Evolutionary Method Based Hybrid Entry Guidance Strategy for Reentry Vehicles

4th IFAC International Conference on 4th IFAC International Conference on 4th IFAC Intelligent Control andConference Automationon Sciences 4th IFAC International International Conference on 4th IFAC International Intelligent Control and andConference Automationon Sciences Intelligent Control Automation Sciences Available online at www.sciencedirect.com June 1-3, 2016. Reims, France Intelligent Control and Sciences Intelligent Control and Automation Automation Sciences June 1-3, 1-3, 2016. 2016. Reims, France June Reims, France June 1-3, 2016. Reims, France June 1-3, 2016. Reims, France

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Evolutionary Method Based Hybrid Entry Evolutionary Method Based Hybrid Entry Evolutionary Method Based Hybrid Entry Evolutionary Method Based Hybrid Entry Guidance Strategy for Reentry Vehicles Guidance Strategy for Reentry Vehicles Guidance Strategy for Reentry Vehicles Guidance Strategy for Reentry Vehicles

∗ ∗ Gangireddy Sushnigdha Joshi ∗ ∗ ∗ Ashok ∗ Gangireddy Sushnigdha Joshi ∗ Ashok ∗ Gangireddy Sushnigdha Ashok Joshi ∗ Gangireddy Sushnigdha Ashok Joshi Gangireddy Sushnigdha Ashok Joshi ∗ ∗ ∗ Indian Institute of Technology, Bombay, 400076 India (e-mail: ∗ Indian Institute Institute of of Technology, Technology, Bombay, Bombay, 400076 400076 India India (e-mail: (e-mail: ∗ ∗ Indian Indian Institute Bombay, Indian [email protected], Institute of of Technology, Technology, [email protected]) Bombay, 400076 400076 India India (e-mail: (e-mail: [email protected], [email protected]) [email protected], [email protected]) [email protected], [email protected], [email protected]) [email protected]) Abstract: This paper presents hybrid approach combining Pigeon Inspired Optimization Abstract: This This paper paper presents presents aa a hybrid hybrid approach approach combining combining Pigeon Pigeon Inspired Inspired Optimization Optimization Abstract: Abstract: This paper presents a hybrid approach combining Pigeon Inspired Optimization (PIO) with Gauss-Newton method for entry guidance of winged vehicles. The bank Abstract: This paper presents a hybrid approach combining Pigeon Inspired Optimization (PIO) with Gauss-Newton method for entry guidance of winged vehicles. The bank angle angle (PIO) with Gauss-Newton method for guidance of vehicles. The bank (PIO) with is Gauss-Newton method for entry entry guidance of winged winged vehicles. The bank angle angle modulation considered as the primary control. In the hybrid guidance approach, PIO (PIO) with Gauss-Newton method for entry guidance of winged vehicles. The bank modulation is is considered considered as as the the primary primary control. control. In In the the hybrid hybrid guidance guidance approach, approach, angle PIO modulation PIO modulation is considered as the primary control. In the hybrid guidance approach, PIO algorithm is initially used to find a bank angle that satisfies a predefined cost function. In modulation is considered as the primary control. In the hybrid guidance approach, PIO algorithm is is initially initially used used to to find find aa bank bank angle angle that that satisfies satisfies aa predefined predefined cost cost function. function. In In algorithm algorithm is initially used to find a bank angle that satisfies aa predefined cost function. In the second phase, the corresponding bank angle is updated to correct the terminal errors using algorithm is initially used to find a bank angle that satisfies predefined cost function. In the second phase, the corresponding bank angle is updated to correct the terminal errors using the second phase, the corresponding bank angle is updated to correct the terminal errors using the second the bank angle is tothat correct the terminal errors using Gauss-Newton algorithm. Advantages of are it not require an initial the second phase, phase, the corresponding corresponding bank anglealgorithm is updated updated correct the not terminal errors using Gauss-Newton algorithm. Advantages of PIO PIO algorithm aretothat that it does does not require an initial initial Gauss-Newton algorithm. Advantages of are require an Gauss-Newton algorithm. Advantages of PIO PIO algorithm algorithm are that it it does does notfrom require an initial guess and that equality and inequality constraints can be incorporated, apart the fact that Gauss-Newton algorithm. Advantages of PIO algorithm are that it does not require an initial guess and and that that equality equality and and inequality inequality constraints constraints can can be be incorporated, incorporated, apart apart from from the the fact fact that that guess guess and that equality and inequality constraints can be incorporated, apart from the fact that it has global convergence and randomness. Gauss-Newton method, however, is deterministic guess and that equality and inequality constraints can be incorporated, apart from the fact that it has has global global convergence convergence