High-Speed Aircraft Stability and Control Metrics ^†

Abstract

This review paper identifies key stability and control screening parameters needed to design low-risk, general-purpose high-speed aircraft. These derive from MIL-STD-8785C, MIL-STD-1797, and older AGARD reports, and are suitable for assessing conceptual high-speed vehicles. We demonstrate their applicability using published ground test, computation, and flight test data from the Bell X-2, North American X-15, Martin X-24A, Northrop HL-10, Lockheed Blackbird (YF-12/SR-71), and North American XB-70 as well as the Rockwell Space Shuttle Orbiter. The relative success of the X-15 and Blackbird and the performance limitations of the others indicate the need to scrutinize lateral-directional stability at the preliminary design phase. Our work reveals the need for strong bare-airframe static directional stability to obtain favorable flying qualities.

1. Introduction

Hypersonic aircraft, those that fly at least five times the speed of sound, present a challenging design problem. Practical vehicles must demonstrate broad capabilities for controllability across a wide speed/altitude envelope. Therefore, hypersonic aircraft share many design attributes in common with other high-speed aircraft. The Rockwell Space Shuttle Orbiter [1] and the North American X-15 rocket plane [2] are examples of two general-purpose hypersonic aircraft which successfully flew many missions. Their respective designers used different design strategies to achieve satisfactory controllability.

A critical objective of flying qualities research is to find methods that enable engineers to predict the behavior of new aircraft while they are in their preliminary design stage. In the United States, it has been many years since any AIAA journal published even a generalized review article regarding aircraft flying qualities [3,4]. In Europe, it has been more than 15 years since Steer and Cook [5] or Nangia [6] published on generalized supersonic flying qualities. In Asia, many authors developed a significant body of openly published work regarding guidance, navigation, and control (GN&C) strategies to address specific vehicle configurations; these have been collected in a review article by Ding, Yue, Chen and Si [7].

The recent interest in hypersonic aircraft development complicates matters, as no serious proposed concept will be “piloted” in the manner of a 1950s X-plane (for example, the Bell X-1, Bell X-2, or North American X-15). At the same time, it was during the 1950s and 1960s that United States Air Force (USAF) engineers developed many methods to predict the likelihood of unsatisfactory flying qualities. This culminated in the release of MIL-STD-8785 in its various editions [8,9,10]; this document provides guidance to ensure that engineers design aircraft to have specific “equivalent low-order system dynamic” response characteristics: notably frequencies, damping ratios, and time constants for pitching, rolling and yawing motions that correlate with favorable flying qualities.

As flight speeds increased, test pilots experienced unanticipated trouble. Many of these issues arose from “coupling dynamics”—unforeseen exchanges in kinetic energy between pitching, rolling and yawing motions—that were driven by inherent rigid airframe motions. New methods were needed to predict coupling onset [11,12] which were incorporated into MIL-STD-1797A [13]. That document is an encyclopedia of commentary, strategies, and criteria to develop successful “fly-by-wire” aircraft.

For the purposes of this paper, we identify stability and control screening methods that can help support the development of general-purpose “bank-to-turn” high-speed aircraft. These methods are suitable to encompass both powered and unpowered flights across a wide range of dynamic pressure ($\overline{q}$) at flight speeds where inertial effects dominate aircraft dynamics [10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25]. In Section 2, we explain some aerodynamic characteristics associated with “bank-to-turn” flight. In Section 3, we present powerful discriminators that separate low-risk and high-risk aerodynamic designs. In Section 4, we present case studies that demonstrate the predictive capabilities of these metrics when applied to a diverse collection of historical high-speed aircraft.

As this paper presents legacy flight test data, we will present our work in the customary unit system used to design and operate these aircraft: altitudes in feet, speeds in nautical miles per hour true airspeed (VKTAS) or equivalent airspeed (VKEAS), pressures in pounds-force-per-square-foot, reference areas in square feet and lengths in feet.

2. Hypersonic Aircraft vs. Missile Paradigm

“Hypersonic flight” describes operations under a broad tent of speeds, flow physics, and atmospheric properties that begins when a vehicle operates at freestream Mach numbers in excess of five. A defining element of hypersonic aerodynamics derives from the definition of the pressure coefficient $Cp={\displaystyle \frac{\left(p-p_{\infty }\right)}{\left(p_0-p_{\infty }\right)}}$ where p is the local pressure, p_∞ is the freestream static pressure, and p₀ is the freestream stagnation pressure. As the Mach number increases, so does the stagnation pressure. Because of this, the Cp which represents a perfect vacuum becomes a strong function of the freestream Mach number (M): $C{p}_{vac}=-{\displaystyle \frac{1.43}{M^2}}$ [26]. Because $C{p}_{vac}\ll -1$ and $C{p}_{stag}\approx +1$, the majority of lift at incompressible speeds derives from leeward surface suction. Conversely, at hypersonic speeds $C{p}_{vac}\to 0$; windward surface pressures must therefore dominate [27].

This trend also manifests itself in the Mach number dependence of lift; see Figure 1. At high speeds, the loss of leeward surface suction reduces the slope of the lift coefficient with respect to angle-of-attack, dCL/dα. As the freestream Mach number increases, an aircraft will have to trim at an ever-increasing α to maintain a constant CL.

A further byproduct of this trend manifests itself in the change in the magnitude of the directional stability derivative, dCn/dβ. In order to develop positive static directional stability, aircraft have a vertical projected area aft of their center-of-gravity (CG). The vertical tail produces a restoring moment when the aircraft rotates to the left or right. As the Mach number increases, the loss of the ability of the leeward side to develop suction manifests itself as a decline in directional stability; larger and/or wedge-shaped [28] vertical tail surfaces will be needed to stabilize an airframe in yaw.

Traditional aircraft GN&C strategies employ a “bank-to-turn” maneuvering philosophy where airframes have a clear orientation. The “top” is always the “leeward side”; the “bottom” is always the “windward side”. To turn, the GN&C system directs the aircraft to roll to a preferred bank angle (ϕ), tilting the lift vector to change the heading. While bank-to-turn aircraft may fly over a wide range of angle-of-attack (α), they always fly at minimal sideslip (β).

Hypersonic aircraft may operate far from the traditional “lift-equals-weight” ($n_Z$~1) paradigm; see Figure 2. Many X-15 flights took the aircraft to altitudes where the dynamic pressure dropped far below the “1-gee stall speed”; an essentially ballistic ($n_Z$~0) trajectory [29,30]. During decelerating reentry, the X-15 wings could generate so much lift ($n_Z$~3.8) to have the airframe “skip” back out of the atmosphere. Consequently, to enable descending and decelerating flight, the X-15 had to fly portions of its flight in a steep bank (ϕ~75°). Similarly, the Space Shuttle Orbiter also flew its hypersonic reentry at steep bank angles; ϕ > 60° [31]. Not only does this mitigate against atmospheric “skip”, but symmetric or asymmetric scheduled roll maneuvers allow the Orbiter to steer its hypersonic trajectory cross-range. Our subsequent metrics and analysis will demonstrate why the X-15 [29], and the Space Shuttle Orbiter [31] required their aerodynamic control to be augmented by reaction control jets (RCS) in order to obtain satisfactory flying qualities.

3. Flight Dynamics Issues Pertinent to High-Speed Flight

Fundamentally, aircraft have six degrees of freedom in motion: translations in the x, y, and z coordinate frames (either body aligned, earth aligned, or flight path aligned) and rotations about the x, y, and z axes (roll, pitch, and yaw); see Figure 3.

Larson [32] noted that high-speed vehicles exhibit “unconventional, configuration-dependent dynamic issues that impact … flying qualities … associated with configuration features set by propulsion and/or aerodynamic performance considerations essential to achieve hypersonic flight”. Consequently, “these characteristics are “unalterable” even though they may be directly or indirectly adverse from the standpoint of flight control and flying qualities” [32]. Larson [32] notes that high-speed/high-altitude aircraft exhibit novel dynamic modes not seen on slower aircraft and may be prone to both open-loop as well as closed-loop instabilities, which place additional demands on the control system.

Poor open-loop flying qualities may be improved through the use of feedback control. Augmented longitudinal stability arises when the controller schedules the “elevator” as a function of angle-of-attack or the integral of the pitch rate [20]. Augmented longitudinal damping can be implemented when the controller schedules the “elevator” as a function of the angle-of-attack rate, $\dot{\alpha }$, or the dimensionless pitch rate, q. McGruer notes that feedback strategies often have unintended consequences: “washout equalization” at low frequencies is required to prevent the feedback system from resisting pilot inputs, yet such a filter prevents the system from stabilizing a system where the inherent modes overlap pilot modes. Other feedback strategies prove intolerant to external disturbances. Ultimately, the real limitations of finite aerodynamic control power, actuator bandwidth, group-delay and controller noise sources limit the ability of feedback control systems to generally rectify all open-loop deficiencies.

This section documents screening metrics suitable for preliminary design, which will assess (1) longitudinal control power, (2) longitudinal and lateral/directional “equivalent low-order system dynamic” response characteristics, (3) the presence of favorable or unfavorable bandwidth constraints, as well as (4) the propensity to control couple, inertia coupling and/or frequency couple the longitudinal and lateral/directional rigid-body modes.

3.1. Longitudinal Control Power to Trim

A typical performance specification for flying qualities defines desired boundaries in terms of speed and altitude for both steady, level flight (n_z = 1) as well as maneuvering flight (n_z ≠ 1, climb/descend, accelerate/decelerate, and roll into and out of banked turns). The first goal of a control authority specification is to demonstrate trim across the entire flight envelope (a combination of weight, CG position, flight speed (M), altitude (ALT) and attitude (α and β)) [15,33]. Longitudinal trim occurs when the pitch control surface, the elevator, neutralizes the pitching moment about the CG:

$$Cm\equiv 0\approx Cm+{\displaystyle \frac{dCm}{delev}}\partial elev$$

(1)

Any available control power beyond trim may be used to (1) command pitch, roll, and yaw maneuvers, (2) damp oscillatory modes, or (3) augment inherent aerodynamic stability. A typical requirement might require the preliminary design team to establish static trim using no more than 75% of available control power, i.e., on a vehicle where the elevator is mechanically limited to ±30°, the designer needs to limit static trim to ±22.5°. This constraint governs an interplay between the fundamental static stability of the airframe (dCm/dα), which governs the controls-neutral pitching moment, and the size of the control surfaces, which governs the available control power (dCm/delev).

