Optimal Spacecraft Trajectories
The present text is aimed at graduate students in engineering and applied mathematics, and as a reference. As the author states in his preface, “Books on the field of optimal spacecraft design are very few in number and date back to the 1960’s, and it has been nearly 40 years since a comprehensive new book has appeared” [1-4].
The first three chapters lay the foundations for rocket trajectory models and optimal control theory. The exposition of control theory and Pontryagin’s minimum principle is very clear, well done, and very valuable.
The central topic of this book is the equation
$$
\ddot r = g(r) + \Gamma u \,.
$$
The variable \(r\) specifies the spacecraft position, the gravitational acceleration is modeled by the function \(g(r)\), the acceleration due to the rocket thrust is modeled by the term \(\Gamma u\), where the variable \(u\) is a unit vector in the direction of the thrust, and the scalar \(\Gamma\) specifies the magnitude of the thrust. The general problem is to minimize some cost function depending on the initial position and velocity, on the control term, and on the resulting trajectory.
There are two types of control: first where the thrust is continuous in time and second where the thrust is an impulse. Both the direction \(u\) of thrust and its magnitude \(\Gamma\) must be chosen as part of mission design. Pontryagin’s minimum principle is used to choose an optimal thrust term based on cost criteria.
Several chapters are devoted to optimizing trajectories in a central force gravitational field
$$
g(r)=- \mu |r|^{-3} r \,.
$$
Transfer between coplanar circular orbits is the example used to illustrate the theory. For continuous thrust the cost functional is
$$
\Psi = \int_{t_0}^{t_f} \Gamma(t) u(t) dt \,.
$$
Pontryagin’s principle is used to find a “primer vector” \(p(t)\) by solving the differential equation
$$
\ddot{p} = G(r) p \,.
$$
The matrix \(G(r)\) is the matrix of partial derivatives of the acceleration field \(g(r)\).
The thrust direction is \(u(t)= p(t)/|p(t)|\). A switching function \(S(t)\) to turn thrust on or off is given by
$$
S(t)=|p(t)|-1 \,.
$$
The control term applies maximum thrust in the direction \(u(t)\) when the switching function is positive and no thrust when it is negative.
For impulsive control, the cost function used is \(\Psi = \sum_k \tfrac12 \Gamma^2(t_k)\) and impulses are applied only when the switching function is zero.
The fixed time impulsive rendezvous problem is to go from an inner circular orbit to an outer circular orbit at a specific start time and a future arrival time by using two or more impulses. In chapter 5 the author discusses how to improve a non-optimal impulsive trajectory. Continuous thrust transfers are discussed in chapter 6. The most efficient two impulse transfer is called the Hohmann transfer and is explained in Appendix B. Several other topics are covered in the final two chapters and in the appendices.
This book makes a valuable contribution to the literature and belongs on the reference shelves of professionals planning spacecraft missions. It provides a foundation for an advanced analysis of optimal trajectories for a spacecraft interacting with two or more bodies.
References
- J.E. Prussing and B.A. Conway (1993).
Orbital Mechanics, Oxford University Press.
- B.A. Conway (ed) (2010)
Spacecraft Trajectory Optimization,
Cambridge Aerospace Series.
- A. Shirazi, J. Ceberio and J. A. Lozano (2018), Spacecraft trajectory optimization: A review of models, objectives, approaches and solutions, Progress in Aerospace Sciences 102: 76-98.
- R. Chai, A. Savvaris, A. Tsourdos, A. Chai, Y. Xia, (2019),
A review of optimization techniques in spacecraft flight trajectory design
Progress in Aerospace Sciences, in press (2019).