ACCURATE PREDICTORCORRECTOR SKIP ENTRY GUIDANCE FOR LOW LIFT-TO-DRAG RATIO SPACECRAFT

This paper develops numerical predictorcorrector skip entry guidance for vehicles with low lift-to-drag L/D ratio during the skip entry phase of a Moon return mission. The guidance method is composed of two parts: trajectory planning before entry and closed-loop guidance during skip entry. The result of trajectory planning before entry is able to present an initial value for predictorcorrector algorithm in closed-loop guidance for fast convergence. The magnitude of bank angle, which is parameterized as a linear function of the range-to-go, is modulated to satisfy the downrange requirements. The sign of the bank angle is determined by the bank-reversal logic. The predictor-corrector algorithm repeatedly applied onboard in each guidance cycle to realize closed-loop guidance in the skip entry phase. The e ̈ectivity of the proposed guidance is validated by simulations in nominal conditions, including skip entry, loft entry, and direct entry, as well as simulations in dispersion conditions considering the combination disturbance of the entry interface, the aerodynamic coe©cients, the air density, and the mass of the vehicle.


INTRODUCTION
Recent Space exploration missions, such as ORION mission from NASA, require an entry guidance approach can realize a wide range of entry trajectories with a high landing accuracy.Although the Apollo skip reentry guidance has been successfully applied in 1960s, the various approximations and linearization assumptions in the guidance have been found to limit the accuracy when the downrange and crossrange are relative long.With the development of §ight computers, onboard calculation of a trajectory has become available.Some numerical predictorcorrector guidance algorithms appeared in literature research recently [16].These methods are not limited by the validity of the simpli¦cations, approximations, and empirical assumptions necessary for any analytic treatment.Therefore, they hold greater potential to be more accurate and adaptive.The representative of them are the two Orion skip entry guidance: the Numerical Skip Entry Guidance(NSEG) developed at NASA/JSC and PreGuid developed at the Charles Stark Draper Laboratory [1,2].Both algorithms use a numerical predictorcorrector for the skip phase coupled with the Apollo ¦nal phase entry guidance.Putnam et al. also presented a skip entry guidance by introducing a numerical predictorcorrector phase during the atmospheric skip portion with an Apollo guidance baseline algorithm [3].Vernis et al. reported a numerical predictorcorrector entry guidance that had been designed within the frame of the ESA-funded ¢Robust skip Entry£ program [6].A succession of three constant bank angle phases with three roll reversals for the Moon return mode of this guidance is required.Therefore the correction of the guidance algorithm included two parts: the cross-range correction and the downrange correction which may need high performance of onboard processor.A full numerical predictor corrector method for trajectory planning and closed-loop guidance of skip entry was put forward by Christopher and Ping.The problem is formulated as a nonlinear uni-variate root-¦nding problem in downrange control guidance and bank reversal problem in lateral guidance [4,5].
In this paper, a numerical predictorcorrector approach motivated by similar principles of Christopher and Ping is developed for the skip entry problem which can cover the cases of direct entry, loft entry, and skip entry cases in lunar return mission.

SKIP ENTRY GUIDANCE PROBLEM DESCRIPTION
The general purpose of reentry guidance is to guide the reentry vehicle from entry interface (EI) to the speci¦ed landing site in the disturbance circumstance of the Earth£s atmosphere.The typical skip reentry trajectory is illustrated in Fig. 1.For the capsule type reentry vehicle, the guidance algorithms modulate the bank angle to achieve the speci¦ed terminal constraints land point while using a trim angle-of-attack.So, the magnitude and the sign of the bank angle are determined by the guidance algorithm which will be described in the next section.
The §ight of a return capsule is subject to the vehicle dynamics described by the following dimensionless equations of three-dimensional motion for a point mass about a rotating circular Earth: Here, r is the radial distance from the center of the Earth to the vehicle, normalized by the average radius of the Earth R 0 = 6378 km.The Earth relative velocity is V , normalized by V c = g 0 R 0 where g 0 = 9.81 m/s 2 .The longitude, latitude, §ight path angle (FPA), and velocity azimuth angle are θ, ϕ, γ, and ψ, respectively.The heading angle ψ is measured from the north in a clockwise direction.Derivatives of these variables are taken with respect to the dimensionless time τ = t/ R 0 /g 0 .The dimensionless angle velocity Ÿ is the Earth self-rotation rate normalized by g 0 /R 0 .The bank angle is σ acting as one of the main control variables, L and D represent the nondimensional lift and drag accelerations, in terms g 0 , and can be calculated from 3 PROPOSED SKIP ENTRY GUIDANCE APPROACH

