Trajectory Optimization Design for Morphing Wing Missile

，Chao Mingand Chuanjie Sun

（1．Nanjing University of Science and Technology，Nanjing 210094，China；2．Institute of Systems Engineering，China Academy of Engineering Physics，Mianyang Sichuan，621900，China）

Trajectory Optimization Design for Morphing Wing Missile

Ruisheng Sun1?，Chao Ming1and Chuanjie Sun2

（1．Nanjing University of Science and Technology，Nanjing 210094，China；2．Institute of Systems Engineering，China Academy of Engineering Physics，Mianyang Sichuan，621900，China）

This paper presents a new particle swarm optimization（PSO）algorithm to optimize the trajectory of morphing-wing missile so as to achieve the enlargement of the maximum range.Equations of motion for the twodimensional dynamics are derived by treating the missile as an ideal controllable mass point.An investigation of aerodynamic characteristics of morphing-wing missile with varying geometries is performed.After deducing the optimizing trajectory model for maximizing range，a type of discrete method is put forward for taking optimization control problem into nonlinear dynamic programming problem.The optimal trajectory is solved by using PSO algorithm and penalty function method.The simulation results suggest that morphing-wing missile has the larger range than the fixed-shape missile when launched at supersonic speed，while morphing-wing missile has no obvious range increment than the fixed-shape missile at subsonic speed.

morphing wing missile；trajectory optimization；optimization model；particle swarm optimization（PSO）

1 Introduction

Morphing is the ability of a flying structure to achieve certain maneuvers or performance specifications by means of in-flight shape changing.The shape variation may be continuous，smooth and seamless as with variable camber，variable twist and self adapting wings involving the development and research of new smart materials and actuation systems［1-6］.In recent years，morphing aircraft has become a focus because of two aspects.One is that we sostenuto acquire new inspiration from morphing flight biology in the nature. For example，Lentink et al.［7］put forward morphing plays an important role with the future airplane through investigating how swifts control their glide performance with morphing wings.The other one is morphing aircraft agrees with the development direction for the future aircraft.It can obtain the integrated performance with aerodynamics，ballistics and guidance，for satisfying the variation of environment and mission by changing its shape and size neatly.

The quantity of published literatures on morphing missile technology is quite extensive.Most of the previous research focused on a singular optimum shape rather than one that changed shape throughout flight［8-13］.Notable previous work on missile optimization includes Refs.［9，11，13］，where trajectory simulations were integrated into their optimization studies giving them the ability to optimize trajectory characteristics，such as maximizing range. Tanil［11］explored the optimization design space of missiles with discrete fin sets rather than continuous fin geometry variables.The results showed what the capabilities are possible given a particular set of fin geometries.Yang et al.［14］took the optimization one step further and maximized the range of a guided projectile. Although the geometry variation was limited and the speed was subsonic，the addition of canard control exemplified the ability to optimize a more complex design space.Ref.［10］successfully showed that genetic algorithms are capable of designing aerodynamic shapes which perform well in both single and multiple objective applications.Ref.［12］described the design optimization of a variable-span morphing wing to be fitted to a small UAV which flies in the speed range 11 m／s to 40 m／s. Refs.［8-13］all utilized genetic algorithms to find an optimum projectile geometry.

In order to investigate the advantages of a flexible morphing structure，a deep understanding of the interaction among aerodynamics，structures and controls is necessary.So the goal of this research is to optimize the range of unpowered ballistic systems by utilizing direct method based on PSO algorithm. The construction of this paper is as follows.In Section 2，the trajectory optimization model of the morphing-wingmissile is briefly deduced.In Section 3，the PSO optimization algorithm is presented.Then we explore the trajectory optimization design for morphing missile in this section.In Section 4，the simulation example of the trajectory optimization using the PSO algorithm are demonstrated for a missile released by airplane.In order to show the extended-range performance，the conventional fixed-shape missile systems are included in the simulation.Finally，some conclusions are presented in Section 5.

2 Trajectory Optimization Model

2.1 Aerodynamic Model

2.1.1 Governing equations

Three dimensional compressible N-S equations in integral form are defined as：

ρ，p，EandHrepresent the fluid density，pressure，total energy per unit mass and total enthalpy per unit mass，respectively；u，v，wrepresent the fluid velocity components along the coordinate axis in directionx，y，z；Vris the fluid normal velocity relative to the mesh；τijis the viscosity tensor andΘiis the combined items of viscous stress and fluid heat transfer functionality（where subscripti，j，kdenote directionx，y，z）；nx，ny，nzare the unit outward normal vector components of the control volume’s surface，and their specific expressions refer to Ref.［14］，not repeat them here.

N-S equations are solved using Fluent software to simulate the flow field.Spalart-Allmaras turbulence model is also adopted.Missile surface uses no-slip boundary condition，symmetric plane with the symmetric boundary condition and other boundaries with the pressure far field boundary condition.