3.2. Rigid-Body Low-Order System Dynamic Modes

The concept of perfectly steady straight-and-level flight is a mirage; residual oscillations occur at their inherent Rigid-Body modes. The Short-Period mode is a porpoise-like motion in pitch where angle-of-attack varies with minimal impact on airspeed and altitude. Even though aircraft have inherent bilateral symmetry, lateral/directional control strategies are made difficult because aircraft lack inherent pendulum stability. For an aircraft to remain “top-side-up” it needs to be locked in a stable oscillation with simultaneous synchronized motions in yaw and roll, the Dutch-Roll mode.

Depending upon the aerodynamic configuration, surface stability and control characteristics of an airplane may prove satisfactory or unsatisfactory. Over the next few subsections, we show how the Rigid-Body behavior may lead to “wallowing” or “porpoising” flight and preclude holding bank angle and effecting heading change. Some of these deficiencies may be corrected through the use of a feedforward command augmentation system (CAS) or a feedback stability augmentation system (SAS). Larson [32] noted that many of these congenital issues prove unsolvable. Although modern flight control computers are far faster and more precise than human pilots, they are still limited by the finite bandwidth and control power limits of the physical airframe. Cybernetic control systems, whether classical, modern or AI-powered, can neither stabilize nor control a sufficiently unruly airframe.

3.3. Longitudinal Equivalent Low-Order System Dynamic Modes

The inherent Short-Period Rigid-Body oscillatory mode of the aircraft (frequency ${\omega }_{SP}$ in rad/s) at a given speed (M) and angle-of-attack (α) may be approximated as a one-degree-of-freedom system (see Figure 4), [33,34,35] as:

$${\omega }_{SP}\approx \sqrt{{\displaystyle \frac{-57.3\frac{dCm}{d\alpha }\overline{q}Sref\overline{c}}{I_{yy}}}}$$

(2)

where dCm/dα is given in terms of US industry customary units, per deg; $\overline{q}$ is the dynamic pressure; $Sref$ is the reference area; $\overline{c}$ is the reference chord; and $I_{yy}$ is the mass-moment-of-inertia in pitch, given in customary units of slug-ft². The frequency quickens in response to an increase in the negative magnitude of dCm/dα and/or a decrease in the mass-moment-of-inertia or an increase in the dynamic pressure.

For a basic aircraft with irreversible control surfaces, the value dCm/dα represents the inherent aerodynamic properties of the airframe. Aircraft may be fitted with a classical feedback control system that synthesizes an “apparent” static stability; by deflecting aerodynamic surfaces in response to sensed deviations from the desired trimmed angle-of-attack, the control system may synthesize an apparent dCm/dα.

Similarly, the Short-Period damping ratio may be estimated as:

$${\zeta }_{SP}\approx -{\displaystyle \frac{M_q+Z_a/U_1}{2{\omega }_{SP}}}$$

(3)

where

$$M_q={\displaystyle \frac{C{m}_q\overline{q}Sref{\overline{c}}^2}{2IyyVKTAS\left(\frac{6076}{3600}\right)}}$$

(4)

and

$${\displaystyle \frac{Z_a}{U_1}}\approx -{\displaystyle \frac{\overline{q}Sref}{mVKTAS\left(\frac{6076}{3600}\right)}}\left(57.3{\displaystyle \frac{dCL}{d\alpha }}\right)$$

(5)

where VKTAS is the aircraft true airspeed in nautical miles per hour; dCL/dα is given in terms of customary units, per deg; $m$ is the mass (in slugs); and Cm_q represents the variation in the pitching moment coefficient with dimensionless pitching rate. For a basic aircraft with irreversible control surfaces, the value of Cm_q represents the inherent vehicle aerodynamic properties [34,35]. By deflecting aerodynamic surfaces in response to sensed pitch-rate, classical control systems may synthesize an apparent Cm_q. Like dCL/dα, stick-fixed Cm_q is a function of the Mach number and tends to decline with increasing speed; see Figure 5.

The presence of the true airspeed in the denominator of the terms that control the damping ratio is important for high-speed flight because ζ is inversely proportional to VKTAS. Absent any sort of synthetic stability system, holding constant dynamic pressure but increasing true airspeed from 200-KTAS to 4000-KTAS (i.e., M = 0.3 → ~6), reduces the damping factor by a factor of at least 20; see Figure 6 to contrast the time history of a system with ζ~0.5 to ζ~0.025. An aircraft with inherent Short-Period pitch damping of ζ~0.5 would be easy to fly; one with ζ~0.025 will be so lightly damped as to be impossible to control without artificial pitch damping. This “feature” of the equations of motion means that even statically stable hypersonic airframes that inherently develop favorable stick-fixed rigid-body frequencies need closed-loop feedback augmentation.

3.4. Lateral-Directional Equivalent Low-Order System Dynamic Modes

Directional divergence is the byproduct of a statically unstable airplane in yaw. In the absence of a stable Dutch-Roll Mode arising from a strong dihedral effect (dCl/dβ < 0) occurring at high angles of attack sin(α) >> 0, a yaw perturbation will result in a continuing yaw out of the wind if dCn/dβ < 0, i.e., a spin [33,34,35].

The Dutch-Roll mode is typically a lightly damped oscillatory mode coupling roll and yaw [21,33,34,35]. If a statically stable aircraft encounters an initial disturbance that gives it a positive sideslip angle (β > 0) (i.e., a leftward disturbance), the aerodynamic rolling (dCl/dβ < 0) and yawing (dCn/dβ) moments due to sideslip will cause the right wing to rise and the nose will swing to the right. Oscillations will continue until aerodynamic damping causes the motion to die away.

We may estimate the Dutch-Roll frequency in rad/s by:

$${\omega }_{DR}\approx \sqrt{{\displaystyle \frac{57.3C_{n\beta dynamic}\overline{q}Srefb}{I_{zz}}}}$$

(6)

where

$$C_{n\beta dynamic}={\displaystyle \frac{dCn}{d\beta }}cos\left(\alpha \right)-{\displaystyle \frac{dCl}{d\beta }}sin\left(\alpha \right)\left({\displaystyle \frac{Izz}{Ixx}}\right)$$

(7)

where b is the reference wingspan given in terms of customary units in feet. The directional stability derivative, dCn/dβ, and the dihedral effect, dCl/dβ, are given in terms of customary units, per degree, in a body aligned reference frame; $I_{zz}$ is the mass-moment-of-inertia in yaw (typically in slug-ft²); $I_{xx}$ is the mass-moment-of-inertia in roll.

If C_nβdynamic goes negative, the aircraft will not oscillate; it will depart. At low angles of attack, C_nβdynamic is dominated by dCn/dβ; the larger the vertical tail, the more positive C_nβdynamic. As the angle-of-attack increases, the dihedral effect plays an additional stabilizing role provided that dCl/dβ < 0 and ${\displaystyle \frac{I_{zz}}{I_{xx}}}$ >> 1 [21,33,34,35].

In AGARD CP-235 [18], Skow recommends that C_nβdynamic is > +0.004 to ensure adequate stability to prevent yaw departures regardless of the implied Dutch-Roll frequency.

The Dutch-Roll damping ratio [34,35] may be approximated by:

$${\zeta }_{DR}\approx -{\displaystyle \frac{\left(N_r+\frac{Y_{\beta }}{U}\right)}{2{\omega }_{DR}}}$$

(8)

where

$${\displaystyle \frac{Y_{\beta }}{U}}=57.3{\displaystyle \frac{dCY}{d\beta }}{\displaystyle \frac{\overline{q}Sref}{mVKTAS\left(\frac{6076}{3600}\right)}}$$

(9)

and

$$N_r=C{n}_r{\displaystyle \frac{\overline{q}Sref{b}^2}{2I_{zz}VKTAS\left(\frac{6076}{3600}\right)}}$$

(10)

Note that the controls-fixed lateral-directional aerodynamic derivatives dCn/dβ, dCl/dβ, and dCY/dβ are given in terms of per-degree while the dynamic derivative Cn_r representing the variation in the airplane yawing moment coefficient with dimensionless change in the yaw rate is given in rad/s; they, along with all of the mass moments of inertia, are in the body fixed axis. Since the true airspeed (VKTAS) is found in the denominator of both terms controlling the damping ratio, dCY/dβ and/or Cn_r must increase with increasing flight speed to maintain damping. Thus, a classically controlled high-speed aircraft will need to deflect aerodynamic surfaces in response to sensed sideslip and yaw-rate to synthesize favorable Cn_r.

The Roll-Mode is a first-order convergence/divergence seen as a tendency to damp roll rate when the commander executes a bank maneuver. The Roll-Mode is typically defined with the roll time constant, τ_R; this represents the time to achieve 63% of the peak roll rate based upon steady aileron input. τ_R is independent of the magnitude of the roll control input. If ${\tau }_R$ < 0, the aircraft is unstable and behaves unpredictably to roll commands. If ${\tau }_R$ < 0.1 s, the airplane is extremely (possibly excessively) responsive to roll inputs. Conversely, if ${\tau }_R$ is too long, the airplane will respond so sluggishly to commanded roll inputs to degrade maneuvering performance. Per MIL-STD 8785C [10], the LEVEL 1 boundary for flight is ${\tau }_R$ < 1.4 s; for LEVEL 2, ${\tau }_R$ < 3.0 s; for LEVEL 3, ${\tau }_R$ shall not exceed 10 s. We may estimate ${\tau }_R$ by:

$${\tau }_R\approx -{\displaystyle \frac{1}{L_p}}$$

(11)

where

$$L_p=C{l}_p{\displaystyle \frac{\overline{q}Sref{b}^2}{2I_{xx}VKTAS\left(\frac{6076}{3600}\right)}}$$

(12)

Note that the dynamic derivative Cl_p represents the variation in airplane rolling moment coefficient with dimensionless change in roll rate; it is given in rad/s. All aerodynamic and mass moments of inertia are in body-fixed axis.