and and randomness. randomness. Gauss-Newton Gauss-Newton method, method, however, however, is is deterministic deterministic it it has global convergence and randomness. Gauss-Newton method, however, is and ensures global convergence with high accuracy given an initial guess. Thus, hybrid guidance it has globalglobal convergence and with randomness. Gauss-Newton method, however, is deterministic deterministic and ensures global convergence with high accuracy accuracy given an an initial initial guess. Thus, hybrid hybrid guidance and ensures convergence high given guess. Thus, guidance and ensures global convergence with high accuracy given an initial guess. Thus, hybrid guidance algorithm exploits the benefits of both and determines an optimal bank angle profile that steers and ensures global convergence with high accuracy given an initial guess. Thus, hybrid guidance algorithm exploits the benefits of both and determines an optimal bank angle profile that steers algorithm exploits the of determines an bank profile steers algorithm exploits the benefits benefits of both both and and determines an optimal optimal bank angle angle profile that thatresults steers the vehicle to destination accurately, satisfying the path constraints. The simulation algorithm exploits the benefits of both and determines an optimal bank angle profile that steers the vehicle vehicle to to destination destination accurately, accurately, satisfying satisfying the the path path constraints. constraints. The The simulation simulation results results the the vehicle to accurately, satisfying show effectiveness of the proposed algorithm. the to destination destination accurately, satisfying the the path path constraints. constraints. The The simulation simulation results results showvehicle effectiveness of the the proposed proposed algorithm. show effectiveness of algorithm. show effectiveness of the proposed algorithm. show effectiveness of the proposed algorithm. © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Re-entry guidance, Trajectory optimization, Hybrid techniques, Pigeon inspired Keywords: Re-entry Re-entry guidance, guidance, Trajectory Trajectory optimization, optimization, Hybrid Hybrid techniques, techniques, Pigeon Pigeon inspired inspired Keywords: Keywords: Re-entry Re-entry guidance,method. Trajectory optimization, optimization, Hybrid Hybrid techniques, techniques, Pigeon Pigeon inspired inspired optimization, Gauss-Newton Keywords: guidance, Trajectory optimization, Gauss-Newton method. optimization, Gauss-Newton method. optimization, Gauss-Newton method. optimization, Gauss-Newton method. 1. drag rate 1. INTRODUCTION INTRODUCTION drag rate rate and and control control law law has has been been designed designed by by Joshi Joshi et et al. al. 1. drag 1. INTRODUCTION INTRODUCTION drag rate and and control control law law has has been been designed designed by by Joshi Joshi et et al. al. (2007). 1. INTRODUCTION drag rate and control law has been designed by Joshi et al. (2007). (2007). (2007). The third type of approach adopts the direct trajectory (2007). The third third type type of of approach approach adopts adopts the the direct direct trajectory trajectory The atmospheric entry is considered to be the critical The atmospheric atmospheric entry entry is is considered considered to to be be the the critical critical The The third type of approach adopts the direct trajectory optimization techniques like pseudospectral methods and The The third type of approach adopts the direct trajectory The atmospheric entry is considered to be the critical optimization techniques like pseudospectral methods and phase for an entry vehicle. In this phase, vehicle is subtechniques like pseudospectral methods and The is considered to be the critical phaseatmospheric for an an entry entryentry vehicle. In this this phase, phase, vehicle is subsub- optimization optimization techniques like pseudospectral methods and evolutionary algorithms. Optimal guidance based on indiphase for vehicle. In vehicle is optimization techniques like pseudospectral methods and phase for an entry entry vehicle. In this this phase, phase, vehicle is is subsubevolutionary algorithms. Optimal guidance based on indijected to high heat load, aerodynamic decelerations and evolutionary algorithms. Optimal guidance based on indiphase for an vehicle. In vehicle jected to to high heat heat load, load, aerodynamic aerodynamic decelerations decelerations and and evolutionary algorithms. Optimal guidance basedproposed on indiindirect Legendre pseudospectral method has been jected Optimal guidance based on jected to high high heat load, aerodynamic decelerations and evolutionary rect Legendre Legendrealgorithms. pseudospectral method has been been proposed dispersed flight environment. Thus, generating guidance rect pseudospectral method has proposed jected to high load, aerodynamic decelerations and dispersed flightheat environment. Thus, generating generating guidance rect Legendre pseudospectral