Ascent Phase Trajectory Optimization for a Hypersonic Vehicle Using Nonlinear Programming

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1 Ascent Phase Trajectory Optimization for a Hypersonic Vehicle Using Nonlinear Programming H.M. Prasanna,D.Ghose, M.S. Bhat, C. Bhattacharyya, and J. Umakant 3 Department of Aerospace Engineering Department of Computer Science and Automation 3 Aerodynamics Division, DRDL, Hyderabad, India, Indian Institute of Science, Bangalore 56, India prasannahm@rediffmail.com, {dghose, msbdcl}@aero.iisc.ernet.in, chiru@csa.iisc.ernet.in, umakantj@yahoo.com Abstract. In this paper we present a nonlinear programming solution to one of the most challenging problems in trajectory optimization. Unlike most aerospace trajectory optimization problems the ascent phase of a hypersonic vehicle has to undergo large changes in altitude and associated aerodynamic conditions. As a result, its aerodynamic characteristics, as well as its propulsion parameters, undergo drastic changes. Further, the data available through wind tunnel tests are not always smooth. The challenge in solving such problems lies both in the preprocessing of the data as well as in the judicious use of optimization techniques. In this paper we advocate approximation of the infinite dimensional optimal control problem, derived from practical considerations of a hypersonic vehicle ascending from an altitude of 6 kms to an altitude of 3 kms with specified mach numbers, into a set of finite dimensional nonlinear programming problems. This finite dimensional approximation is shown to produce acceptable optimized results in terms of angle-of-attack control histories and state behaviour. A modification, that exploits the ultimate scheme of linear interpolation of the optimal discrete history, is proposed and is shown to produce accurate results with smaller number of grid points. Introduction Ascent phase trajectory optimization of hypersonic vehicles is one of the most challenging real-life optimization problems []. Numerical algorithms for these problems are difficult to develop due to the inherent nature of the problem []. Our paper is an effort in this direction. Numerical solution of optimal control problems has two important components: optimization and numerical integration of differential equations [3, 4]. A naive approach to solve such problems is a cut-and-paste strategy that uses numerical integration and optimization packages in the form shown in Figure (a). A sophisticated O. Gervasi et al. (Eds.): ICCSA 5, LNCS 3483, pp , 5. c Springer-Verlag Berlin Heidelberg 5

2 Ascent Phase Trajectory Optimization 549 u u Integration package u J Optimization package J x~ i+ x i t t i i+ x~ i+ x i+ x i+ t i+ (a) (b) Fig.. (a) The naive cut-and-paste method (b) Direct transcription method approach exploits the intimate relation that exists between the methods used for solving differential equations and the methods used for optimization [3]. In this paper we will describe such an approach based on nonlinear programming (NLP) and illustrate the method with an example of optimization of the trajectory for an experimental hypersonic vehicle during its ascent phase. The basic approach in solving a trajectory optimization problem using NLP is to represent an infinite dimensional optimal control problem by one or more simpler finite dimensional subproblems. While using this technique, several practical issues arise that delineate these problems from textbook problems. In fact, taking care of these issues requires a sound knowledge of the physics behind the problem. Apart from the well understood issues of infeasible constraints, rank deficient constraints, constraint redundancy, and scaling, there are other issues of missing data in the aerodynamic and propulsion parameters as well as non-smoothness (which may be due to engineering errors or real) which may confuse standard spline interpolation methods. These problems require careful pre-processing of data before they can be used in the optimization process. Methods for Solving Optimal Control Problems. The Optimal Control Problem An optimal control problem may be formulated to find the control vector u(t) to minimize the performance index [3, 5] φ[y(t f ),t f ] () evaluated at the final time t f, which may be fixed or free. The dynamics of the system are given by the state equations and initial and final conditions, ẏ = f[y(t),u(t),t], χ[y(t ),u(t ),t ] χ =,ψ[y(t f ),u(t f ),t f ] ψ f = () where, y is the state vector. Also, we have state and control constraints, ξ L ξ[y(t),u(t),t] ξ U, y L y(t) y U, u L u(t) u U (3)

