Adaptive maneuvering entry guidance with ground-track control

Aerospace Science and Technology 11 (2007) 419–431 www.elsevier.com/locate/aescte

Adaptive maneuvering entry guidance with ground-track control M. Sudhir 1 , Ashish Tewari ∗,2 Indian Institute of Technology, Kanpur 208016, India Received 7 June 2006; received in revised form 14 February 2007; accepted 16 February 2007 Available online 27 February 2007

Abstract We demonstrate that a three-dimensional entry guidance with accurate ground-track control can be achieved using a combined strategy of proportional down-range maneuvering and linear quadratic (LQ) technique, thereby precluding the necessity of sophisticated nonlinear programming. It is also shown that heating rate and load-factor constraints can be met by a careful but simple selection of LQ gains. Although the speed, flight path angle, and down-range accuracy are comparable to that of a planar guidance strategy, the present technique also demonstrates an accurate cross-range control, which is not possible with a planar strategy. It is shown that adaptive maneuvering produces a superior performance than that of a nonlinear tracking control derived using feedback linearization. The method can handle much larger deviations in the entry conditions than those published previously, and has the important advantage of a relatively simple implementation in a realistic entry scenario. Furthermore, the LQ technique can take advantage of the good robustness properties of a linear Kalman filter based upon the measurement of longitude, latitude, and speed, wherein little deterioration is observed in the closed-loop performance. © 2007 Elsevier Masson SAS. All rights reserved. Keywords: Lifting entry; Adaptive guidance; Ground-track control

1. Introduction A primary focus in entry technology and development of a reusable launch vehicle (RLV) is entry guidance and control. In the past, several distinct approaches [1] have been considered for entry guidance, which can be grouped into two general categories: (a) path-predicting, where the controller uses a near real-time prediction to generate the trajectory and the required controls in flight, and (b) path-following, wherein the controller tracks a pre-determined reference trajectory. The path-predicting methods have the disadvantage of large on-board computer requirements, and a limited capability of handling off-design conditions. Such methods can be subclassified into approximate closed-form prediction and fasttime prediction. The problem of determining a feasible entry trajectory, that meets the constraints of heating rate, axial * Corresponding author. Tel./fax: +91 512 2597858/2597561.

E-mail addresses: [email protected] (M. Sudhir), [email protected] (A. Tewari). 1 Doctoral Student. Department of Aerospace Engineering. 2 Associate Professor. Department of Aerospace Engineering. 1270-9638/$ – see front matter © 2007 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.ast.2007.02.005

and transverse load-factors results in an optimal control input, which is inherently nonlinear, and requires geometric, nonlinear programming methods. In order to avoid solving the complete optimal guidance problem in real-time, a practical alternative is closed-form, analytical solution based on approximate dynamics [2–4]. However simple closed-form solutions usually do not have the flexibility to handle markedly off-design conditions. The fast-time prediction is based upon repetitive solution of the equations of motion in real-time, in order to determine possible future trajectories. This method has the ability to handle any possible flight condition, but again requires large on-board computing facilities. In contrast to path-predicting techniques, path-following guidance about nominal trajectories provides a simple guidance method that can be designed to accommodate off-design entry conditions. Path-following guidance techniques involve guidance about a nominal trajectory, where the state variables along the nominal path are pre-computed and stored on-board. The variations in the measured variables from the stored values are used in guidance either to track the nominal trajectory (path control) or to establish a new trajectory in order to reach the destination (terminal control). The desired nominal trajec-

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Nomenclature A(t) state dynamics matrix B(t) state input coefficient matrix drag coefficient CD CF skin friction coefficient CL lift coefficient D drag force E[.] expected value operator eo state estimation error g acceleration due to gravity G, H, M coefficient matrices of reduced-order observer h altitude In×n identity matrix of order n J objective function k1 , k2 feedback gains for down-range maneuvering K(t) feedback LQ gain matrix Ko (t) observer gain matrix L lift force m mass n total load-factor P (t) solution of the regulator algebraic Riccati equation q dynamic pressure Q state weighting matrix Qh heating rate r distance from the planet’s center R input weighting matrix √ RC combined range (Rc = X 2 + Y 2 ) RE radius of the Earth Re optimal covariance matrix S reference area Sw total wetted area t time