method has been proposed by Tian and QunZong (2011). A nominal reference trajecdispersed flight environment. Thus, guidance rect Legendre pseudospectral hasreference been proposed dispersed flight environment. Thus, generating guidance by Tian Tian and QunZong QunZong (2011). method A nominal reference trajeccommands for the entry vehicle to safely reach the destinaand A trajecdispersed generating commandsflight for the theenvironment. entry vehicle vehicleThus, to safely safely reach the theguidance destina- by by Tian and generated. QunZong (2011). (2011). A nominal nominal reference trajectory is first A robust state feedback guidance commands for entry to reach destinaby Tian and QunZong (2011). A nominal reference trajeccommands for the entry vehicle to safely reach the destinatory is first generated. A robust state feedback guidance tion is a challenging task. Over the years, entry trajectory tory is first generated. A robust state feedback guidance commands for the entry vehicle destina- tory tion is is a challenging challenging task. Over to thesafely years,reach entrythe trajectory isgenerated first generated. generated. A robust robust state feedback guidance law is in real time using indirect Legendre pseution task. Over the years, entry trajectory tory first A feedback guidance tion is a challenging task. Over the the years, entry of trajectory law is isisgenerated generated in real real time time usingstate indirect Legendre pseuoptimization has been and is still subject interest in using indirect Legendre pseution is aa challenging task. the years, entry trajectory optimization has been been andOver is still still the subject of interest law law is generated in real time using indirect Legendre pseudospectral feedback method. The control variable, bank optimization has and is the subject of interest law is generated in real time using indirect Legendre pseuoptimization has been and is still the subject of interest dospectral feedback method. The control variable, bank for many researchers. In the current literature, three difdospectral feedback method. The control variable, bank optimization has been and is still the subject of interest for many many researchers. researchers. In In the the current current literature, literature, three three difdif- dospectral dospectral feedback method. The control variable, bank angle, has been discretized at a set of Legendre-Gauss colfor feedback method. The control variable, bank for many researchers. In the current literature, three difangle, has been discretized at a set of Legendre-Gauss colferent approaches have been used to solve entry guidance has been discretized at aa set of Legendre-Gauss colfor many researchers. the used current literature, dif- angle, ferent approaches haveInbeen been used to solve solve entry three guidance angle, has been discretized at set of Legendre-Gauss collocation points and is optimized with artificial bee colony ferent approaches have to entry guidance angle, has been discretized at a set of Legendre-Gauss colferent approaches have been used to solve entry guidance location points and is optimized with artificial bee colony problem. location points and is optimized with artificial bee colony ferent approaches have been used to solve entry guidance location problem. points and and isDuan optimized with artificial bee bee colony colony (ABC) algorithm by and Li (2015). problem. location points optimized artificial problem. (ABC) algorithm algorithm byisDuan Duan and Li Liwith (2015). (ABC) by and (2015). problem. (ABC) algorithm by Duan and Li (2015). This paper presents a new approach for solving entry (ABC) algorithm by Duan Li (2015). This paper paper presents newand approach for solving solving entry entry The first category of guidance algorithms are based on presents aaa new approach for The first first category category of of guidance guidance algorithms algorithms are are based based on on This This paper presents new approach for solving entry guidance problem by merging swarm intelligence method The This paper presents a new approach for solving entry The first category of guidance algorithms are based on guidance problem by merging swarm intelligence method generating reference trajectories and tracking them using problem by swarm method The first category guidance algorithms arethem based on guidance generating referenceoftrajectories trajectories and tracking them using guidance problem by merging merging swarm intelligence intelligence method PIO, with the traditional Gauss-Newton method. The adgenerating reference and using guidance by merging swarm intelligence method generating reference trajectories and tracking tracking them using PIO, with withproblem the traditional traditional Gauss-Newton method. The The adcontrol laws. Most often these reference trajectories are PIO, the Gauss-Newton method. adgenerating reference trajectories and tracking them using control laws. Most often these reference trajectories are PIO, with the traditional Gauss-Newton