In this paper, a new guidance problem with the impact time constraint is investigated, which can be applied to salvo attack of anti-ship missiles. The closed form solution based on the linear formulation is derived, suggesting an additional loop for adjusting the impact time in addition to the traditional optimal guidance loop. This solution is a combination of the well-known PNG law and the feedback of the impact time error, which is the difference between the impact time by PNG and the prescribed impact time. The new guidance law called ITCG (Impact-Time-Control Guidance) can be used to guide multiple missiles to hit a stationary target simultaneously at a desirable impact time. Nonlinear simulation of several engagement situations demonstrates the performance and feasibility of ITCG. In addition, the similarity of the closed form solution and APNG is investigated and the switching rule for practical implementation is discussed.

Impact-time-control guidance law for anti-ship missiles

Contents 1 Introduction References 2 The Generalized Missile Equations of Motion 2.1 Coordinate Systems 2.1.1 Transformation Properties of Vectors 2.1.2 Linear Vector Functions 2.1.3 Tensors 2.1.4 Coordinate Transformations 2.2 Rigid-Body Equations of Motion 2.3 D'Alembert's Principle 2.4 Lagrange's Equations for Rotating Coordinate Systems References 3 Aerodynamic Forces and Coefficients 3.1 Aerodynamic Forces Relative to the Wind Axis System 3.2 Aerodynamic Moment Representation 3.2.1 Airframe Characteristics and Criteria 3.3 System Design and Missile Mathematical Model 3.3.1 System Design 3.3.2 The Missile Mathematical Model 3.4 The Missile Guidance System Model 3.4.1 The Missile Seeker Subsystem 3.4.2 Missile Noise Inputs 3.4.3 Radar Target Tracking Signal 3.4.4 Infrared Tracking Systems 3.5 Autopilots 3.5.1 Control Surfaces and Actuators 3.6 English Bias References 4 Tactical Missile Guidance Laws 4.1 Introduction 4.2 Tactical Guidance Intercept Techniques 4.2.1 Homing Guidance 4.2.2 Command and Other Types of Guidance 4.3 Missile Equations of Motion 4.4 Derivation of the Fundamental Guidance Equations 4.5 Proportional Navigation 4.6 Augmented Proportional Navigation 4.7 Three-Dimensional Proportional Navigation 4.8 Application of Optimal Control of Linear Feedback Systems with Quadratic Performance Criteria in Missile Guidance 4.8.1 Introduction 4.8.2 Optimal Filtering 4.8.3 Optimal Control of Linear Feedback Systems with Quadratic Performance Criteria 4.8.4 Optimal Control for Intercept Guidance 4.9 End Game References 5 Weapon Delivery Systems 5.1 Introduction 5.2 Definitions and Acronyms Used in Weapon Delivery 5.2.1 Definitions 5.2.2 Acronyms 5.3 Weapon Delivery Requirements 5.3.1 Tactics and Maneuvers 5.3.2 Aircraft Sensors 5.4 The Navigation/Weapon Delivery System 5.4.1 The Fire Control Computer 5.5 Factors In.uencing Weapon Delivery Accuracy 5.5.1 Error Sensitivities 5.5.2 Aircraft Delivery Modes 5.6 Unguided Weapons 5.6.1 Types of Weapon Delivery 5.6.2 Unguided Free-Fall Weapon Delivery 5.6.3 Release Point Computation for Unguided Bombs 5.7 The Bombing Problem 5.7.1 Conversion of Ground Plane Miss Distance into Aiming Plane Miss Distance 5.7.2 Multiple Impacts 5.7.3 Relationship Among REP, DEP, and CEP 5.8 Equations of Motion 5.9 Covariance Analysis 5.10 Three-Degree-of-Freedom Trajectory Equations and Error Analysis 5.10.1 Error Analysis 5.11 Guided Weapons 5.12 Integrated Flight Control in Weapon Delivery 5.12.1 Situational Awareness/Situation Assessment (SA/SA) 5.12.2 Weapon Delivery Targeting Systems 5.13 Air-to-Ground Attack Component 5.14 Bomb Steering 5.15 Earth Curvature 5.16 Missile Launch Envelope 5.17 Mathematical Considerations Pertaining to the Accuracy of Weapon Delivery Computations References 6 Strategic Missiles 6.1 Introduction 6.2 The Two-Body Problem 6.3 Lambert's Theorem 6.4 First-Order Motion of a Ballistic Missile 6.4.1 Application of the Newtonian Inverse-Square Field Solution to Ballistic Missile Flight 6.4.2 The Spherical Hit Equation 6.4.3 Ballistic Error Coef.cients 6.4.4 Effect of the Rotation of the Earth 6.5 The Correlated Velocity and Velocity-to-Be-Gained Concepts 6.5.1 Correlated Velocity 6.5.2 Velocity-to-Be-Gained 6.5.3 The Missile Control System 6.5.4 Control During the Atmospheric Phase 6.5.5 Guidance Techniques 6.6 Derivation of the Force Equation for Ballistic Missiles 6.6.1 Equations of Motion 6.6.2 Missile Dynamics 6.7 Atmospheric Reentry 6.8 Missile Flight Model 6.9 Ballistic Missile Intercept 6.9.1 Introduction 6.9.2 Missile Tracking Equations of Motion References 7 Cruise Missiles 7.1 Introduction 7.2 System Description<7.2.1 System Functional Operation and Requirements 7.2.2 Missile Navigation System Description 7.3 Cruise Missile Navigation System Error Analysis 7.3.1 Navigation Coordinate System 7.4 Terrain Contour Matching (TERCOM) 7.4.1 Introduction 7.4.2 De.nitions 7.4.3 The Terrain-Contour Matching (TERCOM) Concept 7.4.4 Data Correlation Techniques 7.4.5 Terrain Roughness Characteristics 7.4.6 TERCOM System Error Sources 7.4.7 TERCOM Position Updating 7.5 The NAVSTAR/GPS Navigation System 7.5.1 GPS/INS Integration References A Fundamental Constants B Glossary of Terms C List of Acronyms D The Standard Atmospheric Model References E Missile Classi.cation F Past and Present Tactical/Strategic Missile Systems F.1 Historical Background F.2 Unpowered Precision-Guided Munitions (PGM) References G Properties of Conics G.1 Preliminaries G.2 General Conic Trajectories References H Radar Frequency Bands I Selected Conversion Factors Index