tories (altitude, speed and ground-track) are selected and stored apriori by solving the two-point, boundary-value problem (TPBVP) where the two boundaries being the entry and TAEM (Terminal Area Energy Management) point. For the three dimensional Apollo-type re-entry problem, Williamson and Tapley [5] use a successive linear solution, Stoer and Bulrisch [6] employ a multiple shooting method, Yeo and Singh [7] adopt a quasi-linearization approach for a problem with control bounds, whereas Trent et al. [8] propose a modified simple shooting method for a planar problem. The path-following guidance methods [9–17] can be classified as either linear, or nonlinear based upon the kind of trajectory control used. The guidance methods include path control with fixed feedback gains, terminal control having influence coefficients (closed-form, bang-bang, etc.), and optimal feedback gains. A closed-form solution of the equation of motion is used in the US space shuttle orbiter guidance [9,10], where the down-range is controlled by issuing bank angle magnitude commands to track a drag acceleration profile. The profile is also re-planned onboard during entry to correct for tracking errors and to nullify the predicted down-range error. The

td u(t) v(t) V x(t) x1 (t) xo2 (t) X Y z(t) zk (t) Zk (t) α γ δ(.) θ μ ρ σ φ ψ ω

duration of down-range maneuvering input vector process noise relative speed state vector measured state vector estimated state vector down-range cross-range reduced order observer state vector measurement noise power spectral density matrix of zk (t) angle of attack flight path angle deviation from nominal trajectory longitude gravitational constant atmospheric density bank angle latitude heading angle planet’s angular velocity

Subscripts com f max o s r

commanded variables terminal values maximum allowable values observer surface of planet nominal/reference variables Overdot represents time-derivative

cross-range is controlled by issuing bank reversal commands to keep the heading error within certain bounds. De Virgilio et al. [11] apply linear optimal (LQ) guidance to ballistic type vehicles. Both Refs. [11] and [12] address the determination of the weighting matrices required for LQ regulator design. Archer [13] formulates the linear control problem with a trajectory variable instead of time. Roenneke and Cornwell [14] design a tracking controller for a low-lift re-entry vehicle, where the reference trajectory is defined by altitude, velocity, and flightpath angle. Position and heading angles are not included in the controller design. Roenneke and Markl [15] schedule the drag profile as functions of energy for the entire trajectory for accurate range prediction, and the reference profile is parameterized by cubic splines. For trajectory control, they employ a linear feedback control law whose gains are scheduled to vary with the energy. Lu [16] uses a linear receding-horizon control law for the X-33 vehicle, while Shen and Lu [17] employ an LQ based proportional-integral (PI) control law for lateral entry guidance. They have found that the cross-range is a more suitable parameter than the heading error for lateral guidance.

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A nonlinear control law based on the feedback linearization method is compared with shuttle trajectory control law in Ref. [18]. Cavallo and Ferrara [19] propose a control strategy which is based on the (LQ) design technique and on the variable structure systems (VSS) mathematical machinery applied to assured crew return vehicle (ACRV). They employ a slidingmode controller in conjunction with the LQ technique to meet the ground-track requirements. Several authors [20–22] have proposed and evaluated variations to the shuttle entry guidance for medium to high L/D vehicles. All of these retained the basic assumption of the shuttle drag planning in which the entry guidance is designed by analytically defining a desired drag acceleration profile and commanding the Orbiter attitudes to achieve the desired profile. Lu et al. [20,21] however, included a final pre-TAEM phase to increase the capability of meeting the TAEM conditions. Hanson and Jones [26] examined several approaches and proposed an automatic scoring method based on the various performance results obtained from previous works. Based on their scoring method, LQ technique would be the best choice for a simple, robust guidance that does not involve onboard trajectory design. Ref. [26] have also shown that LQ technique fails to meet the high cross-range requirement and our paper addresses this issue and it is shown that high crossrange control can be achieved by using LQ technique. This paper presents the trajectory control of lifting re-entry vehicles using a combined strategy of down-range maneuvering and an LQ design technique. The present adaptive strategy is to be contrasted with previously published techniques wherein a terminal control for the slow dynamics given by the longitude, latitude and heading angle is carried out independently of the fast dynamics given by altitude, velocity and flight-path angle. Such an approach – based upon tracking only a few variables at a time – is unable to meet the ground-track accurately, especially in the presence of moderate deviations from the nominal entry point. Our approach is based upon a simultaneous penalty on all the state variables applied continuously from entry to TAEM, resulting in an accurate ground-track control. In the presence down-range deviations, an adaptive maneuvering algorithm is devised such that either a turn-back, or a pull-up is carried out, depending upon whether the vehicle has overshot, or under-shot the nominal entry point. The maneuvering is carried out with proportional banking (or lifting) for a predetermined duration arrived at after careful simulation with the given nominal entry conditions. The new strategy is applied to two vehicles: a low-lift capsule where the bank angle is the only control input, and another an RLV, which requires modulation of both lift-coefficient and bank-angle. The results compare favorably with those obtained from a nonlinear geometric tracking law derived using feedback linearization [22] in which a nonlinear state transformation and a state-dependent control transformation are performed in order to linearize the input-output dynamics. Results for entry deviations much larger than those previously published show the efficacy and simplicity of the scheme. Thus, it is demonstrated that a well designed, simple control concept is sufficient for re-entry trajectory control, even in the presence of large deviations in entry conditions, without exceeding the maximum

