Near optimal guidance law for descent to a point using inverse problem approach

Aerospace Science and Technology 12 (2008) 241–247 www.elsevier.com/locate/aescte

Near optimal guidance law for descent to a point using inverse problem approach Abolghasem Naghash, Reza Esmaelzadeh ∗,1 , Mehdi Mortazavi, Reza Jamilnia Aerospace Engineering Department, Amirkabir University of Technology, Tehran, Iran Received 27 April 2006; received in revised form 12 June 2007; accepted 12 June 2007 Available online 28 June 2007

Abstract An explicit guidance law that maximizes terminal velocity is developed for a reentry vehicle to a fixed target. Motion is constrained to an optimal, three-dimensional Bezier curve. Acceleration commands are derived by solving an inverse problem related to Bezier parameters. An optimal Bezier curve is determined by solving a real-coded genetic algorithm. For online trajectory generation, optimal trajectory is approximated by fixing the second control point of the Bezier curve. The approximated trajectory is compared with the pure proportional navigation, genetic algorithm and direct transcription’s solutions. The near optimal terminal velocity solution compares very well with these solutions. The approach robustness is examined by Monte Carlo simulation. © 2007 Elsevier Masson SAS. All rights reserved. Keywords: Explicit guidance; Optimal guidance; Bezier curve; Genetic algorithm; Reentry

1. Introduction Generally, the design of optimal guidance algorithms may be defined loosely as the art of finding the correct acceleration commands to move between two given points that maximizes a performance index while satisfying some constraints. Many different techniques have been suggested for the design of guidance algorithms. These range from the earliest algorithms derived using physical insight, for example, pursuit, proportional navigation (PN), and their variants to those derived from a systematic application of mathematical techniques. Most current guidance design methods may be classified into two main categories [13] (1) nominal trajectory based techniques and (2) on-line trajectory generation, reshaping and prediction schemes. In the first approach, an (optimal) reference trajectory is defined prior to the mission, and during the flight, controller keeps the vehicle close to the nominal trajectory. The predictive and/or reshaping approaches propagate the future trajectory based on current flight state by means of onboard numerical * Corresponding author. Tel.: +98 21 77467440; fax: +98 21 66404885.

E-mail address: [email protected] (R. Esmaelzadeh). 1 PhD Candidate.

1270-9638/$ – see front matter © 2007 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.ast.2007.06.006

integration to calculate the control input during the remaining flight. This technique may be combined with optimization algorithms to meet performance criteria. A combination of both techniques has also been attempted. The general solution of the optimal flight of a reentry vehicle (RV) to an arbitrary, but specified, final condition has been of interest for some time. Contensou was the first to consider optimal control and proposed the problem for unconstrained range in terms of flight path angle as the independent variable [5]. Eisler and Hull [9], using neighboring optimal control, devised a sampled-data feedback control method to obtain unconstrained, approximate, maximum-terminal-velocity descent trajectories at a designated target. This maximization has the dual advantage of reducing the time to reach the target as well as maximizing the kinetic energy level. In this paper, inverse problem approach is used to develop an explicit guidance law for guiding a hypersonic untrusted RV to a fixed point on the ground (Eisler’s problem [9]). This approach, which uses presetting of external trajectories, gives a huge calculation advantage and can provide a near optimal solution. This is applicable to systems that are linear in the acceleration and aims at constructing guidance algorithms with specified desired dynamics, i.e. solving an inverse problem. In

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an inverse problem, we are asked to predict the controls that are compatible with a desired trajectory [3]. Inverse methods are of great interest in the context of synthesizing nonlinear autopilots [16,17,26] and guidance algorithms [14,18,19,21,29]. A survey about inverse problem approach in optimal trajectory generation, both in Russia and in the United States, can be found in Yakimenko’s paper [31]. In guidance applications, the variable guidance gains are correlated with the shape of the trajectory that will follow and the satisfaction of particular terminal constraints. Although, with an extension of Cameron [4], Page [23], and Taranenko’s [29] methods, the use of this approach in the guidance algorithms design has been developed by Hough [14] and Yakimenko [31], it still suffers from serious flaws: relatively many optimization parameters (OPs) (Taranenko: 20, Mortazavi [22]: 12, Hough: 8), depending on the vehicle’s velocity vector, relatively difficult numerical calculations, accuracy dependence on the number of segments used in the approximation, and offline application, that make it impossible for using in this problem. The present paper deals with a new, in some sense, simplified method that provides near optimal spatial trajectories being presented analytically and completely defined by minimum OPs. This method combines a number of advantages over methods presented by Taranenko, and Hough. Although guidance law is designed for an RV, it can also be applied to any vehicle at any phase. The remainder of this paper is organized as follows. The optimal guidance problem as well as RV’s dynamics is described in Section 2. Section 3 introduces the computational algorithm, and Section 4 deals with simulation results.

