The study addresses the problem of space debris motion determination using a nanosatellite with video equipment that allows making stereo images. An original algorithm to determine a spatial orientation of the space debris is proposed. The algorithm is based on a pair of simultaneous stereo images and uses random samples of triangulated points to match the sequentially conducted stereo images. The Monte Carlo method is used to estimate the probability of the proposed algorithm effectiveness. The study offers the results of solving the model problem of a nanosatellite approach to the space debris and proposes recommendations on adjusting an algorithm and choosing the measurement program.