method. The advantages of PIO and Gauss-Newton method are exploited control laws. Most often these reference trajectories are PIO, with the traditional Gauss-Newton method. The adcontrol laws. Most often these reference reference trajectories are vantages vantages of PIO and Gauss-Newton method are exploited generated offline and are preloaded before flight. Roenneke of PIO and Gauss-Newton method are exploited control laws. Most often these trajectories are generated offline offline and and are are preloaded before before flight. flight. Roenneke vantages vantages of PIO PIO and and Gauss-Newton method are exploited in the proposed hybrid algorithm. The PIO algorithm is generated of Gauss-Newton method exploited generated offline andpresented are preloaded preloaded beforecontrol flight. Roenneke Roenneke in the the proposed proposed hybrid algorithm. The The PIO are algorithm is and Markl (1994), a linear law that hybrid algorithm. PIO algorithm is generated and are preloaded before flight. Roenneke and Markl Markloffline (1994), presented a linear linear control law that that in in the proposed hybrid algorithm. The PIO algorithm is independent of initial guess requirement and facilitates and (1994), presented a control law in the proposed hybrid algorithm. The PIO algorithm is and Markl (1994), presented a linear control law that independent of initial guess requirement and facilitates achieves local tracking of drag-vs-energy reference. Saraf of guess requirement and facilitates and Markl (1994), presented a linear control law Saraf that independent achieves local tracking of drag-vs-energy reference. Saraf independent of initial initialand guess requirement and apart facilitates inclusion of inequality equality constraints, from achieves local tracking of reference. of initial guess requirement and facilitates achieves local has tracking of drag-vs-energy drag-vs-energy reference. Saraf independent inclusion of of inequality inequality and equality constraints, apart from et al. (2004), proposed an entry guidance algorithm inclusion and equality constraints, apart from achieves local tracking of drag-vs-energy reference. Saraf et al. (2004), has proposed an entry guidance algorithm inclusion of inequality and equality constraints, apart from ensuring global convergence. Gauss-Newton method is et al. (2004), has proposed an entry guidance algorithm inclusion of inequality and equality constraints, apart from et al. (2004), has proposed an entry guidance algorithm ensuring global convergence. Gauss-Newton method is a a called evolved acceleration guidance logic for entry (EAensuring global convergence. Gauss-Newton method is et al. (2004), has proposed an entry guidance algorithm called evolved evolved acceleration acceleration guidance guidance logic logic for for entry entry (EA(EA- ensuring ensuring globalapproach convergence. Gauss-Newton method is a deterministic and ensures global convergence called global convergence. Gauss-Newton method is aa called evolved acceleration guidance logic for entry (EAdeterministic approach and ensures global convergence GLE), which has trajectory planner, that generates referdeterministic approach and ensures global convergence called evolved acceleration guidance logic for entry (EAGLE), which which has has trajectory planner, planner, that that generates referrefer- deterministic approach and ensures global convergence with high accuracy given an initial guess. These methods GLE), approach ensures global convergence GLE), which has trajectory trajectory planner, that generates generates refer- deterministic with high high accuracy accuracy givenand an initial initial guess. These methods ence drag acceleration and heading profiles onboard. Then, with given an guess. These methods GLE), which has trajectory planner, that generates reference drag drag acceleration and heading heading profiles onboard. Then, Then, with high accuracy given an initial guess. These methods are integrated to solve entry guidance problem owing to ence acceleration and profiles onboard. with high accuracy given an initial guess. These methods ence drag acceleration and heading profiles onboard. Then, are integrated to solve entry guidance problem owing to a tracking law based on feedback linearization has been are integrated to solve entry guidance problem owing to ence drag acceleration heading profiles onboard. tracking law based based and on feedback feedback linearization hasThen, been are are integrated to solve solve entry guidance guidance problem owing to their advantages. The hybrid guidance method uses PIO aaa tracking law on linearization has been integrated to entry problem owing to tracking law based on feedback linearization has been their advantages. The hybrid guidance method uses PIO employed to follow reference drag and heading profiles. their advantages. The hybrid guidance method uses PIO a tracking law based on feedback linearization has been employed to follow follow reference reference drag drag and heading heading