Thrust-vectored differential turns

0005-1098/821050565-04503.00/0

Automatica, Vol. 18, No. 5, pp. 565-568, 1982 .

Pergamon Press Ltd. © 1982 International Federation of Automatic Control

Printed in Great Britain.

Brief Paper Thrust-vectored Differential Turns* HENRY J. KELLEY,t EUGENE M. CLIFFt and LEON LEFTONt Key Words--Game theory; optimal control; navigation; perturbation techniques; hovercraft. Abstract--Differential-turning games played with thrust vectoring as an aid to maneuvering are examined in an extension of the research of a companion paper which reports a study of thrust-vectored maneuvering in energy approximation. The benefits are found to be major when there is a sufficient margin of thrust over weight to permit hover. Families of turning duels between a conventional aircraft and a thrust-vectored opponent are described in a computational example.

Vehicle modeling The two aircraft are modeled in 'energy' approximation, the state of each described by specific-energy and heading variables. The difËerential equations of state are equations (2) and (3) of the companion paper (Cliff, Kelley and Lefton, 1982). The lift coefficient and angle of attack are limited to onset-of-stall values (~L and 6; linear CL versus a is assumed. The omission of nonlinear post-stall aerodynamic characteristics is not important in energy modeling with thrust vectoring because high-a maneuvering coincides with near-hover and attendant low dynamic pressure. Turn capability in near-hover is idealized as instantaneous in this and the companion paper. The essence of the grasshopper-like ascent with quick near-hover turn at the top, which has been noted in Harrier mock-combat tests (Brown, 1973; Anonymous, 1974), is thought to be caught in the energy-state instant-turn model, however. More refined modeling featuring a turn-rate bound might be of interest in future work; so would thrust-direction bounds, omitted in this exploratory study.

Introduction Tins PAPER deals with differential-turn studies of aircraft incorporating thrust vectoring. The research is based upon modifications of existing computer programs to incorporate this feature. One of these generates 'hodograph' data for conventional aircraft (Lefton and Kelley, 1975). The other is an 'energy-turn' program that calculates optimal energyheading transients (Kelley and Lefton, 1972; Kelley, 1973; Lefton and Krenkel, 1976), families of which are central to the computation of differential-turn maneuvering of aircraft in circling encounters (Kelley, 1975a, b, 1976; Kelley and Lefton, 1977). The modifications are complete and represent technical capability equivalent to that previously existing for the computation of energy-turn and differential-turn solution families for conventional aircraft configurations. The dynamical modeling of the differential-turn approach is intermediate in complexity between the energy-rate/turnrate overlay technique of comparison and digital simulation of combat between two point-mass-modeled aircraft. The hodograph and dual-hodograph devices in conjunction with energy modeling can be thought of, in fact, as extended energy-maneuverability schemes. The construction of differential-turn trajectory families is superior, with respect to control logic, to conventional all-digital simulations employing ad hoc control logic, although approximate in respect to vehicle kinematics. For configurations featuring an additional control (i.e. thrust vectoring, variable sweep), the optimization approach has particular appeal in determining the setting of the additional control; hence, it contrasts with its competitors even more than usual, as these all suffer more or less from the curse of too much freedom. A companion paper (Cliff, Kelley and Lefton, 1982) discussed optimal turning flight in energy approximation; this is an extension of Kelley (1973) to include thrust vectoring. In the following, the construction of barrier surfaces for thrustvectored versus conventional aircraft will be discussed and illustrated with an example. For perspective on the differential-turn model, the reader is referred to the excellent review paper of Ardema (1981).