Fig. 1. Geometry of problem.

index J = −Vf , subject to the state equations (1); known initial conditions, V0 , γ0 , ψ0 , ξ0 , η0 , and h0 ; and known final conditions, ξf , ηf , and hf (equivalent to fixed target position). The solution must satisfy the following in-flight constraint:  2 a (2) ac = a 2vc + ahc max The amax can be related to the limitations of angle of attack, loading, etc. 3. Computational algorithm 3.1. Inverse problem approach

2. Problem definition Assuming a spherical, nonrotating Earth and a gravitational field with g = μ/r 2 , we have three-dimensional point mass equations of motion for the RV (Fig. 1): V˙ = (mg sin γ − D)/m   γ˙ = avc + (g − V 2 /r) cos γ /V ψ˙ = ahc /(V cos γ )

ahc = −V 2 cos2 γ cos ψψ 

tan γ = cos ψ dh/dξ (1)

.

where () = d()/dt, V is the velocity, γ is the flight-path angle, ψ is the heading angle, ξ is the range, η is the cross range, h is the altitude, m is the mass, r is the radius from the center of the Earth to the vehicle, μ is gravitational parameter, and avc , ahc are the vertical and horizontal acceleration commands, respectively. For bank-to-turn control configuration (BTT): avc = L cos σ/m,

(3)

where primes denote differentiation with respect to ξ . On the other hand, with geometrical considerations, flight path γ and heading ψ angles are obtained:

ξ˙ = −V cos γ cos ψ η˙ = −V cos γ sin ψ h˙ = −V sin γ

To apply the concept of inverse problem approach, we first change the independent variable from t to ξ in the system equations (1). After that, we solve for acceleration commands:   avc = V 2 /r − g cos γ − V 2 cos γ cos ψγ 

ahc = L sin σ/m

where L is aerodynamic lift, and σ is bank angle. The optimal guidance problem is to find acceleration commands (or equivalently angle of attack (α), and σ for BTT), which maximize the final velocity or minimize the performance

tan ψ = dη/dξ

(4)

In Eq. (3), the desired trajectory shape enters through the curvature terms γ  and ψ  , obtained by implicit differentiation of Eq. (4) with respect to ξ : γ  = (h cos ψ − h ψ  sin ψ) cos2 γ ψ  = cos2 ψη

(5)

These functions introduce second derivative terms h and η . Therefore, the guidance commands are related to the shape of trajectory. Admissible trajectory must satisfy relation (2). Actual acceleration a lags the acceleration command a c , whose components are specified by Eqs. (3) and (5). For three

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degree-of-freedom (3DOF) simulations, noninstantaneous response could be modeled by a first order lag:

η1 = η0 + λ tan ψ0 /3

1 da 1 + a = ac dt κ κ where the time constant κ approximates the dominant closed loop pole of rigid body model, autopilot and actuator. In the sequel, instantaneous response (κ → 0) will be assumed, and it follows that the acceleration commands of Eqs. (3) are the actual acceleration components (a = a c ).

where λ = (ξf − ξ0 ). The second derivatives in the beginning of trajectory may be written as: d2 η 6(η2 − 2η1 + η0 ) = 2 dξ 0 (ξf − ξ0 )2 d2 h 6(h2 − 2h1 + h0 ) = (10) dξ 2 0 (ξf − ξ0 )2 as:

Many methods have been used for trajectory generation [14,15,27,31]; all of them have many parameters, and need specific conditions. In this paper, the Bezier curve [25] is suggested for trajectory generation. This curve, for its properties, has been used in various fields of studies, such as computer graphics, robotic guidance, airfoil design. Desideri and et al. [8] applied this curve to trajectory optimization in a planar case. Mathematically, a parametric Bezier curve of order n is defined by n 

