The Ascent Trajectory Optimization of Two-Stage-To-Orbit Aerospace Plane Based on Pseudospectral Method

Transcription

1 Available online at ScienceDirect Procedia Engineering 00 (014) APISAT014, 014 Asia-Paciic International Symposium on Aerospace Technology, APISAT014 The Ascent Trajectory Optimization o Two-Stage-To-Orbit Aerospace Plane Based on Pseudospectral Method Jing Song a *, Haoqin Su a a China Academy o Aerospace Aerodynamic, Beijing, ,P.R.China Abstract Aerospace plane can ly in the Earth s atmosphere with hypersonic and run in the low earth orbit. According to the tactical and technical index, by adjusting power, control, pneumatic components relating to trajectory optimization, we get the optimal ascent trajectory o minimal uel consumption at a speciic separation state (height 30km, Mach 6) with the Radau pseudospectral method. The terminal state can accurately meet the requirements o the ellipse orbit o height 300km. On the basis o optimal trajectory we need ind the best separation condition. The relationship between the separation height, Mach number and uel consumption is studied. The relationship between the light path angle and the uel consumption o the speciic separation states is also researched. 014 The Authors. Published by Elsevier Ltd. Peer-review under responsibility o Chinese Society o Aeronautics and Astronautics (CSAA). Keywords: Aerospace plane; Pseudospectral method; minimal uel consumption; seperation states 1. Introduction Aerospace plane can ly in the Earth s atmosphere with hypersonic and run in the low earth orbit. Aerospace plane which can take o and land horizontally, and climb two stages to orbit is researched. Its light process is as ollows: The combination body o carrier at the irst stage and orbiter at the second stage takes o at ground. When the aerospace plane climbs to a certain altitude and reaches a certain speed, the carrier and the orbiter separate. Then the carrier saely returns to the ground and the orbiter quickly climbs into the low earth orbit by the thrust vector * Corresponding author. Tel.: ; ax: address: songjing1a@163.com The Authors. Published by Elsevier Ltd. Peer-review under responsibility o Chinese Society o Aeronautics and Astronautics (CSAA).

2 Jing Song,Haoqin Su / Procedia Engineering 00 (014) control. The orbiter, with the ability to maneuver in orbit, changes its orbit by using power device and releases the eective payloads. When needed, the orbiter returns to the atmosphere to carry out combat missions. Finally the orbiter returns saely by unpowered gliding or powered cruise. Spacecrat trajectory optimization is based on the light conditions and technical indexes to ind out a light trajectory which can make a certain perormance index optimal and satisy various constraints. Two-stage-to-orbit aerospace plane trajectory optimal design is a class o ree terminal time, ixed terminal states, multiple constraints, and multiple stages, nonlinear optimal control problem. In general, numerical method or solving optimal control problem can be divided into direct and indirect method. When indirect method used to dealing with complex problems with multiple constraints and phases, it is very complicated and has small convergence domain, and it is diicult to estimate the initial values o conjugate variables. The direct method has been widely applied in the trajectory optimization ield because the direct method overcomes many shortcomings o the indirect method. In recent years, the pseudospectral method, one o direct method, has become a hot spot or solving trajectory optimization problems due to its less discrete points and higher accuracy. The Radau pseudospectral method adopts global polynomial interpolation on a series o LGR (Legendre-Gauss-Radau) points to approximate state variables and control variables. In addition, the transorm o optimal control problem by pseudospectral method meets the co state mapping theorem. The KKT(Karush-Kuhn-Tucher) conditions o LP(nonlinear program) problem rom pseudospectral method agrees with the irst-order optimal necessary conditions o the optimal control problems. The stage separation technology o two-stage-to-orbit aerospace plane ascent, as a key technique o multi-stage reusable spacecrat, has become one o the urgent key technologies o aerospace plane s research or developing country [1]. The separation o carrier and orbiter in the atmosphere is a complex and challenging work, separation conditions solution is essential. In the climbing process, the separation in good condition not only can minimize take-o weight, reduce costs and requirements o various subsystems, but also provide the necessary data or subsystems design. The good separation condition is an important part o the system design o vehicle. Optimal control problem ormulation.1 Equation o motion Among the processing o the aerospace plane climbing into orbit, the carrier is controlled by the pneumatic rudder and the orbiter is controlled by the thrust vector. Although aerodynamic action is weak in the upper atmosphere, we also consider the aerodynamic eect. Thereore, we assume that the angle o attack is always kept in the value where the lit to drag ratio is maximum. In order to acilitate the solution, we only consider the longitudinal light and have ollowing assumptions:1)no considering the earth rotation, )no considering the longitude and latitude. In the geocentric inertial coordinate, the dynamic model is modeled using (1). h vsin Tcos T D sin v m r Tsin T L v cos mv r vr T m gi 0 sp where, h is the altitude, v is the velocity, γ is the light path angle, m is the mass, μ is the gravitational constant, L and D are lit and drag, T is the thrust, I sp is the engine speciic, g 0 is the gravitational acceleration, α is the angle o attack, η is the thrust regulation, T is the thrust vector angle. At the irst stage the vehicle is lit by aerodynamic orce, where T 0. At the second stage the vehicle is controlled by the thrust vector, where the aerodynamic eect is gradually weaker and the angle o attack is ixed on the value o the maximum lit to drag ration(about α=5 ). The states variables x h, v,, m, the control variables u, at the irst stage or the carrier, the control u at the second stage or the orbiter. variables T, (1)