The differential-turning game model Since only heading difference, AX = X2-Xl, is of consequence, the turning game state vector is a three-vector, Et, E:, AX. Capture consists of closing angularly (driving AX to zero) with sufficient pursuer energy to follow upward in an evasive zoom. In energy approximation, this amounts to equaling or exceeding the evader's 'loft-ceiling', the highest altitude at a given specific energy for which vertical equilibrium can be maintained. If weapon-envelope effects are modeled, gaps in heading and loft-ceiling can be partially filled by the upward reach and angular reach of weaponry. When capture is inevitable, assuming no pursuer errors, minimax time to capture is the performance index. Trajectory pairs for the limiting case of time-open capture are of particular interest, for their family forms a so-called 'barrier surface', separating regions of capture and escape. It is known from previous studies that diferential-turning tactics tend to be rather sharply tailored to differences in the characteristics of the opposing craft and their weaponry, and are sometimes rather intricate. The simplest results emerge for so-called 'fast-evader' modeling, in which the evading craft is assumed to have an advantage over the pursuer in terms of the sum of information processing and control actuation lags, so that the pursuer is ineffective in altitude-matching and cannot drive his opponent aloft after closing angularly (Kelley, 1975a, b, 1976; Kelley and Lefton, 1977). Some computations using this modeling, in which two similar aircraft of conventional and familiar design (F-5A) are pitted against one another, one of them having been granted thrust-vectoring capability, will be discussed.

Barrier surfaces for fast-evader modeling

*Received 23 January 1981; revised 27 October 1981; revised 18 April 1982. The original version of this paper was presented at the Joint Automatic Control Conference which was held in San Francisco, California, U.S.A. during August 1980. This paper was recommended for pubfication in revised form by associate editor E. Kriendler. tOptimization Incorporated, Blacksburg, VA 24060, U.S.A.

With static-thrust-to-weight ratio well under unity ( ~ 0.69 for the F-SA), maneuvering takes place mainly in the conventional energy range and the benefits of thrust-vectoring turn out to be unspectacular. There is what might be termed an 'efficiency' effect, on the order of 2--4% improvement in maximum energy rate and maximum sustainable turn rate resulting from allocating normal force optimally between lift 565

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and thrust component. The increase in maximum instantaneous turn rate is more substantial, being on the order of 10% in the conventional range and as much as 15% at low energies. This is realized by use of fully-vectored (900) thrust, to the accompaniment of energy losses at high rates. Gains similarly obtained in loft-ceiling amount to a few hundred feet. No account has been taken, in these values, of weight or drag penalties incurred by the thrust-vectoring feature. Vertical-take-off-and-landing (VTOL) capability comes with static-thrust-to-weight in excess of unity, and this is, of course, often the principal reason for incorporating thrust vectoring into a design. With (T/W)> 1 comes a spectacular increase in maneuverability at energies low enough to permit hover; in energy approximation, this amounts to instantaneous turn capability. When the thrust-vectoring-system design permits smooth and continuous transition between hovering and forward flight, this effect completely dominates maneuvering at low energies and influences tactics at high energies as well. In the differential-turn results to be presented for conventional versus thrust-vectored F-5A, the thrust levels for both aircraft have been increased 60% uniformly over the entire Mach number/altitude regime, with the result that static thrust exceeds weight at altitudes up to 3800 ft.* The estimation of the weight and drag penalties incurred in the acquisition of thrust-vectoring capability is difficult, since 'simple' thrust vectoring (Kestrel, Harrier) imposes drastic restrictions upon design. So does the alternative of thrust-augmentation-of-lift via blowing, sucking, and/or ejector-jet effect. In the example at hand, this problem will be avoided by means of an arbitrary assumption: the award of a weight advantage to the unvectored opponent, 5% in the first instance. It will be seen that this arbitrariness does not much obscure the main points of the comparison to be drawn. Designate the various aircraft configurations as follows: CI: C2: C3C4:

basic F-5A F-5A F-5A

F-5A; with thrust increased 60%; with thrust increased 60% and vectorable; with thrust increased 60%, weight 95% of basic.

