## 2 Motion Model and Failure Effect  
The launch vehicle power system consists of turbopumps, combustion chambers, piping, valves, and other important parts. Common power failures include turbopump failure, thrust chamber failure, piping and valve failure, etc. Since the generation of all these faults reduces engine thrust, the power failures can be equivalented to engine thrust loss. This paper establishes the kinetic equations in the launch inertial coordinate system [18] (the origin \(O_{n}\) is the launch point; the \(O_{n}X_{a}\) axis is in the horizontal plane of the launch point, pointing in the aiming direction; the \(O_{n}Y_{a}\) axis points upward, perpendicular to the horizontal plane of the launch point; the \(O_{n}Z_{a}\) axis and the other two axes constitute a right-handed system, and the coordinate system remains motionless in inertial space). \[\left\{ \begin{array}{l}\dot{\mathbf{V}}=\frac{\kappa\mathbf{T}_{\rm vac}}{m}+\mathbf{g}( r)+\frac{\mathbf{R}}{m}\\ \dot{\mathbf{r}}=\mathbf{V}\\ \dot{m}=-\frac{\kappa\|\mathbf{T}_{\rm vac}\|}{l_{\rm sp}\delta_{0}}\end{array} \right. \tag{1}\]  
where \(\mathbf{r},~{}\mathbf{V}\) denote the position and velocity vector of the launch vehicle, respectively; \(\mathbf{g}\) denotes the gravitational acceleration vector; \(\mathbf{g}_{0}\) is the sea level gravitational acceleration value; \(\mathbf{R}\) denotes the aerodynamic vector; \(\kappa\) is the equivalent power system failure factor; \(\mathbf{T}_{\rm vac}\) denotes the engine thrust vector; \(m\) is the launch vehicle mass; \(\dot{m}\) is the second consumption; \(l_{\rm sp}\) is the fuel-specific impulse. The injection stage of new-generation launch vehicles usually has a secondary switching ability to adapt to more complex missions. However, different failure modes have different influences on the secondary switching ability of launch vehicle engines, and mission requirements and engine performance will jointly determine the injection mode of launch vehicles. For example, in the event of turbopump wear failure, speed drop failure, or partial engine failure, the remaining normal parts of the engine and even the affected engine itself can still switch on and off again. When fire and thrust chamber failure occurs, the engine should be shut down immediately and not allowed to rework. When the turbine pump bearings are damaged, the opening of pipes and valves is not enough, and the closing is not strict, the engine can continue to work in the current mode to avoid the deterioration of the fault or failure to start up when restarted. In addition to causing decreased engine thrust, the different failure modes will also affect its ability to perform the orbital transfer. Therefore, the authors artificially divided the launch vehicle orbit-injection mode under power system failure into direct orbiting and transfer orbiting. The target orbit is usually described using the orbital elements, including semimajor axis \(a\), eccentricity \(e\), inclination \(i\), longitude of the ascending node \(\Omega\), argument of periapsis \(\omega\), and true anomaly \(\theta\). Among them, \(a\) and \(e\) determine the shape of the orbit, \(i\), \(\Omega\), and \(\omega\) determine the spatial position of the orbit, and \(\theta\) determines the position of the vehicle in orbit. When the target orbit is circular, the definitions of the argument of perigee and true anomaly are meaningless, and the geocentric magnitude angle \(\phi\) formed between the ascending nodal line and the geocentric vector of the injection point has physical significance in any case, which can be used to describe the target orbit. The target orbit can be described as follows [19]:  
\[\left\{