Online trajectory planning and guidance for reusable launch vehicles in the terminal area

Acta Astronautica 118 (2016) 237–245

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Acta Astronautica journal homepage: www.elsevier.com/locate/actaastro

Online trajectory planning and guidance for reusable launch vehicles in the terminal area Xue-Jing Lan a,b, Lei Liu a,b,n, Yong-Ji Wang a,b a b

National Key Laboratory of Science and Technology on Multispectral Information Processing, China School of Automation, Huazhong University of Science and Technology, Wuhan 430074, China

a r t i c l e i n f o

abstract

Article history: Received 5 August 2015 Received in revised form 16 September 2015 Accepted 23 October 2015 Available online 2 November 2015

A guidance scheme has been proposed based on a new online trajectory planning algorithm for an unpowered reusable launch vehicle (RLV) in the terminal area energy management (TAEM) phase. The trajectory planning algorithm is able to rapidly generate a feasible path from the current state to a desired state at approach and landing interface (ALI) based on the dynamic pressure profile and new ground track geometry. Simple guidance laws are used to keep the RLV flying along the reference path which can be adjusted online by five related parameters. Then, the effectiveness and adaptability of the proposed TAEM guidance scheme is demonstrated by numerical trials with variations in the initial energy, position and aerodynamic performance. & 2015 IAA. Published by Elsevier Ltd. All rights reserved.

Keywords: RLV TAEM Online planning Guidance

1. Introduction Recently, advanced reusable launch vehicles (RLV) have been developed to reduce the costs of the space transportation and make significant improvement of vehicle flexibility and reliability [1,2]. In addition, the future space transportation systems will likely emphasize the robustness in terms of variations in mission objectives, dispersions and other uncertainties. Therefore, it is necessary and important to develop an automated mission-planning algorithm which can potentially serve as an onboard guidance scheme with the ability to rapidly generate the various trajectories and guidance commands by replanning the trajectory from the current vehicle state to a desired target state. The descent flight of the RLV commonly consists of the entry phase, the terminal area energy management (TAEM) phase and the approach and landing (A&L) phase. The n Corresponding author at: School of Automation, Huazhong University of Science and Technology, Wuhan 430074, China. E-mail addresses: [email protected] (X.-J. Lan), [email protected] (L. Liu), [email protected] (Y.-J. Wang).

http://dx.doi.org/10.1016/j.actaastro.2015.10.019 0094-5765/& 2015 IAA. Published by Elsevier Ltd. All rights reserved.

TAEM is a critical flight phase that brings the unpowered vehicle from the terminal entry point (TEP) to the approach and landing interface (ALI) and it is mainly responsible for aligning the vehicle with the runway while dissipating the vehicle's surplus energy. The traditional TAEM guidance strategy is based on the space shuttle and consists of several segments: energy dissipation S-turn phase (exists only when the energy surplus at TEP), acquisition phase, heading alignment phase, and pre-final phase [3]. In spite of that the space shuttle TAEM guidance strategy has been demonstrated to be effective, it nevertheless mainly depends on a set of fixed, pre-determined reference trajectories which only account for a limited number of off-nominal scenarios. Therefore, some improvements have been made in the TAEM trajectory guidance scheme. A great effort has been made with the aim of developing advanced guidance algorithms capable to deal with drastic off-nominal conditions. Hanson had argued that advanced guidance and control technologies can successfully return an RLV suffered from aero-surface failures, poor vehicle performance and larger-than-expected flight dispersions [4,5]. Grantham presented an adaptive critic neural network based guidance methodology that can