C3 has 10.6% more thrust than weight at sea level and can hover at altitudes up to 3800 ft, which is the maximum hover energy. In energy approximation, C3 has instantaneous turn capability at specific energies up to 3800 ft. A comparison of maximum sustainable turn rates of C3 and C4 is shown in Fig. 1 and a comparison of maximum instantaneous turn rates in Fig. 2, both with specific energies matched for equal loft-ceilings. C4 is seen to have a slight sustainable-turn-rate superiority over most of the conventional range as a result of the assumed weight advantage; it disappears abruptly and completely below C3s maximum hover energy of 3800 ft. C3 has a substantial ( ~ 12%) maximum-instantaneous-turn-rate advantage over the conventional range and an overwhelming one in C3s hover range. With C3 assigned pursuer and C4 evader, the situation in the joint state-space Ec4, Ec~, AX is as shown in Fig. 3. Capture must take place in the subspace AX = 0 on or above the matched-loft-ceiling curve, the target set. The usable part of the target set, caded the capture set, is that portion in which the maximum instantaneous turn rate of the pursuer equals or exceeds that of the evader. The shaded region of Fig. 3 is the capture set; the black lower-triangular portion of it has instant capture for any AX; thus it extends along the AX axis, making the capture set, in essence, three-dimensional. Trajectory pairs emanating from the matched-loftceiling curve above C3s maximum hover energy form a barrier surface, dividing the joint state space into regions of successful pursuit and successful evasion. The dual hodograph figures drawn at matched-loft-ceiling energies intersect at least once, furnishing at least one root of H = 0, thus spawning barrier trajectories (Kelley, 1975a, b; Kelley and Lefton, 1977). The trajectories are single-arc affairs, having * I000 ft ~ 305 m.

only ordinary (Erdman) corners, such as thrust switchings, for the conventional opponent. The barrier surface generated, represented by contours of AX, is of conventional type for pursuer energies above and to the right of a curve in energy space, the projection on the plane Ax = 0 of the limiting trajectory which emanates from the maximumhover-energy point on the matched-loft-ceiling locus. Below and to the left of this curve (shaded region), capture takes place no matter what the initial Ax separation; however, not instantly--there is a delay until the trajectory pair in ques-

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FIG. 1. C3 and C4 sustainable turn rates versus C3 specific energy at matched loft-ceilings.

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FIG. 2. C3 and C4 maximum instantaneous turn rates versus C3 specific energy at matched loft-ceilings.

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FIG. 3. Barrier surface. Pursuer C3; evader C4.

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0

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FIG. 4. Capture

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set and barrier surface. evader C3.

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tion enters the black triangular instant-capture region. The thrust-vectored pursuer can close angularly and capture at high energies only with a substantial energy advantage, and then only through small angles, fractions of a radian. Thrust vectoring contributes to his instantaneous-turn-rate advantage, but not decisively. At low energies, however, the advantage of thrust vectoring is overwhelming. The evasion tactics of the conventional aircraft C4 emphasize gaining energy for a zoom disengagement, although some turning is involved. It is worth noting, in this connection, that the modeling for these computations was ‘point-capture’, i.e. heading and altitude match, no weaponry-allowance mismatch of either. With C4 as pursuer, the capture set lies above the matched-loft-ceiling curve and is the very small darkened triangular region (Fig. 4). The cross-hatched plus darkened regions of Fig. 4 would comprise the capture set were C3 prohibited from hovering and escaping by instant turn, as he can do in the cross-hatched region. Note that the energy scales of this figure extend to 60 000 ft and that there is considerable action at high evader energies. The basic evasion technique is rapid energy loss with thrust vectored 180”thrust reversal. The conventional aircraft’s basic pursuit tactic is hard turning at his sea-level corner-velocity energy, 6400 ft-singular control (Cliff, Kelley and Lefton, 1982). A preliminary adjustment of energy to this value is carried out via full or zero throttle depending upon whether the initial energy is lower or higher, respectively. The thrust-vectored aircraft’s main vulnerability is to be caught at high energy by an opponent whose energy is already in his own favorable range for maneuvering. The dashed curves in Fig. 4 denote loci of jump discontinuities in first partial derivatives occasioned by transition between various generators; the left side of the capture-set triangle spawns a family, as do the upper and lower vertices; so does the intermediate point on the left side corresponding to singular pursuer energy. For thrust-to-weight ratio 1.61 for both aircraft and C4 as the pursuer again, the triangular capture set has shrunken to virtually a single point, yet the barrier surface has changed but very little from the one for 1.60 shown in Fig. 4. At a value of thrust-to-weight ratio slightly exceeding 1.61, however, (1.6107, in fact) the capture set has disappeared completely, and so has the barrier surface (i.e. there are no captures at all, except at initial time)! Weapon-reach effects Some computations were performed for the example just presented taking account of weapon-envelope effects, as in Kelley and Lefton (1977). The weapon envelope is idealized as * 10” off-boresight (semi-apex angle) and a reach of 2 statute miles.* The upward component of the reach, some *l Statute mile = 1069 m.

