Uncertainty Requirement Analysis for the Orbit, Attitude, and Burn Performance of the 1^st Lunar Orbit Insertion Maneuver

Two-body equations of motion for the spacecraft after the 1^st LOI maneuver can be expressed as (Vallado 2013)(1)(2)

with the initial position, r₀, and velocity, v₀, vectors of the spacecraft after the 1^st LOI execution (Vallado 2013).(4)

Here, μ is the gravitational constant of the moon, r and v denote the position and velocity vectors, respectively, and r_hyp and v_hyp denote the hyperbolic arrival position and velocity vectors of the spacecraft at the periapsis, respectively. Δv_LOI is the delta-V of the applied 1^st LOI maneuver, assumed to be an impulsive burn in the current study to capture the spacecraft in orbit around the moon. As shown in Eq. (3), r_hyp can be obtained directly from r₀, and v_hyp can be obtained as follows:

where M is a transformation matrix that converts the states expressed in the inertial frame into the radial-tangentialnormal (RTN) frame, and ν_rtn is the periapsis velocity vector of the hyperbolic approach trajectory with its vector components of v_r, v_t, and v_n. And v_r, v_t, and v_n are the radial, tangential, and normal components of the periapsis velocity vector, respectively; they can be defined as v_t = v_app, and v_r, v_n = 0, where v_app is the magnitude of the hyperbolic approach velocity at periapsis. M can be computed using r₀ and v₀, and these vectors can be directly obtained using the six orbital elements (a₀, e₀, i₀, ω₀, Ω₀, ν₀) of the 1^st capture orbit, which are all predesigned parameters. The semi-major axis, a₀, of the 1^st capture orbit can be computed using the desired orbital period of the 1^st capture orbit, P; the eccentricity, e₀, can also be derived using the mission-dependent radius of the closest approaching periapsis, r_p, as shown in Eq. (6) (Brown 1998).

In Eq. (6), R_m is the mean radius of the moon, and h_p is the periapsis altitude at the moment of closest lunar approach, which is also a pre-defined value. Other elements such as inclination, i₀, argument of periapsis, ω₀, right ascension of ascending node, Ω₀, and true anomaly, ν₀, are missiondependent parameters that are determined during mission definition or during the LOA phase design. The definitions shown in Eqs. (3)-(5) are valid as the current study assumed that the 1^st LOI maneuver was executed at the periapsis with an impulsive maneuver.

To consider uncertainties in the initial conditions of the 1^st LOI maneuver (pre-maneuver uncertainty for both position and velocity) and maneuver performance during execution (attitude and burn magnitude), statistical dispersion analysis was performed using Monte Carlo simulations. The perturbed initial position, $r_{hyp}^{pert}$, and velocity, $v_{hyp}^{pert}$, of the spacecraft at the moment of the 1^st LOI maneuver execution are constructed as follows:

Here, randn(0,1) is a Gaussian random number generator with zero mean and a unit standard deviation (Ross 2009); ${\sigma }_x^{pos},{\sigma }_y^{pos},{\sigma }_z^{pos},{\sigma }_x^{vel},{\sigma }_y^{vel}\text{\hspace{0.17em}and\hspace{0.17em}}{\sigma }_z^{vel}$ are the expected pre-maneuver position (indicated by superscript pos) and velocity (indicated by superscript vel) uncertainties. $\Delta v_{LOI}^{pert}$ is the perturbed 1^st LOI maneuver with both three-dimensional attitude and burn magnitude uncertainties and can be modeled as

where Δv_LOI is the magnitude of the nominally planned 1^st LOI maneuver, σ_dvm is the expected magnitude error of that maneuver, Q is a 3-by-3 transformation matrix to reflect the three-dimensional attitude uncertainty as a function of the expected errors in the radial (${\sigma }_{\psi }^{att}$) , transverse (${\sigma }_{\theta }^{att}$), and normal (${\sigma }_{\varphi }^{att}$) directions. Again, M is a transformation matrix that converts states expressed in the inertial frame into the RTN frame; finally, ${\widehat{u}}_{rtn}^{pert}$ is the unit vector of the perturbed states expressed in the RTN frame, which can easily be determined from the initial states of $r_{hyp}^{pert}$ and $v_{hyp}^{pert}$. In Fig. 1, the exaggerated geometry of the 1^st LOI maneuver, nominal vs. the case with an initial error condition, is shown. Using the initial perturbed states shown in Eqs. (7a) and (7b), the 1^st capture orbit is numerically propagated using the Runge– Kutta–Fehlberg 7–8^th order variable step size integrator. In addition, the uncertainties in the apoapsis and periapsis approach conditions of the 1^st lunar capture orbit (due to the perturbed initial conditions) are determined using the root-finding algorithm described in Brent (2002) with implementation of the proper objective functions.