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generate TAEM trajectories in a wide range of vehicle states during landing [6]. Burchett developed a TAEM trajectory guidance method based on fuzzy logic which is capable of compensating for control surface failures by restricting the allowable bank angle [7]. Hall presented the first application of sliding mode disturbance observer driven sliding mode control to improve RLV flight control performance [8]. Kluever proposed a TAEM guidance method for a scenario with bank constrains, which determines the best feasible path by iterating on the bank reversal switch time and downrange location of the heading alignment circle (HAC) [9]. Morio et al. designed a robust TAEM guidance scheme based on a non-linear dynamic inversion technique so as to circumvent offnominal flight conditions [10]. The online trajectory planning algorithm has been newly investigated as an effective way to fundamentally compensate for the defect of the pre-determined reference trajectories. Horneman and Kluever determined an optimal TAEM trajectory among a set of feasible trajectories that were propagated according to different geometric parameters [11,12]. Hull et al. described an online trajectory generator for both the TAEM and A&L phase by periodically updating six parameters online in response to changes of the aerodynamic characteristics of the vehicle [13]. Mayanna et al. developed a TAEM guidance method with vertical guidance relying on a pre-specified dynamic pressure profile and lateral guidance relying on the new proposed ground track geometry [14]. Kluever proposed an algorithm capable of rapidly generating a feasible TAEM trajectory by iterating on the downrange location of the HAC and its radius [15]. Ridder and Mooij presented a planning algorithm based on the energy-tube concept, which can be used in an on-board planner [16,17]. Jiang and Yang proposed a TAEM trajectory planning algorithm by energy-to-range ratio with a new lateral approaching strategy method in the situation of energy surplus [18]. In this paper, a TAEM trajectory guidance strategy is proposed based on an online trajectory planning algorithm with the ability to rapidly generate a feasible path from the current state to the desired ALI state. Unlike the existing way of fixing the vertical profile [12,14,15,18,19], the dynamic pressure profile used to determine the vertical profile in this paper is adjustable online, which greatly increases the design freedom. The proposed lateral design approach based on a new ground track geometry makes it unnecessary to execute the energy dissipation S-turn phase and choose the approaching mode (direct mode or overhead mode) [9,12,15,16], which reduces the design complexity, and similarly, this ground track geometry can be adjusted online. In addition, as the ground track distance predicted along fixed geometric segments is inaccurate because of the existence of coupling between vertical and lateral motion, this online trajectory planning algorithm can obtain the actual ground track distance by numerically propagating the trajectory rather than adjusting the downrange according to a table with correction information stored in advance [14].

2. System model The earth is assumed to be flat, nonrotating, and inertial during the derivation of the equations of motion for an unpowered RLV in the TAEM phase. The RLV is considered as a point mass, so its motion is defined as: D V_ ¼   g sin γ m

ð1Þ

γ_ ¼

L cos σ g cos γ  mV V

ð2Þ

χ_ ¼

L sin σ mV cos γ

ð3Þ

h_ ¼ V sin γ

ð4Þ

x_ ¼ V cos γ cos χ

ð5Þ

y_ ¼ V cos γ sin χ

ð6Þ

where þx axis points along the runway centerline in the direction of approach and the þy axis points right of the runway on approach, the þz axis points downward along the local vertical, and the origin is at the runway threshold. V is the velocity magnitude, and γ is the flight path angle measured from the horizonal plane to the velocity. The runway-relative heading angle χ is measured clockwise in the horizontal plane from the runway centerline to the velocity. h represents the altitude, and x and y are the horizontal position coordinates of the RLV. σ is the bank angle, m is the vehicle's mass, and g is the gravitational acceleration. The simplified aerodynamic model is used in the formulation as follows: L ¼ qSC L

ð7Þ

D ¼ qSC D

ð8Þ

where L is the aerodynamic lift force, D is the aerodynamic drag force, q is the dynamic pressure, and S is the reference area of the RLV. The aerodynamic lift coefficient C L and drag coefficient C D are computed by a two-dimensional table look-up with angle-of-attack α and Mach number M as the independent variables.

