Sage Journals: Discover world-class research

Abstract

This article improves an enhanced predictor–corrector entry guidance method for hypersonic flight vehicle. To compensate for the shortcoming that the enhanced predictor–corrector guidance method sacrifices guidance precision to meet path constraints, this article develops an improved predictor–corrector guidance method. Through consuming more energy in the middle section of entry, the guidance method greatly reduces the load in the end section of entry, earning more time for the guidance bank angle which aims at ensuring entry precision to function in the end section, and thus, the entry precision is improved. The simulation results on the CAV-H flight vehicle show preliminarily that the guidance method effectively enhances the guidance precision for both orbital and suborbital entry missions and strictly constrains distance errors within the stipulated range. Furthermore, this method is also fairly effective for flight vehicles with lower lift-to-drag ratio (X-33). Then, the flight vehicle with a higher lift-to-drag ratio is chosen as the simulation target to further examine the guidance method’s effectiveness and precision.

Keywords

Predictor–corrector equilibrium heating rate and load entry precision

Introduction

Entry flight of hypersonic flight vehicle is quite a demanding task due to the high speed and long distance flying which lead to large load, heat, and dynamic pressure. Thus, how to guide hypersonic flight vehicle to fly along a reasonable trajectory to guarantee entry precision and other process constraints is crucial in the whole entry. Recent years saw the rise of the predictor–corrector entry that has advantages in many aspects than the conventional trajectory-planning method.

Predictor–corrector entry guidance method derives from the onboard trajectory planning which does not need to pre-design a reference trajectory. This further leads to its advantages such as greater flexibility in reducing dispersions and the uncertainty of initial and terminal entry conditions. Most of predictor–corrector algorithms are less sophisticated and require less memory space for the onboard computer. Due to the above merits, this method has gone through a great and chronological development. Gamble et al.,¹ Braun and Powell,² Powell,³ Fuhry,⁴ Kaluzhskikh and Sikharulidze,⁵ Putnam et al.,⁶ Lu,⁷ and Brunner and Lu⁸ present a series of representative literatures studying in predictor–corrector entry guidance method. All these methods base on a similar fundamental algorithm which is to eliminate the errors of range-to-go in every guidance cycle, while their differences mainly lie in the ways when they deal with the path constraints and other details.

In spite of the advantages of predictor–corrector method, there is a long-standing shortage which is the lack of effective and widely applicable means to integrate inequality trajectory constraints into guidance algorithm (such as those on the heating rate and aerodynamic load). This shortage confines the application of this method to low-lifting entry vehicles. Xue and Lu⁹ and Lu¹⁰ aim at compensating for this shortage. In Xue and Lu,⁹ a constrained predictor–corrector guidance method is developed. It introduces the upper and lower bounds for the magnitude of the bank angle into the guidance method by resorting to quasi-equilibrium glide condition (QEGC), without increasing the complexity of the algorithm and decreasing the robustness. Lu¹⁰ presents a unified entry guidance method, on which this article is mainly based on. The unified method further overcomes the problem of enforcing path constraints and makes it viable in both orbital and suborbital entry missions. Plus, the method from Lu¹⁰ is more robust and widely applicable; it is a rather new method which can resolve the extant problem satisfactorily.

While Liu et al.,¹¹ and Wang et al.^12,13 all mainly derive from the guidance logic from Xue and Lu⁹ and Lu,¹⁰ they make their own contribution to addressing problems not discussed in Xue and Lu⁹ and Lu.¹⁰ For instance, Liu et al.¹¹ focus on altitude control; it increases altitude precision while allowing for the path constraints and other terminal conditions simultaneously. Wang et al.¹² combine Xue and Lu⁹ and Lu¹⁰ and develop an entry guidance method which could also be used for high-lifting hypersonic vehicles and reduce trajectory oscillations effectively.

Li et al.¹⁴ take perturbation of the atmospheric density and aerodynamic parameters into consideration and resorts to Kalman filter to develop an adaptive predictor–corrector guidance law which is able to decrease the influence of these perturbations. Gu et al.¹⁵ further use adaptive controller to suppress trajectory oscillations.