(6)

Bi Jn,i (τ )

i=0

ahc = f1 (η2 , η1 , η0 , X0 ) avc = f2 (h2 , h1 , h0 , η2 , η1 , η0 , X0 )

f1 = −6V02 cos2 γ0 cos3 ψ0 (η2 − 2η1 + η0 )/λ2   f2 = V02 /r0 − g0 cos γ0 − V02 cos3 γ0 cos2 ψ0  × 6(h2 − 2h1 + h0 ) − 9 sin 2ψ0 (η2 − 2η1  + η0 )(h1 − h0 )/λ /λ2 and X0 is navigation quantities at the beginning of flight. We know, |ahc |  amax

and τ denotes the parameter of the curve taking values in [0, 1], Bi are the control points. So, as seen from Eq. (6) the Bezier curve is completely determined by Cartesian coordinates of the control points. The derivative of order r of a Bezier curve can be derived as:

g1 (η1 , η0 , X0 )  η2  g2 (η1 , η0 , X0 )

dr n!  r P (τ ) =

Bi Jn−r,i r dτ (n − r)!

therefore from Eqs. (3), (5), and (8), we get:

(7)

(12)

where g1 = −

n−r

g2 =

i=0

amax λ2 + 2η1 − η0 6V02 cos2 γ0 cos3 ψ0

amax λ2 + 2η1 − η0 6V02 cos2 γ0 cos3 ψ0

With a value in this bound, ahc may be determined, then from Eqs. (2) and (11): g3 (h1 , h0 , η2 , η1 , η0 , X0 )  h2

for k = 0, . . . , r

 g4 (h1 , h0 , η2 , η1 , η0 , X0 )

It is clear that the derivative of order r of a Bezier curve at one of its end points only depends on the r + 1 control points nearest (and including) that end point. It follows that, at τ = 0

where

P  (0) = n(B1 − B0 )

g3 =



(11)

where

where the Bezier or Bernstein basis or blending function is   i i i Jn,i (τ ) = CN τ (1 − τ )n−i , CN = n!/ i!(n − i)!

where, for i = 0, . . . , n,  0

Bi = Bi ,

k Bi = k−1 Bi+1 − k−1 Bi

(9)

Using Eqs. (9), (10), and (5), then Eq. (3) may be rewritten

3.2. Trajectory generation

P (τ ) =

h1 = h0 + λ tan γ0 sec ψ0 /3

P (0) = n(n − 1)(B2 − 2B1 + B0 )



− (a 2 − a 2 ) + (V 2 /r − g ) cos γ 0 0 0 max hc 0 V02 cos3 γ0 cos2 ψ0

(8)

For this problem, the parameter τ is equal to the normalized range ξ¯ = (ξ − ξ0 )/(ξf − ξ0 ) and Bezier approximation of the trajectory is determined by coordinates (hi , ηi ) of the control points Bi . With allowable assumption n = 3 for reentry trajectories, the first point B0 = (h0 , η0 ) and the last point B3 = (hf , ηf ) will be fixed. Now we have to determine the middle control points B1 = (h1 , η1 ) and B2 = (h2 , η2 ). In the beginning of trajectory, the second control point, B1 , can be set by using Eqs. (4) and (8):

+ 9 sin 2ψ0 (η2 − 2η1 + η0 )(h1 − h0 )/λ

g4 =

(13)

 3

λ2 6

+ 2h1 − h0 

(a 2 − a 2 ) + (V 2 /r − g ) cos γ 0 0 0 max hc 0 V02 cos3 γ0 cos2 ψ0 + 9 sin 2ψ0 (η2 − 2η1 + η0 )(h1 − h0 )/λ3 + 2h1 − h0