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allowable aerothermal loads. Furthermore, it is shown that a linear Kalman filter based upon the measurement of longitude, latitude, and speed, results in an accurate tracking of all state variables, including the ground-track 2. Entry dynamics The translational equations of motion of an unpowered reentry vehicle of mass, m, flying inside the atmosphere of a spherical, rotating planet are used to simulate the guiding controller’s performance. The state vector is chosen to comprise the standard planet fixed variables, x(t) = (r, θ, φ, V , γ , ψ)T , while the control vector is u(t) = (σ, CL )T . The kinematic equations of motion can be written as follows [23]: r˙ = V sin γ V cos γ cos ψ θ˙ = r cos φ V cos γ sin ψ φ˙ = r

(1) (2) (3)

whereas the dynamical equations of motion are D V˙ = − − g sin γ m + ω2 r cos φ(sin γ cos φ − cos γ sin φ sin ψ)   g V L cos σ − − cos γ + 2ω cos φ cos ψ γ˙ = mV V r ω2 r cos φ(cos γ cos φ + sin γ sin φ sin ψ) V V cos γ cos ψ tan φ L sin σ − ψ˙ = mV cos γ r + 2ω(tan γ cos φ sin ψ − sin φ) +

−

ω2 r sin φ cos φ cos ψ V cos γ

(4)

(5)

(6)

The aerodynamic lift and drag have the following standard form based upon non-dimensional coefficients: 1 L = ρV 2 SCL 2 1 D = ρV 2 SCD 2

(7) (8)

The Newtonian gravitational law g = μ/r 2 is employed for a spherical planet, while the stagnation point (maximum) heating rate, the load-factor, and dynamic pressure are given by 1 Qh = CF ρSW V 3 4  L2 + D 2 n= mgs 1 q = ρV 2 2

(9) (10) (11)

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3. Trajectory control

where z(t) is the reduced order observer state vector satisfying the following state-equation:

The design of LQ guidance laws requires a linearization of Eqs. (1)–(6) about the nominal trajectory. A perturbed state δx = (δr, δθ, δφ, δV , δγ , δψ)T is defined such that x = xr +δx, and correspondingly, a perturbed control is given by δu = δσ such that δu = u − ur . Here δCL = 0, as the LQ guidance law is used mainly to bring the cross-range and velocity deviations to zero at the terminal time. For small perturbations, Eqs. (1)– (6) can be linearized into ˙ = A(t)δx + B(t)δu δx

(12)

The time-varying linearized system’s coefficient matrices A(t) and B(t) are obtained at discrete time intervals from the reference trajectory. The standard LQ objective function [24] is given by tf J (t, tf ) =

 T  δx (τ )Qδx(τ ) + δuT (τ )Rδu(τ ) dτ

(13)

t

where Q is positive semi-definite matrix and R > 0 are state weighting and input weighting factors, respectively. Minimization of J in Eq. (13) as tf → ∞, results in an algebraic Riccati equation given by P A − P BR −1 B T P + Q + AT P = 0

(14)

Eq. (14) is solved for constant P (t) at each time instant, t, and the optimal feedback gain matrix is given by K(t) = −R −1 B T (t)P (t)

(15)

for the full-state feedback control-law δu(t) = −K(t)δx(t).