The Outline of the Guidance Logic
There are four phases in this guidance law both in trajectory planning and closedloop guidance (see Fig. 1): Phase I: Initial entry phase which is from the EI to the skip altitude; Phase II: Skip phase which is from the end of phase I to the atmosphere exit point; Phase III: Kepler phase which is from the exit point to the reentry point; and Phase IV: Final entry phase which is from the reentry point to the parachute deploying point.
The basic scheme of the proposed guidance is a numerical predictorcorrector algorithm in which the equations of motion are integrated with a fourth-order RungeKutta method adopted by the onboard predictor.Further more, there are di¨erent forms of predictorcorrector algorithms used in the di¨erent phases of skip guidance process in order to deal with the challenges in the lunar return cases including the initial velocity of about 11 km/s, varied entry range requirements from 2500 to 10 000 km, and the possibility of bouncing o¨or diving steeply into the atmosphere in the disturbance circumstances.
The guidance phase transition logic and the algorithm description of each phase are illustrated in Fig. 2. The initial entry phase begins at the entry interface.The guidance algorithm regulates the magnitude of σ d which is a design parameter of the bank angle (see Fig. 2 and Eq. ( 8) below) to make the ¦nal range-to-go less than the given ¦nal phase range (F stogo ) until the altitude below 80 km.The guidance algorithm, in this phase is bene¦t to avoid the case of bouncing o¨the atmosphere and too much computational burden if using a high accuracy predictorcorrector algorithm which should propagate the nearly whole skip reentry trajectory in every prediction cycle.Then, the guidance logic is transferred to the skip phase in which the magnitude of σ d is iterated by a scant method to satisfy the range requirement with a relatively high accuracy in every guidance cycle.If the g-load under the value 0.1, the guidance automatically transit to the Kepler phase.In this phase, the previous designed bank angle |σ d | is stained and the current magnitude of bank angle is computed by Eq. ( 8).When the g-load exceeds the value 0.1 again or the range-to-go is less than the given ¦nal phase range, the guidance law is switched to the ¦nal phase guidance in which a new model of bank angle (see Fig. 4 below) and equations of motion (see Eqs. (13) below) for prediction are adopted.Then, the initial bank angle of the ¦nal phase σ f0 is the design variable of the ¦nal predictorcorrector algorithm.
The above transition logic of guidance is for the normal skip entry case.There are two other branches you can see in Fig. 2 which is for the loft entry and director entry cases.After the initial entry phase, if the range-to-go is already less than the given ¦nal phase range, the guidance law is directly transferred to phase IV.That is a direct entry case.Otherwise, if the range-to-go is less than the F stogo in the process of skip phase, it also transits to phase IV.This case is called loft entry.Then, the guidance logic presented here is adaptive to both the long and the short range cases which are required by the near-future Moon return projects.