For slender vehicles, the mass moment of inertia $I_{xx}$ tends to be small as mass is concentrated near the vehicle centerline, and the flight speed (VKTAS) is large. The roll time constant will depend upon the flight dynamic pressure; at low dynamic pressures, ${\tau }_R$ may prove to be objectionably long, whereas at very high dynamic pressures ${\tau }_R$ may prove to be objectionably short.

The Spiral-Mode is another first-order convergence/divergence that manifests itself as a tendency to roll into an ever-tightening spiraling turn (unstable) or roll out of a turn back to wings-level flight (stable) [33,34,35]. The unstable mode is a potentially dangerous divergence; the aircraft, when disturbed in yaw, displays a bank angle increase which locks the aircraft into an ever-tightening spiral. We may estimate ${\tau }_{\mathrm{s}}$ by:

$${\tau }_{\mathrm{s}}\approx -{\displaystyle \frac{1}{\mathrm{s}}}$$

(13)

where

$$s\approx {\displaystyle \frac{L_{\beta }{N}_r-N_{\beta }{L}_r}{L_{\beta }+N_{\beta }\left(\frac{I_{xz}}{I_{xx}}\right)}}$$

(14)

$$L_{\beta }=57.3{\displaystyle \frac{dCl}{d\beta }}{\displaystyle \frac{qSrefb}{I_{xx}}}$$

(15)

$$L_r=C{l}_r{\displaystyle \frac{\overline{q}Sref{b}^2}{2I_{xx}VKTAS\left(\frac{6076}{3600}\right)}}$$

(16)

$$N_{\beta }=57.3{\displaystyle \frac{dCn}{d\beta }}{\displaystyle \frac{\overline{q}Srefb}{I_{zz}}}$$

(17)

$$N_r=C{n}_r{\displaystyle \frac{\overline{q}Sref{b}^2}{2I_{zz}VKTAS\left(\frac{6076}{3600}\right)}}$$

(18)

Note that the dynamic derivative Cl_r represents the variation in airplane rolling moment coefficient with dimensionless change in yaw rate; it is given in rad/s. Similarly, the dynamic derivative Cn_r represents the variation in airplane yawing moment coefficient with dimensionless change in yaw rate; it too is given in rad/s.

When s < 0, the Spiral-Mode is stable; this is good. If s > 0, the Spiral-Mode is unstable. However, an aircraft can still have a satisfactory Spiral-Mode as long as this mode is not too unstable; ${\tau }_{\mathrm{s}}$ < 4 s if s > −0.25 [10,13]. While high-speed (VKTAS > 1000) high altitude (q < 35-lbf/ft²) flight should lead to small s regardless of the aerodynamic derivatives, we must be vigilant against spiral divergence issues at high dynamic pressures.

3.5. Bandwidth Concerns

To ensure that a pilot may direct the aircraft to fly as desired, the aircraft should be able to successfully execute a set of “Standard Evaluation Maneuvers” [20]. Relevant maneuvers include: (1) significant (i.e., 50-KTAS) accelerations and decelerations, (2) rolls from rest through specified bank angles, and (3) pitch from rest to stabilize at new angle-of-attacks. In order to execute these commands without objectionable phase lag, maneuvering bandwidth specifications set a floor to the lower bound of acceptable “equivalent low-order system dynamic” response characteristics. These were incorporated into MIL-STD-8785C [10] and MIL-STD-1797A [13]. In Mitchell’s proposed revisions for a forthcoming MIL STD-1797C that better address high-speed aircraft, including a “generic hypersonic aerodynamic model”, the SR-71, the TU-144L and the F-16XL, he noted that the “results of these studies … are not sufficiently mature to directly affect … current requirements” [25]. He continued, stating that there is a lack of “strong evidence that any of the existing criteria should be modified or new criteria adopted” [25].

The MIL-STDs [10,13] categorize aircraft based on their size and intended purposes. CATEGORY A flight implies active maneuvering. CATEGORY B includes “Climb” “Cruise” and “Loiter” with gentle command inputs. CATEGORY C includes takeoff and landing in adverse weather, this may be the most demanding to the pilot. The MIL-STDs also define three levels of “handling qualities” representing pilot workload: LEVEL 1 where the qualities are clearly adequate, LEVEL 2 where flying qualities are adequate but requires a higher workload, and LEVEL 3 where the aircraft is still safe but requires excessive workload. Below LEVEL 3 minimums, the aircraft is functionally uncontrollable.

The MIL-STDs for flying qualities [10,13] find airframe pitch responsiveness an important metric:

$${\displaystyle \frac{n_Z}{\alpha }}\approx {\displaystyle \frac{57.3\frac{dCL}{d\alpha } q Sref }{W}}$$

(19)

We may re-arrange this equation to estimate the required angle-of-attack to achieve a given load factor: $\alpha \approx {\displaystyle \frac{57.3}{\left(\frac{n_Z}{a}\right)}}$. Turning to Figure 7, we see that if the pitch responsiveness is too low (${\displaystyle \frac{n_Z}{\alpha }}$ << 10), excessive angle-of-attack will be needed to attain steady, level flight. Thus, most maneuvering aircraft need to be designed to operate where ${\displaystyle \frac{n_Z}{\alpha }}$ >> 10.

The MIL-STD’s [10,13] outline desired Stick-Fixed Short-Period frequencies for satisfactory flight; see Figure 8. The upper bound of the Rigid-Body Short-Period mode is one where the frequency is below that of primary structural resonance; typically, this will be a single-digit frequency (i.e., 3 Hz) as predicted by structural finite-element analysis. It is possible for this condition to develop when a very stable aircraft flies at extremely high dynamic pressure (KEAS >> 1000).

The MIL-STD’s [10,13] stipulate LEVEL 1 qualities when the Control Anticipation Parameter (CAP), i.e., ${\displaystyle \frac{{\omega }_{SP}^2}{\left(\frac{n_z}{\alpha }\right)}}$, falls between 0.28 < CAP < 3.6; with a floor of ω_sp = 1-rad/s (0.16 Hz or a 6.3 s Short-Period mode). LEVEL 2 qualities if 0.16 < CAP < 3.6; with a floor of ω_sp = 0.6-rad/s (0.095 Hz or a 10.5 s Short-Period mode). LEVEL 3 qualities exist so long as CAP > 0.16. If ω²_sp/(n/α) < 0.16 the aircraft is unacceptably unresponsive. The MIL-STD’s [10,13] also stipulate desired Short-Period damping; ${\zeta }_{SP}>0.35$ for LEVEL 1 flying qualities and should never fall below ${\zeta }_{SP}<0.15$.

Because Hypersonic Boost-Glide (X-15) and Reentry Vehicles (Shuttle) operate at extremely high altitudes, we must therefore consider the lower bounds of longitudinal responsiveness since both $\overline{q}$ and ${\displaystyle \frac{n_z}{\alpha }}$ trend towards zero as altitude increases. At the highest altitudes, where the aircraft may operate ${\displaystyle \frac{n_z}{\alpha }}$ < 10, we see from Figure 8 that the minimum permissible LEVEL 3 Short-Period frequency is ~0.4-rad/s (a 15 s period). When the rigid-body frequencies drop below that, the aircraft becomes hopelessly unresponsive to both pilots and autopilots using aerodynamic control. Under such circumstances, GN&C systems need to utilize reaction control systems (RCS); such control strategies will no longer display conventional, damped oscillatory behaviors amenable to MIL-STD-1797 [13] style screening.

The requirement to examine the product of frequency and damping was first noted by Koven and Wasico, [15] who observed that the pilot preference for Short-Period damping-ratio correlated to the inherent frequency. They realized that the important metric was driven by energy dissipation rate rather than the number of oscillations after a disturbance; the faster the frequency, the weaker the damping could be before pilots would deem an aircraft unsafe to fly. They also noted the need for adequate pitch damping to “minimize ‘hunting’ and overshoots which occur when the pilot … changes pitch attitude”. We observe that MIL-STD-8785C [10] does not specify a minimum energy dissipation rate (ω∙ζ) criteria for the longitudinal Short-Period but does specify one for the Dutch-Roll.

Mitchell [25] notes that “lateral-directional flying qualities are usually defined in terms of the basic lower-order modal characteristics of the free response, with some additional limits on time-domain measures of forced responses”. The MIL-STDs [10,13] suggest that high-speed combat aircraft have preferred (i.e., LEVEL 1) capabilities to roll from stable flight to accelerate in roll through a 90° change in a bank angle in no less than 1.4 s. It also suggests that non-maneuvering transport category aircraft have a preferred (i.e., LEVEL 1) roll capability to re-orient from wings level through a 30° bank angle in no less than 2.0 s. To prevent excess phase-lag, Mitchell [25] suggests that even large transport aircraft have a roll-mode time constant, ${\tau }_R$, < 0.8 s; MIL-STD-8785C [10] advises ${\tau }_R$ > 0.1 s to avoid an “over-control” situation from forming.

In terms of the Dutch-Roll Mode, the MIL-STD’s [10,13] suggest that the frequency, ω_dr, and damping ratio, ζ_dr, of the lateral-directional oscillations following a yaw disturbance input shall exceed the following minimums (for LEVEL 1 controllability): ω_dr > 0.4-rad/s for cruise and >1.0-rad/s for maneuvering flight (i.e., the period should not exceed 15 s), ζ_dr > 0.19, as well as a metric of net energy dissipation (ζ_dr∙ω_dr > 0.35). Existing MIL-STD’s [10,13] stipulate that under no circumstances should residual limit-cycle oscillations exceed β = ±0.17° (±3 mils).

3.6. Lateral-Directional Control-Coupling

Lateral-directional control-coupling occurs when commanded lateral rolling moments (dCl/dail) due to “aileron” deflection are accompanied by unintentional yawing moments (dCn/dail); this is an innate byproduct of aerodynamic control surfaces. Ideally the unintentional yawing moment is of the same sign as the rolling moment; allowing a stable equilibrium to develop where the static directional stability (dCn/dβ > 0) at the developed sideslip angle (β < 0) opposes the yawing moment associated with the roll. In the favorable case, the innate static dihedral effect (dCl/dβ < 0) at the developed sideslip angle (β < 0) augments the commanded rolling moment. In the unfavorable case, the yawing moment is of opposite sign to the rolling moment; this is known as adverse yaw-due-to-aileron. This fosters an unstable equilibrium, where the static directional stability (dCn/dβ > 0) leads to a positive sideslip angle (β > 0) opposing the adverse yaw. If the innate static dihedral effect (dCl/dβ < 0) is strong enough at the developed sideslip angle (β > 0) it may overwhelm the commanded rolling moment and lead to an apparent “control reversal”. A commanded right roll thus leads to a dynamic yawing motion which eventually results in a left roll. This dynamic mode is exceedingly difficult to arrest and is likely to lead to a loss of control of the vehicle.