(16)

δ x(t) ˙ = A(t)δx(t) + B(t)δu(t) + F (t)v(t)

(17)

δy(t) = C(t)δx(t) + D(t)δu(t) + zk (t)

(18)

The state-vector of the plant, δx(t), is partitioned into measured state variables, δx1 (t), and the estimated state variables, δxo2 (t) as follows:  δx1 (t) δx(t) = (19) δxo2 (t) The estimated state-vector is given by (20)

(21)

The optimal Kalman filter gain is thus given by [24] Ko (t) = Re (t, t)C T (t)Zk−1 (t)

(22)

where   Re (t, t) = E eo (t)eoT (t)

(23)

is the optimal covariance matrix obtained from the solution of the associated Riccati equation [24], Zk is the power spectral density of measurement noise zk , and the estimated error eo is given by eo (t) = δx2 (t) − δxo2 (t)

(24)

The LQ technique presented above is based upon linearized equations of motion, Eq. (12), consisting of small deviations from the nominal trajectory. Hence, while the LQ strategy can handle longitudinal and cross-range deviations, it is incapable of handling even moderate deviations in the down-range, which require a large maneuver (turn-back, or pull-up) in order to return to the nominal trajectory. Therefore, a separate adaptive down-range maneuvering algorithm is used to minimize the ground-track error resulting from initial perturbations in downrange. The algorithm consists of different proportional control laws for either positive, or negative perturbations in the downrange: u = ur + δucom

The weighting factors Q and R are chosen from simulation with the nominal entry conditions such that the terminal errors in the important state variables are small, and the specified aerothermal load limits are never exceeded at any point in the trajectory. In order to avoid measuring and feeding back all the state variables, a reduced-order Kalman filter is designed. Apart from the resulting reduction in the implementation cost, Kalman filter possesses well known parametric robustness and noise rejection properties [24]. In the design of a reduced-order Kalman filter, the state and the output equations are

δxo2 (t) = Ko δy(t) + z(t)

z˙ (t) = Mz(t) + H u(t) + Gy(t)

(25)

where

⎧  k1 δRc  ⎫ ⎪ , δX > 0, t  td ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪  ⎪  

⎨ ⎬ 0 δσcom = , δX < 0, t  td δucom = ⎪ ⎪ δCLcom δRc ⎪ k2 ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ ⎪ −Kδx ⎪ ⎪ ⎩ ⎭ , t > td 0

(26)

k1 and k2 are constant gains, δRc is the combined range error given by  δRc = δX 2 + δY 2 (27) and td is the duration of down-range maneuvering that is predetermined from the down-range deviation at re-entry. The down-range maneuvering is carried out for 0 t  td , while the LQ control law is applied for t > td . In the case of positive down-range perturbations, the lift-coefficient is unchanged from the nominal value while the bank angle is modulated to execute a horizontal turn back towards the nominal trajectory. For negative down-range perturbations, the bank angle is unchanged from the nominal value while the lift co-efficient is increased in order to pull-up the vehicle towards the nominal trajectory.