3 55 H.M. Prasanna et al.. Transcription Method The optimal control problem is an infinite dimensional extension of a finite dimensional NLP problem. This infinite dimensional problem can be transcribed to a finite dimensional approximation. The transcription approach treats the state and control variable at discrete times as optimization variables. The time points are defined as, t I = t <t <...<t M = t F. These points are referred to as node, mesh, or grid points. The values of the state and control variables at these intermediate steps are treated as a set of NLP variables. The differential equation is thus replaced by a finite set of constraints. The transcription method decomposes the dynamic trajectory into several phases or intervals. If there are n number of intervals, the number of grid points are n +. This method uses the value of both state and control at each grid point as NLP (nonlinear programming) variables. We may include initial or final time as one of the NLP variables. The continuity in the states across the phase boundary is achieved by forming an NLP constraint such that, the propagated value of the state from previous phase matches with the value of the state variable at current phase. In the direct transcription method the propagation is done by using algebraic formulas derived from numerical integration schemes. This makes the propagation function approximate (with increasing levels of accuracy depending on the order of the integration scheme). However, as the interval size decreases the accuracy will increase. Let the propagation scheme be given by a function P (.) so that the propagated state at t i+ be defined as x i+. Then P (x i )= x i+ (4) where, x i is the assumed state at time t i. The constraint at t i+ is, x i+ x i+ = x i+ P (x i ) = (5) Figure (b) explains the concept of NLP formulation. The choice of P is important. The control which is obtained from solving NLP problem is interpolated to make it continuous. The accuracy of discrete trajectory, obtained from solving the NLP problem, is assessed by using the interpolated control to integrate the state equations (ODEs). It is expected that as the number of intervals increases, the error between the simulation result and optimization will reduce..3 Discretization Methods Euler Method: This is a first order method and the constraints are given by, =y k+ y k h k f k c(x) (6) Trapezoidal Method: This is a second order method and the constraints are, =y k+ y k (h k /)[f k + f k+ ] c(x) (7) Hermite-Simpson (Compressed) (HSC): This is of order four and the constraints are given by, =y k+ y k (h k /6)[f k+ +4f((y k+ + y k )/+h k (f k f k+ )/8, ū k+,t k + h k /) + f k ] c(x) (8)

4 Ascent Phase Trajectory Optimization 55 The NLP variables are given as x T =(t k,y,u, ū,y,u, ū 3,...,ū M,y M,u M ). Hermite-Simpson (Separated) (HSS): The constraints are given by, =ȳ k+ (/)(y k+ + y k ) (/8)h k (f k f k+ ) (9) =y k+ y k (/6)h k (f k+ +4 f k+ + f k ) () Eqn. (9) defines the hermite interpolation for the state at the interval midpoint, while () enforces the Simpson quadrature over the interval. The NLP variables are, x T =(t k,y,u, ū,y,u, ȳ 3, ū 3,...,ȳ M, ū M,y M,u M ). In general, the duration of a phase may be variable, in which case, the set of NLP variables must be augmented to include the variable time(s) t I and/or t F. Also, we must alter the discretization such that the step size is h k = τ k (t F t I )= τ k t, where, t =(t F t I ) with constants <τ k < chosen so that the grid points are located at fixed fractions of the total phase duration. 3 Basic Models for Ascent Phase Trajectory Optimization of a Hypersonic Vehicle The experimental hypersonic vehicle is air dropped from a carrier aircraft and boosted to Mach number 3.5 at an altitude of 6 km using solid rocket motors. The boosters are separated upon achieving this condition. Then the ascent phase commences and the final condition of Mach number 6. and altitude 3 km is achieved. The problem is to optimize the ascent phase trajectory so that the least fuel is expended (so that maximum remaining fuel can be utilized during the subsequent cruise phase). The constraints and other parameters of the problem are angle of attack α 8, maximum range during ascent phase 5 km, maximum time to reach the cruise conditions 5 s, maximum longitudinal acceleration during ascent phase < 4g, maximum lateral acceleration < 5g, weight of empty vehicle= kg (excluding fuel), fuel weight = 6 kg. The aerodynamic characteristics [6] of the vehicle are usually available in tabular form as values of the normal force coefficient (c N ) and axial force coefficient (c A ) which are functions of Mach number (M) and angle of attack (α). The lift and drag coefficients are computed using the relations c L = c N cos α c A sin α and c D = c N sin α + c A cos α. The resulting variation of lift and drag coefficients with mach number and alpha are shown in Figure. Mass Flow Rate: The mass flow rate of air (ṁ air ) as function of Mach number and α is given in Table (a), for density corresponding to altitude 3.5 km. The mass flow rate of air at any altitude is, ṁ air = ρva, where, A is nozzle area, v is free stream velocity at the nozzle, and ρ is density at given altitude. At 3.5 km, let the density be ρ 3.5, speed of sound be S 3.5, and the speed corresponding to a given mach numbers M is V 3.5 = MS 3.5. We non-dimensionalize Table (a) and determine ṁ air as,

