In this chapter, we present the theory that is used in the analysis and design of ballistic guidance systems, with an emphasis on the principles rather than on the implementation. The trajectory of a ballistic missile typically consists of three phases: a powered lift-off, a free flight (this typically lasts about 80% of the engagement), and an aerodynamic reentry (see Figure 6.1). The purpose of ballistic guidance is then to determine the boundary conditions between the first and second phases that lead to a hit, and to analyze the miss due to navigation errors. Note that the optimization of the first phase leads to interesting “rocket staging” problems that can be treated with methods introduced in Chapter 8.

Section 6.1 describes the restricted two-body problem. Section 6.2 deals with the two-dimensional hit equation, whereas Section 6.3 contains the in-plane error analysis. Section 6.4 deals with three-dimensional error analysis, Section 6.5 with accounting for effects of the Earth's rotation. Section 6.6 considers the effect of the Earth's oblateness and geophysical uncertainties. Section 6.7 presents a general framework for numerical solution of general ballistic guidance problems. Sections 6.8, 6.9, and 6.10 present a summary of the key results in the chapter, bibliographic notes for further reading, and homework problems, respectively.

The Restricted Two-Body Problem

Assume that a ballistic missile, modeled as a particle, moves above the sensible atmosphere (i.e., above an altitude of 80 km). We assume that the Earth is nonrotating and perfectly spherical.

This chapter presents the fundamentals of deterministic systems theory that we use in the analysis and synthesis of aerospace navigation and guidance systems. The emphasis is on linear, time varying dynamic systems. This emphasis is justified in Section 2.1, where it is shown that when the trajectory of a nonlinear system undergoes a small perturbation, the time history of that perturbation can be approximated using a linear system. Thus, this linear approach is appropriate when the navigation and guidance errors are small. This, of course, implicitly assumes that we use successful navigation and guidance laws to ensure small errors.

In addition to system linearization, Section 2.1 surveys properties of linear dynamic systems: the state transition matrix and its features, various notions of stability and their criteria, the variation of constants formula, and the impulse response. Section 2.2 studies observability, which is a central concept in navigation. Section 2.3 considers the particular case of linear, time invariant dynamic systems. Section 2.4 presents the method of adjoints, which is invaluable in terminal guidance because it yields the miss distance due to a perturbation of the trajectory as a function of the time at which the perturbation is applied. Finally, Section 2.5 discusses controllability and the duality between controllability and observability. Sections 2.6, 2.7, and 2.8 present a summary of the key results in the chapter, bibliographic notes for further reading, and homework problems, respectively.

During this 10-chapter journey through Fundamentals of Aerospace Navigation and Guidance, the book endeavors to give an orderly account of the lay of this land. This orderliness should now be apparent through synergies between many items. By this we mean that, often, a newly introduced fundamental relies on past fundamentals, sheds new light on some of them, and foreshadows future ones. Let us give examples of such synergies.

1. Inequality (1.2) establishes that, in static systems, good navigation combined with good navigation-based guidance guarantees good guidance. This principle is extended to deterministic linear dynamic systems through inequality (7.54) under the umbrella of the deterministic separation principle. Then, it is further extended to systems with linear dynamics, quadratic cost, and Gaussian noise processes in Section 9.5.6 under the umbrella of the stochastic separation principle.

2. The controllability Gramian is introduced in Proposition 2.22 to settle the binary question of controllability of a state. However, after deriving optimization results in Section 8.4 and extending them to optimal control in Section 9.3, we show in Example 9.6 that the same controllability Gramian can also be used as a quantitative measure of controllability through (9.54).

3. The implicit function theorem, introduced in Section 6.3 to analyze the accuracy of ballistic shots, is also used in Proposition 8.6 to establish the existence of Lagrange multipliers in constrained optimization. This result is extended in Sections 8.4 and 9.3 to establish the existence of the time varying co-state vector.

In this chapter, we present the fundamentals used in the analysis and design of terminal homing guidance systems. The purpose of terminal guidance is to cause a pursuer (typically a missile) to hit, or come close to, a preselected target.Moreover, we require that this interception take place despite unpredictable maneuvers of the target, disturbances, and navigation uncertainties.Here wemake a rough distinction between three types of terminal guidance: homing, ballistic, and midcourse. This distinction is based on the amount of control authority that is applied during most of the flight and does not lead to categories with sharply defined boundaries. Ballistic and midcourse guidance are studied in Chapters 6 and 7, respectively.