profiles. their advantages. The hybrid guidance method uses PIO PIOa algorithm initially to find a bank angle that satisfies employed their advantages. hybrid guidance method uses employed to to follow follow reference drag and and heading heading profiles. profiles. algorithm initiallyThe to find find bank angle that satisfies satisfies algorithm initially to aaathe bank angle that aaa employed to reference drag and profiles. algorithm initially to find bank angle that satisfies predefined cost function. In second phase, the correalgorithm initially to find a bank angle that satisfies a predefined cost function. In the second phase, the correThe second category of guidance algorithms employ precost function. In phase, the correThe second second category category of of guidance guidance algorithms algorithms employ employ prepre- predefined predefined cost angle function. In the the second second phase, the corresponding bank is updated to correct the terminal The predefined cost function. In the second phase, the correThe second category of guidance algorithms employ presponding bank angle is updated to correct the terminal dictor corrector algorithm. This algorithm predicts the sponding bank angle is updated to correct the terminal The category of guidance employ predictorsecond corrector algorithm. This algorithms algorithm predicts predicts the sponding bank angle to the terminal errors using Gauss-Newton algorithm. This hybrid guiddictor corrector algorithm. This algorithm the bank angle is is updated updated to correct correct terminal dictor corrector algorithm. This algorithm predicts the errors using using Gauss-Newton algorithm. This the hybrid guidtrajectories based on current state and updates the control errors Gauss-Newton algorithm. This hybrid guiddictor corrector algorithm the sponding trajectories basedalgorithm. on current current This state and and updatespredicts the control control errors using Gauss-Newton algorithm. This hybrid guidance scheme satisfies equality and inequality constraints, trajectories based on state updates the errors using Gauss-Newton algorithm. This hybrid guidtrajectories based on current state and updates the control ance scheme satisfies equality and inequality constraints, variable to correct the terminal errors. Quasi Equilibrium scheme satisfies equality and inequality trajectories based on state and updates the control ance variable to to correct thecurrent terminal errors. Quasi Equilibrium Equilibrium ance scheme satisfies equality and variables inequalitytoconstraints, constraints, with minimum number of control be found. variable the errors. Quasi ance satisfies equality and inequality variable to correct correct the terminal terminal errors. Quasi Equilibrium with scheme minimum number of control control variables toconstraints, be found. found. Glide Condition (QEGC) has been used to reduce infinite with minimum number of variables to be variable to correct the terminal errors. Quasi Equilibrium Glide Condition (QEGC) has been used to reduce infinite with minimum number of control variables to be found. This method is computationally simple and it has flexibilGlide Condition (QEGC) has been used to reduce infinite with minimum number of control variables to be found. Glide Condition (QEGC) has been been used toconstraints reduce infinite infinite This method is computationally simple and it has flexibildimensional problem of meeting the path into This method is computationally simple and it has flexibilGlide Condition (QEGC) has used to reduce dimensional problem problem of of meeting the the path path constraints constraints into into This method is computationally computationally simple and and it has has flexibility to incorporate additional equilibrium glide constraint dimensional is simple it flexibildimensional problem of meeting meeting the path constraints into This ity to tomethod incorporate additional equilibrium equilibrium glide constraint a one-parameter search problem by Shen and Lu (2003). ity incorporate additional glide constraint dimensional problem of meeting into one-parameter search problem the by path Shen constraints and Lu Lu (2003). (2003). ity to incorporate additional equilibrium glide constraint or further reduction in heat rate by using altitude rate aaa one-parameter search problem by Shen and ity to incorporate additional glide constraint one-parameter search problem byinto Shen andon Ludrag (2003). or further further reduction in heat heat equilibrium rate by by using using altitude rate The path constraints are converted limits and reduction in rate altitude rate a one-parameter search Shen and Lu (2003). The path constraints constraints are problem convertedbyinto into limits on drag and or or further reduction in heat rate by using altitude rate The path are converted limits on drag and or further reduction in heat rate by using altitude rate The path constraints are converted into limits on drag and The path constraints are converted into limits on drag and

