NUMERICAL CALCULATION OF OPTIMAL3D-TRAJECTORIES FOR THE MANEUVERABLE AIRCRAFT

A.M. Shmatkov∗, A.E. Utemov∗, Yu.N. Zhelnin∗∗, F.L. Chernousko∗∗Institute for Problems in Mechanics (IPMech), Russian Academy of Sciences,

∗∗Central Aerohydrodynamic Institute (TsAGI), Russia

Keywords: Maneuverable Aircraft, 3D-Trajectories, Optimal Control Problem

Abstract

A problem of how to move the center of gravityof the maneuverable aircraft from one given pointin the three-dimensional space to another point asquickly as possible with the fixed vectors of therespective velocities is considered. The exam-ple is given for the case when the velocity vectorequals the initial one at the end of motion.

1 Introduction

A great number of works published both in Rus-sia and abroad deal with aircraft flight control.Most elaborated are the problems related to au-tomatic control methods used to stabilize themotion parameters set by the pilot (see, for in-stance, [1, 2]). The involved problems of choos-ing the optimal flight trajectory using classicalvariational calculus methods, the maximum prin-ciple and direct numerical optimization methodsare mostly [3, 4] connected with searching forthe best parameters of cruising flight regimes, in-creasing the flight range, reducing the fuel con-sumption, finding optimal flight profiles, 4-Dnavigation, etc. All these issues are first ofall critical for performance optimization of non-maneuverable aircraft [5].

Much fewer works deal with optimization ofcharacteristics of a maneuverable aircraft, basi-cally limiting to a rather narrow problem of op-timizing the angular velocity of the bank in thehorizontal plane with no restrictions on the coor-dinates of the aircraft at the end of the maneu-ver (the problem with the free right end). A suf-

ficiently complete solution to this problem wasobtained in [6–8]. Trajectories that are lying inthe vertical plane and cannot be reduced to con-ventional civil aviation trajectories (change of theflight height, approach, etc.) were principallystudied for the space industry (see, for instance,[9]).

Current aircraft development and improve-ment needs require solving problems to improveaircraft characteristics for a wider class of trajec-tories both plane and 3-D. The latter is for theclass of maneuverable aircraft that are to perform,according to their specialization, various maneu-vers [10], with optimality of the latter being sig-nificantly important, for instance, in air combat[11–14]. This, in turn, makes it significantlymore difficult to state the optimal control prob-lem. It becomes 3-D while the positions of coor-dinates of the aircraft and its flight direction at theend of the maneuver should satisfy the particu-lar, given conditions (the problem with fixed rightend). The results of the respective studies are ofabsolute interest since they help find whether theoptimal solution is efficient for typical maneuverssuch as loop, oblique loop, split S, half-roll, upline, etc. as compared to conventional ways oftheir execution and estimate whether the optimalcontrol can be implemented in practice.

One of the principal research objectives inthis direction is to form control algorithms forthe aircraft when it is maneuvering based on so-lutions to the optimization problem that can beused in the onboard intellectual pilot support sys-tem. The latter must provide the pilot with thepossibility of optimal control of the aircraft when

1

A.M. SHMATKOV, A.E. UTEMOV, YU.N. ZHELNIN, F.L. CHERNOUSKO

it performs a maneuver. At present, such systemsare becoming increasingly common in aviationsince onboard hardware and software of modernaircraft allow implementing sufficiently compli-cated control algorithms. The experience of thelatest conflicts [15] shows that during air fights“enemies approached at almost double soundspeed. The increased pace of combat still had therequirement that, if not met, made success impos-sible, viz. forestalling the enemy as they attack.The pilot should have thought and acted muchfaster. The rapid nature of fight contradicted theabruptly increased scope of work the pilot didin a combat flight.” In-cabin actions needed tobe combined with piloting, air situation evalua-tion, and tracking the enemy. As a result, thepilot reached the limits of his psychophysiologicabilities. The conflict between thinking and therough work [15] required “developing expertiseon-board systems to assist the pilot in fight plan-ning and decision-making in the challenging airsituation.” In particular [15], the surprise attackof the enemy “makes him instinctively, withoutthinking it over” perform defensive maneuverspracticed during trainings. By [15], in this case,the best are the “loop” or slanted turn (“obliqueloop”), and one can make next steps in the fight“not before they drastically evade the enemy’s at-tack”.

