Space Vehicle Maneuvering, Propulsion, Dynamics and Control

This chapter summarizes the concepts underpinning celestial mechanics, the space environment, the environment around the Sun, the intervening environment between the Sun and the Earth, the environment closer to the planet Earth as well as the environment around the launch site. It also covers the Keplerian orbit resulting from the solution of the two-body problem and its features, energy-based characterization of the orbits leading to the formulation of the vis-viva equation for the conservation of energy, the special features of elliptic, circular, parabolic and hyperbolic Keplerian orbits, Lambert’s problem based on the two-body problem, relative motion of a satellite in a rotating frame and the Hill-Clohessy-Wiltshire equations, real orbits arising from the solutions to the three-body problem and their equilibrium and stability features and finally some applications of the orbits arising from the solutions to the three-body problem.

This chapter focusses on the perturbations that are present when the primary orbit is considered to be Keplerian. All the major perturbations that exist and that must be considered in simulating the trajectory of a spacecraft are identified, and this is followed by a discussion and presentation of the Gauss perturbation equations which may be considered to be generalization of the Lagrange planetary equations with perturbations expressed as forces rather than potentials. The primary perturbations and methods of estimating the perturbing forces including the forces due to various gravitational perturbations such as the non-spherical nature of the central body and third-body perturbations are discussed in some detail. In particular, the method of orbital averaging to identify the contributions of the perturbing forces to the secular variations in the orbital elements is introduced and applied to obtain the secular variations in the line of apsides and the line of nodes. This is followed by a discussion of the non-gravitational perturbation forces acting on the satellite in its orbit, aerodynamic forces in low Earth orbits, solar radiation and photonic pressures and the solar wind, magnetic forces, forces due to the impact of the space debris and other celestial bodies like meteorites and finally the influence of control forces and moments.

This chapter is about the use control forces and the impulses to manoeuvre the spacecraft into specific and desired orbits. Following the introduction of the principles of space vehicle manoeuvring and conservation of momentum, the equivalence of applying an impulse to the velocity increment it generates is discussed. The components of the velocities of a spacecraft in an orbit in a planeto-centric frame of reference are derived. This is followed by a detailed presentation of the velocity increments required to make specific changes to the orbit, including the trigonometric laws that govern the vector changes to the space vehicle’s velocity, changes to the inclination and other orbital elements, rotating the lines of apsides and nodes and making changes to the orbital geometry and orbital transfers from circular and elliptic orbits to elliptic, circular, parabolic and hyperbolic Keplerian orbits and vice versa. This is followed by a study of the primary methods of two and multi-impulse transfers, leading to coplanar and noncoplanar Hohmann orbital transfers and Oberth’s and Edelbaum’s multi-impulse manoeuvres. Finally, the application of the solution to Lambert’s problem to space vehicle manoeuvring is briefly discussed.

In this chapter, following a preliminary discussion of interplanetary trajectories, the concept of the sphere of influence, where the ratio of disturbing accelerations of the third bodies to the acceleration due to the central body, from the perspective of each of the central bodies, is equal, is introduced. The patched conic approximation is the first approximation that is usually considered in the mission planning as it provides an estimate of the velocity increment required for a particular mission and the estimate is quite accurate in a majority of the situations. These estimates provide an upper bound for the velocity increment required over the entire mission, which are useful for planning purposes. The methodology is largely based on an application of Hohmann two-impulse transfer. Also considered is the estimation of the phasing requirements and the times taken to complete the mission, although these may not be too accurate. Application examples to the Moon, a typical outer planet such as Mars and an inner planet such as Venus are also considered.

In this chapter, following an introduction to the concepts and practices of mission planning, their relationships to the space propulsion systems used to generate the propulsive forces are discussed. Space propulsion systems are then broadly classified, and the fundamental concepts and metrics used to characterize space propulsion forces and impulses are presented. The dynamics of launch vehicles is then introduced, and this is followed by a brief discussion of the differing requirements for propelling launch vehicles into orbits and manoeuvring satellites in orbits. The celebrated rocket equation is then introduced and expressed in different forms for various applications. The concept of multistaging is then introduced followed by its use in optimizing the number of stages and maximizing the velocity increment that can be generated. Electric propulsion is briefly discussed including its applications to mission design, and this is followed by chemical propulsion systems from the perspective of mission design.

Following an introduction and an enunciation to the basics of the calculus of variations and Pontryagin’s minimum principle, the fundamental of optimal control is presented. After illustrating the methodology with a couple of examples, the application to optimal trajectory tracking is discussed. This is followed by a number of application examples. These include an example of a spacecraft orbiting a central planet and the motion defined in a set of space fixed or planeto-centric coordinates, an example of a spacecraft orbiting a central planet and the motion defined in a relative motion frame of reference coordinates, an example of a spacecraft orbiting a central planet and the motion defined in a set of planeto-centric spherical coordinates and an example of a spacecraft orbiting a central planet with the motion defined in a rotating frame as in the analysis of the circular restricted three-body problem. The principles of linear optimal feedback control and its application to the synthesis of optimal trajectories as well as trajectory tracking are briefly revisited. Finally, the methods of Edelbaum and Petropolous, including the Q law, as well as other methods are discussed.

