Collision Probability Based Avoidance Control For Satelites In Formating Flying

Satellites formation flying is defined by the several satellites consisted of certain configurations to complete some space missions together. Satellites in formation flying have many advantages. It is convenient, small amount of energy, strong performance maintain, et al. But, it is easy to collide when some satellites fly in the close orbit. The problem to be considered in this paper is based the collision probability for satellites in the formation and then we can determine the avoidance strategy of collision satellites. Thus the collision probability will be reduced to an acceptable safety level.In formation flight, each of the satellite position vector and velocity vector and error covariance can be obtained by analyzing large amounts of data, and then we can get the relative position vector and the relative velocity vector of two satellites, these data can be converted to the meet coordinates using coordinate system. According to the linearized relative motion equation of the formation flight(C-W equation), any time the relative state covariance matrix and state transition matrix can be got, thus, any time the relative vector covariance matrix will be able to calculate. And the probability density of relative position error between satellites is a three-dimensional Gaussian distribution. So the collision probability is the integration of the probability density function in the envelope ball. In the meet coordinate system, we can get the integration of the probability density function along the relative velocity direction, so the three-dimensional probability density function convert to a two-dimensional probability density function, the collision probability is the integration of the two-dimensional probability density function in the circular domain which is the projection of the envelope bal in the meet coordinates. In order to simplify operation, we can rotate the meet coordinate system, and then the collision probability is the integral of a two-dimensional probability density function in the elliptic region.In this paper, we compare the collision probability that is calculated by the formula with threshold which is prior to be defined, if the collision probability is larger than the threshold, the satellites are possible to collide. So we need to develop a suitable strategy to avoid collision. In the paper, according to the collision probability's variation of time in the meet coordinates, we design a strategy, which are included two major factors, the avoidance velocity's direction and magnitude. We can analysis to get the best avoidance velocity direction which is the opposite direction of gradient, as it is the fast change direction of satellite velocity. Then, the avoidance velocity of collision probability that equals to the safety threshold is the solution of a nonlinear equation, we can obtain the velocity by numerical solution. Finally, a numerical example to show probability error ellipse changes with time clearly in the y direction of meet system, and some typical results for the planar to show the accuracy of avoidance velocity's calculation. In the paper, the avoidance strategy is simple to get, we not only can save computing time, but also calculate the right result.