As the current study focused on an early design phase uncertainty analysis, two-body equations of motion are used, and the 1^st LOI burn is assumed to be an impulsive maneuver. Furthermore, the direction of the 1^st LOI maneuver is assumed to always be in the direction opposite to the velocity vector at the periapsis of the orbit. The spacecraft is assumed to have already approached the periapsis with a hyperbolic arrival trajectory around the moon, and the orbital element of the 1^st target capture orbit is assumed to be i₀ = 90°, with values of 0° for ω₀, Ω₀ and ν₀. The other orbital elements, a₀ and e₀, are determined from the pre-defined values of v_app, P, and h_p. The current study assumed v_app to be within the range of 2.4–2.6 km/s, h_p is assumed to be 100, 200, and 500 km, and P is assumed to be 12 or 24 hr, considering the human factors that may be a major consideration while establishing operation schedules.

First, the minimum maneuver burn performance requirement to achieve capture around the moon was investigated. As the current study assumed an impulsive maneuver, the maneuver burn performance can simply be regarded as the magnitude of the executed maneuver. Among the various error sources described in Subsection 2.2, only σ_dvm was considered for the current simulation, while the other errors were set to zero. For σ_dvm, it was assumed that a maximum of 0–80 % of the error (3σ) occurs with respect to the magnitude of the nominally planned delta-V maneuver; given that error is increased in 5 % steps during the simulation, a total of 1,000 Monte Carlo runs were performed for the statistical analysis. The main purpose of this simulation is to evaluate the sensitivities of the maneuver burn performance that directly affect the shape of the 1^st capture orbit. Through this analysis, the minimum burn requirement to achieve capture around the moon can be estimated, which may aid in establishing additional plans for contingency trajectory options during the early system design phase. Tables 1 and 2 show the maximum allowable maneuver magnitude errors for cases in which the 1^st elliptical capture orbits have an orbital period of 12 hr (Table 1) or 24 hr (Table 2). If the maneuver magnitude errors are greater than those shown in Tables 1 and 2, the spacecraft is not captured, and it will fly past the moon. As expected, Tables 1 and 2 clearly show that the maximum allowable maneuver magnitude errors decrease with an increase in the assumed magnitude of the hyperbolic arrival velocity, at periapsis, with its altitude. For example, if the spacecraft is targeted to approach the moon with a periapsis arrival velocity of 2.4 km/s at a 100 km altitude, and if a 12 hr orbital period is desired for the 1^st capture orbit, then the spacecraft can still be captured around the moon, even if maneuver magnitude errors reach approximately 65 %. If the hyperbolic arrival velocity at periapsis is increased to approximately 2.5 km/s, the maximum allowable maneuver magnitude errors decrease to approximately 50 %. Although the spacecraft can still be captured in orbit around the moon in both cases, the resultant shape of the captured orbit due to the maneuver burn errors should be carefully considered. For example, a resultant capture orbit could have a minimum and maximum apoapsis altitude of approximately 2,324 km and 501,628 km, respectively, (approximately 65 % of the maneuver burn error case) and a minimum and maximum apoapsis altitude of approximately 2,386 km and 264,667 km, respectively, (approximately 50 % of the maneuver burn error case). In Fig. 2, the resultant capture orbits with a maneuver burn magnitude error of approximately 70 % (targeted to approach the moon with a periapsis arrival velocity of 2.4 km/s at a 100 km altitude) are shown for the case in which the orbital period of the nominal capture orbit is assumed to be 12 hr. Fig. 2 clearly shows that there exists a perturbed trajectory not captured by the moon owing to the maneuver burn magnitude error; even if captured, the apoapsis altitude of the same orbit appears to reach approximately 50 LU (1 Lunar Unit = approximately 1,738.2 km). These results indicate that even if capture around the moon is achieved, the orbit could have an extremely high apoapsis altitude (even more than the Earth-moon distance), which decreases the suitability of the derived maximum allowable errors for the maneuver burn performance. Considering diverse combinations of periapsis arrival velocity, target periapsis altitude, and desired orbital period for the 1^st capture orbit, we can conclude that the 1^st LOI maneuver should be executed within errors less than approximately 25 % of the magnitude with respect to the nominally planned delta-Vs, as clearly shown in Tables 1 and 2. This result indicates that it is still possible to achieve capture around the moon in the case where one of the four main onboard engines malfunctions during the LOA phase. This is the current baseline design of the KPLO propulsion system. Similar analysis results can be found in studies performed by Houghton et al. (2007), which reported that an error of less than approximately 50 % in the nominally planned delta-V is required for a successful LOI maneuver and that larger maneuver errors must be avoided as they weakly capture or fly by the moon. Based on the currently estimated minimum burn performance analysis, the early LOA phase design strategy can be further improved, including a contingency trajectory design plan.