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compensation in the second phase of the algorithm. The proposed algorithm is applied to Common Aero Vehicle (CAV-H) (Phillips, 2003). Simulation results show that constraints are met accurately. This paper is organised as follows. Section 2 describes the entry dynamics and the constraints involved. Problem statement is defined in section 3. Section 4 formulates the cost function based on terminal and path constraints. Section 5 discusses the proposed hybrid entry guidance algorithm. Section 6 has results validating the proposed algorithm. Section 7 concludes the paper.

V ϒ

Z

ψ O

r

φ θ

Y

2. ENTRY DYNAMICS X

The 3-DOF point mass dynamics of re-entry vehicle gliding over a spherical, rotating Earth in terms of nondimensional variables considering energy as independent variable are dr sinγ = (1) de D cos γ sin ψ dθ = (2) de rD cos φ cos γ cos ψ dφ = (3) de rD dV 1 = de DV



−D −



sin γ r2



+ Ω2 r cos φ(sin γ cos φ − cos γ sin φ cos ψ)



   1 1  cos γ  dγ 2 = + 2ΩV cos L cos σ + V − de DV 2 r r  φ sin ψ + Ω2 r cos φ(cos γ cos φ + sin γ cos ψ sin φ  dψ 1 L sin σ V 2 = + cos γ sin ψ tan φ−2ΩV (tan de DV 2 cos γ r  Ω2 r γ cos ψ cos φ − sin φ) + sin ψ sin φ cos φ cos γ cos γ ds =− de rD

(4)

(5)

Fig. 1. Nomenclature used in equations of motion considered as a state. Final time of flight can be found by integrating equation 9 from initial energy to final energy. 1 V2 (8) e= − r 2 dτ = 1/DV (9) de The dimensionless variables are obtained by scaling them with appropriate factor as given in Lu (2014). The terms L and D are the non-dimensional aerodynamic lift and drag acceleration(in g0 = 9.8m/s2 ), respectively. 1 ρV 2 CL Sref (10) L= 2mg0 1 D= ρV 2 CD Sref (11) 2mg0 The aerodynamic coefficients CL and CD are functions of angle of attack α and Mach number. The bank angle σ is the roll angle of the vehicle about the relative velocity vector, positive to the right. Ω is the dimensionless Earth self rotation rate. 2.1 Path constraints

(6)

(7)

where, r is the radial distance from the Earth center to the vehicle O, θ and φ are the longitude and latitude, V is the Earth-relative velocity, γ is the flight-path angle, and ψ is the heading angle of the velocity vector, measured clockwise in the local horizontal plane from the north as shown in Fig. 1. s denotes the range to go(in radians) on the surface of a spherical Earth along the great circle connecting the current location of the vehicle and the site of the final destination. The gravity force is based on Newtonian gravity law. The differentiation in the previous equations are with respect to the dimensionless energy e. e is defined as negative of the specific mechanical energy used in orbital mechanics as mentioned in (Lu, 2014). It is monotonically increasing with time. Time of flight is also 340

Entry flight has allowable limits on maximum heat rate Q˙ on surface of vehicle, load factor a and dynamic pressure p as given by equations (12), (13), (14) respectively. They, together form path constraints. √ Q˙ = kQ ρV 3.15 ≤ Q˙ max √ where kQ = 9.4369 × 10−5 × ( g0 R0 )3.15  a = L2 + D2 ≤ amax 2

p = (g0 R0 ρV )/2 ≤ pmax

(12)

(13) (14)

Quasi equilibrium glide condition is considered to be soft constraint. Equilibrium glide refers to aerodynamic lift balancing the gravitational and centrifugal forces as given by equation (15). Lcosσ = (1/r2 ) − (V 2 /r) (15) where, σ is specified bank angle. In equilibrium glide, the flight path angle should be constant. But, it usually varies with time. Hence, it is called quasi equilibrium glide condition.

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2.2 Terminal constraints Terminal conditions refer to the final altitude, velocity and range to go as desired by the requirement of Terminal Area Energy Management (TAEM) phase. The final time of flight is unknown, whereas the final and initial energy can be calculated from desired final and initial altitude and velocities respectively. Hence, energy is considered to be independent variable. This ensures that the terminal conditions on altitude and velocity are strictly met. 3. PROBLEM FORMULATION The entry guidance problem is to obtain complete bank angle profile and corresponding trajectories from entry to the terminal phase at every instant of time based on the current state so as to satisfy terminal conditions and the path constraints accurately. 3.1 Control profile The primary control variable is chosen as the bank angle. A fixed angle of attack profile for CAV-H vehicle is considered as given in Lu (2014). The bank angle magnitude profile is considered to be linear function of current energy e e − e0 |σ(e)|= σ0 + (σf − σ0 ) (16) ef − e0 where, e0 and ef are energies at entry point and TAEM phase respectively. σ0 ≥ 0 is the parameter to be found. It is the control variable that optimizes the cost function such that equality and inequality constraints are satisfied. σf is a given constant as per the requirement of TAEM phase. Once, σ0 is obtained entire control profile is obtained. The magnitude of σ0 is determined by the hybrid guidance algorithm. The sign of bank angle is obtained by using bank reversal logic as described in Shen and Lu (2004). A crossrange parameter which is a function of current heading offset is defined. If this parameter exceeds a predefined velocity dependent deadband the bank angle is commanded to change its sign to reduce the current heading offset. 4. COST FUNCTION AND PATH CONSTRAINTS The main goal of optimization is to determine the initial bank angle σ0 in the assumed control profile. Once, σ0 is obtained, the complete control profile and state trajectories from entry point to terminal point are determined. The cost function consists of terminal conditions as described in equation (17). J = |s(ef ) − sf | (17) where, s(ef ) is the range to go at final energy ef . sf is according to the desired TAEM conditions. The cost function J has to be minimized to satisfy the terminal conditions accurately. Whenever the path constraint equations (12), (13), (14) are violated a very high penalty factor is added to cost function. Thus, PIO ensures that the path constraints are strictly satisfied. The terminal conditions on altitude and velocity are indirectly satisfied, as the terminal energy is achieved. Further, the Gauss-Newton 341

