Missile Roll Research Articles

Adjoint Method for Missile Performance Analysis on State-Space Models

The adjoint method is a method widely used in the preliminary design of guidance loops to obtain quick estimates of the performance of a guided weapon while avoiding time consuming Monte Carlo simulation experiments. Traditionally, the adjoint method is presented as a set of rules for transforming the original (linearized) model of the guidance loop to an adjoint model that is used to obtain the performance estimate for the original model in a single simulation run. An attempt is made to derive the adjoint method in the general setting of state-space models. This does not only lead to an elegant and clear exposition of the adjoint method, but it also extends the area of application of the adjoint method to more general situations that were not previously covered in the literature. One possible extension is illustrated with the analysis of the performance of a rolling missile against a target that performs a maneuver with a random start time. I. Introduction T HE adjoint method has a well-established place in the arsenal of tools available for guidance-loop performance analysis, in particular in preliminary phases of guided missile design. The success of the method is mainly because it is relatively simple to use and because it enables a quick performance assessment for a wide range of engagement conditions. The method as a general analysis method for control systems under stochastic disturbances was introduced by Laning and Battin. 1 However, some authors, such as Zarchan, 2 place the origin of the method back to the work of the 19th century mathematician Vitto Voltera. Zarchan mentions the application of the method in computing ballistic dispersions as early as the 1920s. Although the method is also applicable for the analysis of systems under deterministic disturbances, it is the stochastic case that demonstrates the real power of the method. In this case, the method can be efficiently used to avoid time consuming Monte Carlo simulations when linear approximation results can deliver sufficient accuracy. The traditional presentation of the adjoint method in the literature 2,3 is based on the input‐output (transfer function-type) system representation with a strong emphasis on the procedural aspects of the method: inverting the sense of the signal flow, substituting time with time-to-go, etc. In this work, we pursue in detail the derivation of the construction rules for the adjoint system based on state-space models. We consider both the deterministic and the stochastic case for continuoustime models. In the stochastic case, our approach is shown to extend the potential of the method beyond applications currently reported in the literature that assume uncorrelated inputs and look only at the variance of the chosen output signal. We show here that the adjoint method can perfectly accommodate correlations between the different inputs and that a possible outcome is the influence of the random inputs on the covariance of two different outputs at the a priori fixed moment of time. In this sense, it is shown that the analysis power of the adjoint method is almost identical to that of another method for preliminary design analysis called the covariance ma

유도탄 성능분석을 위한 실시간 병렬처리 시뮬레이터 연구

In this paper, we describe the real-time parallel processing simulator developed for the use of performance analysis of rolling missiles. The real-time parallel processing simulator developed here consists of seeker emulator generating infrared image signal on aircraft, real-time computer, host computer, system unit, and actual equipments such as auto-pilot processor and seeker processor. Software is developed from mathematic models, 6 degree-of-freedom module, aerodynamic module which are resided in real-time computer, and graphic user interface program resided in host computer. The real-time computer consists of six TIC-40 processors connected in parallel. The seeker emulator is designed by using analog circuits coupled with mechanical equipments. The system unit provides interface function to match impedance between the components and processes very small electrical signals. Also real launch unit of missiles is interfaced to simulator through system unit. In order to apply the real-time parallel processing simulator to performance analysis equipment of rolling missiles it is essential to perform the performance verification test of simulator.

Geometric Approach to Three-Dimensional Missile Guidance Problem

The Frenet–Serret formula in classical differential geometry curve theory and the characteristics of a Ž ctitious missile pointing velocity vector are used to design missile guidance curvature command. Qualitative analysis is conducted to study capture capability of the designed guidance command in three-dimensional engagements. The region that miss can occur is derived in terms of the tangential component of the kinematics equation. Then, sufŽ cient initial condition is derived, which, with target’s maneuvering information, can guarantee capture for arbitrary targetmaneuver. To validatethis capture conclusion,two simplemissile torsion commandsare introduced to rotate direction of themissile curvature commandto ensure that the curvature command formula iswell deŽ ned. Such missile roll strategy may also help to improve capture capability in the Ž nal phase of an engagement.

Aerodynamic characteristics of a canard-controlled missile at high angles of attack

A low-speed wind-tunnel investigation was conducted to examine the aerodynamic characteristics of a one-thirdscale model of a canard-controlled missile at high angles of attack using force and moment measurements. The data were taken at a nominal Mach number of 0.2 for angles of attack up to 50 deg at three different canard deflection settings. The test runs were limited to 0and 45-deg missile roll angles (symmetric configurations) and two sets of tails (aft fins), one with the full area including the roll damping tabs (rollerons) and the other without the rollerons. The data indicate that the rollerons act as an effective fin area at low speeds and high angles of attack, and make the missile more stable. The test data were also used to validate the aerodynamic characteristics of the missile as predicted by the Missile Datcom program. The agreement between the Datcom predictions and the test data is fairly good, with the latter indicating a slightly higher static stability.