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4. Simulation results To demonstrate the trajectory control concept two vehicle models are chosen, an entry capsule with low-lift (Model I), and an RLV with adequate lifting capability (Model II). The US standard atmosphere 1976 is used to model the Earth’s atmosphere, while the integration of nonlinear equations of motion is performed using a variable step, fourth-order RungeKutta method. The effect of off-nominal entry conditions on the relative speed and flight path angle are considered in Model I, whereas for Model II, perturbations in initial longitude, latitude and altitude are also considered. Wind is not considered in the simulations as it is absent in the upper atmospheric strata to which an entry trajectory is confined. 4.1. Model I Model I represents a small re-entry capsule [14], which has a sphere-cone configuration with a base diameter 0.5 m, height 0.8 m, and mass 80 kg. To provide aerodynamic lift, one side of the cone is flattened, resulting in a constant supersonic liftto-drag ratio of 0.52. The capsule’s bank angle is varied by a moving mass. For the following simulation, a reference trajectory with a constant 45 deg bank angle is chosen. In Ref. [14], the transfer orbit is modeled as a Hohmann ellipse with a fictitious perigee altitude of 200 km. De-orbit is induced by a single impulse opposing the orbit velocity vector. To represent entry condition dispersions, the nominal de-orbit impulse is offset by ±10%, which produces offsets in the relative speed (δV ) and flight path angle (δγ ) as shown in Table 1. The choice of weighting factors (Q, R) is crucial in the design of the LQ controller. The pair (Q, R) is chosen based upon the simulated terminal errors in the states, control input limitations, as well as the aerothermal loads. In the calculation of maximum heating rate for the Model I, we assume CF = 4CD . Table 2 lists the various cases tried in the selection of weighting matrices, while Table 3 presents the resulting terminal errors in each case. Clearly, Case 20 has the smallest terminal state error, of which the ground-track error is very small (0.2066 km). When aerothermal loads are considered in Table 4, Case 2 has the smallest stagnation heating rate among all cases. By increasing the bank angle, the aerothermal loads increase. Hence, there is a trade-off between the terminal state errors and aerothermal loads.

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We shall consider two alternative strategies: planar guidance (similar to that of Ref. [14]) without considering ground-track, and nonplanar guidance, wherein deviations in ground-track are penalized in addition to the other state variables. Note that nonplanar guidance is covered in Cases 1–22 of Tables 2 and 3, while the planar strategy is limited to Case 23. Fig. 1 shows the time history of altitude, velocity, and flight-path angle due to a +10% (Case A) and −10% (Case B) deviations in de-orbit impulse with the planar guidance law. Although the planar strategy successfully tracks the profile of Ref. [14], it fails to meet the nominal ground-track (not considered in Ref. [14]), resulting in unacceptable terminal range deviation. The bank angle profile and the aerothermal loads for perturbed entry with planar guidance are compared with those of the nominal trajectory in Fig. 2. The closed-loop trajectory obtained by the nonplanar approach of Case 20 (Table 3) is plotted in Fig. 3, demonstrating a convergence to the desired terminal state, including groundtrack. The largest bank angle for nonplanar guidance (Fig. 4) is smaller than that required for planar guidance (Fig. 2). As a reTable 2 Weighting matrices for Model I Case

Q

R

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

I6×6 I6×6 I6×6 I6×6 I6×6 I6×6 I6×6 diag[1 2.30 × 1010 2.30 × 1010 1 1 1] diag[1 2.30 × 1011 2.30 × 1011 1 1 1] diag[1 2.20 × 1011 2.20 × 1011 1 1 1] diag[1 2.10 × 1011 2.10 × 1011 1 1 1] diag[1 2.00 × 1011 2.00 × 1011 1 1 1] diag[1 1.90 × 1011 1.90 × 1011 1 1 1] diag[1 1.80 × 1011 1.80 × 1011 1 1 1] diag[1 1.70 × 1011 1.70 × 1011 1 1 1] diag[1 1.90 × 1011 1.90 × 1011 1 1 1] diag[1 1.70 × 1011 1.70 × 1011 1 1 1] diag[1 1.60 × 1011 1.60 × 1011 1 1 1] diag[1 1.70 × 1011 1.70 × 1011 1 1 10] diag[1 1.70 × 1011 1.70 × 1011 1 2 10] diag[1 1.70 × 1011 1.70 × 1011 1 3 15] diag[1 1.650 × 1011 1.650 × 1011 1 3 10] diag[1 0 0 0.1 100 0]

1.00 × 1010 2.00 × 1010 3.00 × 1010 2.10 × 1010 2.20 × 1010 2.30 × 1010 2.40 × 1010 2.30 × 1010 2.30 × 1010 2.20 × 1010 2.10 × 1010 2.00 × 1010 1.90 × 1010 1.80 × 1010 1.70 × 1010 1.80 × 1010 1.80 × 1010 1.80 × 1010 1.80 × 1010 1.80 × 1010 1.80 × 1010 1.80 × 1010 1.00 × 1009

Table 1 Nominal and off-nominal entry conditions for Models I and II State variable x