341

method considers constraint on range to go as cost function to be minimized as in equation (18). 1 1 f = z 2 = [s(ef ) − sf ]2 (18) 2 2 5. HYBRID GUIDANCE ALGORITHM 5.1 Phase 1: PIO The PIO is a swarm intelligence method inspired from a natural phenomena. It mimics the motion of a flock of pigeons, reaching their home using magnetic field, the sun, and landmarks as described by Duan and Qiao (2014). In PIO algorithm, each pigeon uses the experience of entire flock in the search space rather than its own experience Duan and Qiao (2014). Pigeons use map, compass and landmark operators for navigation. The map and compass operators are used in their initial phase of their journey and then shift to landmark operator. In map and compass operator, all the pigeons try to adjust and follow the best position in the flock. Since PIO algorithm is used only to generate a reasonable initial guess for Gauss Newton method, only map and compass operators are used till the predefined cost function is achieved in the hybrid algorithm. PIO algorithm can be summarized as following (1) Set the population of pigeons Np , the number of control variables to be optimized n and The number of iterations in the algorithm i. (2) Generate Np pigeons with random position X(q) and velocity V (q) given by X(q) = [x1 (q), x2 (q), ...., xn (q)]

(19)

(20) V (q) = [v1 (q), v2 (q), ....., vn (q)] where, q = 1, 2, ...Np . (3) Each pigeon represents a possible solution and corresponds to a cost function given by equation (17). In each iteration update the position vector X(q) and the velocity vector V (q) using the following equations V (t) (q) = V (t−1) (q).e−Rt + rand.(P (t−1) − X (t−1) (q)) (21) X (t) (q) = X (t−1) (q) + V (t) (q) (22) where, t is given iteration, R represents the map and compass factor that influences the velocity of each pigeon, P (t−1) denotes the best position in the pigeon flock, and rand is a random number within [0,1]. (4) Evaluate the cost function of each pigeon. Determine the current best position. (5) At the end of iterations, determine the global best position. Flowchart of PIO is given in Fig. 2. In each iteration, calculate the local best and global best. At the end of each iteration, condition given by equation (23) is checked. If the condition is satisfied, the second phase begins. P IO J P IO ≤ Jmax (23) P IO where, J is the the cost function corresponding to P IO is the the global best solution achieved so far and Jmax predefined cost function value.

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(k−1)

1)th iteration, σcmd is the magnitude of modified bank angle. The equations of motion are integrated from initial (k−1) energy e0 to ef . The value of cost function f (σcmd ) using equation (18) is evaluated. If the stopping criteria in equation (28) is not met, bank angle is updated using equation (27), as shown below. (k)

(k−1)

σcmd = σcmd − λ

(k−1)

∂f (σcmd )/∂σcmd (k−1)

[∂z(σcmd )/∂σcmd ]2    ∂f (σ (k) )   cmd  ≤   ∂σcmd 

Fig. 2. Flowchart of the PIO using mass and compass operator 5.2 Phase 2: Equilibrium glide constraint (EGC) and Gauss-Newton method The σbase is the converged bank angle obtained at the end of first phase. This σbase is further modified to σcmd by equation (24) given below. The altitude compensation technique described in Lu (2014) has been obtained considering energy as independent variable and is employed to avoid phugoid oscillations in altitude profile of high L/D ratio vehicles. L cos σcmd = L cos σbase − k1 (h˙ − h˙ QEGC ) (24) ˙ where h = V sin γ is the current altitude rate from the navigation system and h˙ QEGC = V sin γQEGC is the altitude rate required to follow the QEGC. The gain k1 > 0. It is considered to be linear function of velocity. The value of sinγQEGC is calculated using equations (25), (26)   1 1 sin γQEGC = (25) (V 2 /2)(βr cos σQEGC ) CL /CD where ∂ρ/∂r (26) βr (r) = ρ ρ is atmospheric density and is a function of the dimensionless r. βr is called scale height and is normalized with respect to radius of the Earth. The resulting magnitude of σcmd is then used to predict the trajectory considering its sign from bank reversal logic. The cost function in equation (18) is then evaluated and minimized. The σcmd is further updated using GaussNewton method till the stopping criteria is met. At (k − 342