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FIG. 5. Barrier surface with weapon offsets. Pursuer C3; evader C4.

1834 ft, can be used to close the loft-ceiling gap for capture. Results for the thrust-vectored aircraft as pursuer are shown in Fig. 5. The heading-difference figures marked on the contours are in addition to the 10”. The barrier-surface contours are sketched from fragmentary trajectory data and are somewhat approximate. However, the substantial increase in effectiveness associated with weapon-envelope effects is clear. The main effect is the altitude offset provided in loft-ceiling match. Results for the thrust-vectored aircraft as evader are similar to those of Fig. 4 with 10” (0.1745 rad) added to the Ax values. An effect of the trend toward all-aspect weaponry is worth noting: the frequency of occurrence of ring-around-a-rosie turning duels is somewhat reduced since an increased percentage of nearly headon encounters are terminated before they can develop into turning duels. Conclusions The turning-game analysis indicates that any advantages realized from thrust-vectoring are minor unless near-hover maneuvering is permitted by T/W > 1, in which case they are major at low energies and affect tactics at high energies as well. Thrust-over-weight margin permitting hover at altitudes of a few thousand feet is an important parameter, values above a certain reasonable threshold afIording a sweeping improvement in defensive capability. Weaponreach effects are substantial, resulting in increased effectiveness for both conventional and thrust-vectored aircraft, but more so for the latter.

Acknowledgement-This research was supported under contract NAS 2-10261 for NASA Ames Research Center, Moffett field, California, U.S.A. References Anonymous (1974). Harrier in-flight thrust vectoring honed. Aviation Week and Space Technology, 10 June, pp. 32-35. Ardema, M. D. (1981). Air-to-air combat analysis:, review of differential-gaming approaches. Proceedings of the 1981 Joint Automatic Control Conference, Charlottesville, Va., 17-19 June, Paper TP-1B. Brown, D. A. (1973). Thrust vectoring to aid combat. Aoiation Week and Space Technology, 15 January, pp. 46-49. Cliff, E. M., H. J. Kelley and L. Lefton (1982). Thrustvectored energy turns. Automatica, 18, 559. Kelley, H. J. (1973). Aircraft maneuver optimization by reduced-order approximation. In C. T. Leondes (Ed.) Control and Dynamic Systems: Advances in Theory and Applications, Vol. 10, Academic Press, New York, PP. 131-178.

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Kelley, H. J. (1975a). Differential-turning optimality criteria. J. Aircra[t, 41. Kelley, H. J. (1975b). Differential-turning tactics. J. Aircraft, 930. Kelley, H. J. (1976). Dilferential-turn maneuvering. Autoraatica, 12, 257. Kelley, H. J. and L. Letton (1972). Supersonic aircraft energy turns. Automatica, 8, 575. Kelley, H. J. and L. Lefton (1977). Computation of differen-

tial-turning barrier surfaces. J. Spacecra[t and Rockets, 87. Lefton, L. and H. J. Kelley (1975). A user's guide to the aircraft energy-turn hodograph program. Analytical Mechanics Associates, Inc. Report No. 75-7. Lefton, L. and R. Krenkel (1976). A user's guide to the aircraft energy-turn and tandem-motion computer programs. Analytical Mechanics Associates, Inc. Report No. 75-26, revised.