In homing guidance, we make the following assumptions:

1. The pursuer is actively controlled during the entirety of the engagement.

2. The pursuer has a velocity with constant norm (this is typical of the terminal phase of an engagement in atmospheric flight).

3. The pursuer is equipped with a passive seeker (e.g., a heat seeker), or a semipassive seeker (such as in laser-guided, smart bombs).

Under the preceding assumptions, we study solutions to the terminal homing guidance problem. We also allow the possibility of limiting the radius of curvature of the pursuer's trajectory.

Section 5.1 presents the fundamentals of planar homing guidance and identifies the main candidate strategies for homing.

Game theory is an extension of control theory that deals with situations where several players exercise authority over a system. In general, each player may pursue its own objective. The players are not necessarily adversaries; they may choose to cooperate if it is to their advantage. In addition, the players may not know everything there is to know. They may act to learn what they do not know, or exploit the situation based on what they do know. This is a typical exploration-exploitation trade-off.

Game theory has applications to economics (several Nobel Prizes were awarded in this field), international diplomacy, guidance and pursuit evasion, warfare, and sports. Game theory is also applicable to control (see Figure 1.4), where the exogenous and endogenous inputs are two players in a game.

Section 10.1 presents a taxonomy of two-player games. Section 10.2 describes an example of a simple game of pursuit evasion in a two-player football scrimmage. Section 10.3 describes the Bellman-Isaacs equation. Sections 10.4 and 10.5 present modeling and features of the solution for the homicidal chauffeur game. Section 10.6 describes a game-theoretic view of proportional navigation. Sections 10.7, 10.8, and 10.9 present a summary of the key results in the chapter, bibliographic notes for further reading, and homework problems, respectively.

Taxonomy of Two-Player Games

We consider two-player games, in which the two players have control actions u and v, respectively.

The next two chapters treat optimization and optimal control for the purpose of their application to guidance. Chapter 8 focuses on optimization as a stepping-stone toward optimal control, which is treated in Chapter 9. Optimization is concerned with finding the best option from among several to solve a problem.

Optimization is often necessary in aerospace engineering because of the merciless requirements that physics and chemistry impose on flying systems. For instance, it is well known that to lift 1 kg of payload from Earth's surface to orbit, using chemical rocket propulsion, it is required to use at least 80 kg of rocket structure, engine, fuel, and propellant [78]. This staggering 80/1 ratio is one of many stark reminders that, when it comes to flight, optimization is of the essence.

Throughout this chapter, we discuss necessary and sufficient conditions for optimality. Let us clarify what we mean by these. A necessary condition for optimality is a statement of the form: “If item x is optimal (i.e., is the best), then condition NC(x) must be satisfied.” Typically, condition NC(x) provides enough information to determine x. A sufficient condition for optimality is a statement of the form: “If item x satisfies condition SC(x), then item x must be optimal.” Here also, typically condition SC(x) provides enough information to determine x. In view of these statements, the following caution is in order: necessary conditions guarantee neither optimality of x, nor even existence of a solution to the optimization problem.

In this chapter, we present the theory that is used in the analysis and design of navigation systems for aerospace vehicles, with an emphasis on the fundamentals rather than on their hardware implementation. The purpose of navigation is twofold: to estimate the position and velocity of a vehicle based on the output of imperfect sensors and to assess the accuracy of these estimates. Mathematically, this corresponds to computing the first and second moments (expected value and covariance) of a particular random variable. Note that one should not neglect the importance of the accuracy assessment. Indeed, for a vehicle traveling in the vicinity of Earth, the question “Where is the vehicle?” can always be answered by “On the Sun, to within one light-hour.” (Recall that the Earth is 8 light-minutes away from the Sun.) Such an answer, although it is correct, is totally useless for guiding the vehicle in the vicinity of the Earth. It is therefore very important to quantify the navigation error.

We start, in Section 4.1, by considering the navigation problem under the most restrictive assumptions: perfect sensors, nonredundant measurements, static estimation, nonrecursive processing, and perfect clock. Then, in subsequent sections, we remove these assumptions sequentially to build up the theory. Specifically, Section 4.2 considers position fixing with imperfect, nonredundant measurements, and Section 4.3 treats the case of imperfect redundant measurements.

In this chapter, we present the fundamentals used in the analysis and design of midcourse guidance systems. These systems are predicated on the following assumptions:

1. In mission planning, a set of nominal trajectories that meet mission specifications is determined.

2. During flight, corrections are continually applied to the trajectory to return it to nominal. This sustained application of control authority is the distinguishing feature of midcourse guidance.