Adaptive controller can resist uncertainties of the model and increases the robustness of the entry process; yet, it requires more computing capability which will burden the structure design of hypersonic flight vehicles. Methods which share the same basic logic as Xue and Lu⁹ and Lu¹⁰ need much less calculation and have to great potential to be put in practical use. Between the two, the unified entry guidance method designed in Lu¹⁰ is superior in adaptiveness and calculation.

This article presents an improved predictor–corrector entry guidance method on the basis of Lu.¹⁰ As in the process of utilizing the method proposed by Lu,¹⁰ the method’s shortcoming of sacrificing the entry precision to meet the path constraints is revealed; thus, this article aims at proposing an improved method to compensate for this shortcoming. First, the basic entry guidance method will be introduced and then the method in Lu¹⁰ will be presented briefly in order to give rise to further discussion on this method. Then, the improved method is developed by taking the general characteristics of entry trajectory into consideration to overcome the method’s shortage. Simulation tests are carried out and the results show the improved method could promote the entry precision effectively while the path constraints are met and the merits of previous method remain.

The entry problem

Dimensionless 3-DOF equations of motion

The dimensionless 3-degree of freedom (DOF) equations of entry motion for entry vehicles are as follows¹⁶

\overset{\cdot}{r} = V \sin γ

(1)

\overset{\cdot}{θ} = V \cos γ \sin ψ / (r \cos ϕ)

(2)

\overset{\cdot}{ϕ} = V \cos γ \cos ψ / r

(3)

\begin{array}{l} \dot{V} = - D - \sin γ / r^{2} + Ω^{2} r \cos ϕ (\sin γ \cos ϕ \\ - \cos γ \sin ϕ \cos ψ) \end{array}

(4)

\overset{\cdot}{γ} = \frac{1}{V} [L \cos σ + (V^{2} - \frac{1}{r}) (\frac{\cos γ}{r}) + 2 Ω V \cos ϕ \sin ψ + Ω^{2} r \cos ϕ (\cos γ \cos ϕ + \sin γ \sin ϕ \cos ψ)]

(5)

\overset{\cdot}{ψ} = \frac{1}{V} [\frac{L \sin σ}{\cos γ} + \frac{V^{2}}{r} \cos γ \sin ψ \tan ϕ - 2 Ω V (\tan γ \cos ψ \cos ϕ - \sin ϕ) + \frac{Ω^{2} r}{\cos γ} \sin ψ \sin ϕ \cos ϕ]

(6)

where $r$ is the distance from the mass center of the flight vehicle to the earth center; $θ$ and $ϕ$ are longitude and latitude, respectively; $V$ is the flight vehicle’s velocity relative to the earth; $γ$ is the flight path angle; $ψ$ is the heading angle, namely the angle between the projection of the flight vehicle’s velocity vector on the local horizontal plane and the due north, the clockwise rotation being positive; $σ$ is the bank angle; $Ω$ is the earth’s rotational rate. Equation (1) is a dimensionless equation set; the dimensionless length is obtained by dividing the earth’s radius; the dimensionless time is obtained by dividing $\sqrt{R_{0} / g_{0}}$ , where $g_{0}$ equals to $9.81 m / s^{2}$ ; it can thus be deduced that the dimensionless velocity is obtained by dividing $\sqrt{R_{0} g_{0}}$ ; the dimensionless mass is obtained by dividing $m_{0}$ ; the dimensionless $L$ and $D$ are obtained by dividing $g_{0}$ .

The entry constraints

The path constraints

The typical trajectory inequality entry path constraints include heating rate, load, and dynamic pressure,⁹ which are expressed as equations (7)–(9)

\overset{\cdot}{Q} = k_{Q} \sqrt{ρ} V^{3.15} \leq {\overset{\cdot}{Q}}_{\max}

(7)

a = \sqrt{L^{2} + D^{2}} \leq a_{\max}

(8)

\bar{q} = (g_{0} R_{0} ρ V^{2}) / 2 \leq {\bar{q}}_{\max}

(9)

where $ρ$ is the atmospheric density; $k_{Q} = 9.4369 \times 10^{- 5}$ . The unit of $\overset{\cdot}{Q}$ is $w / m^{2}$ ; the unit of $a$ is $g_{0}$ ; the unit of $\bar{q}$ is $N / m^{2}$ .