In this subsection, the effect of the combined uncertainties (including orbit, attitude, and maneuver burn performance) at the moment of the 1^st LOI maneuver execution on the characteristics of the 1^st capture orbit is analyzed. For the initial error conditions, σ^pos and σ^vel are given within a range of 0–10 km and 0–10 m/s (increased in 2 km and 2 m/s steps, respectively) with respect to its nominal position and velocity, σ_dvm is given within a range of 0–10 % (increased in 1° steps) with respect to its nominally planned delta-V maneuver magnitude, and finally, σ^att is given within a range of 0–10° with a 2° step, respectively. Note that these uncertainties are all in three-sigma and given under conditions related to the worst cases of the KPLO system design configuration and with adequate margin. With the assumed initial condition uncertainties, 1,000 Monte Carlo runs were performed for the statistical analysis. For the orbital period constraints of the 1^st capture orbit, a maximum difference of 1 hr is assumed to be allowed with respect to the nominally planned 12 or 24 hr for the current study as the orbital period with regard to the pre-scheduled ground station contact time. In Tables 3 and 4, the maximum allowable delta-V magnitude errors are shown with respect to the orbit (position and velocity) and attitude errors to meet the given constraints of a 1 hr orbital period difference for the nominally planned 12 hr (Table 3) or 24 hr (Table 4) of the orbital period of the 1^st capture orbit. In Tables 3 and 4, the allowable delta-V magnitude errors are shown as percentage values with respect to the nominal delta-V magnitude; the magnitude of the hyperbolic arrival velocity at periapsis is assumed to be 2.4 km/s for this simulation. A status of “Not Available” in both tables indicates that the orbital period of the 1^st capture obit cannot be achieved within 1 hr from the nominally planned orbital period, due to uncertainties posed upon the position, velocity, and attitude at the moment of the 1^st LOI maneuver, even if the maneuver burn is performed perfectly. As expected, Tables 3 and 4 clearly show that smaller errors in the delta-V magnitude are required if the 1^st capture elliptical orbit is intended to remain within a period of 24±1 hr rather than 12±1 hr; this phenomenon originates from the orbit periapsis velocity characteristics. Regardless of the periapsis altitude, we can conclude from Table 3 that the delta-V burn should be executed with an error of approximately 2 % or less, with position errors of less than approximately 4 km, velocity errors of approximately 4 m/s, and attitude errors of approximately 8° to maintain the orbital period of the 1^st capture orbit within 12±1 hr. However, if the periapsis altitude is fixed at 100 km, the maximum allowable delta-V magnitude error is increased to 4 % for the same attitude, with position and velocity errors of approximately 2 km and 2 m/s, respectively.

Unlike the cases shown in Table 3, the maximum allowable delta-V magnitude errors and the position, velocity, and attitude errors are greatly reduced if the spacecraft is intended to be captured around the moon with an orbital period of 24±1 hr. We can conclude from Table 4 that the delta-V burn should be executed to have approximately 1 % error, with position errors of less than approximately 2 km, velocity errors of approximately 2 m/s, and attitude errors of approximately 6° to limit the orbital period of the 1^st capture orbit to within 24±1 hr. Moreover, the allowable periapsis altitude for this case is only 100 km, and the spacecraft cannot be captured within 24±1 hr of elliptical orbit at a higher periapsis altitude, even with smaller initial errors. From the findings discussed above, the following factors are reemphasized and confirmed: the importance of considering diverse errors during the early design phase of LOA, allocating proper system requirements in the orbit, attitude, and onboard propulsion system, and considering human factors regarding the operation schedule. LOI burns in the real world may be executed with a finite burn, and associated results may differ from the results in the current study. However, a more detailed and profound analysis can be further conducted using the current results as a baseline.