The Lateral Control Departure Parameter, LCDP, predicts control coupling:

$$LCDP={\displaystyle \frac{dCn}{d\beta }}-{\displaystyle \frac{dCl}{d\beta }}{\displaystyle \frac{\left(\frac{dCn}{dail}\right)}{\left(\frac{dCl}{dail}\right)}}$$

(20)

when LCDP < 0, roll command inputs are likely to lead to a spin.

Since slender, swept vehicles inherently have significant aerodynamic dihedral (dCl/dβ < 0), substantial static directional stability (dCn/dβ >> 0) is needed to resist any adverse yaw from the roll control aerodynamic surface to keep LCDP positive [17]. In the case study portion of the paper, we will see that LCDP is an effective discriminator to predict the presence of debilitating controllability problems.

To reduce the amount of adverse yaw from the roll control surface, the CAS may implement complex “stick-to-surface” feedforward gain schedules; for example, an Aileron-Rudder-Interconnect. Depending upon the configuration, a fractional movement of the yaw control surface may be able to neutralize the adverse yaw from the roll surface. High-speed vehicles, particularly if short-coupled, may well feature substantial adverse roll from their rudder surfaces. On such a configuration, poor LCDP cannot be salvaged by aileron-rudder-interconnect because the ensuing “force-fight” degrades the attainable rolling moment to render other control metrics (such as time to roll) unacceptable.

Johnston [19] notes that although aileron-rudder-interconnect can produce more favorable LCDP, when static aerodynamic cross-coupling is very strong, ARI can saturate the control system driving the vehicle into divergence. A related issue discussed in AGARD 336 [15] relates to the implied equilibrium sideslip angle (β) that develops in response to the application of full roll control power). Koven and Wasico’s discussion turns on the ratio, ${\displaystyle \frac{{\beta }_{max}}{{\dot{p}}_{max}}}$. We suggest, given the finite linearity of weathercock stability, that the appropriate screening parameter be written (provided Cnp, the yawing moment, due to dimensionless roll rate, is nearly zero) as:

$${\beta }_{max}=\left|{\displaystyle \frac{\frac{dCn}{dail}\delta ai{l}_{max}}{\frac{dCn}{d\beta }}}\right|$$

(21)

For most aircraft, β_max should be limited so that the sideslip does not stall the vertical stabilizer; β_max < 10°. For high-speed air-breathing vehicles with inlets sensitive to unstart, β_max, due to control inputs, should be limited to a much smaller value, perhaps less than ±1°.

A related cross-coupling metric is the Carter’s ${\displaystyle \frac{\varphi }{\beta }}$ ratio, which represents the physical motion of the aircraft in response to a sideslip as the dihedral effect (dCl/dβ), and weathercock stability (dCn/dβ) are viewed through the prism of the mass moments of inertia [14]. If ${\displaystyle \frac{\varphi }{\beta }}$ >> 1, the response to sideslip will present as a rolling (ϕ) motion; if ${\displaystyle \frac{\varphi }{\beta }}$ << 1, the response to sideslip will present as a yawing (β) motion.

$${\displaystyle \frac{\varphi }{\beta }}=\left|{\displaystyle \frac{\frac{dCl}{d\beta }{I}_{zz}}{\frac{dCn}{d\beta }{I}_{xx}}}\right|$$

(22)

The Bihrle–Weissman criteria have evolved over the years. This approach categorizes lateral-directional departure resistance in terms of both Dutch-Roll stability (C_nβdynamic) and LCDP. Weissman [17] found departure free characteristics for his studied aircraft (the F-4, F-111, A-7, and F-5) so long as LCDP was positive. In subsequent years, the chart has been modified to follow MIL-STD-1797A, [13] which added a requirement for inherent Dutch-Roll stability (C_nβdynamic > 0). We also Skow’s criteria from AGARD CP-235 [18] to insist that C_nβdynamic > +0.004 to fully guard against departure; see Figure 9.

3.7. Inertia Coupling and Frequency Coupling

When an airplane rolls about an axis which is not aligned with its longitudinal axis, inertial forces tend to swing the fuselage in sideslip. While classical theory decouples the longitudinal (i.e., Short-Period and Phugoid) from the lateral-directional (i.e., Dutch-Roll, Roll, and Spiral) modes reality proves to be more complex; energy can exchange between the longitudinal and lateral-directional Rigid-Body modes as well as between any Rigid-Body mode and an elastic structural mode.

In 1948, Phillips [11] noted that slender high-speed aircraft which “include short wing spans, fuselages of high density and flight at high altitude” tend to have rolling mass-moments-of-inertia ($I_{xx}$) much smaller than their pitching ($I_{yy}$) or yawing ($I_{zz}$) mass-moments-of-inertia. It is always good practice to compute the ratio ($I_{zz}/I_{xx}$). If ($I_{zz}/I_{xx}$) >> 1, mass moments of inertia in yaw overwhelm those in roll; we deem the vehicle “body heavy”.

Since energy can exchange between the longitudinal and lateral-directional Rigid-Body modes as well as structural frequencies and control inputs, we must pay attention to congenital frequency coupling issues endemic to vehicle design [11]. We may track the ratio, $\left|{\displaystyle \frac{I_{xx}-I_{yy}}{I_{zz}}}\right|$, to gain insight as to how easy it is for motions in the longitudinal plane to cross over into the lateral-directional [11]. When this term is near zero, there is no energy transfer between modes. The larger the magnitude of this term, the more pitch motions will excite yaw motions and vice versa.

In practice, slender high-speed vehicles typically have $I_{xx}$ less than 1/5 the magnitude of $I_{yy}$ and $I_{zz}$; this makes them extremely body heavy and extremely inertia coupling prone. An example of inertia coupling is when a lateral control input excites oscillatory motions in the pitch plane; thus, a command to roll leads to lagging unintended changes in angle-of-attack. Phillips [11] notes that a vehicle with $I_{xx}$~0.4 $I_{yy}$ where $I_{yy}~ I_{zz}$ is highly prone to inertia couple; this corresponds to $\left|{\displaystyle \frac{I_{xx}-I_{yy}}{I_{zz}}}\right|~0.6$.

Both Open-Loop and Closed-Loop control systems may also demonstrate frequency coupling; these issues may manifest when:

• The Short-Period lies too close to the Dutch-Roll mode; inertia coupling prone configurations may exhibit considerable energy exchange between modes [11]
• Pitch commands have spectral content near the Short-Period mode; pitch inputs may excite the Short-Period [11]
• Pitch commands have spectral content near the Dutch-Roll mode; inertia coupling prone configurations may excite Dutch-Roll from pitch command inputs [11]
• Roll commands have spectral content near the Short-Period mode; inertia coupling prone configurations may excide Short-Period from roll command inputs [11]
• Roll commands have spectral content near the Dutch-Roll mode; this can lead to a “ratcheting” rather than a smooth response to roll command [11]
• The Spiral-Mode lies too close to the Roll-Mode; this excites the Lateral-Phugoid mode. MIL-STD-1797 holds that Lateral-Phugoid behavior is impermissible in CATEGORY A flight [13,23].

A quick look at a generic Bode pilot demonstrates these issues; see Figure 10. The characteristic frequencies for Short-Period and Dutch-Roll represent turnover points for low-pass filters. Below ${\omega }_0$ = 1 the system passes inputs with near unity gain and only modest phase lag; for example, the exemplar system has response lagging the forcing function by 45° at ~90% of ${\omega }_0$. Above ${\omega }_0$, the system response declines at 6 dB per octave. If an aircraft featured a ${\omega }_{SP}$ ~2-rad/s (i.e., ~3.1 s period), the command could give control inputs with no more than ~1.8-rad/s peak spectral content (i.e., nothing faster than a ~3.5 s period) before encountering objectionable phase lag. The exemplar system also features a noticeable gain over a fairly broad range of frequencies centered on ${\omega }_0$. Given (1) that the equations of motion inherently cross couple the Short-Period mode to the Dutch-Roll Mode and (2) that high-speed aircraft are inherently poorly damped, it is easy to see how a longitudinal oscillation (${\omega }_{SP}$) can provoke a lateral-directional oscillation (${\omega }_{DR}$) that could drive the aircraft beyond allowable sideslip limits. Similarly, when ${\tau }_S$ and ${\tau }_R$ align, closely control systems may drive Lateral-Phugoid oscillations.

Equations (1)–(22) represent metrics that can be computed directly from aerodynamic data available during preliminary design. In subsequent sections, we will show how they predicted stability and control successes and failures across a variety of high-speed aircraft.

4. Discussion—Case Studies

In this section, we review wind tunnel and flight test data for diverse vehicles (see Figure 11) in the context of the criteria developed above. We consider the rocket-propelled Bell X-2, North American X-15, Martin X-24A, Northrop HL-10, and Rockwell Space Shuttle Orbiter. We also consider the jet-propelled Lockheed YF-12 (SR-71) and North American XB-70. We note that the X-2, YF-12/SR-71, and XB-70 were high supersonic aircraft. The X-24A and HL-10 were configured as viable atmospheric reentry vehicles and flew a successful envelope expansion flight test program, albeit at subsonic and low supersonic speeds. We also admit that very limited open forum data exists for the X-43A, HTV-2, and X-51. However, all of these airframes are bank-to-turn “airplane-style” vehicles.

4.1. Mass Properties

Reported mass properties from aircraft with complex flight test programs vary from flight to flight. In

Table 1. Mass Moment of Inertia for High-Speed Vehicles.