The entry terminal constraints

The entry mission requires the flight vehicle to meet not only the above path constraints but also a series of entry terminal constraints at the terminal area energy management (TAEM) interface. They include terminal energy constraint, altitude constraint, velocity constraint and entry distance constraint, and so on. One important constraint is distance constraint

s (t_{f}) = s_{f}^{*}

(10)

where $s$ is the range to go and $s_{f}^{*}$ is the specified terminal value. If the entry azimuthal error is sufficiently small, then the expression of entry range to go is

{\overset{\cdot}{s}}_{togo} = - \frac{V \cos γ}{r}

(11)

The improved predictor–corrector method

As the lateral guidance is only used to determine the symbol of bank angle so as to keep the flight vehicle’s boresight error within the pre-set boundary and it is simple and mature, this article focuses on the longitudinal guidance method. The development of improved guidance method is based on the method in Lu,¹⁰ which we call the enhanced predictor–corrector method in this article. First, the enhanced method is to be introduced briefly and discussed.

The enhanced entry guidance method

Lu¹⁰ presents a guidance method, which is based on the basic predictor–corrector guidance method and then enhanced with its algorithm by introducing feedback items, thus called the enhanced entry guidance method. The basic predictor–corrector method calculates each moment that can enable the entry vehicle’s terminal distance to meet its constraint (see Brunner and Lu⁸ for details on its principles).

In each guidance cycle, $σ_{s}$ is the bank angle calculated with the basic predictor–corrector guidance method. Then the corresponding component of aerodynamic lift in its vertical direction is $L \cos σ_{s}$ . A feedback compensation of altitude change rate ${\overset{\cdot}{h}}_{ref}$ is introduced to regulate the component of aerodynamic lift so that the vehicle can entry as the way we want. Equation (12) gives the guidance law

L \cos σ_{cmd} = L \cos σ_{s} - k (\overset{\cdot}{h} - {\overset{\cdot}{h}}_{ref})

(12)

where $σ_{cmd}$ is the flight vehicle’s bank angle command to be used finally; $\overset{\cdot}{h}$ is the actual altitude change rate; $k$ is the coefficient to be determined. The feedback compensation ${\overset{\cdot}{h}}_{ref}$ should be designed according to the entry process characteristics so as to enable the flight vehicle to meet particular flight statuses and constraints during its entry process.

Feedback compensation items for balanced glide

First, the concept of QEGC is introduced to eliminate the oscillations of the altitude profile to make the entry trajectory smoother. The definition of QEGC is that the vehicle’s flight path angle and its change rate are approximately zero during entry process: $γ = 0$ , $\overset{\cdot}{γ} = 0$ . With the earth’s rotational rate neglected (which is accurate enough and proper for preliminary entry design), equation (13) is obtained with equation (5)

1 / r^{2} - V^{2} / r = L \cos σ_{QEGC}

(13)

where $σ_{QEGC}$ is the bank angle obtained under QEGC. Hence, equation (14) gives the flight path angle under QEGC

\sin γ_{QEGC} = 1 / (V^{2} / 2) ((\partial ρ / \partial r) \cos σ_{QEGC} / ρ) \cdot (L / D)

(14)

Equation (12) can be expressed as

\begin{array}{l} L \cos σ_{Q E G C} = L \cos σ_{s} - k (\dot{h} - V \sin γ_{Q E G C}) \\ = L \cos σ_{s} - k (\dot{h} - {\dot{h}}_{Q E G C}) \end{array}

(15)

The QEGC feedback item ${\overset{\cdot}{h}}_{QEGC}$ can make the entry vehicle fly under QEGC. Because QEGC item should function during the early stage of entry, the flight vehicle will not have an excessive oscillation after it enters QEGC. Therefore, the coefficient k should be designed to be large in the early stage and small in the later stage so as to make QEGC compensation item to function less in later entry process. Then, on the basis of QEGC, load and heating rate feedback items are further introduced to make the vehicle meet path constraints.