2.1.2 Calculation mesh

To calculate the aerodynamic properties of morphing-wing configurations rapidly and efficiently，the space is discretized by tetrahedral mesh.The geometry of the calculation model of morphing wing is given in Fig.1，which is seen as mirror symmetry.So the unstructured tetrahedral mesh can be used in half of the flow field.Figs.1-3 show the surface mesh of missile corresponds to sweepback angle of the anterior wingχ＝35°，55°，85°separately.Whenχ＝35°，half of the flow field is discretized by unstructured tetrahedral meshes：mesh points are 970 772；quantity of tetrahedral elements is 5 486 101；quantity of surface mesh is 149 018（as shown in Fig.1）；mesh quantity of symmetry plane is 58 174.Whenχ＝55°，half of the flow field is discretized by unstructured tetrahedral meshes：mesh points are 897 336；quantity of tetrahedral elements is 5 046 604；quantity of surface mesh is 147 538（as shown in Fig.2）；mesh quantity of symmetry plane is 59 630.Whenχ＝85°，half of the flow field is also discretized by unstructured tetrahedral meshes：mesh points are 696 415；quantity of tetrahedral elements is 3 912 204；surface mesh quantity is 126 659（as shown in Fig.3）；mesh quantity of symmetry plane is 41 315.

Fig.1 Calculated surface mesh of missile withχ＝35°

Fig.2 Calculated surface mesh of missile withχ＝55°

Fig.3 Calculated surface mesh of missile withχ＝85°

2.1.3 Numerical calculation result

Figs.4-5 showCDandCLchange in relationships withαandχwhenMa＝1.4.It can be seen thatCDincreases with the decrease ofχ，and there is an approximately linear relationship between them whenαis a fixed value.CDis an approximately quadratic function ofαwhenχis a fixed value.CLis a quadratic function ofχ．Whenαin the range of 0°-4°andχ＝55°，CLreaches the maximum value as variable swept wing is in a semi-expanded state.However，when α in the range of 4°-8°，with the increase ofα，the maximum value ofCLhas an offset to small sweepback angle state.

Fig.4 Surface chart of drag coefficient（Ma＝1.4）

Fig.5 Surface chart of lift coefficient（Ma＝1.4）

As shown in Figs.6-7，lift-drag ratio is in nonconvex domain distribution withαandχ，and the maximum value of lift-drag decreases with the decrease ofMa．And the greaterMais，the more obvious of nonlinearity degree of lift-drag ratio is.In addition，whenMaandαare fixed values，the maximum value of lift-drag ratio andχform the down-opening parabola law relationship.

Fig.6 Surface chart of lift-drag ratio（Ma＝0.8）

Fig.7 Surface chart of lift-drag ratio（Ma＝1.8）

2.2 Flight Dynamical Model

2.2.1 Motion equations

For the periodic optimal flight problem under consideration，a mathematical model based on point mass dynamics can be used for determining the trajectory，yielding

Here the Earth is assumed to be flat，and the model for the drag（D）and the lift（L）can be formulated as

whereqis the dynamic pressure andSis the wing reference area.2.1 Aerodynamic Model is used for modeling the drag-lift characteristics of a morphing missile.The models are graphically presented in Figs.4 and 5，showing the lift and drag curves of the configurations.The coefficients are to be used in Eq.（2）for the morphing configuration read.

2.2．2 Flight path constraints

The constraints corresponding to flight path optimization for a given morphing-wing missile are as follows：

1）The constraints on states，parameters and control are

These numbers are completely based flight envelope which is probable maximum and minimum values of state variables in the problem.

2）Path constraint of missile being at attitude hold till 2.5 s after launch from airplane.

3）The nonlinear state inequality constraints are 0°≤α≤10°during complete trajectory 35°≤χ≤85°during complete trajectory 0≤ay≤50 m／s2during complete trajectory

3 Trajectory Optimization Algorithm

3.1 Cost Function

The objective of the optimization problem is to maximize the cross range，which is equivalent to maximizing the rangextfrom initial timet0to final timetf．Therefore，the performance index used for maximization of missile launch-range is

3.2 Discretization Methods

With the advent of computers and evolution ofmodern theories of optimal control，the numerical computation techniques for optimal atmospheric trajectories continue to be an active research area since the seventies.The approaches on trajectory optimization are of two distinct categories，such as：i）direct method based on mathematical programming and parameterization of state and control histories；ii）indirect method based on solution of two point boundary value problem（TPBVP）using optimal control principle［15］.The direct method is more popular in application work over the indirect method because of relative robustness.In the direct approach，the model equations of the considered system are discretized，and the control trajectories are parameterized to obtain a finite-dimensional parameter optimization problem［16］.

So the direct method is derived for the trajectory optimization of guidance missile with morphing swept wings.Considering the following continuous-time stateinequality path constrained optimization problem on the domainwhereIhas been divided intoKmesh intervalsFurthermore，without loss of generality we can transform the independent variable in each mesh interval fromvia the affine transformation．Then at the discrete time pointare derived for replacing the continuous control variables.The variables betweencan be derived by Lagrange interpolation algorithm.