Control decoupling analysis for gyroscopic effects in rolling missiles

The complex summation method is used to represent the pitch/yaw dynamic equations for a missile with autopilot. The particular forms of complex motion variables that are used enable cross coupling to be identified with imaginary coefficients in the resulting equations. A designer who wishes to produce uncoupled responses to commands for a maneuver then has a clearcut objective when choosing control laws, namely, to eliminate the imaginary coefficients from the response equations. This paper illustrates the use of this technique with an application to gyroscopic coupling resulting from a rolling airframe. It leads via transfer functions to the proposal of a zero-starred filter as an alternative to a pole/zero cancellation technique for eliminating the major source of coupling when the command is in a nonrolling frame of reference.

Control of missile dispersion via roll rate modulation

Abstract : Feedback control via roll rate modulation to limit cross-range dispersion of spinning missiles is investigated. The dispersion to be controlled is assumed to result from lift variations that do not average out over a roll cycle when the missile roll rate remains constant. By modulating the roll rate in a prescribed manner through feedback control, lift nonaveraging and the resulting cross-range dispersion are minimized. Two different approaches to the feedback control are investigated. In one approach, the roll rate is modulated by feedback of the state variables to prevent the occurrence of lateral velocity increments that produce dispersion. In the other approach, the roll rate is modulated harmonically about a steady roll rate, and the amplitude of the harmonic variation is controlled to limit dispersion. Keywords: Angle of attack; Equations of motion; Linear and nonlinear differential equations; Time dependence; Missile aerodynamics; Charts. (Author)

Halphen Puiseux inequalities in the precessional motion of a rolling missile

In the theory of precessional motion of the rolling missile, [5] observed that the aspidal angle is always bounded below and above by π/2 and π respectively, provided the roots of the libration polynomial satisfy certain launching conditions. In this paper, we have obtained the same results without these conditions. We have also proved that the bounds are sharp, so far as the purely retrograde and direct motions are concerned.

Comparison of Analytical and Experimental Supersonic Aerodynamic Characteristics of a Forward Control Missile

Techniques to predict the aerodynamic characteristics of slender cruciform missiles have been developed and are constantly being updated and improved. This paper presents comparisons between analytical and experimental supersonic aerodynamic data for a class of canard-controlled missile configurations similar to the sidewinder missile. Three aerodynamic prediction computer codes, including program MISSILE2, a recently improved version of program MISSILE, are evaluated by comparison with the test data to assess their accuracy. The major emphasis is placed on the roll control characteristics. In addition, tail span optimization, longitudinal and lateral control, induced roll, and missile roll orientation effects are addressed.

On the apsidal limits of a rolling missile

It is proved that a rolling missile whose initial angular oscillations are nonlinear will have the same librations as its equivalent common top provided that q > 0 q > 0 and Z 1 + Z 4 ≤ − 2 {Z_1} + {Z_4} \le - 2 where q q is certain aerodynamic parameter contained in the nonlinear overturning moment of the missile and Z 1 {Z_1} , Z 4 {Z_4} are respectively the least negative and the largest positive zeros of a certain quartic polynomial.

Determination of Allowable Tolerances for Asymmetries of a Free Rolling Missile

Covers advancements in spacecraft and tactical and strategic missile systems, including subsystem design and application, mission design and analysis, materials and structures, developments in space sciences, space processing and manufacturing, space operations, and applications of space technologies to other fields.

A missile adaptive roll autopilot with new Dither principle

A missile adaptive roll autopilot with new Dither principle

Remarks on the inertia instability of a rolling missile

Lyapunov’s second method is applied to the question of stability when dynamic cross coupling is considered. The main result is a condition on the maximal non-symmetry that gives stable performance for any constant or non-constant angular velocity in roll. Methods are outlined for the treatment of some related questions.

Progress report on a solid state FM-FM telemetering system

Program progress of hypersonic missiles and satellites is dependent upon advances in the telemetering art, for instrumentation must be functional under the imposition of severe environmental extremes. Recently, a solid state FM-FM telemetering system has been developed that is immune to shock and vibration of considerable magnitude. Transistorized units of the solid state system described in this paper are inductance, resistance, and voltage controlled subcarrier oscillators, a missile roll indicator, and a mixing amplifier.

