The Mid-resolution InfRAreD Astronomical Spectrograph (MIRADAS) is a near-infrared multi-object echelle spectrograph for the 10.4-meter Gran Telescopio Canarias. The instrument has 12 pickoff mirror optics on cryogenic probe arms, enabling it to concurrently observe up to 12 user-defined objects located in its field-of-view. In this paper, a method to compute collision-free trajectories for the arms of MIRADAS is presented. We propose a sequential approach that has two stages: target to arm assignment and motion planning. For the former, we present a model based on linear programming that allocates targets according to their associated priorities. The model is constrained by two matrices specifying the targets' reachability and the incompatibilities among each pair of feasible target-arm pairs. This model has been implemented and experiments show that it is able to determine assignments in less than a second. Regarding the second step, we present a prioritized approach which uses sampled-based roadmaps containing a variety of paths. The motions along a given path are coordinated with the help of a depth-first search algorithm. Paths are sequentially explored according to how promising they are and those not leading to a solution are skipped. This motion planning approach has been implemented considering real probe arm geometries and joint velocities. Experimental results show that the method achieves good performance in scenarios presenting two different types of conflicts.