Kinematics based guidance laws like the proportional navigation (PN) and many other linear optimal guidance laws perform well in near-collision course conditions. These have been studied thoroughly in the literature from all aspects, ranging from optimality to capturability, for planar or two dimensional interceptor-target engagements, and to a lesser extent, for three dimensional engagements. But guidance in widely off-collision course conditions like high initial heading errors has been relatively less studied. This is probably due to the inherently high nonlinearity of the problem, which makes it a far more difficult problem to solve. However, with increasing speed and agility of interceptors and targets, solutions of such problems have acquired an increased urgency, as has been reflected in the recent literature. This thesis proposes a guidance law based on differential evolution (DE), a member of the evolutionary algorithms (EA) family. While EAs have been applied extensively to static optimization problems, they have been considered unsuitable for solving dynamic optimization or optimal control problems, due to their computationally intensive nature, and their consequent inability to produce solutions online in real-time, except for systems with very slow dynamics. This thesis proposes an online-implementable optimal control for interceptor guidance, a problem with inherently fast dynamics. The proposed law is applicable to all initial geometries including those that involve high to very high heading errors. While interception by itself is a challenging task in the presence of high heading errors, an additional requirement of optimality is also imposed. The first part of the thesis considers only the 2-D kinematic model with high heading errors. In the second part, a 3-D realistic dynamic model, which includes a time-varying interceptor speed, thrust, drag and mass, besides gravity in the vertical plane of motion, and upper bound on the lateral acceleration, is considered, in addition to high heading errors. It is shown that the same structure of the law that is proposed for the 2-D kinematic model can also be used for the 3-D realistic model, if the rest of the complexities of moving from 2-D space to 3-D space, and from kinematics to dynamics is duly addressed. The guidance law proposed does not require time-to-go, the estimation of which can be a difficult problem in high heading error scenarios in which the closing velocity can be negative. Easy to compute and simple to implement in practice, the proposed law does not need any of the techniques or methods from classical optimal control theory, which are complicated and suffer from several limitations. The empirical pure PN (PPN) law is augmented with a term that is a polynomial function of the heading error. The values of the coefficients of the polynomial are found by using the DE. The computational effort required for this low dimensional polynomial optimization problem is shown to be low enough to enable online implementation in real-time. The performance of the proposed law in nominal and off-nominal conditions is validated through several simulations for the 2-D kinematic model, and the 3-D realistic dynamic model. The results are compared with the PPN, augmented PPN and the all-aspect proportional navigation (AAPN) laws in the literature, as per several criteria like optimality, peak latax required and robustness to off-nominal conditions. A successful online implementation of the proposed law for application in practice is also demonstrated. An obvious limitation of optimization by soft computation methods like differential evolution is that no rigorous proof of either convergence or optimality exists for such methods. A fallback option in the form of a conventional guidance law is included in the scheme in case of failure of convergence, and an indirect proof of optimality is provided in the third and final part of the thesis. The same guidance problem is solved by direct multiple shooting method, and it is shown that the numerical results of the two methods compare favourably. The solution by the shooting method is optimal, but computationally far more intensive and takes a computation time of an order of magnitude that is at least one or two times that of the simulation time of the plant. It also needs a good initial guess solution that lies within the region of convergence, which can be a difficult task by itself. Moreover, the shooting method solution is only open loop, and hence applicable for the given model and initial conditions only. Whereas, the simplicity of the method proposed in the thesis makes the solution or guidance law computable in a fraction of the flight time of the engagement, thereby making it online implementable. Equally important, is the fact that it is closed loop, and hence robust to off-nominal conditions like variations in the plant model and parameters assumed in its design.