In this work, we calculate the minimum timeloop maneuver based on the Pontryagin max-imum principle [16]. This problem for two-dimensional case was studied in [17] (see also[18] and references in [19]). Another thing thatmakes this work different from [17] is that all cal-culations in [17] were performed given the as-sumption that the speed value at the end of themaneuver can be arbitrary. As a result, this valuewas almost always significantly lower than theinitial one since the kinetic energy storage wasused to complete the motion as soon as possible.However, it is critical not to lose speed both whenevading the enemy and when tracking the target[20].

2 Motion Equations

Since the form of the motion equations of the air-craft we use in this work is not very common,we briefly describe the way they are derived fol-lowing [21, 22]. We also introduce the values ofconstants needed in calculations.

Neglecting the wind, the Earth rotation andits surface curvature, we consider the centerof gravity of the aircraft moving in the three-dimensional space with respect to a terrestrialfixed right Cartesian rectangular inertial coordi-nate system OXY Z, with its axes OX and OZ ly-ing in the horizontal plane. We have

mr = G+R+P, m = const

r = v(t) ∈ R3, |v|=V ≥ const > 0(1)

Here, m is the mass of the aircraft, which we con-sider constant, r is the radius vector of the centermass of the aircraft in the system OXY Z with thecomponents x, y and z, v is the velocity vector,G is the gravitation force, R is the principal vec-tor of aerodynamic forces, and P is the tractiveforce of the engines. By the above said, we con-sider the value V to be always sufficiently highand one can neglect the restriction on it.

We put the gravitation field to be homoge-neous so that

G = col(0,−mg,0), |G|= mg

g = 9.81 m/s2(2)

where g is the acceleration of gravity and colstands for the column vector. We consider theatmosphere to be isothermal with the followingapproximation dependence of the mass density ofthe air ρ on the height y

ρ = ρ0 exp(−y/h)

ρ0 = 9.81 ·0.125kg/m3 = 1.22625kg/m3

h = 104 m

(3)

The vector R in (1) is generally considered as thesum of three vectors

R = Rx +Ry +Rz (4)

2

NUMERICAL CALCULATION OF OPTIMAL 3D-TRAJECTORIESFOR THE MANEUVERABLE AIRCRAFT

where Rx is the drag force, Ry is the lifting force,and Rz is the lateral force. The vector Rx is oppo-site in direction to the velocity vector v, the vectorRy lies in the vertical symmetry plane of the air-craft and is orthogonal to Rx, and the vector Rz isorthogonal to Rx and Ry. We consider the aircraftto move with no slip, i.e., v is always in the sameplane as Ry. Then, we can put Rz ≡ 0.

We assume that the lifting force Ry can bechanged instantaneously both in its value and di-rection, and |Ry| can be expressed via a dimen-sionless scalar lift coefficient Cy

|Ry|=CyqS, 0≤Cy <=Cmaxy = 1.5

q = ρv2/2(5)

where q is the dynamic pressure and S is the wingarea of the aircraft. We explain the physical senseof the variable Cmax

y . During the flight, the pilotcan change the variable Ry by varying the angleof attack α, i.e., the angle between the longitu-dinal axis of the aircraft and the velocity vectorprojected onto its vertical symmetry plane. Oneknows that the coefficient Cy is linear with re-spect to α for subsonic speeds and |α|. 15◦. For|α|& 15◦, this dependence becomes significantlynonlinear, which is caused by flow separation onthe wing that worsens stability and controllabil-ity of the aircraft with the subsequent stall. Thismeans that the value Cy should not exceed somemaximal value Cmax

y during the aircraft operation.The module of the drag force can be written

using the dimensionless scalar drag coefficient Cxas

|Rx|=CxqS, Cx =Cx0 +Cxi (6)

The scalar Cx0 is the zero-lift drag coefficient.Its value depends on air viscosity and compress-ibility and corresponds to passive drag, i.e., whenit is independent of the lift. Hence, the part of thedrag force that depends on the lift is described us-ing Cxi , which is the lift-induced drag coefficient.For subsonic speeds and symmetric aircraft con-figuration and linear dependence of Cy on α thefunction Cx(Cy) called a polar line is a parabola.We put

Cx =Cx0 +AC2y ,

Cx0 = 0.025, A = 0.14(7)

where A is the airplane efficiency factor. In otherwords, by (6), we put Cxi = AC2

y . Note that thequadratic dependence can be applied in engineer-ing practice for non-symmetric configurations ofthe aircraft for transonic speeds.