5 55 H.M. Prasanna et al..5 c L.5 α α = = 8 8 α = 6 c D α = 8 α = 6 α = 4 α = 4.5 α =.5 α = α =.4.3 α = Mach number Mach number.5 M = 3.5 M= 4. M = 5. M=6. M = M = 4. M= c L c D M = 6.5 M = 6. M = Fig.. Variation of lift and drag coefficients with α and Mach number Table. (a) Mass flow rate of air (-kg/s) corresponding to altitude 3.5 km (b) Specific impulse (I sp in seconds) M/α (a) Altitude (km) M (b) ṁ nond = ṁ air /(ρ 3.5 V 3.5 A) () ṁ }{{} air = ṁ nond ρva = ṁ }{{} air /(ρ 3.5 V 3.5 ) ρv () }{{} at given altitude at given altitude at given altitude Specific Impulse: Specific impulse is a function of free stream Mach number and altitude (Table (b)). For Mach number between 3. to 4., specific impulse

6 Ascent Phase Trajectory Optimization x Specific Impulse (s) Altitude (m) x 4 Altitude (m) Mach number (a) Temperature (k) (b) Fig. 3. (a) Specific impulse using spline interpolation (b) Temperature variation in Indian standard atmosphere is constant. So linear interpolation is used for this region and spline interpolation for the rest of the data (Figure 3(a)). Thrust: Thrust is calculated as T = I sp ṁ f g. Assuming equivalence ratio φ= and stoichiometric ratio for fuel as 5, we have ṁ f = ṁ air /5, where, ṁ air at given altitude is obtained from the procedure described above. Standard Atmosphere: The Indian standard atmosphere is assumed. The variation of temperature with altitude is shown in Figure 3(b). The atmospheric pressure at sea level is P = 5 (N/m ) and specific gas constant is R= (J/kgK). 4 Problem Formulation and Computational Results During ascent phase the objective is to minimize fuel consumption (or maximize mass at final time). Thus, the optimization problem is formulated as, max m f (3) s.t., ḣ = v sin γ, ẋ = v cos γ, ṁ = T/(I sp g) (4) v =(/m)[t cos α D] µ sin γ/(r e + h) (5) γ =(/(mv))[t sin α + L] + cos γ[v/(r e + h) µ/v(r e + h) ] (6) where, h is the altitude (m), v the velocity (m/sec), α the angle of attack (rad), γ the flight path angle (rad), m the mass (kg), µ the gravitational constant, g the gravitational acceleration, and R e the radius of the earth. The trajectory related constraints t f 5 s, x f 5 m, α 8, a long 4g, a lat 5g

7 554 H.M. Prasanna et al. are also imposed. where, a long and a lat are the longitudinal and lateral accelerations, respectively. defined as a lat =(/m)[t sin α + L] + cos γ[v /(R e + h) µ/(r e + h) ] (7) a long =(/m)[t cos α D] µ sin γ/(r e + h) (8) The lift (L) and drag (D) forces are defined by, D =(/)ρsv c D, L =(/)ρsv c L (9) where, S is the aerodynamic reference area of the vehicle, and ρ is the atmospheric density. The following constants complete the formulation of the problem: h() = 6. m, h(t f ) = 3. m, v() = m/s v(t f ) = 847. m/s, γ() =.698 rad, γ(t f )=rad m() = 36. kg, x() = m, µ = m 3 /s g =9.8 m/s, R e = m, S =m The optimization was started with 5 grid points initially and subsequently higher number of grid points (i.e., 9, 7, 33, 65) were considered. For 5 and 9 grid points, Table. Results of ascent phase trajectory optimization Grid method h f (m) v f (m/s) γ f ( o ) m f (kg) m f(fuel) x f ( 5 m) time (s) points (kg) 5 NLP Step Linear Hermite NLP Step Linear Hermite NLP Step Linear Hermite NLP Step Linear Hermite NLP Step Linear Hermite