3. Vertical guidance 3.1. Dynamic pressure profile Generally, the linear, quadratic or cubic curve will be used to parameterize the dynamic pressure profile [20,21]. As Ridder and Mooij have concluded that it is not possible to use a single fixed dynamic pressure profile in the case of abnormal conditions [17], in this paper, we design the dynamic pressure profile with respect to altitude as a straight line combined with two third order polynomials to make it easier to modify the profile. The dynamic pressure

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profile is divided into three sections as follows: 8 2 3 > ðhMID1 o hÞ > < a0 þa1 h þ a2 h þa3 h qðhÞ ¼ qMID ðhMID2 rh r hMID1 Þ > > : b þ b h þ b h2 þ b h3 ðho h Þ 0

1

2

3

ð9Þ

MID2

where the demarcation altitudes hMID1 and hMID2 are predesigned and do not change in the planning process. The constant dynamic pressure qMID in the second section is the only variable parameter to be designed for different vertical trajectory: qMID ¼ qMIN þ kq  ðqMAX qMIN Þ

ð10Þ

where qMAX and qMIN are the maximum and minimum dynamic pressure respectively constituting the dynamic pressure boundary to satisfy the physical feasibility of trajectory. kq A ½0; 1 is the adjustable proportional coefficient. The four polynomial coefficients in the first section can be determined from the knowledge qTEP and hTEP (start of TAEM), qMID and hMID1 (demarcation point), and the derivative of the reference dynamic pressure profile at both ends of this section: dq 2 ¼ a1 þ2a2 hTEP þ 3a3 hTEP ¼ q0TEP dh h ¼ hTEP

ð11Þ

dq 2 ¼ a1 þ2a2 hMID1 þ3a3 hMID1 ¼ 0 dh h ¼ hMID1

ð12Þ

where q0TEP is determined by the initial trajectory information. The derivative of the reference dynamic pressure profile at hMID1 is set to zero for a smooth connection. Similarly, the four polynomial coefficients in the third section can be determined from the knowledge qALI and hALI (end of TAEM), qMID and hMID2 (demarcation point), and the derivative of the reference dynamic pressure profile at both ends of this section: dq 2 ¼ b1 þ 2b2 hMID2 þ3b3 hMID2 ¼ 0 dh h ¼ hMID2

ð13Þ

dq 2 ¼ b1 þ 2b2 hALI þ 3b3 hALI ¼ 0 dh h ¼ hALI

ð14Þ

The derivative of the reference dynamic pressure profile at hMID2 is set to zero for a smooth connection. And the constraining Eq. (14) is enforced so that the TAEM trajectory ends on the equilibrium glide-slope for the A&L phase trajectory planning [22]. The initial A&L phase begins with a quasi-equilibrium glide at the proper constant flightpath angle γ ALI such that dynamic pressure qALI remains constant. 3.2. Tracking the dynamic pressure profile In this section, we will derive the guidance command required to track the reference dynamic pressure profile. Using the chain rule, we obtain the derivative of dynamic pressure with respect to altitude:   dq dq dh 1 dρ dh 2 dV dh ¼ = ¼ V þ ρV = ð15Þ dh dt dt 2 dh dt dt dt If we assume a simple exponential model for atmospheric density, ρ ¼ ρ0 expð  h=hs Þ, then dρ=dh ¼ ð  1= hs Þρ0 expð  h=hs Þ ¼  ρ=hs , where hs is the scale height of

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the density model. Substituting Eq. (1) for dV=dt and Eq. (4) for dh=dt leads to:   dq 1 ρSC D ¼   ð16Þ q  ρg dh hs m sin γ Based on Eq. (16), we can estimate the dynamic pressure qest in next height approximately according to the current state of the RLV with the only unknown variable CD:   dq qest hcur þ Δh ¼ qcur þ h ¼ hcur  Δh ¼ f ðC D Þ dh

ð17Þ

where the subscript cur indicates the current state and Δh is the height step. On the other hand, we can obtain the reference dynamic pressure qref in next height based on Eq. (9). Then, Eq. (18) specifies the drag coefficient C D required to track the reference dynamic pressure profile: qest ðhcur þ ΔhÞ ¼ qref ðhcur þ ΔhÞ

ð18Þ

We use a secant iteration method to determine the angle of attack α that results in the drag coefficient that satisfies Eq. (18). The secant iteration begins with the angle of attack in previous height, and the search typically converges in 2–3 iterations. As the lateral maneuver will affect the flight path angle, thereby affecting the range, we calculate the range according to different dynamic pressure profiles by ignoring the lateral maneuver (with σ ¼ 0) so as to analyze the impact of dynamic pressure on the range simply. The three dynamic pressure profiles are shown in Fig. 1 with different proportional coefficients kq , and the corresponding ranges are presented in Fig. 2. It can be seen that the dynamic pressure has a great influence on the range. The larger the constant dynamic pressure qMID , the shorter the range. Therefore, the variations of the initial conditions are allowable, as the total range can be appropriately adjusted by tracking different dynamic pressure profiles.

