (added moment of inertia). Force components comprise \(X\) (longitudinal), \(Y\) (transverse), and \(N\) (yaw moment), while \(H\) (hull), \(P\) (propeller), and \(R\) (rudder) denote subsystem-specific forces/moments. Environmental disturbances, \(E\), encapsulate wind, current, and shallow-water effects on hull dynamics. When a ship is sailing at a constant velocity, the effective thrust, \(Y_{P}\), of the propeller can be expressed as Equation (2).  
\[Y_{P}=\left(1-t_{p}\right)T \tag{2}\]  
\[T=\rho(np)^{2}D_{p}{}^{4}K_{t} \tag{3}\]  
\[t_{p}=\left\{\begin{array}{l}0.04+\frac{(t_{p}^{\prime}-0.04)u}{u_{0}}\ \ \ u<u_{0}\\ t_{p}^{\prime}\ \ \ \ \ \ \ \ \ \ \ u>u_{0}\end{array}\right. \tag{4}\]  
\[t_{p}^{\prime}=f+t_{p} \tag{5}\]  
Figure 1: The process of ship encounter and collision avoidance actions.  
Here, \(T\) denotes propeller thrust, and \(t_{p}^{\prime}\) represents the thrust deduction coefficient, defined as the ratio of thrust reduction to propeller thrust. Its magnitude depends on factors including ship type, propeller dimensions, propeller loading, and the relative position between the propeller and hull. And \(f\) is the function of the effect of ship motion, which is determined as Equation (6).  
\[\left\{\begin{array}{l}f=k_{t}\beta_{R}\\ k_{t}=0.00023\big{(}\gamma_{A}\cdot L/D_{p}\big{)}-0.028\\ \gamma_{A}=(B/d)\{1.3(1-C_{b})-3.1l_{cb}\}\\ l_{cb}=x_{c}/L\cdot 100\\ \beta_{R}=\beta-l_{R}\tilde{\
u}\end{array}\right. \tag{6}\]  
\(\beta_{R}\) indicates the rudder drift angle, while \(B\) (beam), \(d\) (draft), \(L\) (length), \(D_{p}\) (propeller diameter), and \(x_{c}\) (longitudinal coordinate of the center of flotation) are key ship parameters. The coefficients \(k_{t}\) (thrust correction factor), \(\gamma_{A}\) (hull wake correction factor), \(C_{b}\) (block coefficient), \(l_{cb}\) (longitudinal center of buoyancy), and \(l_{R}\) (rudder drift angle coefficient) characterize hydrodynamic adjustments. Finally, \(r\) (yaw angular velocity) and \(V\) (longitudinal velocity) describe the ship's kinematic state.