3.3 Penalty Function Method

The basic form of an equally constrained problem can be expressed as

A new unconstrained problem can be obtained by appending the constrained to the object function with norm functionrepresents the vector’sp-norm．When the designing variables go off the feasible regionthe penitentiary functionswill penalize the object functions.So the final object function can be written asis the range of guidance missile.

3.4 PSO Optimization Algorithm

Particle Swarm Optimization（PSO）is a population based algorithm，but the search mechanism leans on each particle dynamics correction at each iteration.Each particle stays for a possible solution and moves into the search domain looking for the optimum. The main parameters to be considered while applying the PSO are theand thewhich represent the best position assumed by a particle within its life，and the best position assumes up to the current instant throughout the whole particles population respectively. The dynamics of the particle，in charge of accomplish the search in the variables domain，is ruled by.

where，r1n，r2nare randomly generated in the［0，1］range，while social coefficientc1and acknowledgement coefficientc2must be tuned according to the problem；Wis termed“inertia”weight，and it is used to control the impact of a particle’s previous velocity on the calculation of the current velocity vector.A large value forWfacilitates global exploration，which is particularly useful in the initial stages of an optimization.A small value allows for more localized searching，which is useful as the swarm moves toward the neighborhood of the optimum.

In addition to the basic PSO algorithm，significant enhancements aboutW’s decreasing policy have been proposed such as：a）linear decreasing policy，b）index decreasing policy，c）polynomial decreasing policy.Considering the convergence rate and the convergence precision，linear decreasing policy is adopted，whereWdecrease from 1.0 to 0.4 linearly. Combined control variable discretization method，the entire iterative algorithm for the basic process is as follows：

1）Initialize model，such as flight dynamics model，particle swarm algorithm parameters and so on.

2）According to Eq.（5）update the particle’s velocity and position.

3）According to Eq.（4）generate an interpolated curve as the control variable.

4）Solve the flight dynamics model（1）.

5）Calculate the objective functionfu（x），takepinandpgninto comparison.

6）If the termination condition is satisfied，stop the calculation，and output the best individual fitness.

7）Otherwise，return to step 3）for continue.

4 Results and Analysis of Simulation

In this section，the simulation results of trajectory optimization by using the PSO algorithm are demonstrated for a missile released by airplane.In order to show the extended-range performance，the conventional fixed-shape missile systems are included in the simulation，whose nominal values of the aerodynamic coefficients are equal to that of the modeof the morphing-wing missile with swept angle 35°.

The simulation parameters are as follows：the initial values ofy，θ，Maare set to bey（0）＝12 km，θ（0）＝0°，Ma（0）＝0.6，1.2 and 2.0.In PSO algorithmN，c1，c2are set to be 100，1.8 and 1.8. AndWis decreased from 1.0 to 0.4 linearly.

Corresponding to the optimized trajectory between fixed-shape missile and morphing missile，yvs.xhistory，αtime history，χtime history andvtime history along longitudinal trajectory are shown in Figs.8-11，respectively.And morphing-wing missile has the larger range than the fixed-shape missile.Moreover the increment of the range is incremental fromMa（0）＝0.6 to 2.0 in Table 1.Simultaneously we find morphing-wing missile has more range increment released at supersonic speed.From drag forceDtime history and lift forceLtime history depicted in Figs.12 and 13，it is clear that during supersonic speed，drag force of morphing-wing missile is less than fixed-shape missile，while lift force of morphing missile is more than fixed-shape missile.

Fig.8 y vs.x history

Fig.9 α time history

Fig.10 χtime history

Fig.11 v time history

Fig.12 Drag force D time history

Fig.13 Lift forceLtime history

Table 1 Parameters of terminal in different released velocity

5 Conclusion

In this paper，PSO algorithm is applied for trajectory optimal design for a morphing wing missile released by airplane.We transform the optimal control problem into a nonlinear dynamic programming problem，avoiding solving two-point boundary problem. Through integrated morphing structure and flight dynamics，the optimal trajectory is obtained by using PSO algorithm.The result of the optimal trajectory for morphing structure is also validated in the simulation by taking into account 0.6，1.2，2.0Maas well as the fixed-shape structure.Furthermore，compared with the conventional fixed shape structure，morphing structure has the capability of external range，especially when it is released at the condition of supersonic.

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TJ760.12

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：1005-9113（2015）05-0025-06

10.11916／j.issn.1005-9113.2015.05.004

2014-05-05.

Sponsored by the Natural Science Association Foundation（NSAF）of China（Grant No.11176012）and the Research Innovation Project for Graduate Student of Jiangsu-Provincial Ordinary University（Grant No.KLYX15-0394）.

?Corresponding author.E-mail：srscom＠163.com.