In this thesis the cubic spline guidance law and its variants are derived. A detailed analysis is carried out to find the initial conditions for successful interception. The results are applied to three dimensional guidance design and for solving waypoint following problems. The basic cubic spline guidance law is derived for intercepting a stationary target at a desired impact angle in a surface-to-surface engagement scenario. The guidance law is obtained using an inverse method, from a cubic spline curve based trajectory. For overcoming the drawbacks of the basic cubic spline guidance law, it is modified by introducing an additional parameter. This modification has an interesting feature that the guidance command can be obtained using a single cubic spline polynomial even for impact angles greater than π/2, while resulting in substantial improvement in the guidance performance in terms of lateral acceleration demand and length of the trajectory. For imparting robustness to the cubic spline guidance law, in the presence of uncertainties and acceleration saturation, an explicit guidance expression is also derived. A comprehensive capturability study of the proposed guidance law is carried out. The capturability for the cubic spline guidance law is defined in terms of the set of all feasible initial conditions for successful interception. This set is analytically derived and its dependence on various factors, such as initial engagement geometry and interceptor capability, are also established. The basic cubic spline guidance and its variants are also derived for a three dimen- sional scenario. The novelty of the present work lies in the particular representation of the three dimensional cubic spline curve and the adoption of the analytical results available for two dimensional cubic spline guidance law. This enables selection of the boundary condition at launch for given terminal boundary condition and also in avoiding the singularities associated with the inverse method based guidance laws. For establishing the feasibility of the guidance laws in the real world, the rigid body dynamics of the interceptor is presented as a 6 degrees-of-freedom model. Further, using a simplified model, elementary autopilots are also designed. The successful interception of the target in the presence of the rigid body dynamics proves practical applicability of the cubic spline based guidance laws. Finally, the theory developed in the first part of the thesis is applied to solve the waypoint following problem. A smooth path is designed for transition of vehicle velocity from incoming to outgoing direction. The approach developed is similar to Dubins’ path, as it comprises line–cubic spline–line segments. The important feature of this method is that the cubic spline segments are fitted such that the path curvature is bounded by a pre-specified constrained value and the acceleration demand for following the smooth path obtained by this method, gradually increases to the maximum value and then decreases. This property is advantageous from a practical point of view. All the results obtained are verified with the help of numerical simulations which are included in the thesis. The proposed cubic spline guidance law is conceptually simple, does not use linearised kinematic equations, is independent of time-to-go es- timates, and is also computationally inexpensive.

This thesis deals with development of guidance laws for advanced applications. Two class of guidance problems, namely, impact angle constrained guidance and pulsed guidance for exo-atmospheric engagements, are considered here. Three impact angle constrained guidance schemes are developed using (i) Proportional navigation guidance (PNG), (ii) State Dependent Riccati Equation (SDRE) technique and (iii) geometric concepts, respectively. A collision course based pulsed guidance law is presented for exo-atmospheric interceptors. Proportional Navigation Guidance (PNG) law is the most widely used guidance law because of its ease of implementation and efficiency. However, in its original form, it achieves only a limited set of impact angles. A two stage PNG law is presented for achieving all impact angles against a stationary target. In the first phase of guidance, an orientation PNG command is used. The orientation navigation constant (N ) is a function of the initial engagement geometry and has a lower value (N less than 2). It is proved that following the orientation trajectory, the interceptor can switch to N = 2 and achieve the desired impact angle. Simulations, with a constant speed and with a realistic interceptor model, show successful interception of the target with all desired impact angles. Feedback implementation of the guidance law results in negligible errors in impact angle with uncompensated autopilot delays. The idea of a two-stage PNG law with impact angle constraint is further used to develop a guidance law for intercepting moving targets. Following the orientation trajectory, the interceptor can switch to N = 3 and achieve the desired impact angle. It is proved that the guidance achieves all impact angles in a surface-to-surface engagement scenario with receding and approaching targets, respectively. In a air-to-surface engagement scenario, it is proved that the guidance law achieves all impact angles in a deterministic set. Constant speed and realistic interceptor models are used for simulations. Results show negligible error in impact angle and miss distance for moving targets. The guidance law, in its feedback implementation form, achieves the desired impact angle for interceptors with delay and with a maneuvering target. The impact angle errors are low with negligible errors in miss distance. Next, the impact angle constrained guidance problem against a stationary target is solved as a non-linear regulator problem using the SDRE technique. The interceptor guidance problems are of finite time nature. As the main contribution of this part of the work, we solve a finite time interceptor guidance problem with infinite horizon SDRE formulation by choosing the state weighting matrix as a function of time-to-go. Numerical simulations are carried out both for a constant speed interceptor model and a realistic interceptor model. Simulations for both the models are carried out for various impact angles and firing angles. Robustness of the proposed guidance law with respect to autopilot lag is also verified by simulations. Results obtained show the efficiency of the SDRE approach for impact angle constrained missile guidance. A geometric guidance scheme is proposed for lateral interception of targets in a planar engagement scenario in the absence of line-of-sight rate information. A kill-band is defined for target initial positions capturable by an arc maneuver, followed by a straight line path by the interceptor. Guidance law for capturing targets inside the kill-band is presented and is further modified for targets outside the kill-band. Based on analytical studies on the kill-band, a guidance law is proposed for lateral interception of maneuvering targets. Simulations are carried with for typical low speed engagements. The concept of kill-band provides an inherent robustness to the proposed guidance law with respect to uncompensated system delays and target maneuver. As the final part of the work, an interceptor endgame pulsed guidance law for exoatmospheric engagements is derived by using the notion of collision heading. The proposed guidance law is derived in steps by (i) Obtaining the collision heading based on the collision triangle engagement geometry and then (ii) Computing the width of the pulse fired by the divert thruster to attain the collision heading. It is shown that this strategy is more effective than the existing zero effort miss (ZEM) based guidance laws for intercepting targets with higher heading angles off the nominal head-on collision course. A result on pulse firing sequence is also presented showing that firing pulses in quick succession results in minimum pulse widths and hence minimum control effort for a desired miss distance. Simulations are carried out for various engagement scenarios. Results show better miss-distance and divert thrust performance as compared to the existing ZEM based law.