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Missile Guidance and Control Systems

OVER the past few years, there have been significant efforts devoted to the research and development of cooperative unmannedsystems [1–3].The formationflyingofmultipleunmanned aerial vehicles (UAVs) has been studied for radar deception, reconnaissance, surveillance, and surface-to-air-missile jamming in military operations. An example of a cooperative operational scenario of multiple vehicles is that of a small UAV flying over an urban area, dispensingmultiplemicro aerial vehicles to examinepointsof interest fromclosedistances [4].Agroupofwell-organized low-costmultiple vehicles can be far superior to a single high-technology and high-cost UAV in effectiveness. Tactical missile systems as well as UAVs provide more capabilities when they are organized as a coordinated group than when they are operated independently. Modern antiship missiles need to be able to penetrate the formidable defensive systems of battleships such as antiair defense missile systems and close-in weapon system (CIWS). CIWS is a naval shipboard weapon system for detecting and destroying incoming antiship missiles and enemy aircraft at short range. These defensive weapons with powerful fire capability and various strategies seriously intimidate the survivability of the conventional antiship missiles. Hence, antiship missile developers have made great efforts to develop a high-performance missile system with ultimate sea-skimming flight and terminal evasive maneuvering capabilities despite a huge cost. On the other hand, cooperative attack strategies have been studied to enhance survivability of the conventional ones. Here, a cooperative attack means that multiple missiles attack a single target or multiple targets cooperatively or, in a specific case, simultaneously [5,6]. Clearly, it is difficult to defend a group of attackers bursting into sight at the same time, even though each member is the conventional one in performance. So the simultaneous attack ofmultiple missiles is a cost-effective and efficient cooperative attack strategy. A simultaneous attack of a group of missiles against a single common target can be achieved by two ways. The first approach is individual homing, inwhich a common impact time is commanded to all members in advance, and thereafter each missile tries to home on the target on time independently. The second is cooperative homing, inwhich themissiles communicate among themselves to synchronize the arrival times. In other words, the missiles with larger times-to-go try to take shortcuts, whereas others with shorter times-to-go take detours to delay the arrival times. The first concept requires determination of a suitable common impact time before homing, but the second needs online data links throughout the engagement. Despite a number of studies on guidance problems related to timeto-go [7–10], studies on guidance laws to control impact time for a simultaneous attack are rare, except a few recent works by the authors. An impact-time-control guidance law (ITCG) for antiship missiles was developed in [5] and, as an extension of this study, a guidance law to control both impact time and angle (ITACG) was presented in [11]. These individual homing methods are based on optimal control theory, providing analytical closed-loop guidance laws. Herein, the desired impact time is assumed to be prescribed before the homing phase starts. Alternatively, this Note is concerned with a new guidance law based on the second approach, cooperative homing, for a simultaneous attack of multiple missiles. Proportional navigation (PN) is a well-known homing guidance method in which the rate of turn of the interceptor is made proportional with a navigation ratio N to the rate of turn of the line of sight (LOS) between the interceptor and the target. The navigation constant N is a unitless gain chosen in the range from 3 to 5 [12]. Although PNwithN 3 is known to be energy-optimal, an arbitrary N > 3 is also optimal if a time-varying weighting function is included into the cost function of the linear quadratic energy-optimal problem [13,14]. In general, the navigation ratio is held fixed. In some cases, however, it can be considered as a control parameter to achieve a desired terminal heading angle [15].Although PN results in successful intercepts under a wide range of engagement conditions, its control-efficiency is not optimal, in general, especially for the case of maneuvering targets [16]. Augmented proportional navigation, a variant of PN, is useful in cases in which target maneuvers are significant [12]. Biased proportional navigation is also commonly used to compensate for target accelerations and sensor noises or to achieve a desired attitude angle at impact [17]. Even if PN and its variants are alreadywell known andwidely used, they are not directly applicable to many-to-one engagements. This Note proposes a homing guidance law called cooperative proportional navigation (CPN) for many-to-one engagements: CPN has the same structure as conventional PN except that it has a time-varying navigation gain that is adjusted based on the onboard time-to-go and the times-to-go of the other missiles. CPN uses the time-varying navigation gain as a control parameter for reducing the variance of times-on-target of multiple missiles. This Note begins with the formulation of the homing problem of multiple missiles against a single target, subject to constraints on the impact time. Next, preliminary concepts such as the relative time-togo error and the variance of times-to-go of multiple missiles are introduced and a new guidance law is proposed. Then the major property of the law is investigated and the characteristics of the law for the case of twomissiles are examined in detail. Finally, numerical simulation results illustrate the performances of the proposed law.