4. Lateral guidance 4.1. Ground track geometry Unlike the ground track geometry used in [9,11,12,15,16], which has two turning modes with different flight range, Mayanna et al. designed a ground track geometry by three circles and two straight lines [14], and Jiang and Yang designed a circumscribed circle tangent to the inner HAC [18]. In this paper, based on the inspiration that comes from the segmented snack analogy [23,24], a new ground track geometry is proposed and presented in Fig. 3. An energy dissipation circle (EDC) is designed tangent to the HAC to dissipate surplus energy. In the TEP, the RLV turns to align its heading tangent to the EDC, and then flies in a straight-line towards to this tangent point. Afterwards, the RLV turns to follow the EDC until the tangent point with HAC, and then turns to follow the HAC in the opposite direction to the pre-final phase. This TAEM ground track geometry is uniquely determined by the x-axis coordinate of HAC center xHAC , HAC radius RHAC , EDC radius

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Fig. 3. Ground track geometry.

Fig. 1. Reference dynamic pressure profiles.

(2) The EDC has the ability to dissipate energy by adjusting the radius and angle of EDC appropriately according to the amount of surplus energy. Therefore, even in the situation of energy surplus, the S-turn phase is not needed. In addition, this lateral strategy can also work when there is no surplus energy by designing appropriate parameters. This new design of EDC forms a more unified and easier approach to plan TAEM trajectory online.

4.2. Tracking the ground track

Fig. 2. Range for different reference dynamic pressure profiles.

REDC , and the angle of EDC corresponding to HAC θEDC . The turning directions and the y coordinate of HAC center yHAC are adaptively determined once the initial position of RLV in y-axis yTEP is given. If yTEP is positive, then the RLV turns right to track the EDC and turns left to track the HAC, otherwise the RLV turns in the opposite direction. For the convenience of explanation, we define an identifier Γ to indicate the sign of yTEP , where Γ ¼ 1 if yTEP Z0 and Γ ¼ 1 if yTEP o 0. Then, yHAC can be determined as yHAC ¼  Γ RHAC , and the coordinates of EDC center are set as follows: xEDC ¼ xHAC  ðREDC þRHAC Þ  sin ðθEDC Þ

ð19Þ

yEDC ¼ yHAC þ Γ ðREDC þ RHAC Þ  cos ðθEDC Þ

ð20Þ

where θEDC A ½20; 180 is limited to guarantee the proper implementation of lateral guidance scheme. The characteristics of this ground track geometry are concluded as follows: (1) The selection of approaching mode is unnecessary, which simplifies the trajectory planning algorithm. Once the initial conditions of RLV at the TEP are determined, there is only one turning mode in no need of selecting the direct or overhead HAC mode, and the initial positions of EDC and HAC are computed automatically, which makes the planning algorithm adaptive.

The lateral trajectory is determined based on the ground track geometry. At the TEP, the ground track attempts to acquire the EDC, therefore, the bank angle in the EDC acquisition phase is simply proportional to the heading error:

σ ¼  Gχ _EDC  Δχ cur_EDC

ð21Þ

where Δχ cur_EDC is the difference between the current heading and a desired heading tangent to the EDC. It should be noted that the desired heading angle is calculated in real time based on the current position, EDC position and EDC radius. Gχ _EDC is the proportional gain. During the energy dissipation phase, the bank angle for the EDC turn consists of open-loop and closed-loop terms, where the open-loop command is the bank angle required to follow an EDC turn with radius REDC , and the closed-loop command is a simple proportional-derivative (PD) scheme:

σ ¼ Γ tan  1

! ! V 2 cos γ þGR EDC ΔREDC þGR_ EDC ΔR_ EDC gREDC ð22Þ

where ΔREDC is the difference between the distance to the EDC center and the EDC radius, and ΔR_ EDC is the change rate of ΔREDC . GREDC and GR_ EDC are the feedback gains. The identifier Γ determines whether a left or right bank is required for the EDC turn. In the heading alignment phase, the RLV turns to track the HAC by the same PD scheme in a

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different turning direction: ! ! 2  1 V cos γ _ σ ¼  Γ tan þGRHAC ΔRHAC þ GR_ HAC ΔR HAC gRHAC ð23Þ where ΔRHAC is the difference between the distance to the HAC center and the HAC radius, and ΔR_ HAC is the change rate of ΔRHAC . GRHAC and GR_ HAC are the feedback gains. The open-loop term is the bank angle required to follow an HAC turn with radius RHAC . When the heading error with respect to the runway becomes less than 5°, the RLV goes to the pre-final phase with the bank angle simply in a PD scheme:

σ ¼  Gy y Gy_ y_

ð24Þ

where Gy and Gy_ are the feedback gains. In all cases, the bank angle command computed by Eqs. (21)–(24) is limited to a maximum magnitude of 70°.

5. Trajectory propagation As it is inaccurate to predict the ground track distance along fixed geometric segments [11,12,18,19], this online trajectory planning algorithm can obtain the actual ground track distance so as to adjust the related designing parameters efficiently and accurately. Therefore, the TAEM trajectory planner must have the ability to rapidly propagate the trajectory online by numerically integrating the equations of motion. The governing equations (1)–(3), (5) and (6) are divided by Eq. (4), and numerically integrated with altitude h as the independent variable from the initial altitude hTEP ¼ 28 km at the TEP to the final altitude hALI ¼ 3 km at the ALI. The angle of attack α during the trajectory propagation is determined based on Eq. (18) so as to track the reference dynamic pressure profile. As the propagation is terminated at the desired altitude with successfully tracking the reference dynamic pressure, the RLV can certainly achieve the desired velocity at ALI. In addition, shaping the reference dynamic pressure profile (set dq=dh ¼ 0 at ALI) guarantees the desired flight-path angle at ALI. And the bank angle σ is determined to track the ground track geometry defined by the EDC and HAC with corresponding guidance laws (Eqs. (21)–(24)). Therefore, if the trajectory propagation can successfully perform according to this planning algorithm, five of the six states (h; V; γ ; χ , and y) at the end of the numerical propagation will match the desired target value at ALI. Only the final position in x-axis along the runway centerline will potentially not match the expectation xALI . Therefore, the reference dynamic pressure profile and ground track geometry should be corrected so that the propagated trajectory terminates at x ¼ xALI . The correction method will be presented in detail in the next section.

241

a feasible TAEM trajectory for the given RLV state at the TEP. Therefore, the algorithm begins by propagating a TAEM trajectory to ALI with minimum values for all the designing parameters. If the propagated trajectory does not overshoot the downrange location xALI , then there is no feasible TAEM path because the RLV is either too low on energy or too far from the runway to meet the ALI. Otherwise, the RLV can arrive at the desired xALI by adjusting the planning parameters to increase the total ground track. The downrange distance error to ALI at the end of trajectory propagation is defined as:

Δs ¼ xALI  xf

ð25Þ

Note that Δs is positive if the propagated trajectory undershoots ALI and negative if it overshoots ALI. Since the correction method involves adjusting qMID , REDC , θEDC , RHAC , and xHAC , we will make reasonable arrangements for the correction of Δs. The proportional coefficient of the dynamic pressure qMID can be adjusted during each iteration cycle (the superscript indicates the iteration cycle) as follows: kþ1

kq

¼ kq  εkq  k

Δs qs

ð26Þ

where qs represents the error correction capability of the proportional coefficient kq . And the correction factor εkq A ½0; 1 is determined empirically. The change of the EDC radius ΔREDC and angle ΔθEDC has an influence on the ground track of the EDC turn ΔsEDC which can be approximately computed as:

ΔsEDC ¼ ΔREDC Δχ EDC_HAC þ ðREDC þ RHAC ÞΔθEDC

ð27Þ

where Δχ EDC_HAC means the total turning angle along the EDC. As the ΔsEDC contributes to Δs, the parameters of EDC can be adjusted during each iteration cycle using a Newton–Raphson method for the correction of Δs: þ1 ¼ RkEDC  εREDC RkEDC

Δs Δs ¼ RkEDC  εREDC ∂Δs=∂REDC Δχ EDC_HAC

þ1 k θkEDC ¼ θEDC  εθEDC

Δs Δs k ¼ θEDC  εθEDC ðREDC þ RHAC Þ ∂Δs=∂θEDC

ð28Þ

ð29Þ where εREDC A ½0; 1 and εθEDC A ½0; 1 are the correction factors. The parameters of HAC are adjusted during each iteration cycle using the same method: þ1 RkHAC ¼ RkHAC  εRHAC

Δs Δχ HAC_0

þ1 ¼ xkHAC þ εxHAC Δs xkHAC

ð30Þ ð31Þ

where Δχ HAC_0 represents the remaining heading angle from the tangency point on the HAC to the runway. εRHAC A ½0; 1 and εxHAC A ½0; 1 are the correction factors. Additionally, all the correction factors are usually constrained as:

6. Online trajectory reshaping

εkq þ εREDC þ εθEDC þ εRHAC þ εxHAC ¼ 1

Fig. 4 presents the detailed logic of the TAEM trajectory planning algorithm. The first step is to determine if there is

A new TAEM trajectory is numerically propagated after the parameters have been corrected by Eqs. (26), (28)–(31),

ð32Þ

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explain that the original point in the ground track diagram is represented as the ALI position to facilitate the observation, and the same as the following illustration of ground track. Fig. 7 depicts the dynamic pressure profiles for different initial energy. The lower the initial energy, the deeper the dynamic pressure profile so as to increase the total range for reaching the ALI successfully. Furthermore, in the case of the same initial energy, the middle dynamic pressure correspondingly increases as the initial altitude increases. With all these trials achieving adequate final states at ALI, it perfectly shows that the proposed guidance scheme is effective in the energy management. 7.2. Trials with initial position and heading dispersions

Fig. 4. Logic for TAEM trajectory planning algorithm.

and the iteration continues until the absolute magnitude of Δs is less than 200 m. Numerical trials have shown that this iteration scheme typically converges in 2–5 iterations. Divergence has never been observed, since the trajectory propagation has firstly identified scenarios where no feasible TAEM trajectory exists.

Several TAEM trajectories are obtained for cases involving a wide distribution of the initial position and heading angle without any change in the initial energy or flightpath angle. As the ground track presented in Fig. 8, the initial positions of the RLV are set around the ALI with the same distance to the ALI but different heading angles towards the ALI respectively. Fig. 9 shows the final position errors  for these 73 cases, where the final downrange errors Δx are less than 200 m and the final cross-range errors   Δy are less than 100 m. The good performance of all cases with acceptable errors at ALI successfully demonstrates the adaptability of the trajectory guidance scheme for various initial positions and heading angles. In addition, Fig. 10 presents the ground track of the TAEM trajectories where the initial positions of the RLV are placed closer or farther from the ALI with the same heading angle towards the ALI. As the initial distance to ALI is reduced, the EDC radius REDC and the angle of EDC corresponding to HAC θEDC are increased in order to increase the total ground track followed by the RLV before it reaches ALI. The change of the HAC is not obvious because we set the correction factors εRHAC ¼ 0:002 and εxHAC ¼ 0:02, which is relatively small to ensure the stability of the algorithm.

7. Numerical results

7.3. Trials with aerodynamic deviations

This section demonstrates the high effectiveness and adaptability of the TAEM guidance scheme by presenting the results of numerical trials with variations in the initial energy, position and aerodynamic performance, and also a baseline TAEM trajectory with nominal initial condition for comparison.