Load and heating rate feedback items

Take the load constraint in equation (8) as an example. $δ$ is a rather small assigned time increment, and then the objective is to make equation (16) valid

a (t + δ) \leq a_{\max}

(16)

And equation (17) is obtained

\sin γ \geq \frac{a_{\max} - a (1 - \frac{2 D δ}{V})}{aV (\partial ρ / \partial r) / ρ \cdot δ} = U_{a}

(17)

To meet load constraint, each position along the entry trajectory requires the following

{\overset{\cdot}{h}}_{ref} = V \sin γ_{ref} = V \cdot \max {\sin γ, U_{a}}

(18)

$U_{\overset{\cdot}{Q}}$ and $U_{\bar{q}}$ , the feedback items of heating rate and dynamic pressure, respectively, are obtained with the similar method. To implement all constraints simultaneously, we have

\sin γ_{ref} = \max {\sin γ, U_{a}, U_{\overset{\cdot}{Q}}, U_{\bar{q}}}

(19)

Thus, the use of the guidance command in equation (19) can help implement the above three constraints.

The improvement on enhanced method

The shortcomings of enhanced predictor–corrector guidance method

We take generic high performance Common Aero Vehicle (CAV-H) as the simulation model and use the enhanced method to analyze the simulation results. Its entry heating rate constraint is ${\overset{\cdot}{Q}}_{\max} = 4, 315, 280 W / m^{2}$ ; load constraint is $a_{\max} = 1.8 g$ . As ratio between load and dynamic pressure is static, meaning when load constraint is met the dynamic pressure constraint is also met, we omit dynamic pressure constraint here. To investigate the adaptation of guidance law to entry missions with different styles, we simulate orbital entry and suborbital entry, as shown in Table 1.

Table 1.

Simulating entry missions.

Missions	$h_{0} (km)$	$V_{0} (m / s)$	$s_{0} (km)$	$h_{f}^{*} (km)$	$V_{f}^{*} (m / s)$	$s_{f}^{*} (km)$
Orbital	120	7400	11,000	28	2000	92.6
Suborbital	60	7000	6533	28	2000	92.6

In Table 1, $h_{0}$ , $V_{0}$ , and $s_{0}$ are initial values of altitude, velocity, and range-to-go, and $h_{f}^{*}$ , $V_{f}^{*}$ , and $s_{f}^{*}$ are their terminal target values. Here, among the results, we emphasize the load factor. Figures 1 and 2 are the amplified end sections of load change curves for orbital and suborbital entry missions, respectively.

Figure 1.

The amplified end section of load curves for orbital entry.

Figure 2.

The amplified end section of load curves for suborbital entry.

The figures show that the guidance law with QEGC item makes the curve smoother and the heating and load item constrains the load at the end section of entry where there is a tendency that the load exceeds its constraints, thus enabling the load to stabilize under the constraint (1.8 g). This is also true for suborbital entry mission, as shown in Figure 2.

The figures show that there is an overall trend that the changes in both orbital entry and suborbital entry loads gradually increase. When the heat flow load item is not used, the load at the end section of entry may exceed a constraint value. Therefore, the time when the load item functions is mainly during the end section of entry, while the heating rate operator $U_{\overset{\cdot}{Q}}$ in equation (19) functions mainly at the middle section of entry.

