To design efficient flight paths for spacecrat in space is called trajectory design.ETrajectory design is indispensable to any space mission, so research on it has been performed for many years. This article introduces trajectory design for interplanetary cruising and formation flight of spacecraft orbiting the earth, from a slightly different viewpoint than the conventional one.
To process huge amounts of information efficiently
With the discovery of the asteroid Apophis, etc., in recent years we have seen more survey observations of asteroids that could approach and collide with the earth, and research to prevent such collisions. In terms of trajectory design, these missions and research forced us to face new problems. Specifically, in past planetary exploration missions, target object and number of objects to be explored were limited, e.g., Jupiter or Venus. In addition, the scope of exploration was defined in advance. If the spacecraft was required to visit and observe as many asteroids as possible near earth orbit as time and fuel permit, we had to design the orbit considering which objects,Ein which order,EwhenEand how.ETo meet the requirements, we need a trajectory-design methodology that can process huge amounts of information efficiently. The traditional design method is unsuitable to handle huge amounts of information in an efficient manner because the object to visit was predetermined. To tackle this issue, I noted trajectory design using the generating functionEand am now engaged in its research.
What is trajectory design using the generating function?
First, I would like to explain briefly the conventional design method. Designers first try to determine an orbit in which spacecraft moving only by the gravity of the main celestial bodies can travel from an initial position to a target position in a given time interval. The spacecrafts trajectory is given by determining positionEand velocityEat a certain time. When an initial position and a target position are given, the problem is to calculate the velocityEfor the spacecraft to pass accurately through these positions. Known as the Lambert Problem, this is the basic process for trajectory dynamics. In the classical method, velocity satisfying requirements was obtained by repeating calculations based on the trajectorys geometric shape. When designing a trajectory for a mission to visit many asteroids in turn, however, we need to determine the orbits repeatedly, every time the initial and target positions change as the asteroids continue to move. This is time-consuming work. It would be very convenient if there is a formula where a required velocity was provided by inputting initial and targeted positions. In other words, if we have a functionEto show velocity when initial and target positions are variables, the work of examining velocity change in response to position change and to determine positions for making velocity maximum or minimum can be processed as a function.EThis is trajectory design using the generating functionEintroduced below.
The method proposed by Dr. Scheeres, et al., of the University of Colorado uses a motion of equation called Hamiltons equation to represent motion. This equation is basically equivalent to Newtons equation of motion, but with a very different approach. In Newtons equation, velocityEis defined as one obtained by differentiation of positionEby time. Meanwhile, in Hamiltonian Formalism, motion is processed on condition that positionEand momentumE(i.e., velocity) are equivalent independent variables. In Hamiltonian formulation, we can also use variables other than position and velocity as independent variables. Motion can be represented by regarding constants not governed by time (e.g., initial position, target position) as independent variables. When representing the same motion, however, new variables must have a specific relation to the original variables, i.e., positionEand velocity.EGenerating function can provide this relation. If this relation is determined, the velocity necessary for deciding trajectory is represented as a function of the initial and target positions.
In the conventional method, trajectory design is made according to the order of trajectory; meanwhile, in the generating-function method, the order is reversed, i.e. boundary condition. Fig. 1 shows an example where trajectories are determined using the generating function. In the classical design methodology, trajectories are calculated one by one after determining conditions such as initial position. With the generating-function method, each trajectory is not calculated individually. Once a generation function is set in advance, all trajectories to meet the conditions are obtained by giving the necessary boundary conditions. In Fig. 1, trajectories are given per various initial positions so that terminal velocity becomes zero.