λ2 6

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By selecting the second control point, B2 , the initial trajectory will be generated, and the RV will follow it so long as satisfying the constraints. When the acceleration commands of Eq. (6) exceed maximum allowed acceleration, acceleration command saturation causes the actual value of h, η, γ , and ψ to deviate from the desired values along the initial trajectory. Holding the terminal conditions fixed, Bezier control points should be continuously updated with instantaneous λ values. Using C0 (position), C1 (angle), and C2 (acceleration) continuity conditions, the new trajectory’s control points can be obtained automatically. These conditions imply the position, angles, and acceleration at the beginning of the new trajectory must be equal to position, angles, and acceleration at the end of previous trajectory. i.e.: C0 :

hnew0 = holdf ,

ηnew0 = ηoldf

C1 :

hnew0

= holdf ,

  ηnew = ηold 0 f

C2 :

hnew0 = holdf ,

  ηnew = ηold 0 f

51 discrete nodes with their corresponding times, states, and controls. 4. Simulation results and discussion To demonstrate the effectiveness of this guidance law, it has been used in a 3DOF (point mass) simulation containing a standard atmosphere, and aerodynamics coefficients as functions of Mach number, angle of attack, and Reynolds number in tabulated form for an RV model based on Ref. [20]. This model is an aeroshell of sphere-cone-cylinder-flare configuration with the maximum lift to drag ratio in the order of 1.7, mass of 1500 kg, reference area of 1.77 m2 , and the ballistic coefficient m/(CD S) = 7349 kg/m2 . Results are shown in Figs. 2–7 for a sample period of t = 0.01 s. It is understood that the optimum selection of B2 via RGA approach requires a very large computer resource. For this reason, we have used the Near Optimal Inverse Problem Approach

Therefore, for guiding the RV, the only necessary task is to select the second control point, B2 , for initial Bezier trajectory. It must be noted that all choices in the bounds of Eqs. (12), and (13) guarantee that the RV reaches the target while satisfying the constraints. 3.3. Optimal solution For this problem, we seek optimum selection of B2 to generate optimal trajectory which leads to maximum impact velocity. It is clear that we face a parameter optimization problem. Many different methods have been suggested to solve these problems, especially in space trajectories applications, that can be found in Betts’s excellent survey [2]. The use of genetic algorithms (GAs) to determine optimal space trajectories has recently become popular. The applications range from trajectory planning for launch vehicles to the trajectory design of interplanetary missions [6,8,30,32]. In this paper, a form of GA, known as floating-point or realcoded GA (RGA), is used [7,12,24]. The RGA used in this study is similar to that described in [11], and simulated with the Genetic Algorithm and Direct Search Toolbox V.2 (with some modifications) in MATLAB 7.3. Stochastic uniform selection with elitism, scattered crossover with 0.8 probability, uniform mutation with 0.1 probability, population size 100, and 50 generation for termination were used. The calculations were repeated several times using different seeds to check the repeatability of the optimal parameters. On the other hand, for comparison, this problem was solved by direct transcription method (DTM) [1,2] as well. The basic idea behind direct transcription involves discretizing the state and control representation of a continuous Optimal Control Problem (OCP). This technique allows the OCP to be transcribed into a Nonlinear Programming Problem (NLP). The OCP can be thought of as an NLP with an infinite number of controls and constraints. For this problem, trajectory is divided into 50 segments. These 50 segments can be represented by

Fig. 2. Altitude comparison.

Fig. 3. Cross range comparison.

A. Naghash et al. / Aerospace Science and Technology 12 (2008) 241–247

Fig. 4. Total acceleration command comparison.

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Fig. 6. Bank angle comparison. Table 1 Trajectory boundary conditions

ξ km η km H km V m s−1 γ deg ψ deg

Fig. 5. Angle of attack comparison.

(NOIPA). As first seen by Eisler and et al. [10] and can be seen in Fig. 2 by RGA approach, the optimal trajectory steers the RV to fly at higher altitudes as far as possible. This is equal to h2 = g4 , i.e. avc = amax = 3.5 g and selection of η2 in such a manner that ahc = 0 in the beginning of flight (for BTT it is equal to Lc = Lmax , and σ = 180◦ ). Also shown in these figures are the optimal trajectory (RGA) and the trajectory obtained from pure proportional navigation (PPN) with N = 3 [28]. A fourth-order, fixed step, Runge– Kutta integrator is used in all simulations. The trajectory boundary conditions for the example problem are shown in Table 1. Figs. 2 and 3 display the simulation flight path profiles. NOIPA and RGA have the same vertical path, but slightly different horizontal paths, whereas PPN, which turns quickly to line up with the target. As just said, the DTM, RGA, and NOIPA simulations shift the majority of flight time to the higher altitudes, where drag is low. Since DTM is not limited to a cubic polynomial, using maximum allowable acceleration command