This thesis addresses several aerospace guidance and decision making problems using both no cooperative and cooperative game theoretical solution concepts in the differential games framework. In the first part of the thesis, state dependent Riccati equation (SDRE) method has been extended to a zero-sum nonlinear differential games setting. This framework is used to study problems of intercepting a manoeuvring target, with and without terminal impact angle constraints, in the zero-sum differential games theory perspective. The guidance laws derived according to the proposed method are in closed from and online implementable. In the second part of the thesis, cooperative game theoretic concepts are applied to make a group of unmanned aerial vehicles (UAV) achieve rendezvous, in a given finite time horizon. An algorithm has been proposed that enables the UAVs to realize Nash bargaining solution. In this context, criteria for time consistency of a cooperative solution of nonzero-sum linear quadratic differential games have been studied. The problems where the UAVs try to achieve rendezvous by implementing cooperative game theoretic strategies, based on local information structure only, is also addressed.

This thesis proposes and evaluates a new cooperative guidance law called General Vector Explicit - Impact Time and Angle Control Guidance (GENEX-ITACG). The motivation for GENEX-ITACG came from an explicit trajectory shaping guidance law called General Vector Explicit Guidance (GENEX). GENEX simultaneously achieves design specifications on miss distance and terminal missile approach angle while also providing a design parameter that adjusts the aggressiveness of this approach angle. Encouraged by the applicability of this user parameter, GENEX-ITACG is an extension that allows a salvo of missiles to cooperatively achieve the same objectives of GENEX against a stationary target through the incorporation of a cooperative trajectory shaping guidance law called Impact Time and Angle Control Guidance (ITACG). ITACG allows a salvo of missile to simultaneously hit a stationary target at a prescribed impact angle and impact time. This predetermined impact time is what allows each missile involved in the salvo attack to simultaneously arrived at the target with unique approach angles, which greatly increases the probability of success against well defended targets. GENEX-ITACG further increases this probability of kill by allowing each missile to approach the target with a unique approach angle rate through the use of a user design parameter. The incorporation of ITACG into GENEX is accomplished through the use of linear optimal control by casting the cost function of GENEX into the formulation of ITACG. The feasibility GENEXITACG is demonstrated across three scenarios that demonstrate the ITACG portion of the guidance law, the GENEX portion of the guidance law, and finally the entirety of the guidance law. The results indicate that GENEX-ITACG is able to successfully guide a salvo of missiles to simultaneously hit a stationary target at a predefined terminal impact angle and impact time, while also allowing the user to adjust the aggressiveness of approach.

Autonomous aerial vehicles like missiles and unmanned aerial vehicles (UAVs) have attracted various military and civilian applications. The primary guidance objective of any autonomous vehicle is to reach the desired destination point (target or waypoint). However, many practical engagements impose additional constraints like minimum control effort, a desired final velocity direction or a predefined engagement time. This thesis addresses engagement time constrained guidance problems pertaining to missiles and UAVs. The first part of the thesis discusses a nonlinear guidance law for impact time control of missiles against stationary target. The guidance law is designed with a particular choice of missile heading error variation as a function of ran to-target. The proposed heading error variation leads to an exact closed-form expression for the impact time. controlling the impact time, a closed-form relation is derived relating the control parameter to the desired impact time. A new Lyapunov based guidance law with a monotonically decreasing lateral acceleration is proposed in the next part of the thesis. An exact expression for impact time with minimum and maximum achievable impact times is derived. A control parameter is proposed with a closed-form relationship to the desired impact time. Using the concept of predicted interception point, the two guidance laws are extended for impact time control against non-maneuvering and moving targets. The proposed guidance models are extended to three-dimensional engagements by deducing yaw and pitch lateral accelerations satisfying the desired heading error profile. Extensive simulation studies are carried out for single missile and salvo attack scenarios. The last part of the thesis presents a guidance methodology governing the arrival time of a UAV at a waypoint. A specific arrival angle is considered as an additional constraint. The arrival constraints are satisfied by varying the navigation gain of the proportional navigation guidance law. The methodology is applied for simultaneous and sequential arrival of UAVs at a waypoint.

