## 5 Thrust Distribution Layout
### Thrust Coupling Projection Analysis of the Propulsor
#### 5.2.3 Total Thrust of the Propulsion System  
After the above derivation, the total thrust \(T_{X}\), \(T_{Y}\), and \(T_{Z}\) of the six thrusters in the new ROV propulsion system on the three coordinate axes is  
Figure 16: Thrust decomposition of the horizontal thruster. (**a**) Thrust decomposition of No. 3 and No. 4 thruster. (**b**) Thrust decomposition of No. 3 thruster.  
\[T_{X}=T_{X1}+T_{X2}=T_{1}\cos\alpha+T_{2}\cos\alpha+T_{3}\cos\eta+T_{4}\cos\eta-T_{5 }\cos\delta-T_{6}\cos\delta \tag{13}\]  
\[T_{Y}=T_{Y1}+T_{Y2}=T_{1}\cos\beta-T_{2}\cos\beta+T_{5}\cos\varepsilon-T_{6}\cos\varepsilon \tag{14}\]  
\[T_{Z}=T_{Z1}+T_{Z2}=T_{1}\cos\gamma+T_{2}\cos\gamma-T_{3}\sin\eta-T_{4}\sin \eta-T_{5}\cos\varsigma-T_{6}\cos\varsigma \tag{15}\]  
The total thrust torque \(M_{X}\), \(M_{Y}\), and \(M_{Z}\) of the six thrusters in the three coordinate axes is  
\[\begin{array}{l}M_{X}=M_{X1}+M_{X2}=T_{1}\cos\beta z-T_{1}\cos\gamma y_{1}-T_ {2}\cos\beta z+T_{2}\cos\gamma y_{1}+T_{3}\sin\eta y_{2}\\ -T_{4}\sin\eta y_{2}+T_{5}\cos\varepsilon z+T_{5}\cos\varsigma y_{3}-T_{6} \cos\varepsilon z-T_{6}\cos\varsigma y_{3}\\ \\ M_{Y}=M_{Y1}+M_{Y2}=-T_{1}\cos\alpha z-T_{1}\cos\gamma x_{1}-T_{2}\cos\alpha z -T_{2}\cos\gamma x_{1}-T_{3}\cos\eta z\\ -T_{3}\sin\eta x_{2}-T_{4}\cos\eta z-T_{4}\sin\eta x_{2}+T_{5}\cos\delta z-T_{ 5}\cos\varsigma x_{3}+T_{6}\cos\delta z-T_{6}\cos\varsigma x_{3}\\ \\ M_{Z}=M_{Z1}+M_{Z2}=T_{1}\cos\alpha y_{1}+T_{1}\cos\beta x_{1}-T_{2}\cos \alpha y_{1}-T_{2}\cos\beta x_{1}\\ +T_{3}\cos\eta y_{2}-T_{4}\cos\eta y_{2}-T_{5}\cos\delta y_{3}-T_{5}\cos \varepsilon x_{3}+T_{6}\cos\delta y_{3}+T_{6}\cos\varepsilon x_{3}\\ \end{array} \tag{16}\]  
\[\begin{array}{l}M_{Y}=M_{Y1}+M_{Y2}=-T_{1}\cos\alpha z-T_{1}\cos\alpha z-T_{ 2}\cos\alpha z-T_{3}\cos\eta z\\ -T_{3}\sin\eta x_{2}-T_{4}\sin\eta x_{2}+T_{5}\cos\delta z-T_{5}\cos\delta z-T _{6}\cos\delta z-T_{6}\cos\varsigma x_{3}\\ \\ M_{Z}=M_{Z1}+M_{Z2}=T_{1}\cos\alpha y_{1}+T_{1}\cos\beta x_{1}-T_{2}\cos \alpha y_{1}-T_{2}\cos\beta x_{1}\\ +T_{3}\cos\eta y_{2}-T_{4}\cos\eta y_{2}-T_{5}\cos\delta y_{3}-T_{5}\cos \varepsilon x_{3}+T_{6}\cos\delta y_{3}+T_{6}\cos\varepsilon x_{3}\\ \\ \end{array} \tag{17}\]  
The above three equations can be generalized. The six degrees of freedom thrust and thrust torque generated by the six thrusters on the ROV are  
\[T=\begin{bmatrix}T_{X}\\ T_{Y}\\ T_{Z}\\ M_{X}\\ M_{Y}\\ M_{Z}\end{bmatrix}=B(\theta)\begin{bmatrix}T_{1}\\ T_{2}\\ T_{3}\\ T_{4}\\ T_{5}\\ T_{6}\end{bmatrix}=B(\theta)U \tag{18}\]  
In the formula,  
* \(U\)--thrust vector matrix of the thruster output, \(U=\begin{bmatrix}T_{1}&T_{2}&T_{3}&T_{4}&T_{5}&T_{6}\end{bmatrix}^{T}\);
* \(B(\theta)\)--vector placement matrix of the thruster. The vector arrangement matrix of the ROV propulsion system is \[\begin{array}{l}B(\theta)=\begin{bmatrix}\cos\alpha&\cos\alpha&\cos\eta\\ \cos\beta&-\cos\beta&0\\ \cos\gamma&\cos\gamma&-\sin\eta\\ \cos\beta z-\cos\gamma y_{1}&-\cos\beta z+\cos\gamma y_{1}&\sin\eta y_{2}\\ -\cos\alpha z-\cos\gamma x_{1}&-\cos\alpha z-\cos\gamma x_{1}&-\cos\eta z-\sin \eta x_{2}\\ \cos\alpha y_{1}+\cos\beta x_{1}&-\cos\alpha y_{1}-\cos\beta x_{1}&\cos\eta y_{2} \\ \cos\eta&-\cos\delta&-\cos\delta\\ 0&\cos\varepsilon&-\cos\varepsilon\\ -\sin\eta&-\cos\varsigma&-\cos\varsigma\\ -\sin\eta y_{2}&\cos\varepsilon z+\cos\varsigma y_{3}&-\cos\varepsilon z-\cos \varsigma y_{3}\\ -\cos\eta h-\sin\eta x_{2}&\cos\delta z-\cos\varsigma x_{3}&\cos\delta z-\cos \varsigma x_{3}\\ -\cos\eta y_{2}&-\cos\delta y_{3}-\cos\varepsilon x_{3}&\cos\delta y_{3}+\cos \varepsilon x_{3}\\ \end{array}\] (19)