Although both the QEGC item and the heating rate and load item are introduced on the basis of equation (12), the selection of the $k$ in equation (12) for the two items varies a lot. The simulation results show that the QEGC item functions mainly at the initial stage of the entry process; after the orbital path becomes sufficiently smooth in the later stage, the bank angle in equation (12) may return to $σ_{s}$ due to the decrease to 0 of item $k (\overset{\cdot}{h} - {\overset{\cdot}{h}}_{QEGC})$ , thus ensuring the entry precision. Since the QEGC item $k (\overset{\cdot}{h} - {\overset{\cdot}{h}}_{QEGC})$ aims at eliminating the fluctuation of entry trajectory rather than enabling the altitude change rate to change with ${\overset{\cdot}{h}}_{QEGC}$ strictly, it mainly acts as damping. However, the flight vehicle is required to strictly follow heating rate and load constraint; therefore, the selection of $k$ for the heating and load item requires the altitude change rate to strictly follow ${\overset{\cdot}{h}}_{ref}$ .

The above characteristics of the enhanced predictor–corrector guidance method and the characteristics of entry trajectory that result from the use of this guidance method bring about the following problems. $σ_{s}$ in equation (12) is the guidance bank angle whose function is to ensure the precision of entry distance; $σ_{cmd}$ is the bank angle generated by introducing factors such as QEGC and process constraints into the guidance algorithm, while it cannot ensure the precision of the distance. Besides, during the end section of entry, there exits greater influence on guidance precision than during its front and middle sections. But as the load value is rather big in the end section of the entry and tends to transcend the constraint causing operator $U_{a}$ to function frequently, the precision greatly declines correspondingly.

Improving the enhanced predictor–corrector guidance method

To solve the above problems of the enhanced guidance law, we develop an improved predictor–corrector guidance method according to general characteristics of entry trajectory. First of all, QEGC item mainly functions at the initial entry stage; the heating rate item functions mainly during a small period of time in the middle section of entry. Thus, the compensation item $k (\overset{\cdot}{h} - {\overset{\cdot}{h}}_{ref})$ functions little and weakly at the front and middle section of entry; instead, it frequently and strongly functions at the end section when load increases to a great amount. We can see that the basic item $σ_{s}$ in equation (12) always functions. Thus, if we can ensure that the compensation item $k (\overset{\cdot}{h} - {\overset{\cdot}{h}}_{ref})$ stops functioning or its functional time is greatly reduced during most of entry (particularly at the end section of entry), then $σ_{s}$ can act as the ultimate output guidance law and entry precision could be ensured.

Second, during the entry, the larger the bank angle of the entry vehicle is, the faster its altitude drops and consequently the faster its kinetic energy decreases as the flight vehicle receives greater aerodynamic drag during the subsequent entry. According to the observation of entry process under the enhanced guidance method, the compensation item does not function during most of the time in the middle section of entry. Plus, the middle section of entry has a rather weak influence on entry precision. Therefore, it is feasible to take advantage of the middle entry part to consume more kinetic energy.

Consequently, the starting point for improving the enhanced guidance law is to try to enable the flight vehicle to increase its bank angle in order to consume more energy during the middle section and to reserve enough time for the end section which has larger influence on entry precision to execute the bank angle based on the entry precision ( $σ_{s}$ ).

After the entry starts, at the moment $t$ , when compensation item $k (\overset{\cdot}{h} - {\overset{\cdot}{h}}_{ref})$ in equation (12) is not functioning (namely, $σ_{cmd} = σ_{s}$ ), we employ the following guidance logic. In every guidance cycle, our destination is to find a new bank angle $σ$ ( $σ = σ_{s} + Δ σ$ , $Δ σ > 0$ ) as guidance bank angle and we resort to the following iterative steps to calculate $σ$ :

In guidance algorithm, the flight vehicle carries out its entry flight with bank angle $σ$ until its load value $a$ increases to $a_{\max} - ε_{a}$ , namely $a = a_{\max} - ε_{a}$ , where $ε_{a}$ is a given small positive value.

After the flight vehicle’s load value reaches $a_{\max} - ε_{a}$ , we stop using the bank angle $σ$ and start using bank angle $σ_{s}$ calculated by the basic guidance law. Calculate the maximum load value $a_{\max}^{σ_{s}}$ the flight vehicle receives from the moment it uses the bank angle $σ_{s}$ to entry till the end. Then we assume $Δ a = a_{\max}^{σ_{s}} - (a_{\max} - ε_{a})$ .