We assume that the thrust vector P is alwaysdirected along the velocity vector v. The variable|P| is bounded by the minimal and maximal ad-missible power of the engines

Pmin ≤ |P| ≤ Pmax,Pmin

|G|=

12

Pmax

|G|=−B1y+B2V +B3 (8)

B1 = 10−4 m−1, B2 = 0.002s/m, B3 = 1

Here, we use the linear approximation of the de-pendence Pmax on the altitude y and the value ofthe speed V . Note that Pmin is chosen to equalabout half of Pmax. We introduce the scalar u bythe formulas

Up =g

2|G|(Pmax +Pmin)

Um =g

2|G|(Pmax−Pmin) (9)

|P|m

=Up +uUm, |u| ≤ 1

This transformation allows us to move fromthe variable |P| with bounds that depend on thephase variables to the variable u whose boundsdo not depend on the phase variables.

The ratio |G|/S is called the unit wingload. We consider it constant |G|/S = 9.81×300 N/m2. We divide both parts of the differen-tial equation for r in (1) by m. Using expressions(2) and (4)–(9), we have

r = v

v =Gm+(Up +uUm−

(Cx0 +AC2

y)

qk)

τ+Cyqkν

τ,ν ∈ R3, τ =vV, ν =

Ry

|Ry|, (ν,τ) = 0 (10)

k =Sm

=1

300m2/kg

where (·, ·) stands for the scalar product of twovectors.

3

A.M. SHMATKOV, A.E. UTEMOV, YU.N. ZHELNIN, F.L. CHERNOUSKO

We introduce the load factor

ny =|Ry||G|

=CyqSgm

=Cyqk

g

0≤ ny ≤ nmaxy = 8

(11)

It is bounded by the constructive features ofthe aircraft and physiological abilities of the pi-lot. Note that the latter are quite significant (see,for instance, [23]) therefore the condition nmax

y =8 actually reflects the limit capabilities of the air-craft. Similarly, we introduce the tangential loadfactor

nx =|P|− |Rx||G|

(12)

Using (11) and (12), we can write differentialequations (10) as

r = v, v = fg +gnxτ+gnyν

fg =Gm, | fg|= g

(13)

3 Optimal Control Problem in the Three Di-mensional Case

Taking into account (5), (8), (11) and (13), wecan re-write (10) as

r = v

v =(Up +uUm−qkCx0−qk|cl|2A

)τ+qkcl + fg

|u| ≤ 1, cl =Cyν (14)

|cl|=Cy ≤Cmaxy , |cl|qk ≤ gnmax

y

Choosing the vector cl and the scalar u, weneed to spend minimal time T to move the centerof gravity of the aircraft from its initial state

r(0) = col(x0,y0,z0), v(0) = col(v0x,v0y,v0z)(15)

to its final state

r(T )= col(xT ,yT ,zT ), v(T )= col(vT x,vTy,vT z)(16)

Without loss of generality, we can alwaystake x0 = z0 = 0.

We introduce the vectors ψv and ψr conjugateto the vectors v and r, respectively. We decom-pose ψv into the components collinear to τ and ν.We have

ψvτ = (ψv,τ) , ψvν = ψv−ψvττ (17)

We compose the Hamiltonian

H =(Up +uUm−qkCx0−qk|cl|2A

)ψvτ+

+qk (ψvν,cl)+(ψv, fg)+(ψr,v)+

+λ(|cl|qk−gnmax

y)

(18)where λ is the additional scalar undeterminedLagrange multiplier introduced due to the phaseconstraint on the control in (14). By the Pontrya-gin maximum principle [16], we find the max-imum of (18) with respect to u and cl . Sincelater in this section we use only the functionsthat satisfy the necessary optimality conditions,we can preserve the designations for them thatwere used above. In particular, the controls foundas a result of searching for the extremum willbe still denoted by u and cl . Substituting therespective expressions into the phase constraint|cl|qk ≤ gnmax

y , we have

λ =2Agnmax

y

qkψvτ−|ψvν| (19)

The optimal values cl should be calculated by thefollowing algorithm

if λ < 0 andgnmax

y

qk≤Cmax

y , then |cl|=gnmax

y

qkelse

χ =|ψvν|2Aψvτ

if χ < 0 or χ >Cmaxy , then |cl|=Cmax

y

else |cl|= χ

cl = |cl|ψvν

|ψvν|(20)

We have for the optimal u

u = sign(ψv,v) (21)

4

NUMERICAL CALCULATION OF OPTIMAL 3D-TRAJECTORIESFOR THE MANEUVERABLE AIRCRAFT

Since the differential equations for ψv and ψrare very cumbersome, we cannot give them here.The respective C++ algorithm was obtained us-ing the MAPLE system by the known formulas[16]. The exception is

ψrx = 0, ψrz = 0 (22)

since by (18) the coordinates of the center ofgravity x and z are not included in the functionH explicitly.