8 Ascent Phase Trajectory Optimization 555 the NLP constraints are formed using trapezoidal discretization and for higher number of grid points Hermite-Simpson discretization is used. Each variable is scaled by its upper bound to make the ranges, over which they vary, similar in their order of magnitude. This helps better numerical conditioning of the NLP. The optimization is carried out only at the grid points and the shape of the trajectory between the grid points is not known. To assess the accuracy of the result, we integrated (4) - (6) using control histories obtained from interpolation of the NLP generated discrete control. We used three interpolation schemes (step, linear, and hermite). The initial conditions for n grid points were obtained by linearly interpolating the optimized values from n grid points. The results of optimization and integration (using different interpolation schemes for control) are given in Table. From Table, we observe that the error between the optimization and integration results are significantly large when number of grid points are few and the error reduces as the number of grid points increases. The integration using control generated by step interpolation has large error as compared to linear and hermite interpolations. The results for 33, and 65 grid points show very little improvement in the optimization results (see Figure 4). Figure 5 shows variation of altitude, velocity, flight path angle, mass, and range with time using linearly interpolated control for 65 grid points (a) (b) (c) (d) (e) (f) Fig. 4. (a) (e) Angle of attack vs. time for 5, 9, 7, 33, and 65 grid points in the original NLP (f) Angle of attack vs. time for 65 grid points with modified method

9 556 H.M. Prasanna et al x Altitude (m) Velocity (m/s) x 5 Flight path angle ( γ in deg) 6 4 Overall mass (kg) Horizontal range (m) Fig. 5. State histories (for linearly interpolated control from 65 grid points) 5 Improving Computational Accuracy Since the final step in the transcription method is implemented by linearly interpolating the control history between grid points, we use this knowledge by assuming a linear dynamics for control variable. So, we assume, α = β () Table 3. Results of ascent phase trajectory optimization (with modified method) Grid method h f (m) v f (m/s) γ f ( o ) m f (kg) m f(fuel) x f ( 5 m) time (s) points (kg) 5 NLP Linear NLP Linear NLP Linear NLP Linear NLP Linear e

10 Ascent Phase Trajectory Optimization 557 where, β is now used as the control variable. Then, the equations of motion will include () in addition to (4) - (6). The set of NLP variables will consequently include an additional variable β at each grid point. With this modification, the results of optimization and integration are given in Table 3. Figure 4(f) shows variation of angle of attack against time for 65 grid points. Comparing with the α history in the earlier scheme we see a flat region between 7 and 9 seconds in the new scheme. The modified method also gives better accuracy. In fact it does so for just 33 grid points, whereas the original scheme produces similar accurate results at 65 grid points. However, the additional NLP variable and constraint increases the computational time slightly. 6 Conclusions In this paper, we addressed the challenging application problem of ascent phase trajectory optimization of a hypersonic vehicle. It was shown that the linear interpolation scheme used to create the continuous control history for verification can itself be exploited while forming the dynamical equations to get faster convergence and higher accuracy. Acknowledgements. This work was partially supported by a grant from DRDL. References. P.F. Gath and A.J. Calise, Optimization of Launch Vehicle Ascent Trajectories with Path Constraints and Coast Arcs, Journal of Guidance, Control, and Dynamics, Vol. 4,, pp J.T. Betts, Survey of Numerical Methods for Trajectory Optimization, Journal of Guidance, Control, and Dynamics, Vol., 998, pp J.T. Betts, Practical Methods for Optimal Control Using Nonlinear Programming, The Society for Industrial and Applied Mathematics, Philadelphia,. 4. E. Gill, Murray, and H. Wright, Practical Optimization, Academic Press, A.E. Bryson, and Y.-C. Ho, Applied Optimal Control, John Wiley, New York, J.D. Anderson, Introduction to Flight, McGraw-Hill, Singapore,.