Model I

Model II

Nominal xr

Case A xr + δx

Case B xr − δx

Nominal xr

Case I δx

Case II δx

h (km) θ (deg) ϕ (deg) V (m/s) γ (deg) ψ (deg)

120 0.0 0.0 7440.3 −2.97 0.0

120 0.0 0.0 7422.5 −3.30 0.0

120 0.0 0.0 7458.3 −2.60 0.0

60.96 0.0 0.0 5181.6 0.0 0.0

±0.3048 ±0.1 ±0.1 0.0 0.0 0.0

0.0 ±0.45 0.0 0.0 0.0 0.0

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Table 3 Terminal state errors for Model I (Case B) Case

Terminal groundtrack error (δRc ) (km)

Terminal altitude error (δh) (km)

Terminal velocity error (δV ) (m/s)

Terminal flight-path angle error (δγ ) (deg)

Terminal heading angle error (δψ) (deg)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

30.9397 16.5049 26.2114 14.9712 14.6089 14.2912 14.5152 11.5747 9.6908 7.0877 5.7877 3.4374 1.1186 0.8708 1.7639 1.1678 0.2622 0.8840 0.7992 0.2066 0.8173 0.5508 182.4179

−0.0135 0.0228 0.0184 0.0050 0.0305 0.0194 0.0390 −0.0049 0.1379 0.0098 −0.0002 0.0044 0.0180 0.0066 0.0017 −0.0003 0.0040 0.0005 0.0279 0.0007 0.0126 0.0258 −0.2459

−0.2929 0.1387 0.1687 0.0315 0.2919 0.1866 0.2983 −0.0809 −0.2115 0.2625 0.0552 0.1475 0.0201 0.0134 −0.0281 −0.0251 0.0378 0.1464 −0.2492 −0.0083 0.1041 0.2746 2.0808

0.0001 −0.0007 0.0274 −0.0183 0.0201 0.0096 0.0171 0.2422 4.1167 −0.2843 0.0148 −0.3079 0.3563 0.1292 0.2302 0.0429 −0.0129 −0.1711 1.2256 0.1010 −0.0580 −0.3039 0.8442

5.0577 −0.2642 −1.6415 1.4490 −1.0498 −0.2257 −1.5305 −8.6021 108.1844 4.8072 0.7297 1.5196 1.3922 −4.5346 8.2360 −0.0504 −8.9254 9.0986 0.8310 1.1323 −0.2104 −0.5335 −25.1387

Table 4 Control input and aerothermal loads for Model I (Case B) Case

Maximum bank angle, σ (deg)

Maximum dynamic pressure, σ (N/m2 )

Maximum heating rate, Qh (W/m2 )

Maximum load factor, n (g)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

78.5915 69.8871 64.4860 69.2427 68.6029 67.9788 67.4069 68.4513 146.5194 77.5533 69.8919 70.3848 70.8409 71.4197 71.8214 71.2144 71.1761 71.0942 76.2617 71.1782 71.1807 71.1090 103.6322

7.43 × 103 6.62 × 103 7.06 × 103 6.61 × 103 6.68 × 103 6.68 × 103 6.72 × 103 7.36 × 103 8.70 × 103 8.72 × 103 8.74 × 103 8.80 × 103 8.83 × 103 8.89 × 103 8.97 × 103 8.92 × 103 8.84 × 103 8.79 × 103 8.84 × 103 8.86 × 103 8.90 × 103 8.83 × 103 7.93 × 103

7.91 × 106 6.69 × 106 7.25 × 106 6.71 × 106 6.77 × 106 6.78 × 106 6.84 × 106 7.39 × 106 8.74 × 106 8.77 × 106 8.81 × 106 8.87 × 106 8.92 × 106 8.95 × 106 9.02 × 106 8.98 × 106 8.88 × 106 8.84 × 106 8.91 × 106 8.93 × 106 8.94 × 106 8.88 × 106 9.73 × 106

3.07 2.97 3.06 2.91 2.90 2.91 2.91 3.15 3.69 3.70 3.72 3.74 3.76 3.78 3.81 3.80 3.76 3.74 3.76 3.77 3.77 3.76 3.53

sult, the peak heating rate is smaller for the nonplanar method, as shown in Fig. 4, although a slight increase in the load-factor is observed, compared to that of the planar method. Thus, the nonplanar guidance law outperforms the planar one in terms of ground-track error, bank angle input and aerothermal loads.