	${\mathit{I}}_{\mathit{x}\mathit{x}}$ (slug-ft2)	${\mathit{I}}_{\mathit{y}\mathit{y}}$ (slug-ft2)	${\mathit{I}}_{\mathit{z}\mathit{z}}$ (slug-ft2)	$\frac{{\mathit{I}}_{\mathit{z}\mathit{z}}}{{\mathit{I}}_{\mathit{x}\mathit{x}}}$	$\left|\frac{{\mathit{I}}_{\mathit{x}\mathit{x}}-{\mathit{I}}_{\mathit{y}\mathit{y}}}{{\mathit{I}}_{\mathit{z}\mathit{z}}}\right|$
Bell X-2 [37]	5000	25,500	29,000	5.7	0.70
North American X-15 [38]	3600→5200	85,000→108,000	86,500→110,500	21→24	0.93→0.94
Martin X-24A [39]	1450→1900	8300	9000→9400	5.0→6.2	0.68→0.76
Northrop HL-10 [36]	1350	6400	7400	5.5	0.68
Lockheed YF-12/SR-71 [40]	~220,500	~955,000	~1,172,000	5.3	0.63
North American XB-70 [41]	~2,600,000	~22,000,000	~24,000,000	11.9	0.61
Rockwell Space Shuttle Orbiter [42]	1,200,000→1,225,000	8,905,000→9,435,000	9,305,000→9,845,000	7.7→8.1	0.82→0.83

, we summarize nominal values from a range of reported values found in flight test. The reader should focus on the key ratios ${\displaystyle \frac{I_{zz}}{I_{xx}}}$ and $\left|{\displaystyle \frac{I_{xx}-I_{yy}}{I_{zz}}}\right|$. All of these aircraft are “body heavy”; their mass-moments-of-inertia in pitch and yaw are much greater than roll. In contrast, a large transport like a Lockheed C-5 has ${\displaystyle \frac{I_{zz}}{I_{xx}}}$~2 [36]. All studied vehicles also have an innate propensity to inertia couple, as $\left|{\displaystyle \frac{I_{xx}-I_{yy}}{I_{zz}}}\right|$ lies between 0.6 and 0.9. Whereas a large transport like a Lockheed C-5 has as $\left|{\displaystyle \frac{I_{xx}-I_{yy}}{I_{zz}}}\right|~$0.07 [36]. Thus, slender vehicles, unlike transport aircraft, will crosstalk any energy found in a lightly damped longitudinal mode into the lateral directional. Similarly, inertia coupling prone configurations will crosstalk any energy found in a lightly damped Dutch-Roll Mode into the longitudinal mode.

Although details of the aerodynamics and mass properties of the X-43A, HTV-2, and X-51A remain closely held, a cursory inspection reveals a slender geometry similar to other high-speed vehicles. Thus, they also appear to be strongly “body heavy” and otherwise prone to inertia coupling.

4.2. Longitudinal Trim

None of the aircraft studied needed a re-design due to a lack of longitudinal control power. By the time that the outer mold line was finalized, engineers correctly sized the longitudinal control surfaces to provide reasonable pitch authority throughout the flight envelope. This does not mean that the ground test and computation precisely matched flight test experience, as we noted three discrepancies in our survey. First, Wolowicz and Yancey [41] note pilots reported elevator saturation issues during approach, landing flare, and touchdown during the NASA XB-70 flight test program; see Figure 12. Second, Layton [43] describes moderate pitch trim anomalies on the HL-10; see Figure 13. Third, Kirsten, Richardson and Wilson [44] document how the body flap deflection needed to trim the Space Shuttle Orbiter differed from pre-flight expectations; see Figure 14.

Kirsten [44] noted that the Orbiter pre-flight “big-picture” aerodynamic performance assessments were satisfactory; “the hypersonic lift-to-drag ratio (L/D) data obtained from pushover-pullup maneuvers… showed excellent agreement with predictions”. At low speeds, the Orbiter over-performed with a “higher-than-predicted L/D out-of-ground effect … due primarily to the lower-than-predicted drag coefficient” arising from an “overprediction of the drag due to surface irregularities in the Thermal Protection System (TPS)” [44].

For the Orbiter, Kirsten [44] found “significant error in longitudinal trim in the hypersonic Mach regime” between pre-flight and flight test. These anomalies were “apparent on all five Orbiter reentries. For example, during STS-1 the trim body flap was 16° rather than 7° at Mach numbers greater than 17” [44]. They found that “the major contributor to the trim error was an error in the basic pitch curve Cm₀” [44]. The Orbiter stability generally matched ground-test estimates; “the Orbiter was statically stable, and the slopes indicated that the combined elevator effectiveness/pitch static stability was close to predictions” [44]. The need for additional longitudinal pitch control power was “attributable to an error in basic pitching moment (Cm₀)” [44]. They believe that the “primary cause of the error in the predicted hypersonic values of Cm₀ is felt to be real gas effects … [which] were not fully simulated in wind tunnel tests” [44].

4.3. Longitudinal Stability

Wind-tunnel and flight test data for the Bell X-2, North American X-15 (see Figure 15), the Martin X-24A, Northrop HL-10, Lockheed YF-12 (SR-71), North American XB-70 (see Figure 16), and the Rockwell Space Shuttle Orbiter confirm designs with inherent longitudinal static stability; dCm/dα < 0 [36,45,46].

We next consider how well these high-speed aircraft satisfy the MIL-STD-8785C “Control Anticipation Parameter” criteria.

First, we considered the X-15, noting that it was flown at high speeds both within the atmosphere and outside the atmosphere on its “reentry” missions; recall Figure 2. The CAP chart (see Figure 17) contains direct flight test data points from a variety of low altitude missions sourced from flight test [2,36,38] as well as data prepared by Griffin and Takahashi [30]. While the basic low-altitude properties of the X-15 are firmly within the LEVEL 1 region, on an exo-atmospheric mission as the vehicle leaves and initially re-enters the atmosphere the dynamic pressure drops so low as to render both pitch responsiveness and the Rigid-Body frequencies unacceptable despite maintaining the control anticipation parameter (CAP) within LEVEL 1 guidelines. For this flight, it appears that the pitch responsiveness becomes unacceptable (${\displaystyle \frac{n_z}{\alpha }}$ < ~3) before the Rigid-Body frequencies drop below acceptable minimums. In other words, the X-15 has an insufficient wing area to “glide” at n_z = 1 before the longitudinal flying handling qualities become hopelessly unresponsive.

Next consider the X-2; see Figure 18 [48]. Its stick-fixed inherent longitudinal flying qualities are firmly in LEVEL 1.

The X-24A, while largely sharing its outer mold line with the atmospheric reentry X-23, was flown 28 times at speeds up to M = 1.6 and at ALT < 71,400-ft. Longitudinal data flown with the pitch SAS intentionally disabled (or “adjusted” based on SAS enabled data) found inherent frequencies and pitch responsiveness well within LEVEL 1 guidelines [23].

The HL-10 outer-mold-line, while being designed for atmospheric reentry, was flown 37 times at subsonic, transonic, and supersonic speeds (M < 1.86). The flight test explored low-dynamic-pressure limits with one supersonic flight to ALT = 90,300-ft. Longitudinal data, flown with the pitch SAS intentionally disabled, revealed inherent frequencies and pitch responsiveness well within LEVEL 1 guidelines; see Figure 19; see Reference [36] for the basis data. However, the flight test program was not without issues as a “lack of longitudinal and lateral-directional control” plagued its first flight [49]. After extending and re-cambering the outboard fins, revising SAS gains and control laws, the HL-10 gained satisfactory longitudinal flying qualities; see Figure 19.

Over many years, NASA conducted “extended flight tests” of the YF-12 as well as SR-71 “Blackbird” aircraft. For the YF-12, pilots experimented with supersonic flight with “pitch SAS off and with roll and yaw SAS off, but never with pitch and yaw SAS off at the same time” [50]. They note that with the “pitch SAS off, the Short-Period is not as well damped”, as it was with the SAS enabled [50]. However, they found that the “decrease in damping is not immediately apparent to the pilot during cruise conditions” [50]. Once again, longitudinal data, flown with the pitch SAS intentionally disabled, documented inherent frequencies and pitch responsiveness well within LEVEL 1 guidelines; see Figure 20. With the SAS enabled, the Blackbird “will hold speed and altitude well if not disturbed … [but] small pitch attitude changes not immediately apparent to the pilot occur, and by the time the pilot notices it, a moderate altitude change is underway” [50].

Stitch, Sachs, and Cox [22] explain these motions as a result of an interaction of the Phugoid with a previously neglected dynamic mode called the Height-Mode. The Height-Mode drives an instability in altitude and airspeed driven by the altitude dependence of propulsion system performance. If the Height-Mode time constant lies close to the Phugoid mode it destabilizes even gentle maneuvers such as a simple heading change at constant altitude and KEAS. Suppressing this mode requires a complex control algorithm including pitch and throttle.

NASA ran a foreshortened flight test program of two XB-70 aircraft. The XB-70-1 had an irreversible, powered flight control system, actuating a variable incidence canard with trailing edge flap, fourteen elevons on the trailing edge of the main wing, twin all moving vertical tails with 45° hinge lines, and variable dihedral “droopable” wingtips. NASA found the XB-70s “inherent longitudinal stability and control characteristics [to be] generally satisfactory” [41]. The major deficiency was available elevator control power during landing. Otherwise, NASA found the un-augmented Short-Period dynamics to be satisfactory, and “the correlation between flight data and predicted results to be generally good” [41]. Figure 21 shows the frequencies and pitch-responsiveness to be a bit lower than ideal for LEVEL 1, but still within LEVEL 2 guidelines. Pilots reported that “response in pitch was quite slow” [16].

The Space Shuttle Orbiter has a complex flight control system including reaction-control thrusters; during gliding flight, “the body flap is the predominant longitudinal trim device, while the wing-mounted elevons are used for longitudinal stability” and control [46]. “Aerodynamically, during the major portions of the flight from entry to touchdown, the vehicle is longitudinally … stable” [46]. With such a complex algorithm, the Orbiter engineering team did not call out oscillatory frequencies or damping ratios.

At high flight speeds, aerodynamic pitch damping declines precipitously. The effective damping ratio, ζ_SP, falls far below established norms in the absence of some form of stability augmentation system. We believe that the X-2 lacked an electronic pitch SAS, but probably included a mechanical bob-weight/down-spring mechanism as such was common design practice during its era. Day wrote that “the damping of both the longitudinal and lateral modes was poor” [37]. We believe that the other survey airframes all featured some form of fly-by-wire control systems with pitch rate (q or $\dot{\theta }$) feedback controls.