We performed numerical integration of dif-ferential equations for phase and conjugate vari-ables in the neighborhood of the regular pointsby the typical fourth-order accurate Runge-Kuttamethod with the variable step and the check termin the England form [24]. We applied the first-order Euler method with the step 10−13 s on thetime intervals of about 10−13 s during which thefirst derivative of the functions to be calculatedjumped. For additional accuracy control, we usedthe values of the Hamiltonian H that, by [16],should preserve its initial value. As a result, wemanaged to ensure conditions (16) are met up tothe accuracy of less than 0.1 m with respect tospatial variables and less than 0.1 m/s with re-spect to projections of the velocity vector for therelative integration accuracy of about 10−8 andabsolute accuracies with respect to time and com-ponents of the vectors r and v of about 10−13 s,10−13 m, and 10−13 m/s, respectively.

Two-dimensional results can be found in[25].

4 Example of Numerical Calculation

As an example, we consider the case with the fol-lowing initial and final phase points (15) and (16)

r(0) = col(0,5000m,0)v(0) = col(250m/s,0,0)r(T ) = col(0,5000m,500m)v(T ) = col(250m/s,0,0)

(23)

The results of calculations are shown in thefigures.

5 Conclusion

We studied the two-point three-dimensional op-timal performance problem for (14). We found

Fig. 1 The obtained 3D-trajectory. The optimaltime for considered case is T = 24.98 s

Fig. 2 The projection of the trajectory to the ver-tical plane OXY

Fig. 3 The projection of the trajectory to the hor-izontal plane OXZ

5

A.M. SHMATKOV, A.E. UTEMOV, YU.N. ZHELNIN, F.L. CHERNOUSKO

Fig. 4 The time dependence of the module of thespeed of the aircraft center of gravity V

Fig. 5 The tangential load factor nx and load fac-tor ny depending on time

Fig. 6 The scalar control u depending on time

Fig. 7 The lift coefficient Cy depending on time

the respective optimal controls (20) and (21). Weconstructed the numerical solutions for boundaryconditions (23).

References

[1] J. H. Blakelock, “Automatic Control of Aircraftand Missiles,” J. Wiley & Sons, 2nd Ed, NewYork, 1991.

[2] V. G. Vorob’ev and S. V. Kuznetsov, “AutomaticControl of Aircraft Flights,” Transport, Moscow,1995 [in Russian].

[3] A. Miele, T. Wang, H. Wang, and W. W. Melvin,“Optimal Penetration Landing Trajectories inthe Presence of Windshear,” Journal Optimiza-tion Theory and Applications, vol. 57, no. 1, pp.1–40, 1988.

[4] A. Benoıt and S. Swierstra, “A Simulation Fa-cility for Assessing the Next Generation of 4-DAir Traffic Control Procedures,” in Proc. of the15th Congr. Intern. Council of the AeronauticalSciences, London, 1986, ICAS-86-3.4.1.

[5] A. A. Umnov, “Designing Onboard ControlComplexes,” SPbGUAP, St. Petersburg, 2000[in Russian].

[6] V. P. Kuz’min, “Optimal Turn of the Aircraftin the Horizontal Plane,” Uch. Zap. Tsentr.Aerogidrodin. Inst. (Proc. of Zhukovskii CentralInstitute of Aerohydrodynamics) vol. 8, no. 1,pp. 70–78, 1977.

[7] B. Järmark, “Minimum Time Turning,” in Proc.of Atmospheric Flight Mechanics Conf., 12th,Snowmass, Colorado, 1985, Technical PapersA85–43826 21–08, AIAA Paper 1985–1780(New York, 1985).

[8] B. Järmark, “Optimal Turns with Altitude Vari-ations,” in Proc. of 11st AIAA Applied Aero-

6

NUMERICAL CALCULATION OF OPTIMAL 3D-TRAJECTORIESFOR THE MANEUVERABLE AIRCRAFT

dynamics Conf., Monterey, California, 1993,AIAA Paper 1993–3658 (New York, 1993).

