Automation of transportation will play a crucial role in the future when people driving vehicles will be replaced by autonomous systems. Currently, the positioning systems are not used alone but are combined in order to create cooperative positioning systems. The ultra-wideband (UWB) system is an excellent alternative to the global positioning system (GPS) in a limited area but has some drawbacks. Despite many advantages of various object positioning systems, none is free from the problem of object displacement during measurement (data acquisition), which affects positioning accuracy. In addition, temporarily missing data from the absolute positioning system can lead to dangerous situations. Moreover, data pre-processing is unavoidable and takes some time, affecting additionally the object’s displacement in relation to its previous position and its starting point of the new positioning process. So, the prediction of the position of an object is necessary to minimize the time when the position is unknown or out of date, especially when the object is moving at high speed and the position update rate is low. This article proposes using the long short-term memory (LSTM) artificial neural network to predict objects’ positions based on historical data from the UWB system and inertial navigation. The proposed solution creates a reliable positioning system that predicts 10 positions of low and high-speed moving objects with an error below 10 cm. Position prediction allows detection of possible collisions—the intersection of the trajectories of moving objects.