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

A method for computing isovalue or contour surfaces of a trivariate function is discussed. The input data are values of the trivariate function, F/sub ijk/, at the cuberille grid points (x/sub i/, y/sub j/, z/sub k/), and the output of a collection of triangles representing the surface consisting of all points where F(x,y,z) is a constant value. The method is a modification that is intended to correct a problem with a previous method. >

/pdf/the-asymptotic-decider-resolving-the-ambiguity-in-marching-49khalh3zk.pdf

The asymptotic decider: resolving the ambiguity in marching cubes

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 novel sliding mode-based impact time and angle guidance law for engaging a modern warfare ship is presented in this paper. In order to satisfy the impact time and angle constraints, a line-of-sight rate shaping process is introduced. This shaping process results in a tuning parameter that can be used to create a line-of-sight rate profile to satisfy the final time and heading angle requirements and to yield acceptable normal acceleration values. In order to track the desired line-of-sight rate profile in the presence of uncertainties, a novel robust second-order sliding mode control law is developed using a backstepping concept. Due to the robustness of the control law, it can be applied to many realistic engagement scenarios which include uncertainties such as target motion. Numerical simulations with different warship engagements are presented to illustrate the potential of the developed method.

Impact Time and Angle Guidance With Sliding Mode Control

https://escholarship.org/content/qt9x4542q1/qt9x4542q1.pdf?t=ptt2jl

A data reduction scheme for triangulated surfaces

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

Triangular interpolants are widely used to construct smooth surfaces from scattered data in 3D. To apply smooth triangular interpolants, the input 3D points have to be triangulated. This paper presents an algorithmic procedure for 3D triangulation. Relevant algorithms and datastructures are described in detail. Existing methods of 2D triangulation are based on Thiessen polygonization, which is an optimal partitioning of a geometric domain. To obtain an optimal triangular grid on the domain, the max-min angle criterion is frequently used. As there is no domain for 3D points, a new criterion called the smoothness criterion is proposed.

Triangulation of scattered data in 3D space

The proportional navigation (PN) guidance law is derived for an ideal pursuer with a stationary target, by obtaining an optimal feedback solution minimizing a performance index of the range-weighted control energy. Without any linearization of system equations, the optimal PN with an arbitrary navigation constant is obtained. Thus the exact analyses on optimality of PN based on nonlinear formulation is presented here.

Optimality of Proportional Navigation Based on Nonlinear Formulation

This paper explores the feasibility of roll-pitch-yaw integrated autopilots for a high angle-of-attack missile. Investigation of the aerodynamic characteristics indicates strong cross-coupling effects between the motions in longitudinal and lateral directions. Robust control techniques based on H ∞ synthesis are employed to design roll- pitch-yaw integrated autopilots. The performance of the proposed roll-pitch-yaw integrated controller is tested in high-fidelity nonlinear 5-degree-of-freedom simulations. The proposed controllers are scheduled as a function of total angle of attack in a linear parameter varying framework with proportional navigation guidance laws. The integrated controller demonstrates satisfactory performance that cannot be achieved by the controller designed in a decoupled manner.

Jin-Ik Lee

Papers

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

Guidance law to control impact time and angle

Triangulation of scattered data in 3D space

Optimality of Proportional Navigation Based on Nonlinear Formulation

Roll-Pitch-Yaw Integrated Robust Autopilot Design for a High Angle-of-Attack Missile