The X-15-1 and X-15-2 (serial numbers 56-6670 and 56-6671) used a pilot-selectable “fixed-gain” three-axis SAS with rate gyro feedback having a range of ten preset gains in each axis available for pilot selection during flight [51]. Pilots were expected to “adjusting SAS gains during flight to maintain acceptable handling qualities” [51]. The later X-15-3 was fitted with the MH-96 adaptive gain feedback controller for its aerodynamic surfaces [51]. For normal “stick-and-rudder” flight, this system synthesized pitch-rate damping with or without acceleration feedback. It could also command pitch-attitude and angle-of-attack-attitude holds [51]. The MH-96 also enabled automatic blending of the reaction controls and aerosurfaces during atmospheric exit and reentry [51]. When the autopilot aerodynamics gains reached 90% of their maximum sum, the system would enable the reaction control jets. It would discontinue reaction control augmentation when the aerodynamics gains fell below 75% of maximum. Failure of this system appears to be the proximate cause to the fatal crash of the X-15-3 on Flight 3-65 [52].

Layton [43] generalizes the lifting bodies longitudinal flying qualities with the remark that “conventional handling-qualities criteria … apply reasonably well to these vehicles”.

NASA flight tests [39] document that the X-24A had satisfactory damping with the SAS off, ζ_SP~0.4 → 0.8, at low speeds. At high speeds, ζ_SP~0.1 → 0.15 with the SAS disengaged which was clearly inadequate. With the SAS engaged, pilots reported that the “longitudinal handling characteristics of the X-24A … were generally well-behaved. Short-Period frequency and damping were adequate for all configurations flown” [39].

The HL-10 had a simple pitch rate feedback controller. SAS disengaged, it had poor pitch damping; ζ_SP~0.17 at low speeds and altitudes declining to ζ_SP~0.03 at M = 1.5 and 72,000-ft [36]. With the revised SAS engaged, pilots deemed the HL-10 to have the “best flying of the lifting bodies” [49]. Pilot comments on executing a pushover-pullup maneuver were ecstatic; “it was just so straightforward and pretty … extremely smooth and comfortable … pitch damping was fantastic” [39].

The YF-12 was “normally operated with a stability augmentation system (SAS) engaged to provide artificial stability in pitch and yaw, and damping in pitch, yaw, and roll” [50]. With the SAS disabled, the inherent Short-Period damping ratio fell below ~0.1 at M = 3 cruise; see Figure 22. With the SAS enabled, the airframe develops enough synthetic Cm_q so that it exceeds MIL-STD-8785C Short-Period damping requirements at all speeds and altitudes.

For the XB-70, NASA flight test [53] found solid damping in the subsonic region (ζ_SP~0.5) and light damping of the order of ζ_SP~0.10 → 0.15 in the high supersonic region; see Figure 23. NASA [53] found that the pitch augmentation system further enhanced the Short-Period damping of the airplane in the subsonic Mach number region. Time-histories of disturbances made at M = 2.5 indicate ζ_SP > 0.5 at high speeds with the pitch SAS engaged [53].

The Space Shuttle Orbiter Flight Control System (FCS) provides augmentation for both longitudinal and lateral directional axes; angle-of-attack and pitch-rate feedback provide stability augmentation and damping for the pitch axis [54]. The flight control gains are scheduled as a function of Mach number, angle-of-attack and dynamic pressure and are designed to provide “good flying qualities” throughout entry [54]. As with the X-15-3, the flight control system blends the use of aerodynamic and reaction control jets [31]. While early Orbiter documentation discusses a need to engineer the airframe to conform with MIL-STD-8785C [10] LEVEL 1 longitudinal pitch responsiveness and damping standards during terminal maneuvers, [55] little discussion relating to compliance has been found in post-flight data reduction reports [1].

Taken together, these experiences indicate that a vehicle which meets existing MIL-STD-8785C [10] longitudinal Short-Period frequency and damping guidelines exhibits low-risk behavior in flight test. Inherent with high-speed flight comes the need for synthetic pitch-damping. While no survey airframes exhibited satisfactory flying qualities at high speeds with pitch SAS disabled; all could successfully implement simple rate (q or $\dot{\theta }$) feedback controls. X-15 and Shuttle Orbiter demonstrated blending of reaction control and aerodynamic surface movement during flight at low dynamic pressure.

4.4. Lateral-Directional Stability and Control

Lateral-Directional stability and control issues prove much more challenging to the high-speed design community. Many programs required significant redesign or imposed envelope limitations due to lateral-directional aerodynamic deficiencies. Because increasing Mach number leads to an intrinsic loss of directional stability, aircraft configured for subsonic stability became unstable at high speeds. When the design teams accepted marginal lateral/directional stability, aircraft often crashed. Future designs should learn from history and accept the need for strong static directional stability at all Mach numbers.

The X-2 represents an excellent case study. Its flight test program terminated after a fatal crash on a Mach 3+ flight attempt. The proximate cause of the crash was control-coupling, leading to a supersonic spin. Strong propulsion performance, Inertia Coupling, and a lack of pilot familiarity with the airframe were contributory reasons [37,48]. The Bell X-2 exhibited noticeable effective dihedral (dCl/dβ < 0) while maintaining some level of positive directional stability (dCn/dβ > 0); see Figure 24. However, as the speed increases from M~1.2→3.2, the static directional stability declines from dCn/dβ~0.18/radian to 0.01/radian; ~0.003/° to ~0.0002/°. Thus, C_nβdynamic falls far below Skow’s recommendation to exceed +0.004.

Consider next the Bihrle–Weissman chart [22] for the X-2; see Figure 25. Due to adverse yaw from the ailerons and weak static directional stability, the vehicle operates in the “F” region throughout its planned flight profile: “weak departure resistance heavily influenced by secondary factors”. Careful examination of the yaw-to-roll ratio of the aileron reveals adverse yaw trends rising with both angle-of-attack and Mach number; see Figure 26. At α = 10° and M~3, differential aileron produces approximately 40% as much yaw as roll [48]. In light of the mass properties noted in Table 1, we can infer that the Dutch-Roll will express itself as wing rock at many flight conditions since ${\displaystyle \frac{\varphi }{\beta }}$ >> 1 for all supersonic conditions.

Generally speaking, engineers would employ aileron-rudder-interconnect to reduce the adverse yaw to more manageable levels. Figure 27 shows the challenge facing engineers in the era predating “fly-by-wire” control systems; the yaw-to-roll ratio of the rudder shows equally strong angle-of-attack trends. In order to cancel the adverse yaw of the ailerons, the flight control system would need to schedule rudder as a function of both Mach number and angle-of-attack. Since this level of flight control complexity was beyond the state-of-the-art in the late 1940s when Bell engineered the X-2, engineers decided to “lock” the rudder during high-speed flight and only enable its use for turn-coordination and sideslip trim during the terminal subsonic glide [11].

This leads to the “fatal flaw” of the X-2 flight control system. With the rudder disabled, the adverse yaw from the ailerons causes any roll control inputs to drive the airframe to sideslip. Across a wide range of speeds and altitudes, 10° of aileron input implies a steady-state sideslip trim angle α > 10°; see Figure 28. It is no surprise that a pilot, when faced with a need to bank to turn at a flight speed above Mach 3, would accidentally induce a spin [48].

The X-15 flight test program revealed additional issues concerning lateral-directional flight dynamics. As originally configured, the X-15 had a large vertical tail volume with dorsal and ventral fins. This provided extremely strong static directional stability at the expense of dihedral effect; see Figure 29 [12]. As the flight test program continued, and the X-15 was flown to greater speeds and over a wider range of angle-of-attack, pilots and engineers noted the weak Dutch-Roll stability; “poor handling qualities at the high angles of attack was due primarily to the large negative dihedral effect (positive dCl/dβ) caused by the presence of the lower ventral fin” [56]. Planned reentry missions could easily see the pilot lose control of the airframe given the combination of declining Dutch-Roll stability, the change in sign of the dihedral effect and weakening lateral-directional damping; see Figure 30 [57].

Beginning in late 1962, NASA flew the X-15 with the lower rudder removed. With the lower rudder removed but speed brakes deployed; dCn/dβ~+0.008. Figure 31 compares the C_nβdynamic trend with Mach number between large ventral, small ventral and small ventral with speed-brake deployed. Omission of the lower rudder doubles C_nβdynamic despite the reduction in static directional stability; even with speed-brakes closed the X-15 now satisfies Skow’s criteria (C_nβdynamic > +0.004). Dutch-Roll stability improves even more at hypersonic speeds with the small ventral and deployed speed-brake.

Turning next to the Weissman chart (see Figure 32) which demonstrates that the X-15 has strong departure resistance at all speeds and attitudes; all data are firmly in the “A,” “highly departure and spin resistant” region. The strong static directional stability (even with the lower rudder removed, so long as the speed-brakes are deployed above Mach 3) masks any adverse yaw from the “aileron” effect of differential horizontal tailplane. These changes increased the envelope limits of controllable flight substantially; consider Figure 33, in contrast with Figure 30. With these configuration changes, the X-15 went to have a largely successful flight test program of 199 flights including many astronaut wings flights and speed records.

The X-24A and HL-10 lifting body configurations have many similar lateral-directional characteristics. They have relatively high dihedral effect (dCl/dβ << 0); they are also “wing heavy” ($I_{zz}/I_{xx}$ > 5) [58]. Even though these configurations are festooned with many vertical and/or canted fins, they have relatively weak static directional stability; Figure 34 and Figure 35.

The operational challenge with these aircraft stems from a need to fly the aircraft at relatively high angles of attack. Turning next to Figure 36, we see that much of the X-24A flight test program had the airframe operate at attitudes above its N_β = 0 boundary but did not exceed the C_nβdynamic = 0 limit. As with the X-15, as the angle-of-attack increases, the dominant contributor to C_nβdynamic arises from their effective dihedral, not from their static directional stability. On an ordinary, swept wing airplane this would not pose any problem but the HL-10, like other lifting bodies, has unusual aerodynamic dihedral characteristics.