The guidance algorithm is to find the bank angle $σ$ to satisfy $Δ a = 0$ . With the Newton–Raphson method, the iterative equation of $σ$ can be expressed as

σ^{(k + 1)} = σ^{(k)} - \frac{Δ a (σ^{(k)})}{\partial Δ a (σ^{(k)}) / \partial σ^{(k)}}

(20)

Meanwhile, $σ < σ_{\overset{\cdot}{Q}_\max}$ should be ensured; $σ_{\overset{\cdot}{Q}_\max}$ is the bank angle at which the flight vehicle may exceed the heating rate constraint in the remaining entry process. When Step 3 fails to find a solution, it means that there is no bank angle that can meet $Δ a = 0$ . Then we make the output bank angle $σ = σ_{\overset{\cdot}{Q}_\max} - ε_{\overset{\cdot}{Q}}$ , where $ε_{\overset{\cdot}{Q}}$ is also a given small positive value. With the improved method, we carry out simulations with the same simulation conditions in section “The shortcomings of enhanced predictor–corrector guidance method.” The simulation results, given in Figures 3 –5, show that with the improved guidance method, the bank angle curve is a bit larger in the middle section of entry than it is with the enhanced guidance method; the altitude curve is a bit lower in the middle section of entry. Furthermore, in the end section of entry, the bank angle curve is a bit smaller and the altitude curve is a bit higher. Most of energy is greatly consumed in the middle section of entry; the load value in the end section of entry, where the guidance law that ensures entry precision is functioning, is far less than the load constraint value, thus reducing entry distance error at TAEM interface.

Figure 3.

Comparing the orbital entry load curves.

Figure 4.

Comparing the orbital entry altitude curves.

Figure 5.

Comparing the orbital entry bank angle curves.

The simulation results on suborbital entry mission, given in Figures 6 –8, show that the improved guidance method is still applicable to suborbital entry mission, and that the general characteristics of altitude curve and bank angle curve agree with those for orbital entry mission. The only obvious difference is that because the conditions for suborbital entry mission are harsher (the vehicle is required to consume great energy in a shorter period of time), the time for consuming energy in the middle section of entry is shorter than that in orbital entry. Although it is not as obvious as it is in orbital entry mission from the figures, the improved guidance method effectively reduces the load in the end section of entry and yet tremendously enhances entry precision in suborbital missions.

Figure 6.

Comparing suborbital entry altitude curves.

Figure 7.

Comparing suborbital entry load curves.

Figure 8.

Comparing suborbital entry bank angle curves.

To quantitatively investigate the improved predictor–corrector guidance method, we establish a set of suborbital entry missions whose initial conditions for entry (such as initial longitude and latitude) and entry distances are different while whose terminal conditions are the same (all at longitude 279.504°, latitude 28.611°) (see Table 2 for details). All the initial altitudes are 60 km; the initial velocities are 7000 m/s. The distance error range in the end of entry is required to be 9.27 km.

Table 2.

The initial longitudes and latitudes and entry distances of entry missions.

Entry missions	1	2	3	4	5	6
Longitudes (°)	242.99	248.99	234.99	218.28	222.28	201.69
Latitudes (°)	−18.255	−22.510	−12.223	−2.305	−8.446	22.99
Entry distances (m)	6,519,000	6,554,000	6,589,000	7,360,000	7,382,000	7,625,000

Table 3 shows that with the enhanced predictor–corrector guidance method, the distance error in the end section of entry often exceeds the stipulated distance error range of 9.72 km. This is because the enhanced method makes the bank angle focus on reducing load to meet load constraints and unable to guarantee entry precision in time before end of entry. Table 3 also shows that with the improved guidance method, the error at TAEM interface greatly decreases, thus being able to fully satisfy given distance error ranges. Comparing the two methods, the improved method improves the precision by 68.79% averagely for the six entry missions.

Table 3.

Distance error for CAV-H with the enhanced and improved guidance method.