Hence, there is no need of a separate dedicated ground-track control claimed in Ref. [14]. The down-range maneuvering (Eq. (26)), is not necessary in this case. The next example is more interesting, because it includes down-range deviation at entry.

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Fig. 1. Trajectory profiles and ground-track for Model I (planar guidance).

Fig. 2. Control inputs and aerothermal loads for Model I (planar guidance).

4.2. Model II Model II represents an RLV [22] corresponding to the SX-2 vehicle of former McDonnell Douglas Corp., with a maximum

lift-to-drag ratio of 1.6. A polynomial fit of the aerodynamic data for CL (α) and CD (α) over the range 5  α  25 deg, including the effects of flap deflections for trimming the vehicle, is given by [22]

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Fig. 3. Trajectory profiles and ground-track for Model I (nonplanar guidance).

Fig. 4. Control inputs and aerothermal loads for Model I (nonplanar guidance).

CL (α) = 2.5457α − 0.0448

(28)

CD (α) = 3.7677α 2 − 0.1427α + 0.197

(29)

The mass during descent is constant at 37335 kg, and S = 65.9 m2 . Since in this case δψ = 0, we have δX = RE δθ , con-

sequently the duration of down-range maneuvering, td , depends solely on the initial perturbation in the longitude, δθ . An empirical relation for td in terms of initial perturbation in longitude has been derived from careful simulations with off-nominal entry conditions in the range of 0.05  |δθ |  0.45 deg to be the

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427

Fig. 5. Altitude, ground-track, and control inputs for Model II (Case I).

following: δθ > 0.05:

Table 5 Comparison of range errors with Ref. [22] for Model II (Case I)

td = −24615δθ 5 + 36946δθ 4 − 21862δθ 3

Case

+ 6530.9δθ 2 − 1035.7δθ + 87.917

(30)

Terminal Ground track errors δRc (km)

1

δθ < −0.05: td = 303.03δθ 4 + 252.53δθ 3 + 56.818δθ 2 − 5.5339δθ + 31.107

Description

(31)

where δθ is in deg. For |δθ | < 0.05, td = 0. With this strategy, an initial down-range deviation of ±50 km can be handled. The reference trajectory for the simulation is obtained by open-loop integration of a selected reference control vector [22] ur = (σr , CLr )T . Simulations were performed for a range of initial offsets from the reference trajectory and some of them are presented in this section. The weighting matrices (Q, R) are chosen by a similar procedure presented above for Model I. The altitude profile, ground-track, and the required control inputs are plotted in Fig. 5 for both positive (Case A) and negative perturbations (Case B) in longitude, latitude, and altitude (Case I of Table 1), which is the largest perturbation reported in Ref. [22]. The controlled trajectory follows the reference trajectory after a short period of time as shown in Fig. 5. In Ref. [22], a nonlinear control strategy is used for tracking the ground-track with a terminal error less than 1 n.m. (1.85 km), with only one variable perturbed at a time. Our control strategy is able to track the ground-track more accurately without exceeding 1 km terminal error in ground-track (Fig. 6), with simultaneous perturbations in all state variables. The difference in the perturbations and

One state variable perturbed at δRc < 1.85 a time (Ref. [22]) 2 All state variables perturbed δRC < 0.83 simultaneously (present method) For both the cases δx = [304.8 m, 0.1 deg, 0.1 deg, 30.48 m/s, 0.1 deg, 0.1 deg]

final ground-track error tolerance of Ref. [22] and the present method is highlighted in Table 5. The heating rate for the Model II is calculated using the empirical expression [25] √ Qh = 1.1969 × 104 ρV 3.15 (32) In Fig. 7, it is demonstrated that the present strategy performs well even in the presence of much larger deviations in the initial conditions (Case II of Table 1) than those previously reported [19,22]. (In Ref. [19], a sliding mode control is used for δX = ±28 km, while Ref. [22] considers δX = ±11 km; we reiterate that the present method can handle δX = ±50 km). Table 6 lists the required values of td for various down-range perturbations. The value of td is chosen such that the terminal ground-track error is less than 1 km which can be observed in Table 6. This data is used for obtaining the empirical relations of Eqs. (30) and (31). It is also evident in Table 6 that by using only the LQ technique (no down-range maneuvering) results terminal ground-track errors several orders of

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Fig. 6. Range errors and aerothermal loads for Model II (Case I).