Trajectory Planning Algorithm
In trajectory planning, the magnitude of the bank angle is parameterized as a linear function of the range-to-go until a speci¦ed threshold s f0 to-go , which is the start of the ¦nal entry phase and the range of this phase depends on the total range.The choice of distance of ¦nal entry phase (F stogo ) is such that it represents typical downrange at the initiation of the ¦nal phase guidance.According to simulation experience, it is taken the value of 2000 km when the total range is longer than 5000 km; otherwise, it is taken as 1000 km.These values do not vary widely for low L/D reentry vehicles.When the current rangeto-go is less than s f0 to-go , a constant bank-angle magnitude of σ f is used.Figure 3 shows the bank angle pro¦le.Then, the function of the bank angle magnitude |σ| is described: , s d to-go ≤ s to-go ≤ s 0 to-go ; where The trajectory planning algo- The integration of the equations of motion is end up at the parachute deploying altitude 10 km.The ¦nal range-to-go when arrive at this altitude will be positive if the reentry vehicle undershoots and negative if there is overshoot.Here, a relatively simple secant method is utilized to iteratively solve Eq. ( 10) for saving computational resource onboard.The iteration equation is: In this equation, where s i f denotes the ¦nal range-to-go using the value of |σ d | i to propagate the equation of motion.When the iteration succeeds, a feasible skip entry trajectory can be obtained and the value of |σ d | also can be used as an initial value in the closed-loop guidance.

Closed-Loop Guidance
To cope with the uncertainties of the varied range requirement, the closed-loop guidance is carried out after trajectory planning at the EI which repeatedly generates a feasible skip trajectory using the current state as an initial state in each guidance cycle.
The closed-loop guidance imposes the same algorithm to modulate the bank angle until the end of Kepler phase.In the ¦nal entry phase, the longitude equations of motion and a new bank angle function are implemented in the closed-loop guidance to improve the ¦nal accuracy and the convergence rate.The bank angle function is a linear function of the dimensionless energy e which is de¦ned as [7] Figure 4 shows the bank angle pro¦le of phase IV in closed-loop guidance.Therefore, the bank angle magnitude |σ| at any energy e in ¦nal entry phase is: where e f0 is the current dimensionless energy; and e f is the speci¦ed dimensionless energy in parachute deployment condition with a velocity of 200 m/s and an altitude of 10 km [4].
With the assumption of great circle, the range-to-go s satis¦es the following di¨erential equation: One can see that from the equations of motion, the longitude motion is decoupled from the lateral motion variables when the Earth£s rotation e¨ects are ignored.Therefore, the longitude motion with Eq. ( 12) can be described as follows: Then, eliminating the equation for V by using the dimensionless energy (Eq.( 11)) as the independent variable, one can deduce the dimensionless longitude equations [7]: where Set the initial values of the state variables to be the current condition.For a given magnitude pro¦le of the bank angle |σ|, the integration of Eqs.(13) from the current condition to the speci¦ed ¦nal energy gives the ¦nal range-to-go s f (e f ).Then, the predictorcorrector guidance problem in phase IV can be formulated as: the initial magnitude of bank angle in ¦nal phase |σ f0 | is to be found to satisfy the ¦nal range requirements.Through the integration of the equations of motion (13), the ¦nal range-to-go can be regarded as a function of |σ f0 |.A scant method is still adopted to solve the one-parameter problem, the iterate equations on |σ f0 | are given by: where the partial derivative ∂s f (|σ f0 | (i) )/∂σ is obtained by a ¦nite-di¨erence approximation.

Lateral Guidance Logic
The lateral guidance logic utilizes a bank reversal logic as usual in reentry guidance law with a cross range variable de¦ned as Here, ā LoS is the line-of-sight azimuth angle along a great circle to the landing site de¦ned by where θ fT and ϕ fT are the target parachute deployment longitude and latitude, respectively.Bank reversal logic is employed to determine the sign of bank angle for trajectory prediction both in trajectory planning and in closed-loop guidance.Then, the sign of bank angle is reversed whenever the cross range exceeds the velocity-dependent dead band [8] whose width is de¦ned by where C 1 and C 0 are two constants.This Apollo-like lateral logic regulates the cross range to ensure a §ight direction toward the landing site.The lifting capability, size, and weight of a vehicle determine how quickly it can change its §ight direction and unnecessary bank reversals could prematurely deplete the propellant of the reaction control system.Therefore, the constants C 1 and C 0 may need to be tuned for di¨erent vehicles for a good balance between crossrange regulation tightness and reasonable number of bank reversals.In this paper, the following values have been used: C 0 = 1.7 • 10 −5 and C 1 = 5.2 • 10 −3 .