3 number o nodes Jing Song, Haoqin Su/ Procedia Engineering 00 (014) Objective unction The uel reduction can make the plane easier to reach to a predetermined height and reduce the cost. The problem we study is to ind the optimal control variables to make the uel consumption minimum. The objective unction o this optimization problem is J m min () 3 The Radau Pseudospectral Method The Radau pseudospectral method discretizes the states and controls on a series o Legendre-Gauss-Radau(LGR) points. Then construct the Lagrange interpolation polynomial to approximate the state and control variables. The polynomial derivative equals to the state variables derivative, so that the dierential equation constraints are converted into a set o algebraic constraints. Ater above transormation, the optimal problem is transormed into a parameter optimization problem with a series o algebraic constrains. Finally, numerical optimal solution can be obtained by the constrained nonlinear programming algorithm []. For the trajectory optimization problem, it is ormulated over the time interval t, 0 t. The Radau pseudospectral transcription need change the time interval o optimal control problem rom t, 0 t to [-1,1]. This is done using the mapping t t t0 (3) t t t t 0 0 The Radau pseudospectral method is based on interpolating unctions on the LGR quadrature nodes which are distributed over the interval [-1,1) or (-1,1], namely the roots o the polynomial Pk Pk 1 or Pk Pk 1. Usually we called the irst one containing -1 standard LGR(Figure 1) and another one lipped LGR. In this paper, we adopt standard LGR LGR Fig 1 Location o Standard LGR State variables and control variables are approximated by using the Lagrange interpolation polynomial, x X X L (4) i0 * i i i i (5) i1 u U U L i Where, L i 0,1,, is a basis o Lagrange interpolating polynomials. The derivative o (4) can be represented as ollows i i (6) i0 x X X L Thus, dierential equation constraints o aircrat motion equations on locations can be transormed into algebraic equation constraints t t0 Dki X i X k, U k, k 0 (7) i0

