## 4 Energy Consumption Modeling  
### Subsystem Model  
An inertial reference frame \(\{\mathbf{e_{1},e_{2},e_{3}}\}\) is initially established, followed by consideration of the fuselage of the FPTLIR as a rigid body to establish a rigid coordinate system \(\{\mathbf{b_{1},b_{2},b_{3}}\}\), with the origin positioned at the center of gravity. Here, \(\mathbf{b_{1}}\) and \(\mathbf{b_{2}}\) denote the forward and upward directions of the FPTLIR, respectively, while \(\mathbf{b_{3}}\) can be determined using the right-hand rule. Figure 4 shows the coordinate system and force analysis of the FPTLIR, while its rigid-body dynamics equations in 3D space are presented as Equations (1)-(3). \[m\ddot{\mathbf{s}}=\mathbf{T}_{n}\mathbf{Re_{3}}-mg\mathbf{e_{3}}-\frac{1}{2}\rho S_{E}\| \dot{\mathbf{s}}\|\dot{\mathbf{s}} \tag{1}\]  
\[\dot{J}\dot{\mathbf{\Omega}}=M-\mathbf{\Omega}\times J\mathbf{\Omega} \tag{2}\]  
\[\dot{R}=R\dot{\mathbf{\Omega}} \tag{3}\]  
where \(m\) is the mass of the FPTLIR, \(\mathbf{T}_{n}\) is the net thrust of the FPTLIR, \(R\) is the rotation matrix of the rigid-body coordinate system mapped to the inertial reference system, \(g\) is the acceleration of gravity, \(\rho\) is the density of the air at 25 \({}^{\circ}\)C, \(S_{E}\) is the equivalent planar area of the fuselage (\(S_{E}\) is equal to \(S_{HE}\) for horizontal flight of the FPTLIR, and is equal to and \(S_{VE}\) for vertical flight), \(\mathbf{s}\) is the path position of the FPTLIR in the inertial coordinate system \(\{\mathbf{e}\}\), \(J\) is the inertia matrix and \(J=diag[I_{x},I_{y},I_{z}]\), and \(\mathbf{\Omega}\in\mathbb{R}^{3}\) is the angular velocity of the FPTLIR in the rigid-body coordinate system. Operator \(\wedge\) is defined as follows: for any \(x,y\in\mathbb{R}^{3}\), there exists \(\dot{x}y=x\times y\). The operator \(\vee\) is the inverse of operator \(\wedge\). The net thrust and combined moment required for the FPTLIR to follow a given path \(\mathbf{s}_{g}\) can be determined using Equations (4) and (5), respectively. \[\mathbf{T}_{ng}=m\ddot{\mathbf{s}}_{g}+mg\mathbf{e_{3}}+\frac{1}{2}\rho S_{E}\|\dot{\mathbf{s }}_{g}\|\dot{\mathbf{s}}_{g} \tag{4}\]  
\[\mathbf{M}_{g}=\dot{J}\dot{\mathbf{\Omega}}_{g}+\mathbf{\Omega}_{g}\times J\mathbf{\Omega}_{g} \tag{5}\]  
where \(\mathbf{\Omega}_{g}=(R_{S}^After calculating the first and second order derivatives of the rotation velocity matrix, \(\dot{R}_{g}\) and \(\dot{R}_{g}\) can be obtained. Substituting them into Equation (5), the required combined moment \(M_{g}\) is determined. Through this deduction, the net thrust and combined torque required to achieve the corresponding motion can be obtained from the path. The FPTLIR adjusts the magnitude of the angular velocity of all six propellers to change both the generated thrust and torque accordingly. Neglecting the precision errors due to elastic deformation and friction between blades and fuselage, Equations (7)-(10) demonstrate how the net thrust and the torque are calculated through rigid-body dynamics of the FPTLIR. \[T_{n}=\sum_{i=1}^{6}F_{Ti} \tag{7}\]  
\[M_{1}=\frac{1}{2}l(-F_{T1}-F_{T3}+F_{T4}+F_{T6})+l(-F_{T2}+F_{T5}) \tag{8}\]  
\[M_{2}=\frac{\sqrt{3}}{2}l(-F_{T1}+F_{T3}+F_{T4}-F_{T6}) \tag{9}\]  
\[M_{3}=F_{Q1}-F_{Q2}+F_{Q3}-F_{Q4}+F_{Q5}-F_{Q6} \tag{10}\]  
where \(F_{Ti}(i=1,\ldots,6)\) is the individual thrust generated by each of the six propellers' rotations. \(M_{1}\), \(M_{2}\), and \(M_{3}\) represent moments around the axes \(b_{1}\), \(b_{2}\), and \(b_{3}\), respectively. The combined moment is denoted as \(\mathbf{M}=[M_{1},M_{2},M_{3}]\). \(F_{Q7}(i=1,\ldots,6)\) is the torque produced by each propeller's rotation. Additionally, \(l\) refers to the distance between the center of gravity of the FPTLIR and its propellers' center. The thrust and torque of a single power unit of the FPTLIR can be determined by the propeller aerodynamics. The model consists of two components: one based on the blade  
Figure 4: Schematic diagram of system dynamics model.