Space Vehicle Conflict Probability for Ellipsoidal Conflict Volumes

S PACE vehicles can be protected against collisions with tracked space objects by executing collision avoidance maneuvers [1]. Once a space object is identified as a collision risk, plans can bemade to maneuver the space vehicle to increase the closest approach distance to a safe level. In some cases, a maneuver is also needed to move the spacecraft back into its operational location. The key challenge in this process is identifying those conjunctions that pose the most collision risk. Initially, the metric used to quantify collision risk was the predicted closest approach distance itself. Analysts recognized that the predicted closest approach distance is based on the nominal state vectors and does not include position uncertainty. To improve this situation, position error probability density ellipsoids were calculated based on the position error covariances of the respective space objects. Placing the ellipsoids on the nominal locations of the two objects at the time of conjunction enables one to determine the degree of overlap and hence collision risk. This “touching or overlapping ellipsoid” method can be used to identify risky conjunctions [2]. The next improvement was to actually compute the collision probability based on the nominal positions, position error covariances, and sizes of space objects. Although a significant amount of work has been done in deriving effective methods to compute collision probability [3–6], its effectiveness in identifying risky conjunctions is limited due to uncertainty in the size of the space objects, as well as the large position error uncertainties. Recently, this author proposed using conflict probability to improve the situation [7]. Conflict probability is the probability that a conflict volume, which is centered on one object, will be penetrated by a secondary object. The aviation community uses cylindrical conflict volumes to identify potential midair collisions [8]. A spherically shaped conflict volume was found effective in identifying high-risk conjunctions [7]. However, the best shape for a conflict volume may not be spherical. Operators of the Earthobserving satellites use ellipsoidal-shaped keep-out regions to identify conjunctions requiring further analysis [9]. Therefore, conflict probability for ellipsoidal-shaped conflict volumesmaybe of value in detecting higher-risk conjunctions. This Note presents a method to compute conflict probability for ellipsoidal-shaped conflict volumes. The approach of the current method is to transform the problem to the conflict volume frame, which has its coordinate axes aligned with the ellipsoid principal axes. A scale change is performed along two of the three axes such that all three axes are equal. This scale change symmetrizes the conflict volume, thereby transforming its shape from ellipsoidal to spherical. The combined position error covariance is transformed to the conflict volume frame and modified to account for the scale changes. Likewise, the positions and velocities of the space objects are transformed to the conflict volume frame and adjusted based on the scale changes. Once completed, the problem is reduced to one involving a spherical conflict volume. Because collision probability tools are already developed for spherical hardbodies, these tools can be used for conflict probability involving spherical conflict volumes. The relativemotion can be linear or nonlinear, becausemethods have been developed for both cases [10]. A computer program was created to implement the method. The method was tested by comparing results with known solutions obtained using a completely different method [11]. Once validated, the computer toolwas used to determine howwell ellipsoidal conflict volumes can identify high-risk conjunctions. A method was devised to determine the effectiveness of identifying risky conjunctions using three known space collision events [12,13]. The 23 December 1991 collision between object numbers 13,475 and 18,985; the 24 July 1996 collision between object numbers 18,208 and 23,606; and the 17 January 2005 collision between objects 7219 and 26,207 were investigated. Each collision event involved objects from unrelated launches. The entire unclassified catalog of tracked space objects was screened for conjunctions of less than 5 km over a five-day period centered at the time of each collision. Databases of space object sizes and estimated covariances were used in the calculations. The database was processed to produce collision and conflict probabilities. One would like the conflict probability associated with the actual collision event to rank highest among all conjunctions processed. The relative ranking of the actual collision event is a measure of the utility of a given parameter. The axes of the ellipsoidal conflict volume were varied in an attempt to find the most suitable shape. Results of the study are presented.