Critical Issues in Guidance Scheme
Several critical issues in guidance scheme which can improve the adaptivity and robustness implemented in this paper are as follows.

A. A parameterized aerodynamic model for prediction
A parameterized aerodynamic model is constructed to get the trimmed aerodynamic coe©cients instead of the general looking table method to mostly reduce the prediction computation time.Here, a structured aerodynamic model for parameter identi¦cation of a reentry experimental vehicle is utilized [9].The original model is simpli¦ed in the trim condition as follows: The model allows one to analytically express each aerodynamic coe©cient as a nonlinear function of Mach number and a set of constant aerodynamic parameters.These parameters, C D0 , C Dα , C Dα 2 , C L0 , C Lα , and C Lα 2 , are identi¦ed by the least square method based on the original aerodynamic data of a low L/D return capsule.

B. A Pre-Guid algorithm in guidance cycle
In the Moon exploration mission, the vehicle entry into the atmosphere of the Earth with a near second cosmic velocity, the trajectory is most likely to bounce o¨the atmosphere or dive deeply into the atmosphere in a given bank angle.When the instance happened in prediction process, the range-to-go will be a nonmonotonous function of the iteration parameter.Then, the scant method may fail in solving the root-¦nding problem.This paper introduces a Pre-Guid algorithm to regulate the predict trajectory in a tolerable range which requires the ¦nal range-to-go less than F stogo , which was de¦ned in subsection 3.2, before iteration.This Pre-Guid algorithm decreases the designed bank angle |σ d | when the predict trajectory bounces o¨the atmosphere or overshoots whereas increased the bank angle |σ d | when the predict trajectory is under shoot.The regulate step of the bank angle |σ d | is varied from 5 • to 0.1 • according to the predicted ¦nal range-to-go to insure the fast convergence.

C. Adaptation of the prediction integration steps and convergence accuracy
The primary disadvantage of a numerical predictorcorrector algorithm is the onboard computational burden especially in the long range condition [6].To alleviate the computational burden in prediction, the RK4 (4th-order Runge Kutta) prediction step and convergence accuracy is adapted in the di¨erent §ight phases as described in Table 1.

D. Lift and drag ¦lters
It is recommended in the relative literature that the appropriate use of estimated information on the aerodynamic and density biases is very bene¦cial to obtain high accuracy [1013].A ¦rst-order fading-memory ¦lter is used [4,14]: where X * is the current ratio of the measured variable (lift or drag acceleration) to its nominal value based on the nominal model via Eqs.( 7), X n is the past ¦ltered ration and 0 < β < 1 is the gain chosen to emphasize past values over the most recent value.To initialize the ¦lter, the ¦rst past ¦ltered ratio X 0 = 1.
In each guidance cycle, the output of this ¦lter is used to multiply the nominal lift and drag pro¦les in the integration of the trajectory.

PERFORMANCE OF THE PROPOSED GUIDANCE ALGORITHM
To evaluate the performance of the proposed reentry guidance algorithm, detailed numerical simulations are carried out with the following conditions: (i) nominal condition in di¨erent downrange requirements situation; and (ii) perturbation condition considering the dispersion reentry interface, the disturbance of aerodynamic coe©cients, the air density, and the mass of the vehicle.