We consider three methods for midcourse guidance. In the first method, the set of nominal trajectories is parametrized by initial position and initial time, which specify the required velocity. This leads to the formalism of velocity-to-be-gained guidance, also known as Q-guidance. In the second method, a single nominal trajectory is determined and control is applied to return the trajectory to it. This leads to the formalism of state feedback guidance, also known as Delta-guidance. The third method combines state feedback guidance with navigation in that it uses Delta-guidance based on an estimate of the state vector rather than the true state vector.

Midcourse guidance is related to work presented in the previous two chapters. Indeed, in Chapter 5, the primary purpose of homing is to come close to a target - in other words, homing is a “final-value” problem. However, constant bearing guidance can be viewed as a form of midcourse guidance where the nominal trajectory is defined by β = 0.

This chapter presents the elements of stochastic systems theory that we use in the analysis and synthesis of navigation and guidance systems. Roughly speaking, a stochastic phenomenon is randomly unpredictable but exhibits “statistical regularity,” in a sense defined subsequently. Consistent with our emphasis on linear systems, justified in Chapter 2, the purpose of this chapter is to quantify how linear systems respond to uncertain input signals.

We use stochastic models to account for uncertainty for at least two compelling reasons. The first one is purely pragmatic: stochastic models have proven extremely effective in modeling uncertainty in science and engineering and provide a rational basis for taking and optimizing decisions in the presence of the unknown. The second, deeper reason, appeals to the fundamental postulate of natural science [24], which states that “nature is governed by laws that are uniform.” Although this postulate leads us to expect that natural phenomena be highly and exactly repeatable, the uncertainty we observe in practice seems to violate this expectation. A way to resolve this dilemma is to relax the notion of exact repeatability into that of statistical regularity, leading seamlessly to stochastic systems theory. Hence stochastic systems theory affords a provision for uncertainty within the context of the fundamental postulate of natural science.

The motivation of this chapter is that many phenomena in navigation and guidance can be modeled as stochastic. They include measurement errors in components of acceleration, velocity and position, drift of gyroscopes, wind gusts, and so on, and the navigation and guidance errors that the aforementioned generate.

As explained at the end of Chapter 7, the computational vibroacoustic equation could be solved directly ω by ω, but, as explained in Chapter 1, it is recommended to use a reduced-order computational model, which is constructed as follows. The strategy used for constructing the reduced-order computational model consists in using the projection basis constituted of:

1. • The acoustic modes of the acoustic cavity with fixed boundary and without wall acoustic impedance. Two cases are considered. The first one is a closed acoustic cavity (the internal pressure varies with variation of the volume of the cavity), for which the boundary value problem has been defined in Section 6.2. For an almost closed cavity with a nonsealed wall, which are often encountered (the internal pressure does not vary with a variation of the volume of the cavity), an adapted procedure, derived from the closed cavity case, will be presented. The second one concerns an internal cavity filled with an acoustic liquid with a free surface for which the boundary value problem has been defined in Section 6.3.

2. • The elastic structural modes of the structure, taking into account a quasi-static effect of the internal acoustic fluid on the structure. The constitutive equation of the structure corresponds to an elastic material (see Eq. (5.35)) and consequently, the stiffness matrix of the structure has to be taken for ω = 0.

This chapter is devoted to the description of the equations in terms of the pressure field and the associated boundary conditions for the internal dissipative acoustic fluid of the vibroacoustic system. A wall acoustic impedance can be taken into account. In the last section, the case of a free surface for a compressible liquid is considered.

EQUATIONS IN THE FREQUENCY DOMAIN

As introduced in Section 2.3, the fluid is assumed to be homogeneous, compressible, and dissipative. In the reference configuration, the fluid is supposed to be at rest. The fluid is either a gas or a liquid and gravity effects are neglected (see Andrianarison and Ohayon, 2006, to take into account both gravity and compressibility effects for an inviscid internal fluid). Such a fluid is called a dissipative acoustic fluid. Generally, there are two main physical dissipations. The first one is an internal acoustic dissipation inside the cavity due to the viscosity and the thermal conduction of the fluid. These dissipation mechanisms are assumed to be small. In the model proposed, we consider only the dissipation due to the viscosity. This correction introduces an additional dissipative term in the Helmholtz equation without modifying the conservative part. The second one is the dissipation generated inside the wall viscothermal boundary layer of the cavity and is neglected here. We then consider only the acoustic mode (irrotational motion) predominant in the volume.