If we examine the M = 2.16 wind-tunnel run of the HL-10 with the revised “Mod II” vertical fins [59] (return to Figure 35) we see only weak aerodynamic dihedral. This is due to the vertical disposition of directionally stabilizing elements. Dorsal vertical elements will produce effective dihedral (dCl/dβ < 0) while ventral vertical elements oppose this (dCl/dβ > 0). Thus, the HL-10 being a thick lifting body with multiple short vertical fins, demonstrates a peculiar dihedral effect trend: dCl/dβ~−0.001/deg relatively invariant to α, along with directional stability that declines as α increases.

Layton [43] generalizes that lifting bodies have inherently poor lateral-directional flying qualities; un-augmented, they “are almost impossible to fly”. He continues stating that if the airplane has both (1) effective ailerons and (2) favorable yaw, lateral-directional flying qualities can be improved with artificial roll damping. If the aircraft has substantial adverse yaw due to aileron deflection, it needs additional vertical fin area (i.e., strong positive static directional stability) to permit the implementation of an effective roll damper.

Transforming this wind tunnel data onto a Weissman plot, see Figure 37, we see that the airframe typically operates in region “F; “weak departure resistance heavily influenced by secondary factors”. Like the Bell X-2, the HL-10 does not achieve Skow’s criteria (C_nβdynamic > +0.004) for strong resistance to spin departures. The aerodynamic properties also indicate that the Dutch-Roll will express itself as a strong wing rock; ${\displaystyle \frac{\varphi }{\beta }}\cong 5$ at α = 12°.

Flight test data of NASA’s YF-12 [50] noted ζ_DR~0.8-rad/s at subsonic speeds; ω_DR~1.3 → 2.0 during the transonic and low supersonic and ω_DR~1.00 → 1.36-rad/s at M~3 cruise speeds. With the SAS engaged, ζ_DR~0.4 → 0.6 in the transonic and low supersonic and ζ_DR~0.4 at M~3 cruise [50]. These frequencies and closed loop damping ratios are well within the LEVEL 1 region of MIL-STD-8785C [10].

With its long service record, the YF-12/SR-71 clearly demonstrated acceptable lateral-directional flying qualities at high speed. McMaster and Schenk [60] note that the YF-12 “encounters low directional stability at high Mach numbers … [which] dictate full-time use of the yaw … stability augmentation systems (SAS) to provide … directional static stability”. Moes and Iliff [40] extracted lateral-directional static stability derivatives for NASA’s later SR-71 from flight test; see Figure 38. Considering $I_{zz}/I_{xx}$~5.3 and a typical supersonic cruise at α~3°, we infer that C_nβdynamic~+0.001 which places it in Region “F” of the Weissman plot; incorporating aileron-rudder-interconnect would likely lead it to more to favorable LCDP. The aerodynamic properties also indicate that the Dutch-Roll will express itself as a moderate wing rock; ${\displaystyle \frac{\varphi }{\beta }}\cong 2.5$ at M~3.0.

The XB-70 features more nuanced behaviors due to its scheduled wing-tip droop that reduced Mach effects in longitudinal aerodynamic stability. Turning to Figure 39, we see that even with the 65° drooped wingtips providing additional side facing projected area aft of the CG, we see that dCn/dβ declines from +0.002/° at M~1.4 to +0.001/° at M~2.34. The more troubling byproduct of the wingtip droop is the complete loss of effective dihedral; dCl/dβ is positive at all speeds creating aerodynamic “anhedral” that counteracts formation of a stable Dutch-Roll. Andrews [45] and White and Anderson [16] noted aileron adverse yaw issues. “Pilots commented that at supersonic speeds, the lateral-directional handling qualities are degraded by an adverse yaw experienced as a result of aileron deflection” [45]. The adverse yaw is “approximately zero at low speed, highly adverse transonically [but] approach zero again at speeds above Mach 2” [16]; see Figure 39. Thus, given its body-heavy nature, the XB-70 is clearly in Region “F”, “weak departure resistance heavily influenced by secondary factors”, of the Weissman chart at all supersonic flight test conditions; see Figure 40.

The XB-70 had marginal lateral-directional flying qualities. Wolowicz and Yancey [41] note that “sideslip maneuvers … were adversely affected by a drop in static directional stability … at sideslip angles greater than ~2° for all Mach numbers and all wingtip configurations”. White and Anderson [16] also note that the “aileron becomes a sideslipping control rather than a rolling control”. When the pilot applies aileron to “stop a wing from dropping, sideslip is introduced, and the dihedral effect causes the airplane to tend to roll more and additional aileron is required”. They state that an inattentive pilot is likely to “reach high sideslip angles inadvertently, especially, when flying in turbulence”. Recall that MIL 8785C desires β < ±0.17°; the fact that the XB-70 experienced such large sideslip excursions is worrisome. Response to turbulence is experienced “primarily in roll, with less disturbance in pitch or yaw”. Wolowicz and Yancey note that lateral-directional “maneuvers flown with the augmentation system off were [so] erratic [that they] usually could not be analyzed” [41].

In order to reenter, the Space Shuttle Orbiter [61] needs to fly its initial aerodynamic reentry at high angle-of-attack (α~40°) before reverting to basic gliding flight from M~5 to touchdown; see Figure 41. As with other slender vehicles, it has relatively high dihedral effect (dCl/dβ << 0) which only grows with increasing α. The shuttle is quite “wing heavy”; $I_{zz}/I_{xx}$~8. Despite the visually large vertical tail, which becomes an effective “wedge” when the speed-brake split rudder is opened fully, the Orbiter lacks static directional stability. Nonetheless, due to its mass properties and nose-up flight attitude, the Orbiter demonstrates inherent Dutch-Roll stability throughout its entire reentry profile. The dominant contributor to C_nβdynamic arises from the effective dihedral, not from the static directional stability.

None of the examined reports from the Space Shuttle program explicitly discuss stick-fixed Dutch-Roll frequencies or damping ratio. Like so many other high-speed vehicles, the Orbiter exhibits strong adverse yaw from its ailerons [61]. Figure 42 shows how the rolling moment from differential aileron holds across the entire reentry profile but how they develop substantial (30% to 50%) adverse yaw at all supersonic speeds. With the rudder dedicated to “speed-brake” at high speeds, the Orbiter has dangerous innate aerodynamic qualities if all lateral-directional control were to rely on the ailerons.

The Orbiter features a complex, “fly-by-wire” control system to handle the adverse yaw. As seen in Figure 41 and Figure 42, if the Orbiter were reliant solely upon its ailerons it would have unacceptable control characteristics. At speeds above 5 Mach, 30° of aileron deflection would develop enough adverse yaw to force 10° of sideslip, far beyond the linear limit. The Weissman plot, built upon the basic aerodynamic data, shows the resulting poor departure resistance.

According to Gamble, [54] the Orbiter was conceptualized to fly high cross-range reentry using “an all-aerodynamic control concept”. As aerodynamic data became available; it became clear that LCDP was unfavorable; see Figure 43. At subsonic speeds the Orbiter is in the “F” region: “weak departure resistance heavily influenced by secondary factors”. At supersonic speeds, it lies on the boundary region C, "weak spin tendencies," and B, "spin resistant, but with objectionable roll reversals." For a period of time, the Orbiter development team devised a curious strategy employing “reverse aileron control (negative aileron for positive roll)” for entry control [54]. After they found that this approach lacked robustness, the flight control system was revised to utilize “the yaw RCS to initiate bank maneuver and the ailerons to coordinate the maneuvers prior to activation of the rudder” [54]. The flight controller uses RCS from the very highest Mach numbers and lowest dynamic pressures “down to Mach 1.0, at which time the rudder becomes fully effective for directional control” [62]. This combination of RCS and aileron-rudder-interconnect functionally elevates LCDP so that it has more favorable departure resistance according to the Weissman criteria. Because the RCS operates using pulse-width-modulation control of discrete hydrazine thrusters, it is difficult to define the numerical value for augmented LCDP.

Gamble [54] reiterates that “LCDP was the first controllability criterion that was systematically applied to the Orbiter” followed by C_nβdynamic. RCS jet activation is governed by a flight control feedback loop that senses side accelerations; however, the jets are not strong enough to synthesize true directional stability. Since the fully functioning RCS system can only increase C_nβdynamic by +0.002, the Orbiter is restricted to flight above a critical angle-of-attack where the dihedral effect still dominates the Dutch-Roll stability (see Figure 44). This is an unusual schedule in light of earlier experience with the X-24A where operations were restricted to flight below a critical angle-of-attack; recall Figure 36. However, both schedules derive from the same principles: a controllable aircraft needs to display non-divergent lateral-directional modes and an ability to command roll without excessive sideslip.

Of the more recent high-speed vehicles with proprietary data, we can only note generalities. Neither X-43A nor X-51A were advertised as being maneuvering airframes. Both flight test programs were marred by launch failures but ended with a successful run of their respective scramjet engines over their intended trajectories.

Circumstantial evidence from the failure of the first launch of the HTV-2 indicates lateral-directional controllability deficiencies. The ostensible goal of the HTV-2 was to “develop and test an unmanned, rocket-launched, maneuverable, hypersonic air vehicle that glides through the Earth’s atmosphere up to Mach 20 speed” [63]. After launch from a Minotaur IV booster, the HTV-2 experienced “flight dynamics anomalies” and departed controlled flight during a “pull-up maneuver” [63]. The independent engineering review board identified “higher than predicted adverse yaw coupled into roll that exceeded the available roll control capability” as the proximate cause of the crash [63]. Referring to Figure 45, the board describes a departure consistent with that experienced by the X-2. In the presence of a roll disturbance, the flight control system differentially deflected its body flaps to arrest a developing roll rate. The interplay between adverse yaw from the “aileron” and the dihedral effect leads to a sideslip “run-away” and control power saturation. The remediation plan was to alter the flight profile to fly at a lower angle-of-attack, adjust the vehicle CG, and augment aerodynamic controls with RCS. After following these recommendations, Lockheed’s second HTV-2 test flight was more successful [63].

4.5. Bandwidth and Frequency Coupling

The inherent longitudinal dynamics of an airframe must be well matched to maneuvering expectations so that the airframe follows command inputs without excessive phase lag. If airframe response sufficiently trails command inputs, the airframe may inadvertently oscillate. If a classical design followed MIL-STD-8785C [10] CAP guidelines, it should be resistant to pilot-induced-oscillations.