Initial

Final

80 2 30 4000 20 0

0 0 0 Maximum Unconstrained Unconstrained

(Fig. 4), RV has flown at highest altitude. However, using saturated command is not proposed. For BTT, the required angle of attack for Lc and orientation of Lc (bank angle) are shown in Figs. 5 and 6. The control angles produced by DTM, RGA and NOIPA, are the same although DTM has slight differences due to saturated acceleration command. These differences have a very small effect on velocity profile (Fig. 7). The acceleration command profiles shown in Fig. 4 reconfirm the turning rates between PPN and the other guidance schemes. This figure displays a good agreement in total acceleration command profiles for RGA and NOIPA schemes. Velocity-range profiles in Fig. 7 are grouped in a similar fashion to the flight profiles. Because of the altitude management in the DTM, NOIPA and RGA guidance, the terminal velocities show small differences. The optimal velocity produced by DTM scheme is 1710 m s−1 . The terminal velocity generated by RGA and NOIPA methods differs from it by about 0.3% (1620 and 1625 m s−1 , respectively), while the PPN solution is over 13% less (1410 m s−1 ). Note also that the times of flight for RGA and NOIPA schemes are almost identical. Uncertainties or off-nominal conditions in models characteristics have a profound effect on a guidance performance. The overall goal of guidance algorithm is the successful steering of the vehicle in spite of these uncertainties. The more important are the entry conditions, the aerodynamics characteristics, and the atmospheric density. A Monte Carlo analysis was performed by introducing variations in vehicle aerodynamic coefficients

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absence of “wild” trajectories during optimization; (3) an analytical (parametric) representation of reference trajectory with minimum optimization parameters; (4) applicable to any RV configuration, regardless of its lift-to-drag ratio (L/D) or range of flight Mach number regime; (5) applicable to any control schemes (bank-to-turn or skid-to-turn), and (6) offline nominal trajectory and Time-to-Go independence. Comparison with the direct transcription method for the terminal velocity is excellent and far exceeds the pure proportional navigation solution. References

Fig. 7. Velocity comparison.

(±10%), atmospheric density (±10%), and entry conditions (δξ = δη = δh = ±1000 m, δψ = ±1 deg, δV = ±10 m/s). Also the following wind profile was considered: ⎧ V1 ± δVw1 , h  hw1 ⎪ ⎪ ⎪ ⎨ (V ± δV ) + (V2 ±δVw2 )−(V1 ±δVw1 ) 1 w1 (hw2 −hw1 ) Vw = ⎪ × (h − h ) h < h < hw2 w1 w1 ⎪ ⎪ ⎩ V2 ± δVw2 , h  hw2 where V1 = 15 m/s, V2 = 20 m/s, δVw1 = δVw2 = ±5 m/s, hw1 = 10 km, hw2 = 20 km. Finally, with a uniform distribution in uncertainties, 200 cases were conducted to assess the robustness of the proposed guidance approach. The mean value and standard deviation resulted in the range were mr = −13.65 m and δr = 4.24 m. These values were mc = −0.21 m and δc = 0.32 m in the cross range. The scheme provides robust properties as indicated by results of Monte Carlo flight simulations. This robustness is the characteristic of closed loop explicit guidance laws. 5. Concluding remarks A near optimal explicit guidance method is devised to obtain approximate, maximum-terminal velocity descent trajectories at prescribed destination by inverse problem approach. The guidance commands are related to shape of trajectory, specified by Bezier curve, to minimize the time spent in the denser parts of the atmosphere. During periods of command saturation, the instantaneous Bezier control points vary until sufficient control is available to follow optimal trajectory. Optimal Bezier control points can be determined by using any parameter optimization method, such as RGA, implemented by NOIPA. For comparison, this trajectory was optimized by a direct numerical optimization method, DTM as well. The robustness of the algorithm in the face of worst case perturbations in the entry conditions, the aerodynamics characteristics, and the atmospheric density was examined by Monte Carlo simulation. The proposed method is characterized by the following advantages: (1) a priori satisfaction of the boundary conditions; (2) an

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