Dynamic Stability of a Missile in Rolling Flight

The paper sets down the equations of motion for a symmetric rolling missile with respect to axes attached to the missile. The missile may be jet (or rocket) propelled or coasting under accelerating or decelerating conditions, respectively, wherein the variable rolling velocity is derived from intentionally or unintentionally fins and/or wings. The equations contain a force and moment system that includes, in addition to the usual forces and moments, those due to magnus effects, misaligned surfaces, canted surfaces, jet misalignment, and the linear accelerations in the plane normal to the missile axis. The results present general stability criteria for a rolling missile which are summarized in the Discussion of Stability. LIST OP SYMBOLS (I) English Letters A — moment of inertia in roll about x axis, pd, lb0 sec. A = A/pd\itr B = moment of inertia about y axis, lb. sec. B = B/pd*. hr C ~ moment of inertia about z axis, lb. sec. CD = drag coefficient, dimensionless c = wing or tail chord, d — diameter of missile, ft, G = linear momentum vector H = angular momentum vector i, j = unit vectors along y and z axes, respectively L, M, N = moments about x, y, and z axes, respectively m = mass of missile, lb. sec. ft. m — m/pd, dimensionless P = rolling velocity of missile about x axis, rad. per sec. pz = P / F , r a d . p e r f t . q, r — angular velocities in pitch (about y axis) and in yaw (about z axis) 5 = area, sq.ft. T — thrust force, lbs. T = T/Pd\4 ft./sec. 2 t — time, sec. 11, v, w = components of linear velocity of the missile along x, y, z axes, respectively, per sec. u, v,w = derivatives with respect to time V = scalar velocity of missile along flight trajectory per sec. X, Y, Z — forces acting along x, y, z axes, respectively x, y, z = designations of coordinate axes z = distance along trajectory (II) Definition of Coefficients K = ?/V 1/ft. Kz , Km = dimensionless coefficients of force and moment, a a respectively, due to angle of attack or yaw Kz , Km = Kz ,/d and Km /d, respectively, 1/ft. Q'

A Limiting Case for Missile Rolling Moments

Missile rolling moments are studied by linearized theory for the limiting case in which infinitely many wings are symmetrically arranged around a circular cylinder. This case is physically interesting when interference effects prevent the rolling moments from getting too far out of the range of those practically obtainable with a finite number of wings. Mathematically, the limiting case is convenient because it can be analyzed through knowledge of the rate at which angular momentum is being added to the air, the rolling moment being proportional to this quantity. The analysis is independent of Mach Number, and, since there are few restrictions on the geometry of the system, it should furnish an upper limit for rolling moments for a variety of configurations and flight speeds. Many cases with a finite number of wings are difficult to analyze, and knowledge of a limiting value may help in checking intermediate cases. For a delta wing system with leading edges on the Mach lines, body of zero radius, and each wing acting as an aileron, the rolling moment coefficient for an infinite number of wings is -K times tha t for the planar wing and 1.85 times tha t for the cruciform wing. For narrow delta systems, the above values apply exactly for the planar case and approximately for the cruciform. Wing-tail interference effects on aileron rolling moments may also be studied for the limiting case. Here, all angular momentum is removed from the disc corresponding to the maximum cross section of the tail (unless opposite surfaces of the tail are designed to float relative to each other and are controlled only through the sum of their displacements). If the tail disc is as large as the wing disc, then no resultant rolling moment is produced by wing ailerons in the limiting case. This assumes the discs to be concentric. For a lifting wing, the effective displacement of the disc centers might introduce important nonlinear effects. Damping in roll can be studied by assuming the wing angle of attack proportional to the radius. Equating the damping moment to the aileron moment gives an equilibrium rate of roll which may be expressed as the ratio of tip helix angle to aileron displacement angle. For the delta wing system previously considered (with sonic leading edges), this ratio decreases as the number of wings increases. For an infinite number of wings, the ratio is two-thirds of the planar value or 0.73 times the cruciform value. For a given aileron displacement, the equilibrium rate of roll is therefore decreased by increasing the number of wings. Physically, this corresponds to the fact tha t interference effects on damping in roll are less than on aileron rolling moments. The above figures are approximately correct for narrow delta systems.

A Method of Determining Some Aerodynamic Coefficients from Supersonic Free-Flight Tests of a Rolling Missile

Abstract : Based on the assumption that the aerodynamic forces acting at any point on a lifting surface are linearly dependent upon the local angle of attack at that point, the theory of pure rolling motion is applied to the experimentally observed motion of a finned missile in order to (1) determine how well the actual motion is represented by the theory, (2) determine the suitability of the Aerodynamic Range to the proposed free-flight roll technique, and (3) determine certain aerodynamic coefficients associated with the motion. Values for the aerodynamic coefficients are also derived using the linearized supersonic theory and are compared with those obtained from experiment. The results indicate that the actual motion is well represented by the theory to within the small experimental errors and that excellent re-producibility of the aerodynamic coefficients in roll is obtained. Further - more the results show a fairly good correlation with the linearized theory considering the degree of approximation associated with this theory when applied to aerodynamic surfaces of 16% thickness as employed on the missiles used in the reported tests.