Assumptions and Design Parameters for Simulation Studies
The assumptions and the main design parameters used for veri¦cation of the algorithm are as follows.
A. The guidance algorithm is implemented on a simulation platform of 3DoF (three degrees-of-freedom) described by equations of motion (1) (6).Then, the RK4 integral method is used for numerical integration and with a time step of a half second.The guidance cycle is 2 s.That is to say, in every 2 s, the predictorcorrector algorithm is implemented once.The simulation is executed in the software environment of MATLAB.
B. According to the case of the Apollo-type entry vehicle, it was assumed that the initial altitude of the reentry at EI was 400,000 ft which corresponds to 121.92 km.For simplicity, an equatorial orbit with initial position of zero latitude and zero longitude and a typical lunar return velocity of 11,032 m/s are chosen.The initial FPA and azimuth angle are −5.9 • and 90 • , respectively.
C. The terminal requirements in range-to-go and altitude are ≤ 5 km and equal to 10 km, respectively.
requirements (see Table 2).It demonstrates that the guidance transition logic and algorithms described in Fig. 2 are valid for the varied range conditions in lunar return mission.The bank angle, as the control variable in guidance problem, is in the e©cient scope and without too much bank reversal.In the guidance cycles, the iterative convergence just needs 3 or 4 steps, and the computational time is less than the guidance cycle which is 2 s in these cases.

Performance Under Disturbed Flight Environment
To assess the performance of the guidance algorithm under conditions of dispersed reentry interface, the simulations have been carried out with the assumed dispersion on the reentry initial states, the air density, mass of vehicle, and aerodynamic coe©cients, as given in Table 3.Each of the dispersion is described by a scalar value which is assumed to be a Gaussian distribution and determined by a mean value (µ) and a maximum expected perturbation (3σ).Here, the same perturbation values are adopted as reference [2].For perturbations introduced as a multiplier, the same multiplier is applied during the whole range for one time Monte-Carlo simulation.The corresponding altitude and steering pro¦les are given in Figs.6a and 6b.The §ight path variables, including g-load factor are shown in Fig. 6c.
Figure 7 shows the time property of the guidance algorithm for a random trajectory in the Monte-Carlo simulation in which the guidance cycle is set as 2 s. Figure 7a gives the computation time in every guidance cycle in phases I, II, and III.Every time expense is less than 2 s. Figure 7b gives the computation time in every guidance cycle in phase IV which is also less than the guidance cycle.

CONCLUDING REMARKS
This paper has presented a new predictorcorrector skip entry guidance for vehicles with low lift-to-drag (L/D) ratios which will be potentially used in the near-future Moon exploration mission with next generation onboard computer processor.The guidance scheme includes a trajectory planning algorithm and a closed-loop guidance law executed in each guidance cycle during skip entry.The numerical predictorcorrector method is used to compute the parameterized bank angle to make the trajectory satisfy the downrange accuracy while the bank reversal logic is utilized to make sure the entry vehicle guided to the landing site precisely.The standard condition and the Monte-Carlo simulations of the proposed guidance law are carried out.It demonstrates that the guidance law is valid in the varied reentry situation in lunar return missions, including direct entry, loft entry, and skip entry cases.And the robustness of the algorithm is also evaluated through simulations considering dispersions on the reentry initial states, the air density, the mass of vehicle, and the aerodynamic coe©cients.

Figure 1
Figure 1 Typical skip entry trajectory

Figure 2
Figure 2 Guidance phase transition logic and algorithm description (H denotes the altitude, FPA denotes the §ight path angle, and Fstogo denotes the given ¦nal phase range)

Figure 3
Figure 3 Bank-angle parameterization model in trajectory planning

Figure 4
Figure 4 Bank-angle parameterization model in the ¦nal phase in closed-loop guidance

Figure 6 Figure 7
Figure 6 Monte-Carlo results for 1000 dispersed trajectories: (a) altitude vs. range; (b) bank angel vs. velocity; (c) g-load vs. time; and (d ) landing error distribution (crosses inside the inner dotted curve refer to 2.5 km and inside the outer solid curve to 5 km)

Table 1
Adaption of the prediction integration

Table 3
Parameters used in Monte-Carlo analysis