4 4 Jing Song,Haoqin Su / Procedia Engineering 00 (014) At the terminal boundary the point value can be obtained by Gauss quadrature ormula t t0 X X 0 k X k, Uk, k k 1 Here, k is the Gauss weight. The cost unctional can be approximated with a Gauss quadrature, resulting in t t0 J X 0, t0, X, t k g X k, U k, k k1 Also boundary constraints and path constraints are transcribed as ollows X0, t0 0, X, t 0 C X, U, 0 k 1,, k k k (10) 4 The Optimization Results and Analysis (11) Initial conditions: m0 v0 h t 10 m / s 0m 0 Separation conditions: m1 v1 h1 1 m1 6Ma 30km 0 ; m0 v0 h t 6Ma 30km 0 Terminal conditions: m v h m 7800 m / s 300km 5 In addition, the maximum thrust o the irst state is kept. In the simulation process, the h-p adaptive inite element method [3] is used and each interval has 0 nodes. (8) (9) Fig the optimal results o the irst stage. Fig 3 the optimal results o the second stage. From igure, it can be seen ater the speeds reaches a certain value, the vehicle was o the ground. The light path angle increased, the height grew rapidly, the velocity accelerated with a smaller acceleration. It takes about 100s or the vehicle to reach at 17km. Ater that, the light path angle gradually reduced, the height growth was relatively small, and the velocity accelerated with a larger acceleration. In the climb, the vehicle, maintaining maximum thrust and 3-4 angle o attack, reached the speciied separation height and separation Mach number. From igure, we can know the climbing time 77.57s and uel consumption 75478kg.

5 m/t m/t Jing Song, Haoqin Su/ Procedia Engineering 00 (014) The optimization results o the second stage are shown in igure 3. The climb time is s and uel consumption is 11070kg. The change o height and velocity is relatively uniorm, and the light angle changes a little rom 0s to about 70s. Then, the height increased rapidly, the velocity increased slowly, and the light path angle had a large change range. The thrust vector angle increased and then quickly became smaller. The thrust regulation became small and then increased to the maximum. This is as ar as possible to increase the vertical velocity. According to igure and 3 it can be seen that the optimization results satisy the terminal conditions and path constraints. The light trajectory is very smooth. 5 The optimal separation states By numerical simulation, the overall parameters have a decisive role on the separation state and the design o vehicle. As we can see rom Figure 4 as separation Mach number increases, the uel consumption becomes greater, and the greater separation height, the uel consumption is more. This is because the vehicle takeo weight is very large, and orms the main inluence on the separation states. As we can see rom Figure 5 there is an optimum separation in the separating condition which makes the total uel consumption minimum Ma=7 Ma= Ma=5 Ma=4 Ma= h=30km Ma= h/km /deg Figure 4 the uel consumption at dierent separation height, dierent separation Mach Figure 5 separation height 30km, separation Mach 6, the eect o light path angle to uel consumption When separation Mach number and the height increase, the requirement o the overall parameters o orbiter reduces greatly. The separation Mach number and height mainly determine the dynamic system characteristics and separation point mass o the orbiter. For the orbiter, the bigger the separation Mach number and height are, the easier the orbiter design is. However, the separation Mach number and height are restricted strictly by the carrier (mainly air breathing propulsion, thermal protection system, dynamic pressure, and overload). 6 Conclusions 1) The pseudospectral method can well solve the problem o optimal control or aerospace plane ascent trajectory optimization. It can satisy the terminal state constraints, path constraints, and also ensure a high accuracy. ) For the separation o the two-stage-to-orbit aerospace vehicle in the upper atmosphere, due to various constraints, it is important to choose the optimal separation condition. In turn, the design o the carrier and orbiter can be improved through the state o separation conditions, which makes the design more reasonable. Acknowledgements The authors wish to express their sincere thanks to Mr. Zhang or valuable suggestions which improved the inal manuscript. Reerences [1] HUAG Guoqiang, The Ascent Trajectory Optimization o TSTO space plane, Journal o Astronautics, 010. [] David Benson, A Gauss Pseudospectral Transcription or Optimal Control, February 005. [3] Hong Bei, XI WA-Qing, Rapid Gliding Trajectory Optimization Via hp-adaptive Pseudospectral Method, Proceedings o the 30th Chinese Control Conerence.