Homing Guidance Law for Cooperative Attack of Multiple Missiles

Optimal guidance laws providing the specified impact angle as well as zero terminal miss distance are generalized for arbitrary missile dynamics. The optimal guidance command is represented by a linear combination of the ramp and the step responses of the missile’s lateral acceleration. Optimal guidance laws in the form of the state feedback for the lag-free and the first-order lag system are derived, and their characteristics are investigated. Practical timeto-go calculation methods, which are important for the implementation of the optimal guidance laws, are proposed to consider the path curvature. Nonlinear and adjoint simulations are performed to investigate the performance of the proposed laws.

Optimal Guidance Laws with Terminal Impact Angle Constraint

This paper proposes a new guidance law to control both impact time and impact angle for a flight vehicle's homing problem, which can be applied for an efficient salvo attack of antiship missiles or a cooperative mission of unmanned aerial vehicles (UAVs). The proposed law can lead vehicles to home on a target at a designated impact time with a prescribed impact angle. It comprises a feedback loop and an additional control command, the first to achieve the desired impact angle with zero miss distance, and the second to control the impact time. Numerical simulations demonstrate the performance of the proposed law in the accuracy of impact angle and impact time

Guidance law to control impact time and angle

A new homing guidance law is proposed to impact a target with a desired attitude angle. It is a variation of the conventional proportional navigation guidance (PNG) law which includes a supplementary time-varying bias. The proposed guidance law does not require a time-to-go estimation and has a simpler form. Analytic conditions for fulfilling the guidance goal are also provided. Simulation results demonstrate that the proposed guidance law has wider launch envelopes than the previous one and shows a good performance even against a maneuvering target.

Biased PNG law for impact with angular constraint

Sufficient conditions for the stability of time-varying discrete interval systems are presented. The spectral radius of the maximum absolute matrix of all interval matrices is employed as opposed to utilizing the matrix norm or Lyapunov theorem. For special cases, our result becomes a necessary and sufficient condition for stability. Furthermore, the proposed conditions are inclusive of the previous results of stability of time-invariant interval systems.

Necessary and sufficient conditions for stability of time-varying discrete interval matrices

A Study on the Flow Conditions of the Combustion Air Heater Outlet for the Supersonic Combustion Experiment

Biased PNG Laws for an Underwater Vehicle Guidance

A noncircular engine cross-section could provide great flexibility in the integration of propulsion into the airframe. In this work, a tri-arc RDE was constructed and tested as an example of noncircular cross-sectioned RDE. The operational characteristics of detonation wave propagation and thrust performance were investigated and compared with an equivalent circular RDE under the same operating conditions. High-speed camera images, short-time Fourier transform (STFT), and fast Fourier transform (FFT) were used for the investigation. The tri-arc RDE showed very similar characteristics to the circular RDE but exhibited slightly better stability and propulsion performance than the circular RDE. We consider that repeated curvature changes positively affect the stability of detonation wave propagation. The experimental data show contradicting results from the numerical analysis with a homogeneous mixture assumption in which the detonation pressures at the convex corner were greater than those at the concave corner. It is reasoned that the tri-arc injector design provides a non-uniform mixture composition, resulting in a strong detonation at the convex corner. Overall, the noncircular RDE of a tri-arc shaped cross-section is demonstrated, one which performs slightly better than an ordinary circular-shaped RDE both in detonation stability and performance.

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Hyung Seok Han

Papers

Biased PNG law for impact with angular constraint

Necessary and sufficient conditions for stability of time-varying discrete interval matrices

A Study on the Flow Conditions of the Combustion Air Heater Outlet for the Supersonic Combustion Experiment

Biased PNG Laws for an Underwater Vehicle Guidance

Experimental Proof of Concept of a Noncircular Rotating Detonation Engine (RDE) for Propulsion Applications