In this chapter, in the first instance the circular, planar, restricted three-body problem is first revisited. Following the enunciation of the key features of the three-body problem, including the transformation of the reference coordinate frame to a rotating frame, the calculation of the equilibrium points, the definition of the Hill sphere and its comparison with sphere of influence introduced earlier as well as other features, the transformation from one synodic reference frame to another is considered in some detail with examples. The methods of genera rating trajectories by solving the circular, restricted three-body problem are then discussed and compared with representative solutions to the multi-body problem. Based on the solutions to the circular, restricted three-body problem, the patched three-body approximation method is introduced. In this method the trajectory is first constructed by solving circular, restricted three-body problem using an appropriate set of three bodies including the departure planet. Then, when the contribution of the second primary or the departure planet is insignificant, it is substituted by employing a transformation, to include the arrival planet as the second primary. A trajectory correction manoeuvre is applied at the patch point to switch from one trajectory to the other. The application of this method including the influence of impulsive and continuous control forces is then considered. Optimal control theory is used to construct segments of the trajectories which are patched together to construct a suitable interplanetary trajectory.

In this chapter, the focus is on asteroids. As the shape of most asteroids is quite variable, the gravitational modelling of the asteroids is first discussed in detail, providing several methods of modelling the gravitational field. These include spherical harmonics-based models, MacCullagh’s moment of inertia based modelling of the gravitational potential, geoid models, geometrical modelling, polyhedron model, discrete element models, modelling based on distributed concentrated masses as well as other models.

Following a discussion of the equations of motion of an asteroid, particularly in the presence of a finite second body in addition to the central body and the spacecraft, the applications of these equations to the problem of synthesizing a feasible optimal trajectory is considered.

This chapter is intended to provide a brief introduction in plasma physics, including common definitions, basic properties and typical processes found in plasmas. These include the basic physical laws governing plasmas, electrostatics of plasma, quasi-neutrality of the plasma, plasma oscillations, the Saha ionization equation, the Hall effect, forces on an ion stream in electric and magnetic fields, the probabilistic nature of plasma velocities, plasma velocities in an electric field, collisions between charged particles and Debye shielding, the collective behaviour of plasma particles, motion of a single charged particle in an electromagnetic field, plasma instabilities and drifts, magnetic mirrors and plasma confinement by magnetic fields. These and other related concepts are inherent in contemporary plasma-based spacecraft thrusters and thus provide a foundation for the methods and concepts that will be presented in the following chapters. The reader is only assumed to be familiar with basic electrodynamics and fluid mechanics. The chapter also discusses the modelling of plasmas as a fluid, the magneto-gas dynamic equations governing such fluid flows as well as a number other related concepts associated with such flows.

Following a review of the basics of electric propulsion, the chapter presents the basics of plasma propulsion applied to electrically propelled spacecraft. The primary motivation for the development of electric thrusters is the increase in the specific impulse which is a measure of the utilization of propellant mass. Electric spacecraft thrusters are generally grouped into three categories based on the main heating mechanism: electro-thermal, electrostatic and electromagnetic thrusters. To heat the propellant, “electro-thermal” thrusters use either resistance heating as in “resistojet” or an electric arc as in an “arcjet” as well as microwaves. In an arcjet, a plasma in the form of a high current arc is created in a small region enclosed between a cathode and an anode. In addition to the hot neutral propellant expelled by a thruster which generates thrust, the exhaust stream not only contains ions and electrons but also sputtered material from the cathode or any grids present in the thruster. Plasma arcs are also used in “electromagnetic” thrusters such as the “pulsed plasma thruster” and the “magnetoplasmadynamic” (MPD) thruster, which is also called the “Lorentz force accelerator”. Thus, the principles governing resistojets and arcjets are discussed in some detail.

In this chapter, after reconsidering the motion of plasmas when influenced by electric and magnetic fields, particularly the generation and acceleration of plasmas in such fields and the Child–Langmuir law and its applications to developing relations for the generated thrust, typical plasma thruster technologies such as gridded ion engines, pulsed plasma thrusters, Hall current thrusters, field effect emission–based electric thrusters, colloid propulsion and electrospray-driven thrusters are briefly discussed. The physics of cone and droplet formation is also briefly considered.

In this final chapter, plasma thrusters based on expansion from low-pressure high-density electromagnetically coupled and wave-excited plasma sources, including a radiofrequency (RF) helicon source, are considered. Associated with these electrodeless and gridless thrusters are plasma-focussing magnetic nozzles and magnetic mirrors, which are also briefly discussed. Developments of plasma thrusters such as magnetoplasmadynamic (MPD) thrusters, self-induced and applied field MPD thrusters, air-breathing electric thrusters, pulsed plasma thrusters and air-breathing PPTs for future space travel are discussed in some detail. This is followed by a detailed discussion of radiofrequency-excited plasma thrusters such as those excited by helicon waves, including the physics of helicon waves, and applications resulting in several design examples of the helicon wave–driven thruster. The chapter is concluded by summarizing recent developments of novel thrusters such as the Variable Specific Impulse Magnetoplasma Rocket (VASIMR) and inertial electrostatic confinement thruster and by providing a summary of helicon wave–excited plasma thrusters.