Owing to the initial error conditions, the resultant capture orbit will have a perturbed shape. In Fig. 3, an example of a perturbed capture orbit is compared to the nominal orbit. Again, each of the axes in Fig. 3 is scaled to LUs. In this plot, a 12 hr 1^st capture orbit with a 100-km periapsis altitude is assumed with initial error conditions of 6 km (3σ) and 6 m/s (3σ) for position and velocity error, respectively, 10° (3σ) of attitude error, and 2 % (3σ) of delta-V magnitude error, at the moment of the LOI burn execution. In Table 5, the resultant orbital characteristics of the perturbed 1^st capture orbit are shown. The semi-major axis of the captured orbit is significantly affected (approximately 6,178±99 km) because of the diverse initial error conditions. Thus, the apoapsis altitude is approximately 8,779±198 km. This result is primarily caused by errors in the delta-V magnitude, which is only approximately 2 % from the nominal value.

In planetary missions, one of the most challenging problems is to design a transfer trajectory that takes the spacecraft within a very close vicinity of the target planet. One possible approach for placing the spacecraft close to a target planet is the B-plane, or body plane, targeting method. B-plane targeting can be achieved by performing an MCC during the transfer phase, and then an orbit insertion maneuver can be performed to achieve capture into the orbit of the target planet. To aim the target point, the B-plane coordinate system is used, which can be thought of as a target frame attached to the aiming central body. The B-plane is defined as a plane passing through the center of a target body normal to the incoming asymptote vector of the approach hyperbola; this plane is often described by its two components, B·T and B·R. Here, B is the vector that points to the aiming points, and T and R are the defined unit axes from the orthogonal set to the incoming asymptote vector of the approach hyperbola (Sergeyevsky et al. 1983). Details on the B-plane system are omitted in the current study as the B-plane system is not the main concern of the current work. However, more details on the definition and usage of the B-plane coordinate frame can be found in Kizner (1961), Sergeyevsky et al. (1983), and Jah (2002). Using the initial error conditions of position and velocity at the moment of the 1^st LOI maneuver discussed above, the characteristics of the error distributions on the B-plane coordinate system can easily be analyzed. Therefore, the required targeting accuracy during the lunar transfer phase can be determined. In Fig. 4, the allowable B-plane targeting errors (uncertainty distribution) for a selected lunar capture condition are shown, and Fig. 5 presents a magnified view of Fig. 4. To plot the uncertainty distribution on the B-plane coordinate system shown in Figs. 4 and 5, the periapsis altitude of the 1^st capture orbit is assumed to be 100 km with an orbital period of 12 hr. The initial error conditions assumed at the moment of the LOI burn execution are described as follows: 6 km (3σ) and 6 m/s (3σ) for the position and velocity error, respectively, 10° (3σ) for the attitude error, and 2 % (3σ) for the delta-V magnitude error. With the assumed initial position and velocity errors, distributions of the allowable B-plane targeting errors can easily be derived and used as reference values during the trans-lunar phase operation to determine whether additional error cleanup MCCs are required. These dispersions shown in Figs. 4 and 5 can be regarded as B-plane entry corridor conditions for successful capture around the moon, under given position, velocity, attitude, and delta-V magnitude errors at the moment of the LOI burn execution. In Table 6, the dispersion characteristics shown in Figs. 4 and 5 are summarized. From Table 6, we can conclude that MCC maneuvers during the lunar transfer should target the B-plane within approximately 0.184±6.349 km for B·T and approximately 6,794.914±80.799 km for B·R. Generally speaking, B·T errors may represent errors out of the desired orbit plane about the moon, and B·R errors represent errors in arrival altitudes. If these error conditions are measured as a two-dimensional distance from the aiming point, then the allowable mean error distance from the aiming point will be approximately 65.105±48.194 km. The resulting acceptable error ranges for target aiming points on the B-plane frame are consistent with those used for the Chandrayaan program, which is managed by the Indian Space Research Organisation (ISRO). The errors for this program were approximately several tens of km from the reference target center point on the B-plane coordinate frame (Pitchaimani 2015, private communication).