Ozan Tekinalp

Simulated Annealing for Missile Optimization: Developing Method and Formulation Techniques

Hide-and-seek is a continuous simulated annealing algorithm that uses an adaptive cooling schedule. A number of improvements are proposed for the global optimum estimation required for the cooling schedule. To handle equality constraints, two approaches are examined: the rejection method and augmentation of constraints to cost using penalty coefficients. It is demonstrated that a faster convergence is possible if, in the penalty coefficients approach, equality constraints are replaced with tight inequality constraints. The missile trajectory optimization problem is formulated using nodes equally spaced in time until burnout and equally spaced in energy consumption after burnout. This approach is shown to be superior to the use of all nodes equally spaced in time. Also investigated is the effect of node number on the performance of the algorithm. The problem of combined optimization of design and control variables is also addressed. For this purpose a two-loop approach, where each loop has its own temperature and cooling schedule, is proposed, and its effectiveness is demonstrated.

A new multiobjective simulated annealing algorithm

A new multiobjective simulated annealing algorithm for continuous optimization problems is presented. The algorithm has an adaptive cooling schedule and uses a population of fitness functions to accurately generate the Pareto front. Whenever an improvement with a fitness function is encountered, the trial point is accepted, and the temperature parameters associated with the improving fitness functions are cooled. Beside well known linear fitness functions, special elliptic and ellipsoidal fitness functions, suitable for the generation on non-convex fronts, are presented. The effectiveness of the algorithm is shown through five test problems. The parametric study presented shows that more fitness functions as well as more iteration gives more non-dominated points closer to the actual front. The study also compares the linear and elliptic fitness functions. The success of the algorithm is also demonstrated by comparing the quality metrics obtained to those obtained for a well-known evolutionary multiobjective algorithm.

Tilt Duct Vertical Takeoff and Landing Uninhabited Aerial Vehicle Concept Design Study

¨A new autonomously controlled tilt-duct vertical takeoff and landing uninhabited aerial vehicle concept is proposed. This design combines the vertical flight capability of a helicopter and forward flight performance of a fixed-wing conventional aircraft. The two main engines and propellers are located inside the tilting ducts attached to the wing tips. There is a third engine‐propeller combination located inside the aft fuselage for pitch and yaw control during hover and transition. The advantages and disadvantages of the ducted propellers are discussed. A conceptual design study is performed including airfoil and geometry selection, initial sizing calculations, estimation of stability and control parameters, etc. Drawings of the aircraft in hover, transition and forward flight modes are presented.

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 =-

A numerical comparison of frozen-time and forward-propagating Riccati equations for stabilization of periodically time-varying systems

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.

Dynamic Modeling of Transverse Drill Bit Vibrations

Feedback control of linear time-varying systems arises in numerous applications. In this paper we numerically investigate and compare the performance of two heuristic techniques. The first technique is the frozen-time Riccati equation, which is analogous to the state-dependent Riccati equation, where the instantaneous dynamics matrix is used within an algebraic Riccati equation solved at each time step. The second technique is the forward-propagating Riccati equation, which solves the differential algebraic Riccati equation forward in time rather than backward in time as in optimal control. Both techniques are heuristic and suboptimal in the sense that neither stability nor optimal performance is guaranteed. To assess the performance of these methods, we construct Pareto efficiency curves that illustrate the state and control cost tradeoffs. Three examples involving periodically time-varying dynamics are considered, including a second-order exponentially unstable Mathieu equation, a fourth-order rotating disk with rigid body unstable modes, and a 10th-order parametrically forced beam with exponentially unstable dynamics. The first two examples assume full-state feedback, while the last example uses a scalar displacement measurement with state estimation performed by a dual Riccati technique.

Development of a State Dependent Ricatti Equation Based Tracking Flight Controller for an Unmanned Aircraft

A dynamic model is presented for the transverse vibration of drill bits, and validated experimentally. The model includes” the effects of drill rotation speed and feed rate as well as drill bit length, diameter and material properties The drill transverse natural frequencies are shown to depend on the rotation speed, and transverse instability occurs at a critical speed where the natural frequencies vanish. Simulation results snow the influence or parameters such as drill length, drill diameter, rotation speed, and feedrate on the critical speed. Experimental results confirming the predicted trends, are presented. Potential uses of the model, are discussed, and extensions to the model are suggested to account for the complex drill geometry, torsional vibrations, damping mechanisms, and distributed mass and stiffness properties.

Simulation and Flight Control of a Tilt Duct UAV

A dual loop nonlinear State Dependent Riccati Equation (SDRE) control method is developed for the ight control of an unmanned aircraft. The outer loop addresses the attitude and altitude kinematics, while the inner loop handles the translational and rotational equations of motion. The control strategy utilizes a tracking control problem. The mismatch due to the SDC factorization of the inner loop is handled with a nonlinear compensator again derived from the tracking control formulation. The quadratic optimal control problems of the inner and outer loops are solved at discrete intervals in time. A nonlinear simulation model of the UAV is used to examine the performance of the SDRE controller. Two ight scenarios are considered: a coordinated turn maneuver and a high angle of attack ight. These simulation results show the eectiveness of the proposed nonlinear controller.

Spacecraft energy storage and attitude control

Tilt duct VTOL UAV concept is presented. The equations of motion are given and, trim and simulation code is described. Trim flight conditi ons are given for hover, cruise and forward flight cases. A two loop SDRE control is propos ed and explained. The blended inverse control allocation algorithm is used for allocati ng controllers during the transition flight phase, where there are redundant controls. Sim ulation results during transition phase are presented, and the success of the controller as we ll as the allocation algorithm is demonstrated. Nomenclature ij I = mass moments of inertias

Quaternion Based State Dependent Ricatti Equation Control of a Satellite Camera on Piezoelectric Actuators

The energy storage and attitude control technologies in spacecrafts are considered. The conventional technologies like electrochemical batteries, reaction wheels and control moment gyroscopes are compared to the promising Integrated Power and Attitude Control System with Variable Speed Control Moment Gyros (IPACS-VSCMG) technology. For this purpose, various energy storage and attitude control system conceptual designs are carried out, and the results are compared. Also given a simulation to show the effectiveness of an IPACS-VSCMG to carry out a roll maneuver command while its kinetic energy is converted into electrical energy and drained at the same time.