Simulated Annealing for Missile Trajectory Planning and Multidisciplinary Missile Design Optimization

Missile trajectory planning and multidisciplinary design optimization a missile together with its trajectory is investigated. Direct shooting method and Hide-and-Seek simulated annealing algorithm used in optimization are presented. Two degree of freedom trim flight, flight mechanics model, end burning solid propellant engine model, and structural design models are employed. Maximum range trajectory optimization problem, minimum flight time specified range trajectory optimization problem, and minimum weight missile design optimization problem are addressed and solved. It is shown that the methods used are quite effective, robust, and capable of finding the global optimum. INTRODUCTION Missile trajectory has an important bearing on the accomplishment of its mission. It was shown that for given launch conditions, and impact conditions selected according to a particular mission, the range of an air to surface missile can be extended [ 1, 21. A minimum weight missile design problem was also addressed [ 1, 31. The latter problem was a multidisciplinary design optimization problem with models on the missile trajectory, missile structure, and end burning solid propellant rocket engine. Thus, a minimum weight airto-surface missile that will fly on the best trajectory for a given set of launch conditions, and realize the specified impact conditions at a given range was designed. In the named study a local algorithm, named * Associate Professor, Aeronautical Engineering Department, Member. + Graduate Student, Aeronautical Eng. Dept. Copyright 01999 The American Institute of Aeronautics and Astronautics Inc. All rights reserved. BFGS was employed [iI. During these studies it became clear that gradient type local algorithms had convergence difficulties for such highly nonlinear problems. Additionally, there was no guarantee that the resulting trajectory or design was a global optimum. Trajectory optimization has been applied to aerospace vehicles ranging from rockets to aircraft in the past [ 11. However, to the knowledge of the authors, design optimization together with the trajectory of an aerospace vehicle (i.e. missile), is only addressed in [l 3,5]. Many methods on the trajectory optimization of aerospace vehicles have been proposed. A rather detailed review may be found in a recently published paper WI. Hide and seek is a simulated annealing algorithm developed by Belisle et al [7]. It is capable of finding the global minimum of highly nonlinear functions. It is also claimed to be suitable for optimization of nonconvex functions with disconnected regions [8]. The algorithm is robust and guaranteed to converge to global minimum with probability of one. The purpose of this manuscript is to report on’ the recent work done for missile trajectory planning, and on multidisciplinary missile design optimization together with its trajectory, using direct shooting technique together with Hide-and-Seek simulated annealing algorithm. In the following, first the statements of the problems solved and discussed in this manuscript are listed. Then, mathematical models employed in the optimization are given. The Hide-and-Seek simulated annealing, and its application to the classical Zarmelo’s trajectory optimization problem is reported next. The manuscript continuous with the presentation and discussion on the problems addressed for missile trajectory optimization, and multidisciplinary missile design optimization. Finally, conclusions are given. American Institute of Aeronautics and Astronautics (c)2000 American Institute of Aeronautics & Astronautics or published with permission of author(s) and/or author(s)’ sponsoring organization. STATEMENTS OF THE PROBLEMS In this manuscript three problems related to missile trajectory planning and missile design optimization are addressed: 1. Find the maximum range trajectory of a missile with given launch and specified impact conditions. 2. Calculate the minimum flight time trajectory of the missile such that the missile hits the specified target with given launch conditions and required impact conditions. 3. Redesign the above missile such that, it flies to the prescribed extended range, with given launch conditions and specified impact conditions. The total mass of the new missile shall be as small as possible. Figure 1. Forces acting on the two degree of freedom missile and the coordinate system used. Figure 2. Plots of the axial force coefficient tables during trim flight for fore and aft center of mass locations. Motor is active. MATHEMATICAL MODELS For missile trajectory optimization study missile flight mechanics model (i.e., equations of motion) is needed. For the missile design optimization study, in addition to the flight mechanics model, engine and structural design models are also used. Although during missile design optimization the missile shape changes, no aerodynamics model to predict the aerodynamics coefficients of the new shape is employed. Thus, missile diameter, nose geometry, wing and tail geometry and their distance from the nose are assumed to be the same. To simplify the study, the small changes in missile length due to new engine designs are assumed to have a negligible effect on the aerodynamic coefficients. In the following the flight mechanics, as well as engine and structural design models are introduced. Flbht Mechanics Model In this work a two degrees of freedom missile model, assumed to fly in trim flight condition, is used. The commanded input is the angle of attack, CL. It is assumed that required angle of attack is instantaneously realized by an angle of attack autopilot. Thus, the pitch dynamics is neglected [5]. i=vcosy (1) Ii = V sin y (2) V=J-(Tcosa-D)-gsiny (3) m p =-$Tsin*+L)-m V (4) 8=y+a (5) In the above equations, L, D, T, are the lift, drag and thrust forces respectively. V is the total velocity of the missile, yis the flight path angle, 0, is the pitch attitude, r is the downrange and h is the altitude (Figure 1). To calculate aerodynamic forces, normal and axial force coefficients for the hypothetical missile with a prescribed geometry are generated in a tabular form using Missile DATCOM software [9]. Since the missile is assumed to fly in trim flight condition, the normal and axial force coefficients change with the change in center of mass position until burnout. Consequently two separate tables, corresponding the fore and aft center of mass locations, are used. These tables are plotted Figures 2 and 3. Since axial force changes after burnout, a separate table is used for this purpose. During the integration of the flight mechanics equations, appropriate interpolations are made to calculate the axial and normal force coefficients, C, and C, , for a required Mach number and angle of attack. From these coefficients, drag and lift force coefficients, C, and C, , are calculated. Finally lift and drag forces are found from: American Institute of Aeronautics and Astronautics (c)2000 American Institute of Aeronautics & Astronautics or published with permission of author(s) and/or author(s)’ sponsoring organization.