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This textbook introduces the theories and practical procedures used in planetary spacecraft navigation. Written by a former member of NASA's Jet Propulsion Laboratory (JPL) navigation team, it delves into the mathematics behind modern digital navigation programs, as well as the numerous technological resources used by JPL as a key player in the field. In addition, the text offers an analysis of navigation theory application in recent missions, with the goal of showing students the relationship between navigation theory and the real-world orchestration of mission operations.

### Chapter 1. Equations of Motion

The equations of motion describe the path that a spacecraft, planet, satellite, molecule, electromagnetic wave or any body will follow. In space, the path that a spacecraft follows is called a trajectory and for a planet it is called an ephemeris. For the purpose of navigation, a planet is defined as any object that orbits the sun and thus, includes comets and asteroids. A satellite is any body that orbits a planet. Flight operations are generally conducted using solutions of Newton’s equation of motion obtained by numerical integration. Analytic solutions of Newton’s equation of motion provide some insight into trajectory design and navigation analysis, but these solutions are seldom used in the conduct of flight operations. For spacecraft near the Sun and Jupiter, and for the planet ephemerides, Newton’s equations of motion are augmented with terms from the n-body solution of General Relativity.
James Miller

### Chapter 2. Force Models

The acceleration of a spacecraft is proportional to the vector sum of all the forces acting on the spacecraft. Each component of the resultant force is computed by individual force models. The required accuracy of force models is dependent on the magnitude of the force and the observability of the force in orbit determination software. By far the most important force model is gravity. Gravity force models are formulated as acceleration but this is only a matter of convenience because the mass of the spacecraft factors out of the equations of motion. Force models are generally independent of motion. Even though solar pressure and rocket thrust involve motion of molecules and photons, the force on the spacecraft does not depend on its motion. A notable exception is atmospheric drag forces that are dependent on the velocity of the spacecraft relative to the atmosphere.
James Miller

### Chapter 3. Trajectory Design

The problem of trajectory design requires the determination of spacecraft position and velocity as a function of time that satisfy design constraints. The constraints that must be satisfied are supplied to the trajectory designer as parameters that are generally functions of the Cartesian state. Thus, the main interest in developing solutions of the equations of motion for navigation is to enable computation of parameters that satisfy mission constraints and state vectors that may be used to initialize numerical integration for further refinement of the trajectory design. Analytic solutions of the equations of motion are of intrinsic interest because of their mathematical elegance. However, when applied to trajectory design, solutions are sought that enable the full Cartesian state to be determined with high precision and these solutions are numerical.
James Miller

### Chapter 4. Trajectory Optimization

Navigation operations require the refinement of the design trajectory to obtain a high-precision trajectory for flight path control and science operations. The preliminary trajectory design often involves approximate solutions of boundary value problems that provide sufficient accuracy for mission design but are not accurate enough for flight operations. The final precision trajectory is obtained by driving a high-precision trajectory model with targeting and optimization algorithms that yield the final high-precision solution. The preliminary trajectory design provides an initial guess for starting the targeting algorithm. With experience, the preliminary design may sometimes be omitted and the trajectory design obtained directly by targeting.
James Miller

### Chapter 5. Probability and Statistics

Navigation of planetary spacecraft requires determining a nominal design trajectory that obeys the laws of physics and has a high probability of achieving mission success within the constraints of the mission objectives and the cost of the spacecraft design and mission operations. It is relatively easy to design a trajectory that satisfies all the physical laws but cannot be flown. For example, a trajectory describing the path of a coin that is tossed on the floor and rolls to a stop remaining on its edge is easy to design. However, the perturbations that the coin encounters as it rolls on the floor almost guarantees that it will not remain on its edge. Spacecraft trajectory design encounters this same problem in many forms. Statistical perturbations of the trajectory along the flight path may result in failure to meet mission objectives if not complete failure as in the case of the coin. Therefore, the trajectory designer and navigator must give as much attention to the mathematics of probability and statistics as to the laws of physics.
James Miller

### Chapter 6. Orbit Determination

Determination of the orbit of a spacecraft and all the constant and dynamic parameters that affect the orbit is an application of estimation theory. A model of the spacecraft motion as a function of initial conditions and certain constant and dynamic parameters is developed that may be used to predict the flight path and compute the value of measurements that are obtained during the space flight. Orbit determination involves adjusting the value of the independent parameters that need to be determined until the computed measurements are close to the actual measurements. Since there are many more measurements than independent parameters, the solution is found that minimizes the error in the measurements. This solution will also minimize the error in the estimated parameters.
James Miller

### Chapter 7. Measurements and Calibrations

The measurement system is a collection of instruments on board the spacecraft and on the ground that provide observations of the spacecraft motion with respect to Earth and specific target bodies. Instruments of this kind are the Deep Space Network (DSN), a solid-state imaging (SSI) device and a laser altimeter. The DSN tracking stations transmit radio frequency signals to the spacecraft and receive signals via the spacecraft transponder and antenna. The received signals constitute observations of range and Doppler data by conventional methods, and observations of angles by VLBI methods. A SSI allows optical observations of planets, satellites, comets and asteroids to be made against the background of the fixed stars and direct observation of landmarks. A laser altimeter bounces laser beams off the surface of a body and measures the round trip light time.
James Miller

James Miller