$Δ s_{togo}$ (km)	Missions
	1	2	3	4	5	6
Enhanced method	8.44	7.24	9.67	8.26	11.10	10.7
Improved method	2.15	2.23	2.25	2.84	3.18	2.64

CAV-H: common aero vehicle.

The above simulation results show that even if the CAV-H flight vehicle faces suborbital entry missions, the improved guidance method can still effectively reduce its distance error. To further test the extent of the method’s applicability, we should examine whether it is applicable to the entry flight vehicle that has a larger lift-to-drag ratio. Because the larger the lift-to-drag ratio is, then the larger the load value in the end section of entry is, the longer time the heating and load item functions, and the greater influence it exerts on entry precision.

As the enhanced guidance method is effective to be applied to flight vehicles with different lift-to-drag ratios, the versatility of improved guidance method should also be examined. Thus, we choose X-33 to test the applicability of improved guidance method to flight vehicles with lower lift-to-drag ratio. Simulation conditions are as same as previous ones except that X-33’s heating rate constraint is ${\overset{\cdot}{Q}}_{\max} = 726, 784 W / m^{2}$ and load constraint is $a_{\max} = 2.0 g$ .

According to Table 4, the improved method improves the precision by 46.62% averagely for the six entry missions. We can still see errors are reduced fairly well under the improved method, even though entry distance errors of flight vehicles with lower lift-to-drag ratios are already low enough under the enhanced method. Thus, the improved guidance method proves to be versatile enough to be applied to different flight vehicles.

Table 4.

Distance error for X-33 with the enhanced and improved guidance method.

$Δ s_{togo}$ (km)	Missions
	1	2	3	4	5	6
Enhanced method	2.12	1.86	2.30	1.63	2.91	2.77
Improved method	0.81	0.63	1.29	0.79	1.99	2.09

We then look at the simulation results by factitiously increasing the lift-to-drag ratio of CAV-H by 0.2 time.Figure 9 shows that for the flight vehicle with a higher lift-to-drag ratio, the improved guidance method is still effective in orbital entry mission, and that its load curve in the end section of entry is much smaller than that of the enhanced predictor–corrector guidance method. The former method creates enough time for the bank angle to control entry precision and still contains the entry error effectively. However, the difference for suborbital entry mission is not that clear to see (see Figure 10 for detail).

Figure 9.

Orbital entry load curves for the flight vehicle that has a higher lift-to-drag ratio.

Figure 10.

Suborbital entry load curves for the flight vehicle that has a higher lift-to-drag ratio.

Table 5 shows that as the lift-to-drag ratio increases, the distance error in the end section of entry increases with the enhanced guidance method. However, the improved method increases the precision to a very limited degree (averagely 21.43%) and does not effectively constrain it within the stipulated scope. The reason is that the suborbital entry mission leaves a much shorter middle section for the flight vehicle to consume energy than orbital entry mission. Furthermore, because of the higher lift-to-drag ratio of the flight vehicle, the middle section of entry becomes even shorter, limiting the effectiveness of the guidance method. This problem needs to be further resolved.

Table 5.

The distance errors for the flight vehicle with higher lift-to-drag ratio.

$Δ s_{togo}$ (km)	Missions
	1	2	3	4	5	6
Enhanced method	12.89	9.4	10.73	11.12	15.78	13.85
Improved method	12.33	9.62	7.38	9.86	11.54	10.54

Conclusion

This article proposes an improved predictor–corrector guidance method which compensates for the shortcoming of enhanced predictor–corrector guidance method that the distance precision at TAEM interface for the entry vehicle could not be guaranteed under some demanding entry conditions. The main idea of improved method is to use the middle section of entry to consume a large quantity of energy and to ensure that the load value decreases to a sufficiently small degree, thus enhancing entry precision.

Then the guidance method is used to simulate the orbital and suborbital entry missions of the CAV-H flight vehicle. The guidance law is quite effective and can effectively increase entry precision and satisfy entry distance requirements. To test whether it inherits the versatility of enhanced guidance method, we use X-33 as a low lift-to-drag flight model to carry out simulations. The results show that the method is also applicable under this circumstance. This shows that this method compensates for the disadvantage of enhanced guidance method with preserving its advantage without increasing too much computing burden.