[9] Yu. N. Lazarev, “Controlling Trajectories ofSpace Aircrafts,” Samar. nauch. tsentr RAN,Samara, 2007 [in Russian].

[10] D. C. Sederstrom, N. R. Zagalsky, and R. C.McLane, “Energy/Energy Rate Meter for En-ergy Management in Flight,” Final report pre-pared for Office of Naval Research, AD–763450, February 1973.

[11] R. K. Mehra, R. B. Washburn, S. Sajan, and J. V.Carrol, “A Study of the Application of SingularPerturbation Theory,” NASA CR–3167 (1978).

[12] Yu. N. Zhelnin, S. A. Shelekhov, and V. A.Yaroshevskii, “Calculation of the Probabilitiesof Air-Fight Out-comes,” Comput. Syst. Sci. Int.,vol. 43, p. 827, 2004.

[13] Yu. N. Zhelnin and A. E. Utemov, “Constructionof Barrier Surfaces in a Pursuit-Evasion GameProblem,” Comput. Syst. Sci. Int., vol. 44, p. 753,2005.

[14] K. Virtanen, J. Karelahti, and T. Raivio, “Model-ing Air Combat by a Moving Horizon InfluenceDiagram Game,” J. Guidance, Control, and Dy-namics, vol. 29, no. 5, pp. 1080–1091, 2006.

[15] V. K. Babich, “Air Combat (Origin and Evolu-tion),” Voenizdat, Moscow, 1991 [in Russian].

[16] L. S. Pontryagin, V. G. Boltyanskii, R. V.Gamkrelidze, and E. F. Mishchenko, “TheMathematical Theory of Optimal Processes”,Wiley-Interscience, New York, 1963.

[17] S. Uehara, “Theoretical Investigation of Min-imum Time Loop Maneuvers of Jet Aircraft,”Ph. D. Thesis, California Institute of Technol-ogy, Pasadena, California, 1974.

[18] S. Uehara, H. J. Stewart, and L. J. Wood, “Min-imum Time Loop Maneuvers of Jet Aircraft,” J.Aircraft, vol. 15, no. 8, pp. 449–455, 1978.

[19] S. Uehara, “Application of Optimal ControlTheory to Supersonic Fighter Maneuverability,”J. Japan Society for Aeronautical and Space Sci-ences, vol. 30, no. 340, pp. 238–251, 1982.

[20] V. K. Babich, “Aviation in Local Wars,” Voeniz-dat, Moscow, 1988 [in Russian].

[21] A. Miele, “Flight Mechanics,” vol. 1, Theoryof Flight Paths Addison-Wesley Publishing Com-pany, London, 1962.

[22] A. F. Bochkarev, V. V. Andreevskii, V. M.Belokonov, V. I. Klimov, and V. M. Turapin,

“Aeromechanics of the Aircraft: Flight Dynam-ics,” Mashinostroenie, Moscow, 1985 [in Rus-sian].

[23] N. T. Spark, “The Fastest Man on Earth,” Annalsof Improbable Research, vol. 9, no. 5, pp. 4–26,2003.

[24] O. B. Arushanyan and S. F. Zaletkin, “Numeri-cal Solution of Ordinary Differential EquationsUsing Fortran,” Mosk. Gos. Univ., Moscow,1990 [in Russian].

[25] Yu.N. Zhelnin, A.E. Utemov, and A.M.Shmatkov, “Minimum Time Loop Maneuverwith No Speed Loss,” J. of Computer and Sys-tems Sciences International, vol. 51, no. 6, pp.833–848, 2012.

Contact Author Email Address

• A.M. Shmatkov (shmatkov@ipmnet.ru)

• A.E. Utemov (utemov@yandex.ru)

Acknowledgments

This study was financially supported by the Rus-sian Foundation for Basic Research (projects no.13-01-00180, 14-08-00606, and 14-01-00476)and Program for Support of Leading Scien-tific Schools of the Russian Federation (NSh-369.2012.1).

Copyright Statement

The authors confirm that they, and/or their company or or-ganization, hold copyright on all of the original materialincluded in this paper. The authors also confirm that theyhave obtained permission, from the copyright holder of anythird party material included in this paper, to publish it aspart of their paper. The authors confirm that they give per-mission, or have obtained permission from the copyrightholder of this paper, for the publication and distribution ofthis paper as part of the ICAS 2014 proceedings or as indi-vidual off-prints from the proceedings.

7