Table 6 Case

1 2 3 4 5 6 7 8 9 10

Perturbation in longitude δθ (deg)

No down-range down-range maneuvering δRc (km)

Duration of down-range maneuvering, td (s)

With maneuvering, δRc (km)

+0.10 +0.20 +0.30 +0.40 +0.45 −0.10 −0.20 −0.30 −0.40 −0.45

11.1809 22.3235 33.4852 44.6278 50.1959 11.1108 22.2598 33.4088 44.5450 50.1386

31.0 18.0 14.0 13.0 13.0 32.0 33.0 33.5 34.0 34.5

0.2420 0.2361 0.2927 0.3531 0.5879 0.1741 0.1446 0.2262 0.4134 0.4397

magnitude larger than those possible with down-range maneuvering. The necessary control inputs are plotted in Fig. 7, which shows that the upper limit on CLmax is not exceeded, while the aerothermal loads are shown in Fig. 8, illustrating a slight increase in the peak loads, which are still within allowable limits. A reduced-order linear Kalman filter is designed, based upon the measurement of longitude, latitude and speed, thus avoiding the requirement of measuring the states such as the flight path angle and heading angle, which add to onboard instrumentation requirements. Table 7 shows the terminal state errors and the maximum aerothermal loads for the +50 km deviation in the down-range. It can be observed that the terminal errors increase slightly with Kalman filter, but remain quite small. Fig. 9 compares the altitude profile, ground-track, and con-

Table 7 Terminal state errors and maximum aerothermal loads for Model II (+50 km down-range deviation) for full-state feedback and with Kalman filter Errors at terminal time and maximum aerothermal loads

Full-state feedback

With Kalman filter

δRc (km) δr (m) δV (m/s) δγ (deg) δψ (deg) Qhmax (W/m2 ) qhmax (N/m2 ) nhmax (g)

0.5879 0.2627 24.763 1.3357 0.08067 1.1316 × 1014 1.2569 × 104 1.7958

0.6242 −1.2350 2.8008 −2.7432 −9.5862 1.1595 × 1014 1.1177 × 104 1.7169

trol inputs of the full-state feedback with those of the Kalman filter. Fig. 10 demonstrates that the peak heating rate and maximum load-factor with Kalman filter are within the allowable limits. 5. Conclusions The re-entry trajectory control using a combined strategy of adaptive down-range maneuvering and linear, quadratic optimal control has been successfully applied to two lifting entry vehicles. The new strategy is shown to compensate for much larger deviations from a nominal trajectory, when compared to a nonlinear tracking control derived using feedback linearization. Furthermore, it is shown that the control-law can be combined with a linear Kalman filter based upon the measurement of longitude, latitude and speed. Careful simulations are neces-

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Fig. 7. Altitude, ground-track, and control inputs for Model II (Case II).

Fig. 8. Range errors and aerothermal loads for Model II (Case II).

sary to derive the empirical polynomial for the time of adaptive down-range maneuvering and the linear, quadratic gains from the largest expected down-range deviation, as well as aerothermal load limits. However, for a given nominal entry condi-

tion, the derived polynomial constants – as well as the linear, quadratic gains – can be easily stored and programmed into a flight computer. Thus, it is demonstrated that a simple control concept is sufficiently accurate for re-entry trajectory control

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M. Sudhir, A. Tewari / Aerospace Science and Technology 11 (2007) 419–431

Fig. 9. Altitude, ground-track, and control inputs for Model II (δX = +50 km) with Kalman filter.

Fig. 10. Range errors and aerothermal loads for Model II (δX = +50 km) with Kalman filter.

in the presence of moderate deviations in entry conditions, and can be directly implemented in actual entry vehicles. Scope for future developments includes more realistic simulations using

sophisticated aerothermal models, which allow the variation of aerodynamic force coefficients and rate of heat-transfer with Knudsen number (altitude).

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