A series of simulations with vehicle aerodynamic deviations are obtained in this subsection. Aerodynamic variations may be due to modeling errors (relatively small) and control surface failures (large variations). For example, a failed or damaged control surface might result in excessive trim drag. In this study, we assume that the vehicle is able to estimate its aerodynamic performance based on the accelerometer measurements or the online identification techniques. Therefore, the proposed trajectory planning algorithm can periodically compare the estimated lift and drag with the nominal aerodynamic model and update the reference profile if the deviations become large. Fig. 11 presents the energy profile for six trials with aerodynamic deviations (the normal case is shown for comparison), and all these trials arrive at the ALI successfully with the target terminal energy. As the target flight path angle at the ALI is based on nominal aerodynamics, any aerodynamic deviations will require an updated target γ ALI based on the constraint of Eq. (14). It can be seen from the flight path angle profile shown in Fig. 12 that the target γ ALI changes significantly due to the

7.1. Trials with initial energy dispersions Dispersions in initial energy (altitude and velocity) were applied to the RLV and the corresponding TAEM trajectories were obtained by executing the trajectory guidance algorithm. Fig. 5 shows the energy profile for the nine cases based on three initial energy levels respectively with three different altitudes (normal 28 km and deviations 73 km). They all reach the target energy at the ALI, which means the target altitude and velocity. As the ground track depicted in Fig. 6, it can be seen that the higher the initial energy, the longer the ground track resulted to dissipate energy. Additionally, in the case of the same initial energy, the ground track increases according to the increase of initial altitude. Here, it needs to

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Fig. 5. Energy profiles for various initial energy. Fig. 8. Ground track for various initial positions and heading angles.

Fig. 6. Ground track for various initial energy.

Fig. 7. Dynamic pressure profiles for various initial energy.

aerodynamic deviation: increased lift-drag ratio results in a smaller γ ALI . Correspondingly, Fig. 13 shows the angle of attack profile and indicates that the aerodynamic deviation has a great impact on the guidance command. These six cases with aerodynamic deviations show the excellent tracking performance of the guidance scheme for the updated reference profile.

Fig. 9. Final position errors for various initial positions and heading angles.

Fig. 10. Ground track for various initial distances to ALI.

8. Conclusion and future work A new guidance scheme based on an online trajectory planning algorithm for the TAEM phase of an unpowered RLV has been proposed in this paper. The online trajectory planning algorithm generates the appropriate reference

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Fig. 11. Energy profiles with aerodynamic deviations.

trajectory is performed at the initiation of TAEM, in practice, the trajectory planning algorithm could be performed periodically during flight to adapt to potential failures. Several numerical trials were conducted in order to demonstrate the proposed guidance scheme. Scenarios involving initial energy dispersions, position variations, and vehicle aerodynamic deviations were simulated, and the results show that the trajectories successfully target the desired ALI state for all cases. Most of the cases required considerable changes in the TAEM reference profile (such as changes in the EDC position, EDC radius and middle dynamic pressure), however, these changes seem to be necessary in order to accomplish the TAEM mission while satisfying path constraints. Therefore, the proposed guidance scheme may potentially serve as an onboard guidance scheme with the ability to rapidly generate the guidance commands from the current vehicle state to a desired target state, which will greatly improve the safety and reliability of the RLV. However, further research is needed to improve this guidance scheme. Once the feasible reference profile has been generated by the online trajectory planning algorithm, instead of this simple guidance law, we will further design more advanced guidance algorithms with robustness in mind. On the other hand, in the future work, we will consider the scenarios involving bank limitations, wind disturbance and density deviations to further validate this guidance scheme.

Acknowledgments Fig. 12. Flight path angle profiles with aerodynamic deviations.

This work was supported by National Natural Science Foundation of China (Grant nos. 61473124, 61203081, and 61174079), Doctoral Fund of Ministry of Education of China (Grant no. 20120142120091), and Precision Manufacturing Technology and Equipment for Metal Parts (Grant no. 2012DFG70640).

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Fig. 13. Angle of attack profiles with aerodynamic deviations.

dynamic pressure profile and ground track geometry which can be perfectly tracked by the open- or closed-loop guidance laws. The five designing parameters of the reference path can be adjusted online according to the downrange error and converge quickly with the trajectory reshaping strategy. Note that the task of planning

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