Guidance is defined as the determination of a strategy for following a nominal path in the presence of o-nominal conditions, disturbances and uncertainties, and the strategy employed is called a guidance law. Variants of Proportional Navigation (PN), such as True Proportional Navigation (TPN) and Pure Proportional Navigation (PPN), have been studied extensively in the literature on tactical missile guidance. In the absence of target maneuvers, in a linear interceptor guidance problem, TPN was shown to be optimal. However, the standard PN class of guidance laws per se does not show good performance against maneuvering targets, and was found to be eective in intercepting a maneuvering target only from a restrictive set of initial geometries. Also, since these guidance laws were eectively designed for lower speed targets, they show a degraded performance when applied against higher speed targets. However, in the current defense scenario, two classes of agile targets, which are capable of continuous maneuver, and/or of much higher speed than the interceptor, are a reality. This thesis presents analysis of several variants of PN class of guidance laws against these two classes of agile targets. In the literature, an augmentation of the TPN guidance law, termed as Augmented Proportional Navigation (APN), was shown to be optimal in linearized engagement framework. The present work proposes an augmentation of the PPN guidance law, which is more realistic than TPN for an aerodynamically controlled interceptor, and an-alyzes its capturability in fully nonlinear framework, and develops sauciest conditions on speed ratio, navigation gain and augmentation parameter to ensure that all possible initial engagement geometries are included in the capture zone when applied against a target executing piecewise continuous maneuver. The thesis also obtains the capture zone in the relative velocity space for augmented PPN guidance law. In the literature, a novel guidance law was proposed for the interception of higher speed targets in planar engagement by using a negative navigation gain instead of the standard positive one, and was termed as Retro-PN. It was shown that even though the Retro-PN guided interceptor takes more time than PN guided one in achieving successful interception, Retro-PN performs significantly better than the classical PN law, in terms of capturability, lateral acceleration demand, and closing velocity, when used against higher speed targets. The thesis analyzes Retro-PN guidance law in 3-D engagement geometries to yield the complete capture zone of interceptors guided by Retro-PN guidance philosophy, and derives necessary and sucient conditions for the capture of higher speed non-maneuvering targets with and without a constraint on finiteness of lateral acceleration. Terminal impact angle control is crucial for enhancement of warhead eectiveness. In the literature, this problem has been addressed mostly in the context of targets with lower speeds than the interceptor. The thesis analyzes the performance of a composite PN guidance law, that uses standard PPN and the Retro-PN guidance laws based on initial engagement geometry and requirement of impact angle, against higher speed non-maneuvering targets. Then, to expand the set of achievable impact angles, it proposes a modified composite PN guidance scheme, and analyzes the same. For implementation of many modern guidance laws, a good estimate of time-to-go is essential. This requirement is especially severe in case of impact time constrained en-gagement scenarios. To this end, an ecient and fast time-to-go estimation algorithm for generic 3-D engagement is required. Two time-to-go estimation algorithms are presented and analyzed in this work for the engagement of a PPN or Retro-PN guided interceptor and a higher speed target. The first one is a closed form approximation of time-to-go in terms of range, nominal closing speed and an indicator of heading error, and the second one is a numerical recursive time-to-go estimation algorithm. To improve the odds of intercepting an intelligent target and destroying it, a salvo attack of two or more interceptors could be considered as a viable option. Moreover, this simultaneous salvo attack can also be further improved in eciency by incorporating the shoot-look-shoot approach in making a decision about launching interceptors. This can be considered as the first step towards a layered defense system, which has been described in the literature as a potentially eective strategy against short range or long range ballistic threat. To this end, the present work proposes two PPN and Retro-PN based guidance strategies for achieving simultaneous salvo attack on a higher speed non-maneuvering target. For the implementation of the same the numerical recursive time-to-go estimation technique proposed in this work is utilized

Tato diplomová práce je zaměřena na vývoj simulačního prostředí v Matlab/Simulink zvoleného letadla ve známém letovém režimu. Pozice a orientace letadla pohybujícího se ve vzduchu je popsána pohybovými rovnicemi se šesti stup\v{n}i volnosti. Soustava translačních, rotačních a kinematických rovnic tvoří soustavu devíti nelineárních diferenciálních rovnic prvního řádu. Tyto rovnice lze linearizovat okolo nějakého rovnovážného stavu, který budeme nazývat letovým režimem. Součástí simulačního prostředí je řídící systém letadla založený na PID regulaci. Základem je návrh autopilota, který řídí úhel podélného sklonu a úhel příčného náklonu. Součástí návrhu jsou takzvané „flight director\textquotedblright \phantom{s}m\'dy jako udržení výšky, volba kursu, regulace vertikální rychlosti, změna výšky, zachycení požadované výšky a navigační m\'{o}d založený na nelineárním navigačním zákonu. Optimalizace regulátorů za použití PSO algoritmu a Pareto optimalitě je využita pro nastavení parametrů PID regulátoru. Simulační prostředí je vizualizováno v softwaru FlightGear.