(27) (28)

where,  > 0 is preselected small value and λ is the step (k) (k−1) size chosen such that f (σcmd ) < f (σcmd ). The partial derivatives in equations (27) and (28) are computed by the finite difference method. This process of updating the bank angle is continued till the cost function in equation (18) attains a minimum value and corresponding bank angle is the optimum value. This bank angle ensures that both the equilibrium glide and terminal constraints are satisfied simultaneously. The hybrid guidance algorithm is summarized as flowchart in Fig. 3. This guidance scheme can be repeated at every instant considering current state as the initial state and could be employed as closed loop guidance method. The PIO algorithm has randomness and it does not guarantee convergence within a fixed time. Gauss-Newton method is deterministic and guarantee convergence in a fixed time. But, the disadvantage is that it depends on initial guess. If a reasonable initial guess is not chosen, the Gauss Newton method may fail to converge. The Hybrid guidance algorithm combines both these methods so as to be insensitive to initial guess and have rapid convergence rate. In the hybrid guidance algorithm, PIO has been used to generate reasonable initial guess, by achieving the predefined cost function. The predefined cost function can be chosen to be high so that the inherent randomness in PIO algorithm does not effect the overall time taken to generate optimal bank angle. 6. SIMULATION RESULTS OF HYBRID GUIDANCE ALGORITHM The proposed hybrid guidance, has been applied to common aero vehicle (CAV-H) vehicle during its reentry phase. R . An ideal case, Simulations are coded in MATLAB 2012 where no disturbances and all parameters are certain, considering one guidance cycle from entry point as initial state to terminal state would be sufficient. The initial and final conditions are given in Tables 1 and 2 respectively. Taking energy as independent variable, equations of motion given in section 2, are integrated from initial energy to final energy. The total number of pigeons, the maximum number of iterations in PIO are set to Np = 10 and i = 10 respectively. The map and compass operator has factor R = 0.2. The evolutionary PIO algorithm is used till P IO the cost function Jmax = 0.5. In phase 1, PIO satisfies the equation (23) in 4 iterations. The corresponding bank angle value σbase = 72.3571◦ is given as the initial guess to Gauss-Newton method in the phase 2. The aerodynamic data for CAV-H vehicle is in Phillips (2003). The limits on bank angle boundary are σmin = −90◦ and σmax = 90◦ .

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corresponding to case 1, the sudden decrease in magnitude of bank angle during initial phase results in larger vertical component of lift allowing vehicle to fly at higher altitudes compared to case 2. As density is low at higher altitudes, heat rate is also reduced. Later, the gradual increase in bank angle magnitude beyond the corresponding bank angle in case 2, results in decreasing vertical component of lift. Thus, vehicle flies in denser atmosphere. Once, the vehicle attains equilibrium glide condition it continues to be in that state. From Fig. 4, it is clear that terminal bank angle σf = 60◦ is accurately met in both the cases. Without incorporating EGC, the trajectory has phugoid oscillations as shown in Fig. 5. These oscillations are due to lack of coordination between bank angle control and altitude changes. The Altitude compensation that has been employed for achieving EGC has damped the phugoid oscillations in the entry trajectory. The resulting flight path angles are non-oscillatory and have small negative values as seen in Fig.9. Figs. 6, 8, 7 show that the path constraints are satis100

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REFERENCES

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Fig. 9. Flight path angle variations 7. CONCLUSION The proposed hybrid guidance algorithm makes use of the advantages of PIO and Gauss-Newton method to adhere a better performance. The hybrid algorithm is independent of initial guess and simple to formulate. The number of control variables to be found is kept minimum. This algorithm has additional flexibility to incorporate equilibrium glide constraint by using altitude rate compensation in the second phase of the algorithm. The path and terminal constraints are satisfied accurately. Future work includes implementation of methods that constrain heat rate to lower levels. 344

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