Beginning with the X-2 and Figure 46, we consider how the Short-Period and Dutch-Roll frequencies interact as the aircraft flies its full, planned final mission. Both Short-Period and Dutch-Roll frequencies are strong functions of dynamic pressure and moderate functions of Mach number. The Dutch-Roll, alone, is also strongly dependent upon angle-of-attack. Both frequencies slow for approach and landing as well as during the “over-the-top” ballistic portion of flight at ALT > 65,000-ft. Although the pilot lost control at the beginning of the “pull-up” maneuver at M > 3 and ALT > 70,000-ft, neither the Short-Period nor Dutch-Roll frequency taken in isolation was the cause of the crash. Taken together, we see that frequencies do cross on numerous occasions: (1) during initial powered ascent around 45,000-ft, (2) at the beginning of the ballistic “over-the-top” maneuver and (3) just at the beginning of the “pull-up” maneuver—just where control was lost. Thus, inertia-coupling where lightly damped (due to high speed and altitude) modes crosstalk was a contributory factor to the crash. The coupling between Short-Period and Dutch-Roll Modes explains a source of lateral-directional energy that led to the high sideslip angles associated with adverse-yaw which precipitated loss.

Turning next to the X-15, we see that it is also prone to Inertia Coupling. We can see that the frequencies are distinct, and only get close to one-another while “going over the top”; see Figure 47. Since Inertia Coupling is likely to occur only as the X-15 flies “over the top”, where the Rigid-Body frequencies are already so low as to need supplemental control from reaction-control, this flight does not raise concern.

Among the lifting bodies, the X-24A did not appear to suffer from inertia-coupling as reported longitudinal Short-Period frequencies did not coincide with Dutch-Roll frequencies [39]. Hoey [39] noted that “in the transonic and supersonic flight regime, the roll response did not meet the specification requirements with SAS on”. The roll time constant, ${\tau }_R$, was found to be as much as 50% longer than the LEVEL 3 minimum (2 s to 45°) in the transonic; but exceeded LEVEL 1 requirements (1.2 s to 45°) during approach-and-landing. That said, poor pilot reviews resulted from pilot-induced-oscillation (PIO) sensitivity rather than low roll power [39]. At odds with MIL-STD-8785C specification compliance, pilots felt that “the rolling capability was adequate for all phases of the X-24A mission, except for landing in a crosswind or [flight in] moderate turbulence” [39].

While Hoey [39] did not mention Inertia Coupling, he did mention that the X-24A had a Lateral-Phugoid frequency coupling between the long-period Roll and Spiral-Modes. “All SAS-off conditions exhibited an oscillatory, coupled Roll-Spiral-Mode” which is impermissible under MIL-STD-8785C [10]. Given the >20 s time constant of this mode, it was “never observed in flight … although test maneuvers confirmed values of the derivatives that contributed to [it.]” [39]. Hoey [39] continues stating that under normal SAS gain settings, the Lateral-Phugoid divided into distinct non-oscillatory roll and Spiral-Modes; these gains “were chosen so as to avoid the coupled Roll-Spiral-Mode whenever possible”.

The HL-10 likely had troubling inherent inertia coupling based on its mass properties, $\left|{\displaystyle \frac{I_{xx}-I_{yy}}{I_{zz}}}\right|~0.7$ as, well as closely spaced Short-Period and Dutch-Roll frequencies; both being around 4→5 rad/s [64]. With its weak dihedral effect, the HL-10 probably did not suffer from Lateral-Phugoid issues. Pilot comments, flying the revised SAS programming, did not mention any sorts of flying qualities degradation due to inertia coupling or Lateral-Phugoid [64].

On NASA’s YF-12, pilot reports [50] did not mention inertia coupling. Flight test revealed that the roll time constant was as short as τ_R = 0.27 s during approach and landing, and typically τ_R = ~1.2 s at M > 3 high altitude cruise. Thus, roll responsiveness achieves LEVEL 1 capabilities across much of the flight envelope [50]. The spiral stability was positive and was “well within the military specification requirement of a time to double of no less than 20 s” [50]. Thus, the YF-12/SR-71 family seems immune from Lateral-Phugoid issues.

Bourne and Kirsten [65] note that earlier versions of the Orbiter flight control system were prone to pilot-induced oscillations (PIO) during final approach and touchdown; the primary causes for this behavior were “inadequate pitch attitude visual reference cues” (poor cockpit visibility with the nose high final approach) and poor pitch responsiveness. Pilot bandwidth and phase response had to accommodate a “one-half-second delay … between pitch stick command input and normal acceleration response partly due to the digital control system and partly due to vehicle geometry” [65]. We note that a 0.5 s group delay introduces a minimum of ~45° phase lag on a 4 s period; 1.6-rad/s. Since Orbiter aerodynamic data (subsonic dCL/dα~+0.048/deg, and 200-KEAS final approach flown at W/S~70 lbm/ft²) indicates that n/α~5, our interpretation of MIL-STD-8785C standards would suggest a desired longitudinal bandwidth > ~0.9-rad/s (a 7 s period). Thus, we concur that the Orbiter had sufficient bandwidth to fly approach and landing but was PIO prone due to control system group delay. Indeed, to resolve the PIO issues prior to STS-1, NASA added a PIO suppression filter into the pitch controller that “reduced pilot command inputs as a function of pitch command frequency” [65]. The Orbiter is a good example of how a “fly-by-wire” system can reduce the spectral content of control commands to avoid phase issues associated with fundamental modes.

5. Conclusions

This review paper identifies stability and control screening metrics, which are especially suitable for high-speed airframe design. These metrics may be used to evaluate novel candidate configurations for stability and control deficiencies. Many of these metrics stem from “lessons learned” on prior military aircraft programs; others derive from the development of the Space Shuttle Orbiter. It is important to monitor the controls-fixed un-augmented rigid-body frequencies, damping ratios, and time constants in context with established standards to understand the sort of control augmentation that will be needed for a successful flight. It is also important to track the adverse yaw from the “aileron” control surfaces to understand if lateral control will inadvertently destabilize the aircraft.

Engineers can utilize all of these metrics using widely available aerodynamic data. All of the screening criteria discussed here rely solely upon direct (CL, etc.) or partially linearized (dCn/dβ, Cmq, etc.) aerodynamic data. At the initial design screening stage, this information may derive from semi-empirical (ex. DATCOM) and low-order computation (panel methods). At the preliminary design stage, volume-grid CFD and wind tunnel data will augment the conceptual aerodynamic database. For detail design, the Space Shuttle Orbiter program serves as an excellent role model; the Orbiter engineering team applied these screening techniques to a multi-fidelity database with an associated uncertainty model. As seen on the Orbiter, values for these parameters are refined as a result of a typical comprehensive flight test program.

We demonstrated the broad utility of these metrics by post-processing old wind tunnel and flight-test data to show how flight success and failure can be predicted from data easily available to the modern design team early in the design process.

As the screening criteria relies upon an “equivalent low-order system response”, an airframe with highly nonlinear aerodynamic response will exhibit dynamic response which will differ from that predicted by the low order system. In general, nonlinear aerodynamic response tends to worsen flying qualities; for example, declining yaw restoring moments at high sideslip angles will degrade Dutch Roll response and worsen Spiral Divergence.

Mass property uncertainties in preliminary design add to the challenge facing the vehicle configurator. Longitudinal CG uncertainty impacts longitudinal trim as well as the un-augmented Short-Period frequency; and hence impacts the gain of any feedback system needed to augment such response. The magnitude of the mass moments of inertia also impacts un-augmented frequencies; sufficient error in preliminary design may lead the configurator astray.

To reduce risk, high-speed vehicles should inherently present a stable Dutch-Roll Mode, C_nβdynamic > 0. Since high-speed vehicles have substantial adverse yaw from roll control, significant static directional stability (dCn/dβ >> 0) is desired to ensure that full “aileron” deflection (considering control allocation such as aileron-rudder-interconnect) does not drive the vehicle to significant sideslip angles. The flying qualities of the early X-15, the XB-70, and the lifting body configurations (X-24A and HL-10) were degraded because a significant fraction of their directional stabilizing surfaces were ventrally mounted; dCl/dβ > 0 is associated with elevated flight risk.

The Bihrle–Weissman chart (C_nβdynamic vs. LCDP) is an effective screening tool. Low-risk vehicles are present in the “A” region of the chart. Vehicles which inhabit the “F” region often exhibit performance-degrading flying quality limitations. A lack of attention to Bihrle–Weissman type criteria and lateral-directional control-coupling will lead to operational loss of control of the X-2 and HTV-2. Supplemental RCS can mitigate these issues under limited flight conditions. While successful flights of the Space Shuttle Orbiter and HTV-2 testify to this possibility, an active RCS system adds weight and complexity.

Since high-speed vehicles are inherently slender, they are prone to inertia coupling and control-coupling. Due to the adverse effects true airspeed has on open-loop damping, synthetic pitch, roll and yaw damping will be required even on a statically stable airframe. The ${\displaystyle \frac{\varphi }{\beta }}$ ratio determines if a roll, yaw, or multi-axis damper will prove most effective. The configurator should be cognizant of the control power requirements needed to develop the needed synthetic stability.

Configurations should be screened to ensure that their Short-Period and Dutch-Roll frequencies do not overlap (inertia coupling), and that Roll-Mode and Spiral-Mode time constants do not overlap (Lateral-Phugoid divergence); if they do, the flight control system needs to suppress these modes without inadvertently worsening other modes.

Unlike a technology demonstrator, whose flight profile may be tightly constrained, general-purpose high-speed aircraft need to operate over a wide range of flight conditions and mass properties. While the North American X-15 rocket plane had favorable aerodynamic stability and controllability within the atmosphere, the Space Shuttle Orbiter lacked sufficient static-directional stability to maneuver without the need for supplemental RCS to address its adverse yaw problems.

While a future manned high-speed aircraft would likely have a contractual stipulation to conform to MIL-STDs for flying qualities, the challenge arises when an engineering team designs an unpiloted airframe, as there are no equivalent military standards for the flying qualities of unpiloted aircraft. To succeed, all programs must address flying qualities throughout the development process. We believe that the select metrics shown in this paper should be used to screen candidate designs. An unpiloted vehicle, engineered to conform to these select piloted flying qualities metrics, will reduce risks that, if left unaddressed, could jeopardize program success.