To further investigate the applicability of the guidance method, we factitiously raise the lift-to-drag ratio of CAV-H. The simulation results show that the distance error of orbital entry mission can still be strictly controlled, while that of suborbital entry mission cannot be effectively constrained within the stipulated scope which needs to be further resolved.

Footnotes

Academic Editor: Anand Thite

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (grant nos 61104195 and 6150330).

References

Gamble

Cerimele

Moore

et al . Atmospheric guidance concepts for an aeroassist flight experiment. J Astronaut Sci 1988; 36: 45–71.

Braun

Powell

. Predictor-corrector guidance algorithm for use in high-energy aerobraking system studies. J Guid Control Dynam 1992; 15: 672–678.

Powell

. Six-degree-of-freedom guidance and control-entry analysis of the HL-20. J Spacecraft Rockets 1993; 30: 537–542.

Fuhry

. Adaptive atmospheric reentry guidance for the Kistler K-1 orbital vehicle. In: Proceedings of the guidance, navigation, and control conference, Portland, OR, 9–11 August 1999, paper no. AIAA 99-4211. Reston, VA: AIAA.

Kaluzhskikh

Sikharulidze

. A control algorithm for reentry of a rescue space vehicle into the earth’s atmosphere. Cosmic Res+ 2000; 38: 262–269.

Putnam

Braun

Bairstow

et al . Improving lunar return entry footprint using enhanced skip trajectory guidance. In: Proceedings of the AIAA space 2006 conference, San Jose, CA, 19–21 September 2006, paper no. AIAA 2006-7438. Reston, VA: AIAA.

. Predictor-corrector entry guidance for low-lifting vehicles. J Guid Control Dynam 2008; 31: 1067–1075.

Brunner

. Skip entry trajectory planning and guidance. J Guid Control Dynam 2008; 31: 1210–1219.

Xue

. Constrained predictor-corrector entry guidance. J Guid Control Dynam 2010; 33: 1273–1280.

10.

. Entry guidance: a unified method. J Guid Control Dynam 2014; 37: 713–728.

11.

Liu

Liang

et al . Predictor-corrector guidance for entry with terminal altitude constraint. In: Proceedings of the 35th Chinese control conference, Chengdu, China, 27–29 July 2016. New York: IEEE.

12.

Wang

Ren

. Predictor-corrector entry guidance for high-lifting hypersonic vehicles. In: Proceedings of the 35th Chinese control conference, Chengdu, China, 27–29 July 2016. New York: IEEE.

13.

Wang

Hua

et al . Constrained predictor corrector entry guidance for common aero vehicle. In: Proceedings of the 34th Chinese control conference, Hangzhou, China, 28–30 July 2015. New York: IEEE.

14.

Sun

Shen

. An adaptive predictor-corrector entry guidance law based on online parameter estimation. In: Proceedings of the 2016 IEEE Chinese guidance, navigation and control conference, Nanjing, China, 12–14 August 2016. New York: IEEE.

15.

Zhang

Wang

et al . Trajectory oscillation suppression control for lifting reentry vehicles with predictor-corrector guidance. In: Proceedings of the 2016 IEEE advanced information management, communicates, electronic and automation control conference (IMCEC), Xi’an, China, 3–5 October 2016. New York: IEEE.

16.

. Hypersonic flight vehicle guidance and control techniques. Beijing, China: China Aerospace Press, 2012.

An improved predictor–corrector entry guidance method for hypersonic flight vehicle

Abstract

Keywords

Introduction

The entry problem

Dimensionless 3-DOF equations of motion

The entry constraints

The path constraints

The entry terminal constraints

The improved predictor–corrector method

The enhanced entry guidance method

Feedback compensation items for balanced glide

Load and heating rate feedback items

The improvement on enhanced method

The shortcomings of enhanced predictor–corrector guidance method

Improving the enhanced predictor–corrector guidance method

Conclusion

Footnotes

Declaration of conflicting interests

Funding

References